Engineering Turbulence Modelling and Experiments 4
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Engineering Turbulence Modelling and Experiments 4 Proceedings of the 4th International Symposium on Engineering Turbulence Modelling and Measurements Ajaccio, Corsica, France, 24-26 May, 1999 Edited by W. RODI Institut ftir Hydromechanik Universit~it Karlsruhe Kaiserstrasse 12 76128 Karlsruhe, Germany D. LAURENCE Electricit6 de France Laboratoire National d'Hydraulique 6, Quai Watier, B.E 49 78401 Chatou Cedex, France
1999 ELSEVIER Amsterdam- Lausanne- New York- Oxford- Shannon- Singapore- Tokyo
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SYMPOSIUM
SCIENTIFIC
AND ORGANISING
COMMITTEE
Chairman
Co-Chairman
Professor W. Rodi Institut ftir Hydromechanik Universit/it Karlsruhe Kaiserstrasse 12 76128 Karlsruhe, Germany
Professor D. Laurence Electricit6 de France Laboratoire National d'Hydraulique 6, quai Watier, B.P. 49 78401 Chatou Cedex, France Members
R.A. Antonia, University of Newcastle, Australia
N. Kasagi, University of Tokyo, Japan
G. Bergeles, National Technical University of Athens, Greece
B.E. Launder, University of Manchester Institute of Science and Technology, U.K.
D. Besnard CEA-DTP / Grenoble, France
M.A. Leschziner University of Manchester Institute of Science and Technology, U.K
F. Bontoux Technop61e de Chateau-Gombert, France
F. Martelli University of Florence, Italy
J.-P. Chabard Electricit6 de France, France
W. Merzkirch University of Essen, Germany
C.T. Crowe, Washington State University, USA
Y.Nagano, Nagoya Institute of Technology, Japan
J.H. Ferziger, Stanford University, USA
J. Pantaloni Academy of Corsica, Ajaccio, France
W.K. George, State University of New York at Buffalo, USA
N. Peters RWTH Aachen, Germany
K. Hanj alic, Delft University of Technology, The Netherlands
G. Scheuerer, AEA Technology GmbH, Germany
W.P. Jones, Imperial College, U.K.
J.-Y. Yoo, Seoul National University, Korea
vi
ASSEMBLY OF WORLD CONFERENCES ON EXPERIMENTAL HEAT TRANSFER, FLUID MECHANICS AND THERMODYNAMICS
Officers R.K. Shah, President Delphi Harrison Thermal Systems, Lockport, USA
G.P. Celata, Vice President ENEA Casaccia, Rome, Italy Y. Kurosaki, Member
Tokyo Institute of Technology, Tokyo, Japan A.T. Prata, Member Univ. Federal de Santa Catarina, Florian6polis, SC, Brazil E.N. Ganic, Secretary General, University of Sarajevo, Sarajevo, Bosnia-Herzegovina
General Members
B. Azzopardi, UK D.R.H. Beattie, Australia J. Corberan, Spain P. Di Marco, Italy P. Downar-Zapolski, Poland L.J. Fang, Taiwan F.A. Franca, Brazil M. Giot, Belgium N. Kasagi, Japan D.G. Kr6ger, South Africa Z.G. Ling, China J.R. Lloyd, USA L. Maroti, Hungary
F. Mayinger, F.R. Germany L.F. Melo, Portugal D. Mewes, Germany P.H. Oosthuizen, Canada K. Rehme, F.R. Germany W.Rodi, F.R. Germany M. Shoji, Japan B.S. Sunddn, Sweden H.E.A. van den Akker, The Netherlands P.R. Viswanath, India M.C. Welsh, Australia K.T. Yang, USA I. Zun, Slovenia
vii
CONTENTS Preface
1.
xvii
Invited Lectures
Strategies for turbulence modelling and simulations P.R. Spalart ,Boeing Commercial Airplanes, Seattle, USA
3
Modelling shock-affected near-wall flows with anisotropy-resolvingturbulence closures M A . Leschziner, P. Batten, H. LoyayUMIST, Manchester, U.K.
19
Thermal hydraulics simulations: what turbulence model strategies? D. Besnard, CEA-DTP, Grenoble, France
37
Modelling of production, kinematic restoration and dissipation of flame surface area in turbulent combustion N. Peters, RWTH Aachen, Germany
49
Visualization and measurement of spatial structures in turbulent flow, W. Merzkirch, T. Rettich, F. Schneider, W. Xiong, Universitat Essen, Germany
63
2.
Turbulence Modelling
Three-dimensionalmodelling of turbulent free-surfacejets T.J. Craji, J. W. Kidger, B.E. Launder, UMIST, Manchester, U.K.
73
A-priori tests of Reynolds stress transport models in turbulent pipe expansion flow C. Wagner, DLR, Gottingen, Germany, R. Friedrich, TU Miinchen, Miinchen, Germany
83
Thermodynamically consistent second order turbulence modelling based on extended thermodynamics A. Sadiki, K. Hutter, J. Janicka, Technische Universitat Darmstadt, Darmstadt, Germany
93
A nonlinear stress-strain model for wall-bounded turbulent flows J. Knoett, D.B. Tautbee, University at Buffalo, Buffalo, USA
103
Accuracy and robustness of non-linear eddy viscosity models W.Bauer, AEA Technology GmbH, Otterfing, Germany, 0. Haag, D. K. Hennecke, TU Darmstadt, Darmstadt, Germany
113
A proposal for taking into account the intermittency phenomenon in the calculation of wall-bounded turbulent flows N. Bouquet, J.-B. Cazalbou, P. Chassaing, ENSICA, Toulouse, France
125
...
Vlll
Modelling the turbulent flow subjected to magnetic field S. Kenjerei, K. Hanjalicí, Delft University of Technology, Delft, The Netherlands, R. Duursma, T. van Essen, Hoogovens Research and Development, Ijmuiden, The Netherlands
135
Modeling separation and reattachment using the turbulent potential model B. Perot, H. Wung,University of Massachusetts, Amherst, USA
145
A turbulence model for the pressure-strain correlation term accounting for the effect of compressibility H. Fujiwara, Y. Matsuo, National Aerospace Laboratory, Tokyo, Japan, C. Arakawa, University of Tokyo, Tokyo, Japan
155
Consistent modelling of fluctuating temperaturegradient-velocity-gradient correlations for natural convection M. Worner, Q.-Y. Ye, G. Grotzbach, Forschungszentrum Karlsruhe GmbH, Karlsruhe, Germany
165
On the modelling of the transport equation for the passive scalar dissipation rate P.M. Wikstrom,A. V.Johansson, KTH, Stockholm, Sweden
175
3.
Direct and Large-Eddy Simulations
The direct influence of mean flow on subgrid stresses in LES of turbulent flows L. Shao, Ecole Centrale de Lyon, Ecully, France, S. Sarkur, C. Pantano, University of California at San Diego, La Jolla, USA
187
Large-eddy simulation of non-equilibrium intlow conditions and of the spatial development of a confined plane jet with co-flowing streams A. Meri, H. Wengle,Universitat der Bundeswehr Munchen, Neubiberg, Germany, M. Raddaoui, P. Chuuve, R. Schiestel, Technopale de Chkeau-Gombert, Marseille, France
197
Influence of spatio-temporal inflow organization on LES of a spatially developing plane mixing layer Ph. Druault, J. Delville, J. P . Bonnet, UniversitC de Poitiers, Poitiers, France, P. Sagaut, ONERA, Chatillon, France
207
On the influence of turbulence characteristics at an inlet boundary for large-eddy simulation of a turbulent boundary layer T. Maruyamu, Kyoto University, Kyoto, Japan
217
A dynamic one-equation subgrid model for simulation of flow around a square cylinder A. Sohankar, L. Davidson, Chalmers University of Technology, Goteborg, Sweden, C. Norberg, Lund Institute of Technology, Lund, Sweden
227
ix
Numerical simulation and modelling of a wall-bounded compressed turbulence M. N’Diaye, M. Buffat, L. Le Penven, Ecole Centrale de Lyon, Ecully, France
237
Influence of curvature and torsion on turbulent flow in helically coiled pipes T.J. Hiittl, R. Friedrich, Technische Universitat Miinchen, Garching, Germany
247
Large eddy simulations of stirred tank flow J. Derksen, H. van den Akker, Delft University of Technology, Delft, The Netherlands
257
4.
Applications of Turbulence Models
Higher order turbulence modelling for industrial applications H . Grotjans, F. Menter, R. Burr, M . Gliick, AEA Technology GmbH, Otterfing, Germany
269
Flow and turbulence modelling in a motored reciprocating engine using a cubic non-linear k-&turbulence model S.A. Behzadi, A.P. Watkins, UMIST, Manchester, U.K.
279
A spectral closure for inhomogeneous turbulence applied to the computation of an engine related flow S. Parpais, M. Michard, Ecole Centrale de Lyon, Ecully, France, and Stt MCtraflu, Ecully, France, H. Touil, J.-P. Bertoglio, Ecole Centrale de Lyon, Ecully, France
289
Prediction of turbulent oscillatory flows in complex systems P. Tucker, University of Dundee, Dundee, U.K.
299
Numerical computation of turbulent flow around radome structures S. Majumdar, B.N. Rajani, National Aerospace Laboratories, Bangalore, India
309
Computations of turbulent flows using the V2F model in a finite element code R. Manceau, S. Parneix, Electricitt de France, Chatou, France
319
IntakdS-bend diffuser flow prediction using linear and non-linear eddy-viscosity and second-moment closure turbulence models N.E. May, Aircraft Research Association Ltd., Bedford, U.K.
329
Modelling of the flow in rotating annular flumes R. Booij,W.S.J. Uijttewaal, Delft University of Technology, Delft, The Netherlands
339
X
Second-moment closure predictions of turbulenceinduced secondary flow in a straight square duct B.A. Pettersson-Reif, Kongsberg Defence & Aerospace, Kongsberg, Norway, and Stanford University, Stanford, USA, H.Z.Andersson, Norwegian University of Science and Technology, Trondheim, Norway
5.
349
Experimental Techniques
Experimental investigation of coherent structures using digital particle image velocimetry J.v. Lukowicz, J. Kongeter, RWTH Aachen, Aachen, Germany
361
Measurements on the mixing of a passive scalar in a turbulent pipe flow using DPIV and LIF L. Aanen, J. Westerweel, TU Delft, Delft, The Netherlands
37 1
Wavelet patterns in the near wake of a circular cylinder and a porous mesh strip H. Hangan, G.A. Kopp, R. Murtinuzzi, The University of Western Ontario, London, Canada, A. Vernet, Universitat Rovia i Virgili, Tarragona, Spain
38 1
6.
Experimental Studies
An investigation of the engulfment mechanism in a turbulent wake G.A. Kopp, University of Western Ontario, London, Canada, F. Giralt, J.F. Keffer, University of Western Ontario, London, Canada, and University of Toronto, Toronto, Canada, J.A. Ferre', Universitat Rovira i Virgili, Tarragona, Spain
393
Investigation on boundary effects in jet flows" G.P. Romano, A. Sabatino, University "La Sapienza" Roma, Roma, Italy
403
Influence of shallowness on growth and structures of a mixing layer W.S.J. Uijttewaal, R. Booij, Delft University of Technology, Delft, The Netherlands
415
Influence of stationary and rotating cylinders on a turbulent plane jet J.F. Olsen, S. Rajagopalan, R.A. Antonia, The University of Newcastle, Callaghan, Australia
423
Turbulence measurements of an inclined rectangular jet in a boundary layer X. Zhang, University of Southampton, Southampton, U.K.
433
Pressure velocity coupling in a subsonic round jet C. Picard, J. Delville, UniversitC de Poitiers, Poitiers, France
443
Scalar mixing in variable density turbulent jets J.F. Lucas, M.Amielh, F. Anselmet, L. Fulachier, UniversitCs d' Aix-Marseille I & 11, Marseille, France
453
xi Decay of round turbulent jets with swirl D. Ewing, McMaster University, Hamilton, Canada Correlating structure of tip vortices and swirl flows
46 1
induced by a low aspect ratio rotor blade Y.O. Hun, Y.S. Kim, Yeungnam University, Kyungbuk, Korea
47 1
Effects of adverse pressure gradient on quasi-coherent structures in turbulent boundary layer T. Houra, T. Tsuji, Y.Nagano, Nagoya Institute of Technology, Nagoya, Japan
48 1
The semi-deterministicapproach as way to study coherent structures: Case of a turbulent flow behind a backward-facing step S. Aubrun, CERT-ONERA, Toulouse, France, P.L. Kao, H. Ha Minh, H. Boisson, IMFT, Toulouse, France
49 1
Experimental investigation of the coolant flow in a simplified reciprocating engine cylinder head D. Chartrain, A.-M. Doisy, M. Guilbaud, J.-P. Bonnet, UniversitC de Poitiers, Poitiers, France
50 1
Secondary flow in compound sinuousheandering channels Y. Muto, T. Zshigaki, Kyoto University, Kyoto, Japan
51 1
Diffused turbulence distortion by a free-surface M.A. Atmane, J. George, INP, Toulouse, France
52 1
7. Transition Surface curvature and pressure gradient effects on boundary layer transition B.J. Abu-Ghannam, H.H. Nigim, Birzeit University, Birzeit, Palestine, P. Kavanagh, Iowa State University, Ames, USA
533
Calculating turbulent and transitional boundarylayers with two-layer models of turbulence R. Schiele, F. Kaufmann, A. Schulz, S. Wittig, Universitat Karlsruhe, Karlsruhe, Germany
543
Turbulence modeling and computation of viscous transitional flows for low pressure turbines A. Chernobrovkin, B. Lukshminarayana, The Pennsylvania State University, University Park, USA
555
A general model for transition in wall-bounded compressible flows D. C. Weatherly, University of Kentucky, Lexington, USA
567
The use of a turbulence weighting factor to model by-pass transition J. Steelant, ESTECESA, Noordwijk, The Netherlands
577
xii
Modelling of separation-induced transition to turbulence with a second-moment closure I. Hadfic ', K. Hanjalic', Delft University of Technology, Delft, The Netherlands
587
The effect of a single roughness element on a flat plate boundary layer transition M . Zchimiya, Tokushima University, Tokushima, Japan
597
Features of laminar-turbulent transition in a free convection boundary layer near a vertical heated surface Yu. Chumakov, S. Nikolskaja, State Technical University, St. Petersburg, Russia
607
8.
Turbulence Control
On active control of high-lift flow A. F. Tinapp, W. Nitsche, TU Berlin, Berlin, Germany
619
A demonstration of MEMS- based active turbulence transitioning W. P . Liu, G.H. Brodie, Naval Surface Warfare Center, W. Bethesda, USA
627
Evolution of instabilities in an axisymmetric impinging jet S.V. Alekseenko, A.V. Bilsky, D.M. Markovich, V.I. Semenov, Siberian Branch of RAS, Novosibirsk, Russia
637
9.
Aerodynamic Flows
Assessment of eddy viscosity models in 2D and 3D shockhoundary-layer interactions T. Coratekin, A. Schubert, J. Ballmann, RWTH Aachen, Aachen, Germany
649
Assessment of explicit algebraic stress models in transonic flows T. Rung, H. Liibcke, M . Franke, L. Xue, F. Thiele, TU Berlin, Berlin, Germany, S. Fu, Tsinghua University, Beijing, PRC
659
Detached-eddy simulation of an airfoil at high angle of attack M. Shur, M . Strelets, A. Travin, Federal Scientific Center ,,Applied Chemistry", St. Petersburg, Russia, P.R. Spalart, The Boeing Company, Seattle, USA
669
Transition and turbulence modelling for dynamic stall and buffet W. Geissler, DLR, Gottingen, Germany, L.P.Ruiz-Calavera, INTA, Torrejon de Ardoz, Spain
679
Scrutinizing flow field pattern around thick cambered trailing edges: experiments and computation E. Coustols, G. Pailhas, ONERNCERT, Toulouse, France, Ph. Sauvage, AQospatiale, Toulouse, France
689
...
Xlll
Turbulent structure in the three-dimensional boundary layer on a swept wing M. Ztoh, Nagoya Institute of Technology, Nagoya, Japan, M . Kobayashi, Nippon Sharyo Ltd., Nagoya, Japan
699
Flowfield characteristics of swept struts in supersonic annular flow K.E. Williams, Adroit Systems Inc., Bellevue, USA, F.B. Gessner, University of Washington, Seattle, USA
709
10. Turbomachinery Flows Flow in a radial outflow impeller rear cavity of aeroengines X. Liu, Pratt & Whitney Canada Inc., Mississauga, Canada
72 1
Experimental investigation of turbulent wake-blade interaction in axial compressors A. Sentker, W. Riess, Universitat Hannover, Hannover, Germany
73 1
An experimental study of the unsteady characteristics of the turbulent wake of a turbine blade M. Ubaldi, P. Zunino, Universith di Genova, Genova, Italy
74 1
Experimental guidelines for retaining energy-efficient axial flow rotor cascade operation under off-design circumstances J. Vad, F. Bencze, Technical University of Budapest, Budapest, Hungary
75 1
11. Heat Transfer On prediction of turbulent convective heat transfer in rib-roughened rectangular cooling ducts A. Saidi, B. Sundkn, Lund Institute of Technology, Lund, Sweden
763
The measurement of local wall heat transfer in stationary U-ducts of strong curvature, with smooth and rib roughened walls H. Zacovides, D.C. Jackson, G. Kelernenis, B.E. Launder, UMIST, Manchester, U.K.
773
Studies of turbulent jets impinging on moving surfaces K. Knowles, Cranfield University, Shrivenham, U.K., T.W. Davies, University of Exeter, Exeter, U.K.
783
12. Combustion Systems Turbulence modelling in joint PDF calculations of piloted-jet flames J. Xu,S.B. Pope, Cornell University, Ithaca, USA
795
Towards a general correlation of turbulent premixed flame wrinkling K. Atashkari, M. Lawes, C.G.W. Sheppard, R. Woolley, University of Leeds, Leeds, U.K.
805
xiv Mixing in isotropic turbulence with scalar injection M. Elmo, J.P. Bertoglio, Ecole Centrale de Lyon, Ecully, France, V.A. Sabel'nikov, TSAGI, Moscow, Russia and CEAT, Poitiers, France
815
Modelling turbulent diffusion flames with full second-moment closures using cubic, realizable models W.T. Chan, Y. Zhang, UMIST, Manchester, U.K.
821
Advanced modeling of turbulent non-equilibrium swirling natural gas flames A. Hinz, T. Landenfeld, E.P. Hassel, J. Janicka, Technische Universit~it Darmstadt, Darmstadt, Germany
831
Effects of turbulence length scale on flame speed: a modelling study A. Lipatnikov, J. Chomiak, Chalmers University of Technology, G6teborg, Sweden
841
Application of a Lagrangian PDF method to turbulent gas/particle combustion M. Rose, P. Roth, Gerhard-Mercator-Universit~it Duisburg, Duisburg, Germany, S.M. Frolov, M.G. Neuhaus, Russian Academy of Science, Moscow, Russia
851
Large eddy simulation of a nonpremixed turbulent swirling flame N. Branley, W.P. Jones, Imperial College of Science, Technology and Medicine, London, U.K.
861
Large eddy simulation of a bluff body stabilised flame S. Sello, G. Mariotti, ENEL Ricerca, Pisa, Italy
871
Investigation of the effect of turbulent flow behaviour and mixing conditions on the combustion process in the homogeneous burnout zone of a small scale wood heater by numerical simulations and measurements S. Unterberger, H. Knaus, H. Maier, U. Schnell, K.R.G. Hein, Universit~it Stuttgart, Stuttgart, Germany
881
13. T w o - P h a s e F l o w s Proposal of a Reynolds stress model for gas-particle turbulent flows and its application to cyclone separators O. Kitamura, M. Yamamoto, Science University of Tokyo, Tokyo, Japan
893
A CRW model for free shear flows T.L. Bocksell, E. Loth, UIUC, Urbana, USA
903
Test of an Eulerian Lagrangian simulation of wall heat transfer in a gas-solid pipe flow R. Andreux, P. Boulet, B. Oesterld, Universit6 Henri Poincar6, Vandoeuvre, France
913
XV
Analysis and discussion on the Eulerian dispersed particle equations in non-uniform turbulent gas-solids two-phase flows S. Lafn, Martin-Luther-Universit~it Halle-Wittenberg, Halle, Germany, R. Aliod, University of Zaragoza, Huesca, Spain
923
Experimental research of modification of gridturbulence by rough particles M. Hussainov, A. Kartushinsky, Estonian Energy Research Institute, Tallinn, Estland, G. Kohnen, M. Sommerfeld, Martin-Luther Universit~it Halle-Wittenberg, Halle, Germany
933
Interaction between leading and trailing elongated bubbles in a vertical pipe flow C.A. Talvy, L. Shemer, D. Barnea, Tel Aviv University, Tel Aviv, Israel
943
AUTHOR INDEX
953
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xvii
PREFACE These proceedings contain the papers presented at the 4th International Symposium on Engineering Turbulence Modelling and Measurements held at Ajaccio, Corsica, France, in the period May 24 - 26, 1999. The symposium followed the previous three conferences on the topic of engineering turbulence modelling and measurements held in Dubrovnik, Yugoslavia, in 1990, Florence, Italy, in 1993 and Crete, Greece, in 1996. The proceedings of the previous conferences were also published by Elsevier Science. The purpose of this series of symposia is to provide a forum for presenting and discussing new developments in the area of turbulence modelling and measurements, with particular emphasis on engineering-related problems. Turbulence is still one of the key issues in tackling engineering flow problems. As powerful computers and accurate numerical methods are now available for solving the flow equations, and since engineering applications nearly always involve turbulence effects, the reliability of CFD analysis depends more and more on the performance of the turbulence models. Successful simulation of turbulence requires the understanding of the complex physical phenomena involved and suitable models for describing the turbulent momentum, heat and mass transfer. For the understanding of turbulence phenomena, experiments are indispensable, but they are equally important for providing data for the development and testing of turbulence models and hence for CFD software validation. Research in the area of turbulence modelling and measurements continues to be very active worldwide, and altogether 220 abstracts were submitted to the symposium and screened by experts in the field. 126 abstracts were accepted and 109 final papers were received and each reviewed by two experts. In the end, 86 papers were accepted, and most of these underwent some final revision before they were included in these proceedings. The papers were conveniently grouped in the following sections: Turbulence modelling Direct and large-eddy simulations Applications of turbulence models Experimental techniques Experimental studies Transition Turbulence control Aerodynamic flows Turbomachinery flows Heat transfer Combustion systems Two-phase flows
xviii The contributed papers are preceded by a section containing 5 invited papers covering various aspects of turbulence modelling and simulation, combustion modelling and visualisation and measurement techniques. The conference was organised under the auspices of the Assembly of World Conferences on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics. We should like to thank the President of the Assembly, Dr. R. K. Shah, for his help. The support and cooperation of the following institutions is also acknowledged: Electricit6 de France, DER, France University of Karlsruhe, Germany Commissariat/t L'energie Atomique, DRN, France We are grateful to the members of the Scientific and Organizing Committee for their various efforts in making this conference a success. We also acknowledge the help of many Fluid Mechanics experts from all over the world in reviewing abstracts and full papers for the conference. Finally, we express our sincere appreciation for the good cooperation provided by Mr. Dean Eastbury of Elsevier Science in the preparation of the proceedings.
W. Rodi and D. Laurence
0
Invited Lectures
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
Strategies for turbulence modelling and simulations P. R. Spalart ~ ~Boeing Commercial Airplanes, P.O. Box 3707, Seattle, WA 98124, USA This is an attempt to clarify the many levels possible for the numerical prediction of a turbulent flow, the target being a complete airplane, gas turbine, or car. These levels still range from a solution of the steady Reynolds-Averaged Navier-Stokes equations to a Direct Numerical Simulation, with Large-Eddy Simulation in-between. However recent years have added intermediate concepts, dubbed "VLES", "URANS" and "DES". They are in experimental use and, although more expensive, threaten complex traditional models especially for bluff-body flows, where three-dimensional simulations in two-dimensional geometries are flourishing. Turbulence predictions face two principal challenges: (I) predicting growth and separation of the boundary layer, and (II) providing accurate Reynolds stresses after separation. (I) is simpler, but makes much higher accuracy demands, and appears to give models of higher complexity almost no advantage. (II) is the arena for complex RANS models and the newer strategies. With some strategies, grid refinement is aimed at numerical accuracy; in others it is aimed at richer turbulence physics. In some approaches the empirical constants play a strong role even when the grid is very fine; in others, their role vanishes. For several decades, practical methods will necessarily be hybrid, and their empirical content will remain substantial. The law of the wall will be particularly resistant. Estimates are offered of the grid resolution needed for the application of each strategy to full-blown aerodynamic calculations, feeding into rough estimates of the feasibility date, based on computing-power growth. 1. I N T R O D U C T I O N The turbulence problem is of course far from solved, whether in terms of mathematical and intuitive understanding, or in terms of obtaining engineering accuracy in machines that depend on viscous fluid dynamics. Technological fields of global importance such as the airliner and automobile industries revolve around such devices. This economic stake motivates relentless, imaginative, and expensive efforts at turbulence prediction by any plausible approach. This should not defeat common sense, as argued elsewhere [1], and we must have visibility of when a method may progress from experimental to established and useful in engineering (research uses of simulation are another matter). Chapman made such predictions in 1979 [2], which still carry weight although his view of turbulence prediction in the 1990's is now recognized as optimistic. This paper focuses on the numerical prediction of turbulent flow regions. The equally difficult problem of transition prediction is mentioned only in passing. Physical testing methods in the transportation industry are beset by their own severe transition- and
turbulence-related difficulties. Tests with scale models usually imply both lower Reynolds numbers and higher freestream turbulence levels (in addition to blockage, bracket and mounting issues, and aero-elastic differences). The resulting scale effects can be misleading, with unforeseen reversals of the normal trend (by which higher Reynolds numbers bring better performance), especially as competing companies seek optimal aerodynamic designs; such designs have narrow margins and magnify the sensitivity to viscous effects. Thus, industry demands accuracy from Computational Fluid Dynamics (CFD), but not perfection. Also note that, whether in the airframe, turbine engine, or automotive industry, turbulence is not the only obstacle in CFD. Major numerical challenges remain between the state of the art and the routine calculation of flows over even moderately complex 3D geometries. These challenges relate not only to computing cost, but also to solution quality, particularly in terms of gridding. Presenting turbulence as the only "pacing item" in CFD might benefit research funding, but it is not accurate. On the other hand, many more capable people are engaged in grid generation, solvers, and pre- and post-processors, than in turbulence. Our effort may be unbalanced, although more duplication occurs on the programming side (it is easier to show progress in programming, let alone in code exercising, than in modelling). Sharing large codes is more difficult than sharing turbulence models, for which the equations (normally!) fit on one page. With a few exceptions, models have been freely published. The numerical strengths of CFD increase by the year thanks to the progress of computers and algorithms, whereas turbulence modelling c a n stagnate. If that is the case long enough, modelling will become the pacing item in important types of CFD in a matter of a decade or less, at least in the Reynolds Averaged Navier-Stokes (RANS) mode. It is then very sensible to examine approaches that trade "intelligence" (in the sense of powerful turbulence theories) for computing effort (unfairly described as "brute force"). It is my purpose to provide a viewpoint on such methods, which I predict will proliferate, and make a major contribution. The most stimulating issue may be the share between empiricism and numerical force in the eventually successful methods (w The concrete cost issues are addressed through a table attached to w
2. P H Y S I C A L A S P E C T S 2.1. R A N S m o d e l s The field of classical RANS turbulence modelling is active. At a recent biennial international symposium, about twenty-five papers presented new models or new versions of models [3]. These were offered for outside use, with varying degrees of sincerity and completeness in the description. No student of turbulence has the time to give each of these serious consideration. The full range of RANS methods is receiving work; this unfortunately testifies that no class of models has emerged as clearly superior, or clearly hopeless. Activity is not even restricted to differential methods; isolated groups are refining integral boundary-layer solvers, to allow more three-dimensionality and more separation. The same seems to apply to algebraic models. Eddy-viscosity transport models, being the simplest models that can be applied with a general grid structure, are now used extensively. The step back from two equations to a single equation has clearly not crippled the ap-
proach [4-6], while tangibly reducing the true cost of solutions. Conversely, models with up to four equations are in contention [7,8]. Perot & Moin's is especially intriguing. Having referred in the abstract to "Challenges (I) and (II)", I could add the following "Challenge Zero". Complete configurations often have laminar regions in their boundary layers; it is very helpful if a turbulence model can be "dormant" in such regions, meaning that its transport equations accept solutions with vanishing Reynolds stresses. Similarly, regions of irrotational and non-turbulent fluid, which are large in external aerodynamics, do not physically influence the turbulent regions such as boundary layers (weak freestream activity does have much influence on natural transition, but we leave transition prediction to a separate method). Again it is very helpful if the model accepts zero values in such regions, or small values without influence on the turbulent layers. At the same time, the model should allow the contamination of a laminar shear layer by contact with a strongly turbulent layer (contamination by moderate freestream turbulence is more subtle, and is within reach of only a few models). This all depends on the behaviour of the model at the turbulent/non-turbulent interface. In some models the stress level in the turbulent layer depends demonstrably on either the freestream values of the turbulence variables or, even worse, on the grid spacing at the interface. Few people have devoted attention to this question [9,10,5], and model descriptions sometimes make no mention of recommended freestream values (and also fail to demonstrate insensitivity). However, it happens that the k-e, SST and S-A models, which all three pass the freestream-sensitivity test, can fairly be described as "popular". Possibly, their tendency to give the same answer in different codes is valued by the users. In the perennial question of the choice of a second variable in two-equation models, freestream sensitivity should be given a high priority. It is much more important than the value of some high derivative at a solid wall. The model activation or "transition" from laminar to turbulent boundary layer is in fact more troublesome than the interface with irrotational fluid. Even though the S-A model was designed and tested for it, users still encounter premature transition. This occurs even with first-order upstream differencing, which at first sight should guarantee that unwanted nonzero values of eddy viscosity are "washed out" of the region upstream of the numerical trip. A factor in this is the steep variation of the eddy viscosity at transition [11]: a grid may be fine enough in both the laminar and turbulent regions but much too coarse at transition, creating oscillations which then propagate upstream. This problem can be solved with grid adaptation, but it remains an embarrassment to the model designers, especially since the transition process is not modeled accurately to start with [11]. The failure of most models to predict relaminarisation also causes frustration. While it is not reasonable to expect a model to predict transition in quiet environments, expecting relaminarisation is rather justified. In terms of Challenge (I), the different classes of models are surprisingly even. Within that challenge, we can include the prediction of skin friction and boundary-layer thicknesses (which dictate the parasite drag in the absence of separation), along with separation (which creates pressure drag). Integral methods and algebraic models have been so well optimised that surpassing them with any Navier-Stokes model is difficult. Reasons include the grid needed, the intrusion of artificial dissipation, and the constraints placed on the turbulence model such as locality, performance in free shear flows, and simplicity. It is geometric complexity and the drive towards massive separation, not lack of accuracy,
that are making integral and algebraic methods obsolete.
2.2. Simple R A N S models Appreciable improvements will be made to the simpler transport models, usually by adding new empirical terms such as for streamline curvature or for better anisotropy of the Reynolds-stress tensor (non-Boussinesq constitutive relations) which can, for instance, create secondary flows of the second kind in a square pipe [12]. This work is and should be very cautious, as it is highly desirable for new versions to preserve all or almost all of the past successes of a model. In other words, we hope for gradual progress on Challenge (II) but are unwilling to give up any of the accuracy and experience base in Challenge (I). For this reason, this author is very intent on limiting the number of versions of the S-A model, believing that it best serves the needs of the community. In addition, the rate at which new models or even versions are added to large 3D codes is unfortunately very slow. Codes become so large they are difficult to manage, and frequent changes in computer architecture divert the attention of the code custodians. This could cause a model to become entrenched, if it was first on the "market", and to dominate even after its accuracy has been surpassed. There is little dispute that the ultimate potential of eddy-viscosity models does not include separated flows over 3D geometries. In fact, 2D bluff bodies are sufficient to make them fail, even with sharp corners (certainly in steady mode, see w for the unsteady mode). The models are just too simple and replete with empiricism, and are trained in such a small pool of simple shear flows, that they have no reason to generalize to complex flows. We should however heed a remark of Hunt [13], which I slightly paraphrase: "It is important to note that in most flows (including those over bluff bodies) where the duration of a distortion is smaller than the intrinsic time scale of the turbulence, there is insufficient time for the turbulence to affect the m e a n flow and therefore an erroneous turbulence model has little effect on the m e a n flow. Thus, fortuitously, in most turbulent flows one-point models of turbulence only affect the mean flow calculations where the models are most appropriate (namely in shear flows where the intrinsic time scale is smaller than the distortion time scale)". Hunt appears to place all one-point RANS models, of any complexity, in the class of turbulence treatments that have "no reasonable claim" to provide accurate stresses in complex flows, but in many cases do "little enough damage". I can easily accept this assessment of my own models. An example is given by Ying et al., who compare measured and calculated Reynolds stresses over a multi-element airfoil [14]. This is the type of flow Hunt had in mind. As the shear layers (boundary layer and slat wake) pass over the trailing edge of the main airfoil element, the streamlines have a modest deflection, as part of the abrupt merging with the stream from below the trailing edge. The strain-rate tensor has an excursion that is n o t modest, and propagates to the calculated Reynolds-stress tensor. In contrast the measured Reynolds stresses show no such excursion, and their behavior is consistent with that of a "conserved quantity", with only a slow evolution in the streamwise direction. The anomaly in the computed stresses is due to the eddy-viscosity approach (the eddy viscosity is conserved, instead of the stresses). It certainly makes the experiment-computation comparison more delicate. On the other hand, the velocity profiles downstream show no
clear sequel of the stress excursion, as predicted by Hunt. Another wording of Hunt's line of thought is that quite a few flow regions that appear complex and 3D are dominated by "vortical inviscid" physics. It may well be the case for the "necklace" vortex at a wing-wall junction. Its characteristics may depend more on the upstream growth of the boundary layer, which a simple model can accurately predict, than on the Reynolds stresses inside it. This contrasts with the secondary vortices in a square duct, which are created by the turbulence. These vortices expose straight eddyviscosity models, but their practical importance is modest. Thus, a simple model can "get credit" for the successful calculation of a new flow, merely because the Reynolds stresses it generates in the complex regions are not damaging; usually, it is just as well if the stresses are too weak. T h e chances that the flow feature will be smeared due to insufficient grid resolution are also higher in such regions, because the user's experience base or willingness to manually refine the grid is less than in attached boundary layers. Unstructured adaptive grids will address that problem, but only in the next generation of codes. Hunt's optimism does not extend to primary bluff-body flows such as a stalled airfoil. In view of their limited prospects after separation, it is natural that most of the refinements applied to simple models will be aimed at their accuracy in boundary layers, including short separation bubbles and curvature, and a few thin shear flows.
2.3. C o m p l e x R A N S m o d e l s I conclude that the simpler transport models will remain useful and receive slight improvements, but that "something else" must be found before CFD becomes general. It is a matter of debate whether higher-quality models will provide that answer. I am primarily referring to Reynolds-Stress-Transport (RST) models. Now these models have a much closer connection to the equations, and boast several exact terms. An RST model would remove the anomaly noted in w with the sudden distortion over the trailing edge [14]. With proper attention to invariance, RST models should generalise from their "training ground" to flows with curvature, or vortex and similar flows, much more reliably than eddy-viscosity models. On the other hand, they also contain many empirical terms particularly in the pressure and dissipation areas, adjusted by trial and error. For some of these quantities, data can be obtained only from DNS which has been limited to simple geometries, although progress is being made. In addition, precise term-by-term matching is often too much to ask for; compensating errors, for instance between the anisotropy of the dissipation and pressure-strain tensors, appear both common and acceptable. In terms of the two challenges, RST models have a tentative advantage for Challenge (II), the separated and vortical regions [15]. They are usually far from "user-friendly" in the sense of Challenge Zero. For Challenge (I), incipient separation, no model can succeed without excellent empiricism, and it is no easier to impress such empiricism on a complex model than on a simple one [16]. In fact the exact character of certain terms puts them off limits to empiricism; in that sense, RST models are more difficult to "steer". A classic case is the use of vorticity instead of strain in production terms [17]. This step is neutral in thin shear flows, since both reduce to the shear rate, but it solves the long-standing problem of excessive turbulence levels in the approach to stagnation points. In RST and two-equation models, using vorticity is not legitimate, because the
exact production terms contains the strain rate instead. Typically, vorticity is used as a "temporary" expedient, which does nothing for the implicit hope that the dependence on empiricism will gradually decrease. The quandary seems intact as of 1998 [18]. Assessing true progress is made difficult by the constant modifications made in publications; the reader cannot be sure that the new version of the model can also succeed where the last version ~did. Another concern is the persistence of controversies such as about the use of "wall-reflection" terms or the question of whether RST models reproduce curvature effects without additional empirical modifications. Similarly, Zeman's study of free vortices implies that even RST models need specific curvature/rotation modifications to reproduce the damping of the turbulence [20]. Not only does this make the hope of an elegant resolution to Challenge II seem very remote, but streamline curvature is not a Galilean invariant [21], and therefore Zeman's model for that flow is not applicationready. Separated cases which are problematic for simple models, for instance strong shock interactions are also problematic with complex models [19]. Possibly, solutions with any model suffer from numerical errors in strong interactions. At the risk of minimizing the work of fellow modelers, I deem it unlikely that a single RANS model, even complex and costly, will provide the accuracy needed in the variety of separated and vortical flows we need to predict. The intellectual task of feeding all the available findings into a truly higher and durable version of a complex model is huge, and few model developers seem keen on doing it. Large groups tend to publish along "tentacles". This fits better with educational, institutional and funding needs than with the needs of the code writer who is in need of a robust, stable and understood formulation. It appears that Reynolds averaging suppresses too much information, and that the only recourse is to renounce it to some extent, which means calculating at least the largest eddies simply for their nonlinear interaction with the mean flow. This step appears desperate to observers, especially the mathematically oriented ones, with some reason. Prof. Jameson remarked that "we should not compute 1-centimetre eddies over a Boeing 747". My Boeing colleagues keep wishing for a "first-principle-based" turbulence model. 2.4. U R A N S
The first candidate beyond complex RANS modelling has been called "Very-Large Eddy Simulation" (VLES) or "Unsteady Reynolds-Averaged Navier-Stokes" (URANS). The URANS acronym is more descriptive. Such calculations rely on typical RANS models but are deliberately unsteady; for instance, vortex shedding is allowed past a bluff body [2224,18]. Durbin correctly notes that the Reynolds stresses created by the time averaging of the URANS solution overwhelm those carried by the model itself, in the separated region, and therefore remove much "responsibility" from that model [7]. Nixon's group used the acronym VLES for some very interesting work [25], but I would classify it as LES, and certainly not as URANS. I first note that URANS implies a separation of time scales, between that of the shedding and that of the residual turbulence, which is not indicated in measurements of spectra. This is a little disturbing, although some URANS calculations produce a kind of chaos (non-periodic behavior) which widens the spectral peak. Nevertheless, the approach has plausibility, and certainly improves results for bluff bodies; in the boundary layers, there is no strong additional reason to distrust the models because they operate in quasi-steady
mode. I also note that for bluff bodies, conducting unsteady calculations is optional only in somewhat artificial conditions: those in which a steady solution can be obtained by imposing symmetry, or a large time step, or using a Newton method for convergence. More frequently, a user that is after massive separation will find that the code simply cannot find a steady state, and that the only course is to operate in a time-accurate mode and analyze an unsteady solution, presumably a nearly-periodic one. A 2D URANS recognizes the role of time, but not of the spanwise coordinate z. By now, DNS and LES results for bluff bodies [26-29] have established that ignoring the z-dependence is not safe, although there are examples of successful 2D simulations with rather large separation [30]. At least in LES and DNS, 3D solutions in 2D geometries are highly justified. We denote them by "3/2D" in the table by w A z-dependence obviously belongs in a thorough study at the URANS and higher levels, but its role is clouded by several facts. Let us assume a 2D geometry. Then, the spanwise boundary conditions are arbitrary; periodic conditions are very plausible, but some studies use reflection conditions at the side boundaries. The size of the spanwise domain is also arbitrary. Systematic tests are costly, and a certain finding is that very narrow domains force the flow back to 2D, while very wide ones allow oblique vortex shedding, which is physically correct and has a noticeable effect. Intermediate domain widths could be explored for a long time. This issue will resolve itself in practice, in the sense that actual geometries are 3D, but it is an obstacle to a clear understanding of the nominally 2D flow. 2.5. L E S a n d D E S
Away from boundaries and without chemistry, Large-Eddy Simulation is well understood, and there is probably little to gain by refining the SGS models. In the wall regions, I have described most of the current LES work as Quasi-Direct Numerical Simulation (QDNS) [1]. By this I mean that these simulations resolve the near-wall "streaks", and the grid spacing is limited in wall units. A "true" or "full" LES, meaning that the Reynolds number based on the grid spacing is unlimited, appears to be a difficult goal (apparently the community switched to QDNS, although the pioneering work of Deardorff [44] was full LES, in order to reduce empiricism in the near-wall treatment). The grid spacing in all directions (or at least in the two directions parallel to the wall) would scale with the boundary-layer thickness. In this area, huge gains are expected from improved SGS modelling. However, it is unavoidable that empiricism will be added; at the least, such a treatment would have to imply values for the constants in the logarithmic law. The method reverts to quasi-steady RANS behaviour near the wall, in the sense that grid refinement does not eliminate the influence of the SGS treatment rapidly at all (until an extreme refinement turns the method into QDNS). A robust and accurate treatment of that kind is a plausible target. The streaks seem to be just as "numerous and universal" as the small eddies in free turbulence (the streaks are not isotropic, but calling the small eddies of free turbulence isotropic is misleading: the collection of eddies that are modeled as SGS in one grid cell at one time step obviously has preferred directions). It is only that the streaks have much more leverage than the small Kolmogorov eddies. It will be well worth the effort, for several reasons. First, the law of the wall is quite a robust feature of boundary layers, although we expect an erosion of its domain of validity, expressed as y/5, in strongly stimulated flows (5 is the boundary-
l0 layer thickness). Second, in 3D flows, the skin-friction vector is very unlikely to vanish; thus, the law of the wall could retain its validity even under a separating boundary layer. Finally, most of the difficulty in RANS modelling of strongly stimulated boundary layers resides in the outer region. There, LES can clearly capture effects such as straining, crossflow, and curvature. Therefore, LES addresses both Challenges, (I) and (II). However, it is at a considerable cost over RANS, and wall-bounded LES with the streaks modeled can be described as hybridized with RANS, although the implied empiricism is confined to a very shallow layer. At a recent LES workshop, a variety of QDNS and full LES methods were applied to fairly simple geometries with sharp corners [45]. In spite of these helpful features, the conclusions were particularly mixed, and did not make LES or even QDNS appear very mature. The flow past a circular cylinder, even at Reynolds numbers of a few thousand, has also led to quite different levels of agreement with experiment in the last few years. SGS models also remain quite different between different schools. In w I discuss why LES, even with the best wall-region treatment, is very far from affordable in aerodynamic calculations, and will be for decades. This is due to the large regions of very thin boundary layers, where 5 is of the order of 0.1% of the airfoil chord c. It led us to propose Detached-Eddy Simulation (DES), a further step in the hybridization of LES [1]. The idea is to entrust the whole boundary layer (populated with "attached" eddies) to a RANS model, and only separated regions ("detached" eddies) to LES. It is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not, and that LES is well understood away from walls. We show below that it leads to a manageable computing cost [31]. A typical application~0f DES is to a wing with a spoiler or a landing gear. Large areas of boundary layers are treated efficiently with quasi-steady RANS. Behind the spoiler, the momentum transfer is dominated by large unsteady eddies which are candidates for LES on two counts. First, they are not as numerous as the "horseshoe" vortices in the outer part of a boundary layer (let alone the wall streaks) and second, they are geometryspecific. An additional benefit of DES is its unsteady information. Though useless for many purposes, such as the range of the airplane, it will sooner or later be of great use for structural or noise studies. An attractive feature of DES is that it is simply formulated, and already being tested. This is not the case for similar concepts which have been informally envisioned. Many of them are zonal. DES is not, which I view as much preferable for routine use, and only requires a quick alteration of the S-A one-equation model. On the other hand deriving an efficient unsteady code, as needed for DES, from a steady one is not trivial. In a companion paper we present DES results for an airfoil at high angles of attack, a classical Challenge II example [31]. The agreement with experiment is surprisingly good, but no better than in the best examples of DNS and LES for bluff-body flows [27,28]. DES will require definite skills from the user in directing the grid resolution; however, RANS also benefits from careful grid generation. Presently, a few patient users are refining grids "manually" after exploring preliminary solutions [14], but I am afraid many solutions are under-resolved in the separated regions. Some similarities between DES and past treatments of the wall region in LES [41,42] have led to comments such as "DES contains nothing new". These comments stem from a
11 narrow focus on the classical applications of LES, such as channel flow. There, it is correct that DES is no more or less plausible than methods which blend simple buffer-layer models and simple SGS models. For instance, one could well use an eddy viscosity that is the smaller of the one given by the mixing-length approximation, with Van-Driest damping, and the Smagorinsky model. The capability DES has in addition is hpwever clear: to treat the entire boundary layer in RANS mode, to the standard of the better engineering models. A mixing-length model does not have this capability; the lowest level that does is an algebraic model such as Cebeci-Smith. Algebraic models do not lend themselves to complex geometries, unstructured grids, or to function under detached eddies; therefore, a one-equation model is the simplest type that makes DES practical. DES is young and has yet to be tested in a channel, with an LES grid; I fully expect reasonable results, but cannot predict how accurate the additive constant C in the logarithmic law will be. Note that in such a simulation, DES relaxes the restrictions on the wall-parallel spacing in wall units, such as Ax +, but not the wall-normal spacing which will have to be of the order of
y + - 1. I can formulate, but am not pursuing, still another hybrid concept. It is zonal and would consist in treating the "unchallenging" regions of the boundary layer with RANS, and the Challenge- (I) regions with full LES. The method would switch to LES upstream of separation, which tentatively makes another step in accuracy. The difficulties are: the "artificial intelligence" of identifying where the switch should be located; the generation of quality turbulent eddies for the RANS region to dispatch into the LES region (the regions might have to overlap). This is a concept that would live much more easily in a 2D boundary layer than on a 3D object. I believe that the "eddy seeding" problem is much less severe with DES, because a separated shear layer is exposed to vigorous new instabilities, thanks to both the removal of the wall confinement and what I loosely call "absolute instabilities" [43] (in contrast, the RANS-to-LES switch in the other concept would occur in a region of "convective instabilities"). 2.6. D N S
The value and requirements of Direct Numerical Simulation are well known. Few DNS projects have been conducted at a "full" Reynolds number, but the attachment line of swept wings is an example. DNS was applied at the (local) Reynolds number of the flow on an airliner. This is a case of "microscopic" simulation, in which it is justified to isolate a very small region of the flow (the justification relies on experiments). I once received dubious praise for simulating "a milli-second over a postage stamp"! Simulations of homogeneous turbulence and of other boundary layers could also be described as microscopic. DNS of a whole device is normally out of the question. It is a beautiful research tool; in fact I believe its reach is sometimes under-estimated, due to a misguided insistence on simulating at the "right" Reynolds number. The argument, which has long been a minority one, is the following. When asking a fundamental question in turbulence, assume we have the choice between a DNS and an LES having the same cost. The DNS will have a slightly larger range of scales in each direction (certainly less than double), because of saving the SGS model cost. The LES will have a higher Reynolds number; if a QDNS, the difference will be less than a doubling. The LES will assume that the unresolved eddies have a simple enough behaviour to be modeled;
12 for instance the great majority of the Reynolds shear stress will be resolved. If so, the same-cost DNS can run at a Reynolds number sufficient to sustain turbulence. Extrapolating the DNS results to the LES Reynolds number can also be done with confidence (especially if the DNS is possible over a range of Reynolds numbers). An extrapolation "after the simulation" is inexpensive and can be refined, much more easily than the LES can be re-run with an different SGS model. By that standard, we could have counted one run for a thorough DNS study, as opposed to maybe three runs for a thorough LES study, which changes the cost balance somewhat. Three to four runs is typical in LES studies, many of which are presented as comparative tests of SGS models and/or as tests of LES itself, by comparison with experiments. In contrast I believe a DNS study can be free-standing [32]. In addition, the extrapolation can then reach any Reynolds number (this amounts to the view that turbulence is more predictable, the higher the Reynolds number, which is not shared by all). Atmospheric scientists never consider DNS for the Planetary Boundary Layer, but fundamental PBL questions can very well be asked with DNS and extrapolation [33], and their attitude is counter-productive. One good reason for doing QDNS is the comparison with a laboratory experiment that is moderately out of reach of DNS. This occurs typically when the experiment was designed to allow measurements of the smallest eddies; physical limits restrict the possible range of scales, but not as severely as DNS does. Below, I classify DNS as requiring "no empiricism". This does not imply that the DNS of a complex flow is free of decisions once an accurate DNS code has been created. In the case of channel flow, the decisions consist in the grid spacing and the domain size. For both, the direction of "goodness" is clear: smaller spacing and larger domain. Homogeneous turbulence adds the influence of the initial conditions or stirring system, for which goodness is not simply a direction. There is an art, and people may disagree regarding its "color". Flows containing transition require many further decisions, regarding the freestream-disturbance and wall-imperfection content and the vibrations. This fine information is not found in the CAD file of an airplane or car. Engineering DNS would not be a "black box". Recent Reynolds-number increases in DNS have been almost unnoticeable, partly because the "super-computers" have nearly stagnated, certainly compared with personal computers. For a really attractive new study, for instance to lock the value of the Karman constant, a factor of 5 or preferably 10 in Reynolds number would be needed. Therefore the DNS effort has, correctly, be directed instead at simulating more complex geometries, or simple ones with strong pressure gradients, three-dimensionality, rotation and curvature, complex deformations, heat transfer, combustion, shock waves, and noise [34-40]. Fully successful DNS studies of the supersonic boundary layer should appear very soon. The current standard includes "reasonable results" but not quantitative comparisons, a problem being that low-Reynolds-number supersonic experiments are lacking [35]. A definitive study of the interaction with a normal shock will certainly be of great interest to the airliner industry.
2.7. Role of grid refinement In RANS and URANS, the equations possess a smooth exact solution, and the numerical solution approaches that solution as we refine the grid. The aim of grid refinement is
13 numerical accuracy. In contrast, in an LES, the Sub-Grid-Scale (SGS) model adjusts to the grid so that the smallest resolved eddies match the grid spacing. In a finer grid, resolving eddies to a smaller size gives the large energy-containing eddies more eddies for genuine nonlinear interactions, making them more accurate. The aim of grid refinement is now physical instead of numerical, to use simple words. This distinction is tracked in the table in w (several methods had to be labeled "hybrid", as their aim is different in different flow regions). Another description of it is that when the aim is numerical, the turbulence/SGS model does not depend on the grid spacing but when the aim is physical, it does. A consequence is that in URANS, no amount of grid refinement will override the influence of the empirical content of the turbulence model. In contrast, in a method with the "physical" aim, grid refinement weakens the role of the modeled eddies and thus improves the fidelity of the simulation. A 20% change of the Smagorinsky constant in a well-resolved LES is minor, but a 20% change in the Karmnan constant is not. The character of grid refinement has implications for adaptive grid approaches. When the aim is numerical, error estimates are constructed and guide the refinement. When the aim is physical, error estimates are not simple. In fact the question of the "order of accuracy" of LES has apparently not been asked. In the classical LES, with cut-off in the inertial range, the resolved kinetic energy converges to order 2/3. However the full kinetic energy can be recovered and converges much faster, as does the Reynolds shear stress, because it has a steeper spectrum, possibly - 7 / 3 instead o f - 5 / 3 . The shear stress has more impact in practice, and is probably a better measure (in isotropic turbulence, the low end of the spectrum is a fair model of the "more relevant" quantities, and must converge much faster than to order 2/3). Grid adaptation in LES will be a field of study; we could adapt instantaneously (follow eddies) or gradually, based on the history of a region in creating small eddies. Truly balancing the allocation of numerical effort within a complete flow will be a great achievement. So far, our approach to grid design in LES has been very intuitive. An example is the preference for cubic grid cells, away from walls. This will not be viable in an industrial tool. 3. C O S T A S P E C T S The principal definitions and assumptions which entered the estimates in Table 1 are the following. The target flow is that over an airliner or a car. The acronyms have all been used above. "Aim" refers to the aim of grid refinement, numerical or physical (w The Reynolds-number dependence refers to the number of grid points. The step from "strong" to "weak" Reynolds-number dependence indicates a change from a slow logarithmic dependence similar to that of the skin-friction coefficient, to a strong one similar to that of the boundary-layer thickness in wall units. "3/2D" refers to simulations which are 3D even if the geometry is 2D. When the geometry is 3D, 3/2D means that the grid spacing scales with the shorter dimension of the device, and does not "take advantage" of high aspect ratios. A clear example would be a wing flap: a 3DRANS will cluster points near its tips but use a loose spacing elsewhere, whereas a 3/2D method will space point by a fraction of the flap chord all along. The step from "strong" to "weak" empiricism indicates, quite arbitrarily, that the only remaining adjustable constants are those in the Law of the Wall.
14
For the grid spacing, RANS and DES figures are based on current practice. The requirements are well understood for the spacing normal to the wall. In the other directions, the geometry is assumed to have only a moderate number of features such as flaps and spoilers. Under "DES" I include both strict DES as defined in [1], and other hybrid methods which are likely to be actively developed in the next few years, with the general expectation that they will treat the simple attached boundary layers with RANS. For such methods, a grid block of the order of 643 points appears adequate to resolve a separated region, since we used about this many points on the stalled airfoil [31]. At higher Reynolds numbers, the cost increase is modest, since only the normal grid spacing needs to be reduced. Thus, the grid increase over 3DRANS is plausibly in the millions of points, not tens of millions, and l0 s is fair for the grid count. For LES we had estimated 1011 for a clean wing [1], leading to a few times more for the whole aircraft. The DNS estimate is based on grid patches with an area of 100 wall units, and agrees with that of Moin & Kim [46]. The number of steps uses the same grid information and CFL numbers of order 1, and assume the simulation needs roughly 6 spans of travel for an airplane. The readiness estimates are based on the "rule of thumb" that computer power increases by a factor of 5 every 5 years. This will be disputed, but another rule has been a factor of 2 every 2 years, which is not much faster. You are free to apply your favorite rate of progress, starting from the assumption that a very expensive problem today costs about 1015 floating-point operations. "Readiness" roughly means that a simulation is possible as a so-called "Grand Challenge". Industrial everyday use will come later. 4. O U T L O O K Progress in numerical methods and computers is intensifying the challenge for turbulence treatments, to provide a useful level of accuracy in massively separated flows over fairly complex geometries at very high Reynolds numbers. This is desirable in the near future, between 5 and 10 years, and not only on a research basis; industry is more than ready for this capability. In addition, the needs of non-specialist users and automatic optimizers dictate a very high robustness. Flows with shallow or no separation appear to be within the reach of the current steady RANS methods or rather their finely calibrated derivatives, incorporating modest improvements such as nonlinear constitutive relations. For such flows, transition prediction with generality, accuracy, and robustness may well
Table 1 Summary of strategies Name Aim Unst. 2DURANS num yes 3DRANS num no 3DURANS num yes DES hybr yes LES hybr yes QDNS phys yes DNS num yes
Re-dep. weak weak weak weak weak strong strong
3/2D no no no yes yes yes yes
Empiricism strong strong strong strong weak weak none
Grid 105 107 107 l0 s 10115 1015 1016
Steps 103.5 103 103.5 104 106.7 107.3 107.7
Ready 1980 1985 1995 2000 2045 2070 2080
15 prove more challenging than turbulence prediction. With massive separation, it appears possible we will give up RANS, steady or unsteady. This will probably be the major debate of the next few years. The alternative is a derivative of LES, in which the largest unsteady geometry-dependent eddies are simulated and (for most purposes) "discarded" by an averaging process. We have to balance our ambitions with cost considerations, and I tentatively provided a table summarizing the issue. A major consideration is whether LES is practical for the entire boundary layer, and I strongly argued that it will not be, in the foreseeable future. This forces hybrid methods, with quasi-steady RANS in the boundary layer. I have effectively "defined" LES as a simulation in which the turbulence model is tuned to the grid spacing, and RANS as the opposite. Other more subtle definitions probably exist, but this one seems to classify almost all the studies to date. Speziale's hybrid proposal involves the grid spacing and the Kolmogorov length scale but, surprisingly, not the internal length scale of the RANS turbulence model; thus, it is difficult to classify [47]. Variations on the now-running DES proposal clearly have a wide window of opportunity. The plausible spread of hybrid methods highlights the durability if not the permanence of a partnership between empiricism and numerical power in turbulence prediction at fullsize Reynolds numbers. This demands a balance in funding and in publication space. Since hybrid methods offer much leeway in the boundary between "RANS regions" and "LES regions", the more capable the RANS method is, the lower the cost of the calculation will be. Therefore, the switch to LES in some regions does not remove the incentive to further the RANS technology. This scene also raises the issue of which core of experiments will be the foundation of the empirical component of the system. As ever, we will need simple flows for calibration of the RANS sub-system, and more complex flows for validation of the full CFD system.
Acknowledgements I am grateful for the comments of J. Crouch, W-H. Jou, G. Miller, A. Secundov, and M. Strelets, and for years of informal teaching by P. Bradshaw.
REFERENCES 1. P. R. Spalart, W.-H. Jou, M. Strelets and S. R. Allmaras, 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, 1997, Ruston, LA. In "Advances in DNS/LES", C. Liu and Z. Liu Eds., Greyden Press, Columbus, OH. 2. D.R. Chapman, 1979. Computational aerodynamics development and outlook. AIAA J. 17, 12, 1293. 3. F. Durst, B. E. Launder, F. W. Schmidt, J. H. Whitelaw, M. Lesieur and G. Binder, eds., 1997. l l t h Symp. Turb. Shear Flows. Sept. 8-10, Grenoble, France. 4. A.N. Gulyaev, A. N. Kozlov, V. Ye and A. N. Secundov, 1993. Universal turbulence model ~t-92. Ecolen Science Research Center Preprint No. 3, Moscow. 5. P.R. Spalart and S. R. Allmaras, 1994. A one-equation turbulence model for aerodynamic flows. La Recherche A drospatiale, 1, 5. 6. D.C. Wilcox, 1998. Turbulence modeling for CFD. http://webknx.com.dcw.
16 P. A. Durbin, 1995. Separated flow computations using the k - c - v 2 model. A I A A Y. 33~ 4, 659. B. Perot and P. Moin, 1996. A new approach to turbulence modeling. Proc. 1996 Summer Prog., Ctr for Turb. Res., Stanford. F. R. Menter, 1992. Influence of freestream values on k-w turbulence model predictions. A I A A J. 30, 6, 1657. 10. J. B. Cazalbou, P. R. Spalart, and P. Bradshaw, 1994. On the behavior of two-equation models at the edge of a turbulent region. Phys. Fluids A, 6 (5), 1797. 11. P. R. Spalart, 1995. Topics in industrial viscous flow calculations. Coll: Transitional Boundary Layers in Aeronautics, Royal Neth. Acad. of Arts and Sciences, Dec. 6-8. 12. C. G. Speziale, 1987. On nonlinear K-I and K-c models of turbulence. J. Fluid Mech., 178, 459. 13. J. C. R. Hunt, 1990. Developments in computational modelling of turbulent flows. Proc. ERCOFTAC Work., 26-28 March, Lausanne, Switz. Cambridge U. Press. 14. S. X. Ying, F. W. Spaid, C. B. McGinley and C. L. Rumsey, 1998. Investigation of confluent boundary layers in high-lift flows. AIAA 98-2622. To appear, J. Aircraft. 15. B. E. Launder, 1988. Turbulence modelling in three-dimensional shear flows. AGARD CP-438, Fluid Dyn. 3D Turb. Shear Flows and Transition, Oct. 3-6, Turkey. 16. F. S. Lien and M. A. Leschziner, 1995. Modelling 2D separation from a high lift aerofoil with a non-linear eddy-viscosity model and second-moment closure. Aero. J.. 17. M. Kato and B. E. Launder, 1993. The modelling of turbulent flow around stationary and vibrating square cylinders. Ninth Symp. Turb. Shear Flows, Kyoto. 18. G. Bosch and W. Rodi, 1998. Simulation of vortex shedding past a square cylinder with different turbulence models. Int. J. Num. Meth. Fluids 28, 601. 19. T. HellstrSm, L. Davidson and A. Rizzi, 1994. Reynolds stress transport modelling of transonic flow around the RAE2822 airfoil. AIAA 94-0309. 20. O. Zeman, 1995. The persistence of trailing vortices: a modeling study. Phys. Fluids A 7, 135. 21. P. R. Spalart and M. Shur, 1997. On the sensitization of simple turbulence models to rotation and curvature. Aerosp. Sc. and Techn., 1, 5, 297. 22. S. H. Johansson, L. Davidson and E. Olsson, 1993. Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a k-e turbulence model. Int. J. Num. Meth. in Fluids 16, 859. 23. S. A. Orszag, V. Borue, W. S. Flannery and A. G. Tomboulides, 1997. Recent successes, current problems, and future prospects of CFD. AIAA-97-0431. 24. M. L. Shur, P. R. Spalart, M. Kh. Strelets and A. K. Travin, 1996. Navier-Stokes simulation of shedding turbulent flow past a circular cylinder and a cylinder with a backward splitter plate. Third Eur. CFD Conf, Sept. 1996, Paris. 25. R. E. Childs and D. Nixon, 1987. Turbulence and fluid/acoustic interaction in impinging jets. SAE 87-2345. 26. G. E. Karniadakis and G. S. Triantafyllou, 1992. Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech. 238, 1. 27. F. M. Najjar and S. P. Vanka, 1995. Effects of intrinsic three-dimensionality on the drag characteristics of a normal flat plate. Phys. Fluids 7 (10), 2516. 28. R. Mittal and S. Balachandar, 1995. Effect of three-dimensionality on the lift and .
o
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17 drag of nominally two-dimensional cylinders. Phys. Fluids 7 (8), 1841. 29. S. A. Jordan and S. A. Ragab. 1998. A large-eddy simulation of the near wake of a circular cylinder. J. Fluids Eng. 120, 243. 30. J. C. Muti Lin and L. L. Pauley, 1996. Low-Reynolds-number separation on an airfoil. A I A A J. 34, 8, 1570. 31. M. Shur, P. R. Spalart, M. Strelets and A. Travin, 1999. Detached-eddy simulation of an airfoil at high angle of attack. 4th Int. Symposium on Eng. Turb. Modelling and Measurements, May 24-26, Corsica. 32. P. R. Spalart and, M. Kh. Strelets, 1997. Direct and Reynolds-averaged numerical simulations of a transitional separation bubble. 1lth Symp. Turb. Shear Flows, Sept. 810, Grenoble, France. 33. G. N. Coleman, J. H. Ferziger, and P. R. Spalart, 1990. A numerical study of the turbulent Ekman layer. J. Fluid Mech. 213, 313. 34. M. Alam and N. D. Sandham, 1997. Numerical study of separation bubbles with reattachment followed by a boundary layer relaxation. Parallel CFD 97. Elsevier. 35. N. A. Adams, 1998. Direct numerical simulation of turbulent compression ramp flow. Theor. Comp. Fluid Dyn. 12, 102. 36. G. N. Coleman, J. Kim and R. D. Moser, 1995. A numerical study of turbulent supersonic isothermal-wall channel flow: J. Fluid Mech. 305, 159. 37. S. Gavrilakis, 1992. Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct. J. Fluid Mech. 244, 101. 38. Y. Na and P. Moin, 1998. Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 370, 175. 39. S. Ragab and M. Sreedhar, 1995. Numerical simulation of vortices with axial velocity deficits. Phys. Fluids 7 (3), 549. 40. K-S. Yang and J. H. Ferziger, 1993. Large-eddy simulation of turbulent obstacle flow using a dynamic subgrid-scale model. A I A A J. 31, 8, 1406. 41. U. Schumann, 1975. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annul:. J. Comp. Phys. 18, 376. 42. E. Balaras, C. Benocci and U. Piomelli, 1996. Two-layer approximate boundary conditions for large-eddy simulations. A I A A J. 34, 6, 1111. 43. P. Huerre and P. A. Monkewitz, 1985. Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151. 44. J. W. Deardorff, 1970. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41,453. 45. W. Rod:, J. H. Ferziger, M. Breuer and M. Pourqui~, 1997. Status of Large Eddy Simulation: results of a workshop. J. Fluids Eng. 119, 248. 46. P. Moin and J. Kim, 1997. Tackling turbulence with supercomputers. Scient. Amer. 276, 1, 62. 47. C. G. Speziale, 1998. Turbulence modeling for time-dependent RANS and VLES: a review. AIAA J. 36, 2, 173.
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
19
Modelling Shock-affected Near-wall Flows with Anisotropy-Resolving Turbulence Closures M. A. Leschziner, P. Batten and H. Loyau, Department of Mechanical Engineering, UMIST P.O. Box 88, Manchester M60 1QD, UK
This paper reviews some recent research aiming to assess the performance of advanced forms of second-moment closure and non-linear eddy-viscosity models for compressible flows, with particular emphasis placed on strong shock-boundary-layer interaction involving separation. This topic is of particular relevance to high-speed aerospace and turbomachinery aerodynamics. Relevant closure forms and their merits, relative to simpler liner eddy-viscosity approximations, are reviewed first, mainly in qualitative terms. The performance of some particular models is then examined by reference to solutions for a range of 2D and 3D compressible flows.
1. I N T R O D U C T I O N The aerodynamic behaviour of a high-speed flow over a carefully streamlined aerodynamic body or turbomachine blade operating at its design condition is usually dictated by a balance of inviscid processes, with turbulence playing an important role only in the thin, normally attached layer developing on the component 's surface and in the component's wake. The prevalence of such flows in aerospace and turbomachinery, in which a single shear strain dominates the flow structure, has promoted the formulation and use of 'simple' algebraic, one-equation and two-equation eddy-viscosity turbulence models which are carefully tuned to return the correct level of shear stress - the only one of importance to the mean flow in the shear layer. There are, however, important sets of circumstances in which the near-wall structure of high-speed flows is much more complex, and in which turbulence can materially affect, directly or by indirect interaction, the aerodynamic performance of the component in question. The most challenging conditions arise in transonic and supersonic flows in which shocks interact with boundary layers, causing substantial changes not only to the boundary layers themselves, but also to inviscid portions of the flow. Although shock systems can be highly complex, especially if provoked by obstacles subjected to supersonic flow (Fig. l(a)), the most frequent case involves a single incident shock (Fig. l(b)) - for example, over a wing in transonic flow, on a ramp in supersonic flow or in a passage formed by blades in which the flow is choked.
20
Figure 1: Visualisation of complex shock/boundary layer interactions. (Figs. courtesy J. Ddlery, ONERA) Viewed in terms of boundary-layer properties, a shock may be regarded, essentially, as a region of steep adverse pressure gradient causing rapid deceleration. The details of the interaction itself are not trivial, even in the (nominally) 2D normal-shock conditions shown in Fig. l(b): the shock splits into an oblique leg and a weaker normal leg, giving it the appearance of a 'lambda'. The oblique leg is the one primarily interacting with the boundary layer, causing it to thicken and possibly separate. Following the oblique shock, the flow above the thin region very close to the wall finally decelerates to a subsonic state through the weak normal leg of the lambda-shock structure. If the incident shock is sufficiently strong, the boundary layer separates, and there arises a long and relatively thin recirculation region which has a profound effect on the pressure distribution on the wall and which substantially affects the shock position. Thus, the interaction has important viscous as well as inviscid elements. In 3D conditions, some very complex interaction patterns arise, as will emerge later by reference to computational solutions for a wallmounted fin and in a swept bump in a channel. From a computational and modelling point of view, compressible flows, in general, and shock/boundary-layer interaction, in particular, pose substantial challenges. Although some features of this interaction are also encountered in other boundary layers subjected to smoothly-varying adverse pressure gradient, there are aspects which make the prediction of shock-affected flows especially difficult. There is, first, the question of numerical accuracy and shock-capturing in a mixed elliptic/hyperbolic flow in which the representation of acoustic characteristics is of considerable importance. Modern implicit upwind methods, based on bounded, Riemann-solver technology and higher-order characteristic interpolation, are now generally regarded as offering an accurate and stable numerical framework. However, their complexity, especially when applied to general 3D turbulent flows, has encouraged the use of simpler, arguably less adequate approaches, such as Jameson-type methods, and compressibility-extended SIMPLE-type schemes, the latter originally formulated for incompressible, elliptic flow. As regards turbulence, one important process is the large normal straining associated with the shock in the transonic part of the flow and the steep compression/expansion waves in the subsonic boundary layer. Both cause turbulence to be generated in proportion to the product of streamwise normal stress and normal strain. Hence, turbulence anisotropy, which is observed to be especially high in the interaction region, can be expected to exert a material influence on the turbulence energy. While the response of the boundary layer
21 to the shock is dictated, principally, by the level of the shear stress, this level is sensitive to both the turbulence energy and normal-stress anisotropy. This is especially so if separation is provoked, in which case anisotropy-related curvature/turbulence interaction in the separated shear layer can be expected to influence the reattachment process and post-reattachment recovery. Another potentially influential issue is the sensitivity of the eddy structure (e.g. spanwise boundary-layer streaks), the cascade mechanism and the dissipative processes to compressibility, in general, and shocks, in particular. It is well known that the spreading rate of a shear layer reduces substantially with increasing Mach number beyond about M=0.7, a fact attributed to dilatational processes. While one might expect a similar sensitivity to prevail near a wall, this is not, in fact, born out by DNS studies [33, 22] and modelling efforts [31], and this has been linked to the very different levels of turbulence anisotropy in free and wall-bounded shear layers, especially the low level of wall-normal fluctuations. In view of the above considerations, one might assume that only the most elaborate turbulence models, specifically those resolving anisotropy and its interplay with the shear stresses, would be able to give an adequate representation of the complex interaction processes in shock-affected flows. However, the long history of using simple models in aerospace, the desire for economy and robustness, and the fact that, even in complex separated conditions, the turbulence-affected shear layers tend to remain relatively thin and to occupy a minor proportion of the whole flow, have combined to encouraged an ethos of devising flow-class-specific extensions to models formulated, principally, for nearequilibrium, incompressible, attached shear layers, so as to make the models applicable to much more complex strain fields. The success of this approach has been distinctly mixed. There have been several substantive validation exercises aiming to categorise the performance of a wide variety of models for compressible flows. Several CEC-funded projects are among them, e.g. EUROVAL [26], ETMA [20] and ECARP [27]. Moreover, a recent ERCOFTAC Workshop focused specifically at 2D shock-boundary layer interaction [8]. Perhaps the most definitive- certainly most carefully-controlled- set of studies was undertaken by Bardina et al. [5] (see also Huang [32]). This examined, in great detail, the performance of four substantially different eddy-viscosity models for a range of flows, including several incompressible and compressible thin shear flows, and two transonic flows with shock-induced separation (Bachalo & Johnson's bump flow [3] and the RAE 2822 aerofoil [12]). Of the models investigated, only the shear-stress transport (SST) variant of Menter [52] - a blend between the k-e and the k-co models - performed adequately in respect of shock/boundary-layer interaction. However, like other eddy-viscosity models, this is unable to capture anisotropy and the multiplicity of associated effects, and its favourable performance reflects careful tuning and the introduction of a shear-stress limiter. This latter feature is extremely influential because, as already noted, the shearstress level is the key to predicting the mean flow in what essentially remains, despite the separation zone which is elongated and hugs the wall, a thin shear flow. Extensive experience with modelling a wide range of incompressible flows (e.g. [43, 29]), demonstrates that, in complex strain associated with strong curvature, swirl, separation and impingement, anisotropy-resolving closures often return distinctly superior performance to that of isotropic eddy-viscosity models. Some limited experience with 'simple' second-moment models (high-Re variants using linear pressure-strain proposals)
22 applied to shock-induced separation (e.g. [10, 34, 44, 46, 47, 66]) suggests improvements over conventional eddy-viscosity models similar to those achieved with Menter's SST variant. To a degree, these improvements reflect the fact that second-moment closure implies, in simple shear, a value for the eddy-viscosity coefficient C, which is somewhat lower than the standard value 0.09. Similarly, the SST model partly derives its favourable characteristics from a reduction in C, through its functional dependence on the vorticity magnitude. While this might be interpreted as an argument in favour of using 'corrected' linear eddy-viscosity models, it must be borne in mind that the improved performance of such models is tied up with the peculiar nature of relatively simple, thinly separated flow regions. More complex cases, involving 3D straining and impingement, are far more challenging, and it is arguable that only general approaches, involving higher-order closure practices, are likely to have a wide range of applicability. This paper reviews some recent studies which aimed to assess the predictive performance of non-linear eddy-viscosity and full second-moment-closure models for compressible 2D and 3D flows, with particular emphasis placed on shock-induced separation. Importantly, these efforts did not simply confine themselves to the complex applications of principal interest, but followed a route starting with homogeneous flows and progressing through a sequence of increasingly complex 2D and finally 3D flows. It turns out, as is almost invariably the case with turbulence-modelling studies, that conclusions on predictive performance cannot be stated in simple, monochromatic terms. Nor is it possible to track down unambiguously what precise mechanisms are responsible for particular improvements or predictive characteristics; any model is a 'package' rather than a simple superposition of fragments. All that can be offered, at present, is a cautiously favourable view of anisotropy-resolving models on the basis of a wide-ranging validation. 2. M O D E L L I N G
ISSUES
2.1 C o m p r e s s i b i l i t y effects
Virtually all turbulence models in existence have been derived and calibrated, at least in their original form, by reference to incompressible flows. Hence, an issue which needs to be addressed, before any particular model variant is considered in detail, is the effect of compressibility on the governing equations. The usual starting point for modelling compressible flow is the density-weighted (Favre-averaged) form of the RANS and turbulence-closure equations. Consideration must next be given to the dependence of the (incompressible) closure approximations on mean-density gradients, density fluctuations and the non-divergent nature of the velocity field which leads to additional dissipationand pressure-dilatational terms in the turbulence-energy equation. Huang et al. [31] have considered in some detail the influence of density gradients on the quality with which k-e , k-w and stress-transport models, in their incompressible form, predict the (compressible) law of the wall U+(y +) and have argued that model coefficients, specifically those in the length-scale determining equation, need be made functions of the mean-density gradient. However, the strength of this interaction is modeldependent, as well as being only significant at friction Mach numbers greater than about 0.1, corresponding to a mean Mach number of order 3. As regards the effects of density fluctuations, Huang et al [31] and Coakley et al., [11] demonstrate that turbulent-Mach-
23 number-dependent corrections proposed by Wilcox [69], Zeman [70], Sarkar [58] and E1 Baz et al. [21], while performing qualitatively correctly for free shear layers, actually tend to worsen predictive quality in near-wall flows. This observation is consistent with DNSbased studies (e.g. Huang et al. [33], Friedrich & Bertolotti [22]) which indicate that the interaction between compressibility and near-wall turbulence is substantially weaker than that in free shear layers, as well as being somewhat different in nature. In view of the above considerations, and the fact that attention focuses here on near-wall flows at relatively modest Mach numbers (lower than 2), models used below do not incorporate any specific compressibility-related elements beyond those associated with density-weighted averaging. 2.2 S e c o n d - m o m e n t closure
All second-moment models consist, in essence, of transport equations (or simplified algebraic approximations thereof) for the individual components of the Reynolds-stress tensor" _
1!
I!
Opu i ttj Ot
_~__
i!
1!
u p u i ttj tt k +
Oxk
=
+
+
-
(1)
All share one common key element" the (formally) exact representation of stress generation, Piy, which is of primary importance for predicting anisotropy and the distinctly different levels of sensitivity of turbulence to different types of strains. Despite this commonality, there are important differences among second-moment models, especially in respect of pressure-strain interaction, r and rate of dissipation, Q'j, which can have a decisive impact on their respective predictive performance. The most basic form of second-moment closure is that of Launder et al. [38] or Gibson & Launder [25], which is applicable to fully-established turbulence only and employs a linear approximation for the pressure-strain terms, in combination with influential nearwall corrections, and an isotropic rate of dissipation. This is by far the most intensively investigated form for complex flows, and variants of this model have also been applied to shock-affected flows, either in conjunction with wall functions or low-Re eddy-viscosity models for the semi-viscous near-wall region (e.g. Vandromme et al. [65], Davidson [18], Lien & Leschziner [47], Leschziner et a/[44]). An overall conclusion emerging from these studies is that second-moment closure predicts a higher level of sensitivity of boundary layers to strong shocks, relative to k-e eddy-viscosity models, the former being in closer accord with reality. In itself, this conclusion is perhaps not of pivotal importance, since, as stated in Section 1, linear eddy-viscosity models can be modified to yield correspondingly good predictions (Bardina et al. [5]). What is important, however, are the implications for more complex flows involving impingement, separation, complex 3D strain and high curvature, where all Reynolds stresses are dynamically active and contribute in equal measure to momentum transport. Unfortunately, it is here that limitations of the above simple second-moment models can become constraining - specifically, their reliance on influential near-wall corrections which contain wall-normal distances, and on lower-order near-wall models to represent viscous effects.
24 Recent developments in second-moment modelling have focused on two areas: nonlinear approximations for the pressure-strain and stress-dissipation processes, aiming to correctly steer the stresses and the related dissipation rates towards the two-component wall limits without large, topography-related wall corrections, and on extensions to lowRe conditions. While several low-Re second-moment models have been proposed in recent years, some linear [36, 60, 61] and others non-linear [40, 41, 15, 62], these have not been widely adopted, and only few applications to complex, non-attached flows have been undertaken. Even rarer are studies of shock-affected flows with such closure forms (Batten et a/J7], Vallet and Gerolymos [64]), mainly because of numerical difficulties, associated especially with the near-wall region, and the high resource requirements. Recent work at UMIST (e.g. Launder and Li [41], Craft and Launder [15]) has focused on the design of low-Reynolds-number closures which represent adequately the near-wall state with much weaker explicit wall corrections, owing to the use of a cubic pressurestrain model. In what follows we have used a variant of the cubic pressure-strain model, obtained originally by Fu et al. [23] from imposed constraints such as symmetry, tensorial consistency, zero-trace and satisfaction of the two-component limit. In the construction of this model, extensive use is made of the stress invariants, A2 -- aijaij,
A3 = aijajkaki,
A = 1 - 9/8(A2 - A3).
(2)
The last of these invariants - Lumley's stress-flatness parameter, A [51] - has the especially useful property of assuming the value 1 in isotropic turbulence and 0 in the two-component limit. Whilst much more elaborate than the early linear forms, this non-linear model offers improved predictive characteristics in a variety of complex strain fields. Some inhomogeneity corrections are still included to obtain the best possible prediction of the near-wall anisotropy, but these do not involve wall-normal distances or vectors, relying rather on the local, normalised length-scale gradients [15], =
= 0 1
1
-~ + ( N~ N~ ) ~
(3)
Oxi
The vector, dA, which provides information on both the direction and magnitude of inhomogeneities, is used to modify the 'slow' part of the pressure strain model. In particular, dA become unit 'normal-to-wall' vectors close to a solid surface where the length scale and flatness parameter undergo significant changes in the wall-normal direction. The above model, calibrated by reference to incompressible flow, was initially found to give rise to serious anomalies when applied to shock-containing flows. Specifically, terms relying on A for steering turbulence towards the two-component limit were found to respond to shocks as though they were walls, thus shutting off the redistribution process and causing a prediction of a two-component-limit state of turbulence downstream of strong shock waves. Hence, a number of specific modifications have been found necessary in compressible flow fields [7]. Among them, a modified dissipation equation has been adopted, using the length-scale moderator term proposed by Iacovides and Raisee [35], and inhomogeneity corrections are damped by a function which vanishes as the turbulence approaches an isotropic state, thus avoiding non-physical damping of the streamwise-normal stresses across a shock wave.
25 2.3 Non-linear eddy-viscosity models
Predictive inadequacies of linear eddy-viscosity models observed for many years in computations of complex flows have motivated the introduction of numerous corrections into these models in efforts to extend their applicability, thus avoiding the use of secondmoment closure. Such corrections have included a functionalisation of the coefficients C~1 or C~2 in the dissipation-rate equation to curvature- or buoyancy-related gradient or flux Richardson numbers (e.g. Rodi [55]), sensitising the eddy-viscosity coefficient C, to curvature strain, strain invariants and turbulence production to dissipation ratio Pk/c (e.g. Rodi & Sheuerer [56], Rodi [57], Leschziner & Rodi [45], Cotton & Ismael [14], Menter [52]) and giving different weights to normal and rotational straining in the rate of production of dissipation (Hanjalic & Launder [28]). While these corrections undoubtedly yielded predictive improvements, their generality was somewhat limited, for they simply addressed specific symptoms rather than their fundamental source - the linear stress-strain relations coupled with the isotropic eddy viscosity. While second-moment closure was well understood to offer a far superior modelling foundation, there was (and still is) much reluctance to use this closure form, because of its complexity, the numerical challenges it poses, the variability in performance of different variants and persistent uncertainties in the modelling of the influential pressure-strain and dissipation terms, especially in low-Re flow.
Against the above background, the quest for simplicity and computational economy has motivated, in recent years, the development and use of non-linear eddy-viscosity models which are able to resolve some of the interactions described by second-moment closure. Such models have their origin in the work of Lumley, [50, 51] and in Pope's [53] early explicit form of the Reynolds-stress closure from which transport terms are excluded. Indeed, several much newer forms have been derived along a route involving successive simplifications to Reynolds-stress-transport models (Gatski & Speziale [24], Taulbee [63], apsley & Leschziner [1]). The starting point of several other non-linear models (e.g. Shih et al. [59], Craft et al. [17]), has been the general non-linear, explicit expansion for the Reynolds-stress tensor in terms of the strain and vorticity tensors, the linear truncation of which gives the conventional Boussinesq relations. While expansions of any order can be proposed, in principle, the most general formulation depends on a finite number of tensorially independent groups and its coefficients are functions of a finite number of tensorial invariants, owing to the Cayley-Hamilton theorem. The most general cubic constitutive relationship for the Reynolds stresses may be written in the following canonical form : UiUj
k
~SklStkSij) -525ij - - 2 C . f , Sij + al (SikSkj -- -5
+a2(W kSkj--S kWkj) + a3(W kWkj -- 1Wk W kS j) +(b, SktStk +b Wk Wtk)S j
(4) 2
+ b3 (W~kWktS ~j + S~kWktW~j - 5 Skt Wtn W,~k5~j ) + b4(W~kSk~Stj - S~kSk~Wtj) lk which contains non-linear products of the vorticity and strain tensors Wij--~;(Ui,j-Uj,i) lk and S i j - -~-~(Ui,j-[-Uj i, - 5Uk,kSij), 2 with coefficients being functions of strain and vorticity
26 invariants S v/2SijSij and f~ = v/2Wij Wij. The principal merit of relation (4) is that the quadratic terms allow anisotropy to be described, while the cubic terms allow the model to be sensitised to the effects of streamline curvature and swirl. However, the quality with which these interactions are represented depends greatly on the coefficients, which are typically derived by calibration against experimental or DNS data for key flows. This is well brought out in Fig. 3 and Table 1 which give, respectively, distributions of normal stresses predicted by four models (k-e- Launder & Sharma [39], W R - Wilcox & Rubesin [67], CLS- Craft et al. [17], AL - Apsley & Leschziner [1]) in a fully-developed channel flow at Re= 5500 in comparison with DNS data of Kim et al. [37] and limiting stress values in homogeneously strained turbulence. Clearly, there is considerable variability between models, and this reflects a decisive sensitivity on calibration. An important feature in most non-linear models is the dependence of the coefficient C, (associated with the linear term) on the stress and strain invariants. This dependence, shown in Fig. 2, for four models, is quite strong and plays a decisive role in the response returned by the models to normal straining (e.g. in impinging flow) and is also influential in shear flow, especially at large strain rates. Importantly, as the non-dimensional strain ~OU/Oy exceeds the equilibrium value of 3.33, C, drops significantly. This condition arises especially in flows subjected to normal straining, among them decelerating boundary layers. Hence, with C, reducing, the eddy viscosity declines, and a boundary layer subjected to adverse pressure gradient is more likely to separate. This might be interpreted as suggesting that linear models, incorporating strain-dependent C, functions might be satisfactory substitutes for non-linear models. While this is indeed a defensible proposition in some simple flows in which curvature is weak and in which the normal stresses are not dynamically active, it is not the case in complex strain. Moreover, it may be shown that at least in one model (Apsley & Leschziner, [1]), C, depends on the cubic fragments, even in simple shear, while this interdependence arises in all models in the presence of curvature (apsley et al. [2]). =
3. S O M E A P P L I C A T I O N S
Over the past three years, the writers have been engaged in studying the performance of second-moment and non-linear eddy-viscosity models by reference to a number of shockaffected flows for which well-regarded experimental data exist. Geometries considered included nominally 2D cases, such as D(~lery's channel bump [19], Bachalo & Johnson's axisymmetric bump [3], several transonic jet/afterbody flows, supersonic flow around a plate-mounted fin [4] and Pot et al.'s 3D skewed channel bump [54]. This section presents a small selection of results to underpin some tentative conclusions presented at the end. It is important to point out first that the models used herein for compressible flows had been examined, as part of broad-ranging validation efforts, by reference to a sequence of increasingly complex flows, starting from homogeneous strain (plain, axi-symmetric and shear) and progressing through channel flow to impinging jet flow and finally transonic and supersonic flow. Space constraints preclude, except for one example, results arising from initial incompressible-flow studies to be included herein; the interested reader is referred to Batten et al. [7] and Loyau et al. [48]. These same references also contain much
27 broader expositions of compressible-flow results than can be included herein. Computations presented below have been performed with an implicit HLLC Riemannsolver-based upwind scheme (Batten et al[6]). Both non-linear and second-moment models pose particular stability problems within this low-diffusion numerical framework. Quadratic and higher-order terms in a non-linear tensorial expansion contribute a large and potentially de-stabilising source if added explicitly, whilst a full linearisation of all higher-order terms makes the implementation of the more complex non-linear eddyviscosity models an expensive task. Second-moment models are notorious for being numerically 'brittle'. This is partly due to a natural stiffness in the equations, but is also often due to the prediction (at least in early transients) of non-realisable stress data. One of the most frequent causes of instability is the violation of a normal-stress positivity or a shear-stress Schwartz inequality, and this provides a strong motivation for the use of a non-linear, realisable, pressure-strain model. The present approach solves the set of mean-flow conservation equations implicitly and the turbulence equations implicitly as a separate subset. The subsets are partially (block-) coupled through the use of apparent viscosities (see Huang and Leschziner [30]). Prior to presenting examples for shock-affected flow, attention is directed briefly to an incompressible flow, namely a round air jet discharging from a smooth pipe and impinging on a flat plate fixed perpendicular to the pipe axis (Cooper et al. [13]). The main purpose of presenting this case is to highlight the importance of representing correctly the sensitivity of turbulence to irrotational straining associated with stagnation flow. Fig. 4 shows profiles of normal-stress and mean velocity predicted by four models: the modified Craft-Launder Reynolds-stress model (~MCL')[7], the Launder-Sharma k-c model ('k-epsilon') [39], the SST variant ('SST')[52] and the Jakirlic-Hanjalic Reynolds-stress model ('JH')[36]. The most noticeable defects occur with the linear eddy-viscosity models in the stagnation region. Moving away from the centre-line in the radial direction, the cubic Reynolds-stress model displays better agreement than the linear JH variant. This is due, in part, to known weaknesses with the conventional wall-reflection terms used in the latter model (see, for example, Craft and Launder [16]). Earlier computations by Ince & Leschziner (see ref. [42]) for twin-impinging, under-expanded jets at Mach numbers up to 2.6 with Gibson & Launder's linear second-moment closure [25] show features broadly consistent with those above. The first compressible case, frequently used to assess turbulence models, is that of the transonic flow over a plane channel bump, known as D~lery's case C [19]. The computations were performed over a clustered 121x121 mesh with the wall-nearest gridline being at y+ < 0.5. Wall-pressure distributions are compared to experimental data in Fig. 6, while Fig. 5 compares predicted iso-Mach contours for seven different models. For this test case, the normal stresses do not contribute significantly to the mean-flow behaviour, and as a result the well-predicted shear stresses obtained with the SST model gives a mean-flow solution which is comparable with second-moment closure and the AL-model, the latter being the most sensitive of the non-linear eddy-viscosity models investigated. More comprehensive studies on this case may be found in [8, 26, 48, 49]. This next example was examined experimentally by Bachalo & Johnson [3], and is essentially the axi-symmetric equivalent of the previous flow. The geometry consists of a cylinder, with a circular arc bump subjected to a M - 0.875 flow. A 180 x 110 grid
28 was used, with clustering applied around the shock and near the wall. Computational results for different models are compared in Fig.7 with experimental data. The MCL and the AL models are seen to return a considerably more pronounced pressure-plateau region than do other models, predicting a shock location which is fractionally too far upstream. Here again, the SST model gives a performance very similar to the best nonlinear eddy-viscosity and second-moment models, for reasons discussed in the previous section. Comparisons for velocity or shear-stress profiles may be found in [7, 48] and are, in terms of sensitivity to the shock, consistent with the results for pressure. The final 2D case considered here is the RAE 2822 aerofoil [12]. Attention is restricted to 'case 10', in which the shock induces mild separation. The flow conditions are Re = 6.2x 106, Mo~ = 0.754 and incidence = 2.57 ~ This set is the one adopted in the collaborative EUROVAL project [26], but it must be noted that studies at NASA (e.g. Bardina et al. [5]) have used a higher incidence angle which allows better matching of pre-shock pressure on the aerofoil's suction surface, especially at the leading edge. Calculations were performed on a 364 x 128 grid extending to 30 chords, at which a vortex-corrected far-field boundary condition was applied. Fig. 8 shows comparisons for wall pressure predicted with three models. Here again, the SST model and the MCL second-moment closure return similar predictions. While all models over-predict the post-shock suction pressure and give a shock which is too far downstream, the uncorrected k - e models tend to give far worse disagreement. The first 3D example is that of a Mach 2 flow around a blunt-fin/flat-plate junction. Extensive experimental data for this configuration were obtained at ONERA by Barberis and Molton [4]. An 80 x 80 x 70 grid was used to compute this flow, with the mesh clustered at the walls to ensure y + < 1 everywhere. Fig. 10 gives a general overview of the flow field, as predicted by the SST and MCL models, with stream-tubes identifying virtual-particle paths. The multiple-vortex phenomenon is a key feature of this flow which is missed by the linear SST model. This is most clearly brought out through plots of skin-friction lines (or limiting streamlines) on the wall, which indicate the footprint of the vortices. Such plots may be found in reference [7]. Wall-pressure plots in Fig. 11 show a rise in pressure following separation, followed by a local minimum, caused by the stronger vortex nearer to the fin surface. Predicted peak pressures in front of the fin are around 5.5 times that of the undisturbed free-stream, but these peaks decay and move downstream as the horizontal traverses move away from the fin in the span-wise direction. Although the MCL model under-predicts the upstream extent of separation, the results are in better agreement than the predictions returned by the linear JH Reynolds-stress model, which employs conventional wall-reflection terms. The final case considered here is the flow over a 60 ~ skewed bump in a channel (Pot et al. [54]). This case is highly challenging, both computationally and physically. As shown in Fig. 12, it requires adequate resolution not only of the first throat creating the supersonic flow ahead of the bump, but also a second downstream throat used to control the flow-rate. Physical complexities arise from the strongly 3D, asymmetric nature of the interaction, giving very different flow patterns at all four confining walls. Computations have been carried out with linear two-equation models and one non-linear variant using a grid of 120x55x60. Figs. 13 and 14 compare pressure distributions at one spanwise position and skin-friction lines on the bump and the top walls. At this time, all that can
29 be said is that the calculations indicate that a modest improvement can be gained by using models which are more elaborate than linear eddy-viscosity forms. 4. C O N C L U D I N G
REMARKS
The investigation of turbulence models for compressible flows is subject to many uncertainties arising from numerical issues, the sensitivity of shock position to boundary conditions and the structure of the turbulence-affected parts of flow, the ill-understood effects of compressibility on turbulence and the range and accuracy of experimental data obtained in very challenging flow conditions. These uncertainties militate against definitive conclusions being derived even from wide-ranging investigations which encompass several compressible flows and many turbulence models implemented within a single numerical framework, and which are also supplemented by studies for homogeneous and incompressible flows. The focus of this paper has been the interaction between shocks and boundary layers. In most cases, especially in 2D flow, this interaction is dominated by a single shear stress, while the influence of the normal stresses is of subordinate importance. In such circumstances, some carefully corrected eddy-viscosity models, such as Menter's SST variant, give results which are, in essence, as good as those obtained with more elaborate anisotropyresolving closures. However, this equivalence is unlikely to extend to complex 3D flows in which all Reynolds-stress components are influential, and some evidence for this assertion has been provided. At this stage, the body of experience on the very small number of 3D flows for which reasonably detailed experimental data are available does not suffice to make a more definitive statement. Second-moment-closure calculations have now been done for geometries as complex as a generic single-engine fighter model [9]. However, experimental data for such geometries are limited to surface-pressure profiles, and these do not suffice to assess the relative merits of different turbulence models. Acknowledgements
The authors are grateful to British Aerospace plc and the UK Defence and Evaluation Research Agency (DERA) for their financial support. Some of the calculations presented in this paper were performed with computational resources provided within the framework of the 'Contrat de Plan du Bassin Parisien' (Article 12-PSle interr~gional de mod~lisation en sciences pour l'ing~nieur). REFERENCES
[1] Apsley, D.D. and Leschziner, M.A., 1997, Int. J. Heat Fluid Flow, 19, pp. 209-222. [2] Apsley, D.D., Chen, W-L, Leschziner, M. A. and Lien, F-S., 1998, IAHR J. of Hydraulic Research, 35, pp. 723-748. [3] Bachalo, W.D. and Johnson, D.A., 1986, AIAA J., 24, pp. 437-443. [4] Barberis, D. and Molton, P., 1995, AIAA 95-0227. [5] Bardina, J.E., Huang, P.G. and Coakley, T.J., 1997, NASA TM-110446. [6] Batten, P., Leschziner, M.A. and Goldberg, U.C., 1997, JCP, 137, pp. 38-78.
30
[7] Batten, P., Craft, T.J, Leschziner, M.A. and Loyau, H., 1998, Report TFD/98/02, UMIST, Dept. of Mech. Eng. (submitted to AIAA J.). Is] Batten. P., Loyau, H. and Leschziner, M.A. (Eds.), Proc. of ERCOFTAC Workshop on Shock-Wave/Boundary-Layer Interaction, UMIST, Manchester. [9] Batten. P. and Leschziner, M.A., 1998, Final BAe Tech. Report, UMIST, Dept. of Mech. Eng. [lO] Benay, R., Coi~t, M.C. and D~lery, J., 1987, 6th Turb. Shear Flows, Toulouse, pp. 8.2.1-8.2.6. [11] Coakley, T.J., Horstman, C.C., Marvin, J.G., Viegas, J.R., Bardina, J.E., Huang, P.G. and Kussoy, M.I., 1994, NASA TM-108827. [12] Cook, P.H., McDonald, M.A., and Firmin, M.C.P., 1979, AGARD AR-138. [13] Cooper, D., Jackson, D.C., Launder, B.E. and Liao, G.X., 1993, Int. J. Heat and Mass Transfer, 36, pp. 2675-2684. [14] Cotton, M.A. and Ismael, J., 1995, 10th Turb. Shear Flows, Pennsylvania State University, pp. 26-7- 26-12. [15] Craft, T.J. and Launder, B.E., 1996, Int. J. Heat Fluid Flow, 17, pp.245-254. [16] Craft, T.J. and Launder, B.E., 1992, AIAA J., 30, pp. 2970-2972. [17] Craft, T.J., Launder, B.E. and Suga, K., 1996, Int. J. Heat Fluid Flow, 17, pp. 108-115. [ls] Davidson, L., 1995, Computers & Fluids, 24, pp. 253-268. [19] Delery, J., 1981, AIAA 81-1245. [20] Dervieux, A., Braza, F. and Dussauge, J-P., 1998, Computation and Comparison of Efficient Turbulence Models for Aeronautics- European Research Project ETMA, Notes on Numerical Fluid Mechanics, Vol. 65, Vieweg. [21] E1 Baz, A.M. and Launder, B.E., 1993, Engineering Turbulence Modelling and Experiments 3, Elsevier, pp. 63-70. [22] Friedrich, R., and Bertolotti, F., 1997, Appl. Sci. Research., 57, pp. 165-194. [23] Fu, S., Launder, B.E. and Tselepidakis, D.P., 1987, Rep. TFD/87/5, UMIST, Mech. Eng. Dept. [24] Gatski, T.B. and Speziale, C.G., 1993, JFM., 254, pp. 59-78. [25] Gibson, M.M. and Launder, B.E., 1978, JFM, 86, pp. 491-511. [26] Haase, W., Brandsma, F., Elsholz, E., Leschziner, M.A. and Schwamborn, D., 1993, Results of the EC/BRITE-EURAM Project EUROVAL, 1990-1992, Notes on Numerical Fluid Mechanics, Vol. 42, Vieweg. [27] Haase, W., Chaput, E., Elsholz, E., Leschziner, M.A. and Muller, U.R., 1996, ECARP: II, Notes on Numerical Fluid Mechanics, Vol. 58, Vieweg. [28] Hanjalic, K. and Launder, B.E., 1979, ASME J. Fluids Eng., 102, pp. 34-40. [29] Hanjalic, K., 1994, Int. J Heat Fluid Flow, 15, pp. 178-203. [30] Huang, P.G. and Leschziner, M.A., 1985, 5th Symposium on Turbulent Shear Flow, Cornel, pages 20.7-20.12. [31] Huang, P.G. Bradshaw, P. and Coakley, T.J., 1994, AIAA J., 32, pp. 735-740. [32] Huang, P.G., ASME FED Summer Meeting, June 22-26, 1997. [33] Huang, P.G., Coleman, G.N. and Bradshaw, P., 1995, JFM, 305, pp. 185-218.
31
[34] Huang, P.G., 1990, CTR, University of Standford, Annual Research Briefs, pp. 1-13. [35] Iacovides, H. and Raisee, M., 1997, 2nd International Symposium on Turbulence, Heat and Mass Transfer, K. Hanjalic and T. W. J. Peters (Eds.).
[36] Jakirlic, S. and Hanjalic, K., 1995, 10th Turb. Shear Flows, Pennsylvania State University, pp. 23.25- 23.30.
[37] Kim, J., Moin, P. and Moser, R., 1987, JFM, 177, pp. 137-166. [38] Launder, B.E., Reece, G.J. and Rodi, W., 1975, JFM, 68, pp. 537-566. [39] Launder, B.E. and Sharma, B.I., 1974, Letters in Heat and Mass Transfer, 1, pp. 131-138. [4o] Launder, B.E. and Tselepidakis, D.P., 1993, Turbulent Shear Flows 8, Springer. [41] Launder, B.E. and Li, S-P., 1994, Physics of Fluids, 6, pp. 999-1006. [42] Leschziner M.A., 1995, Computers and Fluids, 24, pp. 377-392. [43] Leschziner M.A., 1994, CFD '94, Wiley, pp. 33-46. [44] Leschziner, M.A., Dimitriadis, K.P. and Page, G., 1993, The Aeronautical Journal, 97, pp. 43-61. [45] Leschziner, M.A. and Rodi, W., 1981, ASME J. Fluids Eng., 103, pp. 352-360. [461 Leschziner, M.A., 1993, ONERA Bumps a and C, (in ref. [26]), pp. 185-265. [471 Lien, F.S., and Leschziner, M.A., 1993, J. of Fluids Engineering, 115, pp.717-725. [48] Loyau, H., Batten, P. and Leschziner, M.A., 1998, Report TFD/98/01, UMIST, Dept. of Mech. Eng., (submitted to J. Flow, Turbulence and Combustion). [49] Loyau, H. and Vandromme, D., 1994, TC5 Synthesis, (in ref. [20]). [50] Lumley, J.L., 1970, JFM, 41, pp. 413-434. [51] Lumley, J.L., 1978, Adv. Appl. Mech., 18, pp. 123-176. [52] Menter, F.R., 1994, AIAA J., 32, pp. 1598-1605. [531 Pope, S.B., 1975, JFM, 72, pp. 331-340. [54] Pot, T., Dfilery, J. and Quelin, C., 1991, ONERA TR-92/7078 AY 116 A. [55] Rodi, W., 1979, 2nd Symp. on Turbulent Shear Flows, London, pp. 10.37-10.42. [56] Rodi, W. and Scheuerer, G., 1983, Phys. of Fluids, 26, pp. 1422-1436. [57] Rodi, W., 1972, PhD Thesis, Univ. of London. [581 Sarkar, S., Erlebacher, G. and Hussaini, M.Y., 1991, ICASE TR 91-29. [59] Shih, T-h., Zhu, J. and Lumley, J.L., 1993, NASA TM-10599g. [60] Shima, N., 1993, J. Fluids Eng., 115, pp. 56-69. [61] Shima, N., 1997, llth Symposium on Turbulent Shear Flows, Grenoble, pp. 712-717. [62] Speziale, C.G., Sarkar, S. and Gatski, 1991, T.B., JFM, 227, pp. 245-272. [63] Taulbee, D.B., 1992, Phys. Fluid, 28, pp. 999-1001. [64] Vallet, I. and Gerolymos, G.A., 1996, CFD '96, Wiley, pp. 167-173. [65] Vandromme, D., Haminh, H., Viegas, J.R., Rubesin, M.R. and Kollman, W., 1983, Proc. 4th Symp. Turb. Shear Flow, Karlsruhe, Germany, pp. 1.1-1.6. [66] Vandromme, D., Haminh, H., 1985, Turbulent Shear-Layer/Shock-Wave Interactions, Ddlery, J. (Ed), Springer, pp. 127-136. [67] Wilcox, D.C. and Rubesin, M.W., 1981, NASA TP-1517. [68] Wilcox, D.C., 1988, AIAA J., 26, pp. 1299-1310. [69] Wilcox, D.C., 1993, Turbulence Modelling fort CFD, DCW Industries, Inc. [70] Zeman, O., 1990, Phys. Fluids, 2, pp. 178-188.
32
Sk (a,1)oo (a12)oo (a22)oo (a33)oo (--~-L 6.0
Exp. k
0.403 -0.284 -0.295 -0.108
e
0
-0.434
0
4.82
0
4.82
WR
0.300 -0.434 -0.300
CLS
0.530 -0.273 -0.307 -0.223
7.66
AL
0.449 -0.276 -0.353 -0.095
6.81
'
'
'
o
~ .0 2.0
Table 1" Equilibrium values for homogeneous shear :flow. 1
o
6.08
0
o
Oo
Ol
..Ooo ~
~
,
o DNS (R%=180) k-s (Launder-Sharma) - - - - - Craft-Launder-Suga (CLS)
0.2
0.4
,
,
Y/h
O0~176176176176
0.6
0.8
,
,
1.0
'
q', o ~
-
-...--3 S::h(eZrhs-sLTumley ß
C~
,
~ .u.~
6.0 -
,
o o o o o
0.0 0.0
0.3 -
'
% o
,~= .+
0.2
'
j
4.0
~
....
o,%,
i
;
o DNS Wilcox-Rubesin (WR) Apsley-Leschziner (AL)
,
!
4 ,,
%" 0 ""%,
%%, ' , Q
~
2.0
0.1
I
"%, "-,
? 0.0
0.0
0.0
2.0
4.0
@
6.0
8.0
10.0
kOU Figure 2: Variation of C. with a - - - -
," "~
i
0.10
i
0.08
~
I ,I
~',
> 0.06 0.04
'
'
0.4
,
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Figure 3" turbulent normal stressesPlane channel flow.
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Figure 4: Impinging j e t - axial normal stresses on stagnation line (left) and mean velocity profile at R / D - 1
0.5
1.0
33
Figure 5: Iso-Mach contours (ONERA bump C).
Figure 6: Lower-wall pressure distributions (ONERA bump C).
Figure 7: Wall-pressure distributions. Axi-symmetric bump.
34
Figure 8: Pressure coefficient distributions. (RAE2822 Case 10).
Figure 9: Iso-Mach contours- k - e . (RAE 2822 Case 10).
Figure 10: Overview of flow-field predictions. (ONERA Blunt-Fin)
Figure 11: Plate pressure distributions- D=2 cm. (ONERA Blunt-Fin)
35
Figure 12" Iso-Mach contours in the Y-middle plane - full geometry. (ONERA Skewed Bump). Y=75mm
Y=6Omm ,
,
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0.1
'
0.2
0.3
0.4
0.5
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Figure 13: bottom-wall pressure distributions (ONERA Skewed Bump).
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X (m)
Figure 14: top-wall pressure distributions. (ONERA Skewed Bump).
This Page Intentionally Left Blank
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
37
T h e r m a l H y d r a u l i c s Simulations : W h a t Turbulence M o d e l i n g Strategies ? D. Besnard Commissariat ~t l'Energie Atomique, Direction des R6acteurs Nucl6aires, D6partement de Thermohydraulique et de Physique, 17 rue des Martyrs, 38054, Grenoble cedex 9, France Abstract Thermal-hydraulic flow modeling is very important for many industries, among which is the nuclear industry. We analyze in this paper the current modeling and numerical issues related to RANS and LES of turbulence. Some example of applications are then presented. Finally, some of the new trends in thermal-hydraulics research are described that may answer industry' s needs. I. INTRODUCTION For several decades, nuclear industry has been instrumental at developing thermalhydraulics modeling and measurements. This field is of upmost importance for the analysis of nuclear reactor cooling systems and safety. In France, this effort has led to very effective nuclear power plants that produce about 80% of this country's electricity. French nuclear reactors, run by Electricit6 de France (EDF) are a mature source of energy. This does not mean that no R&D is necessary any more. On the opposite: competitivity and safety are strong driving forces for design improvements and better margin evaluations. During the course of the past two years, two working groups, respectively R&D (FASTNET, DRN/DTP) and industry (EDF/FRAMATOME/DRN/IPSN) led, identified the deficits in modeling, regarding both single phase and two-phase flows. Addressing these problems means exploring new numerical and modeling techniques, as well as making significant advances regarding instrumentation. Indeed, experimental validation is always required: -the geometry of the flow is too complex to be entirely modeled at a local level (i.e. velocity fluctuation distribution functions) -It is not always possible to measure all the relevant parameters at temperatures and pressures at which reactors are operated. Several types of experiments are therefore used to validate models: -separate effect tests (studied with generic experimental configurations), where the key parameters for a specific physical phenomenon are measured. The results provide a validation base for models; -integral experiments, using realistic geometries, albeit somewhat simplified through the use of similarity criteria; these facilities are operated either at full pressure (150 bars in
38 PWRs) and full temperature (350~ in PWRs), or in less demanding circumstances. In the first case, it is more difficult to set up local measurement techniques, and this is why the corresponding experiments are complemented with the second type of facilities. These integral experiments take coupling effects into account. Their careful interpretation often allows to identify further modeling needs that were not identified through the use of separate effects experiments. From the above comments, the requirements regarding thermal-hydraulics R&D for nuclear reactors are quite clear: Turbulence modeling is one of the key factors in our ability to predict the flow in the different circuits and components of a nuclear reactor. The coolant in a reactor (water in pressurized water or boiling water reactors, liquid metal in fast breeders) extracts heat from the core and transfers it to the turbines through the primary and secondary circuits, then steam generators and heat exchangers. These flows are characterized by large temperature gradients (over 100~ as well as a wide range of flow rates. Making use of the increased power of computers, we need: -for single phase flows, to increase our ability to perform detailed simulations of turbulent flows in complex geometries; this includes the need for access to thermal and velocity fluctuations spectra, to be able to predict both peak thermal stresses (for material fatigue, such as fuel rods cladding, or nuclear reactor vessel) and vibrations (for control rods operation, or steam generator long term reliability); as a measure of the problem, it was shown in [ 1] that a LES simulation of a PWR vessel requires about 109 cells; -for two-phase flows, to reduce the level of empiriscism due to the use of flow regime maps, and averaging modeling techniques. The long term goal is to be able to predict transitions between flow regimes, as well as transient circumstances. This requires access to two-phase flow detailed modeling techniques. The same methodology is used in CEA for other types of flows. Inertial Confinement Fusion is another example whereby direct measurements are impossible. Therefore, one has to rely heavily upon numerical simulation, strongly interlocked with experiments. Simulation is considered here as a design tool. As an example, let us consider the CEA Megajoule Laser project. Several hundred laser beams are to implode capsules, in which the nuclear fuel (a cryogenic deuterium-tritium mixture) is contained. The interface between the outer shell and the inner fuel is Richtmyer-Meshkov and Rayleigh-Taylor unstable during the course of the implosion. Therefore, one idea is to use capsules with rather thick shells, in order to prevent shell break-up due to instabilities. The drawback is that this would in turn imply a more powerful laser, therefore more expensive. There is an optimum to be found, based on simulations and experiments. Simulations require a level of accuracy such that instability growth is adequately described; this translates into the need for detailed modeling techniques for multimaterial compressible flows. From the above, it is clear that the most efficient way to calculate this strongly nonhomogeneous, highly contorted, internal flows is to use differencing in physical space, rather than in spectral space. We will concentrate on this type of numerical techniques thereafter. In Section 2, we show several examples of thermal-hydraulic flows simulations. In Section 3, we present some new techniques, both for single phase and two-phase flow.
39 2. C U R R E N T RANS AND LES OF TURBULENT T H E R M A L - H Y D R A U L I C F L O W S .
2.1. Modeling issues Most circumstances we are interested in are internal flows, where the scales are constrained by the flow geometry itself. The flow in a tee junction is a good example of this. In nuclear power plants, tee junctions of two fluids at very different temperatures may be a source of thermal fluctuations, which in turn might induce thermal fatigue and potential damage in the pipe material downstream. We look here into the case of a tee within a fast breeder. A square duct of size L is swept by a cold natrium flow from left to right. The hot natrium inlet, of width .3L, is located about .7L downstream and centered on the mid-plane of the top face of the duct. The mixing of the two flows is computed using LES. It is unstable and induces temperature fluctuations at scales comparable to the inlet size, as shown in Figure 1. Because we want to accurately reproduce energy containing eddies, so that we can adequately predict peak fluctuations in the flow, we therefore chose to concentrate on LES simulations. Actually, it was suggested in [2] that LES is better suited to such circumstances than RANS. RANS, taking large scale turbulence into account in an approximate fashion, would be more relevant in cases when turbulence is confined to scales much smaller than the mean flow, such as in wall induced turbulence. The drawback is that the resolution requirements near walls appear to be very stringent in the LES case. For example, it was shown in [3] that filter widths have to be chosen so that the subgrid stresses in Smagorinsky-type subgrid models are a negligible fraction of the total. This was also observed in [4]; the simulation of the experiment by Simonin et al [5] of the flow within the mock-up of a nuclear reactor rod bundle was calculated. The first grid point varied from y+=l.5 to 4 wall units on the circumference of the rods. Wall functions were activated whenever needed because of higher local instantaneous Reynolds. This implied a very fine mesh. Other turbulent flow circumstances may also induce severe constraints on the mesh. This is the case of mixing between two fluids, more especially when other important phenomena occur at interfaces (e.g. chemical reactions). To evaluate the accuracy of a given simulation (which might use k-~ or subgrid scale models), we calculate which part of the energetic eddies, as well as which part of the interface area (or volume fraction for miscible fluids) are accurately described.
Figure 1. Flow in a tee junction (courtesy of B. Menant): Temperature map
40 Assume first that the flow spectrum is confined between ki and kd (the Kolmogorov scale). The flow is simulated with a model that calculates explicitly scales between k~ and k~ (grid size wavenumber), and uses a model for the remaining part of the spectrum. The ratio I~ of the "resolved" kinetic energy to the total kinetic energy is:
~E ( k )dk
II(ki'kc'kd)= ~kitE(k)dk Denoting the resolved part of the spectrum as x, and the Reynolds number as Re, we have
e9,4
x
we obtain that 1 - x -2/3
I 1(Re, x) = 1 - Re -9/1~
(1)
Similarly, we estimate the ratio I2 of the "resolved" interface L~c to the total interface area Ltd. Let us consider the case of an unstable interface between two fluids. Denoting the initial length of the interface (in a 2D case) as Lo, we obtain, after the primary instability development, a length L1 such that Ll(L0)=Cte L0. Partitioning L~ into N segments, we follow the same scenario; the total length, after the next interface folding is Cte 2 L0 (we assume here that Cte/N is smaller than 1, so that the interface distortion is somewhat possible). After p iterations, we obtain that Lp= Cte p L0. Parametrizing Lkd by p, and Lkc by m, taking the smallest scale in the flow to approximately be 1/kd (- Lp_,/Np_,), and the smallest resolved scale as 1/kc (=Lm_,/Nm.l), we obtain that
ln~ p=
l +
In kc
cte
m= l +
ln~ N
cte ln~ N
Then,
12 (Re, x) :
cte
ln(Cte) In
(2)
Equation (1) shows that energy can be adequately described with LES, as exemplified in Table 1. Equation (2) shows that the constraints are much stricter in the case of interface
41 rendering. The results obtained with this simple model show that grids adequate for energy rendering are not sufficient enough for interface rendering. Re
x
nx x ny x nz
11
12
103
322
643
0.9025
0.764
103
1024
1283
0.9393
0.85
103
322
643
0.9007
0.44
103
1024
1283
0.9375
049
Table 1./1 and/2 versus R e and x The choice of the subgrid model itself is important. Most of the models dissipate kinetic energy even when the flow is not turbulent, preventing any growth of instabilities: the transition cannot not take place. We chose here to use the so-called "selective structure function model" originally described by David [6], and justified by Lesieur and Metais [7]. In this model, the energy is dissipated only in the region where the flow is developed, i.e. is fully three dimensional. The eddy viscosity is turned off whenever the three-dimensional behavior of the flow is not strong enough. 2.2. Numerical issues
Our current workhorse code for incompressible flows is TRIO-VF. This industrial code was developed for thermal-hydraulics applications at CEA, Grenoble. It uses a finite volume element method on a structured mesh. The discretized variables are harbored on a staggered grid: the pressure and other scalar quantities are located at control volume center, and velocities at cell side midpoints. The time discretization is first order (Euler), and space discretization is 3 rd order (Quick Sharp scheme). A conjugate gradient iterative technique is used to solve Poisson's equation. The need for numerical accuracy is also stressed by Kravchenko and Moin [8]. They calculated a channel flow with several different numerical schemes (spectral, 2nd order finite d i f f e r e n c e , 4 th order finite difference, and 6 th order Pad6). They demonstrated that the ratio of subgrid-scale dissipation to total dissipation-including numerical dissipation- ranged from 48% in the case of spectral simulation, down to 10% with 2no order differencing, and back up to 30% with 6 th order Pad6. Considering now one-dimensional spectra, they found that they were heavily distorted for low-order schemes. In CEA, we also looked [9] into numerical schemes such that their dissipation would be small enough so that they could be used for transitional flows. To do so, we run numerical experiments for a null value of the physical viscosity. The energy is injected in the flow in a spectral zone centered at k~. Energy is accumulated at cut-off frequency (i.e. corresponding at grid size), and up to critical time to. At to, enstrophy goes infinite, as demonstrated for EDQNM by Lesieur [ 10]. 5.9
tc = ~/D(0)' with D(0)=
k2E(k)dk
42 In practice, with a pseudo-spectral code, one can expect tc to be about equal to 4D(0) -'/2 [11]. We used the CEA, 3D, NSMP code, with a second order, unsplit, TVD rendered Mac Cormack scheme (this code is used for simulating highly non stationary and non homogeneous compressible flows). We calculated a homogeneous isotropic turbulent flow, at Re-10 '~ and on a 643 grid. Indeed, the kinetic energy continuously spreads itself towards large wavenumbers, and exponentially increases at cut-off wavenumber (see Figure 2). This simulation gives tc=250 s, to be compared to 3 l0 s and 220 s for EDQNM and a pseudospectral code. Subsequent simulations run with the same parameters as before, but with a variety of advection schemes (of order 2 and 4 in space) and subgrid scale models (structure function and filtered structure function models, respectively denoted as SFM and FSFM [12]) demonstrate that FSFM is a better choice. In all cases, total energy is conserved; Figure (3) shows the time evolution of the energy spectrum for FSFM. Simulations run with 4-order schemes are globally better in terms of enstrophy and spectra behavior, as found in [8].
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le-06 1
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Figure 3. Homogeneous isotropic turbulence: turbulent spectrum, at different times; FSFM
9
43 Another rather surprising finding by Rollet-Miet et al [4] is that what is considered adequate for RANS does not seem to suit LES. Indeed, Rollet-Miet and coworkers found that the best choice was a centered scheme using a P1-P1 element based on collocated velocitiy and pressure nodes. This is to be opposed to the corresponding RANS performed with a scheme based on a P l-isoP2 element. Here again, channel flow simulations were performed (using the EDF N3S-LES code), a common benchmark for turbulent flow, demonstrating the right behavior of LES modeling. The flow in a rod bundle was also analyzed. It was observed that LES succeeds where RANS fails, supporting our own choice for simulating this type of flows.
2.3. Examples 2.3.1. Rod bundle in a reactor assembly Competitivitiveness is of course a key factor in the nuclear industry. One example is nuclear fuel. The fuel is made of small cylinders which are put in zircalloy rods. In PWRs, rods are organized in 17xl 7 rod assemblies, which are held together by a metallic frame. They are also maintained by grids located every 20 inches or so along the rods (which are 4 m long). Due to nuclear reactions occurring within the fuel, heat is produced and is evacuated thanks to the coolant (water in PWRs). Obviously, heat transfer between rods and the coolant is more efficient if the latter is turbulent. Therefore additional grids, denoted as "mixing grids" are introduced at a few locations, whose function is to increase turbulence within the coolant. The net result of this is to prevent any local overheating of the rods, which might lead to a local melting event. In that case, the rod is deteriorated, and must be replaced. Until recently, it was not possible to calculate the flow around and downstream the mixing grid in a very detailed fashion. In particular, it was not possible to zone up the grid itself. Figure (4) shows the temperature map within a nuclear core sub-assembly, resulting from a 3D calculation. The result was obtained using the CEA TRIO-U code [13]; the numerical options used in this calculation are the same as those of the TRIO-VF code. It demonstrates how turbulence is enhanced behind the grid. One can distinguish the grid itself, delineated in black on the figure. With more refined simulations, we will eventually be able to estimate the efficiency of a grid, without fully relying on experiments. 2.3.2. Rayleigh-Taylor Instability induced mixing We consider here the idealized planar geometry problem of two superimposed fluids, submitted to an acceleration. The problem is to estimate the overall mixing layer thickness versus time. An important question is to evaluate the characteristic time at which one may consider that the flow has turned fully turbulent. The calculation presented here was performed on a 64x64x128 grid; the acceleration is equal to 250,000 m/s 2, the density ratio is
44
Figure 4. Turbulent flow within a sub-channel in a reactor core (courtesy of F. Barr6); temperature map along the sub-channel
Figure 5. Rayleigh-Taylor instability induced mixing
1.225/0.169, and the calculation box is 1.75x1.75x3.5 (cm). As for initial conditions, we chose to start with a linearly perturbed interface (with 30 harmonics). Figure (5) shows the .05 and .95 concentration iso-surfaces. The calculation itself shows a late time behavior such that the mixing thickness evolves as Agt 2, with A=0.045. This results compares to those obtained by Youngs [14]. Other simulations demonstrate that A is a function of the turbulent Mach number and the dimensionless ratio of the initial interface perturbation wavelength to its amplitude. 3. NEW TRENDS IN THERMAL-HYDRAULICS RESEARCH From the above results, it is clear that two main questions are not fully answered. 1. How can one deal with large and complex geometries? 2. How can one extend the simulations to two-phase flow turbulence, with and without phase change? In this Section, we present some current attempts for solving these problems. About the first question, we look into the extension of LES to unstructured grids, and coupling methods for RANS and LES, or spectral methods with LES. The second question is explored in several ways: two-phase flow models, two-phase LES, numerical simulations of two-phase flows with phase change. 3.1. Complex geometries One obvious answer to the problem is to use non structured grids, which are well suited for describing complex geometries. Several workers looked into the extension of LES to such grids. For example, Ducros and co-workers [15] solve the complete Navier-Stokes equations in their conservative form. Subgrid models are also implemented. Note that the subgrid model lengthscale is chosen as the 1/3 power of the cell volume, which is certainly questionable for
45 highly distorted grids. They actually demonstrate the ability of this technique to deal with turbulent pipe flows, including in transitional regimes. At CEA, we developed the TRIO-U code, which is based on a finite element volume method. This C++, object -oriented code is also highly modular, which allows us to enrich it very easily. Similarly, we are currently developing the PRICELES code in collaboration with the french electric utility EDF. One preliminary result is shown on Figure (6). It addresses the following problem: Water in a PWR cooling system is not chemically pure. The added chemicals are used to control neutron slowing-down. Under accidental circumstances, chemical variability may be observed in the flow. It is therefore important to evaluate the effect of these variations and we looked here into circumstances by which some clear water is injected in the system. The water slug travels within the system, and one wants to observe its dilution process. This calculation is not fully resolved (400,000 cells). Complex geometries may be dealt with in a different fashion. It was observed in [4] that RANS is very efficient whenever turbulent scales are small compared to scales defined by the flow geometry itself (e.g. wall turbulence). On the opposite, resolving those large scales can be done using LES. Coupling these two techniques may be therefore a faster way for getting an industrial tool than trying to increase the grid size.
3.2. Two-phase turbulence Modeling turbulence in multiphase flows adds a degree of complexity. Many of the models described in the litterature rely on a generalization of k-e models, that can be categorized in two families. The first one is based on the use of ensemble, space, or time averaging. The second uses the model of a particulate fluid carried by the other. These models suffer from the same limitations as k-e models. One of the main difficulties comes from the existence of two spectra of scales, the first due to inclusions of one fluid in the other, the second due to turbulence scales. Therefore, this approach is better suited to circumstances whereby inclusions are somewhat well calibrated. These techniques are widely used in nuclear engineering [ 16].
Figure 6. Simulation of a clear water slug in a reactor cooling system (courtesy of F. Barr6): 3D view of water chemistry (clear water in dark grey)
46
I.!)E+O2 + I
I I I / 1.8E+O2 1-.4.E-06
--
--
----4--3.E-06
4-2.E-~;
4-I.E-06
4---.-O.E+O0
+ I.E-(~;
-t 2.E-{M;
-4 - 3.E-I~;
--
-4.E4N;
Figure 7 (courtesy of D. Jamet). Return to equilibrium of an initially perturbed spherical bubble Another type of technique is to directly track interfaces. It has been widely used for simulating the interaction between non miscible fluids, such as instability development. The corresponding numerical techniques are based on an approximate description of the interface, and a way of allocating mass, momentum, and energy to the fluids present on each side of the interface, within a cell. We may interprete this as a sub-grid two-field model. Numerous techniques have been developed, and we refer here to [ 17]. One drawback of these techniques is that they lead to discontinuous solutions (as physics dictates it), but induce numerical difficulties. One way to avoid them is to artificially enlarge the interface itself. Physical discontinuities across the interface are preserved, and numerical solution may be kept as smooth as wanted [ 18, 9]. An additional difficulty is added when one wants to deal with phase change. Not only phase velocities must be known on each side of the interface, as well as the interface velocity, but the dynamics of the phase change itself has to be accurately described. Jamet et al [20] recently described a technique by which the interface is enlarged through the use of a thermodynamics model. This method is based on the Second Gradient Theory, and makes use of the Van der Waals interfacial model. Surface tension and nucleation are then automatically taken into account. Mass, momentum, and energy equations degenerate into Navier-Stokes equations in each phase, but show additional terms in interfacial zones. Solving these equations for 3D flows is not more difficult than for 2D flows. This technique was tested for 2D isothermal flows, and 1D fully non isothermal flows. Its robustness is demonstrated on Figure (7). A non-equilibrium profile is imposed onto a bubble; the bubble profile goes back to its equilibrium profile after 5.e-8 s.
3.3. Spectral methods LES has a very definite advantage: it can deal with flows which are not in spectral equilibrium. Another approach to take those flows into account is to use the so-called spectral models, by which one keeps information on the turbulence spectrum. To do so, one derives a two-point correlation function equation for the turbulent kinetic energy (see [21] as an example). Then, one Fourier transforms the equation, and after truncating it, one obtains an
47 approximate equation that must be closed. In [21], we used the simplest, dimensionally correct, rotationally invariant, linear or nonlinear expressions satisfying the intrinsic properties of the terms to be modeled. From the resulting model, one can derive either a k-e model, or a subgrid model. Their advantage is that the moment equations that are obtained are fully coherent, and do not rely on independent closures, such as for classical k-e models. These models could be used as boundary conditions for k-e or LES, which lack spectral information, either at large scale (k-e), or small scale (LES). 4. CONCLUSION During the past few years, a good deal of progress was made for simulating single phase turbulent thermal-hydraulics flows. We are on the verge of being able to simulate the major components of the cooling circuits of a nuclear reactor with LES techniques. Feasibility calculations demonstrate this, which should be confirmed within a year. Nevertheless, some real challenges are still in front of us. Complex and large geometries will not be possibly rapidly calculated on an industrial basis, even though some reference simulations are at hand. An avenue of research is to study the feasibility of a coupling between RANS and LES techniques. As for two-phase thermal-hydraulics, with or without phase change, the situation is less clear. Current turbulent two-phase flow models lack full validation, and suffer from a rather high number of coefficients. More detailed simulation is clearly a tempting goal, but the stakes may be even higher. To us, the main avenues of research pertain to: - phase change simulation, which feasibility was recently shown, but which needs to be fully demonstrated; detailed two-phase simulation, which requires still more powerful machines; two-phase LES, whose has not been addressed at this point. R E F E R E N C E S
1. 2. 3.
4. 5.
6. 7.
D. Grand, G. Urbin, B. Menant, M. Villand, and O. Metais, Large eddy simulation in nuclear reactors thermal-hydraulics, J. Hydraulic Research, 35-6 (1997) 831. D. Laurence, S. Pameix, Second moment closure analysis of a DNS backstep flow database, Turbulent Heat Transfer 2, Delft U. Press (Hanjalic, Peeter Eds), (1997). J.S. Baggett, J. Jimenez, and A.G. Kravchenko, Resolution requirements in large eddy simulations of shear flows, Center for Turbulence research, Annual Research Briefs, (1997) 51. P. Rollet-Miet, D. Laurence, and J. Ferziger, LES and RANS of turbulent flow in tube bundles, submitted. O. Simonin, M. Barcouda, Measurements and prediction of turbulent flow entering a staggered tube bundle, Proceedings of the 4 'h Int. Syrup. on Appl. of Laser Anemometry to Fluid Mech., Lisbon, Portugal, (1988). E. David, Modelisation des 6coulements compressibles et hypersoniques: une approche instationnaire, PhD Thesis, Nat. Polytech. Inst., Grenoble, France, (1993). M. Lesieur and O. Metais, New Trends in Large Eddy Simulation of Turbulence, Annual Rev. Fluid Mech., 28 (1996) 45.
48 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18. 19. 20.
21. 22.
A.G. Kravchenko, P. Moin, On the effect of numerical errors in large eddy simulations of turbulent flows, Center for Turbulence Research manuscript # 160, (1996). D. Besnard, et al, Turbulent Compressible Mixing: Where do we stand?, Proceedings of the 5th Int. Work. on the Physics of Turbulent Compressible Mixing, Stony Brook, (1995); see also F. Ducros, CEA Report #CEA-N-2811, (1996). M. Lesieur, Turbulence in Fluids, Kluwer Academic Publishers, (1990). M. Lesieur, Spectres auto-similaires en Turbulence de Rayleigh-Taylor, CEA Report CEA-N-2793, (1995). F. Ducros, P. Comte, and M. Lesieur, Large-eddy simulation of a spatially weakly compressible boundary layer over an adiabatic flat plate, Int. J. of Heat and Fluid Flow, (1995), 341. C. Calvin and Ph. Emonot, The TRIO-U project:a parallel CFD 3D code, Proceedings of ISCOPE'97, California (1997). D. Youngs, Three-dimensional numerical simulation of turbulent mixing by RayleighTaylor instability, Phys. Fluids, A, 3, (1991), 1312. F. Ducros, F. Nicoud, and T. Schonfeld, Large eddy simulation of compressible flows on hybrid meshes, 1 lth Symp. on Turbulent Shear Flows, 3, (1997), 28. See also F. Ducros, F. Nicoud, and T. Schonfeld, All-adapting local eddy-vidcosity models for simulations in complex geometries, Numerical Methods for Fluid Dynamics VI, (1998), 293. R.I. Nigmatulin, Satial averaging in the mechanics of heterogeneous and dispersed systems, Int. J. Multiphase Flow 5, (1979), 353. D.J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Computer Methods in Applied Mechanics and Engineering, 99, (1992), 235. J.U. Brackbill, D.B. Kothe, C. Zemach, a Continuum Method for Modeling Surface Tension, J. Comp. Physics, n~ (1992), 100. B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, G. Zanetti, Modeling Merging and Fragmentation in Multiphase Flow with SURFER, J. Comp. Physics, 113,(1994),134. D. Jamet, O.Lebaigue, O. Coutris, and J.M. Delhaye, Numerical description of a liquidvapor interface based on the second gradient theory and applied to the modeling of evaporation and condensation, 3rd Int. Conf. On Multiphase Flow, (1998). D. Besnard, F. Harlow, R. Rauenzahn, and C. Zemach, Spectral Transport for Turbulence, Theo. Comp. Fluid Dynamics, 8, 1, (1996), 1. J.P. Bertoglio, Simulation Num6rique des Grandes Echelles d'une turbulence homog6ne avec prise en compte des petites 6chelles par un mod61e de sous-maille stochastique et une fermeture en deux points, LMFA Report, (1985).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
49
Modelling of Production, Kinematic Restoration and Dissipation of Flame Surface Area in Turbulent Combustion N. Peters Institut fiir Technische Mechanik, Templergraben 64, 52056 Aachen, Germany The only sound foundation of turbulence models lies in inertial range scaling laws of turbulence. These scaling laws apply to flow and mixing problems, where they establish a link between the integral scales and the dissipation scales. Although dissipation is due to diffusive processes, the at first surprising result is that neither viscosity nor diffusivity appears explicitly in models for high Reynolds number turbulent flow and mixing. This situation changes for premixed turbulent combustion, where an additional quantity, the laminar burning velocity needs to be considered. This quantity results from a convective-diffusive reactive balance within the flame structure. Therefore the laminar burning velocity plays a similar role as the viscosity in non-reacting flows. The existence of a burning velocity leads to a kinematic cut-off scale, the Gibson scale. Smoothing of the flame surface area at that scale is due to kinematic restoration. This scaling is valid in a regime called the corrugated flamelets regime. There is, however, a regime in premixed turbulent combustion, where diffusive processes are dominant and the cut-off occurs at the dissipation scales. This happens at high intensity small scale turbulence when the Kolmogorov scale becomes smaller than the flame thickness. Small eddies enter into the preheat zone of the flame structure and enhance scalar mixings, while the thin reaction zone remains quasi-stationary. This regime is called the thin reaction zones regime. A level-set formulation is used to formulate a field equation in terms of the distance function G. Classical turbulence modelling assumptions and spectral closure is applied to this equation. The resulting expressions for the burning velocity in both regimes are used to derive an equation for the flame surface area ratio. The balance of production, kinematic restoration and dissipation in that equation leads to an expression for the turbulent burning velocity for large scale and small scale turbulence. 1. I N T R O D U C T I O N One of the most important unresolved combustion problems in turbulent combustion is that of the turbulent burning velocity. DamkShler [1] was the first to present theoretical expressions for this quantity. He identified two different regimes which he called large scale and small scale turbulence, respectively. For large scale turbulence he assumed that the interaction between a wrinkled flame front and the turbulent flow field is purely kinematic and therefore independent of lengthscales. This corresponds to the corrugated
50 flamelets regime which has been discussed previously (cf. Peters [2]; Bray and Peters [3]). DamkShler equated the mass flux rh of unburnt gas with the laminar burning velocity SL through the turbulent flame surface area F T to the mass flux through the cross sectional area F with the turbulent burning velocity ST ? : [ t - Pu 8 L F T -
Pu 8T F
.
(1.1)
Here Pu is the density of the unburnt mixture. The burning velocities SL and ST are also defined with respect to the conditions in the unburnt mixture. This leads to
sr
=
SL
Fr
.
(1.2)
F
Using the geometrical analogy with a Bunsen flame, DamkShler assumed that the area increase of the wrinkled flame surface area relative to the cross sectional area is proportional to the increase of flow velocity over the laminar burning velocity FT
8L -~ VI =
(1.3)
.
F
8L
Here v ~is the velocity increase which finally is identified as the rms velocity v ~. Combining (1.2) and (1.3)leads to 8T
V~
= 1 +--.
8L
(1.4)
8L
In the limit of a large ratio of the rms turbulent velocity v / to the laminar burning velocity S L the turbulent burning velocity ST is then proportional to v t (1.5)
ST ~ V'
For small scale turbulence DamkShler argued that turbulence modifies the transport between the reaction zone and the unburnt gas. In analogy to the scaling relation for the laminar burning velocity SL ~
(1.6)
( D / t ~ ) 1/2 ,
where t~ is the chemical time scale and D the molecular diffusivity, he used the turbulent diffusivity D t to obtain ST ~
( D t / t c ) 1/2
9
(1.7)
Therefore the ratio ST ~
(____~)1/2
(1.8)
8L
is independent of to, where it is implicitly assumed that the chemical time scale is not affected by turbulence. Since the turbulent diffusivity D t is proportional to the product vlg where ~ is the integral lengthscale, and the laminar diffusivity is proportional to the
51
Figure 1. A schematic representation of the flame front as an iso-scalar surface of G(x, t).
product of the laminar burning velocity and the flame thickness t~F one may write (1.8) as
8_~T,.o ( UA~ ) 1/2 SL
SL -~F
(1.9)
showing that for small scale turbulence the ratio of the turbulent to the laminar burning velocity not only depends on the velocity ratio V'/SL but also on the lengthscale ratio In the following half century there were many attempts to modify DamkShler's analysis and to derive expressions that would reproduce the large amount of experiment data on turbulent burning velocities. Expressions of the form
8L
=1+C
--
(1.10)
8L
have been proposed. The exponent n is often found to be in the vicinity of 0.7 ([4]). Attempts to justify a single exponent on the basis of dimensional analysis, however, fall short even of DamkShler's pioneering work who had recognized the existence of two different regimes in premixed turbulent combustion. 2. T H E L E V E L - S E T A P P R O A C H
FOR PREMIXED
COMBUSTION
It is useful to formulate the problem of premixed combustion in a general flow field in terms of a partial differential equation that does not explicitly contain a chemical source term. Such an equation may be derived for any well-defined front in a flow field by defining the normal vector to the front as VG n-[VG[
(2.1)
and considering an iso-scalar surface representing the front as
t) = a0
(2.2)
52 where Go is arbitrary. If one considers a flame front, the surface G ( x , t) - Go divides the flow field into two regions where G > Go is the region of burnt gas behind the front and G < Go that in front (figure 1). This is called the level-set approach. If one differentiates (2.2) with respect to t one obtains
dx]
-0.
OG Ot + V G . - - f [ a=ao
(2.3)
In the corrugated flamelets regime a kinematic balance involving the flow velocity v, the burning velocity normal to the front s L n defines the resulting propagation velocity d x / d t of the front as dx dt
(2.4)
=v+nsL.
Introducing (2.4) into (2.5) and multiplying both sides by the density p one obtains the equation derived by Williams [5] OG
+
v a - ps
lVal
(2.5)
which is known as the G-equation in the combustion literature. It contains a local and a convective term but no diffusion term. Instead there is on the r.h.s, a propagation term containing the product of the burning velocity SL and the modulus of G. Since (2.5) was derived at the flame front, it is valid at G ( x , t) - Go only. Therefore the density p and the velocity v are conditional values at the front. If, as in the following, the flame is assumed to be of finite thickness, a location within the flame structure must be defined to assign to G ( x , t) the value Go. If gas expansion effects are taken into account this location is conveniently defined as the midpoint of the flame structure where half the density change has occurred. If the density in the unburnt gas and that in the burnt gas are constant, the conditional density at the midpoint is also constant. Although G represents an arbitrary scalar it is convenient to interpret it as the distance from the flame front by imposing the condition IVGI - 1 for G r Go. Then G has the dimension of a length. It will be called distance function in the following. The burning velocity SL in (2.5) may be modified to account for the effect of flame front curvature and flame strain. In asymptotic analyses employing the limit of a large ratio of the fluid dynamic length scale to the flame thickness resulting in a quasi-steady structure of the preheat zone, first order corrections to the burning velocity due to curvature ~ and straining of the flame may be derived ([6]; [7]) yielding SL -- s ~ - s ~
+ Ln . Vv.n.
(2.6)
Here s ~ is the burning velocity of the unstretched flame and L is the Markstein length. The flame front curvature ~ in (2.6) is defined as -V-n-V.
(VG) IVG]
wh re V(IVGI) - - V ( n . VG) equation may be written as p -oa -~ + pv . VG - ps~
V2G-n.V(n'VG)
--
(2.7)
IVGI been used. If (2.6) is introduced into (2.5) the G-
- pDztca + p S
(2.s)
53 where a-
D -
sO L
is the Markstein diffusivity, S is the strain term in (2.6) and
IVGI
(2.9)
is the modulus of G. In a recent paper [8] an alternative derivation of the G-equation for flames of finite thickness has been derived. Starting from the equation of a diffusive-reactive scalar, it has been shown that the diffusion operator can be split into normal diffusion and curvature (which represents tangential diffusion). Then normal diffusion and reaction are combined to an effective burning velocity s~, which implicitly contains the strain term in (2.8). The diffusion coefficient in the curvature term then is the mass diffusivity rather than the Markstein diffusivity. The resulting G-equation is OG p-~ + pv . VG
-
ps~
pDaa
(2.10)
.
here s}~ has been replaced by the constant value s ~ in order to account for both cases, the infinitely thin flame and the flame structure with finite thickness. The properties of the G-equation for turbulent flow fields have been investigated in a number of papers (cf. [9] for a recent review). In particular Kerstein et al. [10] have performed direct numerical simulations for the constant density (passive) G-equation in a cubic box and have shown that the mean absolute gradient of G may be interpreted as the total flame surface density of the front equal to the ratio of the turbulent to the laminar burning velocity 8T
= --]VGI 9
(2.11)
8L
3.
THE TURBULENT FUNCTION G
MEAN
AND
VARIANCE
OF
THE
DISTANCE
In [11] Reynolds-averaged equations for the mean G and the variance G '2 h a v e b e e n derived. A constant density was assumed and G, a and the velocity component v~ was split into a mean and a fluctuation G -
G + G' ,
a -
a- + a ' ,
v,~ -
-v,~ - + %' .
(3.1)
m
The equation for the mean G is simply m
OG O---t + ~ V - O + V v ' G '
-
s~
-
D
~IVGI 9
(3.2)
The condition G - Go now defines the location of the mean flame front, while the variance G '2 accounts for flame front fluctuations and thereby is a measure of the flame brush thickness. An equation for G '2 may be derived by subtracting Eq. 3.2 from Eq. 2.10 to obtain an equation for G'. Multiplying this by 2G' and averaging one obtains OG'2 O----t + v . V G '2 -
-Dtr~lVG,,21 - 2v'G'
.
VG - & - ~ .
(3.3)
54
10 3 -
Vt/SL
broken reaction zon
l0 2 -
10
-
/ Re
n-e
= 1
~
1
~''~
10
corrugated flamelets
10 2
10 3
10 4
Figure 2. Regime diagram for premixed turbulent combustion.
The terms on the 1.h.s of this equation are the local and convection term, the first two terms on the r.h.s, are a curvature term and the production term. The third term on the r.h.s is defined as -~ -
-2
s~ a'G ' .
(3.4)
This term was called kinematic restoration [11] in order to emphasize the kinematic effect of local laminar flame propagation with velocity s ~ in restoring the flame front corrugations by turbulence. In [8] it was shown that there are two different regimes of importance in turbulent combustion, the corrugated flamelets regime and the thin reaction zones regime. These are shown in the regime diagram in Fig. 2, where the velocity ratio v ' / s ~ is plotted as a function of the length scale ratio t~/t~F. Here g is the integral length scale and t~F is the flame thickness. The line r/ = gY, where r/ is the Kolmogorov scale, separates the two regimes. The upper border line is given by ~ = t~6 where g5 is the reaction zone thickness. If the Kolmogorov eddies become smaller than the reaction zone thickness they can break the structure of the reaction zone. In [11] the focus was on a closure relation for the kinematic restoration, because this is the most important term in the corrugated flamelet regime. This closure was achieved by integrating the scalar spectrum function to obtain _ ~ C w ~c a , 2
(3.5)
with c~ = 1.62. It was emphasized that the kinematic restoration plays a similar role as the scalar dissipation for diffusive scalars. Furthermore it was shown that kinematic
55 restoration is active at the Gibson scale t~a = sO.
(3.6)
g
In the thin reaction zones regime the propagation term ps~ in (2.10) becomes of lower order than the curvature term pD~a. Therefore the dissipation term ~ in (3.3) is more important than the kinematic restoration term. It was shown in [8] by integrating over the scalar spectrum, that for this case the scalar dissipation, defined as
- ~ - 2Da 2
(3.7)
scales as _X - cx-~ e G, 2
(3.8)
where cx - 1.62 again. Scalar dissipation is active at the Obukhov-Corrsin scale ec - - ( D 3 / e ) 1/4 9
(3.9)
Consequences of G-equation modelling and relations to the BML-model were presented in [3] where also further references may be found. These different closures in the two different regimes lead to the at first disappointing result that, while there are different physical mechanisms active in the two regimes, the resulting closure relation for the sink terms in the variance equation is the same. In fact, it was shown in [8] that a model for the sum of ~ and ~, valid for both regimes is
w_ + - ~ -
(3 10)
e G, 2 c~-s
where Cs = 2.0. As a consequence the variance G a and therefore the flame brush thickness does not change, whether turbulent combustion takes place in the corrugated flamelets regime or the thin reaction zones regime. The turbulent velocity, however, is different in the two regimes. In [8] it was shown that (1.5) can be derived from the kinematic restoration in the corrugated flamelets regime, while (1.8) can be derived from scalar dissipation in the thin reaction zones regime. In order to combine these two limiting expressions one must derive an equation that is valid in both regimes. 4. A M O D E L
EQUATION
FOR
THE
FLAME
SURFACE
AREA
RATIO
If we compare Damk6hler's expression (1.2) with (2.11) we identify the flame surface area ratio as FT F = IVG["
(4.1)
Therefore, in order to predict the turbulent burning velocity on a rigorous basis, we must derive an equation for ~. We will again assume a constant density at the midpoint of the flame structure. For illustration purposes we will therefore at first derive an equation
56 for a from (2.10) by applying the V-operator to both sides and multiplying this with -nvv/IvGI O~ -- + pv . Va Ot
-n.
+ V2G) - D [ V . (aVG) + a2a].
Vv . na + s~
(4.2)
Here the terms proportional to D are a result of the transformation n . V(tw-) = - V G - V ~ -
~ V 2 G - ~20 = - V - ( t ~ V G ) -
~2a.
(4.3)
The first term on the r.h.s of (4.2) accounts for straining by the flow field. It will lead to a production of flame surface area ratio. The second term containing the laminar burning velocity will have the same effect as kinematic restoration has in the variance equation. The last term is proportional to D and its effect will be similar to that of scalar dissipation in the variance equation. In principle one could take the Favre average of this equation to obtain an equation for 9. However, there is no standard procedure for the closure of such an equation, as there is none for deriving the s equation from an equation for the viscous dissipation. Therefore another approach was adopted in [8]" The scaling relation between 9, s, k and G '2 for the corrugated flamelets regime can be derived from (3.4) and (3.5) as
C (G--~)l/2
(4.4)
whereas that for the thin reaction zones regime using (3.7) and (3.8) is D&2 ~ k G '2 .
(4.5)
These expressions were used in [8] to derive equations for 9 from a combination of the k, s and G ~2 equations. These equations are linear in 9 but different in both regimes. They contain the local change and convection of 9, a production term by mean gradients and another due to turbulence. Each of them contains a different sink term, which is proportional to (s/k)9. The scaling relations (4.4) and (4.5) were used again to replace s/k. This leads to nonlinear sink terms in both regimes: In the corrugated flamelets regime the sink term is proportional to 8092 and in the thin reaction zones it is proportional to DO-3. Finally, in order to obtain an equation for 9 in both regimes, the two sink terms are assumed to be additive as are the last two terms in (4.2) which also are proportional to s ~ and D. Since the scaling relations were derived for the limit of large v ' / s ~ and large Reynolds numbers only, they account only for the increase of the flame surface area ratio due to turbulence, beyond the laminar value 9 / I V G I = 1 for v' --~ 0. We will therefore write 9 as 9 -IVUI
+ 9t
(4.6)
where 9t is the contribution of turbulence to the flame surface area ratio 9. The model equation for 9t that covers both regimes was obtained as 09t - ~ + iJ . V gt
-
-
(D
+
Dt)klVgt]
o ( -2 SL Tt C2 ~
(G,2)1/2
+
D93t _
C3 - -
G,2
Vt .t~
~.u~)~
C O ~ - - 9 t
k
Ox~
Dt(VG) 2 "~ Cl
G '2
9t
(4.7)
57 The terms on the r.h.s, represent the local change and convection. Turbulent transport is modelled by a curvature term. This is the first term on the r.h.s, of (4.7). The second term models production of flame surface area ratio due to mean velocity gradients. The constant Co -- c~1 - 1 originates from the c-equation. The last three terms in this equation represent the turbulent production, the kinematic restoration and the scalar dissipation of the flame surface area ratio, respectively, and correspond to the three terms on the r.h.s of (4.2). The average of the production term in (4.2) is equal to Sa. Numerical simulations by Ashurst presented in [3] show that the strain rate is statistically independent of a and that the mean strain on the flame surface is always negative. When this term, divided by sOp is plotted over v'/s~ one obtains a linear dependence. This leads to the closure model Vt
-Sa-
b3T~
(4.8)
where b3 = 1.3. Using the definition for the turbulent flame brush
(Gt2)l/2
IV l
(4.9)
with the modelling
gF,t = b2t~
(4.10)
derived in [8] with b2 = 1.78 and Dt = 0.78v~, the modelling constant Cl becomes 5.28. The constants c2 and c3 in (4.7) were determined in [8] as c2 = 0.39cl/b2 and Ca = b4c1. For c2 experimental data by Abdel-Gayed and Bradley [121 in the corrugated flamelets regime leading to ST = 2.0v ~ were used. Damk6hler [1] believed that the constant of proportionality in (1.8) should be unity. That would result in b4 = 1.0. Recent DNS results by Wenzel [13] suggest b4 = 1.3. In order to define the laminar flame thickness unambiguously in the present context we set D-
sOgF.
(4.11)
Then the balance of production, kinematic restoration and scalar dissipation in (1.8) leads with (4.9) to the quadratic equation
~t2
f
~t
0.39gF [VG]
v'e
0"78bns~
= 0
(4.12)
with the solution 1/2
-- =
[VGI
2 ~F
]-
2 ~F
+ 0.78b4
s~
(4.13)
This equation satisfies the limit g/g.F ~ c~ corresponding to ~t = 2.0v'lVUI/s ~ for the corrugated flamelets regime and the limit g/t~F ~ 0 corresponding to ~t = b4(Dt/D)I/21~'--GI
58 20
,
,
,
,
,
,
,
,
,
o
X o
15
,
,
,
,
,
,
,
~
I ..................... -,'...... _X................
x
o
i
[] 10
o
......................... ; . . . . . . . . . . .
i
o
_o_...... i................................ i.........................
o
5
0
5
10
15
20
v'/r Figure 3. Comparison of the burning velocity ratio calculated from (4.13) and (4.14) for R e - 650 (solid line) with data collections by [12] for Reynolds numbers ranging between 500 and 750. The origin of the individual data points may be found in that reference.
for the thin reaction zones regime. Using (2.11) and (4.6) we now express the turbulent burning velocity of a steady planar flame as s~
s~
at ) "
1 + IVGI
(4.14)
In Fig. 3 the ratio of the turbulent to the laminar burning velocity has been plotted using (4.13) with b4 -- 1.0 and with the Reynolds number as a parameter. Such correlations have been plotted in [12] for Reynolds numbers ranging from R e < 25 to R e = 4600. We use Fig. 9 of Abdel-Gayed & Bradley [12] and the average Reynolds number value of 625 for the data in this figure. The comparison between the curves calculated from (4.13) and (4.14) and the experimental data in [12] is shown in Fig. 4. Although the scatter of the data is quite considerable the calculated burning velocities seem to follow the trend in the data.
5. DIRECT NUMERICAL SIMULATIONS OF THE G-EQUATION FOR ISOTROPIC T U R B U L E N C E DNS simulations similar to those by Kerstein et al. [10] were performed by Wenzel [14] to calculate production, kinematic restoration and scalar dissipation in (3.3) and (4.2). The numerical method is described in [13]. A particular numerical scheme has been developed that is able to calculate derivatives even for the case D - 0 very accurately. In Fig. 4 the temporal development of e is shown for the cases V~/SL -- 1 with D - 0 and
59 7
....
I ....
I ....
I ....
I ....
I ....
t -I ~ -
I
I1%
~
I __
rl
i~'l
.t
e
i
r
I ~
~ I v
r
I
iI
-"
I
I "~l
I
I%
Ilk ~
I
v
~
~ II
--
I t |l
-
~
~ /
/I 9
;
-
_
3 2 1 o
. . . . . . . . . . . .
0
5
I . . . . . . . .
10
15
20
[ ....
25
',
30
Figure 4. Temporal evolution of the flame surface area ratio in a cubic box for Vt/SL -- 1.0 (solid line) and v ' / S L - 0.8 (dashed line).
Vt/SL 8 with D - 0.96. The first case with ~./~.F "-'+ CO corresponds to the corrugated flamelets regime and the second with g/f-F = 3.04 to the thin reaction zones regime. The source terms in the variance equation averaged over 25 eddy turnover times are plotted in Fig. 5 over the spatial distance from the mean flame position. At the mean flame front position G - Go the fluctuations are zero and therefore all source terms vanish because they are proportional to G'. It is seen that for the case v'/SL -- 1 and D - 0 the scalar dissipation is zero and production equals kinematic restoration. In the case v'/SL -- 8 and D - 0.96 kinematic restoration is smaller than scalar dissipation and the sum of both nearly balances the production term. A similar balance is shown in Fig. 6 for the three source terms in (4.2) averaged over the same range of turnover times. Here the production term is denoted by II, the kinematic restoration term by ft and the scalar dissipation term by X. There are regions within the turbulent flame brush where production and kinematic restoration is slightly negative whereas scalar dissipation always remains positive. It is also seen that for the case v'/SL -- 8.0 the relative importance of the kinematic restoration term is smaller in the # equation than in the variance equation. This indicates that in the thin reaction zones regime turbulent flame propagation is primarily due to diffusive rather than to kinematic effects. -
-
6. C o n c l u s i o n s There are two competing mechanisms in premixed turbulent combustion: Advancement of the local flame front by a kinematic balance of turbulent velocities and laminar burning velocities or by turbulent diffusion of small eddies and curvature of the thin reaction zone. It was shown t h a t these different mechanisms lead to the same result as far as the flame brush thickness is concerned, but to different scaling laws for the flame surface area ratio.
60 In a modelled equation for that quantity the laminar burning velocity and the laminar diffusivity appear explicitly. This is different from the classical methodology in turbulence modelling where viscosity and diffusivity is contained in the definition of dissipation. Here, in order to obtain an equation that is valid in both regimes, the competition between kinematic restoration and scalar dissipation becomes evident through the use of these laminar quantities in the modelled equation for the flame surface area ratio. REFERENCES
1. DamkShler, G. 1940 Z. Eletrochem. 46, 601-652, English translation NACA Techn. Memo. No.1112, 1947. 2. Peters, N. 1986 Twenty-First Symposium (International) on Combustion, The Combustion Institute, 1231-1250, Pittsburgh. 3. Bray, K. N. C. and Peters, N. 1994 In Turbulent Reacting Flows (P. A. Libby and F. A. Williams Eds.) Academic Press, 63-113. 4. Williams, F. A. 1985 Combustion Theory (2nd Edition), Addison-Wesley, 429ff. 5. Williams, F. A. 1985 In The Mathematics of Combustion (J. D. Buckmaster, Ed.) Society for Industrial and Applied Matehmatics, Philadelphia, 97-131. 6. Pelce, P. and Clavin, P. 1982 Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames, J. Fluid Mech. 124, 219-237. 7. Matalon, M. and Matkowsky, B. J. 1982 J. Fluid Mech. 124, 239-259. 8. Peters, N. 1999 The turbulent burning velocity for large scale and small scale turbulence, to appear in J. Fluid Mech. 9. Ashurst, Wm. T. 1994 Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, 1075, Pittsburgh. 10. Kerstein, A. R., Ashurst, Win. T. and Williams, F. A. 1988 Phys. Rev. A37, 27282731. 11. Peters, N. 1992 J. Fluid Mech. 242, 611-629. 12. Abdel-Gayed, R. G. and Bradley, D. 1981 A two-eddy theory of premixed turbulent flame propagation, Phil. Trans. R, Soc. Lond. A 301, 1-25. 13. Wenzel, H. 1997 Annual Briefs-1997, Center for Turbulence Research, 237-252. 14. Wenzel, H. 1999 PhD thesis at the RWTH Aachen.
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
63
V I S U A L I Z A T I O N A N D M E A S U R E M E N T OF S P A T I A L S T R U C T U R E S IN T U R B U L E N T FLOW W. Merzkirch, T. Rettich, F. Schneider, W. Xiong Lehrstuhl for Str6mungslehre, Universit~it Essen, D-45117 Essen, Germany Vortical spatial ("coherent") structures were quantitatively visualized in turbulent pipe flow by means of particle image velocimetry and subsequent decomposition of the velocity data. The experiments were performed at Reynolds numbers of the order of 10s. The experimental results served for explaining quantitatively the process of signal formation for a particular, correlation-based ultrasound flow meter. It could be shown that the signals are formed by an interaction of the ultrasound waves with the coherent structures in the flow. 1. INTRODUCTION Measurements conducted over decades with mechanical and optical probes (e.g. hot wire, LDA) have made available an abundance of data on temporal characteristics of turbulent flows. Through the application of qualitative and quantitative flow visualization methods it became evident that turbulent flows also exhibit characteristic spatial structures. Many of these structures are dominated by vortical flow, and it is common to designate them as "coherent" structures, although there is no general or precise definition of the term "coherent" in this context [1,2]. We report here on the formation and existence of coherent structures in turbulent pipe flow at high Reynolds numbers and their relevance for ultrasonic flow metering. Evidence of the structures can be given by the use of a 2D measuring or imaging technique, e.g. particle image velocimetry (PIV) [3]. Due to the dominating axial velocity component in pipe flow, the structures are normally hidden in a PIV recording, and a special procedure is required for making them visible, e.g. subtracting from the 2D instantaneous velocity distribution a constant velocity [4], or the mean (time-averaged)velocity profile [5]. The latter case corresponds to a "Reynolds decomposition" of the velocity field. Other types of decomposition, for analyzing quantitatively the coherent structures, are known. We shall present results obtained by applying the method of proper orthogonal decomposition (POD) [6]. The coherent structures are convected in the pipe with the main flow. Later we shall show that this convection is the physical basis for a correlation-based ultrasound method for flow metering [7]. It must be expected that the structures are convected at different speeds, depending on their position in the pipe. According to Adrian et al. [8] this speed can be estimated by subtracting from the instantaneous PIV recording constant velocities of different value. Only those structures become visible that are convected with the respective velocity subtracted.
64 In the following we shall report on PIV measurements in pipe flow at a Reynolds number, based on the pipe diameter D, ReD = 1105. For comparison: The experiments of Westerweel et al. [4] were performed at a Reynolds number that was by more than one order of magnitude lower, because their experiments were accompanied by DNS caluclations. The higher values of ReD as used here are of interest for the application to flow metering. We shall then describe the method for ultrasonic flow metering and present experimental results showing the influence of the coherent structures on the performance of the meter.
2. PIV EXPERIMENTS Planar (2D) velocity measurements in the turbulent flow of air through a circular pipe were performed with digital particle image velocimetry (DPIV). The main elements of the DPIV system are a double-pulsed YAG laser and a CCD camera recording double-exposures of the particle images (Fig. 1). The air flow in the pipe (100 mm i.d.) is seeded with oil droplets having a mean diameter between 2 and 3pm. The optical test section is made of a transparent membrane of 0.3 mm thickness, thus minimizing optical distortions. The velocity distribution in a vertical plane through the pipe axis is determined by using the MQD evaluation method [9]. Such 2D instantaneous velocity fields can be recorded at time intervals 0,16 s. For comparison the integral time scale at the Reynolds number 105 and mean pipe velocity 15 m/s is of the order of 0.01 s.
Fig. 1: Experimental set-up for PIV measurements The time averaged velocity profile, calculated according to the analytical solution of Gersten and Herwig [10] that is valid for high Reynolds numbers, is subtracted from the instantaneous velocity field. The result is the instantaneous distribution of the turbulent fluctuations u', v' (Fig. 2). Here, u' is the axial, and v' is the radial component of the fluctuations in the plane of the light sheet. Vortical structures can be observed in Fig. 2. Their appearance can be emphasized by determining the vorticity component normal to the plane of measurement (Fig. 3). Only ranges of the vorticity value are indicated in Fig. 3 due to the errors arrising in the differentiation of the experimental data.
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Fig. 3: Vorticity contours of the field shown in Fig. 2. Another verification of the coherent structures is the decomposition of the velocity fields by the POD method [5]. We apply this decomposition to a series of 20 successively recorded velocity distributions. Among the resulting eigenfunctions (~i, the first eigenfunction (~ represents approximately the time-averaged velocity distribution. After having found all functions (~, we take a series expansion of the (~ with the exception of (~. The result (Fig. 1) is comparable to a distribution of the fluctuations (Fig. 2). The similarity of the raw patterns shown in Figs. 2 and 4 is obvious. Similarity of the details cannot be expected, because Fig. 2 is a distribution taken at a random instant of time. After we have verified the existence of coherent structures in turbulent pipe flow, we can now study their influence on the propagation of ultrasound and their relevance for an ultrasonic method for flow metering.
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3. A C O R R E L A T I O N - B A S E D METERING
ULTRASOUND
METHOD
FOR
FLOW
The principal set-up of this method is shown in Fig. 5. Two continuous ultrasound waves are radiated through the flow in the pipe in a direction normal to the pipe axis. The two beams at an axial distance Ax form a plane that includes the pipe axis. The two continuously received signals exhibit the frequency of the ultrasound wave (here: 111 kHz) and modulations of this wave with a considerably lower frequency, of the order of 1 kHz. It is expected that these modulations are caused by coherent structures convected with the main flow.
Fig. 5: Principle of set-up for correlation-based ultrasound flow metering.
67 The convection time At needed by the structures for moving along the distance Ax can be determined by cross-correlating the signals received by the two ultrasound sensors. Division of Ax by At gives a velocity u that can be calibrated against the mean pipe velocity Um (volume flow/cross-sectional area). The
u/.. = Um,x/.-.
ratio K =
,
measured for various values of u~, assumes values between 1 and K ~ , where Umaxis the maximum flow velocity on the pipe axis (Fig. 6). Note that these experiments have been performed with air flow in a straight circular pipe of D=100 _
mm i.d., and that the Reynolds number ReD for the range of velocities Um shown in Fig. 6 varies from 2104 to 2.3105. 1,3 K
1,2
U ma.~,
I
1,1 1,0
5
15 25 u~ [m/s]
35
Fig. 6: Ratio of measured convection velocity and mean pipe flow velocity, K=u/Um, for different values of Urn. The ultrasound wave travelling from the emitting to the receiving sensor integrates from wall to wall the results of its interaction with the convected coherent structures. If we suppose that the structures are convected with the local, timeaveraged velocity of the turbulent pipe flow, then the values of K must be expected to lie in a range as seen in Fig. 6; i.e., the modulation of the signal is dominated by the interaction of the ultrasound wave with the structures moving in the central part of the pipe. This was confirmed by experiments in which the position of the plane formed by the two ultrasound beams was not through the axis but excentric, where the average velocity of the structures should be lower than in the "central" plane. Consequently, these experiments gave values for K lower than those in Fig. 6. The results in Fig. 6 were obtained with a non-dimensional distance Ax/D=I. Variation of Ax showed that this is an optimum regarding the signal quality, e.g. expressed by the magnitude of the correlation coefficient. At lower values of Ax/D, the signal quality decreases because of an interference of the two ultrasound waves on the receiving sensors. At values Ax/D>I the signal quality decreases again, most probably due to a decay of the coherent structures. Note that for the Reynolds numbers used here the turbulent integral scale is in the order of D. According to our interpretation the modulation of the ultrasound wave is caused by the superposition of the sound velocity and the radial velocity component occuring in the vortical structure ("radial" here with respect to the geometry of the circular
68 pipe). Depending on the sign of this component, the propagation of the wave is either accelerated or decelerated, depending on the (instantaneous) position of the vortical sturcture relative to the ultrasound beam. The passage of such a structure through the beam can be observed in the recorded signal; an example is given in Fig. 7 where the voltage expressing the signal from one receiving sensor is plotted for a period of 6 ms. The time interval needed by the structure to pass the beam is 1.7 ms. With the mean pipe velocity u=--15m/s (ReD=l105) follows a size (diameter) of the strucure of approximately 2.5 cm, and this is the same order of sizes observed in the PIV experiments (Fig. 2).
75,0 At= 1,7 ms U[mV]
0,0
-75,0 2,0
4,0
6,0
8,0 t [ms]
Fig. 7: Voltage U expressing the signal from one (of the two) receivers recorded during a time period of 6 ms; At is the time needed by a vortical structure for passing the ultrasound beam.
Fig. 8: Real signal (in terms of voltage U) from one receiver and simulated signal obtained with a ray tracing method.
69 Further evidence for the assumed interaction of coherent structures and ultrasound wave results from a calculation of the wave propagation with a ray-tracing model. For this purpose we had traced an ultrasound wave through a turbulent pipe flow with a velocity distribution as measured in the PIV experiments (Fig. 2); for details see [11]. The numerically simulated signal exhibits the same amplitude and frequency characteristics as a measured signal (Fig. 8). Of course, a similarity of the details cannot be expected, because the PIV result reflects a state at a random instant of time. 4. C O N C L U S I O N S We have shown that large-scale vortical or "coherent" structures exist in fully developed turbulent pipe flow at Reynolds numbers in the order of 105. Planar measurement with PIV and subsequent decomposition, here with the POD method, are an appropriate tool for a quantitative visualization of the structures. Evidencing the structures can, in principle, be reduced to a difference of two coordinate systems moving with respect to each other: the system of the physical flow and the system in which the images are recorded. Different structures appear if the relative velocity between the two systems is changed. In the past, coherent structures were mostly visualized in flows with Reynolds numbers lower than in the present experiments, particularly when those flows were also described by direct numerical simulations. The higher Reynolds numbers in our experiments were chosen because it was anticipated that coherent structures in the turbulent flow are dominating the signal formation in a particular set-up for ultrasonic flow metering, and the interest in this method lies particularly at flow conditions with high Reynolds numbers. The hypothesis of signal formation by interaction of the ultrasound waves with coherent structures could be verfied directly and indirectly by various observations. These results now allow to optimize the configuration of the meter and the algorithms for signal processing. Future experiments are also planned for characterizing the coherent structures under not-fully developed flow so that flow metering might become possible at such conditions with disturbed pipe flow.
REFERENCES 1. A.K.M.F.Hussain, Coherent structures and turbulence, J. Fluid Mech. 173 (1986), 303-356. 2. H.E. Fiedler, Coherent structures, in: Advances in Turbulence (eds. G. ComteBellot, J. Mathieu), pp. 320-33, 1987. 3. R.J. Adrian, Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23 (1991), 261-304. 4. J. Westerweel, A.A. Draad, J.G.T. van der Hoeven, J. van Oord, Measurement of fully develped pipe flow with digital particle image velocimetry, Exp. Fluids 20 (1996), 165-177. 5. F. Schneider, W. Xiong, W. Merzkirch, Visualization and measurement of coherent structures in high Reynolds number pipe flow. Proceed. 8th Int. Sympos. Flow Visualization, Sorrento, Italy, 1998. 6. G. Berkoosz, P. Holmes, J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech. 25 (1993), 539-575.
70 7. Rettich, G. Poppen, Ultrasonic flow measurement in turbulent pipe flow using a correlation technique. Proceed. FLUCOME '97, Hayama, Japan, 1997. 8. R.J. Adrian, K.T. Christensen, S.M. Soloff, C.D. Meinhart, Decomposition of turbulent fields and visualization of vortices and turbulent momentum transport. Proceed. 8th Int. Sympos. Flow Visualization, Sorrento, Italy, 1998. 9. L. Gui, W. Merzkirch, A method of tracking ensembles of particle images, Exp. Fluids 21 (1996), 465-468. 10. K. Gersten, H. Herwig, StrOmungsmechanik, Vieweg-Verlag, Wiesbaden, Germany, 1992. 11. T. Rettich, Dissertation, Univ. Essen, 1998.
ACKNOWLEDGEMENT Financial support of this research by Deutsche Forschungsgemeinschaft (DFG, Me 484/29, Forschergruppe "StrOmungsmechanische Grundlagen der Durchflul~messung") is gratefully acknowledged. We also express our thanks to DLR GOttingen for making available computer software for the POD calculations, and to LaVision GmbH for their support in carrying out the PIV experiments.
0
Turbulence Modelling
This Page Intentionally Left Blank
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
73
T h r e e - D i m e n s i o n a l M o d e l l i n g of T u r b u l e n t F r e e - S u r f a c e Jets T.J. Craft, J.W. Kidger and B.E. Launder Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK
Results are presented from a numerical investigation into the effects of a free surface on the profiles of a circular turbulent jet issuing parallel to, and just below, a liquid-gas boundary. Computations have been performed using two Reynolds-stress turbulence models: a "basic" pressure-strain model incorporating free-surface pressure-reflection terms, and a more elaborate two-component-limit model. The latter reproduces the flow development markedly better in the developing region. However, our computations suggest that the flow continues to develop, exhibiting an increasingly asymmetric spreading in the vertical and horizontal directions, far beyond the range of current experimental data. 1. I N T R O D U C T I O N The present study concerns the numerical simulation of turbulent, flee-surface jets;that is, jets which issue parallel to, and just below, a liquid-gas interface. This flow is particularly relevant to the power-generation industry, where a cheap method for removing waste heat involves drawing water from a nearby natural body, and then releasing the heated water back to the source (McGuirk & Rodi 1978). For environmental reasons, it is important to ensure that the discharge jet is properly mixed with the surrounding fluid in order to minimise the presence of localised warm areas. However, it is found that the presence of the free surface greatly alters the structure of the flow from that prevailing in a free submerged jet. This problem has been the subject of several experimental investigations, including those by Rajaratnam (1984), Rajaratnam & Humphries (1984), and Anthony and Willmarth (1992), though all have considered only the relatively near field (x/d .......{....... !....... {. . . . . . . i ........ } ........ } .........}...=~.~ ......i .........i ....... i .......{.TCL---.I
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A CKNOWLEDGEMENTS T.J. University Studentship.
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contribution while names
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82 REFERENCES
Anthony D.G., Willmarth W.W., 1992, "Turbulence measurements in a round jet beneath a free surface" J. Fluid Mech., 243, pp699-720 Celik I., Rodi W., 1984, "Simulation of free-surface effects in turbulent channel flows" Physico-Chemical Hydrodynamics, 5, pp217-227 Craft T.J., 1998, "Second-moment computations of the three-dimensional thermal wall jet" Proc. 2n~ EF Conference in Turbulent Heat Transfer, Manchester UK, pp 115-122 Craft T.J., Ince N.Z., Launder B.E., 1996, "Recent developments in second moment closure for buoyancy-affected flows" Dynamics of Atmos. and Oceans, 23, pp99-114 Craft T.J., Launder B.E., 1992, "New wall-reflection model applied to the turbulent impinging jet" AIAA J., 30, pp2970-2972 Craft T.J., Launder B.E., 1996, "A Reynolds stress closure designed for complex geometries" Int. J. Heat and Fluid Flow, 17, pp245-254 Craft T.J., Launder B.E., 1998, "The self-preserving 3-dimensional, turbulent wall jet", submitted for publication Gibson M.M., Launder B.E., 1978, "Ground Effects on Pressure Fluctuations in the Atmospheric Boundary Layer" J. Fluid Mech., 86, pp491-511 Kebede W., 1982, MSc Dissertation, Dept of Mech. Eng. UMIST Lien F-S., Leschziner M.A., 1994, "A general non-orthogonal finite-volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part 1: Numerical Implementation, and Part 2: Application" Comp. Meth. in Appl. Mech. Engng. 114, pp 123-167 Lumley, J.L., 1978, "Computational modelling of turbulent flows" Adv. in Applied Mech., 18, pp123-176 Launder B.E., Li S.-P., 1994, "On the elimination of wall-topography parameters from second-moment closure" Phys. Fluids, 6-2, pp 999-1006. McGuirk J.J., Papadimitriou C., 1985, "Buoyant surface layers under fully entraining and internal hydraulic jump conditions", Proc. Turbulent Shear Flows 5, Cornell University, pp 22.33-22.41 McGuirk J.J., Rodi W., 1978, "Mathematical modelling of three-dimensional heated surface jets", Sonderforschungsbereich 80, Universitat Karlsruhe Rajaratnam N., 1984, "Non buoyant and buoyant circular surface jets in coflowing streams" J. Hydraulic Research, 22, pp 117-140 Rajaratnam N., Humphries J.A., 1984, "Turbulent non-buoyant surface jets" J. Hydraulic Research, 22, pp 103-115 Reece G.J., 1977, "A generalised Reynolds stress model of turbulence", PhD thesis, Faculty of Engineering, University of London.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
83
A-Priori Tests of Reynolds Stress Transport Models in Turbulent Pipe Expansion Flow C. Wagner a and R. Friedrich b aDLR, Institut fiir StrSmungsmechanik, Bunsenstr. 10, 37073 GSttingen, Germany bLehrstuhl ffir Fluidmechanik, Munich University of Technology, Boltzmannstr. 15, 85748 Garching, Germany The database obtained from a direct numerical simulation (DNS) of turbulent sudden pipe expansion flow with an expansion ratio of E R - 1 . 2 has been used to analyse various Reynolds stress models. The inflow conditions were provided by a DNS of fully developed turbulent pipe flow. The Reynolds number at the inlet thus corresponds to that in the pipe and is 360 based on diameter and friction velocity. In this DNS a mean reattachment length of 10.2 step heights was observed. Various models for pressure strain correlations, turbulent transport and viscous dissipation were evaluated using the DNS database. It was found that none of the investigated redistribution models is able to predict the extraction of fluctuating kinetic energy from the wall normal velocity component within the reattaching flow at least qualitatively correct. Improved models are needed to predict the turbulent transport of the wall normal stress component for separated flows accurately. 1. I N T R O D U C T I O N The derivation of most of the so-called single-point turbulence models is based on the assumption that the turbulent flow is in local equilibrium. In general this does not hold for turbulent flow of practical interest, since changing wall boundary conditions and/or varying pressure gradients can lead to separation and strong deformation of the flow. Therefore it does not surprise that these conventional turbulence models including the second order Reynolds stress models behave poorly in non-equilibrium flows. Besides this most commercial Navier-Stokes solvers make use of wall functions, which are developed for simple shear flows, in which a logarithmic distribution of the streamwise velocity component is valid. Since wall laws are not available for more complex flows the only alternative is the integration to the wall where exact no-slip and impermeability boundary conditions can be applied. In order to do this, more complex turbulence models are needed which properly decribe viscous and anisotropy effects in the vincinity of the walls. It is the aim of this work to evaluate different second-order closure assumptions which include modifications to overcome the above described short comings. For this purpose a DNS data base of the turbulent flow in a sudden pipe expansion at a low Reynolds number is used.
84 2. G E N E R A T I O N
OF DNS
DATABASE
The incompressible Navier-Stokes equations were discretized by a cylindrical (z, ~, r) grid system with second order accurate central finite differences. The spatially discrete momentum equations were integrated in time on staggered grids with a second order accurate semi-implicit scheme in which all convection and diffusion terms containing derivatives in circumferential direction were treated implicitly. The time-step was chosen following a linear stability argument. A fractional step approach provided the coupling between pressure and velocity fields. It led to a Poisson equation for the pressure which was solved using a F F T in ~-direction. The remaining set of 2D Helmholtz problems was treated with the influence matrix technique in combination with a cyclic reduction algorithm. Boundary conditions at the walls were the impermeability condition with respect to wall-normal velocity components and no-slip conditions for the tangential velocity components. The time-dependent velocity vector in the inflow plane was obtained from the DNS of fully developed pipe flow during the whole simulation process. In the outflow plane we used linear extrapolation of the mean velocities and solved a convection equation for their fluctuations. The computational domain consisted of an upstream pipe
'~
('
1
/
-
II
L i - 1.5D _....
i
L = 3.5D
Figure 1. Computaional domain of the sudden pipe expansion section of diameter D and length Li - 1.5D as shown in fig. 1. The downstream section had the diameter D o - 1.2D and the length L o - 2.0D. Simulations were performed on three different grids in order to investigate resolution effects. The finest grid consisted of 7.2.106 cells with refinement of the mesh close to the up- and downstream wall. For the fine mesh the grid spacing in terms of wall units (of the incoming flow) was: Az + -- 3.7
,
(rA~) + -- 0.073,..,4.71
,
Ar + -- 0.185,..,5.56
(1)
For each simulation a great number (~ 1000) of statistically independent realizations of the flow was provided to ensure stable averages. 3. R E S U L T S The highest turbulence activity occured in the mixing layer, with maximum values exceeding those in the wall layer of the incoming flow by a factor of 5. In Wagner and Friedrich [17] we conjectured that these high amplification rates are to a good deal due to the rapid relaxation of the flow from the wall blocking effect. Radial velocity fluctuations can achieve higher values in the free shear layer being unaffected by any
85 kinematic constraint. This immediately increases the production of the Reynolds shear stress. An increased Reynolds stress in turn produces higher axial velocity fluctuations. 3.1. P r e s s u r e s t r a i n t e r m s
The pressure strain terms in the budgets of the Reynolds stress components (Uz2), presented in eq. 2 - 3 in cylindrical coordinates. Modeling this redistribution mechanism respresents one of the most difficult tasks in the development of a Reynolds stress model. It is on the other hand rather easy to evaluate both the exact pressure strain terms and the corresponding models from the DNS-database.
(u~l, @71, (u:uTI ~e
-
--
p'
es,, - 2 (p, Ou',
+
(2)
(Ou', Ou':]
In figures 2 - 9 a-priori tests of different redistribution models are presented and compared 50
:.._..-..L..L..-..J..L..-..:_..L..
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"--':: .5
2'.0
2.5
1 3.0
z/D Figure 3. Pressure strain term in the budget of the axial velocity fluctuations at y+ - 7. Symbols as in fig. 1.
with the pessure strain term evaluated from the DNS at positions r / D - 0.46 and 0.58, y+ - 14 dimensionless wall units from the wall in the inflow section and y+ - 7 from the outflow wall, respectively. The results for y+ - 14 and z / D < 1.0 in figures 2, 4, 6 and 8 practically correspond to fully developed pipe flow. Position r / D - 0.46 is approximately the position of maximum production in fully developed turbulent pipe flow and in the shear layer downstream the expansion. Profiles for r / D - 0.58 reflect the dynamics within the recirculation and reattachment zones. All investigated pressure strain models assume that the stress redistribution between the velocity components can be described by a slow and rapid redistribution term and two additional terms which represent the influence of solid walls on the redistribution mechanism. The redistribution terms according to Hanjalic and Launder (HL) [7], Launder
86
et al. (LRR) [6], Gibson and Launder (GL)[1], Speziale, Sarkar and Gatski (SSG) [3] were computed from the DNS-data. Besides these high-Reynolds number redistribution models, a priori tests of the low-Reynolds number models of Launder and Shima (LS) [4] and Hanjalic et al. (HJD)[11] were performed as well. 150
300 9-
./-'.\
r~
//
200
."
/,.-, x x. \
:;\
..""
100
-
"i.. 50-
100
. . . . . . . . . .
O-
"..:.: 0
1
-50
2
3
.5
,
2.0
...~,,,.-~ 0 I(iOw'i + - ~ '" ' -- '
(16)
where f/has been substituted in the averaged form of (14) and equations (10)1,2 have been considered, vqT is the turbulence coldness defined analogously to the thermal coldness ~M. While the heat capacity coefficient equals C = ~t~M, C T, the corresponding parameter of the heat capacity of the turbulent fluctuations, has been introduced by noting that the
99 coldness of the turbulence is inversely proportional to its kinetic energy [2, 10]. Inequality (16) is the averaged form of the entropy inequality for the mean field fluid. This will help us to restrict the closure assumptions of the turbulent unknown quantities in a thermodynamically consistent way. Thus, it represents the restricting condition for the realizability of turbulent flows in the modeling and thus must hold for all processes satisfying (10) and (12). After evaluating all differentiations to calculate the time derivative of ~bM and ~ T subtituting then all evolution equations in (16) and demanding that the emerging inequality must hold for all independent variations of the time derivatives of thermodynamical variables, it follows that the entropy inequality reduces to the so-called "residual-inequality of entropy" for the turbulence [10]. Based on this residual entropy inequality, thermodynamically consistent closure assumptions are derivable for the turbulent constitutive quantities. Of all detailed results we restrict ourselves to presenting the pressure-strain correlation tensor for illustration. The ansatz (13) reduces to
II j - II j(R j,
(17)
and the evolution equation for the dissipation rate reads Cij
--
--s
--~ s
-- 2 ( C i k a k j -- C j k a i k ) "~- (OL0 "~ OL1Rklr "~ Ol4r~s -
2-
1
,
2
2+o~7(TR -nt- I~T -- 5T<mn>R<mn>~ij) + Aihk,k "t- c~lof;uj ,
(18)
where Ai~jk - ag(Rkleii,l + Rjleki,l + Rilejk,I). All model coefficients appearing in these relations may be functions of the independent constitutive variables and corresponding invariants. To support physical realizable turbulent flows and to guarantee thermodynamic stability, the residual entropy inequality will impose restrictions on these functions. Some related contraints in equation (11)1 are 2p~
#1-
20,
(7
1 ->0,
1 ->0
T
(7
and
0equil.AiA
b-c=
.
(19)
#1
The expressions in (17) and (18) are new and have no counterpart in the common turbulence modeling as they depend on T. A natural question to be asked from the extended thermodynamics is how they are related to the known results in the literature. A first impression can be gained, if differential expressions for the Cauchy-stress tensor and the heat flux vector for Navier-Stokes-Fourier fluids are derived and substituted in (17) and (18). Such relations are provided by a formal Maxwell iterative scheme. 5. T R A N S I T I O N
TO ORDINARY THERMODYNAMICS
AND RESULTS
Navier-Stokes-Fourier equations as first iterates, __ @(1).
--~DiJ
-(1)
qi
-
-- --I'~O i
#1
__
[2 > 0
-
>_ o ,
(20)
100 are calculated by using the zeroth iterates T A0) = 0 on the left-hand sides -~ = 0 and t/i of equations (11). These zeroth iterates are the equilibrium values. With (20) the set of turbulent basic balance equations (10) reduces to the known classical results (Launder et al (1984)) and the balance equation (18) for the dissipation rate takes a thermodynamically consistent form in the framework of the Navier-StokesFourier theory. Inserting (20) in (17), one now obtains the turbulent closure assumption for the pressure-strain correaltion tensor in the framework of the Navier-Stokes-Fourier theory in the form 1-Iij
--
-(1)
I I i j ( R i j , ~ i ,cij , Dij , qi ) - -s --[-/~4E -Jr-/~5c
-1t-
--
2
,~6( RC ~- C R -- -~ R ( m n > C<mn> --
-Jr-~7(Dikr
-[- r
zr- 2A2kbij + A3P
2
5~j)
-
-- 5Dmnr
-t-()~s~j nt- ,~'s~j ~(1)) Qi ,
(21)
where in agreement with Launder et al (1984) and Ahmadi et al (1991) the notations
Eij - -RikUk,j - RjkUk,i , P -- --
(-
1
Pkl -- P + (wkjRjl + a;ljRjk) + -~PSkt,
t~ImDkm ~- t~kmDml - ~ m j O j m ~ k l 2 5
,
P -- t~ijgj,i /2
(22)
ujmT<jm>
vr
have been introduced. The production of the Reynolds stress tensor by the gravitational ! ! fields, noted as Gk~ -- fku~ + f~ Uk, is considered as a given supply term in the Revolution equation. Furthermore, the flux of the turbulent kinetic energy, for instance, reduces to
I(i -- (#uj,iuj - - p ' u i -
1/2puiujuj)
First, we remark that the classical expressions result as a special case from the equations of extended thermodynamics in an approximate manner by means of the first iterates in the Maxwellian scheme. Except for the diffusive transport which could be derived from the evolution equation of a higher moment, all thermodynamically consistent second order closure assumptions appear independent on the rotation of the frame. Consequently, all existing models of the pressure-strain correlation tensor which are depended on the rotation of the frame can be considered thermodynamically inconsistent in the framework of the Navier-Stokes-Fourier theory. These models are mainly based on the general classical expression Ili~j derived by Lumley as follows
IIi~
=
1 bo~aij + bl~(aikakj -- -~IISij) + b2kDij + (b3akiDlk + b4aplazmDpm)kaij --
--
2
-
_ nt- b6aklalmDkm)k(aikakj -- 51 IISij) + bTk(aikDkj + ajkDik -- -~amnDmnSij) nt-(b5akiDlk - + ajkakzDil - - -~aplazm 2 DpmSij) + b9k(aikl~jk + ajkiTVik) +bsk(aikDkl Djl +blok(aikakzlTVjt + ajkakllTVit) + bllklTVpqapk(ajkaiq + aikajq).
(24)
101 Here, the model coefficients hi, ( i - 0,... ,10) may depend on the second (II) and third (III) invariants of the anisotropy tensor aid "- kR. We conclude" 1) To be thermodynamically consistent in the framework of the Navier-Stokes-Fourier theory, the conditions b9 = bl0 = bll - 0 must be imposed on such models irrespective of whether they obey the realizability-constraints by Schuman or not. 2) To guarantee physical realizable turbulent flows, the appearing model coefficients in (18), (21) as well as in (23) and (18) must be restricted by the residual entropy inequality. Some related restrictions are:
a~ > o,
(a~ + ( a ~ - a4)RT) >__0, c~_>o,
(~0 + ~RT + ~4Tck~)_> 0, (25) ,~-c~3-1, c~9_< ~
Second, insertion of (20) in (25) leads to the thermodynamically consistent realizabilty conditions for turbulent flows. By deriving the restrictions (25), the condition k >_ 0 has been taken into account per definition. Furthermore, from the trivial condition on k and the inequality (A~ + ( A 3 - A4)RT)(RR) >_ 0 which has led to (25)2, the Schwarz inequality emerges obviously with (RR) >_ 0 in the form
0
%
uulk
0.5 ,
o.o
,
. -
- .~-
- ~
-~
. . . . . . . . . . .
L . . . . . . . . .
0.0
2.0
4.0
6.0
8.0
1
10.0
.....
vv/k
..................... w w / k
......
(uv) l(u v
r=1:~
Figure 1. Realizability for homogeneous shear flows" Symbols, experimental data by Tavoularis and Corrsin [7], Tavoularis and Karnik [8], and Harris, Graham and Corrsin [9]; Lines, computations for the ARSM.
4. M o d i f i c a t i o n s of t h e A R S M for W a l l - B o u n d e d flows In the vicinity of the wall the pressure-strain components are strongly influenced by the wall reflection of the turbulent pressure field and by viscous effects. These effects combined with non-homogeneities which are not described by the wall-independent pressure-strain correlation need to be modeled in wall-bounded turbulent flows. Thus, following an approach taken by Launder and Shima [13] for second moment closures the total pressure-
106 strain correlation can be divided into two parts, namely a basic wall independent part O i~ and a wall dependent part Oi~ which represents the effects created in the presence of the wall" r
_
+
(5)
Even though the general linear pressure-strain correlation in Eq. (2)is not able to account for the wall effects, second moment closures and therefore the ARSM are also used for predicting inhomogeneous flow fields, such as a boundary layer and the near wall region in a channel flow. As a consequence additional degrees of freedom have to be created by O i~j. Furthermore, the anisotropies in e are grouped together with the wall dependent pressure-strain part O i~. Since the exact processes of the wall reflection of the turbulent pressure field cannot be resolved by the considered turbulence model, empirical functions have to be introduced in the formulation of O i~. Assuming the same functional form for 9i~ and ~i~ yields for the coefficients C~ -
(6)
C~o + C ~ f ~ ,
where i = 1,2,3,4, so that the empirical dependence on the wall is grouped inside a wall function fw. Therefore, the pressure-strain coefficients consist of a wall independent part Cio which has already been calibrated against homogeneous shear flows and the equilibrium region of channel flows and a wall dependent part Ciwfw. Physical arguments support the idea that the effects of the wall have to disappear with an increasing distance from the wall and have to be stronger adjacent to the wall. Therefore, the use of the standard wall function fw = exp(-y+/B +) in terms of the wall coordinate y+ = U~y/v, with U, denoting the friction velocity and B + an adjustable constant, is an obvious choice and frequently used in near wall models. However, these models cannot be applied to flows in complex geometries where a unique wall distance does not exist. As a quantity which is independent of the wall distance, the turbulent time scale r + = k+/e + is implemented in the wall function fw = exp(-r+ /A+ ), with A + = 60, leading to a more general model applicable to all geometries. The same wall function fw is used for all wall coefficients Ciw .
5. C a l i b r a t i o n of t h e W a l l D e p e n d e n t Coefficients Eq. (6) shows that four additional coefficients appear in the wall dependent pressurestrain part. From Taylor-Series expansions of the fluctuation velocities at the wall and continuity constraints it can be deduced that u ,,~ y, v ,,~ y2 and therefore, g-g ~ y3. This argument further yields k ,,~ y2 and e ,,~ y0. The mean velocity goes as U ,,0 y, thus for the gradient: (OU/Oy) ~ 1. The Boussinesque approximation leads to ut "~ y3, so that finally C, ,,~ 1/y, since C, - vte/k2. This asymptotic behavior can only be represented if the function g in the formulation for C, goes as 1/y in the vicinity of the wall. Since the production P disappears and e reaches a finite value, the condition lim C 1 -
y+ ---*O
1
----+
Clw- 1 -
Clo
must be satisfied for consistency of the model at the wall, so that Clw is determined. Cow, C3w and C4w remain to be calibrated against a wall dependent flow field. The set
107 Cow = 0.211, Caw = 0.71 and C4w = 0.275 is the result from the calibration of the coefficients with the fully developed channel flow and obtained by optimizing the mean velocity and the Reynolds shear stress and simultaneous adjusting the level of anisotropies of the normal stresses in the wall region. The same set of constants is kept for the zero pressure gradient boundary layer (ZPG) and the adverse pressure gradient boundary layer (APG) case. The wall dependent ARSM formulation also has to satisfy the realizability constraints. In a general two-dimensional incompressible flow the velocity field depends on three independent strain components which can be grouped together to two nondimensional quantities describing the relative effects of the different strains. Since aij in Eq. (4) depends besides on the strain field and the pressure-strain coefficients also on the non-dimensional time scale r and the value of fw, four independent quantities determine the Reynolds stress tensor in the ARSM. The numerical approach taken is to vary the four quantities and search for the extrema in the four-dimensional space. In this way it was ensured that all energy components are non-negative and that the Schwarz inequality is satisfied for the given set of coefficients. 6. C o r r e s p o n d i n g
k - e Model
The kinetic energy equation is modeled in the traditional way as
Dt
=
Oxj
--+u
+P-e,
crk
(7)
with ~'t - C, k2/e and ~rk = 1.0. The high Reynolds number transport equation for the dissipation e is modified in three ways to include near wall effects. First, the coefficient of the decay term is made Reynolds number dependent with the function f2 following Hanjalic and Launder [14]. Secondly, a secondary source term appearing in the exact transport equation is retained and modeled as
OUi 02Ui 2l"ttl Oxj OXlOXj
( 02Ui ) 2 Ce3l"l/t OXjOXl '
as suggested by Shih [1,5]. Finally, the time scale is changed according to Durbin [16] =
-;CT
,
so that re is identical to the standard time scale k/e over most part of the flow. However, adjacent to the wall where the flow field is dominated by viscous effects, re switches to the finite Kolmogorov time scale. Thus, the dissipation equation reads
Dt = Oxj (u +
)
- eel 7"~ -OXj -
C~2f2-r e + C~auut
(Oxj(~Xl
,
(8)
where f2 = 1 - 2 / 9 e x p [ - ( R e t / 6 ) 2] and ~ = 1.3, CT = 3.0, C~1 = 1.44, C~2 = 1.9, and C~a = 1.0. Eq. (8) works well for channel flows and the ZPG boundary layer. However, as discussed in detail by Rodi and Scheuerer [17] problems arise if a k - e type model is used in an adverse pressure gradient flow. Hanjalic and Launder [18] recommended
108 the artificial enhancement of the production of dissipation originating in the irrotational strain part by a factor C~4 so that the total production of dissipation reads in the APG case
re -- - C e l ~ 0 g
we Oy
Ce4 d (~__~._ ~_.~) 0 g
~
~
(02U~ 2
+ C~3uut Oy2] ,
with C~4 = 3.5 which works better for the ARSM than the original value of Ce4 = 4.44 as suggested by [18]. The irrotational strain term has no influence on the ZPG boundary layer and the channel flow since OU/Ox 50. However, close to the wall the anisotropies of the Reynolds stress tensor are not fully resolved. In this region the turbulent transport is likely to play a dominant role and therefore, most algebraic Reynolds stress closures fail to accurately predict u~uo. Thus, the variable pressure-strain coefficients in ~ij are obviously not sufficient to compute the anisotropies close to the wall. Finally, Fig. 2 shows reasonable agreement of the skin friction computation with the experimental data compiled by Dean [19]. Fig. 3 shows the result of the calculations for the two boundary layer cases. First, the ZPG boundary layer at Re0 = 7700 is computed and compared to experimental results by Klebanoff [20]. As in the channel flow case the mean velocity and the Reynolds shear stress show good agreement with the data while the ARSM is not able to completely reproduce the anisotropies in the normal stresses. Furthermore, an APG boundary layer case according to the experiments by Andersen, et al. [21] is predicted by the model. In agreement with the discussion by Rodi and Scheuerer [17] for standard low Reynolds number k - e models the ARSM also suffers from inaccuracies in the APG case. Close to the wall both the mean velocity and the shear stress differ from the measurements while the agreement is very good towards the outside of the boundary layer. The last plot for the Reynolds normal stresses shows again that the modification in the pressure-strain correlation is insufficient to completely capture the anisotropies in the normal stresses for wall-bounded turbulent flows. It seems that the transport effects in Eq. (1) need to be included in some way in the stress-strain relation to improve the quality of the predictions.
9
~'~-~
r~
~ ' ~
o
.-LO 0
o
~
0 0 0 0 0
0 0
:0g
"~+
I
,
.
~,/
IX5
IX5
,
. . . .
~ ~
0 ~*'-~
0 .
.
.
+
.
.
....
-I~
0
GO
.
, , , , , , , ,
Ob
Cf * 10 3
t
.
O~
B> 0 IZl ~ 0, then an exponential growth of the turbulent kinetic energy corresponding to an unstable flow is calculated (Cel = 1.44 and Ce2 = 1.92). Due to linear stability results [9] this exponential growth is only in the intermediate range 0.0 < ~t/S < 0.5 a stable solution. Outside this interval a decay of turbulent kinetic energy referring to a stabilizing Coriolis effects is observed. It is characterized by ( ~ ) ~ - 0. This exchange of stabilities is illustrated in the bifurcation diagram in figure 1. Due to its material frame indifference the standard k-e model does not account for any effect of Coriolis forces. For homogeneous turbulent flows the GS-model represents exactly the underlying second moment closure calibrated for this flow [3]. It predicts an unstable flow, corresponding to an increased turbulent momentum transfer in the interval - 0 . 0 7 < ~/S < 0.51. Otherwise the flow is stable. The altered turbulent momentum
117 0.30
0.25
0.20 ,o~0
0.15
0.10
0.05
0"001.0
-0.5
o.o
05
1:o
is
20
~/S
Figure 1. Bifurcation diagram for homogeneous shear flow in a rotating frame of reference
transfer predicted by the non-linear SZL- and CLS-model is not limited to the rotation rate interval of the linear stability analysis. The SZL-model predicts a higher dimensionless shear rate for ~ / S - 0 than for ~ / S - 0.5 according to the Richardson number similarity consideration of Speziale [10]. A feature the CLS-model does not obey. The CLS-model can be improved by employing a modified absolute rotation rate tensor for the calibration of the linear term coefficient C, similar to those of the SZL- or GS-model. 4. N U M E R I C A L
METHOD
The three dimensional Reynolds averaged Navier-Stokes equations are discretized by an element based finite volume method in a co-located variable arrangement. The coupled set of algebraic equations is solved by a coupled algebraic multigrid iterative solver (CFX-TfC) and the solution is accelerated by an algebraic multigrid. A proper linearization of the non-linear stress representation is necessary as the momentum equations are strongly coupled through the non-linear terms of stress-strain relation. A purely lagged implementation alone and in combination with a relaxation of the higher order non-linear terms (those terms multiplied with the coefficients C1 - C 7 in (4)) is highly sensitive to the grid aspect ratio. These schemes converge for simple flows such as a plane boundary layer and plane turbulent channel flow but diverge for the calculation of more complex flows. The solver's stability is not significantly increased once the inter equation coupling is implicitly accounted for by a Newton-Raphson linearization. For the more complex test cases reported below, only a quasi-linear truncation of the non-linear turbulence models yields grid independent converged solutions. The dependence of the coefficient C, on the mean strain and vorticity tensor invariants (see table 1) was kept while all other coefficients (C1 - C 7 ) were neglected. This quasi-linear truncation turned out to be a good compromise between turbulence models ability of taking into account effects of extra rates of strain through a variable C, and necessary robustness of the solver. The implemented turbulence models are verified by calculating a plane turbulent bound-
118 ary layer and a plane turbulent channel flow. The model's capability to account for extra rate of strain effects is assessed by calculating three different testcases presented in the next section. In all testcases described a converged solution was obtained if the sum of the maximum values for all residuals was below 10 -4. Grid independence studies ensured grid independent solutions in all cases. Starting from a coarse grid the number of elements in streamwise and wall normal direction was multiplied by the factors 2, 3 and 4 for finer grids. Grid independence was reached for indistinguishable solutions on different grid refinement levels. All solutions were obtained on block structured meshes with hexahedral elements. Boundary conditions for walls, inlets, outlets and symmetry lines had to be specified. At walls the no-slip boundary condition was modeled with standard wall functions. At outflow and symmetry boundaries zero gradient conditions for all variables were applied. At inlets profiles of measured flow variables were prescribed if available. Otherwise constant values for the flow properties had to be prescribed. 5. R E S U L T S Three situations are considered lent flow. A backward facing step a plane U-duct [6]. All flows are data is available and is compared
where streamline curvature has an influence on turbuflow [5], a curved mixing layer [1] and the flow through nominally two-dimensional. In all cases experimental to the predictions of the different turbulence models.
5.1. Backward facing step Driver and Seegmiller [5] investigated the two-dimensional, steady, incompressible and turbulent flow at a backward facing step experimentally. The geometry is sketched in figure 2. The characteristic length of the geometry is the height of the step H = 1.27cm. The characteristic velocity outside the boundary layer is Uref = 44.2m/s resulting in a Reynolds number of Re = 37423.
Figure 2. Backward facing step flow
The computational inlet is located 4H upstream of the step. To discretize the inlet region (20 x 40 x 2) elements were used for the coarsest grid. The discretization of the downstream region consisted of (50 x 60 • 2) elements on that level. Around the region of the measured reattachment point (x/H = 6.26) the grid was refined. The computed reattachment length is an important parameter for the assessment of turbulence models in separated flows. Table 3 shows the comparison of computed and measured reattachment lengths for different turbulence models on the coarsest grid and the next grid refinement level. The CLS and the GS model compute the correct value for the reattachment length.
119 Table 3 Comparison of reattachment lengths
x/H Data 6.3 6.3
(20 x 40 + 50 x 60) x 2 (40 x 80 + 100 x 120) x 2
k- e 5.9 5.8
SZL 7.1 6.8
CLS 6.8 6.4
GS 6.5 6.3
The value for x/H is predicted too small by the standard k - e and too large by the SZL model. All turbulence models predict the main features (primary recirculation and corner vortex) of the flow. Figure 3 shows the development of the pressure coefficient cp along the lower wall in streamwise direction downstream of the step. The standard k - e model predicts the rise of the pressure coefficient too early compared to experimental data. This is a consequence of the computed length of the recirculation zone being too small. All other models are in better agreement with experimental data because of their computed location of the reattachment point being in close agreement with experimental findings. Downstream of the reattachment point the overall pressure loss is computed correctly by the non-linear EVMs but too small by the standard k - e model.
0.4
1
0.2 ...,
9
0.2 0 0
0.1
r / ,,,~~
0.0
l ~ ~
o ~
9Data o k-E
o-
................CL a .... o SZL
9Data
t~~ " 0.0
9-----* GS
" ..............,~ CL . . . . . . SZL
-mm~mre
ee
0
10
20 x/H
Figure 3. Pressure coefficient %
30
-0.2
0
10
x/H
2'0
3'0
Figure 4. Skin friction coefficient cf
The location of the reattachment point in figure 4 is characterized by c/ - 0. Again the differences between linear and non-linear EVMs can be observed in that region. But in contrast to the development of cp the measured and computed development of c/ is not in good agreement for x/H > 10 and inside the recirculation zone. This is mainly a consequence of the standard wall functions used throughout the computations. Grotjans and Menter [11] were able to compute the correct values for c/ using a standard k - e model and a modified wall function approach.
120
5.2. Curved mixing layer Castro and Bradshaw [1] investigated the influence of streamline curvature in a highly curved plain mixing layer experimentally. Figure 5 shows a sketch of the experimental setup. Air is flowing through a nozzle with D = 12.7cm towards a flat plate. The distance of nozzle exit and plate is H = 44.7cm. After the nozzle exit a plain, free mixing layer develops. The presence of the plate causes the oncoming mixing layer to bend away before impinging onto the plate resulting in a curved mixing layer. A boundary layer separation along the inner wall close to the stagnation point region is avoided through the the presence of an outflow of width W = 4 c m in the back plate.
Figure 5. Curved mixing layer flow
In this case no profiles for mean and turbulence properties at the inlet are available. Therefore constant values for velocity Uref - 3 3 m / s , turbulent intensity I - 0.001 and a typical lengthscale for the energy containing eddies It - l c m were prescribed. The first calculations showed however, that these assumptions were not sufficient to approximate the situation at the nozzle inlet properly. Following the approach of Gibson, Rodi and Scheuerer [12,13] the computed distributions of velocity and turbulent kinetic energy of a free mixing layer with a mixing layer thickness 5 - 3 r a m were prescribed at nozzle inlet instead of constant values. Figure 6 shows the development of mixing layer thickness 5 along the length of a reference streamline. For angles ~ > 40 ~ a decreased spreading rate of ~ compared to spreading rates of free mixing layers is observed. This is a consequence of the stabilizing effects of convex streamline curvature on turbulence. Downstream the curved section the spreading
121
,
0.10
,
,
O~
.
90~
,
..
"'"/" 0
,
0.04
0
0~
0.08
--.90~
o
0.03
o
o
%~-/ ..
~g.g
0.04
0.02 ,,," 0.00
-~
/-
0.0
/
/-
~
*- - -* GS
z~--~ CL O Data
0:4
o
0.02
.... szt.
@,
0:~
~ 0
o o Cffk
0:6
0:8
,
1:0
Figure 6. Mixing layer thickness 6 vs. length of reference streamline S
0.00
~" ~
0
.......5~
o/~ %~SY o
0.0
'
0:2
o
0:4
..........
~ S + , ~ ....... sz,
,---, as
: ~"
0.01
S,m
0
] --~
o ~ o
t
freie M.
o 0.06
o
~
'
0:6
'
CL o Data
0:8
1:0
S,m
Figure 7. Maximum k vs. length of reference streamline S
rate increases again and reaches values that are slightly higher compared to values for free mixing layers. Results obtained with the standard k - e model show a slight reduction of spreading rate in the curved section. The non-linear EVMs are generally more sensitive towards curvature effects. The reduction in spreading rate is predicted higher than with the standard k - e model. None of the turbulence models predicts the correct values for the spreading rates downstream the curved section. This is consistent with calculations of free mixing layers where also all models predict too small values for the spreading rate as well. In fact the standard k - e model is closer to experimental data than the nonlinear EVMs. in that region. In figure 7 the maximum values of radial distributions of non-dimensionalized turbulent kinetic energy along the reference streamline can be seen. All models show the stabilizing effect of streamline curvature qualitatively correct. Again the non-linear EVMs are more sensitive towards effects of streamline curvature than the linear EVM. The location of maximum damping of turbulent kinetic energy occurs at ~ 60 ~ which is in good agreement with the predictions of the non-linear EVMs. The standard k - e model predicts the location of maximum damping too early and not as pronounced as the other models. The generally higher values for k predicted with the standard k - e model are the reason for the higher values of spreading rate observed in figure 6. None of the models predicts the measured "overshooting" of turbulent kinetic energy downstream the curved section, but the asymptotic behaviour of k far downstream the curved section towards the value of the free mixing layer ( k m a x / U , . e f 2 = 0.028)is both seen experimentally as well as computationally with all turbulence models. 5.3. P l a n e U - d u c t Monson and Seegmiller [6] investigated a low speed internal flow in a strongly curved U-duct. Figure 8 shows a sketch of the experimental setup. The Reynolds number based on the channel height H = 3.81 cm is Re =1-106. The basic grid is (100 • 10 • 2) hexahedral element grid. Results presented below are obtained on a (400 • 80 • 2) hexahedral element grid.
122
Figure 8. Plane U-duct flow
1.5 ,~--o0
~8o
a_a - 9aew 9 a
~
0.0
1.0 i
. '-'i- e
, iD 9
9
9 J
Cp
-1.0
..... ,,
~ I~'" Ji I
-2.o
- -
9Exp.
....
GS
i;o
SIH
2;o
Figure 9. Streamwise development of pressure coefficient %
9D a t a
,lo']/ ~l'i 9
~0..........OVk-eSMC [] - - -El SZL -9- - ~ GS ~---~ CL
0.0
~ 9
Ck
oo
9
k-i~
............... ..... SMC SZL
II
t/;
:__z 0.5
-0.5
0.0
0.2
0.4
y/H
0.6
0.8
1.0
Figure 10. Spanwise distribution of axial velocity at ~ - 180 ~
Figure 9 shows the pressure distribution at the inner and outer wall. Due to its constant linear term coefficient C, the standard k-e model does not account for the damping of turbulence at the convex, inner wall and consequently no separation is predicted downstream the bend. The non-linear EVMs predict an altered turbulent transport due to streamline curvature. A separation zone (figure 10) is calculated similar to that calculated by the second moment closure. The SZL-model is highly sensitive to adverse pressure gradient conditions. It predicts the highest pressure loss of all models employed as it over-predicts the length of the separation zone shown in figure 9. The GS- and CLS-model results are similar to the second moment closure (SMC) solution. The measured thickness of the separation zone shown in figure 10 is questionable due to transient conditions there. The CPU time of the turbulent U-duct flow calculations is for non-linear EVMs in the range of 0.9 - 1.4 times that necessary for standard k-e calculations in all grid resolutions considered. For SMC calculations a factor 2.1 - 4.6 is observed.
123 6. C O N C L U S I O N S Three non-linear eddy viscosity models have been assessed with respect to accuracy of flow predictions and computational robustness in flows where curvature alters turbulent momentum transport. Due to curvature secondary strain components become more important resulting in high cross-velocity coupling of the momentum equations which finally leads to solver divergence. Underrelaxation of the non-linear terms and different linearization schemes cannot cure the problem. Truncation of the non-linear EVM at first order leaves the quasi-linear models capable in accounting for the altered turbulent momentum transport through the dependence of the linear model term coefficient C, on the strain and vorticity tensor invariants. As these truncated EVMs are of tensorially linear order they can not predict different magnitudes of normal Reynolds stress components, but extra rates of strain effects on the shear components are modeled in a manner similar to cubic non-linear EVMs. The quasi-linear EVMs offer an improvement in accuracy of flow predictions compared to standard k - ~ calculations with a moderate increase in CPU-time. Computational robustness is only slightly decreased compared to standard k - e calculations. Results also indicate that care has to be taken when specifying inflow boundary conditions as well as wall boundary conditions because their influence can exceed the influence of turbulence modeling on accuracy of flow calculations.
REFERENCES
1. Castro, I. P., Bradshaw, P.: The turbulent structure of a highly curved mixing layer, Journal of Fluid Mechanics, Vol. 73, pp. 265- 304, 1976 2. Craft, T. J., Launder, B. E., Suga, K.: Development and application of a cubic eddyviscosity model of turbulence, Int. Journal of Heat and Fluid Flow, Vol. 17, pp. 108115, 1996 3. Gatski, T. B., Speziale, C. G.: On explicit algebraic stress models for complex turbulent flows, Journal of Fluid Mechanics, Vol. 254, pp. 59- 78, 1993 4. Shih, T.-H., Zhu, J., Lumley, J. L., A realizable Reynolds stress algebraic equation model, NASA TM-105993, 1993 5. Driver, D. M., Seegmiller H. L.: Features of a Reattaching Turbulent Shear Layer in Divergent Channel Flow, AIAA Journal, Vol. 23, No. 2, pp. 163- 171, 1985 6. Monson D. J., Seegmiller, H. L.: An experimental investigation of subsonic flow in a two-dimensional U-duct, NASA TM-103931, 1993 7. Speziale, C. G.: Analytical methods for the development of Reynolds-stress closures in turbulence, Annual Review of Fluid Mechanics, Vol. 23, pp. 107- 157, 1991 8. Lumley, J. L.: Computational modeling of turbulent flows, Advances in applied mechanics, Vol. 18, pp. 123- 176, 1978 9. Speziale, C. G.: On the prediction of equilibrium states in homogeneous turbulence, Journal of Fluid Mechanics, Vol. 209, pp. 591 - 615, 1989 10. Speziale, C. G., Mac Giolla Mhuiris, N.: Scaling laws for homogeneous turbulent shear flows in a rotating frame, Physics of Fluids, A 1, pp. 294 - 301, 1989 11. Grotjans, H., Menter, F. R.: Wall functions for general application CFD Codes, ECCOMAS 98, Conference Proceedings, J. Wiley & Sons Ltd., 1998
124 12. Gibson, M. M., Rodi, W.: A Reynolds-stress closure model of turbulence applied to the calculation of a highly curved mixing layer, Journal of Fluid Mechanics, Vol. 103, pp. 161- 182, 1981 13. Rodi, W., Scheuerer, G.: Calculation of curved shear layers with two-equation turbulence models, Physics of Fluids, Vol. 26, pp. 1422- 1436, 1983
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
125
A proposal for taking into account the intermittency phenomenon in the calculation of wall-bounded turbulent flows N. Bousquet, J.-B. Cazalbou and R Chassaing ENSICA, 1 place t~mile Blouin, 31056 Toulouse cedex, FRANCE
1. I N T R O D U C T I O N In unconfined turbulent shear flows, turbulence generated in the core of the shear layers proceeds toward the outer undisturbed flow behind a well-defined but irregular and unsteady boundary. The region where the boundary fluctuates is subject to the intermittency phenomenon" at a fixed point in space and at different times the flow may be either fully turbulent or irrotational according to the instantaneous position of the boundary. As a result, the mean position of the boundary does not, in general, correspond to the conventional shear layer edge. For instance, in the flat plate boundary layer of Klebanoff [ 1], it is located at 0.78 d (where ~ is the boundary layer thickness) and its standard deviation is 0.14 ~ with a Gaussian distribution. Thus one can consider that, at least, the outer 35% of the layer is significantly affected by intermittency. This phenomenon is empirically taken into account in some algebraic turbulence models (Cebeci-Smith [2], Baldwin-Lomax [3]) where the intermittency factor (7) is used to damp the eddy viscosity as the free stream is approached. This approach is not so common however for most of the current models ---based on the solution of transport equations for turbulent quantitiesn for which the mean position of the interface corresponds exactly to the conventional shear layer edge. The problem stems from the fact that incorporating intermittency in the models can be very expensive: the most rigorous way to do so is to perform conditional averages of the flow variables and to solve conservation equations for both turbulent and non-turbulent components. This approach has led to several proposals based on theoretical analysis by Libby [4], Dopazo [51 and Chevray & Tutu [6]. A (k, e) turbulence model based on these ideas has been developed by Byggstoyl & Kollmann [7], and Reynolds-stress transport models by Janicka & Kollmann [8] and Byggstoyl & Kollmann [9]. A different and more economical approach is adopted in the (k, e, 7) model of Cho & Chung [ 10]. The model involves a transport equation for the intermittency factor 3' (defined as the fraction of time for which the flow is fully turbulent) whose effect is explicitly incorporated in the conventional (k, e) model equations. The results obtained with this model are fairly satisfactory over a wide range of free shear flows rain particular, it seems to resolve the round jet/plane jet and jet/wake anomalies~, the model appears therefore as much more general than the conventional (k, e) model. The aim of this study is to evaluate the benefits that could be brought by this approach in the case of wall-bounded flows. In a first step, we shall study a straightforward extension of
126 the model that enables its use in low turbulent Reynolds number flows. Several weaknesses in the original formulation will be evidenced regarding the characteristics of the intermittency profile and the well-known difficulty encountered in the prediction of adverse-pressure-gradient boundary layers. Then, a modified version free from these weaknesses will be proposed. 2. L O W T U R B U L E N T REYNOLDS NUMBER FORM OF THE (k, e, "7) M O D E L
An extension of the (k, e, "7) model suited to the calculation of wall-bounded flows has been devised based on the well-known Launder & Sharma [11] low-Reynolds-number treatment. Applying this treatment is straightforward for the k and e equations, the original Cho & Chung intermittency equation is left unaffected except for the damping of the eddy viscosity. The closure equations, written with the boundary layer approximation for simplicity, can be written in the form:
~rk UN+V
Oy
= Oy (r, + -O'-e) N
+C~ l ~ l]t ~
(i - ' 7 ) - -
g _-~--+ V-a-
(i)
Oy +(C~4F - C ,2 ) f 2 --~ -2r, r,t
c)y2 j
(2)
+ Cgl (i - '7)
+Cg2--s
- Cg3'7(1 - ' 7 ) # r
(3)
1 73 - 3 ' k3e2 (O0-~y)21 k 2 - c " f ~ e [ l + f ( ' 7 ) ] k2e
with:
F = k5/2 ~ OU 07 62 (~-2 "iVV2) 1 / 2 0 y Oy f.--exp
[
]
(I+Rt/50) 2 '
]r
f 2 - [ 1 - 0 . 3 exp(-R~)]
and
R t --
~6
This form reduces to the original Cho & Chung model far from a solid wall except for the Morse and Pope corrections which are unnecessary here and have therefore not been retained. The expression of 1-' remains as in the original reference although not flame-indifferent, a point that will need some future refinement. The values of the model constants are those given by Cho & Chung: C~,=0.09, C~1=1.44,
C~g = 0 . 1 , Crk=l, ~r~=l, cr.~= 1 C~2 = 1 . 9 2 , C ~ 4 = 0 . 1 , C g 1 = 1 . 6 , Cg2 = 0 . 1 5 ,
Cg3 = 0 . 1 6 .
When this model (hereinafter called the base model) is applied to the calculation of a fiat plate turbulent boundary layer in zero pressure gradient, the intermittency profile appears to be significantly in error: the mean position of the interface (defined by 3' - 0.5) is located very close to the boundary layer edge (~ ~ 0.86 compared to the value of 0.78 given by Klebanoff) and the profile falls too quickly to zero at the edge. Considering the first point, it has been found
127 that reducing the generation of ~ due to the mean motion and to the gradient of "7 itself enables to accurately locate the position of ft. Consequently, the values of the generation constant C~1 and C~2 have been lowered to 1.3 and 0.05 respectively. The behaviour of "7 at the edge is the second point of concern, we have found that the (k, e, "7) model admits the same kind of weak analytical solution in this region as the two-equation models in Cazalbou et al. [12], i. e. the dependent variables all go to zero at the edge of the shear layer as powers of the distance from the edge (Y). Performing the same kind of analysis as in [12] we get: 3
3a~
U c< Y ~ - ~ ' - ~ ,
3o'~
k ~x Y ~ ' ~ - ~ , - ~
,
3a-r
~ ~x Y ~ - ~ , - ~
,
7 ~x Y ~ - ~ , - ~
(4)
in the vicinity of the edge, provided that the quantity a = 5or k - 3~r, - cr-~ remains positive. This is the case with the Cho & Chung model and, incidentally, (4) allows to demonstrate that the eddy-viscosity coefficient C~,(1 + f('-/)) goes to infinity at the edge (as y-(o'u+a.~)/a) which was found desirable in [ 10] and is supported by experimental as well as simulation data [13-15]. Relations (4) can further be used to tune the behaviour of 7. We chose to smooth the intermittency profile without affecting the velocity profile, this leads to the following set of turbulent Prandtl number values:
Crk-- i,
~--0.9,
C%--1.3
Lowering the value of or, affects the slope of the velocity profile in the logarithmic layer owing to the classical relation: n 2 - (C,~ - C,1)or, V@-~, (where n is the von Kfix'mS.n constant), so that the difference between the values of the dissipation constants is to be increased, we chose: C,~
1.41
-
and
C ~ 2 - 1.94
With these adjustments the model is accurately calibrated for wall-bounded flows. In Fig. 1 and 2, we show the results obtained for the mean velocity and intermittency factor in the calculation of a fiat-plate zero-pressure-gradient boundary layer at Reo = 7800 together with those obtained with the base model. The velocity profiles are similar, the slight upshift observed with the base model could be easily suppressed with some enhancements of the low-Reynolds-number treatment. The intermittency profile obtained with the modified model marks a significant improvement and the agreement with Klebanoff's formula is remarkable. 0
i
I
Illltll
]
r facile I
I
r a~:~la I
i
I
I
1.0 ;
]~[|T|i
M I
:;2@:~~
!
~'
, ,,5~-"":;'
.............. 1
i
---_.
i
t
[
!
!
,,;
10 ~-v
f Jilrl~ I
,'5.!"......... , . Z " .....
20 ~ U+
f
\\
7 0.5 -
J
'~
I
-!
,//
!
/
I !
10
100
1000
10000
0.0 0.0
t
i
0.5
1.0
+
Y Fig. 1 9Semilogarithmic plot of mean velocity; 9 data of Klebanoff; - - - (k, e, 7) base model; - - (k, e, 7) modified model.
l
1.5
_ y/~i
Fig. 2" Intermittency tactor; - - Klebanoff's formula (7 - ( 1 - e r f ( ) / 2 with ( = 5[(y/3) - 0 . 7 8 ] ) ; - - - (k, e, 3') base model; --(k, c, 7) modified model.
128 A major target flow for low turbulent Reynolds number models is the case of adversepressure-gradient boundary layers. Huang & Bradshaw [ 16] have shown that the choice of the length-scale variable in two-equation turbulence models was of fundamental importance for an accurate representation of the fully turbulent region of the decelerated boundary layer. Their analysis shows that the choice of e results in a significantly reduced slope of the velocity profile in the logarithmic layer in response to the adverse pressure gradient. It also indicates that the (k,w) model of Wilcox [17] (w = e/k) is the best compromise from this point of view. With the (k, e, 7) model, the intermittency only affects the outer region of the boundary layer so that the problem should remain the same as in the (k, e) model. In Fig. 3, the velocity profile obtained for an equilibrium-boundary-layer calculation with/3 = (3" / pU2) (dP/dz) = 5.14 and Ro = 25000 (3" is the displacement thickness) has been plotted against the data of Bradshaw & Ferriss [ 18](Flow a = -0.255, station z = 3.91 ft.) at the same pressure-gradient parameter and Reynolds number. The reduced slope obtained with both (k, e) and (k, e, 7) models is apparent, the corresponding low values reached by U + at the edge of the boundary layer indicate an overestimated friction coefficient. On the contrary, the results obtained with the (k, w) model are convincing with better log law and skin friction coefficient. We shall propose in the next section a modification to the (k, e, 7) model drawn from the (k, ,J) formulation that helps to cure this problem.
, , ,":
I
9 ' tln,~ I
:
i~
~l,, I
,
~
vv~ I
.
icr,,f
[ ......
40
_
/t /: /.i, , ..,, / / /" /' / ...• , /
30 +
.....i
U
.t* ~..,,
20
,
,
I I I _
10 0
. . . .
0
~""'~7/
1
...... [
, ' l
10
I
100
......
I
......
1000 10000
y+ Fig. 3 9Semilogarithmic plot of mean velocity; * data of Bradshaw & Ferriss; - - - - (k, e, 7) modified model; - 9- (k, e) model; ... (k, a~) model.
3. IMPROVEMENTS FOR ADVERSE-PRESSURE-GRADIENT FLOWS Given closure equations for k and e, an equation for ~ - e/k is easily derived. Wilcox [17] has shown that, doing so, the only difference that appears with his model w equation comes from an additional term in the form:
1 Ok 0e/k
,
when ak -- cr,~ -- a.
(5)
129 The presence of this term explains the different response of the (k, e) and (k, co) models to adverse pressure gradient. It would therefore be tempting to devise a (k,co, hi) model on the basis of the (k, e, 3') model without the extra "diffusion" term in the co equation. Unfortunately, such an approach was compromised here by the characteristics of the edge solution to the attempted model: the weak propagating solutions--desirable for a reasonable independence to the free stream conditions--do exist for the derived (k, a;, 7) model but bring untractable constraints on the Prandtl number values (2crk - 3or,, - cr.~ > 0). To circumvent this difficulty and still get a better response to adverse pressure gradient, one can incorporate an extra term in the dissipation equation in the form:
k 2 Ok O~/k
S~ - - 2 C~5 C t, f ~, e Oy
Oy
(6)
"
This term helps to depress the length scale in adverse pressure gradient and recalls the Yap correction [ 19] or a proposal by Ince (see Launder [20]). It is derived from the reciprocal of S,~ in the e equation and essentially differs from it by the absence of the eddy-viscosity coefficient (1 + f(.,/)). This coefficient has been shown above to be going to infinity at the edge and removing it leaves the edge relations (4) unaffected. Such an edge behaviour is actually what makes the difference between our (k, e, y) model supplemented with the S~ term and the regular (k, w) model. The latter does not admit weak solutions in powers of Y in this region, a fact that should explain the sensitivity to free-stream conditions observed by Menter[21 ]. Thus, the use of intermittency appears here as a way to remove the free-stream sensitivity problem while keeping the satisfactory behaviour of the (k, w) model for the length scale in the fully-turbulent region. Note that the molecular viscosity contribution, being unnecessary, is not retained in (6) and also that S~ is damped as ut for wall proximity (ft,). The new model, hereinafter called (k, e~, ~,), can now be summarized:
u ~ + Voy = o-~ (~ + -1~
+~
uN+
+C~l~,
oy - o y
(~+-)~
-~ - 2.
oy
(7)
-c~f~ T
[o v] -2,... Lb-~--p~J - 2 c.~ c~. It.
c
ok o /k Oy Oy
(8)
- ~)gr
(9)
--O~ O[ ('-~)-- .... O~] +GI (, ~)~~.. [O~] ~ v + v o o ; - oy cr.r ~ y "Y -
+G~--~
- c~(1
We note that the term c~4re2/k in the dissipation equation has been discarded: in the outer region of boundary layers it acts as a destruction term of the same order of magnitude as the residual values of S~ there, so that it cannot be retained in addition. Recalibration of the model led to the following set of constants: C~=0.09, Cr - 1.38,
C~g=O.1, Crk=l, crr Cr = 1.98, Cr -- 1.43,
cr~--1.3, Cg2 = 0.05,
Cgl = 1.3,
Cg3 = 0.16.
130 4. R E S U L T S The present model has been tested in the calculation of several cases of turbulent boundary layers. All calculations have been performed using a parabolic space-marching code. Numerical grids involve at least 100 points in the transverse direction and the results have been tested for grid independence. The model behaves likes the original (k, e, 7) model near free stream edges and therefore does not exhibit any odd sensitivity to free stream boundary conditions, in all cases reasonably "small" values of the transported variables suffice (typically : k, = 3/2 zc 10 -6 • U V m a x , Ce - - 10 -6 • -UVmax/O, "~ _ _ 1 0 - - 3 .) In Fig. 4 and 5, results obtained in the case of a flat plate boundary layer without pressure gradient are presented. At Reo = 7800 the velocity profile follows closely the one obtained with the (k, e) model except near the boundary layer edge where it is somewhat smoothed consistently with the use of intermittency. As with the (k, e) model the logarithmic portion of the profile is shifted above the standard "log" law to compensate for a slightly underestimated wake component and give a correct friction coefficient. The latter can be seen in Fig. 5 to be in very good agreement with Wieghardt [ 18] experimental data for a wide range of the Reynolds number. The intermittency factor profile is not presented here but fairly agrees with Klebanoff's f o r m u l a : - ) , - 0.5 ( 1 - e r f
( Y ~ 2 - ~ ) ) . A best fit to this formula gives ~ -
0.79 instead of 0.78
and cr = 0.138 instead of 0.141. 0
i iiiiill
t
11111111
i
i Iiiiiq
i
i iitlii I
i
0.004
ii11111
I
1
I
5000
10000
t 15000
,..
20
.......-,
+
\ '\
...~..4 ''t~
U
".... ......
Cf 0.003 lO
0
,,5..~......"" .
0.002 1
10
100
1000
10000
y§ Fig. 4 : Semilogarithmic plot of mean velocity; * data of Klebanoff; I (k, ~,o, 7) model; - 9- (k, e) model.
0
R%
Fig. 5 : Skin friction coefficient; * data of W i e g h a r d t ; - (k, e~, ~,) model; - 9- (k, e) model; -.. (k, w) model.
The adverse-pressure-gradient boundary layer of Bradshaw & Ferriss has been used to assess the performances of the model in equilibrium boundary layer. The velocity profiles obtained in this case are plotted in Fig 6 and 7. The inner-variables scaling shows the efficiency of the S~ correction term for an accurate prediction of the logarithmic region. In this region, the results mark a significant improvement over that obtained with the (k, e) model and, as expected, come close to those obtained with the (k, w) model. The friction coefficient and shape factor (H = 5*/0) agree well with the experiment: H = 1.6 against an experimental value of 1.59 and
131 C f - - 0.0135 for 0.0138. The linear plot in Fig. 7 also shows a very good agreement between our model and experimental data, it illustrates the benefit of modeling intermittency: none of the other models tested here can reproduce the smooth approach to Ue experimentally observed at the boundary layer edge. The intermittency factor (not figured) still follows Klebanoff's formula with best fit values: ff = 0.81 and cr = 0.102, which means that the mean position of the interface is marginally displaced toward the edge while the standard deviation is clearly reduced. This trends are consistent with the measurements of Fiedler & Head [22] but experimental data on this matter are scarce and it is difficult to draw definitive conclusions here.
i
I fllllll I
i i iiiiii I
i i iiiiii I i L illlll[
10!
t i illlll
40 j /-
30 U
:if: 1,1'
+
-
,.,,.S '-/
~j
~- 0.5
20
,,
j l "/
10 o*"
0
1
0.0
i li
10
+
100
1000 10000
i
0
1
2
3
4
y/8'
Y Fig. 6" Semilogarithmic plot of mean velocity; * data of Bradshaw & Ferriss; (k, e,,, 7) model; - 9- (k, e) model; . . . (k, co) model. 0.003
Fig. 7" Mean velocity; * data of Bradshaw & Ferriss; m (k, e,,, ../) model; - 9- (k, e) model; . . . (k, co) model.
.9
I
i
.......
t 2
L 3
0.002 Cf 0.001
0.000 1
z
X
Fig. 8 9Skin friction coefficient; * data of Samuel & Joubert; u modified model; - . - (k, e) m o d e l ; . . . (k, co) model.
(k, e,7)
The last case presented here is the adverse-pressure-gradient boundary layer of Samuel & Joubert [23]. In this non-equilibrium flow, the pressure-gradient parameter and shape factor
132 increase from values of fl = 0.09 and H = 1.39 up to 7.7 and 1.61 respectively at the last measurement station (z -- 3.40m). The strength of the pressure gradient and its effect on the boundary layer are therefore comparable with the case of Bradshaw & Ferriss previously studied. In this non-equilibrium configuration the results obtained with our model are more mitigated. The skin-friction curve is represented in Fig 8, traditionally the (k,w) model gives good results in this flow while the (k, e) model constantly returns too high values. Results obtained with our model lies in-between and only show a moderate improvement over that obtained with the (k, e) model. Inspection of the logarithmic plot of the velocity at the last station in Fig. 9 shows that the improvement only relies on the inner-layer behaviour. For this value of the pressure-gradient parameter and out of equilibrium, the computed flow has not yet achieved a sufficient development in its wake component. This is confirmed with the outer-layer scaling in Fig. 10 where the profiles obtained with the (k, e) and (k, e,,, ~,) models are barely discernible. I
I II11111
I
I
IIIIII
I
I
I IIIIII
I
I
I I~[lll
I
~
40
I 11IIII
:/
30 U
0
t
I
I
, . ~
.,o
o'....: /" _ ~ /
-10 i
..... ~
+
-20
20
...
-30
10 1
t
1
i lllllll
i
i lllllll
10
i
100
i iiiilll
1000
i
I ILIAI
10000
+
y
Fig. 9 : Semilogarithmic plot of mean velocity ; * data of Samuel & Joubert (x=3.40 m); u (k, e,o, 3') model; - . - (k, e) model; ... (k, w) model.
-40 0
I 2
J 4 y/0
[ 6
Fig. 10 9Defect velocity; * data of Samuel & Joubert (x=3.40 m); ~ (k, e,o, "7) model; (k, e) model;-.. (k, ~v) model.
CONCLUSION A low-turbulent-Reynolds-number model as been constructed starting from the (k, e, 7) model of Cho & Chung. Two major modifications of the latter have been proposed in order to get a better behaviour of the model in the calculation of wall-bounded flows. First, it appeared that obtaining an accurate intermittency profile for consistency needed a significant change in the production constants of the intermittency equation. So that the good generality of the original model in the calculation of free shear flows seems to be lost when dealing with wallbounded flows. In a second step, we have incorporated in the model a correction term in the e equation whose aim is to counter the anomalous increase of the computed length scale in the fully turbulent region of retarded boundary layers. This modification is very efficient as can be seen in the calculation of adverse-pressure-gradient equilibrium boundary layers but also lead
133 to modify the original model by discarding a sink term in the dissipation equation. The fact that the model is fairly accurate in the calculation of retarded equilibrium boundary layers but rather fails in the non-equilibrium flow of Samuel & Joubert for comparable pressure-gradient parameter and shape factor is of concern, it may indicate the limit of a local expression for the eddy-viscosity and the necessity of taking into account the history effects in the model. On the overall, the main benefit of including the intermittency appears as being the possibility to include the extra S~ term in the dissipation equation and thus to reproduce the good response of the (k, a~) model to adverse pressure gradient without the odd sensitivity to free-stream conditions of the latter. This is not possible with the regular (k, e) model which would necessarily return to the (k, ,~) model with such an extra term. Additionally, the use of intermittency allows smoother and therefore more realistic profiles at the edge. Other interesting developments would be to study the behaviour of the model in more complex flows and the way the transition problem could be dealt with in this context. REFERENCES
[ 1] R S. Klebanoff, Characteristics of turbulence in a boundary layer with zero pressure gradient, NACA TN-3178, 1954. [2] T. Cebeci & A. M. O. Smith, Analysis of turbulent boundary layers, Academic Press, 1974. [3] B. S. Baldwin & H. L. Lomax, "Thin layer approximation and algebraic model for separated turbulent flows", AIAA Paper 78-257, AIAA 16th Aerospace Sciences Meeting, Huntsville, Alabama, 1978. [4] R A. Libby, " On the prediction of intermittent turbulent flows ", J. Fluid Mech., Vol. 68, pp. 273-295, 1975. [5] C. Dopazo, "On conditioned averages for intermittent turbulent flows", J. Fluid Mech., Vol. 81, pp. 433-438, 1977. [6] R. Chevray & N. K. Tutu, "Intermittency and preferential transport of heat in a round jet ", J. Fluid Mech., Vol. 88, pp. 133-160, 1978. [7] S. Byggstoyl & Kollmann, "Closure model for intermittent turbulent flows", Int. J. Heat Mass Transfer, Vol. 24(11), p. 1811-1822, 1981. [8] J. Janicka & W. Kollmann, " Reynolds-stress closure model for conditional variables", Turbulent shear flows 4, L. J. S. Bradbury, E Durst, B. E. Launder, E W. Schmidt & J. H. Whitelaw (eds.), pp. 73-86, 1983, Springer. [9] S. Byggstoyl & Kollmann, "A closure model for conditioned stress equations and application to turbulent shear flows", Phys. Fluids A, Vol. 29(5), pp. 1430-1440, 1986. [ 10] J. R. Cho & M. K. Chung, "A k-e-7 equation turbulence model", J. Fluid Mech. Vol. 237, pp. 301-322, 1992. [11] B. E. Launder & B. I. Sharma, "Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc", Letters in heat and mass transfer, Vol. 1, pp. 131-138, 1974. [ 12] J.-B. Cazalbou, R R. Spalart & R Bradshaw, "On the behavior of two-equation models at the edge of a turbulent region", Phys. Fluids, Vol. 6(5), pp. 1797-1804, 1994. [ 13] W. Rodi, The prediction of free turbulent boundary layers by the use of a two-equation model of turbulence, PhD Thesis, University of London, 1972.
134 [14] W. Rodi & N. N. Mansour, "Low-Reynolds number h-e modeling", Studying turbulence using numerical simulation databases -III, CTR, Proceedings of the 1990 summer program, pp. 85-106, 1990. [ 15] J.-B. Cazalbou & E Bradshaw, "Turbulent transport in wall-bounded flows. Evaluation of model coefficients using direct numerical simulation", Phys. Fluids A, Vol. 5(12), pp. 32333239, 1993. [ 16] E G. Huang & E Bradshaw, "The law of the wall for turbulent flows in pressure gradients", AIAA J., Vol. 33, pp. 624-632, 1995. [17] D. C. Wilcox, "Reassessment of the scale determining equation for advanced turbulence models", AIAA J., Vol. 26, pp. 1299-1310, 1988. [18] J. Kline, B. J. Cantwell & G. M. Lilley (eds.), Stanford Conference on Complex Turbulent Shear Flows: Comparison of Computation with Experiment and Computors' Summary Reports Stanford University Press, Stanford, CA, 1982. [ 19] C. Yap, Turbulent heat and momentum transfer in recirculating and impinging flows, PhD Thesis, Faculty of Engineering, University of Manchester, 1987. [20] B. E. Launder, "Second moment closure: present.., and future?", Int. J. Heat and Fluid Flow, Vol. 10(4), pp. 282-300, 1989. [21 ] E R. Menter. "Influence of freestream values on k-co turbulence model predictions", AIAA J., Vol. 30, p. 1657, 1992. [22] H. Fiedler & M. R. Head, "Intermittency measurements in the turbulent boundary layer", J. Fluid Mech., Vol. 25, pp. 719-735, 1966. [23] A. E. Samuel & E N. Joubert, "A boundary layer developing in an increasingly adverse pressure gradient", J. Fluid. Mech., Vol. 66, pp. 481-505, 1974.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
135
Modelling the turbulent flow subjected to magnetic field S. Kenjere~ ~, K. Hanjali~ ~, R. Duursma b and T.van Essen b aDepartment of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands bHoogovens Research and Development, IJmuiden, The Netherlands The paper deals with modelling of the effects of magnetic field on turbulent flow of conductive fluid within the framework of second-moment and two-equation turbulence closures. Based on Direct Numerical Simulations (DNS) of turbulent flow of conductive fluid in an infinite plane channel by Noguchi et al. (1998), a proposal is presented for modelling the additional source terms due to Lorentz force in transport equations for turbulent stresses and energy dissipation rate. A priori validation of the new model is presented and compared with the DNS results. The same approach is then applied to the low-Re number k - e model. The new model, together with that of Ji and Gardner (1994), was used to solve the developing three-dimensional flow of mercury in a rectangular-sectioned duct at inlet Re=2 x 105, subjected over a part of its length to a magnetic field of Ha=700 (Hartman number). The predicted development of 'M' shaped velocity profiles show acceptable agreement with the experiments of Tananaev (1978).
1. I n t r o d u c t i o n The application of magnetic field in controlling the flow, heat and mass transfer is regarded nowadays as an important technological advance towards the improvement of quality and increase in productivity in various metallurgical processes. In casting of metals, a stationary magnetic field is used as a brake to suppress instabilities and control the flow conditions in tundish devices, or a nonstationary field is applied to enhance mixing during casting. The other prospective application is in semiconductor crystal growth, where the control and suppression of turbulence is an important prerequisite for achieving a desired quality, particularly in view of a new trend to use larger crucibles. In metal casting applications the Lorentz force generated by the interaction of fluid velocity and applied magnetic field is the major external force. However, in crystal growth the Lorentz force interacts also with the thermal and mass buoyancy, as well as with forces generated by crucible and/or crystal rotation. Predicting the flow of melt, heat and mass transfer within it and at its boundaries under the influence of magnetic force, is an important prerequisite for the design of optimum magnetic field and for the prediction of the product quality. The flow is usually turbulent (metal casting) or transitional (crystal growth), so that the prediction model needs to account accurately for the effect of magnetic force on turbulence suppression or enhancement. In most full-scale applications, the melted metal assumes a complex three-dimensional domain and the process is
136 further complicated by solidification and associated interfacial phenomena, which require accurate prediction of the phenomena near a solidifying interface. The MHD turbulent flows are characterised by the flow reorganisation caused by alignment of the vortex structures with the direction of the imposed magnetic field, which increase the anisotropy of Reynolds stresses. These structural adjustments cannot be reproduced by the Reynolds-averaged approach but its effects on damping the turbulence can be modelled using the available turbulence parameters. This paper deals with modelling the turbulent flow of conducting fluid under the effects of magnetic field. The goal is to introduce and validate adequate modifications for the effects of Lorentz force into engineering-type turbulence models (two-equation eddy-viscosity and algebraic stress/flux models), which would be suitable for application to complex metallurgical problems. Instead of a d - h o c modifications, the 'engineering' model is based on the truncation of a general second-moment closure, extended to account for the Lorentz force through a unified approach in modelling the effects of external forces. The model is based on a term-by-term analysis of DNS results for a fully-developed plane channel flow of liquid metal under a transverse magnetic field (Noguchi and Kasagi, 1994, Noguchi, Ohtsubo and Kasagi, 1998). In addition to this generic flow case, the paper presents also some results of modelling of a more complex three-dimensional case such as flow of conductive fluid in a square-sectioned duct subjected to local finite-length transverse magnetic field.
2. The model The system of equations describing the motion of electrically conducting fluid in a magnetic field consists of the Navier-Stokes equations, Maxwell's equations and Ohm's law for moving media. The coupling of the equations is through the Lorentz force in momentum equation defined as F L - J • B, where J is the total electric current and B is the imposed magnetic field. By applying the Ohm's law for moving media, J - a ( - V f f + U x B), Lorentz force can be written in the following form: Fi L - a
--eijkBk-~z i + UkBiBk - UiBk
(1)
where 9 is the electrical potential. Using Kirchhoff continuity condition VJ=O leads to the numerically convenient Poisson equation for electric potential: V2~ = V. U x B
(2)
In this way, instead of three equations for the components of magnetic field, the system of Maxwell's equation was reduced to one scalar equation for the electricpotential. In addition to direct interactionwith the mean velocity fieldthrough the electromagnetic force (F~), the magnetic field affects also the velocity fluctuations,yielding extra terms in the transport equations for the second moments. This influence can be divided into direct and indirect contributions. The direct contribution comes from the additional source-like terms in the exact equations for Reynolds stress components ~ (as well as in equation for turbulence kinetic energy k = u i u i / 2 ) and in the equation for energy dissipation rate c, Eq. 3, Eq. 4 and Eq. 6 respectively. The indirect contribution is through the pressure-strain due to additional external force (F~) in Poisson equation for pressure. The DNS of a plane channel flow by Noguchi et a/.(1994) can be used to analyse and model these effects. Unfortunately, these data do not provide separate information on slow, rapid and magnetic part of the pressure-strain term, and we confine our attention only to a deriving and validating models for the additional source terms.
137 The exact form of the additional source terms in the transport equations for Reynolds stress, turbulence energy and its dissipation rate can be written as"
Duiuj
Dt
(y
p
or _
_
0r
s
-.,r
n t-
B Bku-
nc-
BjBku-
--
2u,. u,B 2 .
(3)
9
~r Dk a( 0r ) Dt . . . . + S~4 -"'" + -p ~ijkBkUi~ + BiBku-S~- 2kB~
DE
21Jo" l OBkO~.~iOr
Dt . . . . + SM . . . . - -~e~jk
(4)
0Ui 02r
Ox, Oxt Oxj ~ Bk Ox--~Ox,Oxj +
2Va ( Bk OBi uk COui + Bi Bk COui - - OUk B,p Oxz
x
oui- 2(ou)2 - 2 B k
(5)
Oui ) ui ~-~xt (6)
Recently, Ji and Gardner (1997) proposed modifications to the standard low-Re-number k - e model to account for the magnetic field. They proposed additional terms in the k and e equations, supposedly to model Sff and S M, as well as a damping function fM (in addition to f~,) for turbulent viscosity, defined as vt = C , f , f Mk2/e. Both new terms and the additional function in ~'t contain exponential damping exp(-C2MN) which they derived from an approximate analysis of the velocity decay under the action of a transverse magnetic field, U(~-) = Uoexp(-t/tm), by taking tm - p/aBg as the characteristic magnetic breaking time. The modelled equations have the standard forms, Dt = Pk + z : -
-e+S M
DE 0 j [ ( vl i+t )-~ O~xj]Dt CleE -s Pk + -~z
~2 + SM C2~-s
(7)
(8)
except for the extra source terms and the function fM, which are specified in Tab. 1 (denoted by JG). The JG model requires two new coefficients for which the following values were proposed: c M = 0 . 0 5 and cM--0.9. JG proposal uses the bulk flow Stuart number (the interaction parameter) N=aBgL/pU as a basic parameter to define the damping of turbulent quantities due to magnetic field. The use of bulk-flow parameters to model terms in equations for turbulence quantities is not a common practice in turbulence modelling as it makes the model dependent on bulk flow properties. The Stuart number defined above is constant for a particular flow for the imposed magnetic flux B0 and, consequently, the JG proposal is limited to configurations for which it is possible to define integral value of Stuart number, e.g. to relatively simple geometries (pipe or channel configuration) and homogeneous magnetic fields. Also, an ad hoc direct damping of turbulent viscosity due to applied external force has no physical justification, if kand e- equations are properly modelled.
138
Model
SM
sM
fM
JG
-crBg k C1M exp (--C2MN) P
-crBg c C1M exp (-C2MN) P
exp (-C2MN)
NEW
p
p oe
-pBgcexp
(- C MpBg-ek)
1
Table 1: Specification of additional terms in the Ji and Gardner (1997) and NEW model In order to remove the above mentioned deficiency of JG proposal, we replaced this integral value by a local interaction parameter which was obtained by introducing the local time scale (k/c) in its definition (what was also indicated by JG as a possibility). This new definition should reflect the local interaction of the characteristic magnetic breaking time and the characteristic turbulence time scale, so that the model should be able to deal with configurations of arbitrary geometries as well as with inhomogeneous magnetic fields. Additionally, the damping function fM is omitted from our new proposal. These changes lead to the reduction of empirical coefficients to only one, C M, which was evaluated through a priori term-by-term testing on the basis of the DNS database for the plane channel by Noguchi et al. (1994). Fig. 1 shows magnetic source term contributions to the turbulence kinetic energy and dissipation rate budgets. In DNS database this source contribution is decomposed in two parts. The first part is associated with the terms that include velocity-electrical field correlation, uiej = -uiOr and the second, which include the remaining terms (D). The lines show the validation of the here proposed model, denoted by "NEW' and of JG model. As seen, the JG proposal heavily underestimates the total source terms in both the k-equation and in e-equation. The NEW proposal with C1M - 0.025 agrees very well with the total DNS contribution (/%) for turbulence kinetic energy. For dissipation, agreement is qualitatively good, but the DNS value is underestimated in the near wall region. The JG proposal also underestimates the value of dissipation rate obtained by DNS across the entire channel cross-section (not shown here). Both of the above mentioned proposals were then extended to RSM model, by replacing k with ~ in Tab. 1. For a plane channel flow under transverse magnetic fields, only the uu, ww and ~ stress components include directly magnetic source terms. Fig. 2 shows that the model based on JG proposal again largely underespredicts the total magnetic source contribution for all stress components. In contrast, the NEW model shows a qualitatively good behaviour for the normal stress components, but the shear stress is overestimated. The model for Su~ actually reproduces the dissipative contribution, S ~ 2, Eq. 3. In order to improve the model of magnetic source contribution to the shear stress, we decided to model its both contributions separately, i.e. Si~ 1 and S 0M2. For the plane channel flow, the S ~ 2 contribution can be treated exactly (expression reduces to the product of the imposed magnetic field and the particular stress component, -2Ha2/Re ~ B~). Consequently, it remains only to model Sift1. As seen from Eq. 3, the correlation of velocity and electrical field fluctuations (uiej - -uiOr determines the Sift I contribution. After deriving the parent equation for uiej correlation (by substituting Ohm's law for moving media in equation
139
Plane Channel Flow Under Transverse Magnetic Field D N S Noguchi et al. (1994): Haffi6, R e ~ = 1 5 0
~I .010
o
~
SM1
~ A
A
SM2 SM
.005
~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6~1 7~ 6~1 7 69 1o7 691 796 1o 7 6o 1 o7 6 o o o o o o o 9 o o o 9 o o 9 9 o 9 o o b-"-'- --'=--- --'= "-" "-=--- ~ --" --= -" "" "b "~."" . . . . . . . . . . . . . . . . . . . . . . .~. . . . . . . . . . . . . . . . . . .
~b
b
AAAA A
-.005
~
m~
~
v
A~
~
^ ~ ~ ~ m~ ~ ~ m* m ~ ~ ~ g~" ~
........... JG
mm
New
-.010 llllnlllllllltnlllllllttltliilltlJlllitlilltltltlllllJlIIiilli~illllililli
l
0
20
2.E-4
40
60
80
100
120
140
y+
Plane Channel Flow Under Transverse Magnetic Field DNS N o g u c h i e t a l . ( 1 9 9 4 ) : H a - - 6 , R e ~ - - 1 5 0 9 e SM1 9 m SM2
": -
A
SM
A
I.E-4
0.
-I.E-4
-2.E-4
~ - - = - '
V
t--
_d"
~
_
m
i
m
m
A"
20
u
.......
...........
I-JJ~l l l l l l l l l l J l J D i J l l l l r 0
m
40
llJlJ
,I'G
I~lllmlll~JJllllJllllllllllllllllllllllllll 60
80
100
120
140
y+
Figure 1" Plane channel flow under transverse magnetic field, DNS by Noguchi et a/.(1994), Re,=lS0, H a = 6 , a priori testing of JG and NEW proposals for MHD-source terms in k and c equations for magnetic field), and reducing the model to a plane channel flow, the following form of the model for SiM1 emerges"
0r
(9)
where 0 < C o < 1 remains to be determined. The value of Cr was determined by optimising the Suml contribution, since this Reynolds stress component is the largest, Fig. 2a. As seen, the new splitting procedure improves the prediction of magnetic source term contribution for the shear stress significantly, and retains good agreement for normal stress components.
140
Plane Channel Flow Under Transverse Magnetic Field D N S Noguchi et al. (1994): Ha=6, Rev=150
11 .02
o n
o o
A
,
SMI SM2
SM
.01 d ~ # ~ ~
0-*.o O.o o ~
,
" ..~ee
I~_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
--.01
l~
mm
m
r--
el
........... J G
l~m
", I
; --~ -'-~-~-.-~.--.-~~,~ *-,-
o.O" ~
.......
g::,-,ro~l
. . . . . .
Newl-SM1
--.0~ [r-I 1 I l l l l l [ l l l l l 0
.0015
IIItlllllllllllllll
20
-
40
Illlllllltl 60
IIltlllltllllllllllll~llllli
80
100
1~0
l 140
y+
Plane Channel Flow Under Transverse Magnetic Field D N S Noguchi et al. (1994): Ha=6, Rev=150 9 9 SMI " " SM2 A A SM
.0010
.0005 . /
-
i.
.o .....
~"
,,,,,=,ooo,,ooo
oo o,,o~,~
;-g;~---T;
.oo.~,.e ,o~176
O.
-.0005
I,
-.0010 -.0015
~ ~ " ~ " " '~ " m
4 A A A A A A A a a6Am
mmmmmmmm mmmmm mmm ~m mm m m m -'m m" mm ........... JG
......
,
It II I I I I I I I L I
0
till
[ I I l l !t t I I l l l l 1 1 1
20
.0010
40
60
New Newl-Total Newl-SM1
! ~ ~ll !l ! ! , I I t I ' ' : III I I I 1I It I It: I I I , I I J 80
100
120
140
y+
Plane Channel Flow Under Transverse Magnetic Field D N S Noguchi et al. (1994): Ha=6, Re~=150 o o SMI
~r~
" A
" A
SM2 SM
.0005 oo o
o /
i
-.0005
.
.
.
.
~_~
.
o o o ~
.
o
~ _ _ _ _ _~
t
o o o
...........
~D
JG New
Newt-Total NewI-SM1
-.0010
,ttttJtt'tltlllJtttllt~'f! 0
20
I'lttltttlt!l!lt'ttl!llt~:t'ltrtt[tt'tlt'lllt'tl, 40
60
/30
100
120
140
y+
Figure 2: Plane channel flow under transverse magnetic field, DNS by Noguchi et a/.(1994), Re,=lS0, Ha=6, a priori testing of JG, NEW and NEW1 proposals for the MHD-source term (SiM) in expanded RSM
141
3. Duct flow of liquid metal in an inhomogeneous magnetic field The majority of liquid metal flows of practical relevance occur in presence of inhomogeneous magnetic fields. This inhomogeneity is mainly due to a limited size of the magnet. The effect of inhomogeneity of the magnetic field on the flow of conductive fluid was examined in the configuration shown in Fig. 3. Liquid metal (Hg-Mercury) with velocity (U) enters a perpendicular magnetic field of strength (B0). As a result, the current is induced in the liquid metal. This induced current leads to electric potential (~) which generates a current due to the electrical conductivity of the liquid metal. This conductive current counteracts the induced current, resulting in a relatively small difference between them in form of the total current (J). The total current combined again with the magnetic field (B -- B0) results in strong Lorentz force. In order to investigate the validity of numerical implementation of N-S-M equations, we considered the above mentioned flow of liquid metal through a rectangular duct with insulating walls entering and leaving a uniform magnetic field. The magnetic field induces characteristic Mshaped velocity profile in the plane normal to the imposed magnetic field. This test case is based on the experimental investigation of Tananaev (1978), and is defined by the characteristic nondimensional parameters: R e - U L / u - 2 • 105, H a - v / N R e - 700, N - aB~L/pU- 2.45. Numerical computations were performed by a finite volume Navier-Stokes-Maxwell solver for three-dimensional flows in structured non-orthogonal geometries, with Cartesian vector and tensor components and colocated variable arrangement and with wall functions. The grid resolution was 102x43x62 grid nodes in x, y and z direction respectively. The fine grid distribution was applied in the direction perpendicular to the imposed magnetic field in order to resolve Hartmann boundary layers which develop along the duct walls. The diffusion terms were treated by CDS and the convective terms by using QUICK scheme. At the inlet of the channel, the velocity was assumed to be uniform with a small uniform turbulence level (Fig. 5). Fig. 4 shows a streamwise velocity profiles at three different locations. Despite a very high value of Re number, very strong magnetic field suppresses the turbulence and almost laminar profiles are obtained in the central part of the magnet. The velocity profiles obtained by calculations in laminar regime with F~ in momentum equation, show a very similar behaviour with experiment. The formation of the characteristic M-shape of the velocity is captured in accordance with experiment, but results show a constant overestimation of the velocity profiles indicating that turbulence is not completely damped. This is probably due to a relatively short length of the magnet as well as high value of Re number of the incoming flow. The calculations with the standard k-e model with F~ in momentum equation produced velocity profiles which show lower peak values (due to a better mixing) than the corresponding experimental profiles, indicating that the produced level of turbulence kinetic energy is too high. Only application of the NEW model which includes also the additional magnetic source terms in k- and e equations, brings the predicted values in close agreement with experiment (except at the entrance of the magnet). This means that the ability of the model to predict a low level of turbulence is essential for this type of the flow. The calculations were performed also with JG model, and surprisingly, similar agreement with experimental values was obtained as with the NEW model. In order to clarify this behaviour, we deactivated first the additional damping function in the turbulent viscosity, by specifying fM--1, and the final results were identical to the standard version of k-e model. By deactivating the additional magnetic source terms and retaining the damping of turbulence viscosity, the effects on the final solutions were negligible in comparison with the full JG model. This leads to the conclusion that JG model affects turbulence mainly through
142 the additional damping function in turbulent viscosity. This is in full accordance with our earlier findings which resulted from a priori testing where the magnetic source contributions were heavily underestimated in the JG model. Fig. 5 shows the development and formation of the characteristic M-shaped velocity profiles as well as velocity vectors, electrical potential and turbulent viscosity contours in the central plane (x,y=0,z) of the rectangular duct, shown in Fig. 3. It is interesting to note that the velocity profiles becomes slightly deformed even before the fluid reaches the channel part exposed to the magnetic field, due to the back pressure effects. In addition to the strong deformation of the mean velocity field at the magnet entrance, the second interesting behaviour is visible at the magnet exit, where the velocity profile is additionally deformed (due to strong gradient of the magnetic field in the horizontal direction). The electrical potential exhibits a linear distribution in z-direction with a positive value in the lower part, and also positive ones in the upper part of the channel. The turbulence viscosity profiles indicate that the turbulence is strongly damped in the magnet region, but it does not vanish totally.
4. Conclusions It has been demonstrated that the effect of magnetic field can be accounted for in turbulence closure models by introducing source terms due to Lorentz force in the same manner as is practiced to model other body forces, e.g. thermal buoyancy. The new terms in the second-moment closure showed the same behaviour as those obtained by DNS for a flow of conducting fluid in a plane channel. This a priori validated enabled also to determine one new empirical coefficient. The standard low-Re-number k - ~ model with the same model terms reproduce well the development of the 'M' shaped velocity profile in a high-Re-number developing flow of liquid metal in a rectangular-sectioned duct subjected to a finite-length uniform magnetic field.
References Branover, H., 1978, "Magnetohydrodynamic Flow in Ducts", John Wiley, New York, USA Branover, H. and Lykoudis, P. S., 1983, "Liquid-Metal Flows and Magnetohydrodynamics", Progress in Astronautics and Aeronautics, Ed. Summerfield, M., American Institute of Aeronautics and Astronautics, New York, U.S.A. Ji, H. C. and Gardner, R. A., 1997,"Numerical analysis of turbulent pipe flow in a transverse magnetic field", Int. J. Heat Mass Transfer, Vol.40, pp.1839-1851 Moreau, R., 1990, "Magnetohydrodynamics", Kluwer Academic Publishers, Dordrecht, The Netherlands Noguchi, H. and Kasagi, N. ,1994, "Direct Numerical Simulation of Liquid Metal MHD Turbulent Channel Flows" (in Japanese), Preprint of JSME, No. 940-53, pp. 365-366. Noguchi, H. and Ohtsubo, Y. and Kasagi, N., 1998, "DNS database of turbulence and heat transfer", ftp. thtlab, tu-tokyo, ac.jp/DNS
143
!:
c~l
U
t'"" _
- l:.., ""N
0.2
. _1_
.
0.304 .
.
I.
0.2
J
Figure 3" Schematic representation of the flow (U) entering and leaving magnetic field (B0)
o.
o.
-
I
N
N
-.2
-.2 ~
-.6
i
i
9
\o
\.
",o
l,,, 'r
\
~ ~
o
-laminar-
~\A '\ A
-.4
A~
,
-standard\
\o
,
' \ ,, ,.\
m
-
"\a
\
i \\
-.8
'
\A '\
-.8
-1.0
-I.0
,
1
1
,.5
uIun~
,
-t
N
Figure 4: Velocity profiles in a 3D MHD flow in a square duct, Re=2 x 105, Ha=700, N=2.45, experiment by Tananaev (1978), ( - - , o ) magnet inlet (x=0.2 m), ( - , [ ] ) magnet half (x=0.352m), ( - . , A ) magnet outlet (x=0.504m), a) laminar, b) standard k - ~ model with FL in momentum equation, c) new k model;
1
1
1.~
U/Un~
0.
-.2 o
-.4
!
~,
/"
'o
'
~
-new-
t~ \
-.6 ',,| ',
~
'\ A 9\
-.8 -1.0
1
1
1
1.5
u/u~
144
Figure 5" The flow of liquid metal through a uniform magnetic field in rectangular duct with isolating wall; R e - 2 x 105,Ha=700,N=2.45; the velocity vectors, electrical potential and turbulent viscosity distributions
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
145
Modeling Separation and Rezttachment Using the Turbulent Potential Model Blair Perot and Hudong Wang Department of Mechanical and Industrial Engineering University of Massachusetts, Amherst, MA 01003, USA. ABSTRACT A new type of turbulence model has recently been proposed by the first author which involves transport equations for the scalar and vector potentials of the turbulent body force (the divergence of the Reynolds stress tensor). Theoretical analysis of this turbulent potential model suggests that the predictive accuracy of Reynolds stress transport equation models might be obtained at a cost comparable to state-of-the-art two-equation turbulence models. Initial tests of the model showed promising results for simple turbulent flows such as channel flow at various Reynolds numbers, boundary layers, mixing layers, rotating channel flow, and even transition. In order to understand the model's behavior in more complex flow situations it is now tested on a number of more complex flows involving flow separation, reattachment and stagneation. Model predictions for two adverse pressure-gradient boundary layers are presented, one mild and one on the verge of separation. In addition, predictions for the backward facing step and a turbulent impinging jet are presented. 1. INTRODUCTION Reynolds Averaged Navier-Stokes (RANS) turbulence models are usually concerned with modeling the Reynolds stress tensor. An alternative approach to RANS turbulence modeling has been proposed 1'2 where the primary modeled quantities are the scalar and vector potentials of the turbulent body force - the divergence of the Reynolds stress tensor. This approach has been found to have a number of attractive properties, most important of which is the ability to model non-equilibrium turbulence situations accurately at a cost and complexity comparable to the widely used two-equation models such as k-e. Like Reynolds stress transport equation models, the proposed model does not require a hypothesized constitutive relation between the turbulence and the mean flow variables. This allows non-equilibrium turbulence to modeled effectively. However, unlike Reynolds stress transport equation models, the proposed system of partial differential equations is much simpler to model and compute. It involves roughly half the number of variables, no realizability conditions, and removes the strong coupling between the equations. A analysis of the turbulent body force potentials and their physical significance 1 has revealed that they represent the relevant information contained in the Reynolds stress tensor and are fundamental turbulence quantities in their own right.
146 Many existing RANS models require a constitutive algebraic relation between the Reynolds stress tensor and the mean flow gradients. The most common relation is the eddy viscosity model, R=-~kI-vT(Vu+VuT), where k is the turbulent kinetic energy and v T is the eddy viscosity. More complex constitutive relations are certainly possible 3-5 and these nonlinear eddy viscosity relations fix a number of deficiencies of the standard linear model, but they still assume that the turbulence is close to equilibrium and has had time to adjust to any changes in the mean flow. Unfortunately, many turbulent flows of practical engineering significance are not close to equilibrium. A classic example is the adverse pressure gradient boundary layer. Other examples include rapidly strained flows and three-dimensional boundary layers. The equilibrium assumption imbedded in any constitutive relation for the Reynolds stress tensor is emphasized here because the proposed model avoids such a relation and therefore has the potential to predict non-equilibrium turbulent flows more accurately. There is some prior evidence that models which avoid a constitutive relation for the Reynolds stress tensor outperform other models of the same general class. Both examples of this phenomenon come from models developed for nearly parallel shear flows (where the Reynolds shear stress is the important Reynolds stress). For example, the zero-equation model of Johnson & King 6 solves an ordinary differential equation for the maximum turbulent shear stress. As a result it often performs better than other zero-equation models which use the traditional approach of defining an eddy viscosity. A similar result is also obtained with one and two-equation models. The model of Bradshaw, Ferriss & Atwell 7 was widely accepted to be the most accurate model of the 1968 Stanford competition 8. This model differed from the competitors in that it solved an equation for the shear stress directly, rather than using a constitutive equation involving the mean shear. The principal drawback of both these methods (and probably the reason that they are not more popular) is that they can only be applied to nearly parallel shear flows. In some sense, the proposed model can be viewed as a way to generalize the model of Bradshaw et. al. to arbitrary flows. In the past, for arbitrary flows the only alternative to using a constitutive relation was to solve modeled transport equations for the Reynolds stress tensor itself. Reynolds stress transport models can potentially contain more physics than eddy-viscosity based models, however the equations are significantly more difficult to solve. In three-dimensions one must solve six highly coupled transport equations for each Reynolds stress. The equations are stiff, and none of the Reynolds stresses are universally dominant, so uncoupling the equations numerically is very difficult. In addition, the Reynolds stress tensor is a positivedefinite tensor but the modeled equations often do not preserve this property (realizability9). The proposed model does not suffer from these difficulties. It involves half the equations of a Reynolds stress transport model. The equations are not strongly coupled and are not numerically stiff. The key to developing a model which avoids the use of a constitutive relation and yet does not involve the complexity of a full Reynolds stress transport closure is to note that the Reynolds stresses contain more information than required by the mean flow. Only the divergence of the Reynolds stress tensor (a body force vector) is required to solve for the mean flow. With this in mind, the potential turbulence model defines two new turbulent quantitiesthe scalar and vector potentials of the body force vector. The advantages of a model that uses these turbulent potentials, rather than the body force vector itself, are twofold. Firstly, this
147 allows the momentum equation to remain a conservative equation. Secondly, and more importantly, these potentials have a very clear physical interpretation which will facilitate the construction of models for their evolution. Turbulence modeling based on the force vector itself (or its rotational c o m p o n e n t - the Lamb vector) have been proposed by Wu, Zhou & Wu l~ and Marmanis l~, but the author is not aware of any model results based on these ideas. 2. T U R B U L E N T P O T E N T I A L S The scalar potential, r and vector potential, ~, of the turbulent body force are defined mathematically by the following equations. VC+Vxv=V.R
(la)
V.v=0
(lb)
The second equation is a constraint on the vector potential. Other constraints are possible but this is the simplest for the purposes of our analysis. These equations can be rewritten to express the turbulent potentials individually. V:r
V.(V. R)
V2V = -V x (V. R)
(2a) (2b)
The boundary conditions on these elliptic equations are constructed intuitively. Both potentials are required to go to zero at infinity, at a wall, or at a free surface. Note that by its very definition (Eqn. 1a) the scalar potential is the part of the turbulence that contributes to the mean pressure but does not effect the mean vorticity. Only the vector potential has the ability to effect the mean vorticity, and it only moves the vorticity around (enhanced transport), it does not create or destroy mean vorticity. Physically, we sometimes find it useful to regard the scalar potential as a measure of the average pressure drop in the cores of turbulent vortices, and the vector potential as a measure of the average vorticity magnitude of the turbulent vortices. In flows with a single inhomogeneous direction (say the y-direction), Eqns. (2a) and (2b) simplify to r R22, ~r =-R23, V~ = 0, xr = R,~. For this reason, it is also reasonable to view the vector potential as a conceptual generalization of the shear stress (u'v') to arbitrary geometries and three dimensions. In two-dimensional mean flows the vector potential is aligned perpendicular to the flow (like the vorticity) and has only a single nonzero component (tr The scalar potential (in combination with the turbulent kinetic energy) gives a good indication of the anisotropy of the turbulence and is fundamental to modeling the presence of walls and/or surfaces without using wall functions. The scalar potential is a positive semi-definite quantity in flows with a single inhomogeneous direction. It is hypothesized that this is also true for arbitrary flows. In three-dimensional flows the presence of the divergence free constraint on the vector potential implies that the vector potential can be computed at a cost roughly equivalent to the scalar potential. Since the k and e transport equations are also solved with the model, the
148
overall complexity and cost of solving the potential model is four transport equations. This is roughly half that of a full Reynolds stress transport equation closure (seven equations), with little, if any, loss in the overall predictive capacity. 3. T U R B U L E N T P O T E N T I A L M O D E L The transport equations that constitute the turbulent potential model are summarized below. Dk Dt
= V.(v + v T/o~)Vk + P-e
DE E = V. (v + v T/ o~ )Vs + 13-~(C~,P - C a s ) + C~3(1 -002eV .-*Vk Dt
D, = V.(v + VT/O~)V~ - 20~--~+ Dt k
2v(V~''2.v~)"2) s 2 ) -~ _~_------~ ~ k (1 + Cp3
+Cp E (1-00 ~) ~( 14/'V k (1 + 25/Re) (-~k-~)+Cp2k-P+ (C~ +Cp4)2~ VT(1+25/Re )
k
D~-
(k(~)'n
-Pl
+ k (1 + Co3-~-)
+ (1- Cp2)(~r Cp' t~ (1-o0 Vp ~( V'V k (1 + 25/Re) v+CP2 k +(CP2+Cp')2~ ~ , VT(I+2-5}Re)
P/
V.R = V r
where P=~.co,
v T=C.
E
[3 = (,+r,+Cw, .... T,R~R~)~'2 e ,
0~= 1/(1+1.5~-), and Re = k 2/(ve).
The constants are: O k =0.8,O~ = 1.2,CE1 = 1.5,CE2 =1.83,C,3 =0.17,Cp1 = 4.2,Cp2 =3,Co 3 =.12,Cp4 = 6 .
The two constants given by fractions are determined by matching Rapid Distortion Theory (RDT) in the case of strongly sheared turbulence. A detailed derivation of these equations is found in Reference [2]. Initially these equations appear daunting. In fact, they represent a fairly simple extension of the classic k-e equation system, and are relatively simple compared to Reynolds stress transport equation models. The second source term in the potential equations (in parentheses) is a dissipation-like term. This term is a standard dissipation model with two nearwall/surface modifications, one for the dissipation and one for the near wall pressure correlation term. These modifications are active in the laminar sublayer and allow the model to obtain the correct asymptotic behavior in the sublayer. The source terms involving the constants Co, and C02 are pressure-strain redistribution terms. The slow pressure-strain is
149 based on return-to-isotropy and the fast pressure-strain is based on isotropization of the production model. The constants are set to common values for these models. The effect of system rotation can be explicitly included in the model, (as with a Reynolds stress transport equation model) but is not necessary in this context. While the model includes transport equations for k and e it should be emphasized that the proposed model is a significant departure from standard two-equation models, k and e are now auxiliary quantities that are only used to help model the source terms in the turbulent potential evolution equations. They are not used to determine the Reynolds stress tensor or the resulting mean flow. The elimination of the constitutive equation for the Reynolds stresses is an important departure that removes one of the weaker modeling assumptions. A ko9 implementation could easily be substituted for the current choice of k and e. The dissipation rate was chosen because some of the test cases have this quantity available for comparison. If computational time is a serious issue, algebraic models for either or both of these variables can be used. In particular, for shear dominated flows, k=~(~+ E~u .~u/~) and e=C,P-~-r are good approximations.
The latter expression is equivalent to the linear eddy
viscosity hypothesis (though used in a different context). Computations of turbulent channel flow with these algebraic expressions and E=I.1 showed a reasonable agreement with the DNS data of Kim, Moin & Moser 12. Some of the important theoretical properties of the model are summarized below: 9 Correct decay of homogeneous isotropic turbulence. 9 Correct behavior in the log layer. 9 Correct behavior for homogeneous shear flows at early times or after the sudden introduction of mean shear along a streamline. 9 Correct behavior for homogeneous shear flows at long times. 9 Exact asymptotic behavior near walls. 9 Exact asymptotic behavior at free surfaces. 9 No ad hoc damping functions or functions of the wall normal distance (which is poorly defined in complex geometries). 9 No ad hoc manipulation of the model constants for different flows. 9 Explicit dependence on system rotation if it is present. 9 Natural relaminarization, and precise control of turbulence growth during transition. 9 Stability/Numerical robustness (for the flows tested to date). 9 A complexity roughly twice that of two-equation models and half that of Reynolds stress transport equation models. Note that most modem two equation models (see Durbin 13 and also Craft et al. TM)add one or two additional PDEs making this model directly comparable in complexity to 'enhanced' two equation models. 9 Exact transport equations (albeit unclosed) from which to derive the model terms. When adding additional physical effects (compressibility, particle interactions, buoyancy, etc.) the exact equations give a concrete analytical expression from which to construct the model extensions. 9 No algebraic constitutive relations relating the turbulence to the mean flow.
[50
4. ADVERSE PRESSURE GRADIENT BOUNDARY LAYERS
Adverse pressure-gradient boundary layers represent a situation where the classic assumptions of turbulence modeling are not well approximated. In particular, the turbulence is not in equilibrium with the mean flow, and the eddy viscosity hypothesis is a poor approximation. Two equation models (even the more elegant models, such as Durbin's elliptic relaxation model) tend to have some problems predicting adverse pressure boundary layers. However, models which predict the shear stress directly (Johnson & King, Bradshaw, Ferriss & Atwell, and full Reynolds stress closures) generally show considerably more success with these types of flows. Since the potential model also directly predicts a quantity akin to the shear stress, it is expected to perform well in these situations. A common adverse pressure-gradient flow for tests of turbulence models is the experiment of Samuel & Joubert 15. The experimental and computed velocity profiles at two downstream locations (Samuel & Joubert's station 9 and station 12) are shown in Figure 1. Only the nondimensional profiles are available for this experiment. A more difficult test of the model's ability to capture separating flows is given by the experiments of Schubauer & Spangenberg 16. This experiment has a very strong adverse pressure-gradient which comes very close to causing separation. The data is dimensional, which means that the boundary layer growth must also be predicted correctly for this test case. Figure 2 shows velocity profiles for the initial condition and three downstream locations of this flow compared with the experiments. The poorer agreement at the last station, close to separation, is thought to be a result of the boundary layer approximation, not necessarily the model. 1.25
0.25 0
0
V
V
A
x=1.87 m x=2.55 m
.
0.75
0.2 .
.
.
.
.
A
E
y / 899
Exp. (x=1.016 m)
/
0.15
u
v
0.1
0.5
[3
0
0.05
0.25 0
/8
o Exp.(x=2.032 m) / [] [] Exp.(x=3.048 m) / [] o Exp.(x=0 m) l D s
0
0.25
0.5
0.75
u/u
1
1.25
Figure 1. Experimental (symbols) and calculated (lines) velocityfor the Samuel & Joubert adverse pressure gradient boundary layer.
0
0.25
0.5
0.75
Lvu.
1
1.25
Figure 2. Experimental (symbols) and calculated (lines) velocity profiles for the adverse pressure gradient boundary layer of Schubauer & Spangenberg.
151 5. B A C K W A R D F A C I N G STEP The first backward facing step case is for a step with an expansion ratio of 1.2 and a Reynolds number of 5100 (based on the maximum velocity and step height, H). This geometry corresponds to the DNS simulations of Le & Moin ~7 and the experimental results of Jovic & Driver 18. The step is located at y/H=l and x/H=0. The calculation domain extends f r o m - 3 H upstream of the step to 27H downstream and 6H in the vertical direction. The mesh consisted of 120x120 quadrilateral cells stretched so as to resolve the boundary layers, shear layer, and the reattachment zone. The computed reattachment point was found to be 6.36 step-heights downstream of the step. This corresponds very favorably with the value of 6.28h found by the DNS simulations and 6. lh found by the experiments. DNS data and model predictions for the mean velocity are shown for a number of downstream positions in Figure 3. This figure focuses on a reduced area of the computational domain where the flow is most complex. Squares indicate the DNS results and solid lines indicate the model predictions. There appear to be some differences in the inlet velocity profile, and the magnitude of the velocity in the recirculation bubble is slightly underpredicted, but the overall agreement is reasonable. The higher Reynolds number backstep experiment of Driver & Seegmiller ~9 was also calculated. The Reynolds number is an order of magnitude larger (37,500) and the expansion ratio is somewhat smaller (1.125). The mesh was increased near the walls (to 140x140) to
4
4
x/H=1 J 3
I
I
2 1
0 -0.25
0.25
4
0.75
1.25
0.25
0.75
1.25
3
i
0 -0.25 4
4
x/H=8 J i
0 -0.25
.
.
.
.
.
.
.
.
0.25
0.75
1.25
0.75
1.25
I .1o I
.
1
-0.25
0.25
0.75
1.25
0 -0.25
0.25
0.75
1.25
0 -0.25
0.25
Figure 3. Velocity profiles at several downstream positions for the low Re number backward facing step. Symbols, experimental data of Le & Moin. Solid lines, model predictions.
152 capture the thinner boundary layers. Figure 4. shows the experimental and predicted velocity profiles at a number of downstream positions. The results are typical of low Reynolds number Reynolds stress transport equation models 2~ The reattachment point is well predicted but the strength of the recirculation zone is underpredicted, and the boundary layer does not recover quickly enough. It is believed that corrections to the vector potential equation could correct both of these deficiencies. 4
3
4
I
4
3
j
2
2
1
1
0 -0.25
#" 0 -0.25
4
0.25
0.75
1.25
4
I .'H71
I x,H- I
0.25
0.75
I
I
0 -0.25
0.25
1.25
0 -0.25
4
0.25
r
0.75
1.25 t
I"H-'0 I
/ 0 -0.25
0.25
0.75
1.25
0.75
1.25
-0.25
0.25
0.75
1.25
Figure 4. Velocity profiles at several downstream positions for the high Re number backward facing step. Symbols, experimental data of Driver & Seegmiller. Solid lines, model predictions. 6. IMPINGING JET Calculations of an impinging slot jet were performed to test the accuracy of the model in stagnation point flows. Comparisons of the local Nusselt number on the base plate were made at a Reynolds numbers of 2900 (based on the jet width and average jet exit velocity) and a jet height of two nozzle widths (H/W=2). The local Nusselt number as a function of the distance from the jet centerline is shown in Figure 5. The peak value compares very well with the experimental data of Martin 21. It is suspected that coarse mesh resolution towards the domain exit reduces the quality of the predictions far from the jet centerline. Standard two-equation models incorrectly predict the maximum Nusselt number by as much as 100%. This is due to the spurious production of kinetic energy on the portion of the plate directly beneath the jet (in the stagnation region). The success of the turbulent potential model is attributed to the fact
153 that it does not produce kinetic energy in the stagnation region. Contours of the turbulent kinetic energy are displayed in Figure 6. to demonstrate this fact. Due to symmetry considerations only one half of the domain is displayed. The jet enters at the top left of the domain and exits to the right.
Figure 5. Local Nusselt number as a function of distance from the jet centerline. Re=2900.
Figure 6. Contours of turbulent kinetic energy. Note the lack of kinetic energy at the stagnation point.
7. C O N C L U S I O N Calculations of relatively complex turbulent flows were performed using the turbulent potential model. The model produced predictions that were superior to many two-equation models and comparable to Reynolds stress transport equation models. However, we estimate the computational cost of the turbulent potential model to be very similar to enhanced two equation models. It was found that the transport equations for the potentials could be uncoupled and updated individually and that the system of equations was no stiffer than a standard k/E implementation. The proposed transport equations can be integrated up to a wall or surface; they do not require wall functions. In addition, no ad hoc functions of the wall normal coordinate have been used, so the model can be implemented easily into existing flow solvers and complex geometries. ACKNOWLEDGEMENTS This work was funded, in part, by NASA-Ames Research Center in the form of a Phase I SBIR Grant. Their support is gratefully acknowledged.
154
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
B. Perot & P. Moin, "A new approach to turbulence modeling", Proc. of the 1996 Summer Program, Center for Turbulence Research, Stanford University/NASA Ames, 35-46 (1996). B. Perot, "Turbulence modeling using body force potentials", Submitted to Phys. Fluids, Sept. (1998). W. Rodi, "A new algebraic relation for calculating the Reynolds stresses", ZAMM 56, 219-221 (1976). S.B. Pope, "A more general effective-viscosity hypothesis", J. Fluid Mech. 72, 331-340 (1975). T.B. Gatski & C. G. Speziale, "On explicit algebraic stress models for complex turbulent flows", J. Fluid Mech. 254, 59-78 (1993). D.A. Johnson & L. S. King, "A mathematically simple turbulence closure model for attached and separated turbulent boundary layers", AIAA Journal, 23 (11), 1684-1692 (1985) P. Bradshaw, D. H. Ferriss & N. P. Atwell, "Calculations of boundary-layer development using the turbulent energy equation", J. Fluid Mech. 28, 593-616 (1967). S.J. Kline, M. V. Cockrell, M. V. Morkovin & G. Sovan, eds. Computation of Turbulent Boundary-Layers, Vol I, AFOSR-IFP-Stanford Conference. (1968). U. Schumann, "Realizability of Reynolds-stress models", Phys. Fluids 20, 721-725 (1977). J. Wu, Y. Zhou & J. Wu, "Reduced stress tensor and dissipation and the transport of the Lamb vector", ICASE Report No 96-2, (1996) H. Marmanis, "Analogy between the Navier-Stokes equations and Maxwell's equations: applications to turbulence", Phys. Fluids 10 (6), 1428-1437 (1998) J. Kim, P. Moin & R. Moser, "Turbulence statistics in fully developed channel flow at low Reynolds number", J. Fluid Mech. 177, 133-166 (1987). P. A. Durbin, "A Reynolds stress model for near-wall turbulence", J. Fluid Mech. 249, 465-498 (1993). T. J. Craft, B. E. Launder & K Suga,. "A non-linear eddy-viscosity model including sensitivity to stress anisotropy", Proc. 10th Symp. Turbulent Shear Flows, Pennsylvania State University, (1995). A. E. Samuel & P. N. Joubert, "A boundary layer developing in an increasingly adverse pressure gradient", J. Fluid Mech. 66, 481-505 (1974). G. B. Schubauer & W. G. Spangenberg, "Flow A", Computation of Turbulent Boundary Layers, VolumeH Compiled Data., D. E. Coles & E. A. Hirst, eds., 416 (1968). H. Le, P. Moin & J. Kim, "Direct numerical simulation of turbulent flow over a backward-facing step", J. Fluid Mech. 330, 349-374 (1997). S. Jovic & D. M. Driver, "Reynolds number effects on the skin friction in separated flows behind a backward facing step". Experiments in Fluids 18, 464-467 (1995). D. Driver & H. L. Seegmiller, "Features of reattaching turbulent shear layer in divergent channel flow", AIAA J., 23, 162-171 (1985). S. H. Ko, "Computation of turbulent flows over backward and forward-facing steps using a near-wall Reynolds stress model", CTR Annual Research Briefs, Stanford University/NASA Ames, 75-90 (1993). H. Martin, "Heat and mass transfer between gas jets and solid surfaces", Adv. Heat Trans., 13, 1-59 (1977).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
155
m o d e l for the p r e s s u r e - s t r a i n correlation t e r m accounting for the e f f e c t o f compressibility A turbulence
H. Fujiwara ~, Y. Matsuo ~ and C. Arakawa b ~National Aerospace Laboratory, Jindaiji-higashi-Machi 7-44-1, Chofu, Tokyo 182, Japan bThe Department of Mechano-Informatics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
1.
Introduction
Compressibility effect on turbulence has been studied extensively and some turbulence models for compressible flows have been proposed[i, 2, 3, 4]. In view of engineering application, a compressible turbulence model is inevitable for numerical simulation of high-speed mixing layers. It is well known that the growth rate of high-speed mixing layers is critically reduced with the increasing convective Mach number (for example Ref.[5]). The reduced growth rate is believed to be due to the effect of compressibility on turbulence. Direct numerical simulation(DNS) of compressible turbulence has been performed to clarify the compressibility effect with special focus on the dilatational terms such as the dilatation dissipation and the pressure-dilatation correlation [6]. A turbulence model for the dilatational terms has been already proposed by Sarkar et all7, 8], which can well perform in predicting the reduced growth rate of the high-speed mixing layer. However, recent work by Sarkar et al.[9] confirmed that such dilatational terms cannot be regarded as essential in causing the reduced growth rate. They also pointed out that compressibility was found to affect the production term more than the dissipation term. Vreman et al.[10] concluded that reduced pressure fluctuations are responsible for the changes in growth rate via the pressure-strain term. This conclusion is based on the examination of DNS databases of compressible mixing layers. Recent experimental research on high-speed mixing layers show that the anisotropy of the Reynolds stress tensor increases with the increasing Mach number[Ill. This is an interesting phenomenon in view of turbulence modeling. The compressibility of turbulence may affect the energy redistribution mechanism among the components of turbulent energy, which finally reduce the growth rate. In this paper, the effect of compressibility on the pressure-strain correlation is first investigated by performing large eddy simulation. The results of the LES show that the anisotropy of the Reynolds stress increases due to the suppression of the pressurestrain correlation in the compressible mixing layer. Next, a turbulence model for the pressure-strain correlation including the effect of compressibility is derived. Finally, the derived turbulence model is used to simulate the compressible mixing layer to check the performance of the model.
156
2.
Large Eddy simulation
Large eddy simulations of compressible mixing layers were performed. The convective Mach numbers Me, defined by eq.(53), are 0.1, 0.3, 0.6 and 0.9. Three dimensional NavierStokes equations were solved with Smagorinsky subgrid scale model[12]. A TVD high resolution scheme was used to evaluate the spatial difference [13]. For time advancement, the third-order Runge-Kutta explicit method was adopted. The number of grid points are 200 for the streamwise direction, 100 for the transverse direction and 20 for the spanwise direction. Figures 1 and 2 show the vorticity distributions obtained in the simulations with the convective Mach numbers equal to 0.3 and 0.9. The simulations show that the growth of the mixing layer is clearly suppressed at high convective Mach number. The rms values of the streamwise and transverse grid scale velocity fluctuations, u ~ and d, are examined. The maximum value of u ~ and that of v ~ in the flow field are obtained in each simulation and are plotted as a function of Mc in Fig.3. As the Mach number increases, the streamwise fluctuations are amplified while the transverse fluctuations are suppressed. This implies that the intercomponent energy transfer from u' to v' is suppressed due to compressibility effect. The maximum value of the pressure-strain correlation II12 and that of the turbulent energy production P in the flow field are obtained in each simulation. The definitions of II~j and P are given by Eqs.(3) and (4), respectively. Figure 4 shows how the ratio of max II12 to max P decreases with increasing M~. This reduction of the pressure-strain correlation in high Mach number flow should be included in a turbulence model. In the following sections, a turbulence model for the pressure-strain correlation accounting for the effect of compressibility is derived and the model is used for a simulation of compressible mixing layers.
157 1.4
1.4
.... ç ....
f 1.0
c~ 1.0
"B. "El,
0.6
~0.6
U'
"-)(
0.2
0.2
0
0'.2 014 0:6
ois
i
112 1:4
0
0.2 0.4 0.6 0.8
Mc
A turbulence
I
1.2 1.4
Mc
Figure 3: The maximum values of u' and v' (normalized by the values at Mc - 0.1)
3.
Pressure-Strain / Production
)Q .
Figure 4: The ratio of maxl-I12/ max P (normalized by the ratio at Mc - 0.1)
model
Various elaborate turbulence models for the pressure-strain correlation have been proposed. Among those, the simplest models, the Rotta model[14] and the isotropization of production model(IP model[15]), are chosen as a basis for modification because we focus on including compressibility effect. The Rotta and IP models for compressible flows are expressed as
~k --
-C1
1-Iij
--
pt 8ij,
Pij
=--
- p u~-j u"k"
-ilk
-
-2 [')=U "-" i Ui
aij
----
p a" "" i "uj" -ilk
s~
__
3
T- - ~'~
'
(1) (2)
-fie aij - 6 2 P bij,
(3) "
Ok ~ti
--
p- "u i" ~ Uk" Ok~tj,
P -- P k ~ / 2 ,
1
(4)
(5)
'
2 6ij, 3
bij -- Pij - 2 5 . . P 3 ,3,
It II u~.~ + uj.~,
(6)
(7)
where e is the dissipation rate of turbulent energy k. The overbar denotes the conventional Reynolds average, while the overtilde is used to denote the Favre mass average. A double superscript " represents fluctuations with respect to the Favre average, while a single superscript ' stands for fluctuations with respect to the Reynolds average. The model may be rewritten as a more general form" 1-Iij /-p6. -- -
(
C1 + C2 ~-~
F ( aii , bij , ... ).
(8)
158 where F(aii, bij, ...) is a function of non-dimensional anisotropy tensors. Before including compressibility effect, the model (8) is explained with special attention to the correlation coefficient between the pressure and strain rate fluctuations. The pressure-strain correlation can be written as a product of three terms: the root mean square of the pressure fluctuation, that of strain rate fluctuation and the correlation coefficient between them 1
1-Iij - gsij - P Vp '2. ~/s~j. G(ReT)F(aij,bij, ...).
(9)
(correlation coef.) The correlation coefficient is assumed to be a function of both the turbulent Reynolds number RT(--k2/ue) and anisotropy tensors. The magnitude of the strain rate fluctuation, is approximated by
while the magnitude of the pressure fluctuation is related to turbulent energy in low Mach number turbulence: ,'- C, ~ k,
Cp --
(11)
P C1 - } - C 2 _ . pe slow rapid
(12)
The first term on the RHS of Eq.(12) corresponds to the so-called "slow pressure", while the second term corresponds to the "rapid pressure". The coefficient Cp is an order of one. Substituting Eqs.(10-11) into Eq.(9), we obtain
l-Iij/--fie. - Cp ~/ReT G(ReT) Y(aij, bij, ...).
(13)
(correlation coef.) If the correlation coefficient is O(1), the pressure-strain correlation is ~/RT times as large as the dissipation rate. However, this is not the case because RT is usually much larger than one in boundary free turbulence. The pressure fluctuations are related to large eddies, while the strain rate fluctuations are related to small dissipative eddies. In high Reynolds number turbulence, the pressure and strain rate fluctuations cannot interact strongly because they are not tuned to the same frequency range. The correlation coefficient should scale with the ratio of the time scales of these fluctuations, which is of order /~1/2116].
1
(ReT ~ co).
(14)
Appling the relation (14) to Eq.(13), the general form is obtained
IIij/~e = -Cp F(aij, bij, ...).
(15)
159 In the case of compressible turbulence, Eq.(11) should be revised. This can be explained as follows: the pressure in compressible turbulence is the thermodynamic pressure which is always positive: p > 0.
(16)
This means that the pressure fluctuation should always satisfy the relation:
p' > -~.
(17)
Dividing both sides of Eq.(11) by p, we have
p Mt
~ C.My,
~/~
--
_
a
(is)
,
(19)
where Mt is the turbulent Mach number and ~ is the mean speed of sound. When Mt (< 1, which corresponds to incompressible turbulence, the pressure fluctuations are much smaller than the mean value p. On the other hand, the turbulent Mach number Mt may become nearly as large as one in compressible turbulence. However, even in such compressible turbulence, ~ cannot become so large as p due to the strict limitation (17) that any pressure fluctuation can never be smaller than -p. In other words, the eddy shocklets appear when the velocity fluctuations become as large as the mean speed of sound, which prevent the pressure fluctuations from becoming too large. In compressible turbulence, therefore, the coefficient Cp is a function of Mr, which should, at least, satisfy the condition
c . ~ ~ (M,--.0), c~ --+ o ( M , - - . ~ ) .
(20)
The simplest way to express this compressibility effect is to multiply the RHS of Eq.(12) by a dumping function f (Mr)
c. = / ( M , )
(
Cl + C~ ~
.
(21)
Assuming that the LHS of Eq.(18) ~ / p becomes asymptotically constant when Mt --* co, the damping function/(Mr) should satisfy the condition
f(Mt)-
{~ 1/M2
(M, --. 0), (Mr ~ oc).
(22)
For example, the following function satisfies the above condition"
f(Mt) - 1 - exp(-C//MZt ).
(23)
where C / i s a model constant. Finally, a turbulence model for the pressure-strain correlation accounting for the effect of compressibility is obtained"
IIij/-~e - - f ( M , )
(
C1 + C2 ~-~ F(aij, bij, ...).
We assume that the turbulent Reynolds number is large enough.
(24)
160 4.
Application
of the turbulence
model
The above turbulence model was used to simulate the compressible mixing layers. The governing equations are the Reynolds averaged Navier-Stokes equations with the additional equations for the Reynolds stresses and the turbulent dissipation rate. The equations for conservation of mass, momentum and energy[17] are o,~ + 0 k ~ k O,-~ti + Ok-p~ti~tk
--
o,
(25)
--
--Oip + Ok ( - - p u " u ~ + Tik) ,
(26)
Ot --fi'd+ Ole( -#e + P ) ~tle -
Ok ( - Cp ~,,,T ' ' ,,. le'' - -qle) + Ok (, - P U "i U" ~ti + u i ~ i le)
1 " Uj " u ~ -~- U "i Tile) , +Ok ( ---~puj
T,j
2
-5#Ok~,k~,j + #(0,% + %~,,),
=
_
cp# n ~
(27)
(~8) (29)
qle = ~ r ~,le~. The following relations are assumed to evaluate ~ and ~:
"~ -
-pnT,
(a0)
c v T + -~uiui + k,
(31)
where k is the turbulence kinetic energy 1 l' l/ --pk - -~pui ui .
(32)
In the above relations, the Favre mass average is used for simplicity. Therefore the density-velocity correlation p'u~ does not appear explicitly. Smits[18] pointed out the density-velocity correlation cannot be neglected in hypersonic turbulent boundary layers. A turbulence model for the correlation has been already proposed by Yoshizawa[19]. You should note that such correlation is not included in the following simulations. In the tt tt tt ) energy equation, the last term on the RHS, 0le (1- ~ p a j.tt u j u l e + Ui Tile , is considered small and is neglected[17]. To close the above equations, a Reynolds stress turbulence model was used to determine the Reynolds stress - p u- i ""uj". The equation for the Reynolds stress[17] is Ot
p u iI t ujI f + O k p u •Ii u jl lu- -k
(33)
-
Pij + Tij --~ rIij + D i j ,
Pij
--
- p u j UkOkUi -- iju i a k uk ~tj
(34)
Tij
-
~ "~ " " " ~u"~ - u " ~ '~ Ok -- p u ~ u j u k + uj 7ik -+- U i Tjk - P i jk - 1) j i k ) ,
(35)
1-Iij
_
pt (Oj?~:t_.~_ O i u ~ ) ,
(36)
D~j
=
-- r~kOku~ - ~3kOk u",
(a7)
161 Several terms in the transport equation (33) should be modeled. Here, based on the LRR model[15], we derive a simplified turbulence model which can be easily used in engineering application. The diffusion t e r m T/j is simply modeled using the gradient diffusion hypothesis
\ Ok
/
where #t is the turbulent eddy viscosity ,, - c.
(39)
E
The model for the pressure-strain correlation in compressible turbulence was already derived in the equations (23) and (24) with the undefined function F of anisotropy tensors. Adopting the Rotta model[14] for the slow term and the isotropization of production model[15] for the rapid term, II~j is modeled ~k f(Mt)
-~ 5ij
- C2P
---fi-- - -~Sij
,
(40) (41)
=
1 - exp(-C//M2t
),
M,
-
v'5-il-a,
(42)
P
--
Pkk/2.
(43)
The dissipation term Dij is determined by using an isotropic dissipation model: 2
(44)
Dij - - -~e6ij ,
where e is the dissipation rate of turbulence kinetic energy k. The dilatation dissipation and the pressure-dilatation correlation are not included in order to evaluate the performance of the present model clearly. The dissipation rate e is obtained by solving the following transport equation: e
pE2
0kE}
The turbulent heat flux is modeled using a gradient diffusion hypothesis ," ,~ "~ " = cP P #t 0iT, rT
-...pt.T ,~i
(46)
where P r T is the turbulent Prandtl number which is assumed constant. The values of the constants are C~-
1.4, C~2- 1.8, C~,- 0.09, a k - 1.0, a ~ - 1.4,
C1 - 1.8, C2 - 0.6, C f - 0.02, P r T -- 0.8 Note that the wall effect is not considered in this model.
(47) (48)
162 The above relations are used in the 2D simulation of the compressible mixing layer(Figure 5). The inlet ftow conditions of the upper and lower streams are Ul
--
2 U2,
(49)
Pl
--
P-'2,
(50)
Pl
--
P--2,
(51)
al
--
a2.
(52)
The subscript 1 denotes the value of the upper stream while the subscript 2 denotes the value of the lower stream. Fifteen cases are simulated with the convective Mach number changing from 0 to 2.0. The convective Mach number is defined by m
(53)
M c _ Ul_ - u2.
al +a2
Figures 6 and 7 show the Mach number distribution obtained in the simulations with Mc - 0.15 and 1.0. The two figures illustrate that the above turbulence model can predict the reduced growth rate at high convective Mach number. The growth rate dS/dx is calculated in each simulation. The width of the mixing layer ~ is defined by the transverse distance between the two positions where the mean velocities are equal to ~2 + 0.1 (~1 - ~2) and ~2 + 0.9 (~1 - ~2). The calculated growth rates are normalized by the value of the incompressible case
G-
d~/dx (d~/dX)Mc=O
(54)
and are compared with an experimental result [5] (figure 8). The calculated growth rate decreases drastically with increasing the convective Mach number. This phenomenon was often observed in experimental studies of the compressible mixing layers.
= Ul
-
_
-
-
=
-
_
-
:
U2 Figure 5: the compressible turbulent mixing layer
163
Figure 7: Math number distribution(Me - 1.0)
1.2
w
~
w
w
w
v
w
w
1.0~
0.8
................... .................... i................ i. . . . . . .i..... . . .i..... . . .i..... . . .i..... . . .i..... . . .i.................. ... i
;
i
i
i
i
i
!
i
G 0.6 ...................i............+i ..................i....................i....................i....................i....................i....................i...................~.................. 0.4 0.2
i
i
+i
i
i
i
J
i i i i i+ i i i....................i..................# ..................i.+ .............i....................i....................i;
0:2
0:4
0.6
0:B
110
1'.2
1:4
j
i
i i
.............i......... O
00
i
..............................
1.6
1.'B 2.0
Mc
Figure 8" The growth rate G with Me, -e-: simulation; +: Exp.(Ref.[5])
164 5.
Summary
and future study
The present large eddy simulation shows that the growth rate of high-speed mixing layers is reduced due to the suppression of intercomponent energy transfer in velocity fluctuations. In order to include this compressibility effect, a dumping function f(Mt) is introduced into a model for the pressure-strain correlation. The turbulence model is applied to the simulation of the compressible mixing layer, showing the growth rate decreases with increasing the convective Mach number, which are often observed in experimental studies. For future study, the wall effect should be taken into account in the present model. The pressure-strain correlation is suppressed due to the wall effect as well as the compressibility effect. Recent direct numerical simulation shows that the compressibility effect in the compressible turbulent boundary layer is fairly small[20]. Therefore, the dumping function f(Mt) should be modified in the near wall region.
References .
2. 3. 4. 5. ~
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Viegas, J.R. & Rubesin, M.W., AIAA 91-1783, 1990. Sarkar, S. & Lakshmanan, B., AIAA J., Vol.29, 1991, pp.743. Wilcox, D.C., AIAA J., Vol.30, 1992, pp.2639. Yoshizawa, A., et al., Physics of Fluids, Vol.9, 1997, pp.3024. Papamoschou, D., & Roshko, A., J. Fluid Mech., vo1.197, 1988, pp.453. Blaisdell, G.A., Mansour, N.N. and Reynolds, W.C., NASA TF-50, 1991. Sarkar, S., et al., J. Fluid Mech., vol.227, 1991, pp.473. Sarkar, S., Physics of Fluids A, vol.4, 1992, pp.2674. Sarkar, S., J. Fluid Mech., vol.282, 1995, pp.163. Vreman, A.W., et al., J. Fluid. Mech., Vol.320, 1996, pp.235. Goebel, S.G. & Dutton, J.C., AIAA J., Vol.29, 1990, pp.538. Smagorinsky, J., Mon. Weather Rev. Vol 91, 1963, pp.99. Chakravarthy, S.R. & Osher, S., AIAA 85-0363, 1985. Rotta, J.C., Zeitscr Phys., Vo1.129, 1951, pp.547. Launder, B.E., Reece, G.J. & Rodi, W., J. Fluid. Mech., Vol.68, 1975, pp.537. Lumley, J.L. & Tennekes, H., "A First Course in turbulence", MIT Press, 1972. Zha, G.C. & Knight, D., AIAA 96-0040, 1996. Smits, A.J., "Turbulence in Compressible Flowg', AGARD-Report-819, 1997. Yoshizawa, A., Phys. Review A, Vol.46, 1992, pp.3292. Huang, P., Coleman, G.N. & Bradshaw, P., AIAA 95-0584, 1995.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
165
Consistent modelling of fluctuating temperature-gradient-velocity-gradient correlations for natural convection M. WSrner, Q.-Y. Ye t and G. GrStzbach Forschungszentrum Karlsruhe, Institut fiir Reaktorsicherheit, Postfach 3640, D-76021 Karlsruhe, Germany
A model is proposed for the buoyant production term in the dissipation rate equation of turbulence kinetic energy and the molecular sink term in the turbulent heat flux equation. Based on an analytical decomposition by the two-point correlation technique the model consists of an inhomogeneous and a homogeneous part. The inhomogeneous part involves the Laplacian operator of the turbulent heat flux and needs no further modelling. For the homogeneous part a model is derived which incorporates Pr/R as key parameter, where Pr is the Prandtl number and R is the ratio of thermal to mechanical turbulent time scales. The model is shown to obey the correct wall-limiting behaviour without further wall corrections. Comparisons with DNS data for Rayleigh B6nard convection in air and sodium and for convection in an internally heated fluid layer confirm its excellent near-wall performance for a wide range of Prandtl numbers. Utilising these DNS data, the performance of the model in the bulk region is improved by slightly modifying the homogeneous part of the model. 1. I N T R O D U C T I O N In nuclear engineering, knowledge of the heat transfer capabilities of buoyant flows is of importance for a variety of applications. Examples related to reactor safety are the passive decay heat removal by natural convection, the in-vessel cooling of a decay-heated pool of molten core material, and, supposed the vessel fails, the long-term sump cooling of the core melt by a recirculating water layer, driven by natural convection. For all these applications, computational fluid dynamics (CFD) is an important tool to explore the velocity and temperature field and extend the safety capabilities. In engineering CFD codes, turbulent momentum and heat transfer are modelled by statistical turbulence closures, which need to be validated and improved to be reliably applicable to strongly buoyant flows. In the present paper, we focus our attention on two terms which are of importance in statistical modelling of buoyant flows: (i) the buoyant production term P~b in the dynamic equation for the dissipation rate c of turbulence kinetic energy k = ~ / 2 and (ii) the tpresent address: Fraunhofer IPA, Nobelstrat3e 12, D-70569 Stuttgart, Germany
166 dissipation term eoi in the dynamic equation for the turbulent heat flux Oui. Here, ui and 0 denote the fluctuating velocity and temperature. Introducing the correlation
O00ui Ti = Oxl cOxt
(1)
both terms of interest can be expressed as
P~b = -2v/39iTi,
eoi = (v + n)Ti,
(2)
where v= kinematic viscosity, /3= thermal expansion coefficient, n=thermal diffusivity, and g - (0, 0 , - 9 ) is the gravity vector. In Ye et al. [8], where the two-point correlation technique and the invariant theory are used to develop models for the e-equation for natural convection, also a model for P,b was obtained as a preliminary result by using direct numerical simulation (DNS) data. This model together with the consistent model for eoi [9] was implemented in the engineering CFD code FLUTAN [6]. Calculations for the buoyant flow along a heated vertical plate were performed and compared with experimental results. The models of [8] and [9] clearly yielded improved results as compared to standard models for both terms. Models for eoi were also recently proposed by Dol et al. [2] and Shikazono & Kasagi [4]. In the present paper, the development of the model proposed in [8] for P~b is put on an analytical and physical basis. Furthermore, the analysis is extended to develop a consistent closure relation for both, P~b and eoi which adequately accounts for the Prandtl number Pr = v/n, turbulence level, and wall effects. 2. D e v e l o p m e n t of m o d e l The turbulent dissipation rate e can analytically be decomposed by the two-point correlation technique [1] in an inhomogeneous and a homogeneous part [3]"
-
Ous OUs
_luAxusu s - u(A~u--~)0 -
o
4
O x--7 =
,
1
uAxk + eh.
(3)
-i
inhomogeneous homogeneous Here, A~ - 02/OxlOxl is the Laplace operator with respect to the variable x; ~ represents a local coordinate system relative to two arbitrary points (see [1]). The prime' in Eq. (3) indicates a value for the two-point correlation function and the subscript 0 represents the zero separation ~ - 0 between the two points. Similar to e, Ye et al. [8] used the two-point correlation technique to decompose Ti"
O00ui _ 1AxOui 1 Zi = OXl OXl -- ~ ~.~-2 [(/~(0~t~)0 -~- (A~Ui0')0],, -- Ti,ih -t" Zi,h. Ti,ih
(4)
Ti,h
In correspondence to Eq. (3) Ti,ih is denoted as the inhomogeneous and Ti,h as the homogeneous part of Ti. While the term Ti,ih is known by second moment closure of the turbulent heat flux, only the term Ti, h needs to be modelled for closure of P~b and eoi. The advantages of the above decomposition will become clear in this paper. At the moment we only note that Ti,ih will adequately account for wall effects.
167 In deriving a closure for Ti,h w e proceed similar to Shikazono & Kasagi [4] who developed a model for eoi. We define a correlation coefficient for Ti and relate it to the correlation coefficient for the turbulent heat flux via a functional coefficient C which has to be determinedO00ui
o~,ox,
~_C
Ou i
(5)
We note the definition of the thermal variance dissipation rate
oooo (oo)2
co - ~Ozl Oz~ = ~ ~
(6)
and assume that the ratio of kinetic energy to dissipation rate of velocity component i equals the ratio of the total turbulence kinetic energy to total dissipation rate: u~/)
( Ou(~)"~2 Oxz ]
=
2k
~.
(7)
1]
Making use of definition (6) and approximation (7) and introducing the ratio of thermal to mechanical turbulence time scale 02 c R - 2co k'
(8)
we can rewrite (5) to yield C T i ~-
e
(9)
- Oui.
2v/V~ k
In their model for eio Shikazono & Kasagi [4] introduced two rather complicated wall functions to account for wall effects. In the present approach, the wall effects in modelling Ti are already taken into account by the inhomogeneous part of decomposition (4), i.e. by Ti,ih. To model the homogeneous part Ti,h w e make use of (9) but replace Ti by Ti,h and the total dissipation rate c by the homogeneous one, Ch 1 l,,mxk" For the moment we leave open whether R defined by Eq. (8) involving the total dissipation rates shall be used, or whether the homogeneous dissipation rates shall be involved, i.e. --
Rh-
C -
02 ch 2e0h k '
(10)
where eOh -- e 0 - l~cA~0-ff. With RT beeing either R or Rh we obtain
C eh Oui. Ti,h ~ 2V/U~CRT k
(11)
This eventually results in our basic model for P~b and eoi"
1 OOui P~*b -- ---~ u~9i OxtOxt
/ Pr e h /59iC V-~T T Oui,
(12)
168
r
- -~
OxtOx= + -~C 1 + -~r
V-R~T--k Oui - r
+ r
(13)
Beside the selection of RT also the functional coefficient C has to be determined. To do so, in the next section the near-wall behaviour of the above basic model is analysed. In a second step, DNS data will be used to determine C. Below, however, we first compare our basic model with the model of Ye et al. [8] proposed for P~b and r [9]:
Iu OOui P~*b,r~ = --~ /3g~Ox, OxL
,
1(.+~)
r
= -~
OOu,
fig'
(P_~! ~ r --~ Ou,,
1(
(14)
1 ) (~)0"7r
OxlOxl + -2 1 + ~
--~ Oui,
(15)
where the term ( P r / R ) ~ was obtained by fitting DNS data. We see, that (12) and (14) and (13) and (15) are quite similar and will be identical for RT = R and C = ( P r / R ) ~ 3. N e a r - w a l l b e h a v i o u r of basic m o d e l The wall-limiting behaviour of the basic model can be evaluated by expanding the temperature and velocity fluctuations near the wall:
0 ui --
ao + boxa + cox I + doxl + - - ' , ai + bix3 + cix 2 + dix 3 + ' " ,
(16) (17)
where x3 is the wall-normal coordinate. Here, we consider no-slip boundary conditions (hi = 0), an incompressible fluid (b3 = 0) and isothermal walls (a0 = 0). By this, we obtain the limiting behaviour
Ou3 - boc3x~ + - - - ,
O00u3 T3 = Oxl Oxl = 2boc3x3 + ' " .
(18)
For the near-wall behaviour of the analytical term T3,h it follows T3,h -- T3 - T3,ih --(2boc3x3 + "
6 " ) - (-~boc3x3 + "
1 ") - -~boc3x3 + ' " .
(19)
To determine the wall-limiting behaviour of our model for T3,h we consider the walllimiting behaviour of the individual variables on the r.h.s, of Eq. (11):
-
u
1 + bl)+---, k - i(b + bl) l
+-..
,02
--
2 2
box + . . . ,
-
ibl + . . . .
(20)
Thus, we obtain the well known result
R = Pr +...,
Rh = P r + ' " ,
(21)
which is valid only for isothermal walls. Introducing these results in Eq. (11) we get g C -2(b2 -~-bl)Jr-''' 9(boc3xl -Jr-...) -- -~boc3x3 -Jr-.... (22) + . . . ) 1 (b 2 _jr_bl)x I _.~_... , To establish the correct near-wall behaviour of model (11), the comparison with Eq. (19) requires the wall-limiting behaviour of the functional coefficient C to be C = 1 +---.
(23)
169 0,15 ONS --
0,10
--
data
inhomog,
part
....
homog,
part (R)
.....
homog,
part (R
O 9
basic
model
(R)
basic
model
(R
t~03 n
o o ~ - . - 7 s"
n
-.
o~
.) []
0,05
~176 ........ ~176 ooO~176 9 ~ 9 o~176 0,......-- ..............
.)
.
~.o
0"
"'~'0
0
i
'
I-I
0
i
'
9
0
o.: s I-I
e.-;-"
2".-"
oO~
,p ~." ..~ o,f. " ~
0,00
,,.-o
0,0
"
.
i
0,1
'
.,- .,...
i
0,2
'
0,3
0,4
i
0,5
X3
Figure 1. c03 for Rayleigh B~nard convection in sodium ( P r - 0.006, R a - 12,000).
4. C o m p a r i s o n of basic m o d e l w i t h D N S d a t a In this section we investigate the performance of the basic model for c03 with R x beeing either R or Rh and C = 1 by comparison with DNS data. For this purpose, all turbulence quantities appearing on the r.h.s, of Eq. (13) are taken from the DNS database. Two types of natural convection in a horizontal layer bounded by isothermal top and bottom walls are considered. For the Rayleigh B~nard convection, where the lower wall is heated and the upper wall is cooled, a series of simulations has been performed with the TURBIT code at the Research Centre Karlsruhe [10]. The fluids investigated encompass sodium ( P r = 0.006), mercury ( P r = 0.025), and air ( P r = 0.71) and cover a wide range of Prandtl numbers. In this paper the DNS data for sodium at R a = 12,000 and for air at R a - 630,000 are utilised. The Rayleigh number is given by R a = u2oD/(un), where D is the channel height, u0 = v / g / 3 A T D is the buoyant velocity scale, and A T is the overall temperature difference. The second type of natural convection is a fluid layer heated internally by a volumetric heat source qv. Both walls have the same temperature and the fluid is over a wide area stratified thermally stable. The direct numerical simulations, documented in [7], are for P r = 7 and internal Rayleigh numbers up to R a i = 109, where R a i = (g/3qvD~)/(unA) with A=thermal conductivity. Here, we use the DNS data for Rag = 10 s. In the present paper all data displayed in Figures are dimensionless; the normalisation is by D, u0, and AT. In Figure 1 we analyse the perfomance of the basic model for Rayleigh B~nard convection in liquid sodium for RT = R and R x = Rh. Because of symmetry, only the lower half of the channel is displayed. It appears, that the inhomogeneous part is of importance in the entire channel. The homogeneous part is predominant in the channel centre. In the near wall region we find a good agreement of the basic model with the DNS data if RT = Rh is used. In the channel centre the discrepancy between the basic model and the DNS data is considerable. This holds for both, RT = R and RT = Rh, where the latter performs somewhat worse. Due to its superior wall-behaviour we choose RT -- Rh. To investigate the influence of Prandtl number and turbulence level, we show in Figure
170
0,006 "
~
DNS data i n h o m o g , part
"".
0,004
[
",,
o
h o m o g , part (R
h)
basic model (R
h)
"~.. "~b" " ~ " - O -
S03
O ....
0
O- - - -
....
0
0,002 t
0,000
-0,002
,, ~
.
0,0
.
....
i
.
.
0,1
.
.
.
.
,
0,2
.
'
X3
i
0,3
'
.
0,4
'
0,5
Figure 2. co3 for Rayleigh B~nard convection in air ( P r - 0.71, R a - 630,000).
- - - - - - - D N S data
0,008
- - - inhomog, part ......
0,006
O
h o m o g , part ( R basic model ( R
h) h)
0,004 E;03
0,002
' ~1 9
I I
0,000 "
~,
.
-0,002
l
0,0
, ~1
'
'
0,2
0,4
0,6
0,8
1,0
X3
Figure 3. ce3 for internally heated convection ( P r - 7, R a l
-
108) 9
2 the perfomance of the basic model for Rayleigh B~nard convection in air. The inhomogeneous part of the model is predominant inside the thermal boundary layer, but zero in the channel centre. This is because the bulk region is isothermal and the vertical turbulent heat flux takes a spatially constant value. The basic model for the homogeneous part yields an almost constant value in the bulk region. Figure 2 shows an excellent agreement between the basic model and the DNS data near the wall, while the performance in the channel centre is also good, but may be further improved. In Figure 3 a comparison of the basic model with DNS data for internally heated convection and Pr - 7 is given. The profiles are not symmetric, thus the complete channel is shown. A g a i n , we find a good agreement for both boundary layers. In the region 0.1 < x3 < 0.25 a negative value of c03 is predicted by the model, which is not
171 realistic. This failure can directly be attributed to the linear relation between c~3,hb and 0u3. Indeed, the DNS data show that for 0.1 _< x3 _< 0.25 the vertical turbulent heat flux is negative. While for 0.35 , is given by: Rapid >
.-
A2k ~
12
< z ij
< Uj > - - [ - O ( A 4 )
< Ui > b
k
k
(9a)
which implies that, to leading order, the rapid SGS stress depends on the square of the m e a n v e l o c i t y g r a d i e n t and varies as Az. Similarly, Slow
< ~ ii
>
=
A2k
12
< b
' b kU' > + O ( A 4) kU ~ J
(9b)
189 As an example, consider the DNS test case used in the present study (the temporally evolving mixing layer), O k < u i >= S, for i=1 and k=2 and zero otherwise. The only important mean rapid SGS stress component is the 11 component in this flow, A2
2 dr O(A4) (10) 12 From this simple example of the temporally-evolving mixing layer, it is clear that the rapid component has an anisotropy that is directly related to the mean flow gradients. The scaling given by Eq. (10) is checked later in the results section.
~ ~ S
How large is the rapid term relative to the slow term? After some algebra it follows that: Rapid
9
sl..... = O < xq > t~eo~
u"
(1 1)
e
Here, U denotes the mean velocity difference, u denotes the rms value of the turbulence, and the Reynolds number, Re o, -- U~5~~, is defined using the vorticity thickness. Thus, for nonv equilibrium turbulence ( P / e > 0(1)), the rapid part is comparable to or larger than the slow part. In practice, the filter size in LES of complex flows may not be much smaller than the integral length scale. In such a situation a low-order Taylor-series expansion of the velocity fluctuation does not apply and alternative scaling arguments give, Rapid > $212 A2 < T, ij =
u
"
Here l is a characteristic integral scale of the turbulence. The magnitude of the rapid component can be relatively large in rapid distortion flows with Sl/u >> 1. Even in equilbrium turbulent flows with Sl/u = 0(1), the rapid part can become comparable to the slow part when the filter size increases such that All = O(I).
3. THE SUBGRID E N E R G Y TRANSFER It is important to identify clearly the energy transfer mechanisms between the different scales of motion and ensure that the model for the subgrid-scale tensor "c,:/represents these mechanisms 9We follow the approach of Haertel, Kleiser, Unger and Friedrich [6]. The energy transfer term, also called the SGS dissipation, is split into a mean and fluctuation part
!
=
equation shows that they are of similar importance.
(a)
(b)
0.40
0.401 O---~in i , 0.30
, to, the compression ratio is defined by (2)
r(t) = L~(to)/Lc(t)
As usually for low Mach number flows, we make the assumption that the fluctuating field is divergenceless, which is physically relevant. In addition, temperature and density are assumed to be uniform throughout the flow. The turbulent flows obtained by simulation are statistically homogeneous in the planes parallel to the solid boundaries. Thus, the different statistical quantities will be calculated by using both ensemble averaging over different realizations and spatial averaging in the planes of homogeneity. 2. N U M E R I C A L
METHOD
The spatial approximation is based on a divergence-free spectral Galerkin formulation on a moving mesh similar to the method of Moser, Moin and Leonard [1]. The basis functions are the Fourier modes in (x2, Xa) directions and, in normal direction Xl, linear combinations of Chebyshev polynomials which are divergence-free and satisfy the no-slip boundary conditions. The pressure is thus eliminated from the momentum equation. The time-advancement is carried out by a second order semi-implicit scheme. In order to calculate the pressure correlations that appear in Reynolds stress budgets, the fluctuating pressure is retrieved using the method proposed by Pasquarelli et al. [2] and decomposed into a linear term (or rapid part) and a non linear term (slow part). Following Mansour, Kim and Moin [3], we have isolated the effect of the boundary condition on the solid walls in a separate contribution, the Stokes pressure. 3. F L O W P A R A M E T E R S Two cases of compression have been studied, the main parameters of which being indicated below (Table 1). For the first case, labelled as RDT, non-linear and viscous terms are removed from the equations so that the corresponding results are virtually those of Rapid Distortion Theory (RDT). For the second one, the full Navier-Stokes (case NS) are simulated. For both cases, Ni is the number of grid points in the xi direction, and N is the number of realizations. The initial flows is first generated on a uniform grid following the method of Rogallo [4] and then projected on the MML basis in order that no-slip boundary conditions be satisfied at the walls. At this stage, the triple correlations are equal to zero. Thus, for the NS case, it is necessary to let the turbulence decay freely in a first step and the triple correlations grow before the compression is applied. In Table 1, R e = k 2 / c p is the Reynolds number based on kinetic energy k - ~uiuil-- and Ly is the integral length scale of ux component in the Xl direction at t = to. We have also indicated the value of L y / L c which a measure of the non-homogeneity at the beginning of compression. The distance between the walls is given during the compression (to < t < 7r/w) by
_()
Lc(t) 1 L~(to) - r(t) -
1 1 - -~
1
1 + c o s ( w ( t - to)) + _ _ 2 rm
(3)
239 in which rm = r(~/w) is the compression ratio at the end of compression. The non-dimensional parameter R = 5k/c(to) is characteristic of the rapidity of the compression. It is defined as the product of the time-scale of the turbulence by 5, averaged value of the mean velocity gradient c, during the compression. A high value of R corresponds therefore to a rapid compression for which the effects of the mean strain are expected to be more important than viscous and the non-linear ones. In the RDT case R, is infinite.
Re - k2/cu (to) Lf /Lc (to) R -
RDT
N-S
cc
90 0.15 o.s3 10 72 • 4s2 6O
0.1
(to)
rm N1 • 2/2 • N3 N
8.33 643 20
Table 1 Characteristic parameters
4. R E Y N O L D S
STRESS BUDGETS
Fig.(2) and (3) show Xl variations for the kinetic energy k and for the principal values of the Reynolds tensor at t - to just before compression, and for r - 3 and r - 7. The abcisses Xl correspond to the two walls 1 In homogeneous turbulence, a one-dimensional compression is known to increase the velocity fluctuations in the direction of compression (here u12). In the presence of boundaries, this tendancy is inhibited partially by the blockage effect on the normal component. The following comments can be made. First, consider the R D T case. Although no viscous mechanism is active, it is clear that the wall layers, in which the Ul fluctuation is damped, are growing with time with respect to the reduced coordinate Xl/(2Lc). This indicates that the length scale of wall effect is not simply reduced in proportion to the compression ratio. Moreover, we observe peaks near the walls on the profiles of u-~ and k, whose relative levels increase with time. In the NS case, diffusive layers are developping from the walls. In that case, the maxima are no longer attached to the boundaries, but at some distance in the flow. However, the picture is qualitatively very similar. To explain the growing of the Ul-boundary layer in the reduced coordinate, we may refer to the analysis of the shear-free boundary layer by Hunt and Graham [5]. This analysis, based on inviscid R D T (without mean strain), shows that the boundary layer thickness is comparable the integral length scale of Ul in the transverse direction (x2 or x3). This integral scale does not vary much during the compression and, in any case, does not decrease as faster as r(t) -1. This may be the reason for the increase of the blocking 1In these figures and the following ones, data have been normalized using Lc/2 for the length scale and (2k(to)/3) 1/2 for the velocity scale, k(to) is the averaged value of k at t -- to.
240 effect. The other important feature, which is relative to the maxima of k near the walls, proves the existence of some energy transfer from the core of the flow in direction of the walls. This is examined now in greater detail from the Reynolds stress budgets. For the present flow, the transport equations for the Reynolds tensor reduce to the equations of Ulz and uzz - u~. These equations, together with the equation for k read DDut2 - - -
2CU2
--(U3),l - -
-t-V(Ul2),ll
--(P--~"),I -~
Pll
Tll
Dll
II/11
--(?tlU2), 1
- -
,1
-- 2P'?-tl,kUl ,k -Cll
-[-/2(U22), ix
-kTPU2,2
--2V'U2,k?'t2, k
T22
D22
(I)22
-c22
-- ~1 (?-tl ?-tk?-tk),l T
q--pk,11
- -
2
2
-~-ppUl
(I)ll
m
Du~ Dt
-DD- tk
--
__ --
C U'-~.~
P
--
D
2
-- p1 (PU'T) , 1 9
(4)
--YUi,kUi, k
-c
More precisely, considering the different terms in the budget of k in the NS case (Fig. (6)). In the core of the flow, the most important contributions are the production P, the dissipation e and the pressure transport 9 - p1 ~oxl 9Negative values for 9 indicate that energy is transported towards the walls. This energy is supplied around the peaks of k where 9 has positive values. In these parts of the flow, the diffusion D is also an important term contributing to transfer energy in the viscous layer. Close to the wall, D and e are in balance, as in more general wall-bounded shear flows. Fig.(7) represents the pressure terms in the Ul2 equation: the pressure transport ~11 and the pressure-strain correlation (I)11, together with the different linear (L) (or rapid), non linear (NL) (or slow) and Stokes (S) contributions. The linear contributions are shown to be the most important ones. In addition, it can be seen that (bll and ~11 cancel together in the peaks regions (the sum of these terms must vanish exactly at the walls), indicating that, since cbij is a traceless tensor, the energy is redistributed to the lateral components. So, the energy of the normal fluctuations supplied by the mean velocity gradient in the core of the flow is transported by the pressure-velocity correlation in direction of the walls where it is redistributed to the lateral components by the pressure-strain correlations. In the R D T case, the non-linear transport T and the viscous terms D and c vanish. Moreover ~ij and ~ij reduce to their linear parts. In that case, Fig.(4) and (5) show that the same picture is retrieved qualitatively. i
5. A M O D E L F O R L I N E A R P R E S S U R E
TRANSPORT
We have seen in the previous sections that pressure transport contributes significantly to Reynolds stress budgets. In standard turbulence models, when it is not neglected, this correlation is represented generally as a non linear term comparable to the triple velocity moments. Both terms are generally considered together and modeled using a pseudodiffusive formulation. Accordingly, the linear pressure-velocity correlation pLui, which is, in the present case, the most important contribution to pui is not modelled explicitely. It is now our purpose to derive a specific model for this quantity.
241 5.1. G e n e r a l f o r m u l a t i o n The mean velocity Ui considered in the following is a linear function of the space coordinate xi, which is more general than the compression flow considered so far. The turbulence will be supposed to be homogeneous in x2 and x3 directions, as in the previous sections. We will suppose moreover that the flow is only weakly inhomogeneous, that is to say that the scale of variations of the statistical quantities is much larger than the integral length scale of the turbulence. Thus, introducing the 'slow' variable X - ex, where is a small non-dimensional parameter characteristic of the non-homogeneity, we shall represent the fluid velocity in terms of Fourier components ui(x) - f s
k)eikXd3k
(5)
In this expression, the amplitudes fii(X,k) are slowly varying in space. But locally, at a given x, these amplitudes will be supposed to be statistically independant, as in homogeneous turbulence. Thus, starting from the equation for the linear pressure
ApL P
= -2Ul,mUm,t
(6)
we can use the Green function of the laplacian in infinite space to obtain the pressure velocity correlation f
d3x,
ppnU 1 1 (X) -- 2gm,l Ul (x)U~,m(X') 4 llx' - xl[
(7)
The next step is to calculate the spatial derivative of ul (Eq. (5)), replace in Eq. (7) and use the local homogeneity of the flow. After integration over x', the final result is
ppLu 1 1 -- 2Um,l k -2 ikmU~Ul+ s u~OXm d3k
(8)
In this expression, the correlations are taken at the same point. This general expression contains two contributions, one is o(1) and the other, o(e), but only the second one is specific to non-homogeneous turbulence. We now examine three particular cases. First, we consider the case of an irrotational mean flow. The mean velocity gradient will be supposed to be axisymmetric, as in the case of compression, and we shall derive a simplified model from the general expression Eq.(8). Then, the case of a one-dimensional mean shear and the case of a solid body rotation will be considered briefly. 5.2. I r r o t a t i o n a l a x i s y m m e t r i c s t r a i n In case of irrotational axisymmetric strain, the non-vanishing components of the mean gradient are given by /-/1,1 - Dll, U2,2 - Ua,3 - D22 - D33. It is found that the contribution o(1) vanishes and Eq.(8) yields 1
ppLUl --
1 0 r ~(3DllDkkl~x1 ]
k-21721]2d3k
(9)
242 In order to model the pressure transport in the frame of second-order moment equations, it is necessary to represent the integral in Eq.(9) in terms of known single-point moments. This integral is a positive quantity, and since [~tll2 is the spectral density of u~, a possible choice is to replace the integral by the quantity L2u~ where L is a characteristic length scale of the turbulence. We can now evaluate this model in the compressed flow described in the previous sections. Since, in that case, the unique non-vanishing component of Ui,j is U1,1 = Dll = - c , our model reads 1 ;pLu I
-
-
0 (L2~12) --C~x 1
(10)
m
Near the walls, the quantity Ul2 is a decreasing function of the distance to the wall and L2u~ is expected to behave much the same. Thus, the model is expected to lead to an energy transfer directed to the walls, as in the numerical simulations. At this point, an interesting consequence of the linearity of pLUl m u s t be underlined. If the sign of the mean velocity gradient is reversed (one-dimensional expansion" c < 0), the sign of pLUl also changes. In that case, the energy is transported in direction of higher values of u~, which is clearly different from standard diffusion. Eq.(10) is now compared to the simulations. Different choices can be made for L. We have taken L - L~2), integral scale of Ul in direction x2. The reason for this particular choice is that, during the compression, much of Ux fluctuation in the integral of Eq.(9) is due to wavevectors parallel to the walls. As shown on Fig.(8) and (9), the agreement is qualitatively correct, for a large variation of r.
5.3. Solid b o d y rotation We now consider the case of a turbulence transported by a solid body rotation with x l as rotation axis. In that case the non-vanishing components of the gradient are: /-/3,2 - -U2,3 - 12. The contribution o(1) to pLu1 does not vanish and we obtain 1 L Ul--p
P
__
fk-: ~lg;~d3k J
2~-~
(11)
in which O21 denotes the Fourier transform of the Xl- component of the vorticity V • u. This expression does not refer explicitely to the non-homogeneity of the flow and, thus, may be non-zero even in homogeneous turbulence. Actually, the integral in E q . ( l l ) vanishes in isotropic turbulence and more generally whenever the correlations ui(x)uj(x') are even functions of x r - x. Rotating inhomogeneous turbulence is an exception to this general class of flows and non zero values of pLul may be generated in that case. The reason is that, rotating turbulence can be viewed as the interaction of plane, inertial waves. These waves are dispersive and propagate with their own group velocity [6]. More precisely, it can be shown that waves having positive (resp. negative) values for Ux&~ propagate in the direction of Xl < 0 (resp. Xl > 0). Hence, at a given x, the net value of pLu1 is the result of imbalance between waves propagating towards Xl > 0 and those going towards Xl < 0. In simple flows, like the plane shearless mixing layer [7], it can be shown that the energy flux is oriented as in standard diffusion.
243 5.4. P l a n e s h e a r The plane shear flow is defined by U2 - 8 X l . Associated to a direction of nonhomogeneity along x l, this flow configuration is typical of many inhomogeneous shear flows (boundary layers, wakes ...). The simple result obtained in that case 1
--pnul -- 0
(12)
P may be a possible explanation for the fact that the pressure transport has not been yet identified as a major source of energy transport from the data available in the literature. 6. C O N C L U S I O N From numerical simulations of wall-bounded compressed turbulent flows, it has been shown that the mean strain may be the source of important energy transfer by pressure effect in direction to the walls. A general expression for the rapid part of the pressurevelocity correlation has been derived, relying on local homogeneity assumption. In the case of irrotational mean flow, this expression can be simplified and leads to a model that compares favourably with the numerical data. The cases of a mean shear and solid body rotation have also been considered. Further developments are needed to incorporate the effect of pressure transport in this more general class of turbulent flows. This research has been funded in part by the European Communauty in the framework of the Joule III programme in cooperation with Renault. We would like to thank our partners in this project: K. Hanjali5, C. Tropea, S. Jakirlid, I. Hadji6, M. Michard and M. Volkert for stimulating discussions during the course of this research and H. Pascal is for his important contribution to the numerical code. REFERENCES
1. R. Moser, P. Moin and A. Leonard. A spectral numerical method for the NavierStokes equations with applications to Taylor-Couette flow. J. Comput. Phys., 52 : 1983, pp 524-544. 2. F. Pasquarelli, A. Quarteroni and G. Sacchi-Landriani. Spectral Approximations of the Stokes Problem by Divergence-Free Functions. J. Sci. Comput., 3 (2) : 1987, pp 195-225. 3. N . N . Mansour, J. Kim and P. Moin. Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech., 194: 1988, pp 15-44. 4. R.S. Rogallo. Numerical experiments in homogeneous turbulence. NASA TM81315, Ames Research Center, Moffett Field, CA. 5. J . C . Hunt and J. M. R. Graham. Free-stream turbulence near plane boundaries J. Fluid Mech., 84 : 1978, pp 209-235. 6. H.P. Greenspan. The theory of rotating fluids Cambridge University Press., 1968. 7. L. Le Penven, J.P. Bertoglio and L. Shao. Dispersion of linear inertial waves in a rotating shearless turbulence mixing layer. 9th Symp. on Turbulent Shear Flows, Kyoto Japan, August 1993.
244 2.0
6
1.5 4 1.0 2 0.5
0.0
-1.0
-0.5
0.0
0.5
1.0
0
' -0.5
-1.0
2Xlfl--~(t)
' 0.0
' 0.5
.. 1.0
2xl/Lr m
Figure 2. (a). R D T case. Ul2, u 2 and k profiles at t = to; compression ratio r = l
15
Figure 2. (b). R D T case. Ul2, u 2 and k profiles: r = 3 .
2.0
1.5 I0 1.0 5 0.5
0--
-1.0
-0.5
0.0
-
0.5
1.0
2Xl/Lo(t)
0.0
-1.0
-0.5
0.0
0.5
1.0
2Xl]Lc(t) m
Figure 2. (c). R D T case. u 2, u 2 and k profiles: r=7.
m
m
Figure 3. (a). NS case. u~, u~ and k profiles at t - to; compression ratio r=l
245 1.5
,
,
,
2.0
1.5 1.0 1.0
~--~k I
0.5
0.5
0.0
-1.0
-0.5
0.0
0.5
1.0
0.0
-1.0
-0.5
2Xl/L (t)
0_._0
--
~t/
"
0.5
1.0
Figure 3. (c). NScase. Ul2, u 2 and k profiles: r=7.
Figure 3. (b). NScase. Ul2, u 2 and k profiles" r=3
3000
0.0
2x/L (t)
5000
|
|
dk/dt
~--~p
3000
2000
1000
9
f
-
//
1000 1000 0
3000
H
-1000 I
-1.0
,
l
-0.5
,
2x/L (t)
Figure 4. R D T case. Terms in the budget of the kinetic energy" r=3.
I
0.0
5000
lkI/11
---d(d-~/dt &-~ Pll
t
|
-1.0
I
-0.5
2x/L (t)
|
0.0
Figure 5. R D T case. Terms in the budget of the normal stress u12: r=3.
246
I - %-?Lot [~
IH%
T _k
IO----OD
I
I~176 I I,--,%~ I
r
i
I
-2
i
-1.0
-0.5
0.0
-2
.0
2x,/Lc(t)
-0.5
0.0
2x,/L_(t)
Figure 7. NS case. Split of the pressure transport term II/ll into a non-linear term, ~NL, a linear term, ~L1, and a Stokes term, ~lSl 9r=3.
Figure 6. NS case. Terms in the budget of the kinetic energy: r=3.
0.06
0.1 0
i
|
|
O.C8 0.04
I
I - - - - - model O----ODNS data I
O.C 6 O.C4 /
0.02
I-----m-~l I IO~-ODNS data -r~'--"~%
O.C 0.00 (
O.C -O.C
.0.02
-O.C -0.r
0.04
il ""
-O.C ,0.06 o
.0
|
I
-0.5
i
I
0.0
i
I
0.5
-0.1
1.0
-1.0
2Xl/Lc(t)
Figure 8. NS case. Model (Eq. 10) for the linear part of the pressure transport piul" r=3.
I
-0.5
|
I
0.0
2xa/Lo(t)
m
I
0.5
1.0
Figure 9. NS case. Model (Eq. 10) for the linear part of the pressure transport 2~-L u 1 9 r ~ 7 . p/3
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
247
Influence of Curvature and Torsion on Turbulent Flow in Helically Coiled Pipes T.J. H/ittl and R. Friedrich Lehrstuhl ffir Fluidmechanik Technische Universitgt Mfinchen Boltzmannstr. 15, 85748 Garching, Germany Fully-developed, statistically steady turbulent flow in straight, curved and helically coiled pipes is studied by means of direct numerical simulation for a Reynolds number of Re~- = 230. The incompressible Navier-Stokes equations, written for orthogonal helical coordinates, are integrated numerically by means of a second order accurate finite volume method. It is shown that pipe curvature which induces a secondary flow has a strong effect on the flow quantities. Turbulence is significantly inhibited by streamline curvature and the flow almost relaminarizes for high values of the curvature parameter (~ = 0.1). The torsion effect is much weaker, compared to the curvature effect. Nevertheless it cannot be neglected. It influences the secondary flow and leads to a small increase in turbulent kinetic energy and dissipation rate. 1. I n t r o d u c t i o n Turbulent flow in pipes is of great importance in industrial applications in many branches of engineering. Heat exchangers, chemical reactors, exhaust gas ducts of engines, or any kind of pipelines, tubes and conduits transporting gases and liquids consist of straight, curved and coiled pipes. Due to the simple geometry, the turbulent flow in straight pipes has been investigated in detail by means of direct numerical simulation, see [12,5,6,13]. Curved or helically coiled pipes are frequently used in practice in order to save space, to satisfy geometric requirements or to take benefit of the characteristics of the induced secondary flow (influence on heat and mass transfer). The direct numerical simulation of fully developed turbulent flow of an incompressible, newtonian fluid in straight, curved and helically coiled pipes with circular cross section, constant curvature ~ and torsion ~- is the subject of the present investigation. Although the systematic theoretical and experimental investigation of flow in pipes with curvature and torsion is just of recent origin, this flow configuration has always been classified as even more complex than the flow through straight ducts. Until now only few investigations have been made to predict the turbulent flow in curved or coiled pipes. Boersma and Nieuwstadt performed a DNS of fully developed turbulent flow in a toroidal pipe for Re~ = Ru~/u = 230,~ = 0.1, see [3,4], and large-eddy simulations (LES) for higher
248 Reynolds numbers, [1,2].
2. G e o m e t r y a n d g o v e r n i n g e q u a t i o n s A helical pipe is constructed by winding a pipe of radius R around a cylinder of radius 1). With the pitch ps, defined by the increase in elevation per revolution of coils hg = 27rps, the curvature ~ and the torsion T of the helical pipe axis can be calculated from
(ra- R) (see fig.
-
ra
and
+
T
--
Ps
(i)
(d +
/ i
/
/x " x2 ,
Figure 1. Helically coiled pipe
1
Figure 2. Description of the orthogonal helical (s, r, 0)-coordinate system, as introduced by Germano [7,8].
As introduced by Germano [7,8], an orthogonal helical coordinate system can be established with respect to the master Cartesian coordinate system. By using the helical coordinates s for the axial direction, r for the radial direction and 0 for the circumferential direction, the Navier-Stokes equations in helical coordinates have been derived in [9].
3. N u m e r i c a l m e t h o d and b o u n d a r y c o n d i t i o n s A finite volume method on staggered grids is used to integrate the governing equations. It leads to central differences of second order accuracy for the mass and momentum fluxes across the cell faces. A semi-implicit time-integration scheme treats all those convection and diffusion terms implicitly which contain derivatives in 0-direction. The remaining convection terms are integrated in time with a second order accurate leapfrog-step. An averaging step all the 50 time steps avoids possible 2At-oscillations. Diffusive terms with derivatives in s- and r-directions are treated with a first-order Euler backward step. It has been proven that this kind of second order accurate finite-volume method and the Euler-leapfrog time integration scheme is capable to predict turbulent pipe flow, [6,12].
249 Table 1 Geometrical parameters of the pipe configurations: case g 7 ra Ps DP 0.0 0.0 ~ 0.0 DTSC 0.01 0.0 100.0 0.0 DT 0.1 0.0 10.0 0.0 DKH 0.1 0.0275 9.297 2.557 DH 0.1 0.055 7.678 4.223 DXH 0.1 0.11 4.525 4.977 DXXH 0.1 0.165 2.686 4.433 DHSC 0.01 0.006875 67.90 46.68
hg = 27rps 0.0 0.0 0.0 16.06 26.53 31.27 27.85 293.3
hg/ra 0.0 0.0 0.0 1.72 4.45 6.91 10.39 4.31
The size of the time step is selected according to a linear stability argument. The use of a projection step leads to a 3D Poisson problem for the pressure correction, which is solved by a Conjugate Gradient method for unsymmetric matrices, [9]. Boundary conditions are required for all boundaries of the computational domain. At the walls impermeability and no-slip boundary conditions are realized. Velocity components which are needed on the pipe axis, are obtained by interpolation across the axis. In the circumferential direction all variables are periodic by definition. In axial direction periodic boundary conditions are used, too. For helically coiled pipes this means that the rotation of the coordinate system along the pipe axis must be taken into account and the perfect matching of the cells at the in- and out-flow boundaries has to be ensured by choosing a suitable combination of axial length and number of grid points in 0-direction (see [11,10]). The flow is driven in the axial direction by a pressure gradient A P / A s , which must balance the viscous friction along the pipe wall.
4. C o m p u t a t i o n a l d e t a i l s On order to analyse the influence of curvature ~ and torsion v on the turbulence structure several pipe configurations are computed for the same Reynolds numbers: one straight pipe DP, two toroidal pipes DTSC, DT and five helically coiled pipes DKH, DH, DXH, DXXH, DHSC, see Table 1. Note that all parameters have been non-dimensionalized by R. The toroidal case D T has also been computed by Boersma and Nieuwstadt [3,4]. The Reynolds number Re~ based on friction velocity and pipe radius has been taken as 230 for all cases. Therefore the Reynolds number Reb = 2ubR/u, based on bulk velocity Ub and pipe diameter 2R, varies between 5576 and 6926. Table 2 shows the flow parameters of all cases: Besides the Reynolds numbers Re~ and Reb, the Dean numbers De~ - v/-~Re,, Deb = x/~Re~, the Germano numbers Gn~ = vRe~, Gnb = rReb and the friction factors A are shown, with:
(2)
250
Table 2 Flow parameters: case Re~Reb
De~
Deb
Gnu-
Gnb
Ub
~
Ub DP
DP 230.0 DTSC 230.0 DT 230.0 DKH 230.0 DH 230.0 DXH 230.0 DXXH230.0 DHSC 230.0
0.0 23.00 72.73 72.73 72.73 72.73 72.73 23.00
0.0 692.60 1779.2 1779.6 1779.1 1773.8 1763.3 690.62
0.0 0.0 0.0 6.325 12.65 25.3 37.95 1.58
0.0 0.0 0.0 154.76 309.44 617.01 920.04 47.480
14.8095 15.0565 12.2313 12.2338 12.2306 12.1939 12.1218 15.0135
0.036476 0.035289 0.053474 0.053452 0.053480 0.053803 0.054445 0.035492
1.0 1.01668 0.82591 0.82608 0.82586 0.82338 0.81852 1.01377
6812.4 6926.0 5626.4 5627.5 5626.1 5609.2 5576.0 6906.2
u~
Table 3 P a r a m e t e r s of the numerical grids used:
i s 0 r
li 15.23196 2~r 1.0
ni 256 180 70
hi 1 + ~r sin(0 - ~-s) r 1
h i A Zmin "+ 12.33 0.O94 1.038
hiAi+a~ 15.07 8.01 5.38
For all simulations the same length of the computational domain li and the same n u m b e r of grid points have been taken. The grid has ns - 256, no - 180 and nr - 70 points in axial, circumferential and radial direction, respectively. The grid is equidistant in s- and 0-directions. In the wall-normal direction, close to the wall a clustering of grid points is achieved according to
A r = R
tanh(Tk) tanh (Tnr) '
k-1 '
2,3 . . . , n r '
,
7-0.0210604
(3)
The distribution ensures that 5 points are below z + - ( R - r) + - 5 and that the first point is located below z + - ( R - r) + - 1. In table 3 li, ni and the m i n i m u m and m a x i m u m mesh size hiAi + in wall units are listed. The simulations were started from a 3D turbulent data set for straight pipe flow and Re~ - 180. 5. N u m e r i c a l
Results
A secondary flow is induced by centrifugal forces, when flow passes through a curved pipe. Starting from the pipe core region the fluid is driven outward towards the wall where it bifurcates feeding two recirculation zones with a flow along the pipe walls. A pressure difference between the inner wall and the outer wall is built up and the m a x i m u m axial velocity moves from the pipe's center to the outer wall. This behaviour is well known for laminar flow, [11,10], and it is also shown by the mean quantities of turbulent flow through a curved pipe. With increasing value of the curvature ~ the m a x i m u m axial velocity {us) moves from the axis (r - 0) to the outer wall. The mean axial velocity profiles
251
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Figure 3. Mean axial velocity component along a horizontal cut (left, with the inner wall at r = - 1 and the outer wall at r = +1) and along a vertical cut (right, with the lower wall at r = - 1 and the upper wall at r = +1): DP, - DTSC, DT, . . . . DKH, DH, . . . . . . DXH, DXXH, . . . . . . DHSC, rq: Experiment, [14].
DTSC
Figure 4. Vector plot of the mean secondary flow in two curved pipes and one helically coiled pipe (s - 3).
are in a good agreement with experiments by Webster and Humphrey for I~eb = 5400 and Deb = 905, [14]. Velocity profiles along vertical cuts become rectilinear between two small local maxima near the upper and lower walls, see Figure S. Near the inner walls the axial velocity (us) is very low and therefore the radial velocity gradient becomes very low there. The mean axial velocity component in helically coiled pipes is similar to that of the corresponding toroidal pipes. Only the maximum near the outer wall becomes lower with increasing value of the torsion r. The mean velocity components in radial (u~) and circumferential (uo) direction are not zero like in straight pipe flow, because a secondary flow is induced by centrifugal forces. Two symmetrical recirculation zones are established for toroidal pipe flow. With increasing curvature, the secondary flow becomes stronger and the circumferential veloc-
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2 ~ + U o2, r ~ ~ + U~,~m 2 ~) along a horizontal cut Figure 5. Turbulent kinetic energy k - ~1 ( U~,rm (left) and along a vertical cut (right)" For legend see Figure 3. 1
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Figure 6 9 Reynolds shear stress i' U s"U o"2' along a horizontal cut (left) and along a vertical cut (right)" For legend see Figure 3.
ity component near the upper and lower walls becomes very high, see Figure 4. Due to this secondary flow, fluid with high axial velocity near the upper and lower wall is transported to the left which is shown by the local maxima of the axial velocity there, see Figure 3. Two secondary flow cells rotate the fluid at the left hand side of the cross section and the slow fluid is forced to the center. The vector plot of the mean secondary flow shows two strong vortex like structures at the left hand side, that stretch far to the right hand side of the cross section. The mean pressure distribution (not shown here) is almost linear over the whole cross section with the minimum at the inner wall and the m a x i m u m at the outer wall. No local pressure minima are observed at the centers of the recirculation cells and therefore these cells cannot be called vortices. Big differences exist between the secondary flows of toroidal and of helical pipe flow. While in the upper half of the cross section a recirculation still exists, the lower half does not show such a zone any more. The lengths of the vectors in the core region of the secondary flow of case DXH are much bigger than for case DT and it can be seen that the intensity of the secondary flow is higher at the lower wall. The flow near the center of the cross
253
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Figure 7. Reynolds shear stress tu~u~) ' " "' along a horizontal cut (left) and along a vertical cut (right)" For legend see Figure 3.
/
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Figure 8. Flatness of the fluctuating velocity components along a horizontal cut" For legend see Figure 3.
section rotates clockwise around a point near the axis and is not driven outward any more. Figure 5 shows the turbulent kinetic energy among a horizontal and a vertical cut. For low curvature the profiles of k only show differences with respect to straight pipe flow in the core region of a cross section and near the inner wall. In strongly curved pipes the turbulent fluctuations are drastically reduced and only some small peaks can be seen near the upper and lower wall. Turbulence is strongly inhibited near the inner and outer walls. With increasing torsion, the turbulent kinetic energy increases. Near the outer wall a local m a x i m u m appears. Near the axis, k can become bigger than for straight pipe flow. The effect of curvature and torsion on turbulent kinetic energy is similar to that on the Reynolds shear stress ( us%), " "' see Figure 7. It can be seen, that for strongly curved pipe flow, the values of lusu~) ' " "' are very low near the inner and the outer wall and some small peaks appear at the upper and lower wall. In contrast to straight pipe flow, the Reynolds shear stresses (u~u~o') and (u~u'~~) do not become zero. (u'~'u~o~) can even get bigger than II II\ u~u~) and it has an extremely big negative peak near the lower wall for strong curvature
254 o.25 -1 -
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Figure 9. Flatness of the fluctuating radial velocity component along a horizontal cut (left) and rms-values of the vorticity component in axial direction along an vertical cut (right)" For legend see Figure 3.
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Figure 10. Rms-values of the vorticity components along vertical cuts" For legend see Figure 3.
and strong torsion, see Figure 6. '~,UoU,~ " "~~ is lower than the other Reynolds shear stresses (not shown here). Figures 8 and 9 (left) show the flatness factors F of the fluctuating velocity components. Near the outer wall F becomes extremely high for big values of the torsion r. This seems to indicate intermittency near the outer wall, which could explain the two different solutions of Boersma's DNS of toroidal pipe flow, [4]. Like the velocity fluctuations, the vorticity fluctuations are reduced by the effect of curvature. Figures 9 (right) and 10 show the rms-values of the vorticity components along vertical cuts. They become low in the center of the pipes and bigger near the walls. With increasing curvature the rms-values of the vorticity components decrease. With increasing torsion, the rms-values of the vorticity components decrease near the upper and increase
255
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Figure 11. Enstrophy (E~) _ 71 (w~,~m~2+ co2 0,~m~ + co~,~ms)2 along a horizontal cut (left) and along a vertical cut (right): For legend see Figure 3.
near the lower wall. aJ0,~ms is the biggest component for flow in pipes. For toroidal and helical pipe flow a double-peak appears near the upper and lower walls, cv~,~msdrops near the wall and co~,~ shows a local maximum near the walls, drops down to a local minimum and rises up to high values at the wall. In the center of the pipe all the rms-values have almost the same values. Finally, the enstrophy which is a measure for the TKE dissipation rate, is shown in Figure 11. With a logarithmic scale the influence of torsion on the enstrophy can be seen. Along a vertical cut the profiles of (E~) for high curvature are almost rectilinear in the core region. Increasing torsion leads to a higher level of enstrophy. Near the upper wall (E~} decreases and near the lower it increases with increasing torsion. Along a horizontal cut, (E~) is no longer rectilinear, but it also increases with increasing torsion.
6.
Conclusions
The effect of curvature and torsion on fully developed turbulent flow in pipes has been studied by means of direct numerical simulation. Due to curvature, a secondary flow is induced and the profiles of the flow quantities differ considerably from those for straight pipe flow. For high values of the curvature parameter (~ = 0.1) turbulence is significantly inhibited and the turbulent kinetic energy becomes much lower than in a straight pipe. In helically coiled pipes, a torsion effect on the flow quantities can be observed. Strong torsion in a geometrical sense (r > 0.01, hg/ra > 1) has, however, a comparatively weak physical effect. While the mean axial velocity component is almost uninfluenced by torsion, the secondary motion becomes stronger and its pattern changes. Turbulent kinetic energy increases slightly with increasing torsion. While the DNS technique has extremely high requirements on CPU time (ca. 3000 h per case on a Fujitsu VPP700), memory (600 MByte RAM) and disk/archive space (150 GByte per case), the simulations yield a valuable data base that helps to test and improve
256 models for LES and statistical simulations. REFERENCES
1. Boersma, B.J. and Nieuwstadt, F.T.M. - Large Eddy simulation of turbulent flow in a curved pipe, In: Tenth s y m p o s i u m on t u r b u l e n t shear flows, The Pennsylvania State University, 1, Poster Session 1, P1-19 - P1-24 (1995). 2. Boersma, B.J. and Nieuwstadt, F.T.M.- Large-Eddy Simulation of Turbulent Flow in a Curved Pipe, Transaction of the ASME, Journal of Fluids Engineering, vol. 118, pp. 248- 254 (1996). 3. Boersma, B.J. and Nieuwstadt, F.T.M. - Non-Unique Solutions in Turbulent Curved Pipe Flow, In: J.-P. Chollet et al. (eds.), Direct and L a r g e - E d d y S i m u l a t i o n II, Kluwer Academic Publishers, pp. 257-266, (1997). 4. Boersma, B.J.- Electromagnetic effects in cylindrical pipe flow, Ph.D. Thesis, Delft University Press, (1997). 5. Eggels, J.G.M., Direct and Large Eddy Simulation of Turbulent flow in a Cylindrical pipe geometry, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands (1994). 6. Eggels, J.G.M., Unger, F., Weiss, M.H., Westerweel, J., Adrian, R.J., Friedrich, R. and Nieuwstadt, F.T.M.- Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment. J. Fluid Mech. 268, 175-209 (1994). 7. Germano, M. - On the effect of torsion on a helical pipe flow., J. Fluid Mech., vol. 125, pp. 1-8 (1982). 8. Germano, M.- The Dean equations extended to a helical pipe flow., J. Fluid Mech., vol. 203, pp. 289-305 (1989). 9. Hfittl, T . J . - S i m u l a t i o n s p r o g r a m m HELIX: Ein F i n i t e - V o l u m e n Verfahren zur LSsung der inkompressiblen 3D-Navier-Stokes Gleichungen ffir Rohrgeo m e t r i e n m i t Krfimmung und Torsion, Lehrstuhl ffir Fluidmechanik, TU Mfinchen, Bericht TUM-FLM-96/29 (1996). 10. Hfittl, T.J. and Friedrich, R. - Fully Developed Laminar Flow in Curved or Helically Coiled Pipes-In: J a h r b u c h 1997 der D e u t s c h e n Gesellschaft ffir Luft- und R a u m f a h r t - L i l i e n t a h l - O b e r t h e.V. ( D G L R ) , Tagungsband "Deutscher Luftu. Raumfahrtkongress 1997, DGLR-Jahrestagung, 14.- 17. Okt. in Mfinchen", Band 2, DGLR-JT97-181, pp. 1203-1210 (1997). 11. Hfittl, T.J., Wagner, C. and Friedrich, R.- Navier Stokes Solutions of Laminar Flows Based on Orthogonal Helical Coordinates-In: N u m e r i c a l m e t h o d s in l a m i n a r and t u r b u l e n t flow, C. Taylor, J. Cross (eds.), Pineridge Press, Swansea UK, Vol. 10, pp. 191-202 (1997). 12. Unger, F.- Numerische Simulation turbulenter RohrstrSmungen, Ph.D. Thesis, Technische UniveritS~t Mfinchen (1994). 13. Wagner, C. - Direkte numerische Simulation turbulenter StrSmungen in einer Rohrerweiterung, Ph.D. Thesis, Technische UniveritSot Mfinchen (1995). 14. Webster, D. R., Humphrey, J. A. C. - Traveling wave instability in helical coil flow, Physics of Fluids, vol. 9, pp. 407-418, (1997).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
257
L a r g e e d d y s i m u l a t i o n s of stirred t a n k flow Jos Derksen and Harry Van den Akker Kramers Laboratorium voor Fysische Technologie, Department of Applied Physics, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands, e-mail: jos @klft.tn.tudelft.nl Large eddy simulations have been performed on the flow in a baffled tank, driven by a disk turbine, at Re=29,000. The Navier-Stokes equations were discretized using a lattice-Boltzmann scheme on a uniform, cubic computational grid with 1803 nodes. The impeller was represented by an adaptive force field, acting on the flow. A conventional Smagorinsky subgrid-scale model with Cs=0.10 was applied. The results on the phase-resolved, average flow field, and on turbulence characteristics are compared with experimental data.
1. I N T R O D U C T I O N Stirred tanks, operated in the turbulent regime, are used in industry as mixing devices. Good predictions of the average flow as well as of turbulence characteristics have some relevance in the optimization of industrial processes. The levels of the turbulent kinetic energy, and its dissipation rate throughout the tank are generally assumed to be important for the small-scale motion and deformation, and therefore for the creation of interfacial area between two chemical species, or for the way particles, or drops, or bubbles, collide and deform. A realistic simulation of the flow system at hand is, however, not an easy task. The flow structures are highly three-dimensional and complex, and cover a wide range of spatial and temporal scales. Furthermore, the flow is inherently time-dependent due to the rotating motion of the impeller with respect to the baffled tank wall. As a result, fine grids, and a computationally efficient means of modeling the revolving impeller have to be applied. The inherent time-dependence of the flow makes a large eddy simulation (I_ES) approach of turbulence modeling favorable over solving the Reynolds averaged Navier-Stokes (RANS) equations in conjunction with a closure model. In LES, the distinction between the resolved and unresolved scales is clearly defined by the size of the grid spacing and the time step. Therefore, the relatively slow, coherent fluctuations, induced by the impeller, can be resolved explicitly. In a RANS simulation, it is less clear which part of the fluctuations (in terms of energy content as well as spectral distribution) is resolved, and which part is represented by the Reynolds stresses. As a result, part of the coherent fluctuations due to blade passage might get mistakenly contained in the Reynolds stresses. Eggels [1] was the first to report on LES in a stirred tank. In his paper, he gives an unprecedented view of the turbulent flow structures in the tank. There was good agreement with phase-averaged experimental data. In the vicinity of the impeller, however, the most relevant flow details can only be studied in a frame of reference rotating with the impeller. This asks for an analysis in terms of phase-resolved data.
258 In the present paper, the simulation procedure as proposed by Eggels [1], is adopted and somewhat refined with respect to modeling the revolving impeller. The procedure consists of a lattice-Boltzmann disretization of the Navier-Stokes equations, a standard Smagorinsky subgridscale model [2], and an adaptive force field technique to mimic the action of the impeller on the fluid [1,3,4]. The lattice-Boltzmann scheme was chosen because of its computational efficiency on parallel computer platforms. Its major disadvantage, at least in the present implementation, is the uniformity of the grid, which inhibits a local grid refinement at high-gradient flow regions (e.g. the volume swept by the impeller). The adaptive forces allow for an efficient means of modeling moving objects in the flow domain. These objects can have arbitrary shape, which makes the method at hand very suitable for studying the effects of geometrical adaptations. This is important for industrial applications aiming at process efficiency improvement. Results for the flow field driven by a Rusht0n turbine will be presented and compared with experimental results from literature.
2. S I M U L A T I O N P R O C E D U R E The lattice-Boltzmann method provides an efficient Navier-Stokes solver. The basic idea is that fluid flow, which is govemed by the laws of conservation of mass and momentum, can be simulated by a many-particle-system obeying the same conservation laws [5]. In the latticeBoltzmann approach to fluid flow, the particles reside on a lattice and are allowed to move from one site to the other during time steps. The evolution of the many-particle system can be written in terms of the lattice-Boltzmann equation [6]
Ni (x -]-.ci,t -~-1)- Ni (x,t)+ Zi (N )
(1)
with Ni the mass of a particle traveling with velocity r +), and F/the collision operator, which depends in a non-linear way on all particles involved in the collision. The symmetry properties of a three-dimensional projection of the face-centered-hyper-cube (FCHC) lattice allow for the simulation of the incompressible Navier-Stokes equations [7]. The primitive flow variables, such as density (p) and momentum concentration (,ou), are related to the particle properties:
10 -" Zi Ni
(2)
Iou a = Z i c i a N i
(3)
The boundary conditions can be stated in terms of reflection rules at the domain boundaries. E.g., a no-slip wall is a wall on which particles bounce back. To represent the revolving impeller, however, we need a way to smoothly model a moving object. For this, the impeller is viewed as a force-field acting on the fluid. The distribution of forces is then (iteratively) calculated in such a way that, at points on the impeller surface, the fluid velocity closely approximates the (prescribed) velocity of the impeller. This can be achieved with a control algorithm, in which the mismatch between the actual velocity and the imposed velocity at a point is opposed by a force.
§ The set of velocities is a discrete set due to the lattice.
259 In the implementation in the computer code, the impeller geometry is defined as a set of M control points rJ n) (j=I...M), where the superscript (n) indicates the moment in time. There is basically no restriction on the position of these points in the flow domain; they do not need to coincide with lattice sites. At these points, we require a velocity equal to wJn)=g2xrJn), with ,(2 the angular velocity of the impeller. First, the deviation d j~n) of the actual flow velocity from the prescribed velocity is determined by a second-order interpolation of the flow velocities at the lattice sites: ) Uk d~.n ) - w j ( n ) - ~ k G k (r:n)(n)
(4)
where the sum is over the lattice sites in the vicinity of rj (n), Gk is an interpolation coefficient, and the velocity vector at lattice site k. The coefficients Gk also serve to distribute the forces that oppose the deviation d~") over the lattice sites:
uk,
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Figure 9.__:..Contribution of the Individual terms in the Non-linear Stress-Strain Relation in the u'32 prediction, Vp=200 rpm, CA=144 ~
288 5. C O N C L U S I O N The non-linear k - e model of Suga has already shown superior performance compared to the standard k - e model in many apphcations [1]. In order to obtain further insight into the capabilities of the model in handhng engine flow problems, it was implemented in two codes and then applied to three engine-like flow geometries. The results were also compared with those from the hnear model, DSM, and LDA measurements in which detailed quantities of velocity and turbulence at some CAs have been reported. Compared to the experimental data, mean velocities predicted by all models are satisfactory. Best results are obtained for the non-hnear model during the exhaust and compression stroke, especially for the locations further downstream of the cylinder. Good agreement with the data during the intake stroke can be obtained, if tangential velocities on all solid walls are considered as zero. In general, the non-hnear model appears to be the best model for predicting the trend of changes in turbulent intensities with better results for the locations away from the cyhnder head and in the jet region. Finally the CPU time required for the non-hnear model appears to be only 1.2 to 1.3 times greater than that for the standard k-e model. This in addition to the better results from the non-linear model encourages its application in more realistic engine-flow problems. Research is underway to investigate the capabihties of the model in more complex geometries using the most recent version of the KIVA computer code [10]. REFERENCES
1. K. Suga, Development and Application of a Cubic Eddy-Viscosity Model of Turbulence, Int. J. Heat and Fluid Flow, Vol. 17 No. 2 1996 108. 2. C.G. Speziale, On Non-Linear k - l and k - e models of Turbulence, J. Fluid Mech. Vol. 178 1987 459. 3. R. Rubinstein, J. M. Barton, Non-Linear Reynolds Stress Models and the Renormalization Group, Phys. Fluids A, Vol. 2 No. 8 1990 1472. 4. A.A. Amsden, KIVA-3: A KIVA Program with Block-Structured Mesh for Complex Geometries, Report LA-12503-MS, Los Alamos National Laboratory(LANL), New Mexico, 1993. 5. C . J . Lea, A. P. Watkins, Differential Stress Modelling of Turbulent Flows in Model Reciprocating Engines, Proc. I. Mech. E., Vol. 211 Part D 1997 59. 6. A.D. Gosman, A. D. MeUing, J. H. Whitelaw, A.P. Watkins, Axisymmetric Flow in a Motored Reciprocating Engine, Heat and Fluid Flow, Vol. 8 No. 1 1978 21. 7. B. Ahmadi-Befrui, C. Arcoumanis, A. F. Bicen, A. D. Gosman, A. Jahanbakhsh, J. H. Whitelaw, Calculations and Measurements of the Flow in a Motored Model Engine and Implications for Open-Chamber, Direct Injection Engines, Three_Dimensional Turbulent Shear Flows, Proc. AIAA/ASME Conf., 1982 1. 8. A.C. Gonzalez, Application of Non-Linear k - e Models to In-Cylinder Flows, MSc Dissertation, Faculty of Technology, University of Manchester, Manchester, 1997. 9. K.Y. Chung, Modelling Steady Turbulent I.C. Engine In-Flow, Final Year Report, UMIST, Manchester, 1998. 10. A. A. Amsden, KIVA-3V: A Block-Structured KIVA Program for Engines with Vertical or Canted Valves, Report LA-13313-MS, LANL, New Mexico, 1997.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
289
A spectral closure for inhomogeneous turbulence applied to the computation of an engine related flow S. Parpais*, H. Touil, J.-P. Bertoglio and M. Michard* LMFA, UMR CNRS 5509 - Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69 130 Ecully, France A spectral model for inhomogeneous turbulence ("S.C.I.T." for Simplified Closure for Inhomogeneous Turbulence) recently developed and implemented in a Navier Stokes solver [1,2], is applied to the prediction of a flow representative of the intake stroke in a car engine. The results are compared to experimental data [3] as well as with the predictions of a classical k - c model and of a Reynolds stress closure. 1. I N T R O D U C T I O N Mixing and combustion processes in internal combustion engines are strongly connected to the properties of the turbulent field at the end of the compression stroke. The turbulent flow at the time of injection in diesel engines or at the time of ignition in spark ignition engines is characterized by a complex interaction between a small scale fully developed turbulent field and large vortical structures induced by the geometry. The prediction of this type of flow remains a challenging problem for turbulence models. The compression stroke in itself induces specific phenomena which are difficult to tackle in the framework of turbulence closures (e.g. the compression of a tumble flow), but it is also necessary to predict accurately the flow before the compression, that is to say to describe precisely the intake stroke. This is the problem addressed in the present paper. Engineering computations of turbulent flows are generally relying on turbulence models, known as one point-closures, namely the (k, c) models, and the second order or Reynolds stress models ( ~ , ~). One of the weaknesses of these models is related to the fact that an information on a turbulent length-scale is required to predict the turbulent kinetic energy, and that this information has to be provided by the use of an e equation. During the last few years, we started developing a new type of model, called S.C.I.T. (for Simplified Closure for Inhomogeneous Turbulence). This model is based on a statistical spectral approach, that is to say on a description of turbulence by correlations at two points. It is known that two-point models directly take into account information on different length scales, up to the Kolmogorov scale. Consequently no e equation is required. Before being applied to real flows, the complex formulations of two-point closures for inhomogeneous turbulence must however be simplified. This is the approach followed when developing the S.C.I.T. model. The main feature of this model is that the transport equation for the *also St~ M~traflu, 64 Chemin des Mouilles, 69 130 Ecully, France
290 turbulent kinetic energy spectrum E(K, Y() is solved at each wave-number. Recently the S.C.I.T. model was implemented in a Reynolds Averaged Navier Stokes code, and applied to inhomogeneous and wall bounded flows such as diffusive turbulence, boundary layers, pipe flows, flow over a backward facing step and wall jet [4,2,5]. This model can be used with the same grid and numerical scheme as a k - e model. In terms of computational cost, it requires an increase in c.p.u, time of a factor approximatively equal to 5 compared to a basic k - ~ model. The model is described in section 2. In section 3 it is applied to the prediction of the flow corresponding to the experiment of Belmabrouk et al. [3] originally devised to study, in a simplified geometry, the intake stroke in a car engine. The geometry under consideration is axisymmetrical and consists in a cylinder head with a unique axial valve. 2. T R A N S P O R T E Q U A T I O N FOR T H E S P E C T R U M
The basic equation for the present model is the transport equation for the turbulent kinetic energy spectrum E(K, X). This equation reads:
~Y) = P(K,.;~) + T(K,X) + D(K,X,)
(1)
It is obtained from the equation for the double velocity correlation at two points A and B, by using a central point description X -" - - XA +2 X n ~ This is then Fourier transformed with respect to the separation variable/~ = XA - X'B and integrated over spherical shells of radius K and finally, index contracted, in equation (1), P ( K , f ) is the production of turbulent kinetic energy by the mean velocity gradient. T(K,X) is the non-linear transfer term which takes into account the energy cascade from the large eddies to the small scales. D(K, X) is the inhomogeneous transport term corresponding to transport by triple velocity correlations as well as transport by pressure-velocity correlations. This equation is used together with the mean flow equation :
_oK
0-7 +
oxj =
--
p ox,
(2)
oxj
2.1. N o n - l i n e a r t r a n s f e r t e r m The closed form for the non-linear transfer term is identical to the classical expression resulting from the Eddy Damped Quasi-Normal Markovian theory [6]"
" = f O :q, xy + ? E T(K,Y,)
K E(Q, .~'~)- Q2E(K, X)} dQdR
(3)
This is the form obtained in the case of homogeneous isotropic turbulence, but use of this expression for inhomogeneous turbulence was justified by Laporta [7]. in equation (3) A(K, Q, R) denotes the integration domain as the part of the Q, R plane where the wave-vectors K, Q and/~ can form a triangle and x, y, z are the cosines of angles in this triangle. OKQRis the usual E.D.Q.N.M. characteristic time[8].
291 2.2. P r o d u c t i o n t e r m The production term in the E equation is:
P(K, f, ) =-gii(K,.V,) Ox, ff~j
(4)
where 7~j is the spectrum of the Reynolds stress tensor u~:uj-. ~ij is modeled using a spectral extension of the approach proposed by Reynolds [9], Shih and Lumle)' [10]. ~ij(K,X)is expressed in terms of E(K, f,), Ox~-~j and K. This leads
to: 2.
=
5~,, -
3"
ox., +
)
-.
:::,:,:,:,:,:,:,,:,,,E(K,X) al ~/E 3E(K, .Y) + a2.v/SzmS,m + f ltmf~,m ,
:, .....
,~,:
,,,
.....
:::,:
.......
(5)
with
S~j = (ax,~ + Ox~'~)/2
and
ft,j
=
(Ox,~jj -
axj~.')/2
Values for al and a2 for incompressible flows are found by comparisons with the experiments of Tavoularis & Karnik [11]: al = 0.40 and a2 = 3. Replacing (5) in (4) gives a closed form for the production term in (1). 2.3. I n h o m o g e n e o u s t r a n s p o r t t e r m The inhomogeneous transport term for the turbulent kinetic energy spectrum is modeled introducing a diffusive expression:
D(F., X) = Ox, ui (X )cOx., . . with yb(,.,7)
foo~
E(K, f()dK ..... This form is an extension of the a~v/K3E(K":~'i"+ a~V/SiiS,j + f2,jf~,j
expression originally proposed by Besnard et al [12], in which the mean velocity term is added. The effect of the mean velocity term is to reduce the transport when a mean velocity gradient is present. Such a reduction was shown to occur in the case of a solid body rotation [13] and in the case of a plane strain [14]. In the case of a shear flow, this effect is usually referred to as shear sheltering, a~ and a~ are chosen respectively equal to
1.66 and 3/~/~ [I].
2.4. Wall effects To take into account wall effects, a low wave-number cut-off Kw(~Y) is introduced, as proposed in Bertoglio and Jeandel [15]. The energy spectrum E(K, ~) associated with wave-numbers smaller than the cut-off is forced to zero. The wave-number cut-off is
0.5
where 5(~Y) is a distance from the wall. The value 0.5 is found for compatibility with the logarithmic-law.
292
2.5. Mean flow equation and usual one point quantities The Reynolds stress tensor:
uiuj(.,~) = o ~ij(K, X ) dh" is expressed by wave-number integration of (5). It is found: 1----
--.-
is the relation used to close the Reynolds averaged Navier Stokes equation (2), where u~(,Y) is a turbulent viscosity defined as:
This
l/t (.~7)
E(K, 2 ) dK
f
a2 /s js u
3
s
The one-point quantities, such as the turbulent kinetic energy, the dissipation rate, etc..., are obtained by wave-number integration of the spectra. So one gets the turbulent kinetic energy: q2 = /2oo E(K,..Y) dK k(f, ) = -~ For the dissipation rate : e(s
_ ~oo 2uh'~E( K, 2 ) dh"
The model can also directly predict various length scales (e.g. integral , Taylor and Kolmogorov scales) without having to introduce assumptions on the turbulent spectrum. Of interest for the comparison with experiment proposed in the next section is the integral length scale defined as :
.-, 3YI foo E ( K , X ) n(x)=--~jo '~
K
'
dK "
3. RESULTS 3.1. Presentation of the computation In the present paper, the spectral model is applied to the computation of the turbulent field in an axisymmetrical steady flow which reproduces some basic features of the flow in a real engine during the intake stroke. A continuous inflow is imposed and the valve lift is kept constant. The right-hand vertical side of the domain is open. The geometry of the domain is shown in figure 1, as well as the non structured grid on which the equation for the mean velocity and for the energy spectrum are solved using a finite element technique. As shown in figure 1, the x reference, x = 0, is the left-hand vertical side of tile domain, x = 20mm is a particular section that will be studied later. The different physical parameters have been chosen such as to simulate the experiment made by Belmabrouk and al. [3]. The flow structure downstream of the cylinder head reproduces the basic features of the flow in real engines during the intake stroke (high velocity valve
293 jet, recirculating zone behind the valve,..). The valve lift is l Omm, the valve diameter is 30ram and the cylinder bore is 120mm. The Reynolds number built on the inlet duct velocity is 92 400. At each grid point of the computational domain represented in figure 1 the spectra are discretized on 40 wave-numbers.
Figure 1. Computational domain- 12692 nodes. 3.2. B o u n d a r y c o n d i t i o n s The boundary conditions at the wall are computed at a distance 5b from the wall. Since the model can take into account low Reynolds number effects [1], there is no need to check whether the first point of the grid is in the logarithmic region or not. For the mean velocity, a logarithmic profile is assumed for high values of the ratio y+ - 5buf/~, :
L
"l
= •247 ui g
+ c
where uf is the skin friction velocity deduced from the mean velocity gradient on the first cell of the grid. For lower values of y+, we use : U
1
+
-- 7.4-tanh(C3y ) ~f ~'3 with C3 = 0.0722 for a = 0.414 and C = 5.45. This expression leads to the usual viscous sublayer profile for very small y+, and to a matching with the log law. For the spectrum, its boundary values are obtained by solving, with a time marching method, the equation:
0 = -2vK2E(I 1,000 a complex oscillatory flow is found. Ghaddar et al. [2] and Patera and Mikic [3] make predictions, illustrating the increased heat transport caused by the oscillatory nature of the flow. Nigen and Amon [4], Amon [5] and Nigen and Amon [6] further extend the above to include realistic electronic component thermal modelling. The modelling of internal laminar oscillatory flows has also been addressed by Pulicani et al. [7]. There has been less consideration of the economical prediction of turbulent, internal, oscillatory flows in complex
300 engineering geometries. This will be considered here for the system shown in Figure 1, which is representative of a central processor unit.
~
F
a
n
i
GrillI
t
Cut-out
Fan2 -------"
//Grill
Row outletinlet/-'-- ~ l y
~
I ~"/
]/l
. I / Grill
z
Row
inlet/outlet
Figure 1. Geometry to be considered.
2. NUMERICAL METHOD 2.1 Governing Equations Conservation of momentum for the system considered can be expressed as
OUi + O U i U j Ot Oxj
10p
90xi
t g 0 ( OU i ~ U j + 9 0 x j ~ , O x j + Ox i )
o u;u;.
(1)
Oxj
m where U is the mean fluid velocity, u' is the averaged fluctuating component, 9 density, gt viscosity, p static pressure, t time and x the spatial co-ordinate. The corresponding continuity equation is
0 Uj _ 0 OXg The kinetic energy k of the turbulence is evaluated here using
(2)
301
Ok OUjk "}-~ Ot Oxj
=
1 0 (( ~ 9 3xj
Jr"
l't-~k! O-~xjl '-7~-"OUi-a
(3)
-- UiUj O X j
where (3"k is a constant. Three turbulence model variants are considered. For the rate of dissipation of turbulence kinetic energy e in the above is defined using
k-I model,
the
k~ = ~
(4)
L where lt = C~oy(1-neAy+/C~"),
y
is a normal wall distance and y
+
=
ypk 1/2 C.1/4 /~t
is the
corresponding dimensionless distance. For the k-~ model the following ~ equation is used
Oe at
+
OUj e . 1 0 ~ Oxj = 9 Oxj
~tt OG ~t +
~. ~
og i
- G, -; u, uj axj -
Cc 2
k
(5)
Once the dissipation scales have been found, the turbulent viscosities can be expressed as
(6)
~'t ---~D Cla lla k i/2
where l, =C,0y (1
Ft,= 9 C.
-
ne&Y*/C~/4),and
k 2
(7)
for the k-l and k-e models, respectively. The k-e model is standard [8] using computationally economical logarithmic functions at walls. This model is popular for electronics design. For the k-I model, when n - 1, the length scales l~ and l~ have near wall damping functions and again the model is standard [9]. For comparison, predictions are also made with n - 0 and logarithmic wall functions [8]. The following standard constants are used: (Yk = 1, (~ = 1.3, A~ = 0.263, A~, = 0.016, C~0 = 2.4, C~l = 1.44, C~2 = 1.92, C,0 = 2.4, C, = 0.09. 2.2 C a l c u l a t i o n
of Normal
Wall Distances
For cluttered geometries y can be calculated using a method derived by Spalding [10]. This involves the solution of a Poisson equation of the following form,
axj where S - -1. Here y is related to L using a slightly modified expression to Spalding's presented by Tucker [11] and given below
302
y = -
+
+2
(9)
Equation (9) has negative and positive roots corresponding to nearest (Ymin) and furthest (Ymax) normal wall distances. Therefore, the distance between two surfaces is equal to Y~n + Ym~x" For the k-I model y = Ymin is used in the length scale equations. Solution of the Poisson and fluid flow equations is accelerated using the multilevel scheme described later.
2.3 General Numerical Features The governing equations are discretized using the structured staggered grid technique outlined by Patankar [12]. Upwind, HYBRID [13] and the second order CONDIF [14] convective term treatments are all compared in the results. The pressure field is computed using the SIMPLE [15] method and for transient predictions an implicit time scheme is used. Solution of the governing equations is accelerated using non-linear multi-level convergence [ 16] with a fixed V cycle. Turbulence equations can sometimes show poor convergence when subjected to multilevel convergence acceleration [17]. Therefore, here they are solved on the finest grid level, turbulence properties being restricted onto coarser levels. 2.4 Evaluation of Turbulence Intensities The flow to be considered is unsteady. Consequently, the measured instantaneous resolved
velocity q =x/u 2 + v 2 + w 2 consists of Q, the time averaged velocity and fluctuations due to turbulence q' and unsteadiness q" q = Q + q' + q"
(10)
Measurements of turbulence intensity to be compared with are not corrected for flow unsteadiness. These can be expressed as
To enable comparisons with measurements predicted values of stochastically by expressing
q = Q + N ~/2~3 k + q"
Ti' can be evaluated
(12)
where N is a Gaussian random number with a standard deviation of unity and ~ = ~]2k/3. Following [18] the concept of an eddy of size l is used where
303
l
=
C~/4k3/2
(I3)
~3
l/q' along with the approximate time for an ed0y to traverse a measurement probe t p = l/q. It then follows that the maximum eddy/pr'obe An eddy life time tr is defined using
te =
interaction time Atmax is given by At max = min(te, t p )
(: ~: i t"
To approximate the nature of the measured turbulence and hence numerically evaluate 2~ , N is re-computed at intervals of Atmax = min(te,tp).
2.5 Boundary Conditions At inflow boundaries the total pressure is fixed. At flow outlets the pressure is fixed arid the gradients of all other variables set to zero. Grills are modelled using loss terms of the fenn
KpU] / 2. For the present predictions K = 1, which is a reasonable approximation [~9i As a sensitivity check K is doubled and found to have no significant effect on results. Fans 1 and 2 are modelled using quadratic momentum sources based on manuiacturers characteristic curves. The sensitivity of predictions to assumed values of i~] a~ flow inlets is tested by varying Ti between 0 and 10 %. Profiles presented are ft~tmd i~sr to this variation. For the wall distance equation, at solid walls L - 0 and at flow boundaries the condition c3L/c3y = 0 is used. Grids with between 99 x 89 x 37 and 105 x 97 x 51 control volumes in the x, y and z directio~s~ respectiwly are risen. T~c~se a~'~ c~,~,~;tr~cted such that y a v e ,,t,~md z and 15 tb~ the low a]~.d hi!..,!~Reyv.~lds m ~ b e r mrbuIe~ce models, respectively. For unsteady predictions time steps of At = 0.001 s are used. 3. DISCUSSION OF R E S U L T S Figure 2 compares the drop in mass residual against number of iterations when both single and multilevel convergence is used with the diffusive upwind scheme. $
31.4.6
~
Multilevel convergence .~ - 2 . 0 -
..
~ -2.5-
- ....
\.
Single-level convergence
.....
~-3.0-
,2 |
9......
-
-s.s . . . . . . . . .
9
I
.~
.
-~-
I 9 ..
-4.o-
I
I _iT,-
I "...... ..
! ... .........
.4.s -s.o
!
o
500 Number
,o'oo
9
I
............ .
,5'0o
of fine grid iterations
I I I
t
'
y x
Figure 2. Multi-level convergence acceleration
Figure 3, Regions where comparisons are made.
ito
304 The multi-level convergence gives around a factor of a half CPU time saving. Because of geometry changes between grid levels, the mass residual could not be reduced much below 3 x 10-4 kg/s unless the solution is switched to a single grid. It can be seen the convergence path is smooth during multilevel to single level solution transitions. When less diffusive convective schemes are used, most predictions to be shown here have significant flow unsteadiness where relatively small time steps are required. Generally, for such flows multilevel convergence can be inefficient and so single grid convergence is used, the multilevel convergence being used to provide good starting conditions.
3.1 Spatial Velocity Variations Comparisons are made with measurements along the lines 1-7 shown in Figure 3. Table 2 establishes the locations of these lines. Figures 4 and 5 show the variation of the x velocity component U at locations 1-7 for the k-I and CONDIF schemes when using wall functions. The predicted flow is unsteady. The full lines represent the time averaged predictions. Predicted temporal velocity extremes are given by the dashed lines. The solid black and open circles represent the time averaged Pitot static tube and hot wire anemometry measurements, respectively. Encouragingly, except for regions 1 and 2 the predictions lay between the hot wire and Pitot static tube measurements. The accuracy of the measurements is expected to be no better than + 25 % for this low speed flow. Predictions for the low Reynolds number k-I model are similar to those for the high Reynolds number model. Line
X 0.533 0.373 0.413 0.413 0.373 0.413 0.373 Table 2 Profile
1 2 3 4 5 6 7
10~
...." ,,"''.
0.0
0.2
"...... "....... 0.4
Z
0.6
0.8
Z
Y 0.734 0.734
0.060 0.100 0.575 0.962 0.914 co-ordinates
~. -1.0.5.0 1.0
0.0
0.0
~ 0.2
......
' 0 o."" 0.4
Z
. 0.6
"....... 0.8
1.0
1
2
.I
-2
-
=--"--"__"._.
o.,
0.0
0.2
0.4
Z
0.6
0.8
1.0
Figure 4. Variation of U with Z in regions 1,2 and 7.
305 3.3 Spatial T' i Variations Figures 7 and 8 show spatial T' i variations. The k-l model values are stochastically evaluated. The dotted and dashed lines represent the upwind and CONDIF predictions, respectively. The k-e model significantly over-predicts turbulence intensities and these results are not shown. As can be seen it appears the k-I model under predicts T'i. However, this error may be due to
. -0.95 t ~" -1.0 -1.05
1.Io
015
o.o
115 t (s)
210
215
3.0
. -o.95 Co) ~
-1.0 -1.05
6
8
10
12
14
16
t (s)
18
20
22
24
9
-0.16
(c) ~"
-0.18
6
8
10
12
14
16 t (s)
18
20
22
24
0.06 "~ ~,
0.0
(d)
-0.05 6
8
10
12
14
16
18
20
22
24
t (S)
Figure 6. shows velocity time traces for point 1.
Point
Q (m/s)
q"ax,, (m/s)
q"ax,m (m/s)
fn (l/s)
fm (I/s)
........
1 0.99 0.063 0.33 0.33 0.50 2 1.20 0.069 0.36 0.33 0.50 3 1.50 0.100 0.17 0.33 0.80 4 2.65 0.365 0.17 0.33 0.75 5 2.23 0.285 0.17 0.33 0.65 6 1.48 0.100 0.24 0.33 0.10 Table 3. Comparison amplitudes and frequencies at different points inaccurate modelling of the temporal flow behaviour. To check if this is the case, the predicted amplitude of the flow unsteadiness is approximately scaled to match the measured values. These scaled results are represented by the full lines. As can be seen agreement is considerably improved.
306 -4 . -3-
::)
9--.
(3)
3
~~
.
.
.
fOOWoooo
-1-
o- I 0.0
I 0.2
I 0.4
I 0.6
I 0.8
1.0
.
I 0.8
0.7
I 0.9
Y
21
(6)
-2-00000000
9 9 0000~ ".... ...
9 ..................
21 0.7
O--
I 0.8
I 0.9
1.0
0.0
0.2
0 4
0.6
0.8
Y
Figure 5. Variation of U with Y in regions 3-6.
3.2 Temporal Velocity Variations Temporal velocity variations are considered in detail for six points on lines 1-6. Points 1, 2 and 7 are located at Z = 0.562, 0.562 and 0.895, respectively. Similarly, points 3-6 are at X = 0.300, 0.914, 0.914 and 0.300. Figure 6 shows velocity time traces for point 1 when using the k-I model with wall functions. Figure 6a is for the Upwind scheme, U velocity component. Frames b to d give the HYBRID scheme U, V and W velocity components. There is no detectable unsteadiness in the diffusive upwind scheme trace and a danger of using first order schemes is clearly illustrated. In channel regions, the HYBRID scheme, will tend to second order accuracy in the cross stream direction reducing false diffusion. Consequently, an oscillatory flow is predicted as shown in flames b-d. Use of the CONDIF scheme and also wall damping functions does not greatly alter the unsteady flow behaviour shown. For the k-~ model, k and ~t are over predicted and, incorrectly, as for the Upwind scheme, the flow becomes steady. Because of the low Reynolds numbers for this flow it is found impractical to position first off-wall grid nodes in the preferred 30 < y+ < 100, range when using wall functions. Importantly, the k-~ model results proved highly sensitive to near wall y+ values. Around 20 % variations were found in channel region, centre line k values for different node positions in the 0.1 < y+ < 15 zone. As y+ became smaller k increased. The k-I model with its constrained length scale did not suffer from this sensitivity. Table 3 gives the following approximate values for points 1-6: Q,, the predicted local mean resolved flow velocity, q'~ax,n and qmax,mthe dominant predicted and measured amplitudes due to unsteadiness and the associated predicted and measured frequencies f~ and fro. As can be seen there are significant differences between the predicted and measured frequencies and amplitudes. These are perhaps to be expected. The temporal behaviour would be best modelled using a spectral technique.
307 6O 50
(1)]
40
~--~Ooooo ~
[-:-- 30
... P "" ""
oo 0
I
_1
60 50 40 t ~-- 30
(2)
20 10 0
0 0
0.2
0.4
Z
0.6
0.8
1
60 50
0
0.2
0.4
Z
0.6
0.8
(7)
40
o o
20 10
0
'
~
~:
o
0 0
0.2
0.4
0.6
Z
0.8
Figure 7. Variation of Ti' with Z in regions 1,2 and 7. 60 50 40 ~-- 30 20 10 0
6o
4o
~-- 3o
l, , i
201100 0
0.2
0.4
Y
0.6
60 50 ~-- 30 20 10 0 0.65
0.8
0.65
(s)l
o
I
1
~;
~'-''"," / ~ ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 0.75
0.85 Y
0.95
60 50
(6)
~- 30 20
Y
0.85
10 0 0
,
,
,
0.2
0.4 Y
0.6
0.8
Figure 8. Variation of Ti' with Y in regions 3-6. 4. CONCLUSIONS For complex geometries and transitional Reynolds numbers, oscillatory turbulent flows can arise. The flow is then a superposition of fluctuations due to unsteadiness and also turbulence. Directly measured un-filtered turbulence intensities will contain an unsteady flow component. Comparisons are made with un-filtered measured total intensities using a stochastically based time history reconstruction technique. When using logarithmic wall functions in the prediction of complex geometry, transitional flows, imposition of the preferred (30 < y+ < 100) near wall
308 grid distribution can be impractical. For the k-~ model this results in over prediction of k and damping of flow unsteadiness. The high Reynolds number k-I model's constrained length scale diminishes the near wall grid sensitivity giving similar results to those for the low Reynolds number variant. Consequently, when higher order connective term treatments are used (for the upwind scheme flow unsteadiness is damped out) flow unsteadiness is not suppressed. However, predicted amplitudes and frequencies show significant differences when compared to measurements. The temporal flow behaviour would be better captured using spectral techniques. If the predicted flow unsteadiness amplitudes are scaled to match the measured and incorporated into the stochastic reconstruction technique, encouraging agreement is found with total intensity measurements. The k-~ model is not recommended for use in systems similar to that considered her. However, multi-level convergence acceleration gives significant time savings for steady flows and the novel differential equation based wall distance algorithm is effective. ACKNOWLEDGEMENTS I am grateful to acknowledge the support of the Engineering and Physical Sciences Research Council (grant number GRJL05600). I would also like to thank Dr Hector Iacovides for his countless helpful discussions and also Dr Zhiqiu PAN. REFERENCES
[1 ] N. K. Ghaddar, K. Z. Korczak, B. B. Mikic and A. T. Patera, J. Fluid Mech, 163 (1986) 99. [2] N. K. Ghaddar, M. Magen, B. B. Mikic and A. T. Patera, J. Fluid Mech, 168 (1986) 541. [3] A. T. Patera and B. B. Mikic, Int. J. Heat Mass Transfer, 29, No. 8, (1986) 1127. [4] J. S. Nigen and C. H. Amon, ASME J. of Fluids Engineering, 116 (1994) 499. [5] C. H. Amon, J. of Thermophysics and Heat Transfer, 9, No. 2 (1995) 247. [6] J. S. Nigen and C. H. Amon, Int. J. Heat Mass Transfer, 38, No. 9, (1995) 1565. [7] J. P. Pulicani, E. Crespo Del Arco, A. Randriamampianina, P. Bontoux and R. Peyret, Int. J. for Num. Meths in Fluids, 10 (1990) 481. [8] B. E. Launder and Spalding, D. B., Comput. Meth. Appl. Mech. Eng., 3 (1974) 269. [9] M. Wolfshtein, Int. J. Heat Mass Transfer, 12 (1969) 301. [ 10] D. B. Spalding, Proc. 10th Int. Heat Transfer Conf., Brighton, UK, (1994). [ 11 ] P. G. Tucker, 'Assessment of geometric multilevel convergence and a wall distance method for flows with multiple intemal boundaries', Accepted for Publication in Applied Mathematical Modelling (1998). [12] S. V. Patankar, Numerical heat transfer and fluid flow, Hemisphere, New York, 1980. [13] D. B. Spalding, Int. J. Numer. Meth. Eng., 4 (1972) 551. [14] A. K. Runchal, Int. J. for Num. Meths in Engineering, 24 (1987) 1593. [15] S. V. Patankar and D. B. Spalding, Int. J. Heat and Mass Transfer, 15 (1972). [16] A. Brandt, Mathematics of computation, 31, No. 138, (1977) 333. [17] P. G. Tucker, Technical Report 90/TFMRC/TN91, Thermo-Fluid Mechanics Research Centre, University of Sussex (1990). [18] A. D. Gosman and E. Ioannides, J. of Energy, 7 (1983) 482. [19] E. Fried and I. E. Idelchik, Flow resistance: A design guide for engineers, Hemisphere, New York, (1989)
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
Numerical computation Sekhar Majumdar
of T u r b u l e n t flow a r o u n d R a d o m e
309
Structures
B.N.Rajani
Computational and Theoretical Fluid Dynamics Division National Aerospace Laboratories, Bangalore 560 017, INDIA
A finite volume algorithm is presented for prediction of turbulent flow around radome structure through numerical solution of the Reynolds Averaged Navier Stokes (RANS) equations. The algorithm based on the pressure-velocity solution strategy, employs nonorthogonal body-fitted grids, collocated variable arrangement coupled to the low-diffusive upwind schemes for convective flux discretisation and the eddy-viscosity based standard k - e turbulence model for simulation of the turbulence effects. Computation for typical application shows that most of the gross features of the radome aerodynamics are captured by the present prediction procedure. 1. I N T R O D U C T I O N Radomes are large dome-shaped structures which protect the radars from the bad weather conditions but at the same time allow the electromagnetic signals to be received by the radar without any distortion or attenuation. A large radome consists of a dome in the form of the frustum of a sphere mounted on a cylindrical tower of circular, hexagonal or octagonal cross-section. During the service life Of a radome, wind load is the most important load for the design and analysis of strength and stability of the structure. On the other hand, good electromagnetic performance of a radome also requires optimum electrical characteristics at the operating frequency band. The contradictory requirements of the electromagnetic and aerodynamic design often require a very judicious decision regarding the choice of the amount and the arrangement of the composite material of the radome. Some important factors which determine the wind load are the radome geometry, its location relative to the adjacent buildings, the kind of local terrain and also the local wind velocity profile. This paper presents an efficient finite volume method for prediction of steady-state radome aerodynamics through numerical solution of the governing RANS equations for three-dimensional turbulent flows. The algorithm was proposed earlier by the first author [1-4] and developed further by his research group [5-8] at NAL Bangalore during the last ten years. Capability of the flow algorithm is demonstrated by the validation test against some wind-tunnel measurement data [9] for turbulent flow around a radome. The algorithm is also used for prediction [10] of flow around a (DSpler Weather Radar)
310 DWR radome designed at NAL under the joint sponsorship of the Indian Space Research Organisation and the Indian Meteorological Department
2. N U M E R I C A L
GRID G E N E R A T I O N
The radome geometry is simplified as the frustum of a sphere sitting on the top of a circular cylinder with its axis normal to the flat horizontal ground surface. The vertical mid-symmetry plane parallel to the flow direction is assumed to form one computational boundary dividing the flow domain in two symmetric halves. The far field surface, assumed to be a spherical quadrant forms a O-O grid topology. A boundary orthogonal body-fitted 2D grid is generated on a typical azimuthal plane ( r - r employing a differential-algebraic hybrid procedure [11]. Similar network of points are then stacked at each of the azimuthal planes (0 = constant) to form a quasi 3D volume grid. The coordinate system used for the radome, the boundary conditions on the mid-symmetry plane and a close view of the grid near the radome surface (160 x 50 x 40 control volumes) are shown in Fig. 1.
Fig. I Numerical grid and boundary conditions for flow analysis
311
3. FLOW SOLUTION A L G O R I T H M 3.1 Governing Equations The RANS equations for turbulent flow in non-orthogonal coordinates using cartesian velocity components may be written in a generalised form as follows:
0 Oxi (6'/r162 + D i e ) - JSr
(1)
where,, the terms Cir , Die and Sr related to convection, diffusion and source terms for different variables r are given in Table 1.
General Transport Equation:
r
Die
1
0
S~
0
(Gr162 + Die) = JSr
Term Definition Constants
Oyj
/3} =cofactor of ~
and
Model
in the transfor-
mation matrix for which the Jacobian is J
( OUl ~r(BjOV2
89
~r ( ~OV3
) ~ J)
-)- j xj+Z 4
)
1 0 (p/3j) J Oxj o J Oxj
1 0 (p~j) J Oxj
kOV~
~j - 9j Oxk i
i
r = #t + #t = #t + pC.k2/e yr Ok Jak B} oxj
ttr Oe Ja~ Bj oxj
G + Ga -- pe
s Clog
~2 -- C2p--~
C,=0.09, crk = 1.0,
C1 =1.14, a~ = 1.32
Table 1 Governing equations in nonorthogonal coordinates
C2=1.92,
312
3.2 Turbulence Model The eddy-viscosity based k - r turbulence model [12] has been used in the present work where the additional stresses arising out of turbulent fluctuations in the momentum equations are replaced by viscous type stresses analogous to their laminar counterpart. The turbulent or eddy viscosity #t is defined by the following relationship: = pc.k
(2)
l
The field distribution of the turbulence kinetic energy k and its dissipation rate r (a measure of turbulence length scale) are obtained from the solution of the relevent transport equations. Since these equations for the turbulence scalars are not strictly valid in the viscosity dominated near wall region, standard Wall Function approach [12] has been used in the present algorithm for the near wall cells.
3.3 Flux Balance Equations Integration of the conservation equations (Eq. 1) over the control volumes, using the Gauss theorem, yields the following flux balance equation for each control volume and each variable: Ie - Iw + I n - Is q- I t - Ib -- [ SedV (3) .IAV where, I~ is typically the sum of a convective, a normal diffusive and a cross diffusive component of flux for any scalar r through the west-face 'w' of a control volume. Using simple linear internodal variation of r for diffusive fluxes, the second-order accurate Composite Upwind Scheme [13] for the convective fluxes and linearisation of the source terms, the balance equation (Eq. 3) for each momentum component can directly be cast into the following quasilinear form i.
(~--~Ai- S P ) r
- ~~ Air +
(4)
where the summing index i denotes each of the six neighbouring nodes and the coefficient A's represent the combined effect of convection and normal derivative diffusion and SU and S P are the components of the linearised source terms. The detailed derivations are provided elsewhere [1,4]. 3.4 Special Treatment of t h e C o n t i n u i t y Equation The cell-centered velocity components (V/p) are computed from the solution of the momentum equations using a guessed pressure field as the predictor step. In order to avoid the chequerboard splitting of velocity and pressure solution, the principle of Momentum Interpolation [1-4] is used to compute the velocity components (V~w) at the cell faces. In the corrector step, to be followed, the continuity equation is transformed to an equation for pressure correction using the momentum equation as a link between the corrections of velocity and pressure. Ignoring second order correction terms, the corrected convective mass-flux Cw, typically for the west face, for example, may be written as :
Cw - biwp~,(Vi~ 1 + ~ D~ (Pw ' _ PP)) ,
(5)
313 where c~v is an underrelaxation parameter, D~ is an area coefficient and Pw ' and pp are the pressure corrections at the nodes P and W on either side of the face 'w'. Substitution of the cell-face velocities and their corrections in terms of nodal pressure-corrections transform the continuity equation into a system of linear equations of pressure-correction to be solved. Appropriate corrections are then added to pressure and the momentum-satisfying velocities at cell centre and cell faces to satisfy the cellwise continuity.
3.5 Boundary Conditions and Solution Algorithm Fig. 1(c) shows the boundary conditions schematically on the vertical mid-symmetry plane. The azimuthal section (0 = 0 ~ at the left of the symmetry axis shows that the free stream wind conditions are prescribed at the far field boundary whereas outflow conditions are prescribed at the far-field boundary at 0 = 180 ~ since the wake velocity field is not known apriori. The fluxes are delinked at the symmetry axis. At the far-field outflow surface, the radial gradient of the variables are made to zero alongwith satisfaction of overall continuity. On the wall surface, all the velocity components are set to zero. For near wall zones, the velocity component locally parallel to wall is assumed to obey the Logarithmic Law of Wall [12] and the total wall shear force is appropriately resolved along the three cartesian velocity directions.The present method uses an iterative decoupled approach and the system of linear equations (Eq. 4) is solved for the three velocity components, pressure-correction and the two turbulence scalars (k and e) sequentially using the strongly implicit procedure of Stone [14].
4. RESULTS A N D D I S C U S S I O N 4.1 C o m p u t a t i o n a l Details Computations have been carried out for different radome flow s i t u a t i o n - viz., the DWR Radome to be designed at NAL for which no measurement data is available and also a few tunnel test situations for the purpose of code validation. The first near wall grid point in radial direction is so chosen that the local value of y + is around 100-200. Fine grid computations are always started with field values interpolated from the coarse grid solutions. The wind velocity and the corresponding profiles of the turbulence energy and dissipation at the far-field inflow boundary have been assumed to be uniform with a constant eddy viscosity level of 100 times the laminar viscosity.
4.2 Numerical Convergence and Grid Sensitivity The convergence of the numerical process is shown in Fig. 2(a) where for a calculation with 40 x 40 x 40 control volumes, the L2 norm of the residue of the continuity equation is observed to be reduced by nearly six decades in 688 iterations requiring about 5 hours on a Silicon Graphics machine P O W E R CHALLENGE L. Similar convergence history is observed for each of the equations solved. Fig. 2(b) shows the effect of grid refinement in the vicinity of the singular symmetry axis around which all the control volumes degenerate to a singular line. Undesirable kink, observed in the computed meridional pressure distribution near the dome apex for the
314
coarse grid, gradually disappears as the spatial resolution is refined near the dome-apex zone through appropriate circumferential stretching. The effect of the location of the far field boundary is shown in Fig. 2(c) where the computed value of Cp at the dome apex point ( r = 0 ~ as well as at a point on the sphere frustum base ( r = 134.34 ~ and 0 = 87.75 ~ are plotted against Ro, the far field radius non-dimensionalised by the radius of the radome sphere. The results appear to be independent of the far field location beyond Ro = 20. Final computations are carried out for 160 x 50 x 40 control volumes with a far-field placed at Ro = 20 and the circumferential grid size near symmetry axis as 0.010 and the first near wall y+ is of the order of 200. 2.0
.
.
.
.
10
.
1.o .
G r i d size ricer
a)
1.0
>,,
0.0
I~ "~ 10
.~o
,~,,""~
-"
"~
--
0.1008
...................................................................... 0 1 O0 200 300 400 500 600 700
Iteration N u m b e r
- 1.5-
0.0273 0.0113
a) Convergence history
0 (Deg)
. ,,.#
'
sp~-~ ( P o ~ l ] '
-
'
r
i -zs i
,
.
Grid : 90 x 50 x 40 CV
/'x
2.0 . . . . . . . . . . - 140 - 70
. ' -"~At
~s
-1.o
o ~0
10
.
...... 70
140
b) Effect of resolution near symmetry axis
-3.0 i 5
,-l~z,( o..sT.." , ............................. 10 15 20 Ro
. . . . . . ,. . . . . . 25 30
c) Effect of location of farfield
Fig. 2 Convergence & grid sensitivity for radome flow calculation 4.3 P r e s s u r e d i s t r i b u t i o n on R a d o m e Surface The computed isobars and the azimuthal variation of pressure on the spherical surface of the radome are shown in Fig. 3. The figures clearly show the formation of the high pressure zone near the front stagnation point, followed by an accelerating flow region ending near the dome midplane ( 0 = 90 ~ and consequently an adverse pressure gradient zone ending in a pressure plateau indicating flow separation on the radome surface. The potential flow pressure distribution for free flow around a sphere and flow around a cylinder, plotted on the same figure, justify the prediction to be physically realistic for a sphere-cylinder combination where the negative pressure peaks lie between 1.25(sphere) and 3(cylinder). 4.4 F l o w S e p a r a t i o n a n d W a k e S t r u c t u r e The computed particle traces on the mid-symmetry plane and the surface streamlines on the radome surface are shown in Fig. 4. Fig. 4(a) shows how on the symmetry plane, the flow divides itself into two halves about the front stagnation point. In the top half, the flow bends upwards and remains attached to the sphere upto a certain point beyond the dome-apex, whereas in the lower part in front of the cylindrical tower, the boundary layer slowly grows on the terrain surface till it separates near the stagnation zone and a
315 small clockwise vortex is generated which finally wraps around the cylinder to form the so-called horse-shoe vortex. In the downstream region of the cylinder, the reverse flow zones created due to the strong interaction between the sphere wake and the wake of the cylinder standing on the terrain surface are clearly observed. The particle traces on the surface nearest to the radome presented in Fig. 4(b), as viewed from the downstream end of the radome, clearly show the converging streamlines on both the spherical and the Level P K
1.07
H G F E D C
0.49 0-30 0.10 -0.09 -0.28 -0.48
B A
-0.67
-0.86
9 8 7 6 5 4 3
-1.06 - 1.25 -1.44 -1.64 -1.83 -2.02 -2.22
2 1
2.0
0.g8
-2.~1
-2.60
(DEE,)
1.0
----- 134.3o . . . . . ~~0.24
'! ~ 9."~ ]=~,.,~"~'~l&
:l
..... ~
.....
~
5~39
//
//
..,
0.0 ,.a. -1.0
-2.0
(Potential)/ , -5.0
a) Surface pressure contours
_S~r
(Pdteati=d)
91.23 67.37
o
.
.
.
.
.
.
30
.
.
.
.
.
.
/ .
.
.
.
.
.
.
.
.
.
90 12o 1~o 1 180 Azimuthol Angle, o (Deg) 6o
b) Azimuthal variation of surface pressure
Fig. 3 Surface pressure distribution for flow around radome (Re=4.789x107 )
Fig.
4
Particle trace for flow around radome (Re=4.789x107 )
316 cylindrical part of the radome, as separation lines in three-dimensional flows. The traces on cylinder surface showing the large downward entrainment of fluid all along the cylinder height, may possibly be attributed to the strong negative pressure near 0 = 90 ~ Further, the longitudinal vortex formed due to the complex three-dimensional interaction of the near surface flow around the spherical frustrum and the flow past the cylinder is also clearly observed in the same figure near the sphere-cylinder junction. 4.5 C o m p a r i s o n w i t h W i n d T u n n e l M e a s u r e m e n t s Wind Tunnel measurements for large radome structures have been reported by Bicknell & Davis [9] where the salient dimensions of the radome are available; but the size and the wall conditions of the tunnel or the free stream turbulence level of the test flow are not reported anywhere in the literature. 3D RANS computations have been carried out for this measurement condition assuming a free stream boundary condition at a far field 20 Dome Radius away from the body. The computed surface pressure distribution as Cp versus 0 is compared to measurement data in Fig. 5 for four different values of r The computed negative pressure peak value, for a far-field placed at 20 dome radius, is -1.31 against the analytical potential flow value of -1.25 whereas the measured value with unknown blockage effect however is approximately -1.52. At higher values of r beyond 60 ~ both measurement and calculation demonstrate typical separated flow in the lee side of the sphere. But the k - e computation always predicts late separation leading to a major difference in the location and the extent of the pressure plateau in the post separation region. Significant difference is therefore observed between the calculated and the measured base pressure at the leeward side of the sphere. Besides the inadequacy of the k - c model coupled to the logarithmic Wall function for flows with adverse pressure gradients, the other reason for the discrpancies may also be the assumption of a vertical mid-symmetry plane that prohibits transport of mass or momentum across the same plane. 4.6 Lift Coefficient The lift coefficient for a radome, calculated from the numerical integration of the pressure and the shear forces around the spherical part of the dome, is observed to be a very strong function of the extent of the Meridional Angle r the spherical frustum subtends at the radome centre. An approximate assessment about the effect of this frustum angle on the lift coefficient may easily be worked out from the well-known analytical pressure distribution for potential flow around a sphere. The pressure distribution for potential flow around a sphere (axisymmetric), given as: C~ = 0.25(9sin~r
5)
(6)
and integrating this pressure distribution around the spherical frustum, one may obtain the lift coefficient, CL as C L - - 2.5sinCb -- 0.75sin3r (7) where Cb is the value of r at which the sphere is truncated to join the cylindrical part.
317 1.20
1.20
~ = 60o
=45 ~
Frcseat Computation
0.20
.....
0,20
Measurement
-0.80
-0.80
-1.80
.........
f, . . . . . . . . i . . . . . . . . . | . . . . . . . . . | . . . . . . . . ' , i , , , , . . . . 30 60 90 120 160
Azimuthol Angle,
-
1.80
9(Deg)
1.20
0
30
60
150
180
150
180
{~= 90 ~
0.20
0.20
-0,80
0.80
1,80
t20
1 . 2 0 "=
-
90
Azimuthol Angle, e (Deg)
o ........ ~ ....... ~6
.......
6'~ . . . . . .
Azimuthal Angle,
'~'ao
'1%' ..... "~'6' .....
9(Deg)
1.80
0
30
60
90
120
Azimuthol Angle, e (Deg)
Fig. 5 Azimuthal variationof pressure on radome surface at differentmeridional angles 2.00
1.75 1.50
-9 x []
1.25 tJ- 1 . 0 0
rs
Potential Flow analysis Present RANS Calculation Information from ESSCO Wind Tunnel Data (BickneI1 eta/)
0.75 x
0.50 0.25 0.00
0
f|ll|||||l|'|V(r|l,Wl,||,,,l|,ir'l||~l,~|
45
90
135
180
Fig. 6 Variation o f Lift Coefficient with Truncation Angle
318 Fig. 6 compares the variation of Cc with ~bbcomputed from Eq. 7 to the present RANS solution as well as to measurement data or estimates reported by other researchers. Reasonable agreement is observed between the present RANS computation and other measurement data for the important gross aerodynamic characteristics like force or moment coefficients for flow around radome structures. 5. C O N C L U S I O N A generalised pressure-based RANS algorithm employing standard k - e turbulence model is employed successfully for calculating turbulent flow around a sphere-cylinder type radome structure. Physically realistic surface pressure, flow separation lines, aerodynamic lift coefficient and the major features of the radome wake flow are captured by the present prediction procedure. Work is in progress for time-accurate computation of the periodic vortex shedding phenomenon for the structure without any assumption of symmetry on the mid vertical plane. ACKNOWLEDGEMENTS The authors wish to thank the Director NAL for his kind permission to publish this work and also to thank Dr.S.S.Desai and Dr.R.M.V.G.K.Rao for their constant support and encouragement. Thanks are also due to Dr. S.Viswanath for kindly providing the geometrical data and flow conditions of the radome analysed. REFERENCES
~
2. 3. .
5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Majumdar, S., Tech. Rep. SFB 210/T/29, Uni. Karlsruhe (1986) Majumdar, S., Num. Heat Transfer, 13(1988), 125-132 Rodi, W., Majumdar S., SchSnung B., Comp. Meth. App. Mech. and Engg., 75(1989), 369-392 Majumdar, S., Rodi, W., Zhu, J., J. Fluid Engg., ASME (1992), 496-503 Majumdar, S., Proc. Int. Symp. on CFD, Sendai, Japan (1993), 164-171 Pal, S., Majumdar, S., NAL PD CF9505 (1995) Majumdar, S., Proc. First Asian CFD Conf, Hong Kong (1995), 925-930 Rajani, B.N., Aruna, S., Debnath, R., Majumdar, S., NAL PD CF9604 (1996) Bicknell, J., Davis, P., MIT, Lincoln Lab. Group Report No. 76-7(1958) Rajani, B.N., Majumdar, S., NAL PD CF9802 (1998) Rajani, B.N., Majumdar, S., Proc. Conf. on Ship and Ocean Tech.(SUOT97), IIT Kharagpur (1997), 75-80 Launder, B.E., Spalding, D.B., Comp. Meth. in App. Mech. & Engg. 3(1974) Zhu, J., Comm. App. Num. Meth., 7(1991), 225-232 Stone H., SIAM Journal of Numerical Analysis, 5(1968), 530-538
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
319
Computations of turbulent flows using the V2F model in a finite element code R. Manceau and S. Parneix* Laboratoire National d'Hydraulique, Electricit~ de France, 6 quai Watier, BP 49, 78 401 Chatou, France The V2F model has been implemented and used for the first time in an industrial finite element code, N3S, developed at the LNH. The implementation has been validated on the channel flow at Re~ = 395 and the backward-facing step at Re = 5,100. Very satisfactory results have been obtained for the test case, proposed at the 7th E R C O F T A C / I A R H workshop, of the ribbed-channel flow at ReH = 37,200 and heat transfer at ReH = 12,600. In particular, the present V2F model yielded very accurate predictions for the Nusselt number distribution. 1. I n t r o d u c t i o n In industrial applications, turbulence modelers must conciliate two constraints: computational efficiency and solution accuracy. In the past, since the computational resources were very limited, wall functions were widely used. Unfortunately, there is a lack of accuracy for wall heat transfer applications, actually one of the most important quantity in nuclear engineering. Thus, in order to improve numerical predictions, turbulence models valid down to the wall are requested. The standard way to extend high-Reynolds models is to use so-called "damping functions" in order to reproduce the influence of the molecular viscosity and the wall-blocking effect on the mean quantities of the flow. This kind of approach has been widely used since it was introduced by Jones and Launder [1], though these functions are mainly derived from experimental or DNS results and are not based on theoretical grounds. Moreover, from a numerical point of view, they introduce dangerous non-linearities and lead to numerical instabilities for complex configurations. To avoid this type of problems, Durbin [2] introduced an elliptic relaxation method, which allows the derivation of low-Reynolds models from all high-Reynolds models. This approach, shortly presented in the first section, is related to a theoretical analysis. The simplest model which can be derived with such a technique is the V2F model; it has been successfully applied in a number of different cases, including heat transfer cases [3-7]. It seems therefore to show a good compromise for engineering applications where lowReynolds models, implying fine meshes, can be used. It is indeed simpler than full Reynolds stress models and very efficient. *Present address: Heat Transfer, ABB Corporate Research Ltd., CH-5405 Baden-D/ittwil, Switzerland
320 The distinctive feature of this work is the use of a finite element code, N3S, whereas previous V2F computations used finite difference or finite volume codes. The second section of this paper presents the implementation of the model, in particular how the boundary conditions are handled, as well as validation cases. The third section describes the test case of a 2-D periodic ribbed-channel, which was proposed at the 7th E R C O F T A C / I A R H workshop in Manchester [8]. A 2-D unstructured triangular mesh has been generated for this purpose. For this case, two sets of experiments are available: the first one, with flow measurements, from Drain & Martin [9], at ReH = 37,200; the second one, with heat flux measurements on the ribbed-wall, from Liou et al. [10], at ReH = 12,600. Both cases were computed separately. Results showed that the V2F model accurately predicts mean flow properties. In particular, the prediction of the Nusselt number distribution is very satisfactory. 2. T h e V 2 F m o d e l The standard way to develop low-Reynolds versions of eddy-viscosity models is to introduce a damping function in the Prandtl-Kolmogorov formula, such that the eddy-viscosity will match experimental data. To avoid this ad hoc modification, Durbin [2] introduces a velocity scale called v 2 in order to naturally damp the eddy-viscosity ut = C, v2T, where T is the time scale (evaluated in a classical way as k/e, but bounded by the Kolmogorov time scale to avoid singularities in the e-equation; note the latter is not acting as a damping function, it's active only in the laminar sublayer, below y+ _< 5, where the molecular viscosity becomes dominant). An additional transport equation is solved for the velocity scale v 2, directly derived from the equation of the Reynolds stress v 2 in a channel. In order to preserve the non-local effect in this equation, the pressure term is modeled by 422 = k f, where f satisfies the following Yukawa equation: 1
f-
L2V2 f - ~r
h
(1)
In this equation, the slow and rapid part of the source term r 2 are respectively given by Rotta and IP models. The dissipation in the transport equation for v-ffis modeled by supposing that e 2 2 - c / k v 2 also follows an elliptic relaxation equation. The difference between 522 and e/k v 2 has been included in the term k f and hence, the source term 1/k (elk v 2 - 2/3 e) is added in equation (1). The total dissipation becomes isotropic far from the wall. 1 ( v2 2 L 2 V 2 I - -s ,;b~2+ --~-c - ~c
f-
)
(2)
The use of the elliptic equation (2) allows the integration of the model down to solid boundaries, since v 2 and the eddy-viscosity are correctly damped, when the appropriate boundary condition for f is provided (Durbin [2]): f =
20 u 2 v~ -
c y4
The resulting model, called V2F, consist in the following equations:
(3)
321 9 Equations:
DtU - - V p + V . ( (u + ut)(VU + v t u ) )
(4)
Dtk = P - r + V - ( ( u + ~'t)Vk)
(5)
Die =
c'elP-Q
+ V" ( (// + --)V~ )
T
v2
m
Dtv 2 - k f f-
(6)
o's
~-r + V . ((u + u t ) V ~ )
L2V2 f -
~t = Cuv2T
(C1
-
-
1)(2/3 - v2/k)
;
T
+
(7) P
(S)
P = 2aS~jS~j
(9)
9 Length and time scales:
(/~3/2 Q_~) 1/4) ; C, s
L-CLmax
(E~_ - (~) 1/2) ; T-max
;6
(10)
9Coefficients: C;1- 1.4 (1 -ql-0.045g/k/v-ff/ C.=0.22;
C I = 1.4; C 2 = 0.3; C~2 = 1 . 9 ; c r ~ = 1 . 3 ;
C L = 0.25; C , = 8 5 . 0
(11)
9 Boundary conditions at walls:
Ui-O;
k-0;
v2-0;
e=
2uk y2 ; f -
20 v,2 7 ~ ey4
(12)
This model was successfully tested in a lot of different situations: 9 Flows with adverse pressure gradient: backward-facing step and Obi et al. separated diffuser [3]; 9 Flows around bluff bodies with deterministic unsteadiness: vortex shedding behind a triangular cylinder (flame holder)[3]; 9 3-D boundary layers: wMl-mounted swept bump at 45 degrees and square cross-section U-bend duct with strong curvature [4]; 9
3-D flows with horse-shoe vortex: appendage-body junction [4];
9 Aerodynamic flows: high-lift aerofoil and prolate spheroid [5]; 9 Impinging flow heat transfer: axisymmetric jet impinging on a flate plate [6] and on a heated pedestal [7]. The model proved to reproduce accurately the near-wall turbulence and appeared to be very efficient in predicting heat transfer. Therefore, it has been chosen to be used in the industrial code N3S, as presented in the next section. The last part of this work is concerned with the test of the model in a relevant case for turbine cooling, namely the periodic ribbed channel flow.
322 5.0 &
4.0 +
A V
3.0
2.0
1.0 0.0 0
Figure 1. Evaluation of the limiting values involved in the boundary conditions (12) of e and f. o point where the BC is applied 9point where the limiting value is evaluated
100
200
y+
300
400
Figure 2. Validation of the implementation for a channel at Re~ - 395 [12]. o DNS 9 1D-code N3S
3. I m p l e m e n t a t i o n and validation 3.1. T h e N3S code N3S is an industrial code developed by the LNH based on a finite element discretization. Elements are tetrahedra in 3-D cases and triangles in 2-D cases. The basis and test functions are the same (Galerkin method). The pressure is defined at the nodes of the element with a linear interpolation inside the element (P1 interpolation). For all other variables, the element is divided into sub-elements by adding a node at the middle of each edge. Linear interpolation is used on each sub-element (isoP2 interpolation). Partial derivatives over time are kept in the transport equation. The time integration is performed by a fractional step method. First, all the variables excepting pressure and f are convected using the characteristic method: the characteristic is followed in upwind direction by a Runge-Kutta scheme, the convective velocity being taken at the previous time step. Thus, the LHS in (4)-(7) can be replaced by ( X n + ! - X n ) / A t , where X ~ is the convected value of X ~. A diffusion step for the scalar variables is then performed, i.e., equations (5)-(8) are solved. In order to impose the coupled boundary condition for c and f (cf. w coupled systems are solved for k and c in one hand, and for ~-7 and f in the other hand. Thus, in the k - e system, only the production, the time scale and the turbulent viscosity are explicit. In the v-if- f system, the transport equation for v 2 is completely implicit and the RHS of the elliptic equation for f is explicit. The coupling comes from the implicitation of the boundary condition on f. The resolution of both systems is performed by a BICGSTAB algorithm [11]. Finally, the remaining generalized Stokes problem, i.e., the momentum equation coupled with the continuity equation, is solved by the iterative Uzawa algorithm.
323
0.003
0.002
0.001
0.000
-0.001 -0.002
-0.003
9
9
-0.004
-0.005
o
10 20 10 U/U + x/h
Figure 3. Validation test case: backwardfacing step at Re = 5,100. [] DNS [12] 9 NASA code solution [3] N3S
3.2.
Boundary
. . . .
0
'
5
. . . .
'
. . . .
10
'
15
. . . .
'
20
. . . .
25
x/h
Figure 4. Backward-facing step at R e - 5,100. Friction coefficient on the lower wall. See Fig. 3 for legend.
conditions
Boundary conditions in N3S are usually imposed by a projection method. In the particular case of the V2F model, those for c and f involve respectively k and v 2. Hence, they are directly included in the coupled systems k - e and v 2 - f. To this day, the V2F model has only been implemented for structured meshes in finite difference or finite volume codes, and the limiting values k / y 2 and v2/y 4 entering the boundary conditions are then evaluated at the first point inside the domain. In finite elements with unstructured meshes, these limiting values were evaluated at the point where a side of an element is encountered while following the direction normal to the wall (Fig. 1), though it is a finite difference approximation. 3.3. V a l i d a t i o n 3.3.1. C h a n n e l flow As a first validation, the channel flow at Re~- = 395 was chosen in order to compare solutions given by N3S and a 1D-finite difference code developed at the CTR. Results k and v-if, shown in Fig. 2, also compared with DNS data [12], are exactly the same the two different codes; this validates the implementation and in particular the way boundary conditions are imposed.
the for for the
3.3.2. B a c k w a r d - f a c i n g s t e p The second validation case is the 2-D backward-facing step at Re = 5,100, for which DNS data are available, as well as a V2F solution from a NASA finite difference code [3]. Comparisons of mean streamwise velocity and friction coefficient between the latter and N3S solution are shown in Figs. 3 and 4. The solutions are very similar, which gives a high confidence in the V2F implementation in N3S. The backflow and the friction coefficient are very well predicted as found by Durbin [3]. This point will be crucial for the following heat transfer test case: the ability of the model to reproduce accurately the near-wall region allows the prediction of the right heat exchange between the wall and the flow.
324
Figure 5. Geometry of the ribbed-channel test case (2H=5e; L=7.2e).
Figure 6. Mean streamwise velocity. Ribbed-channel flow at ReH = 37,200. o Experiments (Drain & Martin [9]) V2F model ............... k - c model with wall functions
4. T e s t case" p e r i o d i c r i b b e d - c h a n n e l 4.1. P r e s e n t a t i o n
of t h e t e s t c a s e
This 2-D test case was proposed for the 7th ERCOFTAC/IARH workshop [8]. A sketch of the channel is shown in Fig. 5. Two sets of experiments are available. In the first one, from Drain & Martin [9], velocity measurements were performed for the Reynolds number ReH = 37,200, based on the bulk velocity Ub and the hydraulic diameter. The second one, from Liou et al. [10], contains only heat transfer data, without velocity measurements. The Reynolds number for the latter is /~eH = 12,600. Thus, two different calculations have been performed. In both cases, the V2F model described in w is used. Periodic boundary conditions are applied at inlet and outlet, with a splitting of the pressure and, in the second case, of the temperature into linearly growing part and periodic part. 4.2.
Case without
heat transfer
Profiles of the streamwise velocity are shown in Fig. 6. The solution obtained with the standard k - e model with wall functions is plotted for comparison. The velocities are very similar for both models in the main part of the domain. The flow is indeed mainly determined by the geometry induced pressure field and the influence of the eddy-viscosity is significant only where the shear is important, near the walls and in the shear layer between the recirculation bubble and the rest of the flow. With both models, the velocity seems to be over-predicted near the upper-wall and under-predicted in the region between y/e = 1 and y/e = 2. Actually, all the results showed at the workshop by different teams using a number of different models presented the same feature. This might be interpreted as a 3-D effect in the experiment, creating contra-rotating eddies in y - z plane. These eddies diverge near the upper-wall, adding a "loss" of fluid in this area and by continuity a deceleration, and an acceleration where they converge. Streamlines are shown in Fig. 7. Two different recirculation zones exist between consecutive ribs. In the solution given by the V2F model, the two bubbles merge. Experimental
325
0 00
0.04
0
r
0.02
V I 0.00
_
0.0
Figure 7. Streamlines obtained with the V2F model. Ribbed-channel flow at ReH 37,200. -
-
1.0
2.0
y/e
3.0
4.0
5.0
Figure 8. Shear stress given by the Boussinesq equation at x/e = 4.18. Ribbedchannel flow at ReH = 37,200. O Experiments (Drain & Martin [9]) V2F model
data are too scarce to confirm this feature, but it was also exhibited in a Large Eddy Simulation performed at the LNH [8]. Anyway, Fig. 6 shows that the k - e model predicts a too early reattachment before the location x/e = 5.32, where some back-flow still appears in the experiment. The V2F model reproduces quite well the intensity of the back-flow, even if it is slightly under-predicted at x/e = 4.18 and over-predicted at x/e = 5.32. However, the solution is globally qualitatively and quantitatively correct. Actually, reproducing the mean flow is not the most difficult task; indeed, the flow is mainly driven by the geometry induced pressure field. The challenge in this case concerns the level of turbulence and the shear stress near the lower wall, on which will depend heat transfer predictions. Unfortunately, the experimental database [9] only contains data for the Reynolds stresses u 2, v 2 and ~-g. Since the V2F model is an eddy viscosity model, it does not provide directly the Reynolds stresses. However, they can be calculated by using the Boussinesq equation u~uj = -v't (OUj/Ox~ + OU~/Oxj)+2/3kS~j. Figs. 8 and 9 show comparisons between the experimental Reynolds stresses and those reconstructed by using the Boussinesq equation. The shear stress is very well reproduced, except between y/e = 1 and y/e = 2, where a slight disagreement with the experiment exists. However, the anisotropy is very badly predicted, the diagonal component u 2 and v-g being almost equal everywhere, which shows that the Boussinesq approximation is not valid in this case for these components; if these have to be modeled, Reynolds-stress or non-linear eddy-viscosity models would be needed [5]. The velocity scale v 2 given by the transport equation (7) has been plotted in Fig. 9, though it can only be regarded as the component of the Reynolds stress tensor normal to the wall in the particular case of the simple channel flow. For instance, it is close to the experimental profile in the upper part of the ribbed-channel, where the flow is very similar to a non-ribbed-channel flow. In the recirculation bubble, the two quantities cannot be rigorously compared. But results show that v 2 is a much better scaling parameter than 2/3 k. Thus, even if no conclusions can be drawn concerning the level of turbulent energy because of the lack of experimental data (w 2 has not been measured), the correct prediction
326 9
,
.
,
.
4.0
,
0.15
I
eq
,~'~ 0.10
,,o"
-~,..~
,,;,, I,~ ~~
ft. "-
2.0
,.(......,, o 00o%0 o
oo
",.',-, OSoo "-.'-...,,. ~
1.0
2.0
3.0
4.0
5.0
y/e
Figure 9.
,
oor
f,,~
--%~Ooo~
y
0.0
0.00 0.0
o
-2.0 ~ 0.0
i
2.0
410
610
i 8.0
s/e
Diagonal Reynolds stresses at Ribbed-channel flow at
x/e = 4.18. ReH = 37,200. O []
Exp. u 2 from Drain & Martin [9] Exp. v 2 from Drain & Martin [9] u 2 from the Boussinesq equation .......... v 2 from the Boussinesq equation v 2 from the model
Figure 10. Heat transfer enhancement. Ribbed-channel flow at ReH = 12,600. s is the curvilinear coordinate, s = 0 corresponding to the upstream protruding corner of the rib. o Experiment from Liou et al. [10] Computation with resolution of the conduction equation in the rib Computation with imposed flux on the rib walls
of the shear stress is encouraging for the computation of the heat transfer case, since a turbulent diffusivity hypothesis will be used; the shear stress is the predominant turbulent term in the mean flow equations. 4.3. H e a t t r a n s f e r case The experiments for this case were performed at ReH = 12,600. Only the Nusselt number is available. It is defined by Nu = @D~/(kf(Tw - Tb)), where @ is the local wall surface heat flux, D~ the hydraulic diameter, kf the conductivity of the fluid, T~ the wall temperature and Tb the bulk temperature. The latter is defined as:
Tb --
T IUl y dy
IUl y dy
(13)
The Nusselt number is normalised by the Nusselt number Nu~ for fully turbulent flow in smooth circular tubes, given by the Dittus-Boelter correlation Nu~ = 0.023 Re~ ~ The ratio Nu/Nu~ characterizes the heat transfer enhancement due to the presence of the ribs. In the experiment, an uniform heat flux is imposed on the lower wall as shown in Fig. 5. However, it is not clear in [10] how the Nusselt number given in the database is calculated on the rib walls. The heat flux is probably considered as a constant which value is one third of the imposed heat flux. Thus, for the computations, two different boundary conditions for the energy equation have been used. In both cases, an uniform heat flux ~ is imposed on the lower wall between consecutive ribs. In the first one, the uniform heat flux ~/3 is imposed on the walls of the rib as well, whereas in the second one, the heat flux 0 is imposed on the channel wall and the conduction equation is solved in the rib, using the SYRTHES tool developed at the LNH to couple solid and fluid heat
327 transfer computations. The turbulent fluxes are modeled by a turbulent diffusivity hypothesis: r't O T
(14)
uiO (7"t O x i
with the turbulent Prandtl number given by the Kays & Crawford [13] correlation 1
crt = 0.5882 + 0.228(r,t/r,) - 0.0441(r,t/u)2[1 - e x p ( - 5 . 1 6 5 / ( r , t / u ) ) ]
(15)
Results are shown in Fig. 10. The Nusselt numbers obtained using different boundary conditions are almost exactly the same on the lower wall. They are different only on the rib walls and very close to its lower corners. Since the way the experimental Nusselt number has been calculated on the rib walls is not clear, only the results on the lower wall of the channel are relevant. It is even so interesting to notice that for the coupled resolution, the heat flux through the downstream wall of the rib is negative. This phenomenon is due to the fact that the upstream upper corner is cooled by the flow, whereas the very low velocity just downstream of the rib leads to a heat accumulation. Therefore, the heat is conducted through the rib from the downstream lower corner up to the upstream upper corner. The results on the lower wall of the channel are excellent in the first half of the wall between two ribs, whereas in the second half, the Nusselt number is slightly under-predicted. However the results are very close to the experiment. This is a confirmation that the V2F is a very efficient and predictive model for heat transfer problems. 5. C o n c l u s i o n This paper has presented the application of the elliptic relaxation approach to the eddyviscosity concept, leading to the V2F model. The main contribution of this work has been to test the possibility to implement this model in a finite element code in an industrial context. The most questionable point was the use of a finite difference approximation to handle the coupled boundary conditions. This approach has been found to be totally successfull and validated for the channel flow at Re~ = 395 and the backward-facing step at Re = 5,100. The model has been applied to the 2-D test case of the periodic ribbed-channel at Re = 37,200 without heat transfer and at Re = 12,600 with heat transfer. In both cases, the results are excellent, especially in the second case concerning the Nusselt number prediction. In the context of nuclear engineering, the success of the implementation of the V2F model in a finite element code, allowing computations of complex geometries with unstructured meshes, and its ability to predict correctly the Nusselt number, is an important result. Heat transfer is indeed one of the most important phenomenon to take into account in nuclear power stations and the V2F model emerges as a very good compromise between the simplicity imposed by the cost limitation and the accuracy needed to reproduce the physics.
328 REFERENCES
1. Jones, W., P., Launder, B., E., The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transfer, 15,301-314, 1972 Durbin, P., A., Near-Wall Turbulence Closure Modelin9 Without "Damping Functions", Theoret. Comput. Fluid Dynamics, 3, 1-13, 1991 Durbin, P., A., Separated Flow Computations with the k - c - v 2 Model, AIAA Journal, 33, 659-664, 1995 Parneix, S., Durbin, P., A., Computation of 3-D turbulent boundary layers using V2F model, Flow, Turb. and Comb. 3., 60-1, 19-46, 1998 Lien, F.-S., Durbin, P., A., Parneix, S., Non-linear-~7_ f modellin9 with application to aerodynamic flows, Eleventh symposium on turbulent shear flows, Grenoble, France, 6, 19-24, September 8-10, 1997 Behnia, M., Parneix, S., Durbin, P., Prediction of heat transfer in a jet impinging on a fiat plate, Int. J. Heat Mass Transfer, 41, 1845-1855, 1998 7. Parneix, S., Behnia, M., Durbin, P., Predictions of turbulent heat transfer in an azisymmetric jet impinging on a heated pedestal, to be published in J. Heat Transfer, 1999 Proceedings of the 7th ERCOFTA C / I A R H workshop on refined turbulence modelling, UMIST, Manchester, UK, 28/29th may 1998 9. Drain, L., E., Martin, S., Two-component velocity measurements of turbulent flow in a ribbed-wall flow channel, Int. Conf. on Laser Anemometry-Advances and Applications, Manchester, UK, 99-112, 1985 10. Liou, T.-M., Hwang, J.-J., Chen, S.-H., Simulation and measurement of enhanced turbulent heat transfer in a channel with periodic ribs on one principal wall, Int. J. Heat Mass Transfer, 36-2,507-517, 1993 11. Van der Worst, H.A., BI-CGSTAB: A fast and smoothly converging variant of BICG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, 631-644, 1992 12. Center for Turbulence Research database, Stanford University 13. Kays, W. M., Crawford, M. E., Convective heat and mass transfer. Third Edition, McGraw-Hill, 1993 .
.
.
o
.
.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
329
Intake/S-Bend Diffuser Flow Prediction Using Linear And Non-Linear Eddy-Viscosity And Second-Moment Closure Turbulence Models. N E May* *Aircraft Research Association Ltd, Manton Lane, Bedford, MK41 7PF, U.K. l
Abstract The predictions provided by several turbulence models, ranging in complexity from linear two-equation to second moment closure, have been compared for three-dimensional separated flow in a diffusing s-duct. At a high mass flow rate, improved surface pressure predictions are obtained using the higher-order models, but a comparison of engine-face total pressure predictions with experiment reveal some deficiencies in these models. At a low mass flow rate, all turbulence models fail to predict any significant secondary flow in apparent contradiction to experiment.
1.
INTRODUCTION
The use of geometrically complex diffusing engine inlet ducts is common practice in modem military aircraft. S-shaped ducts are often used, since the airframe-mounted inlet is not usually located on the central axis of the engine-face. For high engine performance, an important requirement of the design of the duct is that the total pressure at the engine-face is as uniform as possible, i.e. there is low engine-face distortion EIIE2j. However, engine frame weight and space limitations demand as short a duct as possible, resulting in high centre-line curvature and rapid changes in cross-sectional area. These geometric features can be responsible for the development of separation and strong secondary flow features, both of which increase the engine-face flow distortion. Thus aircraft designers have the difficult task of attempting to design s-ducts with low engine-face distortion within severe geometric constraints. The ability to quickly predict the complicated s-duct flow phenomenon without experimental testing would clearly be desirable for designers. However, the highly threedimensional nature of such flows presents a substantial challenge to computational fluid dynamics, and to turbulence modelling in particular. The purpose of the present contribution is to assess the predictive capabilities of several turbulence models for an idealised intake/sbend diffuser geometry. 1 This work has been undertaken with the support of the Procurement Executive, United Kingdom Ministry of Defence.
330 The investigation is performed using the SAUNA (Structured And Unstructured Numerical Analysis) grid generation and flow simulation system for aircraft components and complex configurations. For the present application, the structured grid mode, based on the multi-block approach E3JL41has been used. The predictions obtained using several turbulence models, including linear and non-linear two equation and second moment closure models, are compared with experimental data for surface pressure and engine-face total pressure. 2.
GEOMETRY AND GRID GENERATION
The duct geometry used is the DERA inlet model M2 129, denoted Test case 3 from the AGARD Fluid Dynamics Panel Working Group 13 ESj . The geometry consists of an intake cowl of circular cross-section joined smoothly to an s-bend diffusing duct of circular crosssection, which terminates at a circular engine-face with an axisymmetric bullet at the centre. The downstream end of the duct is offset by 0.3 times its length from the upstream end. The geometry has been modified here by extending both the external cowl and engine-face further downstream, thus moving the position at which an outflow boundary condition is applied to a more appropriate position. Block-structured grids are generated around the geometry, thus allowing for component adaptive topologies to be created for each part of the configuration. A C-grid topology is used for the cowl and bullet and a polar O-grid is used throughout the diffuser. A region of polar O-grid around the duct centre-line has been replaced by a cartesian topology to avoid cells of very small volume along the centre-line which, it is known, would adversely effect the convergence rate of the flow code [2]. Grids with differing near-wall spacings have been generated suitable for both wall-function based turbulence models and low Reynolds number turbulence models. In addition, to conduct a grid dependency study (see section 5.2) a number of low Reynolds number turbulence model grids have been generated. A view of the default surface grid, which also serves as an illustration of the geometry is shown in Figure 1. Note that the configuration and flow domain is symmetrical about the port-starboard plane.
3.
TURBULENCE MODELS
A wide range of turbulence models have been used in the present study ranging in complexity from linear two-equation models to second moment closure. The models used are: the high Reynolds number k-e model f61, the Wilcox k-o~ model L71,the Menter SST model Esl,
331 the Wilcox multiscale second moment closure model t9] and non-linear cubic versions of the k-c0 and SST models. These cubic models have been derived from a straightforward modification to the Craft, Launder and Suga model [~~ which was originally developed to be used in conjunction with a low Reynolds number k-e model. For the present application the model may be written as
a0 =
Cl ~-'k( Sik Sjk --
"~-Ske ~ Ske/
k]tIt Sjl, "k"~"~JkSik) + C2--~-(D'ik
k~t ( ~ij ~,~ ) [Lttk2 "!- C 3 7 ~. ~'-~jk--T kg~"~k, + C4 (p2 (Ski~"~gJ"l-Skj~"~gi) Skg [LI,tk2 gt k2 -b C5 q02 Sij SkgSkg + C67 Sij~"~kg~"~kg
whereSij
~c)xj-t Oxi
3
-~Xk)'
f~ij=
()xj
~)xi
(1)
"
The tensor a o is added to the usual eddy-viscosity Reynolds stress formulation, i.e.
o~Uk )
~)U~
- Puiuj = ~t ()xj
~xi
3 8ij ~'-~k
2 -- ~- 8 0 p k - aij
(2)
In equations (1) and (2), PUiUj is the Reynolds stress tensor, ~t is the eddy-viscosity (= 9k/co), Ui, xi are the cartesian mean velocity and spatial components, p is the mean density, k is the specific turbulence kinetic energy, 8 0 is the Kronecker-delta and =0.09kcowhere m is the rate of dissipation per unit turbulence kinetic energy. The coefficients CI-C6 in equation (1) take the values tm~ C~= -0.1, C2=0.1, C3=0.26, C4= -0.081, C5=-0.0405, C6=0.0405. 4.
SOLUTION M E T H O D
4.1
MeanFlow Equations
The three-dimensional, time-dependent, compressible Reynolds-averaged Navier-Stokes equations are solved using an explicit, time-marching, 5-stage Runge-Kutta scheme, based on the method Jameson et altill. The flow variables are stored at grid vertices and the equations are spatially discretised used a second-order accurate central differencing finite-volume technique. The equations are augmented by 2nd and 4th order artificial dissipation terms to prevent odd-even decoupling and oscillations near shock waves. For improved accuracy in highly stretched regions of grid, an anisotropic artificial dissipation formulation is used, similar to that devised by Radespiel tl21. Local time-stepping and implicit residual smoothing at odd stages tl2~ are used to enhance the convergence of the basic time-stepping scheme.
332 Further convergence enhancement is achieved by using the W-cycle multi-grid technique tl31
4.2
Turbulence Model Equations The spatial discretisation of the turbulence transport equations follows the same approach adopted for the mean flow equations. The general solution strategy adopted is to march the discretised turbulence equations in time to a steady state simultaneously with the mean flow equations. The same five-stage explicit Runge-Kutta scheme is used, but it has been found beneficial to use a separate time-step for the turbulence variables. This time-step is evaluated by considering a von-Neumann stability analysis of the convective and diffusive terms and using an implicit treatment of the source terms (14'15). The same time-step is used to advance each turbulence variable, both for two-equation and second moment closure models. As is common practice, the source terms and diffusive terms are only evaluated at stage one of the scheme and then frozen at these values for the remaining stages. This practice both increases stability of the scheme and reduces the computational cost of each time-step. Numerical experiments have shown that residual smoothing and multi-gridding are not as effective for the turbulence equations as the mean flow equations. Current practice is not to use residual smoothing and to solve the turbulence equations on the finest grid level only. 4.3
Boundary Conditions The treatment of the engine-face boundary condition has been found to be extremely important, since it directly affects the mass flow rate within the duct. The approach used is to fix the static pressure, whilst using zero-order extrapolation for the other variables. However, the value of the static pressure must be carefully chosen so as to produce a mass flow rate within the duct which is consistent with experiment. In practice an iterative procedure is used, whereby the pressure is up-dated in response to a mis-match between the computed and experimental mass flow rates. This iterative procedure is typically invoked every 200-300 multi-grid cycles. The change in engine-face pressure does have a detrimental affect on the rate of convergence. For the k-co and multiscale models, co is determined at no-slip boundaries using the procedure of Menter N. The remaining boundary conditions are standard and details may be found in Peace and Shaw [131and Rudy and Strikwerda tl61. 5.
5.1
R E S U L T S AND DISCUSSION
Test Cases The flow code described in section 4 has been thoroughly validated for a number of flows, utilising all of the turbulence models which are compared in this study. Details of some of these previous validation studies which include a fiat plate boundary layer, transonic aerofoil flow, the Delery bump (17) and a 180 ~ u-bend (18), may be found in references 15 and 19-21. In this study, the predictions are compared with experiment for the s-bend diffuser configuration at two different mass flow ratios (MFR' s), defined as MFR=A~/Ah, where Ah is the area of the cowl at the highlight and A~ is the area of the captured streamtube at upstream infinity. For both the high MFR=2.173 and the low MFR=l.457, the free-stream Mach number =0.21 and the Reynolds number based on maximum diameter =8.6x 105.
333 5.2
Grid-dependency study In order to make meaningful comparisons of turbulence model predictions, it is essential to conduct a thorough grid dependency study. For three-dimensional simulations this can be a time-consuming and expensive task, especially if a separate study is conducted for each turbulence model. For these reasons, a grid-dependency study was only conducted for the multiscale model for the high MFR case. This model was chosen, since it involves the solution of an extra six equations compared with the other models, and would therefore be expected to be the most grid-sensitive. The grid-dependency of a default grid solution has been assessed by generating three further grids, each of which has an increased number of grid points in one of the three spatial dimensions. The characteristics of the four grids are summarised in Table 1. Table 1 Characteristics of the grids used for the grid-dependency study Number of grid points Grid default grid_stream grid_azimuthal Internal streamwise 73 147 73 Azimuthal 33 33 65 Normal across diameter 89 89 89 Total 255046 342094 518174
grid_normal 73 33 113 320386
The near-wall grid spacing was the same for all four grids and was specified such that at the near-wall grid points, y§ -~ 2 in the upstream, straight portion of the duct, and y§ 1 downstream. A comparison of the surface static pressures on the symmetry plane along the port and starboard side (refer to Figure 1) is shown in Figure 2(a). Just downstream of the highlight, i.e. x/Dmax=0, where Dmax is the maximum diameter of the diffuser, the solution obtained on grid_stream differs from the other three. On further examination, it was evident that this solution had separated around the highlight, which probably indicates a turbulence modelling deficiency around this relatively sharp leading edge. Further downstream, apart from slightly differing resolutions apparent around the suction peak on the starboard side at
1IF
p/Po _
PolPo, _
1.02
0.95
_
1 .(X)
-
0.90
_
0.98
--
0.85
_
0.96-
.q--] ~ I i',
/i i
O. 8 0
0.94
-
0.75
0.92
-
0.70
0.90
!t' ~--- b u l l e t - - - ~
0.88
0.65
0.60
/
0.55
0.86 ~-- starboard
_
0.84 0.82
0.50 0.80
0.45
- -
0.40
default
................ g r i d . s t r e a m grid.azimuthal
0.35
_
g~d.normal i
0.00
i
1 .(X)
i
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334 x/Dmax=l.00, the four solutions are very similar. The engine-face plane total pressure predictions, on the symmetry plane, traversing from the starboard side (negative radial distance, r) to the port side (positive r) are compared in Figure 2(b). Although the figure does illustrate some differences in the predictions, the discrepancies were considered to be sufficiently small to enable meaningful comparison with experiment. Overall, it was decided that the default grid provided sufficient resolution. It was therefore decided to use the default grid for the remaining calculations for all turbulence models except the high Reynolds number k-e model. For this model, a grid of the same dimensions as the default grid in Table 1 was used, but the first grid point was positioned so that y + - 60 in the upstream straight portion of the duct and y+ -~ 40 downstream. 5.3
High MFR Surface pressure predictions are compared with experiment along the port and starboard sides on the symmetry planes in Figures 3(a),(b). Note that data on the external cowl surface is included in the figures. In the upstream part of the duct, i.e. x/Dmax 0 prevails far beyond the near-wall region. 5. C O N C L U D I N G
REMARKS
A low-Re SMC model has been applied to predict fully developed flow inside a straight square duct. The low-Re approach is strongly motivated by the fact that reliable compu-
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rations of complex wall-bounded flows require a proper representation of wall boundary conditions. This cannot be achieved in general by the wall-function approach since 'wallboundary-laws' of turbulence do not formally apply to complex turbulent flows which involve e.g. flow separation or effects due to body forces, which often is the case in engineering flows. Near-wall effects are modelled by elliptic relaxation, used in conjunction with the quasi-linear pressure-strain model due to Speziale et al. [8]. This novel approach makes the model directly applicable also to complex geometrical boundaries of engineering interest since no wall topography parameters are involved. Although the secondary mean fluid motion generated solely by the turbulence is relatively weak it still has a profound effect on lower-order flow statistics. The model predictions agree fairly well with the reference data, but it is noteworthy that the strenght of the secondary mean fluid motion is significantly underpredicted at both Re. However, the present predictions of the secondary mean flow are in better agreement with the reference data than the high-Re SMC predictions reported by Demuren and Rodi [1] in the fully developed flow regime. A successful prediction of the secondary mean fluid motion is strongly dependent on the ability of the model to reproduce the delicate imbalance betw-
358 een gradients of the secoi;dary Reynolds shear stress and the normal stress anisotropy. Since the same deficiency has been observed in the present study at both Re, it is anticipated that a viable approach to future research is to make use of low-Re DNS data to scrutinize the model and thereby improve the possibility for successful predictions not only at low Re but also in the high-Re, engineering flow regime. This approach offers the possiblity to scrutinise not only the important secondary Reynolds shear stress component (u2u3), which is difficult to measure in an U l U 3 2 experimental setup, but also the modelled Uc higher-order turbulence correlations which plays a decisive role in the prediction of the secondary flow field.
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REFERENCES
0.2 1. A. O. Demuren and W. Rodi, J. Fluid -1.5 ' I Mech., 140 (1984) 189. 0.4 0.6 0.8 1.0 0.0 0.2 2. S. Gavrilakis, J. Fluid Mech., 244 (1992) Y~ 101. 3. C.G. Speziale, ASME J. Fluids Engng, Figure 13. Primary shear-stress distribution 108 (1986) 118. at t ~ e b - - 65000. Lines: SMC; Symbols: Exo P. A. Durbin, J. Fluid Mech., 249 (1993) 1. periments. 5. B. A. Pettersson and H. I. Andersson, Fluid Dyn. Res., 19 (1997) 251. 6. G. Mompean, S. Gavrilakis, L. Machiels and M. O. Deville, Phys. Fluids, 8 (1996) 1856. 7. B.A. Pettersson, H. I. Andersson and A. S. Brunvoll, AIAA J., 36 (1998) 1164. 8. C. G. Speziale, S. Sarkar and T. B. Gatski, J. Fluid Mech., 227 (1991) 245. 9. P. G. Huang, Ph.D. Thesis, Dept. Mech. Engng, UMIST, Manchester, UK (1986). 10. H. Yokoshawa, H. Fujita, M. Hirota and S. Iwata, Intl J. Heat and Fluid Flow, 10 (1989) 125.
0
Experimental Techniques
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
361
Experimental Investigation of Coherent Structures using Digital Particle Image Velocimetry JOACHIM V. LUKOWICZ a), JORGEN KONGETER b)
a) Research Engineer, Institute of Hydraulic Engineering and Water Resources Management, Kreuzherrenstr., Aachen University of Technology, 52062 Aachen, Germany b) Director, Institute of Hydraulic Engineering and Water Resources Management, Kreuzherrenstr., Aachen University of Technology, 52062 Aachen, Germany ABSTRACT In the present study large scale coherent structures in compound open-channel flows are investigated experimentally using Digital Particle Image Velocimetry (DPIV). In a straight flume images of particles are recorded in a horizontal plane using a CCD-camera and evaluated to reveal instantaneous velocity fields. Afterwards, data is validated using an advanced dynamic filter algorithm. To obtain fluctuating velocity fields different decomposition techniques are applied and compared with respect to their capability to detect turbulent structures. The mean-bulk Galilean decomposition proves best and is used for further analysis. Instantaneous fluctuating velocity fields are examined regarding occurrence of vortical structures and their spatial extension, direction of rotation, movement and interaction with other vortices. 1.
INTRODUCTION
Natural rivers typically consist of a main channel with a continuous discharge and adjacent flood plains which take part during flood events. These compound open channels are characterised by a variable flow depth and roughness across the cannel width. Especially in the lateral direction mass and momentum transport during flood events is dominated by interaction mechanisms between main channel and flood plain. Therefore, compound open-channel flow differs fundamentally from simple open channel flow. Since flood elevation schemes are the focus of much engineering work the hydrodynamic behaviour of compound open-channel flow has important application in river management, flood control planning and irrigation. The objective of the experiments conducted in this study is to identify turbulent structures which play an important role regarding the exchange of mass and momentum in lateral direction. Transport as well as interaction with other structures are examined analysing fluctuating velocity fields.
362 2.
COHERENT STRUCTURES IN COMPOUND OPEN CHANNEL FLOWS
The governing feature regarding the flow field and its associated parameters is turbulence with the build-up of coherent structures. The term coherence is not used standardised in literature. A common description is given by Blackwelder [2] who defines coherent structure as a parcel of vortical fluid occupying a confined spatial region such that a district phase relationship is maintained between the flow variables associated with its constituent components. Nezu & Nakagawa [1] distinguish between bursting phenomena and large scale vortical motion. Whereas bursting occurs in the near wall region large scale vortical motion appears in the outer flow field. In the study presented in this paper, as compound open-channel flow is the subject, the focus is on large scale vortical motion. Compound open channel flow is dominated by two mechanisms. First, a shear layer at the junction develops due to the difference of streamwise velocity between main channel and flood plain. As a result horizontal vortices with vertical axis are generated. These horizontal vortices are superimposed by secondary currents of Prandtl's second kind. Due to anisotropy of turbulence near the junction vortices with longitudinal axis develop. They occur intermittently. Both processes influence each other and generate coherent structures which are characterised by a complex three-dimensional time dependent behaviour. Up to today, experimental research on turbulence in compound open channels has been dominated by a statistical description of fluctuating velocities. Shiono & Knight [3] carried out measurements with Laser-Doppler Velocimetry (LDV) in a flume revealing turbulence characteristics. The three-dimensional flow field for different geometries has been determined by Nezu [4] using a two component LDV-system. Lukowicz & K6ngeter [5] conducted turbulence measurements in the flume described in this study by a single probe 3D LDV-system (5 beam). However, with the use of such point measurement techniques only the time-averaged flow field can be determined, but not the time dependent behaviour of turbulent structures. '~x~ Horizontalvortex ~ Developing s The development of quantitative visualisation techniques like Particle-Tracking Velocimetry (PTV) and Particle-Image Velocimetry (PIV) enables to overcome this shortage. Because instantaneous velocity fields can be deterFiood-Plaln11 ~ ~ ~,~ ~ " Detached mined Taylors frozen turbulence hypothesis "" coherent has no longer to be adopted. Nezu & ~ 1 "~.." vortex "H/~//'#'///I Main-ChanneI Nakayama [6] used an innovative PTVsystem to reveal coherent structures in compound open channel flows. Based on their studies they developed a conceptual model of FIG. 1" Conceptual model of coherent structure 3D coherent vortices between main channel and flood plain, shown in Fig. 1. in compound open channel/low [61
363 3.
EXPERIMENTAL SET-UP
3.1 Open-Channel Flume and Experimental Conditions Experiments are conducted in a 30 m long, 1.0 m wide, and 0.9 m deep flume. The compound cross section of the flume represents one half of a symmetrical trapezoidal channel with a slope between main channel and flood plain of 26 ~ (1:2) as shown in Fig. 2. Side walls as well as the bottom are made of optical glass to realise optical access from all directions.
FIG. 2:
Experimental Set-up and hydraulic conditions
A steady fully developed and uniform flow is established at a longitudinal bed slope of 0 . 1 % with a discharge of 13 1/s. This yields a flow depth of 20.4 cm and 8.0 cm in the main channel and on the flood plain respectively.
3.2 DPIV-System Instantaneous velocity fields are measured using a quantitative flow visualisation technique, i.e. Digital Particle-Image Velocimetry (DPIV). Its schematic arrangement is given in Fig. 3. For visualisation the fluid is seeded uniformly with Polyamid particles (1.01 specific gravity, nominal diameter 28 gm) and illuminated by a horizontal laser light-sheet. The laser light-sheet is generated by a 3W Argon-Ion-Laser and a cylindrical lens connected through a fibre optic cable. The thickness is of approximately 2 mm. Images of illuminated particles are recorded using a high sensitive CCD-camera with a frequency of 25 images per second. Every image with a resolution of 768x572 pixel represents an area of 26x20 cm. The bottom made of optical glass enables to install the CCD-camera below the flume to avoid influence of the fluctuating free surface. The camera as well as the cylindrical lens are mounted on traverse mechanisms. Therefore any arbitrary horizontal plane within the flume can be investigated. Recorded images are stored on a S-VHS video-tape and digitised using a commercial frame grabber.
364
FIG. 3: DPIV-system
To consider distortion due to refraction, the digitised images are corrected and afterwards evaluated using the cross-correlation method. The interrogation area is 64x32 pixel which corresponds to regions of2.33xl.17 cm. The use of asymmetric correlation windows is based on the consideration that velocity in the streamwise direction is much larger than in the spanwise direction. With the resolution applied here, smallest scales of fluid motion can not be resolved. However, it is sufficient to detect large scale vortical motion in terms of coherent structures due to interaction processes between main channel and flood plain. Spacing between the interrogation areas is 32 and 16 pixel respectively which implies an overlap of 50 %. This meets the Nyquist sampling criterion. Data is post-processed to remove vectors which have a maximum displacement of more than 1/4 of the interrogation area (Keane & Adrian [7]). Raw vectors obtained by interrogation are validated using a dynamic filter algorithm. Vectors are removed if they do not fit within a limit defined by an user assigned deviation. The deviation tolerated is calculated based on static and a dynamic contributions. The static part considers that grey values in the original image have to be converted into discrete positions in the digitised image. It is chosen according to the actual size of a pixel. The dynamic contribution is calculated as a multiple of standard deviation from the mean of the surrounding area in space as well as in time. Vectors exceeding this limit are replaced by mean values of the surrounding area. Using this algorithm it is possible to repeat the filtering procedure until no incorrect vectors can be detected, see e.g. Leucker [8]. Since the quality of the images in this study was good, the filter algorithm had to be run only once. Within the chosen tolerance limits only 8% of the data had to be corrected.
365 4.
RESULTS AND DISCUSSION
Sequences of 80 sec of length are recorded in a horizontal plane located at about half of the flow depth on the flood plain, i.e. 4 cm. The sequences consist of 2000 images which is assumed to be sufficient for time-averaging.
4.1 Decomposition of instantaneous flow field Using DPIV it is possible to obtain instantaneous velocity fields and therefore analyse turbulent behaviour in space as well as in time. Velocity fluctuations as the characteristic property of turbulent motion are calculated by decomposing the instantaneous velocity into a mean and a fluctuating component. Different approaches of decomposition techniques exist to reveal turbulent fluctuations, see e.g. Adrian et al. [9]. The most common technique which is widely used for statistical analysis, is Reynolds decomposition. Mean velocity is calculated by time-averaging for each grid point. Thus, it separates the flow field into a component with infinite time scale and fluctuations with all other time scales. As shown in Fig. 4 (top) Reynolds decomposition reveals a rather unorganised flow field and therefore does not clearly visualise regions of organised vortical motion. The existence of a vortex (labelled A) can only be assumed. This is due to the fact that Reynolds decomposition removes features which are associated with mean flow, i.e. as large scale structures. Therefore this decomposition technique is not appropriate for compound open-channel flows. The simple Galilean decomposition, in literature sometimes referred to as movable co-ordinate system, is based on the idea of removing translation caused by the large scale field. Instantaneous velocity consists of a constant convection velocity plus the fluctuating velocity. Because the constant convection velocity can be selected arbitrary no scale decomposition is possible. Only structures having a mean velocity equal to the chosen convection velocity can be evaluated. Fig. 4 (centre) presents the flow field with a constant convection velocity of 8.5 m/s removed which is the mean velocity at the junction. Obviously, only structures near the junction are exposed. One can clearly identify a vortex (labelled B) on the main-channel side of the junction and again may detect another one (A) directly at the junction. To overcome the shortage of simple Galilean decomposition, the constant convection velocity is calculated by averaging over the whole flow field. This is called mean-bulk Galilean decomposition. It implies that the scale velocity corresponds to a moving average. The characteristics are similar to those of a low-pass filter. Results using mean-bulk Galilean decomposition are shown in Fig. 4 (bottom). Regions of organised motion can be identified with high-speed streaks and low-speed streaks in between. Two vortices (A and B) can be detected unambiguously occupying a distinct spatial region. Comparing to Reynolds decomposition large regions of uniform momentum can be observed. Therefore, for further analysis of the flow field meanbulk Galilean decomposition is used.
366
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FIG. 4:
Decomposition techniques
367
4.2 Fluctuating velocity field Velocity fluctuations are evaluated using the mean-bulk Galilean decomposition. Because velocity in the main flow direction is of at least one order of magnitude higher than in the lateral direction streamwise fluctuations are much larger than spanwise fluctuations. Generally, fluctuations become positive and more uniform towards the main channel due to higher mean velocity and decreasing influence of the shear layer. Near the junction, on the main channel side as well as on the flood plain side, the fluctuating velocity field is dominated by the shear layer and its associated interaction processes. Distribution of fluctuations is non-uniform and highly time-dependent. The most striking feature is the occurrence of horizontal vortices both on the main chal~ael side and on the flood plain side of the junction. They consist of regions of increased and decreased velocities concerning mean-bulk velocity. The spatial extension of the observed vortices vary from approximately 8 cm which is about the flow depth on the flood plain down to scales beyond resolution of the recorded images. Regarding the occurrence of the vortices no regularity or periodicity can be observed. Single vortices are generated as well as pairs of vortices which are counter-rotating. Whereas the direction of rotation on the flood plain is mainly positive in terms of the chosen co-ordinate system it appears to be negative on the main channel side of the junction. In addition, regions of large uniform flow develop either as high-speed streaks or as low-speed streaks.
4.3 Vortical motion Vortices are transported with the mean flow. It appears that vortices in the main channel move mainly m ~r~,~,,w ~,. direction. In contrast, vortices oa she flood plain have momentum in the lateral direction, too. They usually tend to be transported towards the junction. Moving into a streak they become disintegrated which illustrates the influence of the shear layer at the junction. Although they are convected some vortices appear to be relatively stable in space and time which indicates organised motion. In some cases, this can be determined observing the fluctuating velocity field. However, a more sophisticated method is to calculate space-time correlations. This approach indicates coherent behaviour in terms of related parameters. Normalised space correlation coefficients Ruu are computed for streamwise turbulent fluctuations u for a fixed point. By calculating correlation coefficients for several time-steps vortical motion in space and in time can be analysed:
Ru. (Ax, Ay, At) =
U(Xo,Yo,t o) 9u(x o + Ax, y o + Ay, t o + At) u' (Xo,Yo,t o) 9u' (x o + Ax, y o + Ay, t o + At)
368 Fig. 5 shows an example of a vortex on the flood plain and the associated correlation coefficients. The vortex is convected with the mean flow and slightly transported towards the junction. Typically, the spatial extension and shape of the vortex is relatively constant over a period of 0.5 s. Only a light stretching in the streamwise direction can be observed which is typically again for vortical motion on the flood plain. Afterwards, the vortex rushes into a lowspeed streak and is disintegrated.
FIG. 5:
Vortex motion in terms of turbulent fluctuation and space-time correlation
369
Interaction processes between turbulent structures is another important aspect. Mainly, turbulent kinetic energy is dissipated through the energy cascade. An exception is the pairing p h e n o m e n o n where two or more structures fuse together to one larger structure. In this case turbulent kinetic energy is transferred to larger structures. The appearance o f several pairing processes can be observed, like the one presented in Fig. 6. A vortex on the flood plain with a high lateral m o m e n t u m m o v e s into a vortex at the junction. Both vortices build up a n e w vortex with spatial extension only slightly larger than the single ones. The n e w vortex is transported with the mean flow, m o v e s into a low-speed streak and is disintegrated.
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370 5.
CONCLUSIONS
In compound open-channel flows turbulence with the development of coherent structures is the governing feature. Large scale vertical motion with vertical axis near the junction between main channel and flood plain have been studied quantitatively using DPIV linked to a filter algorithm as a post-processing tool to verify data. The obtained instantaneous velocity fields are examined using different decomposition techniques. For the detection of large scale vortical motion the mean-bulk Galilean decomposition has proven to be superior to Reynolds and simple Galilean decomposition, because it resolves in scale as well as in time. Appearance and behaviour of coherent structures indicate strong interaction processes between main channel and flood plain. Coherent structures which build up appear to be relatively stable in space and time. Transport towards the junction, pairing processes and disintegration when moving into streaks indicate a complex three-dimensional behaviour. Therefore, it is necessary to obtain additional information of the instantaneous vertical velocity component, e.g. by stereoscopic PIV. ACKNOWLEDGEMENT The research on coherent structures in compound open channel flows is supported by the Deutsche Forschungsgemeinschaft (DFG); Ko 1573/6-1
REFERENCES
[I] [2]
[3]
[4] [s] [6]
[71
[81 [9]
BLACKWELDER, R. F. (1987): "Coherent structures associated with turbulent transport", Proc. of 2nd Int. Sym. on Transport Phenomena in Turbulent Flows, Tokyo, 1-20 NEZU, I.; NAKAGAWA, H. (1993): "Turbulence in open-channel flows", IAHR Monograph Series, A. A. Baklema, Rotterdam SHIONO, K.; KNIGHT, D. W. (1991): "Turbulent open channel flows with variable depth across the channel", Journal of Fluid Mechanics, 222, 617-646 NEZU, I. (1996): "Experimental and numerical study on 3D turbulent structures in compound open-channel flows", Flow Modelling and Turbulence Measurements (eds. C. J. Chen et al.) Balkema, 65-74 LUKOWICZ, J. VON; KONGETER, J. (1998): "Determination of turbulent structures in compound open channel flows using 5 beam 3D LDV", Int. Conference on Flow Diagnosis Techniques, 30 June-3 July, St. Petersburg, Russia NEZU, I.; NAKAYAMA, T. (1997): "Space time correlation structures of horizontal vortices in compound open channel flows by using particle-image velocimetry", Journal of Hydraulic Research, 35 (2) KEANE, R. D.; ADRIAN, R. J. (1990): "Optimisation of particle image velocimeters, Part I: Double pulsed systems", Meas. Sci. Technol. 1, 1202-1215 LEUCKER, R. (1995): "Analyse instation~irer Str6mungsph~inomene zur Vorhersage des Kavitationsbeginns", Reports Institute of Hydraulic Engineering and Water Resources Management, Aachen University of Technology, Band 101, Academica Verlag St. Augustin ADRIAN, R. J., et al. (1998): "Decomposition of turbulent fields and visualisation of vortices", Proc. 9th Int. Symp. on Application of Laser Techniques to Fluid Mechanics, Lisbon
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
371
Measurements on the mixing of a passive scalar in a turbulent pipe flow using DPIV and LIF. L. Aanen, J. Westerweel ~ ~Laboratory for Aero and Hydrodynamics Rotterdamseweg 145, 2628 AL Delft, The Netherlands In order to gain a better understanding of the mixing process, simultaneous measurements of velocities and concentrations would be helpful. Therefore experiments were carried out by means of simultaneous PIV and LIF measurements on the mixing of a point source placed at the centreline of a turbulent pipe flow. The PIV and LIF measurements do not influence each other. The results are used to determine the concentration velocity correlation term present in the time averaged Reynolds equation of the concentration field. In addition we study the effect of coherent structures on the mixing process. 1. I N T R O D U C T I O N Mixing is one of the fundamental properties of turbulence and it has many applications in science and engineering. Points of interest from a point of view of studying turbulent mixing are: m a x i m u m concentration levels, micromixing in relation to chemical reactions, and the influence of coherent structures on mixing. The mixing process can be described as the interaction between a flow and a concentration field. The flow field follows from the Navier-Stokes equation, and is in many cases independent of the concentration field, whereas mass transport can be described in terms of a conservation equation where the molecular diffusion is usually modelled. Consider the case of a time averaged steady and axisymmetric flow field. The Reynolds decomposed equation for the concentration then reduces to:
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(1)
As can be seen, this equation contains two correlation terms between velocity and concentration fluctuations, that include radial and axial velocity components. To study these terms experimentally it is necessary to obtain simultaneous measurements of velocity and concentration fields. This type of measurements is the objective of this study. To measure the correlation terms in 1 we apply Particle Image Velocimetry [1] for the observation of the two velocity components, whereas the concentration field is measured with Planar Laser Induced Fluorescence [2]. In our experiment these two techniques are combined and applied simultaneously to a steady and axisymmetric turbulent pipe flow with a scalar point source at the centreline. The results will be compared with the results of a direct numerical simulation (DNS) and can for instance be used for validation of PDF-models.
372 Before we can use these measurements to study turbulent mixing, we should first check whether the PIV and the LIF measurements do not influence each other, and determine the accuracy of our observations. The accuracy of the measurements can be estimated by measuring the mean velocity and concentration profiles, which appears on left hand side of equation 1, and compare them with the measured turbulent fluxes present on the right hand side of this equation. As our measurement technique is in principle two-dimensional, an axisymmetric flow is an ideal test case for the evaluation of this combined measurement technique, because in that case we are able to measure all terms present in equation 1. After we have determined the accuracy of our measurements, they will also be used to study the effect of coherent structures on the mixing process.
2. M E A S U R E M E N T
METHODS
Particle Image Velocimetry (PIV) is used to measure the instantaneous velocity field in a planar cross section of the observed flow. With PIV the fluid velocity is determined by measuring the displacement of small tracer particles over a small time interval. The particles have to be small so that they follow the flow accurately. To measure the particle displacement the particles are illuminated two times with a laser light sheet. The two exposures are recorded by means of a high resolution CCD sensor array. Each image frame collects both exposures. Given this double exposed picture the autocorrelation of the image is computed in small areas, so called interrogation windows. The off-centre peak in this autocorrelation determines the particle displacement. In our flow all displacements have the same sign and overlapping of particle images only occurs near the pipe wall. Hence no image shifting is required for resolving the directional ambiguity that occurs for double exposed PIV recordings. Sub-pixel displacements are estimated using a Gaussian peak fit through the displacement-correlation peak in the autocorrelation function [3]. The time interval used in the measurements presented is 3ms. The Kolmogorov timescale of the flow is 7ms, so the velocities are measured instantaneously. For measuring the concentration distribution planar Laser Induced Fluorescence (LIF) is used. The concentration of a fluorescent dye is observed by measuring the amount of light emitted by the dye when the dye is illuminated by a light source with known intensity. The amount of emitted light is measured with a CCD camera. The intensity of the light source is measured by recording the emitted light intensity distribution of a uniform concentration field. This is done in the same setup in which the measurements are done. To determine the light intensity distribution we record 50 images and average them for each pixel. Besides that a series of 50 dark images is recorded to determine the grey value offset for each pixel. Subtracting the dark grey value distribution from the grey values obtained from the uniform concentration gives an estimation for the light intensity distribution of the light sheet. Finally, the concentration distributions are measured by recording an image, subtract the dark grey value distribution and normalise the result with the light intensity distribution obtained by the method described above. The exposure time of the measurements is 1.5 ms, which is again much smaller as the Kolmogorov time scale, so also the concentration fields are frozen. During the combined measurements first a PIV image, then a LIF image is recorded. The two images are taken within 6 ms so within a Kolmogorov time scale, and can there-
373 for be seen as recorded instantaneous. To determine the fluctuating components of the measured quantities an ensemble average of a series of measurements is subtracted from the frames. In this way the instantaneous fluctuations of the velocities and concentrations can be computed. A linescan camera was used for doing concentration line measurements with a high spatial and temporal resolution. This gives us detailed information on all spatial concentration scales at a discrete number of positions behind the injection point. 3. E X P E R I M E N T A L
SETUP
To be able to do these measurements, an experimental facility was designed and built. The flow facility consists of a 6 meter long perspex pipe with an inner diameter of 50 mm. In the pipe an injection device for fluorescein is placed. The injection mechanism consists of a syringe driven by an electro motor. The syringe is connected to a thin needle with an inner diameter of 0.8 m m and an outer diameter of 1.0 mm. The needle is mounted at the centreline of the pipe. A separate section for the PIV and LIF measurements is mounted just behind the injection mechanism. In this measurement section the pipe wall is replaced by a thin glass cylinder, with a wall thickness of 1.8 mm, placed in a rectangular box filled with water and with glass windows. This reduces the optical abberation by the curved pipe wall far below the measurement accuracy in the centre region of the pipe. An overview of the total setup is given in Figure 1.
A w
reservoir
auxiliary pump dye reservoir ~inj pump
"~
6m.
ection device
out I
0.5 m. return line
Figure 1. A overview of the flow facility used. (not to scale)
The experiments are done in water, where fluorescein is the scalar with a Schmidt number (i.e the ratio of viscosity to molecular diffusion) of 2075. Except for a small disturbance due to the injection mechanism, the flow can be considered as a fully developed turbulent pipe flow with a steady and axisymmetric mean flow field. For the illumination of the PIV images a twin Nd:YAG pulsed laser system (Spectra Physics PIV 400) was used. A frequency doubler is used to convert the 1064 nm laser beams to visible 532 nm laser beams. The lasers have a fixed pulse rate of 30 Hz, with an adjustable time separation between the pulses of each laser. The lasers have a beam
374
diameter of approximately 6 mm. The light source for the LIF measurements is the 488 nm line of an Argon-ion laser. This wavelength is selected because it is very close to the position of the peak in the absorption spectrum of fluorescein. The beam of the Argon-ion laser, which has a beam diameter of 1.5 m m has to be expanded to be able to combine it with the YAG beams. Both planar and line measurements can be done with help of a 992 x 1004 pixel camera and a linescan camera respectively. The 992 x 1004 pixel cameras (Kodak ES 1.0) can be operated in 'free run' at a frequency of 29 Hz or externally triggered with any frequency between 0 and 30 Hz. To obtain good statistics of the flow, the cameras are operated at a low frequency. When the evolution of coherent structures will be studied, the cameras will be operated at 30 Hz. The cameras are connected to D a t a c u b e MV200 pipeline processors, which read out the cameras and store the frames in a R A M of 267 Mb. This enables us to store 268 frames. After the acquisition the frames are stored on a normal hard disk of a workstation. The workstation is used to process the raw frames. The linescan camera is manufactured by E G & G Reticon, has 2048 pixels and can be operated with a line time between 0.104 and 37.2 ms. The camera is connected to a MaxVideo 200, which enables us to record 7168 scans in one run. The linescan c a m e r a will be used to measure concentration probability density functions and spectra at fixed positions behind the fluorescein injection. For the combined PIV and LIF measurements the laser beams of the Nd:YAG and the Ar + lasers have to be combined along the same optical path. The optical setup creates a parallel light sheet with a minimal thickness. The setup is sketched in Figure 2. The light sheet has a Rayleigh length of 50 mm, so the light sheet is as thin as possible and almost parallel over the complete pipe diameter.
f ~
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~-'igure 2. A top and a side view of the optics used to create the light sheet.
375 ~
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Figure 3. Statistical properties of the velocity field as a function of the radial distance from the centreline: (a) mean axial velocity, (b) axial (u') and radial (v') velocity fluctuations; (c) Reynolds stress (u-~v'). All data are compared with a DNS, DPIV data and LDV data. [1]. All velocities are scaled with the friction velocity u,.
376
4. R E S U L T S
The PIV system and the LIF system were tested first for a flow on which we have alternative data from other studies. This is a fully developed pipe flow at a Reynolds number of 5300, for which we have results obtained by means of LDV and PIV measurements, and results from a DNS [1]. All profiles are scaled with the friction velocity, which is in our case 0.0068 m/s. The results for mean profiles, rms profiles and the turbulent stress are shown in Figure 3. The different profiles agree within the measurement uncertainty. The results confirm that our DPIV system is able to perform accurate measurements. Next we consider the concentration measurements. The main problem in measuring statistical properties of the concentration field is the intermittency of the fluorescein concentration. This intermittency has two reasons. The first reason is that the diameter of the injected fluorescein plume is small with respect to the pipe diameter of 50 mm. The second reason is the high Schmidt number of fluorescein, due to which almost all fluorescein will remain in small structures which are deformed by the flow. The result is that most images are almost completely black except for some small regions containing the concentration structures. An example of an inverted LIF picture is given in Figure 4. This is also the explanation for the high rms values of the concentration profiles (see Figure 5).
Figure 4. An example of the measured concentration field. The flow direction is from left to right. The grey values are inverted, i.e. dark regions represent high dye concentration
377
(a.) m e a n concentration profile I
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5. C O M B I N E D
MEASUREMENTS
WITH
PIV AND
LIF
Preliminary tests with the combined PIV and LIF techniques were done. The first results show that the LIF measurements are not influenced by the PIV measurements and that the PIV measurements are not hindered by the LIF measurements. An example of a combined measurement is given in Figure 6. In the figure only a detail of a measurement frame is given. To be able to present statistics more measurements need to be done.
378
Figure 6. A detail of a combined PIV-LIF measurement.
These are currently in progress. As discussed above the convergence of LIF data is slow due to the intermittency of the fluorescein concentration.
6. C O N C L U S I O N S
AND FUTURE WORK
DPIV and LIF measurements were applied simultaneously to study mixing in a turbulent pipe flow. The results for the velocity measurements are found to be accurate within 0.8% of the mean velocity. The results for the concentration measurements are influenced by the intermittency of the fluorescein concentration which leads to large statistical errors. In the near future complete simultaneous LIF/PIV measurements will be done. After the axisymmetric measurements off-axis injection experiments will be done. In the future we plan to measure the mixing of two point sources, in which one source emits acidic fluid and the other alkaline fluid. The pH dependency of the fluorescent dye is then used to determine the influence of turbulent mixing on the chemical reaction. [4] [5].
ACKN OWLED G EMENTS The authors would like to thank professor F.T.M. Nieuwstadt for his help and support during this study. This project is sponsored by grant DWT.3296 of the STW Technology Foundation.
379 REFERENCES
1. J. Westerweel, A.A. Draad, J.G.Th. van der Hoeven, and J. van Oord. Measurement of fully-developed turbulent pipe flow with digital particle image velocimetry. Ezperiments in Fluids, 20:165-177, 1996. 2. D.A. Walker. A fluorescent technique for measurement of concentration in mixing liquids. J. Phys. E. Sci. Instrum., 20:217-224, 1987. 3. J Westerweel. Digital particle image velocimetry. Theory and application. PhD thesis, Technische Universiteit Delft, Delft, 1993. 4. M.M. Koochesfahani. Ezperiments on turbulent mizing and chemical reactions in a liquid mixing layer. PhD thesis, California Institute of Technology, Pasadena, 1984. 5. H Stapountzis, J. Westerweel, J.M. Bessem, and F.T.M. Nieuwstadt. Measurement of product concentration of two parallel reactive jets using digital image processing. Appl. Scient. Research, 49:245-259, 1992.
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
381
W a v e l e t p a t t e r n s in the n e a r w a k e o f a c i r c u l a r c y l i n d e r a n d a p o r o u s m e s h strip H. Hangan a, G.A. Kopp a A. Vemet b, R. Martinuzzi a aFaculty of Engineering Science, University of Western Ontario London, Ontario, N6A 5B9, Canada bDepartament d'Enginyeria Mecanica, ETSEQ, Universitat Rovira I Virgili Carretera de Salou, s/n, 43006 Tarragona, Catalunya, Spain A pattern recognition technique in wavelet space (wavelet pattern recognition - WPR) is introduced and applied to the turbulent near wake field of a circular cylinder and a porous mesh strip. The results of the WPR compare well with results based on a "classical" pattern-recognition method. It is suggested that pattern recognition in wavelet space has enhanced capabilities to detect multiple patterns in wake flows. 1. I N T R O D U C T I O N Historically, turbulence has been treated as either a superposition of waves (in Fourier space) or a superposition of "elementary" vortices (in physical space). Wavelet decomposition is a new approach to understanding and modelling turbulence, based on phase-space structures, allowing the determination of "local" spectra I . The overall objective of the proposed work is to develop a tool to identify "wavelet patterns" and their role in turbulence dynamics for the purpose of investigating the relationship between the smaller (incoherent) and larger (coherent) turbulent scales. This work is motivated by the need to: (i) improve the subgrid turbulence models used with LES and (ii) find new solutions in the active control of turbulence. In the present work, the velocity fields in the near region of two wake generators, a (solid) circular cylinder and a porous mesh strip, are investigated. A "wavelet pattern recognition" (WPR) technique is presented and applied for the two data sets. This novel method defines templates based on the wavelet transform coefficients and uses these templates in a pattern recognition search in which individual events, similar to the template, are identified in wavelet space. Based on these sets of events (or wavelet patterns) ensemble averages are determined for the Reynolds stresses. Results of the WPR method agree well with results via a "classical" pattern-recognitiOn (PR) method (applied in physical space). It is suggested that the advantage of the WPR over the PR methods comes from the fact that, in wavelet space, the templates can be easily identified based on the sharp modulus maxima lines thus improving the capability to detect multiple patterns in wakes. This aspect can be important in developing a better understanding of how the differences in individual structures affect the "incoherent" or "random" turbulence in wake flows.
382
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396 4. FEATURES OF THE LARGE SCALE TURBULENT BULGES 4.1 Identification of Large Scale Bulges In order to identify the large-scale turbulent bulges we developed a technique based on spatially averaged intermittency functions. In particular, two spatial averages are used; one of a larger scale to identify when there is a large scale bulge present,
nls Ils(~X, t) - nlsnz 2~
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and a smaller scale one to determine the approximate location of the edge of the bulge (with I~s and nss replacing I~s and nls in equation 3), where Ils is the large scale spatially-averaged intermittency factor, nl~ is the number of data points used in the streamwise direction and nz is the number of probes used in the spanwise direction. I,s and nss are defined similarly but are of a smaller scale. The intermittency function was determined using the technique and detector function of Tabatabai et al. (1989),
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It is assumed that there is a large scale bulge present when Ils is near unity and that a quiescent (irrotational) region is present when Ils approaches zero. We thus use L to determine when a bulge is present with a threshold set to an appropriate value. We found that the actual value of nls (= 19 = 1.3 integral scales) was not critical as long as it was chosen to represent the smallest of the large scale bulges but was not so large that I~s approached the global time average. We used Ls to identify the edges, with nss set to 3 (= 0.2 integral scales). . . . . . . . . . . . . . . . . . .
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397 of a size greater than 0.7 integral scales and 35% of the flow was classified as quiescent, with 30% of the flow being quiescent regions of a size greater than 0.7 integral scales. Table 1 gives additional statistics relating to these regions. Other details can be found in Kopp et al. (1997), where the identification technique is described in more detail. Table 1 Some statistics relating to the turbulent bulges and quiescent regions turbulent bulges
quiescent regions
fraction of flow fraction of intermittency
41.9% 74.2%
35.1% 3.5%
fraction of u 2
87.2%
3.2%
fraction of (au/&) 2
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4.2 Positioning of Large Scale Bulges Relative to the Coherent Structures Figure 4 shows a sample of the results obtained with the present experimental data. All bulges were grouped according to their respective sizes. Figures 4a-c show ensemble-averages for bulges of a streamwise extent of lo, 3 lo and 4 lo, respectively. In these figures the contour levels are at y = lh and are unambiguously associated with the bulges identified. The vectors are + <w*> at y = lo. It must be emphasized that these motions occur at the same time, but are physically separated in the y-direction by 0.6 lo (or 1.1 integral scales). Note that the lateral location y = lo is at the edge of the fully turbulent zone. Several observations can be made from these figures: 1. When there is a bulge present at the edge of the wake, the interior motions consist of negative streamwise velocity fluctuations. 2. When there is a larger bulge present, there are velocity fluctuations of a larger magnitude within the fully turbulent core. 3. Within the bulges, at y = lh, the largest negative streamwise fluctuations occur nearer the backside (or upstream edge) of the bulges while the largest positive streamwise fluctuations occur nearer to the fronts (or downstream edge). 4. The larger the bulge, the more energetic it is. These observations indicate that the motions in the fully turbulent core of the wake associated with outer bulges are the same motions that occur in the mixing jet region of the double roller structure. In other words, this is the region with negative in Fig. 3. These motions are well correlated with outward lateral velocity fluctuations (Vernet et al., 1997). This is not suprising as it has long been expected that the double roller portion of the ring-like vortex forms the base of the turbulent bulges (e.g., Ferr~ and Giralt, 1989; Giralt and Ferr6, 1993). Vernet et al. (1998) have shown that these coherent motions contain fine-scale turbulence near the back of the mixing jet region which is being ejected outwards. In addition, it seems likely that the larger bulges have emerged further from the core of the wake and, therefore, are relatively more energetic with correspondingly higher levels of fine scale turbulence and dissipation. In this case, engulfment occurs further from the wake core. Similarly, the smaller bulges are ones which have not emerged quite as far. These are less energetic and are really the tops of the bulges. In this case it seems likely that engulfment
398 occurs closer to the wake centreline.
Figure 4. Velocity fluctuations associated with the turbulent bulges in two planes with (a) to (c) being top to bottom. Contours are at y = lh, max and n~n are, respectively, (a) 0.62 and -0.01, (b) 0.34 and -0.67, and (c) 0.41 and -1.13. Vectors are + <w*> at y = lo.
4.3 Positioning of Large Scale Quiescent Regions Relative to the Coherent Structures Figure 5 shows a sample of the results obtained from the identification of the quiescent zones. Again, these zones were grouped according to their size and ensemble-averaged. Figure 5 shows the results for zones of a streamwise extent of 2 lo. This figure is characteristic for all sizes. The most important observation that can be made is that when there is a zone of irrotational flow at y = lh, there are positive within the fully turbulent core at y = lo. Comparing Fig. 3 and Fig. 5 it is clear that the quiescent zones are occurring simultaneously with the outer spanwise edges of the double roller structure. A second observation is that at
399 y = lh there is little variation of , consistent with this being (external) irrotational flow.
Figure 5. Velocity fluctuations associated with quiescent zones in two planes. Contours are at y = lh, max and ,~, are, respectively, 0.35 and -0.23. Vectors are +<w*> aty = lo. 4.4 Some Kinematics of the Engulfment Process Figure 6 shows a sketch summarizing the present results, together with those in Vernet et al. (1998) and Ferre-Gine et al. (1997). The heart of the figure is the double roller eddy, which we have already shown to be a horizontal slice of a large, shear-aligned, ring-like vortex. In its centre is Grant's (1958) mixing jet with a net outflow of fine-scale turbulence. This outflow is originating from the core of the wake (out of the page). Inflow, or engulfment, must occur primarily at the outer edges of the double roller. Since the ring-like vortex is shear-aligned, streamwise velocity fluctuations are correlated with lateral velocity fluctuations. It is important to note that the scale of the structure is large, and the motions relatively slow, so that the structure must be significantly evolved before this newly engulfed fluid is ejected. Inflow
Figure 6. Sketch of the engulfment process in a horizontal plane. Upstream of the double roller is a saddle point (see Fig. 3; also Ferre-Gine et al., 1997) and upstream of the saddle point is more inflow. This inflow is part of another vortex, either a double roller or single roller. Ferre-Gine et al. (1997) found that there is either a double roller, or single rollers of either clockwise or counter-clockwise rotation associated with this inflow. However, the ensemble-averages of these motions lead us to believe that these single rollers
400
are part of double rollers which fall outside of the width of the data (see also Ferr6 and Giralt, 1989 for a discussion of single rollers and double rollers). There is also evidence for a downstream saddle point, although this evidence is weaker, possibly because the downstream edge is not as sharp as the upstream edge (Ferr6 and Giralt, 1989; and others). There is also no apparent shear-alignment at the downstream edge. Downstream of this saddle point there is also inflow into another vortex structure. The same orientation issues described in the paragraph above come into play here. Finally, as pointed out by Ferre-Gine et al. (1997), the double roller/ring-like vortex configuration is a real instantaneous flow structure and is not just an artefact of the averaging procedure. One reason that this configuration is stable may be because the external irrotational flow moves around it on either side of the ring-like vortex as it emerges (as a bulge). -,r
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401 is shown in Fig. 7. The term involving is ignored because no simultaneous lateral and spanwise velocity measurements have been made in the present study. It is suspected that this term is quite small because the coherent strain rate 3<w*>/~y*+3/~z* is also small (not shown). The contours of are well organized and discussed in more detail by Vernet et al. (1998). It is observed that, in the vertical plane, is largest near the saddle point at the upstream end of the mixing jet region, consistent with the results of Antonia et al. (1987). The positive values of production are shear-aligned like the coherent motions causing them. Interestingly, there is also negative production of incoherent turbulence energy at the downstream end of the mixing jet region (in the centre of the ring-like vortex). This negative production appears to be due to recently engulfed fluid moving inwards from the spanwise edges of the structure, as sketched in Fig. 6. In this way, the engulfed/entrained fluid contributes to the maintenance of the coherent structure. This should not be mistaken for energy transfer from the fine turbulence scales to the large-scales, but should rather be viewed as an effect of the bulk (coherent) motions. In this case, this leads to a transfer from the incoherent to coherent turbulence. This would be consistent with the view of negative production put forward by Beguier et al. (1977). Finally, this negative production also means that there is little n e t transfer between the coherent and incoherent turbulent motions. 6. CONCLUSIONS Large-scale bulges and quiescent zones have been identified in the far wake region of a circular cylinder and related to the prototypical coherent structure in this region; a shear-aligned, ring-like vortex. Some details pertaining to the process of engulfment have been found and related to the coherent structure. The momentum transfer due to this structure is rather complicated and negative production of incoherent turbulence energy was found in the core of the structure. This negative production is due to newly entrained fluid from the spanwise edges of the coherent structure that contributes energy to the coherent motions. Two equation turbulence models tend to work well in free shear flows such as plane turbulent wakes in spite of the relative complexity of the transfer processes in this "simple" flow. This complexity includes the fact that the incoherent turbulence (i) does not exist in the same region as it is produced (Vernet et al., 1998), and (ii) contributes to the production of coherent turbulence. The reason that these models function well in simpler flows may be because there is very little net transfer between the coherent and incoherent motions, leaving only transfer from the mean flow. In the far region of a plane turbulent wake, this occurs over a single length scale. It seems likely, that when a significant strain is placed on a flow such as a turbulent wake, the relative magnitudes of turbulence production are shifted, owing to the structural changes which occur (e.g., Kopp et al., 1995). In this case there will be different net transfer involving the mean flow and the coherent and incoherent motions. In these situations the turbulence models tend to fail (Hanjalic, 1994). ACKNOWLEDGMENTS Thanks are accorded to Dr. A. Vemet for his assistance with the plots and analysis and to J. Ferre-Gine, Dr. A. Arenas and Dr. R. Rallo for assistance with the analysis. This work was
402 financially supported by DGICYT project PB96-1011, NATO Collaborative Research Grant 960142 and the National Sciences and Engineering Research Council of Canada. REFERENCES
Antonia, R.A., Browne, L.W.B., Bisset, D.K. and Fulachier, L., 1987, A description of the organized motion in the turbulent far wake of a cylinder at low Reynolds number, J. Fluid Mech., vol. 184, pp. 423-444. Beguier, C., Giralt, F., Fulachier, L. and Keffer, J.F., 1977, Negative production in turbulent shear flows, Lecture Notes in Physics, vol. 76, pp. 22-35. Brown, G.L. and Roshko, A., 1974, On density effects and large structures in turbulent mixing layers, J. Fluid Mech., vol. 64, pp. 775-816. Corrsin, S. and Kistler, A.L., 1955, Free-stream boundaries of turbulent flows, NACA report 1244. Falco, R.E., 1977, Coherent motions in the outer region of turbulent boundary layers, Phys. Fluids, vol. 20, pp. S124-S132. Ferr6, J.A. and Giralt F., 1989, Some topological features of the entrainment process in a heated turbulent wake, J. Fluid Mech., vol. 198, pp. 65-77. Ferre-Gine, J., Rallo, R., Arenas, A. and Giralt, F., 1997, Extraction of structures from turbulent signals, Artificial Intelligence in Engineering, vol. 11, pp. 413-419. Grant, H.L., 1958, The large eddies of turbulent motion, J. Fluid Mech., vol. 4, pp. 149-190. Hanjalic, K., 1994, Advanced turbulence closure models: a view of current status and future prospects, Int. J. Heat Fluid Flow, vol. 15, pp. 178-203. Hussain, A.K.M.F., 1983, Coherent structures - reality and myth, Phys. Fluids, vol. 26, pp. 2816- 2850. Jeong, J. and Hussain, F., 1995, On the identification of a vortex, J. Fluid Mech., vol. 285, pp. 69-94. Kopp, G.A., Giralt, F., Keffer, J.F. and Ferrr, J.A., 1997, On the configuration of three-dimensional intermittent turbulent bulges in a far wake, Proc. l l t h Symp. Turb. Shear Flows, Grenoble. Kopp, G.A., Kawall, J.G. and Keffer, J.F., 1995, The evolution of the coherent structures in a uniformly distorted plane turbulent wake, J. Fluid Mech., vol. 291, pp. 299-322. Mumford, J.C., 1983, The structure of the large eddies in fully developed turbulent shear flows. Part 2. The plane wake, J. Fluid Mech., vol. 137, pp. 447-456. Tabatabai, M., Kawall, J.G. and Keffer, J.F., 1989, Choice of the threshold in a velocitybased conditional sampling procedure, Phys. Fluids (A), vol. 1, pp. 307-311. Vernet, A., Kopp, G.A., Ferrr, J.A. and Giralt, F., 1997, Simultaneous velocity and temperature patterns in a homogeneous plane of a cylinder-generated turbulent far wake, ASME J. Fluids Eng., vol. 119, pp. 493-496. Vemet, A., Kopp, G.A., Ferr6, J.A., and Giralt, F., 1998, Three-dimensional structure and momentum transfer in a turbulent cylinder wake, submitted to J. Fluid Mech.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
403
I n v e s t i g a t i o n on b o u n d a r y effects in j e t flows G.P. Romano, A. Sabatino Mechanics and Aeronautics Department, University "La Sapienza", Roma, Italy
The near flow field of an axisymmetric jet is investigated by using LIF, LDA and PTV. Primary and secondary instabilities are detected on a longitudinal plane and on cross-sections. The attention is focused onto the effects of rigid or free boundaries sideways to the nozzle outlet (no-slip or free-slip conditions), and in particularly on the start up, growth and interaction of large vortical structures. On the average, higher ambient fluid entrainment (which means higher mixing) is detected in free-slip rather than in no-slip conditions.
1. I N T R O D U C T I O N
The investigation of jet flows is a fundamental item of fluid mechanics and has several applications concerning mixing phenomena: the results obtained on this flow field can be used to improve the design and efficiency of several devices for engineering applications as burners and mixers. The attention recently has moved from the far to the near flow field where the major part of mixing takes place. Simultaneously, the interest on the production, growth and interaction of the large scale vortical structures observed in this region (compared to the small diffusive scales) has increased [1,2]. There is a large variety of parameters which influence these phenomena: upwind and inlet conditions, boundary conditions, geometry of the nozzle (including tabbed nozzles), external forcing, swirl. Several studies have been devoted to the investigation of the geometry of the nozzle and on evaluating the effect of the external forcing and/or swirl, while only recently the attention is also given to the study of upwind conditions. Therefore, there is a general lack of detailed analysis on inlet and boundary conditions, although they strongly influence the average and instantaneous evolution of the downstream flow field. Moreover, the investigation of the effects of different conditions is extremely important also when considering numerical simulations of jet flows and in particular for Large Eddy Simulations (LES): small differences in inlet conditions give rise to discrepancies in comparison to measurements [3,4]. Usually, the attention is focused onto the effect on the near flow field of different mean and rms velocity at the inlet [5]. However, the effect of boundary conditions
404 seems to attain a similar importance: Direct Numerical Simulations (DNS) of the formation of vortex rings have shown that the boundary conditions strongly modify the mixing behavior of a high Schmidt number scalar [6]. A few experimental effort has been spent along this direction: the co-flowing condition (in which the flow from the nozzle enters into an ambient fluid with velocity different from zero) has been investigated in comparison with the situation in which the ambient flow is at rest [7]. The results indicate a higher instability in the position of the jet axis and that the average concentration at the centreline decreases more quickly for the co-flow condition t h a n for the ambient fluid at rest [8]. Therefore, a higher mixing should be expected for the former over the latter. In this paper, the interest is focused onto the effects on the velocity field due to the presence or absence of a wall sideways to the nozzle. In the past, passive devices as cylinders or plates, with limited spatial extension, were placed sideways to the nozzle: here the considered wall is just that of the discharging tank. Therefore, the effects are expected to involve a large region in the vicinity of the nozzle. A similar geometrical configuration was investigated numerically in [6]: the presence or absence of the wall results in a substantial modification of the large scale vortices both in regards of their shape and convection velocity. In particular, the convection velocity is higher for the no-slip boundary condition in comparison to free-slip due to the advection (in the direction orthogonal to the mean flow) of a wall layer that restricts the nozzle area in the no-slip case. Therefore, when comparing the two flow fields at the same axial distance, the free-slip field is much more developed than the no-slip, thus resulting in a higher mixing. However, these results are obtained in the very near field (x/D" o
(a)
\
\
~
x
Figure 1. (a) Schematic of flow field. (b) Schematic of coordinates system.
2. E X P E R I M E N T A L
-40
,
o
,
,
I
5
,
i
I
I
i
10
ww (m 2/s2)
,
,
,
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Figure 2. 95% uncertainties in the normal stress, uu, measurements downstream of the jet at z/D=0.25.
ARRANGEMENT
Wind tunnel model tests were conducted in the R.J. Mitchell wind tunnel which has a low speed closed circuit with a 3.5m (wide)x2.6m (high) test section. A 2.43m long, 10ram thick aluminum fiat plate with an elliptic leading edge was installed across the span of the test section. The freestream velocity, Uo~, was set to 20m/s, and boundary-layer transition was fixed by a 10mm wide sand strip located 100mm downstream of the leading edge. The thickness of the boundary layer in front of the jet exit at x=-100mm was 25mm (Fig. 1). The rectangular jet has a 28ram by 5.5ram exit, being equivalent to a hydrodynamic diameter, D, of 14mm, which is the same as that of a round jet employed in the same series of studies[5,10]. The nozzle was connected to a compressed air supply and pressure controlled by a Fisher valve. The pitch angle of the jet, c~, was fixed at 30 ~ The skew angle of the jet, /3, ranged from 0 ~ to 135 ~ in 15 ~ intervals. The velocity ratio (jet speed/freestream speed), A, was fixed at 1.0. For each flow setting, laser doppler anemometry (LDA) measurements were conducted on cross planes at x - 5 , 10, 20, 30,
435 and 40D. The uncertainty in the position measurements was +0.16mm. The uncertainty in the pitch and skew angles of the .jet was +0.25 ~. The tunnel speed was maintained to an accuracy of +0.05m/s. The uncertainties in the velocity measurement were estimated at i 0 . 0 9 m / s , +0.3Ira/s, and +0.09m/s for the three velocity components, U, V, W, in the z, y, z directions respectively at the freestream speed of 20m/s. For the turbulent stress measurements, estimations of the 95% uncertainties were obtained following the procedures given by Benedict and Gould[8]. The number of samples collected for each position was about 1500. In Fig. 2, an example is given of the uncertainties in the normal stress, w w , measured at two streamwise locations. Typical values are: uu, w w < +13%; v v , u v , v w < +16%; and ~ < +13%. The above numbers represent the worst cases. The smoothness of the results suggests that the actual uncertainties were better than the numbers quoted above. 3. R E S U L T S A N D D I S C U S S I O N 3.1. M e a n F l o w The focus of the present paper is the turbulent stress distribution. As such, a detailed discussion of the mean flow velocity and vorticity distribution is not presented and can be found elsewhere[7]. Examples of mean velocity field are given in Figs. 3 and 4. Selected cross-plane secondary velocity vectors are given at /3 = 0 ~ 15 ~ and 90 ~ representing three typical skew angles. The velocity vectors clearly show that the wall-bound flow is dominated by streamwise vortices. Without skew, the inclined rectangular jet produces two contra-rotating vortices. The two vortices have a small common 'up-wash' region in between them. The strengths of the vortices are rather weak. When skew is introduced, a main streamwise vortex will eventually form as the flow evolves downstream of the jet exit. This feature is apparent at/3 = 15 ~ where a main (primary) vortex is observed. The strength of this vortex is substantially higher than that of an inclined jet without skew[7]. In Fig. 3, only one primary vortex is observed at /3 = 15 ~ Immediately downstream of the jet exit, though, there does exist another, weaker, secondary vortex. This feature is illustrated in Fig. 1. Soon the secondary vortex disappears and only the primary vortex remains and continues downstream. It is this vortex which plays an important role in flow control. Between/3 = 15 ~ and 120 ~ the flow experiences only qualitative changes and the single vortex dominant flow remains. A typical flow is shown in Fig. 3 at/3 = 90 ~ The effect of the vortex on the mean flow is illustrated in Fig. 4 where the streamwise velocity contours are plotted. The vortex generates downwash and upwash of the nearwall flow. The basic features of the streamwise velocity are similar to those of a round jet [3,5].
3.2. Turbulent Kinetic Energy, T K E The normalised turbulent kinetic energy, TKE, distributions at two skew angles are presented in Figs 5 and 6. The 45 ~ skew jet was identified as the optimal angle for this flow based on the mean velocity field survey[7]. Between ~ = 0 ~ and 45 ~ the strength of the vortex experiences a substantial rise. Above ~ = 90 ~ the strength of the vortex was found to drop quickly. Therefore 45 ~ and 90 ~ represent the two limiting skew angles for an effective vortex generation.
436
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y/D
Figure 4. U/U~ contours at x=20D" (a) /3 - 0~ (b) 15~ (c) 90 ~ | of the vortex. Q - p e a k vorticity.
The normal stresses, uu, vv, ww, behave rather similarly. The magnitudes and distributions of the normal stresses are comparable with each other. Only slight variations between them are observed. As such, we will only discuss the TKE distribution 1. Several interesting features of the flow are present. The modification of the turbulent boundary layer is quite obvious. The downwash and upwash effects of the vortex create changes to both sides of the vortex. The effect of the inclined jet is illustrated by the comparatively high T K E level and the concentric distribution of the TKE contours. The downwash induced by the vortex introduces a high TKE distribution near the wall. The TKE level experiences a substantial drop as the flow evolves downstream. At /3 = 45 ~ the T K E peak drops from 0.0176 at x = 5 D to 0.0110 at x=10D, 0.0081 at x=20D, 0.0070 at x = 3 0 D and 0.0066 at x=40D. The corresponding values at/3 = 90 ~ are 0.0421, 0.0229, 0.0121, 0.0107 and 0.0092. After x=20D, the peak TKE value experiences a smaller drop than that before x = 2 0 D , supporting the observation made in the mean velocity field study that x = 2 0 D is the demarcation line between the near-field and far-field vortex development[7]. Differences in the TKE distribution exist in the two flows. The/3 = 90 ~ jet introduces larger changes to the boundary layer than the/3 - 45 ~ jet. The initial TKE level is higher at /3 - 90 ~ the peak value being 58% higher than that of the/3 - 45 ~ jet at x = 5 D and 79% higher at x=10D. Farther downstream at x=20D, 30D and 40D, the peak TKE value at f~ = 90 ~ is consistently about 28% to 34% higher than that at /3 = 45 ~ which also 1A complete database is available from X. Zhang
437
Figure 5. TKE contours at/3 - 45 ~ Contour variables normalised by U~.
Figure 6. TKE contours at/3 - 90 ~ Contour variables normalised by U~.
supports the assertion that x=20D is the demarcation line between the near-field and far-field vortex development. The center of the vortex is seen to be in a location close to the peak TKE at x=5D and 10D at both skew angles. Farther downstream, the center of the vortex departs from the peak TKE location. For example, at /3 = 45 ~ and x=30D, the peak TKE is located at roughly the same height as the center of the vortex and slightly to the upwash side. At = 90 ~ and x=30D, the peak TKE is located below the center of the vortex and slightly to the downwash side. The turbulence production and convection[9] of stresses are examined in this study. Using the cartesian tensor notation, the turbulence production of stress ~-ij is given as --7-~kO~j/Oxk -- ~-jkO~i/Oxk and the convection term is given as ~kOT-ij/OXk. In calculating these terms we assumed a 'slender flow' and omitted the contribution due to the streamwise flow variation. Figs. 7 and 8 show, at a typical streamwise location, the turbulence production and convection of TKE at/3 - 90 ~ The major physics are dominated by the inclined jet. The production and convection terms are of the same order. However, the
438 3
3
x=lOD
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o.
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Figure 8. Turbulence convection of T K E a t / 3 = 90 ~ Contour variables normalised
by uL/D.
peak production value is 50% higher than that of the peak convection at the streamwise distance shown in the figures. The peak production value is 0.0133 at x = 5 D , 0.034 at x = 1 0 D and 0.0013 at x=20D. The corresponding values for convection is 0.0059, 0.0017 and 0.0005. It can be observed that there exists a large area of T K E production associated with the jet development, the center of the vortex being close to ~:he peak production. However, the downwash of the vortex brings a negative region of T K E production in the near-wall area. The turbulence convection contours show more complex features. There are fbur distinct areas; two of positive value and two of negative value, with the center of the vortex at the center of the four areas. Beneath the center of the vortex, there exists a positive area to the downwash side and a negative area to the upwash side. Above the center of the vortex, there is a negative area to the downwash side and a positive area to the upwash side. It is interesting to review the effect of the high levels of the turbulent mixing on the effectiveness of the vortex. The strength of the vortex in a wall-bound shear layer will decrease as it develops downstream, unlike the situation in a free stream where the turbulent dissipation and diffusion play a relatively small role and a streamwise vortex is able to maintain its strength. Measurements suggest that the cross-plane circulation level (normalised by U ~ / D 3) at ~ = 45 ~ is-0.681 at x = 5 D , and-0.426 at x=40D. At ~ = 90 ~ the circulation level is -0.815 at x = 5 D and -0.379 at x=40D. It is clear that, although the vortex is initially stronger at /3 = 90 ~ the /3 = 45 ~ jet is stronger farther downstream. The high levels of turbulent mixing at/3 = 90 ~ contributed to the development.
3.3. P r i m a r y Shear Stress The measured primary shear stress - u w also shows distortion to the boundary layer. Examples of the cross-plane -'uw distribution are given for t h e / 3 = 45 ~ and 90 ~ jets in Figs. 9 and 10. An important feature is the existence of a negative stress area near the wall. According to Shizawa and Eaton[9], this feature was caused by the mean velocity distribution across the vortex which introduces sign changes in the velocity gradient. It is also rather similar to that observed using a round jet[10]. The negative shear stress occupies an area between the center of the vortex and the wall. It is interesting to observe t h a t at x = 5 D a n d / 3 = 45 ~ the negative area does not exist; further supporting the observation t h a t this skew angle is a better one t h a n / 3 = 90 ~ Generally, the level of the peak negative shear stress is lower a t / 3 - 45 ~ than that a t / 3 - 90 ~ For example, at /2 - 45 ~ the peak negative value is -0.015 at x = 1 0 D and -0.0014 at x=20D. A t / 3 - 90 ~ the values are -0.0040 and 0.0017 respectively. Above the center of the vortex, there
439 exists an area of large, positive shear stress distribution. This area is characterised by the concentric contours. Again, the peak positive value is larger at/3 = 90 ~ At/3 = 90 ~ this value is 0.0085 at x=5D, 0.0060 at x=10D, 0.0037 at x=20D, 0.0032 at x=30D and 0.0031 at x=40D. At/3 = 45 ~ the peak positive value is 0.0034 at x=5D, 0.0024 at x=10D, 0.0018 at x=20D, 0.0018 at x=30D and 0.0017 at x=40D. Generally, distortion to the boundary layer is larger at/3 = 90 ~
Figure 9. -~--~/U2~ contours at/3 = 45 ~
Figure 10. -~--~/U~ contours at/3 = 90 ~
Figure 11. Turbulence production of uw at/3 = 90 ~ Contour variables normalised by U~/D.
Figure 12. Turbulence convection of uw at/3 = 90 ~ Contour variables normalised by UL/D.
The turbulent production and convection of the primary shear stress were calculated. Here examples are given in Figs. 11 and 12. In Fig. 11, the production of ~-~ is shown to be characterised by two distinct areas. There is an area of negative production of uw above the center of the vortex, corresponding to the high - u w in Fig. 10. Beneath the
440
center of the vortex, an area of positive turbulent production of ~ exists, corresponding to the negative shear stress in Fig. 10. The turbulent convection of ~ is characterised by four areas around the center of the vortex. The peak negative convection value i s 0.0011 compared to-0.0028 for the production term. The peak positive convection value is 0.0014 compared to 0.014 for the production term, being an order of magnitude lower. A comparison with Fig. 10 suggests that the existence of the negative shear stress in Fig. 10 is closely associated with the turbulent production of the shear stress and the wake region following the jet. The characteristics of the positive shear layer in Fig. 10 above the center of the vortex is influenced by both the turbulent production and convection of the shear stress; showing the effect of the vortex.
3.4. Skew A n g l e Introducing a skew angle to an inclined jet has the eventaul effect of producing a single streamwise vortex instead of two, weaker ones. The strongest vortex is produced between /~ = 45 ~ and 90 ~ for the rectangular jets employed in this study[7]. The effects of skew were measured at x=10D for angles up to 135 ~ In Figs. 13 and 14, the effects on T K E and - u w are shown. 3
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|
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Figure 14. -~--~/U~ contours at x=10D.
At/3 - 0 ~ the presence of two contra-rotating vortices is indicated by the two areas of high T K E and - u w value in Figs. 13 and 14. B e t w e e n / ~ - 15 ~ and 90 ~ only one area
441
of high T K E value exists. W h e n the skew angle is increased to a value above fl = 90 ~ the size of the distorted area is increased. The presence of the wake of the jet is again illustrated by the concentric contours. The peak T K E value experiences a continuous increase with skew. At x = 1 0 D , it increases from 0.0059 at fl = 0 ~ to 0.0071 at fl = 15 ~ 0.0075 at ~ = 30 ~, 0.0110 at fl = 45 ~ 0.0138 at fl = 60 ~, 0.0198 at fl = 75 ~ 0.0229 at f l = 90 ~, 0.0258 at f l = 105 ~ 0.0340 at f l = 120 ~, and 0.0338 at f l = 135 ~. The shear stress distributions in Fig. 14 show that, at fl = 0 ~ and 15 ~ negative - u w does not exit. The negative shear stress area appears at 30 ~ and its peak value reaches m a x i m u m at 90 ~ The t u r b u l e n t p r o d u c t i o n and convection contours of ~ at x = 1 0 D are given in Figs. 15 and 16 at selected skew angles. The basic features of the t u r b u l e n t p r o d u c t i o n and convection are similar to those described earlier. This is particularly true for the flow between ,~ = 45 ~ and 90 ~ suggesting t h a t the m a j o r flow physics r e m a i n unchanged.
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0
y/D
2
r - 105~
4
'
'
' 6
Figure 16. Turbulence convection of u w at x = 1 0 D . C o n t o u r variables normalised
by uL/D.
442 4. S U M M A R Y Turbulence measurements were performed for streamwise vortices induced by a rectangular jet in a turbulent boundary layer, using laser doppler anemometry in a wind tunnel experiment. The study provided a database for validating numerical models of an inclined rectangular jet in a boundary layer. When properly arranged, the rectangular jet is able to produce a superior vortex for flow control than a round jet at the same mass flow rate. Both TKE (and normal stresses), and primary shear stress distributions suggest complex flow physics, particularly in the areas around the center of the vortex and between the vortex and the wall. The TKE is characterised by the wake of the jet and its crossplane distribution is characterised by concentric contours, the peak value of which drops sharply as the vortex develops. The turbulent production and convection of T K E are characterised by distinct areas of negative and positive values. A high level of turbulent mixing was found to contribute to the relatively quick drop of the cross-plane circulation level at 3 = 90 ~ The cross-plane primary shear stress distribution features an area of positive - u w above the center of the vortex and an area of negative - u w beneath it. The center of the vortex was found to represent the location of the vortex better than the peak vorticity. When the skew angle was varied, only qualitative changes were observed between fl = 45 ~ and 90 ~ Between 3 - 30 ~ and 135 ~ the area of negative - u w was seen to be present between the center of the vortex and the wall. At fl = 0 ~ and 15 ~ no negative primary shear stress was observed.
Acknowledgement The study is supported under EPSRC grant GR/J17722. Mr Ian McKnight assisted in the LDA study.
REFERENCES 1. 2. 3. 4.
Wallis, R.A, 1956, ARC, CP-513, London, U.K. Johnston, J.P. and Nishi, M., 1990, AIAA J., Vol.28, No.6, pp. 989-994. Compton, D.A. and Johnston, J.P., 1991, AIAA Paper 91-0038. Selby, G.V., Lin, J.C. and Howard, F.G., 1992, Experiments in Fluids, Vol. 12, No. 6, pp. 394-400. 5. Zhang, X. and Collins, M.W., 1997, ASME Transcation, Journal of Fluids Engineering, Vol. 119, pp. 934-939. 6. Freestone, M.K., 1985, Research Memo. Aero. 85/01, Dept. of Aero., City University, London, U.K. 7. Zhang, X., 1998, Submitted to Experiments in Fluids. 8. Benedict, L.H. and Gould, R.D., 1996, Experiments in Fluids, Vol. 22, No. 2, pp. 129136. 9. Shizawa, T. and Eaton, J. K., 1992, AIAA Journal, Vol. 30, No. 1, pp. 49-55. 10. Zhang, X., 1998, ASME Transcation, Journal of Fluids Engineering, Vol. 120. To appear.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
443
Pressure Velocity Coupling in a Subsonic Round Jet C. Picard ~* and J. Delville ~ ~Laboratoire d'Etudes A~rodynamiques, UMR CNRS 6609/Universit~ de Poitiers, CEAT, 43 route de l'a~rodrome, F-86036 Poitiers, France An experimental investigation involving simultaneous measurements of the radial distribution of velocity within the shear layer of a jet and the longitudinal distribution of pressure surrounding a jet is performed. The use of two statistical approaches (Proper Orthogonal Decomposition and Linear Stochastic Estimation) permits the analysis, in terms of vortical structures, of the pressure fluctuations surrounding the jet. These structures are found to be responsible for far the field noise emission. This method seems then promising to provide a "structural model" of the turbulent flow field. 1. I N T R O D U C T I O N In 1952, starting from the momentum and continuity equations of fluid, Lighthill [1] proposes to consider the sound emitted by a turbulent flow as the solution of an equation governing sound propagation in an acoustic medium containing a distribution of quadrupole sources: the so called Acoustical Analogy. Then, using the generalized Green's formula, the far field radiated noise is calculated through a volume integration of sources which are basically the turbulent velocity fluctuations for low Mach number flows. Later, in the early 70's, some authors (Bishop et al. [2] for example) show that the large scale instabilities, which can be interpreted as a coherent structure model, contribute mainly to the jet noise production process. In order to improve the knowledge of these mechanisms and finally to reduce their influences on the far field acoustic radiation of high speed hot jets, we try to develop a non intrusive experimental method of large scale structure identification adapted to jet flows. Indeed, direct velocity measurement can hardly be obtained in such jets and to overcome these experimental difficulties, we use two approaches. The first one, is based on the Proper Orthogonal Decomposition (POD), first introduced by Lumley [3] for characterizing the coherent structures (Glauser et al. [4]), and is applied to the near field hydrodynamic pressure surrounding a jet as performed by Arndt et al. [5]. The second one, first introduced by Adrian [6] is based on the Linear Stochastic Estimation (LSE) (Cole et al. [7]) and uses the correlation between the fluctuating pressure field and the fluctuating velocity field in a jet as outlined by Arndt et al. [8] and performed in Nithianandan [9]. This leads to a velocity field model useful both in the physical interpretation of the pressure POD modes in terms of structures and in the calculation of the far field noise induced by the estimated vortical field. *Region "Poitou-Charentes" fellowship.
444
Some results obtained in the simplest case of a subsonic round jet are presented here. 2. E X P E R I M E N T A L
CONFIGURATION
2.1. S e t - U p
Experiments are carried out in a round jet with an exhaust velocity Uj = 15.6m/s and a diameter D=50mm. The radial distribution of both the longitudinal and radial velocity components and the streamwise distribution of the hydrodynamic pressure at the outer edge of the jet mixing layer are measured simultaneously (Fig. 1). A rake of N=16 condenser microphones (1/4 in. Brfiel & Kjaer)is used and aligned at 4.4 degrees relatively to the jet axis, following the longitudinal expansion of the jet and located radially close to y/5~ = 0.5, where 5~ is the local vorticity thickness of the jet shear layer. The microphones are equally spaced and measure the longitudinal distribution of the near field pressure fluctuations p(x, t), x E [D, 5.5 x D]. The axial velocity u(x, y, t) and the radial velocity v(x, y, t) are obtained thanks to a radially aligned (Y direction) rake of 12 X-wires probe (wire length and diameter are 0.Tmm and 2.5#m). The separation between probes remains constant: 5y=4mm and the rake extent is 0.88 x D. Sixteen experiments are performed where the rake of microphones remains at the same location and where the rake of hot-wires is located successively beneath each of the microphones. By this arrangement it becomes possible to relate the space time evolution of the hydrodynamic pressure to the organized part of the velocity field. A particular attention has been paid to the interaction between the rakes. Particularly, spectra and space-time correlations measured with only one rake are found quite similar to those measured in presence of the other rake.
Figure 1. Sketch of the experimental arrangement
The signal conditioning system is composed of 24 T.S.I.1750 Constant Temperature Anemometers and a microphone polarization unit. The signals are low-pass filtered at
445 2.5kHz and are simultaneously sampled at 5kHz. Thirty acquisition sets of 12k samples are successively performed. The spectral quantities are obtained for 512 frequency points and averaged using 720 records. 2.2. J e t c h a r a c t e r i s t i c s From the measured data sets, a first characterization of the shear layer of the turbulent jet is performed. All measured quantities, mean longitudinal velocity field U(z, y) (Fig. 2), Reynolds stress tensor components u ' 2 ( z , y ) , v'2(z,y) and u'v'(x, y) (Figs. 3, 4 , 5) are found in agreement with classical results (Rodi [10]). Dashed lines on Fig. 2 correspond to the two extreme positions of the hot-wire rake. Note that the rake always cover the jet shear layer extent. Points outside the rake are plotted using appropriate symmetries.
u/Q
100 x u~2/U 2
00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1,
I,
,
I,
I,
I,
I,
I,
I,
I,
I,
I,
I,
I,
I,
I,
I, . . ~ _ . . ~
,
1
0.5
0.5
~-
0 -0.5
o
-0.5 I I
-1'
I
1 1.6 2.3 2.9 3.5 4.2 4.8 5.4
I
-1
X/D
Figure 2. Mean longitudinal velocity
Figure 3. Mean squared longitudinal velocity fluctuations
100 x v121U.2
I
I
I
I
I
I
I
I
I
I
I
1 1.6 2.3 2.9 3.5 4.2 4.8 5.4
X/D
1oo x u~v'/U 2
0000000000060500
0 0 0 0 0 0 0 0 0 0 0 d0~)00
I
0.5
0.
c~
~-
ooooooooooo6oJooo
C~
o
-0.5
-0. I
I
I
t
I
i
i
ill
I
I
I
I
I
1 1.6 2.3 2.9 3.5 4.2 4.8 5.4
X/D
I
I
I
I
i 196 2.3'219'315'
412'4'.8'5. 4 X/D
Figure 4. Mean squared radial velocity fluc- Figure 5. Mean shear stress tuations
The longitudinal decay of the standard deviation of pressure fluctuations plotted on
446 Fig. 6 follows also a classical behavior (Nithianandan [9]). Typical spectra are plotted on Fig. 7. The pressure spectrum Ep(f) measured at x/D - 4.5 exhibits a f - ~ - decrease, indicative of the dominance of noise induced by the interaction of turbulence with the mean shear (Jones et al. [11]). The longitudinal velocity spectrum Eu(f) and the radial velocity spectrum Ev(f) measured on shear layer axis at x/D = 4.5 are also plotted on Fig. 7. The inertial subrange ( f - ~ ) covers more than one decade. Spectra Ep and Ev exhibit maxima around f=70Hz, corresponding to a Strouhal number St = 2fS,,/Uj = 0.4, that is the mark of large scale structures.
6
f--.. c-i
~5 ~4 ~3
''''''''
' -'Present ' ' an s ~a~a' ' Nithianan
0.1 -~ p ( f ~ , , - l l / 3
.i..,
...
......
,.,
0.01!~Ev(f) "'"",,
.......
J
....
-"
""
0.001 X
0
I
I
I
1 1.
I
I
I
I
I
I
213 219 315 412 418 514
X/D
Figure 6. RMS pressure fluctuations
3. P R O P E R
ORTHOGONAL
i
0.0001
,
,,--I
,
0.1
,
,
,
0
, , , I
,
1
s~ - 2 f s ~ / u j
10
Figure 7. Typical spectra
DECOMPOSITION
3.1. G e n e r a l i t i e s First introduced by Lumley [3], the Proper Orthogonal Decomposition (POD) consists in finding among an ensemble of realizations of the flow field, the realization which maximize the mean square energy. This leads to a Fredholm integral eigenvalue problem whose kernel is a two-point cross-correlation tensor. The resolution of the POD problem provides a complete set of orthogonal eigenfunctions which can be interpreted as the most representative realizations of the flow. The POD can be applied only in the directions which are "inhomogeneous". Generally in mixing layers, the POD is applied in the transverse direction y and to velocity (e.g. Delville [12]). In the present study, the POD is applied, in the longitudinal direction x, to the near field pressure as in Arndt et al. [5]. From the near field longitudinal pressure distribution p(x, t) in the jet and its two-point spatial cross-correlation tensor Rpp(x, x'), the Fredholm equation is written:
') d x ' x
where (~(n)(x) are the real valued eigenvectors at mode n. Dx is the microphone rake extent. Note that N=16 POD modes are obtained. Eigenvalues A (n) correspond to the
447 energy contained in mode n and form a convergent decreasing series. The overall fluctuating pressure energy is given by the sum of the N eigenvalues. The original fluctuating pressure field can be reconstructed from the eigenfunctions by:
N
p(x, t) - ~ p~n)(x, t)
where
p~n)(x, t) - A(n)(t)o(n)(x),
(2)
n=l
with the help of the random projection coefficients"
(3)
A(n)(t) - fDx p(x, t)~(n)(x) dx. So, each POD mode contributes independently to the original field through first application of the POD method is hereafter called physical POD" PODp.
p(pn). This
Considering the time to be stationary, the kernel of integral equation (1) becomes the two-point cross-spectral tensor Spp(x,x'; f) that is the time Fourier Transform of the space-time correlation tensor. The Fredholm equation to be solved is then:
fD Spp(X,X'; f)r
f ) d x ' - ,~(~)(f)r
(4)
f).
x
The Fourier Transform i~(x; f) of the original field
p(x, t)
can be reconstructed using:
N
i~(x; f) - ~ ~n)(x; f) where ~n)(x;
f) - a(n)(f)r
f),
(5)
n=l
with:
a(n)(f) - fnx i~(X; f)r
f)dx,
(6)
and where (*) corresponds to the complex conjugate. In these last three equations, the eigenvectors and the eigenvalues are frequency dependent and the eigenvectors are complex valued. Note that, due to a phase indetermination problem, the eigenvectors r can be described only in the spectral space. This particular application of the POD is hereafter called spectral POD: PODs. There is no direct mathematical relationship between the eigenvectors and the eigenvalues arising from these two types of POD. However, for a given realization of the longitudinal pressure distribution p(x, t), results of the application of these two POD methods can be compared: p!n)(x, t) (inverse Fourier Transform of equation (5)) and p(n)(x, t).
448 3.2. P O D
results
A typical space-time sample p(x, t) is plotted on Fig. 8. On this figure, where isovalues are plotted, the horizontal direction corresponds to time t E [0,102]ms (from left to right) and vertical direction corresponds to the streamwise location x E [D,5.5 x D] (from bottom to top). Pressure waves are clearly exhibited by this plot: they are tilted towards the positive time. This inclination can be linked to a global convection velocity. The averaged slope dx/dt ~_ 0.6 is in agreement with the conventional value of the convection velocity Uc = 0.6 x Uj in a jet. On the other hand, the noticeable temporal wave length increase with the downstream location can be related to the spatial expansion of the jet shear layer. The spatio-temporal evolution of the first two modes of this sample" t's ..(1+2)(x, t) resulting from the PODs and p(pl+2)(x, t) resulting from the PODp are respectively plotted on Figs. 9 and 10. Clearly by keeping two modes, the spatio-temporal pressure evolution is filtered. For PODp and PODs as well, the averaged inclination due to convection is preserved. However the temporal wave length variation is enhanced by the PODs (Fig. 9) while only an averaged wave length is sorted out by the PODp (Fig. 10). The better representation provided by the PODs is confirmed by the amount of energy contained in the first mode: 14% and 42% for PODp and PODs respectively, or contained in the first two modes: 28% and 60%. This indicates that the PODs can be a good filter to enhance the detection of the passage of large scale structures from the microphone rake signals.
Figure 8. p(x, t)
Figure 9. p!l+2)(x, t)
Figure 10.-(1+2)(x t) rp
As stated before, due to the phase indetermination, no description of the pressure modes can be obtained in the physical space for the PODs. The analysis of the modes is then restricted to the PODp. The longitudinal evolution of the first three eigenvectors of the PODp are plotted on Fig. 11. For all the eigenvectors, the wave length is increasing with the abscissa x while the amplitude is decreasing. Moreover, the wave length becomes shorter when the mode number is increasing. The phase difference between the modes 1 and 2 can be related to the structure convection (aubry et al. [13]). The next step is to find a physical interpretation of the PODp modes in terms of vortical structures.
449 I I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
mode 1
I
I
I
l
I
I
I
I
I
I
I
I
I
I
~
I
i
!
I
I
..........
.~~_..----~ -- - - ~ 0.5i
mode 3
I
....... ; ~ ( . . . ) . . . ~ c~
mode 2
I
0 -0.2
t
-
1
't
,-
""
9
/
/
"
-
4
"
"
~ -~. /
/
"
~
-
t
. . . . . . .
I
I
I
I
I
I
I
0
1
2
3
4
5
6
X/D
Figure 11. First three eigenvectors of the Figure 12. Estimated velocity vector field physical PODp induced by the instantaneous pressure distribution
4. L I N E A R
STOCHASTIC
ESTIMATION
In order to relate the longitudinal pressure distribution to the velocity distribution within the shear layer, the Linear Stochastic Estimation (LSE) can be used. In the present approach, the velocity fluctuations are estimated as a linear combination of the longitudinal fluctuating pressure distribution: N
g(x, y, t) - ~ Ai(x, y)p(xi, t)
(7)
i=1 N
~(x, y, t) - ~ 8~(x, y)p(x~, t), i=1
where xi correspond to the microphone locations, and (x,y) covers all the shear layer extent. By multiplying this equation by the pressure at location xj and applying a temporal average, the steady coefficients Ai and Bi can be calculated from the two point spatial v e l o c i t y - p r e s s u r e and p r e s s u r e - p r e s s u r e cross-correlations. This method requires high correlation levels which is the case here. Once the coefficients Ai and Bi are known, the velocities on the whole measurement domain can be estimated only from the near field pressure. A typical example of an estimated velocity field is plotted, in a convected frame of reference on Fig. 12, at the given time t = to (see Fig. 8). The plotted vectors are:
{~(x, ~)+ U(x, y ) ~(x,y)
u~
(s)
The estimated velocity field corresponds to a succession of aligned vortices whose size increases with downstream distance. Note that the abscissa of the negative pressure peaks coincide exactly with the vortex centers. In equation (7), the pressure p(xi, t) can be replaced by a physical POD eigenfunction v/A(n)O(vn)(xi ). The resulting velocity fields are plotted on Figs. 13 and 14 for modes
450 n = 1 and 2. The spatial growth of the eigenfunction wave length, previously observed, is clearly related to the augmentation of the size of the structures. Then the use of the LSE technique provides a physical interpretation of the pressure POD modes. At this stage, a vortical model is available and can be useful to the calculation of the noise generated by the structures.
I I
I
I
I
I
I
I
.
.
.
i
.
.
.
.
I
I
I
I
I
I
I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
0.5 I
0I
-0.2 i]
~--'~_.-~.~_~_-r--~_.._.~_~~~ ~_._
0
I
I
I
I
I
I
1
2
3
4
5
6
X/D
.---,-
.~....~--.~'-.~'~
0 J- . . . . . . . . . . . . . . . . /
-0.2' ' 0
' 1
' 2
-.---....,.~"
_. - ~ . _ - z ' r . _ _ , . . - - e - _ :
I 3
X/D
.....
, 4
""-
x
~
\
.-~_._-~. _'-,,_~_ . ~ _ .
, 5
, 6
Figure 13. Velocity vector field induced by Figure 14. Velocity vector field induced by the first spatial POD mode the second spatial POD mode
5. A E R O A C O U S T I C A L
RESULTS
A physical interpretation of this model can be obtained by calculating the far field acoustic radiation of the estimated velocity field. As pointed out in the introduction, Lighthill [1] rearranges the Navier-Stokes equations and expresses the acoustic pressure generated at an observation point X through an integration over a turbulence finite volume l; containing the source terms and called the Lighthill tensor. At low Mach numbers, for isentropic flows, the Lighthill tensor depends only on instantaneous velocities pou~(t)uj(t). Then the acoustic pressure can be written:
-.
po [ ]_ [ 02u~uj~Jdr,
(9)
r [ OyiOyj J
with r - IX - )~J, )P is the source point vector and where the terms between brackets are taken at the retarded time t - r/co. The density p0 and the sound velocity Co are those of the fluid at rest. The Lighthill's theory is established for a finite turbulent field containing all the sources and surrounded by a medium at rest. Generally, as in the present case, only a truncated part of the turbulent field is known. Witkowska and Juv6 [14] show for a truncated domain V that the use of the following equation leads to a correct evaluation of the radiated noise:
p(X, t) -
Ox Oxj
;
451 Starting from 2D slices of the truncated turbulent estimated velocity field: Ul(f,t)
-- U ( x , y )
-1l- ~ t ( x , y , t )
u2(Y, t) - V(x, y) + F(x, y, t),
(11)
equation (10) is used to calculate the sound generated at a large distance, the third velocity component contribution being overlooked. The velocities (g and g) are estimated with the help of equation (7) from the measured near field pressure distribution. The instantaneous temporal far field acoustic pressure is calculated at 17 observation points defined by IXI - 120D and 0 - 10 to 170 degrees, where 0 is the angle between the observation vector 3~ and the jet axis. The directivity of the jet (i.e. mean squared pressure in dB, rel. to 2.10-5 Pa, function of 0) is plotted on Fig. 15. The level drop around the angle 0 = 90 ~ corresponds to the dominance of the noise due to the interaction of turbulence with the mean shear (called shear noise) over the noise induced by the turbulence itself (called self noise). The calculated directivity is compared to the shear noise directivity model suggested by Goldstein and Rosenbaum [15] : the mean square pressure is proportional to (cos 4 0 + cos 2 0)/(1 - Mc x cos0) 5 where the convection Mach number Mc = U~/co. Clearly, because of the relatively small Reynolds number of the flow under study (R~ = 4.5 x 104), the self noise component contributes weakly to the global noise generation process. In fact the spectrum of the near field pressure given on Fig. 7 already points out this information since the spectral decrease following the f - ~ law is typical of the shear noise. The good representation of the acoustic phenomena of the estimated field is also illustrated by the 1/3 octave spectral density. A typical spectrum estimation taken at 0 = 90 ~ and IX I - 120D is given on Fig. 16. The obtained frequency shape is classical with a maximum around the Strouhal number Ste = 1. Nevertheless, the high frequency behavior (following the co-1 law) indicated by Blake [16] is not really followed. It may be due to the weak contribution of the self noise which radiates at higher frequencies than the shear noise. Actually, the relative contribution of shear noise and self noise on the global noise emission can only be known by measuring the acoustic far field. It will determine whether the dominance of the shear noise corresponds to the real acoustic radiation of the jet or to a particular behavior of the model. 6. C O N C L U S I O N S By measuring simultaneously the longitudinal distribution of the near field pressure and the radial velocity fluctuations in a jet, a 2D vortical model is obtained. This model allows a physical interpretation, in term of structures, of the eigenvectors stemming from the Proper Orthogonal Decomposition (POD) applied to the pressure distribution. Moreover, the simple measurement of the near field pressure instantaneous distribution provides an estimate of the temporal evolution of the spatial velocity field. Thus the acoustic far field generated by these structures can be calculated using the Lighthill equation. It is shown that these structures produce sound which is basically a shear noise. Even if many developments and checks of this method have to be carried on, it would
452 lead to a promising way of acoustical identification of large scale structures. In particular, future work will be devoted to the relation between the POD modes of the near field pressure with the induced far field noise. Azimuthal dependency will also be investigated.
60
40
0.0001
~ le-06I ...........i.....] ~ ~ : : - i : : ] . ~ .
30
20 ....................:........... ................:!-!...... i...........i...........i......... 10 .........i...........:...........i...........i...........i.ExiS.~i5qii}iil.//i~ . 0; i tv~oaelr-B--] 20 40 60 80 100 120 140 160 18(] Angle 0 in degrees
Figure 15. Directivity at 120D
~ 1e-07IL",2- 2 ~
.....................................i
~lu
1e-08~4"--- i..............................................i.................................."~ le-09 ~ i ~ i 4 0.1 xD1 S t d ~-
fui
Figure 16. Estimated 1/3 octave spectral density at 0 - 90 ~ and IX I - 120D
REFERENCES
M.J. Lighthill, Proc. Roy. Soc London, A 211 (1952) 564. 2. K.A. Bishop, J.E. Ffowcs Williams and W. Smith, J. Fluid Mech. 50 (1971) 591. 3. J.L. Lumley, Atm. Turb. and Radio Wave Prop., Moscou, (1967) 166. 4. M.N. Glauser, X. Zheng and W.K. George, Studies in Turbulence, Gatski, Sarkar, Speziale edts, Springer-Verlag (1991) 207. R.E.A. Arndt, D.F. Long and M.N. Glauser, J. Fluid Mech., 340 (1997) 1. 6. R.J. Adrian, Turbulence in Liquids, Science Press, Princeton, NJ (1975) 323. 7. D.R. Cole, M.N. Glauser and Y.G. Guezennec, Phys. fluids A Vol 4 No.1 (1992) 192. 8. R.E.A. Arndt and W.K. George, Proc. of the second Interagency Conf. on Transportation Noise, North Carolina State University Press (1974) 142. C.K. Nithianandan, PhD Dissertation, Univ. of Illinois at Urbana-Champaign (1980). 10. W. Rodi, Studies in Convection- Theory, Measurements and Applications, Vol. 1 B.E. Launder Edt. Academic Press (1974) 79. 11. B.G. Jones, R.J. Adrian, C.K. Nithianandan and H.P. Planchon, AIAA vol.17 No. 5 (1979) 449. J. Delville, Applied Scientific Research 53 (1994) 263. 12. 13. N. Aubry, R. Guyonnet and R. Lima, J. Nonlin. Sci. 2 (1992) 183. 14. A. Witkowska and D. Juv~, C. R. Acad. Sci. Paris, t. 318, S~rie II (1994) 597. 15. M. Goldstein and B. Rosenbaum, J. Acou. Soc. Amer. 54 No. 3 (1973) 630. 16. W.K. Blake, Applied Math. and Mech., Academic Press vol. 17-I (1986) 182. .
.
.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
453
Scalar mixing in variable density turbulent jets
J.F. Lucas, M. Amielh*, F. Anselmet and L. Fulachier
Institut de Recherche sur les Ph6nom6nes Hors Equilibre 12, Avenue du G6n6ral Leclerc, 13003 Marseille, France. *Corresponding author :
[email protected], fr
The present paper is concerned with a turbulent jet flow in which the density varies due to exchanges of mass. Experimental results about the turbulent mass fluxes are given for the mixing of an axisymmetric turbulent helium jet issuing into a low velocity air coflow in the region covering the first twenty diameters from the nozzle exit. Our measurements are performed by an original method combining 2D laser-Doppler velocimetry and a single hot-wire. Estimations of the eddy diffusivity and the associated turbulent Schmidt number, which are used in variable density turbulent flow modelling, are also deduced.
1. I N T R O D U C T I O N Turbulent flows in which the density varies due to exchanges of mass are important from many viewpoints and constitute therefore an attractive subject of research. Firstly, numerous industrial applications are concerned with them. Secondly, the coherent structures responsible for large-scale mixing may have specific characteristics associated with the presence of strong density gradients. In addition, even though the flows presently considered do not involve any chemical reaction, our results are important for flows with combustion since the local density variations caused - among other things - by the exothermic effect then create linkage between phenomena due to the chemical reaction and to aerodynamics. If quite a few studies have already been concerned with such flows, it clearly appears that the near field region of variable density turbulent jets, where the inertial forces are predominant and strong gradients of velocity and mass are present, has not been investigated extensively in past years [1]. Indeed, the developments of second- or third-order models which use turbulent flux balances, and in particular those for the turbulent mass fluxes, provide a good description of the mixing in the far-field region [2,3]. However, these models still have to be improved in the near-field and, thus, some complete experimental data bases are needed. To our knowledge, only So et al. [4] report detailed measurements of longitudinal and radial turbulent mass fluxes in the near-field region of a helium-air jet where the initial density ratio between the jet and the ambient air is 0.64 and the exit Reynolds number is R;=4300. Another interesting work concerning simultaneous measurements of velocity and mass fraction (or density) is that of Panchapakesan and Lumley [3], also in a helium jet mixed with the ambient air, but these authors only investigated the far-field and gave the asymptotic or 9
J
454 self-similar reference behaviour. Most other works give only partial results about either the dynamic or the scalar field in gas flows, or results for scalar mixing in liquid flows with large Schmidt numbers Sc and without any density variation. Our main objective is to fill this lack of experimental data in the near-field region by performing simultaneous measurements of two components of velocity and of the helium mass fraction in a helium turbulent jet mixed with a low velocity air coflow, in order to deduce the radial and longitudinal turbulent mass (or mass fraction) fluxes.
2. EXPERIMENTAL SET-UP The facility consists in a fully developed turbulent vertical pipe flow of pure helium (mean velocity Uj=32m/s, Reynolds number Rj=7000) discharging into ambient air (mean velocity Ue=0.9m/s) in a slightly confined configuration (jet diameter Dj=26mm, enclosure section 285x285 mm2). The initial density ratio is 0.14. The mixing is studied in the near-field region, which extends from the exit section up to 20Dj. Velocity measurements are performed with a two-component laser Doppler system (Argon 4W) fitted with fibre optics and two Burst Spectrum Analysers operated by a Dantec acquisition software. Backscatter is used. The primary and secondary flows are simultaneously seeded using diffusers of silicon oil particles (approximately l~tm in diameter). A single hot-wire, sensitive to both the velocity and the helium mass fraction, is placed in the downstream vicinity (only few multiples of the Kolmogorov length 11) of the laser probe volume. This tungsten hot-wire (DANTEC, 55P11) is 1.2mm long and its diameter is 51am. It is operated by a constant temperature anemometer (CTA Streamware, DANTEC) with a 0.8 overheat coefficient. The wire is maintained at an around 523K temperature, so that the seeding oil particles are burnt as soon as they hit the wire. By taken caution of maintaining the heating of the wire during the whole experiment, no effect of the seeding arises as shown by the spectral analysis of figure 2. The hot-wire is separately calibrated in terms of both velocity and helium mass fraction using suitable binary mixing flows obtained in a specially designed facility (Fig.3). This calibration gives an implicit, but bijective, relation between a pair (hot-wire voltage E, velocity U) and a pair (helium mass fraction C, velocity U). In practice, the calibration surface given in figure 3 is fitted by a 2Dpolynom of second order. Therefore, by combining simultaneous LDV and hot-wire measurements, the instantaneous and simultaneous longitudinal (U), radial (V) velocity components and helium mass fraction (C) are deduced (Fig.4). This technique is rather similar to that we have previously developed for studying a slightly heated air jet where turbulent heat fluxes were measured [5].
3. RESULTS AND DISCUSSION From the instantaneous and simultaneous measurements of (U,V,C) many results may be deduced concerning the different moments associated with these quantities. Our development will be here limited to the second order moments including the mass fraction. Measurements were performed in several sections from the exit (X/Di=0.2) up to 20Dj and on the axis. When this is possible, some comparisons are presented v~ith the case of slightly heated jet where temperature acts as a passive contaminant. The comparison of the transfer of these scalars (enthalpy or mass fraction) is here interesting, and the slightly heated jet is a reference flow where the velocity field is not submitted to strong density variations.
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Figure 1. Experimental set-up.
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Figure 4. Instantaneous and simultaneous velocity and helium mass fraction (C) deduced from the simultaneous LDV (velocity U) and hot-wire measurements (voltage E), on the axis, at x/Dj=5.
Figure 5 presents the radial profiles of the mean mass fraction of helium versus the radius r non-dimensionalised by the half mass fraction radius L c. The comparison with the similarity law /Cc-eX p [-(r/Lc)2 ln2 ] (e.g. Hinze, [6]), where C c is the centreline mean mass fraction, shows that the mean mass fraction profile is self-similar from the section X/D.-5 j 9 This result confirms previous results obtained in the same set-up by Djeridane [7] with another technique, an aspirating probe measuring the mean oxygen partial pressure. As
456 the turbulence levels detected by the LDV and the hot-wire are not the same -there is an intrinsic turbulence level in LDV- a simplistic treatment of the raw data (E,U) may lead to C 1. The profile plotted with the open symbols corresponds to a treatment where data are truncated because the data where CI are discarded. For the profiles with the closed symbols, these data are conditioned by: if C 1), then C=0 (respectively C=I). The comparisons with Djeridane's data [7] and the similarity law validate this choice for the next results. The comparison with the profiles of mean temperature obtained in a slightly heated air jet [7] shows that the similarity is reached at X/Dj=5 in this case too. The initial density ratio 9j/Pe=l/7 in the case of the helium jet strongly influences the development of the jet. The mixing is more rapid when the central jet is lighter than the surrounding environment. In the case of the helium jet, the potential core is about 3 diameter long whereas it is about 5 diameter long in the air jet for Reynolds numbers in the same range (around 104). This result was deduced from the axial decrease of the mean mass fraction [7], [8], and it is confirmed by the measurement on the axis of the mass fraction turbulence intensity (figure 6). The mass fraction turbulence intensity of 2% detected at the exit section is the noise level of the present technique since the exhausting gas is pure helium with ideal uniform mass fraction, C=I and 1/2/Cc gives a plateau level of 23-25% between X/Dj-5 and 15 in a good agreement with results reported by Pitts [9] or Panchapakesan and Lumley [3]. Downstream of this station, a decrease of 50)
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Figure 7. Radial profiles of the mass fraction turbulence intensity in the helium jet, comparison with the temperature turbulence intensity in the slightly heated jet [5].
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Figure 9. Radial profiles of the radial turbulent fluxes of mass fraction in the helium jet, comparison with results in the slightly heated jet [5].
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Figure 10. Radial profiles of the turbulent Schmidt number in the helium jet. Comparison with the turbulent Prandtl number in the slightly heated jet [5].
458 Figure 7 presents the radial profiles of the mass fraction turbulence intensity. Similarity is reached at X/Di=15 in the helium jet. Comparison with the turbulence intensity of temperature confirms the-more rapid development of the scalar field when the central jet is lighter. Turbulent fluxes and are presented in figures 8 and 9. As for the case of the Reynolds stress , the maximum of the flux is always located around r/Lu=0.8 and associated to a maximum of turbulence production. The non-dimensionalised turbulent fluxes of mass fraction are always larger than the turbulent fluxes of temperature in the slightly heated jet in the same section. The dispersion with respect to the similarity profiles proposed by Panchapakesan and Lumley [3] does not allow any assessment of a self-similar behaviour of these quantities from X=I 5Dj in the helium jet. It must be noted that, even if the turbulent fluxes only represent a small percentage (o
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3-2. Swirl c h a r a c t e r i s t i c s in far w a k e ages. Swirl flow of the rotor wake has an attraction to inform how strongly the tip vortex influences the slipstream formation. The mixed flow of the tip vortex and the swirl flow with their two distinctive rotating axes perpendicular to each other provides an excellent illustration as to how 3-dimensional vortical motion develops in the rotor wake without any artificial effort. To observe such mixed field another set of measurements were carried out at six different ages, extending up to 2.67X/R, in the far wake.
Their time
averaged induced velocities are plotted in Figure 7. Peaks of the induced velocities appeared near 0.7 of the normalized radius and moved towards the root. The induced velocity accelerated near the root of the blade as the wake age grows. Swirl velocities are also plotted in Figure 8.
Their orders of magnitudes competed with about 20% of
the reduced ones. On the farthest data, 1.60, 2.13, 2.67 of X/R where R is the rotor radius, of this experiment in which the influence of the counter weight could be minimized, the swirl showed a weak similarity of the 3 rdorder polynomials.
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The induced and swirl turbulence components were also obtained in time averaged domains as in Figures 9, 10, respectively. As expected, the induced turbulence has the most peaks near the slipstream boundary and their tm'bulence levels were reduced rather drastically downstream while the mean behaviors stay unchanged. Similar profiles were obtained for swirl turbulences, but their strength became stronger as the wake got older. This contrasting behavior implies that beyond one time of the rotor diameter both turbulences become comparable orders or isotropic near the wake boundaries. Therefore, both rotating components established a hilly mixed mode, which means the correlation between the tip vortex and the swirl flow became most active in this stage. 0.06
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4. CORRELATION B E T W E E N TIP VORTICES AND SWIRLS 4.1 Vortex s h e e t on polar plots For the positive collective pitch, the rotor blade generated the strong induced flows and the weak swirls in the rotating direction of the rotor. The vortex sheet generated
477 by the bound circulation over the blade span built the helical space plane[11]. It is generally granted that once the sheet is formed, it moves fast to the wake, compared to the tip vortex trails, and the distance between the sheet and the tip vortex trail becomes greater even though they were generated by the same blade. But even though the vortex sheet comes away from the tip vortex with wake age, its boundary is still maintained by the vortex trails. Figure 11 was constructed by use of only the induced velocity components (or, circumferential velocities of the tip vortex coordinate) in the polar coordinates. They were obtained by the phase-average process at all azimuthal angles on the three fixed parallel planes to the tip path plane so that the tip vortices at the wakes, 48 o, 340 o and 670 o, respectively could be captured on the corresponded planes. Probe volumes were traversed along the fine grids in the radial direction of the rotor coordinate at the fixed rotor axial positions. Because only the induced or the circumferential velocity components were selected to build these sequential polar plots, the tip vortices were located at the angles where the maximum contrast of the circumferential velocity of the tip vortex appeared on each planes.
Figure 11. Polar plots for vortex sheet.. The 48 o tip vortex plane had the vortex sheet crossmg this plane near the angle of 20 o. It can be interpreted that the sheet arrived at the fixed probe just after the blade rotated 20 o (the angle tells time length) while the tip vortex arrived at the same position when
478 the blade swept 48 o. It tells that the sheet passed the measuring position faster than the tip vortex did. So the gap between them becomes wider as the wake age got older. At the 670 o plane the tip vortex met the vortex sheet generated by the blade passing about two revolutions ahead. We can easily perceive that the similm-phenomena will be repeated continuously downstream. The tip vortex was able to be captured until the wake age of 850 o by rare data samples as shown in Fig. 12, but it could not be discerned afterwards in the data set because it became fully mixed in the wake. This makes it possible to see how tip vortex contributes to the swirl motion at least before this age. To see how the tip vortex correlated to the swirl formation, the phase averaged, unsteady behaviors of individual swirls were also measured as shown in Figure 13. These phase averaged swirl velocities included the axial velocities of the corresponding tip vortices. The subtracted data by the time averaged swirls show solely the pure tip vortices as shown in Figure 14. Note that most tip voz~ices were appeared near 0.7 of the rotor radius and the tip vortex of 670 o was the strongest. This phenomenon also implies that at this wake age the tip vortex was strongly cozTelated to the swirl formation by meeting the vortex sheet. i.a
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9 a
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9
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N o r m a l i z e d radial location,
n -0.2
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Figure 14. Tip vortex effect on swirls.
,
,
." 0.0
0.0
9, % . 2 ~ -1.0
. . . . . . . . . . . . .0.8
Normalized
.0.6
-0.4
radial l o c a t i o n ,
, .0.2
0.0
Yr I R
Figure 15. Correlation of mean velocities.
479 Correlation rates In far downstream, two kinds of rotating flows, whose axes are perpendicular, became fully mixed in a three-dimensional way. Hence, their mixing rate can be quantified by the correlation function as, 4. 2
2
>< W 2 >
where the parenthesis means the time averaged values. The calculation focused on the sole tip vortex of 6700 at which angle the communications between two rotating components were most active(figure 14). With the circumferential velocity component of the tip vortex, U and the swirl velocity component of the rotor, W the calculation results of the above equation can be used to unveil how the tip vortex contributes or degenerates the swirl formation. The rates calculated by their mean data along the rotor radius is shown in Figure 15. Within 0.7 of the rotor radius, the two rotating components mixed so energetically that they developed a fully 3-dimensional vortical wake at this wake age. Similar flow behavior can be seen easily afterwards. The profile of Reynolds stress for both components had the maximum gradient near the wake boundary as shown in Figure 16, which convinces that the turbulent mixing at the boundary of the slipstream was the source of the wake turbulence, and that tip vortex stimulates the turbulence level near the wake boundary by coupling the swirl motion. By such an action, the turbulent correlation of both components got higher rates near the boundary m contrast to the mean actions. It was shown in Figure 17. 0.4
0.0006 J
N
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I
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.
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Figure 16. Reynolds' stress at 670 ~.
0.0
0.0 -1.2
.
.
.
- I .0
.
.
.
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9 -0.6
9 -0.4
Normalized radial location,
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Figure 17. Correlation of turbulences.
"_
0.0
480 5. CONCLUSION The tip vortex and vortex sheet generated by one blade rotor and their evolving and correlating behavior with respect to wake age was examined using the LDV system. The following conclusions have been drawn from this work. 1) A strong similarity based on the n = 2 profile exists for the circumferential component of the tip vortex until the wake age of 850 ~ Also, a weak similarity to Gaussian profiles with the base-width being about 3 r~(core radius) for the axial component was observed within one revolution of the blade. 2) Profiles of turbulences and higher order moments of the tip vortex generated by the blade with the aspect ratio of 4.2 widened radially in extent while their peaks diminished in magnitudes, compared to those of a long span blade. Hence, turbulence became more isotropic with a short span blade. It also confirms that they are dependent on the aspect ratio of the blade. 3) The vortex sheet generated 670 ~ ahead met the tip vortex and encouraged a threedimensional vortical mixing near 0.7 of X/R. The correlation of the circumferential and the swirl turbulence components were a higher rate near the wake boundary, whereas the mean velocities obtained higher correlation reside the wake. It explains that most downstream turbulence of the rotor slipstream was produced near the wake boundary. This work was supported by the academic research fund of Ministry of Education, Republic of Korea(ME97-B-08). The support is greatly acknowledged. REFERENCES
1. F. H. Schmitz, NASA Reference Publication 1258(1991) Chap.2. 2. G. L. Crouse, J. G. Leishman and N. Bi, J. of AHS Vo137, No1(1992)55. 3. S. Dawson, A. Hassan, F. Straub and H. Tadghighi, NASAR~port195078(1995).. 4. P.I. Singh and M. S. Uberoi, Physics of Fluids, Vo119, No12(1976)1858. 5. T. Sarpkaya and D. E. Neubert, AIAA Journal, Vo132, No3,(1994)594. 6. A. D. Curtler and P. Bradshaw, Experiments in Fluids, Vo112(1991)17. 7. C. Tung, S. L. Pucci, F. X. Caradonna and H. A. Morse, Vertica, Vo17, No1, (1983)33. 8. Y. O. Han, J. G. Leishman and A. Coyne, AIAA Journal, Vo135, No3, (1997)477. 9. G. H. Vatistas, V. Kozel and W. C. Mih, Experiments in Fluids, Vo111, No1, (1991)73. 10. W. J. Devenport, M. C. Rife, S. I. Liapis and G. J. Follm, JFM, Vo1312(1996)67. 11. K. Ramachandran, C. Tung and F.X. Caradonna, J. of Aircraft, Vo126, No12 (1989) 1105.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
481
Effects of adverse pressure gradient on quasi-coherent structures in turbulent boundary layer T. Houra a, T. TsujP and Y. Nagano b ~Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan bDepartment of Environmental Technology, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan Statistical and kinematic characteristics of turbulent boundary layer flows subjected to adverse pressure gradients are found to differ significantly from those of zero-pressure-gradient ordinary boundary layers. The turbulent energy and shear stress transport v u 2 and v u v occurs in the direction toward the wall from the regions away from the wall, as the pressure gradient parameter P+ increases. This event is completely opposite to that in zero-pressure-gradient cases. The instantaneous waveforms of third-order moments are markedly altered in the nearwall region. The quadrant splitting and trajectory analyses reveal the obvious changes do occur in large-amplitude sweep motions (Q4) and outward interactions (Q1). On the other hand, the contributions from other coherent motions, especially the ejection motions (Q2), significantly decrease and their durations become longer, i.e., these motions are dull and less active. Moreover, multi-point simultaneous measurements with five X-probes are made to depict the kinematic pictures of the effects of the adverse pressure gradient on the eddy structures. 1. I N T R O D U C T I O N The efficiency of fluid machinery, such as a diffuser and turbine blades, is often restricted by the occurrence of separation due to a pressure rise in the flow direction. Therefore, it is very important to elucidate the effects of adverse-pressure-gradient (APG) and establish a flow control method to avoid flow separation. It has been confirmed that near-wall quasi-coherent structures in a zero-pressure-gradient (ZPG) flow play a key role in the turbulent transport mechanism (Robinson [1 ]), and there has been an increasing number of literature dealing with so-called boundary-layer-control in such a flow. However, we have only scanty information on the quasicoherent structures indispensable for controlling turbulence of APG flow. In the present study, to obtain useful information on controlling turbulence, we have investigated the effects of the APG on the quasi-coherent structures, by using the quadrant splitting (Lu and Willmarth [2]) and trajectory analysis techniques (Nagano and Tagawa [3]). Moreover, to gain a deeper insight into the effects of the APG on the eddy structures, multi-point simultaneous measurement with five X-probes has been carried out. A very accurate interpolation method by utilizing the Karhunen-Lobve expansion is developed to construct an instantaneous flow field from the multi-point information.
482 Table 1 Flow parameters (U0 = 11.0 m/s)
ZPG APG
X
Ue
(~99
UT
mm 538 937 536 734 933
rrgs 11.0 11.0 8.81 8.02 7.33
mm 13.4 20.2 17.5 25.5 36.7
m/s 0.490 0.470 0.365 0.297 0.231
Ro
P+
/3
1210 1780 1430 2110 2950
0 0 1.14• -2 2.04• 10 -2 3.08• -2
0 0 1.05 2.62 5.18
2. EXPERIMENTAL APPARATUS The experimental apparatus used is the same as in Nagano et al. [4, 5]. The test section is composed of a fiat-plate on which an air-flow turbulent boundary layer develops, and a roofplate to adjust pressure gradients. The aspect ratio at the inlet to the test section is 13.8 (50.7 mm high • 700 mm wide). Under the present measurement conditions, the free-stream turbulence level is below 0.15% and velocity non-uniformities in the y-(normal to the wall) and z-(spanwise) directions are within 0.17% and 0.63%, respectively. Therefore, nearly ideal, twodimensional uniform inflow is obtained. To generate a stable turbulent boundary layer, a row of equilateral triangle plates is located at the inlet to the test section as a tripping device. It is confirmed that even at the end of the test section the boundary layer on the pressure-adjusting roof-plate is separated by the uniform free-stream from the objective turbulent boundary layer developing on the fiat-plate. Thus, there are no interactions between the two (see Nagano et al. [41). Velocity measurement was done with hot-wire probes, i.e., a handmade subminiature (Ligrani and Bradshaw [6]) normal hot-wire (diameter: 3.1 #m; length: 0.6 mm), and a specially devised X-probe for measurement of two velocity components in the streamwise (z) and wall-normal (y) directions (diameter: 3.1 #m; length: 0.6 mm ~ 7.5 v/u~-; and spacing" 0.30 mm ~ 3.8 v/u.,-). An array of five X-probes aligned in the wall-normal direction is employed for simultaneous measurement in a flow field. To convert the hot-wire outputs into the velocity components, we used the well-established look-up-table method (Lueptow et al. [7]). Also, the bias error, which is ascribed to the finite separation of the wires in using an X-probe, was removed in accordance with the procedure indicated by Tagawa et al. [8]. As a result, the measured velocity fluctuations near the wall in the ZPG flow show good quantitative agreement with the DNS data (Spalart [9]) of the ZPG flow (see Nagano et al. [4]). The important flow parameters of the present measurement are listed in Table 1. In the APG flow, the pressure gradient dCp/dx [Cp --= ( P - P0)/(pU~/2), and P, P0 and Uo are the mean pressure, the reference inlet pressure, and the reference inlet velocity, respectively] keeps a nearly constant value of 0.6 m -1 over the region 65 mm _< z _< 700 mm and then decreases slowly (x is the streamwise distance from a tripping point). On the other hand, the pressure gradient parameter normalized by inner variables P + [ = v(dP/dx)/p@] and the Clauser parameter/3 [= (6*/Tw)d-P/dx] increase monotonously, thus yielding moderate to strong adverse pressure gradients.
483 3.0 ,,, ~'~ 2.0 [~
........ , [] P~=O o P~=1.14x10 -2 p4=2.04xlO -2
,
,~ .....
'
'
'
'
' ' I
0 0 0
~%
~ ~ ~
~Oo ~_
O0
'
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-1.0 -2.0
" ""
-1.0
'
9
9 9
" p ' =3.08 xlO 2
1.0
'
''1
1.0
0
[] P~=O o P~=I.14xlO -2 P~ =2.04 xlO 2
~
9 P~=3.08xlO -2
-3.0 ' " ........ ' ........ 10 102 103 10 102 10 3 y~ y~ (a) (b) Figure 1. Distributions of turbulent transport (third-order moments) in adverse-pressuregradient flows: (a) v u 2 ; (b) v u v . i iI
i
i
i
i
i
i i iI
i
i
i
i
i
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3. RESULTS AND D I S C U S S I O N 3.1. T u r b u l e n t t r a n s p o r t
The structural differences in quasi-coherent motions reflect on higher-order turbulent statistics, especially third-order moments (Nagano and Tagawa [10, 11 ]). Figures 1(a) and (b) show the time-averaged turbulent transport, i.e., third-order moments, of turbulent energy component u 2 and Reynolds shear stress u v , respectively (u, v: streamwise and wall-normal fluctuating velocities). The definite effects of the APG are clearly seen on both third-order moments. As shown in Fig. l(a), the positive region of v u 2 in the ZPG flows for y+(-- u ~ y / u ) > 15 disappears partly as P + increases. Since third-order moments are predominately sensitive to the change of coherent structures such as ejections and sweeps (Nagano and Tagawa [10]), this result indicates that internal structural changes do occur in the APG boundary layers. Negative values of v u 2 in the near-wall to outer regions demonstrate the existence of turbulent energy transport toward the wall from the regions away from the wall. This important characteristic of the APG flows conforms to our previous result (Nagano et al. [4, 5]), and is also consistent with the results of Bradshaw [ 12], Cutler and Johnston [ 13], and Sk~re and Krogstad [ 14]. From Fig. 1(b), it can be seen that a similar inward transfer takes place in the turbulent transport of the Reynolds shear stress. It should be noted that, as P + increases, turbulent transport in the APG boundary layer occurs in the direction completely opposite to that in the ZPG case. 3.2. I n s t a n t a n e o u s c h a r a c t e r i s t i c s o f t u r b u l e n c e q u a n t i t i e s
To understand the above features of the APG flows in more detail, we have investigated the instantaneous characteristics of the coherent motions. Figures 2(a) and (b) show instantaneous signals of u v , v u 2 and v u v together with u- and v-fluctuations at y+ _~ 30 where the timeaveraged values of triple products v u 2 and ~ become nearly maximum in the ZPG flow [see Figs. 1(a) and (b)]. In this figure and also in what follows, a circumflex denotes the normalization by the respective r.m.s, value. The time on the abscissa is normalized by the Taylor F
/
time scale ~-E ~= V/2 u 2 / ( O u / O t )
2 ,
which is the most appropriate for scaling the period of the
L
coherent motions, irrespective of pressure gradients (Nagano et al. [5]). The mean burst periods have nearly the same value of 10 7-E in both ZPG and APG flows at this location. In the ZPG flow, large-amplitude fluctuations of the triple products are generally associated with the fluid motions categorized as the second- and fourth-quadrant events in the (u, v)-plane,
484
z2
4 ~u_%/~,~ 0 -4 4 ~.,
,.......
-4
4~ 0 .~x~-~,_ . . . . %4~,-,rtt~A~-,_ . . . . . . . .A,,^. . r -4 4 [- ./t ,^^ . . . . . ~4. . . . A .L~. . . . . . .~, ^. . . . 13 0 ff'~/'~l"v"-"~ '~ ,new, ~y-,'~, ,'-'v" "w,v--"
. . . . ,_-~vv ^_ m,. -,,s. k/.~-x..b,-Wv
z2
A . . . . A., %%.r ~.,,t~,A,
-4
10E.
S~b
10
30 -30 L 30 o
v
F
-'-"
- .....
- ......
~
-.
. . . .
-30 I'-
'b~,~3
o E
,,_,,_ . . . . . . .
o.%.
~o.21 INto. 0
--~--i=l
.10_.
-30
----o-
3
S----0"3~. t~0"2[-~.
H
3
~,
........
,.....
,_10_. (a) . . . . . (b) . . . . . Figure 2. Simultaneous signal traces of third-order moments ~3122and ~3~3, time being normalized by Taylor time scale TE (y+ "~ 30)" (a) P + - 0; (b) P + - 3.08 x 10 -2.
'
Time
.
-10 ~ v',,1-,, . . . . . . . " ~ t . . . . . . . . . . . . 30 E -30 . . . . . . . 30
1
2
-~- i-1 ~-2--'--
1 2 (b) Figure 3. Mean frequencies of events normalized by 7-E in the log region (y+ -'~ 50)" (a) P + - 0 ; (b) P + - 3.08 • 10 -2.
(a)
0
Time
-o-
3 4
H
3
i.e., the ejection (Q2) and sweep (Q4) motions. In Fig. 2(a), very large-amplitude fluctuations of v u 2 and v u v are skewed toward the positive and negative sides, respectively, thus indicating that the ejection (Q2) is the principal contributor here. On the other hand, as shown in Fig. 2(b), in the APG flow the sweep motions (Q4) occur much more frequently than the ejections (Q2); and besides, the large-amplitude outward interaction (Q1) is no longer negligible. Thus, the fluctuations of v u 2 and v u v a r e respectively skewed to the negative and positive sides, which is quite opposite to the ZPG flow case. Next, we investigate the temporal differences between the ZPG and APG flows. Figures 3(a) and (b) show the dimensionless mean frequencies ( - T i / T E ) - l o f each event normalized by the Taylor time scale TE in the log region (y+ -'~ 50), as a function of a threshold level H ( = At H - 0, the frequencies of each motion are essentially equivalent. As the threshold level H becomes larger, the frequency of the Q2-motion (ejection) exceeds that of the Q4-motion (sweep) in the ZPG flow. In the APG flow, however, the sweep-type Q4-motion is dominant at any threshold level and the Q l - m o t i o n (outward interaction) occurs more frequently than the Q3-motion (wallward interaction). This corresponds to the fact that the large-amplitude Q4- and Ql-motions of shorter duration occur in the APG flow, as will be discussed later.
3.3. Fractional contribution The fractional contributions of different quadrant motions to v u 2 and v u v , normalized by the friction velocity u~-, are shown in Figs. 4 and 5, respectively. Lines in the figures represent the theoretical predictions based on a sophisticated cumulant-discard method (Nagano and
485 3.0
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-1.0 " ' ' ~ ' - 7 " 2 + "3 -2.0 _ %" "Experiment a, 9 o ,, Prediction , 7 :, :-, : -3.0 10 102
0
s ~S
O~O.lk '0~ -.0, .0.
i
9
,
O~
-5.0
4
: ....
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O.
_ 4"
-10.0 ' " 102 y~ 103 lO (a) (b) Figure 4. Fractional contributions of different quadrant motions to the third-order moment v u 2" (a) P+ - O; (b) P+ - 3.08 x 10 -2. 1.5 ,,,
~ ~"
....
i'--~1'"2
y,
:3
'4 . . . . . . I
Experiment G, @ o 9 Prediction . . . . . . . . . . .
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,
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,
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,
,,
i
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I
ip,..,1..o,- 1"... - ,,,.ill
. , r . . . ~ 'r"
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0
-2.0
-
-1
-4.0 -
-1.0
| | , , ,, , , i , , || -6.0 "~ 102 103 10 y' Y' (b) Figure 5. Fractional contributions of different quadrant motions to the third-order moment v u v : (a) P+ - O; (b) P+ - 3.08 >< 10 -2.
-1.5
,i
10 (a)
,
,
,
,
, ,,,i
102
. . . . . . . .
i
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1
103
Tagawa [10]), which may be written as: 1
lTp ~v,i .,,.m+q Cpq Bt,p B m q (~t~m)i = 27r p~4' - J ou,i p,q--O
with Bj,+ =
/o + ~JH~(~)
(1)
exp(-x~/2)d~,
where cru,i and av,i are sign functions which represent the signs of u and v of the ith quadrant in the (u, v)-plane. Cpq are coefficients determined from the measured correlation up to the fourth order, and Hn (X) is an Hermite polynomial. The theoretical predictions correspond well with the experimental data in both ZPG and APG flows. In the ZPG flow, as is apparent from Figs. 4(a) and 5(a), the turbulent transport of u 2 and u v in the wall-normal direction is dominated mainly by the Q2- and Q4-motions. Since the net values of the triple products are determined by the disparity in contributions between these two types of motions, v u 2 and v u v result in the positive and negative values, respectively, in the log region (y+ _~ 50) of the ZPG flow where the ejections (Q2) become larger than the sweeps (Q4). However, in the APG flow [Figs. 4(b) and 5(b)], the contributions from the coherent motions, especially the ejections (Q2), significantly decrease. Thus, the triple products v u 2 and v u v in the log region of the APG flow have net values different from those in the ZPG flow, i.e., v u 2 and ~ become negative and positive, respectively [see Figs. 1(a) and (b)].
486
3.4. Weighted p.d.f.s of triple p r o d u c t s The analysis of fl'actional contributions in the (u, v)-plane alone is not sufficient to ascertain the detailed changes of quasi-coherent structures in the APG flow. Hence, we investigate the weighted p.d.f.s of triple products vu ~ and ~ in the (u, v)-plane (Nagano and Tagawa [10, 11])
Iu
~) =
xP(,~, ~),
x = ~2
and ~ ,
(2)
where P(~, ~) is the joint p.d.f, for u- and v-fluctuations. The integrated value of W~(~, ~) in each quadrant becomes the fractional contribution (x)~, and integration over the whole (u, v)plane reduces to the conventional time-averaged value ~. The typical distributions of W~ for x = ~ and Ofi~ in the log region (y+ _~ 50) are shown in Figs. 6 and 7, respectively. In the contour maps of Figs. 6 and 7, solid and broken lines respectively represent positive and negative values, and the interval between successive contour lines is 0.01. In the APG flow, the amplitude of sweep motions becomes much larger than that of ejections in the log region. Furthermore, the extent of the p.d.f, in the outward interaction (Q1) also becomes larger in the APG case. In the outer region, however, these striking differences are not seen between the ZPG and APG flows (not shown here). 4
(
4
) , < ~ >, , and of the identified key patterns (Q2-Q1-Q4 and Q4-Q3-Q2) and the sub-patterns (Q4-Q1-Q4 and Q2-Q3-Q2) in the log region (y+ _~ 50), respectively. In the APG flow, the amplitude of the third-order moments decreases in the Q2-motion and increases in the Q4- and Ql-motions of each pattern. And the duration of the Q4-Q1-Q4 pattern and that of the Q2-Q3-Q2 pattern, become shorter and longer in the APG flow, respectively. Moreover, the frequencies of occurrence differ among these representative patterns. Compared with the ZPG flow, patterns in the clockwise order in the (u, v)-plane (i.e., Q2-Q1-Q4 and Q4-Q3-Q2) do not occur so frequently. However, the Q4-Q1-Q4 pattern is found to occur much more frequently in the APG flow.
1 f ' " ' - ~ . ' . ' . ~ 0 ', ' ' -1 ~ " " - .... "" t /r?
03~~"
t I~I/u,, = 1
are : < p > < Ur, U,.k > -
2
--zany: < p > < k > +#t < &k > O
The turbulent viscosity is: # t -
Ct, ft, 2 27S
Equations for the turbulent kinetic energy < k > and the dissipation rate < e > are identical to classical ones. It is worth pointing out that Jones and Launder's k - c model was originally developed for the classic approach. However, the incoherent Reynolds stresses < Ur~Ur~ > to be modelled are only a part of the classical Reynolds stresses u~u~, since u~u~ - < u,.,u,.~ > + ~]i~]j. For this reason it is necessary to modify the model. In this work, we simply adjust the constant C, which is directly linked to the turbulent viscosity to 0, 06, a value obtained by numerical tests [7]. The other constants in the equations are kept the same as their original values. This kind of modification is the simplest one. It must be understood as
494 the first stage of fitting classical models to this new decomposition. Nevertheless, we will check that coherent values obtained with this slight modification are close to experimental results. 3. N U M E R I C A L
METHOD
The computation code used is based on the explicit MacCormack scheme which is second order accurate in time and in space. The grid contains 1 2 0 . 100 points, and is non-uniform both in x and y directions. Since the development of the Kelvin-Helmholtz type instabilities in the mixing layer depends directly on the boundary layer in the inlet of the backward facing step, it has to be refined in this region. The grid is also stretched near the solid wall in order to allow the application of the near-wall damping functions in the turbulence model. In the inlet section (x = - 1 H ) , the velocity profile < U(y) > is obtained by interpolating the experimental results. However, From the wall (y = 0) to the first measurement point it is obtained from the simulation of a flow over the flat plate by a parabolic code. The profiles for < k > and < c > are also constructed in the same way. An example of space map of coherent vorticity deduced from semi-deterministic modelling is given in figure 1. 4. E X P E R I M E N T A L
APPROACH
4.1. E x p e r i m e n t a l a p p a r a t u s The backward facing step model is fixed in the middle of the wind-tunnel $10 of C.E.A.Toulouse. The test volume is 1 m wide by 2.2 m height by 2 m long. The plate where initial boundary layer is developing is 0.45 m, the reattachment plate is 1.2 m and the step height is 65 mm. The aspect ratio is more than 10 and so, ensures that we simulate a flow in an infinitely large tunnel. Reference velocity is 40 m/s (Reynolds number based on step height is about 170000, Mach number 0.12). Some preliminary Laser Doppler Velocimetry measurements gave access to mean velocity fields and help to check that the flow is two dimensional as well as to find reattachment length (Xr = 5.7 H) and
Figure 1. Example of space map of numerical coherent vorticity
495 adequate position of the rakes in the flow. Then, an experimental study was developed using 8 X hot wires rakes in the mean-gradient direction behind the backward facing-step. The rake is located in the upper part of the separated layer, where the longitudinal velocity is positive for two reasons : the former, some visualisations (in initial laminar flows) show that CSs are confined in this upper part (Zaffalon [9]) and the latter, hot wires do not allow to determine the velocity sign in the separated bubble. The measurements of instantaneous streamwise and transverse velocity with a sampling frequency of 12 kHz during about 40 seconds allow a good definition of the CS's transit and provide a number of detected events high enough for statistical treatments. The Table 1 presents features of measurements: location and extent of rake (AY), intervals between wires (Ay) and vorticity thickness (5~): Table 1 Characteristics of measurements
AY(mm)
Ay(mm)
5~(mm)
25 25 50 75
3.2 3.2 6.4 9.6
15 20 35 ?
x/H-1.2 x/H=2.2 x/H=4.2 x/H=6.2
4.2. T h e v o r t i c i t y - b a s e d conditional m e t h o d This procedure is based on the instantaneous spanwise vorticity (using the Taylor hypothesis in the flow direction and finite difference schemes for evaluating derivative terms). It gives us a time display of the vorticity along the transverse length of the rake. We apply a numerical filter (LP without phase difference) to smooth out high frequencies and improve the definition of high vorticity areas. Practically, we use a method very close to the vorticity-based conditional sampling technique developed by Hayakawa [6] : from a discrete time series of simultaneous U and V signals at N locations separated by Ay, during T with a time step At, we define the instantaneous vorticity :
1 z(t, y) -
= E
Vi+l,j -- Vi-l,j
2At
Ui,j+l
-
-- Ui,j-1
2/ y
(5)
U~ is the average convection velocity of CS's in the longitudinal direction (U~ - 0.5 x Um~• in our study). We obtain spatio-temporal maps of vorticity fields. To isolate CS features from the complete motion, we impose a threshold Th on vorticity to select areas where their magnitude is strong. Positions (t~, y~) of maximum amplitude in these areas are supposed to be the centers of CS's:
~z(tc, yc) > Th
and
wz(t~, y~) - Max,o~a, Wz(t, y)
(6)
Then, we obtain a spatio-temporal matrix I that indicate center positions:
if
Ii,j - 1,
t ~ - i x At
and
yc-j
x Ay
else
I i , j - O,
(7)
496 The application of phase average on instantaneous signals allow to extract the coherent motion (f) and the pure random motion ft. Furthermore, the coherent motion is assumed to be the CS's motion. We choose not to class as the same event structures centered on different transverse positions. These CS's indeed have distinct signatures (Aubrun [4]). So, to determine the mean CS centered on a fixed transverse location, we respectively align original unfiltered realizations of all transverse positions with respect to each center of reference position and then, we apply ensemble-average on U and V. For instance, to educe the mean CS centered on the probe 4, the detection signal is reduced to vector Ii,4. For 7- time delay with respect to the center of structure, the phase-average operator is: 1
tN
f~(r, y ) = (f(t, y ) ) - (fi,j) - -~ ~ fi+~,j
(8)
t--t1
Two criteria are also implemented : - the shape criteria which consists on using the first-step phase-average vorticity wcz as reference scheme to eliminate all events too different of the expected shape. - the size criteria which eliminates too small events. Too large events are naturally smoothed out by filtering. 5. R E S U L T S 5.1. p r e l i m i n a r y r e m a r k s It is important to mention that even though coherent structures inferred from Semideterministic modelling and from space-time conditional sampling are comparable, they are not completely identical: - Solving ensemble-averaged Navier-Stokes equations assumes that the unsteadiness of the motion is periodic and enables us to access to the global 2-D spatial field of coherent motion. - Experimentally, we have to build a detection criteria to detect the spatio-temporal center of each structure (reference time) and apply, a posteriori, the phase average operator. Because, in reality, the unsteadiness is pseudo-periodic, the coherent motion obtained possesses a bias as soon as one moves away from the reference instant. Futhermore, to correctly compare with numerical results, we convert the time direction into a longitudinal direction with the Taylor hypothesis: y) --
v)
(9)
5.2. C o m p a r i s o n of t i m e - a v e r a g e d r e s u l t s To validate the mean velocity, turbulent intensities and Reynolds stresses fields deduced from numerical approach, we compare some profiles (in sections multiple of the step height H) with the ones obtained by Laser doppler velocimetry technique (figure 2). The figure (a) shows the velocity < U >/Uo from X / H = 1 to X / H = 6.
It is obvious that the numerical results are similar to the experimental ones in the first two sections which are still far from the reattachment zone. From X / H = 3 to X / H = 6 the profiles reveal that the shear layer develops too slowly due to the reduction of the
497
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b)
2
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0
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d)
2
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--
1 0.5 0 0
2
4
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6
8
0
2
4
6
8
Figure 2. comparison between numerical (solid line) and experimental (dotted line) results about a) mean velocity ~/Uo, b) and c) turbulent intensities u"---2/U~and -V-~/U~,d) Reynolds stresses u'v'/U~
constant C u. And for this reason the reattachment length predicted numerically(6.15H) is larger than the experimental result(5.7H). The figures (b)-(d) represent the -~/Uo 2, -~/Uo 2 and u'v'/Uo 2 profiles. The underestimation of these quantities comes also from the diminution of Cu. In fact, when Cu is reduced, the quantity < Ur~U,.~> reduces, too. In contrary, the quantity ~ j which represents the coherent motion should increase. In this work, the coherent motion indeed increases due to the diminution of the turbulent viscosity but not sufficiently. This explains that the simulation results are much smaller than the experimental ones.
5.3. Comparison of unsteady results The first stage is to compare characteristic frequencies of the unsteady motion which are associated with the transit of coherent structures. Experimental and numerical results are given in Table 2. Frequencies are very close to experimental data. It means that the semi-deterministic modelling enables to educe the real unsteady motion. We now look more precisely at the respective distribution of coherent motions in both cases (numerical and experimental). We observe that, whatever the studied section, the qualitative distribution of coherent motion for an isolated (not in pairing process) coherent structure is similar. However, the growth of coherent structures is more important in the numerical case. Consequently, we choose to describe the coherent motion only in the fixed section X/H = 2.2, where the sizes of the numerical and experimental structure are
498 Table 2 Comparison of characteristic frequencies X / H - 1.2 X / H = 2.2 experimental 372 285 numerical 390 230
X / H = 4.2
X / H - 6.2
164 170
78 79
comparable (figure 3). We compare numerical and experimental coherent vorticity 9 • h/Uo, unsteady coherent longitudinal and transversal velocities (t/Uo and ~/Uo , the incoherent kinetic energy restricted to 2D 1/2(u'--~ + -v--~)/U~ and the production of incoherent motion from coherent motion [-(Ur~)~ Ox - ( v ~ ) ~Oy - (u~v~)(~ Oy + ~ Ox) ] • h/U 3 " Reader has to keep in mind that the experimental longitudinal direction is obtained by the Taylor hypothesis. At first, it is important to mention that these distributions are typical of vortical structures in shear layer (even in plane mixing layer [5]). So, as expected, in each case, the concentration of coherent vorticity (definition of a coherent structure in a shear layer) is well defined and is associated with a distribution of coherent velocities characterized by over and under-velocity areas. These ones are indicative of vortical events and assert that the peak of vorticity is not only due to monotonous shear effect. Quantitatively, numerical magnitudes are noticeably lower experimental ones. The incoherent kinetic energy and the production are both characterized by maximum of magnitude at saddle-points, at each sides of coherent structure and have a minimum value at the center. Similar distributions have been validated in [4]. Contrary to vorticity and velocities, their experimental and numerical magnitudes are identical. It seems that the semi-deterministic modelling is well adapted to predict the unsteady motion (see characteristic frequencies), the qualitative distribution of coherent and incoherent motions and that it correctly verifies magnitudes of incoherent energy distribution and coherent to incoherent energy transfers. Only the magnitudes of unsteady coherent velocity and thus of vorticity are under-estimated. This fact entails that in time-averaged results, when numerical turbulence intensities and Reynolds stresses(u~u~) are lower than experimental ones, only the coherent contribution (uiuj) is really under-estimated. 6. C O N C L U S I O N Our study had two goals: Using a conditional method to educe experimental coherent motion and separately, developing the semi-deterministic modelling that directly provides the numerical coherent motion in the same configuration of the backward facing step flow. - Comparing precisely numerical and experimental results. Concerning the differences with plane mixing layer, despite an ambiant turbulent level much higher, we have checked, that similar vortical coherent structures can be identified
-
499 and that the higher turbulence level in the separated flow does not alter their coherence. The semi-deterministic approach is a rather rough approach based on a simple generalization of k - c from steady to unsteady case, but it correctly predicts the coherent motion in this flow. Characteristic frequencies and morphologies of coherent structures are in agreement with experimental ones. In contrary, two differences exist about the evolution of the size of coherent events when they go towards the reattachment area (the growth of numerical coherent structures is higher) and the magnitudes of coherent vorticity and velocities (experimental structures are more amplified). Nevertheless, the comparison between numerical and experimental coherent motions has not been made previously for this situation and it shows that the semi-deterministic approach is able to predict the coherent motion, even though some modelling improvements are necessary to improve the introduction of the characteristics of incoherent turbulence. In the work of Kourta [3] modelling coefficients are not constant values but explicit timedependant functions of both ensemble-averaged strain and rotation. First results are promising. REFERENCES 1. H. Ha Minh, Order and disorder in turbulent flows:their impact on turbulence modelling. Osborne Reynolds Centenary symposium. UMIST- Manchester, May, 24, 1994. 2. A.K.M.F Hussain, Coherent structures, reality and myth!, Phys. Fluids 26 (10), october 1983. 3. A. Kourta, Computation of vortex shedding in solid rocket motors using timedependent turbulence model. Journal of Propulsion and Power, Vol. 15, No. 1, januaryfebruary 1999. 4. S. Aubrun, Etude exp~rimentale des structures coh~rentes dans un ~coulement turbulent d~coll~ et comparaison avec une couche de m~lange. Th~se de I'INPT. Toulouse, France, january, 28, 1998. 5. E. Vincendeau, Analyse conditionnelle et estimation stochastique appliqu~es k l'~tude des structures coh~rentes dans la couche de m~lange. Th~se soutenue s l'universit~ de Poitiers, France. 1995. 6. M. Hayakawa, Vorticity-based conditional sampling for identification of large-scale vortical structures in turbulent shear flows. Dans Bonnet J.P. and Glauser M.N. Eds., 1993, "Eddy structure identification in free turbulent shear flows", Proceedings of the IUTAM Symposium- POITIERS Octobre 1992, Kluwer Academic Publishers. 7. P.L. Kao, Etude num~rique des instabilit~s convectives et des structures coh~rentes dans des couches de m~lange libres ou d~coll~es. Th~se de I'INPT. Toulouse, France, may, 29, 1998. 8. W.P. Jones, B.E. Launder, The prediction of laminarization with a two-equation model of turbulence. Int. Heat and Mass Transferts, No. 5, pp 301-314, 1973. 9. P. Zaffalon, Contribution l'~tude a~rodynamique d'un ~coulement de marche. M~moire de diplSme d'ing~nieur C.N.A.M. Centre R~gional associ~ de Toulouse. 1993
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
501
Experimental investigation of the coolant flow in a simplified reciprocating engine cylinder head D. CHARTRAIN, A.-M. DOISY, M. GUILBAUD and J.- P. BONNET CEAT- Universit6 de Poitiers Laboratoire d'Etudes A&odynamiques-UMR CNRS n~ 43 rue de rA&odrome, 86036 Poitiers Cedex, France The objective of this study is to develop an experimental study of the coolant flow in a simplified transparent model of an internal combustion engine cylinder head in order to build a data bank to validate CFD codes. Simple laws of engineering relevant for pressure losses are provided. A set of two-dimensional cylindrical tubes, perpendicular to the flow models the cooling circuit of a 16 valves cylinder head. The test fig-set-up consists in a closed circuit with a pump. Two measurement techniques, Laser Doppler Anemometer (LDV) and Particle Image Velocimeter (PIV) have been used. Mean values, moments and vorticity have been determined. Pressure on the various cylinders and head pressure losses have also been measured. Moreover, the effect of separation on downstream tubes have been investigated. As a simple test, comparison with a conventional k-e code is provided. 1.INTRODUCTION For car builders, the major research topics for reciprocating engines concern the improvement of the performances and the decrease of the pollution due to exhaust emissions from motors. These improvements require the optimisation of every domain involved in the engine conception such as internal aerodynamics, acoustics, strength of material, hydraulics, aeraulics, ...with different constraints in each domain. For most of the design divisions of the factories, the constraints concerning the engine cooling are in general not considered of major interest and are only partially taken into account, in general only after having def'mitely chosen the engine geometry. Consequently, engineers in charge of the cooling circuit design generally use only a cut and try approach based on the knowledge acquired during previous conceptions and on defects revealed during endurance tests, as for example observation of cracks. Thus, the modifications proposed are only of weak importance and involve small change of the initial concept: for example modifications of the size of the pipes in the cylinder head, of the cylinder head gasket or by locating some pipe restrictions in the remaining space. The objective of these improvements can be to force the flow in region where the tests have shown the thermal exchanges to be too low. This methodology is no longer satisfactory for the engineers facing more and more severe conflicting economical and performance constraints. Recent advances in experimental methods and numerical predictions make more rational approaches possible. Several studies deal with engine coolant, Hoag[1], Finlay[2], French[3], Shalev[4], Aoyagi[5], Davis[6], Bederaux-Cayne[7], and Priede[8]. The conclusions of most of the authors point out that it is necessary to develop a new and rationale methodology for a better integration of the cooling problems in the engine design. Due to the progress of computational fluid dynamics, new approaches can be now used to optimise the coolant system, giving accurate informations about the flow structure in the cylinder head/block. These computations can be associated with the more advanced experimental techniques. As the geometry is 3D
502 and quite complex, numerical codes have to be checked on simplified models. This has been done, for example by comparing computations and measurements, Sandford [9] using LDV and PIV, or Coll6oc[10], [ 1]. As usual for the computation of such internal flows, the results are very sensitive to the turbulence model and the wall function chosen. Some papers give results on real models, [1], Liu[11] or Arcoumanis[12] using transparent engine and as cooling fluid, a fluid having same index of refraction that the model walls. These authors use the refractive index matching technique for optical methods. The study presented here is aimed to define a specific experimental methodology. For the cooling purpose, the objective is, indeed, to insure an optimal circulation of the cooling fluid in order to have an efficient heat rejection of the heat developed in the cylinder block and the cylinder head in order to improve the thermal fatigue fife. The purpose of this work is to analyse the flow characteristics, the heat exchange being directly deduced from the knowledge and predictions of the hydrodynamic field. In order to simplify the experiments, we choose a quasi 2D configuration; however, the design of the tubing is chosen in such a way that the essential geometrical characteristics of a realistic configuration are kept. We first present the model and the experimental set-up associated. Then, we present a global flow analysis and a freer description of the mean and turbulent velocity fields. Velocity measurements have been performed using Particle Image Velocity and been associated with some Laser Doppler Anemometry to check the accuracy of the measurements, particularly concerning the flow statistics. These results will enable a physical interpretation of the phenomena leading to a better description of the heat exchange. This flow appear to be more complex that the flow observed by Watterson et al. [13] in the tube bundle of a heat exchanger. The flow is not established in the downstream cylinders and varies from one cylinder to another. In order to test the ability of CFD codes to simulate such a complex flow, even in a simplified geometry, computations both in 2D and 3D configurations have been achieved with the FLUENT software. 2. TEST RIG SET-UP DESCRIPTION f
Y
J
___~x z
,
i ]
Tank
300 litres
~Filter r Isolating \ 7 Valve /X
4 ,~
Cylinderaxis i --
] By-Pass [
i ..........
Model
RegulationValve
PressureTaos
Figure lb Details of the experimental model
~
7
0
mm
X=250mm
X=334m m
9 Flowmeter
Figure la Test-rig experimental circuit
Figure lc Distribution of pipes and cylinders
503 The test fig-set-up consists in a closed circuit with a centrifugal pump driving the water into the model. A schematic of the test circuit is presented on figure l a. The flow is driven to the model, figure lb, via a long pipe followed by a small divergent inlet pipe to limit the pump effects; the exit is a simple divergent. A by-pass system, associated with two valves, is used to adjust the flow rate. Temperatures and pressures are measured in several locations of the circuit. A 3001 tank is used as plenum chamber. The flow rates used during the tests are measured with a turbine flowmeter, and are respectively 130 l/mn and 240 l/mn. The model is 340mm long, 150mm high and 100 mm wide. It is a simplified representation of the cooling circuit of 16-valve cylinder head of a four-cylinder engine. A two-dimensional bundle of staggered cylindrical tubes, as presented on figure lc, has been machined in acrylic. The characteristic dimensions (tube diameter, size of the coolant passage) have realistic values. The heights have been enlarged to obtain a quasi two-dimensional flow in the horizontal plane of symmetry of the model. Each simplified cylinder is composed of 5 cylindrical tubes: 4 of each (0ffi26mm) representing the two inlet and the two exhaust pipes surrounding the fifth tube (0ffi20mm) of smaller diameter, representing of the spark plug. The distance between the two adjacent large tube axes of one cylinder is 40mm and the minimum distance between small and large tubes (the so-called port bridge) is 5.3mm. Cylinder 1 corresponds to pipes 1 to 4, cyl.2 to pipes 5 to 8, cyl.3 to pipes 9 and 12 and cyl. 4 to pipes 13 to 16. The walls of the model are made of Altuglass in order to make use of optical methods. The configuration of the cylinders and numbers attributed to the cylinders are given on figure lc. The frame of reference XYZ is def'med on the figure. The X axis lies in the direction of the flow. The Y axis is vertical; Z is perpendicular to the cylinder axis of the model. The flow restrictions imposed by the cylinder head gasket and the influence of the cylinder block are not included in this model. Most of the measurements have been performed for a flow rate of 2401/min (=14m3/h). The flow enters and exits from the model by two rectangular sections with area 26mm*127mm. A Reynolds number based on the hydraulic diameter Dh of these sections can be def'med as ReDh=(Dh.U)/v (U is average velocity in a section computed from the flow rate and v the kinematic viscosity). With Dhffi4.3cm for 2401/min, we obtained ReDh=50800. Some tests have been also performed at the reduced flow rate of 1301/min, then leading to ReDh=25400. 3. GLOBAL CHARACTERISATION OF THE F L O W 0 3.
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504 After a first qualitative description of the flow by visualisations using laser sheets (video and photographs), input/output pressure measurements have been performed to study the variation of the head pressure loss in the model giving two types of information about the cooling circuit. First, this is an important parameter for the design of the thermo-hydraulic circuit (radiator, pump .... ). Secondly, it can be also considered that there is an analogy between head losses and global heat exchanges between engine walls and the fluid. Then, the head losses can be considered as a global indicator of the thermal exchanges. The head pressure losses have been measured for flow rates ranging between 30 and 2401/min. Two pressures taps are located in the middle section of the model in plane Y---0, see figure 2, in the inlet and outlet sections. Figure 2 shows the variation of the head pressure loss coefficient Cp-Ap/(0.5pU 2) versus the Reynolds number (p is the fluid density). Inaccuracies are estimated less than +_5%. The different curves obtained have similar shapes. These measurements can be interpolated with a simple power law: ACp--a(ReDh)b. A value of-1/5 (.192) for the exponent b is obtained with an accuracy of 3.3%. The head pressure loss coefficient can be considered as decreasing as ReDh-1/5. This result is quite close to the power law obtained for the friction coefficient in several classical configurations such as turbulent boundary layer, pipe flow .... Indeed, for a pipe flow, the experimental Blasius law is i x C r . - - - ~ I C f d x ~ Reyx 115, and for a turbulent boundary layer, the total friction coefficient /k
-0
follows a law proposed by Falkner C F , , ~ ReYx1/7 . Even if it is difficult to assert formal links between these relations and the results observed in our experiments, we can therefore assume that the variation of the head pressure loss coefficient is a dissipative phenomena linked to viscous friction of turbulent type. 4. DETAILED FLOW ANALYSIS Measurements have been performed by Particle Image Velocimetry giving an instant vector map of the flow. The laser sheet is produced by a double cavity 200mJ Nd-YAG laser working at reduced power (about 15mJ). Two images are recorded on a CCD-camera with 768*484 pixels. The flow is seeded with Rilsan particle (diameter ~_25~rn after filtering). Each picture is divided into interrogation areas in which a vector is computed. An overlapping window technique with a 50% ratio has been used; giving about 1363 vectors (2 to 5 vectors per mm2). Each field is the average of 500 or 50 instant fields. These fields have the typical dimensions of 100mm*70mm. A FFF based on cross-correlation is performed between two successive images: once the location of the correlation peak is determined, the velocity vector is computed giving the time step between these pictures. A Dantec Flowmap software is used. Some complementary measurements have been performed by Laser Doppler Anemometry to check the accuracy of the measurements. Agreement is quite good except some alteration of the results in PIV due to shadow regions and to an increase of the laser light intensity produced by the various tubes acting as lenses. Furthermore, LDV is able to give accurate measurements of the velocity fluctuations. The flow is almost homogeneous as soon it enters the model. In the last plane (close to the exit), a converging effect can be observed. The measurements show that there is an area with approximately 100mm width in the middle of the model where the flow pattern can be considered as two-dimensional. In the following sections, only the results corresponding to the central OXZ planes, located in this zone, will be given. Figure 3 presents the mean velocity magnitude (from the X and Z components
505
FIG. 3 -
Velocity iso-values in the Y -
0 plan (L.D. V).
FIG. 4 - U rms ( x/~u'2) iso-values in the Y - 0 plan.
FIG. 5 -
Velocity iso-values in the Y -
0 plan ( R N G , k - e).
506 measured in planes Y=0). The observation of these plots shows that the flow is almost established in the middle of the first cylinder. The velocity is quite low between the large pipes (~0.15m/s) and show weak variation outside of the tubes (between larger tubes and outer walls), from 0.45 and 0.7m/s. Finally, except in the last cylinder, the velocity inside the cylinder (in the port bridges between the larger and small ones) is always greater than 0.7m/s, with larger values in the direction of the cylinder exit. For example, for the first cylinder, the maximum velocity is 1.9m/s (1.5*V, where V is the entrance mean velocity) but only 0.7m/s (0.55"V) for the last one. The flow is deviated toward the lateral walls between the entrance and the first cylinder. The flow becomes then more uniform in the Z direction before being accelerated again at the exit. In the first cylinder, high velocity magnitude is observed for ct----.45~ on pipe 1 and 315 ~ on pipe 2 and in the port bridge: the angle {x is defined on figure 1c. Figures 6a, b,c show the streamlines in Y=0 plane deduced from the previous velocity measurements respectively for cylinders 1-2, 2-3 and 3-4. The large deviation of the walls at the entrance is responsible for the creation of the lateral eddies located between X=0 and 20mm and for a strong deviation of the streamlines at the entrance section. The first cylinder is responsible for a strong obstruction of the flow. Large size eddy systems can be also observed between the cylinders, and, with a weaker intensity, inside the larger pipes of one cylinder. The fluid coming from the outer part of the model is directed into the eddies but does not seem to pass inside the port bridges. The flow is quite similar in the various port bridge cylinders. Figures 3 or 6 show separation for {x-~90~ (~, angular position defined in figure l c) on the large pipes (with non-symmetrical flow with respect to OX) and for et=135 and 225 ~ for the small ones. Other measurements (not presented here) show that the mean flow structure is nearly independent of the flow rate.
Figure 6a Streamlines deduced from LDV measurements for cylinder 1-2
Figure 6b Streamlines deduced from LDV measurements for cylinders 2-3
Figure 6c Streamlines deduced from LDV measurements for cylinder 3-4
Figure 7 Streamlines computed for cylinders 1-2
507
Uniformity o f the flow rate distribution is often used as criteria to evaluate the quality o f cooling in various sections of the cylinder head. F r o m the measurements, the flow rate has been calculated for various sections normal to X axis (shown on figure 1): 8 central sections and 8 lateral ones corresponding to X - 3 0 - 7 0 - 1 1 0 - 1 5 0 - 1 9 0 - 2 3 0 - 2 7 0 and 300mm. The c o m p u t e d flow rates correspond to a unit height of the model. Figure 8 show the axial distribution o f the inner flow rates deduced from measurements in the central sections. 0.006
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O0
Cyl.1
100
Cyl.2
x mm
200
Cyl.3
Cyl.4
Figure 8 Distribution o f the central flow rate
-
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Figure 9 Distribution o f the lateral flow rate
The variation with X is quite weak inside the cylinders but the decrease is about 50% between cylinders 1 and 2 and weaker for the last ones. An increase o f the flow rate appears in the last cylinder, due to the converging effect near the exit. Figure 9 shows the corresponding values for the outer flow, between a row of pipes and the outer walls. The flow rate increases slightly with X, due to fluid transfer from the outer part to the inner part o f the model. The transverse flow rate (through sections parallel to OX) has been computed between the cylinders. A negative flow rate is observed in cylinder 1 (from outer to inner part) becoming almost null in the following ones. This implies that the eddies present between the pipes induced a very small fluid transfer. By adding the inner and outer flow rates (multiplied by a factor o f two to take into account the symmetry), the value of the total flow rate is lower than the entrance flow rate. This means that the flow is not perfectly two-dimensional and that a part o f the flow is deviated toward the upper and lower parts of the model (along the tubes). The deviation begins just at the model entrance. Thanks to the numerous measurements obtained, a large amount o f qualitative and quantitative data on the flow structure are available. It is thus possible to discuss the flow topology: location of the stagnation points, areas with separation or recirculating zones. Wall pressure measurements around the larger circular pipes will complete the results obtained on the velocity field. A pipe element has been instrumented with twelve pressure taps, at midheight, every 30 ~. By rotating this instrumented pipe and locating it at the place of the various tubes, pressure measurements have been performed every 10 ~ on the 16 larger pipes. Examples o f the results are presented on figures 10-a and b, where the pressure difference between pressure on the cylinder and the static pressure in the entrance section has been plotted versus the angular position (x.
508 Plots relative to pipes 1 and 2 (figure 10a) give the location of the two stagnation points on each pipe: the first one for ~-326 ~ for pipe 1 and for ~t-34 ~ for pipe 2; the second ones are located for a - 1 8 0 ~ (pipe 1) and tx-178 ~ (pipe 2). If we consider that the measured pressure is equivalent to a dynamic pressure (-1/2pU2), the corresponding velocity is U-1.34m/s for a pressure difference of 900Pa. This value is in agreement with the values obtained from the corresponding local external velocity measurements (see figure l c, for example).
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a) Pipes 1 and 2 b) Pipes 15 and 16 Figure 10 Wall pressure distribution It can also be observed that there is a zone with very low pressure on pipe 1, between ot-355 ~ and or-150 ~ This interval corresponds to the part of the flow which is accelerated along the pipe being deviated on the model sides. Downstream, along the pipe, the flow suffers separation generating a small recirculation zone connected to the second stagnation point (ct=180~ On figure 10-b, for pipes 15 and 16, the pressure values are systematically shifted toward the negative pressures. This can be attributed to an increase of the head pressure loss when the distance to the entrance section increases. To help the interpretation of the graphs presented on figure 10, the pressure values have been added to the velocity measurements presented in the previous sections: on the figure 6 presenting the streamlines, the wall pressure distributions on the 16 pipes have been plotted as colour part of the tube walls. For pipes 3 and 4, in the port bridges, it is easy to recognise the zone with acceleration of the fluid at the cylinder exit associated with the low-pressure zone (in blue). The pressure peaks due to the presence of a recirculating zone between the pipes is clearly obvious on the figure. Similarly to pipes 1 and 2, stagnation points for ct-310 ~ (respectively 50 ~ are located on pipes 5 and 6, corresponding to the impact of the central jet. This time, the dynamic pressure (160mbar) corresponds to the velocity U-~0.57m/s (result here also in good agreement with the velocity measurements). A zone with a small high pressure due to the recirculating zone for ~ - 4 0 ~ (respectively 325 ~ can also be observed. The similarity of the pressure distribution on pipes 3/4 and 7/8 (for instance location of the peak and minimum values) has to be noticed. The turbulent RMS velocity in X direction obtained by LDV has been plotted on figure 4 as a contour plot. The turbulent kinetic energy is very high at the first cylinder exit. It appears that the jet issuing from the first cylinder is highly turbulent, therefore the jets issuing
509 from the others cylinders seem to be less turbulent. This high turbulence level is of prime interest for the thermal exchange ratios. Turbulence intensities can be very high, RMS values can reach 20% of the local velocity. For the prediction of such an internal flow, a simple gradient model will be probably poorly efficient. Then, we perform, as a preliminary study, a simple computation with a closure k-e model. 5. TEST OF A k-g MODEL In order to check the ability of a k-e model to predict such a flow, we have performed computations both for a 3D configuration on half a configuration (using symmetry property of the flow). The purpose is not to improve predictions of models and codes, but is to analyse the behaviour of a conventional closure model in such a relatively simple geometry. Then no attempt for optimisation have been tried. We use a computational domain including both the divergent and convergent pipes located at the inlet and outlet of the model. The inlet boundary conditions have been determined from the measured corresponding conditions (turbulence intensity of 17%, turbulent characteristic length of 10mm and mean velocity of 2m/s). A conventional RNG k-e model with a non-equilibrium function close to the walls and a second order discretisation proposed by the Fluent code, UNS Version 4.1.9 have been used. The mesh consists of 72230 nodes. The smaller size mesh size (close to the cylinder walls) is 0.7mm along the pipe and lmm radially. In the core of the flow, in one cylinder, the meshes are 0.Smm by 0.5mm. Hexahedral grids typically 10mm is used in the z direction. The convergence have been reached for 594 iterations, the residual being decrease from l i f E to 10-5 during the calculations. The results are presented on figure 5 which corresponds to the experimental configuration given on figure 3. In the first cylinder, the predicted boundary layer separation is close to the one observed in the tests. However, the transverse flow rate is not correctly predicted. To assert this conclusion, figure 7 presents the calculated streamlines in the first cylinder and is to be compared with the experimental ones of figure 6a. The computations show more intense vortex motions, particularly between the pipes corresponding to the same cylinder. Between two cylinders, the predicted motion is more correct, with no transverse flow-rate, corresponding to the observation during the tests. Into the first cylinder, the recirculation area computed behind the first pipe has a too large extent and interacts with the second; this last one has not been observed in the experiments. So the calculations predict a positive flow-rate (from the central part to the sides) contrary to the negative one measured. This inversion of the sign of the flow between experiments and computations is still observed into the second cylinder. 6. CONCLUSIONS AND PERSPECTIVES We have developed a simplified model representing the essential hydrodynamic features of cooling circuit which is more complex than a regular tube arrangement. The model can be considered as two-dimensional and easy to study by computational fluid dynamics. The measurements methods used are complementary and a relatively complete database has been obtained. A head pressure loss coefficient varying as the -1/5 power of the Reynolds number has been proposed. The sensitivity of this law to the geometric location of the pipes is yet to be analysed. Due to the asymmetric repartition of the tubing, the internal flow in the model is quite complex with eddy systems specific to each cylinder. Between the cylinders 2 to 4, the flow seems to be relatively established and independent of the X position. The first cylinder
510 plays a particular role because it corresponds to the impact of the entrance jet but also is responsible for the major part of the pressure loss. The exit is also associated to the strong turbulent intensity. The intricate nature of the problem, which strong asymmetric interactions such as the coupling ~tween separations and walls, strong accelerations and sudden expansions, etc.., represents a challenge for optimisation and computation. A crude test of a commercial code has been performed. It shows that a RNG-k-e model predicts with some details most of the overall features of the flow. However, a detailed analysis shows, as it was expected, that the separation zones and their interactions with inner obstacles are not predicted with a sufficient accuracy. For the purpose of heat transfer evaluation, this drawback can be dramatic. With the present data base, higher order codes, such as ASM or RSM, can be tested with better chances of success. ACKNOWLEDGEMENTS The authors thank gratefully the Renault Company (Direction de la strat6gie et des Avants Projets) for the financial support of the experimental part of the work presented and for an efficient collaboration during this study. REFERENCES 1. K.L.Hoag and S.Brasmer, SAE Technical Paper 891897, (1989). 2. I.C.Finlay, D.Harris, D.J.Boam and B.I.Parks, Proc. Instn. Engrs., 199, 3, (1985) 207. 3. C.C.J.French and K.A.Atkins, Proc. Instn. Engrs., 187, 3, 1973. 4. M.Shalev, Y.Zvirin and A.Stotter, Int. J. Mech. Sci., 25, 7, (1983) 471. 5. Y.Aoyagi, Y.Takenaka, S.Niino, A.Watanabe and I.Joko, SAE Technical Paper 880109, 1988. 6. G.D.Davis and R.J.Christ, SAE Technical Paper 960883, 1996. 7. W.S.Bederaux-Cayne, SAE Technical Paper 960881, 1996. 8. T.Priede and D.Anderton, Proc. Instn., Mech. Engrs., 198D, 7, 1984. 9. M.H.Sandford and I.Postlethwaite I., SAE Technical Paper 930068, 1993. 10. A.Coll6oc, Mellat and J.M.Boyer, Note technique 1096/93/1847, 4 th AACHEN Colloquium Automobile and Engine Technology, 1993. 11. C.H.Liu, C.Valfidis and J.H.Whitelaw, Experiments in Fluid, 10 (1990) 50. 12. C.Arcoumanis, J.M.Nouri, J.H.Whitelaw, G.Cook and D.M.Foulkes, SAE Technical paper 910300, 1991. 13. J.K.Watterson, W.N.Dawes, A.M.Savill and A.J.White, 3rd ERCOFFAC-IAHR Workshop on Refined Flow Modelling, Lison, Portugal, 1995.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
511
S e c o n d a r y flow in c o m p o u n d sinuous/meandering channels Y. Muto and T. Ishigaki Ujigawa Hydraulics Laboratory, DPRI, Kyoto University Yoko-Oji Shimomisu, Fushimi, Kyoto 612-8235, Japan
Secondary flow and its related problems in compound sinuous/meandering channels are discussed. The structure of secondary flow in sinuous/meandering channels during floods is illustrated by detailed velocity measurements using a fibre optic laser Doppler anemometer (FLDA) and an advanced flow visualisation technique using a small submergible video camera. Some data analyses were carried out in order to estimate energy expenditure by secondary flow and its effect on the macro flow structure. The results clearly shows relatively large effect of secondary flow on estimating conveyance of the channel system.
1. INTRODUCTION Rivers during floods often inundate their adjacent plains and show the behaviour of socalled compound channel flow. Compound channel flow is known to have a distorted 3dimensional nature mainly due to the velocity difference between the main channel and the flood plain. A shear layer is formed at the junction of the main channel and the flood plain, and fluid exchange takes place through this junction region (see e.g. Knight and Shiono, 1990; Tominaga and Nezu, 1991). In case of meandering channel such flow structure is more complex. Velocity distributions in those channels are much distorted because directions of the main stream are different between the main channel (lower layer) and the flood plain (upper layer) (see e.g. Toebes and Sooky, 1967; Stein, 1990). Detailed velocity measurements in a meandering channel with flood plains using a laser Doppler anemometer were carried out by Schr6der et al. (1991) and Shiono and Muto (1993). They could successfully illustrate growth and decay processes of secondary flow under overbank conditions, but in a rather limited meandering geometry condition. Muto et al. (1997) also discussed based on velocity measurements unique features of compound meandering flow as to secondary flow and the shear layer instability. They pointed out that these features can take quite important roles when considering conveyance capacity of this sort of channel. This paper deals with secondary flow and its related problems in compound sinuous/meandering channels. Detailed velocity data measured by a fibre optic laser Doppler anemometer (FLDA) were mainly used. Direct visualisation for secondary flow cell in a compound sinuous channel was also carried out by a small submergible video camera. Contribution of secondary flow in energy expenditure mechanism, mainly that for the channel conveyance, was estimated by spectrum analysis for fluctuating velocity data and 1-D type loss coefficient analysis.
512
2. E X P E R I M E N T A L SETUP Figure 1 shows a plan view of the experimental flume and meandering channel. The flume was made of perspex with a rectangular cross section, 10.8m long, 1.2m wide and 0.35m deep. The valley slope of the flume was set at 0.001. The main channels and the flood plains were formed of polystyrene boards. The channel had a rectangular cross-section of 0.15m wide and 0.053m deep. The channel sinuosity s were 1.09, 1.37 and 1.57 corresponding the arc of channel bend q9 of 60 ~ 120 ~ and 180 ~ respectively with the bend central radius rc of 0.425m. s=1.37 .
~
.
.
.
.
FLOW .
.
.
.
.
.
.
.
TEST SECTION .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(Dimensions in rnmi .
.
.
--
INLET
. _ 1
l
OUTLET
,
,
J
H~8oo
"1
Figure 1. The experimental flume and meandering channel s=1.37. A TSI 2 component fibre optic laser Doppler anemometer system was employed for velocity measurements. Three components of the velocity at each measuring point were obtained by installing the beams in two different, which is the same ways as Shiono & Muto (1993) took. In addition measurements for the lateral (V) and vertical (W) components were also carried out to obtain the shear stress - v w directly. The measurements were carried out with the aid of aluminium powder as a seeding agent. The averaged data sampling rate was about 100Hz. The measuring duration for one point was 60sec. A half length of meander channel was devided into 9 or 13 sections and the measurements were performed at every other section. About 100 measuring points were set in each section, whose number depending on the depth condition. The hydraulic conditions are summarised in Table 1. Three typical depth conditions were selected, i.e. the bankfull flow and Dr(---(H-h)/H, where H is water depth at the main channel and h is the flood plain height)---0.15 and 0.50. All measurements are carried out in the pseudo-uniform flow condition. Table 1. Hydraulic Conditions Depth Discharge condition Q Dr (• 1.09 bankfull 1.876 0.15 3.102 0.50 25.755 bankfull 1.556 1.37 0.15 2.513 0.50 19.996 bankfull 1.382 1.57 0.15 2.204 0.50 19.881
Water depth H (m) 0.0525 0.0633 0.1078 0.0519 0.0630 0.1059 0.0532 0.0631 0.1087
Mean velocity Us (m/s) 0.23"7 0.157 0.352 0.197 0.129 0.282 0.170 0.113 0.268
Friction velocity u, (m/s) 0.0166 0.0121 0.0225 0.0148 0.0120 0.0221 0.0140 0.0120 0.0226
Reynolds number Re (• 2.63 0.82 6.26 2.19 0.66 4.92 1.95 0.62 5.16
Froude number Fr 0.431 0.412 0.495 0.359 0.340 0.401 0.307 0.299 0.374
The other relevant information on the experimental setup can be found elsewhere (Muto, 1997) together with the possible error factors and their minimising efforts.
513
3. FLOW STRUCTURE 3.1. Depth Dependency Figure 2 shows secondary flow behaviour in a vector form watching from the upstream for the bankfull and Dr=0.15 flows in the s=1.37 channel. Figure 2(left), for the bankfull flow, shows that the dominant secondary flow cell is developed through a bend section. The fully developed clockwise cell can be observed at the bend exit (Section 5). On the other hand, for the overbank flow, Figure 2(right), an anticlockwise cell recognised at Section 1 near the inner wall suddenly collapses in the latter half of the bend, synchronous with the appearance of a new clockwise cell along the inner wall from Section 3. This new cell immediately grows and occupies most of the cross section in the crossover sections. No. 13
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3.2 percent). At the lower turbulence level of 0.8 percent produced by the fine screen, measurements failed to
537
X 9 r = L80 cm 90 c ~
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ang and mon [13]
90 and 180 cm .
r = ~ and r = 30 in r = 30
Abu-Ghannam and Shaw [ 12]
el
0
1
2
3
4
5
6
Turbulance level, Tu%
Figure 4. Effect of surface curvature and turbulence level on Rexs in zero pressure gradient
, Reee=Z667Re0s
,
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n 9
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Figure 5. Relationship between Rees and Re0e for curved and flat surface
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Figure 6. Relationship between Rexs and Rexe in different flow conditions
538
6007oot
6.4.2 3Tu% Flat O 52 surface
curved 99surface
600 ~[. I ~-
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.,
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Figure 7. Effects of pressure gradient parameter X0 at Transition on Rees.
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Figure 8. Effects of acceleration parameter K t at transition Re0s.
detect any transitional flow anywhere over the entire length of the curved surfaces tested under both zero and favorable pressure gradient conditions. In fact transition did not occur at T u = 0.8 and r = 1520 mm when the pressure gradient was made slightly adverse. These transition results suggested that curvature effects are more evident in flows at low turbulence level. In Figure 4, the Abu-Ghannam and Shaw zero pressure gradient transition modell is shown in terms of Rexs (rather than Re0s ) calculated using the Blasius laminar boundary layer relation, Re 0 = 0.644 (Rex)0.5. In addition to present results, Figure 4 shows results obtained by others [12, 13]. From this figure and Figures 5-6, which relate start of transition momentum thickness and length Reynolds numbers to their counterpart at end of transition, it is clear that for zero pressure gradient, turbulence level dominates the transition process such that the effect of curvature is evident only in flows with T u < 2 percent. However, the curvature effect increases quickly as the turbulence level decreases below 2%. As the start of transition is delayed by convex curvature and favorable pressure gradients, the end of transition is also delayed and the length of the transition zone is stretched. However, it can be seen from Figures 5 and 6 that even with these changes in the start and length of transition, the relationships between momentum thickness and length Reynolds numbers at the start and end of transition remain basically the same as for the flat surface case with zero pressure gradient. The transition results obtained in a favorable pressure gradient and high turbulence plotted in Figure 7 show that for the same value of X0 and T u, the momentum thickness Reynolds number Re0s is on average, 20 percent greater for the curved surface than for the flat surface, while Figure 10 shows that the length Reynolds number Rexs is almost doubled. This
539
Tu%
Flat s u r f a c e
curved surface
T
[]
3.2
6.4 9 9 1.2 + A b u - G h a n n a m [ 14 As s h o w n A b u - G h a n n a m and S h a w [12]
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-4-
T u = 1.2
Tu = 3.2
Tu = 1.2 6
|-
/
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o
~
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2
z,
Tu = 6.4
0 ~-00
o
L
0
1,
1
',
J
,
5
i
r
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~0
lO -4 LX
Figure 9. Effect of pressure gradient parameter ~.X at transition on Re0s.
o I 0
.,
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1
,
!
:
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~
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i0
5
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Figure 10. Effect of pressure gradient parameter )~X at transition on Rexs.
contradict previous conclusion that curvature effects are only significant for low turbulence flows. To resolve this paradox, it seems that the pressure gradient parameter 2~0may not be the appropriate parameter to describe the acceleration of the flow. Furthermore, Figure 8 shows curved and flat surface results in accelerated flow plotted against the acceleration parameter K t. From Figures 7 and 8, it can be seen that the present results agree with the Abu-Ghannam and Shaw modell, but the question of both Re0s and Rexs being larger for the curved surface cases (for the same ~.0 and Tu) remains unanswered. However, a possible explanation can be drawn from examining figure 2 which shows the pressure distribution in the three test cases. It is clear from figure 2 that the pressure gradient dP/dX in the curved surface case is larger than that in the two flat surface cases, especially between the plate leading edge and the start of transition point. On the other hand, ~.0 and K t are not always larger at the point of transition on the curved surface. It follows that neither 2~0nor K t is a suitable parameter to express the flow acceleration at transition since either may increase or decrease as dU/dX changes. The fact remains that flow acceleration should be expressed in terms of X, U, dU/dX, and v . A suitable non-dimensional pressure gradient parameter was sought and defined as Z,x = (X2/v) dU/dX. This parameter calculated at the start of transition always increases as dU/dX increases, since X s increases with dU/dX. The advantage of using ~.x is demonstrated in Figures 9 and 10 where points of higher acceleration have larger values of ~.x and Re0s, thus clearly demonstrating the effect of a favorable pressure gradient in delaying transition. One also notices the improvement of the distribution of the data points in Figures 9 and 10. This improvement is also evident when
540 additional transition data from Abu-Ghannam [ 14] are plotted in terms of)~x. Figures 9 and 10 also show that using )~x provides a smooth and logical departure from the widely accepted values for Re0s measured in constant pressure flows. With this in mind, it is possible to explain some of the conflicting results reported in the literature. For example, Vijayaraghavan and Kavanagh [15], working on the suction side of the turbine airfoil cascade, detected transition at a streamwise distance significantly larger than that predicted by the Abu-Ghannam and Shaw flat plate modell.
4. C O N C L U S I O N S The results presented and discussed in this paper led to the following conclusions about the surface curvature and pressure gradient effects on boundary layer transition. These are:
,
Transition is delayed and stretched by surface convex curvature. This effect is only significant in low turbulence (< 2%) flows and increases as a turbulence level decreases. Transition is significantly delayed by fluid acceleration even in high turbulence flows. To demonstrate this effect, it is important to use the correct acceleration parameter which was found to be )~x = (X2/v) dU/dX . Using )~x improved the relationships between data in constant pressure flows and in accelerated flows. The relationships between the relative location of the start and end of natural transition are not affected by turbulence level, pressure gradient or surface convex curvature.
NOMENCLATURE CD
H Kt r
Rex Re0 Tu U UR X 0 )~x V
P
Pressure Coefficient Shape factor Acceleration parameter, (v / U 2) dU/dX Radius of surface curvature Reynolds number based on streamwise distance Reynolds number based on momentum thickness Turbulence intensity Local free-stream velocity Reference velocity Surface coordinate in streamwise direction Boundary layer momentum thickness Pressure gradient parameter (x2/v) dU/dX Pressure gradient parameter, (02/v) dU/dX Kinematic viscosity Density
Subscripts e s
End of transition start of transition
541
REFERENCES 1 Narasimha, R., The Dynamics of Transition Zone and its Modelling, Lecture series 1991-06, Von Karman Institute of Fluid Dynamics, 1991. 2 Liepmann, H. W., Investigations on Laminar Boundary-Layer Stability and Transition on curved boundaries, NACA ACR 3H30 (NACA-WR-W-107), 1943. 3 Sharma, O. P., Wells, R. A., Schlinker, R. H., and Baily, D. A.," Boundary Layer Development on Turbine Airfoil Suction Surfaces", J. Eng. Power, Vol. 104, (1982), pp. 698-706. 4 Mayle, R. E., The Role of Laminar-Turbulent Transition in Gas Turbine Engines, ASME paper 91-GT-261, 1991. 5 Emmons, H. W., The Laminar-Turbulent Transition in the Boundary Layer-Part 1, J. Aero. Sci., Vol. 18, (1951), pp. 490-498. 6 Volino, R. J., and Simon, T. W., Bypass Transition in Boundary Layers Including Curvature and Favorable Pressure Gradient Effects, NASA Report 17187, (1991). 7 Walker, G. J., The Role of Laminar-Turbulent Transition in Gas Turbine Engines: A Discussion, paper submitted for 1992 ASME Turbo Expo, Cologne, 1992. 8 Halstead, D. E., The Use of Surface-Mounted Hot-Film Sensors to Detect Turbine-Blade Boundary-Layer Transition and Separation, M.Sc. Thesis, Iowa State University, 1989. 9 Young, A. D., and Maas, J. N., The Behaviour of a Pitot Tube in Transverse Total Pressure Gradient, ARC R&M No. 1770 (1937). 10 Goldstein, S., A Note on the Measurement of Total Head and Static Pressure in a turbulent stream, Proc. Roy. Soc. A, Vol. 155, (1936). 11 Vijayaraghavan, s. B., Effects of Free Stream Turbulence, Reynolds Number, and Incidence Angle on Axial Turbine Cascade Performance, Ph.D. Dissertation, Iowa State University, 1987. 12 Abu-Ghannam, B. J., and Shaw, R., Natural Transition of Boundary Layers: the Effects of Turbulence, Pressure Gradient and Flow History, J. Mech. Eng. Sci., 22, (1980), pp. 213-228. 13 Wang, T., and Simon, T. W., Heat Transfer and Fluid Mechanics Measurements in Transitional Boundary Layers on Convex-Curved Surfaces, ASME paper 85-HT-60, (1985). 14 Abu-Ghannam, B. J., Boundary Layer Transition in relation to Turbomachinery blades, Ph.D. Thesis, University of Liverpool, England, 1979. 15 Vijayaraghavan, S. B., and Kavanagh, P., Effect of Free Stream Turbulence, Reynolds Number, and Incidence on Axial Turbine Cascade Performance, ASME paper 88-FT-152, (1988).
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
543
Calculating Turbulent and Transitional B o u n d a r y - L a y e r s with T w o - L a y e r Models of Turbulence R. Schiele, F. Kaufmann, A. Schulz and S. Wittig Institut ftir Thermische Str6mungsmaschinen, Universit~it Karlsruhe (TH), 76128 Karlsruhe, Germany
A two-layer model of turbulence combining a two-equation model in the fully turbulent region with a one-equation turbulence model near the wall is used to calculate turbulent and transitional boundary-layers. By simulating turbulent boundary-layers the validity of the model and the correct coupling of the two layers are reviewed. A new intermittency function is introduced into the one-equation model enabling the application of the so called TRANSIC-T model ('TRANSitional Intermittency Controlled Two-Layer' model) to transitional boundary-layers. The model is verified against transitional boundary-layer flows under varying conditions. Its performance is discussed by comparison to experimental data, a revised version of the two-layer model by Fujisawa et al. [ 1] and four low-Reynolds number k-e turbulence models capable of simulating transitional flows reasonably well.
1.
INTRODUCTION
Two-layer models of turbulence have recently gained popularity (e.g. [2]). Most of the models of this type proposed in the literature use the same standard k-e turbulence model of Launder and Spalding [3] in the bulk of the flow, but different one-equation turbulence models in the viscosity affected near-wall region. The reduced number of required grid points and the associated lower computing time and storage demand, a higher numerical stability in comparison with low-Reynolds number k-E models as well as their potential capability to cope with transitional boundary-layer flows make their application very attractive. Due to their favourable properties two-layer models have been used by various authors for the simulation of attached and separated fully turbulent flows. Combining the one-equation model by Wolfshtein [4] with the standard k-E model successful flow simulations were reported by Chen and Patel [5,6], Patel et al.[7], Iacovides and Launder [8], Abou Haidar et al. [9] and Arman and Rabas [10]. Fujisawa et al. [1], Franke [11] and Cordes [12] were able to predict separated and unsteady flowfields in good agreement to experimental data by applying the one-equation model of Norris and Reynolds [13] in the near-wall region while using the standard k-e model in the core flow. Lakehal et al. [ 14] used the same model combination to simulate the flow- and temperature field behind a film cooling injection. The two-layer model presented here employs a slightly modified version of the oneequation k-1 model developed by Rodi et al. [ 15] in the viscosity-affected near-wall region and
544 the standard k-e model in the core flow. The model by Rodi et al. [ 15] was developed with the aid of direct numerical simulation data (DNS), taking advantage of information not being available from experiments. It therefore shows an improved performance when compared to older one-equation models. When two-layer models are applied coupling of the two domains is of major concern. Correct matching implies that neither the one- nor the two-equation model are used outside their limits of validity. Therefore, a criterion which takes the local properties of turbulence into account is the best means to locate the matching region. Accordingly, in the present model, the two layers are connected with reference to the ratio of eddy viscosity l.tt and molecular viscosity kt. Following a suggestion in [15] the standard k-e model is used when this ratio is greater than 16, while for l.tt/l.t less than 16 the one-equation model is employed. Even by considering this physically sound criterion discontinuities may occur in the matching region due to the different equations used for the calculation of the turbulent length scales. In the first part of the paper the possible impact of these discontinuities on the simulation is assessed by comparing calculated turbulent boundary-layer profiles with data from measurements and direct numerical simulations. Calculating transitional boundary-layers with two-layer turbulence models is possible by introducing an intermittency function T into the eddy viscosity relation near the wall. Through transition 3, is increased from 0 for laminar flow to 1 in fully turbulent flow. At the same time the standard k-e model is used in its original version describing the turbulent core flow. This approach reflects the typical flow situation under which transitional boundary-layers occur i.e. a turbulent main flow and a boundary-layer which changes from a laminar to a turbulent state. Fujisawa et al. [ 1] employed a similar concept to simulate transitional boundary-layers by altering the eddy viscosity relation of the one-equation model of Norris and Reynolds [13] while keeping the standard k-e model unchanged. Using the same model combination Cho et al. [16] performed promising calculations of the two-dimensional unsteady flow field in a linear turbine cascade including transition on the blade surfaces. In a former work [ 17] the authors of the present paper used a new intermittency function to model transition for the first time with a two-layer model employing the one-equation model by Rodi et al. in the near-wall region. In continuing this successful work a new set of correlations for describing transition progress is presented in the second part of the current paper. These correlations improve the model's performance even further. This is verified by calculating different transitional boundary-layer flows with and without pressure gradients. All calculations were performed with an implicit and forward marching finite-volume boundary-layer procedure. See [ 18] for a detailed decription of the code. 2.
TURBULENCE MODELS
2.1
TRANSIC-T
The TRANSIC-T model employs the one-equation model by Rodi et al. [ 15] in the nearwall region and the standard k-e model by Launder and Spalding [3] in the core flow. Using the standard values for the empirical constants appearing in the transport equations for the turbulence parameters k and e the two-equation model is utilized in its form widely applied in the literature.
545 The one-equation model proposed by Rodi et al. uses the normal fluctuation ~/v -~- as velocity scale of the turbulent motion instead of ~ employed in most other models e.g. Norris and Reynolds [ 13]. Therefore, the eddy viscosity relation reads
~l,t = p 4V-72 l~t,v
(1)
where l~v is an appropriate length scale described by l~v = Cl,/~ y. Rodi et al. [15] quote a value of 0.33 for Cl,u. This value has been revised to 0.30 during the present investigation resulting in an improved model's performance. The one-equation model solves the same semiempirical transport equation for the turbulent kinetic energy k as the standard k-E model, but uses an algebraically prescribed distribution for the turbulent length scale. The relevant relations are deduced from DNS data simulating boundary-layer flows fairly well up to y+--50. For a more detailed description of the model the reader is referred to [ 15].
2.2
TLK-T
Fujisawa et al.[1] described a two-layer model for the prediction of fully turbulent and transitional boundary-layers by combining the standard k-e model with the one-equation model of Norris and Reynolds [ 13]. It is denoted as TLK-T as it is a Two-Layer model using k as velocity scale being capable to simulate Transition. To the authors' knowledge besides [ 17] this work has been the only investigation in which a two-layer model is used to simulate transitional, steady state boundary-layer flows. It is therefore interesting to compare the so called TLK-T model to the TRANSIC-T model, giving a good impression of the relative peformance of the models and the validity of the two-layer approach in general. In the TLK-T model coupling of the two calculation domains is realized with reference to the damping function f~t in the eddy viscosity relation of the one-equation model. For values of fu less than 0.95 the one-equation model is applied while the two-equation model is used when f~t becomes greater than 0.95. In turbulent equilibrium boundary-layers this matching criterion is equivalent to a ratio of lxt/lx of approximately 36 which in tern corresponds to a dimensionless wall distance y+ between 80 and 90 [ 12]. Details of the model may be found in [ 1].
3.
CALCULATING TURBULENT BOUNDARY-LAYERS
As stated before when two-layer models of turbulence are used correct coupling of the calculation domains is of major concern. In the following, boundary-layer profiles will be looked at in detail being the best means to quantify the possible impact discontinuities may have on the simulation. To assess the overall quality of the solutions results are compared to data from experiments and direct numerical simulations. Additionally, profiles obtained by applying typical lowReynolds number k-(~ turbulence models are shown in the figures, namely the models proposed by Launder and Sharma [ 19] and the model by Lam and Bremhorst [20], abbreviated subsequently as LS and LB model. Fig. 1 shows velocity profiles in a zero pressure gradient fully turbulent boundary-layer at a momentum thickness Reynolds number Re0 of 15000. Besides measured and calculated values the law of the wall is presented using standard values for )c and C being 0.41 and 5.2,
546 respectively. All models lead to results which are in good agreement with the experimental data with the two-layer models being slightly advantageous. Coupling of the two calculation domains is performed in the TRANSIC-T model at y+ of 43.2 and in the TLK-T model at y+ of 74.4. Discontinuities in the velocity profiles could not be detected, even when looking at the profiles in strong enlargement. Considering nonisothermal boundary-layers, similar results were obtained showing no visible effect of the matching of the models on the temperature profiles.
30
30 LsTRANSIC-T (9
20
TLK-T LB
/~ .-""
exp. resul
(9
20
_~...'"
I i
i
+
10
U+=y + ' 10 . / ~
U+= y+ ,,'./;" ~ 7
Iny§ + C
U+=I/~: Iny+ + C
1/ '
I0 ~
Figure 1.
'
' ' , ' " 1
,
,
101
,,
....
I
10 2
'
y+ [ - ]
'
' ' " " 1
I0 s
'
. . . . . . . .
10 4
100
'
'
....
'1
101
. . . . . . . .
i
102
. . . . . . . .
y+ [ - ]
I
. . . . . . .
10 ~
10 4
Velocity profiles in a zero pressure gradient turbulent boundary-layer at Re0=15000: experimental data from Coles and Hirst [21]
Discrepancies due to the use of different turbulence models should be most obvious when boundary-layer profiles of the turbulence parameters k and e and the Reynolds stresses - u ' v ' are examined. Analyzing the profiles of the turbulence parameters no visible effect on k and only a minor effect on e were found. The largest discontinuities were observed when Reynolds stresses were considered. Therefore, Fig. 2 shows Reynolds stress profiles at a momentum thickness Reynolds number of 1410. Besides the current calculations results from direct numerical simulations [22] are presented. The Reynolds stresses are given in dimensionless form as - u ' v '+ = - u ' v ' / u 2 Both two-layer models describe the Reynolds stress profiles in ,C.
excellent accordance to the DNS data outperforming the low-Reynolds number models especially close to the wall. The TRANSIC-T model leads to the best results of the models considered for y+ less than 10. The models are matched at y+=46 in the TRANSIC-T model and at y+=82.7 in the TLK-T model. Around these coordinates discontinuities of the Reynolds stress profiles are visible. Still, discontinuities remain small in size as well as in extension bearing no detectable influence outside of the matching region. Looking at the results it becomes clear that by applying the two-layer models profiles in fully turbulent boundary-layers are excellently predicted and coupling of the models leads to neglectable discontinuities of the relevant quantities.
547
1.00
t
-~
TRANSIC-T LS
.
I I 1.00
TLK-T LB
0 0.75
0.75
0.50
0.50
0.25
0.25
+>
?
0.00
0.00
0~
Figure 2.
101
y+ [ - ]
10 2
10 3
, ~
10 ~
jy
l,
. . . . . . . . . . . . . . . .
101
y+ [ - ]
10 2
10 3
Reynolds stress profiles in a zero pressure gradient turbulent boundary-layer at Re0=1410: DNS-data from Spalart [22]
4.
M O D E L I N G OF T R A N S I T I O N
4.1
TRANSIC-T
4.1.1
Intermittency Function
Transition cannot directly be simulated by applying the two-layer turbulence model, since it is originally intended for use in fully turbulent boundary layers. In order to extend the model for the calculation of transitional flows, an intermittency function T is therefore introduced into the eddy viscosity relation of the one-equation model. Eq. (1) then becomes =
(2)
By describing the relative fraction of time for which the flow is turbulent T is equal to zero in laminar and attains a value of one in turbulent Hows. Through transition T gradually increases from zero to one. Being part of the eddy viscosity relation (eq. 2) the intermittency controls the increase of IXt from zero to its value in fully turbulent flow. It should be emphasized that intermittency is always set to one in the domain of the core flow thus reflecting the situation under realistic conditions e.g. in gas turbine engines. The intermittency f T=
0.0 1-exp(-4.65r15)+q(1-rl)(r12-rl +0.5)
; Re 0 < A- Re0,s ; A. Ree, s < R e e Reo,E
(3)
is given as a function of the dimensionless coordinate 11 =
Re e - A. Reo,s
(4)
Reo, E - A. Reo,s In this Reo is the local momentum thickness Reynolds number and Re0,s and Reo,E denote its values at start and end of transition, respectively.
548 1.00
r-----I
I
0.75
r-
0.50
New I n t e r m i t t e n c y Narasimho (1957)
function
i
//
iI i
/I
E L_ (D
"~
0.25
Iominori
pretronsitionol
o.oo
0.00
'
~ i
XK A 'Ree.s
Surface
'
I
0.25
'
Distance
'
i
i
turbulent
!
Figure 3.
///
-
/ '
I
X"S 0.50 Ree.s
'
x, M o m e n t u m
'
'
'
I
0.75
'
~
XE Ree.e
Reynoldsnumber
'
'
1.00
Ree
Intermittency function
Fig. 3 shows this new intermittency function together with a typical dependence from the literature (Narasimha [23]). The comparison clearly indicates that the new function differs from conventional approaches by incorporation of a pretransitional zone in which intermittency grows from 0.0 to 0.1. This allows for the first time for the consideration of the unstable laminar region upstream of the actual transition zone. The pretransitional domain starts at the location at which the local momentum thickness Reynolds number reaches the value of A'Re0,s. For predicting transitional boundary-layer flows correlations are needed to determine start and end of transition as well as the location of the beginning of the pretransitional region. 4.1.2
Pretransitional Flow
Strong negative pressure gradients in laminar boundary layers hinder the production of turbulent spots. The dimensionless acceleration parameter K = v / U ~ .dUe/dx is used to describe this effect. If K is greater than its critical value of 3x10 -6 [24], no new turbulent spots form and existing fluctuations are damped. Accordingly, the intermittency in this region is set to zero and the calculation is made under laminar conditions. If K is lower than this critical value, however, the possibility of turbulent spots in the flow can no longer be excluded and the beginning of transition is in principle possible. Due to this consideration the pretransitional flow has to start at the point where the acceleration parameter K drops below its critical value: T~
0.0 ]13.0;1.0]
9 K > Kkrit -- 3.0x 10-6 9 K < Kkrit = 3.0X10 -6
4.1.3
S t a r t of T r a n s i t i o n
and A calculated by" A = Re~ Re0,s K-K~,,
i@
(5)
The best way to characterize the location at which the laminar boundary-layer starts transitioning to its turbulent state is by specifying the value of the momentum thickness Reynolds number Reo,s at start of transition.
549 A detailed examination of a representative database consisting of almost 50 different test cases shows in agreement with the results of earlier investigations (e.g. [25], [26]) a dependence of Re0,s upon the free stream turbulence level in the core flow. An increase of the free stream turbulence causes the transition zone to begin at a lower value of Re0,s. In accordance with Dunham [27], to correlate this effect a turbulence parameter Xs is used which is defined as the average value of inlet and local free stream turbulence level. By inserting this value in percent the start of transition may acceptably predicted by the relation Re0,s = 500 Xs0"65 4.1.4
(6)
End of Transition
With the begin of the pretransitional zone and the start of transition described above, the only additional requirement for the model is a correlation for the end of transition. Again, the momentum thickness Reynolds number serves as the decisive parameter. As Re0,E appears to be dependent on Xs in a similar manner as Re0,s, the correlation may be expected to be of the form Re0,F, = D. Re0,s (7) with D being a constant. However, assuming a constant value for D leads to unsatisfactory predictions and is therefore replaced by a more sophisticated relation which takes the influence of the pressure gradient on transition progress into account. Positive pressure gradients shorten the transition length while negative gradients lead to a stretched transition zone. The Pohlhausen parameter ~,0, which is a function of local pressure gradient and momentum thickness turned out to be the best measure for the correlation of this effect. Careful examination then revealed that D is best obtained by: D=
0.04 ~s +2.2
" ~s > 0.0
0.055~s +2.2
9 ~s 0.4
0
i
F
-0.04 0 0 0 0.04 0.08 -0.02 0.02 0.02 0.02 0.06 0.1 W/Ue WT/Ue WN/U e
X = 50
Z= 4 ' ~8'
1 0.8 20.6 >0.4 02
4.5 ' &~'
~
5 '~'
/
%0
o~ ~oO
9
0
9
,~o
~'
-0.04
_0 0.04 0.02 0 0.04 O.O:Z- 0.02 0 0.04 0.02 W/U e WT/U e WN/U e
X=200 1 0.8 ~o0.6
>0.4 0.2 0
Z=15
17.5 OOI I I I
20 I I 0 o
I
I
I
I
-0.04 0 0.04 0.02 0 0.04 -0.02 0.02 0 0.04 0.02 W/U e WT/U e WN/U e
Figure 2. Distribution of the mean velocity component in the streamwise direction
600 0.041
,
0.04
o og
,
,
,
,
I
,
,
,
,
0.02 0.01
, I
f
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I
I
I
O0
I
I
Y-2
0.02 0.01
0
D 0.02
. oolo o.oo o. o.o4 0.02 -0.04 -0"02
0.03 0.02 0.01
_ -
i
i
i
i
i
i
i
i
i
i
i
i
-0.01 -0.02
I
I
I
I
I
I
I
I
I
I
I
I
I
0 5 10 15 20 25 30 35 5 10 15 Z Z Figure 3. Distribution of the mean velocity component in the wall-normal direction -15-10-5
0
Figure 3 show the distributions of wall-normal mean velocity components V in the spanwise direction. At X - 50 the distribution near the wall at -5 < Z < 5 varies markedly with Z. A comparison of this distribution with the streamwise mean velocity U/Ue in the previous paper [2] helps to understand the behavior of the fluid motion. At the position where U/Ue is minimal, V/U~ takes positive values, whereas at the position where U/Ue is maximal, V/U~ takes negative values, namely, where fluid rises, low-velocity fluid near the wall is elevated and U / U~ is minimal, whereas where fluid falls, high-velocity fluid far from the wall is suppressed and U/U~ is maximal. This suggests that in this streamwise position, many streamwise vortices exist, and that they are litted up or pushed down at their respective positions. Outside the wedge, the velocities are almost zero. Far from the wall, at Y= 2, the maxima or minima cannot be seen (unlike distributions near the wall). At X = 200, like U/Ue and W/U~, extreme change are not seen, so it may be said that the many streamwise vortices at X = 50 disappeared. At Z = 10, values are slightly smaller than the surrounding values, whereas at 16.5 x - 50
5
w/u.=o.05
V/U,--O.05 ' ~ X = 200
W/U,,=O.05
1
i
'L[ i'
3 2
0
......~...........~ ........ .....~ ....... ..........~,iil
-6
-4
-2
0
2
4
6
0
0
5
10
15
Z
...........J.............t ......................... r......................."
20
25
30
35
Figure 5. Velocity vectors In this position, which corresponds to the interface between the wedge and the outer laminar region, a streamwise vortex exists which rotates from the outside to the inside of the wedge in this position. At X - 200, as Figs. 2 and 3 suggest, a positive vorticity exists in the region of Z < 0, and this vorticity makes a pair with the negative vorticity in Z > 0. Thus, at X - 200, a pair of streamwise vortices exists at both interfaces, although the vorticity is only about one-tenth of that at X - 50. Figure 5 show the velocity vectors on the y-z plane. These vectors were obtained from the mean velocities W / Ue and V/Ue in Figs. 2 and 3, respectively. At X - 50, the direction of the vector coincides well with the sense of vortex rotation in Fig. 4. The vectors also confirm the existence of the many streamwise vortices.
3.2. Fluctuating Velocity Figure 6 show rms values of fluctuating velocity in the streamwise direction u' / Ue with Y as a
602
0.040"06I-)( -'5'0 ' ~ ~ '
' ' '-I
o
0.04 0.02 :z)o 0.04o
0.08 0.04006
OO2o
0.04 ,,>0.02 ~o.o o
Y=l-
0.04
0.04
-
0.04 0.02
0.04 0.020
o o~
!
-~ o o~
o
OO2o7f
, -15-10-5
0 Z
5
1015
, , , , , , , , 0
5
Y~O.7[
101520253035 Z
Figure 6. Distribution of the fluctuating velocity component in the streamwise direction
0.04 0.03 0.02 0.01 0 0.03 0.02 221 0.05 ::~0.04 -=0.03 0.02 0.01 0 0.01
3.0
x 10 -3
X-~e2-~/Ue2 '
2.5
c)
2.~ 1.5
C)
~
'
I''"1
'
50
A
z~
900
"
[]
200
s (D
'
~
o
2
I'"'1
A--A ,tL ~
Z--
,a, A
A
-
1.0
o I
1
'
A"
,,~
4'
Klebanoff;-Q-q- ,~
0.5
0.1 y/6
'
0.01
'EI'~/"'~,, ,el
0.1 y/6
e
A
_
, [] , I - ' ~
1
2
Figure 11. Streamwise distribution of the fluctuating velocities and the Reynolds stresses on the wedge center the many streamwise vortices within the wedge might make fluctuating velocities and stresses larger than the value in the ordinal flat-plate boundary layer. After the many vortices disappear, the fluctuating velocities and the stresses decrease gradually and become values near the Klebanoff profiles. The mixing length is a well-known classical concept [11,12]. In this study, a streamwise variation in the mixing length distribution is examined, and the downstream development of the wedge will be considered. This study estimates the mixing length 1using the well-known calculation in equation (1).
605 0.12
I o.o7O.Eo8O./' O, I
i
i
j
i
i
i
i
i
t
i
/
1
o.o6 "" 0.05 0.04 I- ~g~ 0.03 d 0.02 0.01 0 0 0.2
X --~-- 50 '~
0.4 0.6 y/6
0.8
1
1.0
Figure 12. Distribution of the mixing length
l -
-uv
(1)
Figure 12 shows the distributions of the mixing length. The chain line in Fig. 12 shows an equation of I = 0.4y ~ W (K is the Karman constant). At X = 50 and 200, the gradients of the mixing length in y-direction almost coincide with the Karman constant K near the wall, whereas away from the wall 1 ~ 6 t a k e s values around 0.1. It is interesting that although at X = 50 and 200, where the turbulence wedge has not attained fully developed turbulence [3], and the fluctuating velocities and the Reynolds stresses are larger than the values in the fully developed turbulence, the gradient of the mixing length near the wall is almost 0.4, as it is at X= 900.
4. CONCLUSIONS The mean and fluctuating velocities and the Reynolds stress components in the turbulence wedge developing downstream from a single roughness element in a laminar boundary layer on a flat plate were examined. The following conclusions can be drawn. (1) Just behind the roughness, the mean and fluctuating velocity vary markedly in the spanwise direction, and from the isopleths of the vorticity many streamwise vortices exist within the turbulence wedge. The direction of rotation of vortices next to each other is opposite. The distribution of the velocities farther downstream, on the other hand, do not vary markedly, and many streamwise vortices disappear to be replaced by a pair of streamwise vortices on both interfaces between the wedge and the outer laminar region. (2) From the conditional averages of the mean velocity in the spanwise direction, it is found that the wedge expands outwardly since the turbulent fluid within the wedge overflows the wedge. (3) At both sections just behind the roughness and farther downstream, the fluctuating velocities
606 and the Reynolds stresses are larger than the values in the fully developed turbulent boundary layer over the whole height. It may be considered that the effects of the many streamwise vortices still remain even in the section in which a pair of streamwise vortices exist. (4) The gradient of the mixing length near the wall almost corresponds to the Karman constant.
5. ACKNOWLEDGEMENTS
The author wishes to express his thanks to Prof. I. Nakamura of Nagoya University, Prof. S. Yamashita of Gifu University and Profs. Y. Nakase and J. Fukutomi of the University of Tokushima for their kind guidance and constant encouragement throughout the course of this investigation, as well as to Mr. M. Fukunaga and S. Kondoh of the University of Tokushima for their generous cooperation.
REFERENCES
1.
D.S. Henningson, A. Lundbladh and A. V. Johansson, A Mechanism for Bypass Transition from Localized Disturbances in Wall Bounded Shear Flows, J. Fluid Mech., 250(1993), 169207. 2. B.G.B. Klingmann, On Transition due to Three-Dimensional Disturbances in Plane Poiseuille Flow, J. Fluid Mech., 240(1992), 167-195. 3. M. Ichimiya, Y. Nakase and J. Fukutomi, Structure of a Turbulence Wedge Developed from a Single Roughness Element on a Flat Plate, in Engineering Turbulence Modelling and Experiments 2, W. Rodi and F. Martelli(eds.), Elsevier Science, (1993)613-622. 4. S. Yamashita, M. Ichimiya and I. Nakamura, Experiments on the Effects of a Single Protrusion on the Boundary Layer Mound a Cylinder Spinning in an Axial Flow - Mean and Fluctuating Velocities in a Turbulence-Wedge Region, AIAA paper, No. 88-3762-CP (1988). 5. S. Yamashita, M. Ichimiya, I. Nakamura and K. Ogiwara, Effects of a Single Protrusion on the Boundary Layer Mound a Cylinder Rotating in an Axial Flow (Change in Flow Properties in the Turbulence Wedge with the Speed Ratio), Proc. 2nd KSME-JSME Fluids Eng. Cons (1990) 1-190-1-195. 6. S. Yamashita, M. Ichimiya and I. Nakamura, The Effect of a Single Protrusion on the Boundary Layer Around a Cylinder Rotating in an Axial Flow - Conditional Measurement in the Intermittent Region of a Turbulence Wedge -, Proc. 1lth Australasian Fluid Mech. Conf., Hobart (1992) 231-234. 7. I. Tani, H. Komoda, Y. Komatsu and M. Iuchi, Boundary-Layer Transition by Isolated Roughness, Aeron. Res. Inst. Univ. Tokyo Rep. No. 375 (1962). 8. M. Mochizuki, Smoke Observation on Boundary Layer Transition Caused by a Spherical Roughness Element, J. Phys. Soc. Jpn, 16-5 (1961) 995-1008. 9. L.S.G. Kovasznay, V. Kibens and R.F. Blackwelder, Large-Scale Motion in the Intermittent Region of a Turbulent Boundary Layer, J. Fluid Mech., 41-2 (1970) 283-325. 10. P.S. Klebanoff, Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient, NACA Tech. Rep., No. 1247 (1955). 11. L. Prandtl, Bericht fiber Untersuchungen zur ausgebildeten Turbulenz, ZAMM, 5 (1925) 136139. 12.L. Prandtl, Bemerkungen zur Theorie der freien Turbulenz, ZAMM, 22 (1942) 241-243.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
607
FEATURES OF LAMINAR-TUBULENT TRANSITION IN A FREE CONVECTION BOUNDARY LAYER NEAR A VERTICAL HEATED SURFACE Yu. Chumakov a and S. Nikolskaja b aDepartment of Hydroaerodynamics, State Technical University, St.-Petersburg, Russia bDepartment of Hydroaerodynamics, State Technical University, St.-Petersburg, Russia
This article presents the results of an experimental study of the free convection boundary layer formed on a vertical isothermally heated plane surface. Three regimes of the flow were studied. They are laminar, transition and turbulent. A method to measure the surface shear stress and heat flux on the plate is described. New empirical dependencies of these values within a wide range of the Grashof number are obtained. Some features of the flow in the transition area which are appropriate only to the free convection flow are described.
1. E X P E R I M E N T A L UNIT The generator of the free convection flow is a vertical aluminium plate 90 cm wide and 4.95 m high. On the back side of the plate there are 25 heaters which are controlled by the electronic system which is able to keep the set heat regime for a long time (6-8 hours). When setting any regime to each of 25 sections the different laws of the surface heating, in particular the constant surface temperature, can be simulated by the surface height. Due to the considerable height of the plate one can obtain all the three flow regimes such as laminar, transitional and developed turbulent up to a Grashof number of 4.5 x 10 ll . The sensor was traversed in the air flow with a traverse gear. The precision of the movement along the vertical coordinate is about 1 cm and along the normal coordinate with regard to the surface (i.e. across the boundary layer) about 1 micrometer. The movement along the normal coordinate is remote controlled. In addition, the measurement of the flow parameters in one section of the boundary layer is totally automated thanks to the use of the specially designed equipment and computer software. In the course of the experiment the average and fluctuation temperature components were measured. All the measurements were made by using a resistance thermometer and a hot wire anemometer. As sensor a tungsten wire of 5 micrometers diameter and 3-4 mm length was used. It is known that when using the hot wire anemometer to measure the velocity in nonisothermal flow the influence of the temperature should be taken into account to process the hot wire anemometer readings. In our opinion, almost all existing methods of thermal compensation cannot be used for this free convection flow which has
608
low average velocities and high turbulence degree. Therefore, a new method of thermal compensation by the actual temperature was developed. Without describing this method in detail it should be mentioned that unlike other methods of the thermal compensation by the average temperature, in this method the readings of the hot wire anemometer which correspond to the actual temperature in the given point of the area are processed taking into account the actual temperature in this point. All the measurements were made at constant surface temperature which is 70~
The
air temperature at the exterior border of the boundary layer was constant and equal to 25-26~ up to the height 2 meters and then it was increased and reached 27-28~ reached 5 meters.
when the height
2. C A L I B R A T I O N UNIT To carry out a new method of the velocity determination a special calibration unit was designed. The main principle of the unit operation is a regular sensor movement at the set velocity in the stationary non-isothermal air. The sensor goes about a horizontal tube axis. The carriage speed is determined by the time during which the carriage passes the basis distance. To create non-isothermal flow, the air in the tube is heated with the use of infrared emission This unit can calibrate the sensors at velocities from 1 to 50 cm/s and for temperatures from 20 to 80~ For the simultaneous measurement of velocity and temperature the two wire sensor was used: the hot wire measures the velocity and the cold one measures the temperature. These two tungsten wires were located 3.5 mm from each other and perpendicular to the axis of the tube. Such a design of the sensor eliminates the influence of the hot wire on the cold one. The calibration consisted in obtaining the measurements sufficient for the statistic treatment at different sensor movement velocities by the unevenly heated stationary air. The sensor calibration time is about 2 hours. After recording all the required measurements the data processing by computer is carried out. For the further use of the obtained data the calibration results are provided as a ratio of the hot wire anemometer voltage to the flow velocity. The parameter is the air temperature. It is very important to note that when using a measurement procedure like this, there is a thermal compensation of the readings of the hot wire anemometer by the actual temperature value. This is one of the key features of the measurement in this work.
3. ANALYSIS OF E X P E R E M E N T A L DATA The transition region is still seldom studied, mainly because of the measuring difficulties which are due to big intermittency and intensive fluctuations. A method of the thermal compensation of the hot wire anemometer by the actual temperature used in this work does not depend on the intensity of the fluctuations. This made it possible to make measurements in the transition region.
609 3.1. Profiles of average and fluctuation values of longitudinal velocity and temperature within the transition region It is commonly known that the profiles of average velocities and temperature depends a lot on the flow regime in the boundary layer. For example, based on our data, in the case of the turbulent flow regime the profiles of the longitudinal velocity component become more filled compared to such profiles in the laminar flow regime, and the exterior region of the boundary layer (i.e. the region from the maximum of the average velocity to the exterior layer boundary) is more than 90% of the total layer thickness. The thickness of the boundary layer at the unit was varied within the range 2-3 cm in the lower part of the plate (Grashof number Gr x
was about 105 + 108 ) to 20 cm in the upper part of the plate (Grx ~ 10l~ + 1011). Here
GrX = g~ATx
3 / v 2 is the Grashof number where g - gravitational acceleration, 13 - volume
expansion coefficient, AT =
T w -T~
-
difference of surface temperature
Tw
and air
temperature T~ beyond the boundary layer, x - longitudinal coordinate along the surface. The analysis of the obtained data shows that the average velocity and temperature profiles are very close to the laminar profiles even at Grashof numbers 3.109 . At the same time, based on other flow characteristics, the transition processes have already started developing. Especially, the increase of the fluctuation intensity is evident. I.e. if the start of the profile characteristics change is used as a criterion of the transition start, then the transition slows down. More intensive fluctuations do still not result in a change of the average characteristics. In the middle of the transition region the boundary layer becomes thicker very fast, the velocity maximum is decreased and the slope of the temperature profile near the surface is quickly increased. The value of Grashof number corresponding to the end of the transition region when the profiles become like for the turbulent regime turns out to be lower than the value of Grashof number determined based on other characteristics. In other words, at the beginning of the transition region the average velocity and temperature profiles start changing slightly with significant conservatism, but coming down to the end of this region the profiles become
0.4-
I IIIII11
m
!
]U
0.3-
i l llilll
~
,~.
~, .,
0.2-
i
z:~
~ l li~ll
Grx*10"10 0
%. ~%/,,,,
I--
3.7
I
IT
i
i l llllli
Grx*10-10
1 I I IIllll
I
J l lilll
I
I II IIII
,~
02
/X 4.0
/x
4.2
~ '
"'5~v'~~,'
0.10.0-
' Y" 1
'
10
,, ,,ll 100
Figure 1. Distribution of the intensity of longitudinal velocity fluctuation.
0.0
I
I ,, i,,, i
1
,Y'I II Iiii I
10
Figure 2. Distribution of the intensity of temperature fluctuation..
100
610 very fast like those for the developed turbulent flow regime. In this case the fluctuation characteristics are relaxed at some distance along the plate to the constant value corresponding to the developed turbulent regime. The results of the measurement of the intensity profiles of longitudinal velocity I U (y)
(I U =(uZ)l/Z/Um ) and temperature IT(Y) (I T =(t-Z)l/Z/AT ) component fluctuation in the transition region of the boundary layer are shown on Fig. 1,2. These figures show that at the beginning of the transition region the intensity profiles have two maxima: one is near the external boundary of the viscous sublayer and the other one is in the average velocity maximum zone. Then, while the boundary later develops downstream, two maxima become one big maximum, and the profiles of fluctuation intensity I U (y) 6 I T (y) becomes like for the turbulent regime. Unfortunately the available literature provides only two references about work when similar measurements in the transition region were made. In [1] when measuring in air the authors also noted the formation of two maxima on the profile I U (y) in the transition region. The authors of [2] observed the similar phenomena when making the experiments in water. However, the information in [1,2] is not sufficient to make any conclusions on the location of these maxima with regard to the boundary layer regions. 3.2. Results of The Measurement of Wall Heat Flux and Shear Stress
The results of the measurement of the average temperature T and velocity U profiles are used to determine the wall heat flux qw and shear stress ~w. Also the temperature and velocity ratios obtained by integrating the equations of the boundary layer are used. These ratios are represented with the use of Boussinesq approximation. The equations are integrated in the near wall region but the turbulent shear stress -9(uv) and heat flux -OCp(Vt) as well as the left parts of the initial equations are neglected. Hence, as a result of the integration of simplified boundary layer equations the followings ratios of average temperature T and average velocity U are obtained under respective boundary conditions. Linear temperature profile:
T =Tw--~-~y
(1)
and cubic velocity profile:
U=
xw g[3(Tw - Too) y2 + 9gqw y3 yI ~t 2o 6~o
(2)
where X - heat conductivity coefficient, ~t- dynamic viscosity coefficient, 9- gas density. It should be noted that in case of laminar flow regime the ratios of the temperature and the velocity in the near wall area will be the same. That's why in this work a term "viscous sublayer" which is usually applied to the turbulent flow, will denote that part of the boundary layer where the equations (1) and (2) are valid irrespective of the flow regime. For the use of the equations (1) and (2) to obtain qw and Xw the values of the derivatives dU/dy and dT/dy on the surface must be known. These values are determined by extrapolating the equations (1 and 2) to the wall. The extrapolation will result inevitably to a
611 number of difficulties such as a determination of the coordinate of the first measuring point, the determination of the wall 1000 _'ll'll ,llllllll l li,i,lll l lltlllll l llllllll l li,lli,l t lllll zone influence on the readings of the neighboring hot wire Nu~ anemometer sensor. In the _ course of this work an _ extrapolation method was developed. This method was checked many times and 100 carefully by comparing to the results which were available in the literature. The results of the determination of the heat flux on the surface are shown in Gr~ 10~ l" Fig.3 as a relation of Nusselt number N u x ( N u x = h x / ~ , where h - local heat exchange 1E+6 1E+7 1E+8 1E+9 1E+10 1 E + l l coefficient) to Grashof number G r x. The approximation of Figure 3. Dependence of Nusselt number on Grashof number. the experimental points in the laminar flow area gives the following relation (curve # 1)" -
m
m
Nu x
= 0.279 x G r x 0.262
at
Gr x
=5x105 +2.8x109,
in the developed turbulence area (curve #3)" Nu x
1000
= 0.0547 x G r x o.361 _ ,,,,,I
, ,,,,J.I
, ,,,,.d
at
, ,,,~.,I
, ,,~,J,,I
NUx
_ _
_
_
Gr x =
, ,,,,,.I
f
1.4 x 10 lO +5x1011
, ill
~
_
Change of N u x in the transition region can be presented by the following relation (curve # 2)"
_
"~"
100--
present work
.
~
Nu x
at
/ llili]
/
Grx I l lillii I
I l lillilJ
Gr x
=3.5x109+6x109.
This figure shows that there is a local maximum Nux in the
/
10--
=3.75x10 -ll xGrx 1.3041
i l llllil]
I l llilli]
I l lllillJ
I Iiii
IE+6 IE+7 IE+8 IE+9IE+I0IE+11
Figure 4. Comparison of the data of the present work with other data.
transition area. This phenomenon is explained below. Fig.4 shows the comparison of the data of the present work with the data of other authors. It can be noted that the results are very close in the laminar
612
and turbulent regime. The agreement is less good in the transition regime. In the present tests there were more than 100 "~w profiles of the average ouo 2 velocity. In Fig.5 the results 3 * 10-2_ . of these tests are shown as a relation of Grashof
to
Z w / ( p U 2)
number,
U b = ( g [ 3 A T u ) 1/3 .
where In the
laminar flow area the change of the shear stress can be expressed by the following equation:
**
1;___E_w _
, ~ 0 0954
pU~ - 0.743 x ~ r x"
Gr _
_
iiiii I
i i iiiill I
1E+6
i i iii1111
1E+7
i i 1111111
1E+8
i i iiiiii I
1E+9
x
i i illiii I
1E+10
at
I i
1E+11
Gr x = 5 . 9 x 1 0 5 + 2 •
.
Let's note that the value of the initial Grashof number is very low. In the turbulent flow area the Grashof number is considerably higher than in other references. The variation of the wall shear stress can be expressed by Figure 5. Dependence of wall shear stress on Grashof number.
~w
- 0.0752 x GrO183
at
G r x = 7.9x 109 +4.5 x 1011 .
,,,,.,I ,t,,,,,,.l ,l,,,,,,l ,,,,,,,,I ,,,,,,.I ,,,,,,,,I ,,, present work
"cw
0--
A,
[5]
-9
t+]
@ +
[31 [
The transition area turns out to be very short. In addition, we obtained a few experimental points. However, the following equation may be given for the transition area:
+
Xw - 8.45 x 103 x G r ; ~
at
Gr x - 2 . 5 x 1 0 9 + 5 . 5 x 1 0 9 . Ra I IIIIII I
I I IIIII11
1E+6
1E+7
I IIIIII11
I I IIIII11
1E+8
I I IIIII11
1E+9
I I IIIII11
1E+10
X II
1E+11
Figure 6. Comparison of the data of the present work with the other data.
For the free convection flow the wall shear stress in the transition area is decreased. For the forced convection flow the wall shear stress in the transition area is increased.
613 Hence, there is a radical difference between free convection flow and forced convection one. Fig.6 shows the comparison of the present data with the results of other authors. The Rayleigh number Ra x is used as abscissa in order to compare the results obtained in diverse physical media. A quite wide spread in the experimental data of different authors is noted: different values of ~w/(Pf U~) and a different distribution, particularly a different inclination of curves as well as a diverse location and extent of the transition area can be noted. The curve inclination Tw/(PfU 2) = f(Rax)
obtained in the current work in the
laminar area is close to the corresponding theoretical value, and the curve inclination in the turbulent zone is the intermediate value between the data of [3,6]. To our mind, the results of the current work are close to work [3]. In both works the velocity was measured by hot-wire anemometer, and special attention was given to the investigation of the near-wall region. In conclusion, let us get back to the analysis of the Z IT m a x characteristics, given in Fig.3 and bring up one of the 0.2-possible explanations of the A local maximum appearance in ~x A 0.1 number Nux dispersal at the Z~ end of the transition area. Gr X Such a phenomenon was noted A Z~ Z& Z~ Z~ A Z~& 0.0 I I I I I I l I I,,,,, I I lllllll I I llll~llI I lllJ~llI I~lll,ll I I J in some other references, but 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 without explaining the reasons of its appearance. In our work, Figure 7. Distribution of the maximum intensity of the the careful and detailed temperature fluctuations along the surface. measuring of diverse characteristics in the transition regime zone enabled us to explain this interesting and unusual phenomena, typical only for the free convection flow. Figs. 7, 8 show the i l l , , I , i J,,,,,I, ,,llJlll i ~lllllll i , , , , , , , I , ,,,lllll II 0.6 _ dependencies of the A Z maximum intensity of the temperature and velocity 04-Z fluctuations along the A Z ZX Z section. Fig.9 shows the Z Z _----0.2maximum of the average Z /k Z Z velocity (Um) from Grashof Z Z ~ ~ ~ ~ Grx Z number along the section. It 0.0 IIIII] I I I IIIII I I I I IIII1[ I I IIIIII I I I I IIIll I I I I IIIII I I II is perfectly shown that in the 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 transition area (from 0.3
_
Figure 8. Distribution of the maximum intensity of the velocity fluctuations along the surface.
Grx= (2 + 3)
• 109
(0.8 + 1.3) x 10 l~
to the
maximum velocity decreases from (0.48 + 0.52) m/s to 0.4 m/s, which is 20%. The alteration of the maximum velocity
614 along the section occurs practically simultaneously with falling of the wall shear stress, i.e. with decreasing of the derivative d U / d y y = 0 . This process causes a sharp increase in the fluctuation level and a fast growth of the boundary layer thickness. It can be suggested that the sharp increase of ,,,,,I , ,,,,,,,I , ,,,,,,,I ,,,,,,,,I , ,,,,,,,I , I,,,,,ll I IIm Z Z the fluctuation velocity m 0.6 t ~ , m/s is the main reason of A A these phenomena: A A A "i~ "-'A A A Lt~il~i'-" firstly, the average flow O.4 mA A~-~t~~ A energy, spent up to the A AA A A beginning of the A A transition only to 0.2 accelerate the laminar flow longitudinal Grx 0.0 (growing of x w a n d IIIII I I I I III111 I I IIIIII I I I I IIIII I 1E+6
1E+7
1E+8
1E+9
'"'""1
''"'"'1
1E+10
"
1E+11
the laminar area), begins "to transfer" to the Figure 9. Distribution of the maximum of the average velocity fluctuation flow; at the along the surface, same time there occurs capturing of new portions of the cold air from the external boundary layer (increasing of the layer thickness, intermittency). Therefore, there is an increase of the air mass, participating in the longitudinal flow. Because of the terminal velocity of heating, the air mass temperature decreases and, as a result, a buoyancy force ,which is the unique source of the flow, decreases likewise. This process causes the decrease of the maximum value of the velocity, and also the decrease of the inclination of the velocity profile on the surface (i.e. decrease of x w). On the other hand, it promotes increasing the intensive heat exchange, i.e. the local maximum in equation N u x ( G r x ) is formed (see Fig.3). The recovering of the previous maximum value of Um
in
the velocity occurs in the large space within the range of Grashof numbers (1 + 5) x 10l~ .
CONCLUSIONS 1. The method of the measurement of the velocity with the use of hot wire anemometer is modified to measure the velocities in non-isothermal flow of high turbulence degree. While studying the processes of the transition from the laminar regime to the turbulent one a number of features were discovered. These features distinguish this flow from the forced convection flow. 2. At the beginning of the transition area the profiles of the velocity and temperature fluctuation have two maxima. While the flow develops downwards, these maxima are combined together, and the value of this maximum is higher than in the developed turbulent flow area. 3. The transition area shows the decrease of the wall shear stress value and average velocity which is maximum by the section. At the end of the transition area the intensity of the heat transfer from the wall is non-monotonously changed..
615 4. The shear stress and the heat flux on the surface for the three flow regimes were measured. New approximation dependencies on Grasshof numbers are proposed. This work is conducted thanks to the financially assistance by the Russian Funds of the Fundamental Research Study (project: 96-02-19461).
REFERENCES
1. Miyamoto M., Katoh Y., Kurima J., Taguchi Y., Trans. JSME, Ser.B, V.60, N 571, (1994) 971. 2. Jaluria Y., Gebhart B., J. Fluid Mech., V.66, N 2, (1974) 309. 3. Tsuji T., Nagano Y., Int. J. Heat Mass Transfer, V.31, N 8, (1988) 1723. 4. Cheesewright R., Mirzai M.H., Proc. 2nd U.K. National Conf. Heat Transfer, Glasgow, C 140/88, (1988) 79. 5. Smith R.R., Ph.D. Thesis, Queen Mary College, Univ. of London, (1972). 6. Cheesewright R., Ierokipiotis E.G., Proc. 7th Int. Heat Transfer Conf., Munich, FRG, V.2, NC31, (1982) 305.
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0
Turbulence Control
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
619
On active control of high-lift flow F. Tinapp, W. Nitsche Department of Aeronautics and Astronautics, Technische Universit~it Berlin, Marchstr. 14, 10587 Berlin, Germany The active control of airfoil flows is of outstanding importance in future technological applications. For example, maintaining a boundary layer on a wing as much as possible in the laminar state reduces the skin-friction drag. A turbulent boundary layer however, is more resistant to separation than a laminar one. By shifting the onset of separation towards higher angles of attack by means of active flow control, lift is further enhanced and the drag is reduced. Recent investigations have shown that it is highly effective to excite the flow through a narrow spanwise suction/blowing slot, either to damp the turbulent structures in the boundary layer, or to enhance them. Combining the excitation mechanism with a control device, enables the system to operate automatically and to achieve optimal effects under varying flow conditions. In this paper an example of flow control on a simple high-lift airfoil will be presented. Periodic blowing and suction trirough a narrow slot is applied to the separated flow in order to achieve reattachment and thus to increase lift for high angles of attack.
1.
INTRODUCTION
Modern transport aircraft wings have to provide very high lift-coefficients in low speed flight during take-off and landing. This leads to good payload/range capabilities for a given field length and a reduction of the noise footprint in the airport area. Therefore high-lift systems are of complex mechanics, generally consisting of a combination of leading-edge slats and multiple trailing-edge flaps. At high angles of attack the flow over high-lift wings may separate, resulting in a lift reduction and in an increase of drag. If the onset of separation could be delayed towards a higher angle of attack, it will either be possible to achieve a higher lift or to reduce the complexity of the high-lift system (Figure 1). Recent investigations showed clearly that periodic excitation of the separated shear layer results in a partial reattachment and therefore in an increase of lift and decrease of drag (Hsiao et a.1990, Dovgal 1993 and Seifert et a.1996). The present investigation treats this problem and deals with experiments aimed on the separation control via excitation of the separating boundary layer on the flap.
620
2
E X P E R I M E N T A L APPARATUS
2.1 Test model Two different test models for use in wind tunnel and water tunnel experiments were necessary. They are geometrically similar but different in scale. The first test model for wind tunnel experiments, called model A is a generic two element high-lift configuration, consisting of a 180 mm chord length NACA 4412 main airfoil and a NACA 4415 flap with 72 mm chord length, both of 400 mm span. The flap is mounted at a fixed position underneath the trailing edge of the main airfoil, thus forming a gap of 6.3 mm height with an overlap of 4.9 mm (Figure 2). The tested configuration was chosen in accordance with the experiments made by Adair & Home (1989). The angle of attack of the whole configuration o~ can be varied between 3 ~ and 20 ~ while the flap-angle q can be adjusted between 3 ~ and 50 ~ To ensure turbulent separation of the flow, turbulator strips are placed close to the leading edge of the main airfoil and of the flap. For dynamic excitation of the separated shear layer, the flap was equipped with a 0.3 mm wide spanwise slot at 3.5% chord length (figure 3). Static pressure taps were distributed along the midsection of upper and lower surface of the test model and the configuration was mounted on a 3-component balance to measure the aerodynamic forces on the test model. The second test model for water tunnel experiments (model B) is identical to model A but differs in scale. Due to the smaller testsection of the water tunnel, the dimension of model B is reduced on 55% of model A, thus resulting in a chord length of the main airfoil of 100 mm and 40 mm of flap chord length. This test model is also equipped with a excitation slot at the same relative position as model A.
2.2 Excitation system The periodic oscillating pressure pulses are generated externally by a electrodynamic shaker driving a small piston. The excitation signal generated this way is brought into the flap resulting in an oscillating jet emanating perpendicular to the chord from the narrow slot near the flap leading edge (see figure 3). To present the two excitation parameters, frequency and intensity, the Strouhal number ,nd a nondimensional impulse coefficient can be calculated: F St
" l h,t,.
with the flap chord as characteristical length lchar.
--
b/oo
c~, = 2.
9
, with H=slot width, c=chord of main airfoil, v'=velocity fluctuation at the slot exit
621
2.3 Test bed Pilot tests were carried out in a closed water tunnel with a cross-section of 330 x 255 mm. Therefor the smaller test model B was used and a rotating valve, producing periodic pressure pulses, was applied as excitation mechanism. LDV-measurements around the whole test configuration, using a Polytech LDV-580 were conducted at a free stream velocity of uo~=1.6 m/s resulting in a Reynoldsnumber based on the chord length of the main airfoil of 160000. The wind tunnel, used for the main experiments, is a closed one, with a 1.4 x 2 m 2 crosssection. To achieve again a Reynolds number of 160000, the free stream velocity is fixed at u==14 m/s. Due to the short span of the test model A, two side walls reaching from top to bottom of the testsection are installed. All external installations for the test configuration are hidden inside these walls to minimize the influence on the measurement results.
3
EXPERIMENTS
3.1 Pilot tests in the water tuanel To gain an insight into the flow behaviour in the case of natural flow and excited flow, LDV-measurements are carried out around the small test model in the water tunnel. In figure 4 the results are plotted for the case of ~=8 ~ and r1=35 ~ In the upper graphic (figure 4a) the streamlines, calculated from the LDV results are plotted around the whole test configuration. At these conditions, the flow separates over the flap, forming a big, closed recirculation area, while the flow over the main airfoil remains nearly completely attached. In the vector plot (figure 4b) it can be recognized that there is just a small recirculation area very close to the trailing edge of the main airfoil. When the excitation is activated (figure 4c) the recirculation over the flap disappears and the flow reattaches almost entirely. The changed flow field over the flap also has an effect on the flow behaviour over the main airfoil: the small separated region close to the main airfoil trailing edge vanish in case of excitation. The influence of excitation frequency and intensity on reattachment behaviour was also investigated (Tinapp & Nitsche 1998). As an example for the influence of excitation intensity on the reattachment behaviour, three velocity profiles at a fixed position for three different excitation intensities are plotted in figure 5. The dotted line represents the velocity profile of the separated flow for the case of no excitation. It is recognizable, that a weak excitation (a) is not able to suppress the recirculation area over the flap completely, but clearly reduces it in size. A stronger excitation (b) eliminates the backflow, resulting in a much better reattaching of the flow to the flap. The reattached jet is more pronounced and closer to the wall than in the case of weak excitation, nevertheless the velocity profile in case (b) still has a clear tendency towards separation. Increasing the excitation intensity further more, does not change the flow behaviour remarkably but still enhances the velocity profile very close to the flap surface. To better understand the mechanism of reattachment, phase averaged measurements in the flap region were undertaken. In figure 6a the vertical velocity of the excitation jet at the slot outlet is plotted against the positions of the rotating valve used for excitation. Three points of interest are marked in the graphic: In case of a closed valve (b) the velocity of the excitation jet is zero, when the valve opens (c) the jet starts to emanate from the excitation slot and reaches a maximum when the valve is completely open (d).
622 The lower graphics of figure 6 show the flowfields in the flap domain at different moments of excitation that correspond to the three points marked in figure 6a. The positions of the reattached jets are marked in the graphics by the thick arrows. It can be seen, that the flow is attached during the whole excitation cycle (even for the case of a closed valve), but does oscillate in phase with the periodic excitation, resulting in a slight up and down movement of the attached jet. In the moment of starting excitation (figure 6c) a small separation bubble in the vicinity of the excitation slot is formed by the onset of the excitation jet and the flow over the flap moves slightly upwards. The separation bubble disappears immediately afterwards, when the excitation pulse reaches a maximum and the flow reattaches completely to the flap surface.
3.2 Wind tunnel experiments To determine the lift coefficient CL, the wind tunnel test model was mounted on a 3component balance that measures the aerodynamic forces. In figure 7a the lift behaviour of the test model is plotted versus the angle of attack for different flap angles in the absence of excitation. The results show a typical behaviour of highlift configurations. For a fixed flap angle, the lift raises with increasing angle of attack of the whole configuration t~. At a certain angle of attack flow separation on the flap and on the main airfoil inhibits further lift increase and CL drops down to low values. Raising the flap angle shifts the whole lift curve towards higher values due to the enhanced camber of the test model, while flow separation now occurred at lower angles of attack. Maximum lift is obtained in the case of 1"1=35~ with a very peaked lift characteristic at ct=4 ~ For flap angles higher than q=35 ~ the flow over the flap is always separated resulting in a completely changed lift characteristic, as shown in figure 7a for the case of 11=39~ The lift behaviour of the high-lift configuration can be changed by introducing periodic disturbances through the slot on the flap. In figure 7b two exemplary cases (r1=35 ~ and r1=39 ~ are plotted without excitation (dotted line) and with excitation (solid line). The excitation parameters were chosen as F=80 Hz = St=0.4 and %=4010 -5. It can clearly be seen, that the lift is enhanced by flow excitation. Especially in the case of q=39 ~ the recovered flow over the flap yields a strong increase i~, lift up to a value higher than the maximum lift that could be achieved without flow excitation. To depict the enhancement of lift due to flow excitation, the increase of lift as a percentage of the lift coefficient without excitation(cL,exi, ' --CL.b,,.,.ic)/CL,b,.,i, is plotted in figure 8. The achievable lift enhancement is best (about 30%) at a flap angle of q=39 ~ and ix=4 ~ for other configurations the lift improvement obtained by flow excitation is less. It can be seen, that the excitation works best at post-stall conditions. The former separated flow over the flap is reattached, resulting in a recovering of lift. To investigate the influence of excitation frequency and intensity, further measurements are carried out. The test model is set at high angles of attack (ix=7 ~ q=41 ~ to cause flow separation over the flap. In figure 9 the required excitation intensity to achieve reattachment is plotted against excitation frequency marked by the solid line. It can be seen that at low frequencies around F=30 Hz (St = 0.15) minimum excitation intensity is necessary to achieve reattachment of the flow. If reattachment of the flow occurs the excitation intensity can be reduced down to a certain value at which the flow re-separates again (marked by the dashed
623 line). The hysteresis is very small at low frequencies but grows with rising excitation frequency. This indicates a different mechanism between achieving reattachment of the former separated flow and remaining the reattached flow in it's attached state. In the first case, a big recirculation area has to be influenced, which requires big vortices to be introduced into the flow field to achieve a reduction of the separated area. If the flow is attached over the flap, relatively small turbulent structures (St = 1) have to be produced to enhance the mixing process between the shear layer over the flap and the outer flow. This explains, why high frequencies are more effective to remain the flow attached. 3.3 Conclusions Improvement of the lift behaviour of a simple high-lift configuration can be achieved by periodic blowing and suction through a narrow slot at the flap leading edge. This method works best at post-stall conditions, that means, if the flow over the flap is yet separated. The wind tunnel experiment yields, that by introducing periodic disturbances of a certain frequency and intensity into the separated flow, reattachment of the flow can be achieved, resulting in a enhancement of the lift up to 30% of the lift. Low frequencies around St -- 0.15 are best to achieve reattachment of the separated flow, while higher frequencies (St --- 1) work better to remain the flow attached over the flap. LDV measurements of the flow field in a water tunnel around a test model with periodic blowing have shown, that the excited flow over the flap is constantly attached and does oscillate strongly in phase with the excitation frequency.
REFERENCES Adair, D., Horne, W.C. 1989, Turbulent Separating Flow Over and Downstream of a TwoElement Airfoil, Experiments in Fluids, Vol. 7, pp. 531 Dovgal, A. 1993, Control of Leading-Edge Separation on an Airfoil by Localized Excitation, DLR-Forschungsbericht DLR-FB-93-16 Hsiao, F.B., Liu, C.F., Shyu, L.Y. 1990, Control of Wall-Separated Flow by Internal Acoustic Excitation, AIAA Journal, Vol. 28, No.8, pp. 1440 Seifert, A., Darabi, A., Wygnanski, I. 1996, On the Delay of Airfoil Stall by Periodic Excitation, Journal of Aircraft 33, No.4, pp. 691 Tinapp, F., Nitsche, W. 1998, LDV-measurements on a high-lift configuration with separation control, Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Conference Proceedings Vol. 1, pp. 19.1
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
627
A Demonstration of MEMS-based Active Turbulence Transitioning W a y n e P. Liu and G e o r g e H. Brodie Naval Surface Warfare C e n t e r - Carderock, MD U S A 1.0
INTRODUCTION The majority of turbulence research has been targeted at reducing turbulence for o b v i o u s drag and noise benefits. H o w e v e r , there are a few situations where turbulence or an accelerated transition to turbulence would be desirable. Laminar separation and the resulting stall about the leading edge of an airfoil can sometimes be prevented with a transition to turbulence which can provide the needed m o m e n t u m to delay separation. Such an application could be useful for sailplane wings, wind turbine blades, Remotely Piloted Vehicles, and other airfoils in low Reynolds number flows. Model testing is another application where turbulent flow fields are desirable. If the flow field about a full-scale submarine is nearly turbulent, then it w o u l d be a more faithful simulation to provide turbulent or at least transitional flows about a scale model rather than laminar flows. For that reason, straight-ahead, steady drag testing has traditionally e m p l o y e d trip strips, studs or grit to produce a turbulent b o u n d a r y layer about the model. H o w e v e r , how could this be accomplished for a f r e e - s w i m m i n g model such as a r a d i o - c o n t r o l l e d submarine model? Slow speed m a n e u v e r s conducted by a cylindrical model could suffer a laminar cross flow separation which w o u l d p r o d u c e drastically higher drag values than a separation achieved with a turbulent b o u n d a r y layer. One immediate and significant p r o b l e m is that the flow field about m a n e u v e r i n g body is unsteady and not fully u n d e r s t o o d . A second p r o b l e m is that even with a predictable flow field, it would still be impractical to cover the body with studs or grit without irnpacting the -'~0.3 m total drag characteristics of the model. s~bwoofer A future approach to this O.050m --problem could be to cover the model hull with an array of tiny MicroElectro-Mechanical-System (MEMS) sensors and actuators. O n-body 0.2m I sensors could identify critical flow i field characteristics such as velocity, 0.9m p r e s s u r e and shear stress fields to 1.2m reveal stagnation and separation Flow P regions. This flow field information, when combined with tiny hull splitter alate mounted MEMS actuators, could be ! used to excite the natural flow instabilities about the hull to achieve Fig. 1. Wind tunnel set-up of cylinder with an accelerated transition to MEMS sensors, span-wise slit and sub-woofer. turbulence. This s e n s o r - a c t u a t i o n
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~
628 effect could also be used on low Reynolds number airfoils to prevent laminar separation on an active basis. It is the goal of this investigation to demonstrate the use of an on-body sensor-actuation effect in achieving active turbulence transition about a cylindrical body. As shown in figs. 1 and 2, state of the art MEMS sensors mounted on the cylinder perimeter will be used to evaluate classic flow phenomena such as stagnation, separation and the vortex shedding. This data will provide the necessary information to properly tune and locate internal acoustic forcing. Acoustic disturbances aimed at the separation region from a thin span-wise slit on the cylinder, will then be amplified by the shear layer instability to accelerate the transition to turbulence. 2.0
BACKGROUND The effect of flow perturbations matched to instability frequencies has been studied since the 1960's. Klebanoff, Tidstrom and Sargent [1] induced 3-D flow perturbations amidst a field of Tollmien-Schlicting (TS) waves on a flat plate with a vibrating ribbon. Inherent or natural instabilities of the flow then amplified these disturbances to achieve a rapid growth in span-wise velocity irregularities, eventually developing turbulent flow. Later, Bloors [2], Gerrard [3], and Peterka and Richardson [4] investigated the effects of a uniform acoustic field imposed externally upon a cylinder. Their research showed a strong correlation between changes in cylindrical flow characteristics and perturbations which matched instability and vortex shedding frequencies. They concluded that small levels of acoustic forcing enhanced the flow entrainment at strategic points about a cylinder and achieved variable lift and separation characteristics. The amplification of velocity fluctuations through instability frequencies has been called a natural tripping device by Mueller [5]. For low Reynolds number airfoils with a leading edge separation bubble, Mueller showed how the instability frequencies of the bubble's shear layer amplified natural incoming Fig. 2. Cross-sectional view of flush mounted disturbances to achieve turbulent MEMSsensors and acoustic slit on cylinder. mixing and subsequently gain the additional momentum needed for reattachment. Other evidence of initiating turbulence with acoustic forcing was demonstrated by Ahuja [6] and Hsiao [7]. Ahuja transitioned flows to turbulence using an external source of sound, while Hsiao used internal acoustic excitation from a span-wise slit on the cylinder. Again, it was shown that external or internal acoustic excitations, performed at instability and
629 vortex s h e d d i n g frequencies, could amplify velocity fluctuations to transition flows to turbulence. Hsiao and other a e r o d y n a m i c i s t s such as N i s h i o k a [8], B a r - S e v e r [9], and W y g n a n s k i [10], then s h o w e d how local forcing about the leading edge separation point of an airfoil could prevent stall at high angles of attack. They found that the p r o p e r l y tuned and located flow p e r t u r b a t i o n s , w h e t h e r acoustic, mechanical or pneumatic based, were amplified by the separated shear layer to increase flow fluctuations, thereby reattaching the separated flow. Forcing frequency, amplitude and location were found to control the effectiveness of such perturbation a r r a n g e m e n t s . Unlike many previous experiments which used o f f - b o d y sensors to track flow fluctuations, this study will use an o n - b o d y array of MEMS sensors to determine stagnation, separation and vortex s h e d d i n g characteristics about a cylinder. MEMS t e c h n o l o g y has recently surged to the forefront of s e n s o r research and offers a sub-millimeter m e a s u r e m e n t resolution which is critical for s t u d y i n g the d e v e l o p m e n t of flow phenomena. Research by Ho and Tai [1 1,12, 13] has d e m o n s t r a t e d the success of MEMS devices in controlling and evaluating b o u n d a r y layer phenomena. 3.0
EXPERIMENTAL SETUP This i n v e s t i g a t i o n was conducted at the Low T u r b u l e n c e Wind Tunnel of the Naval Surface Warfare Center in Carderock, MD (USA). The tunnel is 1.2 m high by 0.6 m wide and can reach m a x i m u m wind velocities of a p p r o x i m a t e l y 43 m/s. As s h o w n in fig. 1, a 0.9 m long test cylinder of 50 mm diameter was mounted vertically in the test tunnel; a splitter plate was held 0.3 m off the floor of the wind tunnel to allow for sensor cable runs and cameras. The middle 0.2 m section of the e x p o s e d test cylinder was specially c o n s t r u c t e d to house 4 MEMS s e n s o r skins which were divided by thin 0.25 mm s p a n - w i s e slit. The four s e n s o r skins were staggered in the s p a n - w i s e direction to minimize each o t h e r ' s d o w n s t r e a m disturbance and p r o v i d e d a c o v e r a g e of 180 deg about the cylinder perimeter. The cylinder could be rotated by a stepping motor to aim the slit at any angle from the stagnation point. F i g u r e 2 s h o w s that the MEMS s e n s o r skins are flush mounted onto the cylinder over a range of 180 deg, with the slit dividing the c o v e r a g e into 64 s e n s o r s per 9 0 d e g . The s e n s o r skins each measured 2 cm long by 1 cm wide by 0.1 mm thick. On each skin, 32 tiny sensors were arrayed in the streamwise direction over a length of 22 mm--a spatial resolution of almost 0.5 mm or 1.4 deg per sensor. Although a total of 128 s e n s o r s on a span of 180 deg were available for the experiment, only a limited n u m b e r of amplifier channels were p u r c h a s e d to due to funding constraints. As a result only 33 s e n s o r s were on line for the experiment; the sensors were picked to maximize r e s o l u t i o n about the separation region. S e n s o r outputs r e p r e s e n t raw voltages as calibration was not required to determine the flow diagnostics of stagnation, separation and vortex shedding.
4.0
DISCUSSION OF RESULTS If active turbulence transitioning is to be a c c o m p l i s h e d on an u n s t e a d y p l a t f o r m equipped with a skin of sensors and actuators, then the platform s e n s o r s must be able to detect stagnation, separation, and vortex shedding characteristics. Plots will presented to 1) d e m o n s t r a t e how these critical flow characteristics could be detected by s e n s o r outputs and 2) how this information
630 could be applied to actively transition the flow to turbulence. Unless o t h e r w i s e noted, all results were achieved at a Reynolds number of 2 5 , 0 0 0 . 4.1
DETECTION OF C R I T I C A L C H A R A C T E R I S T I C S Stagnation The high spatial resolution and temporal r e s p o n s e of the MEMS s e n s o r s allows the identification of the stagnation point to a high degree of accuracy. The stagnation point is required as a reference and check on the separation region. Figure 3 shows the time traces of 3 c o n s e c u t i v e s e n s o r s located over a 2.9 deg (1.2 mm) span at the stagnation point. As s h o w n by Mangalam [14], the two outer sensors displayed strong oscillations which are 180 deg out of phase with each other while the middle s e n s o r has an oscillation of precisely twice the frequency of the two outer sensors. The two outer s e n s o r s are on o p p o s i n g sides of the cylinder diameter and therefore show out of phase oscillations due to the alternating vortex shedding. The middle sensor is located precisely at the stagnation point and thus s h o w s twice the frequency of the outer sensors. Stagnation can thus be detected by comparing the outputs of consecutive sensors until an out-ofphase or double frequency situation is detected. Separation. Many investigations of the separation Fig. 3. Shear stress sensor outputs about region on cylinders have been based the stagnation point of a cylinder, upon the average and rms (standard deviation) values of hot films located on the surface. For laminar b o u n d a r y layers, separation is typically identified by a m i n i m u m in the average shear stress s e n s o r outputs, along with a c o r r e s p o n d i n g jump in rms s e n s o r outputs ( B e l l h o u s e [15], Coder [16]). Figures 4 and 5 presents such average and rms shear stress s e n s o r outputs for 33 MEMS sensors placed about the perimeter of a 50 mm diam. cylinder in air flow at Re = 2 5 , 0 0 0 . Figure 4 s h o w s average sensor outputs for baseline conditions as the s p a n - w i s e slit is rotated to 82 and 102 deg from stagnation. Slit positions of 82 and 102 deg are s h o w n to produce nearly the same peak and m i n i m u m and compare favorably against data from Bellhouse [15], which s h o w s a m i n i m u m of 88 deg for laminar separation. Comparable data from the 82 and 102 deg slit position show that premature separation is not caused when the slit is p o s i t i o n e d near the typical separation region. Figure 5 presents rms sensor outputs for baseline conditions at slit p o s i t i o n s of 82 and 102 deg from stagnation. Again, the d o w n s t r e a m position of 102 deg compares well with the 82 deg position and proves that the peak and m i n i m u m rms values at 82 deg are naturally occurring and are not induced by the slit.
631 1.2
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F r o m figs. 4 and 5 it can be seen that the separation zone for this test cylinder occurs in the same general vicinity, 80 to 90 deg, as identified in a t h o r o u g h compilation of cylinder separation data by Coder [16]. Figure 6 compares time traces for 7 individual sensors located about the separation region for baseline conditions at Re - 2 5 , 0 0 0 . The individual traces s u p p o r t the trends in fig. 5 - - s t r o n g and regular oscillations at 76 and 79 deg from stagnation are followed by a rapid drop and gradual rise in r a n d o m fluctuations. The sudden change in strength and regularity of fluctuations about the separation region is better s h o w n in fig. 7, which plots a y-direction time history mapping of the shear stress voltages as mapped out by 14 individual sensors clustered about a 9 mm region. Moving from left to Fig. 6. Time histories of MEMS shear stress right, it can be seen that regular, sensor outputs about the cylinder separation high energy fluctuations upstream region, of separation are quickly diluted into more r a n d o m and lower energy fluctuations as separation is encountered. Such a rapid change in frequency strength and regularity can be quickly determined with a spectrum analyzer. Results from such a spectrum analysis are s h o w n in fig. 8, which plots spectral analyses from the individual time traces s h o w n on fig. 6. It is seen
632 that the solitary vortex shedding frequency peak (30Hz) is immediately diluted at the separation region with more energy distributed to higher frequencies. Separation can thus be detected through the minimum of mean and rms shear stress sensor outputs, as well as through the change in spectral energies of the sensor outputs. Instability Frequencies. The instability frequencies which amplify incoming disturbances are predictable and typically constitute a brand band of frequencies. As shown by Hsiao, Bloors, Peterka and Richardson, there exists a ratio between the instability and vortex shedding frequencies which varies with Reynolds number. For Fig. 7. Time history mapping of 14 shear Reynolds numbers of 20,000, a stress sensor outputs over separation region. ratio of about 10 exists between instability and shedding frequencies. At higher Reynolds numbers of about 50,000, the ratio becomes 20. This ratio of instability and shedding frequencies varies as a function of Reynolds number and was shown to be a linear-log relationship. Hsiao also showed that the range of instability frequencies which can provide disturbance amplification was broad; effective frequencies were typically found at Strouhal numbers (St) of 1.0 to 3.0 for Re = 20,000. S trouhal number is defined as" St = [frequency (f)*characteristic length(D)]/freestream velocity (U)
Fig. 8. Spectral analysis of shear stress sensor outputs about cylinder perimeter with no
forcing.
633 An estimate can therefore be made of the instability frequencies by measuring the vortex shedding frequencies just upstream of separation. Shedding frequencies on a cylinder with a known flow velocity can also be estimated with a Strouhal number of 0.21. 4.2
APPLICATION OF FLOW CONTROL
Fig. 9. Spectral analysis of shear stress sensor outputs about cylinder perimeter with forcing at St=2 (275 Hz).
As shown in figs 3 to 8, sensor data were used to identify: 1) the stagnation point, 2) the separation region and 3) vortex shedding frequencies. These inputs were then used to tune acoustic disturbances at 275 Hz (St=2) and to aim the slit at the separation region, 82 deg beyond stagnation. With properly tuned and located acoustic forcing, dramatic changes are achieved in the spectral energies as shown in fig. 9. Prominent energy peaks are observed at the disturbance frequency of 275 Hz, its first harmonic at 550 Hz, and the vortex shedding frequency of 30 H z . It appears that flow forcing at the instability frequency has re-energized the spectral energies at the vortex shedding frequency to extend the life of the peak from 85 to 127 deg beyond stagnation. By contrast, the baseline spectral analysis of fig. 8 displays a permanent drop in the vortex shedding energies for all points beyond 85 deg. The emergence of the forcing frequency and extended vortex shedding peaks are shown in the time traces of the corresponding sensors on fig. 10. Downstream of the forcing at 85 deg, the regular large scale disturbances due to vortex shedding are immediately replaced by rapid fluctuations tuned to the input disturbance of 275 Hz. Between 90 and 120 deg beyond stagnation, a wave form matching the vortex shedding frequency is eventually superimposed upon the forcing disturbances. Figure 11 compares the average shear stress values about the cylinder for Re = 25,000 with and without forcing at 275 Hz. It is seen that the average values are approximately equivalent for the baseline and forcing conditions up to the forcing location (82 deg). Downstream of the forcing between 80 and 127 deg, average shear stress increases significantly for the forcing situation. At 140 deg, average values for both the forcing and baseline conditions re-
634
Fig. 10. Time history traces of shear stress sensors on cylinder perimeter with forcing at St=2 (275 Hz). 1.2
~ I::1~Forcing ~ at 82 ~ ' i 9No forcing
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converge. The trends seen with forcing are c o m p a r a b l e to the traces seen in Bellhouse [ 15] for transition flows. F i g u r e 12 plots the rms shear stress values about the c y l i n d e r for Re = 2 5 , 0 0 0 with and w i t h o u t forcing at 275 Hz. Equivalent values for both the baseline and forcing situations are seen up to the forcing point at 82 deg. H o w e v e r , between 80 and 127 deg, m a x i m u m rms values for the forcing situation are almost three times larger than those of the baseline condition. At 140 deg, rms values for both the forcing and baseline conditions reconverge. As predicted by Hsiao, Bloors, Peterka and R i c h a r d s o n , the most effective forcing frequency matched the instability frequencies, which were found to be a p p r o x i m a t e l y ten times greater than the m e a s u r e d vortex s h e d d i n g frequency at Re=25,000. Other testing at Re = 5 0 , 0 0 0 s h o w e d that an effective f r e q u e n c y of about 950 Hz, or a p p r o x i m a t e l y 20 times the vortex shedding f r e q u e n c y , was required to yield increased fluctuations. Effective forcing frequencies were found at St = 1 to 3.5. 0.25
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Angle from stagnation (deg)
Fig. 12. RMS (Std. Dev.) of MEMS shear stress sensor outputs about the cylinder perimeter with and without forcing.
635 Figure 13 shows a wake profile taken 5 diameters 9No forcing downstream of the cylinder for [] Forcing at St=2 "l 2.5 both baseline and forcing rag .i AForcing at St-1.27 situations. It is shown that disturbances at St=1.27 and 2 9 produced reductions in the wake velocity deficit. While it is understood that the wake is 9 & Q asymmetric with forcing from only in & tV 1 one side, this should represent 9rrl A | adequate evidence to prove that m & m--i "ql flow control has been achieved 0.5 ,," >. with the acoustic forcing at the nn instability frequencies. It would appear that 6 7 8 9 properly tuned flow disturbances Velocity (m/s) were amplified by the separated Fig. 13. Wake profile of cylinder with and shear layer to entrain higher without flow forcing. momentum flows; this entrainment of higher momentum flows re-energized the boundary layer to delay separation. A delay of separation could be inferred from the reduced cylinder wake and the extended spectral energies at the vortex shedding frequency--with flow forcing, the vortex shedding peak is extended from 85 to 127 deg beyond stagnation. r -
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
=
#%p
IIm
8&~k
5.0
CONCLUSIONS An array of on-body MEMS shear stress sensors and internal acoustic actuation was used to actively transition the flows to turbulence about a cylinder. On-body sensors identified stagnation, separation and vortex shedding data; this information was then used to direct an internal acoustic disturbance at the separation region with a disturbance frequency of St=2. Acoustic disturbances tuned to the instability frequencies were aimed at the separation region to produce dramatically higher shear stress fluctuations downstream of the forcing and a reduced cylinder wake. It would appear that properly tuned flow disturbances were amplified by the separated shear layer to entrain higher momentum flows; this entrainment of higher momentum flows re-energized the boundary layer to delay separation. Other characteristics of forcing at the instability frequency included spectral energy peaks at the forcing frequency and re-energized spectral energies at the vortex shedding frequency from 85 to 127 deg. Without forcing, spectral energies at the vortex shedding frequency were typically dissipated at the separation point. 6.0 o
,
REFERENCES
Klebanoff, K. D. Tidstrom and L.M. Sargent, "The three-dimensional nature of boundary layer instability," Journal of Fluid Mechanics, Vol. 12, 1962, pp. 1-35. Bloors, M. Susan, "The transition to turbulence in the wake of a circular cylinder," Journal of Fluid Mechanics, Vol. 19, June 1964, pp. 290303.
636 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15. 16.
7.0
Gerrard, J. H., "A Disturbance-Sensitive Reynolds Number Range of the Flow Past a Circular Cylinder," Journal of Fluid Mechanics, Vol. 22, May 1965, pp. 187-196. Peterka, J. A. and Richardson, P. D., "Effects of Sound on Separated Flows," Journal of Fluid Mechanics, Vol. 37, June 1969, pp. 265-287. Mueller, T. J., "Low Reynolds Number Vehicles," AGAR-Dograph No. 288, Neuilly sur Seine, France, 1985. Ahuja, K. K., Whipkey, R.R. and Jones, G.S. "Control of Turbulent Boundary Layer Flows by Sound," AIAA Paper 83-0726, 1983. Hsiao, F.-B., Shyu, R.-N., and Chang, R.C., "High Angle of Attack Airfoil Performance Improvement by Internal Acoustic Excitation," AIAA Journal, Vol. 32, No. 3, March 1994, pp. 6 5 5 - 657. Nishioka, M., Asai, M., Yoshida, S., "Control of Flow Separation by Acoustic Excitation," AIAA Journal, Vol. 28, No. 11, November 1990, pp. 1909- 1915. Bar-Sever, A., "Separation Control on an Airfoil by Periodic Forcing," AIAA Journal, Vol. 27, 1989, pp. 8 2 0 - 821. Katz, Y., Nishri, B., and Wygnanski, I., "The Delay of Turbulent Boundary Layer Separation by Oscillatory Active Control," AIAA Paper 89-1027, March 1989. Jiang, F., Y.-C. Tai, K. Walsh, T. Tsao, G.-B. Lee, C.-M. Ho, "A Flexible MEMS Technology and its First Application to Shear Stress Sensor Skin," IEEE-MEMS, Jan 1996. Jiang, F., Y.-C. Tai, B. Gupta, R. Goodman, S. Tung, J. B. Huang, an C.-M. Ho, "A Micromachined Shear Stress Sensor Array," Proc. IEEE MEMS-96 Workshop, San Diego, pp. 110-115, 1996. Ho, C.-M., S. Tung, G.-B. Lee, Y.-C. Tai, F. Jiang, T. Tsao, "MEMS - A Technology for Advancements in Aerospace Engineering," AIAA paper 07-0545, Reno, 1997. Mangalam, S. M., and Kubendran, L.R., Experimental Observations on the Relationship between Stagnation Region Flow Oscillations and Eddy Shedding for Circular Cylinder. Instability and Transition, ICASE NASA LaRC Series, Hussaini and Voigt, eds., Springer-Verlag, 1990. Coder, D. W., "Location of Separation on a Circular Cylinder in Crossflow as a Function of Reynolds Number," NSRDC Report 3647, November 1971. Bellhouse, B. J., D.L. Schultz, "Determination of mean and dynamic skin friction, separation and transition in low-speed flow with a thinfilm heated element," Journal of Fluid Mechanics, 1966, Vol. 24, part 2, pp. 379-400.
ACKNOWLEDGEMENTS The authors would like to acknowledge the funding and encouragement provided by Dr. Bruce Douglas, Director of Research at NSWC. Dr. Douglas manages the In House Laboratory Independent Research (ILIR) program at NSWC which provides funding for a wide variety of critical basic research. The authors would also like to thank Mr. Jack Gordon for his insight and technical assistance during the experiment. As he has so generously done during his 33 year career with NSWC, Mr. Gordon provided critical mentoring, advice and humor during the experiment. Mr. Gordon is now retiring from NSWC, but the results of his mentoring will continue to produce long after his retirement.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
637
E v o l u t i o n o f instabilities in an a x i s y m m e t r i c i m p i n g i n g jet S.V. Alekseenko, A.V.Bilsky, D.M. Markovich and V.I. Semenov Institute of Thermophysics, Siberian Branch of RAS, Lavrentyev Ave., 1, Novosibirsk, 630090, Russia
The experimental study of the evolution of axisymmetrical and spiral instabilities has been performed for the round impinging jet flow. In order to create strictly periodical coherent vortices in the jet shear layer the external low amplitude excitation was applied. Digital analysis of flow visualisation patterns at low Reynolds numbers allowed to obtain the values of phase velocity of the large scale structures along their trajectory. For the large Reynolds numbers the electrodiffusion method and Fourier analysis were applied to determine the flow characteristics. It is stated that attenuation in structure's velocity occurs nonmonotonically along the impingement surface. The manifestations of subharmonics resonance were found. The conditions for transition from axisymmetric mode of the large scale instabilities to the spiral one were determined.
1. INTRODUCTION It is well known that the development of flow instabilities in shear layers influences essentially the mixing processes. The initial instability waves roll up into the discrete vortices which can further merge with the creation of more large structures. In free axisymmetric jet or wake flow the initially axisymmetric vortex structures lose their symmetry downstream and the spiral instability modes appear. An active control of the flow can be provided by the external excitation which can lead to resonant amplification of large-scale vortex structures. First comprehensive study of free forced axisymmetric jet was done by Crow & Champagne (1971). In the recent work, Paschereit et al. (1995) generated resonant subharmonic interaction between the two axisymmetric travelling waves in the jet shear layer and they observed the strong influence of the initial phase difference between the subharmonic and the fundamental harmonic during external excitation of the flow. On the other hand, as it was stated by Gutmark & Ho (1983), the spatial disturbances of individual facilities can change the characteristics of jet flow and so the scatter of obtained characteristics reaches 100% in literature. Thus it is necessary to control as more parameters of experimental facility as possible. The spiral instabilities in free shear layers have been studied by many authors. Browand and Laufer (1975) discovered the growth of energy of spiral disturbancies further downstream in free axisymmetric jet, Cohen et al. (1983) found that with increasing jet's velocity the transition to spiral modes occurs to be located more close to the nozzle exit. The impinging jets have been studied extensively for the last three decades. The turbulent structure of impinging round jet was studied in detail by Donaldson et al. (1971). The
638 characteristics of plane impinging jet were presented in the work by Gutmark et al. (1978). The instability analysis for confined jet flows was done only by Ho & Nosseir (1981) who supposed that the decrease of vortices frequency occurs as a result of collective interaction of large-scale structures in the near-wall region of high speed impinging air jet. A structural image of the impingement region was statistically derived by means of analysis of the surfacepressure fluctuations in the work by Kataoka et al. (1985). Kataoka et al. (1987) associated the heat transfer intensification near the stagnation point with the periodical renewal of the surface by large-scale vortex structures penetrating into this region. The unsteady separations of radial wall jet flow in the vicinity of an impingement region were observed by Didden & Ho (1985) and Ozdemir & Whitelaw (1992). The first measurements of the instant velocity field in the round impinging jet were made by Landreth & Adrian (1990) with the use of PIV technique which, however, have not been provided with high spatial resolution. The recent work by Meola et al. (1995) shows the presence of equidistant azimutal structures which appear in impinging nonisothermal round jet. The present work is devoted to the study of instabilities evolution during the round jet impingement under the action of periodical forcing.
2. EXPERIMENTAL TECHNIQUE AND PROCEDURE
The sketch of impinging jet flow is shown in Figure 1. The experimental set-up consisted of a test section representing the rectangular channel made of Plexiglas, with the dimensions of 86xl 62xl 600 mm 3, the system of pumps and flow meters, a reservoir, connecting tubes and apparatus for measurements. A well-profiled round nozzle was inserted through the side wall of a channel. The submerged round jet issuing from the nozzle impinged normally on the opposite wall (measuring plate) of the channel. The skin friction probes were placed at the measuring plate which could be shifted and this allowed to change the radial position of each probe with an accuracy of 0.1 mm. To measure the wall shear stress and liquid velocities the electrodiffusion method was applied. The details of this technique are described in the author's work (Alekseenko and Markovich, 1994). The electrical signals from the probes passed to the a.d. transformer through the d.c. amplifiers. A complete data processing was accomplished by a personal IBM computer. The computer program allowed us to determine the mean values of the velocity and skin friction, its rms. pulsations, spectral density of the velocity and friction pulsations and mutual spectra also. For the spectral density Figure 1. Sketch of the measuring sell of the impinging jet flow. Re = 1000; H / d = 2
639
Figure 2. The visualisation of impinging jet flow. Re = 1000; Sh = 0.5; (a) - H/d = 2; (b) - 3
estimations the Fast Fourier Transform technique was applied. Each array of the experimental data, which was processed by FFT, consisted of about 100 segments of 2048 points. By the analysis of the signals phase difference between two skin friction probes which were placed close to each other ( Ar = 0.5 mm), it was possible to measure the phase velocity of the largescale structures. The visualisation of flow was performed with the aid of hydrogen bubbles which were produced on the surface of platinum wire during water hydrolysis. The excitation of the jet was provided by a standard electrodynamic vibration exciter connected by the instrumentality of the silphone with the plenum chamber. The sinusoidal excitant signals conveyed from the generator through the power amplifier to the exciter. The initial oscillations of flow embodied the axisymmetric mode (m = 0) and their rms. value changed from "ff/U0 = ~ / ~ 2 / U o = 0.0001 to 0.001 depending on the experimental conditions. The forcing frequency fl , was characterized by the Strouhal number, Sh d = f f . d / U 0 . The velocity measurements near the nozzle exit have shown that the imposed oscillations of level mentioned above do not influence the initial flow characteristics. The level of natural turbulence measured in the vicinity of nozzle was in the range of u ' / U o = ~ - ~ / U o = 0.005 § 0.008 at the nozzle axis and 0.05 + 0.06 at the centre of shear layer. The value of momentum thickness 0 at the nozzle exit, obtained from the velocity profile, equals to 0 ~ 0.1 mm. During the experiments the three values of Reynolds number were tested: Re = 1000 (lowest one), 12700 and 25200. Here Re = U o 9d / v , Uo is the mean flow rate velocity at the nozzle exit, d is the nozzle diameter equal to 10 mm and v is the kinematic viscosity of electrochemical solution equal to 1.04.10 -6
m 2 / s . The distance H between the edge of a
nozzle and the plate could be changed in a wide range - from 10 to 60 mm (H/d = 1 + 6).
640 EXPERIMENTAL RESULTS AND DISCUSSION
The flow visualisation pattems are shown in Figures 2 a, b for different nozzle-to-plate distances which were equal correspondingly to H/d = 2 and 3. It can be observed from the plots that the large scale vortex structures are developing in the mixing layer and interacting downstream with the barrier. In order to analyse the dynamics of the vortices along their trajectory the measurements of phase velocities of the large scale structures were fulfilled. With the aid of digital processing of the instantaneous flow patterns the values of structure's velocities were calculated beginning from the instant of originating the nonlinear wave formations in the free jet shear layer. These data are shown in Figure 3, a, b for the lowest Reynolds number studied, Re = 1000, and for H/d equal correspondingly to 2 and 3. In both cases the phase velocity of structures natively falls up to a certain minimum value in the location of the maximum structure's deceleration. Further downstream the large-scale vortex insignificantly accelerates under the action of negative pressure gradient and then, with increasing the radial distance, phase velocity for corresponding frequency decreases again. Phase velocity dependencies for free jet flow are shown also in Figure 3. The attenuation of velocity of the structures occurs by another way in this case. Similar consistent patterns of large scale instabilities persist also in impinging jet for more high Reynolds numbers. The measurements of local wall shear stress values were performed for Re - 12700 and 25200 for the conditions of external excitation of flow in order to provide the information both for the development of large scale vortex formations and the broad-band turbulence structure. In Figures 4 a, b the radial distributions of mean wall shear stress and rms. pulsation's level are shown versus the forcing frequency fi . Here 1:'= X/'~'2/'Cmax, "Cmax is the maximum value of mean friction. The amplitude and frequency of initial sinusoidal perturbations were determined by measuring the local velocities with electrochemical probe "blunt nose" at the centre of the nozzle exit. For presented data the dimensionless amplitude
1.0 4~ k O ~
0.8
0 9 0
1.0 ,i }
freejet impingingjet, H/d=2 0
Vy
O freejet 9 impinging jet, H/d=-3
0.8
~ r
0.6
0.6
(a)
0.4 0
(b)
0.4 |
I
2
x/d
|
4
0
I
'
2
x/d
I
4
Figure 3. Phase velocity of large scale structures in impinging jet. Re = 1000, Sh = 0.5, (a) H/d = 2, (b)- 3.
641 50 fr (Hz) Sh
0.40. 40 0.30
,
~
~,
90 150 210 260
0.34 0.57 0.8 1.0
k
0.20
!
0
9 t [] a
30
20
O
(b)
Ca) 0.10
P
0.00 0
2
4
r/d
6
0
2
r/d
4
6
Figure 4. Distributions of mean wall shear stress (a) and its turbulent intensity (b) for the different frequencies of excitation. Impinging round jet, Re = 25200, H/d = 2 1E+2
lIE+2
i
i
!
1 I
11~+1
1E+I
I
I:
I
'
~
;
i1
I J
1
lIE+0
1E+0 #L~..,,
1
J . . . . i,,
,,
1
1E-1
.~
,J
....
.f~
~Ii
I
I
1E-1
-" ,ii
,
~.~_i
1E-2
i ! i
~ " 1E-2 ~u
i. 1E-3
iJ
i
1E-3 |m
II
II 1E-4 1E-5 0.01
i
I
J . ~
I
l
i
1E-4 1E-5 0.10
Sh (a)
1.00
0.01
O.lO
Sh
1.oo
(b)
Figure 5. Spectral distributions of wall shear stress pulsations at different locations on the impingement plate. Re = 25200; dashed lines - unexcited jet, solid lines excited jet. (a) - r / d - 1.1, (b) - 2.2.
of velocity pulsations was constam and equal to e = u*/u o = 0.001. At forcing frequency J) = 90 Hz, lying outside the range of the natural frequency of coherent eddies, see below, (let's call the centre of this range as the most probable frequency fm), the mean friction distribution changes weakly, whereas the pulsation level grows more significantly with simultaneous displacement of its maximum towards the stagnation point. In the range of forcing frequency J) = 130 - 180 Hz (Sh - 0.5 - 0.68) close to fm, the mean skin friction at any point on the surface decreases, the second maximum of mean friction
642
disappears and the pulsation's level attains the greatest amplification (up to 1E+O % 42%) at r/d ~ 1. Further increase in J) OA ~OA ..... ~O~j sDv vCl..~ leads to reverse changes. In our previous ) o O0 0 1 0 work (Alekseenko et al., 1996) we ) 9 2 b qualified such an effect as | 3 D 0 0 "quasilaminarisation" of the flow, 1E-1 ~ ~ ~_~-J~ because similar distributions for the wall shear stress and its pulsations can be 1E+2 .,-r q-. observed in the low-Re impinging jet --; (b) flows. One more confirmation of the ---O---O 1E+I " o O --~ effect of turbulence suppression can be O _,,w" I found in Figure 5, a, where the spectral 1E+O -~ ' ~ - ---distributions of the wall shear rate pulsations are presented for the unforced O| 9 and forced conditions. This spectrum 1E-2 ~ | C corresponds to the point at the C IE-3 impingement surface where coherent 0.0 1.0 2.0 3.0 4.0 5.0 vortices penetrate from the free jet shear r/d layer with highest intensity. In the absence of forcing (grey line) one can Figure 6. Distributions of phase velocity (a) and observe the pronounced hump in the intensity (b) of fundamental and subharmonic spectral distribution. This hump, along the impingement surface. Re = 25200, however, is not sharp enough and thus H/d = 2. Excitation is carried out at frequency comprises the contributions of the f/ = 150 Hz (Sh = 0.57). 1 - fundamental natural coherent structures with certain harmonic, 2 - subharmonic, 3 - most probable frequency range. When the excitation is frequency in unforced case. applied by the frequency corresponding to the centre of natural hump or its neighbourhood (region of sensitivity of the jet), the sharp amplification of the coherent structures appears in such a way that they become much more powerful and strictly periodical (see black line in Figure 5, a). At the same time the vortices of neighbouring scales are suppressed strongly in the wide range of frequencies. In Figure 5, b the spectra are presented for certain location in the far field of radial wall jet flow (r/d = 2.2). In this point the vortex merging occurs with high degree of probability, i.e. the two sequential vortices are pairing with the creation of more powerful and slow structure. The maximum on the spectral distribution in Figure 5, b shows the subharmonic's intensity. In order to analyse the dynamics of fundamental harmonic and subharmonic during jet impingement let's consider the evolution of corresponding structures. Figure 6, b demonstrates the development of instability waves along the impingement surface. The data were obtained from the measured spectra of the wall shear stress pulsations. Both the intensity of harmonic with most probable frequency for the unforced jet and the intensity of main harmonic for the forced jet develop similar to the rms. friction pulsations. The most probable (preferable) frequency fmp corresponds to the maximum of power spectrum of skin friction pulsations in the absence of forcing (Figure 5, a). If the forcing frequences lie in a range mentioned above, fl. = 130 + 180 Hz (Sh = 0.5 + 0.68), one can observe a sharp amplification of pulsations at response frequency f~ (which is equal to the forcing frequency f (a)
643
2.5
~
]
',
...........
1
2.0
~
....
2
1.5
". -',
,,
1.0
"{,i
71:
-
r/d
3
II 1 0 2 II~ 3
,,;.....,,........, _ ~
""
~a-~_
I ,i
iz"
0.80
,
0.3
.
0.5
,
,
0.7
0.9
0.30
0.3
I
I
0.5
0.7
R/Rtip
1
'
0.9
R/Rtip
a) Pressure
b) K factor in the middle of the cavity
Fig.4 Radial distribution of flow parameters with different C w Re0=2x 107, K at inlet A= 1.5
1.00
1.00 .
-
.-
0.80
0.90 CI. "-
ix. 13.
O
.- -"
........... - - R e 0 = 1.2x 107 ......... - - Re0=2.0x 107 ....'~ .. "..... Re0=2.9xl0 7
q')
0.80
*i, \
~ Re0= 1.2x107 - - Re0=2.0xl 0 7
':~.{,..,..,..~...:
..... 9
0.90
O
,,,I
0.70 0.60 0.50
= "
'i
0.40 '~..~ ...... :.'"
0.70 0.3
i
I
0 5
0.7 R/Rtip
a) Pressure
'
1
0.9
0.30 0.3
I
I
I
0.5
0.7
0.9
R/Rtip
b) K factor in the middle of the cavity
Fig.5 Radial distribution of flow parameters with different Re 0 Cw= 1.7x 104, K at inlet A-- 1.5
The overall flow characteristics are summarized by the pressure ratio across the cavity, the moment coefficient and the temperature change due to windage. The windage work generated by the rotating surface is absorbed by the mass flow exiting at exit D. This produces a AT, i.e. the temperature change due to windage. Figs. 6 to 8 show the influence of K factor at inlet A, C w, and Re 0 on the above three flow parameters.
728 1.00
9e-04
.+
|
~.
0.90 -
'-
8e-04
8
0.80 0.70-t 0.0
. . . . . I 0.5 1.0 1.5 2.0 2.5 K factor at inlet A
5e-04
i
0.0
9
!
i
I
0.5 1.0 1.5 2.0 K factor at inlet A
2.5
b) Moment coefficient
a) Pressure ratio across the cavity
0.13
6e-04
-
0.12 "tO
.c_
0.11
Fig.6 The influence of K factor at inlet A Cw= 1.7x104, Re0=2x 107
.~ o.lo (s
and usually )'L > 7'0 and (RcamberL) -1 > (RcamberD) -1 9
(4a)
Such modified elemental cascade geometry can come "virtually" into being only if the flow entering the rotor at r0 fits to a conical stream tube of e L r e D. For example, if (g/t)D, YD and (RcamberD)-I decreases with r a d i u s - a n d this is the case for rotors BUP-26, BUP-29 and
BUP-103, see Table 1 - , s L < c D (i.e. r03L < r03D) must be performed in order to ensure that the stream tube passes the rotor along sloping, "quasi-optimum" blade sections of increased mean solidity, stagger angle and camber curvature. If the machine is operated at a flow rate higher than design (condition H), the train of thought is just the opposite (cp. Fig. 6): ~03H -- ~003D-at-Ab03 and ~r
-- ~r
-- A ~r and the corresponding optimum geometry
(5)
(g/t)H < (t/t)D , and usually ?'n < 7'o and (R~amberH)-'< (Rc~mberD)-' (5a) ,giving that eg. for the rotors discussed in this paper, e H > c D (i.e. r03H > to3D) must be ensured.
759 Of course, an extensive research work and thorough quantitative analysis is still needed on the possibilities for design utilization of the above effects, to be discussed in another paper. Now an experimental, rather qualitative justification is attempted at present state of the concept, through a simultaneous evaluation of hydraulic efficiency curves in Fig. 1 and stream tube cone half-angle distributions in Fig. 7. Rotor BUP-26 has an acceptably high hydraulic efficiency at the design flow rate, which suggests that it represents a quasi-optimum geometry for the throughflow at tPD. Nevertheless, the hydraulic efficiency is rapidly reduced for off-design flow rates (the efficiency curve has a peak). This experience can be explained with analysis of Fig. 7: though condition c L < e D is fulfilled within the dominant part of the main flow, e D - e L has relatively small values, and has even a zero transition, representing effects which act contrary to the concept of energy-efficient conical cascades, e H > e D is fulfilled for a limited range only, and an undesirable zero transition occurs in e H - e D as well. Rotor BUP-29 has the highest hydraulic efficiency at ~ = 0.5 and also over a wide surrounding flow rate range (the efficiency curve has a "high plateau"). Fig. 7 shows that e L < e D is fulfilled for the entire main flow, with e D - e L values higher than those valid for BUP-26. Though condition e H > e D is failed to be fulfilled for rotors BUP-29 and BUP-103 (Figs. 7b, c), the reader is notified now that s values for these rotors are less than optimum [7] (s was kept constant with span for simplification in manufacturing) and thus, failure in fulfilment of condition e H > e D may be advantageous in these particular cases (preventing the virtual s value from further reduction). Small but positive e D - e L values are present in the case of BUP-103 (Fig. 7). This experience, together with consideration in the previous paragraph suggests why the efficiency curve of also BUP-103 has a plateau, though lower than that for BUP-29, which lag is probably due to the considerable reduction of cascade solidity relative to the optimum values determined in design. 5. SUMMARY The aerodynamic behaviour of high pressure axial flow fans has been investigated through global and LDA measurements. Secondary flow vector plots, turbulence properties (velocity fluctuations) and profiles of pitchwise-averaged velocity components have been studied in detail. The generally applicable concept of energy-efficient conical cascades has been sketched out and illustrated through evaluation of the experimental results. After elaboration of appropriate quantitative models regarding the influence of geometrical and aerodynamic parameters on radial fluid re-arrangement within the rotor, this concept can be built in the design methods of axial flow rotating cascades in order to retain energy-efficient rotor operation even under off-design circumstances. ACKNOWLEDGEMENT The experimental work has been carried out and the concept of energy-efficient conical cascades has been outlined and was discussed in the paper by J. Vad, whose work was supported by a J~.nos Bolyai Hungarian National Research Grant, Ref. No. BO/00150/98. Authors acknowledge the support of OTKA (Hungarian National Foundation for Science and Research) program T 025361, under the co-ordination ofF. Bencze.
760 NOMENCLATURE c absolute velocity h rotor width g blade chord N blade number r radius R = r/rc dimensionless radius Reamber camber radius of curvature t = 2 r n / N blade pitch Tug axial velocity fluctuation ur reference velocity (rr co) s conical stream tube half-angle y blade stagger angle (measured from the circumferential direction) flow coefficient (area-averaged axial velocity in the annulus divided by ur cp = c x/uc local axial flow coefficient (Pr -- C'r/Uc
local radial flow coefficient
~3 = 2 R C3u/Uc local ideal total head rise
coefficient (assuming zero inlet swirl) ~t average total head rise coefficient (total head rise in the annulus normalised by
m /2)
p fluid density v hub-to-casing ratio r average tip clearance corotor angular speed r/h hydraulic efficiency SUBSCRIPTS AND SUPERSCRIPTS c casing wall L,H,D lower/higher/design r, u, x radial, tangential, axial 0 rotor inlet plane 3 rotor exit plane ^ pitch-averaged
REFERENCES 1. 2.
Wallis, R. A.: Axial Flow Fans. Newnes, London (1961) Somly6dy, L., Berechnung der Beschaufelung und der Str6mung axial durchstr6mter Wirbelmaschinen mit Anwendung von Gittermessung. Periodica Polytechnica, No. 3. Budapest (1971) 3. Dring, R. P., Joslyn, H. D., and Hardin, L. W., An Investigation of Axial Compressor Rotor Aerodynamics, ASME J. Eng. for Power, Vol. 104, pp. 84 - 96. (1982) 4. Inoue, M., and Kuroumaru, M., Three-Dimensional Structure and Decay of Vortices Behind an Axial Flow Rotating Blade Row, ASME J. Eng. Gas Turbines Power, Vol. 106, pp. 561 - 569. (1984) 5. Meixner, H. U., Vergleichende LDA-Messungen an ungesichelten und gesichelten Axialventilatoren, Ph.D. Thesis, University of Karlsruhe, Germany (1994) 6. Lakshminarayana, B., Zaccaria, M., Marathe, B., The Structure of Tip Clearance Flow in Axial Flow Compressors, ASME J. Turbomachinery, Vol. 117 (1995) 7. Vad, J., and Bencze, F., Three-Dimensional Flow in Axial Flow Fans of Non-Free Vortex Design, Int. J. Heat Fluid Flow (accepted in 1997, Ref. No. HFF 6038) 8. Vad, J., Bencze, F., Fiiredi, G., and Szombati, R., Fluid Mechanical Investigation on Axial Flow Fans, 10th Conference on Fluid Machinery, Budapest, pp. 500 - 509. (1995) 9. Vad, J., and Bencze, F., Secondary Flow in Axial Flow Fans of Non-Free Vortex Operation, 8th Int. Symp. Applications Laser Techniques to Fluid Mechanics, Lisbon, Portugal. Vol. 1, pp. 14.6.1. - 14.6.8. (1996) 10. Vad, J., and Bencze, F., Laser Doppler Anemometer Measurements Upstream and Downstream of an Axial Flow Rotor Cascade of Adjustable Stagger, 9th Int. Conf. Flow Measurement (FLOMEKO), Ltmd, Sweden, pp. 579 - 584. (1998)
11. Heat Transfer
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
On p r e d i c t i o n of t u r b u l e n t c o n v e c t i v e rib-roughened rectangular cooling ducts
763
heat transfer
in
Arash Saidi and Bengt Sund~n Division of Heat Transfer, Lund Institute of Technology, 221 00 Lund, Sweden
1. ABSTRACT A numerical investigation has been carried out to predict local and mean thermal-hydraulic characteristics in rib-roughened ducts. The Navier-Stokes and energy equations, and a low-Re number k-~ turbulence model are solved. Two methods for determination of the Reynolds stresses, eddy viscosity model (EVM) and explicit algebraic stress model (EASM), are used. The numerical solution procedure uses a collocated grid, and the pressure-velocity coupling is handled by the SIMPLEC algorithm. The computations are performed with the assumption of fully developed periodic conditions. The ribbed duct configuration is identical to the experimental setup of Rau et al. (1998) with only one wall having ribs, rib to hydraulic diameter ratio equal to 0.1 (e/Dh=0.1) and pitch to rib height ratio of 9 (P/e=9), see Figure 1. The Reynolds number is 30,000. The calculated mean and local thermal-hydraulic values are compared with these experimental data and prediction capabilities of the two turbulence models (EVM and EASM) are discussed.
2. INTRODUCTION To increase the performance of gas turbine cycles it is desirable to raise the temperature of the gas entering the turbine. However, this increase is limited by the mechanical and thermal properties of the materials used in gas turbine blades. In the 60s, cooling methods for blades were introduced. More recently internal cooling ducts have been applied. There are various types of such ducts, but in this report rectangular ducts with embedded ribs are considered. In the past both experimental and numerical investigations have been carried out but many features of the flow and heat transfer at various conditions still remain to be found. Experimental investigations by Han (1988) gave some information concerning influence of rib pitch and height on the overall Nusselt number and friction factor. Effects of rib shape were studied by Liou and Hwang (1993). The importance of model orientation was considered by Johnson et al. (1994) and Parsons et al. (1995). Detailed investigations of local flow and thermal fields have only recently become available and the studies by Hwang and Liou (1997), and Rau et al. (1998) have provided
764 interesting and useful results. Some experimental investigations including rotation effects are also available. Numerical investigations appear frequently nowadays but more research is needed to improve the reliability and accuracy in the predictions. Recent works are those of Acharya et al. (1993), Prakash and Zerkle (1995), Iacovides (1996), Rigby (1998), and Saidi and Sund6n (1998). In the present investigation, prediction of local and mean thermal-hydraulic characteristics is considered for a particular ribbed duct. The ribbed duct geometry is identical to that of Rau et al. (1998) and thus experimental data are available for comparison. Two different turbulence models are considered and one purpose of this work is to reveal the performance and evaluate these models. 3. P R O B L E M S T A T E M E N T
The problem to be solved is the fluid flow and heat transfer in a ribbed duct but only one wall has ribs (Figure 1). In this paper the pitch to rib height ratio is 9, (P/e=9).
Z
I'.
tY
Flow direction
Dh
X
Figure 1. One module of the one-sided ribbed duct, aspect ratio is one. 4. M A T H E M A T I C A L MODEL
The governing equations are the steady state continuity, the time averaged Navier-Stokes and the energy equations for turbulent flow, i. e., 0 Ox-~ (oUj) - 0 (1) 0
-0Xj - (oUjUi)-
0 [ OU i OUj ] 0 OP - 0x i + ~jxj L~t( 0xj + -V----)| Oxi J + Oxj ( - P u i u J )
(2)
765
axj (pUjT)=
(3)
~jxj P-r Oxxj- p u j t
In this case incompressible flow is considered and the density is thus constant. In many cases, the assumption of fully developed periodic conditions is applicable in the ribbed ducts and mean velocity and thermal fields field are periodically repeated, and in the considered experimental case this point is explicitly mentioned by the investigators (Rau et al., 1998). The procedure for handling fully developed periodic flow and heat transfer was introduced already by Patankar et al. (1977) and has then been applied extensively. This procedure is followed in this study. One introduces, p* P =-~x + (4) where ]3 resembles the non-periodic pressure gradient and P* is the periodic part of pressure in the main flow direction. In the energy equation a constant heat flux boundary condition is applied because this prevailed in the experimental setup.
4.1. T u r b u l e n c e m o d e l Several models have been suggested for calculation of turbulent flow and heat transfer, see e.g., Wilcox (1993). For engineering calculations, two equation models accompanied with the eddy viscosity concept have become popular because they provide reasonable overall results in many cases although not providing correct local distributions of turbulence generated secondary flows in ducts. Abe et al. (1994) proposed a low-Re number version of the k-~ model. Their model has the following advantages: 1) it uses a velocity scale near the wall other than the friction velocity and this velocity scale solves the singular point problem close to the reattachment in the separated flow cases, 2), numerical experimentation showed that this model does not have the initial value problem as other models in periodic boundary conditions, see Rokni and Sund~n (1997). This model can be summarized as,
ax---j(pUjk)- ~jxj (la + ~
---pe -OUiUj Oxj
' 'E .t ';1 -C l Oxj (pUjl3)= ~jxj (
~2 OUi C~2f~P k puiuj ~-xj
(5)
(6)
k2 ~tt = C~f~p
(7)
766
f~ = 1 - e x p ( - ~-~)
5 Re 2 1 + Ret3/4 exp - ( 2 - ~ )
(8)
fE= 1 - e x p ( - 3 - ~ )
Ret )2 1-0.3exp-(-6-ff
(9)
where f~ and f~ are d a m p i n g functions. Two different approaches for d e t e r m i n a t i o n of the Reynolds stresses are adopted in this study. The first one is the eddy viscosity model (EVM) and the second one is the explicit algebraic stress model (EASM) of Speziale and Xu (1995). In EVM the "Reynolds stresses" are calculated as: 2 ~tt u i u j = ~ kSij - 2 ~ Sij (10) P . In EASM these stresses are calculated as : 2 , k2 , k3 uiuj = ~ kSij -(Zl ~( z 2~jij g - ~ (Sik 0)kj + Sjk 0)ki) (11) , k3
1
+0~3 -~-2-(Sik Skj - ~ Skl SklSij ) la3k ~1- 2 (z2 g
(~jijS~ij)1/2
_ 1 a2 k (co--~co--~)m/2 2a 1 (~i = O~i(11,~)
(12) (13) (14)
where Sij and coij are the m e a n rate of strain and m e a n vorticity tensors, and al.2.,3 are model variables. For detailed description of the EASM, see Speziale and Xu (1995). 4.2. B o u n d a r y
conditions
Periodic b o u n d a r y conditions are applied at the inlet and outlet of every periodic module (which is equal to one pitch in the ribbed wall duct). This condition is defined as, (U,V,W are velocity components, P, k and e are pressure, t u r b u l e n t kinetic energy and t u r b u l e n t energy dissipation, respectively) 9 q)(x,y,z)=r +L,y,z) (15) r The b o u n d a r y conditions at walls are imposed as:
(16)
767 U=V=W=k=O
(17)
and qw=COnstant ~w=2 ~ p (~) k~/~n) 2
(18) (19)
where n represents the normal wall distance. 5. N U M E R I C A L S O L U T I O N P R O C E D U R E
A computer code based on the finite volume technique is modified and applied to solve the governing equations. The code uses a collocated grid arrangement and employs the Rhie and Chow (1981) method to interpolate values of velocity at the control volume faces. The SIMPLEC algorithm handles the coupling between pressure and velocity. Convective fluxes are determined by the hybrid scheme in all equations. An interpolation technique similar to that used by Obi et al. (1991) is employed to avoid oscillatory solutions that might otherwise appear from inappropriate coupling of the mean velocity and Reynolds stress fields in an EASM case. A non-uniform grid has been chosen, and grid refinement has been applied in the near wall regions. As a low Reynolds number formulation is applied, it is important that the n § -value of the grid points closest to the walls is of the order of unity. In this work the n+-values were in the range of 0.3-0.8 for all considered cases. The calculations were terminated when the absolute residuals of all the variables became less than 10 .6. The under-relaxation factors for all calculations were set to values in the range of 0.5-0.6. The non-periodic pressure gradient is specified to create a flow field. Through the rate of mass flow, the Reynolds number is coupled to ]3. Thus various values of [3 correspond to various Reynolds numbers. Grid refinement studies have been carried out, in both normal and axial directions. The presented results are obtained by a 60X 28X 58 points grid and the results (mean Nu and f values) changed less than one percent as further refinement to 70X 28X 58 points was applied (grid refinement was already applied in the normal to wall directions, namely Y and Z directions). The CPU times for calculations with EASM were 70% higher than those of EVM. 6. R E S U L T S 6.1. M e a n v a l u e s : The case considered is a one-sided ribbed duct of pitch to rib height ratio of 9 and rib height to duct hydraulic diameter of 0.1 for a square-sectioned duct at Re=30,000, see Figure 1. This case is identical to t h a t of Rau et al. (1998). The obtained mean values of the friction factor and the Nusselt numbers are
768 compared with the experimental data for the two considered models as shown in the Table 1. ( f and Nu values are normalized with respect to the experimental correlation for the same duct without ribs). As is evident in the table, the friction factor prediction becomes more accurate by using the EASM method. The heat transfer predictions show that both models have the same prediction quality. They have predicted the heat transfer increase in the floor between ribs within an acceptable error but over-predicted the Nusselt number on the smooth side wall.
Table 1. C o m p a r i s o n of m e a n t h e r m a l - h y d r a u l i c p r e d i c t i o n ( c o m p a r e d to R a u et al. (1998)) Turb. Model
fifo, error
EVM EASM
3.79,-27% 5.34, +2%
Nu/Nuo. ,error (Floor between ribs) 2.10,-4% 2.14,-2%
NtffNuo,error (Side wall) 1.90, +22% 1.97, +27%
6.2. Local v a l u e s and flow fields: In order to provide a detailed comparison between the two models, the flow fields and local values are examined. Figure 2 shows the vector representation of the velocity components (floor section) in an XY plane located at Z/e=0.25. In both methods the entrained mean stream flow turns towards the side wall in the front of the next rib. As can be seen in the experiment the m a x i m u m of the U-velocity component downstream the recirculation zone and upstream the next rib does not occur at the symmetry line but somewhere closer to the side wall. This feature is not reflected at all in the EVM but it exists in the EASM prediction of the flow field. x
m
q
Experiment EASM EVM Figure. 2. Velocity vectors in the floor section between ribs, XY plane, Z/e=0.25. This m a x i m u m can be explained by consideration of the secondary flow pattern downstream the recirculation zone behind the rib. The secondary flow vectors in a cross-sectional plane are shown in Figure 3 (a YZ plane). The secondary flow has a sort of impingement at the corner, and then turns towards the symmetry line but before reaching it, disappears, or in other
769 words amplifies the primary flow or U-velocity component. This explains the shift in the maximum point of the U-velocity profile shown in Figure 2.
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~ % ~. % % 'l,%lllWitl
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---
'
..........
~. \
% 't.'%l,l,',lt~.
,~ t ~ ,i ~, I llltillfWI
.
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x
~Q9
,,I
........
,.
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~
x
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t
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I 1 i llllr1~111
,
q
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~
rltltM~
I i I 1 llllllt~
l llj t lJtt , tl
/_
Figure 3. Secondary flow vectors (EASM). Figure 4. Vorticity iso-lines (EASM) Experimental investigations proved the existence of a large vortex cell in each half cross sectional plane of a one-sided ribbed duct (Rau et al., 1998). In Figure 4 the vorticity isolines are depicted in a half cross section of the duct (a YZ plane) and this vorticity plot certainly shows a large vortex cell in the EASM calculation (all calculations were done for a half duct considering existing symmetry line, the left side end of the Figure 3 and Figure 4). On the other hand, the EVM method could not predict this large vortex cell. The vertical velocity component (W) is measured at the symmetry line (see Figure 5). This velocity component (W), which is negative just downstream the rib, can be related to the flow entrainment from the main-stream to the recirculation zone. This flow brings the cold air from the center in to contact with the wall and increases the heat transfer. As can be seen in the figure, both methods have almost the same ability to predict these values but the location of minimum of this velocity component is not correctly predicted.
770 W/U r o2
........... EASM 9
0.1
.'""Ira
EVM EXPERIMENT (Rau et al., 1998)
0.0 .....9
........ ""
-0"ii,I .... , .... I , , ,I, I I , I,, .... , .... , .... , .... -02
I
2
4
3
5
6
7
X/e
Figure 5. Normalized W-velocity component at symmetry line at Z/e=l plane. The Nusselt number enhancement factor along the symmetry line on the ribbed floor is shown in Figure 6. The EASM method presents a relatively better prediction capability, although both methods failed to predict the correct location of the maximum enhancement. N~4Uo
3.0 2.5 n
n
n
•
2.0 1.5 1.0 0.5
1
2
3
4
5
6
7
•
Figure 6. Nusselt number enhancement at the symmetry line between ribs. CONCLUSIONS A numerical investigation is carried out for prediction of the thermalhydraulic characteristics in a ribbed duct. Two different methods for
771 determination of the Reynolds stresses are used. The considered case was chosen similar to the experimental run ofRau et al. (1998). The investigation has shown that EASM can considerably improve the mean friction factor prediction. The superiority of EASM over EVM in the prediction of the friction factor could be partly explained by considering the calculated flow field and comparing it with the experimental one. Then, it is evident that there are some flow field phenomena existing in this problem, which were successfully predicted by EASM, but not by EVM. A displacement of the maximum streamwise velocity toward the smooth side wall is predicted by EASM, which is in a qualitative agreement with the experiment, but the EVM did not capture this shift. Another interesting difference between the two prediction methods is the existence of a large vortex cell in the half cross section of the duct in the EASM prediction. This large vortex cell was experimentally observed. The mean values of the heat transfer enhancement were successfully predicted by both methods on the wall between ribs (rough wall) but they both failed, i.e., over-predicted these values over the smooth side wall. REFERENCES
1. Abe, K., Kondoh, T. and Nagano, Y., 1994, "A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching FlowsI. Flow Field Calculations", Int. J. Heat Mass Transfer, Vol. 37, pp. 139-151. 2. Acharya, S., Dutta, S., Myrum, T.A. and Baker, R.S., 1993, "Periodically Developed Flow and Heat Transfer in a Ribbed Duct", Int. J. Heat Mass Transfer, Vol. 36, pp. 2069-2082. 3. Han, J.C., 1988, "Heat Transfer and Friction Characteristics in Rectangular Channels with Rib Turbulators", ASME J. Heat Transfer, Vol. 110, pp. 321-328. 4. Hwang, J. and Liou, T., 1997, "Heat Transfer Augmentation in a Rectangular Channel With Slit Rib-Turbulators on Two Opposite Walls", ASME J. ofTurbomachinery, Vol. 119, pp. 617-623. 5. Iacovides, H., 1996, "Computation of Flow and Heat Transfer Through Rotating Ribbed Passages", Biennial Colloqium on Computational Fluid Dynamics, UMIST, pp. 3-19:3-24. 6. Johnson, B. V., Wagner, J. H., Steuber, G. D. and Yeh, F. C., 1994, "Heat Transfer in Rotating Serpentine Passages with Selected Model Orientations for Smooth or Skewed Trip Walls", ASME J. Turbomachinery, Vol. 116, pp. 738744. 7. Liou, T. and Hwang, J., 1993, "Effect of Ridge Shapes on Turbulent Heat Transfer and Friction in a Rectangular Channel", Int. J. Heat Mass Transfer, Vol. 36, pp. 931-940.
772 8. Obi, S., Peric, M. and Scheuerer, G., 1991,"Second-Moment Calculation Procedure for Turbulent Flows with Collocated Variable Arrangement", A/AA J., Vol. 29, pp. 585-590. 9. Parsons, J. A., Han, J. C. and Zhang, Y., 1995, "Effect of Model Orientation and Wall Heating Condition on Local Heat Transfer in a Rotating Two-Pass Square Channel with Rib Turbulators", Int. J. Heat Mass Transfer, Vol. 38, pp. 1151-1159. 10. Patankar, S. V., Liu, C. H. and Sparrow, E. M., 1977, "Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-periodic Variations of Cross-sectional Area", ASME J. Heat Transfer, Vol. 99, pp. 180-186. 11. Prakash, C. and Zerkle, R., 1995, "Prediction of Turbulent Flow and Heat Transfer in a Ribbed Rectangular Duct with and without Rotation", ASME J. Turbomachinery, Vol. 117, pp. 255-264. 12. Rau, M., Cakan, M., Moeller, D. and Arts, T., 1998, "The Effect of Periodic Ribs on the Local Aerodynamics and Heat Transfer Performance of a Straight Cooling Channel", ASME J. of Turbomachinery, Vol. 120, pp. 368-375. 13. Rhie, C. M. and Chow, W. L., 1983, "Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation", A/AA J., Vol. 21, pp. 15251532. 14. Rigby, D. L., 1998, "Prediction of Heat and Mass Transfer in a Rotating Ribbed Coolant Passage with a 180 Degree Turn", ASME paper 98-GT-329. 15. Rokni, M. and Sund~n, B., 1997, "Improved Modeling of Turbulent Forced Convective Heat Transfer in Straight Ducts", ASME HTD-Vol. 346, pp. 141-149. 16. Wilcox, D. C., 1993, "Turbulence Modeling for CFD", DCW Industries Inc., USA. 17. Saidi, A. and Sund~n, B., 1998, "Calculation of Convective Heat Transfer in Square-Sectioned Gas Turbine Blade Cooling Channels", ASME paper 98GT-204. 18. Speziale, C.G. and Xu, X-H., 1995, "Towards the Development of SecondOrder Closure Models for Non-Equilibrium Turbulent Flows", 10 ~h Syrup. on Turbulent Shear Flows, Penn. State University, pp. 23-7:23-12.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
773
The m e a s u r e m e n t o f local wall heat transfer in stationary U-ducts o f strong curvature, with smooth and rib roughened walls. H. Iacovides, D.C. Jackson, G. Kelemenis, and B.E. Launder Department of Mechanical Engineering, UMIST, Manchester, UK
Abstract
The paper presents some of our recent experimental investigations of convective heat transfer in flow through stationary passages relevant to gas-turbine blade-cooling applications. Local Nusselt number measurements in flows through round-ended U-bends of square cross-section, with and without artificial wall roughness are presented. Our earlier LDA measurements of flows through these passages are first briefly reviewed and then the liquid-crystal technique for the measurement of local wall heat transfer inside passages of complex geometries is then presented. Tightly curved U-bends generate strong secondary motion and cause flow separation at the bend exit, which substantially raise turbulence levels. Wall heat transfer is significantly increased, especially immediately downstream of the U-bend, where it is over two times higher than in a straight duct. The local heat-transfer coefficient around the perimeter of the passage is also found to vary considerably because of the curvature-induced secondary motiol~ The introduction of surface ribs, results in a further increase in turbulence levels, a reduction in the size of the curvature induced separation bubble and a complex flow development after the bend exit with additional separation regions along the outer wall. Heat-transfer levels in the straight sections are more than doubled by the introduction ofn'bs. The effects of the bend on the overall levels of Nusselt number are not as strong as in the smooth U-bend, but are still significant. The effects of the bend on the perimetral variation of local heat-transfer coefficients within the ribbed downstream section are also substantial. 1.
INTRODUCTION
The demand for in~rovements in the efficiency and power output of gas-turbines, has led engine designers continuously to seek to raise operating temperatures. In order to preserve the structural integrity of rotating blades, elaborate cooling systems have been evolved in which relatively cool air is extracted from the compressor and, through the central shaft, is fed to cooling passages inside the turbine blades and nozzle guide vanes. The flow inside these cooling passages is complex and highly three-dimensional, influenced by the presence of sharp U-bends, artificial rib-roughness and also by the rotation of the blades. In order to optimise the cooling process, the engine designer needs to have accurate information on the detailed flow development and its effects on local wall heat transfer. Because of the geometrical complexities of the cooling passages and the presence of rotation, most of the experimental heat-transfer data available until recently, though of considerable value, were nevertheless confined to averaged values. Due to the
774 lack of detailed data on local flow and heat transfer, it has neither been possible to gain a clear understanding of the hydrodynamic and thermal behaviour in such passages, nor to develop and validate numerical flow solvers that can be reliably used for the simulation of blade cooling flows and the consequent thermal stresses that arise. It is the second of these two issues, the provision of local flow and thermal data for validation of numerical flow solvers, that provides the motivation for the work reported here.
D
II
I
] P "
"
.
e
.
I
-*FI
~
~
.
.
Rib removed
"
~
.
Figure 1. Cases investigated III
m
m
~
m
This investigation forms part of a more extensive experimental effort in our group [ 1], aimed at producing such CFD-validation data for flows through idealised blade-cooling passages under stationary and rotating conditions. As shown in Figure 1, we here focus on the thermal development through stationary, round-ended U-bends of strong curvature of square crosssection, both with smooth walls and with the inner and outer walls of the straight sections roughened with staggered ribs of square cross-section. Though the geometries of these passages are somewhat idealised, the presence of strong curvature and rib roughness leads to the 180"
I
2
3
4
_
Symmetry
~ ii wI'
u"
Plane 9
,
w
w
w
?
Near-wall
Plane ~
O= 0 e
Figure2.
- I
-2
,11
~l
9
qll
Ok
I
11 q l l
-3 z/D
LDA measurements for case I, from reference [2].
Figure 3. LDA measurements for cases II and III, [4] and [3].
775 development of most of the important flow features found in real blade-cooling passages. The local flow development through these passages has been the subject of earlier investigations in our group, [2-4]. As can be seen in Figure 2, for the passage with smooth walls the LDA measurements [2], revealed that the flow development is dominated by the strong streamwise pressure gradients at the bend entry and exit. A separation bubble is formed along the inner wall, half-way around the bend, which extends to about two diameters in the straight downstream sectiorL Because of the flow separation, turbulence levels are greatly increased at the bend exit. As can also be seen in Figure 2, due to the curvature-induced secondary motion, the flow on the mid-plane and on a parallel plane close to the flat wall are very different. The introduction ofrl'bs, [3] and [4], naturally increases turbulence in the straight sections. This results in the delay of flow separation along the inner wall of the bend. As seen in Figure 3, Case III, downstream of the bend, as the faster fluid along the outer wall encounters the first rib, a large separation bubble is formed along the outer wall, behind the first rib. Consequently the length of the separation bubble along the inner wall is reduced. Removal of the first downstream rib from the outer wall, Case II in Figure 1, displaces the outer-wall separation bubble further downstream and reduces its size. This is accompanied by an increase in the length of the separation bubble along the inner wall, however. Here heat-transfer measurements that we have recently obtained using the liquid-crystal technique are reported and, through comparisons with the earlier LDA measurements, the mechanisms by which the flow development influences wall heat transfer are discussed. 2.
APPARATUS
The thermal measurements have been obtained using air as the working fluid. As shown in Figure 4, on open-loop system is employed, with a fan drawing air into the experimental model through a contracting inlet section and a combination of fine wire meshes and a honeycomb section, placed there to ensure uniform and symmetric entry conditions. The working section is connected to the fan through a long orifice plate section, used to measure |- BLEED CONTROL PUMP \ 9 VALVE FLOW the mass flow rate. , ..... STRA~.'rENE,~ The experimental model is made ~"~ - ' -of 10mm thick perspex. The passage ........................... -I ~,\ has a square cross-section with a .~. / ~' " ORIFICE PLATE //" ; diameter of 50mm. The ratio between j /- / the bend mean radius and the duct diameter, Rc/D, is 0.65 and the upstream and downstream straight / - / / y CONTRACTION / .--~-~[ ./"I ~ ~ A I R FILTER sections are 10 diameters long. The ~.7"-// .. fibs, for cases II and III, are of a square : ---/" ~'~qJI FLOW IN cross section. The ratio between the rib HONEYCOMB TmLET height and the duct diameter, e/D, is 0.1 and the rib spacing, P/e, is 10. In Figure 4. Apparatus both the upstream and the downstream sections the ribs closest to the bend are at a distance of 0.45D from the bend entry and exit respectively.
776 3.
HEAT-TRANSFER MEASUREMENTS.
As is well documented [5] and [6], the molecular structure of thermochromic liquid crystals, over a certain temperature range, depends on temperature. Changes in molecular structure in turn affect the wavelength of visible light absorbed by the liquid crystals and hence, over a certain temperature range, the colour of the liquid crystals can be used to determine their temperature. In heat-transfer experiments where air is the working fluid, most groups have adopted the transient liquid-crystal technique, in which the surface under investigation is covered with a layer of liquid crystals and is exposed to a hot air stream. As the surface temperature gradually rises from the initial ambient conditions, the movement of the colour contours of the liquid crystals along the surface is monitored. Then, at each location along the surface, from the time needed for the wall temperature to rise from its initial value to that of the crystal-colourchange temperature, the solution of the one-dimensional transient heat-conduction equation produces the value of the heat-transfer coefficient, h. In this study, however, because of our intention to employ this technique subsequently in experiments in rotating passages using water as the working fluid, the steady state technique was employed. This is because in water, due to the higher coefficients of heat transfer involved, the thermal time constant (a;= PwCpJqflla2) is too short to measure accurately. The internal surfaces of the inner and outer walls of the experimental model are covered with a 0.013mm thick, electrically heated, stainless steel foil which provides a constant-heat-flux thermal boundary condition. The ribs were made of perspex and the heating foil underneath the ribs was short circuited. The rib surfaces could thus be considered as thermally insulated. A thin layer of liquid crystals is then applied over the surface of the heating foil. Once steady thermal conditions are reached, the resulting contours of the yellow colour are also contours of known wall temperature, determined through a prior calibration, under conditions of uniform wall heat flux. Electrical measurements provide the wall heat flux, qw and, from the overall energy balance, the fluid bulk temperature, TB, can also be determined : m Cp (dTB/dZ) = P qw
(1)
where m is the mass flow rate, z the streamwise direction and P the heated perimeter. The local Nusselt number along each colour contour can then be calculated, since : Nu - (qw Dh)/[ k (Tw-TB)]
(2)
By repeating the above procedure for a number of heating rates, detailed mapping of the local Nusselt number over each heated wall can be constructed. The liquid crystals employed, were micro-encapsulated crystals manufactured by Merck, with a nominal colour-change band between 29.5~ and 31 ~ for the entire visible spectnma, and a nominal change of 0.1~ across the yellow colour, which was used to determine the wall terr~mttme. At each heating rate, the image of the resulting colour contours was recorded using a CCD camera at a 45 ~ angle as shown in Figure 5, digitized and stored on a PC. In order to minimise heat losses, the experimental model was covered with thermal insulation. The thermal
777 insulation of the top wall was removed for only the few seconds needed for the camera to record the image. The digitized image was then converted into a hue-saturationand-intensity format and the pixels with a hue value corresponding to that of the yellow colour were identified. Software developed for this study then corrected the co-ordinates of the selected pixels, accounting for distortions caused by the ~ ' ~ tion camera lens and angle. The information from each image was thus reduced to provide the co-ordinates of the constantwall-temperature contour. The value of h along the contour line was then calculated, using equations (1) and (2). Finally, the Perspex Wall Black Paint contour lines produced by different heating Figure 5. Heating and Viewing arrangements. rates were brought together and, through interpolation, the continuous variation of the Nusselt number over the heated safface was produced. Side-averaged values of the Nusselt number were also subsequently computed. Error analysis indicates that the uncertainty in the local Nusselt number is +7.4% and in the side-averaged values +5.5%. While in real blade-cooling passages the cooling fluid would remove thermal energy from all four walls, in these experiments only the inner and outer walls have been heated. This does not however diminish the usefulness of the data in CFD validation, or invalidate any conclusions reached on how the flow development influences wall heat transfer. Ideally the heat transfer pattern should be symmetric about the centre-plane of the duct, y=0. While contour lines, discussed in the next section, are not perfectly symmetric, the main anomalies occur in regions where Nu is not changing rapidly, so discrepancies associated with the asymmetric contours are not important. 4.
RESULTS AND DISCUSSION
Three sets of data are presented, one for each of the three geometries shown in Figure 1, all at a Reynolds number of 95,000. 4.1
Case I
The local Nusselt number measuremems in the U-bend with smooth walls, Figure 6, provide an opportunity to assess the effects of strong curvature on heat transfer. In the upstream section, Nusselt-number levels along both walls are fairly uniform and close to the value returned by the Dittus-Boelter correlation: Nu=0.023 Re ~ Pr ~
(3)
778 Within the U-bend heat transfer levels rise along the outer wall, with the maximum level at each axial location occurring at the centre line of the outer wall. The general rise in Nusselt number is consistent with the flow acceleration along the outer wall, as shown in Figure 2. Within the bend the higher heat transfer levels along the centre line of the outer wall, are probably caused by the secondary motion, which causes relatively cool fluid fxom near the duct centre to impinge along the cemre-line of the outer wall. This fluid heats up as it moves out to the comer regions, leading to a reduction in the heat transfer coefficient in the two comers. Downstream of the bend, along the outer wall, heat-transfer coefficients continue to rise, reaching a maximum value more than twice that prevailing upstream, at about three diameters after the bend exit. As can be seen in [2], the location of the maximum Nusselt number coincides with the region where the highest turbulence levels along the outer wall have been measured. Beyond this point, as the flow along
/ ......... v.7.~---~ioo
t
I
__ 00A, --
^~ ~ ,
r'P"q
2oo"
-
::
2ao
:
~u
---"
~,oo
D
F~
c
o
,so.
Figure 6. Nusselt number contours for the smooth U-bend, Case I. Re=95,000 and Pr=0.71 the outer wall starts to slow down and the turbulence levels begin to reduce, Nusselt numbers also start to diminish. Even after six diameters from the bend exit, however, the Nusselt number is still more than 50% higher than that upstream. Along the inner wall, Nusselt number levels rise very rapidly after the bend exit, reaching their maximum values, similar to those along the outer wall, about two diameters after the exit, which is very close to the reattachment point, Figure 2. In contrast to the measured distribution along the outer wall, along the inner wall, the Nusselt numbers in the corner regions are higher than those on the centre line. This is also consistent with the curvature-induced secondary motion which, along the thermally insulated top and bottom walls, transports relatively cool fluid to the comer regions of the inner wall. As this fluid moves towards the duct centre, it is heated and also slowed down by the inner wall, causing a strong reduction in wall heat transfer. At about three diameters downstream of the bend, the variation in Nusselt number between the centre line and the comer regions, at the inner wall, is about 60%. As the flow begins to recover after r e - a t t a c ~ Nttsselt number levels along the inner wall begin to decrease, though, as for the outer wall, even alter six diameters from the bend exit, they are
779 still significantly higher than those of the upstream region. The strong streamwise changes in the mean flow and the rise in turbulence levels caused by the tight U-bend thus lead to substantial increases in the heat-transfer coefficient. Moreover, curvature-induced secondary motion, as well as contributing to the increase in wall heat transfer, also influences the distribution of the local Nusselt number on the inner and outer walls. The influence of the bend on heat transfer is still significant after six diameters from the bend exit. 4.2 Case II
The effects of rib-roughness on heat transfer can be seen in the Nusselt number contours of Figure 7. In the upstream section, there appear to be enough rib intervals for thermal conditions to repeat themselves over each interval. Nusselt numbers become highest in the middle of each fib interval, by which point, according to the LDA data of Figure 3, the flow has re-attached, after separating behind the upstream rib. Nusselt number levels in the comer regions are higher than in the centre of each wall, probably because the top and bottom walls are thermally insulated. The actt~ levels of the heat-transfer coefficient are typically twice as high as those in the smooth Ubend (Case I). This substantial increase must be related to the high levels of turbulence caused by the presence of fibs. Within the bend, heat-transfer coefficients display a monotonic rise along A
t
............................................
E
F
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C, Figure 7. Nusselt number contours for the fibbed U-bend, Case II. Re=95,000 and Pr=0.71 the outer wall but, in contrast to the smooth U-bend, Nusselt number levels along the centre-line are higher than those in the comers. The levels of Nu along the outer wall are higher than the corresponding levels for the smooth U-bend, but not as high as the highest levels encountered in the ribbed upstream sections. Downstream of the bend, along the outer wall, Nu levels fall after the first diameter, where, as shown in Figure 3, the flow along the outer wall starts to decelerate.
780 Over the first downstream rib interval along the outer wall, where a large separation bubble is generated, the Nusselt number rises to levels higher than those encountered upstream of the bend. Heat transfer coefficents gradually fall, over the subsequent rib intervals, tending towards the levels measured in the upstream section. Along the inner wall, over the first downstream rib interval, which is partially within the region of bend-induced separation, heat-transfer levels are especially high in the comer regions. Over the two subsequent rib intervals, the Nusselt number rises even further, to levels higher than those along the outer downstream wall. Beyond the third downstream rib, heat-transfer coefficients begin to fall towards those of the upstream sections. The overall effect of the U-bend and of the ribs on wall heat transfer, can be more easily assessed through the plots of the side-averaged Nusselt numbers, shown in Figure 8. As expected, the introduction of ribs raises the levels of the heat-transfer coefficient in the upstream section by more than a factor of 2. The round-ended U-bend still influences heat transfer levels downstream of the bend, especially along the inner wall, but its effects are not as strong as in the smooth U-bend. 600 500 400
2110 100
,.nerWall
600
l
500 400 z 300 200 Fully-
100
Developed Flow I
0
0
-10 x/D
Figure 8.
BEND
Outer W all
I
I
I
I
-5
t
I
I
I
I
o x/D
0
5
Axial variation of side-averaged Nusselt number of a smooth and a rib-roughened Ubend, Cases I and II. : Smooth Duct, 9Ribbed Duct.
4.3 Case III Finally the local Nusselt measurements of Figure 9 show how the inclusion of an additional rib along the outer downstream wall, at a distance of 0.95 diameters from the bend exit, affects wall heat transfer. Some of the images for the higher heating rates were, unfortunately, lost and consequently some regions have not been fully mapped. In these regions, shown as shaded in Figure 9, the Nusselt number reaches values that are higher than those indicated by the levels of the contours present. Along the outer wall, differences in the thermal behaviour between Cases II and III begin to appear over the second half of the bend, where the additional downstream rib causes an earlier rise in the Nussett number. After the bend, because the first downstream rib is now closer to the bend, the reduction in Nusselt number after the first diameter si eliminated. Over the first rib interval, now one diameter closer to the bend, the Nusselt nmnber is higher, than for Case II, and the distribution is different, with the highest levels occurring at the downstream end of the
781 interval This must be caused by the larger separation bubble over the first rib interval in Case III. Beyond the first rib interval along the outer wall, the thermal behaviour becomes similar to that of Case II.
Figure 9. Nusselt number contours for the ribbed U-bend, Case III. Re--95,000 and Pr=0.71 Along the inner wail, while the Nusselt number dism"aution is similar to that for Case II, over the first two rib intervals the levels are about 10% higher. The changes observed are consistent with the changes that the additional rib causes to the flow development, Figure 3.
5.
CONCLUDING REMARKS
The results presented provide data for local wall heat-transfer in idealised blade cooling passages that should be useful for CFD code validation. The data have also revealed how, in strongly curved and ribbed passages, the flow development influences wall heat transfer. In flows through smooth U-bends the main flow features are the flow separation from the inner wall at the bend exit and the curvature induced secondary motion. Flow separation causes a substantial increase in turbulence levels, especially over the first three downstream diameters. The secondary motion and the higher turbulence levels increase the Nusselt number through the bend, with the peak values, more than twice as high those upstream, along both the inner and outer walls, reached between two and three diameters after the bend exit. Moreover, the secondary motion induces strong perimemfl variations in Nusselt number, around 60%, along both the inner and outer walls. Bend effects on heat transfer remain appreciable even beyond seven diameters l~om the exit. The introduction of fibs substantiaU~ raises turbulence levels which, for the case of the round-ended U-bend, reduces the size of the separation bubble along the inner wall. The flow development downstream of the bend becomes very complex, as the flow, highly distorted by the bend, encounters ribs along the inner and outer walls. It is also found to be sensitive to the distance of the first downstream ribs to the bend exit. Wall heat transfer is substantially raised within the bend and also within the upstream and downstream sections. While the effects of the U-bend on heat transfer levels are not as strong as in the case with smooth walls, they are
782 nevertheless still significant. Furthermore, the effects of the bend on the perimetral variation in Nusselt number after the bend exit are also highly evident. Moreover, the rather steep streamwise variation of Nusselt number in the space between ribs suggests that, even though the blade as a whole is more effectively cooled by surface ribbing, important thermal stresses may nevertheless be encountered. The non-linear nature of the laws that govern the fluid motion lead to the development of complex flow patterns in blade cooling passages, which can neither be anticipated nor deduced by simply combining the separate effects of rotation, strong curvature and rib roughness. Detailed experimental data, or use of reliable flow solvers is thus necessary to determine local Nusselt numbers of the quality needed for modem design. ACKNOWLEDGEMENTS
Funding for the work presented has been provided by the EPSRC, ABB(Switzerland), EDF(France), EGT, and Rolls-Royce pie. The authors gratefully acknowledge both the financial support and expert technical advice received. The authors also acknowledge technical support provided by Mr D. Cooper, Mr J Hosker and Mr M. Jackson. Authors' names are listed alphabetically. REFERENCES
1.
2.
3.
4. 5. 6.
H. Iacovides, D.C. Jackson, G. Kelemenis, B.E. Launder and Y-M Yuan,1998, Recent progress in the experimental investigation of flow and local wall heat transfer in internal cooling passages of gas-turbine blades, Proc 2nd EF Conference on Turbulent Heat Transfer, pp 7.14-7.28, Manchester, UK Cheah S C, Iacovides H, Jackson D C, Ji H H and Launder B E, 1996, LDA Investigation ofthe Flow Development Through Rotating U-ducts, ASME Journal of Turbomachinery, 118, 590-596. Iacovides H, Jackson D C, Kelemenis G, Launder B E and Y-M Yuan, 1996, LDA Study of the flow development through an orthogonally rotating U-bend of strong curvature and n"a-roughen~ walls, Engineering Turbulence Modelling and Experiments 3, 561-570,Ed W Rodi and G Bergeles, Elsevier, H. Iacovides, D. Jackson, B.E. Launder and Y-M. Yuan, 1998, An experimental study of a rib roughened rotating U-bend flow, In preparation Baughn J W and Yan X, 1992, Local Heat Transfer Measurements in Square Ducts with Transverse Ribs, ASME HTD-Enhanced Heat Transfer, 202, 1-7 Jones T V, Wang Z and Ireland P T, 1992, Liquid Crystal Techniques, ICHMT, Int Symp. on Heat Transfer in Turbomachinery, Athens, Greece.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
783
Studies o f Turbulent Jets impinging on M o v i n g Surfaces K Knowles a and T W Davies b aHead, Aeromechanical Systems Group, Cranfield University, RMCS, Shrivenham, Swindon, Wilts., SN6 8LA, UK. Tel +44 1793 785 354 Fax +44 1793 785 260 bprofessor of Thermofluids Engineering, School of Engineering and Computer Science, University of Exeter, Exeter, Devon, EX4 4QF, UK.
Measurements have been made of the flow produced by a single, round, turbulent jet impinging on a flat moving surface. The wall jet profile formed on the retreating side of the impingement region (where the belt surface was travelling in the same direction as the wall jet) had a very thin inner boundary layer region. On the approach side (where the surface was moving towards the impingement point, against the wall jet) the initial development seemed very similar to the case with the surface stationary. At a radial position of about r/Dn = 4, an increase in the wall jet half-thickness (the height above the surface to half peak velocity) was noticeable over the stationary case and at r/Dn = 7 there was a sudden rapid increase. This appeared to be due to separation of the wall jet inner boundary layer due to the increased shear stress. Measurements of wall jet momemum flux were consistent with this. Numerical modelling, using a commercial CFD package (PHOENICS v 2.1 and also v 3.1) and the low Reynolds number Lam-Bremhorst turbulence model with Yap correction, has shown similar trends. Heat transfer characteristics derived from these CFD studies have agreed with previous experimental data, including the skewing of the radial distribution of local Nusselt number, with the peak values shifted to the approach side of the impingement region. NOTATION
Dn I-I. Mf r
V Vg Vn Vm Y Y1/2
Nozzle exit diameter Height of nozzle above ground Wall jet momentum flux (in V-direction) Radial co-ordinate measured from nozzle centre-line Wall jet velocity Moving ground velocity Nozzle exit velocity Local maximum wall jet velocity Co-ordinate distance above ground Height to half peak velocity (Y at V = Vm/2)
784 1. INTRODUCTION Impinging jets are widely used in many engineering applications involving heating, cooling or drying. Some of these applications involve jets impinging on moving surfaces, such as belts or rotating drums. There is relatively little data in the literature on such flows, other than some global studies of heat or mass transfer characteristics e.g. Martin, (1977), Polat, (1993). There seem to have been few previous studies of the detailed flowfields involved with jets impinging on moving surfaces. The present study has been motivated by two requirements: to be able to design an optimum array of impinging jets for heat transfer control; and to be able to understand the mechanisms involved under vertical take-off aircratt moving over the ground (which produces a jet flowfield equivalent to an impinging jet on a moving surface - Knowles et al., 1992). 2. EXPERIMENTATION For the experimental phase of this work a dedicated jet impingement rig was set up (Myszko, 1997). This consisted of a centrifugal fan driven by a 3.73kW electric motor which provided air at a pressure of 55mbar(g) to a small settling chamber (340mm wide, 140mm deep and 150ram high, externally) containing baffles and filter materials. On the lower face of this settling chamber was mounted a simple conical nozzle of exit diameter (Dn) 12.7ram. The centre-line of the nozzle was vertical and the entire nozzle/settling chamber assembly was mounted on a steel framework (which was 2m high by 3m square) in such a way that it could be varied in height. Below the nozzle was a rolling road (as used for racing car model ground simulation) which consisted of a 1.219m-wide rubber belt driven over a 1.524m-long platen or table via two rollers. The drive roller was powered by a 20kW electric motor and a feedback control system maintained the belt speed to 0.25% under the conditions used here. Flapping of the belt was prevented by the application of suction through holes in the platen under the belt; this could be varied in four sections (front, le~ side, right side and middle and back) using butterfly valves. This same arrangement of compressor and settling chamber had previously been used for studies of a jet impinging on a fixed surface (a varnished wooden board of 3m x 2m - see Myszko & Knowles, 1996 or Knowles & Myszko, 1998). These earlier studies had confirmed the symmetry and top hat profile of the nozzle exit flow and had shown the nozzle exit turbulence intensity to be about 5% (based on nozzle exit velocity). It should be pointed out that this level of turbulence is typical of the sort of industrial applications in which we are interested. Further details of the free jet characteristics, based on crossed-wire hot-wire anemometry measurements, are contained in Myszko (1997). For the present study, only the wall jet flow was investigated using, initially, a pitotstatic probe to measure mean velocities. The probe was traversed using a three-axis linear traverse frame which was found to give an accuracy of +0.050mm in each direction. The total traverse area available was lm x lm x 0.Sin. To allow for possible slight mis-alignments the traverse axis system was calibrated with respect to the nozzle and rolling road centre-lines. The nozzle, rolling road and probe traverse system are shown in Fig 1. Probe measurements were taken in the flowfield plane of symmetry (i.e. along the centre-line plane of the nozzle and rolling-road) on either side of the jet impingement region, both with the road stationary and moving at 10ms-1. Where the road and wall jet are moving in the same direction is referred to as the retreating side (Vg=-10ms-1);
785 where they are in opposite directions is the approach side (Vg = +10msl). Wall jet mean velocity measurements were taken for nozzle heights of 4, 8 and l0 D, at the radial positions shown in Table 1. Table 1 - Radial locations of probe traverses for fixed and moving ground (H/D, = 4, 8, 10) r/Dn V~ 10 0 -10
1 1.5 2 2.5
3 3.5
x x x x x x
x x x
x x x
x x x
x x x
10.5 12 14 16 18 20
4
4.5
5
6
7
8
9
9.5
10
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Throughout these tests the jet was run at a nozzle pressure ratio of 1.05 (NPR is the ratio of nozzle stagnation pressure, p0, to atmospheric pressure, pa; NPR = 1.05 gives a nozzle exit velocity of about 90msl). The nozzle had an exit diameter (Dn) of 12.7mm, giving a jet Reynolds number (based on nozzle exit conditions) of 9 x 104. Detailed heat transfer measurements for single and multiple cold air jets impinging on a hot rotating cylinder have been reported by Journeaux, (1990), and are used here to test the plausibility of the CFD predictions of this system using PHOENICS. Much of the data relates to an air jet emerging from a one inch ( 2.54 cm) diameter round nozzle at 20 C with a speed of around 60 ms 1 giving a jet Reynolds number based on nozzle diameter of 105. These conditions are typically encountered in North American paper manufacturing plants. A range of cylinder surface speeds and nozzle-to-surface distances were used during the study but here we use only data for the case when the dimensionless distance between the nozzle and the surface was H/D, =2 and when the ratio of surface speed to jet exit speed (Vg/Vn) was either 0.3 or 0.6.
3. N U M E R I C A L M O D E L L I N G We report here only two examples of our modelling work. Firstly a single round isothermal air jet impinging on a fiat stationary hot plate was modelled in Cartesian coordinates. The computational domain and objects of the Cartesian model are shown in Figure 2, where full use is made of symmetry to reduce computational effort. The quadrant used for the nozzle was approximated in the Cartesian system as shown in Figure 3. The impact plate was divided into sections to allow some spatial resolution of the local heat transfer rate. In the y direction, the sections W 1 to W4 have a width of 89nozzle diameter and the sections W5 to W9 have a width of 2 nozzle diameters. Plate section W l 0 covers the remainder of the domain where very little heat transfer occurs and is necessary to reduce computational end effects. (More than 90% of heat transfer occurs within 10 nozzle diameters of the stagnation point). The computational grid consisted of (40, 130, 40) cells in the (r,z,y) directions, also nonuniformly distributed according to the complexity of the flow and concentrated around the jet boundaries and plate surface. The turbulence models used were the Lam-Bremhorst with and without the Yap correction. Secondly the same air jet system was modelled but this time the impact surface was in motion normal to the centreline of the jet flow. This required a different model arrangement which is shown in Figure 4, once again making full use of symmetry.
786 Here the plate is divided into 10 strips each 2D wide distributed symmetrically about the stagnation point, and two end strips each 6D wide to reduce end effects. The computational grid had (40,196,40) cells with heavy cell concentration around critical areas. Two impact surface speeds were considered; 30% and 60% of the jet exit speed. Computation was carried out on a Pentium II personal computer and run times for satisfactory convergence were around 40 hours. 4. RESULTS AND DISCUSSION Wall jet velocity measurements revealed, for the fixed ground case, the familiar nondimensional shape, with an inner boundary layer, a peak velocity at about Y/Y1/2 - 0.2 and then a free shear layer with a decaying velocity. At low values of r/D, these profiles were affected by proximity to the impingement region but by r/Dn = 4 (for Hn/Dn = 10, earlier at the lower nozzle heights) the mean velocity profiles had become self-similar. This was consistent with the earlier work of Knowles and Myszko (1998, 1996). With the rolling road running at 10ms-1, the wall jet profiles on the retreating side of impingement revealed a much thinner boundary layer than with the fixed ground (Fig 5a). For most of these data the peak velocity measured was the lowest traverse point: at low radii this meant that the true peak velocity was probably not being resolved due to the very thin boundary layer region, whereas at larger radii (typically r/D,>8) this was consistent with the wall jet peak velocity having decayed to the ground speed. On the approach side of the impingement zone the high wall shear stresses caused the wall jet to be much thicker than with the fixed ground. Typical wall jet profiles for this region are shown in Fig 6. At large radii it appears that the wall jet has separated; this is seen more clearly in the detailed profiles of Fig 7. From this figure it can be seen that a point of inflection develops in the wall jet velocity profile between r / O n - - 8 and 9.5. The thickness of the wall jet can be quantified by the distance from the ground to where the outer shear layer velocity has decayed to half the peak wall jet velocity. This "half-thickness" is plotted against radial position in Fig 5(a). This clearly shows an apparent increase in wall jet thickness at r/D, = 8 on the approach side. Fig 5(b) plots the radial decay of peak wall jet velocity for the three cases of moving ground approach side, moving ground retreating side and fixed ground. The approach side shows a more rapid decay of peak velocity, with an increasing decay rate aiter r/D, = 8. On the retreating side it can be seen that the peak velocity is heading asymptotically to the ground speed of 10 ms"1 as radial position increases. The lower recorded values of Vm at low r/Dn reflects the point made earlier about not being able to resolve the very thin boundary layer on the retreating side close to the impingement region. For this reason the recorded values of Yv2 are too high in this area. The separation of the wall jet on the approach side of impingement in the case of the moving ground plane is further emphasised by the momentum flux plots of Fig 8. This Figure also shows that separation occurs at about the same radial position for all the nozzle heights tested and that the lower the nozzle height above the ground the lower the wall jet momentum flux. For CFD model validation purposes values of local Nusselt number were predicted for the case of impingement on a flat stationary surface. The spatial resolution of the local Nusselt number was fixed by the choice of surface strip width as shown in Figure 2. Surface averaged values of Nusselt number plotted in Figure 9 were obtained from the local values by integration and it can be seen that the Lam-Bremhorst with Yap correction model gives slightly better agreement with the experimental data of
787 Journeaux (1990) than the Lam-Bremhorst model. With the rather coarse surface divisions chosen it was not possible to resolve the finer detail of the Nusselt number distribution in the vicinity of the stagnation point, but for use in design the predicted average value of the Nusselt number over a region of 10Dn (eg for this jet Reynolds number, Nuav = 100) is quite acceptable. The second example of heat transfer rate prediction is that where the surface is in motion under an impingement flow ( Figure 4). The radially averaged predicted Nusselt number distributions for impingement on a stationary plate, and on a plate moving at 30% and 60% of the nozzle exit velocity shows an increasing skewness of the distributions with surface speed show the same trend as the experimental data of Journeaux (1990). At low plate speeds the overall effect of surface motion on total heat transfer is generally small; the heat transfer peak is shifted towards the direction of the approaching plate. At plate speeds greater than half the jet exit speed measureable increases in total heat transfer appear and the flow on the approach side of the stagnation zone is predicted to separate in the same position as that detected experimentally in the aerodynamic study of the wall jet region, as shown in Figure 11. CONCLUSIONS Aerodynamic measuremems on a single, isothermal jet impinging on a moving surface (with a surface to jet velocity ratio of about 0.11) have confirmed a strong asymmetry between the advancing and retreating sides of the impingement zone. On the centreline of the retreating (where the wall jet and the impingement surface are moving in the same direction) the wall jet was extremely thin and beyond r/Dn = 8 to 10 the peak velocity decayed to the surface speed. On the approach side the wall jet was thicker than with a fixed ground, and between r/Dn = 8.0 and 9.5 the wall jet separated from the surface this separation position did not vary significantly with nozzle height (in the range Hn/D, = 4 to 10). For many practical applications of impingement heating, cooling or drying, the estimation of the average heating or drying rate over the surface of interest is often carried out on the basis of the correlation between surface average Nusselt number and jet exit velocity Reynolds number for a stationary surface, i.e. Nuav = 0.133.Re 0.71 (r/D,) -0.63 Davies et al (1997). However these practical applications often involve multiple interacting jet arrays impinging on moving surfaces and these systems create heat transfer rates which are higher than those associated with single jets. Here we have shown that it is possible to predict the heat transfer performance of the basic single jet impingement system with reasonable accuracy and in work to be published we will extend these predictions to the more practical systems involving jet arrays. We have also shown that the model predicts the measured effects of surface motion and that for most practical values of surface/jet speed ratio the effect of surface speed on average heat transfer rate for round jets is negligble. Note that this is not the case for slot jet impingement systems, Pekdemir et al, (1998) because of the major flow disruption which is created by the incidence of an air curtain on the flow induced by plate motion. REFERENCES Davies, T.W. (1997), "Convective heat transfer from a hot rotating cylinder with jet impingement", Proc. 5 th UK Heat Transfer Conference, Journeaux, I.J. (1990), "Impinging Jet Heat Transfer and Thermal Deformation for Calender Rolls", PhD Thesis, McGill University.
788 Knowles, K., Bray, D., Bailey, P.J. and Curtis, P. (1992), "Impinging Jets in Crossflow", Aeronautical J., 96 (952) 47-56, February. Knowles, K. and Myszko, M. (1998), "Turbulence Measurements in Radial Wall Jets", Experimental Thermal and Fluid Science, 17, 71-78. Martin, H. (1977), "Heat and mass transfer between impinging gas jets and solid surfaces", Advances in Heat Transfer, 13, 1-60. Myszko, M. (1997), "Experimental and Computational Studies of Factors Affecting Impinging Jet Flowfields", PhD Thesis, Cranfield University, RMCS. Myszko, M. and Knowles, K. (1996), "Radial Wall Jets - Turbulence Measurements", in: Engineering Turbulence Modelling and Experiments 3, eds. W. Rodi and G. Bergeles, 453-462. Pub. Elsevier Science B.V., Amsterdam. Pekdemir, T. and Davies, T.W. (1998), "Mass transfer from stationary circular cylinders in a submerged slot jet of air", Int J Heat Mass Transfer, 41 (15), 23612370. Polat, S. (1993), "Heat and mass transfer in impingement drying", Drying Technology, 11(6), 1147-1176.
Figure 1. Jet Impingement Rig, showing rolling road, probe traverse flame, nozzle, settling chamber and air supply pipe.
789
~._
20Dn
Noz.z.l~
Opening
/!
Y
H/Dn Openings
k21 _L ....., -1_2 71 ...... r-~.
9~ / w 2 / w 4 wt w3
Plate
Sections
Figure 2. Segment of computational domain
Figure 3. Nozzle model
2.00
28.00 r Y(mm) ~
+r/D,
i
1.20
o.,o
/
= 6.0
r/D n -
7.0
o- r/D a =
9.5
---a- r / D , = 10.0 +
r/D a -
10.5
20.00
0.00
,
J
0.0
. . . .
i
4.0
80.0r
,
8.0
,
,
L
,
--A- V = -10.0 , , El . , ,
12.0
r/I
16.0
Co) -'~
9m
n'B 60.0
i 20.0
-It
YI =
--o-
12.00
-10.0
+v,:
16.00
00
10.0
vz =
-
8.00 40.0
..'--
4.00 ~o.o :30.0.I
?~
--2_~_ ~- --e _~
10.0
0.0 I 0.0
~
,
~
I 4.0
.
~
i 8.0
,
. ,...,
r/t
,
I 13.0
~
,
I 16.0
i
~
J
I 20.0
Figure 5. Wall jet development varying surface speeds at HdD.=IO (a) growth of half thickness; (b) decay of peak velocity
0.00 _~J~~~A 0.00 4.00 8.00 12.00 16.00 20.00 V (ms -])
Figure 6. Detailed wall jet velocity profiles on approach side of impingement zone for Hn/Dn= 10 showing separation
790
Opening/[
Opening
"~/I
Y
~
Nozzle ~
~
l
T,H/Dn /_~Openingsl /1
~
6Dn Z
32Dn
Figure 4. Segment of computational domain for impingement on moving plate
2.50 r/D u r/D= r/D n r/D u r/D.. r/D u r/D u r/D u r/D u r/D, r/D, - ~ - r/D u - e - r/D= r/D u --X-- r / D u
Y/Yv2 2.00
1.50
= = = = = = = = = =
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0
-- B,O
9.5 = 10.0 - 10.5
1.00
0.50
0.00
I
0.00
,
..
,
0.20
,
- J l m m = , , ~
0.40
I-
0.60
,
it~'~,vw'~5~-
0.80
,
1.00
I
1.20
V/V,,
Figure 7. Non-dimensional wall jet profiles on the approach side of the impingement region for Hn/Dn= 10
791 1.0
(kg ms "2)
Mf
Vg = 10ms -1
0.8 x=
0.4
J~P---- -
~::I---B_
_
~ -~) 250 200 150
,=
100
~.~,
50
!
r,/ \x
L-'L'C -
\ !
-15 -50-
-10
-5
0
5
10
15
-.... rlDn
VgNn = 0 ....
Vg/Vn = 0.3 . . . . . . .
V g N n = 0.6 . . . . .
exp. ]
2o
The experimental data are referred to Vg/V. = 0.64 and Re = 21400
Figure 10. Radial distributions of average Nusselt number for a single round jet impinging on a stationary and a moving plate.
Figure 11. Predicted velocity vectors on the approach side of a moving plate with jet impingement, showing flow separation.
12. Combustion Systems
This Page Intentionally Left Blank
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
795
Turbulence modelling in joint PDF calculations of piloted-jet flames J. Xu and S.B. Pope ~ ~Sibley School of Mechanical and Aerospace Engineering, Cornell Univesity, Ithaca, N.Y., 14853, U.S.A. The objective of this work is to investigate the pressure transport modelling in PDF methods. Calculations are conducted on a piloted-jet non-premixed flame. The pressure transport model is analyzed and explored from the aspects of the model constant and its influence on the scalar flux. Also, the importance of modelling the pressure transport in PDF methods is discussed in comparison with the modelling of turbulence frequency. 1. I n t r o d u c t i o n Computational methods based on theoretical grounds for turbulent combustion have become common tools for engineering designs of gas turbine combustors and furnaces [1,29]. Among these methods, the notable advantage of PDF methods that they can treat convection and reaction without any closure models makes them very attractive [19]. The foundation of PDF methods is the probabilistic description of turbulent flows. Consider a low Mach number reacting flow without radiation and gravity, and with a constant reference pressure P0. The one-point joint PDF f of velocity and compositions at location x and time t is defined by f ( V , r x, t) = (5(V - U ) 5 ( r - r
(1)
where r and V are the sample spaces for compositions r - {r a = 1, 2 , . . . , or} and velocity U, respectively. Here, angled brackets ' (w)),
(11)
798 Table 1: Model Constants
Cwl Cw2 C3 0.44
0.9
1.0
C4
Cr
Cf/
0.25
2.0
0.6893
where the model constant is chosen such that f~ equals (co} for fully developed homogeneous turbulence (Table 1). Hence, more precisely, the mean dissipation becomes e = kf~. Since the task is to investigate the turbulence modelling associated with PDF methods, and the chemistry of flame considered is near equilibrium (section 4.1), the particle thermochemical state is characterized only by the mixture fraction defined by Zi-
Zi2
(12)
= Z i l - Zi2'
where the subscript 1 and 2 refer to fuel and oxidant respectively, Zi is the mass fraction of element i. Density, temperature and mass fraction are retrieved as functions of the mixture fraction. Accompanying with this model, the IEM mixing model d~* - - ~1Ccf~(~* -
(~))dt,
(13)
can give reasonably accurate results. Hence, 4~ reduces to a single scalar ~ with Sr = 0. 3. P r e s s u r e T r a n s p o r t o~[i(~lv,r The term T* in Eq.(5) models the transport due to the fluctuating pressure ov~ox~ O(puj) i, Eq.(4) which corr po.ds to - -[ol,o,I qu tio.. Ozj + Ox~ ] i. the a ynola - tr Traditionally, the modelling of the pressure transport term is considered as a minor topic both in RANS models and PDF methods. It is either neglected or modelled together with turbulent transport process in RANS models [9]. Recently, the pressure transport has been discussed by Fu [6], and Demuren et al. [3] for RANS models. They proposed models based on the model first suggested by Lumley [12]
1 (UiUkUk)
(pu~) - -~
,
or
1 [O(uiukuk) + O(UjUkUk)] , Tij- -~ Ozj
Oz~
(14)
which is derived from homogeneous turbulence. Early in the development of PDF methods, Pope put forward the model [17] 1
(15)
This model is consistent with Lumley's model Eq.(14). However, it has not been applied in any PDF calculations, because a tractable numerical implementation of the modelled term has not been devised [25]. Recent calculations of TML using PDF methods by Van Slooten et al. [25] show that the pressure transport becomes dominant at the edge of free shear layer. Ignoring this term leads to very poor predictions of even the mean velocity profiles. Since PDF methods
799 express the turbulent transport in exact form, modelling of the pressure transport becomes crucial at the edge of free shear layer where the turbulent transport is less important. Therefore, Van Slooten et al. suggested the following model [25]
T: - Cpt ( u~u~ 2k
1) Ok Oxi '
(16)
where the model constant Cpt = 0.2 is recommended corresponding to the LIPM model. Application of this model in TML does improve the comparison of PDF calculations with DNS results. But, its performance, such as the influence on scalar fields, needs to be examined further in reacting flows. Given Eq.(16), the pressure transport term in Eq.(4) becomes
02 OViOxi If (pl V, r
0 ~-~ ((T~I V, ~ ) f ) -
1 Ok 0 -Cpt2kOx~OV~ If ( ( 2 k - ukuk)] V, r
.(17)
o This model is similar to Pope's model Eq.(15) except for that ~-~7~ ( ) is taken place by 1 ok Ox, ( )" The reason is, as shown by Van Slooten et al. [25], that Eq.(15) is not able to be implemented in the velocity model of the Langevin equation type for high Reynolds number turbulence where the viscous effect is neglected. The counterpart of the pressure transport model for Reynolds stress is derived from Eq.(17) with the use of Eq.(2)
Tij = Cpt
Ok
2k
Ok +
-
0 (UiUkUk} + -
3
~
Oxj
k3/-'7-- + ~
Ox,
k3/2
"
It might be concerned that Tij does not process a form of true transport process. However, the decomposition in the second line of Eq.(18) implies that T~j actually stands for the outcome of two processes: the first term is a true transport process; the second term is a transport-like process of the normalized triple correlation with coefficient OF'3k3~2 If the second term is approximately homogeneous (similar to the algebraic stress model (ASM) assumption [23]), T~j is identical to the Lumley's model (with Cpt = 0.6). In PDF methods, the velocity model combined with a particular mixing model yields a particular model for scalar flux [20]. This connection also provides a way to justify the velocity model: the velocity model should lead to a consistent model for the scalar flux [5,18,22]. The influence of the pressure transport model Eq.(16) on scalar fields can be understood in conjunction with the transport equation for the scalar flux 0 (u~C') + (Uj)
Ot
Oxj
= Pfi +
~
-ef
"
(19)
The four terms on the right-hand side represent production, turbulent transport, pressure scrambling, and dissipation, respectively. For the convenience of discussion, the pressure scrambling term is expressed explicitly. The modelling of the pressure scrambling is subject to much more uncertainty than is the case for Reynolds stresses. It is appreciated in SMC models that there is very little
800 advantage to decompose it into two parts as it is done for the Reynolds stress equations. Also, it is common to model this term as a linear function of (ui~'}. From the velocity model and Eq.(13), Pope has shown that the LIPM model leads to the following model for pressure scrambling (in absence of T*)
-I~ 'Op }~
- ~ (\ C r1
3C0) ft (ui~'}-~-~
[~Sij-~-~-~ij-- ~f~jlbli](Uj~'} .
(20)
On the other hand, T~* yields an additional term in the scalar flux equation
Z~ -- Cpt(~k~tk~'}2kOXiO__k Cpt 03 (UkUk~'}OXi Cptk3/2Oxi ((Uk~k~'})k3/2'
(21)
which apparently can not be included in the Eq.(20). To seek for an alternative interpretation, we decompose the pressure scrambling into two parts
_{~,op / o( }, b-~z~}- nf + T~, nf- \P~-~z~
T[--( ~oxi )}"
(22)
Then, it is appropriate to state that 1-If is modelled by the right-hand side of Eq.(20), and the transport-like term T/~ is modelled according to Eq.(21). Two observations are worth emphasizing for T[: 1. T[ is in a linear form of ~', and thus satisfies the consistent conditions for scalar field modelling [18]. This then justifies the validity of velocity pressure transport model T~* in terms of consistency.
2. T[ tends
to transport the scalar fluctuations against the energy gradient as well. It affects the scalar variance @'~'} through the influence on the scalar flux (ui~'}.
4. R e s u l t s and D i s c u s s i o n s 4.1. Test Case
Piloted-jet methane flames have been the target of several numerical modelling studies using PDF methods [14,15,24]. The flame considered here is the L flame of which the jet, pilot and coflow velocities are, respectively, Uj = 41re~s, Up = 24m/s and Uc = 15m/s. Detailed description of the experiment is given by Masri et al. [13]. As experiment indicates, this flame is blue up to 60Rj (Rj is the jet radius) with very little extinction. This favors the usage of IEM mixing model and flamelet model. Following the discussion of Xu and Pope [31] about the numerical accuracy in the joint PDF calculations, we use total 1600 cells and about 3.2 z 105 particles in the computational domain (80Rj z 15Rj). Boundary conditions are also specified according to [31]. 4.2. M o d e l T e s t i n g s The calculation results using the standard model constant (Table 1) without pressure transport are compared to the available experimental data (neither (uv) nor scalar fluxes are measured) in Fig.1. For abbreviation, only profiles at x/Rj = 40 are presented. Overall, the comparison seems satisfactory except for (~'~'} whose disagreement is mainly due to the simple mixing model used here.
801
~
3.5
~"
3.0
....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
0.12
ms, 9
""
0.I0 "
.~
a~P 1"0 t 0
t
.
.
t
~I"
0.04 ~r,
0.6
2.01.5
;/
""
_
"
-. . ~ ~. ~ . ~ . . . ~ " t_)
0.5
5
"
0.06
,o
0
0.8
0.08
2.5
"
....................................... 0 1 2 3 4
y/Rj
04
0.02
0.2
-ooo
oo~
9
0.04
0.0
I-I "
9 "
. "---
1~
1 ~176
'%.-, 9".'%-.
I
- x..x.~ O
-0.02
5
6
7
8
00l
0
1
2
3
4
y/Rj
5
6
7
8
Figure 1. Comparison of mean profiles at x / R j - 40. Lines: PDF calculation, solid, Cpt - 0.0, C~1 - 0.44; dashed, Cpt - 0.2, C~1 - 0.44; dash-dotted, Cpt - 0.0, C~I - 0.56; Symbols: experimental data of Masri et al. Attention, however, is drawn to the noticeable difference between the calculation and the experiment at the edge of shear layer: the calculation overpredicts the mean velocity, the mean mixture fraction, and in particular the turbulence energy k. Similar problems have also been detected in previous calculations of the L flame using the more advanced EMST mixing model and the ISAT algorithm [24]. This deficiency may cause significant error in the prediction of composition mass fractions since the stoichiometric value of mixture fraction in the L flame is about 0.035 which lies in the edge region of the turbulent shear layer. It is then conjectured that the inclusion of pressure transport is able to remedy this problem [24]. This argument is based on the observation that in the TML calculation, by doing so, the mean velocity profile at the edge of shear layer is dramatically improved. The mechanism behind this is argued to be that the pressure transport Eq.(16) pushes the high energy particles up the energy gradient such that the turbulence excursions into the non-turbulent region is prevented [25]. A calculation is then conducted first by including the pressure transport with the recommended model constant Cpt = 0.2 suitable for LIPM [25]. Results are plotted in Fig.1. There is almost no improvement over the calculation of Cpt = 0.0. The scalar flux is also insensitive to pressure transport model for Cpt = 0.2. In fact, even in the TML calculation, it is found that modelled pressure transport is relatively much smaller than the estimate from DNS data [25]. But, the similar performance does not appear here. Therefore, we investigate the performance of the model with larger values of the model constant. Calculations with Cpt = 0.2, 0.5 and 1.0 are then compared in Fig.2. It is apparent that there is no significant difference between Cpt = 0.2 and Cpt = 0.5, while Cpt = 1.0 gives rise to unacceptable results. To inspect the behavior of the pressure transport model more carefully, we turn to examine the effect of pressure transport on the energy transport equation which reads
Dk = 7~ - ~ + D + 7-, Dt
(23)
where besides neglecting viscous effect, the pressure transport term 7- appears in addition
802 to the standard model of k [11]
l Ok _CptO 3. The latter result has been critically discussed by Kuznetsov and Sabel'nikov [7]. Recent experiments performed by Leisenheimer and Leuckel [10] in two fan-stirred bombs have shown that Ut, evaluated from the pressure diagrams, increases markedly with bomb size. Ting et al. [11] have varied L between 2 and 8 mm by using different grids to generate turbulence and have observed the decrease of St with L. The above brief review of experimental data shows that the influence of turbulence length scale on flame speed is an intricate issue. The goal of this paper is to contribute to a clarification. For this purpose, simulations of planar and statistically spherical premixed turbulent flames were performed at various L by using the Turbulent Flame Speed Closure Model (TFSCM) put forward by Zimont [12] and developed recently [13-16]. The model is briefly summarized in the next section but the detailed discussion of it is given elsewhere [13-15]. Then, the results of the numerical simulations are reported and analyzed in order to clarify the issue under consideration. 2. T U R B U L E N T F L A M E S P E E D C L O S U R E M O D E L 2.1. Model equations The model employs the reduction of combustion chemistry to a single reaction and characterizes the combustion process by a single progress variable (c = 0 in the unburned gas and c = 1 in the products) following the well-known Bray-Moss method [17]. The model yields the following closed balance equation for the mean progress variable 0~
at
0
0 [
) 0~]
+ -~xj (fi~j~) = Oxj fi(a + Dt
#(1-~) 0 + t~(X + Dt/ab) e x p ( - ~ )
Ap,,u' [uLr~] ~/4 { 1 + Tr' [exp ( - ~ )
- 1] }~/~{ (j=lk
+
(11
~Xj Oq~2)2} 1/2
where the well-known approximation [18]
Dt = D r , 0 [ 1 - e x p ( - ~ ) ]
(2)
of time-dependent turbulent diffusivity Dt is used. Here, the t is time counted beginning with ignition; xj and uj are the coordinates and flow velocity components, respectively; p is the gas density; r~ = tr o and r' = Dt,o/U n are the chemical and turbulent time
843 scales, respectively; subscripts u and b label the unburned and burnt gas, respectively; the Reynolds averages are denoted by overbars and the Favre averages, such as ~ = ~-~, are used. The r.m.s, turbulent velocity u' = ~/2k/3, integral turbulent length scale L = CDU'3/~, and the steady turbulent diffusivity Dt,o = Cuk2/(~ac) are evaluated using, for example, the standard k - e turbulence model [19], where Co, Cu, and ac are constants, k and e are the turbulent kinetic energy and its dissipation rate, respectively. The Favre averaged temperature is linked with the progress variable as follows [17] T = T~(1 - ~
+7~),
(3)
where 7 = P,,/P~ is the heat release parameter. In addition to the two turbulence characteristics (k and e or u ~ and L), the TFSCM includes a single constant A and a set of physico-chemical characteristics, such as SL,O, ,r and the activation temperature O of a single global combustion reaction. The time scale t~ of this reaction is calculated so that it yields the known value of SL,O for the planar steadily propagating flame at u ~ = 0. 2.2. M o d e l f e a t u r e s
To show the basic features of the model in a clear manner, let us consider the limit behavior of Eq. 1 in the simplest case of a planar, one-dimensional flame. For the limit of weak turbulence (u' -+ 0), Eq. 1 is reduced to the standard balance equation of thermal laminar flame theory [20] 0 0 0-7 (ze) + ~ ( p u c )
0 [~ 0E] ~(1-E) exp(_O ) = ~ ~ + t~
(4)
For the opposite limit case of strong turbulence (u ~ >> UL and Dt >> x), the first source term on the right-hand side (RHS) of Eq. 1 is reduced by the ratio of Dt/xb and the last source term dominates. Then, Eq. 1 is reduced to
o--i (fiE) + ~ (pilE) = ~
Dt-~x
+ p,,St I V~[
(5)
in the laboratory coordinate system, or to
0 i) i) 0-7 (ze) + ~ (z~e) = ~
D t . ~x
(6)
in the coordinate system moving from x = +c~ to x = - c ~ with a speed of
{ "fex ( )lJ}
s, = S,,o 1 + T
(7)
Here, ~ = fi - St and St,o is associated with the fully developed turbulent flame speed ,
=
.
(8)
Two points are worth emphasizing. First, Eq. 6 yields a permanent growth of the turbulent thickness St, controlled by the turbulent diffusion law. This feature is the core
844 of the TFSCM: in fact Eq. 5, was proposed by Zimont [12] in order to model a regime of turbulent combustion, characterized by the growing $t. Experimental data reviewed elsewhere [16,21] shows that such a regime occurs in many combustion devices. Second, the turbulent flame speed St is incorporated into the model through Eqs. 7 and 8, so the model uses a certain submodel for St in order to close the balance equation [12] (a similar idea was employed recently by other authors [22-24] in a different manner). This feature of the model has given its name TFSCM although, as is shown below, the flame speed predicted by the model for non-planar flames can differ from Eqs. 7 and 8 even qualitatively. Equation 7 accounts for the development of St due to the fact that as a kernel grows after ignition, it experiences a wider range of the turbulence spectrum. This submodel is discussed in Ref. [15]. Equation 8 accounts for the effects of both turbulence and mixture characteristics on fully developed flame speed. The same expression has been suggested by various authors [21,25-29] using substantially different approaches. A similar expression has been found to be the best fit of an extensive experimental data base associated with moderate turbulence [1]. A close expression (St ", u ' g a -~ where g a ,,, (u'/SL,o) 2 R e t ~/2 and Ret = u ' L / u are the Karlovitz and Reynolds numbers, respectively, u is the mixture viscosity), well approximates the data bases of Bradley et al. [5] and Karpov et al. (see Ref. [4]) in the case of Le = tolD ~_ 1, where D is the molecular diffusivity of the deficient reactant and Le is the Lewis number. Equation 8 predicts that the turbulent flame speed is controlled by the only physicochemical characteristic, that is the chemical time scale re. This feature is supported by the experiments performed by Kido et hi. [3] and offers the opportunity to account for the following important effect. Both old experiments discussed elsewhere [27] and recent investigations [30] show that St increases with pressure P despite the substantial decrease in SL,O. Numerous models employing SL,O as the only physico-chemical characteristic of the mixture cannot predict such pressure effects. On the contrary, the TFSCM is able to do so. For example, Kobayashi et hi. [30] have shown that St is roughly constant in the range of P = 1 - 30 bar, despite the strong decrease in SL,O. For small u'/SL,o, this effect is associated with laminar flame instabilities [30] but it has not been explained for high u'/SL,o. According to Eqs. 7-8 applicable to high u'/SL,O, pressure may affect St only through rc = a,,/S~, o. Since SL,O ": p-~/2 [30] and to, ,-, p - l , Eq. 8 predicts the constant St,o, in agreement with the measurements. 2.3. Validation The TFSCM has been reliably validated [15,31] for statistically spherical kernels expanding in a premixed turbulent gas. Tests performed on the basis of the data of Karpov and Severin [2], Bradley et al. [32], Kido et hi., Mouqallid et al. [33], Groff [34], and Hainsworth (see Ref. [35] and references therein) have shown that the model predicts the growth of flame radius and the development of flame speed at various initial pressures, temperatures, mixture compositions, and u' with the same value A = 0.4. Each aforementioned experimental study used to test the model was performed at a constant L, whereas the objective of this paper is the influence of L on St. For this reason, we have singled out an experimental data base amassed by the different teams but at different values of L. The data base, collected in Fig. 1, encompasses the experimental
845 results of Ting et al. [11] (CH4/air mixture with the equivalence ratio F = 0.70), Bradley et al. [32] (CH~/~ir, F = 0.83), Mouqallid et al. [33] (C3Hs/air, F = 0.75), Groff [34] (C3Hs/air, F = 1.0), and Hainsworth (CH4/air, F = o.80- see Ref. [351). All the measurements were performed for expanding, statistically-spherical flames during a period characterized by a slight pressure rise. Flame kernels were ignited by a spark in homogeneous, isotropic, stationary [32,34] or decaying [11,33,35] turbulence. The flame radius was evaluated using Schlieren movies. All the experiments were modeled by unsteady, spherically symmetrical balance equations for the mass fractions of the fuel and oxidant, closed by the TFSCM and supplemented by the mass conservation equation, the enthalpy balance equation, the k - e turbulence model, and the ideal gas state equation. The full set of governing equations, boundary and initial conditions, input parameters, and the ignition submodel are reported, in detail, elsewhere [15]. It is worth emphasizing that we simulated all the experiments using the same governing equations and the same value A = 0.4 for the only constant of the TFSCM. Since the Schlieren images are associated with the leading edge of flames, the computed flame radius F! was equal to the radius of the surface where ~ = 0.1. The flame speed was calculated to be st = d F ! / d t .
,I"
0.04
~ 0.03
/
/
24
i~
~16
E o~
.
t /
'
E
.4:1
/ /
- -
2
,-r
0.01
'
A L---8 mm L--20 mm 9 L---2 m m - - - L--4 mm
o
'
_
9 "
..I
........
-""
1
t..~.....~ .st," . - - -
~-"....":.-'"
I
L--8mm
0.02
0.00 0.001
o L--2 m m o L---4 mm
""
J
i
8
04
0.006
0.011 Time,
0.016
s
Figure 1. Turbulent flame radius growth. Symbols show experimental data. Curves have been computed. 1 - u ' = 1.73 m/s, L = 20 mm [32]; 2 - u ' = 0.8 m/s, L = 5 [33]; 3 - u' = 2.0 m/s, L = 25 mm [34]; 4 - u' = 1.93 m/s, L = 3 mm [35]; 5 - u' = 1.0 m/s, L = 8 mm [11].
0
I
2
3
4
R . M . S . turbulent velocity, u', rrVs
5
Figure 2. Flame speed vs r.m.s turbulent velocity at different length scales L. Open and filled symbols have been computed for the spherical flames and correspond to F! = 20 mm and F/ --4 cr respectively. Solid lines show second-order fits to the open symbols. Broken lines have been computed for steady-limit planar flames.
Figure 1 shows that the model well predicts the results of all the measurements performed at substantially different values of L, ranging from L = 3 mm [35] to L = 25 mm [34]. These tests, supplemented with an extensive set of tests discussed elsewhere [15,31], support the use of the TFSCM for the purposes of this study.
846 3. R E S U L T S A N D D I S C U S S I O N To simplify the problem and to focus on the discussion on the influence of L on St solely, most of the following simulations have been performed under a constant pressure and for frozen turbulence. In other words, k - e balance equations are not solved and u' and L are assumed to be stationary and uniform. Numerical tests have shown that this simplification does not alter the qualitative trends discussed below. The following results have been obtained by varying u' and L, other things being equal. The initial conditions correspond to the stoichiometric iso-octane/air mixture at temperature To = 358 K and pressure P = 1 bar. The input physico-chemical characteristics of the mixture are as follows: O = 15000 K, tr = 0.525 #s, Sz,o = 0.43 m/s, to= = 0.257 cm2/s. For each set of the initial conditions, the simulations were performed as long as the flame moved the distance equal to 0.5 m, approximately. The simulations were performed both for spherical and planar flames. In the latter case, the mixture was ignited at the left boundary and the symmetry conditions were set at this boundary. For the planar flames, the speed reached the steady-limit value close to 7(SL,0 + St,o) where the term 3' = p,,/pb resulted from the hot product expansion and St,o was determined by Eq. 8. The computed steady-limit planar flame speeds are presented vs. u' for various L by the broken lines in Fig. 2. In line with Eq. 8, a higher St is associated with a larger L, other things being equal. On the contrary, the flame speeds computed for moderately small (~I = 20 mm), statistically spherical flames under the same initial conditions show the opposite trend, that is a higher St is associated with a smaller L (see open symbols or solid curves in Fig. 2). A comparison of the open symbols and broken curves shows that both the value of St and the behavior of St when varying the turbulence length scale, depend substantially on flame geometry and size. This dependence explains, in part, the scatter of the measured data on St and, on the face of it, questions the usefulness of the turbulent flame speed concept. However, some arguments supporting the concept are discussed below. The opposite effects of L on St, predicted for the moderately small, statistically spherical flames and for steady-limit planar flames, appear to be in line with the aforementioned experimental data. When processing the extensive experimental data bases of Karpov and Severin [21, Kido et al. [3], and Bradley et al. [5] in terms of St/u' (or Ut/u') as a function of u'/SL,o and 8L/L, the resulting dependence of St/u' on L follows the dependence of St/u' on 8L by virtue of dimensional reasoning (it is worth keeping in mind that L was not varied in these experiments). Since there are no reasons to suggest that the effect of the laminar flame thickness on the turbulent flame speed depends on the geometry and transient behavior of the turbulent flame; the resulting dependence of St/u' (or Ut/u') on 8L, and, hence, on L should be associated with fully-developed planar flames. For such flames, the computations predict the increase of St with L, in line with the aforementioned empirical approximations. On the contrary, Ting et al. [11] have investigated moderately small, statistically spherical turbulent flames and have reported a decrease of St with L. For such flames, the above computations predict the same behavior. Finally, Leisenheimer and Leuckel [10] have investigated large statistically spherical flames and have reported an increase of St with vessel size and, hence, with L. As we shall discuss in the following, for statistically
847 1,5
,
9
,
1.2
0.05
,
"" -'-~-"~":"~'~
~-~:-'2.-~; ~'~--
E
...... 0.04 . . . . . .
d
._~ 0.03
'o., I/.;,.;/;~
......
/ ~.'~/.~
o.~~
/ ,:~,~"
0.0
o.ooo
,--4 mm
"o 0.02
~_=io;~;~
"~ 0.01 I--
L--8 m m
r
L=IO mm
o.~o2 o.;o,
o.~o,
Time, t, s
o.~
o.olo
Figure 3. Turbulent flame speed development calculated from Eqs. 7 and 8 at u' = 2.36 m/s. Other initial conditions correspond to the results shown in Fig. 2.
L-2 mm L=4 mm L--8 m m L=IO mm L=20 mm
.~4fi
0.00 0.000
0.002
..--"~ . ..-'~.~
.....
0.004
0.006
Time, t, s
0.008
0.010
Figure 4. Turbulent diffusivity Dt v s . time t. The results have been calculated from Eq. 2 at u' = 2.36 m/s. Other initial conditions correspond to the results shown in Fig. 2.
spherical turbulent flames, as the kernel grows, the flame speeds tend towards the values presented by broken lines in Fig. 2. As a result, a higher St is associated with a larger L if the kernel is large enough, in line with the results of Leisenheimer and Leuckel [10]. Thus, for statistically spherical, turbulent flames, the dependence of St on L reverses as ~1 increases. What physical mechanisms can control this effect? Variations in L can affect the predictions of the model: (1) through St calculated from Eqs. 7 and 8 and used in the last source term on the RHS of Eq. 1, and through (2) the turbulent diffusivity calculated from Eq. 2. The dependencies of St(t) and Dr(t) calculated from Eqs. 7-8 and Eq. 2 for various L are presented in Figs. 3 and 4, respectively. The turbulence length scale affects St in two opposite directions. On the one hand, St,o is increased by L according to Eq. 8. On the other hand, the turbulent time scale r' = Dt,o/u '2 ,-, L/u' is increased by L. Hence, the ratios of t/r' and St/St,o are reduced by L. The latter effect dominates at small t but relaxes with time. For the conditions of Fig. 3, the range of 20 mm < ~1 < 40 mm, typical for the laboratory experiments, is associated with 4 ms < t < 8 ms. In this range, the latter effect dominates but the former effect tends to compensate it (see dotted and short-dashed curves in Fig. 3). For L = 2 and L = 4 mm, the results shown in Fig. 3 cannot explain the increase of St with L shown in Fig. 2 (compare open diamonds and circles), especially as the increase of St with L has been computed at F! = 40 mm, too. Similarly, the turbulence length scale affects Dr(t) in the same opposite directions. However, the dependence of fully-developed turbulent diffusivity on L (Dt,o " L) is much stronger than the dependence of fully-developed turbulent flame speed (St,o " L1/4). As a result, for diffusivity, this effect prevails over the decrease in t/r' and Dt/Dt,o by L; and a higher Dr(t) is associated with a larger L. The effects become more pronounced as the flame develops. The increase in Dr(t) by L can affect flame speed by means of the mean flame brush curvature and finite flame thickness mechanisms well-known for laminar flames. The speed Sb = drs/dt of spherically expanding laminar flames is known to differ from
848 0.04
flatne mdius,'c~=0.9
'
/
'/
flame radius, cn=O.5 // / / E flame radius, c~=0.1 ./ / / : . . . . flame brush thickness // 0.03 / / / / , / // ./ 0
"~ 0.02
,
//
#
0.01
/
.~m 15 E
///
/..,
/ "~
/"
lO
,/
o/~
= u'=0.2 m/s
.............
u. s
~~;---
0.00 0.000
0.002
0.004
A u'=0.6 m/s * u'=1.2 m/s
. .-- o - " "
0.006
Time, s
0.008
0.010
Figure 5. Turbulent flame brush thickness
6t and the radii of various iso-scalar surfaces (?:1 =const) vs. time t elapsed after ignition. u' = 2.36 m/s, L = 20 mm, other initial conditions correspond to the results shown in Fig. 2, k - ~ turbulence model has been employed in these computations.
0
0.0
o u'---3 m l s o u'--4 m l s 0.1
0.2
Thickness/Radius
0.3
0.4
Figure 6. Turbulent flame speed St vs the ratio of the mean flame brush thickness 6t to the mean flame radius ~I. Symbols have been computed,curves linearly approximate the computed results. L = 8 mm, other initial conditions correspond to the results shown in Fig. 2.
7SL,O and this difference is well approximated by the following linear relation [36] Sb = 7SL,0 (1 -- 71n77 -12~L)r!
(9)
if Lewis number is equal to unity. The difference between Sb and 7SL,o is caused by the effects of (1) the flame curvature, and of (2) a higher (as compared with Pb) gas density averaged over the enflamed volume. Both of these effects reduce flame speed at the finite 6L/~1 but they relax as 6L/~1 tends to zero. Typically, the difference is small enough because the thickness 6L of the laminar flames is much less than the flame radius r I. The same physical mechanisms, the mean flame brush curvature and higher averaged density, affect turbulent flames too but the corresponding variations in the turbulent flame speed should be much stronger as compared with the laminar case because the mean flame brush thickness 6t, controlled by turbulent diffusivity, is substantial even for quite large flames (see Fig. 5). These mechanisms will reduce flame speed at the finite ~t/r I but this effect will relax a s ~t/r I tends to zero. By analogy with laminar flames, one may assume that turbulent flame speed increases linearly with ~t/r I due to the influence of the emphasized mechanisms. In fact, such an assumption corresponds to the first order approximation when expanding turbulent flame speed with respect to a perturbation parameter, such as the dimensionless curvature, ~t/rl, of the mean flame brush. The results of the simulations, processed in terms of St vs the ratio of ~t/rl, support this hypothesis (see Fig. 6) and show that the effect can be strong enough even for sizeable flames. For example, when ~1 = 20 mm (the right edges of the curves in Fig. 6) the flame speed is approximately less by 2 times as compared with the largest flames. The simulations have shown that the contribution of the time-dependence of St determined
849 from Eq. 7 to the increase of flame speed is of minor importance when f / > 20 mm. Based on the substantial increase of flame speed with 6t/rl, shown in Fig. 6, three points are worth emphasizing. First, these results clarify the effect of L on the speed of statistically spherical, turbulent flames. For moderately small flames, the effect of $t/r! on St is of substantial importance. An increase in L results in increasing Dt (see Fig. 4), enhancing the effect, and decreasing St. As the flame grows, the effect relaxes and the dependence of St on L reverses and is controlled by Eq. 8. Second, similar to the well-elaborated methods for spherical laminar flames, the approximating straight lines shown in Fig. 6 can be used to evaluate fully developed turbulent flame speed that is assumed to be equal to the intersection between these lines and the ordinate axis. The values evaluated using this method are indicated by filled symbols in Fig. 2 and agree very well with the results computed for steady-limit planar flames. This agreement supports the proposed method. Thus, although the speeds measured for expanding turbulent flames substantially depend on flame geometry and size, such experimental results can be employed to evaluate fully developed turbulent flame speed, provided that flame thickness is also measured. Third, the effect under discussion can contribute to the leveling-off of St(u') (see open squares in Fig. 2), observed in numerous measurements [2,3,5]; in addition to the contribution made by local stretching and quenching phenomena, discussed elsewhere [5]. Indeed, if L is large enough, then, Dt and 6t are also large, the decrease in the flame speed by 6t/fl will be well-pronounced and augmented by u', whereas St(t) given by Eq. 7 will always be increased by u'. The counter-action between these two effects results in a leveling-off of the computed data presented for rl = 20 mm by open squares in Fig. 2. 4. C O N C L U S I O N S The results of simulations of expanding, premixed, turbulent flames, performed using the Turbulent Flame Speed Closure Model, highlight the importance of transient phenomena and, especially, of the effects associated with the substantial thickness of the flame brush (flame brush curvature and density averaged over the enflamed area). These effects substantially reduce the speed of moderately large, statistically spherical flames and can result in decreasing the flame speed with L and leveling-off the dependence of the flame speed on u'. As the flame kernel grows, the effects relax and the dependence of flame speed on the turbulence length scale reverses. A method of evaluating fully-developed turbulent flame speed in experiments with expanding kernels is proposed. REFERENCES
1. O.L.G/ilder, 23rd Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1000) 743. 2. V.P.Karpov and E.S.Severin, Combust. Explos. Shock Waves 16 (1980) 41. 3. H.Kido, T.Kitagawa, K.Nakashima, and K.Kato, Memoirs of the Faculty of Engineering, Kyushu University, 49 (1989) 229. 4. A.N.Lipatnikov, in Advanced Computation and Analysis of Combustion, G.D.Roy, S.M.Frolov, and P.Givi (eds.), ENAS Publisher, Moscow, (1997), 335.
850 5. D.Bradley, A.K.C.Lau, and M.Lawes, Phil. Trans. R. Soc. London, A338 (1992) 359. 6. K.Whol and I.Shor, Ind. Eng. Chem., 47 (1955) 828. 7. V.R.Kuznetsov and V.A.Sabel'nikov, Turbulence and Combustion, Hemisphere Publ. Corp., New York, 1990. 8. D.R.Ballal and A.H.Lefebvre, Proc. R. Soc. London, A344 (1975) 217. 9. D.R.Ballal, Proc. R. Soc. London, A367 (1979) 353. 10. B.Leisenheimer and W.Leuckel, Combust. Sci. Technol., 118 (1996) 147. 11. D.S.-K.Ting, M.D.Checkel, R.Haley, and P.R.Smy, SAE Paper 940687 (1994) 1. 12. V.L.Zimont, in Chemical Physics of Combustion and Explosion Processes. Combustion of Multi-Phase and Gas Systems, OIKhF, Chernogolovka, (1977) 77 (in Russian). 13. V.L.Zimont, A.N.Lipatnikov, Chem. Phys. Reports 14, (1995) 993. 14. V.P.Karpov, A.N.Lipatnikov, V.L.Zimont, 26th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh, (1996) 249. 15. A.N.Lipatnikov and J.Chomiak, SAE Paper 972993 (1997) 1. 16. A.N.Lipatnikov and J.Chomiak, Modeling of Turbulent Flame Propagation, Annual Report 98/2, Chalmers University of Technology, 1997. 17. K.N.C.Bray and J.B.Moss, Acta Astronautica 4 (1977) 291. 18. J.O.Hinze, Turbulence, 2nd Edition, McGraw-Hill, New York, 1975. 19. B.E.Launder and D.B.Spalding, Mathematical Models of Turbulence, Academic Press, London, 1972. 20. Ya.B.Zel'dovich, G.Barenblatt, V.Librovich, and G.Makhviladze, "The Mathematical Theory of Combustion and Explosions," Plenum Publ. Corp., New York, 1985. 21. A.G.Prudnikov, M.S.Volynskii, and V.N.Sagalovich, Mixing Processes and Combustion in Jet Engines, Mashinostroenie, Moscow, 1971 (in Russian). 22. M.Wirth and N.Peters, 24th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1992) 493. 23. H.G.Weller, S.Uslu, A.D.Gosman, R.R.Maly, R.Herweg, and B.Heel, Symp. COMODIA 94, JSME, Yokohama (1994) 163. 24. H.P.Schmidt, P.Habisreuther, and W.Leuckel, Combust. Flame, 113 (1998) 79. 25. V.R.Kuznetsov, Izv. Akad. Nauk SSSR, Mekh. Zh. Gaza 5 (1976) 3 (in Russian). 26. V.A.Frost, in Combustion and Explosion, Nauka, Moscow, (1977) 361 (in Russian). 27. V.L.Zimont, Combust. Explos. Shock Waves 15 (1979) 395. 28. O.L.G/ilder, 23rd Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1990) 835. 29. W.T.Ashurst, M.D.Checkel, and D.S.-K.Ting, Combust. Sci. Techn., 99 (1994) 51. 30. H.Kobayashi, T.Tamura, K.Maruta, and T.Niioka, 26th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1996) 389. 31. A.Lipatnikov, J.Wallesten, and J.Nisbet, Symp. COMODIA98, JSME, Tokyo, (1998) 239. 32. D.Bradley, M.Lawes, M.J.Scott, and E.M.J.Mushi, Combust. Flame, 99 (1994) 581. 33. M.Mouqallid, B.Lecordier, M.Trinite, SAE Paper 941990 (1994) 1. 34. E.G.Groff, Combust. Flame, 67 (1987) 153. 35. S.B.Pope and W.K.Cheng, 21st Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1986) 1473. 36. P.Clavin, Progr. Energy Combust. Sci., 11 (1985) 1.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
851
Application of a Lagrangian PDF Method to Turbulent Gas/Particle Combustion M. Rose, P. Roth ~ and S.M. Frolov, M.G. Neuhaus b* ~Institut f/ir Verbrennung und Gasdynamik, Gerhard-Mercator-Universitfit, 47048 Duisburg, Germany bN.N. Semenov Institut for Chemical Physics, Russian Academy of Science, Moscow, Russia A mathematical model for two-phase turbulent reactive flows is presented and applied to calculate the combustion of a dust jet under conditions of homogeneous, isotropic turbulence. Contrary to commonly used Euler/Euler or Euler/Lagrange methods this model is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of two kinds of "particles" representing the gas and the dispersed phase. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is restricted to the vicinity of solid particles by the definition of an "action-sphere" which is attached to every solid particle. Applications of the method indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase. 1. I N T R O D U C T I O N Many flow phenomena in nature and engineering applications involve the interaction between a fluid carrier phase and a dispersed solid or liquid phase. In general, this interphase processes include a multifacetted exchange of mass, momentum, and energy, the latter process being of fundamental importance in chemically reacting flows, involving highly energetic materials. Due to the geometrical size of the flow system and the condensed phase, and the velocity range most of these flows are turbulent. A large variety of different mathematical models for two-phase turbulent reactive flows are known, based either on considering two penetrating and interacting continua in Eulerian manner, or on considering the carrier phase in Eulerian and the dispersed phase in Lagrangian manner. In the models, chemical energy release is usually provided by both homogeneous reactions in the gas phase and heterogeneous reactions at the surface of *This work was supported by the German Science Foundation and by the German-Russian Scientific Exchange Program on Physical Chemistry of Combustion.
852 the combustible condensed phase (CP-) particles. Turbulence is represented by enhanced transport coefficients in the gas phase as compared to lalninar flow. A standard k-e model for homogeneous flow is often used for these purposes. Attempts were made to involve statistical approaches into the models. Berlemont et al. [1] apply a presumed PDF approach to model gas-phase velocity fluctuations in the vicinity of CP-particles. Among other approaches, discrete vortex methods [2] and LES [3] are worth to be mentioned. Fundamental aspects of (liP-particles dispersion dynamics in a turbulent flow have been studied by DNS [4-6]. This paper deals with test implementations of a new method for modelling two-phase turbulent reactive flows. The development of this method is described in detail in [7]. Contrary to existing approaches, the method is based oil considering both interacting continua in Lagrangian manner. It combines the benefits of particle methods for cap culating the joint velocity-composition PDF of a turbulent reactive gas phase with the possibility to follow the processes of heat and mass transfer in the vicinity of the trajectories of either individual CP-particles or groups of CP-particles. The method therefore incorporates the effect of CP-particles on the local flow structure, as there are physical aspects of CP-particle evaporation (volatilization), heterogeneous chemistry, and radiation absorption.
2. G E N E R A L E Q U A T I O N S FOR T W O I N T E R A C T I N G P H A S E S 2.1. Modelling A s s u m p t i o n s In this formulation, the two phases of the gas/particle mixture have their own density, velocity, and temperature and interact due to the exchange of mass, momentum, and energy. For the sake of computational capability, some assumptions are made, concerning the thermochemical properties of the gas phase, the particle phase and the interphase fluxes.
For the gas phase it is assumed that the ideal gas law holds and that heat flux is governed by molecular diffusion. Volume forces are neglected in the gas phase and the mean pressure is assumed to be uniform all over the volume, but fluctuations in pressure do exist. For the condensed phase it is supposed that the volume fraction of particles is neglibibly small and that the number of CP-particles is constant. They are assumed not to interact with each other and CP-particles can volatilize resulting in release of CO. Also volume forces are neglected for the condensed phase. In the interphase fluxes, heat transfer by radiation is neglected, and it is assumed that drag force is the only interphase force.
2.2. P s e u d o C P - P a r t i c l e s and G P - P a r t i c l e s As a matter of fact, the properties of CP-particles are not smeared over the volume, as it is implied when the condensed phase is considered as a pseudo-continuum. Instead, every CP-particle has its own individuality. Due to the large number of condensed phase particles in realistic two-phase systems, it is impossible to simulate each CP-particle on its own. To make the problem computationally capable, we apply the concept of pseudo CP-particles Pk (k - 1 , . . . , Np), as introduced in [7]. Pseudo~CP-particles represent groups of npk individual CP-particles. They move in physical space and interact with its gaseous surrounding like real particles, but the local effect of a pseudo CP-particle on the
853 gas is amplified by the number npk of individual real particles in the pseudo particle. The dynamics of the reactive gas phase is also calculated by a particle method, which is used efficiently for solving a high dimensional transport equation for a joint velocitycomposition probability density function (PDF). This method involves a large number of gas phase particles (GP-particles), Gi (i = 1 , . . . , Na), which represent local realizations of the turbulent flow field. The evolution of the gas phase properties, which are represented by averages over GP-particle properties, is given by the modelled conservation equations in Lagrangian form given below. 2.3. A c t i o n S p h e r e The intrinsic feature of interphase exchange processes is the finite dynamic depth of interphase fluxes. For example, temperature and vapor concentration profiles around a quiescent, evaporating droplet have definite widths, which dependent on time [5,8]. Under flow conditions, the dynamic effect of a particle on the surrounding gas is localized in the vicinity of "perturbed" streamlines. Clearly, mass, momentum, and energy fluxes to/from the gas phase vanish at some finite distances from the CP-particle surface. This implies that the interphase fluxes should be localized in the vicinity of CP-particles, rather than smeared out over the volume. The characteristic depths of the fluxes are different for mass, momentum and energy exchange processes, their proper estimation is a special task. In the present case we assume the characteristic depths of these processes to be the same and independent of location and time. To describe the vicinity of a pseudo CP-particle/Sk, an action sphere f~a (/Sk) of action radius ra is defined, which is attached to the particle, as shown in Fig. la. Interphase
Figure 1. (a) Action sphere attached to a pseudo condensed phase (CP) particle (b) Interaction of three pseudo CP-particles with GP-particles located inside their action spheres.
fluxes are localized in this action sphere. A pseudo CP-particle/Sk is influenced by a GPparticle Gi only if Gi is inside the action sphere of/sk. The interaction of three pseudo
854 CP-particles with GP-particles is illustrated in Fig. lb.
2.4. Conservation Equations in Lagrangian Form In the Lagrangian terminology, the gas phase is represented by Na GP-particles Gi of volume Va~, with mass pa~ Va,. The partial mass of species l in GP-particle Gi is P~a~ Va~. Furthermore, each GP-particle has its own velocity vc;,, density pa,, and enthalpy ha,. The condensed phase is represented by Np pseudo CP-particles, each having its own mass mp~, velocity vpk, and internal energy h p . The number of individual real particles, represented by a pseudo CP-particle/5~ is n p . The trajectories of pseudo CP-particles and the evolution of mass, momentum, and energy along their trajectories become in the Lagrangian frame [7]: D pk xpk Dt = v pk , mPk
__fdrag
DP~vPk D--------~=
Pk
D pk mpk vol _ Dt = -aJPk '
mPk
h~t vk '
(1)
D pk hp~ .co~v ~ o l (~vol_ hp~) -a~ h-~t ~h~t ,(2) Dt =-qPk + pk Pk
where wvol h-et are the rate of mass release due to volatilization and heterogeneous P~ and w P~ reactions respectively. The term fdrag pk represents drag forces acting on the surface of the condensed phase particle, @COYtV is the heat flux, and the terms tzr o t -hp~, and tzh~t account for heat of volatilization an~ heterogeneous reactions, respectively. The governing equations describing the GP-particle trajectories and the conservation equations of mass, species, momentum, and energy in Lagrangian form are: D a' xa, Dt
=
__~7
Dt
pa~
Da'va' Dt
pa~
D a~ hG, Dt
D a' (pa, Va, ) Dt
v~,,
"l + 3~, "l + 9J~,
~
E
,~
~
['t,~, volt~
(cdvol
~
~
+ ~ het ,~),
(3)
.hett~), + ~,
(4)
Pk aien(Pk)
=
V(pE-T)+
E
( fdr a 9 vol ~p~,_~,~ +v~,~( aj ~,~ + ~ het ~)),
(5)
Pk cien(Pk)
-
(0con~ § fd,'%g
L hom - V ' q a , di f f +'~a~ +
Pk ai~a(Pk)
+
~_. npkwa~lpk (~vot
- ' o at,. vol -
va,
. v ~
lv2p~ + -~
lv2a , -~
)
,
(6)
Pk Giea(Pk)
where V.j~, for 1 - 1,... , Ns, is the diffusion flux of gas species l, and 2a,'l and coa,hetzpkis the mass production rate due to homogeneous and heterogeneous reactions, respectively. The .het Pk - ~2~=1 Ns wa~ .hettPk and ojvol Ns a~:~k account for the total mass production terms wa, a, Pk - ~21=1 rate due to all heterogeneous reactions and volatilization, respectively. The fluctuating pressure is p. E and ~- are the unit tensor and the shear stress tensor, respectively. The
855 fdrag
t e r m - c , Pk denotes the drag force between gas particle Gi and pseudo CP-particle /Sk. hom are the diffusioanl heat flux in the gas phase and the energy source due V - q adi~f f and h G, :cony to homogeneous reactions in a gas particle Gi, respectivly. The term qG, Pk accounts for convective heat transfer between gas particle Gi and pseudo C,P-particle Pk. Summations in Eqns. (3) and (6) are taken over all pseudo CP-particles s which are influencing GPparticle Gi.
3. I N T E R P H A S E FLUXES Equations (1), (2) and (4)-(6) for pseudo CP-particles and GP-particles are coupled through the interphase fluxes of mass, momentum, and energy. Simple explicit relationships for modelling these fluxes are adopted as outlined in [7] . For simplicity, the dispersed solid particles are assumed to have thermochemical properties similar to those of coal.
3.1. Mass Flux, Drag Force, and Energy Flux Devolatilization of particles and heterogeneous reactions are both described by simple one-step mechanisms of the form" Coal -+ CO
(devolatilization),
Coal + 2 02 + C02 + 2 CO
(het. reaction).
The rates of pseudo CP-particle volatilization/heterogeneous reaction is given by the commonly used Arrhenius-type expressions. In this study, momentum fluxes are represented by drag forces. The drag force between a pseudo CP-particle/Sk and the surrounding gas in Eqn. (2) is taken in the form: fdrag
7r d2- 1
I%-vb l,
(7)
where fia and 9a are the density and velocity of the gas, averaged over all gas particles inside the action sphere of pseudo CP-particle/Sk. The mean drag coefficient ~D is given by the Stokes law. Interphase energy exchange is represented by convective heat flux between a pseudo CP-particle Pk and GP-particles Gi inside the action sphere of/Sk. It is taken in the form" v
d
-
,
(s)
where Nu is the Nusselt number, taken Nu - 2.0 for simplicity, and To and ~a are the average gas temperature and heat conductivity of gas in the vicinity of pseudo CP-particle Pk, respecitevly. ".conv The drag force fdra9 -c, Pk and heat exchange rate qa~ Pk between pseudo CP-particle /Sk and GP-particle Gi in Eqns. (5) and (6) are also calculated by formulas (7) and (8), respectively, but using the averaged values of density, velocity, and temperature in the vicinity of GP-particle Gi.
856 4. M O L E C U L A R
FLUXES IN GAS PHASE
The basic difficulty in solving Eqs. (1) to (6) under turbulent flow conditions lies in the treatment of the molecular fluxes and the fluctuating pressure gradient. The effects of convection in physical space and chemical reaction require no modelling and can be treated straightforward. For modelling the molecular fluxes V . jlGi and V " "~'::f the model of Dopazo [9 10] is ~tGi used, which is based on the simple representation of relaxation to the local mean values of species concentration and energy. The terms in Eq. (5), representing viscous stress V(T) and fluctuating pressure gradient V ( p E ) , are modelled by a simple stochastic Langevin equation similar to the approach of Pope [10]. The gas-phase reaction is taken in the form [11]: 2 CO + 02 -+ 2 CO2 , with rates of species formation/depletion in Arrhenius type form. The term h h~ a~ in Eq. (6) is calculated with regard to the heat of reaction. 5. R E S U L T S OF T E S T I M P L E M E N T A T I O N 5.1. Initial and B o u n d a r y C o n d i t i o n s The goal of the current study is to simulate two-phase combustion in a uniform pressure reactor under simple conditions of constant, isotropic, homogeneous turbulence. The computational set up used is given in Fig. 2.
Figure 2. Sketch of boundary conditions: open volume of air enriched with CO-devolatilizing reactive particles.
Preheated air of initial temperature T O - 800 K and mean pressure p0 _ 1 bar flows with constant mean velocity of 2.5 ms -1 through an open volume of size 5 cmx2.5 cmx0.1 cm. The airflow contains in its lower part (z k
\ k _
9Exp. u' X / D = 6 . 2 [] Exp. v '2 X / D = 6 . 2 o Exp. u '2 X / D = 1 2 . 4 5 + Exp. v '2 X / D = 1 2 . 4 5
"", \'\ \
Normal Particle Stresses (m2/s2)
'\,
- u '2 X / D = 6 . 2 - v '2 X / D = 6 . 2 - u '2 X / D = 1 2 . 4 5
3.00
9
9
2.00
9 9
9o,@"
"'"
o
'
o ~
9
',
""'~'"'O:~
;5 ~ ' ' ' ' ~> o
0.10
"
i\
o','\ ,& , i
~,~~=~-~=-=__~..
..... 1":-~ . . . . . .
0.02 0.04 0.06 Radial Distance (m)
0.08
0.00
.
0.q )o
,
.
--
,
,
T
U'2 X / D = 2 0 v '2 X / D = 2 0
. . . . .
9
,p
1.00
0.00 0.00
9Exp. u '2 X / D = 1 0 [] Exp. v '2 X / D = 1 0 * Exp. u '2 X / D = 2 0 + Exp. v '2 X / D = 2 0 . . . . . . u '2 X / D = 1 0 .... v '2 X / D = 1 0
9
00
9149
9
% 0 '%
i \,
0 0
1 " : ~ - ~ +
O.Ol
0
o 9 +
+
+
+
0.02
Radial Distance (m)
Figure 3. Normal particle Reynolds stresses for the experiments of Mostafa et al. [5] (left) and Hishida et al. [8] (right) based in the formula (10). u '2 corresponds to the axial streamwise direction and v '2 to the transversal direction.
In Fig. 3 the normal components of the Reynolds stresses are compared with the refered values provided for Mostafa et al. [5] (left plot). The same is showed in the right picture for another experiment performed by Hishida & Maeda is a gas-solids jet with coflow. The details are given in [8]. In both diagrams the same values for the anisotropy parameters presented in section 3 are used. A reasonable agreement is found in both cases, especially in the sections far downstream, where the jet is completely developed. This concerns the
932 axial and radial particle velocity fluctuations. At this stage, it could be suggested that the values for the parameters C (i), i - x, r, ~ are roughly the same for all jet flows laden with high inertia particles. This would be similar to what is generally accepted for the gas rms velocities, which keep the approximate relation u x,2 "u r,2 "u~, 2 _ k . k / 2 . k/2. Nevertheless, the validity of this assertion should be confirmed by additional experimental measurements. 6. C O N C L U S I O N S A study on the continuum equations that describe the particulate phase in a nonuniform dilute turbulent gas-solid jet has been performed. It is concluded, that in the case of great inertia particles the modelling of discrete elements normal stresses drives the spreading of the solid volume fraction along the jet. If a consistent expression is given for the former, based on the anisotropy parameters, a confirmation of several closures previously used for the radial relative velocity is found. Moreover, this work reveals the importance of the formulation of the interaction terms (i.e., forces and fluctuating work exchanged between both phases) for achieving the correct evolution of the mean quantities involved.
Acknowledgement We want to acknowledge to Dr. G. Kohnen his useful comments and suggestions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
M.W. Reeks, Phys. Fluids A 5 (1993) 750. M. Ishii, Thermo-fluid dynamic theory of two-phase flow, Eyrolles, 1975. S.L. Lee and M.A. Wiesler, Int. J. Multiphase Flow 12 (1986) 99. O. Simonin, Proc. 5th Workshop on Two-Phase Flow Predictions, Erlangen (Germany), 1990. A.A. Mostafa, H.C. Mongia, V.G. McDonell and G.S. Samuelsen, AIAA J. 27 (1989) 167. K. Hyland, O. Simonin and M.W. Reeks, Proc. 3rd Int. Conf. Multiphase Flow, Lyon (France), 1998. P. F~vrier and O. Simonin, Proc. 3rd Int. Conf. Multiphase Flow, Lyon (France), 1998. K. Hishida and M. Maeda, Japanese J. Multiphase Flow 1 (1987) 59. R. Aliod and C. Dopazo, Part. Part. Syst. Charact. 7 (1990) 481. A. Prosperetti and D.Z. Zhang, J. Fluid Mech. 267 (1994) 185. S. Lain, Ph.D. University of Zaragoza, Spain (1997).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
933
E x p e r i m e n t a l r e s e a r c h o f m o d i f i c a t i o n o f g r i d - t u r b u l e n c e b y r o u g h particles. M. Hussainov a, A. Kartushinsky a, G. Kohnen b and M. Sommerfeld b aEstonian Energy Research Institute, Paldiski mnt. 1, Tallinn, EE0001, Estonia bInstitut fur Mechanische Verfahrenstechnik und Umweltshutztechnik, Martin - Luther Universit~it Halle - Wittenberg, D-06099 Halle (Saale), Germany
Abstract
Experimental results of the modification of the grid-generated turbulence by rough solid particles in a vertical downstream channel flow are presented here. Glass particles with an average diameter of 700 ~tm are used in the experiments, while the mass loading was set to 0.05 kg dust&g air and 0.1 kg dust/kg air. The two grids with a mesh size of 4.8 mm and 10 mm made from circle rods with a sizes of 2 mm and 5 mm, respectively, were generating isotropic turbulence in the initial period. The mesh Reynolds numbers for these grids, based on a mean air velocity of U=9.5 m/s, were ReM=3045 and ReM=6340, respectively. The results of the mean velocity distribution of the carrier flow, the turbulence intensity expressed in the root-mean-square along the flow and in the different cross-sections downstream, which were obtained by using LDA, are presented. The turbulence decaying along the flow and the spectra in different centerline points downstream are shown as well. 1. I N T R O D U C T I O N For gas - solid two-phase flows as well as for gas - liquid systems the influence of the dispersed phase on the continuous phase is very important with respect to the objective of the whole process under consideration. The mechanism of turbulence modification induced by the motion of solid, liquid or gaseous particles within a flow are not very well understood in most situations. Due to the variety of parameters involved with this issue it is very difficult to extract a reasonable amount of information from a single experiment. Moreover, the quality (or the accuracy) of the results obtained by experimental investigations has a tremendous impact on the expected success of any model used to predict turbulence modification in such two-phase flow systems. Basically, there are two classes of reviews available in the literature, where an attempt was made to classify turbulent two-phase flows. One of them is due to Gore & Crowe [2], where the most relevant experimental investigations are analysed according to the change in turbulence intensity depending on a length scale ratio Dp/LE, where Dp represents the particle diameter and LE the integral length scale of the flow, respectively. This analysis resulted in a criterion for Dp/LE in order to decide, if the particles attenuate the single phase turbulence (values below Dp/LE-0.1) or augment it (values above Dp/LE -- 0.1). In a later publication of
934 Gore & Crowe [3] different parameters were suggested, which may have an influence on the change of turbulence intensities, such like the density ratio 9p/9, the flow and particle Reynolds numbers, Re and Rep, Dp/LE, the relative turbulence intensity, and the volume concentration. There are no quantitative results available until now concerning the parameters listed above with respect to turbulence attenuation and augmentation. Hetsroni [5] in his review argued, that such a criterion for determining turbulence attenuation/augmentation could be found solely in the particle Reynolds number Rep in such a way, that for Rep>400 wake effects become predominant leading to an enhanced turbulence production and hence to turbulence augmentation. Besides, different results are obtained for wall-bounded turbulent shear flows and free shear flows. Schreck & Kleis [10] experimentally studied the mechanisms of turbulence decay behind the grid in water flow loaded by two types of particles, namely plastic and glass beads with a particle size about 650 lam. In this work the fluid velocity was around 1 m/s with the mesh Reynolds number ReM=l.56"104 based on a mesh size of M=17.8 mm. Their experiments have shown, that both kind of particles did attenuate the turbulence of the water flow for various volume fractions of the dispersed phase within the range of 0.4-1.5%. The spectral energy density behaved different depending on the velocity component under consideration. Comparing the results for the laden two-phase flow with the corresponding one for the singlephase flow they found, that the growth of the energy took place at large wave numbers for the axial velocity component. For the transversal component the opposite could be observed, namely a reduction of the spectral density of the energy at this range of wave numbers for both types of particles (plastic and glass spheres). The distribution of the total turbulent energy, namely E=Eu+2Ev (Eu and Ev are the energy of the axial and transversal component of the velocity fluctuations, respectively), has shown a decrease of the spectral density of the energy E in this wave number range. Kulick et al. [8] studied a two-phase flow in a vertically downward channel flow configuration. They used comparatively small particles, namely copper particles with an averaged diameter of 70 lam and two charges of glass particles each having 50 and 90 lam, respectively. For mass loadings up to 40 % a reduction in turbulence intensity was observed. The work of Lance & Bataille [9] was devoted to the investigation of the influence of particles (air bubbles) on the turbulence intensity in a water flow behind the grid. They observed enhancement of turbulence caused by the bubbles. The distribution of the spectral energy density reveals a reduction in the small wave number range due to the bubbles, whereas the large wave numbers were fed with energy and took on higher values compared with the single phase flow. 2. EXPERIMENTAL SET-UP The experiments were carried out in a disconnected vertical two-phase wind-channel with closed test section (Figure 1). The wind-channel consists of an intake (1) with a diameter of l m, followed by a contraction (2) in order to lower the streamwise turbulence level. Its contour was profiled by the Vitoshinsky formula according to Gorlin [4] characterized by an area ratio of 6.25, a transition duct (3), that smoothly changed from a round inlet cross-section to a square outlet cross-section, the test section (5), and the diffuser (6). The test section was made of plexiglass and had a length of 2 m. At the initial upper cross-section its dimensions
935
9\ 8
7
E
?C/AT
! [L. . . . 2 i m W ~
Optics
+
r -i
Figure 1. Flow facility: 1-intake; 2-contraction; 3-transition duct; 4-turbulence grids; 5-test section; 6-diffuser; 7-seeding generator; 8-particle feeder; 9-cyclone separator. were 200 x 200 mm, downstream it increased gradually up to 205 x205 mm at the exit. Such a geometry of the test section allowed to compensate partially the growth of the boundary layer along the walls and, thus, to keep the flow core to be uniform along the test section. There was a possibility to install the optical glass windows along the full length of the test section in order to have the access of the LDA-technique. Air was flowing in the downward direction with a mean velocity of U=9.5 m/s. This downflowing air was created with the help of a suction fan. The particles of the dispersed phase were brought into the flow by a special particle feeder (8) installed in the intake (1). They were separated from the gas flow in the diffuser (6) and then conveyed by a vertical pneumatic transport system into the cyclone separator (9) within the circuit for recovering the solid phase. The reservoir of feeding the solid phase, (8), was loaded by particles taken from the cyclone separator directly before the beginning of the experiments. The capacity of this reservoir was 40 liters. In this way 90 kg of glass beads were available in order to carry out the experiment in a continuous way for 20 minutes, while the mass loading of the solid phase was about 0.1 kg/kg. Particles made of titanium dioxide TiO2 having a particle size in the range of 0.5-2 gm were used as particle
936 tracers for visualizing the air flow. The particle-tracers were brought into the flow through the intake (1) by using the seeding generator (7). The operation principle of this generator was based on the principle of fluidized bed. A turbulence generating grid (4) was placed at the inlet of the test section (5). Two types of grids with square meshes were used. Their characteristics are presented in Table 1. Table 1 Characteristics of the turbulence producin~ ~rids used in the experiments Grid Mesh size M [mm] M/d ReM Solidity Sh/S 1 4.8 2.53 3045 0.487 2 10 2.5 6340 0.49
3. MEASUREMENT TECHNIQUE The forward-scattering Laser Doppler Anemometer (LDA) was used to measure the instantaneous velocities of the gas and particles in the two-phase flow. The LDA consists of two channels, each channel for the respective phase of the dispersed flow. The transmitting unit of the LDA channels with a 26 mW and 10 mW laser formed two different measurement volumes in order to distinguish between the velocities of the two groups of particles, the TiO2 seeding particles and the glass particles with an averaged diameter about 700 lam. The channels were organized by the module principle. Hence, it was possible to easily get access to the system depending on the peculiarities of the problem to be solved. The tuning of each LDA channel for registration of its own particles was based on an amplitude discrimination of the Doppler signals by means of changing the parameters characterizing the measurement volume and the geometrical conditions of the receiving optics, and by varying the sensitivity of the photomultiplier as well. The parameters of the LDA system for studying the gaseous flow laden with solid particles are summarized in Table 2. During the experiments the gaseous phase channel and the channel of the dispersed phase were tuned by the maximum data rate of Doppler signals from tracer and glass particles. The experiments carried out in a single-phase wind-channel have shown a continuous registration of Doppler signals from particle-tracers by the channel of the carrier phase. In the case of the two-phase flow, when 700 gm glass particles were brought into the flow, micron tracer particles sedimented on the surfaces of the glass particles coating their surfaces and resulting in the modifying their optical properties. This means, that we dealt with a situation, where the scattering properties of the glass particles have been modified during the experiments leading to a degradation of the Doppler signals in the direction of the signal detection. As a result, a continuous registration of signals in the channel of the dispersed phase took place only in the case, when the glass particles were still clean. This situation has slightly improved after decreasing the fringe spacing of the interference field to 24.83 gm and, correspondingly, increasing the total number of fringes in the measurement volume to 66. Consequently, only clean glass particles with an averaged diameter of 700 gm were used. The instantaneous velocities of the gaseous phase were measured by the LDA system using the parameters presented in Table 2. 15 series or measurements were carried out in order to obtain the data rate for calculations of the power spectra of fluctuation velocities of the
937 gaseous phase at a certain distance from the generating turbulence grid along the axis of the flow. Each series of the measurements included 2000 samples of the axial component of the instantaneous velocity of the gas. Those series were registered only, if the registration times were less or equal 0.4 seconds. The initial assembly of samples of the instantaneous velocities included 30000 samples with an average interval between each single sample of 150 laS, which results in the average frequency of 6.6 kHz for the registration. Each series of the instantaneous velocities was filtered by amplitude and time discrimination. Table 2 Parameters of the optical system of LDA for studying the gas flow laden with solid particles The channel of the Gaseous phase Dispersed phase
Transmitting optics Wavelength of the laser [~m] Focal length of the front lens [mm] Beam separation [mm] Diameter of the measuring volume [lam] Fringe spacing [~tm] Fringe number
0.6328 300 35.0 87.9 5.43
0.6328 300 3.85 1651 49.30 (24.83)
16
33 (66)
3.0 o 58
5.8 ~ 58
250
422
Receiving optics Off-axis angle Focal length of collimating lens [mm] Distance from the measuring volume to the receiving optics [~m]
4. RESULTS AND DISCUSSIONS The main investigations were performed in different cross-sections of the channel at a distance from the grid X=63, 95, 126, 251, 383, 503, 692, 1264 mm. These distances normalized with the respective grid mesh size correspond to the following values: 9for grid 1 (M=4.8 mm): X/M= 13.1, 19.8, 26.3, 52.3, 79.8, 104.8, 144.2, 263.3; 9for grid 2 (M=10 mm): X/M=6.3, 9.5, 12.6, 25.1, 38.3, 50.3, 69.2, 126.4. The energy spectra for the streamwise velocity fluctuations of the gas have been determined at the axis of the channel for the laden and unladen flow in the above mentioned cross-sections.
4.1 Mean velocity distributions The basic series of experiments were carried out, when the mean velocity of the gaseous flow (air flow) was about 9.5 m/s. Small deviations from this mean flow velocity were taken into account during processing of the experimental data in the way, that all averaged values of fluctuations U~ms and power spectra of velocity fluctuations were corrected to the mean velocity 9.5 m/s. Figure 2 depicts the profiles of the averaged streamwise velocities and their fluctuation values above the grid for the unladen and laden flow. It is evident, that the non-uniformity of the flow for the averaged velocities was less than or equal to 0.5% and for the fluctuation
938
values less than or equal to 6% for the two-phase flow. Figure 3 presents the profiles of the above-mentioned parameters and the particle mass distribution in the cross-section X=293 m m of grid 1 (M=4.8 mm).
1.1~ ,
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i
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.
.
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.
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75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
u~,2 1
I
Um Um 100 -83 -50 -40 -30 -23 -10 0 10 33 30 43 93 83 y, mm
o71
-
8 ............~ - : . - . - ' ~,2
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o
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i
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..........
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02. Q1
o
-if) JE) -43 -33 -33 -10 0 10 33 33 43 93 83 y, mm
Figure 2. Profiles of the averaged velocities Figure 3. Profiles of the averaged velocities, and their fluctuations in the initial cross-section, their fluctuations and the particle mass loading at X=293 ram, grid 1. 4.2 Intensity of turbulence distributions U 2
Figure 4 and 5 present the behaviour of U
m
*2
U 2 2 U rms
along the channel behind the two grids
for the laden and unladen flow.
600
~e = 0.051 9 500
a~ = 0
U2
4o0
.
~e = 0.087 9 0
a~ = 0.099 ~e = 0.047
36O
~e
~/
a~=O
u2____ ~
9
UP2 300 200
~
20
40
60
5O 0 80
0 . 0 ~ = 0 ~
l
100
120
X/M
140
160
180 200 220
Figure 4. Modification of the turbulence intensity behind grid 1. Unladen flow conditions (thin line) versus laden flow conditions (bold line).
~
9
::::
~e= 0.087 = 0.064 Ee= 0.089
100
~e = 0.085 0
~e =
= 0.066
150
~e = 0.087
~ ~e = 0.063
lOO
=0.051
~
0
10
20
30
40
50
X/M
60
70
80
90
100 110
Figure 5. Modification of the turbulence intensity behind grid 2.
The decay curves show, that the turbulence intensity of the flow behind the grid decreases up to the fixed cross-section (X/M--90 for grid 1 (M=4.8 mm) and X/M--40 for grid 2 (M=10
939 mm) and it remains almost constant more downstream. This constant level of turbulence intensity approximately corresponds to the turbulence of the flow in a channel without a gridgenerating turbulence and is conditioned by the initial turbulence. From these figures one also can conclude, that the loading ratio of the flow about 0.05 kg/kg leads to a decrease of the turbulence intensity of the gas phase. This decrease is more pronounced for grid 1 (M=4.8 mm) in comparison with grid 2 (M=10 mm). According to these experimental results the turbulence intensity of the single-phase was about 3.5% at the final stage of the turbulence decay. This is essentially larger than it was found by other experimentalists (e.g. Hinze [6]) and may be caused by the high initial turbulence level of the flow, which was in front of the grid due to the presence of the special particle feeder in the intake (8, Fig.l). The energy decay curves are adequately represented by: 2 -C
-A
.
(1)
U rms
The corresponding constants for the different flow conditions can be found in Table 3. Table 3: Constants in the deca~, curves for the single-and two-phase flow for different l~rids Single-phase flow Two-phase flow A C Grid A C 1 (M=4.8mm) 0.22 8.19 1.24 8.92 2 (M=10 mm) 3.09 11.42 3.7 12.55 Unfortunately, the mass loading was different for each data point on Figures 4 and 5. There were some reasons for that. Each series of experiments were conducted separately in laboratory. In fact, the circuit for recovering the solid phase was disconnected. Besides, a capacity of the receiving bin of a cyclone separator (9, Fig. 1) was limited and therefore each test series had a time limit which were characterized by its own loading ratio of the flow. Some time was required for the flow adaptation in order to reach the same magnitude of mass loading of flow, but, in this case we would lost a definite mass of the glass particles and have no time to measure the fluctuating velocity properly. During the measurements we skipped the flow adaptation process and, therefore, there was some difference in mass loading between various test series. 4.3. Energy spectra The initial energy spectra of the streamwise velocity fluctuations of the gaseous phase were smoothed with the averaging procedure over ensemble of neighbouring values. For the given energy values they are calculated as follows: 1 i+4
E'(f i ) -
~s
E(f i ), i - 5, N - 4,
(2)
2i-1 s t ( f i ), i -
(3)
j=i-4 1
E'(fi ) - 2 i - 1 j=,
1,2,3,4,
N
1
E'(fi ) - 2(N - i ) +
~_ E(fi),i _ N _ 3,N,
1 j=2i_ N
(4)
940 where E(f i ) and E'(f i ) are the initial and smoothed energy density at the frequency fi, respectively. Such smoothing allows to imagine the influence of particles on the energy spectra as a whole, although at that time the information about the minor details of the spectrum vanishes. Figures 6 and 7 show the smoothed energy spectra for the single and twophase flows for grid 1 at X=383 mm and X=1264 mm. For comparison the respective smoothed energy spectra are shown in Figures 8 and 9 for grid 2 at the same positions. One can recognize, that the influence of mass loading in the case of the grid 2 is not as strong as for the grid 1. In the high-frequency range of the spectrum (frequencies higher than 1000 Hz) a decrease of the turbulence energy was observed in the case of grid 1 almost in all considered cross-sections. 1.E-(~ 84
1.ECs
'~": ''"~'::'"+?I":": :"":h"i:iii i!!'-:-:-i: :-'"::-'--:":-:""::--:":F":' I-!!!!t :F-I":--H: '-":"-' ~'i-+'::"!:i"!' 1 '::-":i":"::-i:"4:' '-"i?i: :[ 1.E-03 84
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1.EO4
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1.E-07 10
100
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1000
1(3330
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1.E-04
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..............~:,:-:::-:,:.::-:~,:,:,:,:,:~-::,:,:i ................. ............. i:::;,:.:i ................ -~,...-,~.:-~,~.:~,~,:~:~h~.~!+~,!~+:!-~,~i ~-
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....
...............r........i i.......iiii -
1.Eo3 iiiiiiiiiiiiill~ ~ ~
I.E-05 ............... i......... iT!iii ,........................... iii ....... i ~: ............ ii ]iii![
ii
.......... ..... 100
Figure 7. Smoothed energy spectra for the single and two-phase flows (X=1264 mm) grid 1 (M=4.8 mm).
Figure 6. Smoothed energy spectra for the single and two-phase flows (X=383 mm), grid 1 (M=4.8 mm) 1.E-02
.......... 10
~?~:~i~}~i~}~i~?~:~:~:~:~..:~?~i~i~!~)?
...............i.........,Iiiiigl ..........il......iiiil~I~i~lW
10
100 f,Hz
1000
10000
Figure 8. Smoothed energy spectra for the single and two-phase flows (X=383 mm), grid 2 (M= 10 mm)
10
103 f, Hz
1030
10330
Figure 9. Smoothed energy spectra for the single and two-phase flows (X= 1264 mm) grid 2 (M= 10 mm).
For a detailed analysis of the spectrum shape in the frequency range of the energy containing eddies, 400-2000 Hz, the spectra were approximated by the power function y - Ax -n . The dependencies of the exponent n on the distance from the grid are presented in
941 Figure 10 for grid 1 and in Figure 11 for grid 2. For comparison a value for n=5/3 (dotted curve) related to Kolmogorov's law (the inertial subrange) is included there as well. As follows from Figure 10, the pronounced decrease of the turbulent kinetic energy of the carrier fluid takes place also in the range of the energy containing eddies for grid 1 (M=4.8 mm). Figure 11 shows, that in the case of grid 2 (M=10 mm) there is no significant effect of mass loading on the modulation of the turbulent kinetic energy in the given range.
2.5
[] ,,
,,
~e=O ,
2.5
--I--ae>O
I
2
_.
1.5 n
n
1.5 -~ I
1 0.5
0.5
-I I
0
200
400
600
800
1000
1200
X, m m
Figure 10. Dependence of exponent n in the power function on the distance from grid 1.
1400
ol 0
200
400
600
800 X, m m
1000
1200
1400
Figure 11. Dependence of exponent n in the power function on the distance from grid 2.
Next, the most relevant parameters related to the dispersed phase are presented in Table 4. The Stokes number was based on the turn-over time. To determine the volume concentration let us use the formula [3- pa~U/(ppUs), where ~e is the integral mass loading of the flow. Further, one can find the ratio between the inter-space particle distance between the particles and the particle size ~ / 8 - (n/6[3) 1/3 - 1 by the results of Kenning & Crowe [7]. Table 4 Relevant parameters characterizin~ the dispersed phase Grid 1 Grid 2 ae [kg/kg] 0.09 0.09 U-Us [m/s] 1.5 2 Rep 70 93 Iu [S] 4.5 4.5 StE 2.47.102 1.34.102 ~,/8
118
118
8/qK
6.31
5.22
According to Gore & Crowe [2], the character of the influence of the particles on turbulence modulation of the carrier fluid in depends roughly on the parameter 8/LE, which represents the ratio of the particle size to the integral length scale of the turbulence. If 8/LE>0.1 turbulence enhancement takes place, otherwise, 8/LE
1 0 0
10
20
30
40
50
60
D i s t a n c e b e t w e e n the b u b b l e s , D
Figure 10. Variation of the trailing bubble velocity with the separation distance between the bubbles The velocity of propagation of the trailing Taylor bubble, normalized by that of the leading one, is presented in fig. 10 as a function f the separation distances between the bubbles. The
952 model prediction of Moissis and Griffith (1962), as well as the correlation of the experimental data by Pinto and Campos (1996) are shown in this figure. At the distance of few pipe diameters, a considerable acceleration of the trailing bubble is obtained. In this region the present results are close to those obtained earlier. The current measurements, however, extend to substantially larger distances than those investigated before. It appears that interaction between the bubbles exists at distances exceeding the generally assumed stable liquid slug length (16D for a vertical flow). The present experimental results indicate an average rate of approach of-~10mm/s for distance between bubbles in the range of 60 to 30D. In other words, two Taylor bubbles separated initially by a liquid slug of length below 60D will end up colliding if the pipe is sufficiently long.
4. SUMMARY Interaction between two consecutive Taylor bubbles rising in stagnant liquid is studied quantitatively using the image processing technique. Certain conclusion regarding the process of the bubbles approach and merging are drawn on the basis of the flow visualization. The 2-D flow field in the liquid between the two bubbles is then measured using the PIV technique, and the variation of the vortex structure of the flow is studied up to a separation distance of about 60D. The variation of the propagation velocities of the two bubbles is measured. It is found that the trailing bubble does not affect the leading one. The trailing bubble, on the other hand, is sensitive to the velocity distortion in the wake of the leading bubble even at distances exceeding 50 pipe diameters. The information regarding the propagation velocity of the trailing bubble as a function of the separation distance from the leading one is essential for modeling slug length distribution in pipes (Barnea and Taitel 1993). REFERENCES
Barnea, D. and Taitel, Y. (1993) Int. J. Multiphase Flow, 19, 829-838 Collins, R, de Moraes, F.F., Davidson, J.F., and Harrison, D. (1978)J. Fluid Mech., 89, 497-514. Grace and Clif~ (1979) Chem. Eng. Sci.34, 1348-1350. Davies, R.M. and Taylor, G.I. (1949) Proc. R. Soc. London, Ser. A 200, 375-390. Dumitresku, D.T. (1943) Z Angew. Math. Mech., 23, 139-149. Mao, Z.-S. and Dukler, A. (1990) J. Comp. Phys., 91, 132-160. Moissis, D. and Griffith, P. (1962)J. Heat Transfer, Trans. ASME, Series C, 2, 29-39. Nieklin, D.J., Wilkes, J.O., and Davidson, J.F. (1962) Trans. Inst. Chem. Eng., 40, 61-68. Polonsky, S. (1998)Ph.D. Thesis, Tel-Aviv University. Polonsky, S., Barnea, D., and Shemer, L. (1999) To appear in Int. J. Multiphase Flow. Pinto, A.M.F.R. and Campos, J.B.L.M. (1996) Int. J. Multiphase Flow, 51, 45-54. Shemer, L. and Barnea, D. (1987) Physicochem. Hydrodyn., 8, 243-253. Taitel, Y. and Barnea, D. (1990) Advances in Heat Transfer, 20, 83-132.
953
Author Index Aanen, L. 371 Abu-Ghannam, B. J. 533 Akker, H. van der. 257 Alekseenko, S. V. 637 Aliod, R. 923 Amielh, M. 453 Andersson, H. I. 349 Andreux, R. 913 Anselmet, F. 453 Antonia, R. A. 423 Arakawa, C. 155 Atashkari, K. 805 Atmane, M. A. 521 Aubrun, S. 491 Barnea, D. 943 Ballmann, J. 649 Batten, P. 19 Bauer, W. 113 Behzadi, S. A. 279 Bencze, F. 751 Bertoglio, J. P. 289, 815 Besnard, D. 37 Bilsky, A. V. 637 Bocksell, T. L. 903 Boisson, H. 491 Bonnet, J. P. 207, 501 Booij, R. 339, 415 Boulet, P. 913 Bousquet, N. 125 Branley, N. 861 Brodie, G. H. 627 Buffat, M. 237 Burr, R. 269 Cazalbou, J.-B. 125 Chan, W. T. 821 Chartrain, D. 501 Chassaing, P. 125 Chauve, P. 197 Chemobrovkin, A. 555 Chomiak, J. 841 Chumakov, Yu. 607 Coratekin, T. 649 Coustols, E. 689
Craft, T .J. 73 Davidson, L. 227 Davies, T. W. 783 Delville, J. 207, 443 Derksen, H. 257 Doisy, A. M. 501 Druault, Ph. 207 Duursma, R. 135 Elmo, M. 815 Essen, T van. 135 Ewing, D. 461 Ferr6, J. A. 393 Franke, M. 659 Friedrich, R. 83, 247 Frolov, S. M. 851 Fujiwara, H. 155 Fu, S. 659 Fulachier, L. 453 Geissler, W. 679 Gessner, F. B. 709 George, J. 521 Giralt, F. 393 Gltick, M. 269 Grotjans, H. 269 Gr6tzbach, G. 165 Guilbaud, M. 501 Ha Minh, H. 491 Haag, O. 113 Had~.i6, I. 587 Han, Y. O. 471 Hangan, H. 381 Hanjali6, K. 135, 587 Hassel, E. P. 831 Hein, K. R. G. 881 Hennecke, D. K. 113 Hinz, A. 831 Houra, T. 481 Hussainov, M. 933 Hutter, K. 93 Htittl, T. J. 247
954 Iacovides, H. 773 Ichimiya, M. 597 Ishigaki, T. 511 Itoh, M. 699 Jackson, D. C. 773 Janicka, J. 93, 831 Johansson, A. V. 175 Jones, W. P. 861 Kao, P. L. 491 Kartushinsky, A. 933 Kaufmann, F. 543 Kavanagh, P. 533 Keffer, J. F. 393 Kelemenis, G. 773 Kenjere~, S. 135 Kidger, J. W. 73 Kim, Y. S. 471 Kitamura, O. 893 Knaus, H. 881 Knoell, J. 103 Knowles, K. 783 Kobayashi, M. 699 Kohnen, G. 933 K6ngeter, J. 361 Kopp, G. A. 381,393 Lain, S. 923 Lakshminarayana, B. 555 Landenfeld, T. 831 Launder, B. E. 73,773 Lawes, M. 805 Le Penven, L. 237 Leschziner, M.A. 19 Lipatnikov, A. 841 Liu, W. P. 627 Liu, X. 721 Loth, E. 903 Loyau, H. 19 Ltibcke, H. 659 Lukowicz, J. V. 361 Lucas, J. F. 453 Maier, H. 881 Majumdar, S. 309 Manceau, R. 319 Mariotti, G. 871 Markovich, D. M. 637
Martinuzzi, R. 381 Maruyama, T. 217 Matsuo, Y. 155 May, N. E. 329 Menter, F. 269 Meri, A. 197 Merzkrich, W. 63 Michard, M. 289 Muto, Y. 511 Nagano, Y. 481 N'Diaye, M. 237 Neuhaus; M. G. 841 Nigim, H. H. 533 Nikolskaja, S. 607 Nitsche, W. 619 Norberg, C. 227 Oesterl6, B. 913 Olsen, J. F. 423 Pailhas, G. 689 Pantano, C. 187 Parneix, S. 319 Parpais, S. 289 Perot, B. 145 Peters, N. 49 Pettersson-Reif, B. A. 349 Picard, C. 443 Pope, S. B. 795 Raddaoui, M. 197 Rajagopalan, S. 423 Rajani, B. N. 309 Rettich, T. 63 Riess, W. 731 Romano, G. P. 403 Rose, M. 851 Roth, P. 851 Ruiz-Calavera, L. P. 679 Rung, T. 659 Sabatino, A. 403 Sabel'nikov, V. A. 815 Sadiki, A. 93 Saidi, A. 763 Sagaut, P. 207 Sarkar, S. 187 Sauvage, Ph, 689
955 Schiele, R. 543 Schiestel, R. 197 Schneider, F. 63 Schnell, U. 881 Schubert, A. 649 Schulz, A. 543 Sello, S. 871 Semenov, V. I. 637 Sentker, A. 731 Shao, L. 187 Shemer, L. 943 Sheppard, C. G.W. 805 Shur, M. 669 Sltick, M. 269 Sohankar, A. 227 Sommerfield, M. 933 Spalart, P. R. 3,669 Steelant, J. 577 Strelets, M. 669 Sund6n, B. 763 Talvy, C. A. 943 Taulbee, D.B. 103 Thiele, F. 659 Tinapp, A. F. 619 Travin, A. 669 Touil, H. 289 Tsuji, T. 481 Tucker, P. 299
Ubaldi, M. 741 Uijttewaal, W. S. J. 339, 415 Unterberger, S. 881 Vad, J. 751 Vemet, A. 381 Wagner, C. 83 Wang, H. 145 Watkins, A. P. 279 Weatherly, D. C. 567 Wengle, H. 197 Westerweel, J. 371 Wikstr6m, P.M. 175 Williams, K.E. 709 Wittig, S. 543 Woolley, R. 805 W6mer, M. 165 Xiong, W. 63 Xu, J. 795 Xue, L. 659 Yamamoto, M. 893 Ye, Q.-Y. 165 Zhang, X. 433 Zhang, Y. 821 Zunino, P. 741
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