ADVANCES IN LES OF COMPLEX FLOWS
FLUID MECHANICS AND ITS APPLICATIONS
Volume 65 Series Editor: R. MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d'Hères Cedex, France
Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a funda mental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hy personic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other scien ces and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are mono graphs defining the current state of a field; others are accessible to final year undergradu ates; but essentially the emphasis is on readability and clarity.
Advances in LES
of Complex Flows
Proceedings of the Euromech Colloquium 412,
held in Munich, Germany
4–6 October 2000
Edited by
R. FRIEDRICH Technische Universität München,
Fachgebiet Strömungsmechanik, Garching, Germany
and
W. RODI Universität Karlsruhe,
Institut für Hydromechanik, Karlsruhe, Germany
KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBook ISBN: Print ISBN:
0-306-48383-1 1-4020-0486-9
©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:
http://kluweronline.com http://ebooks.kluweronline.com
v
CONTENTS Preface List of Participants
ix
xiii
Invited Lectures and Contributions 1. Modelling and analysis of subgrid scales
Invited lecture: On the physical effects of variable filtering lengths and times in LES M. Germano
3
Invited lecture: How can we make LES to fulfill itspromise? B. J. Geurts
13
The approximate deconvolution model for compressible flows: Isotropic turbulence and shock-boundary-layer interaction S. Stolz, N.A. Adams and L. Kleiser
33
A new mixed model based on the velocity structure function C. Brun, R. Friedrich, C.B. da Silva and O. Métais
49
On the effect of coherent structures on grid/subgrid-scale interactions in plane jets: The transition and far field regions C.B. da Silva and O. Métais
65
2. Numerical issues in LES Phase-error reduction in large-eddy simulation using a compact scheme
H.-J. Kaltenbach and D. Driller
83
DNS and LES of turbulent backward-facing step flow using 2nd and
4th-order discretization
A.Meri and H. Wengle
99
Parallel multi-domain large-eddy simulation of the flow over a backwardfacing step at Re=5100 115
E. Simons, M. Manna and C. Benocci
vi 3. Cartesian grids for complex geometries
LES of flow around a circular cylinder at a subcritical Reynolds number with cartesian grids F. Tremblay, M. Manhart and R. Friedrich
133
A numerical wind-tunnel experiment
M. Pourquié, C. Moulinec and A. van Dijk
151
4. Curvilinear and non-structured grids for complex geometries
Invited lecture: Towards large eddy simulation of complex flows
A.Fureby
165
Structured and non-structured large eddy simulations of the Ahmed reference model R.J.A. Howard, M. Lesieur and U. Bieder
185
Large eddy simulation of fluid flow and heat transfer around a matrix
of cubes, using unstructured grids
and
199
Large eddy simulation for flow analysis in a centrifugal pump impeller R.K. Byskov, C.B. Jacobsen, T. Condra and J.N. Sørensen
217
5. DES and RANS-LES coupling
Invited lecture: Detached-eddy simulation, 1997-2000
P.R. Spalart
235
Physical and numerical upgrades in the detached-eddy simulation of
complex turbulent flows
A.Travin, M. Shur, M. Strelets and P.R. Spalart
239
Detached and large eddy simulation of airfoil flow on semi-structured grids S. Schmidt and F. Thiele
255
A multidomain / multiresolution method with application to RANS - LES coupling P. Quéméré, P. Sagaut, V. Couaillier and F. Leboeuf
273
vii 6. Aircraft wake vortices Aircraft wake vortex evolution and decay in idealized and real environ ments: Methodologies, benefits and limitations F. Holzäpfel, T. Hofbauer, T. Gerz and U. Schumann
293
VLES of aircraft wake vortices in a turbulent atmosphere: A study of decay H. Jeanmart and G.S. Winckelmans
311
7. Combustion and magnetohydrodynamics
Invited lecture: Subgrid combustion modelling for LES of single and two-phase reacting flows S. Menon
329
Experimental study of the filtered progress variable approach for LES of premixed combustion 353 R. Knikker and D. Veynante Three-dimensional large eddy simulation of decaying magneto-hydrodynamic turbulence W.-C. Müller, B. Knaepen, O. Agullo and D. Carati
367
This page intentionally left blank
PREFACE This volume in the series Fluid Mechanics and its Applications contains a selection of reviewed articles based on presentations given at the EUROMECH Colloquium 412 on LES of complex transitional and turbulent flows. This Col loquium was held at the Munich University of Technology on October 4-6, 2000 with the intention to provide a forum for presentation and discussion of new de velopments in the field of large-eddy simulation of complex flows. Complexity could be due to the geometry of the flow or the physical mechanisms involved, like compressibility, heat transfer, combustion or magnetic induction. The Colloquium was attended by 73 delegates from 12 countries. Its pro gramme covered invited lectures by C. Fureby, M. Germano, B.J. Geurts, S. Menon and P. Spalart and thirty-three original papers within eleven sessions. The Colloquium was financially supported by the following companies: AEA Technology, Astrium, Audi Deutschland, BMW, Fairchild Dornier, Hitachi Eu rope, MTU München, SGI München. Their help is gratefully acknowledged. The Proceedings contain the invited lectures and eighteen contributions related to the topics Modelling and analysis of subgrid scales Numerical issues in LES Cartesian grids for complex geometries Curvilinear and non-structured grids for complex geometries DES and RANS-LES coupling Aircraft wake vortices Combustion and magnetohydrodynamics. In his invited paper M. Germano addresses a problem especially related to inhomogeneous flows. For differential filters he shows analytically that dif fusion effects and modified convective velocities result from variable filtering lengths and times, respectively. The invited paper of B.J.Geurts covers three issues which are important in the context of LES of complex flows: The need to develop subgrid scale (sgs) models based on rigorous properties of the sgs stress tensor, the estimation of commutation errors and their direct similar ity modelling and finally the contamination of LES predictions due to spatial discretization errors. The contribution of Stolz, Adams and Kleiser demon strates the feasibility of the approximate deconvolution model (ADM) in con junction with low-order finite difference schemes often used for flow simulation in complex geometry and discusses the successful application of ADM to LES order compact of the turbulent supersonic compression ramp problem with schemes. Brun, Friedrich, da Silva and Métais propose a new mixed model for LES consisting of a scale similarity component based on velocity increments and a variant of the structure function model and show its effectiveness in LES of transitional plane and round jets. The paper of Da Silva and Métais focusses on the role which coherent vortices play in the energy exchange mechanisms ix
x between resolved and unresolved scales of plane jets. The implications of these findings on sgs modelling are worked out. Numerical issues are the main subject of the three following contributions. The first by Kaltenbach and Driller contrasts high order implicit and explicit finite difference schemes in order to show their effect on the phase error in LES of turbulent channel flow. This error seems to be the cause for the failure of predicting adequately the wake region of the mean velocity profile. Meri and Wengle evaluate second and fourth order spatial discretizations in LES of turbulent backward-facing step flow against fourth order accurate DNS data and observe that high order discretization has to be combined with proper spatial resolution and proper sgs modelling in order to guarantee reliable LES data. Simons, Manna and Benocci report on parallel multi-domain LES of a turbulent boundary layer separating from a backstep and especially focus on an efficient parallel solution of the Poisson equation for the pressure. A common feature of the next two papers is the use of Cartesian grids for LES of flows whose complexity is caused by the geometry. In the contribution of Tremblay, Manhart and Friedrich an immersed-boundary technique of second order accuracy is presented and applied to LES of vortex-shedding flow around a circular cylinder at a subcritical Reynolds number. Results for two different sgs models are contrasted with DNS and experimental data to show the poten tial of the method in terms of efficiency and reliability. Pourquié, Moulinec and van Dijk apply a body-force technique to satisfy no-slip boundary conditions along the walls of a wind-tunnel contraction in an LES of wind-tunnel turbu lence with a second-order accurate cartesian Navier-Stokes solver. Incoming turbulence is generated via LES of the flow around grid bars of square cross section. The combination of grid upstream and contraction downstream forms an interesting tool for generating homogeneous turbulence with a prescribed level of anisotropy. The invited paper of C. Fureby presents a recent update on LES of complex flows, complex in terms of geometry and physics. The general principles are discussed for incompressible flows, but extensions to compressible and reacting flows are also made. For flows of engineering interest which are usually associ ated with high Reynolds numbers and complex geometries, the development of improved, more sophisticated subgrid and wall models is identified as an impor tant research goal. A comparison between LES results obtained on structured and non-structured grids for flow around a car type bluff body (the Ahmed model) is presented by Howard, Lesieur and Bieder for a low Reynolds number case. It shows the advantages and disadvantages of both methods. LES of flow and heat transfer around a wall-mounted matrix of cubes on unstructured grids is the topic of the paper by and Only one cube is heated and kept at constant temperature. Hexahedral and tetrahedral grids are tested showing that hybrid grids with hexahedral cells in the wall-layer provide op timum results in terms of accuracy and computational efficiency. LES of a typical industrial flow has been performed by Byskov, Jacobsen, Condra and Sørensen. They applied LES with a localised dynamic Smagorinsky model to
xi analyse the flow field in centrifugal pumps and obtained encouraging results particularly at partial load where the flow is highly separated. The invited lecture of P. Spalart on ’Detached-eddy simulation, 1997-2000’ appears in the form of an introduction to the history and literature of this tech nique in these Proceedings since a review of the author has recently appeared in Int. J. Heat Fluid Flow 21 (2000). A new DES formulation based on Menter’s SST model is discussed by Travin, Shur, Strelets and Spalart and tested on sev eral flows ranging from homogeneous isotropic turbulence to various separated flows. Furthermore, a new numerical scheme adjusted to the hybrid nature of the DES approach and the demands of complex flow is also presented. DES and LES are applied to predict the flow around a NACA 4412 airfoil at 12 ° angle of attack and the flow around a square cylinder by Schmidt and Thiele. LES results are shown to provide improved results when the near-wall grid is refined. Further improvements would apparently require a larger spanwise flow domain, higher resolution and hence a considerably bigger numerical ef fort. This fact disqualifies the LES technique in the authors’ opinion compared to the DES approach which predicts all flow properties with remarkably good quality. The paper by Quéméré, Sagaut, Couaillier and Leboeuf proposes a new zonal RANS/LES technique as a generalized multidomain problem with interface variables that are either extrapolated from the LES subdomain or extracted from auxiliary computations. The method is successfully applied to turbulent plane channel flow. The dynamics of aircraft wake vortices are thoroughly investigated in two papers. These vortices form a potential hazard during aircraft landing and takeoff. Since they are difficult to investigate experimentally, LES constitutes the tool to analyze their physics with the aim of finding means for alleviating their intensity or for accelerating their decay. The paper by Holzäpfel, Hofbauer, Gerz and Schumann addresses the problem of adequate resolution of coherent turbulent structures in the trailing vortices under different ambient conditions like turbulence level, stable temperature stratification and shear. A new sgs closure is proposed to account for strong streamline curvature effects. Its application to a spectacular situation at London Heathrow Int’l Airport where the upwind vortex of a Boeing 747 in terminal approach was lifted up to flight level shows excellent agreement with measurements when the effect of trees and crosswind shear is modelled. The contribution of Jeanmart and Winckelmans investigates the decay of a pair of wake vortices without ground effect under the influence of the intensity of the surrounding isotropic turbu lence field. A mixed model consisting of a tensor-diffusivity and a dynamic Smagorinsky term is applied in most LES along with explicit filtering. A new exponential decay model for the global vortex circulation is proposed which accounts for an initial delay process caused by the development of transversal coherent structures. Combustion and magnetohydrodynamic effects are the topics of the remain ing three contributions. The invited paper of S. Menon provides a survey of subgrid modelling approaches for single and two-phase reacting flows in practi
xii cal systems. A subgrid mixing and combustion model (LES-LEM) is described based on one-dimensional reaction-diffusion processes for chemical species in the direction of the scalar gradients. The model is a variant of Kerstein’s lineareddy model (LEM) and has the potential of dealing with finite-rate kinetics in high Reynolds number flows of full-scale gas turbine engines. Its two-phase ver sion is applied to droplet vaporization and combustion in a temporal mixing layer and a gas turbine configuration of General Electric. The contribution of Knikker and Veynante deals with LES of premixed combustion and uses the fil tered progress variable approach which is particularly suited when the reaction takes place in thin layers, called flamelets. The flame surface density appear ing as a key parameter in this approach contains an unresolved part that needs modelling. A new model is proposed based on the curvature of the resolved flame which leads to very promising results especially concerning the localiza tion of high-intensity unresolved contributions. The decay process of isotropic magnetohydrodynamic turbulence is predicted by LES in the contribution of Müller, Knaepen, Agullo and Carati. Consistent with the physical nature of kinetic and magnetic energy dissipation, the applied dynamic eddy-viscosity models are based on the deformation tensor, whereas the eddy-resistivity mod els are based on the curl of the magnetic field. These models are combined with tensor-diffusivity models to obtain mixed formulations. LES results for the temporal and spectral energy evolution reveal the need to determine model coefficients dynamically. While the kinetic energy evolution turns out to be acceptably well predicted with gradient-diffusion models, the tensor-diffusivity term is beneficial for the prediction of magnetic field dynamics. The editors wish to express their sincere appreciation to the authors for preparing and carefully revising their contributions and to the Kluwer Aca demic Publishers for producing this volume in a short time.
October 2001,
Rainer Friedrich Wolfgang Rodi
LIST OF PARTICIPANTS
Ms. Caroline Ackermann CEA Grenoble 17, rue des Martyrs 38054 Grenoble Cedex 9 FRANCE
[email protected] Tel.: +334 76 88 33 86 Fax.: +334 76 88 50 36
Mr. Ignace Befeno IRPHE (Marseille University) 38 rue Joliot-Curie 13451 Marseille cedex20 France
[email protected] Tel.: +33(0)491118530 Fax.: +33(0)491118502
Dr. Jean-Pierre Bertoglio CNRS LMFA Ecole Centrale de Lyon 69130 Ecully France
[email protected] Tel.: +33 4 78 29 20 74
Mr. Giuseppe Bonfigli Universität Stuttgart Pfaffenwaldring 21 70550 Stuttgart Germany
[email protected] Tel.: +49 711/6853422 Fax.: +49 711/6853438
Dr. Norbert Brehm BMW AG Hufelandstraße 80788 München Germany
[email protected] Tel.: +49 89-382-33661 Fax.: +49 89-382-32513
Dr. Michael Breuer Institute of Fluid Mechanics University of Erlangen-Nürnberg Cauerstr. 4 91058 Erlangen Germany breuer@lstm. uni-erlangen. de Tel.: +49 9131-761-246 Fax.: +49 9131-761-275
Dr. Christophe Brun Fachgebiet Strömungsmechanik TU-München Boltzmannstr. 15 D-85747 Garching Germany
[email protected] Tel.: +49-(0)89-289-161 34 Fax.: +49-(0)89-289-161 39
Mr. Tobias Buchal Electricité de France 6, quai Watier, B.P. 49 78401 Chatou Cedex France
[email protected] Tel.: +33 (0)1 30 87 76 95 Fax.: +33 (0)1 30 87 79 49
xiii
xiv Ms. Rikke Kau Byskov Grundfos Management A/S
Poul Due Jensens Vej 7
DK-8850 Bjerringbro
Denmark
[email protected] Tel.: +45 8750 1400
Fax.: +45 8740 1491
Dr. Daniele Carati Université Libre de Bruxelles Campus Plaine CP231 1050 Brussels Belgium
[email protected] Tel.: +32-2-650.58.13 Fax.: +32-2-650.58.24
Prof. Pierre Comte Institut de Mécanique des Flu-
ides
2 rue Boussingault
F 67000 Strasbourg
France
[email protected] Tel.: +33 3 88 41 65 78
Fax.: +33 3 88 61 43 00
Dr. Pascale Domingo CORIA/LMFN - UMR 6614
INSA - Av. de l’Université BP 8
76801 Saint-Etienne-du-Rouvray
France
[email protected] Tel.: +33(0)2 32 95 97 93
Fax.: +33(0)2 32 95 97 80
Mr. Michael Düsing Institute of Energy and Power-
plant Technology
TU Darmstadt
Petersenstr. 30
64287 Darmstadt
Germany
[email protected] Tel.: +49 6151 16-5334
Fax.: +49 6151 16-6555
Mr. Sven Eisenbach Fachgebiet Strömungsmechanik,
TU-München
Boltzmannstr. 15
D-85747 Garching
Germany
[email protected] Tel.: +49-(0)89 289 161 34
Fax.: +49-(0)89 289 161 39
Mr. Frank Freikamp Institut für Technische Mechanik
RWTH Aachen
Templergraben 64
D-52056 Aachen
Germany
[email protected] Tel.: +49 241 80 7592
Fax.: +49 241 80 8888 223
Prof. Rainer Friedrich Fachgebiet Strömungsmechanik
TU-München
Boltzmannstr. 15
85748 Garching
Germany
[email protected] Tel.: +49 89 289 16144
Fax.: +49 89 289 16145
xv Dr. Jochen Fröhlich Institute for Hydromechanics University of Karlsruhe Kaiserstrasse 12 76128 Karlsruhe Germany froehlich@ifh. uni-karlsruhe. de Tel.: +49 721/608-3118
Dr. Christer Fureby FOA Defence Research Estab lishment Warheads and Propulsion SE-17200 Stockholm Sweden
[email protected] Tel.: +46 8 706 4036 Fax.: +46 8 706 4144
Prof. Massimo Germano Politecnico di Torino Dip. Ing. Aeronautica e Spaziale C.so Duca degli Abruzzi, 24 10129 Torino Italy germano@polito. it Tel.: +39 01l 564 6814 Fax.: +39 01l 564 6899
Prof. Bernard J. Geurts Faculty of Mathematical Sci ences University of Twente P.O.Box 217 NL-7500 Enschede The Netherlands
[email protected] Tel.: +31 53 489 4125 Fax.: +31 53 489 4888
Mr. Volker Gravemeier Institute of Structural Mechanics Pfaffenwaldring 7 70550 Stuttgart Germany
[email protected] Tel.: +49 711/685 6120 Fax.: +49 711/685 6130
Dr. Günther Grötzbach Forschungszentrum Karlsruhe, IRS Postfach 3640 76 021 Karlsruhe Germany
[email protected] Tel.: +49-7247-82-4477 Fax.: +49-7247-82-3718
Delft University of Technology Lorentzweg 1 2628CJ Delft The Netherland hanjalic@ws. tn.tudelft .nl Tel.: +31 15 278 1773 Fax.: +31 15 278 1204
Mr. Thomas Hofbauer NE-PA, DLR Oberpfaffenhofen 82234 Weßling Germany
[email protected] Tel.: +49 8153 28-2506 Fax.: +49 8153 28-1841
xvi Dr. Frank Holzäpfel DLR-Oberpfaffenhofen
82234 Weßling
Germany
Frank.Holzä
[email protected] Tel.: +49 8153-282529
Fax.: +49 8153-281841
Dr. Richard Howard CEA/LEGI
1 Grande rue
38000 Grenoble
France
[email protected] Tel.: +33(0)4 76 01 86 12
Dr. Suad Jakirlic FG Strömungslehre und Aero
dynamik
TU Darmstadt
Petersenstr. 30
64287 Darmstadt
Germany
[email protected] Tel.: +49 6151-163554
Fax.: +49 6151-164754
Dr. Harm Jonker Thermofluids Section
Delft University of Technology
PO box 5046
2600 GA Delft
The Netherlands
[email protected] Tel.: +31 15 2786157
Fax.: +31 15 2781204
Mr. Nikola Jovicic Institute of Fluid Mechanics
University of Erlangen-Nürnberg
Cauerstr. 4
91058 Erlangen
Germany
[email protected]. de
Tel.: +49 9131-761-233
Fax.: +49 9131-761-275
Dr. Hans-Jakob Kaltenbach TU Berlin, HFI, Sekr. HF 1
Strasse des 17. Juni 135
10631 Berlin
Germany
[email protected] Tel.: +49 30-31423463
Fax.: +49 30-31421101
Prof. Leonhard Kleiser ETH Zürich Institute of Fluid Dynamics CH-8092 Zürich Switzerland
[email protected] Tel.: +41-1-632-2648 Fax.: +41-1-632-1147
Dr. Markus Kloker Institut für Aerodynamik und
Gasdynamik
Universität Stuttgart
Pfaffenwaldring 21
D-70550 Stuttgart
Germany
[email protected] Tel.: +49 711 685-3427
Fax.: +49 711 685-3438
xvii Dr. Bernard Knaepen Université Libre de Bruxelles Boulevard du triomphe, Campus Plaine, CP.231 1050 Brussels Belgium bknaepen@ulb. ac.be Tel.: +-32-2-6505918 Fax.: +32-2-6505824 Prof. Satoru Komori Dept. of Mech. Engineering Kyoto University Yoshida-Honmachi, Sakyo-ku 606-8501 Kyoto Japan
[email protected] Tel.: +81-75-753-5244 Fax.: +81-75-753-5245
Mr. Ronnie Knikker Laboratoire EM2C - CNRS Grande Voie des Vignes F-92295 Châtenay-Malabry France
[email protected] Tel.: +33 1 41 13 10 81 Fax.: +33 1 47 02 80 35
Dept. of Thermo and Fluid Dy namics Chalmers University of Technol ogy
S-412 96 Gothenburg Sweden sinisa@tfd. Chalmers. se Tel.: +46 31-772 1400 Fax.: +46 31-18 09 76
Mr. Emmanuel Labourasse ONERA 29, avenue de la Division Leclerc F-92320 Châtillon France
[email protected] Tel.: +33 1 46 73 42 89 Fax.: +33 1 46 73 41 66
Mr. Cyrille Lesieur LMFN-CORIA Avenue de l’Université - BP 08 76800 Saint Etienne du Rouvray France
[email protected] Tel.: +33 (0)2.32.95.98.15 Fax.: +33 (0)2.32.95.97.80
Dr. Michael Manhart Fachgebiet Strömungsmechanik TU-München Boltzmannstr. 15 82747 Garching Germany
[email protected] Tel.: +49-89-289-16142 Fax.: +49-89-289-16139
Mr. Olaf Marxen IAG, Universität Stuttgart Pfaffenwaldring 21 70550 Stuttgart Germany
[email protected] Tel.: +49-711-6853443 Fax.: +49-711-6853438
xviii Dr. Matthias Meinke Aerodynamisches Institut Wüllnerstr. 5-7 52062 Aachen Germany matthias @aia. rwth- aachen. de Tel.: +49-241-805417 Fax.: +49-241-8888257
Dr. Florian Menter AEA Technology GmbH Staudenfeldweg 12 83624 Otterfing Germany
[email protected] Tel.: +49 8024 905415 Fax.: +49 8024 905417
Prof. Suresh Menon School of Aerospace Engineering Georgia Institute of Technology 270 Ferst Drive NW 30332-0150 Atlanta, Georgia USA suresh.menon@aerospace. gatech.edu Tel.: +1 404 894 9126 Fax.: +l 404 894 2760
Mr. Adnan Meri Universität der Bundeswehr München Institut für Strömungsmechanik 85577 Neubiberg Germany
[email protected] Tel.: +49 89 6004-2115
Mr. Takenobu Michioka Dept. of Mech. Engineering Kyoto University Yoshida-Honmachi, Sakyo-ku 606-8501 Kyoto Japan
[email protected] Tel.: +81-75-753-5263 Fax.: +81-75-753-5245
Mr. Charles Moulinec Lab Aero-en-Hydrodynamica Leeghwaterstraat 21 2628 CD Delft The Netherlands
[email protected] Tel.: +31-15-278-2861
Mr. Wolf-Christian Müller Physique statistique et plasmas CP 231, Boulevard du Triomphe 1050 Bruxelles Belgique
[email protected] Tel.: +32-2-650.57.99 Fax.: +32-2-650.58.24
Mr. Jens Neumann Universität der Bundeswehr München Institut für Strömungsmechanik 85577 Neubiberg Germany
[email protected] Tel.: +49 89 6004-2115
xix
Mr. Bojan Niceno
Dr. Michael Pfitzner
Delft University of Technology Lorentzweg 1 2628 CJ Delft Holland
[email protected] Tel.: +31.15.278.6125 Fax.: +31.15.278.1204
Rolls-Royce Deutschland GmbH Eschenweg 11 D-15711 Dahlewitz Germany
[email protected] Tel.: +49-(0)33708-6-1145 Fax.: +49-(0)33708-6-3086
Mr. Mathieu Pourquie Laboratory for aero- and hydrodynamics Delft University of Technology Leeghwaterstraat 21 2628 CB Delft Netherlands M.J.B.M.Pourquie@wbmt. tudelft.nl Tel.: +31-15-2782997 Fax.: +31-15-2782947
Mr. Patrick Quéméré ONERA 29, avenue de la Division Leclerc F-92320 Châtillon France
[email protected] Tel.: +33 1 46 73 42 89 Fax.: +33 1 46 73 41 66
Mr. Benjamin Rembold ETH Zürich Sonneggstraße 3 8092 Zürich Switzerland
[email protected] Tel.: +41-1-632-5272 Fax.: +41-1-632-1147
Dr. Ulrich Rist
Prof. Wolfgang Rodi Institute for Hydromechanics University of Karlsruhe Kaiserstraße 12 76128 Karlsruhe Germany
[email protected] Tel.: +49/721/608-3535 Fax.: +49/721/608-7712
Dr. Thomas Rung TU Berlin Müller-Breslau Str. 8 10623 Berlin Germany
[email protected] Tel.: +49-30-314 24416 Fax.: +49-30-314 21101
IAG, Universität Stuttgart Pfaffenwaldring 21 70550 Stuttgart Germany
[email protected] Tel.: +49 711 685 3432 Fax.: +49 711 685 3438
xx Dr. Roland Schiestel IRPHE/CNRS IMT, Technopôle de Château Gombert 38 rue F. Joliot Curie 13451 Marseille Cedex 20 France
[email protected] Tel.: +33(0)4.91.11.85.29 Fax.: +33(0)4.91.11.85.02
Dr. Stefan Schmidt Hermann-Föttinger-Institut Müller-Breslau-Str.10 10623 Berlin Germany
[email protected] Tel.: +49-30-314-26283 Fax.: +49-30-314-21101
Prof. Dr.-Ing. habil. Ulrich Schumann DLR Institut für Physik der Athmo sphäre 82234 Weßling Germany
[email protected] Tel.: +49 8153 - 282 521 Fax.: +49 8153 - 281 841
Mr. Emmanuel Sergent LMFA, Ecole Centrale de Lyon 36, av Guy de Collongue 69131 Ecully Cedex France
[email protected] Tel.: +33 4 72 18 61 63
Dr. Carlos Neves Betten court da Silva Equipe MOST/LEGI BP 53 38041 Grenoble Cedex 9 France
[email protected] Tel.: +33 4 76 82 50 19 Fax.: +33 4 76 82 52 71
Mr. Erwin Simons von Karman Institute for Fluid Dynamics Waterloose steenweg 72 1640 Sint-Genesius-Rode Belgium
[email protected] Tel.: +32 2 3599611 Fax.: +32 2 3599600
Dr. Philippe Spalart Boeing Commercial Airplanes P.O. Box 3707 WA 98124-2207 Seattle USA
[email protected] Tel.: +1 425 234 1136 Fax.: +1 425 237 8281
Dr. Jörg Stiller Institut für Luft- und Raum fahrttechnik Technische Universität Dresden 01062 Dresden Germany
[email protected] Tel.: +49 351 463 8328 Fax.: +49 351 463 8087
xxi Mr. Steffen Stolz Inst. of Fluiddynamics, ETH
Zürich
ETH Zentrum, Sonneggstr. 2
CH-8092 Zürich
Switzerland
[email protected] Tel.: +41-1-6325273
Fax.: +41-1-6321147
Prof. Michael Strelets Russian Scientific Center
”Applied Chemistry”
14, Dobrolyubov Ave.
197198 St.-Petersburg
Russia
[email protected] Tel.: +7-812 238-9763
Fax.: +7-812 238-9251
Mr. Frederic Tremblay Fachgebiet Strömungsmechanik
TU-München
Boltzmannstr 15.
85748 Garching
Germany
[email protected] Tel.: +49 89 28916142
Dr. Denis Veynante Laboratoire E.M2.C.
Ecole Centrale Paris - C.N.R.S
92295 Châtenay-Malabry
FRANCE
[email protected] Tel.: +33 1 41 13 10 80
Fax.: +33 1 47 02 80 35
Prof. Peter Voke University of Surrey MME GU2 7XH Guildford U.K.
[email protected] Tel.: +44 1483 879688
Fax.: +44 1483 306039
Dr. Claus Wagner DLR Göttingen
Bunsenstrasse 10
37073 Göttingen
Germany
[email protected] Tel.: +49 551/709 2261
Fax.: +49 551/709 2404
Prof. Hans Wengle Universität der Bundeswehr
München
Institut für Strömungsmechanik
85577 Neubiberg
Germany
hans. wengle@unibw-
muenchen.de
Tel.: +49 89 6004-2131
Mr. Wolfgang Wienken TU Dresden
Technische Universität Dresden
01062 Dresden
Germany
[email protected] Tel.: +49 351 463 2552
Fax.: +49 351 463 5246
xxii Prof. Grégoire S. Winckelmans Université Catholique de Louvain CESAME 2, place du Levant 1348 Louvain-la-Neuve Belgium
[email protected] Tel.: +32 10 47 22 14 Fax.: +32 10 45 26 92
1. Modelling and analysis of subgrid scales
This page intentionally left blank
ON THE PHYSICAL EFFECTS OF VARIABLE FILTERING LENGTHS AND TIMES IN LES
M. GERMANO Dip. di Ing. Aeronautica e Spaziale
Politecnico di Torino, Italy
Abstract. The Large Eddy Simulation of turbulent flows is based on the application of a filtering operator that separates the resolved eddies from the unresolved ones. If the flow is inhomogeneous this resolution is different from region to region, and different filtering lengths or times are required. Variable filtering operators do not commute with differentiation, and this problem is presently the subject of many studies. In the present paper this problem is addressed by considering a particular class of filters, the differential ones, that are provided with simple properties and can be easily applied to the Navier-Stokes equations. The results are read physically, by considering the effect that a variable filtering length or time has on simple evolutionary equations. It is shown in particular that a differential elliptic filter with a variable filtering length produces a diffusive effect, while a differential low pass filter provided with a variable filtering time is responsible for a modification of the convective velocity.
1. Introduction
One basic ingredient of the Large Eddy Simulation of turbulent flows is the filter that separates the resolved from the unresolved part of a generic chaotic quantity Following Leonard (1974) this filter can be generally expressed as an integral convolution in space and time
3 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 3-11. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
4
M. GERMANO
that provides a smoothed filtered quantity We remark that are a characteristic filter length and time, and that
and
Well known filters are the top-hat and the Gaussian, and it is easy to see that if the filtering length and the filtering time are constant, the smoothing operator commutes with the derivatives. In practical applications filtering scales that are constant everywhere are not the best. In highly inhomogeneous turbulent flows the dissipative is different from region to region, and tunable filters of variable scale length could work better in order to give rationally and economically a constant resolution everywhere in terms of a constant ratio
This point is numerically very important, particularly in the framework of hybrid LES/DNS strategies where the filtering length should change optimally from zone to zone, but the related problem is that in this case the commutation between filtering and derivations is lost. We recall that a central problem in LES, as in RANS, is to express the generalized central moment associated to two generic quantities and
in terms of the filtered quantities and and we notice that the new term produced by non commuting filters can be formally written in terms of a new commutative generalized central moment given by
There is presently a big effort in finding commuting variable filters, or at least in minimizing the commutative moment (5). Family of filters with nonuniform filter widths that commute with differentiation up to any given order have been recently proposed by van der Ven (1995), and another class of filters that commute with differentiation to any specified power of the mesh spacing has been examined in detail by Vasilyev et al. (1998). In the present paper we address the same problem, but more than to find par ticular classes of commuting filters provided with variable filtering lengths we try to express the commutative moment (5) exactly in terms of the filtered quantities. In other words the problem that we address is the fol lowing : given a filtering operator, which is the physical effect of varying its
VARIABLE FILTERING LENGTHS AND TIMES
5
scale ? In the filtered equations the commutative moment produces as the usual moments new terms that can be read as new stresses, or directly new forces applied to the filtered quantities. This approach follows the physi cal attitude towards LES introduced by Muschinski (1996), and mainly it consists in reading the LES equations as equations of motion of specific hy pothetical fully turbulent non-newtonian fluids, called LES fluids. We define new filtered quantities, and we want to write and to understand physically their evolution. Physics and numerics are, at least theoretically, unmixed and from a basic point of view this is the same philosophy that is usually applied to the interpretation of the RANS equations. 2. The effect of a variable filtering length. The elliptic filter
The first filter that we have chosen for our study is a differential filter that the present author proposed some time ago, Germano (1986). In the differential form it reads as
where is a filtering length. If the following
is constant, the inverse integral form is
and this elliptic differential filter is interesting because it is very easy to
obtain an exact expression for the generalized central moment In deed if we apply the relation (6) to as well as and separately, we can both write that
and that
and always by applying the relation (6) to and (9) that
it follows finally from (8)
M. GERMANO
6
Formally this is the exact subgrid scale model associated to the elliptic differential filter proposed, and it is interesting to notice that as a first approximation we recover the well known Clark model, Clark et al. (1979)
In order to see now which is the physical effect of a filtering length variable in space, at least for this particular filter, let us now calculate As for it is the commutative moment (5) in the case of very easy to derive an exact expression for this term. Following the same procedure let us write that
and that
Always by applying the relation (6) it results finally that
and as a first approximation we can write
Provided with this result we can now examine the physical effect of a variable filtering length Let us consider a very simple transport equation for the variable
where is a constant velocity. This equation is linear, so that in the case of a constant we have simply
but if
we obtain
VARIABLE FILTERING LENGTHS AND TIMES
7
As a first approximation we can write
and it is interesting to notice that the effect of a variable filtering length is given by equivalent in this case to a diffusive term with a diffusivity
We notice that the diffusivity associated to the variation of the filtering length is positive when increases with the motion. A wave is attenuated when running towards increasing and that seems physically plausible. Obviously this result is peculiar of this particular filter, but probably, in the limit of filtering lengths going to zero, this result is more general. As remarked in Carati et al. (1999) the expansion of any symmetric filter with a well defined second moment in real space always starts with the Clark term (11). Clearly when the filtering length goes to zero its effect, homogeneous or not, disappears. It is however reasonable to think that, at least in the limit of a vanishing effect, also the general physical properties of the commutative moment explored in this paper could be extended to other similar elliptic filters of variable width. 3. The effect of a variable filtering time. The low pass filter Let us now analyze the effect of a variable filtering time in another very simple differential filter, the low pass filter. In the differential form it is given by the expression
where now is a filtering time. The inverse integral form in the general case in which is a function both of and is given by the convolution
and the commutation with the time and the space derivatives is lost. As in the case of the elliptic filter we can write that
and that
M. GERMANO
8
and also in this case it is very easy to derive an exact expression for these commutative moments. From the definition (21) of the low pass differential filter we can write both that
and that
and we finally obtain that
A similar relation can be obtained for the time derivative
and as in the case of the elliptic filter we can now very easily explore the physical effect of a filtering time variable in space and time. Let us notice that as a first approximation we can write
and let us apply this result to the same simple equation for the variable
where is a constant velocity. As previously remarked this equation is linear, so that also in the case of a low pass filter with a constant filtering time we have simply
but if
we obtain as a first approximation
where the new convective velocity
is given by
VARIABLE FILTERING LENGTHS AND TIMES
9
and it is interesting to remark that in the case of a variable filtering time convectively transported
there is no commuting effect on the equation. It is reasonable to think that also in this case the general physical properties of the commutative moment explored in this paper for the low pass filter could be generalized to other causal time filters extended to the past and provided with a variable time scale. 4. The effect of variable filtering lengths and times. The convec tive Lagrangian filter Let us now extend these observations to more complex differential filters. In the filtering operator order to simplify the notation let us indicate with and with the differential one. A differential filter can then be defined as
where is the identity, and it is interesting to remark that the differential operator is by definition the inverse of the filtering operator
A general differential form of the second order is the following
where
and if the parameters are not constant this operator does not commute with the derivatives and We remark that if is a generic operator we can write by applying the relation (36)
and that By comparing this relation with the relation (36) we can write
and as a first order approximation
10
M. GERMANO
An interesting particular case is the Lagrangian convective filter, Germano (2000), where the averaging operator follows the particle in the mean. In this case we can write that
and by applying the relation (43) we can obtain a first approximation to
the commutative moment. In the case in which we get
while in the case
we can write
and we remark that in both cases we have assumed constant the low pass filtering time and the filtering length 5. Conclusions In this paper we have considered the effect of variable filtering lengths and times on the averaged transport equations. The filters chosen are the differ ential ones, that are particularly simple in the applications, and the effects produced by the non commutation of the derivatives have been examined for very simple transport equations in the case of a low pass filter, an ellip tic filter and a Lagrangian filter. The results are read physically and they show in particular that in the case of a differential elliptic filter a variable filtering length produces a diffusive effect, while a differential low pass filter provided with a variable filtering time is mainly responsible for a modifi cation of the convective speed. We remark finally that these results refer to an explicit nonuniform filter, while many LES approaches are based on explicit models, as the subgrid scale eddy viscosity. For incompressible flows this last model is given by
where
and where K is the subgrid scale turbulent kinetic energy
VARIABLE FILTERING LENGTHS AND TIMES
The subgrid scale eddy viscosity
11
is given by
where and where C is the constant of Smagorinsky. We can see that in the case
of a variable filtering length the subgrid scale production P
has the same local expression, while the subgrid scale forces
are given by
where the last term is the explicit contribution of a nonuniform filter length. Presently a lot of work in subgrid scale modeling is devoted to coupling ex plicit filters and explicit models, as in the dynamic or the mixed approaches, in order to explore new possibilities for the representation of the subgrid stresses. In these cases the analysis of the effects due to a variable filter is more difficult, and should be the object of an extended work due to its practical interest. References CARATI, D., WINCKELMANS, G. S. & JEANMART, H. 1999 Exact expansions for filtered scales modelling with a wide class of LES filters. Direct and Large Eddy Simulation III, Voke P. R., Sandham N. D. and Kleiser L. Editors, Kluwer, 213–224 CLARK, R. A., FERZIGER, J. H. & REYNOLDS, W. C. 1979 Evaluation of subgrid scale models using an accurately simulated turbulent flow. J. Fluid Mech., 91, 1–16 GERMANO, M. 1986 Differential filters for the large eddy numerical simulation of turbu lent flows. Phys. Fluids, 29, 1755–1757 GERMANO, M. 2000 Fundamentals of Large Eddy Simulation. Advanced Turbulent Flows Computations, Peyret R. & Krause E. eds., CISM Courses and Lectures 395, Springer, 81–130 LEONARD, A. 1974 Energy Cascade in Large-Eddy Simulations of Turbulent Fluid Flows. Adv. in Geophysics, 18A, 237–248 MUSCHINSKI, A. 1996 A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky type LES. J. Fluid Mech., 325, 239–260 VAN DER VEN, H. 1995 A family of large eddy simulation (LES) filters with non uniform filter widths. Phys. Fluids, 7, 1171–1172 VASILYEV, O. V., LUND T. S. & MOIN P. 1998 A general class of commutative filters for LES in complex geometries. J. Comput. Phys., 146, 82–104
This page intentionally left blank
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
BERNARD J. GEURTS Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands‡‡
Abstract. We review three important issues in LES which need to be addressed in order to systematically advance LES towards flows of realis tic complexity. First, we consider rigorous modeling consequences arising from the analytic – and algebraic structure of the LES modeling problem. In this context, we develop a new generalized similarity model and ap ply this to a spatially developing mixing layer. Second, we estimate the commutation error arising in LES for flows in complex geometries which involve non-uniform filter-widths, and propose direct similarity modeling of these contributions. Third, we consider the numerical contamination of a ‘Smagorinsky fluid’ in order to illustrate the role of spatial discretization errors in LES. Suitable ratios between filter-width and grid-spacing are identified. Some guidelines for developing LES are proposed.
1. Introduction The main promise of large-eddy simulation (LES) is to provide the ‘same’ predictions for statistical quantities and generic unsteady flow features as arise from a direct numerical simulation (DNS), however at strongly re duced computational effort. Stated differently, LES is supposed to provide turbulent flow predictions at flow conditions and/or geometry complexities which are presently not accessible to a fully resolved simulation. Of course, such a promise has a ‘price’. Major problems associated with LES of turbu lent flow are encountered in the form of a closure problem for the subgrid scale stresses, geometry complexities, e.g. wall proximity or strong spatial inhomogeneity, introducing commutation errors, and numerical treatment ‡‡
Also: Department of Engineering, Queen Mary College, University of London, Mile
End Road, London E1 4NS, U.K. 13 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 13-32. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
14
BERNARD J. GEURTS
at marginal resolution. In order for LES to be computationally effective, dynamical consequences arising from each of these three (interrelated) ar eas are likely to be large and can give rise to interactions and accumulated contamination of the predictions. In the absence of a comprehensive theory of turbulence, empirical knowl edge about subgrid-scale modeling is essential but very incomplete. Since in LES only the dynamical effects of the smaller scales need to be rep resented, the modeling is supposed to be relatively simple and somewhat ‘universal’, compared to the setting encountered in statistical modeling such as in RaNS or VLES. To guide the construction of suitable models we ad vocate the use of rigorous properties of the LES modeling problem such as realizability conditions [1] and algebraic identities [2, 3]. Using approximate inversion techniques [4, 5, 6], a generalized similarity model [7] is obtained and illustrated for turbulent flow in a mixing layer [8]. Efficient extension of the LES approach to turbulent flows in complex ge ometries or to cases with strong spatial variation of turbulence intensities, calls for the introduction of non-uniform grids and filter-widths [9, 10, 11, 12]. The non-commutation terms that consequently arise in complex flows, are analyzed in some detail, establishing the dynamic importance of the commutation errors in boundary layers. The (dynamic) similarity model ing of these contributions is put forward, based on the fact that the noncommutation terms satisfy a Germano identity [2]. Apart from the problem of modeling the subgrid-scale stresses, any actual realization of LES inherently is endowed with (strongly) interacting errors arising from the required use of marginal numerical resolution [13, 14]. We consider in some detail numerical contamination of a ‘Smagorinsky fluid’ arising from the use of a fourth order accurate spatial discretization method [15]. We consider the role of the numerical method at various resolutions and various ratios of the filter-width compared to the grid-spacing and identify suitable ratios.
The organization of this paper is as follows. In section 2 we briefly formulate
the filtering approach to LES and identify some basic properties of the LES
modeling problem. Recent developments in subgrid modeling which aim at
incorporating information from the scales that are directly available in an
LES will be considered and applied to a spatially developing mixing layer.
Section 3 describes additional complications to the LES equations that
arise from extension to complex flows. This involves the use of nonuniform
filters and gives rise to additional terms in the filtered equations which
have a specific contribution to the inter-scale energy transfer that will be
interpreted and estimated. The numerical treatment of the LES equations is
another source of unavoidable errors. Section 4 is devoted to the interaction
of numerical and modeling errors for a fluid modeled by the Smagorinsky
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
15
model. All sources of error in LES can be controlled to some degree at the expense of adding to the computational effort. Some guidelines for LES which aim at keeping the mixture of errors within reasonable bounds will be suggested in section 5 where we also collect some concluding remarks. 2. Structure of the modeling problem in LES In this section we consider the filtering of the Navier-Stokes equations. Realizability and algebraic properties of the turbulent stress tensor will be reviewed. Moreover, the use of approximate inversion in order to arrive at generalized similarity modeling will be considered and applied to a spatially developing mixing layer. The starting point in the filtering approach is the introduction of a filter L to filter the Navier-Stokes equations:
where H denotes the normalized filter-kernel and the flow-domain. The filter-kernel H is assumed to have a filter-width and is usually of convo
lution type, i.e.
It is well-known that the Reynolds stress tensor in RaNS is positive
semidefinite (also called ‘realizable’) [16]. Following ref. [1] we will investigate under what conditions the turbulent stress tensor is ‘realizable’. These conditions imply:
It can readily be proved that in LES is realizable semidefinite if and only if the filter kernel is positive for all x and To show this, we introduce
with which the turbulent stress tensor can be written as:
with If we now suppose then eq. (5) defines an inner-product and consequently the tensor forms a 3 × 3 Grammian matrix of inner products which is always positive semidefinite and hence satisfies the realizability conditions. For the proof of the reverse statement we refer to [1].
16
BERNARD J. GEURTS
The turbulent stress tensor in LES is preferred to be realizable for a number of reasons. For example, if is realizable, the generalized tur is a positive quantity. This bulent kinetic energy quantity is required to be positive in subgrid models which involve the which would become ill-defined otherwise. Several further ben efits of realizability and positive filters can be identified [1]; here we only mention that the kinetic energy of is bounded by that of for positive filter-kernels:
The conditions on realizability can be extended to so-called higher-order filters which are defined by the conditions that ; for [11, 19]. General N-th order filters possess the invariance property: where is a polynomial of degree N. It can readily be shown that i.e. higher order filters will the filter-kernel can only be positive if violate realizability. If, in addition, the filter-kernel is not symmetric, one may show that upper-bounds exist on the ‘skewness’ of the filter, in order to maintain realizability of Realizability, places some restrictions on subgrid models. For example, if models for should be realizable. Consider e.g. an eddy-viscosity given by model
In order for this model to be realizable, a lower bound for in terms of the arises, i.e. where and is the eddy-viscosity rate of strain tensor. For the dynamic model we have and we obtain a lower bound constraint Next, we turn to algebraic properties of the turbulent stress tensor. We and write the turbulent introduce the product operator stress tensor as [3]:
Here we introduced the central commutator [L, S ] of the filter L with the product operator S. The commutator defining the turbulent stress tensor shares a number of properties with the Poisson-bracket in classical me chanics. Leibniz’ rule of Poisson-brackets is in the context of LES known as Germano’s identity [2]
where and denote any two filter operators and is the
turbulent stress tensor associated with a filter K. Similarly Jacobi’s identity
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ? holds for S,
17
and
The expressions in eq. (10) and eq. (11) provide relations between the tur bulent stress tensor corresponding to different filters and can be used to dynamically model The success of models incorporating eq. (10) is by now well established and was applied in many different flows. In the tradi tional formulation one selects and where is the so called test-filter. In this case one can specify Germano’s identity [2] as
The first term on the right hand side involves the operator acting on the resolved LES field and during an LES this is known explicitly. The remaining terms need to be replaced by a model. In the dynamic modeling [17] the next step is to assume a base-model corresponding to filter-level K and optimize any coefficients in it e.g. in a least squares sense [18]. The operator formulation allows to readily identify ‘generalized’ similarity models which involve approximate inversion defined by for With this operation it is possible to partially reconstruct the unfiltered solution from the filtered solution and use this information in the definition of a subgrid model. Without any inversion, the original simi i.e. applying larity model by Bardina [7] can be written as the definition of the turbulent stress tensor directly to the available filtered field. A direct generalization of this arises from using the approximate inversion. Recently the approximate inversion was combined with dynamic modeling and for which [20]. This was based on the choice Germano’s identity can be specified to
Compared to the traditional formulation eq. (12), which involves the mod and this exten eling of terms corresponding to length scales sion incorporates the (approximate) inverse of the test-filter and re quires modeling of terms on the scale of as before and Since the terms that require modeling are smaller and at the same time it is easier to maintain modeling assumptions, e.g. involving properties of an inertial range. Dynamic inverse models have been applied successfully in mixing layers [20]. The usefulness of generalized similarity models in actual LES will be con sidered next. Clearly, if the reconstruction of the unfiltered solution would
18
BERNARD J. GEURTS
be accurate, the generalized similarity model would provide a good rep resentation of the turbulent stress tensor. It was observed that the gen eralized models display a high correlation coefficient [21]. A more detailed assessment of the suitability of generalized similarity models is shown in fig ure 1. Here we compare the Bardina model B with the generalized similarity model based on second order inversion G and the combination Clearly, the rough features of are captured by all models to some degree. However, Bardina’s model tends to underestimate fluctuations while sec ond order inversion appears to over-predict these fluctuations. Averaging the two models lead to a significant improvement in capturing High correlation and correct amplitude in a priori tests are not sufficient and it is necessary to perform a posteriori tests. We turn to this next. In figure 2 we compare snapshots of vorticity fields taken from filtered DNS and LES using the dynamic mixed model and the generalized similar ity model. We use a fairly high resolution in the LES which is only a factor of three coarser in each direction, compared to the DNS. At this resolu tion the generalized similarity model performs well and unlike the dynamic mixed model, it seems to capture also some of the finer details contained in the filtered DNS. At coarser resolutions, the well-known lack of dissipation in similarity approaches, leads to somewhat too high levels of small scale features in the flow. Additional damping with an eddy-viscosity contribu
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
19
tion eliminates this problem. Generalized mixed models were successfully applied in LES at a 10 times higher Reynolds numbers, showing clear signs of self-similar development in the turbulent regime [8]. Next to subgrid-modeling, the application of LES to more complex, spa tially strongly inhomogeneous flows requires the treatment of additional
20
BERNARD J. GEURTS
subgrid terms arising form the use of non-convolution filters. This will be considered in the next section. 3. Estimating and modeling commutation-errors The desire to extend LES to complex flows in an efficient way implies that one has to deal with situations in which the turbulence intensities vary considerably within the flow domain. In certain regions of the flow a nearly laminar, smoothly evolving flow may arise while a lively, fine scale turbulent flow can be present in another region. This calls for a filtering approach with a non-uniform filter-width. Here we identify and estimate the additional terms that arise from using a variable filter-width. Since the identification of which scales are ‘subgrid’ and which are ‘re solved’, depends on the local filter-width, it is clear that any variations in the local filter-width will imply additional inter-scale energy-transfer. This is the main dynamic contribution of filter-width variations. If a (coher ent) flow structure would be propagated from a region of small filter-width into a region with strongly increased filter-width, it would appear as if this structure would turn from a ‘resolved’ to a ‘subgrid’ feature, merely by translation. The reverse can also be imagined, leading to the apparent emergence of resolved structures from a collection of subgrid scales. This process suggests additional sources of local energy drain or backscatter, depending on whether the local filter-width increases or decreases in the direction of motion of the (coherent) flow structure. This qualitative picture will be investigated further. We consider the effect of applying a general, compact support filter which is not of convolution type. This implies that the filter does not commute with partial derivatives, i.e. As before, though, we can derive the LES equations for nonuniform filters and identify a ‘mean’ term associated with the Navier-Stokes operator acting on the filtered solution and several new terms which are related to commutators containing the filter L. For incompressible flow the governing equations are
where denotes the pressure, Re the Reynolds number and the summation convention is adopted. Application of a general filter yields:
This shows that application of a non-convolution filter to the continu
ity equation in general violates the local conservation form. Filtering the
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
21
Navier-Stokes equations yields
The turbulent stress tensor arises in the first term on the right hand side. The remaining three terms on the right hand side are related to commu which also violate the conservation property in general. tators of L and The commutators can be shown to obey the same algebraic identities as obeyed by [L,S]. In order to establish the importance of these new commutators we analyze the commutators for general high order filters acting on sufficiently smooth and define the filter operation signals [11], We consider a signal by:
in which denote the upper- and lower filterwidths respectively, is the filterwidth and the filter-kernel. We arrive at a more convenient formulation after a change of coordinates This leads to: in which we put
where Here we introduced the ‘normalized skewness’ in terms of the skewness We introduce N-th order filters by requiring:
in which denotes the Kronecker delta. These filters have the property that for any polynomial P of order N – 1. Application of this filter yields:
This shows that the leading order term of scales with denotes the derivative of and we introduced
Here
22
BERNARD J. GEURTS
If we turn to the decomposition of a typical term
one finds for the commutator
in which the dots denote higher order terms in This commutator is written in the usual way in terms of higher order derivatives of where now a factor appears. The traditional commutator can likewise be expressed as:
The scaling with is readily verified for N > 1. In case N = 1 lowest order term in the summation equals 0 since and commutator scales with with contributions from the term in summation and an additional contribution from Combining these expressions one finds the leading order behavior of flux terms as:
the the the the
where A, B, and are smooth functions containing combinations of derivatives of the solution We infer that a constant filter-width implies and the leading order term of the turbulent stress tensor equals For non-uniform filters which is the same leading order behavior as for Hence, unlike findings in [19, 22] it is not possible to remove the commutators by a careful selection of the filter alone. In fact, all filters that would reduce are of higher order and consequently will also reduce [L, S] with the same order. A more detailed analysis of the commutation error can be obtained in a single-wave analysis in which we assume a solution For a symmetric top-hat filter we find explicitly
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
23
where the characteristic flux function C is given by
The two contributions to the flux in eq. (27) have a ‘weight’ and are sufficiently respectively from which we infer that if variations in slow, i.e. then filter-width non-uniformity can be disregarded. In general, for second order filters one may show that
and again is required for the dynamics of a structure with wave number not to be significantly altered, due to filter-width irregularities. The dynamic effect of the commutation error is related to the sign of which is consistent with the qualitative picture sketched before. To reduce the effect of these terms should remain small. This is difficult to maintain in boundary layers. In a priori estimates using DNS of temporal boundary layer flow [10] it appeared that close to solid walls the flux contribution is about half as large as that arising from from the commutators [L, S]. Hence, one cannot avoid modeling the commutation error near walls since at high Reynolds numbers these are automatically associated with strong grid clustering. If, for efficiency reasons or due to e.g. wall-proximity, a smooth variation of is not possible, one has to resort to modeling of In [10] a high correlation of with (generalized) similarity models was observed which suggests efficient and accurate ways to model these contributions. We which comply with the main list a number of ways to parameterize dynamic features of the non-commutation terms as described above. Similarity approach: In this case one uses as model:
Instead of the filtered solution one may also use an (high-order) approximate inversion on the right-hand side of eq. (30). Taylor expansion: Using the notation expansions of the commutator based on an N-th order filter yield [11]:
For the nonlinearity that arises in the convective flux, i.e. in combination with second order filters one effectively finds
with
and A coefficients.
24
BERNARD J. GEURTS Dynamic modeling: one may use the fact that satisfies a Germano identity, and that it is composed of two linear operators. This may allow to adapt locally, constants that emerge in base models. and the local explicitly filtering or inverting: depending on sign flow direction one may introduce additional local smoothing or decon volution in order to account for changes in the local filter-width.
These models are presently under investigation in a study of turbulent mixing layers and will be published elsewhere. In the next section we turn to the combined effect of adding a subgrid model and performing a simulation at marginal resolution. We will investigate grid-independency of LES of a Smagorinsky fluid at different filter-widths and resolutions. 4. Contamination of a Smagorinsky fluid due to numerical errors The role of numerical methods in relation to LES has not yet been suffi ciently clarified to identify unambiguously whether the accuracy of predic tions is restricted because of shortcomings in the subgrid model or whether this is due to spatial discretization effects. This also makes it difficult to find the best strategy of employing computational resources in an LES. On one hand it may be attempted to achieve a grid-independent solution of the modeled LES equations at fixed filter-width and only assign computa tional resources to eliminate numerical errors. On the other hand one may increase the physical detail that potentially is contained in the numerical solution, and allow for a reduction of the filter-width while simultaneously reducing the mesh-size which should asymptotically give rise to a direct numerical simulation.
In principle a grid-independent solution can be obtained in either strategy.
How this convergence arises, which of the two strategies is most likely to
yield grid-independence and in what way this is influenced by the values
of and will be illustrated here, following [15]. For this purpose we consider flow in a temporally developing mixing layer at a convective Mach number of 0.2 and Reynolds number of 50, following [23]. Visualization of the DNS data demonstrates the roll-up of the fundamental instability and successive pairings. After two pairings a complex 3D flow field emerges (figure 3), with many regions of positive spanwise vorticity. The large-eddy simulations have been started from a suitably filtered and grid points. Resolutions of injected DNS field which was available on and and a variety of filter-widths and ratios were chosen in the LES. The Smagorinsky model with model parameter was used. We start the simulation from a filtered DNS field corresponding
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
25
to and focus on a single pairing process from two developed spanwise rollers at to a single one around In order to quantify the numerical effects we monitor two flow properties.
26
BERNARD J. GEURTS
First, the resolved kinetic energy,
This quantity is representative of mean flow properties. To obtain an indi cation of the sensitivity of predicted properties of small-scale flow features, we also consider the spectrum at We use a filter-width (case A) and (case B) where L denotes the box-length, at reso lutions Refining the grid at constant filter-width
In case A we adopted a small filter-width and use 3/2, 2, 3 48, 64 and 96 respectively. The variations of E corresponding to for exhibit a systematic convergence towards the results at displays too rapid temporal variations The solution at which do not appear to be related to the single pairing process which oc and In case B we use and a curs between correspondingly smoother solution arises that can be better resolved with the available grids compared to case A. The inter-grid variations in the decay of the kinetic energy are roughly a factor two smaller. The spectra of the velocity fluctuations are shown in figure 4. We observe strong inter-grid This indicates that variations, even at low wavenumber at even the larger scales are strongly contaminated at the lower resolutions in case A. This is a clear indication of the effects of discretization errors. Even though the filterwidth is constant and the physical subgrid-activity is the same in all these simulations, the resolved scales are different at different resolutions. The accumulated effects of discretization errors in the resulting computational dynamical system are particularly clear at this small value of The spectra corresponding to the case are quite close to each other for each of the considered resolutions. Variations in the tail of the spectra are also smaller as is larger and a wider range of scales is captured numerically reliably. It appears that gives rise to at least qualitative agreement of both mean and fluctuating properties of the flow while a larger value of up to 6 seems required to more or less remove the discretization error from the computed fluctuating properties of this flow. Refining the grid at constant filter-width to mesh-spacing ratio
We next consider simulation results at constant In this strategy we reduce (and consequently also ), so that not much subgrid activity has to be modeled and the burden lies with the resolution and the ‘proximity’ of a DNS.
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
27
In figure 5 we show the decay of E at together with the cor responding unfiltered DNS-results. The predictions on the coarser grids at do not appear to approach the limit in a systematic manner. However, the proximity of a DNS renders the results for E at quite acceptable, despite the numerical contamination. This fortuitous situation of course does not apply in more demanding simulations of realistic we observe a systematic approach towards the turbulent flow. At results of a DNS. The larger filter-width at the coarser resolutions, gives
28
BERNARD J. GEURTS
rise to a sizeable difference with the unfiltered DNS findings.
The corresponding spectrum at is shown in figure 6. We ob serve that both the large and the small scales are changed considerably by changes in the resolution using whereas a clearer convergence
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
29
towards the unfiltered DNS spectrum can be observed for If we compare the convergence of the spectra for intermediate values of we observe that is to be preferred.
These findings indicate that discrepancies between LES and filtered DNS arise mainly from shortcomings of the model and from numerical discretiza tion on a relatively coarse grid. In LES these sources of error interact which complicates testing. By combining (filtered) DNS results with LES at dif
30
BERNARD J. GEURTS
ferent resolutions and fixed we can arrive at an approximate separation of the effects of modeling and discretization error [3, 14]. An analysis of indicates that the modeling and discretization errors if whereas if then is even larger than at Moreover, the discretization – and modeling error effect have opposite sign for the kinetic energy, which implies that the discretization error assists the subgrid-model: the total error in E is considerably smaller than the modeling error [14]. This issue is presently studied in more detail and will be published elsewhere. 5. Concluding remarks
In this paper we reviewed three issues in LES; subgrid modeling guided by rigorous properties of the turbulent stress tensor, extension of LES to complex flows with non-uniform filter-widths and the contamination of LES predictions due to spatial discretization errors at marginal resolution. All three issues are under considerable development at present and topic of ongoing research. We compile a list of ‘good practices’ in the study and development of LES, in this section. The development of rigorous modeling guidelines, related to mathematical properties of the turbulent stress tensor has proven to be very helpful in LES, of which Germano’s identity and dynamic modeling is a prominent ex ample. We indicated the use of realizability conditions in subgrid modeling and arrived at generalized similarity models using approximate inversion. These models show good results in LES of a spatially developing mixing layer, provided the resolution is adequate. The similarity modeling was also put forward to directly parameterize the commutation error that arises in LES using non-uniform filter-widths. We estimated the commutation error and showed that it is a priori of equal order of magnitude as the turbulent stress tensor. It is not possible to re move the commutation error by a special filter, since such filters also reduce the dynamics of a structure with the turbulent stress tensor. If wave number are not significantly altered due to filter-width irregularities. Finally, we addressed the effects of numerical method in LES of a Smagorin sky fluid. We considered two different strategies. The first one is commonly used in literature and implies that is fixed. In the second one at tempts to obtain the solution to the modeled LES equations and allows to increase. Numerically reliable solutions can be obtained, in partic This does not imply that the solution is also ular at larger values of physically reliable since the adopted subgrid model may be significantly flawed. It does, however, imply that one may address the issues of model ing adequacy and numerical effects separately. The elimination of numer
HOW CAN WE MAKE LES TO FULFILL ITS PROMISE ?
31
ical effects strongly increases computational effort. We observed that the memory-usage roughly scales with and the CPU-time scales closer to The findings and experience gathered are finally summarized in the form of a number of LES-guidelines. Obviously, these guidelines will develop with time, but at present they appear to give a workable approach to systemat ically develop LES. Validate the code against simpler theory and DNS databases.
Use smoothly varying near-orthogonal grids and avoid dissipative nu
merical methods.
Vary numerical and physical parameters (grid, computational box size,
numerical method etc.).
Use dynamic modeling or realizable base-models (mixed models in
volving a generalized similarity part are recommended).
Incorporate LES predictions at different
and different resolutions into the flow analysis and do a cross-comparison of the findings to assess sensitivity. References 1. Vreman A.W., Geurts B.J., Kuerten J.G.M.: 1994 ‘Realizability conditions for the turbulent stress tensor in large eddy simulation’ J.Fluid Mech. 278, 351 2. Germano, M.: 1992 ‘Turbulence: the filtering approach’ J. Fluid Mech. 238, 325 3. Geurts B.J.: 1999 ‘Balancing errors in LES’ Proceedings Direct and Large-Eddy sim ulation III: Cambridge. Eds: Sandham N.D., Yoke P.R., Kleiser L., Kluwer Academic Publishers, 1-12 4. Geurts, B.J.: 1997 ‘Inverse modeling for large-eddy simulation’ Phys. of Fluids 9, 3585 5. Domarakzki, J.A., Saiki, E.M.: 1997 ‘A subgrid-scale model based on the estimation of unresolved scales of turbulence’ Phys. of Fluids 9, 1 6. Stolz, S., Adams, N.A.: 1999 ‘An approximate deconvolution procedure for large-eddy simulation’, Phys. Fluids ]11, 1699 7. Bardina, J., Ferziger, J.H., Reynolds, W.C.: 1983 ‘Improved turbulence models based on large eddy simulations of homogeneous incompressible turbulence’ Stanford Uni versity, Report TF-19 8. de Bruin, I.C.C., Geurts, B.J., Kuerten, J.G.M.: 1999 ‘Direct numerical simulation of the spatially developing turbulent mixing layer’ Proceedings TSFP1, September 1999, Eds: Banerjee S., Eaton J.K. Begell House Inc: 615 9. Ghosal, S.: 1999 ‘Mathematical and physical constraints on large-eddy simulation of turbulence’ AIAA J. 37, 425 10. Geurts B.J., Vreman A.W., Kuerten J.G.M.: 1994. ‘Comparison of DNS and LES of transitional and turbulent compressible flow: flat plate and mixing layer’ Proceed ings 74th Fluid Dynamics Panel and Symposium on Application of DNS and LES to transition and turbulence, Crete, AGARD Conf. Proceedings 551, 51 11. Geurts B.J., Vreman A.W., Kuerten J.G.M., Van Buuren R.: 1997 ‘Non-commuting filters and dynamic modeling for LES of turbulent compressible flow in 3D shear layers’ Proceedings Direct and Large-Eddy simulation II: Grenoble. Eds: Voke P.R., Kleiser L., Chollet J.P., Kluwer Academic Publishers, 47
32
BERNARD J. GEURTS
12. Germano, M.: 2001 ‘On the physical effects of variable filtering lengths and times in LES’, Proceedings EUROMECH-412 (This volume), Kluwer Academic Publishers 13. Ghosal, S.: 1996 ‘An analysis of numerical errors in large-eddy simulations of tur bulence’ J. Comp. Phys. 125, 187 14. Vreman A.W., Geurts B.J., Kuerten J.G.M.: 1996 ‘Comparison of numerical schemes in Large Eddy Simulation of the temporal mixing layer’ Int. J. Num. Meth. in Fluids 22, 297 15. Geurts, B.J., Fröhlich, J.: 2001 ‘Numerical effects contaminating LES; a mixed story’, Modern simulation strategies for turbulent flow: Ed: B.J. Geurts, Edwards Pub lishing, 309 16. Schumann, U.: 1977, ‘Realizability of Reynolds-stress turbulence models’ Phys. of Fluids 20, 721 17. Germano, M., Piomelli U., Moin P., Cabot W.H.: 1991 ‘A dynamic subgrid-scale eddy viscosity model’ Phys.of Fluids 3, 1760 18. Lilly, D.K.: 1992 ‘A proposed modification of the Germano subgrid-scale closure method’, Phys. of Fluids A 4, 633 19. Vasilyev, O.V., Lund, T.S., Moin, P.: 1998 ‘A general class of commutative filters for LES in complex geometries’ J. Comp. Physics 146, 82 20. Kuerten J.G.M., Geurts B.J.: 1999 ‘Dynamic inverse modeling in LES of the tem poral mixing layer’ Proceedings Second AFOSR Conference on DNS and LES, 359 21. Geurts, B.J., Sarkar, S.: 2000 ‘Rapid and slow contributions to the turbulent stress tensor and their inverse modeling in a turbulent mixing layer’ Proceedings European Turbulence Conference 8, Barcelona 22. Van der Ven, H.: 1995 ‘A family of large eddy simulation filters with nonuniform filter widths’ Phys. of Fluids 7, 1171 23. Vreman A.W., Geurts B.J., Kuerten J.G.M.: 1997 ‘Large-eddy simulation of the turbulent mixing layer’ J. Fluid Mech. 339, 357
THE APPROXIMATE DECONVOLUTION MODEL FOR COMPRESSIBLE FLOWS: ISOTROPIC TURBULENCE AND SHOCK-BOUNDARY-LAYER INTERACTION
S. STOLZ, N. A. ADAMS & L. KLEISER ETH Zürich, Institute of Fluid Dynamics, CH-8092 Zürich, Switzerland Abstract. A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation of flows in complex geometries is detailed and applied to compressible turbulent flows. The paper considers two different issues. First, we study the feasibility of low-order schemes with ADM for large-eddy simulation. As test case compressible decaying isotropic turbulence is considered. Results ob tained with low-order finite difference schemes and a pseudospectral scheme are compared with filtered well-resolved direct numerical simulation (DNS) data. It is found that even for low-order schemes very good results can be obtained if the cutoff wavenumber of the filter is adjusted to the modified wavenumber of the differentiation scheme. Second, we consider the application of ADM to largeeddy simulation of the turbulent supersonic boundary layer along a compression ramp, which exhibits considerable physical complexity due to the interaction of shock, separation, and turbulence in an ambient inhomogeneous shear flow. The results compare very well with filtered DNS data and the filtered shock solution is correctly predicted by the ADM procedure, demonstrating that turbulent and non-turbulent subgrid-scales are properly modeled. We found that a computationally expensive shock-capturing technique as used in the DNS was not necessary for stable integration with the LES.
1. Introduction Recently we have proposed a new model for subgrid-scale closure in large-eddy simulations (LES) based on a deconvolution of the filtered solution (Stolz and Adams, 1999; Stolz et al., 2001a, 2001b). With the ADM approach an approxi mation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly com 33 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 33-47. © 2002 Kluwer Academic Publishers. Printed in the Netherlands,
34
S. STOLZ, N. A. ADAMS & L. KLEISER
pute subgrid-scale closures. The effect of non-represented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. The formulation of ADM for compressible flow in complex geometries is de tailed in section 2. We refer to Stolz et al. (2001a; 2001b) for the construction of suitable filters on non-uniform meshes and a description of the dynamic estima tion procedure for the relaxation parameter. In section 3 we assess the feasibility of low-order discretization schemes for LES with ADM. The performance of ADM is assessed in section 4 for turbulent shock-boundary-layer interaction.
2. The compressible conservation equations and the approximate deconvolution model For simplicity we perform the subsequent analysis with a generic one-dimensional transport equation for which represents the continuity, momentum and energy equation, respectively:
In this equation the corresponding functional expressions for the flux are to be used, and additional terms of the flux divergence have to be added for higher dimensions. On a domain we define a primary filter operation with compact sup port by
where and is the constant filter width in the computa is tional space. The constant grid spacing in the computational space of the primary filter is and the cut-off wavenumber, non-dimensionalized with is a filter kernel with compact support which may depend explicitly on For a three-dimensional domain, the filter is applied in each direction succes sively. Applying the filter operator (2) to the one-dimensional transport equation (1) and collecting non-closed terms as subgrid-scale terms on the right-hand side gives for the filtered generic transport equation
Note that for filter kernels G with smooth Fourier transforms (transfer functions) the definition of the non-dimensional cut-off wavenumber is to some . Wavenumextent arbitrary. Here we choose generally the criterion bers close to are, however, not well represented by the numerical discretization. For finite-difference schemes the relevant error-measure is the modified wavenum ber (Vichnevetsky and Bowles, 1982; Lele, 1992) which for the resolved
ADM FOR COMPRESSIBLE FLOWS
35
scales should be close to For these reasons should be chosen smaller and the range of wavenumbers can be used to model the than on the resolved scales Resolved effect of the non-resolved scales scales can be recovered by an approximate inversion of the filter (2) resulting in an approximation of the unfiltered solution . the filtered flux term Using the approximate deconvolution of can be approximated directly by replacing the unfiltered quantities in the filtered flux term with
The approximate deconvolution convolution operator to
is given by applying the approximate de-
Assuming that the filter G has an inverse, the inverse operator can be expanded as
an infinite series of filter operators. Filters with compact transfer functions are noninvertible, but a regularized inverse operator can be obtained by truncating the series at some N, obtaining a regularized approximation (Stolz and Adams, 1999) of
where I is the identity operator (Stolz and Adams, 1999; Stolz et al., 2001a). For all tested applications, we find that truncating (6) at N = 3 gives already acceptable results, and that choosing N larger than 5 does not improve the results significantly. In the LES we therefore set N = 5. Using (6), can be computed accordingly by repeated filtering of Resolved wavenumbers contain physically valuable information and The underlying prin need to be recovered accurately from the filtered solution ciple of ADM is illustrated in Fig. 1. If we consider for simplicity a one-dimensional homogeneous primary-filter kernel then its Fourier transform (transfer function), being a function of the nondimensional wavenumber is written as The Fourier transforms of the approximate-deconvolution operator (6) for N = 5, and of the filter kernel used for an equidistant mesh is shown in Fig. 1. is positive semi-definite and bounded by as can be seen from is a measure of the approximation error. Exact eq. (6). The product of and which is well satisfied for see inversion is achieved where Fig. 1. The effect of non-represented scales on the resolved scales cannot be in the flux term. In order to modeled by just replacing the unfiltered with to scales energy is drained model the energy transfer from scales by subtracting a relaxation term from the range from the right-hand side of equation (3) which gives, using (4): with
36
S. STOLZ, N. A. ADAMS & L. KLEISER
Since is positive semi-definite, the relaxation term is purely dissipative. The use of the relaxation term can also be interpreted as applying a secondary filter to every time steps, being the time-step size of the numerical integration, where the secondary filter is generated from the primary filter by defining its kernel as The transfer function of such a secondary filter is shown in Fig. 1. According to eq. (6), this secondary filter is of order in terms of the primary filter width is the order of the primary filter The transfer function of the secondary filter is close to unity in which ensures that resolved scales are practically unaffected by secondary filtering. The relaxation regularization employing the secondary filter is substantially dif ferent from an eddy-viscosity regularization and due to the secondary-filter prop We found that erties it affects practically only the range of scales the effect of the relaxation regularization is rather insensitive to the relaxation parameter see also Stolz et al., 2001a. The parameter can be determined dynamically from the instantaneous filtered solution (Stolz et al., 2001b), removing the arbitrariness of an a priori parameter choice.
3. ADM employed with low-order finite-difference schemes for com pressible isotropic decaying turbulence Industrial CFD codes commonly employ low-order discretization schemes such as order finite differences. So far we have used ADM with high-order compact finite difference schemes or spectral schemes with good results for compressible decaying isotropic turbulence (Stolz and Adams, 1999; Stolz et al., 2001b), in compressible turbulent channel flow (Stolz et al., 2001a), and shock-turbulentboundary-layer interaction, see section 4 or Stolz et al., 2001b. In order to assess the feasibility of ADM for implementation in industrial CFD codes, we study the effect of the spatial discretization order on the results of ADM in this section.
ADM FOR COMPRESSIBLE FLOWS
37
As test case we consider compressible decaying isotropic turbulence. We have performed DNS and LES for case 9 of spbl96 and use the same notation and ref erence quantities. Spyropoulos and Blaisdell provide data from LES computations grid points and DNS with using the dynamic Smagorinsky model with grid points. Our DNS data and the data of Spyropoulos and Blaisdell, 1996, differ slightly due to a different initial random seed and our higher numerical resolution. The three-dimensional initial energy spectrum for the simulations is of the form
and the initial parameters of the simulations are and where is the root-mean-square value of the velocity fluctua tions, the speed of sound and the dissipation rate. The density, of the initial flow field is constant. We solve the full Navier-Stokes equations in conservation form. The resolu tion for the pseudospectral DNS is points and it is varied between and points for the LES. An implicit filter with an adjustable cutoff wavenumber and being the grid spacing, and a deconvolution order of N = 5 is used. Combinations of numerical resolution and cutoff wavenumber used for simulations are indicated by a ” × ” in table 1. LES (normalized by the results obtained with either constant cutoff wavenumber grid spacing) or constant cutoff wavenumber for differentiation schemes of various orders are compared with filtered DNS data. For time advancement a order Runge-Kutta scheme is employed. For spatial discretization we use a pseudospectral discretization scheme, and or oder central finite differences. Figure 2(a) shows the modified wavenumber for these differentiation schemes. The solid line corresponds to spectral or exact dif ferentiation. The results for LES with the Padé scheme, which was used, e. g., for the simulation of the shock-turbulent-boundary-layer interaction, differ only marginally from those obtained with spectral differentiation. For this reason we do not show the results for the Padé scheme here. Since scales up to the cutoff wavenumber are to be well resolved by the , the differentiation scheme, we choose the cutoff wavenumber such that is small. For example, given error between exact and modified wavenumber at the fact that the difference between and is approximately the same for the for the order scheme at and for the order order Padé scheme at the appropriate choice of the cutoff wavenumber is for the scheme at order Padé scheme, for the order scheme, and for spectral and the
38
S. STOLZ, N. A. ADAMS & L. KLEISER
the
order scheme. The transfer function of the filters with is shown in figure 2(b). First we use the same resolution and filter cutoff wavenumber for the pseu dospectral LES and for the LES with the and order differentiation scheme. The corresponding results are shown as curves in figure 3, while filtered DNS data are displayed as symbols. Figure 3(a) shows the turbulent kinetic energy and figure 3(b) the root-mean-square density fluctuations as a function of the time normalized with the initial eddy-turn-over time These quantities are overpre order scheme by almost 10%, whereas the order scheme gives dicted by the almost as good results as the spectral scheme for these quantities. For the results shown in figure 4 the filter cutoff is kept constant. The di mensionless cutoff wavenumber is adjusted to the modified wavenumber by varying and accordingly. The filtered DNS data the numerical resolution between for these different filters almost coincide. Also the turbulent kinetic energy and root-mean-square density fluctuations obtained with the different discretization schemes coincide for these cases, and are in very good agreement with the filtered DNS data. This study demonstrates that low-order differentiation schemes employed with ADM give good results, provided that the cutoff wavenumber is adjusted such that scales up to the cutoff wavenumber are well resolved by the differentiation scheme, which requires a proper adjustment of the number of grid points.
4. ADM applied to shock-turbulent-boundary-layer interaction The turbulent boundary layer along a compression ramp with a deflection angle at a free-stream Mach number of M = 3 and a Reynolds number of of 1685, based on free-stream quantities and mean momentum thickness at inflow, is computed by LES with ADM as detailed in section 2. A sketch of the compression ramp flow configuration is shown in Fig. 5. The oncoming turbulent boundary layer is deflected at the compression corner, and for sufficiently large deflection angles the adverse pressure gradient due to the shock can cause the boundary layer to separate. The resolution of the LES is 334 × 31 × 91, whereas 1001 × 81 × 181 grid points were used for the DNS. The LES resolution was determined by the requirement to accurately predict the skin friction. A good prediction of statistically averaged mean flow quantities and correlations can be achieved with even fewer grid points. Unlike for the DNS, flow discontinuities are explicitly filtered in the LES, and the filtered discontinuities can be resolved on the mesh. Accordingly, no shock-capturing scheme is needed. Instead of the high-order compact-upwind-ENO scheme (Adams and Shariff, 1996) employed in the DNS, a symmetric 6th order (at inner points) compact finite-difference scheme (Lele, 1992) is used for the spatial discretization in the LES in order to avoid that artificial dissipation is introduced by the numerical scheme. For symmetric finite-difference schemes the imaginary part of the modified wavenumber which is associated with the dissipative er rors, vanishes (Lele, 1992). Since the computational cost of the symmetric compact
ADM FOR COMPRESSIBLE FLOWS
39
40
S. STOLZ, N. A. ADAMS & L. KLEISER
scheme is less than that of the hybrid compact-upwind-ENO scheme, the compu tational cost of the subgrid-scale approximation procedure is more than compen sated, and the computational effort for simulations with ADM is approximately as large as for an underresolved DNS with the compact-upwind-ENO scheme on the same coarse grid. In comparison with the well-resolved DNS at the same flow parameters performed by Adams, 2000, the computational effort for the LES is reduced approximately by a factor of 30. The conservation equations for filtered density, filtered momentum, and fil tered total energy are solved in curvilinear coordinates, see also Adams, 1998. For time advancement an explicit low-storage 3rd-order Runge-Kutta scheme is used (Williamson, 1980). At the inflow we prescribe all filtered variables in time, using the same but filtered inflow data as for the DNS, which were obtained from a separate flat plate boundary layer DNS. At the outflow a sponge-layer technique (Adams, 1998) is used and at the upper truncation plane free-stream conditions are imposed. The wall is assumed to be isothermal, and no-slip conditions are imposed on the velocity. The mapping of the computational domain onto the physical domain is nonconformal and the orthogonal coordinates in the computational domain are mapped onto non-orthogonal coordinates in the physical domain. The mapping consists of with a uniformly two steps: (1) the computational domain spaced orthogonal partitioning is mapped onto an intermediate space with non uniform but still orthogonal partitioning; (2) the intermediate space is mapped (Adams, 1998). The metric terms are onto the physical space calculated with the same finite-difference scheme which is used in the simulation. An overview of the instantaneous flow behavior is provided by an imitated Schlieren visualization of the instantaneous density-gradient magnitude averaged over the spanwise coordinate in Fig. 6. As can be seen in Fig. 6(b), the largescale behavior across the interaction is similar to that of the DNS (Adams, 2000), compare Fig. 6(c). The shock foot is the origin of a high density-gradient interface at the boundary-layer edge, which extends downstream. For comparison we also show the corresponding imitated Schlieren visualization for a mean flow field,
ADM FOR COMPRESSIBLE FLOWS
41
averaged in time over 370 in Fig. 6(a). To obtain the mean-flow data statistical averaging is performed in time over a period of 370 and in space over the homogeneous spanwise direction. The simulation was started with filtered DNS data and sampling was performed after an initial transient when a statistically steady state was reached. We compare the data obtained with ADM with filtered DNS data at 10 different downstream stations along computational mesh lines, as shown in Fig. 7. Station 1 is located in the oncoming boundary layer, stations 2 to 9 are clustered around the corner, and station 10 is located downstream of the corner within the recovery region of the reattached boundary layer. Figure 7 shows each 9th grid line of the curvilinear mesh used for the LES. At the different downstream stations the mean flow profiles of the mean con
42
S. STOLZ, N. A. ADAMS & L. KLEISER
travariant downstream velocity the temperature pressure are shown in Fig. 8. Angles < > indicate statistical and density . Resolved
averages, while Favre filtered quantities are denoted by quantities, denoted by we call those which are computed according to their defi nition but from filtered variables, e.g. the pressure We refer to the contravariant velocities
and
with the coordinates in computational space and in physical space. The difference between contravariant components and longitudinal components, where the velocity vector is rotated into a Cartesian system aligned with the wall, is small. The results obtained with ADM are shown with dotted lines and the filtered DNS data with solid lines. The profiles for the last stations show that the position of the shock predicted with ADM is in good agreement with the shock location of the DNS. The most difficult quantity to predict correctly is the skin friction, see Fig. 9(a). Although for the correct prediction of the mean quantities and the fluctuations a coarser mesh resolution would be sufficient a higher resolution in streamwise and wall-normal direction is needed for a correct prediction of the skin friction. The simulation with ADM gives very good results for the boundary layer ahead of the shock and also recovers the flow-separation correctly, indicated by in Fig. 9(a). The maximum error of is approximately 4 % in the reattaching boundary layer behind the shock, where the wall-normal mesh resolution is gradually decreasing due to the given mesh topology. Using the computed skin friction, the mean velocity scaled in local wall units is plotted in Figs. 10(a) and (b) for stations 1 and 10, respectively. The van Driest
ADM FOR COMPRESSIBLE FLOWS
transformed contravariant mean velocity profile
43
which is computed from
and the law of the wall are shown in Fig. 10(a) and (b). Note that the canonical law of the wall for zero-pressure-gradient boundary layer does not hold at station 10 since the mean pressure has not yet levelled out to its constant post-shock value. The surface pressure and its derivative are displayed of 4 % is observed. in Fig. 9(b). After the shock the maximum error for The derivative of the surface pressure distribution indicates a sequence of three
44
S. STOLZ, N. A. ADAMS & L. KLEISER
inflection points for the surface pressure (maximum-minimum-maximum of the derivative) in the mean-flow separated area (the first and last of which are indicated by arrows), which is one criterion of mean-flow separation (Adams, 2000). Overall, we find a very good agreement between the LES and filtered DNS data for all mean quantities including the skin friction. The Reynolds-fluctuations and the Favre-fluctuations are computed from and respectively. The maximum differences between the LES and filtered DNS results are less than 5 % for the root-meansquare of the Reynolds-fluctuations of the contravariant momentum
ADM FOR COMPRESSIBLE FLOWS
45
the Reynolds-fluctuations of the density the Favre-fluctuations of the contravariant velocity and the Favre-fluctuations of the temperature as shown in Fig. 11 for the downstream stations according to Fig. 7. For the oncoming boundary layer, see station 1 in Fig. 11, the agreement is within the statistical sampling accuracy.
5. Conclusions We have formulated the approximate deconvolution model (ADM) for the LES of compressible flows in complex geometries and have applied the model to LES of supersonic compression ramp flow using a order compact Padé scheme and to LES of compressible isotropic decaying turbulence using low-order finite difference and pseudo-spectral schemes. The model is based on an approximate deconvolution of the filtered data by a truncated series expansion of the inverse filter. Since an approximation of the unfiltered field is computed, an approximation for all SGSterms is available. A relaxation term which acts only on the represented but nonresolved scales is used to model the interaction of the resolved scales with the non-represented scales. The coefficient of the relaxation term can be estimated by a dynamic procedure from the instantaneous solution. The approximate deconvolution procedure can be applied for any underlying filter with positive transfer function and can be formulated with minor differences for both compressible and incompressible flows. Also, all filter and deconvolution operators are defined in real space and can be computed either in real or in spectral space. We point out that the model does not invoke a priori assumptions and, given the primary filter, is parameter-free except for the deconvolution order N. According to our experience N = 5 is sufficient for all flow configurations which we have investigated so far. We have demonstrated the low-order schemes which are often employed for
46
S. STOLZ, N. A. ADAMS & L. KLEISER
simulation of flows in complex geometries are feasible for ADM. Furthermore, LES for the physically complex shock-turbulent-boundary-layer interaction shows a very good agreement between LES and filtered DNS. The model also performed well in a posteriori tests for incompressible turbulent channel flow (Stolz et al., 2001a).
Acknowledgments This work was supported by the Swiss National Science Foundation. Calculations have been performed at the Swiss Center for Scientific Computing (CSCS).
ADM FOR COMPRESSIBLE FLOWS
47
References Adams, N. A.: 1998, ‘Direct numerical simulation of turbulent compression corner flow’. Theor. Comp. Fluid Dyn. 12, 109–129. Adams, N. A.: 2000, ‘Direct simulation of the turbulent boundary layer along a compres sion ramp at M = 3 and = 1685’. J. Fluid Mech. 420, 47–83. Adams, N. A. and K. Shariff: 1996, ‘A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems’. J. Comp. Phys. 127, 27–51. Lele, S. K.: 1992, ‘Compact finite difference schemes with spectral-like resolution’. J. Comp. Phys. 103, 16–42. Spyropoulos, E. T. and G. A. Blaisdell: 1996, ‘Evaluation of the dynamic model for simulations of compressible decaying isotropic turbulence’. AIAA J. 34, 990–998. Stolz, S. and N. A. Adams: 1999, ‘An Approximate Deconvolution Procedure for LargeEddy Simulation’. Phys. Fluids 11, 1699–1701. Stolz, S., N. A. Adams, and L. Kleiser: 2001a, ‘An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows’. Phys. Fluids 13, 997–1015. Stolz, S., N. A. Adams, and L. Kleiser: 2001b, ‘The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent-boundary-layer interaction’. Submitted. Vichnevetsky, R. and J. B. Bowles: 1982, Fourier Analysis of Numerical Approximations of Hyperbolic Equations. Philadelphia,PA: SIAM. Williamson, J. H.: 1980, ‘Low-storage Runge-Kutta schemes’. J. Comput. Phys. 35, 48–56.
This page intentionally left blank
A NEW MIXED MODEL BASED ON THE VELOCITY STRUCTURE FUNCTION
CHRISTOPHE BRUN AND RAINER FRIEDRICH Fachgebiet Strömungsmechanik Technische Universität München Boltzmannstr. 15, 85748 Garching Germany AND
CARLOS B. DA SILVA AND OLIVIER MÉTAIS LEGI/MOST
Institut de Mécanique de Grenoble
B.P. 53, 38041 Grenoble cedex 09, France
Abstract. We propose a new mixed model for Large Eddy-Simulation based on the 3D spatial velocity increment. This approach blends the non linear properties of the Increment model (Brun & Friedrich (2001)) with the eddy viscosity characteristics of the Structure Function model (Métais & Lesieur (1992)). The behaviour of this subgrid scale model is studied both and Large Eddy-Simulation via a priori tests of a plane jet at of a round jet at This approach allows to describe both forward and backward energy transfer encountered in transitional shear flows.
1. Introduction The velocity increment of order between two points separated by a distance is known to be an important variable for the characterization of turbulent flows as illustrated in its extensive use in theoretical, experimental and numerical turbulence studies. In particular, the Structure function of order is directly related to the kinetic energy transfer between large and small scales within the inertial range region (Frisch (1995)). This fact suggests the use of this quantity for subgrid-scale (SGS) modeling. 49 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 49-64. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
50
CHRISTOPHE BRUN ET AL.
In this article, a new mixed model based on the velocity increment, the scale similarity and the eddy viscosity concepts is proposed and analysed both by a-priori and a-posteriori tests. The eddy viscosity concept was used by Métais & Lesieur (1992) to design the Structure function model. The scale similartity ideas were employed by Brun & Friedrich (2001) in the development of the Increment model. The new mixed model presented here blends these two approaches. This article is organized as follows. The next section (2) describes briefly the numerical method and the numerical and physical parameters of the simulations to be reported here. Section 3 details the formalism of the new mixed model proposed here. The results from a-priori tests made using a Direct Numerical Simulation (DNS) of a turbulent plane jet will be analysed in section 4. Finally, a-posteriori tests for the new model carried out through Large-Eddy Simulations (LES) of round jets will be presented in section 5. 2. Numerical simulation details
2.1. NAVIER-STOKES SOLVER All simulations reported here were performed with the same code. It is a highly accurate finite difference, incompressible Navier-Stokes solver which order-compact schemes for spatial uses combined pseudo-spectral and discretization. The time advancement is made with an explicit, 3 step, order, low storage Runge-Kutta time stepping scheme. Pressure-velocity coupling is achieved via a fractional step method that insures incompress ibility at each sub-step of the Runge-Kutta time advancing scheme. The in let condition is made by prescribing a velocity profile at each time step. This inlet velocity is composed of a mean velocity profile plus a three-component fluctuating velocity with prescribed spectra with statistical characteristics of isotropic turbulence. The outlet condition is of a non-reflective type. The high precision of the numerical methods used here implies very low numer ical diffusion allowing an accurate study of subgrid-scale model effects. 2.2. DIRECT AND LARGE-EDDY SIMULATIONS The a-priori tests were conducted using a DNS of a turbulent plane jet at Reynolds number The total number of grid points is about 2 million, which allowed for a domain box of where is the inlet slot-width. This simulation was extensivley validated by da Silva & Métais (2001a), and accurately represents the flow in the far field of a fully turbulent plane jet. The a-posteriori tests were carried out using two LES of turbulent round The computational domain con jets at Reynolds number
A NEW MIXED MODEL
51
tained about 2 million points and had a spatial extent of 12D × 6D × 6D, where D is the diameter of the jet inlet nozzle. The first LES is used as a reference case and was performed by da Silva & Métais (2001b) using the Filtered Structure Function model (Lesieur & Métais (1996)). The second LES was carried out using the new mixed model proposed here. 3. SGS modeling based on the spatial velocity increment 3.1. THE STRUCTURE FUNCTION MODEL Basic formulation :
In the framework of the eddy-viscosity concept, a scale analysis yields
is the mesh size). Métais & Lesieur the eddy-viscosity
(1992) have adopted this approach to develop the Structure Function model (SF):
where
is the subgrid-scale (SGS) stress tensor, is the strain rate tensor, is a model constant and represents the filtered velocity field, explicitly computed in LES. Mean shear effect : A more efficient version of this model is the Filtered Structure Func tion model (FSF) (Ducros, Comte & Lesieur (1996)). This variant was designed to remove large scale effects from the original SF formulation. da Silva (2001) showed that this works particulary well in the transi tion region, as the model succeeds in displaying realistic (almost zero) values of the turbulent viscosity in the early stages of the transition region. This is achieved in practice by the use of a high pass filter, in the form of a three times iterated Laplacian, to filter out the resolved field before computing the velocity structure function :
low Reynolds number effects : In order to take into account low Reynolds number effects we propose
to integrate the molecular viscosity effect that has been neglected in the
52
CHRISTOPHE BRUN ET AL. original formulation of the eddy-viscosity by Métais & Lesieur (1992). In spectral space the eddy-viscosity is written as :
where is the three-dimensional energy spectrum in isotropic tur bulence and is the relaxation time given by the EDQNM theory (Lesieur (1997)) :
In equation (4) the role of the molecular viscosity terms is to partially inhibit the nonlinear terms In physical space this results in a damping effect that we account for in a ‘viscous’ structure function model (VSF) (see Brun et al. (1997), for details). The final turbulent viscosity is given by,
with and Note that this damping is not an ad hoc function as in the case of the Smagorinsky-Van Driest model. In the low-Reynolds number asymptotic case one gets the eddy-viscosity,
This adjustment is expected to be of critical importance in transitional flows, for which the energy spectrum has a slope steeper than the classical value. This formulation constitutes a hybrid model between the eddy-viscosity structure function concept (equation (1)) and the nonlinear Increment concept (equation (8)) presented below, since the complete SGS model, approximated by equation (6) is written in the low Reynolds number limit as,
3.2. THE INCREMENT MODEL A scale similarity approach based on the velocity increment yields the fol lowing anisotropic nonlinear formulation, referred to hereafter as the Incre ment model (INC) (Brun & Friedrich (2001)):
A NEW MIXED MODEL
53
An interesting aspect of the Increment model is that it has a similar struc tural form as the SGS tensor developed through a Taylor expansion. One may write any filtered variable as a 1D expansion of its unfiltered value (Leonard (1974); Pruett & Adams (1998); Winckelmans, Wray & Vasilyev (1998); Brun & Friedrich (2001)). The fourth order expansion is :
where and are respectively the and moments of the filter F. Introducing the expansion above into the SGS stress tensor yields :
The leading term of equation (11) is of second order with respect to (Pruett & Adams (1998)). The lower order terms vanish because of the A truncated Taylor expansion can be used to Galilean invariance of express the gradient of the velocity field through the increment of in the interval
where the maximal error is Combining relations (11) and (12) we get the final approximation for the SGS tensor :
This expression is exact in the limit which corresponds to the case
of a well resolved DNS. The use of a coarse grid, as it is common in LES,
increases the contribution of the higher order terms in equation (13), but
54
CHRISTOPHE BRUN ET AL.
preserves the increment form of the SGS stress tensor which is therefore a strong basis for SGS modeling. 3.3. PROPOSAL OF A MIXED MODEL Following the previous discussion, it is proposed here to blend the two concepts described above. This is achieved through a new mixed model the form of which will now be detailed. We propose to model the SGS stress tensor as,
This approach is justified by the work of Shao, Sarkar & Pantano (1999). They applied the Reynolds decomposition to the SGS tensor and showed that this quantity can be split into two parts,
The slow part is totally independent of the mean-flow gradients. These are accounted for in the rapid part only. They further demonstrated that the slow part can be accurately modeled with an eddy-viscosity concept whereas the rapid part should be modeled using a scale similarity approach. Apriori tests by da Silva (2001) confirmed the existence of a good correlation between the real and modeled slow parts of the SGS stress tensor, when modeled with any model from the Structure Function family. Since the slow part aims at representing the turbulent dynamics which is independent of the mean flow inhomogeneities, it is consistent to model this part using the eddy viscosity concept of the Structure Function model,
where the constant remains the same as the one used in the basic Structure Function model which is obtained considering the inertial range of an isotropic turbulence energy spectrum i.e. The rapid part is modeled using the Increment concept ideas,
where the model coefficient is determined locally in space and time fol
lowing the classical dynamical approach applied to the generalized Leonard
stress tensor written in a Galilean invariant form (Brun & Friedrich (2001)),
A NEW MIXED MODEL
55
The test filter required here has twice the size of the implicit filter of the computational mesh. The trace of equation (15) yields the coefficient based on SGS kinetic energy considerations (Brun & Friedrich (2001)).
56
CHRISTOPHE BRUN ET AL.
4. Results from a-priori tests: DNS of a plane jet at A DNS of a plane jet at Reynolds number (da Silva & Métais (2001a)) was used as data bank to analyse the capabilities of the new model in a-priori tests. Figure 1 a) shows isosurfaces of low pressure in the lower shear layer corresponding to this simulation. The picture shows two strong Kelvin-Helmholtz vortices centered at about and Between these structures several streamwise vor tex pairs develop. One of these streamwise vortices is also clearly seen in the figure. The real stress is shown in figure 1 b). The same stress component modeled with the Increment model has some degree of correlation with the real ones (see 1 c)). No such correlation exists for both the Structure Func tion (1 d)) and Filtered Structure Function (1 e)) based values, which are aligned with the resolved strain rate tensor element (see figure (1 f ) ) . On the other hand, the eddy-viscosity models of the type of the Structure Function model have the best agreement with the slow part of the mean component, which does not depend on the mean flow gradients. This can be seen in figure 2. The present mixed model is designed to cope with both the above requirements.
A NEW MIXED MODEL
57
5. Results from a-posteriori tests: LES of round jets at 25000
This section analyses the results obtained with the new mixed model in an LES of a round jet at This simulation was compared with results of da Silva & Métais (2001b) using the Filtered Structure Function model in the same configuration and with the experimental data of Hussein et al. (1994), at
58
CHRISTOPHE BRUN ET AL.
A NEW MIXED MODEL
59
5.1. SELF SIMILAR REGIME
Figure 3 shows the centerline velocity decay and root-mean square (RMS) for both simulations. In the potential core of the jet, which extends six diameters downstream, the RMS velocity slowly increases (figure 3). A similar qualitative behaviour was obtained experimentally at by (Crow & Champagne (1971)). Once the annular mixing layer reaches the axis, the potential core disappears, yielding both a decrease in the axial mean velocity verifying the experimental slope from 10% with and an increase in the turbulence intensity to 30%. Qualitatively, the two models behave in the same way, although one can notice that the inverse energy transfer related to the INC model yields a slightly stronger increase in the turbulence level of the jet than pure forward energy transfer due to the FSF model. This difference increases through the transitional zone of the jet and reaches 4% at the outlet part and In this zone, the of the computational box between flow should not have yet reached the self-similar regime. Nevertheless, the mean axial velocity profile (figure 4) fits already very well the experiment of Hussein et al. (1994), for both SGS formulations. Figures 5, 6 and 7 show the second to third order moments of the ve locity fluctuations in nondimensional units. The axial RMS profile (figure 5) and the cross correlation (figure 6) reflect a qualitatively good agree ment with the experiment. Some differences are expected to be related to the early stage of the considered self-similarity. The mixed model yields a stronger turbulence intensity and cross correlation level. The use of a coflow leads to a non-standard non-dimensionalisation which complicates the in terpretation of the results. Therefore, at this stage it is not clear which LES result relects reality best. In figure 7 the skewness of the axial (left) and radial (right) velocity are compared with the self-similar profile for the mixed model. The prediction is relatively accurate and indicates the success of this approach. 5.2. TRANSITIONAL REGIME
We now compare the two SGS formulations in the transitional zone of the jet. As noticed in the previous section the mean velocity profile is not visibly affected by the choice of one or the other model (figure 8). The jet spreads in the outer zone as yielding negative values of the transverse velocity a result of the entrainment property of the flow. As expected, the influence of the model concerns the turbulence intensity which reaches already a higher level on the jet axis at (figure 9). At the end of the potential
60
CHRISTOPHE BRUN ET AL.
core the zone of more intense turbulence level spreads from the axis to the whole jet shear layer. The velocity cross correlation seems not to be affected by the use of one or the other model. While the production reflected by remains constant, the dissipation, reflected by the level of kinetic energy k, decreases. This indicates a redistribution of energy obviously due to the model. As described in the first section, the use of the INC model enhances the inverse energy transfer from small to large scales filling up the energy spectrum at low wave numbers while inhibiting the dissipation process at small scales. This, finally, yields an increase in the turbulent intensity. In such a per spective, the present mixed model appears to be closer to reality than the Filtered Structure Function model.
A NEW MIXED MODEL
61
5.3. COHERENT STRUCTURES Figure 10 shows coherent structures obtained with the Filtered Structure Function model (left) and the present mixed model (right). Isosurfaces of low pressure contours (light grey) indicate vortex rings which tend to develop into alternating vorticity structures (Urbin & Métais (1997)) at the end of the core zone. The use of the INC model to describe the rapid part of the SGS tensor shows some differences with respect to the basic FSF model. In particular, for the topology of the flow at the end of the potential core, the alternating pairing situation, obtained with the FSF model tends to modify the initial axisymmetric mode into strong axial vortical structures. With the INC model, equivalent structures appear closer to the inlet of the jet. This difference can be linked to the lack of backward energy transfer in the near field of the jet caused by the FSF model. Such a behavior of the flow, which cannot be captured with an eddy-viscosity based SGS model, is taken into account in the newly proposed mixed model. Finally, it is interesting to look at the kinetic energy exchanges between grid and subgrid-scales. This can be analysed through the transport equa tion of SGS kinetic energy (da Silva & Métais (2001a)). In particular, the instantaneous transfer of energy between grid and subgrid-scales is given
62
CHRISTOPHE BRUN ET AL.
by the quantity, This quantity represents the ‘production’ of SGS kinetic energy caused by the resolved strain rate. It is interesting to note that once modeled with the INC formulation the SGS kinetic energy is directly related to the second order structure function Positive values of indicate energy transfer from the grid to the subgrid-scales (forward scatter), whereas negative values indicate energy exchanges from the subgrid to the grid-scales (backward scatter). Härtel et al. (1994), considered the behaviour of this term as one possible criterion for the validation of a SGS model. For eddy-viscosity models (equation 1) this term of production is by definition positive and yields always forward scatter,
However, the INC model formulation yields each of the two possible and the strain energy transfers, depending on how the increment tensor rate tensor are correlated :
A NEW MIXED MODEL
63
Figure 11 shows positive (black) and negative (dark grey) values of the INC part of the SG energy flux represented by the term The figure shows that the INC model allows for both forward and backward energy transfers. The zones of energy fluxes are located around the high vorticity structures described by the Q-criterion (light grey), in the initial transition region but this correlation decreases as the flow evolves into a fully turbulent state. Both these trends correspond to the correct topology of the interscale interactions for the term as reported by da Silva & Métais (2001a). The high intensity of these transfers demonstrates the need of accounting for backscatter in transitional flows such as in the present round jet.
6. Conclusions We propose a new mixed SGS model concept based on the similarity be tween and the velocity increment tensor This similarity is analysed It reveals that the eddy by a priori tests of a plane jet at viscosity part of the form of the Structure Function family represents accu rately the small scales of the flow while the non-linear part of the form of the Increment Tensor describes well the large scale interaction. A posteri are validated against experiments. ori tests of a round jet at Comparison with a simulation using the Filtered Structure Function model alone reveals that backward energy transfer to large scales provided by the mixed model yields an increasement in the kinetic energy distribution. This effect has to be accounted for in such a transitional flow.
Acknowledgements This work is supported by the EC TMR project ‘LES of complex industrial flows’.
References HUNT, J. C. R., WRAY, A. A., MOIN, P. 1988 Eddies, streams, and convergence zones in turbulent flows Center for Turbulence Research, Annual research briefs. BORUE, V. & ORSZAG, S. A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence J. Fluid Mech. 366, 1–31. BRUN, C., KESSLER, P., COMTE, P., & LESIEUR, M. 1997 Simulation des grandes échelles de jet rond. Rapport de Synthèse DGA/DRET contrat n° 95–2557A, LEGI Grenoble. BRUN, C., & FRIEDRICH, R. 1999 A-priori tests of SGS stress models in fully developed pipe flow and a new local formulation Direct and Large-Eddy Simulation III, Kluwer Academic Publishers, P.R. Voke et al. (eds.), 249–262. BRUN, C., HÜTTL, T.J., & FRIEDRICH, R. 2000 A-posteriori tests of a new subgrid-scale model: L.E.S. of fully developed pipe flow Advances in Turbulence VIII, CIMNE, C. Dopazo et al. (ed.), 547–550.
64
CHRISTOPHE BRUN ET AL.
B RUN , C., & FRIEDRICH, R. 2001 The spatial velocity increment as a tool for SGS modeling. Modern Simulation Strategies for Turbulent Flow, R.T. Edwards Publishing House, B.J. Geurts (eds.), 57–84. COMTE, P., DUBIEF, Y., BRUN, C., MEINKE, M., SCHULZ, C., & RISTER, TH. 1998 Simulation of spatially developing plane and round jets Notes on Numerical Fluid Mechanics, Vieweg, ed. E.H. Hirschel, vol. 66, 301–318, CNRS-DFG Collaborative Research Programme, Results 1996-1998. CROW, S.C. & CHAMPAGNE, F.H. 1971 Orderly structure in jet turbulence J. Fluid Mech. 48(3), 547–591. DUCROS, F., COMTE, P. & LESIEUR, M. 1996 Large-Eddy Simulation of transition to turbulence in a boundary layer developing spatially over a flat plate J. Fluid Mech. 326, 1–36. FRISCH, U. 1995 Turbulence. The legacy of A.N. Kolmogorov Cambridge University Press. HÄRTEL, K., KLEISER, L., UNGER, F., & FRIEDRICH, R. 1994 Subgrid-scale energy transfer in the near-wall region of turbulent flows Phys. Fluids 6(9), 3131–3143. HUSSEIN, H.J., CAPP, S.P., & GEORGE, W.K. 1994 Velocity measurements in a highReynolds-number, momentum conserving, axisymmetric, turbulent jet J. Fluid Mech. 258, 31–75. KRAICHNAN, R.H. 1976 Eddy viscosity in two and three dimensions J. Atmos. Sci. 33, 1521–1536. LEONARD, A. 1974 On the energy cascade in large-eddy simulations of turbulent flows Adv. in Geophys. A18. L ESIEUR , M. 1997 Turbulence in Fluids Kluwer Academic Publishers, 3rd edition. LESIEUR, M. & MÉTAIS, O. 1996 New trends in large-eddy simulations of turbulence Annu. Rev. Fluid. Mech. 28, 45–82. MÉTAIS, O. & LESIEUR, M. 1992 Spectral large-Eddy Simulation of isotropic and stably stratified turbulence J. Fluid Mech. 239, 157–194. DA SILVA, C. B. 2001 The role of coherent structures in the control and interscale inter actions of round, plane and coaxial jets 2001 PhD thesis, INPG, Grenoble. DA SILVA, C. & MÉTAIS, O. 2001a On the influence of coherent structures upon interscale interactions in turbulent plane jets submitted to J. Fluid Mech. DA SILVA, C. & MÉTAIS, O. 2001b Coherent structures in bifurcating jets: a numerical study, to be submitted PRUETT, D. &; ADAMS, N. A. 1998 On the direct approximation of subgrid-scale stresses in large-eddy simulation Unpublished. SHAO, L., SARKAR, S., & PANTANO, C. 1999 On the relationship between the mean flow and subgrid stresses in large-eddy simulation of turbulent shear flows Phys. Fluids 11(6), 1596–1607. U RBIN , G., & MÉTAIS, O. 1997 Large-eddy simulations of the three-dimensional spa tially developing round jet Direct and Large-Eddy Simulation II, Kluwer Academic Publishers, J.P. Chollet et al. (eds.). WINCKELMANS, G. S., WRAY, A. A., & VASILYEV, O. V. 1998 A new mixed model for L.E.S.: the Leonard model supplemented by a dynamic Smagorinsky term Proc. Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ., 367–388.
ON THE EFFECT OF COHERENT STRUCTURES ON GRID/SUBGRID-SCALE INTERACTIONS IN PLANE JETS: THE TRANSITION AND FAR FIELD REGIONS
CARLOS B. DA SILVA AND OLIVIER MÉTAIS LEGI/MOST
Institut de Mécanique de Grenoble
B.P. 53, 38041 Grenoble cedex 09, France
Abstract. Terms from the transport equations of the grid-scale (GS) and subgrid-scale (SGS) kinetic energy are analyzed both instantaneously and statistically in order to understand the role of the coherent vortices in the GS/SGS interactions in free shear flows. In the end, a new light emerges on the role played by these structures in these complex processes. The implications for SGS modeling are discussed.
1. Introduction The interactions between large and small scales of motion have always been one of the key subjects of turbulence theory, but assume the most impor tant relevance in Large-Eddy Simulations (LES) where only the large scales of motion are explicitly calculated and the effect of the small scales on the large scales of motion has to be accurately modeled. Due to this nature of the problem, development of subgrid-scale models draws heavily upon the present understanding of grid/subgrid-scale (GS/SGS) relationships. An other point that begins to receive increasing attention is the influence of the coherent structures on the GS/SGS interactions (Piomelli, Yu & Adrian (1996); Horiuti (1997); O’Neil & Meneveau (1997)). Coherent structures naturally arise in many turbulent flows and are known to be responsible for most of the transport of mass and momentum, whence their high im portance in turbulence. The present study focuses on understanding the effect of coherent struc tures upon GS/SGS interactions. This problem was analyzed in great detail by da Silva & Métais (2001) in the far field of a fully developed turbulent 65
R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 65-80. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
66
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
plane jet. Here we complete the picture given there, by analyzing also the transition region. Two plane jet Direct Numerical Simulations (DNS) were carried out, upon which a box or top hat filter was applied to separate the GS from the SGS. Using these computations as data bank, each term of the transport equations for the GS and SGS kinetic energy was analyzed, both statistically and topologically to understand their relation with the flow coherent structures. Since the type of vortical structures found in plane jets are not unique to plane jets, the conclusions from the present work can be extended to a large class of free shear flows. The article organization is as follows. Section 2 reviews the main GS/SGS physical relations used as starting point for this work. Section 3 details the numerical method, physical and computational parameters used in the two simulations. The main results from the analysis of the impact of the coher ent structures upon the GS and SGS dynamics in the far field of the plane jet are presented in section 4.1. Finally, the more important results from the same analysis in the transition region of the plane jet are discussed in section 4.2. 2. Problem formulation The effect of the coherent structures upon the local GS/SGS interactions, in the physical space, will be analyzed looking into the several terms of the GS and SGS kinetic energy transport equations. The separation between the GS and SGS part of any flow variable is obtained through a spatial filtering operation defined as is a filter function of width The where is given by, transport equation for (twice) the GS kinetic energy,
and the transport equation for the SGS kinetic energy,
is,
GRID/SUBGRID-SCALE INTERACTIONS
Here
means
stress tensor, and
for convenience,
67
is the SGS the GS rate of deformation tensor.
In equation (1) terms I and II account for the total (local and convective) variation of GS kinetic energy. Terms III and IV account for the diffusion of GS kinetic energy by pressure/velocity interactions and molecular viscosity, respectively. Term V is the local GS kinetic energy dissipation due to the molecular viscosity. The terms VI and VII involve the subgrid-stress tensor and are directly related to the kinetic energy exchanges between GS and SGS. Term VI (GS/SGS diffusion) represents a redistribution of GS kinetic energy by GS velocity and SGS stresses interactions. The GS/SGS transfer (term VII), also called subgrid-scale dissipation, represents the transfer of kinetic energy between GS and SGS. If the term is a sink for the GS kinetic energy as energy flows from GS to the SGS (forward scatter). If term VII is a source and the kinetic energy flows from the SGS to the GS (backward scatter). In equation (2) terms VIII, IX represent the local and convective vari ation of the SGS kinetic energy, respectively. The diffusion caused by the local turbulence level on the SGS kinetic energy is represented by term X (SGS turbulent transport). Term XI is the SGS pressure/velocity interac tions and XII is the SGS viscous diffusion. Term XIII (SGS viscous dissi pation) represents the end of the energy cascade process where molecular viscosity finally dissipates the remaining SGS kinetic energy. It is important to notice that the terms XIV and XV are respectively the counterparts of the terms VI and VII. Since these terms appear in both equations with opposite signs, they represent the kinetic energy exchange between the GS and SGS. Terms VI and VII involve the subgrid-scale stress tensor and an inaccurate modeling of these two terms will yield an incorrect representation of the energetic exchanges between GS and SGS. 3. Numerical method and DNS parameters The code used here is a highly accurate finite difference, incompressible Navier-Stokes solver which uses combined Pseudo-spectral and order compact schemes for spatial discretization. The time advancement is made with an explicit, 3 step, order, low storage Runge-Kutta time stepping scheme. Pressure-velocity coupling is solved with a fractional step method that insures incompressibility at each sub-step of the Runge-Kutta time ad vancing scheme. The inlet condition is made by prescribing a velocity profile at each time step. This inlet velocity contains a mean part based on a hy perbolic tangent profile (Stanley & Sarkar(1999)) plus a three-component fluctuating velocity with prescribed spectra following the statistical charac teristics of isotropic turbulence. The outlet condition is of a non-reflective
68
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
type. Two simulations were carried out in this work. The first simulation (DNS1) aimed at studying the GS/SGS interactions in the far field of the fully developed turbulent plane jet. The second simulation (DNS2) deals with the transition region. In both simulations the Reynolds number based and the initial shear layer mo on the inlet slot-width was mentum thickness was chosen to have In DNS1 the maximum amplitude of the inlet noise was 10% of the mean inlet velocity. This high amplitude noise forces a quick transition to turbulence which makes the jet to attain the self-similar state before the end of the computational domain. In DNS2 the maximum noise amplitude was set to 3% and a sinusoidal forc ing with a frequency corresponding to the Strouhal number was also added to the mean and noise inlet profiles. For DNS1 the total number of grid points was (241 × 216 × 60) which allowed a domain box of size and in the streamwise, normal and spanwise directions, respectively. DNS2 was carried out with (201 × 180 × 50) points in a domain box with and To study local interactions between GS and SGS only a localized filter can be used. A box or top-hat filter was used throughout all this work to separate GS from SGS although it was checked that the Gaussian filter leads to similar results. Two filters were used, with filter widths equal to and 4. Local grid / subgrid-scale interactions A picture of the vorticity modulus for DNS1 simulation is given in figure 1. The flow appears to be quite smooth until about Between the first two Kelvin-Helmholtz vortices can be observed both in the upper and lower shear layers. In agreement with the observations of Thomas & Prakash (1991) for a near top-hat initial velocity profile, the upper and lower Kelvin-Helmholtz vortices appear first symmetrically with respect plane showing a preferential amplification of the to the centerline varicose mode, but further downstream they exhibit an asymmetrical ar rangement due to the growth of a sinuous mode. An extensive validation was carried out by da Silva & Métais (2001) for this simulation. Very good agreement was found between several one-point statistics from DNS1 and experimental values from the far field region in turbulent plane jets and other direct numerical simulations. The character istic frequencies corresponding to the shear-layer mode and the preferred mode were also recovered. The self-similar regime was found to begin at about The validation results showed also that: i) in the far field self-similar regime, the flow is statistically isotropic, as far as second order moments are concerned and ii) the computed values for the Kolmogorov
GRID/SUBGRID-SCALE INTERACTIONS
69
micro scale and Taylor micro-scale are representative of a well resolved direct numerical simulation. Finally, in the far field region, the time spectra of the streamwise velocity component (not shown) exhibits a well defined –5/3 slope extending over about one decade. The Taylor micro-scale based on the Reynolds number is about Therefore, the present sim ulation (DNS1) is accurate both at the large and small scale level, and representative of the far field region of the turbulent plane jet (da Silva & Métais (2001)). 4.1. FAR FIELD TURBULENT PLANE JET Figure 2 illustrates the typical structures found in the far field of the tur is shown bulent plane jet. For clarity, only the lower shear layer here. The low pressure isosurfaces show two big Kelvin-Helmholtz vortices and whereas the positive Q crite situated around rion (Hunt, Wray & Moin (1988)), shows the streamwise vortices stretched between them, as well as the high level of small scale turbulence that lies within the big rollers. This study will focus on what happens in these vor tical structures and not in vortices from the dissipative region (the worms of homogeneous isotropic turbulence). With the two filters used here the ratio of SGS to GS kinetic energy changes across the normal direction, but has values between 2-10% and and respectively. Both filters will be 5-20 % for filter used as they both lead to SGS kinetic energy fractions commonly found in this type of studies (Akhavan, Ansari, Kang & Mangiavacchi (2000)). Before analysing GS and SGS interactions it is interesting to see how the GS and SGS kinetic energies correlate. It turns out that there is no
70
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
correlation between these two quantities as their correlation coefficient is This is what is expected to happen in a well resolved LES where there is an effective separation of scales. But it is surprising to notice how well the SGS kinetic energy is correlated with the vortical structures. Figure 3 shows isosurfaces of SGS kinetic energy for filter One can see clearly that the higher values of SGS kinetic energy are concentrated in the center of the coherent structures. This is apparent from their joint PDF shown in the same figure and by their strong correlation Now, if the SGS are highly correlated with coefficient, the big coherent structures, which are clearly large scale (GS) events, this means that the small scale dynamics are being highly affected by large scale processes. The following analysis sheds some light upon this GS/SGS relationship.
Figure 4 shows profiles of averaged terms from equation (1) in the far field of the turbulent plane jet The averaged value of term I is zero everywhere since the flow is statistically stationary. Here it can be seen that, in the mean, terms II (GS kinetic energy advection), III (GS pressure/velocity interactions), VI (GS/SGS diffusion and VII (GS/SGS
GRID/SUBGRID-SCALE INTERACTIONS
71
transfer) are clearly dominant. Conversely, the terms V (GS viscous dis sipation) and IV (GS viscous diffusion) are, on the average, negligible. This is what one may expect to obtain in a well resolved LES where the effects of molecular viscosity are irrelevant for the dynamics of the GS. Again as expected term VII (GS/SGS transfer) is found to be quite sig nificant and always negative which, with the sign convention used here corresponds to a mean kinetic energy transfer from GS to SGS (forward scatter). Surprisingly, the averaged term VI (GS/SGS diffusion) displays extrema of the same order of magnitude as term VII and changes from average forward scatter in the central flow region, into mean backscat ter values at the edge of the shear layer. This term has always been as sumed to be negligible for the GS dynamics (Meneveau & Katz (2000); Akhavan, Ansari, Kang & Mangiavacchi (2000)) because it represents a diffusion term that integrates to zero over the whole domain. The small importance of this term in wall bounded flows was verified by Balaras & Piomelli (1994). A possible explanation for the high mean values of this term here could be the fact that the transport terms are much more im portant in free flows than in the case of wall bounded flows. A measure of the local intensity of all the terms from equation (1) is provided by their root-mean square (rms) values. The local intensities of terms II and III and are much larger than all the other terms. At a second level of importance come
the terms VI and VII The fact that shows unambiguously that GS/SGS diffusion (VI) has a greater local importance than the GS/SGS transfer (VII). This is
a somewhat surprising result which could not be deduced only by examining
the averaged profiles. Finally, concerning the terms related to molecular
72
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
viscosity (IV and V), one notices that they both have very small rms values
and Thus, in the mean as well as locally, the terms IV and V play a negligible role in the GS dynamics. The skewness factor S, being a measure of the asymmetry of the probability density function, shows that most of the terms exhibit a more or less symmetric behavior (–0.452 < S < –0.316). The exception comes from the terms and VII This asymmetry can be attributed to the fact that the term V is always a sink of GS kinetic energy and that forward scatter dominates over backward scatter for the term VII. The high intermittent nature of term VII can be noticed looking into figure 4 and is shown by its high flatness factor F, The figure shows the PDFs of terms VI and VII. Term VI is not very asymmetric and has a moderate intermittent level 11.97) whereas term VII has a very different shape for the negative tail (forward scatter) and the positive one (backscatter). The PDF of term VII shows that forward scatter is not only a more frequent event than backward scatter (as expected), but that backscatter acts very intermittently. We now examine how the various terms of equation (1) relate to the presence of coherent structures. Figure 5 shows an instantaneous picture of term III (GS pressure/velocity interactions). The values of this term have their stronger intensities in the vicinity of the Kelvin-Helmholtz vortices. Its positive and negative peaks are located ahead and behind (or on top and bottom) of these big rollers. It was seen (not shown) that term II (GS kinetic energy advection) has a similar behavior as shown by the corre lation coefficient It is reasonable to suppose that the GS advection (term II) will be enhanced by the local velocity induced by the presence of these big structures and that the GS pressure-velocity in teractions (term III) are increased by the conjunction of the high level of small scale turbulence and the high pressure gradient located nearby the big rollers.
GRID/SUBGRID-SCALE INTERACTIONS
73
Surprisingly it was observed that term VII (GS/SGS transfer) does not show any sign of correlation with the coherent structures, unlike what was shown by O’Neil & Meneveau (1997) and Horiuti (1997). This may be caused by the fact that the present simulation corresponds to a fully developed turbulent flow configuration. O’Neil & Meneveau (1997) had suggested this possibility. However, in a transitional flow where the level of small scale turbulence is not yet very high, very good correlation between term VII and the streamwise structures could be seen as in the above cited references (see section 4.2).
Figure 6 presents isosurfaces of term VI (GS/SGS diffusion). The figure shows that unlike term VII, term VI is clearly correlated with the pres ence of the vortical structures. Intense regions of term VI are located next to the vortex cores of the smaller more intense vortices. Notice that the correlation between terms VI and VII is only This low value shows that GS/SGS transfer and GS/SGS diffusion are not only quite different physical processes, but also take place at very different flow locations and thus seem to be participating in very different physical pro cesses. For example, as seen here, the passage of a coherent structure does not seem to influence the behavior of GS/SGS transfer (term VII), but has a big influence on the GS/SGS diffusion (term VI). Therefore, because i) term VI has greater local intensities (rms) than term VII and ii) term VI has its higher values right next to the flow vortices (whereas term VII is not correlated with these) it is fair to say that local and instantaneous flow events, such as coherent structures for instance, are likely to have a stronger impact on the GS evolution through GS/SGS diffusion (VI) rather than through GS/SGS transfer (VII). This fact has consequences for SGS modeling analysis: the performance of a given SGS model to correctly de scribe the local flow events (e.g. the flow structures) should be evaluated by comparing the modeled and the real terms VI, and not only the modeled and the real terms VII as usually done in a-priori tests.
74
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
To end this section on GS dynamics some comments are also made about the spatial localization of the remaining terms (IV,V) of equation (1), which, as shown above, are negligible both from a global (mean) and local (instantaneous) point of view. The GS viscous diffusion (term IV) was not found to be correlated with the vortical structures (not shown). On the other hand, the GS viscous dissipation (term V) exhibits some correlation with the vortical structures (not shown). It was observed that this term occurs in two locations. Inside the vortices, where there is no turbulent production (Zeman (1995)) and also in the region of high velocity gradients nearby these vortical structures. Here, the correlation coefficient between the vorticity modulus and term V is only The various terms of equation (2) will now be analyzed. The averaged profiles of all the terms (except term VIII whose averaged profile is zero) computed in the far field of the turbulent jet are shown in figure 7. As expected in an energy cascade process, we verify the mean (global) equi librium assumption, which states that all the kinetic energy arriving at the SGS due to the GS/SGS transfer (term XV - symmetric of term VII) has to be dissipated by the viscous SGS dissipation (term XIII). The second observation concerns the terms XIV (GS/SGS diffusion - symmetric of term VI) and X (SGS turbulent diffusion). These two terms are in almost perfect statistical (and local - as will be seen later) equilibrium. This equilibrium was already observed by Balaras & Piomelli (1994) in a wall flow. It is worth noting that it also holds in a free shear layer, for which, unlike the wall-layer region in a wall flow, there is no statistical equilibrium between production and dissipation of (total) kinetic energy. The last group of importance is constituted by the terms IX (SGS advection), XI (SGS pressure-velocity interactions) and XII (SGS viscous diffusion). Even if in the mean, term IX has some importance in the interval this last group
GRID/SUBGRID-SCALE INTERACTIONS
75
of terms can be considered to be negligible for the dynamics of the mean SGS.
As far as fluctuations are concerned, the most intense terms are IX
X and XIV The high local magnitude (rms) of the term IX (SGS ad vection) is surprising since this term is almost negligible in the mean. This indicates that the local SGS dynamics is ruled by different mech anisms than the global (mean) dynamics. This is confirmed by the rms values of the terms XV and XIII: although these two terms dominate the and mean profiles their rms values are low with The rms values show that the remaining two terms, XI (SGS pressure-velocity interactions) and XII (SGS viscous dif fusion) are negligible for the local SGS dynamics. As expected, the SGS dynamics are therefore mainly commanded by the processes of receiving energy from the GS (term XV) and dissipating it by viscous dissipation (term XIII), while also diffusing (term XIV) and advecting this energy (term IX).
76
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
Turning now into the spatial localization of the dominant terms for the SGS dynamics, visualizations of the spatial distribution of the SGS viscous dissipation (term XIII - see figure 8), show that the higher (negative) values are located right at the core of the flow vortices. This can also be verified with the joint PDF of term XIII and the vorticity norm (figure 7) and with their correlation coefficient Thus, the vortex core region of the coherent structures is a sink for both GS and SGS kinetic energy. Another important question is related to the “local equilibrium assumption”. Within this hypothesis it is supposed that there is a local balance between the GS/SGS transfer (term XV=-VII) and SGS viscous dissipation (term XIII). This hypothesis constitutes the basic assumption of several SGS models such as, for instance, the Smagorinsky and the dynamic Smagorinsky models. As was seen above, and as expected, terms XV and XIII are in statistical equilibrium. However, the comparison of their spa tial localization cleary revealed that this equilibrium is not verified locally. The SGS viscous dissipation acts within the core of the vortices, whereas the GS/SGS transfer takes place almost randomly in space and can be an energy source (forward-scatter) or an energy sink (backward-scatter) in equation (2). Thus, the energy transmitted to the SGS at a given location, is not going to be dissipated by molecular viscosity at that same location. The joint PDF of both terms is shown in figure 10. Their correlation co efficient is only Although there is some correlation between the terms, its relatively low level confirms the failure of the local equilibrium assumption. Visualizations of term X (not presented) show that it has a strong re semblance with term VI (=-XIV). This can be seen in the joint PDF of both terms shown in figure 10. Their correlation coefficient is Therefore, in equation (2) terms X and XIV are in local and statistical equi librium. However, it was observed that this balance is not verified within the transition region (see section 4.2) of the jet. Indeed, the level of small scale turbulence has to reach a certain level for the SGS turbulent diffusion
GRID/SUBGRID-SCALE INTERACTIONS
77
to start acting efficiently. This remark is important because it means that terms X and XIV, which otherwise could be neglected since they cancel out each other, have to be kept in equation (2) when dealing with transitional flows. The topology of term IX (SGS advection) is visualized in figure 9. The picture clearly shows the existence of a connection between the vortex struc tures and the regions of high negative/positive values of term IX. Now, as it was seen before this term has high local values. This gives an explanation for the failure of the “local equilibrium assumption”. Indeed, a very high local SGS kinetic energy advection will transport energy received from the GS to another location before the viscosity will have time to act. A tenta tive demonstration that this is actually what is happening here, will now be given. From the previous discussion concerning the global and local intensi ties of the various terms of equation (2) and since terms X and XIV cancel out (in the far field region only), a good approximation of this equation should be obtained with,
In figure 10 the joint PDF of terms VIII+IX and XIII+XV shows that there is a good local correlation between the two sides of this equation. The coefficient is Given the fact that terms XI and XII can be neglected and that terms X and XIV cancel out this shows unambiguously that (3) is a good local approximation to the lo cal SGS dynamics. This demonstrates that it is mainly by SGS advection (term IX) and eventually local temporal variation of the SGS kinetic energy (term VIII) that compensates the lack of local equilibrium between terms XIII and XV. This illustrates one of the main difficulties of SGS model ing. It would be interesting to investigate the consequences of this (wrong) “local equilibrium assumption” on the vortical structures resulting from Smagorinsky and dynamic Smagorinsky LES computations. 4.2. TRANSITION REGION IN A TURBULENT PLANE JET In this section GS/SGS local interactions will be analyzed during a forced transitional plane jet - DNS2. The forcing was made with a varicose mode in order to get clear Kelvin-Helmholtz and streamwise structures, which are typical of transitional free shear flows, right from the start of our com putational domain. Figure 11 displays isosurfaces of positive Q showing the main features of DNS2. The Kelvin-Helmholtz vortices evolve in symmet ric positions at the upper and lower shear layers. The first merging of a Kelvin-Helmholtz vortex pair takes place at A second merging
78
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
happens at Meanwhile, very strong pairs of streamwise vortices are seen first at between a pair of Kelvin-Helmholtz vortices, after at between a merged Kelvin-Helmholtz vortex pair and finally at the end of the computational domain, near the second merging of the Kelvin-Helmholtz vortices. Only the more important results from this analysis will be discussed here. The first important result is that now, unlike in the fully developed turbulent plane jet, all the terms exhibit a clear correlation with the pres ence of the flow vortices. Figure 12 shows visualizations of terms VI, VII and X. Also here, the magnitude of term VI is around one order of magnitude greater than that of term VII, but now, term VII is very well correlated with the vortical structures as in the experimental work of O’Neil & Men eveau (1997) and numerical simulations of Horiuti (1997). It is interesting to see also that there is much more backward than forward scatter in the initial stages of transition. Figure 13 shows the projection, in the upper shear layer, of the veloc ity vectors on a plane located at and the corresponding contours of strong forward scatter (negative terms VI and VII). The pairs
GRID/SUBGRID-SCALE INTERACTIONS
79
of streamwise vortices are clearly identified through the velocity vector. These create strong local regions of upwash motion into the freestream re gion. Forward scatter from term VII has its higher values right at these ejection regions. Near the same location the GS/SGS diffusion (term VI) reaches its higher values which indicates that the energy arriving from GS/SGS transfer is being rapidly diffused around the same region. From what was seen in section 4.1, it seems there is a natural tendency for term VII (GS/SGS transfer) to be correlated with the vortical structures but,
80
CARLOS B. DA SILVA AND OLIVIER MÉTAIS
as the flow evolves into fully developed turbulence this correlation tends to decrease due to the increasing level of complexity of the background turbulence. Finally, it was observed, as already discussed in section 4.1, that the turbulent transport of the SGS (term X) acts very differently from the case of the far field turbulent plane jet. Not only is its relative magnitude quite small but also has no resemblance with term XV (=-VI). Thus, “local equilibrium” between these terms does not occur in the transition region either. 5. Conclusions This work deals with the effect of the large coherent vortices upon the local grid and subgrid-scale interactions in turbulent and transitional plane jets. It is shown how the presence of these large flow structures greatly enhances most of the kinetic energy exchanges between grid and subgrid-scales. The most important mechanisms of grid-scales kinetic energy transport are ad vection and pressure-velocity interactions, while grid/subgrid-scales diffu sion is more important for the local evolution of the grid-scales kinetic energy than grid/subgrid-scales transfer. The “local equilibrium” assump tion fails to be verified due to the high local values of subgrid-scales kinetic energy advection. References AKHAVAN, R., ANSARI, A., KANG, S. & MANGIAVACCHI, N. 2000 Subgrid-scale interac tions in a numerically simulated planar turbulent jet and implications for modelling J. Fluid Mech. 408, 83–120. BALARAS, E. & PIOMELLI, U. 1994 Subgrid-scale energy budgets in wall layer Bull. Am. Phys. Soc. 39, 1969–1969. HORIUTI, K. 1997 Backward scatter of subgrid-scale energy in wall-bounded and free shear turbulence J. Phys. Soc. Japan 66(1), 91-107. HUNT, J. C. R., WRAY, A. A., MOIN, P. 1988 Eddies, stream, and convergence zones in turbulent flows Center for Turbulence Research, Annual research briefs. MENEVEAU, C. KATZ, J. 2000 Scale invariance and turbulence models for large-eddy simulation Annu. Rev. Fluid Mech. 32, 1–32. O’NEIL, J. & MENEVEAU, C. 1997 Subgrid-scale stresses and their modelling in a tur bulent plane wake J. Fluid Mech. 349, 253–293. PIOMELLI, U., YU, Y. & ADRIAN, R. 1996 Subgrid-scale energy transfer and near-wall turbulence structure Phys. Fluids 8, 215–224. STANLEY, S. & SARKAR, S. 1999 A study of the flowfield evolution and mixing in a planar turbulent jet using direct numerical simulation submitted to J. Fluid Mech. DA SILVA, C. & MÉTAIS, O. 2001 On the influence of coherent structures upon interscale interactions in turbulent plane jets submitted to J. Fluid Mech. THOMAS, F. O. & PRAKASH, K. M. K. 1991 An investigation of the natural transition of an untuned planar jet Phys. Fluids 3(1), 90–105. ZEMAN, O. 1995 The persistence of trailing vortices: A modeling study Phys. Fluids 7(1), 135-143.
2. Numerical issues in LES
This page intentionally left blank
PHASE-ERROR REDUCTION IN LARGE-EDDY SIMULATION USING A COMPACT SCHEME
H.-J. KALTENBACH AND D. DRILLER Hermann-Föttinger Institut für Strömungsmechanik Technische Universität Berlin, Sekretariat HF 1 Straße des 17. Juni 135, D-10631 Berlin, Germany
Abstract. A numerical method for the solution of the incompressible Navier-Stokes equations with staggered variable arrangement is presented. Compact differentiation and interpolation with an truncation er ror is used for discretization of the skew-symmetric form of the advection term. The scheme conserves kinetic energy in the absence of viscosity and the momentum balance is satisfied within acceptable error bounds. Three test cases demonstrate the properties of the scheme, including disturbance growth in Poiseuille flow, direct simulation (DNS) of turbulent channel flow and large-eddy simulation (LES) at The compact at scheme yields an accurate prediction of growth rate and phase velocity of the disturbance wave when the wave is resolved within 8 cells. Third- and fourth order moments in DNS of channel flow are in good agreement with results from a spectral code. LES based on the 6th order scheme accurately predicts the mean flow profile whereas explicit finite difference schemes give erroneous results in the wake region. The compact scheme is less sensitive with regard to a Galilean transformation in the streamwise direction than explicit schemes.
1. Background and goals The relevance of discretization errors for LES has been recognized both from a theoretical [6] and a practical standpoint [10, 14, 17]. For example, satisfactory results for LES of a turbulent channel flow based on 2nd or der finite differences could only be obtained with a rather fine streamwise spacing of in [8] whereas spectral codes yield excellent results for three times coarser grids [10, 20, 22]. Recent results from LES with a 83 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 83-98. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
84
KALTENBACH & DRILLER
near-wall model indicate that the failure of proper prediction of the wake region of the mean velocity profile in a channel flow might be caused by shortcomings of the numerical approximation rather than of the modeling itself [4, 7]. In the past, schemes with strictly energy conserving spatial discretiza tion were found to be superior to methods which introduce numerical dis sipation. However, a drawback of these energy conserving schemes is the phase or dispersion error associated with central differencing. In [10], a 6th-order compact scheme was identified as a possible candidate for reli able LES numerics yielding similar results as a spectral method in LES of turbulent channel flow. The properties of compact schemes – also known as Hermitian, implicit, or Padé approximations – are described in [11]. Be sides their capability to achieve low truncation errors with a narrow spatial support compact schemes exhibit lower dispersive (or phase) errors than explicit schemes with the same order of the truncation error. A thorough discussion of conservation properties of various explicit dif ference schemes for the solution of the incompressible Navier-Stokes equa tions with staggered and with co-located variable arrangement is given in [17]. Despite the excellent conservation properties of the schemes pro posed in [17, 18], it is not entirely clear whether these high-order explicit schemes converge to results from a spectral scheme. Unfortunately, the stencils become very wide in these approximations which makes boundary treatment difficult. The explicit scheme proposed in [23] is a rediscovery of the non-conservative 4th order scheme based on Richardson extrapolation presented in [3]. Compact schemes have been used with success in DNS and LES of com pressible turbulent flows [1, 22]. The few attempts to use compact schemes in the context of incompressible Navier-Stokes solvers include the hybrid spectral/finite difference approach of [21] and the finite volume implemen tation of [16]. Conservation properties were not discussed in the latter work. The central topic of the present paper is development and implemen tation of a 6th-order compact finite-difference discretization for the incom pressible Navier-Stokes equations with staggered variable arrangement. We discuss and provide numerical evidence for the conservation properties and accuracy and compare the outcome of DNS and LES of turbulent channel flow with results obtained by explicit differencing methods. 2. Method Essential ingredients for a spatial discretization being conservative with regard to the kinetic energy include discrete analogies of (i) ‘integration by parts’ as shown in [15], (ii) of continuity, and (iii) of the product rule
PHASE-ERROR REDUCTION IN LES
85
of differentiation [10, 17]. In [15] it was shown that every central finitedifference scheme (explicit or implicit) satisfies criterion (i). Criterion (iii) is responsible for subtle differences in the conservation properties of the divergence and the skew-symmetric forms
of the convective terms in the Navier-Stokes equations. Since the divergence form is a priori conservative with regard to momentum it is the preferred starting-point for discretization. For discretizations that satisfy integration by parts, the skew-symmetric form is conservative with regard to the kinetic energy [10, 17]. For two reasons, conservation of energy is desirable in the context of LES: (1) the lack of numerical dissipation guarantees that the transfer of energy from resolved to sub-grid scales is solely governed by the sub-grid model. (2) Lack of energy conservation can cause non-linear numerical instabilities which in turn might impose severe limits on the Reynolds number range accessible with a particular scheme for a given mesh. Criterion (iii) complicates the construction of conservative finite-difference schemes [17]. Since we were not able to construct a compact finitedifference approximation of the non-linear term in divergence form which satisfies the constraints necessary to guarantee energy conservation, our discretization is based on the skew-symmetric form of the convective term. This term is discretized on a staggered mesh as
Here, and denote compact, 6th-order differentiation and interpo lation, respectively, which read
on an equidistant mesh with the coefficients given as
In the following we will mainly present results where the compact scheme and only whereas the wallwas used in the wall-parallel directions normal direction was approximated with 2nd order explicit differences. We
86
KALTENBACH & DRILLER
also experimented with a 4th order compact approximation of the convec tive and the viscous terms in the wall-normal direction. This yielded sig nificant additional improvements in the Poiseuille flow test case but only minor improvements in the turbulent channel flow cases. We did not pursue this direction further since implicit high-order approximation of the pressure gradient and in the continuity equation increases the cost for solving the Poisson equation drastically. The incompressible Navier-Stokes equations in primitive variables are solved on a staggered mesh. For the semi-implicit time-advancement we combine a low-storage third order Runge-Kutta with a Crank-Nicholson scheme [2]. The basic algorithm consists of the four steps
Here, the index i refers to the coordinate direction where as k denotes a substep of the Runge-Kutta integration cycle for each timestep The operator M involves differentiation with respect to the result ing in tridiagonal systems of equations to be inverted in (6). Computation of right hand sides is explicit with respect to variables and from the preceding substep, respectively timestep. Equation (9) is simply an update of the pressure since the Poisson equation is formulated in terms of the pressure difference in order to retain second order accuracy in time. The discretized version of the Poisson equation (7) depends on the form of the difference stencils used in the approximation of and Ini tially, we used explicit 2nd order central differences for both terms. This discretization of the pressure gradient is globally conservative for kinetic energy [17]. This scheme, for which from now on we use the abbrevation S2E defined in Tab. 1, was tested in turbulent channel flow and the re sults were found to be superior to explicit higher-order approximations described in [17]. However, simulation of disturbance growth in a Poiseuille flow showed some severe deficiencies of the scheme S2E: whereas the phase error could be reduced compared to an explicit scheme, the disturbance growth rate was underpredicted. Also, the overall order of the scheme S2E was found to be only, i.e. the error associated with approximation and continuity dominated the overall truncation error. of
PHASE-ERROR REDUCTION IN LES
87
These findings motivated us to investigate in detail the effects of upgrading the discretization of and with respect to the wall-parallel directions. Four schemes were implemented: 2nd-order explicit, 4th-order explicit, 6th-order explicit, and 6th-order implicit. The explicit schemes are standard approximations with central differences as documented in [17] for which the stencil width for the Poisson problem is 5, 7, and 11, respectively. Compact approximation of the pressure gradient
together with (3) on a staggered mesh in conjunction with the projection step results in a discrete version of the Poisson equation. Unfortunately, the sparseness and band structure of the associated matrix is lost which makes the resulting system of equations difficult to solve in the general case. However, for equidistant meshes in a wall-parallel plane and either periodic or von Neumann b.c. it can be solved using Fourier or cosine series expansion together with the modified wavenumber
The scheme which employs 6th order compact approximations for pres sure gradient and continuity in the wall-parallel directions will be denoted as S6I. Table 1 lists several schemes which differ with regard to the ap proximation of the advection term and/or the treatment of the projection step. Only in scheme SY4 a higher-order approximation for the wall-normal direction was used. 3. Results 3.1. ENERGY CONSERVATION All schemes listed in Tab. 1 except for the last one conserve the kinetic energy in the absence of viscous forces besides a negligible error introduced by the time integration. Scheme SY4 is non-conservative due to the treat ment of the cells adjacent to the walls. This scheme is energy-conserving in the interior of the domain. In viscous flow, a violation of the energy con servation principle can be tolerated for the near-wall cell since the flow is not turbulent there. The conservation properties have been confirmed numerically by inte grating a random initial field in time using a channel-type computational domain of size discretized in cells. The
88
KALTENBACH & DRILLER
boundary conditions were at and periodic with regard to the wall-parallel directions. The solution was advanced over 200 timesteps using a maximum CFL-number of The results are shown in Fig. 1. The slight decrease of energy for scheme S6I results from the time ad the scheme is perfectly conservative and vancement. In the limit the integration could be continued forever without encountering blow-up or zero flow.
It is difficult to achieve the same conservation properties on non-equidistant meshes [17]. Still, it is worthwhile to construct schemes which are strictly conservative on equidistant meshes only since experience shows that these schemes perform much better on non-equidistant meshes than schemes which do not conserve energy at all. The numerical simulation
PHASE-ERROR REDUCTION IN LES
89
showed that scheme S6I performs indeed well on a mesh which was stretched in the wall-normal direction: over 200 timesteps, the energy increased by no more than 2 percent. Thus, scheme S6I will likely produce stable solutions on non-equidistant meshes when viscosity is present. 3.2. DISTURBANCE GROWTH IN POISEUILLE FLOW
A well-defined 2-D test case for wall-bounded flow is the disturbance growth in a plane Poiseuille flow in a gap of width The emphasis of our compari son is on the discretization of the wall parallel direction. Therefore, a rather fine non-equidistant mesh with cells was used in the wall-normal direction whereas only cells were used for the discretization of a sin gle streamwise mode of the disturbance. Results are compared with linear stability theory predictions for the corresponding Orr-Sommerfeld eigen value problem. The eigenfunction and the eigenvalue of the linear problem were obtained by Tschebychev expansion. For the most unstable mode with a streamwise wave-length of has the complex eigenvalue Results are shown in Fig. 2. Here, the time axis is normalized by Linear sta bility theory predicts a growth rate In for the volume integral of the disturbance energy Both explicit schemes DIV2 and DIV4 show substantial deviations from the theoretical growth rate whereas the 6th order scheme S6I comes close to the theoretical value. The remaining deviations from the theoretical growth rate vanish when the discretization in the wall-normal direction is improved as in SY4. An important observation from Fig. 2b is concerned with the role of the discretization of the projection step. Implicit 6th-order discretization of the convective term in skew-symmetric form yields only satisfactory re sults when at least 4th order accuracy is used for the projection step. It is interesting to observe that the hybrid scheme S2E causes a decay of the disturbance wave despite the fact that the scheme is energy conserving. The hybrid 6th/4th-order scheme S4E is already closer to the theoretical growth rate than the explicit 4th-order scheme DIV4 which overpredicts the growth rate. With 6th-order explicit approximation of the projection step the growth rate is nearly as good as in the ‘fully implicit’ scheme S6I. This is very encouraging since – as mentioned earlier – the corresponding Poisson problem is much easier to solve in complex domains for scheme S6E than for scheme S6I. Fig. 2c demonstrates that the scheme S6I is of The capability of the implicit scheme to reduce the phase error becomes evident from Fig. 2d: since the time-axis is normalized by the fundamental time
90
KALTENBACH & DRILLER
scale
of the disturbance wave, minima should occur at integer values of Only the implicit 6th-order treatment of the advection is capable of predicting the correct phase whereas the explicit schemes DIV2 and DIV4 exhibit a phase-lag at the resolution of The accurate prediction of the phase appears to be fairly independent of the chosen discretization of the projection operator since S6I and S4E differ only with respect to the amplitude. 3.3. TURBULENT CHANNEL FLOW: DNS AT Fig. 3 shows a comparison of several finite-difference schemes with the spec tral results from [9]. Results were obtained in a domain of size using a resolution of 128 × 96 × 128 cells. The mesh was stretched in the wall-normal direction based on a tanh-function with the stretching
PHASE-ERROR REDUCTION IN LES parameter set to The first off-wall position for at and the spacing on the channel axis is
91
and
is located
All schemes are capable to predict up to second order statistics in fair
92
KALTENBACH & DRILLER
agreement with the spectral method of [9] except for a slight shift in the log-law intercept for S2E and an underprediction of the rms of the vorticity component by the 2nd order scheme DIV2. For higher moments such as skewness and flatness coefficients the advantage of using higher-order approximations becomes evident. We observe a consis tent improvement with increasing order of truncation error. The upgrade of the projection step from 2nd to 6th order yields further improvement although the scheme S2E does perform much better in this case as might be expected from the Poiseuille flow test. Since the skew-symmetric form does not conserve momentum a priori, it has to be checked whether the momentum balance across the channel is satisfied. Fig. 4a shows that this is the case for scheme S6I. The situation is slightly different for the hybrid compact/explicit scheme S2E: momentum is conserved on fine meshes typ ically used in DNS (not shown) but some deviations are found on coarser meshes as shown in Fig. 4b.
PHASE-ERROR REDUCTION IN LES
93
3.4. TURBULENT CHANNEL FLOW: LES AT
The primary goal of this study is the development of a reliable numerical method for the incompressible Navier-Stokes equations which can serve as a sound base for LES. Therefore, we consider it as important to evaluate the performance of a scheme at grid resolutions which are representative for wall-bounded shear flows where the ‘streak-streamwise vortex cycle’ dominates the turbulence production. Since physics of the near-wall layer scale on inner variables, it is reasonable to evaluate LES performance for mesh sizes scaled by the characteristic length In order to capture the near-wall streaks in LES, the spanwise resolution must be and the streamwise resolution should not exceed
94
KALTENBACH & DRILLER
Fig. 5 compares results for LES of turbulent channel flow at with a mesh spacing at the wall, and as in [18]. The simulations used a domain of size discretized in 32 × 64 × 32 cells. The grid was non-uniform in the wallnormal direction based on tanh-stretching with All results were obtained with the standard dynamic sub-grid model [5] with least-squares contraction after [12]. A test filter was used in the wall-parallel planes based on trapezoidal rule integration together with a filter ratio In contrary to [18] we do not take the wall-normal direction into account in the computation of the filter ratio. See the discussion in [13] for the appropriate value of the filter ratio. It is evident that the 6th order compact scheme is superior to the ex plicit schemes which fail to predict the correct mean velocity profile. The improvement is twofold: (i) the compact scheme yields the correct intercept of the log-region of the mean profile and (ii) the prediction in the wakeregion has considerably improved. This is mirrored in an improvement of the ratio of maximum to bulk velocity and other integral parameters when compared to the spectral results of [19], see Table 2. Note that the shape is not a good measure for the quality of the prediction of the factor mean-flow profile in this case. The improvement in the wake prediction is consistent with the observations by [7] and [14].
As in the Poiseuille flow test case the results from simulations with either implicit or explicit 6th order approximation of the projection step are nearly identical. Fig. 4c and 4d prove that both schemes S6I and S6E conserve the mean momentum on a ‘realistic’ LES grid within an acceptable error margin. For 2nd order explicit approximation of the projection, i.e. scheme S2E, the shear stresses do not sum to a straight line, see Fig. 4b.
PHASE-ERROR REDUCTION IN LES
95
Also, this scheme fails in the prediction of integral parameters and and is therefore not recommended for LES. For the standard dynamic model, the contribution of sub-grid stresses to the total shear stress hardly exceed 10%, see Fig. 5d. The differences among the three simulations do not result from the sub-grid term but rather from the stresses carried by the resolved scales which are susceptible to numerical error. We have confirmed this finding by carrying out simulations without SGS-model, i.e. coarse-grid or under-resolved DNS, which showed very sim ilar behaviour as the LES cases: higher-order approximation improved the predictions considerably. 3.5. SENSITIVITY WITH RESPECT TO GALILEAN TRANSFORM
Further evidence for the crucial role of details of the numerical approxima tion comes from a series of simulations which were carried out in order to check the translational invariance of the results. Finite-difference approx imations lack Galilean invariance and it is therefore important to investi gate how much the results might be affected by this inherent shortcoming. For each of the three schemes DIV2, DIV4, and S6I numerical experiments were carried out in which the velocity of both walls was set to The results are summarized in Fig. 6 and Table 3. Obviously, the change in the frame-of-reference can have a substantial effect on the results which can be of similar importance as modifications of the sub-grid model. Fortunately, the sensitivity of the results with respect to a Galilean transformation becomes less severe with increasing approxi mation order. Whereas the mean velocity profile changes substantially for scheme DIV2 with set to the compact scheme shows a small shift in the log-law intercept only. For scheme S6I the near-wall turbulence does not seem to be affected by the change in frame-of-reference. Table 3 corrob orates the findings: integral flow parameters are fairly robust with regard in scheme S6I whereas to the value of the Galilean translation speed considerable changes are observed for the explicit difference schemes. The Galilean transform in the streamwise direction affects phase errors associated with the advection by the mean flow. Because of in the core of the channel, phase errors associated with transport by the mean flow are expected to be largest. We conjecture that the reason for the relative insensitivity of the 6th order compact scheme with regard to the change in frame-of-reference lays in the ability of the scheme to reduce dispersion errors.
96
KALTENBACH & DRILLER
4. Conclusions High-order compact discretization and interpolation in the wall-parallel co ordinates significantly improves simulation results when compared with ex
PHASE-ERROR REDUCTION IN LES
97
plicit differences. Conservation of kinetic energy is guaranteed by use of the skew-symmetric form for the convective terms. Implementation of the proposed 6th-order compact schemes for the advection requires a 7-point stencil. This is the same width as for the explicit 4th-order scheme of [17] but substantially smaller than the 11-point stencil needed in the explicit 6th-order approximation proposed in [17]. A potential drawback for an extension of the proposed scheme to com plex geometries is an increase in the cost for solving Poisson’s equation for the pressure due to the loss in sparseness of the associated matrix when the projection step is approximated in compact form. However, our results for disturbance growth rates and LES of channel flow indicate that a combina tion of compact approximation of the advection terms in skew-symmetric form together with high-order explicit approximation of the pressure gradi ent term and the continuity equation can be a viable alternative to a ‘fully compact’ approximation. The advantage of this hybrid compact/explicit high-order approximation of type S6E lies in the fact that the structure of the underlying Poisson problem is not fundamentally altered. Acknowledgements
We thank Dr. A. Hauschild and Dr. A. Spille for providing the solutions to the Poiseuille flow eigenvalue problem. Part of this work was funded by Deutsche Forschungsgemeinschaft within grants Sfb 557 and KA 1424/3-1. Computer time was provided by the Konrad-Zuse Zentrum (ZIB) Berlin. References 1. Adams, N.: 2000, ‘Direct simulation of the turbulent boundary layer along a com J. Fluid Mech. 420, 47–83. pression ramp at M = 3 and
98
KALTENBACH & DRILLER
2. Akselvoll, K. and P. Moin: 1996, ‘An efficient method for temporal integration of the Navier-Stokes equations in confined axisymmetric geometries’. J. Comp, Phys. 125, 454. 3. Antonopoulos-Domis, M.: 1981, ‘Large-eddy simulation of a passive scalar in isotropic turbulence’. J. Fluid Mech. 104, 55. 4. Cabot, W., J. Jimenez, and J. S. Baggett: 1999, ‘On wakes and near-wall behavior in coarse large-eddy simulation of channel flow with wall models and second-order finite-difference methods’. In: CTR Annual Research Briefs 1999. pp. 343–354. 5. Germano, M., U. Piomelli, P. Moin, and W. H. Cabot: 1991, ‘A dynamic subgrid scale eddy viscosity model’. Phys. Fluids A 3, 1760–1765, Erratum: 3128. 6. Ghosal, S.: 1996, ‘An analysis of numerical errors in large-eddy simulation of tur bulence’. J. Comp. Phys. 125, 187–206. 7. Jiménez, J.: 2000, ‘Some open computational problems in wall-bounded turbu lence’. In: C. Dopazo (ed.): Advances in Turbulence VIII. Gran Capitan s/n, 08034 Barcelona, Spain, pp. 637–646. 8. Kaltenbach, H.-J., M. Fatica, R. Mittal, T. Lund, and P. Moin: 1999, ‘Study of flow in a planar asymmetric diffuser using large eddy simulation’. J. Fluid Mech. 390, 151–185. 9. Kim, J., P. Moin, and R. Moser: 1987, ‘Turbulence statistics in fully developed channel flow at low Reynolds number’. J. Fluid Mech. 177, 133–166. 10. Kravchenko, A. and P. Moin: 1997, ‘On the effect of numerical errors in large-eddy simulation of turbulent flows’. J. Comp. Phys. 130, 310–322. 11. Lele, S. K.: 1992, ‘Compact finite difference schemes with spectral-like resolution’. J. Comp. Phys. 103, 16–42. 12. Lilly, D. K.: 1992, ‘A proposed modification of the Germano subgrid-scale closure method.’. Phys. Fluids A4 3, 633–635. 13. Lund, T.: 1997, ‘On the use of discrete filters for large eddy simulation’. In: CTR Annual Research Briefs 1997. pp. 83–95. 14. Lund, T. and H.-J. Kaltenbach: 1995, ‘Experiments with explicit filtering for LES using a finite-difference method’. In: CTR Annual Research Briefs 1995. pp. 91–105. 15. Mansour, N., P. Moin, W. Reynolds, and J. Ferziger: 1979, ‘Improved methods for large eddy simulation of turbulence’. In: F. Durst, B. Launder, F. Schmidt, and J. Whitelaw (eds.): Turbulent Shear Flows I. pp. 386–401. 16. Meri, A., H. Wengle, A. Dejoan, E. Vedy, and R. Schiestel: 1999, ‘Applications of a 4th-order Hermitian scheme for non-equidistant grids to LES and DNS of incompressible fluid flow’. In: E. H. Hirschel (ed.): Numerical flow Simulation I. pp. 382–406. 17. Morinishi, Y., T. Lund, O. Vasilyev, and P. Moin: 1998, ‘Fully conservative higher order finite difference schemes for incompressible flow’. J. Comp. Phys. 143, 90–124. 18. Morinishi, Y. and O. V. Vasilyev: 1998, ‘Subgrid scale modeling taking the numerical error in consideration’. In: CTR Annual Research Briefs 1998. pp. 237–253. 19. Moser, R., J. Kim, and N. Mansour: 1999, ‘Direct numerical simulation of turbulent channel flow up to Phys. Fluids 11. 20. Piomelli, U.: 1993, ‘High Reynolds number calcuations using the dynamic subgrid scale stress model’. Phys. Fluids A 5, 1484–1490. 21. Schiestel, R. & Viazzo, S.: 1995, ‘A Hermitian-Fourier numerical method for solving the incompressible Navier-Stokes equations’. Computers & Fluids 24(6), 739–752. 22. Stolz, S.: 2001, Large-eddy simulation of complex shear flows using an approximate deconvolution model, Fortschr.-Ber. VDI Reihe 7, Nr. 403. Düsseldorf: VDI Verlag. 23. Verstappen, R. and A. Veldman: 1998, ‘Spectro-consistent discretization of NavierStokes: a challenge to RANS and LES’. Journal of Engineering Mathematics 34, 163–179.
DNS AND LES OF TURBULENT BACKWARD-FACING STEP FLOW USING 2ND- AND 4TH-ORDER DISCRETIZATION
ADNAN MERI AND HANS WENGLE Institut für Strömungsmechanik u. Aerodynamik, LRT/WE 7, Universität der Bundeswehr München, D-85577 Neubiberg, Germany
Abstract. Results are presented from a Direct Numerical Simulation (DNS) and Large-Eddy Simulations (LES) of turbulent flow over a backward-facing step with a fully developed channel flow uti lized as a time-dependent inflow condition. Numerical solutions using a fourth-order compact (Hermitian) scheme, which was formulated directly for a non-equidistant and staggered grid in [1] are compared with numer ical solutions using the classical second-order central scheme. The results from LES (using the dynamic subgrid scale model) are evaluated against a corresponding DNS reference data set (fourth-order solution).
1. Introduction Two basic numerical simulation concepts are available to calculate the three-dimensional and time-dependent structure of a turbulent flow: the Di rect Numerical Simulation (DNS), being restricted to relatively low Reynolds numbers, and the Large-Eddy Simulation (LES), in principle applicable also to high Reynolds numbers. The ’high Reynolds number objective’ of LES is a central issue of present days efforts of solving turbulent flow problems of practical relevance. However, in LES specific problems arise from the interplay between the magnitude of the truncation errors of the numeri cal solution method on the one hand, and the magnitude of the modeled subgrid scale terms (and related modeling errors) on the other hand. This automatically leads to the question of how useful higher-order numerical schemes could be for LES. For example, in Kravchenko and Moin [2] it has been demonstrated that for low-order finite-difference simulations the truncation error can exceed the magnitude of the subgrid scale terms. 99 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 99-114. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
100
ADNAN MERI AND HANS WENGLE
The motivation behind the development of the so-called compact schemes is to increase the accuracy of the numerical solution method from second to (at least) fourth-order accuracy while retaining the basic tridiagonal form for the matrix system of the governing equations to be solved. The gen eral principle behind the compact collocation technique is to treat not only the functional values but also the first (and if required the second) deriva tives as unknown at three collocation points only. The resulting ’compact’ method can be derived either from appropriate Taylor-series expansions (Hermite), see e.g. Hirsh [3], Krause et al. [4], Adam [5], Goedheer and Potters [6], Lele [7], Carpenter et al. [8], Sabau and Raad [9], Gamet et al. [10], or from appropriate polynomial expansions (splines), see e.g. Rubin and Khosla [11]. The derivation of a fourth-order compact scheme will lead to different formulations on an equidistant or a non-equidistant grid and also on the basis of a staggered or a non-staggered arrangement of the dependent vari ables. Compact schemes derived for an equidistant grid can be used in a properly transformed space which gives a desired non-uniform grid when transformed back to the physical space. However, for the numerical solu tion of practical flow problems with complex geometry, non-uniform grids often must be defined directly in the physical space. Therefore, in Meri et al. [1] the fourth-order compact scheme had to be rederived directly for a non-equidistant and staggered grid, extending the works of Adam [5] (nonequidistant, non-staggered grid) and Lele [7] (equidistant, staggered and non-staggered grid). It is however difficult to test and to apply higher-order numerical schemes, together with different subgrid scale models, in sufficiently complex flow sit uations. For the purpose of this paper, we have selected the turbulent flow over a backward-facing step (bfs) in a fully developed turbulent channel flow. The advantage of this combined test flow case is that a DNS can be carried out for a Reynolds number of about Re=3000, and it is easier to produce a (clearly defined) channel flow as an inflow condition for the bfs flow than producing a boundary layer inflow (with clearly defined proper ties). For the turbulent channel flow well documented and highly accurate DNS results are available from Kim et al. [12] (using a spectral code), and DNS results (using a second order central differencing scheme) from Rai and Moin [13] and Manhart [14]. For the turbulent bfs flow there are DNS and from Le et.al. [15] (DNS) and Akselvoll LES results available and Moin [16] (LES), using a turbulent boundary layer as an inflow. In this paper we present results from DNS and LES of a backward-facing based on the maximum inflow velocity and on the step flow step height), utilizing a fully developed turbulent channel flow based on the wall skin friction velocity, and the half height of the plane
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION
101
channel, as an inflow condition. The DNS result serves as a reference data set for the evaluation of different LES results calculated with spatial discretization of second-order and fourth-order accuracy, respectively. The main objective of this paper is to study the effects of truncation errors in the second-order and fourth-order LES solutions (in comparison to a corresponding DNS solution). 2. Numerical solution strategies and subgrid scale models used
Numerical solution method: The numerical code used in this paper is based on a finite-volume formulation of the Navier-Stokes equation for an incompressible fluid on a non-equidistant and staggered Cartesian grid. The discretization in space used here is either the classical second-order (central) discretization, or the 4th-order compact discretization on nonequidistant grids proposed in Meri et al. [1]. For the time advancement of the momentum equations an explicit second-order time step is used (leap frog with time-lagged diffusion). The solution of the Poisson equation is carried out iteratively. The iterative solver used here is the well known point-by-point velocity- pressure iteration described by Hirt et al. [17]. Direct numerical simulation (DNS): For a DNS the conservation equa tions for mass and momentum must be solved without any additional as sumptions or modeling related to the effects of the turbulent motion, i.e. the original Navier-Stokes equations must be discretized in space and time such that all the relevant scales in a turbulent flow are resolved. For exam ple, referring to spatial scales, the size of the computational domain must be large enough to accommodate the largest turbulent scales, and the grid spacing must be sufficiently small to enable a resolution of the order of the dissipation length scale, Here, is the kinematic viscosity of the fluid and is the dissipation rate in the flow field. Large-eddy simulation (LES): A derivation of the grid scale (GS) equa tions by Schumann [18] leads directly to the integral form of the NavierStokes equations in which time derivatives of cell-volume averages of veloci ties are related to differences of cell-surface averaged stress and momentum flux. Finally, the averages are related to their finite-difference operators (on a staggered computational mesh). In this paper, we follow this approach. From the point of view of the explicit filtering approach we use a filter width equal to the grid spacing and the effects of all the unknown terms are modeled all together by an edddy viscosity model which acts as a sink of energy for the short waves in the flow. Subgrid scale models: The unknown SGS stresses in the GS equations must be related to the GS velocity via a proper SGS turbulence model. In this paper we use the so-called dynamic SGS model, see e.g. in Germano et
102
ADNAN MERI AND HANS WENGLE
al. [19]. This model allows to calculate the model constant as a function of time and space. To avoid numerical instabilities, the model parameter must be properly averaged in time and/or space. 3. Direct formulation of the method on a non-equidistant grid For the implementation of the fourth-order (compact) scheme the given structure of our general program for flow over obstacles of general geome try had to be considered. The basic finite-volume discretization (defined on a non-equidistant staggered grid) started from the integral form of the bal ance equations and therefore, fourth-order interpolations are required for the convection terms and fourth-order approximations for the first deriva tives are needed for the diffusion terms (second derivatives are not needed for the integral form of the balance equations). For example, to discretize the x-momentum equation a control volume centered around the u-velocity component is defined at index i.e at the side surface of the basic grid cell (with index i). Then, interpolations of the velocity components and their derivatives are required at locations where they are needed but not defined. Figure 1 illustrates the problem.
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION
103
3.1. 4TH-ORDER APPROXIMATION FOR THE FIRST DERIVATIVES 3.1.1. First derivatives at interior nodes The discrete values of a function f are given at the boundaries (index of the basic grid cell (with index i at the center of the cell) of size (see figure 1 ). Then we start from an approximation for the of discrete values of the derivatives at the locations the following form:
Note, the triangle hat of the function value signalizes that this value is not defined at the location i and therefore, it must be provided by a cor responding fourth-order compact interpolation (see section 3.2). Relations und are derived by matching the between the coefficients coefficients of the Taylor series for the function values and the values of their derivatives, respectively, see [1]. The coefficients are functions of the stretching factor of the grid, The truncation error in the general case of a non-equidistant grid is:
For an equidistant grid are (see Lele [7]):
the coefficients in the approximation (1)
With these coefficients the truncation error merges to the fourth-order value derived by Lele [7] (App.B, table VI) for an equidistant grid:
3.1.2. Non-periodic boundary conditions: formulation for the first and the last grid point For the first derivative at the first grid point (l) within the computational domain, third-order accuracy is provided by applying the following noncentered formulation:
104
ADNAN MERI AND HANS WENGLE
For the first derivative at the last grid point
centered formulation is used:
a corresponding non-
3.1.3. Tridiagonal system of linear equations The approximations (1), (5) and (6) lead to the following tridiagonal system of equations (for the first derivatives) which can be solved , e.g. by a socalled Thomas algorithm:
On the right-hand side of (7) and of (1) the function values characterized by a triangle hat, e.g. are not defined at the location with index i and therefore, before solving the system (7) they must be interpolated to fourthorder accuracy using approximations given in the following section. 3.2. 4TH-ORDER HERMITIAN INTERPOLATION 3.2.1. Interpolation at interior nodes For the interpolation on a non-equidistant grid we used an approximation of the following form:
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION 105 The
and
are known and defined at the boundaries of the basic
grid cell of size and the unknown is defined at the location i at the center of the cell. Again, the relations between the coefficients and are derived by matching the coefficients of the Taylor series for the function values. The truncation error (of fourth order accuracy) is:
Finally, for an equidistant grid tion (8) are:
the coefficients in the approxima-
With these coefficients the truncation error merges to the fourth-order value derived by Lele [7] (App.C table VIII) for an equidistant grid:
3.2.2. Interpolation for the first and the last grid point For the interpolation at the first grid point (l) within the computational domain fourth-order accuracy is provided by applying the following noncentered formulation:
At the last grid point
a corresponding non-centered formulation is used:
Again, the coefficients and are derived by matching the coefficients of the Taylor series for the functions values. 4. DNS and LES of turbulent flow over a backward-facing step in a plate channel In this section we use a DNS result based on the centerline velocity of the incoming channel flow and on the step height) for the evalua tion of LES results calculated with second-order and fourth-order (compact or Hermitian) discretization, respectively. The use of a higher-order numer ical solution method seems to be useful also for LES. The improved spatial
106
ADNAN MERI AND HANS WENGLE
discretization should give a better representation of the resolved smaller scales and of the steep local gradients in the instantaneous flow field. In addition, the higher-order method should give a better numerical represen tation of the non-linear convective terms. 4.1. COMPUTATIONAL DOMAIN, SPATIAL DISCRETIZATION AND BOUNDARY CONDITIONS In our case, x is the main flow direction (velocity component u), y is the lat eral (velocity component v), z is the vertical direction (velocity component w). Note, in the following all the variables are presented in dimensionless form, i.e. the velocities and the velocity correlations are made dimensionless with the bulk velocity, lengths are made dimensionless with the step height, h. Computational domain: The flow configuration selected is a backwardfacing step (of height h) in a plate channel (of height 2h before the step and of height 3h after the step), see figure 2 . The inflow cross-section for the channel flow part is located at X=x/h=-12.6 and its outflow crosssection is located at X=x/h=-3.0 upstream of the position of the step at X=x/h=0.0. The outflow cross-section of the step flow part is located at X=17.32 (DNS) and X=16.2 (LES), respectively. The expansion ratio of this flow configuration, ER=1.5, is defined as the ratio of the channel height after the step to the channel height before the step. The lateral extension of the computational domain was 6.0 step heights. If all length scales are normalized with the step height, h, the dimensions of the computational domain are for the reference DNS and for the LES
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION 107
Spatial resolution: Different numerical simulations have been carried out for this flow case: a direct numerical simulation (DNS) using about 6 million grid points, and large-eddy simulations (LES) using about 0.37 million grid points. In the homogeneous lateral (Y-) direction of the flow problem the grid spacing was equidistant, in the longitudinal (X-) direction the grid spacing was equidistant in the channel flow part, Then, for DNS a stretched grid was used between (for LES it remained equidistant), and again an equidistant grid was applied downstream of the step. In the wall-normal direction (Z) the grid was stretched away from the wall. Table 1 and table 2 present the number of grid points used, and the grid spacings in wall units (for the channel flow part) and in units of the step height (for the step flow part). Note, for the wall-normal direction the distance of the first velocity grid point on a staggered grid, and respectively, is given in the tables.
Note, the number in parenthesis stands for the number of ’blind’ grid points in the rigid body of the step entrance region.
From an a posteriori evaluation of the local dissipation rate, from the DNS (see figure 3) a maximum (dimensionless) dissipation rate of was evaluated. With this value of the Kolmogorov dissipation length scale could be estimated with Measured in units of this estimated the spatial resolution of the DNS was about for the backward facing step flow. Boundary conditions: No slip boundary conditions were used at the rigid walls and periodic boundary conditions were applied in the homogeneous lateral direction. Between X=-12.6 and X=-3.0 the flow was a fully devel oped channel flow (with with periodic boundary conditions between these two locations. In the same production run, the result from
108
ADNAN MERI AND HANS WENGLE
the channel flow at X=-3.0 was simultaneously used as a time-dependent in flow condition for the step flow problem downstream of this position. As an outflow boundary condition gradients normal to the outflow cross-section have been set to zero in calculations using a second-order numerical solu tion method, and for the flow calculation using a fourth-order (compact, or Hermitian) method the outflow boundary condition corresponded to a third-order extrapolation of the velocity components. 4.2. RESULTS FOR THE TURBULENT CHANNEL FLOW (USED AS TIME-DEPENDENT INFLOW CONDITION)
In our combined flow case, a fully developed turbulent channel flow delivers the time-dependent inflow conditions at X=-3.0 for the step flow case down stream of this location. Figure 4a compares vertical profiles of the rms values from the channel flow part of the DNS runs (using second-order and fourth-order spatial discretization, respectively) with the DNS data from Kim et al. [12] (using a Fourier-Chebyshev spec tral method on a grid with about 4 million grid points). As already shown by Meri et al. [1] there are only significant differences to be observed for the statistics of higher than second order (e.g. for the skewness of the wall normal fluctuations). From Figure 4a it can be concluded that using a suf ficiently large number of grid points (the channel flow part contains about 0.5 million grid points) the fourth-order (compact or Hermitian) discretiza tion and even the second-order (central) discretization delivers sufficiently accurate results up to the second-order statistics. In Figure 4b we com pare the rms profiles from LES (with about 60 000 grid points in the channel flow part, and using the dynamic subgrid scale model) with the DNS ref erence data from Kim et al. [12]. Here, the second-order results as well as the fourth-order results contain the subgrid scale contributions which are derived from the SGS turbulent energy by assuming local isotropy. The peaks of the u-rms values are higher and the peaks of the v-rms and w-rms values are lower than those of the reference data. The fourth-order results are closer to the DNS data. 4.3. RESULTS FOR THE TURBULENT BACKWARD-FACING STEP FLOW
In the following we compare results from LES (using about 245 000 grid points within the step flow part of the flow problem) with a corresponding DNS (using about 5 million grid points within the step flow part). For the DNS and LES cases we used second-order (central) differencing and the fourth-order compact method, respectively. Note, in both LES cases the dynamic subgrid scale model was applied.
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION 109
Instantaneous flow field: Figure 5 shows two snapshots at arbitrary times from DNS using second-order and fourth-order discretization, respec tively. The second-order result after the step shows small ’wiggles’ in the upper part of the flow problem, indicating that the grid spacing in this part of the flow is not sufficiently small to properly resolve the instanta neous velocity gradients in the flow using the second-order method. The results from using the fourth-order method do not show this behaviour. In the following we shall always take the result from the fourth-order DNS as a reference data set. Mean flow and second-order statistics: In Figure 6 we first compare vertical profiles of the first-order statistics (mean flow) and of the secondorder statistics from DNS with the LES also using fourth-order discretiza tion. The agreement is very satisfying. In addition, from the results for the rms-values for the fluctuations of all three velocities in Figure 6 it can be concluded that an assumption of local isotropy in RANS modeling is not a bad approximation in parts of this flow field (away from the walls). there However, for example, in the re-attachment zone (around are strong lateral velocity fluctuations (v-rms) created by the re-attaching shear layer. In Figure 7 a comparison of LES results for the mean < U > and < W > velocity components from second-order and fourth-order cal culations, respectively, are presented. The differences in the mean flow are small. The zero-crossing of the downstream < U >-velocity distribution through the first grid point closest to the bottom wall defines the mean of the major recirculation region behind re-attachment length, the step. There is also a secondary recirculation zone immediately at the Table 3 base of the step, with the mean recirculation length shows the results of DNS and of both LES (2nd- and 4th-order).
Figure 8 shows profiles of the turbulent energy, at two downstream locations (X = 1.9 and X = 4.9) in the recirculation zone, from DNS and both LES results (containing the subgrid scale en Closer to the step (X = 1.9) the fourth-order LES ergy contributions, solution is closer to the DNS in comparison with the second-order LES. However, at the location X = 4.9 it is the other way round, i.e. the secondorder LES results is closer to the DNS result.
110
ADNAN MERI AND HANS WENGLE
Grid scale (resolved) and subgrid scale turbulent energy: Figure 9 com pares second-order and fourth-order LES solutions of the grid scale turbu lent energy, and of the (estimated) subgrid scale turbulent energy, In general, the higher-order method is capable to provide a more accurate so lution for the grid scale and therefore, the grid scale energy will not be the same for different numerical schemes working on an identical grid. Figure 8 already shows that there is a complex interplay between grid resolution, accuracy of the numerical scheme and subgrid scale modeling, and their effects on the downstream development of the flow. It is interesting to see in the fourth-order LES is always smaller that the subgrid scale energy, than in the second-order solution, see Figure 9 (middle). Using a finer mesh in the second-order LES would lead to about the same result. In ad dition, Figure 9 (below) shows profiles of the ratio of (estimated) subgrid to the resolved grid scale energy, This ratio is less than scale energy, about 20% in our flow case. However note that in the whole recirculation zone, in particular in the so-called ’dead-water’ zone close to the step, the amount of the subgrid scale contributions is relatively large. 5. Conclusions The backward-facing step flow is a very useful test case together with a fully developed turbulent plane channel flow used as a time-dependent inflow condition (representing the physically correct spatio-temporal structure). can be utilised as a refer A DNS of this combined flow case ence data set to evaluate results from LES using 2nd-order and 4th-order (Hermitian) spatial resolution, respectively, together with a proper subgrid scale turbulenc model (in our case the dynamic model). Here, the LES cases use about 0.37 million grid points, i.e. a factor of about 18 less grid points than the corresponding DNS. In general, using the fourth-order method (instead of the classical second-order scheme) it is expected to receive more accurate LES solutions for the first-order and second-order statistics. In comparison with a corresponding DNS this is true for the channel flow case. However from the LES solutions for turbulent flow over a backwardfacing step a complex interplay can be observed between grid resolution, accuracy of the numerical scheme and subgrid scale modeling, and their effects on the downstream development of the flow. Acknowledgments This research was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number We 705/5. We also gratefully acknowledge the support by the computing center of the Federal Armed Forces University Munich.
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION 111
References l. A. Meri, H. Wengle, A. Dejoan, Védy E., and R. Schiestel. Application of a 4th order hermitian scheme for non-equidistant grids to les and dns of incompressible fluid flow. In E.H. Hirschel, editor, Numerical Flow Simulation I. Verlag Vieweg, 1998. 2. A.G. Kravchenko and P. Moin. On the effects of numerical errors in large-eddy simulations of turbulent flows. Tech. Rep. CTR-160, Center for Turbulent Research, Stanford University, 1996. 3. R.S. Hirsh. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. of Comp. Physics, 19:90–109, 1975. 4. E. Krause, E.H. Hirschel, and W. Kordulla. Fourth-order ”mehrstellen”-integration for threedimensional turbulent boundary layers. Proceedings of IAA Computational Fluid Dynamics Conference, Palm Springs, Calif., pages 92–102, 1973. 5. Y. Adam. A hermitian finite difference method for the solution of parabolic equa tions. Comp. and Maths, with Appls., 1:393–406, 1975. 6. W. Goedheer and J. Potters. A compact finite difference scheme on a nonequidistant mesh. J. Comp. Physics, 61:269–279, 1985. 7. S.K. Lele. Compact difference schemes with spectral-like resolution. J. Comp. Physics, 103:16–42, 1992. 8. M. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order finite-difference schemes. J. Comp. Physics, 108:272–295, 1993. 9. A. Sabau and P. Raad. Comparisons of compact and classical finite difference solutions of stiff problems on nonuniform grids. Computers and Fluids, 28:361–384, 1999. 10. L. Gamet, F. Ducros, F. Nicoud, and T. Poinsot. Compact finite difference schemes on non-uniform meshes, application to direct numerical simulations of compressible flows. Int.J.Numer.Meth.Fluids, 29:159–191, 1999. 11. S. Rubin and P. Khosla. Polynomial interpolation methods for viscous flow calcu lation. J. Comp. Phys., 24:217–244, 1977. 12. J. Kim, P. Moin, and R. Moser. Turbulence statistics in fully developed channel flow at low reynolds number. J. Fluid Mech., 177:133–166, 1987. 13. M.M. Rai and P. Moin. Direct simulations of turbulent flow using finite-difference schemes. J. Comp. Phys., 96:15–53, 1991. 14. M. Manhart. Zonal direct numerical simulation of plane channel flow. Notes on Numerical Fluid Mechanics, Vieweg-Verlag, 64, 1998. 15. H. Le, P. Moin, and J. Kim. Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech., 330:349–374, 1997. 16. K. Akselvoll and P. Moin. Large eddy simulation of a backward facing step flow. In W. Rodi and F. Martelli, editors, Proceedings of Engineering Turbulence Modelling and Experiments 2, pages 303–313. Elsevier Science Publishers, 1993. 17. C.W. Hirt, B.D. Nichols, and N.C. Romero. Sola – a numerical solution algorithm for transient fluid flows. In Los Alamos Sci. Lab. Report LA 5852, Los Alamos, 1975. 18. U. Schumann. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., 18:376–404, 1975. 19. M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3(7):1760–1765, July 1991.
112
ADNAN MERI AND HANS WENGLE
DNS AND LES USING 2ND AND 4TH ORDER DISCRETIZATION
113
114
ADNAN MERI AND HANS WENGLE
PARALLEL MULTI-DOMAIN LARGE-EDDY SIMULATION OF THE FLOW OVER A BACKWARD-FACING STEP AT RE=5100
E. SIMONS1, M. MANNA2 AND C. BENOCCI1 (1) Environmental and Applied Fluid Dynamics Department von Kármán Institute for Fluid Dynamics
Waterloose steenweg 72, B-1640 Sint-Genesius-Rode, Belgium
(2) Dipartimento di Ingegneria Meccanica per l’Energetica
Università di Napoli ”Federico II”
via Claudio 21, 80125, Naples, Italy
Abstract. The present article describes the recent developments carried out at the von Kármán Institute concerning the use of domain decomposi tion techniques as applied to the space filtered incompressible Navier-Stokes equations for the simulation of turbulent flows in complex geometries. Is sues related to the numerical method, based on a grid staggered velocity pressure representations, are addressed, with special emphasis on the solu tion strategy of the elliptic kernel. Numerical results are presented for the flow over a backward facing step for which extensive and accurate DNS data are available in literature.
1. Introduction
Despite impressive progresses of the Large Eddy Simulation (LES) in the past decade, the application of this technique to complex, separated flows remains a challenging task. The success of this endeavor needs, in the opin ion of the present authors, on a wise combination of numerical algorithms, grid refining and subgrid stress (SGS) modeling. In the case of incompress ible flows the practical requirement for an efficient and relatively cheap solution of Helmholtz equations have to be added. As it will be shown, a promising approach to address these somewhat conflicting needs can be the multi-domain (MD) technique. The MD ap 115 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 115-130. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
116
SIMONS E.A.
proach allows to decouple the original problem in a set of two sub-problems, viz. a sub-domain problem (consisting of a sub-set of decoupled problems) and an interface condition (which ensures the appropriate coupling). The evident advantages of this procedure are the easiness of the description and discretisation of complex geometries, a good grid control, the feasi bility of parallel execution on multi-node computers, and, fundamentally the possibility to work on simple sub-domains where Fast Poisson Solvers (FPS) could be applied. In comparison the capacitance matrix technique, which is currently applied to extend FPS to complex geometries (Le and Moin, 1994), (Akselvoll and Moin, 1995) requires the solution of two Pois son problems over an enlarged domain including the actual computational field. Spectral MD approaches have been widely discussed in the recent lit erature and in this respect we must mention the work of (Patera, 1984), and the successive developments and applications of (Henderson and Kar niadakis, 1995). In the finite difference and finite volume framework the application of the MD technique is less common, and we are not aware of any fluid dynamic oriented development. Part of the reasons may be traced back to the difficulties associated with the staggered variables arrangement which is commonly employed for the numerical solution of the incompress ible Navier-Stokes equations in a finite difference framework. The details concerning the numerical methods can be found in (Simons and Manna, 1998) and are only briefly summarized herein. Emphasis is given instead to an original technique which provides well defined inflow boundary condi tions to simulate the flow over a backward facing step at moderate Reynolds number. Following the LES approach the equations are space-filtered and closed with an algebraic closure. This article is organized as follows: in section §2 the governing equations and the subgrid scale model are briefly recalled. Section §3 describes the numerical method with particular emphasis on the solution of the elliptic equations. Finally, in section §4 LES results obtained with the present MD approach are compared in detail to the Direct Numerical Simulation (DNS) data of (Le and Moin, 1994). Satisfactory agreement was found with DNS and with the trends put in evidence by other LES (Akselvoll and Moin, 1995) and experiments (Jovic and Driver, 1994). 2. Governing Equations
The LES formulation is well known to most potential readers and the re lated theory can be easily found in different review articles (Piomelli, 2000), and will not be discussed in depth in the present frame. Applying a spa tial filter to the incompressible Navier-Stokes equations, the corresponding
MULTI-DOMAIN LES
117
formulation for filtered (large scale) quantities (denoted with an overbar) reads:
where is the Reynolds number and being appropriate velocity and length scales), a body force and the subgrid scale stress tensor to be modeled. It has to be remembered that, in the case of Finite Difference (FD) discretisation, the filtering operation is implicit with the discretisation and takes the form of a top-hat filter. There is a considerable experience at the von Kármán Institute in the field of LES, and closures based upon the sub-grid diffusivity concept have been implemented and tested. Results enclosed herein have been obtained applying the Smagorinsky model:
with
is the resolved rate of strain, the filter width, and the magnitude of is added to the resolved rate of strain tensor. The isotropic part of the pressure. A van Driest-type damping function (Piomelli, 2000) D = with has been applied in the wall-normal direction z to impose the correct turbulence decrease towards the wall. The choice of the Smagorinsky model might appear non optimal, in view of the current trend toward the application of scale-similar or mixed subgrid scale models (Sarghini et al., 1999), and the well known limitations of this model against the dynamic procedure (Piomelli, 1987), but it is justified in the present study, whose purpose is to demonstrate the capability of the sub-domain approach to the LES of complex flows and, ultimately, to practical applications. In this optic, the aim is the prediction of low order turbulent moments at the minimal cost. As pointed out by (Haertel et al., 1994) the primary task of the subgrid scale model is to enforce the correct dissipation at the level of the resolved scales. It is our experience that, once this requirement is satisfied over a well resolved grid, a satisfactory result can be expected. Simulations at higher Reynolds numbers for which coarser grids have to be accepted, would require more sophisticated SGS models. Therefore the Smagorinsky model was retained for the present investi gation. This argument is supported by the results of a recent bench-mark on LES flows around bluff bodies (Rodi et al., 1997), whose conclusion was
118
SIMONS E.A.
that the overall performances of advanced SGS models were not signifi cantly better than those obtained with the Smagorinsky model. More pertinent to the present study, a previous LES of the flow over a backward facing step (Akselvoll and Moin, 1995) has shown that the predic tion of the statistics of the resolved scales obtained with the Smagorinsky model were practically identical to the ones obtained with the dynamic pro cedure. These results support the opinion of the present authors that, in FD implementations, the influence of numerical accuracy and grid refinement is predominant with respect to the effect of the SGS model. In-house tests, where the Smagorinsky model has been compared to the filtered structure function model (Piomelli, 2000), have confirmed this fact. 3. Mathematical Formulation
Following the standard pressure correction scheme (Van Kan, 1986) we have decoupled the velocity and pressure at each time step. Thus the projection step requires the solution of a Poisson equation for the pressure, while the predicted velocity field is explicitly integrated with an Adams Bashforth scheme. The semi-discrete equations read:
and
where the R term is given by:
and leads to:
is the pressure correction. Taking the divergence of (5)
which allows to compute and through (5). All partial derivatives in eqs. (4–7) are discretized with second order accurate centered formulae in a staggered grid arrangement, so that veloc ities are defined at cell faces and pressure unknowns at cell centers. The convective terms in (6) are discretized in a skew symmetric form which is energy and momentum preserving.
MULTI-DOMAIN LES
119
The discretization of (7) in a multi-domain formulation is straightfor and denote the ward. Let us begin with some nomenclature. Let solution vectors corresponding to the sub-domains and to the interfaces; the former represents the numerical approximation of the original problem defined in any of the rectangles whose collection adds up to the original domain. In a collocated formulation the interface is not well defined at discrete level (viz. there are no unknowns on the boundaries separating two sub-domains). We thus define as one (or more) layers of unknowns next to the boundary lines which are therefore eliminated from sub-domain count. (see (Simons, 2000) for additional details). Obviously For reasons of efficiency we have used Fourier expansion of the unknowns and the right hand sides of (7) in the homogeneous spanwise direction so that the original three dimensional Poisson problem is reduced to a set two dimensional Helmholtz problems which differ from each other of exclusively for a parameter which essentially depends on
Note that, because of the symmetry of the Fourier transformed equations,
we only need to solve
problems. If we order the unknowns in a domain wise fashion, followed by the in terfacial unknowns, and we apply block Gaussian elimination to the original system of equations arising from the discretization, we find:
where
In (9) is the matrix representing the discretization of the LHS of (7) in N and M the number of sub-domains and interfaces, and:
120
SIMONS E.A.
The interface unknowns can be determined from the lower right sub-block of equations :
The submatrix S is generally referred to as the Schur complement matrix. Note that the connection between the sub-domain and interface unknowns, which is apparently lost, is embedded instead in the rhs’s of (13). To sum marize the solution algorithm reduces to the following three step procedure: 1. Solution of an elliptic problem in each of the subdomains N separately, using the original boundary conditions at the external boundaries, homogeneous Dirichlet boundary conditions at the inter nal boundaries and its original rhs’s 2. Assembling of the rhs’s of the interfacial problem using the step 1 values, and solution of the Schur complement problem. 3. Solution of an additional elliptic problem in each of the subdomains separately, using the original boundary conditions at the external boundaries, imposing now as Dirichlet boundary conditions at the interfaces and using the original rhs’s Although at a first sight the overall cost of the elliptic kernel appears more than doubled (two elliptic problems plus an interface problem of smaller size), this overhead is largely compensated by the fact that this approach not only allows to recover the use of fast Poisson solvers on rectangular do mains (to tackle step 1 and 3), but also opens the way to parallel executions. Additionally, in an iterative solution context, it provides the opportunity of relaxing the stiffness of the full scale original problem (which is reduced to a set of smaller size, better conditioned sub-problems) and is intrinsically suited for zonal approaches and heterogeneous problems. Keeping in mind that our application is the LES of turbulent flows in complex geometries, which effectively requires the solution of several elliptic equations with varying rhs’s for a large number of times steps, we argue that it is convenient and feasible to compute and store the inverse of the Schur matrix through a pivoted LU factorization procedure in a pre-processing step. The overhead of the factorization will be quickly amortized over the large number of time steps necessary to build the statistics. We only wish to underline that when the original problem is rank deficient because of the kind of boundary conditions selected, the singularity will show up in step 2 of the MD algorithm and has therefore to be tackled. There is no space here to discuss issues related to the numerical solution of the discretized elliptic problem, and more specifically the opportunity of
MULTI-DOMAIN LES
121
carrying out step 1 and 3 in a direct or iterative fashion. Results reported herein were obtained with the fast direct solver of Swarztrauber (Swarz trauber, 1974) which, for rectangular geometries, is optimal both in terms of storage an operation count (order Nlog(N)). Whether this solver is su perior compared to a state of the art iterative Krylov subspace solver like the preconditioned restarted generalized minimum residual method (GM RES), is still an open question. In this respect let us just mention that in order to accommodate for the variation of the turbulent length scales throughout the computational domain, violent stretching was applied in all subdomains both in the streamwise and wall normal directions, which has, usually, a negative effect on the stiffness of the original non preconditioned system.
The solver has been parallelized on a sub-domain basis, and in Table 1 we report the performances of the elliptic kernel for the five sub-domains configuration employed in the simulation of the backward facing step flow described in §4. The timings refer to 100 elliptic solves. The tests have been carried out on an 8 processor symmetric multi-processor Sun HPC 3500 shared memory machine consisting of eight 400 MHz UltraSPARC II processors with 8 Gbytes of shared memory and a 2.6Gb/sec Gigaplane system interconnect running Solaris 2.7 and Sun MPI. The serial calculation performance measurements were obtained on a single processor of the Sun E3500 shared memory architecture, whereas the parallel timings were done using 5 of its processors such that each subdomain is allocated to a different processor. We note that the parallel speedup1 is about fair because of the poor load-balancing achieved with the current domain decomposition. The latter was essentially conceived for minimal computation effort and optimal grid spacing, so that the sizes of the sub-domain differ slightly. Overall, however, the parallel performance is considered more than satisfactory. 1
As customary the quantities S and E denote the speedup and the efficiency defined as S = T(1)/T(P) and E = S(P)/P, where T(1) and T(P) are the execution times on one and P processors, respectively.
122
SIMONS E.A.
4. Results
The flow over a backward-facing step (BFS) is a widely studied benchmark case for CFD (see figure 1). Matter of fact the detailed simulation of the recovery region of the flow still remains a challenging task for any numerical approach. The availability of extensive DNS data of the BFS at moder
ate Re number (Le and Moin, 1994) provides an excellent opportunity to validate and assess the performances of the present LES code. Specifically, the simulation should be able to reproduce the particular effects put in ev idence for the first time by the DNS, such as the extremely low minimum value found for the skin friction within the re-circulation region and the extreme slowness of the recovery towards equilibrium of the boundary layer downstream of the reattachment point. Both observations seem to be the result of low Reynolds number effects. For this purpose the Smagorinsky coefficient was set to Corresponding simulations, performed using the value have con firmed the relative insensibility of the results to the details of the SGS model. 4.1. PROBLEM DEFINITION
The present configuration consists of a single-sided expansion duct with expansion ratio of where H is the total height of the domain behind the step expansion and is the height of the step. The
MULTI-DOMAIN LES
123
Reynolds number used for this simulation was of 5100, based on the freeof the incoming developing turbulent boundary layer stream velocity In the DNS the boundary layer thickness of the incoming flow and was found to be in the order of at the location upstream the corner. In the present multi-domain approach the computational field is divided into three logical parts, made of five sub-domains, as shown in figure 2. The first part is an equilibrium half-channel with no-slip wall at the bottom boundary, a slip wall at the top boundary, and periodic boundaries in stream-wise and span-wise directions. This sub-domain, of length acts as pre-solution to generate, in a concurrent way, the inflow condition for the actual BFS simulation. The first subdomain is followed by a true inlet section, which extends itself down to the corner. This region is subdivided in two consecutive sub-domains and of combined length In this region the flow evolves from the inflow condition towards a developed boundary layer. The following post-expansion downstream of the corner covers a length subdivided in the two sub-domains and In this region the flow separates at the corner, reattaches downstream on the bottom wall and evolves towards a boundary layer in equilibrium. The location of the outflow boundary, at 20 step heights downstream of the expansion, is the same as for the DNS (Le and Moin, 1994). The spanwise size of the computational
field, was also chosen equal to the one used in DNS in order to ensure full comparison between the two calculations, even if this value was found a posteriori to be only marginally sufficient to guarantee an adequate drop of the velocity cross-correlation (Le and Moin, 1994). The total height of the post-expansion domains is
124
SIMONS E.A.
4.2. BOUNDARY CONDITIONS The inflow boundary conditions are provided by the concurrent simula and tion, which allow to produce boundary layer thickness turbulence close to the ones produced by the DNS at the stream-wise lo upstream of the step expansion. It is to be remarked that the cation present inflow condition allows a much physical development of the incom ing boundary layer with respect to the perturbation techniques applied in the corresponding studies (Le and Moin, 1994), (Akselvoll and Moin, 1995). At the outflow, a convective boundary condition, coupled with a mass cor rection technique, is applied. No-slip conditions are applied on all the wall boundaries. This choice entails the adoption of a grid fine enough to resolve the turbulent structures in the wall layer. At the upper boundary of the computational domain the symmetry conditions, also referred to as free slip or no-stress condition, is applied for all the sub-domains. 4.3. GRID DEFINITION As mentioned before, the grid must be refined enough to resolve the turbu lent structures in the wall layer. Therefore the one adopted for the present study is highly stretched in both stream-wise and wall-normal directions. In wall coordinates (on the basis of the computed friction velocity at the exit of the computational domain grid spacings in the stream-wise direction range from a minimum value at the step of to a maximum one at the outflow boundary of In the wall-normal direction the minimal cell size is: against a spacing of at the half height of the step and a maximum of at the upper zero-stress boundary of the domain. The resolution of the present grid is comparable to the one used in (Akselvoll and Moin, 1995). The grid was designed to adequately resolve the wall layer regions both upstream and downstream of the corner. The spanwise cell size ensures that the longitudinal structures which can be expected to exist within the wall layer are well resolved by the grid. Due to the Cartesian grid definition, the fine grid close to the wall in the inlet section is carried on the interface between subdomain and It was found that this set-up introduces a severe time step limitation due to the wall-normal advection term, so that the advective limit becomes stricter than the diffusive one induced by the wall-normal terms. Namely, for the current grid, the convective and viscous time step limits have been found to be:
MULTI-DOMAIN LES
125
Therefore the implicit treatment of the wall-normal diffusive terms, as often applied via a Crank-Nicholson discretisation, would not bring any improvement in the allowable time step size with respect to a fully explicit algorithm, if the wall-normal convective term is not implicit as well. This would involve a complex supplementary linearisation procedure for the wallnormal momentum equation (Akselvoll and Moin, 1995) and therefore was not applied for the current study. The temporal discretisation was thus kept fully explicit and the time step has evolved in time to satisfy the maximum allowed as determined by the CFL and viscous stability limits. The actual grid-size numbers per subdomain are summarized in table 2. The total amount of grid points including the half prechannel subdomain equals about 400,000 points. This value corresponds to less than 5% of the grid-points (8, 290, 304) employed in the DNS study.
4.4. SIMULATION The calculation was started from a uniform flow and advanced over 50000 time steps to get rid of initial transients. Subsequently statistics were gath ered over 85000 time steps corresponding, to a total of 412 time units. Statistics were obtained by averaging over the periodic spanwise direction and time; for processing purposes all staggered variables were interpolated to the pressure points. Computations were done both serially on a Digital-Alpha workstation with a EV6.7 chip operating at 667 MHz and with 2 Gbyte of RAM and on a Sun HPC 3500 shared memory architecture at EPCC with 5 processors available for parallel execution. Each time step took 1.3125 seconds on the Alpha workstation compared to 1.2 seconds for the 5-processor Sun combination putting into evidence the increased performance of the Alpha workstation on a single processor basis. Overall simulations required 28
126
SIMONS E.A.
CPU hours on the 5-processor Sun set-up and 31 CPU hours on the serial workstation. The flow upstream the corner was successfully reproduced: the height of the boundary layer on the corner wall and the features of the turbulence were found close to the results of the DNS, ensuring that the at downstream development will be similar for the two cases. The shear layer is bent down by the low pressure region created by the expansion and reattaches itself on the lower wall approximately 6 stepheights downstream the corner to give birth to a new attached wall-layer, which slowly redevelop itself towards an equilibrium boundary layer flow.
The evolution of the mean wall friction coefficient on the bottom wall is presented in figure 3. Comparison of the DNS result shows that the length of the secondary re-circulation bubble is slightly over-predicted, as made evident by the delayed first crossing of the zero line. The second crossing, corresponding to the closing of the main re-circulation region, also shows a comparable over-prediction. The predicted length, of the separate region is found to be to be compared with the value of the DNS and the ones obtained, with different SGS models, in the correspond ing LES study (Akselvoll and Moin, 1995): the present deviation of +5.1% appears comparable to the one found in the former calculation. The min imum value of is identical for both calculations. This is an extremely low value that had never been observed before the DNS study of (Le and Moin, 1994) and the accompanying experimental study of (Jovic and Driver, 1994). The current results provide further confirmation of this finding. At the outflow boundary the wall friction under-predicts the DNS
MULTI-DOMAIN LES
127
value by about 8 %. Downstream of reattachment a slow recovery towards an equilibrium boundary layer can be observed in figure 3. It is remarkable that, even 20 step heights downstream the corner the profiles of mean ve locity still fall well below the theoretical log-law correlation. Again these findings are in agreement with (Le and Moin, 1994) and (Jovic and Driver, 1994).
Mean, rms velocity and shear stress profiles are presented in figure 4 and and the region for the separated region (locations where the flow has reattached itself (locations and 16). The mean streamwise velocity profiles show overall a very good agreement with the DNS data even if there is a small delay in the reattachment and an over-prediction of the re-circulation length already remarked in figure 3. The presence of a positive velocity close to the wall at is consistent with a minor over-prediction of the size of the secondary re-circulation the predicted bubble with respect to the DNS data. At locations profiles are very close to the reference data. It should be noted that the small over-velocity at the step height predicted by the DNS for position is adequately captured. The streamwise turbulence intensity profiles given in figure 5, show that is well represented. This feature cor the maximum peak value at responds to the inner shear layer emanating from the step. Further down stream this maximum is slightly over-predicted but still close to the DNS value. The presence of a local maximum very close to the wall downstream of reattachment puts in evidence the mechanism of a redeveloping boundary layer.
128
SIMONS E.A.
The comparison of span-wise and wall-normal turbulence intensities, fig ures 6-7, show similar trends. The maximum value at the first downstream position is well found and there is a slight over-prediction of the maximum for most positions further downstream. Nevertheless, overall the agreement between the two simulation is quite acceptable. The discrepancy between DNS and LES is in the same order as the one found in the comparable investigation by (Akselvoll and Moin, 1995) and put in evidence similar trends. Finally the shear stress profiles are compared in figure 8. The shear stress given for the LES represents the total shear stress: the sub-grid contribution is included. The present results match well the DNS data both upstream
MULTI-DOMAIN LES
129
and downstream of reattachment.
5. Conclusions The presently wall-resolved LES results for the flow over a backward-facing step at moderate Re = 5100 show an overall good agreement with the exist ing DNS data set of (Le and Moin, 1994). The largest discrepancies between the two simulations can all be linked to the fact that the length of both primary and secondary recirculation regions predicted by the present LES is larger than the DNS by an amount of about Second-order statis tics such as turbulence intensity and shear stress profiles were convincingly
130
SIMONS E.A.
captured. A further confirmation of the low negative peak value of the skin friction coefficient inside the recirculation region and a particularly slow recovery towards equilibrium of the turbulent boundary layer developing downstream of reattachment for the current Re = 5100 were obtained. The main conclusion to be drawn of the present investigation is that a strategy based on fully explicit parallel multi-domain Schur complement algorithm is a general, flexible and cost-effective approach for LES of complex ge ometry flows. Moreover, it is shown that the concurrent inflow generator procedure is a valid approach for the creation of realistic turbulent flow at the upstream boundary for non-equilibrium flows. Concerning SGS closure models, the present results show that the Smagorinsky model remains a viable option for the LES of complex geometry flows so far as numerical accuracy of the solver and grid refinement have a stronger influence on the final simulation than the details of the SGS model. In this optics, the MD technique which allows to refine the grid on a sub-domain basis appears a very promising approach. References Akselvoll, K. and Moin, P. Large Eddy Simulation of Turbulent Confined Coannular Jets and Turbulent Flow over a Backward Facing Step. Report TF-63, Stanford University, 1995. Haertel, C., Kleiser, L., Unger, F. and Friedrich, R. Subgrid-scale Energy Transfer in the Near-Wall Region of Turbulent Flows. Phys. Fluids, 6(9):3130-3143, 1994. Henderson, R. and Karniadakis, G. Unstructured Spectral Element Methods for Simu lation of Turbulent Flows. J. Comp. Phys., (122):191–217, 1995. Jovic, S. and Driver, D. M. Backward-Facing Step Measurements at low Reynolds Num ber Re = 5000. NASA TM 108807, 1994. Le, H. and Moin, P. Direct Numerical Simulation of Turbulent Flow over a BackwardFacing Step. Report TF-58, Stanford University, 1994. Patera, A. T. A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion. J. Comp. Phys., (54):468–488, 1984. Piomelli, U. High Reynolds Number Calculations Using the Dynamic Subgrid Scale Stress Model. Phys. Fluids, 5(6):1484–1490, 1993. Piomelli, U. DNS and LES, Basic Principles, Current Trends, Expectations. Von Karman Institute, Lecture Series 2000-04, 2000. Rodi, W., Ferziger, J., Breuer, M. and Pourquié, M. Status of Large Eddy Simulation: Results of a Workshop. J. Fluids Eng., (119):248–262, 1997. Sarghini, F., Piomelli, U. and Balaras, E. Scale-Similar Models for Large-Eddy Simula tions. Phys. Fluids, 11(6):1596–1607, 1999. Simons, E. and Manna, M. A Multi-Domain Approach to Large-Eddy Simulation of Complex Turbulent Flows. Numerical Methods for Fluid Dynamics VI, M.J. Baines ed., 1998. Simons, E. An Efficient Multi-Domain Approach to Large Eddy Simulation of Incom pressible Turbulent Flows in Complex Geometries. Ph.D. thesis, K.U. Leuven, 2000. Swarztrauber, P. N. A Direct Method for the Discrete Solution of Separable Elliptic Equations. SIAM J. Numer. Anal., 11(6):1136–1150, 1974. Van Kan, J. A Second-Order Accurate Pressure-Correction Scheme for Viscous Incom pressible Flow. SIAM J. Sci. Stat. Comput, 7(3):870–891, 1986.
3. Cartesian grids for complex geometries
This page intentionally left blank
LES OF FLOW AROUND A CIRCULAR CYLINDER AT A SUBCRITICAL REYNOLDS NUMBER WITH CARTESIAN GRIDS
F. TREMBLAY, M. MANHART AND R. FRIEDRICH Fachgebiet Strömungsmechanik Technische Universität München Boltzmannstr. 15, D-85748 Garching, Germany
Abstract. Large eddy simulations (LES) of turbulent flow around a circu lar cylinder are performed with a novel technique using Cartesian grids. Sev eral test cases investigate the influence of the subgrid-scale model (Smagorin sky vs. dynamic Germano) and the grid resolution. The results show overall fair agreement between LES and a recently performed DNS. An important issue turns out to be the proper prediction of the mean recirculation length. Results are also compared with experimental data for the near and far wake regions. The shape of the mean streamwise velocity profile inside the recir culation bubble remains an open question.
1. Introduction Flow across a circular cylinder is one of the classical flow problems which are not understood in all details. At a Reynolds number based on freestream velocity and diameter of 3900, which is low for most technical applications, the flow is already very complex. It is characterized by laminar separation, transition to turbulence in the free shear layers leaving the body and shedding of large-scale vortices. These features have been the subject of numerous experimental and numerical investigations. There seems now to exist consensus about the fact that the near wake region is dominated by the shear layer dynamics and is very sensitive to disturbances (not those due to the relevant instability mechanisms) and cylinder aspect ratio. Such disturbances may, in the experiment, be transported by the freestream and in the simulation result from underresolution, the numerical scheme and the SGS model. Kravchenko and Moin [8], [9] have shown that the shape 133 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 133-150. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
134
F. TREMBLAY ET AL.
of the mean streamwise velocity profile in the near wake depends on the level of velocity fluctuations in the shear layer and is thus related to transi tion. Systematic spectral DNS and LES of Ma et al.[12] indicate that small spanwise computational domains and high-dissipation LES favor U-shape rather than V-shape velocity profiles. The latter were also observed in PIV-experiments of Lourenco and Shih (published in [1]) in the near wake In view of various possible sources of error both in the experiment and the numerical simulations (boundary conditions, sam pling rates) further numerical and experimental investigations of the flow are certainly desirable to further across a circular cylinder at clarify this issue. Different numerical methods have been applied in the past to perform DNS and LES of this flow at Beaudan and Moin [1] used high order upwind-based schemes to solve the low-pass filtered Navier-Stokes equations on O-type grids. They found that these schemes exhibited sig nificant numerical dissipation which affected the small turbulent scales. In their LES of the same flow on C-type curvilinear grids Mittal and Moin [16] used a second-order conservative central-difference scheme in planes perpendicular to the cylinder axis and a Fourier-spectral method in the spanwise direction. Power spectra of velocity fluctuations obtained with this nondissipative scheme were in agreement with the experiment over a wider range of wave numbers than power spectra obtained with the upwind-biased schemes. These issues motivated Kravchenko and Moin [8], [9] to perform LES of this flow with a new method based on B-splines and zonal grids. This technique provided results superior to those of [8] and [16]. Numerical aspects of LES of this flow were also studied by Breuer [2] and Fröhlich et al.[4] using a technique based on a collocated variable arrangement and general curvilinear coordinates or unstructured grids [4]. Their work con firmed that large-eddy simulations with central difference schemes are closer to reality than those performed with dissipative methods. Ma et al. [12] presented spectral direct and large-eddy simulations on unstructured grids. They used a Fourier expansion in the spanwise direction and triangular elements in planes perpendicular to the axis, filled with Jacobi polynomial modes of order P, variable from element to element (p-refinement). The expansions are constructed so that they retain the tensor-product property and thus the efficiency of spectral methods. Simulations were performed with varying number of elements (h-refinement). A nice feature of spectral/hp methods besides their accuracy lies in the fact that they are very flexible and can be used to predict flows over arbitrarily shaped bodies. A method that is certainly as flexible, but perhaps more efficient than a spectral-element method or a method based on general curvilinear coordi nates, is presented and applied below. It belongs to the class of immersed
LES OF FLOW AROUND A CIRCULAR CYLINDER
135
boundary methods for incompressible flows and uses staggered cartesian grids. It is attractive in the sense that cartesian codes are e.g. between 10 to 30 times more economical in terms of both memory and CPU time requirements when compared to a code for general curvilinear coordinates [14]. One can thus afford to do a computation with higher grid resolution and still achieve appreciable savings in computational resources. Another important aspect is the complete elimination of the need to produce a bodyfitted grid, a task that is not trivial and in general consumes an important amount of time. 2. Numerical approach
The code MGLET, used here, is a parallel finite volume solver for the incom pressible Navier-Stokes equations on staggered cartesian non-equidistant grids. The spatial discretization is central and of second order accuracy for the convective and diffusive terms. For the time advancement of the momentum equations, an explicit second-order time step is used which is central with respect to the convective terms. The pressure solver uses a multigrid method based on a point-wise velocity-pressure iteration [13]. The most straightforward method to represent the no-slip condition on an immersed body surface within a cartesian grid is to apply zero velocity at the cell-face which is the closest to the surface of the body. The differ ence between the actual and exact body geometry is at most half a cell. This was the method employed by Manhart and Wengle [15] among others. The drawback of this method is that it is only first order accurate. This drawback can be removed by modifying the discretization near the body in order to take into account the cells which are cut. While more accurate schemes were obtained in [21] for two-dimensional flows, the extension of the methods to 3D geometries is not easy because a cell can be cut in many different ways. Other researchers [3],[6],[17], make efforts to preserve the same discretization in all the domain, even including the cells inside the body. Boundary body forcing is applied at the location of the body in order to represent its effect on the fluid. A major issue encountered is the interpolation of the forcing over the grid that determines the accuracy of the scheme. Another approach [18] is called diagonal cartesian method on staggered grids. It is quite similar to our approach, discussed in section 3, where we apply Dirichlet velocity boundary conditions on each cell located in the immediate vicinity of the body surface. The boundary conditions are applied in such a way that the physical location of the surface and its velocity are best represented. The cells beyond the body surface are ex cluded from the computation by using a masking array. The discretization remains the same for all cells. A similar approach was derived indepen
136
F. TREMBLAY ET AL.
dently by Gullbrand et al. [7], but in the context of finite differencing on regular grids. 3. Handling of arbitrarily shaped bodies
The first step is to obtain a representation of the surface of interest. A very versatile way of doing it is by using an unstructured mesh made of triangles. All geometries can be represented this way. Once the body is available, a preprocessing step is done. During this phase a masking array is initialized, which has values of one for cells within the flow and zero for cells belonging to the body. This array is used to block the pressure cells inside the body considered so that they do not contribute to the computa tion. Boundary conditions have to be applied to the velocity components at the interface between masked and unmasked pressure cells in order to have a well defined problem. In the second and last phase of the prepro cessing step, these boundary conditions are defined as a linear function of the velocities computed within the domain. The coefficients weighting the computed velocities are determined and need no further updating during the run. 3.1. PREPROCESSING STEP 1: MASKING THE PRESSURE CELLS
Blocking pressure cells depends on whether the intersection between compu tational cells and triangles, representing the body surface, satisfies certain conditions which are described below. These conditions have to be simple and unique. We start specifying a triangle by its 3 vertex points (see figure 1) :
They define a plane in which a point P lies, that is given by its position vector and satisfies:
From eq.(1), we get the equation of the plane
where
LES OF FLOW AROUND A CIRCULAR CYLINDER
137
This plane may have up to three intersection points with coordinate lines passing through the cell center C (the pressure point) and these points may and the intersection lie inside or outside the cell volume. We call points with the (x,y,z)-directions, respectively. Their coordinates are :
where is the cell center. If one of these 3 points lies within the pressure cell and within the triangle, then this pressure cell is blocked out of the computational domain and does not contribute to the numerical solution. Verifying if a point lies in a triangle is done in the following way: Let us consider a point D that has been found to lie on the plane defined by a triangle (see figure 2). The vectors perpendicular to the edges of the triangle and lying in its plane are:
The point D is inside the triangle if and Once all the cells have been checked, we obtain a thin layer of blocked cells along the surface of the body. The interior of the body must also be blocked. This is easily done by a “painting” algorithm, i.e. the user specifies one cell which is in the flow field (open cell), and by iteratively marking the neighbours of the open cells unless they are blocked, we can paint the exterior of the body and thus the interior is easily identified. 3.2. PREPROCESSING STEP 2: DETERMINING DIRICHLET VELOCITY BOUNDARY CONDITIONS
Once we have obtained the blocking array of the pressure cells, the blocking arrays of the velocities can be deduced in a straightforward manner. Each velocity component belonging to a blocked pressure cell is also blocked. In order to represent the effect of the body on the surrounding flow, a few layers (enough to have a complete stencil of the discretization at the first open cell) of blocked velocities are interpolated/extrapolated using the noslip condition that prevails at the surface of the body. We use third order
138
F. TREMBLAY ET AL.
Lagrangian polynomials to achieve this goal. In figure 3, is the location of the point to obtain an extrapolated variable, is the location of the surface of the body and are the positions of the neighbouring points. is the variable to be extrapolated (say a velocity component), is the value at the wall which should be zero in case of a no-slip condition, are the values taken from the neighbouring points. The value of the extrapolated variable will be :
where
We can define the distance from the extrapolated point to the wall as e.g.. If the variable at a point can be extrapolated from more than one direction, each direction is weighted by a multiplication
LES OF FLOW AROUND A CIRCULAR CYLINDER
139
factor
where
This formulation ensures that if a point is exactly on a wall, it will have the wall value. Also, the sum of the multiplication factors is always 1. From these simple formulas, all the coefficients used to set the velocities at the surface of the body are determined and used during the actual computa tion. The explicit time advancement scheme for the momentum equations permits us to treat the blocked cells in the same way as the other cells i.e. all velocity cells are advanced in time in the same manner. For each open pressure cell, a pressure correction is computed so that the divergence is driven to zero. For the blocked pressure and velocity cells, no correction is applied. The total overhead in computational time produced by the in troduction of the present method is less than 10%. The method has been validated for a number of laminar cases, where second order accuracy has been demonstrated. A DNS of subcritical flow over a cylinder has been performed and a very good agreement with experimental results has been found [20]. 4. Discussion of LES results
Besides PIV measurements of Lourenco and Shih, published in [1], we use hot-wire data of Ong and Wallace [19] for comparison with our computa tions. The hot-wire device cannot be used in recirculating flows. Hence, data are available only downstream of the recirculation zone, i.e. for Our computational domain is 20 diameters D long in the streamwise xdirection, with the center of the cylinder being 5D downstream of the inflow
140
F. TREMBLAY ET AL.
plane. In the normal y-direction, the domain size is also 20D. The spanwise extent of the domain was chosen to be which corresponds to the size used by most previous authors. Recently, Ma et al. [12] have pointed out the need to choose bigger spanwise computational domains, in order to obtain the ’correct’ shape of the streamwise velocity profile. This is an interesting aspect which we intend to investigate further in the near future. The boundary conditions prescribed in the present simulations are a uni form, unperturbed inflow velocity and periodic velocity components in the normal and spanwise directions. A zero-gradient outflow condition holds at Rather than using uniform inflow velocity, Kravchenko and Moin [9] prescribed potential flow velocities. The streamwise potential flow velocity in the symmetry plane at differs from by only 1%. We presume this difference to have a negligible effect on the overall flow structure. Two meshes with different resolutions, (case LES1) and (case LES2) cells have been used to investigate the grid dependence in (x,y,z)-directions of the solution. The total number of cells is summarized in Table 1. This table also contains the number of cells needed to discretize the square DxD which contains the cylinder. The last two columns represent the grid stretching factors. The corresponding data of the DNS presented in [20] are included for comparison. Figure 4 shows instantaneous isosurfaces of constant pressure fluctua tion from DNS and LES1 smago computations. The large vortical structures (rollers) and the braids, are easily identified. The LES simulation is able to capture such phenomena, but looses the finer scales as can be seen for the DNS. All the flow statistics of the LES and DNS have been accumulated over 300 problem times which correspond to approximately 60 vortex shedding cycles. The discussion of flow parameters starts with a comparison of the mean drag coefficient base pressure coefficient separation angle length of mean recirculation region and Strouhal shedding fre for three different LES, the LES of Kravchenko and Moin [9], quency
LES OF FLOW AROUND A CIRCULAR CYLINDER
141
case 2, the DNS of Ma et al. [12], case I, and our own DNS [20]. The Smagorinsky sgs model with a model coefficient of 0.1 and a length scale but no wall damping was applied in both cases (LES1, LES2). The dynamic Smagorinsky model of Germano et al. [5] with Lilly’s modification [10] was used along with the fine grid only (LES1 dyn). The coarse grid computation together with the Smagorinsky model (LES2 smago) provides the poorest results. On the fine grid, the difference between both sgs models is unimportant. The separation angle and the shedding frequency agree well with the experimental data. For the drag coefficient and the base pressure coefficient the agreement is fair. The length of the mean recirculation region, however appears to be too small. As we will see later, is a quantity which, if not accurately predicted, is responsible for most of the discrepancies in the mean velocity profiles. Figure 5 shows the mean streamwise velocity in the symmetry plane of the cylinder wake. The maximum backflow velocity is fairly well predicted in all cases, especially by the DNS. The underprediction of the recirculation length by all LES runs is obvious and has been documented in Table 2. There is no explanation for the undulations in the experimental profile for nor for the mismatch in the overlap region of both experiments. The following Figures 6-10 contain profiles of the mean streamwise and crossflow velocities, of variances of the corresponding velocity fluctuations and 1.54 and of the Reynolds shear stress at two locations within the recirculation zone and one in the far wake, We note a fair agreement for the streamwise velocity and its variance between experiment and fine-grid LES (Figures 6 and 8). Obviously, a coarse grid LES does not properly resolve the flow phenomena in the recirculation zone. This is by the way, true for all flow quantities (Figures 6-10). The
142
F. TREMBLAY ET AL.
measured mean crossflow velocity in Figure 7 is obviously erroneous since While the it is asymmetric and does not fall off to zero for LES predicts higher levels for the crossflow velocity fluctuations than the experiment, the Reynolds shear stresses coincide pretty well, at least in the near wake, (Figures 9,10). For most of the variables the DNS naturally provides the best predictions. In order to demonstrate the importance of properly predicting the mean we have plotted profiles of the mean streamwise ve recirculation length locity and its variance at locations which are the same when normalized Hence the location e.g. corresponds with the corresponding for the DNS where is measured from the back of the to cylinder while starts from its center Figure 11 shows pro files of the mean streamwise velocity at three locations in the near wake. There is a surprising collapse of the computed results for the two first loca tions. While we observe the characteristic V-shape profile at in agreement with fully turbulent free shear layers and the experiment, such a Our profiles confirm the shape does not perfectly appear at fine-grid (case 5) simulation of Kravchenko and Moin [9], which leads to a shape intermediate between U and V. We finally recall the argument of Ma et al. [12] that the appearance of a U-shape profile is the consequence of using a spanwise box size of rather than The present results were obtained with They do not seem to support Ma et al.’s argument. We observed that the shape of the velocity profiles develops from U towards V as long as the recirculation length has not settled down to a stable state. This may take many vortex shedding intervals and 7 may not be enough. This point needs further clarification both experimentally and numerically. The streamwise velocity fluctuations in Figure 12 reflect a much better behavior compared to the experiment, when they are taken at physically corresponding locations Again the DNS data seem to be the most reliable. Within the free shear layers surrounding the mean recir culation bubble, the LES data indicate effects of overprediction which are well known from underresolved channel flow simulations. Figures 13 and 14 contain one-dimensional frequency spectra at two downstream locations along the centreline of the wake. The Lomb periodogramme technique was used for spectral analysis. Between 2450 and 8490 samples points in the of the velocity fluctuations were collected at each of the spanwise direction over a time-interval ranging from to 90 (the higher values are for the DNS, respectively). The frequency is non dimensionalized by the Strouhal shedding frequency. This frequency is very pronounced and produces nice peaks at both locations in the crossflow ve locity spectra at In the streamwise velocity spectra this peak appears at twice the shedding frequency, At location
LES OF FLOW AROUND A CIRCULAR CYLINDER
143
it is still visible for one of the LES computations (Figure 14). The crossflow velocity spectra reveal a second frequency which is the higher harmonic of the shedding frequency. All LES data and of course DNS data confirm that there exists an inertial range in the selected region of the wake. The size of that range is properly predicted in the DNS and appears shorter as the field resolution decreases. The DNS data contain more energy than the experimental data in the high frequency range. The same was observed in the DNS of Ma et al. [12]. Overall, we observe good agreement of our LES and DNS data with measured frequency spectra of Ong and Wallace [19].
5. Conclusions
Several LES runs of the transitional/turbulent flow over a cylinder at Re = 3900 have been conducted using a Cartesian grid method which was recently developed. They demonstrate the capability of this method to provide reli able solutions in an efficient way for flows around bodies of arbitrary shape. The resolution and subgrid-scale model study shows a clear effect of the grid refinement, but a rather weak effect due to the subgrid scale model. The prediction of the proper recirculation length appears to be of primary importance, since then only a comparison of data makes sense at fixed lo cations. The issue of the right shape of the mean streamwise velocity profile inside the recirculation zone appears to be still open. Its clarification needs new careful experiments which are definitely free of external perturbations and further computations both on narrow and wide domains and with sam pling times covering many vortex shedding intervals.
144
F. TREMBLAY ET AL.
LES OF FLOW AROUND A CIRCULAR CYLINDER
145
146
F. TREMBLAY ET AL.
LES OF FLOW AROUND A CIRCULAR CYLINDER
147
148
F. TREMBLAY ET AL.
6. Acknowledgement
The first author gratefully acknowledges the financial support of this work in the framework of the EC project Alessia.
LES OF FLOW AROUND A CIRCULAR CYLINDER
149
References 1. P. Beaudan and P. Moin. Numerical experiments on the flow past a circular cylin der at sub-critical Reynolds number. Report No. TF-62, Thermosciences Division, Department of Mechanical Engineering, Stanford University, 1994. 2. M. Breuer. Large eddy simulation of the subcritical flow past a circular cylinder: numerical and modeling aspects. Int. J. Numer. Meth. Fluids, 28:1281–1302, 1998. 3. E.A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusoff. Combined immersedboundary finite-difference methods for three-dimensional complex flow simulations.
150
F. TREMBLAY ET AL.
J. Comp. Phys., 161:35–60, 2000. 4. J. Fröhlich, W. Rodi, Ph Kessler, S. Parpais, J.P. Bertoglio, and D. Laurence. Large eddy simulation of flow around circular cylinders on structured and unstructured grids. In E.H. Hirschel, editor, Notes on Numerical Fluid Mechanics, pages 319–338. Vieweg-Verlag, 1998. 5. M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3(7):1760–1765, July 1991. 6. R. Glowinski, T.W. Pan, and J. Periaux. Fictitious domain methods for incom pressible viscous flow around moving rigid bodies. In Proceedings of MAFELAP, Brunnel University, UK, 25-28th June 1996, 1996. 7. J. Gullbrand, X.S. Bai, and L. Fuchs. High order cartesian grid method for calcu lation of incompressible turbulent flows. Int. J. Numer. Meth. Fluids, to appear, 2000. 8. A. Kravchenko and P. Moin. B-spline methods and zonal grids for numerical sim ulations of turbulent flows. Report No. TF-73, Flow Physics and Computation Division, Department of Mechanical Engineering, Stanford University, 1998. 9. A.G. Kravchenko and P. Moin. Numerical studies of flow around a circular cylinder Phys. Fluids, 12:403–417, 2000. at 10. D.K. Lilly. A proposed modification of the Germano subgrid scale closure method. Phys. Fluids A, 4(3):633–635, 1992. 11. L.M. Lourenco and C. Shih. Characteristics of the plane turbulent near wake of a circular cylinder, a particle image velocimetry study. 1993, published in [1], Data taken from [8]. 12. X. Ma, G-S. Karamanos, and G.E. Karniadakis. Dynamics and low-dimensionality of a turbulent near wake. J. Fluid Mech., 410:29–65, 2000. 13. M. Manhart. Direct numerical simulation of turbulent boundary layers on high performance computers. In E. Krouse and W. Jäger eds, editors, High performance Computing in Science and Engineering 1998. Springer Verlag, 1999. 14. M. Manhart, G.B. Deng, T.J. Hüttl, F. Tremblay, A. Segal, R. Friedrich, J. Piquet, and P. Wesseling. The minimal turbulent flow unit as a test case for three differ ent computer codes. In E.H. Hirschel, editor, Vol. 66, Notes on numerical fluid mechanics, pages 365–381. Vieweg-Verlag, Braunschweig, 1998. 15. M. Manhart and H. Wengle. Large-eddy simulation of turbulent boundary layer flow over a hemisphere. In Voke P.R., L. Kleiser, and J-P. Chollet, editors, Di rect and Large-Eddy Simulation I, pages 299–310, Dordrecht, March 27-30 1994. ERCOFTAC, Kluwer Academic Publishers. 16. R. Mittal and P. Moin. Suitability of upwind-biased finite difference schemes for large eddy simulation of turbulent flows. AIAA Journal, 35:1415, 1997. 17. J. Mohd-Yusof. Combined immersed-boundary/b-spline methods for simulations of flow in complex geometries. In Annual Research Briefs - 1997, pages 317–327. Center for turbulence research, Stanford, 1997. 18. C. Moulinec, M.J.B.M. Pourquié, B.J. Boersma, and F.T.M. Nieuwstadt. Diagonal cartesian method on staggered grids for a DNS in a tube bundle. In Proceedings of the 4th Int. Workshop on Direct and Large-Eddy Simulation, Univ. of Twente, The Netherlands, July 18-20, 2001. 19. J. Ong, L. & Wallace. The velocity field of the turbulent very near wake of a circular cylinder. Experiments in Fluids, 20:441–453, 1996. 20. F. Tremblay, M. Manhart, and R. Friedrich. DNS of flow around a circular cylinder at a subcritical Reynolds number with cartesian grids. In Proceedings of the 8th European Turbulence Conference, EUROMECH, Barcelona, Spain, pages 659–662. CIMNE, 27-30th June 2000. 21. G. Yang, D.M. Causon, and D.M. Ingram. Cartesian cut-cell method for axisym metric separating body flows. AIAA Journal, 37(8):905–911, 1999.
A NUMERICAL WIND TUNNEL EXPERIMENT
M. POURQUIE AND C. MOULINEC Laboratory for Aero- and Hydrodynamics
Leeghwaterstraat 21, 2628 CB,Delft University of Technology,
Delft, Netherlands
AND
A. VAN DIJK IMAU, University of Utrecht, Utrecht, Netherlands
Abstract. In this paper we study the possibility of generating homoge neous, isotropic turbulence by putting a grid in a channel flow. Several statistics are studied to assess the quality of the turbulence. Also, the ef fect of the inclusion of a contraction on the isotropy of the turbulence is considered.
1. Introduction
The generation of isotropic turbulence is an often studied subject, theoret ically and numerically. Numerically, people usually resort to the generation of isotropic turbulence by calculating its evolution in time, with periodic boundary conditions which allow the use of spectral methods. An alternative, followed here, is to try to simulate the turbulence as it is generated in a wind tunnel, which comes closer to reality. In this case we have to simulate the spatial evolution of a flow as it goes past a wind tunnel grid and becomes turbulent. Special about the calculation is, that we are interested in the flow far from the grid, i.e. far from the obstacle in the flow. The flow can be studied in its own right, or be used to generate inflow boundary conditions. The flow generated can be expected to be more like the flow in a real wind tunnel, and can be used to simulate experiments in a wind tunnel. 151 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 151-162. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
152
M. POURQUIE ET AL.
2. Simulation method We use a standard Cartesian model for the simulations, which uses second order central differences in space and Adams-Bashforth time stepping, and a pressure correction procedure to satisfy mass conservation. The pressure correction equation is solved using a direct solver based on fast Fourier transforms in two directions and the solution of tri-diagonal systems in the remaining direction. The subgrid model used is a simple Smagorinsky model with a Smagorinsky constant of 0.065 . The wind tunnel is modeled with the following boundary conditions (see figure 1): in the main stream direction, we prescribe the inflow and use a convective outflow boundary condition. In the other directions, we use either free slip or periodic boundary conditions (wall boundary layers are not taken into account). At the inflow the velocity is put to 0 where the grid bars are, and to 1 where they are not. The inflow boundary conditions have to be chosen with some care, given the fact that we do not have an unlimited number of mesh points. On the one hand, we want to have good mixing to get isotropic turbulence. How ever, many physical grid bars are required to get good mixing of the jets coming from the grid. This means that we do not have many mesh cells per grid bar: we are under-resolving. On the other hand, taking only a few grid bars means that not enough mixing may be performed. An intermediate choice of 4 grid bars in two directions has been made, giving four mesh cells per grid bar. This means, that we still do not have a satisfying resolu tion at the inflow boundary. Compare, for instance, the resolutions used in the calculations for the Tegernsee workshop, see [1]. Here, the flow around one square cylinder was calculated by several groups, and a minimum of 20 mesh points on the square cylinder was used. Moreover, the meshes were
A NUMERICAL WIND TUNNEL EXPERIMENT
153
stretched, increasing the resolution close to the cylinder. Thus, we cannot expect to be able to determine the effect of changes in the details of the inflow conditions in a proper way. This would require additional simula tions. The simulations were done to study the possibility of numerically generating turbulence with a grid in a channel in a qualitative way. The resolution is 600x48x48 for most runs, giving 4 mesh cells per grid bar. The size of the calculation domain is 50 M in the streamwise direction and 4Mx4M in the other two directions, with M the distance between the grid bars (so that there are 12 grid cells per M). The solidity of the grid is then 7/16. (The solidity is the ratio of the surface at the entrance which is occupied by the grid bars to the entire surface of the entrance.) The grid bars are evenly distributed over the inflow boundary. Averaging times are of the order of 10 times the time taken to go from inlet to outlet, and the Reynolds numbers considered are 1000 and 4000, based on the distance between grid bars (M) and flow velocity in the channel. For the run with Reynolds number 1000 no subgrid model has been used. For Reynolds number 4000 results are presented for a standard Smagorinsky model. Results
We show results using periodic boundary conditions in the lateral direction. A contour plot gives a general idea of the flow (fig. 2). Moreover, it gives a hint of the size of the region near the entrance where we cannot expect the turbulence to show isotropy. We see fluid jets, breaking up rather quickly in a more or less random eddy pattern. However, the separate jets are still
154
M. POURQUIE ET AL.
discernible after 16 M ( and distances in the figures are given as and We also give the mean square velocity variances (main stream direc tion) , and (lateral). In the literature, decay laws are suggested which are of the form where K is a constant, x is the distance from the grid measured in grid widths M, and is a virtual origin, and is an exponent. Values for are often between 1 and 1.15. The value is
A NUMERICAL WIND TUNNEL EXPERIMENT
155
believed to depend on the solidity of the grid. A least squares fit suggests a value of for the case studied, with a virtual origin at 4M. Testing the dependence on the solidity can be done, but one has to realise that we are under-resolving the grid region. One test case has been run to check the dependence on the solidity. The solidity has been increased from 7/16 (which it is for the results presented) to 3/4, while keeping the grid spacing M and the Reynolds number the same. Again, an exponent of about 1.5 is found, with a virtual origin at about 3M. According to the literature, we would expect a higher value, but because of the under-resolution it is not clear if the results for found here are meaningful for investigating the effect of solidity. and are equal to each other, and is larger then the other two. The same happens in a wind tunnel. Thus, the turbulence is non-isotropic. However, in a wind tunnel the ratio between the variance of the component in the main stream direction and a component in the other direction is of order 1.1-1.8, depending on the circumstances. In our case, the ratio is more like 2 (after 20 M). In case one is interested in using the velocity field to generate inflow conditions for another flow geometry, other lateral boundary conditions than periodic may be interesting too, since the velocity field used has to be compatible with the (lateral) boundary conditions of the other geometry. Therefore, we have also considered free slip lateral boundary conditions. In figure 4 we have shown a 2D contour plot of the variance of a lateral velocity component. We see that, from a distance of 28M onward, there is an influence of the free slip boundaries. The big structures feel the influence of the no-slip boundary, where lateral motions normal to the wall are suppressed. This
156
M. POURQUIE ET AL.
effect will always be visible if the domain is long enough, since the eddies keep growing with distance x. It will be less important for smaller eddies, i.e. higher Reynolds numbers and more grid bars. A comparison with the periodic BC case has been made for the region between 0.25 and 0.75 of the height and width, which is outside the part influenced by the no-slip walls. The free-slip results for the variances are plotted to the right of the periodic BC results in figure 3.
A NUMERICAL WIND TUNNEL EXPERIMENT
157
Several other quantities can be studied, for instance length scales and time scales. For instance, the dissipation rate can be estimated from the decay of turbulent kinetic energy and Taylor’s hypothesis. The Kolmogorov length scale can then be calculated from with the viscosity. The Kolmogorov length scale is plotted in figure 5 for periodic and free slip BC. The figure indicates that the resolution is only just adequate (the mesh spacing is about three Kolmogorov scales). The figure also indicates that quantities like the Kolmogorov scale, for which derivatives of calculated quantities have to be taken, are much more sensitive to the smoothness of the results. The Kolmogorov scale for the free-slip BC (figure 5 right) is about equal to that for the periodic BC case (same figure left). 3. Inclusion of a subgrid model
The inclusion of a subgrid model allows calculations at higher Reynolds numbers (for the calculation without a model, the smallest scales are re solved and inclusion of a subgrid model makes no sense). The standard Smagorinsky model has been used, with a Smagorinsky constant of 0.065 The choice of an appropriate value for this constant is not clear. Near the entrance we have a developing shear flow, for which one would take 0.065. However, further from the entrance a constant of 0.2 seems a better choice if the flow truly resembles isotropic turbulence (a value of 0.2 was found appropriate in many numerical experiments with isotropic turbulence, for
158
M. POURQUIE ET AL.
instance Antonopoulos-Domis [2].) One should really try a dynamic model here which is able to determine an appropriate value for the Smagorinsky constant by itself. Taking a Smagorinsky constant of 0.2 all through the domain gave a very smooth solution with length scales which very quickly after the grid already became of the order of the lateral dimensions of the geometry. These solutions are not shown. The runs have been done for a Reynolds number of 4000. The results are qualitatively the same as for simulations (not shown) without a subgrid model. They are shown in the left plot in the figure 10. As can be seen, the calculation gives less smooth results (more fluctuations in the averaged results). The fluctuations (in the averaged results) make a calculation of other quantities more difficult. The subgrid contribution has not yet been added in this figure for the run with a subgrid model, but from the resolved quantities the fluctuation level seems to be lower for the run with a subgrid model, which may be due to a too large dissipation by the Smagorinsky model. The relevance of the subgrid model in the developed region may be assessed by comparing the resolved and the subgrid contribution to the turbulent kinetic energy. These are compared in figures 6, 7. From these figure we can conclude, that the subgrid contribution is ac ceptably low (less then 10 % of the resolved). However, this is no guarantee that no relevant turbulence has been removed. Regarding the decay exper iment, for the run with a SG model we find a decay exponent of 1, smaller
A NUMERICAL WIND TUNNEL EXPERIMENT
159
than in the runs without a subgrid model. This may have to do with the fact that the subgrid model removes smaller scales first, leaving the (slower decaying) large scales. A quantitative comparison with a run without a subgrid model at Reynolds number 4000 could be used to further assess the role of the subgrid model but these results were no longer available. In figure 8 we see the Kolmogorov scale for the run including subgrid model. As expected, the Kolmogorov scale is smaller then for the run with the lower Re number without subgrid scale. The Kolmogorov scale for a run at the same Re but without SG model showed almost the same scales. If we assume that LES should be able to handle grids three times as coarse as a DNS grid, this plot suggests that we could go to higher Re, since we have a DNS like resolution (one third mesh cell per Kolmogorov scale) over large parts of the domain. 4.
The effect of a contraction
In real wind tunnels, contractions (fig. 9) are used to force the axial variance and the lateral variance to be equal (see [3]). To simulate a tunnel with a contraction one can use an (expensive) boundary-conforming code. Instead, we use an (inexpensive) Cartesian method, in which the bound ary is represented in an appropriate way on the Cartesian mesh. We have applied the so-called force method (see [4], [5]). In this method, a source
160
M. POURQUIE ET AL.
term is added to the momentum balance which forces the velocity to be zero on a prescribed surface. As shown in figure 9, the variance of all three veloc ity components experiences a change (especially the lateral variance), then some distance after the contraction the three variance components become equal for some distance, and then the axial variance becomes bigger than the lateral variance again (just like in some wind tunnel experiments). The run with subgrid model shows similar trends when a contraction is included (see right plot in figure 10).
A NUMERICAL WIND TUNNEL EXPERIMENT
161
5. Conclusion
Several runs have been performed to assess the possibility of generating wind-tunnel like turbulence. Decay rates of the turbulence show realistic tendencies. Just like in real tunnels, unless a contraction is present, the turbulence is not isotropic but rather axi-symmetric. Although the results look promising, more testing is required. Proper testing might require better
162
M. POURQUIE ET AL.
resolution at the grid bars, since for instance hardly any effect of increasing the solidity on the decay rate of the velocity variances was found. More tests would include more grid bars, better resolution at the grid bars, and a dynamic model. The solidity of the grid at the entrance can be important, the Reynolds number, the way the contraction is implemented. Moreover, the isotropy can be studied more in depth, like in Van Dijk’s thesis [6]. When we want to use the flow calculated as inflow condition for another geometry, the lateral BC have to be compatible. This may mean that we cannot use periodic BC to generate the turbulent inflow. As we saw, there will be an influence of the side walls also for free slip BC in this case. References 1. W. Rodi, J.H, Ferziger, M. Breuer and M. Pourquie (1997) Status of large-eddy simulation: results of a workshop, Journal of Fluids-Engineering. 119, 248-262. 2. M. Antonopoulos-Domis (1981) Large-eddy simulation of a passive scalar in isotropic turbulence, Journal of Fluid Mechanics. 104, 55–79. 3. G. Comte-Bellot and S. Corrsin(1966) The use of a contraction to improve the isotropy of grid-generated turbulence, Journal of Fluid Mechanics. 25, 657–682. 4. D. Goldstein, R. Handler and L. Sirovich (1995) Direct numerical simulation of turbulent flow over a modeled riblet-covered surface, Journal of Fluid Mechanics. 302, 333–376. 5. R. Verzicco (2000) Large eddy simulation in complex geometric configurations using boundary body forces, AIAA Journal . 28, 427–433. 6. A. van Dijk (1999), Aliasing in one-point measurements, thesis, Delft.
Acknowledgment. We would like to acknowledge HPAC at TU Delft for supplying the computational facilities to do the calculations.
4. Curvilinear and non-structured grids for com plex geometries
This page intentionally left blank
TOWARDS LARGE EDDY SIMULATION OF COMPLEX FLOWS
C. FUREBY Weapons and Protection Div., Warheads and Propulsion,
The Swedish Defense Research Agency FOI,
S-172 90, Stockholm, Sweden
Abstract. This paper aims at presenting a recent update on LES focusing on complex flows. The general principles will be discussed based on incompress ible flows, but, when necessary, extensions will be made to both compressible and reacting flows. A few examples will be presented as to facilitate the discus sion and to highlight both the advance of LES and the pacing items still to be addressed. As a background for the discussion a few practical problems relevant to the industries’ needs will be provided.
1. Introduction Fluid dynamics is a wide area involving incompressible and compressible flows, as well as flows involving additional physics such as combustion, electro- and magnetohydrodynamics or solid-fluid interactions. In many practical problems complications arise from the complex geometries of man-made constructions that interact with the fluid. The governing equations are the equations for conservation of mass, balance of momentum and energy supplemented by the fluid’s constitu tive equations. Here, we limit our discussion to Newtonian fluids with Fourier heat conduction and so the governing equations reduce to the Navier-Stokes Equations (NSE). Solving these includes dealing with a wide range of scales. The largest scales are related to the geometry, and the smallest scales are associated with the dissipation of turbulence through viscosity. Chemical reactions or other physi cal processes have their own range of scales, and a variety of interactions between such processes and the turbulence are possible depending on the overlap of scales. The ratio between the most energetic scale and that of the smallest dynamic where Re is the Reynolds scale, the Kolmogorov scale is number. Hence, we need degrees of freedom in order to represent all the 165 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 165-183. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
166
C. FUREBY
scales in a volume For high Re-number applications Direct Numerical Simu lations (DNS) are thus not possible in the near future. In order to handle this wide spectrum of scales we need to reduce the number of degrees of freedom in the simulation. The most common way of doing this is by means of Reynolds Averaged Numerical Simulation (RANS), [1]. In RANS, equations for the statistical average of the variables are obtained by averaging the NSE over homogeneous directions, time or across an ensemble of equivalent flows. Since most engineering flows involve inhomogeneous flow, time averaging is the most appropriate averaging. The turbulent fluctuations are not represented directly by the simulation but are included by way of a turbulence model. This av eraging makes it possible to reduce the number of scales, or degrees of freedom, considerably. The statistical character of RANS prevents a detailed description of the physical mechanisms, and it is thus unsuitable for problems where the fluid dynamic details are vital. On the other hand, RANS is appropriate for analyzing performance characteristics as long as the turbulence models are able to repre sent the turbulent stresses sufficiently. When the flow is dominated by coherent structures related to a narrow frequency band, Unsteady Reynolds Averaged Nu merical Simulation (URANS), [2], can be used. In URANS, equations for the sum of and the coherent modes are derived. These equations are closed by tur bulence models similar to those used in RANS. However, URANS is an unsteady approach and therefore contains more information than RANS, but still needs a deterministic description of the flow. The most advanced turbulence modeling method currently at hand is Large Eddy Simulation, [3-4]. In LES a separation between large supergrid and small subgrid scales is imposed by means of a spatial filtering related to the characteristic size of the grid – The effects of the subgrid scales on the supergrid scales are accounted for by way of a model – a subgrid model. LES contains even more information of the flow than URANS but is also more expensive and is thus not presently suitable for screening of design parameters but rather for detailed studies and to gain qualitative and quantitative understanding of phenomena. Recently, due to the difficulties of LES in handling high Re number wall-bounded flows a hybrid RANS/LES approach is being de veloped by several groups. 2. The Conventional LES Model Formulation In LES, the motion is separated into small and large eddies and equations are solved for the latter. The separation is achieved by means of a low-pass filter, which can be formulated in several ways e.g. [5-6]. Convoluting the NSE with a filter yields the LES equations,
TOWARDS LARGE EDDY SIMULATION
167
where v is the velocity, the pressure, the viscous stress tensor, D = Specific to the rate-of-strain tensor, the viscosity and LES is the resolved components, denoted by an overbar, the subgrid scale stress and the commutation errors and tensor where is the commutation represents the effects of the subgrid flow on the resolved operator. The term and reflects the fact that filtering and differentiation does not flow,whilst generally commute. B must be modeled using information from the GS flow prior to discretization at a resolution near – much more affordable than DNS. Such models must, in absence of a universal theory of turbulence, include rational use of empirical information. Furthermore, and needs to be either modeled or neglected. Following, [7], B can be decomposed as B=L+C+R with being the Leonard stress the cross stress and R = the Reynolds stress. Thus, L represents interactions among the resolved eddies, C between large and small eddies and R between the subgrid eddies. One of the key objectives of modeling in mechanics is to conserve the prop erties of the governing equations, [8]. By requiring that the LES equations (1) should have the same transformation properties as the unfiltered NSE under an and where is an arbi arbitrary change of frame given by trary vector, Q a rotation and an arbitrary scalar, it is evident that B is frame [9-10], and thus an isotropic function, [11]. Neither indifferent, L nor C is frame indifferent individually but the sum, L+C, is frame indifferent. Accordingly, R is frame indifferent. Further mathematical analysis and physical arguments, [9-10, 12], suggests that B is Grammian, provided that i.e. that G is symmetric, which implies that where Psym is the set of all positive definite symmetric tensors. The second-rank tensor B is positive definite if the inequalities and det are satisfied. Hence, ideally, the subgrid model should obey these realizability constraints. As outlined, LES is a technique for reducing the computational cost by means of reducing the number of degrees of freedom by separating scales. Only the re solved scales are retained in LES whereas the subgrid scales are grouped in B which has to be modeled. Presently, two modeling strategies exist: Functional modeling consists in modeling the action of the subgrid scales on the resolved scales. This is basically of energetic nature so that the balance of the energy trans fers between the two scale ranges is sufficient to describe the subgrid scale effects. Structural modeling consists of modeling B without incorporating any knowledge of the nature of the interactions between the subgrid and the resolved scales. Such models can be based either on transport equations, scale similarity, series expan sion techniques or other deterministic approaches.
168
C. FUREBY
3. Subgrid Modeling Subgrid modeling in LES is a very wide subject, and here we concentrate on mod els useful to complex flows. Accordingly, we limit this study to subgrid models de veloped in physical space. A more detailed discussion of subgrid modeling in LES can be found in the textbook of Sagaut, [6], and references therein. For isotropic flows the interactions between the large and small scales can be described by (i) a drainage of energy from the resolved scales to the subgrid scales (forward cascade), and (ii) a weak replenishment of energy from the subgrid scales to the resolved scales (backward cascade). The subgrid energy transfer occurs in regions where the vortex stretching is positive or in regions with negative skewness of the rate-of-strain, cf. [13]. In order to discuss subgrid modeling for complex flows we first need to establish a background for subgrid modeling. 3.1. FUNCTIONAL MODELING IN PHYSICAL SPACE The energy transfer mechanism from the resolved to the subgrid scales is here assumed analogous to that of a Brownian motion superimposed on the motion of the resolved scales. Hence, the (forward) energy cascade mechanism is,
where is the deviatoric part of B and is the subgrid viscosity. The non deviatoric part is added to the filtered pressure and does therefore not require modeling in the incompressible case. To close (2) we need models for and k, and we thus assume the existence of characteristic length and velocity scales, and a total separation of scales. Hence, where is the dissipation and with k being the wavenumber vector, denotes the energy spectrum. For a Kolmogorov spectrum,
where with and is a parameter to be varied. The subgrid kinetic energy dissipation rate can be derived from the energy spectrum E under the assumption of local equilibrium, where Here, local equi [6], so that librium implies an instantaneous adjustment of scales associated with the pro duction of so that equality prevails between production and dissipation. Dif ferent forms of can be obtained for different In particular, for we obtain which in conjunction with yields the Smagorinsky model (SMG) with The SMG model is based entirely on the resolved scales, and the existence and behaviour of the subgrid scales are determined by the resolved scales only. A drawback of SMG becomes non-zero as soon as the velocity exhibits spatial variations – is that
TOWARDS LARGE EDDY SIMULATION
169
even if the flow is laminar and well-resolved. For we have which is formally analogous to the model used in RANS or URANS, [1-2], Hence, this model only uses the integral scales. For and does not involve where This model only uses which is a subgrid quan if the flow is well resolved, and thus offers improved tity and ensures that physical consistency than subgrid models based on the resolved scales. Different ways can be used to determine e.g. in the One-Equation Eddy Viscosity Model (OEEVM), [15-16], a modelled transport equation is solved for k, viz.,
Here, the diffusion term is modeled by a gradient hypothesis whilst the dissipation term is modeled using dimensional arguments, e.g. [15-16]. Another model of this type is the Structure Function Model (SFM), [17], in which k is replaced by the second-order velocity structure function Besides the models discussed here a number of other functional models are in use, e.g. [18-19]. The subgrid models described above are based on a Kolmogorov spectrum which only reflects the existence of an in of the form ertial sub-range in a real spectrum. The preceding approach can, however, be extended to more realistic spectrum shapes, Voke, [20]. Using a Pao spectrum it is straightforward to show that where is the mesh Re-number. Here, is a complicated analytical function of for which a more practical expression is given by where is the conventional the curve-fit Smagorinsky coefficient. Models of this kind can be interpreted as models with scale-dependent coefficients and improves the predictive capabilities for transi tional and wall-bounded flows. To adapt the SMG model better to the local structure of the flow, Germano, [21], suggested a scheme for automatic adjustment of the parameter This prowhere cedure is based on the identity and and denotes two consecutive filter levels. As Here, suming that B and T can be modeled similarly, we have with Since cannot simply be extracted and from the second term it should be calculated by minimizing Hence, in the least-squares sence where In is not bounded which can have consequences on numerical this expression stability and cause the simulations to fail. In practice this can be remedied by where the additional averaging can be performed across homogeneous directions, over time, along streamlines, [22], or over an ensemble of equivalent simulations, [23]. The dynamic procedure has been gen eralized to other subgrid models besides SMG.
170
C. FUREBY
3.2. STRUCTURAL MODELING These models are established with no a-priori knowledge of the nature of inter actions between subgrid and resolved scales. The most straightforward of these models is the Differential Subgrid Stress Model (DSSM), [24], which more re cently has been re-investigated by Fureby et al., [25]. In DSSM a modeled trans port equation for the subgrid stress tensor B is solved in conjunction with (1),
The triple correlation is modeled by a gradient hypothesis, the pressure velocityand a mean strain part gradient by a sum of a return-to-isotropy part and dissipation by [24]. In addition, and The model co whilst k is obtained from its definition efficients are estimated from isotropic turbulence and take the values and The DSSM has the advantage of being non local and be ing able to handle simultaneous grid and flow anisotropies because of its tensorial nature. The principal drawback, however, is its high computational cost. An interesting alternative is to assume that the material time-derivative of the anisotropy tensor is zero, i.e. where is the anisotropy tensor, cf. [26]. Using (5), an Algebraic Subgrid Stress Model (ASSM) results,
where is a parameter and is a characteristic time scale of the subgrid turbulence. Note that (6) is made up of two parts, one eddy-viscosity part and one non-linear part related to the production of B. The ASSM also has the possibility of handling simultaneous grid and flow anisotropies, similar to the DSSM, but to a reduced cost. Following Bardina et al., [27], it can be assumed that the statistical structure of the fields constructed on the basis of the subgrid scales is similar to that of the fields evaluated on the basis of the smallest resolved scales. The most well-known model using this scale-similarity hypothesis is the Bardina Model (BM),
Despite its good level of correlation, [27], experience shows that this model is not sufficently dissipative, i.e. it underestimates the forwardscatter, whilst includ ing backscatter. To improve the BM, the concept of Mixed Models (MM) has been suggested, in which the BM is combined with e.g. the SMG model, [14]. This approach has the advantage of increasing dissipation, whilst improving the numerical stability. Further improvements of scale-similarity models have been suggested by e.g. Liu et al., [28].
TOWARDS LARGE EDDY SIMULATION
171
A fourth approach to structural modeling is by means of formal series expansions, starting with the functional relationship As an extension to the non-linear approach of Lund & Novikov, [29], we may consider a generalization of this functional relationship involving the resolved motion de scribed by the isomorphism so that where we have included the dependence on the scalars and to set the subgrid scales. Using the mathematical features of B, such as frame indifference, together with knowledge gained from modern continuum mechanics, e.g. [8], we may re formulate this functional dependence as,
Here
are the Rivlin-Ericksen tensors whilst denotes the Cauchy-Green tensor history. In the next step rep resentation theorems of isotropic functions, [11], can be used to simplify the model (8) in a systematic manner in order to arrive at a family of models where are functions of the invariants of the generat ing set The only linear model of this class is the SMG model, which is obtained by truncating the series at For a non-linear algebraic model, similar to (6), will be obtained. 3.3. EXTENSION OF FUNCTIONAL MODELING TO ANISOTROPIC FLOWS When LES is applied to practical flows, we usually use anisotropic grids to dis cretize the geometry. Hence, independently of modeling strategy, two factors con tribute to the violation of the basic assumptions of isotropy: filter anisotropy, and flow anisotropy, with a non-uniform mean flow. Thus, the anisotropy is an artifact of the filter (grid) whilst the dynamics of the subgrid scales still corresponds to that of isotropic turbulence. For isotropic flows the use of a scalar valued length is apparent for anisotropic flows however, we either modify or intro scale duce a vector valued length scale The latter would imply the use of e.g. a tensor valued viscosity. The most common definition of the filter width of today involves the cube root of the volume of the filtering cell Based on an improved estimate of Scotti et al., [30], proposed an extension to this definition, involving the cell aspect ratios and so that where is the isotropic filter width. The function is then finally evaluated given a spectrum shape and a filter kernel. The most common model is the SMG model, and experience shows that the performance of this model declines when used in an anisotropic flow. Schumann, [15], proposes a simple modification to this model based on splitting into a locally isotropic part and one inhomogeneous part so that,
172
C. FUREBY
where
designates a statistical average, and The statistical average can, in practice, be a spatial average over local regions or in directions of homogeneity, where is the isotropic filter width and the anisotropic filter width. Following e.g. Moin & Kim, [31], This model has the advantage of improved near-wall behav ior, as compared to the SMG model, due to the separate modeling of the isotropic and inhomogeneous parts. Sullivan et al., [32], proposes a variant of (9) that in corporates an anisotropy factor for better control.
The ASSM model (6) together with some of the models obtained from a for
mal series expansion cf. (8) can be re-interpreted so as to formally contain a tensorial subgrid viscosity. The most general form of such a model is where C is a fourth-rank tensor, defined by 81 independent coefficients. Due to symmetry of B and D the number of coefficients reduce to 25. Models of this type have been proposed, e.g. by Carati & Cabot, [33], but have not yet been exercised on a sufficiently large number of flows for them to be adequately investigated in order to judge their general behaviour. 4. Numerical Methods for Conventional LES of Complex Flows LES requires high-order schemes to avoid masking the subgrid term by the leading order truncation error. In general, is related to the grid, i.e. where is the grid size, which makes the modeled subgrid stresses terms. In LES, spectral and high-order finite volume, element or difference methods are used for spatial discretization, whilst explicit semi-implicit or predictor-corrector methods are used for time-integration. For complex geometries the Finite Volume (FV) method is the most convenient technique. Here, the domain D is partitioned into cells The cell-average of f over so that and the cell is so that Gauss theorem may be used to formulate the semi-discretized LES-equations. By integrating these over time, using e.g. a multi-step method, [34], the discretized LES-equations become,
where m,
and
are parameters of the multi-step method, whereas,
are the convective, viscous and subgrid fluxes. To close the FV-discretization the fluxes (at face f) need to be reconstructed from the variables at adjacent cells. This requires flux interpolation for the convective fluxes and difference approximations for the inner derivatives of the other fluxes. Typically, for second order accuracy,
TOWARDS LARGE EDDY SIMULATION
173
where and represent the leading order truncation errors. The equations (10) can be decoupled by combining and into a Poisson equation for which is to be solved together with The scalar equations are usually solved sequentially with iteration over the explicit terms to obtain rapid convergence. To examine how the leading order truncation errors compare with the subgrid model B the modified equations, or the partial differential equations actually solved, can be derived using the leading order truncation errors of the fluxes, viz.,
where the last terms on the right hand side represent the truncation error. When this term is dominated by odd-order spatial derivatives numerical dispersion may cause unphysical oscillations in the solution, and if even-order derivatives dom inate, numerical diffusion is added to the solution. In (13), odd- and even-order derivatives coexist, with dispersion dominating over the hyper-viscosity terms. 5. Implicit LES Formal drawbacks of LES include the effects related to the commutation errors, masking of the subgrid fluxes by the leading order truncation errors, and problems associated with formulating subgrid models for non-homogeneous flows. An al ternative is Implicit LES (ILES), cf. [35-36], which uses the intrinsic properties of high-resolution schemes, [34], to define implicit (or built-in) subgrid models through the leading order truncation errors. ILES draws on the fact that finite difference, element and volume schemes filter the NSE over cells using an anisotropic top-hat kernel Hence,
where m,
and
are parameters of the multi-step method used, and,
Similar to conventional LES the reconstruction of the viscous fluxes typically in volves linear interpolation (i.e. central differencing), whilst the reconstruction of the convective fluxes is here treated differently since it is intended also for defin ing the implicit subgrid model. Different ILES differ mainly in their distinct functional reconstruction algorithms for Tsuboi et al., [37], uses a third order upwind biased scheme, Boris et al., [35], uses Flux Corrected Transport (FCT),
174
C. FUREBY
Knight et al., [39], uses an unstructured grid compressible Navier-Stokes algo rithm, and Karamanos & Karniadakis, [40], uses a spectral vanishing viscosity method. In Monotone Integrated LES (MILES), [4, 35-36, 38], a non-linear flux limiter is used to blend a high-order flux-function being well-behaved in smooth regions, with a low-order dispersion-free flux-function that is wellbehaved in the vicinity of sharp gradients so that E.g.,
where
and and represent the leading order truncation errors. The choice of is not easy, but is should allow as much as possible of the antidiffusion to be included without violating the principles of causality, monotonicity and positivity. Hence, must be a positive function of the ratios of consecutive variations in the flow variables and should only deviate from unity in parts where these are close to an extremum or have sharp gradients. Simulations have shown that the FCT limiter, e.g. [35], works well and satisfies these constraints, but at present the detailed influence of on the solution is unknown. From the numerical formulation of the ILES model (14-16) the implicit or built in subgrid term B can be derived directly from the modified equations, i.e,
so that
where and The built-in subgrid model is thus the sum of two terms, the first being of generalized eddy viscosity type, whilst the second being of scale similarity type. 6. Current Applications for LES LES is well established for ‘building-block’ flows, including homogeneous isotropic turbulence, [41], free shear flows, [42-43], channel flows, [31], bluff body flows with spatial homogeneity, [44], and three-dimensional bluff body flows, [44]. Most subgrid models have been developed and validated for such flows, and a ma jority of these ‘building-block’ flows have also been investigated by other meth ods. These flows serve as benchmark cases at the same time as our fluid dynamic awareness is evolving as new results become available for higher Re and Ma num bers and for superimosed physics such as e.g. combustion, vaporisation or trans port of droplets. These flows usually form part of more complex flows that are harder to simulate with accuracy.
TOWARDS LARGE EDDY SIMULATION
175
As a representative of a free shear flow we present an isothermal, isochoric unforced round jet of diameter D and velocity U expanding into a uniform outer flow of zero velocity, figure 1a. The Re-number is 4800 and the grid consists of cells. At the inlet, four jet diameters downstream of the orifice, a con stant velocity profile is prescribed, together with a Neumann condition on p. The LES reveal two main regions: (i) A potential core region, which is dominated by a laminar cone that is surrounded by an annular mixing layer. The mixing layer is created by an instability associated with the deficit velocity profile of the bound ary layer on the pipe wall. As the mixing layer thickens with downstream distance the potential core shrinks. After the potential core we have the jet region, in which the flow gradually reaches a regime corresponding to a similarity solution. Fig and its ure 1b shows the spatial evolution of the mean longitudinal velocity rms-fluctuations as compared to experimental data, [45]. During the first 4D the remain fairly constant, confirming the existence of the potential core. After the core disappears increases in magnitude, to reach a maximum at about 8D. However, the length of the potential core is somewhat underestimated, as compared to the experimental data – a common feature of most LES of round jets. The statistics presented in figure 1b is virtually indifferent to the effects of the subgrid model – a common feature for many of the ‘building block’ flows. Jets of different types have been investigated for other purposes, such as flow control, thrust vector control and mixing, [47].
The flow around a surface mounted cube, Martinuzzi & Tropea, [48], is a fa miliar test case for LES, [44]. The Re number is based on the freestream flow and cube height, and the grid used in the LES presented here consists of 120×70×60 cells. Here, we show LES results using SMG with constant and scale dependent SFM and the an- isotropic SMG of Schumann. LES on coarser grids with variants of OEEVM have been performed by Krajnovic et al., [49]. Lateral boundaries are treated as slip surfaces, whilst for the upper, lower and
176
C. FUREBY
cube walls no-slip conditions are used. At the inlet, and at the outlet, and Figure 2a shows a perspective view of the flow, revealing its three-dimensionality and key features, being (i) a horse shoe vortex curved around the cube, (ii) a multiple vortex system on the top of the cube, (iii) a secondary arch-shaped vortex behind the cube and (iv) the threedimensional wake. Comparing limiting streamlines with oil-film visualisations in [49] give a qualitative indication of the accuracy and the differences between the subgrid models. The accuracy is good, but with larger differences between subgrid models than what previously have been reported for ‘building-block’ flows. Figure 2b shows a comparison of and between LES and experimental data. The overall agreement is good, but with interesting differences between dif ferent subgrid models. The statistics is virtually grid independent, as confirmed by running the SMG model on a finer grid. The results are in agreement with the results presented at the workshop, [44], and those of Krajnovic et al., [49].
7. Current Challenges for LES When aiming at LES of high Re-number complex flows various difficulties arise that LES does not currently handle sufficently well e.g. (i) Realistic parameteri zation of the subgrid stress tensor under a variety of non-ideal conditions such as non-equilibrium turbulence, transition etc. (ii) Modeling near-wall flow physics and wall shear stress in high Re-number wall-bounded flows in which the nearwall features cannot be resolved. (iii) Appropriate specification of inlet/outlet boundary conditions. (iv) Modeling additional small-scale physics such as chem ical reactions, phase changes, cavitation, etc. (v) Continued improvement of nu merical methods, and formulation of the LES equations in order to minimize dis cretization errors and commutation errors.
TOWARDS LARGE EDDY SIMULATION
177
7.1. IMPROVED SUBGRID MODELS Subgrid models have been discussed in some detail in Section 3. The behavior of these models in complex flows needs further investigation, cf. [50], before any definite conclusions can be drawn concerning their applicability. It is possible that these conventional models have to be modified or that other models need to be developed.
7.2. MODELING THE NEAR-WALL FLOW PHYSICS For high Re-number wall bounded flows the energy containing scales will be come small compared to any grid as Re increases, and thus, none of the familiar near-wall structures (streaks, sweeps, bursts and ejections) will be resolved. These structures can be captured in a wall-resolved LES, freely defined as and The ‘+’ denotes non-dimensionalization using the viscous length scale where is the friction velocity. The LES wall boundary condition problem is thus to account for the effects of the near-wall tur bulence between the wall and the first node and its transfer of momentum (and energy) to the wall. In LES of high-Re-number flows we may either try to resolve the near-wall dynamics or we may try to model it. The first approach calls for very fine grids in the near-wall region, and requiring also aspect ratios less than O(100) we need to use local grid refinement and unstructured grids. For flows of practical interest this is expensive, but technically feasible. The modeling can be split into three categories: (i) modifying the subgrid model to accommodate integration all the way to the wall, (ii) introducing explicit wall models, and (iii) subgrid simulation models. The first method can be split into four sub-categories: (a) modifying the subgrid model with damping functions D so that (b) using the dynamic procedure, [21], where e.g. to determine the model coefficient(s); (c) evaluating the model coefficient(s) from models of the energy spectrum that includes the viscous sub-range; and (d) us ing subgrid models with structural sensors, [51]. The second method can be split into two sub-categories: (a) to model the wall-shear stress (or friction velocity e.g. by and (b) to position the boundary of the simulation at the first node from the wall, so that the tangential is to be modeled instead. The third method is based on a ‘grid-withinvelocity the-grid’ approach in which simplified one-dimensional momentum equations are solved on a one-dimensional grid, aligned with the wall-normal, embedded into the LES grid.
7.3. INLET/OUTLET OPEN BOUNDARY CONDITIONS The stochastic nature of the resolved flow in LES has consequences on the for mulation of the inlet conditions. The inlet condition can be specified in terms
178
C. FUREBY
of a mean velocity plus a fluctuating component that is to comply with the ve locity rms-fluctuation profile. A common approach is to ignore this – specifying constant resolved and subgrid values. Methods tried for generating realistic inlet conditions can be divided into two categories: those that superimpose a stochastic component on a smooth resolved flow and those that use the results of a separate calculation to generate inlet data. The latter approach results in an inlet condi tion that is closer to real turbulence, but is computationally more expensive. One can also imagine introducing ‘synthetic turbulence’, produced by Fourier series or wavelet decompositions, with the parameters defining these series being stochas tic variables satisfying constraints derived from turbulence energy spectra.
7.4. COMPLEX PHYSICS When considering chemically reacting flows, magnetohydrodynamic flows or flows with particles or droplets, additional information is required for the models. For example, for combustion, in addition to the subgrid stress tensor B also required are the subgrid heat flux vector and the subgrid species mass flux vectors, and the filtered reaction rates, [52]. This is also a very important area for LES-reseach to consider. Models have to be developed for the additional physics within an LES-framework, as well as the LES formulation may have to be adjusted to the particular problems considered. See also the paper by Menon, [53], in this volume conserning LES of chemically reacting flows.
8. Stretching the Limits of LES A complex flow of great interest to the aeronautical and marine communities is the flow past a 6:1 prolate spheroid at angles of attack between and 30 ° at a body length Re-number of e.g. [54-55]. Hedin et al., [56], recently performed LES of this flow using between 400k and 800k cells, clustered towards the hull in an attempt to resolve as much as possible of the boundary layer. For the 800k cell grid the spatial resolution was such that At the inlet bound ary, and at the outlet, and at the windtunnel walls slip conditions are used and at the hull no-slip conditions are used. The LES calculations are initiated with quiescent conditions and the unsteady flow evolved by itself. Different subgrid models, including different ver sions of SMG, OEEVM and MILES are being evaluated. Figure 3a shows a per spective view of the flow revealing its principal features. On the windward side, an attached three- dimensional boundary layer is formed, which detaches from the body on the leeward side because of the circumferential adverse pressure gradient and roll-up into a counterrotating pair of longitudinal vortices on the back of the spheroid. Below these primary vortices a pair of counterrotating secondary vor tices are formed. Fluid from the windward side is advected across the ellipsoid, engulfed into the primary vortices, and finally ejected into the near-wake. The
TOWARDS LARGE EDDY SIMULATION
179
primary vortices are symmetric with respect to the body and appear fairly station ary for different angles of attack. Transition occurs naturally at on the forebody, depending on subgrid model, close to what is actually found in the experiments. The qualitative agreement between the predicted and measured flow is satisfactory. In particular, the trends related to incidence ef fects, i.e. the roll-up of fluid and the subsequent formation of the vortices, and the associated distribution of velocity and turbulent kinetic energy are well cap tured. Figure 3b presents the measured and predicted secondary velocity field The primary vortex is located at 3.4 cm above the hull surface at at for the SMG model, which is to be compared with the experimental val The secondary vortex is located ues of 3.1 cm above the hull surface at in both the simulations and the experiments. The associated separa at tion lines are at and respectively, which is in good agreement with the experimental values of and respectively. By compar at x/L=0.60 and x/L=0.772 ing measured and predicted pressure coefficients it appears that the primary vortex is located too far from the hull. On the wind ward side the subgrid model has little or no influence on the results, whilst on the leeward side significant differences can be observed between LES using dif ferent subgrid models and the measurements. The subgrid model must, explicitly or implicitly, take into account the near wall effects – the most efficient way to accomplish this is by the relation
The ability to accurately predict the fluctuating transonic or supersonic flow over rectangular cavities are not only of interest from a fluid dynamics or nu merical point of view, but also for engineers in their effort to design improved air breathing engines and vehicles for transonic or supersonic speeds. The rapid pressure fluctuations inside the cavity yields undesired acoustic phenomena and time dependent structure loads due to the interactions of vortices, shear layers, and shock waves inside the cavity. In figure 4 we pre- sent LES results of the flow over cavities embedded into flat and curved bodies, respectively. For both sent LES results of the flow over cavities embedded into flat and curved bodies,
180
C. FUREBY
respectively. For both geometries L/D=2, Ma=1.52 and based on the velocity, cavity depth D and length L. The grid uses 750k cells, clustered towards the body and the cavity walls. At the inlet, and where denotes Favre filtered variables, at the outlet, At the body, adiabatic no-slip conditions are used whilst ap and propriate freestream conditions are applied at the far-field boundary. The global flow features are shown in figure 4a and consists of an ondulating shear layer due to the rotational motion inside the cavity, that upon impinging on the rear cavity wall sends strong compression waves upstream. Also, the shedding from the up stream edge results in that interact with each other to form longitudinal vortices that dominate the downstream region. Of special interest is the pressure fluctuation inside the cavity since it can cause structural fatigue. An FFT analy sis of p, at the midpoint of the downstream cavity wall, for the two cases shows different results, figure 4b. Here a comparison is made with the Rossiter, [57], fre quencies, compensated for compressibility, Heller, [58], caused by the standing acoustic waves inside the cavity that are created as the shear layer impinges on the downstream wall. An interesting observation for the case when the cavity is embedded in an arc shaped body compared to ordinary flat ones is that the first Rossiter frequency is the dominant one, and not the second as in the flat body case, c.f. figure 4b. Also, the shedding of two additional transverse vortices from the arc shaped leading edge alters the periodic motion of the shear layer causing two extra peaks around the second Rossiter frequency in the frequency spectra, which is not observed in the case of a flat surfaced body, [59].
TOWARDS LARGE EDDY SIMULATION
181
9. Outlook and Concluding Remarks When LES is used to study ‘building-block’ flows, for which the method origi nally was invented it produces excellent results. The dependency of the results on the subgrid model is weak – almost all subgrid models produce identical resolved flows. For wall-bounded flows, however, the situation becomes more complicated as the scales become smaller and smaller as the wall is approached. For low Re the principal structures can be resolved (wall-resolved LES) and the results are gen erally in good agreement with experiments and DNS. For higher Re the demands on the subgrid model increase, and it needs either to be modified in the near wall region or we need separate wall models. For these cases we can no longer afford resolving the key features of the near wall flow, although the trend is moving to wards adaptive grids and local grid refinement. This is an important research area if we have the ambition to use LES for engineering flows, which usually are asso ciated both with high Re and complex geometries. For more complex flows, many other problems arise concerning both the physical modeling and the numerical methods. For such flows the quality of the results is variable, but the predicted physics is often consistent in that it exhibits generic features also observed exper imentally – differences often being related to quantitative issues. The numerical methods used must be at least second order accurate in space and time. Upwindbased schemes are not recommended since the artificial dissipation overwhelms the effects of the subgrid model. When the flow becomes increasingly complex or the resolution becomes poor a need for improved subgrid models arises. This situation will prevail in engineering applications, such as flows around or inside complete configurations, and it is therefore important to focus on the development of subgrid models for complex anisotropic flows with poor resolution. Models that might work well under these circumstances are the DSSM and the ASM, since they contain more built-in physics than e.g. SMG. References 1. Launder, B.E. & Spalding, D.B.: Mathematical Models of Turbulence, Academic Press, Lon
don, (1972). 2. Ferziger, J.H.: Higher Level Simulations of Turbulent Flow, in Computational Methods for
Turbulent, Transonic, and Viscous Flows, J.-A. Essers (ed), Hemisphere, (1983). 3. Ferziger, J.H. & Leslie, D.C., Large Eddy Simulation – A Predictive Approach to Turbulent
Flow Computation, AIAA paper 79-1441, (1979). 4. Boris, J.P., On Large Eddy Simulations Using Subgrid Turbulence Models, in Wither Turbu
lence? Turbulence at the Crossroads, Lumly J.L. (ed), Lecture Notes in Physics, 357, Springer Verlag, Berlin, 344, 1992. 5. Ghosal, S. & Moin, P.: The Basic Equations for the Large Eddy Simulation of Turbulent Flows in Complex Geometry, J. Comp. Phys., 118, (1995), 24. 6. Sagaut, P.: Large Eddy Simulation for Incompressible Flows, Springer Verlag, Heidelberg, 2001. 7. Leonard, A.: Energy Cascade in Large Eddy Simulation of Turbulent Fluid Flows, Adv. in Geophys., 18, (1974), 237.
182
C. FUREBY
8. Gurtin, M.E.: An Introduction to Continuum Mechanics, Academic Press, Orlando, (1981). 9. Fureby, C. & Tabor, G.: Mathematical and Physical Constraints on Large Eddy Simulations, J. Theoretical Fluid Dyn., 9, (1997), 85. 10. Speziale, C.G.: Galilean Invariance of Sub Grid Scale Stress Models in Large Eddy Simula tions of Turbulence, J. Fluid Mech., 156, (1985), 55. 11. Smith, G.F.: On Isotropic Functions of Symmetric Tensors, Skew Symmetric Tensors and Vectors, Int. J. Engng. Sci, 9, (1971), 899. 12. Vreman, B., Geurts, B. & Kuerten, H.: Realizability Conditions for the Turbulent Stress Ten sor in Large Eddy Simulation, J. Fluid Mech., 278, (1994), 351. 13. Borue, V. & Orszag, S.A.: Local Energy Flux and Subgrid-Scale Statistics in Three Dimen sional Turbulence, J. Fluid Mech., 366, (1998), 1. 14. Smagorinsky, J.S.: General Circulation Experiments with Primitive Equations”, Mon. Weather Rev., 91, (1963), 99. 15. Schumann, U.: Subgrid Scale Model for Finite Difference Simulation of Turbulent Flows in Plane Channels and Annuli, J. Comp. Phys., 18, (1975), 376. 16. Menon, S. & Kim, W.-W.: High Reynolds Number Flow Simulations Using the Localised Dynamic Sub Grid Scale Model, AIAA paper No. 96-0425 (1996). 17. Metais, O. & Lesieur, M.: Spectral Large Eddy Simulation of Isotropic and Stably Stratified Turbulence, J. Fluid Mech., 239, (1992), 157. 18. Yoshizawa, A. & Horiuti, K.: A Statistically-Derived Subgrid Scale Kinetic Energy Model for Large Eddy Simulation of Turblent Flows, J. Phys. Soc. Japan, 54, (1985), 2834. 19. Sagaut, P.: Numerical Simulations of Separated Flows with Subgrid Models, Rech.Aéro, 1, (1996), 51. 20. Voke, P.: Subgrid-scale Modeling at Low Mesh Reynolds Number, J. Theoretical Fluid Dyn., 8, (1996), 131. 21. Germano, M., Piomelli, U., Moin, P. & Cabot, W.H.: A Dynamic Sub Grid Scale Eddy Vis cosity Model, Phys. Fluids A, 3, (1994), 1760. 22. Meneveau, C., Lund, T.S. & Cabot W.H.: A Lagrangian Dynamic Subgrid-scale Model of Turbulence, J. Fluid Mech., 319, (1996), 353. 23. Carati, D., Wray, A. & Cabot, W.: Ensemble Averaged Dynamic Modeling, Proceedings of the Summer Program – Center of Turbulence Research, (1996), 237. 24. Deardorff, J.W.: The Use of Subgrid Transport Equations in a Three-Dimensional Model of Atmospherical Turbulence, ASME, J. Fluids Engng. Trans., 95, (1973), 429. 25. Fureby, C., Tabor, G., Weller, H.G. & Gosman, A.D.: On Differential Sub Grid Scale Stress Models in Large Eddy Simulations”, Phys. Fluids, 9, (1997), 3578. 26. Rodi, W.: A New Algebraic Relation for Calculating the Reynolds Stresses, ZAMM, 56, (1976), 219. 27. Bardina, J., Ferziger, J.H. & Reynolds, W.C.: Improved Subgrid Scale Models for Large Eddy Simulations, AIAA Paper No. 80-1357, (1980). 28. Liu, S., Meneveau, C. & Katz, J.: On the Properties of Similarity Subgrid-scale Models as Deduced from Measurements in a Turbulent Jet, J. Fluid. Mech., 275, (1994), 83. 29. Lund, T.S. & Novikov, E.A.: Parameterization of Subgrid-scale Stress by the Velocity Gradi ent Tensor, Annual Res. Briefs. – Center for Turbulence Research, (1994), 185. 30. Scotti, A., Meneveau, C. & Lilly, D.K.: Generalized Smagorinsky Model for Anisotropic Grids, Phys. Fluids A, 5, (1993), 2306. 31. Moin, P. & Kim, J.: Numerical Investigation of Turbulent Channel Flow, J. Fluid Mech., 118, (1982), 341. 32. Sullivan, P.P., McWilliams, J.C. & Moeng, C.H.: A Subgrid Scale Model for Large Eddy Simulation of Planetary Boundary-Layer Flows, Bound. Layer Meth., 71, (1994), 247. 33. Carati, D. & Cabot, W.: Anisotropic Eddy Viscosity Models, Proceedings of the Summer Program – Center of Turbulence Research, (1996), 249. 34. LeVeque, R.J.: Numerical Methods for Conservation Laws, Birkhüser Verlag, Berlin (1992). 35. Boris, J.P., Grinstein, F.F., Oran, E.S. & Kolbe, R.L.: New Insights into Large Eddy Simula tion, Fluid Dyn. Res., 10, (1992), 199. 36. Fureby, C. & Grinstein, F.F.: Large Eddy Simulations of High Reynolds number Free and Wall
TOWARDS LARGE EDDY SIMULATION
183
Bounded Flows, Submitted to J. Comp. Phys., (2001) 37. Tsuboi, K., Tamura, T. & Kuwahara, K.: Numerical Study of Vortex Induced Vibration of a Circular Cylinder in High Reynolds Number Flow, AIAA Paper 89–1824, (1989). 38. Fureby, C. & Grinstein, F.F.: Monotonically Integrated Large Eddy Simulation of Free Shear Flows, AIAA.J., 37, (1999), 544. 39. Okong’o, N., Knight, D.D., & Zhou, G.: Large Eddy Simulations Using an Unstructured Grid Compressible Navier-Stokes Algorithm, Int. J. Comp. Fluid Dynamics, 13, (2000), 303. 40. Karamanos, G.-S. & Karniadakis, G.E.: A Spectral Vanishing Viscosity Method for LargeEddy Simulations, J. Comp. Phys., 163, (2000), 22. 41. Fureby, C., Tabor, G., Weller, H.G. & Gosman, A.D.: A Comparative Study of Sub Grid Scale Models in Homogeneous Isotropic Turbulence, Phys. Fluids, 9, (1996), 1416. 42. Olsson, M. & Fuchs, L.: Large Eddy Simulation of the Proximal Region of a Spatially Devel oping Circular Jet, Phys. Fluids., 8, (1996), 2125. 43. Grinstein, F.F., Hussain, F. & Boris, J.P.: Dynamics and Topology of Coherent Structures in a Plane Wake, in Advances in Turbulence 3, Johansson A.V. & Alfredsson P.H. (eds), Springer, Heidelberg, (1991), 34. 44. Rodi, W., Ferziger, J.H., Breuer, M. & Pourquiré, M.: Status of Large Eddy Simulation: Re sults of a Workshop, ASME, J. Fluids. Engng. Trans., 119, (1997), 248. 45. Cohen, J. & Wygnanski I.: The Evolution of Instabilities in the axisymmetric Jet. Part 1. The Linear Growth of Disturbances Near the Nozzle., J. Fluid Mech., 176, (1987), 191. 46. Crow, S.C. & Champagne, F.H.: Orderly Structure in Jet Turbulence, J. Fluid Mech., 48, (1971), 547. 47. Grinstein, F.F.: Coherent Structure Dynamics and Transition to Turbulence in Rectangular Jet Systems, AIAA paper 99-3506, (1999). 48. Martinuzzi, R. & Tropea, C.: The Flow Around a Surface-Mounted Prismatic Obstacle placed in a Fully Developed Channel Flow, J. Fluid Engng., 115, (1993), 85. 49. Krajnovic, S., Müller, D. & Davidsson L.: Comparison of Two One-Equation Subgrid Models in Recirculating Flows, Direct and Large Eddy Simulation III, Eds: Voke, P., Sandham, N.D. & Kleiser, L., 63, (1999). 50. Breuer, M.: Large Eddy Simulation of the Subcritical Flow Past a Circular Cylinder: Numeri cal and Modeling Aspects, Int. J. Numer. Meth. Fluids, 28, (1998), 1281. 51. Lesieur, M & Metais, O.: New Trends in Large Eddy Simulations of Turbulence, Ann. Review Fluid Mech., 28, (1996), 45. 52. Fureby, C.: Large Eddy Simulation of Combustion Instabilities in a Jet-Engine Afterburner Model, Comb. Sci. & Tech., 161, (2000), 213. 53. Menon, S.: Subgrid Combustion Modeling for Large-Eddy Simulations of Single and TwoPhase Flows, Proc. of EUROMECH Colloquium 412 on LES of Complex Transitional and Turbulent Flows, Kluwer Press. 54. Volmers, H., Kreplin, H.P. Meier, H.U. & Kühn, A.: Measured Mean Velocity Field Around a 1:6 Prolate Spheroid at Various Cross Sections, DFVLR IB 221-85/A 08, Göttingen, Germany, (1985). 55. Chesnakas, C.J. & Simpson, R.L.: Measurements of Turbulence Structure in the Vicinity of a 3D Separation, J. Fluids Eng., 118, (1996), 268. 56. Hedin, P.-O., Alin, N., Berglund, M. & Fureby, C.: Large Eddy Simulation of the Flow Around an Inclined Prolate Spheroid, AIAA Paper 01-1035, (2001). 57. Rossiter, J.E.: Wind-Tunnel Experiments on the Flow over Rectangular Cavities at Supersonic and Transonic Speeds. Reports and Memoranda No. 3438, (1964). 58. Heller, H.H. & Bliss, D.B.: The Physical Mechanism of Flow-Induced Pressure Fluctuations in Cavities and Concepts for Their Suppression, AIAA paper 75-491, (1975). 59. Lillberg, E. & Fureby, C.: Large Eddy Simualtion of Supersonic Cavity Flow, AIAA paper 00-2411, (2000).
This page intentionally left blank
STRUCTURED AND NON-STRUCTURED LARGE EDDY SIMULATIONS OF THE AHMED REFERENCE MODEL.
R.J.A.HOWARD AND M. LESIEUR Equipe MOST, LEGI, BP 53, INP Grenoble, 38041 Cedex 9, France AND
U. BIEDER CEA Grenoble, 17 rue des Martyrs, 38054 cedex 9, France
Abstract. The thermo-hydraulic code PRICELES is used to carry out large eddy simulations of the flow around the Ahmed et al (1984) simplified car body geometry. The calculations are carried out below the experimen tal Reynolds number of at This allows the simulations to be resolved without the need for very dense grids. This in turn makes it possible to carry out a comparison between a large eddy simulation using a standard structured mesh and a non-structured tetra haedral mesh. The structured grid contains 10 times as many elements as the non-structured grid and takes twice as much CPU time. The total drag estimation from the wake of the structured grid is which is ap proximately 30% smaller than the tetrahaedral grid. Both methods showed unsteady wake flow with oscillations at the same predominant frequencies. The flow structures observed are similar with secondary vortices intertwined with the principle vortices however these do not persist on the tetrahaedral mesh due to downstream coarsening of the grid. This type of observation combined with the larger drag coefficient indicate that the non-structured calculation is more diffusive and hence less energetic.
1. Introduction The Ahmed reference model is a generic car type bluff body shape which is sufficiently simple for accurate flow simulation while retaining some im portant practically relevant features of car bodies. The Ahmed body has 185 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 185-198. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
186
R.J.A.HOWARD ET AL.
been studied as a body in ground proximity and as a body in free space. The latter case is more representative of the car flow problem since a mov ing car has a velocity relative to the ground plane whereas when ground proximity is imposed (such as in the Ahmed 1984 experiment), the body is stationary relative to the ground. However the most rigorous case is when a moving ground plane is imposed. The effect of ground clearance on the flow around bodies in general is clearly an important aspect in itself and some workers such as Geropp & Odenthal (2000) and Garry (1996) have included a moving ground in their studies. For the FLUENT calculations of Makowski & Kim (2000) and Han (1989) and the experiments of Becker et al (2000) a stationary ground plane is applied below the Ahmed body whereas the calculations of Gilliéron & Chometon (1999) and Basara et al (1998,1999) assume the body is in free space. An important result of the Ahmed et al (1984) experiment was that the main contribution to the drag on the body was due to the pressure drag and hence the rear end geometry of the body. From this observation they conducted tests varying the slant angle of the rear end of the body and discovered an angle which minimised the drag. This angle was found to be approximately 30 degrees. It is for this reason that the geometry chosen for this study is for the body with a slant angle at 28 degrees. The wake behind the Ahmed body with a slant angle of 28 degrees nor mally contains contra-rotating trailing vortices. These large structures have been observed both in the experiments of Ahmed et al (1984), Becker et al (2000) and Shaw et al (2000) and all the calculations. However, Shaw et al (2000) and Basara et al (1998,1999) also note the presence of time depen dent behaviour in the wake flow and hence the possibility of non symmetric behaviour in the flowfield. This type of feature cannot be captured by calcu lations which solve the stationary RANS equations and assume symmetry along the body centreline. Clearly, after taking statistics over a long pe riod, the mean flow is symmetric as shown by Becker et al (2000), however this does not discount the presence of non-stationary, non-symmetric inter actions. In a series of studies of this type of flow Basara et al (1998,1999 and 2000) also highlight many important points and conclusions. They note that steady and Reynolds stress models are liable to “only accidentally produce good results” depending on the flow behaviour and a “transient approach is the only approach for such simulations”. Their final observation was that “unsteady Reynolds stress transport equation models continuously produce results which are in better agreement with the measurements”. As a natural progression from this it follows that an LES of the entire body also accounts for these factors and, in the same way as for the unsteady Reynolds stress models, can provide much useful information which was not previously available.
LES OF THE AHMED BODY
187
Drag calculations based on the pressure distribution around the body with a 30 degree slant angle have shown the values from the FLUENT cal culations to be close to the experimental values however as Makowski & Kim (2000) acknowledge, Basara et al (1998, 1999) noted and as can be seen from Gilliéron and Chometon’s (1999) results this is due to fortuitous overprediction of the forebody drag cancelling underprediction of the aftbody drag rather than an accurate flow prediction. Another approach to the drag prediction is used by Geropp & Odenthal (2000) in which they used the momentum deficit in the wake to evaluate the drag. In this paper two approaches to carrying out a large eddy simulation of this problem are explained and then results are presented at low Reynolds number. The two methods are tetrahaedral and traditional structured grids. Simulation at low Reynolds number makes it possible to carry out the tests at relatively low computational cost. The grids are arranged so that the geometry of body is clearly modelled and flow around the body is reasonably resolved at the Reynolds number used. It is rather difficult to select the grids in order to have the same computational cost for example, even if the number of degrees of freedom is the same, it is not possible to equalise the effects of such constraints as the number limit for the time step or the number of iterations required for a divergence free solution in the pressure solver. Despite this, the work here provides an indication of the different problems and benefits in each method. 2. LES approach using PRICELES The flow domain used is derived from CAD data for the body geometry at 28 degrees provided by M. Dufresne-Renexter and P. Gilliéron of Renault Guyancourt. In this configuration the body of length L is in a domain of 8L × 2L × 2L in the (streamwise spanwise and stream-normal directions) at a distance of 2L from the inlet and centered in the spanwise and stream-normal directions as shown in figure 1. The boundary conditions for the problem are symmetry for the upper lower and sides of the domain with uniform flow at the inlet and no-slip for the surfaces of the body. Imposed pressure is used as the outflow boundary condition. PRICELES is an intrinsically parallel, object oriented thermal hydraulic code written in and developed at the Commissariat d’Energie Atomique (CEA) in Grenoble, France. Compressibility effects are assumed to be negli gible for this flow and therefore the incompressible Navier-Stokes equations are solved. The flexibility of the code makes it possible to conduct calcula tions on both structured cuboid and non-structured tetrahaedral meshes. Before beginning to analyse the results, it is important to point out many of the a priori benefits and problems with each method.
188
R.J.A.HOWARD ET AL.
Firstly, the standard structured cuboid method is considered. This method is constrained to model all sloped and curved surfaces by ‘steps’. Thus grid refinement is required in order to better describe the geometry. In addition, the grid refinement must extend to the boundaries of the domain uniformly resulting in an additional cost in regions where the grid is dense but the flow is relatively inactive. The discretisation used is a staggered finite vol ume method with one velocity calculated at each face and pressure at the element centres. This results in a well posed problem, reducing the number of iterations required for calculating the divergence free condition for the pressure correction. Time advancement is made using a compact third order Runge-Kutta scheme with a second order central scheme for the convection. Secondly, the case of the non-structured tetrahaedral method. This method makes it possible to generate grids for highly complex and dis torted shapes. In addition, the grid density can be easily localised depend ing on where high grid concentration is required. The discretisation is a P1 -non-conforming / P1-iso-P1-bubble element with velocity nodes on the element faces and pressure nodes at the element centres and vertices. For each element an overlapping system of three control volumes is used. This is constructed with one at the face centres for conservation of momentum a second at the centre of gravity for mass conservation and a third at the vertices also for mass conservation. This means that there is one veloc ity on each face and one pressure node at each element centre, as for the structured grid, however, in addition, there are extra pressure nodes on the vertices with the corresponding additional control volume (reasons for this are discussed in Heib & Emonot 2001). This adds to the cost per el ement. Large changes in the grid distribution can introduce commutation errors between the grid and the LES turbulence model which are difficult
LES OF THE AHMED BODY
189
to quantify. In addition the grid distribution near flat walls means that many element surfaces are inclined with respect to the wall leading to large degrees of irregularity in the way the control volumes are arranged. Near wall control volumes parallel and normal to the wall are more stable and resolve the new wall flow more accurately. Time advancement is made using a second order centered (predictor/corrector) scheme with a similar second order scheme for the convection. The body geometry at 28 degrees was provided by M. Dufresne-Renexter and P. Gilliéron at Renault Guyancourt and the tetrahaedral grid was gen erated using ICEM cfd (tm) non-structured mesh generator. The structured grid was created by G. Fauchet using a complex geometry structured grid generator developed by G. Fauchet and B. Menant at the CEA Grenoble. In all cases the pressure field is calculated using a conjugate gradient solver which is iterated until the divergence of the velocity field is below a user defined threshold. If this threshold is too high a large residual error is retained in the mass balance distorting the flowfield however if the threshold is too low the number of iterations per time step become large and the cal culation becomes prohibitively slow. Here the threshold is set to 0.000001/s. Subgrid scale turbulent motions are resolved using the Smagorinsky turbu lence model turbulence model and details of the model implementation can be found in Ackermann, 2000. The time step size is chosen using a CFL number of 1. In this section the main approach to this problem has been outlined however before continuing to look at the results it is necessary to include some comments on further code improvements that will eventually con tribute to the quality of this type of study. It is envisaged to extend the PRICELES/Trio-U code system to make use of a non-structured hexahae dral element. This type of element will be able to improve the near wall behaviour of the flow compared to the tetrahaedral element while not being constrained by the grid density requirements for complex geometries of the structured mesh method. In addition the convective Orlansky outflow con dition will be made available for all the different element types to replace the pressure imposed outlet condition. This will allow large flow structures to pass more freely through the outflow without upstream distortion. These two developments are not at this stage fully validated and therefore have not been included here. 3. Results and Analysis The results are studied first from the point of view of resolution and cor responding cost requirements. Then some flow field statistics are examined followed by visualisation. Finally the conclusion brings together the main
190
R.J.A.HOWARD ET AL.
points. 3.1. GRID RESOLUTION AND COMPUTATIONAL COST Since the two methods contain a variety of characteristics the goal is to assess the performace of each method in terms of the computational cost. The computational cost here is defined as the CPU time for the calculation to advance a given duration in physical time. This is clearly difficult com parison to make a priori due to the significant differences in discretisation, grid distribution and the way they affect the flowfield itself. Cross sections of the two grids are shown in figures 2 and 3. The struc tured grid contains 2016916 elements and the tetrahaedral grid contains 228441 elements. From the grid cross sections it can be seen that the grids are not evenly distributed it is thus necessary to provide some indication of the extent to which the flow can be resolved on each grid. To do this it is necessary to look at several characteristic length scales associated with the problem. The length scale associated with the smallest turbulent motions for this flow can be evaluated from an estimate of the Kolmogorov microscale given while the length scale for the large scale energy producing by motions can be obtained from an estimate of the Taylor microscale given by For the Reynolds number of Re = 4290 the length scales are and For both meshes the smallest grid spacings are in the vicinity of the
LES OF THE AHMED BODY
191
body surface in order to resolve the surface shear stress. For the structured grid the wall spacing is 0.006L while for the non-structured case the wall spacing is approximately 0.008L. Further away from the body an effort is made to make the grid more sparce in order to reduce the computational cost. The largest spacings are 0.17L for the structured grid and 0.35L for the tetrahaedral grid. From these values it can be seen that both grids have some meshes much larger than the Taylor microscale margin however in the vicinity of the body both grids tend towards the level of the Kolmogorov scale. Another indicator for the resolution is the non-dimensional wall coor where is the dinate of the first grid point. This is given by the wall friction velocity, the wall-normal distance and the molecular viscosity. This parameter is thus dependent on the local flow behaviour. For a DNS the first grid location is normally of the order 1 and an LES is typically below 10. For both grids the value of is integrated over all the elements touching the body surface. For the structured grid this gives Both these values and for the non-structured grid are close to the range acceptable for a DNS. For reference, the largest grid for the structured grid and spacings in wall units are
192
R.J.A.HOWARD ET AL.
for the non-structured grid. The structured grid takes approximately 24 hours for the flowfield to advance one second compared to 11.5 hours for the tetrahaedral grid. This indicates that the grid density of the tetrahaedral case could be increased in order to have a computational cost at a level comparable to that of the structured case. 3.2. FLOW STATISTICS AND DRAG ESTIMATION
Velocity profiles are plotted from traces taken in the centre of the domain at a distance L behind the body. From figure 4 it is clear that both grids show large amounts of oscillation and the energy spectra plots show these oscillations to have similar frequency ranges with predominant frequen and in both cases. The spectrum of the cies at approximately structured grid has a larger amplitude than the tetrahaedral grid over most frequencies indicating the flow is more energetic.
Two methods are used to examine the behaviour of the drag coefficient. The first is determined from the integrated momentum balance of a control volume around the body
where is the cross-sectional area the body presents to the incoming
flow (its width multiplied by its height), is the cross sectional area of
each element in the wake cross-section, subscript 1 corresponds to a plane
LES OF THE AHMED BODY
193
upstream of the body and subscript 2 corresponds to a plane downstream of the body. Some turbulent effects are included however it is assumed that the planes used for the control volume are sufficiently far from the body that viscous effects are negligible. Two measuring stations are used with 20 equally spaced velocity slices in the spanwise and stream normal directions respectively covering planes at L and 1.5L behind the body. This method makes use of the mean and turbulent velocity profiles and mean pressure profiles in the wake and thus relies on a large statistical sample to avoid statistical scatter. A large statistical sample would typically be the time for the mean flow to traverse the length of the computational domain of the order of 10 times. This corresponds to 80 seconds (this is rather small compared to the sample time used in experiments of Becker et al (2000) who used a sample time of 5 minutes). Here due to time and computational cost constraints the statistics have been accumulated over a sampling period of around 6 seconds for the structured grid and 3 seconds for the non-structured grid (after having run the simulation for more than 8 seconds which is the time to allow the initial transient to pass through the domain). It has been highlighted in previous works and also found in this study that the pressure drag contributes the most significantly to the total drag for this flow. The second method examines the relative contributions of the surface normal pressure and friction drag to the total drag. The surface normal pressure drag
is made up of the pressure coefficient acting on surfaces which have a component of cross sectional area normal to the flow direction for the total area of the surface and for the area per element). The friction drag
is made up of the skin friction force acting in the flow direction, over the surface of the body which is parallel to the flow direction for the total area of the surface and for the area per element). The drag coefficients evaluated using the both methods are summarised in table 1 and the time distributions are shown in figure 5. From these fig ures it can be seen that the non-structured calculation has a significantly higher drag coefficient than the structured calculation. This indicates that the non-structured simulation is insufficiently resolved or rather insufficient
194
R.J.A.HOWARD ET AL.
resolution makes the flow less energetic resulting in a drag coefficient within a laminar regime. The drag coefficients evaluated from the momentum bal ance values are larger in particular for the non-structured grid. This is most probably because of a reduced statistical sample and extrapolation errors. All drag coefficient values calculated are above the found in the FLUENT and experimental estimates. However the drag coefficient has not been evaluated experimentally at the Reynolds number of this study
LES OF THE AHMED BODY
195
so this does not represent an appropriate comparison. 3.3. FLOW VISUALISATION
Since the main contribution to the drag of the body is due to the trailing edge flow behaviour it is therefore important to identify the main flow features in this region. As already indicated in the previous studies there are contra rotating streamwise vortical structures generated at the trailing edge. In order to observe these flow features two flow parameters were examined in particular. These were the streamwise vorticity and the Q The Q criterion has been criterion where shown to indicate more subtle vortices including those not aligned in the streamwise direction. In this particular flow at this Reynolds number the Q criterion did not reveal any additional vortical structures other than those shown by the streamwise vorticity so in this case only the streamwise vorticity is examined. Figure 6 shows the streamwise vorticity around the body on both meshes. In both cases it is possible to observe the main contra rotating vortices in addition both cases show the presence of secondary vortices of opposing sign that are intertwined with the principle vortices. This carries on down stream much further in the case of the structured grid as the non-structured grid is no longer dense enough to resolve the vortices further downstream. In addition a contour plot of the total vorticity for the tetrahaedral grid shown in figure 7 shows that there is some additional weak total vorticity downstream of the body caused by an abrupt change in the grid density. This is an indicator that the grid needs to be more dense or at least have a more smooth change in density in this region. 4. Conclusions and discussion The PRICELES/Trio-U code system is sufficiently flexible to carry out both structured cuboid and non-structured tetrahaedral LES calculations of the flow over the Ahmed body geometry. The structured grid contained many more elements than the non-structured grid since a higher resolution is required in order to map the curved front end and sloping trailing edge of the body. This caused the structured grid calculation to use up twice the CPU time. At the low Reynolds number tested, the trailing edge vortical structures observed in the other studies is reproduced. In addition both simulations show the presence of secondary vortices around the principle vortices and significant time dependence in the wake flowfield which is not reported in many other works and ignored in the FLUENT calculations. These secondary structures did not persist far downstream on the tetrahaedral grid due to coarsening of the mesh. The drag coefficient
196
R.J.A.HOWARD ET AL.
LES OF THE AHMED BODY
197
calculated in the wake is for the structured grid and 0.69 for the tetrahaedral grid as compared to calculated in the experiment for higher Reynolds number. The combination surface pressure for the structured grid and and friction drag coefficients give for the tetrahaedral grid. Energy spectra behind the body show that both calculations produce oscillations with predominant frequencies at 0.5 and (this corresponds to the same values for the Strouhal numbers, due to the normalisation of the calculations) since but the non-structured calculation is at significantly lower energy levels. The structured grid calculation contained more grid points and therefore generated more accurate results. The tetrahaedral grid calculation had a grid too sparce in places which degraded the results. The main effect of this was to make the flow less energetic and therefore closer to a laminar flow regime. It is estimated that careful adjustment of the grid density and inclusion of additional elements (not more than twice the number) will improve these results while remaining at approximately the same cost as the structured grid. 5.
Acknowledgements
The authors would like to thank P. Emonot for indispensable help. Com puter time was provided by the MOST lab at the Industrial and Geophys ical flow Laboratory (LEGI) in Grenoble on the COMPAQ SC 232 parallel cluster of the CEA. The work was funded by the EU transnational network on large-eddy simulation of complex industrial flows. References Ahmed, S.R., Ramm, G. and Faltin, G., 1984, Some salient features of the time averaged ground vehicle wake, Paper No. 840300, Society of Automotive Engineers, Inc. Ackermann, C., 2000, Développements et Validation de Simulation des Grandes Echelles d’écoulements turbulents dans un code industriel, PhD thesis, Institut National Poly
198
R.J.A.HOWARD ET AL.
technique de Grenoble. Basara, B. & Alajbegovic, A., 1998, Steady state calculations of turbulent flow around Morel body, 7th int Symposium on Flow Modelling and Turbulence, Oct 5-7, Taiwan. Basara, B., 1999, Numerical simulation of turbulent wakes around a vehicle, ASME Fluids engineering division summer meeting, proceedings FEDSM99-7324, July 18-22, San Francisco, USA. Basara, B., Przulj, V. and Tibaut, P., 2001, On the calculation of external aerodynamics: Industrial benchmarks, SAE conference , 2001-01-0701. Becker, Lienhart and Stoots, 2000, Flow and Turbulence Structures in the Wake of a Simplified Car Model (Ahmed Model), DGLR Fach Symp. der AG STAB, Stuttgart University, 15.-17.November 2000. Garry K.P., 1996, Some effects of ground clearance and ground plane boundary layer thickness on the mean pressure of a bluff vehicle type body, J. of Wind Engineering and Industrial Aerodynamics, 62, 1-10. Geropp, D., Odenthal, H-J., 2000, Drag reduction of motor vehicles by active flow control using the Coanda effect, Experiments in fluids, 28, 74-85. Gillieron, P., Chometon, F., 1999, Modelling of stationary three-dimensional separated flows around an Ahmed reference model, ESAIM proc, vol 7, 173-182 Han, T., 1989, Computational analysis of three dimensional turbulent flow around a bluff body in ground proximity, AIAA Journal, 27, 9. Heib, S. & Emonot, P. 2001, Convergence of a new non-conforming equal order finite volume element method for Stokes problems, (to be published). Houghton, E.L. & Carruthers, N. B. 1990, Aerodynamics for engineering students, 3rd edition, pub Edward Arnold.. Morel T., 1978, The effect of base slant angle on the flow pattern and drag of threedimensional bodies with blunt ends, Proc of Symposium on Aerodynamic drag mech anisms of bluff bodies and road vehicles, New York, Editors G.Sovran et al., pp 191-226. Makowski F.T., Kim, S-E, 2000, Advances in external-aero simulation of ground vehicles using the steady RANS equations, SAE conference, 2000-01-0484. Shaw C.T., Garry K.P. and Gress T., 2000, Using singular systems analysis to charac terise the flow in the wake of a model passenger vehicle, J. of Wind Engineering and Industrial Aerodynamics, 85, 1-30.
LARGE EDDY SIMULATION OF FLUID FLOW AND HEAT TRANSFER AROUND A MATRIX OF CUBES, USING UNSTRUCTURED GRIDS
B. AND K. Department of Applied Physics, Delft University of Technology Lorentzweg 1, 2628 CJ Delft The Netherlands
Abstract We present an unstructured finite-volume algorithm for Large-Eddy Simulations (LES), aimed at facilitating grid resolution problems in high-Reynolds-number flows bounded by walls of complex configuration. The method uses computa tional cells of arbitrary shapes in colocated arrangement, with variables defined at cell centres and varying linearly in between. The convection and diffusion fluxes are discretized using second-order accurate central differencing. Two test cases were considered to validate the method: fully developed turbulent flow in a plane channel, and flow and conjugate heat transfer around the wall mounted matrix of cubes. For each test case, two types of grids were used: a hexahedral and a hybrid grid. The computed results for all cases considered compare well with previous numerical or experimental findings. The algorithm is potentially suitable for LES of complex wall-bounded flows and heat transfer of industrial relevance. 1. Introduction Large Eddy Simulations (LES) of complex wall-bounded turbulent flows of tech nological relevance are still difficult and rare in the literature. A major challenge is the treatment of the near-wall regions and accurate prediction of wall friction, heat and mass transfer, particularly at realistically high Reynolds numbers in fully three-dimensional flows. Resolving the small-scale structures in the buffer zone requires a very fine numerical mesh not only in the wall-normal direction, but also in the streamwise and spanwise directions. The near-wall phenomena are captured well with LES only if the near-wall grid is sufficiently fine for subgrid scale stresses to be of the same order of magnitude as (or less than) the molecular stresses. In this case the subgrid-scale model (SGS) becomes less important and 199 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 199-216. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
200 its role is only to drain the turbulence energy. Such grid refinement with structured and even with block-structured grids leads to an increase in the number of cells even in the regions far away from the walls where it is not needed. The problem becomes especially serious at high Reynolds numbers when the viscous sublayer and the buffer zone become very thin, imposing exorbitant demands on computing resources. The applicability of LES can be significantly extended and its performance enhanced by using an unstructured numerical grid. Such grids offer several ad vantages. The possibility to use polyhedral elementary control cells of various shapes (e.g tetrahedra, hexahedra, polyhedra, or mixed) makes the discretisation of the complex flow domains easier (Mavripilis, 1997). It permits adaptive mesh ing with more flexibility to control the grid refinement in desired area in a more rational way than the structured grids. This in turn enables to achieve sufficient resolution in regions where it is needed, with comparably less total number of grid cells than with structured and even block-structured grids. A rational local grid refinement is particularly useful in industrial flows with complex wall topol ogy, e.g. with protrusions and cavities, or in flows with high local gradients of flow properties, e.g. in the area around the burner mouth in combustion instal lations. Moreover, the unstructured grid seems naturally suited for LES, because the subgrid-scale model is based on the grid size, which needs to be adjusted to resolve the desired local turbulence scale. We present here an unstructured LES algorithm for finite volume computa tions, based on the approach used currently for Reynolds-averaged Navier-Stokes computations and, with modifications pertinent to LES. The approach targets complex wall-bounded flows which require the use of unstructured computational grids to accommodate in a rational manner complex wall configurations. The method uses the colocated cell arrangement with variables defined at cell centres and varying linearly in between. The convection and diffusion fluxes are discre tised using second-order accurate central differencing and the time marching was performed using the three-level implicit scheme. The paper focuses on the nu merical aspects and on effects of the shape of computational cells in unstructured grid arrangement. The numerical scheme was earlier validated in two simple test cases: laminar flow in a lid-driven cavity and Taylor-Green vortices, both with several types of unstructured grids with regular hexahedral, tetrahedral and mixed cells and 2001). Here we present first some results of direct numerical simulation (DNS) of flow in a plane channel using a hybrid tetrahedral grid with a layer of hexahedral cells near the walls. This is followed by large eddy simulations (LES) of flow and heat transfer over a surface-mounted cube in a ma trix. The latter case was simulated using in parallel two different body-fitted grids: a moderately dense hexahedral grid and a hybrid one with predominantly trian gular cells combined with a thin near-wall layer of hexahedral cells. The standard Smagorinsky model was used to model subgrid-scale stresses in LES. All results
201 show good agreement with data available in the literature - DNS of other authors for the channel flow and experiments for the cube matrix.
2. Governing Equations for LES For the finite volume method it is convenient to integrate a priori the conservation equations over the elementary computational cell. For incompressible flow with constant fluid properties the momentum and enthalpy equations can be written as:
Here V is the control volume bounded by a closed surface S, the fluid density, u the velocity vector, the specific heat, T the temperature and and are the effective viscosity and thermal conductivity comprising molecular and turbu lent part:
For the turbulent viscosity
we adopted at present the Smagorinsky model:
where is the Smagorinsky constant, D the deformation rate tensor and corrected filter width, defined as (Spalart et al., 1997):
the
Here is the distance to the nearest wall and the volume of computational cell. This approach is based on the physical reasoning that the eddy in the near wall region can not be larger then the distance to the wall and also provides a simple way to reduce eddy viscosity in complex geometries where wall-function treatment is not applicable.
3. Numerical Method The traditional methods in LES, such as high order (higher then two) finite dif ferences or spectral methods, are not suitable for complex flow configurations of industrial relevance. On the other hand, the computational methods for flows in complex domains, such as found in commercial CFD finite element or finite vol ume packages, are based on low-order numerical schemes and cannot be used di rectly for LES. Modifications are needed to enhance the accuracy and the time and
202
space resolution, but higher order discretisation schemes need to be compromised with the geometrical flexibility and suitability for parallelisation. This compro mise in the discretisation schemes can be compensated by local grid refinement, for which the unstructured arrangement is very suited. We follow the control-volume rationale used in Reynolds-averaged NavierStokes (RANS) approaches for unstructured grids, et al., 1997). Pos sible computational cells are shown in Fig. 1. Any cell type (hexahedral, trape zoidal, tetrahedral, hybrid) is allowed and local grid refinement can be achieved by locally splitting the computational cells. This makes the method easily appli cable to industrial problems. An important feature of the method is that the grid is colocated, i.e. both velocities and pressure are calculated at the centres of finite volumes, which saves a great deal of memory required for storing the geometrical data. There is a price to pay for such saving, i.e. colocated variable arrangement is more prone to oscillations in the flow field, and blended differencing schemes with a small percentage of upwinding (1 – 2 % ) is usually needed to increase the stability of the method. Nonetheless, the memory saved by storing the cell cen tre values can be used to increase the total number of cells in the domain, thus reducing the cell Peclet number and reducing the amount of upwinding needed. The applied data structure enables to simplify the computational code and makes it suitable for an easy parallelisation. The data are organised around the computa tional cell faces (Barth, 1994), i.e. for each cell face in the computational domain, we store the cell indices adjacent to it. The typical situation is depicted in Fig. 1. The adjacent cell indices for each cell face are stored which allows to deal with any shape and type of cell. The governing equations given by equations (1) and (2) can be conveniently
203 written in a general form of the conservation law for a generic variable
where stands for velocity components or temperature, is the inertial and is the diffusive coefficient of and represent sources or sinks associated with surface or volume respectively. The meaning of all the quantities from equa tion (7) is given in table 1. The fact that all the transport equations can be written in the form of a single prototype equation (7), greatly facilitates the description of the discretisation procedure and coding of the computer program.
The discretisation of convection terms is made with a mid-point rule, i.e. by assuming that the value at the cell face centre is representative for the entire face:
where
is the mass flux defined on the cell face by:
The equation (9), represents the implementation of Rhie and Chow’s (1983) tech nique for arbitrary grids. The discretisation of diffusive fluxes by approximating the integrals with sums, yields the following expression:
where is the value of the stress tensor defined on the cell face. Gradients
of all dependent variables are calculated with the least square method in the cell
centres, and as such are not sensitive to the oscillations having period twice the
204
characteristic size of the computational cell. Therefore, the gradient at the cell face is decomposed by the following expression:
The first term on the right hand side is the second order accurate gradient of ve locity, and the second term is the first order accurate gradient of velocity which is calculated from the velocities surrounding the cell face To recover second order accuracy of the discretisation, the third term on the right hand side is introduced. The second term is treated implicitly, i.e. it is used to compute the coefficients of the system matrix, whereas the first and third terms are treated explicitly, i.e. introduced into the right hand side of the linear system. The advantage of such a treatment of diffusive terms is that it keeps the number of entries in the system matrix low, which reduces the memory requirements. The disadvantage of such an approach is apparent if the unstructured grid contains tetrahedral elements in the near wall regions. Namely, in computation of turbulent flows, the first computational cell must fall within a certain range of To achieve that, cells must be clustered in the direction normal to the wall. A portion of triangular grid clustered in the near-wall regions is shown in Fig. 2. If the finite element method were used for discretisation of the governing equa tions, this would not pose any difficulties. The variables are placed in the nodes of triangles, Fig. 2a, and the numerical molecule of the near wall nodes has the same structure as in the finite difference methods (Claes, 1987), thus greatly improv
205 ing the accuracy of the solver. Some node-centred finite volume methods (Barth, 1994), (Mavripilis, 1997) also feature the same numerical properties in the nearwall region as does the finite element method. In the cell centred finite volume method, however, the numerical stencil is quite deformed, Fig. 2b. The practical implication of such a stencil is that it leads to an increased weight of explicitly treated diffusive terms (equation 11) and to a decrease of the implicitly treated part. This deteriorates the stability of the linearised system of equations making it practically very close to the singular one. This increases the number of iterations in the linear solver, leading to the increase of the computational time, a feature very undesirable for LES. The solution to this problem is the use of hybrid grids. One can generate the grid in such a way that well shaped cells (triangular prisms or hexahedra) are placed on the solid walls, and the interior of the domain is filled with tetrahedra. As can be observed in Fig. 2b, such an approach gives numerical stencils in the near wall regions very similar to those obtained by finite element or node centred finite volume methods. It could be argued that this reduces the flexibility of the cell centred approach. However, near-wall regions are almost invariably restricted to very small portions of the problem domains, and it is always possible to cover the solid wall with few layers of high-quality hexahedral cells. The use of hybrid grids in LES of turbulence is demonstrated below in the computation of turbulent channel flow and the flow around the surface mounted matrix of cubes. The time-integration is based on the fully implicit three-level scheme. We ex plored earlier other time stepping schemes and 2001), but found no significant difference in the final results. The velocities and the pressure are linked via the standard SIMPLE method (Patankar, 1980), (Ferziger and 1996). The code was parallelised by a domain decomposition technique, using the explicit message passing programming paradigm. Since the code is unstructured, the domain decomposition is very flexible and always results in equally balanced sub-domains. More details on the adopted parallelisation strategy can be found in and 2000).
Boundary conditions For the momentum equation, the standard no-slip condition at the walls and peri odic at free boundaries were applied. For conjugate heat transfer involving solid, multi-layered bodies, the enthalpy equation is solved both in fluid and solid. Ad ditional difficulty appears in the computation of the temperature at the solid-fluid (or other inter-material) interface. The most obvious way to compute the tem perature of the material interface is by balancing the fluxes coming from fluid and from solid phase. With reference to Fig. 3, the balance of heat fluxes can be expressed as:
206
where is the flux through the material interface, is the temperature gradient in the fluid, is the temperature gradient in the solid and and are thermal conductivities in fluid and solid respectively. However, the flux expressed by Eqn. (12) is a function of a wider stencil, i.e. :
which implicitly contains the surface temperature The only way around this problem is to solve from Eqn. (12), Eqn. (13) and Eqn. (14) iteratively. In the present work, the iterations for computation of are imbedded within the iterations of the SIMPLE algorithm. 4. Results 4.1. DNS OF FLOW IN A PLANE CHANNEL ON A HYBRID GRID The first test of the effects of numerical grid and cell shapes was performed in direct numerical simulations of fully developed flow in a plane channel for is friction velocity), using is the channel half width and in parallel a hexahedral and a hybrid unstructured mesh. The flow domain was corresponding roughly to the minimum flow unit (Jiménez and Moin, 1991). For the hexahedral mesh the grid resolution was 48 × 96 × 48 cells which results in 221184 cells. The cells were uniform in streamwise and spanwise direction and hyperbolically stretched normal to the walls. The first near-wall and was 0.0392, and the uniform The choice of an unstructured grid with tetrahedral cells (or any other than hexahedral) requires a priori decisions on the near-wall treatment. The required high resolution in the wall-normal direction leads either to too elongated tetrahe dra, or to a very large number of high-quality cells. We compromised these re quirements by adopting a hybrid mesh with six layers of hexahedral prisms along
207
both channel walls, with the first at 0.045, so that five computational points were placed within In order to resolve the closely spaced longitudinal streaks in the buffer zone, the spanwise resolution is crucial for successful simula tions. Unlike in single-block structured grids, where the fine spanwise resolution leads to a large number of cells even far away from the walls, the unstructured grids offer the possibility to vary the spanwise resolution from very fine near the
208 walls to coarse in the core region of the flow. For the present grid, the number of cells in spanwise direction varied from 72 on both walls to only 20 in the core varied from 4 at the walls up to 14.4 in the core of of the channel. Thus, the flow. The resolution in the streamwise direction was governed by the require ment that the total number of cells in this hybrid unstructured grid matches as close as possible the number of cells in the hexahedral grid in order to allow a fair comparison of the results. It turned out that by placing 32 planes of the unstructured grid pattern shown in Fig. 4 the final grid had 220224 cells, a number very close to 221184 cells of the hexahedral mesh. was 18 throughout the computational domain. The computed mean velocity and Reynolds stresses for both the hexahedral and hybrid grids are shown in Fig. 5. The DNS data (Kim et al., 1987) are also plotted for comparison. The mean velocity profiles obtained with two grid types are practically indistinguishable and both show very good agreement with DNS data. The streamwise stress component obtained on both grids agrees also very well with DNS data especially up to Further away from the wall the re sult for the hybrid grid is slightly underpredicted. The spanwise stress component shows and better agreement with the reference DNS data on a hybrid than on a hexahedral grid. This is undoubtedly due to the fact that the hybrid grid has better spanwise and streamwise resolution in the near wall regions than the hexahedral one. Also the near-wall normal stress obtained on the hybrid grid agrees some what better with the reference DNS data than those obtained on the hexahedral grid. Away from the walls all Reynolds stresses are predicted slightly better with the orthogonal hexahedral grid than with the hybrid unstructured one (though the difference is very small), because of a better spanwise resolution. 4.2. MATRIX OF CUBES The performance of the unstructured code in separated three-dimensional flows was tested in the case of an equidistantly spaced matrix of wall-mounted cubes placed on the channel floor of a rectangular wind tunnel section. The dimension of the cubes was and the width of the channel was The dis tance between the cubes in stream and spanwise direction was The heated cube was made of a copper core with built in electric heater keeping the copper core at constant temperature of 75 °C. The copper core was covered by a thin epoxy layer of width The schematic representation of the heated cube is shown in Fig. 6. This flow served as one of the test cases for and ERCOFTAC/IAHR/COST workshop on refined flow modelling and Obi, 1997), (Hellsten and Rautaheimo, 1999). Two groups contributed to the work and 1999) shop with LES compuations (Mathey et al., 1999) and one group with DNS (Van der Velde et al., 1999) and they all showed good agreement with experimental findings.
209
The measurements (Meinders and 1999), (Meinders, 1998) were performed around the cube in the row where the flow is fully periodic, thus allowing to consider only a single sub-channel unit. The computational domain consisted of a sub-channel of dimension encompassing the cube that was placed in the domain centre. Periodic boundary conditions for velocities and pressure were imposed in both the streamwise and spanwise direction. The incoming fluid was at constant temperature of 20° C (only the considered cube was heated). For temperature, periodic boundary conditions could not be applied, because only one cube was heated. At the inflow of the sub-channel unit, the uni form temperature was prescribed, whereas at the outflow, the normal derivative of the temperature was assumed to be zero. The cubes were placed in an insulated channel and therefore the zero heat flux was specified as boundary condition for the energy equation on the lower and upper wall of the domain. Periodic bound ary conditions for temperature were also specified in the spanwise direction. The (based on the channel height) and at simulation was performed at (air). Two different grids were used for computation of this test case. The first grid contained hexahedral cells only, whereas the second grid was hybrid with tetrahe dra in the interior of the domain and triangular prisms in the near wall region. On the hexahedral grid both the heat transfer and fluid flow were computed, whereas the hybrid grid was used for the solution of the fluid flow only. Although the time stepping scheme is fully implicit and there are no numerical limitations on the magnitude of time step, the excessive values of time step may yield to damping of turbulent fluctuations. In all the computations for the matrix of cubes that we re port here, the maximum CFL number was limited to the value of 1.5. The average CFL number was around 0.6.
Hexahedral grid For the computation of the fluid flow and heat transfer a hexahedral grid was gen erated with 427,512 cells (373,248 cells in the flow and 54,264 cells in the epoxy layer). Figure 7 shows the computational grid. Since we deal with a wall-bounded
210 flow, the presence of the solid boundary affects the physics of the subgrid scales in several ways: the growth of the small scales is inhibited by the presence of the wall, the exchange mechanisms between the resolved and unresolved scales are altered, and the subgrid scales in the near-wall region may contain some signif icant Reynolds-stress producing events (Piomelli, 1997). In order to account for these mechanisms the wall region is resolved by clustering the mesh points hyper bolically towards the walls and edges in such way that the first calculation point was at location For the Reynolds number considered this corresponds to for the channel flow without a cube. The wall-nearest around the cube was roughly of the same order of magnitude. In order to prevent high aspect-ratio cells near the wall, with consequent degradation of the resolution and numerical accuracy, the grid spacing in streamwise and spanwise direction was ensured to be (which is found to be sufficient for LES (Piomelli, 1997)). This cor responds to the dimensionless streamwise and spanwise spacing of for a channel flow, though somewhat larger values were detected on the cube. The cube surface mantle (epoxy layer), was covered by 38 cells in each direction, with 6 cells across the epoxy layer thickness. These cells were also clustered towards the edges and outer boundary of the cube, in order to have approximately the same cell sizes at the fluid-solid interface.
Hybrid grid The concept of hybrid grids was already illustrated in the computation of the DNS of the channel flow. In this section, the same approach was applied to the flow around the matrix of cubes. The most convenient way to generate the hybrid grid in flows bounded by solid walls is to generate first the two-dimensional grid on all the solid walls and then to project a few layers orthogonally into the core of the flow. Finally, the remaining of the domain is filled with tetrahedra. This approach was followed in creating the grid for the wall-mounted cube. First the surface of the cube was covered with triangular grids with the average size of Then the upper and the lower bounding walls of the the triangle edge of
211 domain were covered with a triangular grid with the average edge size of This value provides enough resolution for LES. Having generated the grid for all the solid walls in the domain, the prismatic mesh could be generated for the near wall regions. This was accomplished by projecting six layers of the wall grid orthogonally towards the core of the flow, while adopting the distance of or the same as the first computational point approximately at for the hexahedral grid. To design a mesh for the core of the flow, the domain was logically divided into two regions: a finer region, where higher turbulence intensities and steeper gradients of dependent variables are expected, and the rest of the core where coarser cells are used. The finer and the coarser region are visible in Fig. 8. The total number of cells for the hybrid grid was 597643, approximately 60% more than for the hexahedral grid.
Velocity and stress fields. The computed mean velocity profiles compare well with the experiments for both grids. Figure 10 shows a vector plot of the recirculation region in the vertical plane of the results obtained on the hybrid grid compared with measurements (Meinders, 1998). Figure 11 show the profiles of the normalised streamwise ve
212
locity and the normalised Reynolds stresses and at three locations in the central vertical plane, obtained on the hexahedral grid (full lines) and on hybrid grids (dashed lines), compared with the measurements of (Mein 1999) (symbols). It can be observed that the results for the ders and mean velocity obtained with both grids are very close to each other and are in good agreement with DNS data. Slightly superior results - smoother and in better agreement with experiments, especially for the turbulent stresses, are obtained on the hexahedral grid although the hybrid grid had a larger number of cells. This is believed to be the consequence of larger numerical errors associated with hybrid grids. The results obtained using present methodology compare quite well with the results obtained by LES (Mathey et al., 1999) and DNS (Van der Velde et al., 1999) as it can be found in (Hellsten and Rautaheimo, 1999). Temperature field and heat transfer. Experiments (Meinders and 1999) provided liquid crystal and infrared pictures of the surface temperature, but the detailed profiles have been reported only along the cube circumference in the characteristic mid-planes. In Fig. 12 the temperature profiles obtained with LES are plotted together with experimental data. The results are in good agreement with the experiments (with an experimen tal uncertainty of 0.4 °C), except at the locations A and D, Fig. 12b. The sim ulation predicts a higher surface temperature near the channel floor. These overpredictions are due to the fact that a small but unavoidable heat loss of the epoxy layer through the base wall was not included in the LES. The overpredictions are of the order ~ 10 %. This is in agreement with experimental findings (Meinders, 1998) that a conductive heat loss from the heated cube to the base plate was ap proximately 10 %. The temperature profiles show relatively uniform temperature over the central portion of each cube face, with steep gradients and minima at all cube edges, where the temperature drops by 8-15 °C (about 15-25% of the tem perature in the central regions). This sharp decrease is a direct consequence of high local heat transfer rates, caused by intensive heat removal due to flow sepa et al., 2001). In view of the small size of the cube ration, such a strong temperature variation is a good indication of the complex vortex structure. Similar quality of agreement show the LES results reported in (Mathey
213
214
et al., 1999), which were obtained using a structured Cartesian grid, as shown in Fig. 12, though their results show even larger discrepancies at points A and D. The complex unsteady vortical structure around the cube is reflected in a strong time and space variation of the temperature on the cube surface. This is illustrated in Fig. 13, which shows the instantaneous surface temperature viewed from front and from back. Hot spots on the rear surface are visible, originating from the reduced cooling due to warm fluid trapped in the recirculation region behind the cube. Finally, in order to demonstrate the overall accuracy of the predicted heat transfer, we compare in table 4.2 the time- and surface-averaged heat transfer co efficients on each of the cube faces, with the experimental results. The agreement for the front and top faces is excellent, whereas the results for the rear and side faces show some discrepancy. For the cube as a whole, agreement is very good.
215
5. Concluding Remarks An unstructured finite-volume numerical algorithm for large eddy simulation (LES) is presented. The colocated arrangement of variables defined in the cell centres offers the flexibility to use polyhedral cells of any shapes, allowing in principle a rational meshing of flows bounded by walls of complex wall config uration. Second-order accurate discretisation schemes are used for both the con vection and diffusion fluxes, whereas the time integration was performed by the three-level fully implicit time advancing scheme. The algorithm was first tested in direct simulation of a fully developed chan nel flow and then in large eddy simulations of flow and heat transfer over a wallmounted cube matrix. Both cases were simulated with two types of grid of similar density: hexahedral and tetrahedral. The results for the two test cases showed with both grid types good agreement with previous DNS results and experimental find ings. As expected, the high-quality body-fitted hexahedral grids show advantages in terms of accuracy and computational robustness. It is recalled, however, that hexahedral grids are not easily applicable to more complex wall configurations, where the flexibility of triangular grid offers a decisive advantage: it makes the grid generation much easier and enables a rational coverage with local refinement of flow domain with less number of cells. For accurate capturing of near-wall phe nomena, hybrid grids consisting of tetrahedral cells, matched with a wall-layer of hexahedral cells, offers optimum conditions in terms of accuracy and computa tional efficiency.
Acknowledgements This work was supported by AVL List GmbH, Graz, Austria. We also acknowl edged fruitful discussions with Dr. B. Basara from AVL.
216 References Barth, T. J. (1994) Aspects of unstructured grids and finite volume solvers for the Euler and NavierStokes equations, von Karman Institute lecture Series 1994-05. Claes, J. (1987) Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge. I. Muzaferija S. and M. (1997) Advances in computation of heat transfer, fluid flow and solid body deformation using finite volume approaches, Advances in numerical heat transfer 1, pp. 59-96. Ferziger, J. H. and M. (1996) Computational methods for fluid flow, Springer, Berlin. K. and Obi, S. (edts.) (1997) Proceedings, ERCOFTAC/IAHR/COSTWorkshop on Re fined Flow Modelling, 6-7 June 1997, Delft University of Technology, Delft, The Netherlands. Hellsten, A. and Rauatheimo, P. (edts.) (1999) Proceedings, ERCOFTAC/IAHR/COST Work shop on Refined Flow Modelling, 17-18 June 1999, Helsinki University of Technology, Finland. Jiménez, J. and Moin, P. (1991) The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, pp.213. Kim, J., Moin P. and Moser, R. (1987) Turbulence statistics in fully developed channel flow at low Reynolds numbers, J. Fluid Mech., 177, pp. 133-166. Mathey, F., Fröhlich, J., Rodi. W. (1999) Flow in a matrix of surface-mounted cubes - Test Case 6.2: Description of Numerical Methodology, Proceedings, ERCOFTAC/IAHR/COST Workshop on Refined Turbulence Modelling, Helsinki University of Technology, Helsinki, Finland. Mavripilis, D. J. (1997) Unstructured grid techniques, Annu. Rev. Fluid Mech., 29, pp. 473-514. Meinders, E. R. (1998) Experimental study of heat transfer in turbulent flows over wall-mounted cubes, PhD thesis, Delft University of Technology K. (1999) Vortex structure and heat transfer in turbulent flow over a Meinders, E. R. and wall-mounted matrix of cubes Int. J. Heat and Fluid Flow, 20, pp. 255-267. B. and K. (1999) Flow in a matrix of surface-mounted cubes - Test Case 6.2: Description of Numerical Methodology, Proceedings, ERCOFTAC/IAHR/COST Workshop on Refined Turbulence Modelling, Helsinki University of Technology, Helsinki, Finland, 1999. B. and K. (2000) Large Eddy Simulation (LES) on distributed memory parallel computers using an unstructured finite volume solver, in J. Periaux, N. Satofuka (eds.) Parallel CFD 2000, Elsevier. B., and K. (2001) An unstructured finite-volume solver for large eddy simulations, submitted for publication B., Dronkers, A.D.T. and K. (2001) Turbulent heat transfer on a mulit-layered wall-mounted cube matrix: Large Eddy Simulation, to appear in Int. J. Heat and Fluid Flow Patankar, S. V. (1980) Numerical heat transfer and fluid flow, McGraw-Hill, New York. Piomelli, U. (1997) Large-eddy and direct simulation of turbulent flows, von Karman Institute for Fluid Dynamics, Lecture Series 1997-03, Rhie, C. M. and Chow, W. L. (1983) A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA J., 21, pp. 1525-1532. Spalart, P. R., Jou, W. H. Strelets M. and Allmaras, S. R. (1997) Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach, in C. Liu and Z. Liu (eds.) Advances in LES/DNS Greyden Press, Columbus, OH, USA. Verstappen, R. W. C. P., van der Velde R. M., and Veldman, A. E. P. (2000) DNS of turbulent flow and heat transfer in a channel with surface mounted cubes, in: E. Onate, G. Bugeda and B. Suarez (eds.) Proc. ECCOMAS 2000, Barcelona, pp 489-506. Van der Velde, R.M., Verstappen, R.W.C.P. and Veldman, A.E.P. (1999) Description of Numeri cal Methodology for Test Case 6.2, Proceedings 8th ERCOFTAC/IAHR/COST Workshop on Refined Turbulence Modelling, Helsinki University of Technology, Helsinki, Finland.
LARGE EDDY SIMULATION FOR FLOW ANALYSIS IN A CENTRIFUGAL PUMP IMPELLER
R. K. BYSKOV AND C. B. JACOBSEN Fluid Dynamic Engineering,
Grundfos Management A/S,
DK-8850 Bjerringbro, Denmark
T. CONDRA Institute of Energy Technology,
Aalborg University,
DK-9200 Aalborg, Denmark
AND
J. N. SØRENSEN Department of Energy Engineering,
Technical University of Denmark,
DK-2800 Lyngby, Denmark
Abstract. With the objective of gaining improved insight in the local flow behaviour and increasing the accuracy of numerical simulations the flow field in a centrifugal pump impeller has been investigated using LES. The effect of the SGS-scales have been modelled using a localised dynamic Smagorinsky model recently implemented in the commercial CFD code FINE/Turbo. Detailed flow structures are analysed at flow rates of 25 and 100 percent design load. Velocities predicted from LES and steady state RANS simulations are compared with PIV measurements, showing satis factory agreement between LES and PIV.
1. Introduction
The traditional approach to the hydraulic design of pumps is based on steady state theory and depends on empirical methods and the combination of systematic model testing and engineering experience. This approach has progressed far in producing efficient and reliable pumps. The flow field in 217 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 217-232. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
218
R. K. BYSKOV ET AL.
centrifugal pumps, which is influenced by system rotation and curvature, is highly turbulent and unsteady and, due to separation and recirculation, very complex. Further improvements of performance for design and offdesign operating conditions will be extremely difficult with the traditional steady methods because it will depend more on controlling complex physical phenomena as boundary layer separation, vortex dynamics, interactions between rotating and stationary components, vibrations and noise, etc. In order to improve the accuracy of the numerical simulations and to be able to analyse and understand more thoroughly the complex physics of flow in centrifugal pumps, it is essential to advance from a steady-state to an unsteady simulation technique. In this context a promising supplement to standard Reynolds-Averaged Navier-Stokes (RANS) turbulence modelling is Large Eddy Simulation (LES). Little work has been done in the field of LES for industrial applications of practical importance and in particular on pump flows. Some scepticism exists in terms of maturity of present methods and it is necessary to investi gate the limitations of the classical LES approach. The main problem in the integration of LES as an industrial analysis tool is that the methodology is not established as a standard commercial CFD tool. Within this work LES has been implemented in the commercial CFD code FINE/Turbo and val idated for plane and rotating channel flow, see Byskov(2000a). This paper presents the first results from simulations of the flow field in a centrifugal pump impeller. The full data is found in Byskov(2000b). The objective has been to investigate the flow field at different operating conditions and in this context to perform a comparative study of the contributions and accuracy achieved from LES compared to traditional RANS turbulence modelling. 2. Numerical Solution Technique
In contrast to academic LES codes, which are often tailored for analysing simpler flows, the use of a commercial CFD code, such as FINE/Turbo, poses no geometrical restrictions and is thus applicable for simulating the flow in a complex geometry like a centrifugal pump impeller. FINE/Turbo is a well-established finite volume Navier-Stokes solver developed by Lacor et al. (1992) and based on solving the three dimensional compress ible continuity, Navier-Stokes and energy equations. The convective fluxes are treated through a second-order central Jameson et al. (1981) scheme with second- and fourth-order scalar dissipation. In the present work the second-order term is turned off and the fourth-order coefficient is set to which is two order less than the default value. Adjusting is, however, a subject needing further research. A pseudo-compressibility method and a second-order dual-time stepping procedure are implemented
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
219
by Hakimi(1997) and allow for simulating incompressible unsteady flows. The governing equations utilised in the present work are hence the un steady preconditioned spatially filtered continuity and Navier-Stokes equa tions given by
where is a preconditioning parameter, the gauge pressure, the SGSviscosity and the source term vector due to rotation. is the physical time and the pseudo time introduced due to the dual-time stepping. During each physical time step an explicit fourth-order Runge-Kutta scheme is used for advancing Eq. (1) and (2) in pseudo time. The overbars indicate spatial filtering. In the present finite volume approach the numerical discretisation is interpreted as an implicit filter. As the flow is influenced by the rotation of the impeller, Eq. (1) and (2) are formulated in a relative frame of reference. This is allowed when the two and the centrifugal force, inertia forces; the Coriolis force, as compensation. are added to the source term vector The relative velocity, is the velocity experienced by an observer rotating with the impeller and is the angular velocity of the impeller. The absolute velocity is computed by vectorial addition of the local circumferential impeller speed to the relative velocity, 3. Subgrid Scale Modelling
The basic philosophy in LES is to directly simulate the three-dimensional time dependent large scale turbulent motion and to take into account the effect of the unresolved subgrid-scales through appropriate SGS-models. In the few Large Eddy Simulations of turbomachinary flows that have ap peared in literature, e.g. Song and Chen(1996), Chen et al. (1998) and Kato et al. (1999), the Smagorinsky model has been used. Different eddy-viscosity SGS-models have been implemented in FINE/Turbo by Byskov(2000a) and plane and rotating channel flow simulations have revealed a superiority of the localised dynamic Smagorinsky model of Piomelli and Liu(1995) which is used in the present simulations. The SGS-viscosity, is given by
where C is the model parameter, cell volume and
the filter length related to the local is the strain rate tensor based on the
220
R. K. BYSKOV ET AL.
filtered velocities. The model parameter C is computed from
where and is an es timate of C found from a first order extrapolation of C from previous time steps. The model is referred to as ’localised’ as it does not require averaging of the model parameter in directions of homogeneity. For reasons of sta bility the SGS-viscosity has been constrained to be non-negative. Further, an upper constraint limiting the model parameter to be less than 1.0 has proved necessary. With these simple constraints the model remains stable even at high Reynolds numbers. 4. Impeller Model 4.1. PUMP IMPELLER The impeller under study is a shrouded low specific-speed pump impeller representing the rotating part of an industrial multi-stage pump, see Fig. 1.
The impeller, which is seen in Fig. 2, has an outer diameter of 190mm and six simple curvature blades with blunt leading and trailing edges. The impeller is designed to operate at a flow rate of and provides a
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
221
pressure rise equivalent to a head of 1.75 m at design load. Simulations at design load, and at severe off-design conditions, at quarter-load are performed. The Reynolds number typically used for pumps and is for the is based on the outer dimensions of the impeller, impeller under study of the order and thus very large in the context of LES. More relevant when investigating the local flow behaviour is to base the Reynolds number on the local radial velocity and the blade In this case the Reynolds number is at design load, height, however, only in the range The main geometrical data of the impeller and the design load conditions are summarised in Table 1.
4.2. NUMERICAL MODEL When modelling the impeller advantage has been taken of an experimen tally detected correlation between every second impeller passage, see Pedersen(2000). Only two of the six impeller passages are thus included in the numerical model. Keeping the computational effort at a realistic level com pared to the available computer power, a mesh of totally 385,000 cells with roughly 150,000 cells in each impeller passage has been utilised. The grid stretching factor has been chosen to allow the first cell point off the sur corresponding to local values less than 5.0 face to be located in in average. The numerical domain and the mesh structure in the passage mid-height is seen in Fig. 3. The subject of turbulent inflow conditions is of general importance in LES of practical applications. A data base of velocity profiles from a turbu lent channel flow simulation at Re=2900 has been created and utilised as
222
R. K. BYSKOV ET AL.
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
223
inflow velocity profiles for the impeller. Due to geometrical differences the profiles have been mapped from a rectangular to a annular cross-section and as the Reynolds numbers are significantly different, the velocities have been scaled according to the mass flow requirements. At the outlet a uniform pressure is applied, thereby not posing any restrictions on the velocities. Figure 3 illustrates how the axial height of the section after the impeller blade has been decreased in order to keep the outlet area constant. This prevents problems with flow reversal over the outlet boundary. A no-slip condition is utilised at the solid walls and periodic boundaries are used in the circumferential direction. In order to resolve the temporal variations of the flow the time step has which ensures an average number less than been adjusted to 1.0. This time step is equivalent to 160 time steps per impeller revolution. As initial conditions a steady state solution at 50 percent design load has been used. With the turbulent inflow conditions described above the flow field has developed for 2000 time steps, equivalent to 12 revolutions. Time averaged quantities are computed during 2000 additional time steps. As the flow field is statistically stable, this is found to be sufficient to stabilise lower order statistics. The simulations have been carried out on a single Alpha EV6 500 MHz processor of a COMPAQ AlphaServer DS20 situated at the Danish Com puter Center, UNI-C located at Technical University of Denmark. The re quired CPU time has been about 14 minutes per time step resulting in a total CPU time of about 960 hours for one operating condition. 5. Results and Discussion
In the following the results of the LES of the flow field in the impeller are analysed. To ease the discussion it has proved advantageous to distinguish between the two impeller passages modelled, these are denoted passage A and B respectively. For reasons of presentation the results have in the following been replicated and rotated 120 deg and 240 deg, thereby covering the entire impeller. 5.1. FLOW FIELD AT DESIGN LOAD In Fig. 4 the time averaged relative vector field at design load is shown. The flow is seen to follow the curvature of the impeller blades in the predomi nant part of the impeller and no significant separation appears. Along the blade suction side both passages experience at the development of a low-velocity zone. It could be expected being the flow starting to or ganise in a jet-wake structure as reported by other investigators, e.g. Hajem et al.(1998) and Liu et al.(1994). However, at outer radii the low-velocity
224
R. K. BYSKOV ET AL.
zone is suppressed and a nearly uniform velocity profile develops across the impeller outlet.
Figure 5 shows the radial velocity in hub-to-shroud sections in the two the effect of curvature is evident. passages. In the inlet section, High momentum fluid is concentrated in the hub-suction side corner and due to blunt leading edges low velocities are present along the blade sides. At the blade pressure surface is not covered by the suction side of the adjacent blade. As indicated in Fig. 5 and known from the literature, e.g. Ubaldi et al.(1998) and Sinha and Katz(2000), this causes an increase in the velocity along the pressure side and a decrease on the suction side, resulting in an unloading of the blade. From Fig. 4 and 5 the flow in the two passages is found to be similar, revealing no significant circumferential variations at design load. 5.2. FLOW FIELD AT QUARTER LOAD The time averaged vector field at quarter-load is seen in Fig. 6 and shows a substantial departure from the well-behaved non-separated flow observed at design load. In passage A the low-velocity zone along the suction side observed at design load has turned into a distinct zone of recirculation. A close analysis of the flow approaching the impeller passage reveals that the flow initially
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
225
follows the curvature of the impeller blade along both suction and pres sure side in passage A. The recirculation is thus seen as a consequence of which determines the an alteration in the Rossby number, relationship between the centrifugal force due to curvature of the bladeto-blade section and the Coriolis force. The highest Rossby number occurs at the inlet and is but reduces rapidly to in the dominant part of the impeller. At design load in the dominating part of the passage. This indicates that the flow field in the blade-to-blade plane is mostly dominated by rotational effects. It is apparent from Fig. 6 that a significantly different flow field is
226
R. K. BYSKOV ET AL.
present in passage B. This passage is largely dominated by a flow field associated with stall. A significant recirculation zone along 30 percent of the blade suction side is observed which effectively blocks the entrance to passage B and apparently unblocks passage A. In contrast to observations in the literature by Lenneman and Howard(1970) and Visser and Jonker(1995) the stall appears to be stationary, non-rotating and not initiated due to in teraction with stationary components. Due to the stall and the consequent blocking, the throughflow is minimal. Hence, Fig. 6 illustrates the forma tion of a relative eddy covering the remaining part of passage B; high values of the relative velocity along the blade suction side and flow reversal along the dominant part of the blade pressure side. The three-dimensionality and distortion of the flow field becomes more evident from the velocities in the hub-to-shroud sections shown in Fig. 7. In the entry section of the passages the reverse flow along the suction side of passage A and both pressure and suction side in passage B is found to fill the entire passage height. In passage B, flow reversal near the exit is seen to cover more than half the passage height along the pressure side, and it is apparent that the throughflow in passage B is minimal.
5.3. TURBULENT FLUCTUATIONS
Figure 8 shows instantaneous samples of the deviation from the time av eraged velocity profiles discussed above. The deviations give evidence of a highly fluctuating flow field superimposed with eddy structures. Due to the relatively coarse mesh only the larger eddy structures of the true turbulent flow are captured. It is seen that the level of the fluctuations increases as the flow rate decreases and the intensity associated with the stall phenomenon is apparent.
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
227
From Fig. 9 the contours of the turbulent intensity is seen. A relevant portion of the flow at design load is characterised by a turbulence level lower than 3 percent, indicating a relatively stable flow. The turbulence intensity coincides with regions of low velocity and defines a turbulent layer that detaches from the blade suction side. Decreasing the flow rate causes an increase in the turbulent intensity. As far as magnitude is concerned the intensity at quarter-load is almost twice the magnitude at design load with an average of around 5 percent. The stall phenomenon is observed to be associated with high turbulence activity. Also the strong reversed flow in the outlet region of passage B gives rise to significant turbulence intensity of up to 10 percent.
228
R. K. BYSKOV ET AL.
5.4. SGS-MODEL BEHAVIOUR The time averaged SGS-viscosity is shown in Fig. 10. As can be seen a qualitatively different distribution of the viscosity is revealed compared to the smooth distribution generally achieved in RANS simulations and with close to 1.0 at design load and 1.2 at quarter-load the an average of level is significantly lower.
The main advantages of the dynamic localised Smagorinsky model by Piomelli and Liu(1995) is an implicit and dynamic computation of the model parameter C, circumventing a priori specification. Figure 11 shows the time averaged model parameter. An expected correlation between the model parameter C and the SGS-viscosity in Fig. 10 is revealed. Signifi cant areas of negative C are present, illustrating that the SGS-model would predict backscatter had the SGS-viscosity not been constrained to be non negative. From the range of the model parameter it is evident that the is not dominant. Generally, the model pa constraint, specifying rameter is seen to vary strongly in the domain with an average equivalent to and at design load and quarterSmagorinsky constants of load respectively. This difference of 33 percent emphases the difficulty of specifying the model parameter a priori. This study also reveals that the model parameter, when computed dynamically, exhibits a very complex behaviour. 5.5. COMPARISON WITH EXPERIMENTS In order to evaluate the achievements of LES as compared to traditional RANS turbulence modelling the radial velocities predicted by LES and
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
229
steady state Baldwin-Lomax and Chien simulations, all performed on the same mesh, are in Fig. 12 compared with ensemble averaged PIV measurements conducted by Pedersen(2000). At the design load the general impression is a satisfactory agreement be tween the numerical and experimental data for all three simulations though a detailed analysis reveals a slightly more accurate prediction of the details of the flow behaviour in the LES simulation. Due to the dominant effect of curvature in the inlet section the velocities show profiles skewed towards the suction side at This trend is seen to be predicted very accurately in LES whereas the two RANS simula tions predict flatter profiles. Towards the outlet, the unloading of the impeller blade is initiated. This results in increased velocities along the blade pressure side. LES predicts, in agreement with measurements, symmetric variations in the radial velocity with equal peaks at pressure and suction side and the lowest value near mid span. The RANS simula tions predict a stronger influence on the unloading and, thus, a decrease in the radial velocity from the pressure to suction side. At quarter-load the above discussion has revealed that LES predicts a flow field that differs substantially in the two passages. From Fig. 12 this is seen to be confirmed by the PIV measurements. In the inlet section, the PIV measurements reveal a ve locity profile in passage A skewed towards the suction side. Contrary to this, LES predicts a profile displaced to the pressure side. This remarkable difference is mainly attributed to the influence of prerotation. Especially, at partial load the rotation of the impeller generates upstream vorticity, which combined with the effect of leakage flow, causes a substantial swirl in the inlet velocity profile. In the measurements it has not been possible to
230
R. K. BYSKOV ET AL.
estimate the extent of the inlet swirl. It has therefore been disregarded in the simulations, but here seen to have a significant influence on the inflow. In passage B the stall phenomenon is again evident in the velocity pro The files with substantial variations across the passage span at radial velocities illustrate reversed flow along the suction side for both LES and PIV. In the passage center positive radial velocities feeding the recir culation are present. The peak in the positive velocity in the LES results is slightly displaced towards the suction side giving room for a small counter rotating eddy at the pressure side. This eddy is, however, not observed experimentally. At the PIV measurements and LES simulation agrees that the flow is uniform in passage A. In passage B the radial velocity gives evidence of the relative eddy with high velocities along the suction side and reversed flow along the pressure side. The Chien model is observed to predict very accurately the variation in the radial velocity in passage B, however, flow reversal is predicted also in passage A instead of a uniform profile. The Baldwin-Lomax model is seen to predict substantial deviations to measurements in both passages. LES do have substantial deviations from measurements in passage A
LES FOR ANALYSING FLOW IN A PUMP IMPELLER
231
which are mainly attributed to differences in the inlet conditions, however in passage B, LES predicts the stall and the flow reversal along the pressure side for larger radii with a satisfactory accuracy. It is evident that RANS do not capture the stall in passage B and consequently provide an inaccurate prediction of the local flow behaviour. 6. Conclusion
The flow field in a centrifugal pump impeller has been investigated at design load and quarter-load using LES. At design load the LES reveals a well-behaved flow dominated by cur vature in the inlet section, causing a displacement of the flow towards the hub-suction side. The differences between the two impeller passages mod elled are found to be negligible. Decreasing the flow rate to 25 percent design load, significant differences are revealed between the two impeller passages. One passage is dominated by rotational effects causing high velocities along the blade pressure side. The other passage exhibits a highly separated flow field; in the entry section a significant stall is observed blocking the entry, and as the throughflow consequently is minimal, a relative eddy develops in the remaining part of the passage. Averaged turbulence intensity of 3 per cent is observed at design load and increases at quarter-load to around 5 percent with maxima of 10 percent. The results from the LES simulations have been compared with results from steady state RANS simulations based on the Baldwin-Lomax and model, and the numerical achievements are evaluated through Chien comparison with PIV measurements. The velocities predicted from LES compare favorable with experimental data, and the measurements confirm the stall phenomenon observed. The two RANS simulations are, however, not able to predict the stall phenomenon. It is thus found that using LES for analysis of the flow field in centrifugal pumps provides improved insight into the basic fluid dynamic phenomena with a satisfactory accuracy compared with experiments. In particular, at partial load where the flow is highly separated the achievements of LES, as compared to Baldwin-Lomax and Chien are encouraging. The present work demonstrate that performing LES of industrial flows based on second-order accurate numerical methods combined with eddy vis cosity SGS-models do open perspectives towards gaining improved insight of the flow dynamics. 7. Acknowledgements
This work has been supported by the Danish Academy of Technical Sciences under the grant LESPUMP EF663.
232
R. K. BYSKOV ET AL.
References Byskov, R. K. (2000a). Large Eddy Simulation of Flow Structures in a Centrifugal Pump Impeller. Part 2: Code Validation, Ph.D. thesis, Aalborg University, Institute of Energy Technology, Aalborg, Denmark. ISBN 87-89179-31-5. Byskov, R. K. (2000b). Large Eddy Simulation of Flow Structures in a Centrifugal Pump Impeller. Part 1: Theory and Simulation of Pump Flow. Ph.D. thesis, Aalborg Uni versity, Institute of Energy Technology, Aalborg, Denmark. ISBN 87-89179-30-7. Chen, X., Song, C. C. S., Tani, K., Shinmei, K., Niikura, K., and Sato, J. (1998). Com prehensive modeling of francis turbine system by large eddy simulation approach. In H. Brekke, C. G. Duan, R. K. Fisher, R. Schilling, S. K. Tan, and S. H. Winnoto, editors, Hydraulic Machinary and Cavitation, volume 1, pages 236–244. XIX IAHR Symposium, World Scientific Publishing Co. Hajem, E. E., Morel, R., Champange, J. Y., and Spettel, F. (1998). Detailed measure ments of the internal flow of a backswept centrifugal impeller. In 9th International Symposium on Applications of Laser, pages 36.2.1–36.2.6. Hakimi, N. (1997). Preconditioning Methods for Time Dependent Navier-Stokes Euations. Application to Environmental and Low Speed Flows. Ph.D. thesis, Dept. of Fluid Mechanics. Vrije Universiteit, Brussel, Belgium. Jameson, A., Schmit, W., and Turkel, E. (1981). Numerical simulation of the Euler equa tions by Finite Volume Methods using Runge-Kutta time-stepping schemes. AIAA Paper No. 81.1259, page 1. Kato, C., Shimizu, H., and Okamura, T. (1999). Large eddy simualtion of unsteady flow in a mixed-flow pump. In 3rd ASME/JSME Joint Fluids Bngeneering Conference, pages 1–8. Lacor, C., Alavilli, P., Hirsch, C., Eliasson, P., Lindblad, I., and Rizzi, A. (1992). Hyper sonic Navier-Stokes computations about complex configurations. In C. Hirsch, editor, Proceedings from First European CFD conference, volume 2, pages 1089–1096. Lenneman, E. and Howard, J. (1970). Unsteady flow phenomena in centrifugal impeller passages. J. Eng. Power, 92(1), 65–72. Liu, C., Vafidis, C., and Whitelaw, J. (1994). Flow characteristics of a centrifugal pump. J. Fluids Eng., 116, 303–309. Pedersen, N. (2000). Investigation of Flow Structures in a Centrifugal Pump Impeller using Particle Image Velocimetry. Ph.D. thesis, Technical University of Denmark, Copenhagen, Denmark. Piomelli, U. and Liu, J. (1995). Large-eddy simulation of rotating channel flow using a localized dynamic model. Phys. Fluids, 7(4), 839–848. Sinha, M. and Katz, J. (2000). Quantitative visualization of the flow in a centrifugal pump with diffuser vanes - I: On flow structures and turbulence. J. Fluids Eng., 122, 97–107. Song, C. and Chen, X. (1996). Simulation of flow through Francis turbine by LES method. In E. Cabrera, V. Espert, and F. Matinez, editors, XVIII IAHR Symposium on Hydraulic Machinary and Cavitation, volume 1, pages 267–276. Kluwer Academic Publisher. Ubaldi, M., Zunino, P., and Ghiglione, A. (1998). Detailed flow measurements within the impeller and vaneless diffuser of a centrifugal turbomachine. Exp. Thermal and Fluid Science, 17, 147–155. Visser, F. and Jonker, J. (1995). Investigation of the relative flow in low specific speed model centrifugal pump impellers using sweep-beam particle image velocimetry. In 7th International Symposium on Flow Visualization, pages 654–659, Seattle, Wasing ton.
5. DES and RANS-LES coupling
This page intentionally left blank
DETACHED-EDDY SIMULATION, 1997-2000
P. R. SPALART Boeing Commercial Airplanes PO Box 3707, Seattle, WA 98124, U.S.A.
Abstract. This brief paper gives a roadmap to the history and literature of Detached-Eddy Simulation since it was proposed four years ago, to similar hybrid methods, and to the author’s related review papers.
1. DES definitions and applications
The motivation and principle of DES were presented by Spalart, Jou, Strelets and Allmaras (SJSA) in 1997 [1] and have not changed in depth since. The activity since has consisted in validations, and in evolving the strategy for grid generation. The only essential step has been to initiate the coupling of DES with other Reynolds-Averaged Navier-Stokes (RANS) models than the initial one, the Spalart-Alhnaras (S-A) model. The moti vation there is to broaden the user base and to gain fine improvements of the separation prediction, in the RANS region; as expected, the model ap pears to make little difference in the Large-Eddy Simulation (LES) region as shown by Strelets [2]. The constant in the S-A implementation was set in 1999, by Shur, Spalart, Strelets and Travin [3], thus making the proposal complete (the constant is not highly critical, and good results have been obtained with values far below the published 0.65). This paper also had the first true 3D simulations, those of an airfoil at high angle of attack, which were quite promising. A more considered definition of DES is with the circular-cylinder study by Travin, Shur, Strelets and Spalart [4], which is a better entry point to DES than SJSA . That paper also offered a grid-refinement study, and gave separation prediction a much stronger role. Some results were quite good, and longer time samples generated after the paper improved the agreement with experiment even further [2]. Conven tional DES applications are those of Constantinescu and Squires for the sphere [5] and Forsythe, Hoffmann and Dieteker for a missile base [6]. 235 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 235-237. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
236
P. R. SPALART
The studies of Nikitin, Nicoud, Wasistho, Squires and Spalart in a plane channel are not conventional, since they apply the DES model as a pure subgrid-scale model in LES [7]. DES provides “wall modeling” in the sense that the grid spacing parallel to the wall is not limited in wall units; the wall-normal spacing is limited, but the standard stretching results in a com puting cost that increases only like the logarithm of the Reynolds number. Thus, the DES model or a derivative may have a future in simulations that do not treat the boundary layers with RANS (but we maintain that such simulations are un-economical in real-life flows [1]); this will depend on elim inating a shift in the logarithmic layer of the velocity profile, which made the skin-friction coefficient about 15% too low [7]. This study was deliber ately constrained not to adapt the grid to the flow direction, nor to adjust any constants. The principal advantages of the model used, over current wall-modeling efforts, are simplicity, fully local formulation, and stability: no expedients (such as averaging in homogeneous directions) were needed. Current conventional DES applications include a runway on an elevated platform [2]; active flow control on airfoils; a model of jet-fighter forebodies; a landing-gear truck; jet noise; heavy road trucks; and even a complete jet fighter stalling and, soon, spinning. In all cases, DES generates finer turbulent eddies than RANS. The quantitative results are slightly superior to those of RANS, in some cases, and vastly superior in other cases. These findings are gradually increasing our confidence in DES. On the other hand, the generation of quality grids is intensive. 2. Other RANS-LES hybrids
In 1998, Speziale considered hybrid methods with an objective similar to that of DES or in fact wider, that is, less focused on aerodynamics [8]. The paper however did not report results, and sadly he could not extend his work. It is continued by Fasel’s group [9]. Two independent lines of work which are both aimed at aerodynamics are the Limited Numerical Scales concept of Batten, Goldberg, and Chakravarthy [10] and the acronym-less method of Arunajatesan, Sinha and Menon [11, 12]. Their direct motivation is to obtain the unsteadiness of the flow field over a cavity. A hybrid method that is zonal is due to Georgiadis, Alexander and Reshotko [13]; it was created for a mixing-layer simulation with splitter plate included. There will be debate over the originality of the different lines of work, but the charters for these methods and for DES are very close. Separation and high Reynolds numbers combined lead serious researchers to very similar approaches.
DETACHED-EDDY SIMULATION, 1997-2000
237
3. Reviews The author’s recent review [14] is the principal reason not to write a new contribution here. The two would have greatly overlapped. On the other hand, several references have appeared since the review. A noncomprehensive set of comments can be found in a conference paper [15]. Many examples and improvements are given by Strelets [2], a paper which also appears in this volume in revised form [16]. References 1. Spalart, P.R., Jou, W.-H., Strelets, M. and Allmaras, S.R. (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, USA. 2. Strelets, M. (2001) Detached eddy simulation of massively separated flows. AIAA2001-0879. 3. Shur, M., Spalart, P.R., Strelets, M. and Travin, A. (1999) Detached-eddy simulation of an airfoil at high angle of attack. 4th Int. Symposium on Eng. Turb. Modelling and Experiments, May 24-26 1999, Corsica. W. Rodi and D. Laurence Eds., Elsevier, Amsterdam, NL. 4. Travin, A., Shur, M., Strelets, M., and Spalart, P. R. (2000) Detached-Eddy Simulations past a Circular Cylinder. Flow, Turb. Comb. 63, pp. 293-313. 5. Constantinescu, G., and Squires, K.D. (2000) LES and DES investigations of turbulent flow over a sphere. AIAA-2000-0540. 6. Forsythe, J., Hoffmann, K., and Dieteker, J.-F. (2000) Detached-eddy simulation of a supersonic axisymmetric base flow with an unstructured flow solver, AIAA-20002410. 7. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D., and Spalart, P. R. (2000) An Approach to Wall Modeling in Large-Eddy Simulations. Phys Fluids 12, pp. 7-10. 8. Speziale, C.G. (1998) Turbulence modeling for time-dependent RANS and VLES: a review. AIAA J. Vol. 36, No. 2, pp. 173-184. 9. Wernz, S., and Fasel, H. (2000) Control of separation using wall jets– Numerical investigations using LES and RANS. AIAA-2000-2317. 10. Batten, P., Goldberg, U., and Chakravarthy, S. (2000) Sub-grid turbulence modeling for unsteady flow with acoustic resonance. AIAA-2000-0473. 11. Arunajatesan, S., Sinha, N., and Menon, S. (2000) Towards hybrid LES-RANS computations of cavity flowfields. AIAA-2000-0401. 12. Arunajatesan, S., and Sinha, N. (2001) Unified unsteady LES-RANS simulations of cavity flowfields. AIAA-2001-0516. 13. Georgiadis, N.J., Alexander, J.I.D., and Reshotko, E. (2001) Development of a hybrid RANS/LES method for compressible mixing layer simulations. AIAA-20010289. 14. Spalart, P.R. (2000) Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 21, pp. 252-263. 15. Spalart, P.R. (2000) Trends in turbulence treatments. AIAA-2000-2306. 16. Travin, A., Shur, M., Strelets, M., and Spalart, P. R. (2001) Physical and numerical upgrades in the detached-eddy simulation of complex turbulent flows. Euromech Coll. 412, Kluwer, Dordrecht, NL.
This page intentionally left blank
PHYSICAL AND NUMERICAL UPGRADES IN THE DETACHED-EDDY SIMULATION OF COMPLEX TURBULENT FLOWS
A.TRAVIN, M.SHUR, M.STRELETS Russian Scientific Center “Applied Chemistry”, 14, Dobrolyubov Ave., St.-Petersburg, Russia AND
P.R.SPALART Boeing Commercial Airplanes, Seattle, U.S.A.
Abstract. A new formulation of Detached-Eddy Simulation (DES) based RANS model of Menter (M-SST model) is presented, the goal on the being an improvement in separation prediction over the S-A model. A new numerical scheme adjusted to the hybrid nature of the DES approach and the demands of complex flows is also presented. The scheme functions as a fourth-order centered differentiation in the LES regions of DES and as an upwind-biased (fifth or third order) differentiation in the RANS and outer irrotational flow regions. The capabilities of both suggested upgrades in DES are evaluated on a set of complex separated flows.
1. Introduction
The Detached-Eddy-Simulation (DES) of turbulence (Spalart et al., 1997) has been suggested as a response to the computational and physical chal lenges associated with the reliable prediction of massively separated tur bulent flows in practical geometries at practical Reynolds numbers. Recent estimates for the cost of LES of an airplane or an automobile (Spalart, 1999) show that due to the presence of large thin near-wall turbulent boundary layers populated with small (“attached”) eddies whose local size is much less than the boundary layer thickness, that cost exceeds the available com puting power by orders of magnitude. As a result, there is no real prospect 239 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 239-254. © 2002 KluwerAcademic Publishers. Printed in the Netherlands.
240
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
of using LES in complex engineering computations for a very long time (even once the problem of “wall modeling” in LES is solved). On the other hand, the hopes that the conventional RANS turbulence models will soon (if ever) achieve engineering accuracy in the massive threedimensional separation zones typical for vehicles and airplane components, are rather vague and not supported by the rate of their progress over the last twenty years, even with the benefit of the unsteady solutions (URANS). This pessimism is consistent with the consideration that the dominant, “detached”, eddies in massively separated flows are highly geometry-specific and have little to do with the fairly universal eddies typical of the thin shear layers used for RANS turbulence-model calibration. Therefore, LES still is and probably will remain for a rather long time the only defensible tool for a reliable treatment of massive turbulent separation zones. A recognition of this conflict (non-affordable computational cost of LES in the attached boundary layers and inability of RANS models to provide a reliable prediction of large separation zones) resulted in an idea of a hy brid approach (Spalart et al., 1997) that combines the fine-tuned RANS technology in the attached boundary layers with the “raw power” of LES in the separated regions. In that approach, the “attached”, boundary layer, eddies are modeled, while the larger “detached” ones (populating the sep aration regions and wakes) are simulated (small eddies in these regions are also modeled, but have much less influence than the boundary-layer ed dies have). Considering that feature of the approach it was given the name Detached-Eddy Simulation (DES). Though DES is a young technique (only four years have passed since its major idea has been formulated), it has already reached some maturity and attracts more and more attention of the aerodynamic community. On the other hand, the approach cannot yet claim to perfection of course, and many issues of both physical and numerical nature are still to be resolved. For the physics, all the DES applications available till now are based on the Spalart-Allmaras (S-A) turbulence model (Spalart and Allmaras, 1994). That may be considered as a flaw. In DES the turbulence is treated by RANS in the attached boundary layer and, maybe, slightly beyond sep aration. So an accurate prediction of separation remains a RANS respon sibility and, therefore, DES versions based on models other than S-A are desirable to provide a range of models for “problem” flows where S-A can fail to predict separation accurately enough. For the numerics, till now, as far as the authors are aware, DES have been performed with the use of implicit upwind schemes. For instance, for the incompressible flows (Constantinescu and Squires, 2000; Shur et al., 1999; Travin et al., 2000), the time accurate (with dual time stepping) im plicit upwind-biased scheme of Rogers and Kwak (Rogers and Kwak, 1988)
UPGRADES IN DETACHED-EDDY SIMULATION
241
was used (the scheme is of fifth order in space and of second order in time). In the only available example of DES of compressible flow (Forsythe et al., 2000), the computations were performed with the use of the unstructured Cobalt code which is upwind and of second order accuracy both in space and time. The choice of implicit upwind schemes for full DES was dictated by the non-zonal nature of DES and, in particular, by the lack of stabil ity of the less-dissipative centered schemes in the flow regions where DES is operating in RANS mode. The causes are well known. They are: high values of the cell Reynolds number (even based on the eddy viscosity), dis persion (especially at angles to the grid lines), non-uniform grid spacing and coefficients, nonlinearity. On the other hand, in the LES regions of DES the upwind schemes seem to be sub-optimal since they are commonly considered as “too dissipative” for LES (e.g., Moin, 1998). Thus, though a statement such as “upwind schemes are unacceptable for LES” cannot be correct (any numerical method, if consistent, results in an acceptable ac curacy with a fine enough grid), this issue of numerical dissipation in DES does exist, and requires attention. Excessive dissipation does not result in an unstable or meaningless solution, but it prevents the solution from tak ing full advantage of the grid provided. It stops the energy cascade before the SGS eddy viscosity does, or in collaboration with the eddy viscosity but still at scales that are larger than the best possible. The above considerations point to the two specific goals of the present work, namely, development of a new, non-S-A, DES version and a numerical algorithm better adapted to the hybrid DES nature and thus providing less numerical dissipation than the techniques currently used in DES. Section 2 M-SST of the paper presents the new DES formulation based on the model (Menter, 1993), which is consistently considered as one of the best two-equation RANS models particularly for separation prediction. Section 3 outlines the hybrid, upwind/centered, numerical method that claims to be less dissipative in the LES region of DES than the fully upwind algorithms. Finally, Section 4 contains numerical examples illustrating the capabilities of the suggested DES enhancements. 2. DES Formulation Based on the M-SST Turbulence Model Although the DES formulation is immediate only on the basis of the S-A or other models which use a distance to the wall as a turbulence length scale, the DES/S-A link is not fundamental, and other models can be built into DES. Indeed, according to the general DES definition (Travin et al., 2000), a DES model can be obtained from any RANS model by an “appropriate” modification of the length scale which is explicitly or implicitly involved in any RANS turbulence model. Below one of the possible designs of such a
242
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
modification is presented to the M-SST RANS model. The length scale of this model in terms of and reads as
Now, a non-trivial question arises regarding which specific terms of the model this length scale should be replaced in with the DES length scale
This situation is quite different from that with the S-A model, where there is little freedom of choice. Considering that the role of the subgrid scale model should not be crucial, our approach was to keep the formulation as simple as possible with the only restriction that at equilibrium the re sulting subgrid model should reduce to a Smagorinsky-like model. By this we mean that the eddy viscosity is proportional to the magnitude of the resloved deformation tensor, and to the square of the grid spacing. As a re sult, if the grid spacing is in the inertial range of the turbulence, the eddy viscosity scales like the power 4/3 of the grid spacing. Based on these thoughts, the only term of the M-SST RANS model we have modified to extend it into a DES mode is the dissipative term of the equation:
The modification consists in the simple substitution which results in
(2) for
in (3)
Just like the classical, RANS, M-SST model, the above DES formu and Although in the major part of the lation has two branches, region where DES functions in LES mode only the branch is important, since precisely this branch is active there, we still have performed separate calibrations of the constants for the two branches and then blended the values obtained with the use of Menter’s blending function (Menter, 1993): The calibration procedure is similar to that used for the S-A-based DES (Shur et al., 1999). Namely, in order to find the optimal values of and to show that the subgrid-scale versions of the and branches of the M-SST model perform fairly well, the model has been exercised in pure LES mode on decaying homogeneous isotropic turbulence, as studied in the experiments (Comte-Bellot and Corrsin, 1971).
UPGRADES IN DETACHED-EDDY SIMULATION
243
The differencing scheme used in the calculations was centered and fourthorder-accurate, and the time integration was by an implicit three-layer second-order-accurate scheme. Figure 1 shows the spectra at and in the experiment, as computed with the use of the S-A subgrid-scale model at in Shur et al. 1999 and with the M-SST one at and in the present work. As seen in the figure, with those constants, the agreement of the subgrid M-SST model with the experiment is fairly good and at least not worse than that of the S-A model. Also, the spectral slopes are close to –5/3 near the cutoff wave-number, the energy decay is about 75% which is close to the experiment, and the average eddy viscosity scales closely with as expected in the inertial range. Thus the test seems to be quite satisfactory and gives credibility to the subgrid-scale version of the values. It should be noted however that these M-SST model and the values appear optimal for the differencing scheme used; schemes with more numerical dissipation may couple best with somewhat lower values.
3. Hybrid (Upwind-Central) Scheme for DES
An approach that allows to resolve or, at least, to weaken significantly the harm caused by upwinding in the LES regions of DES discussed in the Introduction mirrors the hybrid DES nature and uses the following hybrid central/upwind approximation of the inviscid fluxes, in the governing equations: where and denote respectively the central (fourth order) and up
wind (third/fifth order) approximations of F and is an empirical blending
function. It should be designed so that in the regions treated in the RANS
244
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
mode, would be close to 1.0, resulting in an “almost upwind” scheme, while in the LES regions would be close to zero resulting in an “almost centered” scheme. In addition, the blending function has to ensure the choice of the upwind scheme in the irrotational region of the flow. This is needed to guarantee stability of the scheme with the coarse grids typically used in such regions. A specific form of the blending function satisfying the above demands is as follows:
Here the function A is defined as
the turbulence length scale, is defined via the eddy viscosity and a combination of the magnitudes of the mean strain, S, and vorticity,
where is the characteristic convec tive time, and the parameter is introduced to ensure the dominance of the upwind scheme in the disturbed irrotational flow regions where and S > 0:
The constants of the blending functions are:
In Figure 2 a snapshot of the blending function is presented from the DES of a subcritical flow past a circular cylinder at Re = 50,000 together
UPGRADES IN DETACHED-EDDY SIMULATION
245
with a simultaneous snapshot of the eddy viscosity. It gives a clear idea of the behavior of the suggested device. Namely, in the vortical area of the cylinder wake, where DES performs in LES mode, is close to zero and the scheme is virtually the centered fourth-order one. Conversely, in the nearwall RANS regions and, also, in the irrotational outer part of the flow, is close to 1.0 and the scheme is effectively the fifth-order upwind one. 4. Computational Examples
In this section we are presenting numerical examples supporting the credi bility and efficiency of the DES enhancements outlined above. 4.1. NACA 0012 AIRFOIL BEYOND STALL
DES of this flow with the use of the S-A-based DES (Shur et al., 1999) was the first and fairly successful example of application of the real, 3D, DES technique to a complex massively separated flow. The simulations have been performed at the Reynolds number based on the airfoil chord, c, and angle of attack, ranging up to 90 degrees. In order to avoid arbitrarily adjusting transition points, the model was used in fully-turbulent mode. was equal to 15c, and the The computational-domain size in the spanwise period was 1 × c. The grid used in the computations was of the O-type in with a uniform spanwise step equal to (1/24)c, resulting in a total number of nodes 141 × 61 × 26, which is certainly “modest” by LES standards for this high Reynolds number. The near-wall grid spac following normal practice for RANS at The time ing was step was Beyond stall, S-A-based DES predictions of the airfoil drag and lift turned out to be excellent. Therefore the flow seemed to be a natural candidate for an evaluation of the M-SST-based DES capabilities in the present work. The computational domain and grid used in the M-SST DES were the same as those in the S-A-DES. With that grid, the values of the near-wall step in the law-of-wall units, were varying in the range from 0.3 up to 1.5. The corresponding values of the stream- and spanwise steps were ranging from 3 up to 150 and from 120 up to 600 respectively. Finally, a ”soft” transition of the model from the RANS to the LES mode took place at No qualitative difference in models’ performance has been observed in the simulations although, in contrast with the strictly periodic behavior typical of URANS, both DES models display chaotic features as seen in Fig. 3 where we present the time-dependent lift and drag coefficients for the 60° case from the simulations with the S-A and M-SST models. As far as the quantitative agreement of the two DES versions is concerned,
246
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
one can judge by Fig.4, where a comparison is presented of the mean forces computed with both models and the experimental data. Note that the postbut DES is at only This lower stall measurements are at value helps contain computational cost and, considering the weak Reynoldsnumber dependence after stall, only has a small impact on the flow. A major conclusion based on the Fig. 4 data is that though the disparity of the forces with different models is noticeable, it is well within the range of possible uncertainties caused by numerical inaccuracies associated with
UPGRADES IN DETACHED-EDDY SIMULATION
247
an insufficient grid, spanwise domain and/or time sample and anyway is far less than the difference between the DES and URANS and between the two models in URANS mode. The same is true for the pressure coefficient distribution (Fig.5). 4.2. CIRCULAR CYLINDER
An extensive DES study of this flow in the framework of the original, S-Abased, DES version (Spalart et al., 1997) with the use of the pure upwind scheme (Rogers and Kwak, 1988) is presented in Travin et al., 2000. Results obtained in the course of that study provide quite a sufficient database for evaluation of both the M-SST-based DES formulation and hybrid scheme. A typical (“medium”) grid used in the simulation is shown in Fig.6. It has three blocks (total size about 500,000 nodes) with a coarser spacing in the irrotational region, relative to the near-wall and wake regions and is designed so that in the region near where there is high activity and the solution is clearly of LES type, the cells of the grid are close to a square with the side The spanwise period is equal to 2D (D is the cylinder diameter) and the 3D grids are balanced, in the sense that also. A grid refinement study has been performed with three is close to successive grids with ratio Figures 7–9 give an idea about the effects of numerical scheme and tur bulence model on DES solutions. In particular, Fig. 7 compares spanwise vorticity snapshots computed on the medium grid for the flow regime with laminar separation (LS) at with the use of the upwind (fifth order) and the hybrid (fifth order upwind/fourth order centered) schemes and the S-A and M-SST models. As expected, with the hybrid scheme, the solution displays smaller vortices with both models. However, with the
248
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
M-SST model the effect of the scheme is much more visual. The same trends
are even more striking in the time-dependent forces shown in Fig. 8, and are
observed also in Fig.9, where we present the spectra of the resolved kinetic
energy at the wake centerline (at
Figure 10 summarizes results on the span- and time-averaged pressure distribution along the cylinder for the flow with LS with the two numerical schemes and turbulence models. The upper frame of the figure demonstrates an impressive agreement of pressure coefficient with the experimental data of (Cantwell and Coles, 1982) and strongly suggests grid convergence with
UPGRADES IN DETACHED-EDDY SIMULATION
249
the fifth order upwind scheme (note that such convergence has been elusive in LES of this flow (Breuer, 2000)). The other frames of Fig. 10 show the effects of the numerical scheme, grid, and turbulence model on the pressure distribution. A general conclusion from the analysis of those distributions is that the effect of the model is comparable with numerical inaccuracies caused by a coarse grid or excessively dissipative numerics, which seems to be quite consistent with the premise of LES (with laminar separation, these simulations depend very little on the RANS behavior of the models). We expect a stronger model-dependence for the flow regimes with tur bulent separation (TS). For those regimes simulated with the model in
250
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
fully-turbulent mode (see Travin et al., 2000 for more detail) the RANS mode of DES is responsible for prediction of turbulent boundary-layer sep aration, and so the M-SST model might be expected to perform tangibly better than the S-A one. However, as seen in Fig. 11, due to quite a notice able scatter in the experimental data (Roshko, 1961; van Nuen, 1974), it turned out to be difficult to confirm or deny that expectation. Note that the simulations are performed at Re = 50,000 which is far below what is needed for natural transition to be completed well before separation (we do not know precisely how high a value is needed, but a rough estimate is at least Strictly, they should be compared with tripped experiments at the same Reynolds number. However, the Reynolds-number dependence should be weak once the separation type is specified.
UPGRADES IN DETACHED-EDDY SIMULATION
251
4.3. BACKWARD-FACING-STEP (BFS) OF JOVIC AND DRIVER
This flow is a very appreciated test case for evaluating the capability of RANS turbulence models to capture the major features of a flow with separation and reattachment. The test however confronts the DES user with a crucial decision. In one approach, the incoming turbulent boundary layer has “DES content” or, in other words, its turbulence is resolved. Then we have an LES over the whole domain, much like it was done in a channel (Nikitin et al., 2000). In the opposite approach, we assume that the incoming boundary layer is thin enough relative to the step height that the recirculating bubble is dominated by eddies much larger than the incoming eddies, so that it is not essential to resolve those. Then the incoming boundary layer is treated by RANS. Which approach applies is controlled by the grid spacing in the usual fashion. This second approach is much closer to the reality of DES in aerodynamic flows, and was adopted. It should be noted that some of the RANS models, including S-A and, especially, M-SST model, have been shown to be quite accurate for this specific flow (Shur et al., 1995). The reattachment length is obtained quite well, and the remaining challenge is the Reynolds-number dependence of the peak negative skin friction. This is probably due to the thin-shear-layer character of the flow. Therefore, DES is competing with a rather successful technology. The computational domain in our DES of this flow consists of an entry section of length prior to the step and a post-expansion section, while the spanwise dimension is with periodic boundary conditions. The simulation uses 181 × 76 × 22 grid points in the streamwise, wall-normal, and spanwise direction respectively. The spanwise grid-spacing is uniform and the grid in the is shown in Fig. 12. In the near-wall region the grid spacing is the same as that in our RANS computations of the same flow (Shur et al., 1995). It ensures a maximum near-wall step in wall units not higher than 0.4 and, according to the grid-sensitivity study performed for RANS (Shur et al., 1995), is quite sufficient to obtain a virtually grid-independent solution. In the LES region, just as in the circular cylinder simulations, the grid cells are close to cubic.
252
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
Figure 13 shows the flow visualization from the M-SST DES by means It reveals all the typical flow structures ob of the “swirl” surface served in DNS and LES of the BFS flow (see, e.g., Akselvoll and Moin, 1993; Le et al., 1993; Neto et al., 1993; Fureby, 1999), namely, the quasi-twodimensional Kelvin-Hemholtz rolls convected downstream and then pairing, the streamwise vortices forming farther downstream, and gradual transfor mation of the flow to an essentially 3D one. Thus, in spite of the rudimen tary treatment of the inlet region mentioned above, DES is still capable of reflecting the major 3D unsteady physics of the BFS flow discussed in detail in the literature.
As far as the mean flow is concerned, the accuracy of DES turns out no worse than that of the corresponding RANS and, which seems to be rather important, it is also much less model-dependent than in the RANS approach. Those DES features are clearly seen in Fig. 14, where we com pare time- and span-averaged friction coefficient, distributions from the S-A and M-SST DES with the corresponding steady RANS distributions. Note also that DES superiority is quite visually expressed not only in the recirculation zone but in the region of flow recovery after reattachment,
UPGRADES IN DETACHED-EDDY SIMULATION
253
generally the most challenging for all known RANS models. Therefore, the strength of LES appears to be showing even in this flow region, which is two-dimensional and near-parallel in the mean. 5. Conclusions
A new DES formulation, based on the M-SST RANS model, was developed and tested on homogeneous decaying turbulence in a cubic box, NACA 0012 airfoil beyond stall, circular cylinder in subcritical (LS) and supercritical (TS) flow regimes, and backward-facing step. A major conclusion is that the suggested new DES version is quite feasible but does not show any clearly expressed superiority over the original S-A-based formulation for the flows considered: the model-related differences are the same as or smaller than those associated with the grid/scheme-related inaccuracies. A new, hybrid (upwind/centered), numerical scheme was constructed that functions as an effectively centered scheme in the regions where DES is operating in an LES mode, and as an upwind-biased scheme in the RANS and outer irrotational regions. Quite a bit better space resolution of that scheme is shown versus the upwind one in DES of the circular cylinder with almost no loss of stability. Energy spectra extracted from the simulations also show some superiority of the hybrid scheme, although the fifth-order upwind scheme also turns out to be quite acceptable in terms of the length of the inertial range in the spectra, and even the concrete improvement of the range of resolved scales for the flows considered is not directly reflected by obvious changes of global results such as pressure or drag. This suggests that the grid resolution is quite sufficient already to calculate these quantities. 6. Acknowledgments
The work has been supported by Boeing Operations International Inc. and, partially, by the Russian Fundamental Research Foundation (Grant No. 0002-17184). References Abbott, L.H., and von Doenhoff, A.E. (1959) Theory of Wings Sections, Including Sum mary of Airfoil Data, Dover, New York. Abernathy, F.H. (1962) Flow over an inclined plate, ASME J. Basic Eng. 61, 380–388. Akselvoll, K., and Moin, P. (1993) Large Eddy Simulation of a backward facing step flow, in: Eng. Turb. Modelling and Exp. 2, W. Rodi and F. Martelli Eds., Elsevier. Breuer, M. (2000) A challenging test case for large eddy simulation: high Reynolds num ber circular cylinder flow, Int. J. Heat and Fluid Flow 21, 648–654. Cantwell, B., and Coles, D. (1982) An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J.Fluid Mech. 136, 321–374.
254
A.TRAVIN, M.SHUR, M.STRELETS AND P.R.SPALART
Comte-Bellot, G., and Corrsin, S. (1971) Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated ’isotropic’ turbulence, J. Fluid Mech. 48, 273–337. Constantinescu, G., and Squires, K.D. (2000) LES and DES investigations of turbulent flow over a sphere, AIAA Paper 2000-0540. Forsythe, J., Hoffmann, K., and Dieteker, J.-F. (2000) Detached-eddy simulation of a supersonic axisymmetric base flow with unstructured flow solver, AIAA Paper 2000-2410. Fureby, C. (1999) Large eddy simulation of rearward-facing step flow, AIAA J. 37, No.11, 1401-1410. Hoerner, S.F. (1958) Fluid Dynamics Drag, http://members.aol.com/hfdy/home.htm Jovic, S., and Driver, D. (1995) Reynolds number effects on the skin friction in separated flows behind a backward facing step, Exp. Fluids, 18, 464. Le, H., Moin, P., and Kim, J. (1993) Direct numerical simulation of turbulent flow over a backward-facing step. Ninth Symp. on Turbulent Shear Flows, Kyoto, Japan, Aug. 1993, pp. 13-2-1–13-2-5. McCroskey, W.J. (1987) A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil, AGARD_CP429. McCroskey, W.J., McAlister, K.W., Carr, L.W., and Pucci, S.L. (1982) An Experimental Study of Dynamic Stall on Advanced Airfoil Sections, NASA TM 84245. turbulence models for aerodynamic flows, Menter,F.R. (1993) Zonal two-equation AIAA Paper 1993-2906. Moin, P. (1998) Numerical and physical issues in large eddy simulation of turbulent flows, JSME Int. J,, Series B, 41, No.2, 454–463. Neto, A.S., Grand, D., Metais, O., and Lesieur, M. (1993) A numerical investigation of the coherent vortices in turbulence behind a backward-facing step, J. Fluid Mech.
256, 1–25. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D., and Spalart P.R. (2000) An ap proach to wall modelling in large-eddy simulation, Phys. Fluids 12, No.7. Rogers, S.E., and Kwak, D. (1988) An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations, AIAA Paper 88-2583-CP. Roshko, A. (1981) Experiments on the flow past a circular cylinder at very high Reynolds number, J. Fluid Mech., 10(3), 345–356. Shur, M., Strelets, M., Zaikov, L., Gulyaev, A., Kozlov, V., and Secundov, A. (1995) Comparative numerical testing of one- and two-equation turbulence models for flows with separation and reattachment, AIAA paper, AIAA-95-0863. Shur, M., Spalart, P.R., Strelets, M., and Travin, A. (1999) Detached-eddy simulation of an airfoil at high angle of attack, in W.Rodi and D. Laurence (eds.) 4th Int. Symp. Eng. Turb. Modelling and Measurements, pp.669–678. May 24–24, 1999. Corsica, Elsevier, Amsterdam. Spalart, P.R., and Allmaras, S.R. (1994) A one-equation turbulence model for aerody namic flows, La Rech. A’erospatiale 1, 5–21. Spalart, P.R., Jou, W.-H., Strelets, M., and Allmaras, S.R. (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 & Z.Liu (eds.), Greyden Press, Columbus, OH. Spalart, P.R. (1999) Strategies for turbulence modelling and simulations, in: 4th Int. Symp. Eng. Turb. Modelling and Measurements, pp.3–17. May 24–24, 1999, Corsica, France. Travin, A., Shur, M., Strelets, M., and Spalart. P.R. (2000) Detached-eddy simulations past a circular cylinder, Int. J. Flow, Turbulence and Combustion 63, Nos. 1–4, 293 313. van Nunen, J.W.G. (1974) Pressure and forces on circular cylinder in a cross flow at high Reynolds numbers, in: Naudascher, (ed.), Flow Induced Structural Vibrations. 1974, Springer-Verlag, Berlin, pp. 748–754.
DETACHED AND LARGE EDDY SIMULATION OF AIRFOIL FLOW ON SEMI-STRUCTURED GRIDS
S. SCHMIDT AND F. THIELE Hermann-Föttinger-Institut für Strömungsmechanik Technische Universität Berlin, Müller-Breslau-Str.8, 10623 Berlin, Germany.
Abstract. This study aims at investigating the flow around a NACA 4412 airfoil at high Reynolds numbers and moderate angles of attack on semistructured grids, which allow for a blockwise grid refinement in areas of physical interest (e.g. walls). Unlike other configurations using Large Eddy Simulation (LES), this flow features only a small pressure-induced separa tion bubble at the trailing edge of the profile showing no massive transient motion. Therefore, this flow is very sensitive to both spatial resolution, which might not be sufficient in case of wall-resolving LES, and the em ployed subgrid-scale model, which has to capture all relevant unresolved motion in the transitional boundary layer on the suction side. Since wallresolving LES remains rather unfeasible for flows at Reynolds numbers beyond in the near future, the method of Detached Eddy Simula tion (DES) has recently attracted high interest. DES avoids the high nearwall resolution by applying the Reynolds-Averaged Navier-Stokes equations method (RANS) in the vicinity of the wall and a modified Spalart-Allmaras one-equation turbulence model in the far field. In the overlap region, this model blends automatically from a statistical to a subgrid-scale model with out the use of shape functions. The results of the flow around a square cylinder, used for a basic comparison, and the airfoil show a very good agreement with the experimental data, while only using a fraction of the numerical resources compared to an adequately resolved LES. 255 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 255-272. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
256
S. SCHMIDT AND F. THIELE
1. Simulation of Turbulence 1.1. LARGE EDDY SIMULATION
In LES, only the large scales of the turbulent motion are explicitly resolved by the numerical grid while the small-scale motion is represented by a subgrid-scale model. The motivation for this approach is that the large-scale vortices are dominated by geometrical constraints and boundary conditions. Due to turbulent transport phenomena, these vortices pass their kinetic energy on towards smaller vortices and the orientation of the initial vortices gets lost during this energy cascade. Therefore, the small-scale turbulence is expected to behave more isotropically, without any preferred orientation and should consequently be much easier to model than the whole spectrum of turbulence. The governing set of equations read
and denote the filtered velocities, the coordinates in a Herein, cartesian framework and the time, respectively. The pressure contains the isotropic part of the correlation which has to be modelled. 1.2. SUBGRID-SCALE MODELS
The Smagorinsky model (SM) is based on a simple mixing-length ap which has to proach (Smagorinsky, 1963). It includes one parameter be fixed according to the flow problem and is usually set to for most turbulent flows. Under strong shear and near rigid walls, however, this parameter has to be reduced to damp out the eddy viscosity and give the correct near-wall development (Van Driest, 1956). However, in complex flows, no unique parameter can be determined, as different flow regions claim for their specific value of To overcome the parameter problem, the dynamic procedure (Germano et al., 1991) estimates the model param eter from the resolved velocity field and evaluates a local, time-dependent Contrary to the SM, these types of models are gener value ally capable of predicting backscatter and are able to adapt themselves to the local turbulence structure, i.e. the model is able to distinguish between laminar and turbulent flow regions. Numerical stability problems, however, require rather empirical work-arounds (e.g. averaging of the model estimate
DES AND LES OF AIRFOIL FLOW
257
in homogeneous directions and clipping to enforce positive values of the to tal viscosity), which can be attributed to the poor mathematical derivation of the model and missing realizability conditions. The more elaborate dynamic one-equation model (DOEM) employs the turbulent subgrid kinetic energy to avoid these numerical difficulties while retaining the key features of the dynamic approach (Davidson, 1997). This model remains stable also in complex geometries without any homogeneous directions and is in this sense superior to the standard model. The favorable stability properties as compared to the dynamic model usually outweigh the higher computational costs of solving the additional transport equations. 1.3. DETACHED EDDY SIMULATION DES (Spalart et al., 1997) takes advantage of the RANS method where the mean flow remains attached and steady while, like LES, offering the sensitivity to capture unsteady flow phenomena in areas such as wakes or recirculation zones. Although this technique demands higher computational costs compared to a steady 2D-RANS calculation, it reveals nearly as much information of the flow dynamics as LES. For this reason, DES could be a promising way out of the limitation detaining LES from being applied at high Reynolds numbers. The DES methodology is basically a modification of the dissipation term within the Spalart-Allmaras (SA) one-equation tur bulence model (Spalart and Allmaras, 1994). This term is strongly affected by the wall-normal distance which is substituted by a new length-scale (Shur et al., 1999)
which defines the boundary between the RANS and LES zones. For small values of the wall distance, where (e.g. in boundary lay ers), the original statistical model will hold. In the far field the length-scale approaches the local grid size for which the maximum of and is taken. This makes the SA model act as a subgrid scale model with a mixing-length directly linked to the grid spacing. Note that this is in contrast to the usual practice in LES, where the average is used. As RANS remains attractive to simulate a wide range of turbulent flows, the results of steady calculations with rather commonly used two-equations models (e.g. are included for comparison. 2. Numerical Scheme
The flow solver ELAN3D (Xue, 1998) is based on an implicit pressure-based finite volume Navier-Stokes procedure applying a cell-centred discretisation
258
S. SCHMIDT AND F. THIELE
on semi-structured grids. The scheme uses a pressure correction scheme and a generalised Rhie and Chow (1983) interpolation to avoid a de-coupling between velocity and pressure. The code is second-order accurate in space and time and uses multi-block algorithms. Several upwind-biased limited HOC schemes are available, which are only used in the context of DES and RANS. They ensure a higher approximation order of the convective terms on coarse meshes, which are usually employed by RANS/DES compared to LES. In the far field, the inherent numerical diffusion of the convection scheme retains numerical stability without sacrificing the loss of resolved dynamics in areas close to the airfoil (e.g. wake). For LES, only the sym metric convection scheme CDS-2 is applied. Furthermore, the solver has been parallelised using domain decomposi tion, where the performance has been quantified to roughly 60 MFLOPs per processing element (PE) on a CRAY T3E-900. 2.1. SEMI-STRUCTURED GRIDS Since the computational demands of wall-resolving LES of aerodynamic components at higher Reynolds numbers are tremendous, measures are necessary to substantially reduce time as well as memory requirements. A feasible approach towards this goal is the reduction of mesh points by applying semi-structured grids. As the flow features of interest are com monly located in the vicinity of the configuration wall, the majority of the grid points used are concentrated close to the surface. On structured grids, this fine near-wall resolution can only be reduced in the wall-normal di rection, while the streamwise and spanwise spacing remains dense even in the far field where the flow is almost uniform. One way to alleviate this problem is to use so-called semi-structured grids. The method is based on block-structured grids, which allow for non-matching grid interfaces at block-boundaries (see figure 3). This approach has two major advantages over both structured and unstructured grids with regard to flexibility and performance. On one hand, at block interfaces, it provides the flexibility to reduce the total amount of grid points in two directions in areas of less physical importance (e.g. far field), which usually cover most of the com putational domain. On the other hand, this approach benefits from the structured memory alignment of the data, thereby allowing for a higher performance as compared to unstructured methods. Although in principle nearly every cell face ratio is possible at the block interfaces, large face divisions are generally disadvantageous for LES, since larger approximation errors are introduced into the solu tion. Therefore, in this study, the cell face ratio is restricted to 1:2 in each direction, leading to 1:4 for 3D cases.
DES AND LES OF AIRFOIL FLOW
259
3. Validation
A suitable testcase to work out the differences between LES and DES is the turbulent flow over a square cylinder, which was already investigated in a workshop on LES (Rodi et al., 1997) and is considered to be a key flow configuration for the validation of turbulence models. The oncoming laminar flow is impinging on the cylinder front and separating at the lead ing edges. The flow undergoes transition to turbulence in the separated shear-layers and forms a vortex street in the wake due to periodic vortex shedding at the lee-side of the cylinder. The typical shedding frequency is characterised by the Strouhal number normalised by the cylinder diameter D and the flow speed The flow domain and the Reynolds number (Re = 22 000) are set according to the experiment (Lyn et al., 1995). All simulations make use of the same block-structured grid in the cross-sectional plane which consists of about grid points. The wall normal distance of the first mesh point around all cylinder being smaller than the value walls was set to taken by Sohankar et al. (2000). In order to demonstrate the influence of the spanwise discretisation, different numbers of grid points (NK) to resolve this direction are used (table 1).
While the LES case uses NK=32 points, the DES cases employ NK=20 (DES-A), NK=10 (DES-B) and NK=2 points (DES-C), respectively. Note that the latter is applied to a smaller spanwise domain to prevent the rise of numerical oscillations owing to extreme cell aspect ratios in that direction. This simulation (DES-C) is performed on a typical 2D-mesh and therefore comparable to an unsteady RANS calculation, it requires only 6% of the computing time of the LES. The comparative run with the EASM was carried out on the same mesh as the DES-C and made use of wall-functions.
260
S. SCHMIDT AND F. THIELE
The global parameters (table 1) give an impression of the dynamical behaviour of the flow. A decrease in the spanwise resolution results in a loss of unsteady motion represented by the Strouhal number which drops from 0.130 (LES) to 0.088 (DES-C). At the same time, the predicted size of the recirculation zone increases as can also be seen in figure 1a. This is basically due to a reduction of momentum exchange in the poorly re solved spanwise direction, leading to a quasi-2D flow field. The mean drag coefficient and its fluctuating component increase due to a loss in three-dimensionality, while the lift fluctuations retain their magnitudes In order to assess the performance of the DOEM, compara ble LES results with a DOEM (Sohankar et al., 2000) are shown, indicating
DES AND LES OF AIRFOIL FLOW
261
a good agreement for all given parameters. Remaining deficiencies can be attributed to the local grid spacing in the cross-sectional plane and in the spanwise direction. From the LES and DES results it becomes evident that a larger recircu lation length usually reduces the shedding frequency St. The additional RANS results, obtained with a non-linear eddy-viscosity model, however, show a contrary behaviour: both quantities and St are higher than the corresponding LES and DES values. A closer look at the centreline profiles of the mean velocity (figure 1a) reveals that both LES and DES-A yield nearly the same back-flow component at while the latter overpredicts the velocity in the wake Compared to case DES-A, the coarse-grid
262
S. SCHMIDT AND F. THIELE
simulations DES-B/C result in a much better prediction of both mean re circulation length and wake region, which almost matches the LES and experimental data. From the velocity fluctuations (figure 2a,b), it can be seen that the reduction of the spanwise resolution leads to a strong increase of the streamwise stress component and a decrease of the spanwise component which vanishes completely for DES-C (figure 2a). The dominating cross-flow stress component (figure 2b) is rather in sensitive to the spanwise resolution and therefore predicted at nearly the same magnitude by all simulations, which leads only to a small rise of the turbulent kinetic energy (figure 1b). The EASM shows the typical be haviour of the RANS approach, as the dynamic properties of the flow in terms of Reynolds-stress levels are underpredicted. Compared to the DES C, the EASM achieves only poor results, which lead to distinct differences in all quantities. It turns out that even a DES on a 2D-mesh outperforms a sophisticated non-linear eddy viscosity model, like the EASM (Schmidt et al., 1999). The application of DES to the flow past a circular cylinder, which features pressure-induced separation, has been carried out by Travin et al. (2000). 4. Airfoil Flow Configuration
The configuration considered is the transitional flow around a NACA 4412 airfoil at angle of attack. The Reynolds number based on the is set to chord length and the far-field velocity This configuration was chosen because two sets of experimental data are available from Hastings and Williams (1987) and Wadcock (1987).
The LES cases (LES-A, LES-B) make use of C-type semi-structured grids, which consist of and nodes respectively. The main difference between these grids being the spatial resolution along the suc tion side of the profile and the use of twice as many grid points within the boundary layer for the latter case (see table 2). On the fine grid (LES-B),
DES AND LES OF AIRFOIL FLOW
263
the airfoil surface is discretised by 1200 points in the streamwise direction, using 800 points on the pressure side and 400 on the suction side respec tively. The wall normal distance of the first off-wall mesh point is kept at over the entire airfoil surface. The wing span of about is resolved with 48 cells in the near-wall region, while in the far field only half the points were used in order to benefit from the semi-structured grid approach. Details of the fine LES-B grid near the leading and trailing edges are shown in figure 3.
264
S. SCHMIDT AND F. THIELE
For the RANS and DES simulations, the coarse LES-A grid with half the amount of grid points along the airfoil surface is used and consequently 200 cells on each side. While the RANS calculations were carried out on a small the DES was applied to a span of in order to slice resolve the turbulent structures at the trailing edge which appear to exceed used for the LES cases (Schmidt, the spanwise domain size of 2000). The transition strip, which has been used in the experiments and which plays an important role in the streamwise development of the boundary on the suction side and at layer, is located at on the pressure side. The strip basically acts as a forward/backward facing step on the undisturbed laminar boundary layer and triggers the onset of turbulence downstream. In all current simulations, this essential feature is directly resolved by the mesh (figure 4). Other features of the transition strip, such as the surface roughness were not accounted for. 4.1. RESULTS AND DISCUSSION
The pressure distribution over the entire airfoil gives a good impression of the global simulation results (figure 5a). Differences in the suction peak result from the capability of the models to predict the correct boundarylayer growth along the airfoil surfaces downstream of the stagnation point, where A large suction peak generally indicates too small values of the displacement thickness and often coincides with a loss of trailing edge sep aration. Therefore, distinct differences between all simulation methods are visible (figure 5b). The RANS simulations give fairly similar results but to tally fail to capture the horizontal pressure plateau indicating a separation bubble at the trailing edge (figure 6). This is no surprise, as these two
DES AND LES OF AIRFOIL FLOW
265
equation models cannot account for transitional flow regions and normally require the transition point to be fixed to an experimentally determined value. To allow for a peer comparison, no fixation of the transition point was applied in case of RANS. In contrast to RANS, the LES solutions, based on the Smagorinsky Model show a strong grid influence. Only LES-B is partly capable of predicting a trailing edge separation zone, while LES-A totally misses to capture this important flow feature (figure 6). The flow appears to be very sensitive to an adequate resolution of the momentum-containing motion within the boundary layer. Although the spanwise resolution is still insufficient for both LES testcases, an improved resolution of the boundary layer helps to obtain quite reasonable results.
The DES results benefit from the fact that they do not require an accu rately resolved wall region and are able to capture the unsteady motion and transitional effects outside of this crucial area. Based on DES, a compara tive grid study is undertaken in order to evaluate the influence of the spanwise resolution on the resolved flow properties. In both cases, the RANS grid is taken (table 2), only the number of grid points in the spanwise di rection is varied from NK=10 (DES-A) to NK=30 (DES-B), respectively. This basically leads to a shift of the internal LES-RANS boundary from
266
S. SCHMIDT AND F. THIELE
(DES-A) down to only determined by the resolution of the LES-RANS border in wallunits suction side is given.
(DES-B), which is In table 3, the location at different locations on the
The results show a very good agreement of both DES cases with the ex perimental values, giving the coarser simulation DES-A a slight advantage over DES-B. This is confirmed by the streamlines at the trailing edge, shown in figure 6. Only LES-B and both DES are able to capture the separation, however the latter tend to overpredict the size of the bubble compared to the experimental data showing separation at The boundary-layer parameters and the shape-factor (figure 7) give valuable insight in the failure of both RANS and LES-A simulations. (figure 7a) shows a good agreement for LES The displacement thickness B and the DES cases, while LES-A and the RANS calculation are not able to follow the increase at the trailing edge. As even LES-A is rather close to the experiments, is apparently not a good indicator for the quality of the flow prediction. The momentum thickness (figure 7b), however, reveals a strong overshoot of LES-B which exceeds even the results of LES A. This helps to explain the presence of the separation bubble for LES-B. The strong growth of the momentum thickness downstream of the leading edge separation indicated by large values of the shape factor (figure 7c), leads to a removal of momentum from the LES profiles and drives the flow towards separation. Although this feature is very essential for the overall flow prediction, the reason for this behaviour is certainly a lack of resolving the important flow patterns on the airfoil surface. In con trast, both DES results are able to capture the growth of the momentum thickness (figure 7b). The internal LES-RANS boundary (see table 3) has to be considered to understand the different behaviour of the DES cases. As the spatial resolution increases, this boundary moves towards the wall, which in case of DES-B drops below the size of the momentum thickness (figure 7b). This might be disadvantageous for the solution, as it remains outside of in case of the DES-A and is apparently disturbing the solu tion. Showing a close agreement with the experimental data in the range the RANS results fail to capture the correct development of the boundary-layer downstream of that location. The reason for that might
DES AND LES OF AIRFOIL FLOW
267
268
S. SCHMIDT AND F. THIELE
arise from the 2D-setup, which limits the degree of agreement with the mea surements to a certain extent. But, as RANS is usually applied in steady calculations exploiting homogeneous directions in order to scale down the problem, these results give an impression of the performance of this method, which is known to have difficulties with simple airfoil configurations (Haase et al., 1997). The different physical behaviour is confirmed by the velocity and the Reynolds shear-stress profiles. The mean velocities on the suction side of the airfoil (figure 8) show a rather similar behaviour at the location Further downstream, the three approaches diverge remarkably. Both DES results show a strong back-flow component at DES-B, ob tained with a finer spanwise resolution, yields worse results than DES-A, which compares very well with both experiments. DES-B mainly suffers (figure 7a) which leads to from the too large boundary-layer thickness an increase of the separation bubble in the wall-normal direction. LES-B gives very small negative velocity values only at along with the overestimated boundary-layer thickness, which has been discussed already. The RANS solutions are slightly different with respect to the velocity dis tribution without showing a major advantage for either model. The Reynolds shear-stress values (figure 9) of both LES cases by far exceed the experiments and all other simulations. In LES-B, the influence of the boundary-layer parameters and can be seen. The overpredic tion of the stress level is due to the Smagorinsky model, which removes too much kinetic energy in terms of too large values of the eddy viscosity out of the boundary layer, leading to an enhanced mixing and thus momen tum exchange. For comparison, a LES using the DOEM were carried out, showing only a very small tendency to capture the trailing edge separation at all. The reason for this model failure as well as a detailed subgrid-scale model assessment is discussed in Schmidt et al. (2001). The results of the DES simulations are quite similar. The shear-stress values are predicted very well, showing almost no sensibility to grid refine ment in the spanwise direction. Again, the shift of the separation bubble of DES-B compared to DES-A is visible, however, being of minor importance here. 5. Conclusions LES and DES have been applied to an aerodynamic configuration, namely the flow around a NACA 4412 airfoil. In LES, a near-wall grid refinement can improve the flow prediction yielding a trailing edge separation, which is not captured on the coarser grid. Further improvements can only be achieved by doubling both the spanwise extent of the flow domain and the
DES AND LES OF AIRFOIL FLOW
269
270
S. SCHMIDT AND F. THIELE
DES AND LES OF AIRFOIL FLOW
271
spatial resolution in that direction, leading to four times the numerical effort and memory requirements of the current fine-grid LES-B. This disqualifies this method for aerodynamic flows. In contrast, DES has been successfully applied to a square cylinder flow and the airfoil configuration.
Only DES is capable of predicting all flow properties with remarkably good quality (figure 10). Besides giving nearly the correct size of the trailing edge separation, both boundary-layer parameters and mean velocity profiles show a very good agreement with the reference data. As also the shear-stress profiles are ill-predicted by LES and RANS, DES appears to be the only adequate tool for such aerodynamic flows. Since the computational effort for a DES is only 5-10 times larger than for a 2D-URANS, this approach can be an accurate alternative to pure RANS and LES. Acknowledgements
The authors gratefully acknowledge the financial support of the DFG (Deutsche Forschungsgemeinschaft, SFB 557). The calculations were car ried out at the Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) on a Cray-T3E.
272
S. SCHMIDT AND F. THIELE
References Davidson, L.: 1997, ‘Large Eddy Simulation: A Dynamic One-Equation Subgrid Model for Three-Dimensional Recirculation Flow’. In: 11th Symposium on Turbulent shear Flows. Grenoble, France, pp. 26–1–26–6. Germano, M., U. Piomelli, P. Moin and W. H. Cabot: 1991, ‘A dynamic subgrid-scale eddy viscosity model’. Physics of Fluids A3(7), 1760–1765. Haase, W., E. Chaput, E. Elsholz, M. Leschziner and U. H. Müller: 1997, ECARP European Computational Aerodynamics Research Project: Validation of CFD Codes and Assessment of Turbulence Models, Vol. 58. Vieweg Verlag. Hastings, R. and B. Williams: 1987, ‘Studies of the flow field near a NACA 4412 aerofoil at nearly maximum lift’. Aeronautical Journal 91, 29–44. Lyn, D., S. Einav, W. Rodi and J.-H. Park: 1995, ‘A Laser-Doppler Velocimetry Study of Ensemble Averaged Characteristics of the turbulent Near Wake of a Square Cylinder’. J. Fluid Mech. 304, 285–319. Rhie, C. and W. Chow: 1983, ‘Numerical Study of the turbulent flow past an Airfoil with trailing edge separation’. AIAA Journal 21(11), 1525–1532. Rodi, W., J. Ferziger, M. Breuer and M. Pourquié: 1997, ‘Status of Large Eddy Simulation : Results of a Workshop’. Journal of Fluids Engineering 119(6), 248–262. Schmidt, S.: 2000, Grobstruktursimulation turbulenter Strömungen in komplexen Geome trien und bei hohen Reynoldszahlen. Mensch & Buch Verlag, Berlin. ISBN 3-89820185-6. Schmidt, S., M. Franke and F. Thiele: 2001, ‘Assessment of SGS Models in LES Applied to a NACA 4412 Airfoil’. In: 39st AIAA Aerospace Sciences Meeting and Exhibit. Reno, Nevada. Schmidt, S., H. Lübcke and F. Thiele: 1999, ‘Comparison of LES and RANS for BluffBody Wake Flows’. Poster presentation, GAMM-Workshop, Kirchzarten, September, Germany. Shur, M., P. Spalart, M. Strelets and A. Travin: 1999, ‘Detached-Eddy Simulation of an airfoil at high angle of attack’. In: W. Rodi and D. Laurence (eds.): Turbulent Shear Flows. pp. 669–678, Elsevier Science Ltd. Smagorinsky, J.: 1963, ‘General circulation experiments with the primitive equations’. Monthly Weather Review 91, 99–164. Sohankar, A., L. Davidson and C. Norberg: 2000, ‘Large Eddy Simulation of Flow Past a Square Cylinder: Comparison of Different Subgrid Scale Models’. J. Fluids Engng. 122, 39–47. Spalart, P. and S. Allmaras: 1994, ‘A one-equation turbulence model for aerodynamic flows’. La Rech. Aérospatiale 1, 5–21. Spalart, P., W.-H. Jou, M. Strelets and S. Allmaras: 1997, ‘Comments on the feasibility of LES for wings and on hybrid RANS/LES approach’. In: C. Liu and Z. Liu (eds.): Advances in DNS/LES. Columbus, OH, USA, Greyden Press. Travin, A., M. Shur, M. Strelets and P. Spalart: 2000, ‘Detached eddy simulations past a circular cylinder’. Journal of Flow, Turbulence and Combustion 63, 293–313. Van Driest, E.: 1956, ‘On turbulent flow near a wall’. Journal Aero. Science 23, 1007– 1011. Wadcock, A.: 1987, ‘Investigations of low-speed turbulent separated flow around airfoils’. Contractor Report 177450, NASA. Xue, L.: 1998, ‘Entwicklung eines effizienten parallelen Lösungsalgorithmus zur dreidi mensionalen Simulation komplexer turbulenter Strömungen’. PhD thesis, Technical Universiy of Berlin.
A MULTIDOMAIN/MULTIRESOLUTION METHOD WITH APPLICATION TO RANS-LES COUPLING
P. QUÉMÉRÉ, P. SAGAUT AND V. COUAILLIER ONERA
CFD & Aeroacoustics Department
29 av. de la Division Leclerc
92322 CHÂTILLON cedex
FRANCE
AND
F. LEBOEUF Ecole Centrale de Lyon
LMFA
36 av. Guy de Collongue
BP 163
69131 ECULLY cedex
FRANCE
Abstract. This paper deals with the development of a multidomain/multiresolution method with application to RANS-LES coupling. In order to lower the computational cost, it is proposed to use LES in small subdomains embedded in larger RANS subdomains. This coupled approach results in the definition of a general multidomain problem. The main feature of the method is the interface condition between RANS and LES solutions. An original algorithm is proposed and assessed for the plane channel configu ration.
1. Introduction Direct numerical simulation of turbulent flows is still far out of range for flows of practical industrial interest. In order to get an unsteady highfrequency representation of the solution, Large-Eddy Simulation (LES) has been investigated by many authors (see Ref. [14] for a review). This tech 273 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 273-290. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
274
P. QUÉMÉRÉ ET AL.
nique, which is based on a low-pass filtering of the exact solution of the Navier-Stokes equations, makes it possible to obtain a significant reduction in the complexity of the simulation by reducing the number of degrees of freedom. But LES is still subject to severe constraints when wall bounded flows are considered, because (at least theoretically) the internal region of the boundary layer needs to be quasi-directly resolved, yielding large computational costs. Because an accurate unsteady description of the solution is not needed everywhere when dealing with practical engineering problems, the idea of using zonal approaches has emerged. The idea here is to use LES in small localized subdomains where an accurate description of the flow is wanted, while computing the rest of the configuration with a low-accuracy method. The latter method will be the Reynolds Averaged Numerical Simulation (RANS) approach in the present work. The approach proposed in the present paper is based on a multidomain/multiresolution decomposition of the problem. The full configuration is decomposed into several subdomains, which can be treated with either RANS or LES approach. The problem is now to define adequate interface conditions between RANS and LES subdomains. This problem is a fully general multidomain problem, which is a generalization of the LES multiresolution/multidomain approach devised in Quéméré et al. [13]. The paper is organized as follows. The theoretical framework associ ated to the problem is introduced in section 2. Corresponding governing equations and physical models are discussed in section 3. The RANS/LES interface problem and the proposed interface numerical treatment are de tailed in section 4. Section 5 contains the main features of the numerical method. The subsonic plane channel flow test case is discussed in section 6. Conclusions and perspectives are presented in section 7. 2. Statement of the Scale Separation Problem
Both RANS and LES approaches rely on a scale separation procedure. In the former case, it is obtained via a statistical average, leading to:
where is a space- and time-dependent dummy variable, and denotes the ensemble average operator. It is important to note that the dimension of the averaged problem can be lower than the one of the original problem for a large class of flows: the original problem involves 4 dimensions (3 for space, 1 for time), while the dimension of the averaged problem ranges from 0 to 4.
MULTIDOMAIN/MULTIRESOLUTION APPROACH
275 275
In the case of LES, scale separation is traditionally associated with the use of a spatial convolution filter, yielding
with
where G is the kernel filter function and is the cut-off lengthscale associated with the filter. For sake of simplicity we restrict ourselves to a formal presentation, and the extension of the convolution filter to the non-homogeneous case (see Refs. [18, 14]) will not be detailed here. It is important noting that all the developments presented below are fully gen eral. The dimension of the filtered problem is the same as that of the original problem: three dimensions for space plus one dimension for time. 3. Governing equations and closures 3.1. GOVERNING EQUATIONS FOR RESOLVED MOTION We consider here the case of Newtonian compressible fluids. Applying one of the two scale separation approaches described in the previous section (noted here by a bar) to the Navier-Stokes equations, we get the evolution equations for the resolved motion. These equations, which are formally equivalent, can be written as follows:
where the tilde refers to mass-weighted variables:
The modified total energy
is defined as:
276
P. QUÉMÉRÉ ET AL.
and the modified viscous stress tensor
vector as:
and the modified heat flux
The averaged ideal gas law is
The unresolved terms appearing in the momentum and energy equations are defined as follows:
The specific closures associated with LES and RANS approaches are described in the following sections. 3.2. LES CLOSURE The subgrid models and underlying assumptions used in this study are the same as those employed by Lenormand et al. [8, 9] for subsonic and super sonic flows, based on Vreman’s conclusions. Considering subgrid-viscositytype models relying on the Boussinesq hypothesis, we get:
MULTIDOMAIN/MULTIRESOLUTION APPROACH
277
where the subgrid Prandtl number is set equal to 0.6 and where the deviatoric part of the subgrid-scale stress tensor (with superscript L referring to the LES approach) is modeled by :
The subgrid-scale viscosity is computed using the Selective Mixed Scale Model [14, 8, 9, 15]. This model is given by a non-linear combination of the norm of the vorticity the characteristic length scale and the kinetic of the highest resolved frequencies: energy
with and the selective function. Introducing a test filter denoted by a hat which can be interpreted as a second-order approximation of a Gaussian filter [15], the kinetic energy is evaluated by :
3.3. RANS CLOSURE As in the LES case, fluctuations of molecular viscosity and diffusivity are neglected. Only the terms and the contribution of terms are taking into account (the effects of term are small in regard to the other ones for the applications in mind). As for LES, the RANS closure employed in the present study is based on the definition of an eddy-viscosity (superscript R is related to RANS approach), yielding:
where remains to be defined, and (with ) is the turbulent kinetic energy . Unlike the LES modelling, the latter variable is explicitely taken into account in the RANS approach. We use here the low-Reynolds number two-equation model pro posed by Jones and Launder (see Ref. [7] for a description of the model).
278
P. QUÉMÉRÉ ET AL.
4. RANS/LES coupling at the subdomain interface 4.1. THEORETICAL PRESENTATION OF THE PROBLEM For sake of simplicity, and without loss of generality, the theoretical pre sentation of the problem is carried out in the case of the interface of two subdomains. A RANS simulation is carried out in the first one, referred to while an LES simulation is performed in the second subdomain, as The interface between the two subdomains is noted Data associated to the subdomain will be denoted with the superscript The difference in the scale-separation operator selected in each subdo main leads to a discontinuity of the solution at the interface
preventing the use of classical conservative treatments of the interface. This can be seen as a generalization of the discontinuous interface condition derived for multidomain/multiresolution LES/LES simulations developed by Quéméré et al. [13]. It is important noting that the interface treatment is intrinsically based on the numerical method used to solve the governing equations. We present below an interface condition adapted to the RANS/LES multidomain prob lem within the framework of finite-volume, cell-centered numerical meth ods. In the present approach, boundary conditions for each subdomain are prescribed by defining the values of the unknowns in rows of ghost-cells associated with each domain and overlapping the other one. We now introduce the interface variable defined as the difference between the two fields on the interface:
The basic interface problem is the following: Boundary conditions for the RANS subdomain: evaluate from in the ghost cells, and find the values of turbulence-model related variables This corresponds to removing the interface
variable
from the LES field
in the RANS subdomain
ghost cells. Boundary conditions for the LES subdomain: evaluate from in the ghost cells and define the subgrid-scale viscosity
This is equivalent to the problem of computing
in these cells, and
corresponds to the reconstruction of the high frequency fluctuations at the interface.
MULTIDOMAIN/MULTIRESOLUTION APPROACH
279
4.2. BOUNDARY CONDITIONS FOR THE RANS SUBDOMAIN
Values of in the ghost cells are easily obtained by applying the ensemble average operator to the LES field yielding
In practice, the ensemble average can be associated with (i) an ensem ble average performed using several statistically equivalent simulations, as e.g. Carati’s ensemble-averaged LES [4], (ii) an average over homogeneous space directions, (iii) an average in time, or any combinations of these three possibilities. The most general one is the first one, but it is also the most expensive. In the present paper, the second solution is used. Evaluating and is a more difficult task. Numerical experiments have shown that a direct reconstruction of these variables in the ghost cells using the LES field yields poor results because they are very sensitive to the restriction operator, the number of samples available to perform the ensemble average being generally too low to ensure a good evaluation of A robust and efficient method is to compute and with It is important as a velocity field everywhere in the LES subdomain noting that here and have no feedback on the LES field in the interior of the subdomain, and can be seen as two coupled passive scalars within this domain. This allows the definition of turbulent variables which account for memory effects and the structure of the field at the interface, with a weak dependency on the implementation of the ensemble average operator. 4.3. BOUNDARY CONDITIONS FOR THE LES SUBDOMAIN It was said in section 4.1 that the problem of defining the LES field in the ghost cells which overlap the RANS subdomain is equivalent to evaluating the interface variable in these cells. Because the ensemble average can be mathematically interpreted as a projector, information related to the fluc tuations in this subdomain is lost and cannot be reconstructed from the RANS field alone. The distinction will be made here between outflow and inflow subdomain interfaces. Outflow interfaces correspond to interfaces where the flow is directed from the LES subdomain toward the RANS subdomain. In this case, the quantity is extrapolated from the LES subdomain in the following way:
280
P. QUÉMÉRÉ ET AL.
1. Compute in the LES subdomain in rows of cells located just in front of the interface, using the same restriction operator as for the RANS subdomain interface condition ; 2. Evaluate the interface variable in these cells as follows:
3. Extrapolate
in the ghost cells, yielding the extrapolated In the present work, a weighted first-order accurate extrapolation is used. This weighting factor is introduced to account for the spatial variation of the resolved fluctuating kinetic energy. 4. Reconstruct the total LES field in the ghost cells. Inflow interfaces are the most difficult case, because the information is now convected from the RANS subdomain toward the LES subdomain. This problem is very similar to the problem of defining turbulent inflow con ditions for LES. It is now recognized that the definition of rough boundary conditions, which do not account for the two-point and two-time correla tions of turbulent fluctuations can have deleterious effects [5]. The distinction will be made here between the low- and high-normal velocity inflow interfaces. We define here low-normal velocity (resp. high normal velocity) interfaces as interfaces for which the numerical advection lengthscale (where is the time step and the inflow normal velocity component, being the inward normal vector) of the information across the interface is small (resp. high) with respect to the characteristic lengthscale L of turbulent fluctuations near the interface. For low-normal velocity interfaces, the fluctuations remain strongly cor related in space, and can be extrapolated from the LES subdomain. The numerical treatment is then exactly the same as for outflow interfaces. Results dealing with the case of high-normal velocity interfaces will not be discussed here for lack of space (see Ref.[12]). For high-normal velocity interfaces, turbulent fluctuations will be decorrelated in space near the interface, and the extrapolation technique can no can be defined, relying on longer be used. Several ways to reconstruct : (i) the use of analytical deterministic definitions of the fluctuations, (ii) a long-range extrapolation technique, similar to the rescaling technique of Lund [10] for turbulent inflow conditions, or (iii) the extraction of fluctua tions from a secondary simulation carried out on a simpler geometry or a lower Reynolds number. The best results on a blunt trailing-edge configu ration (not shown here, see Ref.[12] for more details) have been obtained by simulating a plane channel flow, independently of the present calculation. is extracted, rescaled and used to At each time-step, the complement define the inflow condition.
MULTIDOMAIN/MULTIRESOLUTION APPROACH
281
These numerical experiments have shown that the first technique (i) yields the existence of a relaxation zone, where coherent high-frequency fluctuations are regenerated through non-linear interactions from the ini tial random perturbations. The same observations have been made for RANS/LES couplings based on ”universal models”, such as DES. The direct rescaling (method (ii)) of a plane extracted from the same computational domain can only be done if self-similarity is not broken in the streamwise direction, leaving the third solution (iii) as the most general one. The use of method (iii) shows that the fluctuations do not have to be perfectly consistent with the mean velocity to improve the results given by method (i). 5. Numerical Method The basic numerical method is exactly the same as the one described in Refs. [13, 6, 17]. It is based on a second-order accurate finite volume, cellcentered discretization of the compressible Navier-Stokes equations. The skew-symmetric form and a centered non-dissipative scheme is used for the convection term. Spatial derivatives (temperature gradients, velocity gradients) present in diffusive fluxes are computed using staggered cells to evaluate gradients in order to ensure the coupling between odd and even cells, preventing spurious wiggles. Time integration is performed using a third-order three-stage compact Runge-Kutta time stepping scheme. For plane channel flow computations, the same forcing term as in Refs. [8, 9] is used. 6.
Application to plane channel flow
The multidomain/multiresolution technique has first been assessed on a subsonic plane channel flow. The interface has been taken parallel to the solid wall, which corresponds to outflow or low-normal velocity boundary conditions for the LES subdomain, and to inflow and outflow boundary conditions for the RANS subdomain. It is then a relevant test case to vali date the associated treatment based on the extrapolation of the fluctuation
6.1. PHYSICAL PROBLEM AND COMPUTATIONAL PARAMETERS The selected configuration is the isothermal-wall plane channel flow. Peri odic boundary conditions are used in the streamwise and spanwise directions. For notational convenience, all bar and tilde symbols associated previously with the resolved variables in the equations are left out. The restriction operator R is defined as the ensemble average over the homoge
282
P. QUÉMÉRÉ ET AL.
neous directions Statistical moments of the solution are computed by performing a statistical average in time and in the homogeneous directions. and in the streamwise, The size of the computational domain spanwise and wall-normal directions, respectively) was chosen such that the two-point correlations in the streamwise and spanwise directions would be essentially zero at the maximum separation (half the domain size). In the present work, all the computations have been performed with CFL numbers equal to 0.95. This small value makes it possible to assume that the time-filtering effects due to the use of finite time steps will be masked by the implicit space-filtering operation. Uniform mesh spacing is used in homogeneous directions, while a stretched grid following an hyper bolic tangent law distribution is used in the wall-normal direction.
Two configurations have been investigated: Resolution of the internal boundary-layer region using LES, the central part of the channel being described using RANS. That allows a fine description of the near-wall turbulent fluctuations, while ignoring the rest of the flow. This kind of description, which is required for some studies related to aeroacoustics and prediction of aero-optical effects, was the original purpose of the present work. Resolution of the near-wall region using RANS, the external boundarylayer region being resolved with LES. The RANS approach appears here like a wall-model for LES. These computations can then be related to the DES approach [16, 11], or to the two-layer computations of Balaras et al. [1] and Cabot [3, 2]. 6.2. NEAR-WALL LES TREATMENT
All the tests presented in this section have been carried out using a three subdomain decomposition: one LES subdomain near each wall, and one RANS subdomain in the core region of the channel. Classical RANS and LES monodomain computations have been performed in order to have some reference data. Two targeted skin-friction Reynolds numbers have been considered: and The Mach number defined from the bulk velocity and the mean sound velocity at the wall as is set equal to 0.5. Computational parameters for all the simulations are presented in Ta bles 1 and 2. The size of the domain is the same for the monodomain and the multidomain computations at the same Reynolds number. For multido
MULTIDOMAIN/MULTIRESOLUTION APPROACH
283
main simulations, is related to the distance between the wall and the interface, expressed in wall units.
The computed mean values are summarized in Table 3. A good agreement between reference monodomain and multidomain computation is observed. It is seen that the friction Reynolds number is recovered within a 2% error level, which is a very satisfactory result. For the high-Reynolds number case, the multidomain is seen to yield better results than the classical LES calculation. Computed mean velocity profiles and resolved Reynolds stresses are compared with those obtained using an usual LES approach in Figs. 1 to 4. The agreement obtained with the theoretical mean velocity profile is very satisfactory, and it is observed that the hybrid computations yield results which are very close to the LES results. The observed discrepan cies at the highest Reynolds number are usual in LES computations with a second-order accurate numerical method and the present grid resolution. The agreement between classical LES and the hybrid RANS/LES com putations on the resolved Reynolds stresses is very good. The observed
284
P. QUÉMÉRÉ ET AL.
differences occuring near the interface are limited in a 4-5 point wide layer near the interface, and they are seen to not pollute the rest of the computa tion. The fluctuations observed in the RANS region are due to the unsteady character of the 1D RANS computation in that subdomain. The effect of the enrichment procedure is seen in Fig. 5, where onedimensional energy spectra near the interface are shown, and compared with those obtained in classical monodomain LES and hybrid RANS/LES without enrichment. The agreement obtained when using the enrichment procedure at the RANS/LES interface is very satisfactory, giving a new validation of the method. The sensitivity of the method to the position of the interface is then investigated. The resolved kinetic energy profiles obtained for three different positions at are presented in Fig. 6. A good general agreement is obtained with classical LES. It is also observed that some discrepancies appear in the center of the channel, as the cutoff is moved toward the centerline of the channel. Careful tests have demonstrated that this is due to a less satisfactory behavior of the 1D RANS computation in the central subdomain, which is due to the fact that both its extent and its resolution are diminishing, yielding stronger fluctuation levels. But, in all the cases, the RANS/LES coupling procedure remains efficient. A proof is the very good agreement between the classical LES computation and the A2 case, in which one-half of the channel is computed with LES, while the other part is computed by RANS.
MULTIDOMAIN/MULTIRESOLUTION APPROACH
285
6.3. NEAR-WALL RANS TREATMENT We now present the results obtained when using the RANS approach for the near-wall region. Two cases have been considered, which correspond to different positions of the interface (all computational parameters are given in Table 2). Computed mean velocity profiles are compared to classical LES and RANS computations in Fig. 7. The observed perfect agreement between RANS and hybrid computations (in the RANS subdomains) shows that the near-wall behavior in the hybrid computation is governed by the RANS model, and seems to be insensitive to the coupling procedure. In the LES subdomain, a very good agreement is recovered with all the other compu tations, demonstrating the efficiency of the interface condition. Resolved Reynolds stresses are compared to results of usual LES in Fig. 8. A very good agreement is obtained in the case where the interface is located at while more pronounced discrepancies near the interface are observed in the second case This is explained by two facts: (i) in the second case the mesh size near the interface is greater, yielding higher extrapolation errors in the ghost cells and, (ii) the mesh size near the interface being larger, it does not allow a very accurate description of turbulent fluctuations in this region, reducing the efficiency of the coupling procedure. But it is worth noting that, even in the second case, the results are at least as good as those obtained by other authors with other hybrid RANS/LES procedures. 7. Conclusions
A new zonal RANS/LES computational technique was proposed. It is based on the use of a subdomain decomposition, each domain being treated using one of these two approaches. Theoretical analysis reveals that the solution is discontinuous at the RANS/LES interface, and that even the dimension of the solution can vary. As a consequence, the RANS/LES coupling strategy appears as a generalized multidomain problem. The proposed treatment is based on the definition of an interface variable, which is extrapolated from the LES subdomain or extracted from an auxilliary computation, depending on the type of the interface. The proposed procedure was succesfully assessed for the plane channel configuration. Both mean velocity and resolved Reynolds stresses profiles are recovered, demonstrating the efficiency of the method. For interfaces referred to as low-normal velocity interfaces, the extrap olation of the field from the LES subdomain seems to be adequate. For high-normal velocity interfaces, good results were obtained using fluctua tions extracted from an auxiliary LES.
286
P. QUÉMÉRÉ ET AL.
Future work will deal with the use of this coupling strategy on realistic geometries. References 1. Balaras, E., Benocci, C., Piomelli, U. (1996) Two-layer approximate boundary con ditions for large-eddy simulations, AIAA J., 34(6), pp. 1111–1119 2. Cabot, W. (1995) Large-eddy simulations with wall models. Annual Research Briefs - Center for Turbulence Research, pp. 41–49 3. Cabot, W. (1996) Near-wall models in large-eddy simulations of flow behind a backward-facing step, Annual Research Briefs - Center for Turbulence Research, pp. 199–210 4. Carati, D., Rogers, M.M. (1998) Ensemble-averaged LES of time-evolving plane wake, Proceedings of the Summer Program - Center for Turbulence Research, pp. 325–336 5. Chung, Y.M., Sung, H.J. (1997) Comparative study of inflow conditions for spatially evolving simulation, AIAA J., 35(2), pp. 269–274 6. Ducros, F., Sagaut, P., Quéméré, P. (2000) On the use of relaxation procedure for localized dynamic models, Phys. Fluids, 12(12), pp. 3297–3300 7. Jones, W.P., Launder, B.E. (1972) The Prediction of Laminarization with a TwoEquation Model of Turbulence, Int. J. of Heat and Mass Transfer, 15, pp. 301–314 8. Lenormand, E., Sagaut, P., Ta Phuoc Loc (2000) Large-Eddy simulation of com pressible channel flow at moderate Reynolds number, Int. J. Numer. Methods Fluids, 32, pp. 369–406 9. Lenormand, E., Sagaut, P., Ta Phuoc Loc, Comte, P. (2000) Subgrid-Scale Models for Large-Eddy Simulation of compressible wall bounded flows’, AIAA J., 38 (8), pp. 1340–1350 10. Lund, T.S., Wu, X., Squires, K.D. (1998) On the generation of turbulent inflow conditions for boundary-layer simulations, J. Comput. Phys., 140, pp. 233–258 11. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D., Spalart, P.R. (2000) An approach to wall modelling in large-eddy simulation, Phys. Fluids, 12(7), pp. 1629– 1632 12. Quéméré, P., Sagaut, P. (2001) Zonal multidomain RANS/LES simulations of tur bulent flows, Int. J. Numer. Meth. Fluids, submitted 13. Quéméré, P., Sagaut, P., Couaillier, V. (2001) A new multi-domain/multi-resolution method for large-eddy simulation, Int. J. Numer. Meth. Fluids, 36, pp. 391–416 14. Sagaut, P. (2001) Large-eddy simulation for incompressible flows, Springer- Verlag 15. Sagaut, P., Grohens, R. (1999) Discrete filters for large-eddy simulation, Int. J. Numer. Methods Fluids, 31, pp. 1195–1220 16. Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S.R. (1997) Comments on the feasibility of LES for wings, and on hybrid RANS/LES approach, 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, Ruston, LA. 17. Terracol, M., Sagaut, P., Basdevant, C. (2001) A multilevel algorithm for large-eddy simulation of turbulent compressible flows, J. Comput. Phys., 167, pp. 1–36 18. Vasilyev, O., Lund, T.S., Moin, P. (1998) A general class of commutative filters for LES in complex geometries, J. Comput. Phys., 146, pp. 82–104
MULTIDOMAIN/MULTIRESOLUTION APPROACH
287
288
P. QUÉMÉRÉ ET AL.
MULTIDOMAIN/MULTIRESOLUTION APPROACH
289
290
P. QUÉMÉRÉ ET AL.
6. Aircraft wake vortices
This page intentionally left blank
AIRCRAFT WAKE VORTEX EVOLUTION AND DECAY IN IDEALIZED AND REAL ENVIRONMENTS: METHODOLOGIES, BENEFITS AND LIMITATIONS
F. HOLZÄPFEL, T. HOFBAUER, T. GERZ AND U. SCHUMANN Institut für Physik der Atmosphäre, DLR Oberpfaffenhofen D–82234 Weßling, Germany Abstract. After a brief introduction of the governing equations and numerical approaches that are used to simulate wake vortices in the atmosphere asso ciated implications and restrictions are discussed. The complex interaction of turbulence and rotation in the vortex core region is not resolved ap propriately and is controlled by the subgrid scale model. A local Richard son number correction for strong streamline curvature effects is proposed that accounts for stabilizing effects of coherent rotation and reduces vortex core growth rates. Real case simulations demonstrate that LES is capable to reproduce complex wake vortex behaviour as the spectacular rebound observed at London Heathrow Int’l Airport. Various idealized cases with stably stratified, turbulent and sheared environments are used to reveal the mechanisms that control vortex decay.
1. Introduction
As an unavoidable consequence of lift, aircraft generate counter-rotating pairs of trailing vortices which constitute a potential hazard to following aircraft. Therefore, separation distances between consecutive aircraft were established already in the early 1970s. The incentive of today’s wake vor tex research is mainly induced by the continuous growth of air traffic which increasingly congests airports during approach and landing. Since meteo rological conditions have significant impact on wake vortex evolution, one strategy to achieve a relaxation of the separation distances is to predict weather and vortex behaviour along the glide path. Another approach, es pecially appealing for the design of new large civil aircraft, is to develop constructive measures that could alleviate wake vortex intensity or selec tively accelerate vortex decay. 293 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 293-309. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
294
F. HOLZÄPFEL ET AL.
Since the experimental access to decaying trailing vortices under real istic and well-defined weather conditions is extremely difficult in the re quired temporal and spatial resolution, investigations which allow for de tailed physical interpretations are almost only possible with high-resolution numerical simulations. Large-eddy simulations (LES) constitute the tool to analyse wake vortex physics throughout their complete lifespan. A major restriction of the LES, however, is posed by limitations of the resolution. An adequate resolution of both the turbulent flow in the vortex cores and the length scales of the ambient flow exceeds present computer capabilities by far. Therefore, the impact of the subgrid-scale (SGS) clo sure on the under-resolved flow regions is analysed in section 3 after a brief introduction of the applied LES method in section 2. We propose a new SGS closure modification for strong streamline curvature effects and dis cuss its potential in comparison with corrections from literature in section 4. Then, in section 5 the numerical strategies that are applied to simulate a real case with rebounding wake vortices in a turbulent shear flow above complex terrain are presented. Finally, section 6 shows that coherent tur bulent structures which are well resolved by LES play the crucial role in the sequence of relevant decay mechanisms under different ambient conditions (turbulence, stable temperature stratification and shear). 2. Numerical Model and Initial Conditions
We apply the two DLR codes MESOSCOP and LESTUF which originally have been developed for studies in atmospheric boundary layers and of stratified, sheared, and homogeneous turbulence, respectively (Schumann et al., 1987; Kaltenbach et al., 1994). In the meantime, various investigations of wake vortex behaviour (e.g. Holzäpfel et al., 2000; Gerz and Ehret, 1997) have been performed with these codes. In space with coordinates and time both codes solve the mass conservation equation
and the Navier-Stokes equations for the resolved velocity vector
in an unstationary, incompressible, and three-dimensional fluid flow, where denotes the dynamic pressure fluctuation. For simulations of thermally stratified flows we employ the Boussinesq-approximated equations and in where are the volumetric expan clude the buoyancy term sion coefficient, the magnitude of the gravitational acceleration, potential
AIRCRAFT WAKE VORTICES
295
temperature and the Kronecker symbol, respectively. Then, also the equa
tion for the resolved potential temperature
has to be solved which is written as the deviation from the mean background The vertical gradient of the mean temperature of the atmospheric state background, is set constant in space and time. In equations (2) and (3) friction and heat diffusion are represented by the SGS fluxes of and heat, which result from the non-linear terms in (2) moment, and (3) after filtering on mesh scale. They are parameterized following an ansatz by Deardorff (1970)
with the strain rate tensor
The SGS viscosity
is modelled by Smagorinsky’s approach
and the constants and are set to the theoretical values for isotropic turbulence.
Alternatively, an algebraically approximated second order closure method that solves a transport equation for the SGS kinetic energy,
and algebraic equations for the anisotropic part of the subgrid scale fluxes
is applied in section 5 (for details regarding closure assumptions we refer to Schmidt and Schumann, 1989). The physical fields are discretized on a Cartesian staggered grid and integrated in space and time by second-order finite differencing. Time ad vancement is performed by a prognostic step for advection and diffusion using the second-order Adams-Bashforth scheme followed by a diagnostic step which solves the Poisson equation for the dynamic pressure. The inte gration scheme is non–dissipative and only weakly diffusive. The computational grid is equidistant in spanwise direction, and ver tical direction, with a typical resolution of whereas a
296
F. HOLZÄPFEL ET AL.
coarser resolution of is chosen in flight direction, When a fitting formula (Scotti et al., 1993) calculating the effective mesh size, is used to account for the anisotropy of the numerical mesh. A typical do main size corresponds to In the idealized cases (LESTUF) periodic boundary conditions are employed in all three directions whereas in the real cases (MESOSCOP) no-slip and free-slip con ditions are prescribed at the ground and top boundary, respectively. The wake vortex induced initial flowfield displays a small velocity kink across the periodic boundaries. The resulting disturbances are negligible since the boundaries are sufficiently far apart from the primary vortices. The wake vortices are initialized as superposition of two Lamb-Oseen vortices where the tangential velocity profile of one vortex is given by
In most simulations a core radius of a root-circulation of and a vortex spacing of are employed to represent the B-747 aircraft. The type of turbulence initialization in the wake vortices and their environment is different from case to case and will be introduced together with the specific cases. 3. Resolution Requirements and Vortex Core Evolution
The resolution issue is well illustrated with the simulation of wake vortices in a convectively driven atmospheric boundary layer (CBL). Ideally, the resolved length scales should span a range from the order of approximately in the vortex cores to the order of in the atmosphere, where the latter length scale roughly corresponds to the inversion height of a CBL. An mesh points, appropriate equidistant numerical mesh would need whereas our current grids are limited to about meshes. Appropriate compromises have to be found to conduct such simulations nevertheless. Here, the simulation of wake vortices with unrealistically large resolved by four grid points in an evolving initial vortex cores of CBL with an inversion height of enabled to cover the main features of wake vortex interaction with turbulent updrafts and downdrafts in a CBL (Holzäpfel et al., 2000). The CBL simulation was driven by a constant vertical heat flux at the lower surface and three wake vortex pairs were superimposed on the turbulent flowfield after the evolving CBL was well established. Figure 1 illustrates the interaction of convective cells and 10 seconds old wake vortices in a perspective view of an iso-surface of the upwards directed velocity
AIRCRAFT WAKE VORTICES
297
Obviously, any LES of wake vortices in the atmosphere will underresolve the vortex core region. To illustrate the consequential effects, the Smagorinsky eddy viscosity (eq. 6) normalized by
is derived for the Lamb-Oseen vortex (eq. 8) and plotted in figure 2 for dif ferent models and vortex core resolutions. The strain rate in equation (6) is applied for convenience in curvilinear coordinates, because then solely the contributes radial-tangential component of the strain rate tensor, to Note that in equation (9) the resolution of the vortex cores, enters to the power of two. Therefore, in poorly resolved vortex cores en hanced values of SGS viscosity are generated (see figure 2) that cause large vortex core growth rates and a strong reduction of peak vorticity. Figure 2 increases from zero1 at (rigid body rotation) to a delineates that maximum at and then decreases again. As a consequence, the radial velocity profiles deviate from the family of self-similar Lamb-Oseen vortex profiles that are achieved in DNS with constant viscosity2. For example, a slight overshoot of circulation on is produced (see figure 3a). 1 Obviously, when approaching molecular diffusion becomes relevant. Since this is a singular situation molecular diffusion can be neglected in the LES. 2 The DNS used in section 5 reached This indicates that only slightly higher vortex Reynolds numbers are achievable by LES with standard Smagorin sky closure compared to DNS.
298
F. HOLZÄPFEL ET AL.
As long as is determined by mean velocity gradients (due to limited resolution) and not by gradients of velocity fluctuations, the SGS momen tum fluxes should be corrected for streamline curvature effects. 4. SGS model correction for streamline curvature effects The following considerations are based on the “Richardson number” for streamline curvature effects that Bradshaw (1969) derived in analogy to the gradient Richardson number for buoyancy effects in stably stratified flows. Bradshaw’s Richardson number
relates the oscillation frequency squared of a radially displaced fluid element which retains its angular momentum in a flow with curvature of radius to the square of a “typical frequency scale of the shear flow”. However, equa tion (10) is misleading because it relates the oscillation frequency in natural coordinates to the frequency scale of plane shear taken in an inertial ref erence frame. As a consequence, in a vortex the Richardson number would where erroneously maximum stability would be as go to infinity at sumed as done by Cotel and Breidenthal (1999). A consistent formulation is achieved when instead in the denominator the strain rate is employed and both numerator and denominator are expressed in inertial coordinates or, for convenience, in curvilinear coordinates
AIRCRAFT WAKE VORTICES
299
For a Lamb-Oseen vortex the Richardson number (11) increases monoton reaches a value of 2 at and goes to ically from zero at infinity when approaching Proctor extended the Richardson number formulation to three dimensions (see Shen et al., 1999)
where and D are the magnitude of three-dimensional vorticity and de formation tensor, respectively. Since equation (12) cannot discriminate be tween vorticity of a plane shear flow and vorticity of coherent rotation an additional discriminator algorithm has to be applied. Proctor implements the Richardson number by modifying the SGS viscosity according to
where is a constant. We apply instead the corrected ansatz by Bradshaw (11) in natural coordinates (termed NaCoo)
that does not require a discriminator function. It employs directly the tan gential velocity, along a local streamline whose curvature is given by an inscribed circle with radius The curvature radius is calculated following and the approach of Hirsch (1995). In equation (14) the shear vorticity, curvature vorticity, just balance each other in the potential vortex. The correction is implemented according to
For an undisturbed Lamb-Oseen vortex both Richardson number formu lations given in equations (12), (14) yield identical results. The different profiles in figure 2 solely origin from the different implementations (equations (13) and (15)). With both implementations the corrections ef in the vortex core region. With NaCoo smoothly fectively reduce goes to small values when approaching the vortex center whereas Proctor’s implementation gives a relatively abrupt transition to zero at We prefer the smooth transition for the sake of physical plausibility (max imum stability is reached in the vortex center) and numerical stability. We performed LES of the evolution of a single Lamb-Oseen vortex with resolved by four grid points. Figure 3a and shows radial profiles of tangential velocity at and The comparison of the velocity profiles for the Smagorinsky closure and the
300
F. HOLZÄPFEL ET AL.
modifications indicates that with the modifications (i) the peak vorticity is well conserved, (ii) the growth of the core radii is retarded, (iii) an outer vortex core between solid body and potential vortex is formed as seen in experimental studies (Jacquin et al., 2001). Furthermore, (iv) the strongly damped diffusivity in the vortex cores allows for vortex core meandering as often observed in experiments (Holzäpfel et al., 1999; Rokhsaz et al., 2000; Jacquin et al., 2001). Figure 3b delineates the temporal evolution of vortex core radii with and without initially superimposed white noise. In the laminar vortex Proctor’s correction even causes shrinking core radii because the region of the velocity maximum experiences diffusion only from larger radii (see figure 2). The large growth rates achieved with standard Smagorinsky are not affected by turbulence. A disadvantage of a vorticity-based Richardson number for mulation (12) becomes visible in the turbulent case: Turbulent vorticity patches receive no damping even outside the stable core region which is re flected in the unsteady evolution of core radii. NaCoo, that produces small core growth rates, is not affected by the spotty structure of vorticity be cause it is controlled by the only marginally distorted streamline curvature (see figure 10). As recent PIV3 measurements (Vollmers, 2001) proved, the combination of small scale turbulence with coherent rotation, e.g. turbulent vortex cores, typically may occur in wake vortices. 3
Particle Image Velocimetry
AIRCRAFT WAKE VORTICES
301
5. Real Case Simulations
For flight safety reasons, situations of persistent vortices which remain in the flight corridor are of special interest. From LDV4 field measurements conducted at London-Heathrow Int’l Airport, Greenwood and Vaughan (1997) report such a case. As depicted in figure 4, the upwind vortex of a Boeing B747-200 in terminal approach was found to rebound back to flight level after an initial descent. 70 seconds after the aircraft passage, this vortex again attains generation altitude with practically undiminished strength. Apparently, this situation might have been critical to a follower aircraft, considering typical temporal aircraft spacings of 90 – 120 seconds. Greenwood and Vaughan suppose that the trees which are also depicted in figure 4 may well have been important in creating the updraft to carry the vortex. Other reasons might have been stable stratification, gusts or a low-altitude jet layer which could have been intensified by the presence of the trees. In order to find out the possible reasons that might be responsible for the observed vortex behaviour, LES and DNS are conducted. Owing to the complexity of this problem, the simulations have to capture the details of this case, namely the vegetation, the instationarity of the wind, and ambient turbulence. The initial conditions with respect to appropriate and fully developed 4 Laser Doppler Velocimetry – Technique to determine the line-of-sight velocity com ponent by measuring the Doppler shift of a Laser beam.
302
F. HOLZÄPFEL ET AL.
turbulence are obtained by a preparatory run which in turn starts by su perimposing the mean wind with homogeneous isotropic turbulence. Once the turbulence has adapted to the inhomogeneity of the environmental con ditions, e.g. the ground, and a quasi steady-state is reached, the wake vor tices are injected at Figure 5 displays the turbulence field right after vortex injection in a vertical-lateral cross-section of The corresponding domain size is with a resolution of During the field measurement, a second LDV – approximately upstream of the first one – was used to measure the ambient wind velocity. In order to account for the unsteadiness of the measured wind, a relaxation scheme is employed in the downstream region of the simulation domain. In the relaxation domain the mean velocities are forced to gradually adapt to the measured velocities. Figure 6a shows good agreement between measured and modeled vertical crosswind profiles prior to vortex injection. The trees that are suspected to influence the vortex trajectories are modeled according to Shaw and Schumann (1992) by embedding a momen in equation (2). The drag coefficient, is set to 0.15 tum sink, and the leaf area density, is prescribed as displayed in figure 6b. The comparison of measured and LES vortex trajectories in figure 7a shows excellent agreement. Alternatively employed DNS yield almost iden tical trajectories which indicates that at least vortex trajectories are insen sitive to vortex core evolution. Neglecting the axial wind component (figure 7b) also leads to very similar results. However, without the group of trees no rebound is observed. We argue that the combined effects of the trees 5
is a measure for coherent vortex structures according to Jeong and Hussain (1995).
AIRCRAFT WAKE VORTICES
303
acting as elevated ground and the positive crosswind shear that is produced above the trees (see figure 6a) are mainly responsable for the remarkable rebound. On the other hand, the simulations show that some high level of detail is necessary to reproduce such striking real case vortex behaviour.
304
F. HOLZÄPFEL ET AL.
6. Decay Mechanisms
Although the streamline curvature correction (section 4) allows for more realistic modeling of vortex core evolution, only minor deviations in overall wake vortex behaviour are found in simulations of wake vortices in a quies cent or a turbulent atmosphere. Therefore, we conclude that an inadequate resolution of vortex cores constitutes a minor restriction of universality and that, as we will show below, well-resolved coherent secondary vorticity structures determine the decay. Although these vorticity structures originate from different sources in the investigated cases – (i) a stably stratified, quiescent atmosphere, (ii) a turbulent atmosphere, (iii) a stably stratified, turbulent atmosphere, and (iv) a turbulent low-level jet (see Holzäpfel et al. (2001) for cases (i)–(iii), Hofbauer and Gerz (2000) for case (iv)) – the underlying mechanisms that trigger decay are very similar. Different phases that control vortex decay are identified: (1) the specific environmental conditions (i)-(iv) introduce secondary vorticity in the vicin ity of the wake vortices, (2) this vorticity is intensified by vortex stretching in the wake vortex induced velocity field, (3) the produced coherent vortical structures wrap around the primary vortices, and (4) – again by velocity induction – deform the vortex cores, destabilize the vortices and enable vorticity exchange between the vortex pair and its surroundings. Finally, (5) these mechanisms lead to a phase of rapid decay. The interrelation of azimuthal vorticity structures and wake vortex decay was first pointed out by Risso et al. (1994) and is typical for three-dimensional turbulence initialization (Kleiser and Schumann, 1984). In cases (ii), (iii) moderate, anisotropic, and decaying atmospheric tur bulence is superimposed on the whole velocity field. The atmospheric turbu lence, which is described in detail in Gerz and Holzäpfel (1999), obeys pre scribed spectra with rms velocities of in horizontal and in vertical direction. The length scales of the most energetic eddies amount Turbulence induced by the aircraft was taken into account in to 60 to cases (i)–(iii) by adding initially a three-dimensional random perturbation field to the swirling flow such that the perturbations reach maximum rms at the core radius, and decay exponentially for smaller values of and larger radii. In the stably stratified case (iii) the prescribed constant potential temperature gradient of the atmosphere was Figure 8 depicts iso-contours of the lateral and vertical vorticity com ponents that are induced by the wake vortices in a turbulent atmosphere.6 Wake vortices are represented by tubular The iso-lines of the lateral velocity, illustrate the converging flow that is induced by the wake 6
For coloured figures see http://www.pa.op.dlr.de/wirbelschleppe/EUROM.html
AIRCRAFT WAKE VORTICES
305
vortices above. This converging flow is deformed by superimposed atmo spheric turbulence such that the iso-line (bold) meanders along the symmetry plane between the vortices. As described in quantitative detail for case (i) in Holzäpfel et al. (2001), this superposition of turbulent and wake vortex induced velocities produces axial gradients of the lateral ve and, equivalently, vertical vorticity, in a volume locity, above and midway between the vortices. The resulting vertical vorticity (see figure 8) are amplified by vortex stretching due to the ac streaks, celeration of the downwards directed flow between the main vortices. Then, the vorticity is tilted, and wraps around the primary vortices. In the stably stratified case (iii) (figure 9) these effects are even more pronounced and develop faster. Here, it is the baroclinicvorticity, that additionally induces lateral velocities above the vortices that are directed towards the symmetry plane and intensifies the axial gradients of the lateral velocity. (Baroclinic vorticity is produced along the oval-shaped interface
306
F. HOLZÄPFEL ET AL.
between the ambient flow and the adiabatically heated flow that descends with the vortices (Holzäpfel and Gerz, 1999)). Note that the induced vortic ity structures in case (iii) at are much more intense than in case (ii) at They initiate a phase of rapid circulation decay (Proctor and Switzer, 2000; Holzäpfel et al., 2001). The transition to rapid decay coin cides with a transition to fully turbulent vortices as illustrated exemplarily in figure 10 for the evolution of wake vortices in a quiescent atmosphere. Similar mechanisms also control the decay of a trailing vortex pair that immerges into a turbulent low-level jet as seen in figure 11 (for details see Hofbauer and Gerz, 2000). In a preparatory run that was performed in a smaller domain the jet was allowed to develop from an initial isotropic state with root-mean-square velocity fluctuations of 5% of the maximum jet velocity. For the main simulation the jet was composed of two identi cal streamwise segments and the wake vortices were injected above the jet. The visual comparison of the formerly identical vorticity structures in the two parts of the jet allows to identify the interaction of jet turbulence and wake vortex induced velocity field. Clearly visible the jet vorticity is inten
AIRCRAFT WAKE VORTICES
307
sified along the upstream flank of the deformed wake vortex oval. At this flank a “stagnation line” forms and the jet-induced vorticity structures are stretched when torn apart in the wake vortex induced velocity field. Fig ure 12 shows in a perspective view how the intensified vorticity structures enclose the upstream vortex like a breaking wave.
308
F. HOLZÄPFEL ET AL.
7. Conclusions
We elucidate the implications and restrictions that are connected to the unavoidably inadequate resolution of wake vortex simulations in the at mosphere. To alleviate these restrictions a Richardson number correction is proposed that accounts for streamline curvature effects in the subgrid scales and reduces unrealistic vortex core growth rates. However, the improved approximation of realistic vortex core evolution does not modify vortex trajectories and decay essentially. In the atmosphere decay is controlled by well-resolved processes of vorticity intensification caused by stretching of environmental vorticity in the wake vortex induced flowfield. The generality of the involved mechanisms is illustrated with examples comprising stably stratified, turbulent and sheared environments. The example of the impres sive wake vortex rebound measured at London-Heathrow airport demon strates the capability of LES to reproduce complex real cases and to reveal the decisive underlying physical mechanisms. References Bradshaw, P. (1969) The analogy between streamline curvature and buoyancy in turbu lent shear flow, J. Fluid Mech. 36, pp. 177–191 Cotel, A.J. and Breidenthal, R.E. (1999) Turbulence inside a vortex, Phys. Fluids 11, pp. 3026-3029 Deardorff, J.W. (1970) A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech. 41, pp. 453–480 Gerz, T. and Ehret, T. (1997) Wingtip vortices and exhaust jets during the jet regime of aircraft wakes, Aerospace Sci. Techn. 1, pp. 463–474
AIRCRAFT WAKE VORTICES
309
Gerz, T. and Holzäpfel, F. (1999), Wingtip vortices, turbulence, and the distribution of emissions, AIAA J. 37, pp. 1270-1276 Greenwood, J.S. and Vaughan, J.M. (1997) Measurements of Aircraft Wake Vortices at Heathrow by Laser Doppler Velocimetry, Air Traffic Control Quarterly 6, pp. 179–203 Hirsch, C. (1995) Ein Beitrag zur Wechselwirkung von Turbulenz und Drall, Ph.D. Dis sertation, Universität Karlsruhe Hofbauer, T. and Gerz, T. (2000) Shear–layer effects on the dynamics of a counter– rotating vortex pair, AIAA Paper 2000-0758 Holzäpfel, F. and Gerz, T. (1999) Two-Dimensional Wake Vortex Physics in the Stably Stratified Atmosphere, Aerospace Sci. Techn. 3, pp. 261-270 Holzäpfel, F., Gerz, T. and Baumann, R. (2001) The turbulent decay of trailing vortex pairs in stably stratified environments, Aerospace Sci. Techn. 5, pp. 95–108 Holzäpfel, F., Gerz, T., Freeh, M. and Dörnbrack, A. (2000) Wake Vortices in a Convective Boundary Layer and Their Influence on Following Aircraft, J. of Aircraft 37, pp. 1001 1007 Holzäpfel, F., Lenze B. and Leuckel W. (1999) Quintuple Hot-Wire Measurements of the Turbulence Structure in Confined Swirling Flows, J. of Fluids Engineering 121, pp. 517–525 Jacquin, L., Fabre, D. and Geffroy, P. (2001) The Properties of a Transport Aircraft Wake in the Extended Near Field: an Experimental Study, AIAA Paper 2001-1038 Jeong, J. and Hussain, F. (1995) On the identification of a vortex, J. Fluid Mech. 285, pp. 69–94 Kaltenbach, H.-J., Gerz, T. and Schumann, U. (1994) Large-eddy simulation of homo geneous turbulence and diffusion in stably stratified shear flow, J. Fluid Mech. 280, pp. 1–40 Kleiser L. and Schumann, U. (1984) Spectral simulations of the laminar-turbulent transi tion process in plane poiseuille flow, Spectral methods for partial differential equations, SIAM, Philadelphia, pp. 141–163 Proctor, F.H. and Switzer, G.F. (2000) Numerical Simulation of Aircraft Trailing Vor tices, 9th Conf. on Aviation, Range and Aerospace Meteorlogy 7.12, pp. 511–516 Risso, F., Corjon, A. and Stoessel, A. (1997) Direct numerical simulations of wake vortices in intense homogeneous turbulence, AIAA J. 35, pp. 1030-1040 Rokhsaz, K., Foster, S.R. and Miller, L.S. (2000) Exploratory Study of Aircraft Wake Vortex Filaments in a Water Tunnel, J. of Aircraft 37, pp. 1022-1027 Schmidt, H. and Schumann, U. (1989) Coherent structure of the convective boundary layer derived from large-eddy simulations, J. Fluid Mech. 200, pp. 511–562 Schumann, U., Hauf, T., Höller, H., Schmidt, H. and Volkert, H. (1987) A mesoscale model for the simulation of turbulence, clouds and flow over mountains: Formulation and validation examples, Beitr. Phys. Atmosph. 60, pp. 413-446 Scotti, A., Meneveau, C. and Lilly, D.K. (1993) Generalized Smagorinsky model for anisotropic grids, Phys. Fluids A 5, pp. 2306-2308 Shaw, R.H. and Schumann, U. (1992) Large-Eddy Simulation of Turbulent Flow above and within a Forrest, Boundary-Layer Meteorology 61, pp. 47–64 Shen, S., Ding, F., Han, J., Lin, Y.-L., Arya S.P. and Proctor, F.H. (1999) Numerical Modeling Studies of Wake Vortices: Real Case Simulations, AIAA Paper 99-0755 Vollmers, H. (2001) Detection of vortices and quantitative evaluation of their main pa rameters from experimental velocity data, submitted to Meas. Sci. Technol.
This page intentionally left blank
VLES OF AIRCRAFT WAKE VORTICES IN A TURBULENT ATMOSPHERE: A STUDY OF DECAY
H. JEANMART AND G. S. WINCKELMANS Center for Systems Engineering and Applied Mechanics,
Mechanical Engineering Department,
Université catholique de Louvain,
1348 Louvain-la-Neuve, Belgium.
Abstract. Very Large-Eddy Simulations (VLES) of aircraft wake vortices in various atmospheric turbulence conditions are carried out. The turbu lence level is characterized by the eddy dissipation rate and ranges from very strong to weak. The turbulence is assumed isotropic. Two LES models are considered. Most simulations were done using explicit gaussian filtering and a mixed model: the tensor-diffusivity model supplemented by a dy namic Smagorinsky term. As a point of comparison, some simulations were done using the classical dynamic Smagorinsky model alone (thus without explicit filtering). Initially, the wake vortices are assumed to follow the fairly “universal” circulation profile shortly after rollup. The vortex global circulation is investigated: its decay exhibits a similar behavior in all cases, leading to the exponential decay model based on the eddy dissipation rate. The present model differs from others by the presence of a time delay before the exponential decay. The constant defining the decay rate is also found to be quite different from that found in the literature. This lack of agreement is partially explained by the differences in the definitions of circulations.
1. Introduction With the increasing number of aircraft travelers per year, wake vortices have become a major concern as they control the spacing between aircraft on landing and takeoff. So far, the spacing is dictated by tables. In some cases, those tables are too conservative, leading to a high security level but to a decrease in airport efficiency. The aim of systems such as the NASA Langley AVOSS [13, 16, 11] or the Transport Canada VFS [1, 15, 19, 20, 22, 8, 23] is 311 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 311-326. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
312
JEANMART & WINCKELMANS
to efficiently simulate, in real-time, the vortices transport and their decay far from the ground, near the ground and in ground effect, under given atmospheric conditions [6, 14, 18, 25]. Those operational systems could also be used in flight simulators. Proper modeling of decay is still an open issue in these systems [17, 7, 18, 9, 8]. The purpose of this study is to investigate, using large-eddy simulations (LES), the decay of a pair of wake vortices under atmospheric turbulent conditions and without ground effect [18, 7, 9] and to derive, by way of a simple model, a link between the decay rate and the atmospheric turbulence intensity. Other atmospheric conditions, such as the cross-wind, the ground proximity, the non-uniform wind shear, or the stratification, are not considered in the present study. The prediction of wake decay mostly relies on the evaluation of the tur bulence intensity, on the circulation distribution within the vortices and on the model used in the LES. Direct numerical simulations are out of reach for such high Reynolds number simulations; only moderate Reynolds number DNS have been done [12]. The strength of turbulence is here represented defined as by the nondimensional turbulence intensity
where is the turbulence energy dissipation (i.e. the eddy dissipation rate, EDR), is the\wake descent velocity due to the mutual interaction between the vortices,
and is the spacing between the vortex centroids. It is approximately equal to where is the wingspan. is the initial total circulation of each vortex.
The LES equations and models are presented in Section 2. The code and the geometry are described in Section 3. The initial conditions for the turbulence background and for the vortices are then discussed in Section 4. After a description of the cases investigated, results on the circulation decay are presented in Section 5. This section also includes the development of a simple decay model and a comparison is made with another model. The conclusions are finally drawn in Section 6. 2. Large-eddy simulation: equations and models
We consider incompressible equations are then:
turbulent flows. The Navier-Stokes
LES OF AIRCRAFT WAKE VORTICES
313
where is the reduced pressure. Two models have been used to carry out the simulations. The first is derived by considering an explicit filtering operation on top of the LES truncation filter, leading to a dynamic mixed model [24]. The second is the classical dynamic Smagorinsky model used for LES without explicit filtering (only truncation). Both models have been shown to provide essentially the same results in decaying isotropic turbulence [24]. Here, the simulations were carried out using the mixed model. Some computations were also done using the Smagorisnky model alone, as a point of comparison. 2.1. LES WITH EXPLICIT FILTERING
We consider first a regular explicit filtering operation:
We here use the gaussian filter,
where is the filter width. Upon applying the explicit filter to the NavierStokes equations, one obtains:
where stress:
is the explicitly filtered velocity field and
is the filtered-scale
This tensor implies no loss of information: it is the commutator between
the operations “product” and “explicit filtering”. On top of the explicit
filter, the LES grid truncation must also be applied, leading to
where
is the total effective stress: it is the LES truncation of
The first term on the rhs can be reconstructed (at the cost of an infinite series or using an iterative procedure). It is here approximated using the first reconstruction term only: the tensor-diffusivity term,
314
JEANMART & WINCKELMANS
where implies a model. The second term on the rhs of Eq. (7) corre sponds to loss of information due to LES truncation. We use, as complete model [24]:
is the strain rate tensor, where is the grid size and is the explicit filter size. C is determined using an adapted dynamic procedure [24]. The first term corresponds to the LES truncation of Eq. (8). The second term is a purely dissipative Smagorin sky term: it models both the LES truncation effects and the incomplete reconstruction effects. Practically, the explicit filter is applied to the initial velocity fields. Simulations are then carried out using these fields. However, all the results are computed from defiltered fields in order to compare them with other LES results or with experimental results. 2.2. LES WITHOUT EXPLICIT FILTERING
In this case, one only considers the LES truncation filter Navier-Stokes equations become:
with used
The truncated
the subgrid-scale stress. A Smagorinsky model is
where C is computed through the dynamic procedure [3, 4, 10, 5, 2, 24]. 3.
Geometry
We use a three-dimensional, fully dealiased (using shifting and spherical truncation), pseudo-spectral code. The size of the domain in the plane is such that there is negligible interaction between the main vortex pair and the images created by the periodicity. Considering as the reference length, the physical domain size is ap and The length in the proximately axial direction, is long enough to study decay due to turbulence. It is too short for the development of the long wavelength Crow instability (not of interest in the present study).
LES OF AIRCRAFT WAKE VORTICES
315
In the numerical domain, lengths are multiples of because the turbu lent background is generated over a cubic domain of size So, distances in the numerical domain are defined in terms of units instead of meters. For equals and the numer the case investigated, the wake of a DC-10, ical unit equals The size of the numerical domain is thus which is consistent with simulations in [12]. Two resolutions have been used for the simulations: one with 96 × 64 × 32 grid points and one with 144 × 96 × 48 grid points. The grid spacing, in and Initially, the terms of physical size, is then respectively number of grid points between the vortex centroids are respectively 20 and 30: these are thus very large-eddy simulations (VLES). 4.
Initial conditions
4.1. TURBULENCE BACKGROUND
The turbulence background is considered as isotropic. To create such con ditions, with the prescribed level of dissipation, we create an initial random velocity field using an analytical energy spectrum and random phases. The spectrum is
It is derived from that of von Kármán and Pao. In the present study, we set the peak of the spectrum to The phases are then converged to more realistic turbulence by conducting a series of short decay runs. Each decay run is followed by a procedure bringing back the energy to the prescribed spectrum, while keeping the phases. After iterations (typically 50), the velocity field reaches an equilibrium, as shown in Figure 1. Different levels of turbulence intensity are obtained by adjusting the value of The most meaningful parameter to quantify the turbulence level is the In fact our physical domain is much energy dissipation rate (EDR), smaller than the large length scale is the rms turbulence where velocity: with the turbulence kinetic energy (TKE). Thus the largest eddies carrying some fraction of the turbulence kinetic energy are not captured by the mesh. Moreover the LES cutoff also removes a sig nificant fraction of the turbulence kinetic energy: this is especially true in VLES, as is the case here. Thus the resolved energy is not comparable with that of the real wake and cannot be used here as parameter quantifying the turbulence background. Notice that, on the time scale of the simulations, the largest atmospheric eddies do not influence the decay: they can safely be neglected in the present study.
316
JEANMART & WINCKELMANS
The preliminary turbulence background simulations are carried out us ing the same LES model used later to study the decay of the wake vortices. Thus both LES with explicit filtering (and a mixed model) and LES with out explicit filtering have been done. The size of the domain is The turbulence background is thus reproduced six times (3 times in 2 times in and once in to fill completely the numerical domain used in the wake vortex simulations. For the simulations with the vortices, the turbulence background decays a little. The turbulence decay time scale is however much larger than the vortex decay time scale. Except at very high level, the turbulence back ground intensity can thus be considered as roughly constant.
4.2. INITIAL VORTEX WAKE The velocity field induced by each vortex shortly after the rollup process follows a fairly universal profile [13, 7, 18, 9, 21, 22, 8, 23]. The mathematical approximation of this profile is here based on different distributions for the core and outer regions of the vortex, with a smooth transition between them [9, 22, 21, 8]. We use, as best fit to the circulation profile,
LES OF AIRCRAFT WAKE VORTICES
317
with the induced azimuthal velocity; and are the “universal” parameters defining the inner and outer cores: and is a parameter to adjust the maximum velocity in the buffer region between produces fine results, we here use the inner and outer regions: The obtained velocity profile is shown in Figure 2. At the beginning of the simulations, a pair of vortices with given global is placed in the turbulence background. During the sim circulation, ulations, we keep track of each vortex centroid position and of the axial vorticity field, 5.
Results
We considered the decay of the pair of vortices created by a DC-10, in different turbulence conditions. The parameters for this aircraft are: and leading to a value of for the descent velocity Results are presented in dimensionless form with the following definitions of dimensionless time and wake vertical displacement:
Levels of turbulence ranging from very strong to weak were studied. The cases are summarized in Table 1. Both LES models have been used to simulate cases l2 and l3 which are in the range of moderate turbulence. Our major interest is the decay of the vortex pair, under given tur bulence conditions: it is an indication of how the vortices coherence is af
318
JEANMART & WINCKELMANS
fected by the turbulence. To evaluate this decay, we consider the circulation around each vortex, at different distances from its centroid:
where is a surface element of the disc of radius around the vortex cen troid. The circulation is evaluated in each axial plane. The results are then averaged between all planes and between the port and starboard vortices. Circulation decay results, for four different cases, are shown in Figure 3. The circulation evolution for four values of the radius is plotted against time. The results are a bit noisy due to the coarse VLES grids. Nevertheless, the decay history is in good agreement with other LESs carried out on a similar problem [18, 7]. These results were obtained using the mixed model for VLES with explicit filtering. At fixed turbulence level, the circulation results for and 0.750 are essentially the same. The circulation results for are lower but not much lower. Since we are interested in global circulation decay, we use the results at In all that follows, means circulation with Clearly, the circulation decay is affected by the turbulence level. At time T = 3, the circulation is only 60% of its initial value for the strongest turbulence level, while it remains close to its initial value (about 97%) in the case of the lowest turbulence level. It is also important to note that the decay only begins after a delay and that this delay increases as the turbulence level goes down. This behavior is clearly seen for the lowest turbulence level, case l4, where the decay only begins at T = 2. This delay corresponds to the time required to develop strong transversal coherent structures. The appearance of those structures significantly enhances the decay of the vortex pair. Figure 4 illustrates those coherent structures at a later time (T = 5).
LES OF AIRCRAFT WAKE VORTICES
319
The vortex pair descent velocity is also important for operational sys tems. Figure 5 shows, for two turbulence levels, the computed trajectory and the estimated trajectory, using:
The agreement between them is very good. The results are also compared as expected, with the ideal case (no decay) with descent velocity equal to the decay clearly has an effect on vortex transport. A comparison between the decay results obtained with both models is made in Figure 6: the results are not in complete agreement but the behaviors are similar: a delay followed by exponential decay in both cases. Besides the intrinsic difference in the models, the differences in the results can also be explained by the differences in the initial condition: while the mean turbulence dissipation rate is the same in both cases, the fields are
320
JEANMART & WINCKELMANS
not the same as they are converged from different random fields, and using different LES approaches. In order to characterize the decay rate as a function of turbulence level, we propose a simple model based on the observed behavior:
LES OF AIRCRAFT WAKE VORTICES
321
with T* the delay time and the decay time constant. This formulation is similar to the eddy dissipation rate (EDR) model [17]:
where is the time to demise function [17] and is shown in Figure 7. At this time, Crow instability or core bursting has occured. is the constant of the decay model. To assess our model, we have checked the observed decay against the exponential law. The comparison is shown in Figure 8, for four levels of turbulence. The agreement is good for all levels. Comparing the exponential decay part with the EDR decay model, we can deduce a best for each run. The values are reported in Table 2. The variance of is quite important with a mean value of about 0.15. The constant obtained for the highest and the lowest turbulence levels are somewhat suspicious. For the highest turbulence level, the resolution is quite poor; for the lowest turbulence level, it is due to the too dissipative behavior of the model in such still atmosphere. The agreement between the two LES models is also not perfect, as shown in Figure 9 and in Table 2. Still they provide a of the same magnitude. The differences in the models are more important for T * . Further work is required to better quantify the relationship
322
JEANMART & WINCKELMANS
LES OF AIRCRAFT WAKE VORTICES
323
Comparisons with experimental data were made by Sparkaya [17]. He proposed a value of 0.55 for The apparent contradiction with our value conceals, in fact, a good agreement. It is easily explained by the way the experimental circulation is reported. In experiments, the circulation is de termined using
is indeed much lower than the actual total circulation. For the uni versal profile at time T = 0, is only 85% of Figure 10 shows the decays of and It is clear, from this plot, that our results should nor lead to but to a much lower value. If we treat our decay results as was done with experimental data, we obtain proposed by Sarpakaya. this is indeed very close to
324
6.
JEANMART & WINCKELMANS
Conclusions
Large-eddy simulation appears to be a good predicting tool for investi gating the decay of aircraft wake vortices. Mixed modeling with explicit filtering and classical modeling appear to perform equally well. The dif ferences between the results for both models are largely explained by the differences in the initial conditions. A statistical treatment of the simula tion results (meaning more runs) could be implemented in order to reduce the dispersion observed between our results. A new decay model, comparable to the eddy dissipation rate model [17], has been proposed. The main difference lies in the presence of a delay before the exponential decay. This delay is also turbulence dependent. More work is required in order to obtain the effect of the turbulence intensity on the delay to global circulation decay. In our model, the constant of the exponential decay is found as being approximately 0.15. This value, apparently in conflict with the one proposed by Sarpkaya, implies, in fact, the same decay rate: the only difference is in the definition of the “circulation”. Our definition is more coherent with what is needed in operational models such as AVOSS and VFS. Indeed, it which controls vortex descent, not is the global circulation,
LES OF AIRCRAFT WAKE VORTICES
325
References 1. Belotserkovsky, S. M. et al. (1999) Computer Vortex Forecast System, Final Report TP 13373E, Science-Engineering Center SABIGO Ltd., Moscow. 2. Dantinne, G., Jeanmart, H., Winckelmans, G. S., Legat, V. and Carati, D., (1998) Hyperviscosity and vorticity-based models for subgrid scale modeling, Applied Sci entific Research 59, 409–420. 3. Germano, M., Piomelli, U., Moin, P. and Cabot, W. (1991) A dynamic subgrid-scale eddy-viscosity model, Phys. Fluids A 3(7), 1760–65. 4. Ghosal, S., Lund, T. S. and Moin, P. (1992) A local dynamic model for large-eddy simulation, Annu. Res. Briefs, Center for Turbulence Research (Stanford University and NASA Ames), 3–25. 5. Ghosal, S., Lund, T. S., Moin, P. and Akselvoll, K. (1995) A dynamic localization model for large-eddy simulation of turbulent flows, J. Fluid Mech. 286, 229–255. 6. Greene, G. C. (1986) An approximate model of vortex decay in the atmosphere, J. of Aircraft 23, 566–573. 7. Han, J., Lin, Y.-L., Pal Arya, S. and Proctor, F. H. (1999) Large eddy simulation of aircraft wake vortices in a homogeneous atmospheric turbulence: vortex decay and descent, 37th Aerospace Sciences Meeting & Exhibit, Jan. 11-14, 1999, Reno NV, paper AIAA 99-0756. 8. Jackson, W., Yaras, M., Harvey, J., Winckelmans, G. S., Fournier, G. and Belot serkovsky, A. (2001) Wake Vortex Prediction - An Overview, Phase 6 and Project Fi nal Report prepared for Transportation Development Centre and Transport Canada, TP 13629E, March 2001. www.tc.gc.ca/tdc/projects/air/9051.htm 9. Jeanmart, H. and Winckelmans, G. S. (2000) Large-eddy simulations of aircraft wake vortices in a turbulent atmosphere, Proc. 5th National Congress on Theoretical and Applied Mechanics, Louvain-la-Neuve, Belgium, May 23–24, 2000. 10. Moin, P., Carati, D., Lund, T., Ghosal, S. and Akselvoll, K. (1994) Developments and applications of dynamic models for large eddy simulation of complex flows, 74th Fluid Dynamics Symposium on Application of Direct and Large Eddy Simulation to Transition and Turbulence, Chania, Crete, Greece, AGARD-CP-551, 1-1–9. 11. O’Connor, N. (2001) Aircraft Vortex Spacing System (AVOSS), Proc. 5th WakeNet Workshop on “Wake Turbulence and Airport Environnement”, DFS Academy, Langen, Germany, April 2–3, 2001. 12. Orlandi, P., Carnevale, G. F., Lele, S. K. and Shariff, K. (1998) DNS study of stability of trailing vortices, Proc. Summer Program 1998, Center for Turbulence Research, Stanford University & NASA Ames, 187–208. 13. Proctor, F. H. (1998) The NASA-Langley wake vortex modelling effort in support of an operational aircraft spacing system, 36th Aerospace Sciences Meeting & Exhibit, Jan. 12-18, 1998, Reno NV, paper AIAA 98-0589. 14. Proctor, H. H. and Han, J. (1999) Numerical study of wake vortex interaction with the ground using the Terminal Area Simulation System, 37th Aerospace Sciences Meeting & Exhibit, Jan. 11-14, 1999, Reno NV, paper AIAA 99-0754. 15. Rennic, S., Posluns, H., Jackson, W., Winckelmans, G. S. and Belotserkovsky, S. (1997) Transport Canada wake vortex activity and evaluation of the Vortex Forecast System, 1st Wake Vortex Dynamic Spacing Workshop, NASA Langley Research Center, Hampton, Virginia, May 13–15, 1997. 16. Robins, R. E. and Delisi, D. P. (1999) Further development of a wake vortex predictor algorithm and comparisons to data, 37th Aerospace Sciences Meeting & Exhibit, Jan. 11-14, 1999, Reno NV, paper AIAA 99-0757. 17. Sarpkaya, T. (1999) A new model for vortex decay in the atmosphere, 37th Aerospace Sciences Meeting & Exhibit, Jan. 11-14, 1999, Reno NV, paper AIAA 99-0761. 18. Shen, S., Ding, F., Han, J., Lin, Y.-L., Pal Arya, S. and Proctor, F. H. (1999)
326
19. 20. 21. 22.
23.
24. 25.
JEANMART & WINCKELMANS Numerical modeling studies of wake vortices: real case simulations, 37th Aerospace Sciences Meeting & Exhibit, Jan. 11-14, 1999, Reno NV, paper AIAA 99-0755. Winckelmans, G. S. and Ploumhans P. (1999) Prediction of aircraft wake vortices during takeoff and landing - Phase 4, Final Report TP 13374E, Mechanical Engi neering Department, Université catholique de Louvain, Louvain-la-Neuve, Belgium. Winckelmans, G. S. and Ploumhans P. (1999) Prediction of aircraft wake vortices during takeoff and landing - Phase 5, Final Report, Mechanical Engineering De partment, Université catholique de Louvain, Louvain-la-Neuve, Belgium. Winckelmans, G. S. and Jeanmart, H. (2000) Investigation of near wakes shortly after rollup, interim phase 6 report, Mechanical Engineering Department, Université catholique de Louvain, Louvain-la-Neuve, Belgium, March 23, 2000. Winckelmans, G. S., Thirifay, F. and Ploumhans, P. (2000) Effect of non-uniform wind shear onto vortex wakes: parametric models for operational systems and com parison with CFD studies, Proc. 4rth WakeNet Workshop on “Wake Vortex En counter” , National Aerospace Laboratory NLR, Amsterdam, The Netherlands, Oct. 16–17, 2000. Winckelmans, G. S. and Jeanmart, H. (2001) Demonstration of a wake vortex pre diction system: the Vortex Forecast System (VFS), Proc. 5th WakeNet Workshop on “Wake Turbulence and the Airport Environment”, DFS Academy, Langen, Ger many, April 2–3, 2001. Winckelmans, G. S. Wray, A. A., Vasilyev, O. V. and Jeanmart, H. (2001) Explicitfiltering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term, to appear in Phys. Fluids. Yaras, M. I. (1999) Effects of atmospheric conditions and ground proximity on the dynamics of aircraft wakes vortices: a study of the 1994-95 Memphis field mea surements, Final Report TP 13372E, Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, March 5, 1999.
7. Combustion and magnetohydrodynamics
This page intentionally left blank
SUBGRID COMBUSTION MODELING FOR LES OF SINGLE AND TWO-PHASE REACTING FLOWS
S. MENON School of Aerospace Engineering
Georgia Institute of Technology
Atlanta, GA 30332
U.S.A
Abstract. Subgrid modeling approaches for large-eddy simulation of single and two-phase reacting flows are discussed here with the focus on simulating high Reynolds number flows in practical devices. Therefore, issues related to computational expediency and accuracy of the subgrid model on rela tively coarse grids are also discussed. It is suggested that models must not only capture the energy transfer from large to small scales, they must also account for small-scale mixing and molecular diffusion processes in order to predict high Re reacting flows. A subgrid modeling approach that can be used in multi-phase flows is described and sample results are shown to demonstrate its capabilities.
1. Introduction
Recent more stringent emission regulations have pushed for the develop ment of more fuel-efficient and low-emission combustion systems for both ground-based and flight gas turbine applications. Experimental estimate of pollutant formation is limited due to the inability to probe non-intrusively, the highly three-dimensional flame zone. As a result, detailed understand ing of where (and why) the pollutants (primarily, NOx, CO and unburned hydrocarbons, including soot) are forming is difficult, if not impossible to determine using experimental data. This has been a major stumbling block in the design of the next generation, fuel-efficient gas turbine engines. Nu merical prediction could help provide the missing information if accurate codes are available. However, most codes currently in use employ steady state methods and are unable to capture the unsteady dynamics that con 329 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 329-351. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
330
S. MENON
trol the flame zone (and hence, control the pollutant formation process). In addition to the inaccuracy in the prediction of NOx formation, CO emission is also not predicted very well under dynamic conditions. For example, re cent studies (Bhargava et al., 2000) have shown that as the equivalence ratio is reduced, CO emission first decreases and then suddenly increases rapidly. This sudden increase occurs during lean combustion when perturbation to the flame can cause instability and also lead to incomplete combustion. The condition at which this sudden increase occurs is very important to predict and control since it impacts the performance and efficiency of the gas turbine engine. However, so far, no numerical simulation tool has been able to predict this behavior. Accurate prediction of unsteady combustion requires a comprehensive model that can predict flame structure and propagation characteristics, pol lutant formation and transport, and ignition/extinction phenomena over a wide range of flow conditions in high Reynolds (Re) number, threedimensional (3D) flows as in a gas turbine combustor. Direct Numerical Simulations (DNS) are not practical since the resolution and computational resource requirements far exceed the present and even future capability. A simulation approach that has become popular in recent years is large-eddy simulation (LES). In LES, all scales larger than the grid resolution are numerically simulated using a space- and time-accurate scheme while the effect of scales below the grid resolution is modelled using a subgrid model. Although many LES studies have been reported, a validated LES approach for practical systems is yet to be demonstrated although the approaches discussed in this paper have shown serious potential for such studies. Subgrid modelling for reacting flows must simultaneously address clo sure issues for both momentum and scalar transport. For momentum clo sure, the effect of the unresolved small scales (assumed to be mostly isotropic) on the resolved motion can be modelled using an eddy viscosity based subgrid model since (almost) all the energy containing scales are resolved and the unresolved small-scales primarily provide dissipation for the energy transferred from the large scales. However, for combustion to occur, fuel and oxidizer species must mix at the molecular level. This mixing process is dominated by turbulent mixing and molecular diffusion in the small-scales. Thus, extending the eddy viscosity concept to scalar fields by using an eddy diffusivity model can be erroneous except under very specialized cir cumstances. Even in premixed combustion, the small-scale flame wrinkling must be captured accurately in order to predict the turbulent flame prop agation. In addition, molecular diffusion (including differential diffusion) and finite-rate kinetics also occur in the small-scales and ad hoc models for these processes can be significantly in error. This paper discusses modelling approaches for LES of reacting flow in
SUBGRID COMBUSTION MODELING
331
practical systems. Thus, only subgrid closures that can be (and have been) applied to high-Re flows are discussed. Section 2 describes the LES model for both single and two-phase flows and then discusses typical dynamic subgrid models for momentum and energy transport closure. In Section 3, subgrid scalar closure models for high-Re, reacting LES are described. Sec tion 4 addresses numerical and computational issues and Section 5 discusses some application of LES to single and two-phase flows. Finally, Section 6 summarizes some of the unresolved issues and open questions for future development. 2. Simulation Model for Momentum and Energy Transport
For two-phase flows, the most popular approach is the Eulerian-Lagrangian approach (Faeth, 1983) in which the gas phase is simulated using a con ventional finite-volume Eulerian model while the liquid phase is simulated using a Langrangian tracking approach. Both phases are fully coupled due to the presence of mass, momentum and energy sources in both sets of equations. Further details are given elsewhere (Faeth, 1983) and therefore, not included here. The full compressible form of the conservation equations are discussed here, since in most practical devices combustion occurs within confined domains, and significant acoustic-vortex-entropy interactions oc cur. LES studies using the incompressible (i.e., zero-Mach number) reacting flows is also possible and have been described elsewhere (Chakravarthy and Menon, 2000a; Chakravarthy and Menon, 2000b). 2.1. LES GAS-PHASE EQUATIONS
For compressible flows, the flow variables are decomposed into resolved where the (supergrid) and unresolved (subgrid) components: (~) denotes resolved and ('') denotes subgrid quantities. The Favre filtered where the over bar represents spatial variable is then defined as: filtering. Using a low-pass filter (of characteristic grid width the LES equations are
for m =1,N species.
332
S. MENON
In the above equations, is the mass density, is the pressure, E is the total energy per unit mass, is the velocity vector, and is the Kronecker delta. The pressure is determined from the equation of state for a perfect gas where and are respectively, the species gas constant and mass fraction. The total energy per unit volume is determined from where is the internal energy per unit mass given by Here, is the species enthalpy. The caloric equation of state is given by, where is the standard heat of formation at temperature and is the species specific heat at constant pressure. The resolved viscous stress and the resolved heat conduction are approximated in terms of the filtered velocity. The unclosed subgrid terms representing respectively, the subgrid stress tensor, subgrid heat flux, un resolved viscous work and species-temperature correlation are:
The scalar subgrid terms, and also require closure but will be discussed in section 3. Also, the volume averaged inter-phase source terms that appear on the right hand side of the LES equations are given elsewhere (Faeth, 1983; Pannala and Menon, 1998) and therefore, not shown here, for brevity. 2.2. LES LIQUID-PHASE EQUATIONS
At present, only dilute spray is simulated and the effects of droplet-droplet interactions, breakup and coalescence have been ignored (but can be in cluded if reliable models are available). The spray field is computed by integrating the Lagrangian equations of motion, heat and mass transfer for the droplets. In this method, instead of tracking individual droplets, parcels of droplets, called groups, are explicitly tracked (Faeth, 1983; Pannala and Menon, 1998). Each group represents droplets of same size, location, tem perature and velocities. The gas phase properties at the droplet locations are estimated using a eight point volume weighted averaging and similar approach is used to redistribute the spray source terms from the particle position to the gas phase LES equations. Drag effects due to the droplets on the gas phase is explicitly included using semi-empirical models (Faeth, 1983). To include stochastic dispersion of the droplets, a random velocity component is added to the gas phase fluctuating velocity (magnitude is de termined using the subgrid kinetic energy), used for the spray calculations.
SUBGRID COMBUSTION MODELING
333
Usually due to resource constraints, only a limited range of droplet groups are computed. Droplets below a certain ad-hoc cut-off size are as sumed to evaporate instantaneously and get fully mixed with the gas phase. As described later, this assumption can lead to erroneous prediction if the cutoff size is relatively large. 2.3. SUBGRID CLOSURE FOR MOMENTUM TRANSPORT
All subgrid models for momentum closure currently in use employ an eddy viscosity approach. To determine this eddy viscosity a length and a velocity scale are needed. All current models assume that the local grid size is the characteristic length scale. Two models for the velocity scale are quite popular: a model suggested by Smagorinsky in which the velocity scale is obtained using and the local resolved strain-rate and a model suggested by Schumann (Schumann, 1975) in which the velocity scale is determined by the solution of a transport model for the subgrid kinetic Smagorinsky’s approach results in an algebraic model for the energy eddy viscosity which is strictly valid only in the limit that production and dissipation of energy in the subgrid scales are in equilibrium. This require ment implies that the grid cutoff must occur in the dissipation scale. This is computationally impractical if high-Re flows have to be simulated. On the other hand, Schumann’s model allows for non-equilibrium in the subgrid scales and therefore, is more applicable to high-Re LES when the grid scale cutoff is not in the dissipation range (but is still in the inertial range). In addition, the model also provides an accurate estimate for the subgrid turbulence intensity that is needed for both droplet dispersion and the subgrid mixing and combustion model described in the next section. For these reasons, model is considered more applicable to reacting single- and two-phase flows. Therefore, only this model is discussed here. A compressible transport model for can be derived (Menon and Kim, 1996; Kim and Menon, 2000) and is given as:
where is the subgrid kinetic energy and is the tur bulent Prandtl number. The terms on the right side of equation (3) rep resent, respectively, the production, the dissipation, and the transport of the subgrid kinetic energy. The production term is where the subgrid shear stresses is given as Also, is the subgrid eddy viscosity and the dissipation term is The two coefficients appear ing in the above equations, and can be determined using a local
334
S. MENON
ized dynamic approach (denoted hereafter as LDK) that is described else where (Menon and Kim, 1996; Kim and Menon, 1999; Kim et al., 1999; Kim and Menon, 2000)). As in other dynamic models (Germano et al., 1991), the LDK model is also based on the assumption of scale similarity in the inertial subrange and test filtering is employed. In the LDK model, the coefficient is ob where and tained from a relation are both evaluated at the test fil ter level. The derivation of this model (and a similar model for is based on similarity between and that has been observed experimen tally (Liu et al., 1994), and is described elsewhere (Kim and Menon, 1999; Kim et al., 1999). In Germano’s dynamic closure a mathematical identity in terms of model representation at the two filter levels is used which can result in the denominator to become ill-conditioned (i.e., to tend to zero locally) (Cabot and Moin, 1993). As a result, some modifications are typi cally needed (e.g., spatial averaging in a homogeneous direction (Lund et al., 1993; Moin et al., 1991)). The LDK model avoids this problem since the denominator contains a well defined (and non-zero) quantity. Furthermore, the prolonged presence of negative model coefficient (described in (Lund et al., 1993)) is also avoided in the LDK model since the present model (which never becomes negative). Finally, the dynamically is based on determined does not vanish in the limit of high Re. The subgrid energy flux is approximated as:
where is the filtered total enthalpy, The turbulent Prandtl number can be obtained using a dynamic approach as described earlier (Nelson and Menon, 1998). The subgrid term can be approximated as but has been found to be small in flows similar to the ones stud ied here (Nelson and Menon, 1998), and so it is neglected here. The speciesin the equation of state is also neglected in all temperature correlation conventional LES reported here. However, the subgrid approach described in section 3.3 can obtain this correlation directly. 3. Simulation Model for Scalar Transport Scalar fields are affected by three physical processes: advection (due to the velocity field), molecular diffusion and chemical reactions. While the re solved velocity field causes large scale convection of the scalar fields, the subgrid velocity fluctuations lead to fine-scale mixing. Furthermore, molec ular diffusion and reaction processes are also small-scale phenomena and hence, are affected strongly by the fine-scale mixing. Scalar subgrid mod els must take into account all these small-scale processes. Such a model is
SUBGRID COMBUSTION MODELING
335
described in Section 3.3. 3.1. LES FLAMELET AND FLAME WRINKLING MODELS
The scalar LES equations described can be used for both non-premixed and premixed combustion. However, for premixed combustion a computa tionally very efficient model can be used in the flamelet regime which is often encountered in practical combustion devices. A model equation that describes the propagation of a thin flame by convective transport and nor mal burning (self propagation by Huygens’ principle), is the G-equation (Kerstein et al., 1988). Upon filtering the G-equation the LES version can be derived:
The unclosed subgrid terms representing, respectively, the filtered source term and the unresolved transport term are and and must be modeled. Here, is the reference reactant density and is the undisturbed laminar flame speed. The modification to to account for the effect of flame stretch (which includes contributions from flame curvature and tangential strain within the flame surface) in the thinreaction-zones regime has been recently developed (Kim and Menon, 2000). Another approach similar to the flamelet approach (Fureby, 2000) em ploys a flame wrinkling model in which transport equations for a reaction coordinate (which is similar to the G variable), a modelled flame-wrinkling density variable and the burning rate are derived and solved along with the LES transport equations. Details are avoided here for brevity. Although both flamelet and flame wrinkling models are very cost effec tive and are capable of capturing flame-turbulence interactions (including combustion instability) accurately, these methods achieve computational efficiency primarily because finite-rate kinetics are not simulated. Thus, these methods cannot be used to predict pollutant emission and chemistry (including extinction) effects. For LES of flows with finite-rate kinetics, the method described in Section 3.3 may provide an accurate approach. 3.2. SUBGRID CLOSURE FOR SCALAR TRANSPORT
When the LES scalar transport equations are solved on the LES grid, the finite-rate reaction rate has to be closed. For this many approaches are currently being investigated. For example, probability density function (pdf) methods have been proposed (Gao and O’Brien, 1993) but subgrid pdf simulation can be very expensive and yet to be demonstrated. An alternate approach assumes the pdf in the mixture fraction space and has become
336
S. MENON
more popular (Cook and Riley, 1998). However, this method is limited since it requires specifying the pdf shape a priori within the mixture fraction approximation. This can be problematic when multi-species (and complex fuels) combustion with finite-rate kinetics (including radical and pollutant kinetics) is to be simulated. The method described in the next section avoids this problem. For premixed combustion in particular, the mixture fraction approach cannot be used. In the flamelet regime, a subgrid flame speed model that and has proven quite successful (Kim et al., 1999; is a function of Kim and Menon, 2000) to close the source term where is the turbulent flame speed. This speed is not known explicitly and, therefore, must be modelled. Two different turbulent flame speed models (Yakhot, 1988; Pocheau, 1994) have been evaluated in recent studies (Kim et al., 1999; Kim and Menon, 2000) and found to perform reasonably well. For both these models, the subgrid turbulence intensity is obtained directly from Note that this information is not available when the algebraic eddy viscosity model is employed. The flame wrinkling model (Fureby, 2000) also employs a similar ap proach except that modeled equations for the flame surface density and the burning rate are solved. 3.3. SUBGRID COMBUSTION MODEL
In this approach, a subgrid simulation of the scalar field is carried out within every LES cell instead of modelling the filtered scalar equation on the LES grid. The scalar fields within the subgrid field evolve due to smallscale processes of molecular diffusion, turbulent stirring and volumetric expansion due to heat release. The subgrid simulation model is a variant of the linear-eddy model (LEM) developed earlier (Kerstein, 1989). Details of this approach (called hereafter LES-LEM) are given elsewhere (Menon et al., 1993; Menon and Calhoon, 1996; Chakravarthy and Menon, 2000a). This method can be explained mathematically by considering the following generic form of an un-filtered scalar (temperature,species concen trations) evolution equation:
Here, and are the resolved and unresolved velocity fields, respectively, is the diffusion coefficient, and are representative source terms for reaction and phase change, respectively. In the LES-LEM ap proach, a two-scale procedure is used:
SUBGRID COMBUSTION MODELING
337
Equation (6) represents the large scale advection of the scalar field by the resolved velocity field and is modelled by transferring fluid volumes between the control volumes on the 3D grid. Also, in Eq. (6) represents the component of the subgrid velocity field (on the control volume faces) that causes volume (scalar) transport between LES cells. In case of G equation, the first step in the integration scheme (Eq. 6) remains the same (G replaces C) but the reaction and diffusion terms in (Chakravarthy and Eq. (7) are replaced by the propagation term, Menon, 2000a). The subgrid method for solving Eq. (7) involves conducting LEM sim ulations in each of the 3D LES cells using a one-dimensional (1D) approx imation (which is critical to reduce the overall computational cost).The reaction-diffusion equations on the 1D domain for the chemical species can be written as
Here, the term represents a source only in the fuel species equation due to phase change. For premixed flame propagation the 1D model is: Here, s denotes the subgrid 1D domain. For multi-species reacting flows with heat release, a 1D temperature equation is also solved along with the species equation. The reaction-diffusion scalar equations or the 1D G-equation is solved on the 1D domain using a standard finite-difference approach. The orien tation of the 1D domain is in the direction of the scalar gradient (Kerstein, 1991). The number of subgrid LEM cells is determined to ensure that even the smallest subgrid scale (e.g., the Kolmogorov scale is fully resolved (Menon et al., 1993). Thus, the subgrid resolution is fine enough to allow a subgrid “DNS” (albeit in 1D) of the reaction-diffusion equations. This is required. For implies that no explicit closure of the production term the G-equation approach, subgrid flame propagation occurs at The explicit resolution of the reaction-rate term in LES-LEM is similar to the capability in scalar-pdf methods (Pope, 1985). However, the subgrid LEM model also accounts for molecular diffusion (including differ ential diffusion) at the small scales (this feature is absent in pdf models).
338
S. MENON
Thus, LEM has the ability to capture Lewis and Schmidt number dependen cies without requiring any ad hoc modifications to the model (Chakravarthy and Menon, 2001). Since, the subgrid velocity field is assumed to be isotropic its effect on (and represents the 1D the scalar fields is symbolically denoted by analog of the term in Eq. 7) and is implemented independent of the reaction-diffusion process by a stochastic Monte-Carlo type simulation within the 1D domain. In-spite of the 1D subgrid representation, LES-LEM captures realis tic 3D turbulence effects on the scalar fields due to the manner in which turbulent stirring (denoted by is implemented. This is accom plished using a series of stochastic rearrangement events called “triplet maps” (Kerstein, 1989), each of which represents the effect of a single eddy on the scalar field. Thus, stirring interrupts scalar (i.e., reaction-diffusion or laminar flame propagation) processes at instants determined by the stir ring time interval, and is carried out by choosing an eddy size l from a eddy size distribution, f(l), randomly locating this eddy within the 1D domain (using an uniform distribution), and then applying the triplet mapping process. Both and f(l) are chosen using inertial range 3D Kolmogorov scaling laws and thus, the effect of turbulent stirring on the scalar field mimics physically realistic mixing in 3D flows. It is this unique feature of the LEM model that allows it to achieve accurate prediction of scalar-turbulence interaction effects within the 1D model. The advection of the subgrid scalar fields by the LES-resolved veloc ity field is modeled by solving equation (6) using a Lagrangian algorithm (Menon et al., 1993; Menon and Calhoon, 1996; Chakravarthy and Menon, 2000a) that is similar to the volume of fluid (VOF) method. This method involves transfer of fluid (i.e. LEM cells) between LES control volumes to account for the volume flux across the cell faces. Details are given elsewhere. To couple the subgrid processes to the LES large-scale motion, the LEM scalar fields are ensemble-averaged and used in the LES equations to pro vide the Favre-filtered scalar fields. Heat release effect is included by using the heat of formation in the definition of the mixture enthalpy. Since all the physical mechanisms that effect scalar evolution are mod elled either through the advection scheme or simulated in the subgrid do main, the integration of the scalar equations on the LES grid is not required. This is in contrast to the conventional LES approach where the filtered scalar conservation equations are modelled using a numerical discretization scheme similar to that used for the momentum equations.
SUBGRID COMBUSTION MODELING
339
4. Numerical and Implementation Issues
The predictive ability of the LES depends considerably on the numeri cal algorithm used for the simulation. There are two primary constraints that must be addressed: the spatio-temporal accuracy (and in addition, the numerical dissipation inherent in the scheme) and the computational per formance of the numerical model used for LES. Both these requirements cannot be compromised. Based on many past studies fourth-order accuracy in space and secondorder in time is preferred for any LES solver. Such a spatial accuracy needs to be maintained even on stretched grids and this requirement imposes some severe constraints on grid stretching. In flows with complex geometry, unstructured or block-structured grids may be needed and achieving high spatial accuracy may be nearly impossible. Therefore, grid resolution and distribution has to be carefully investigated in order to ensure that regions of high shear are well resolved. The lack of proper resolution or low-order schemes imply that the numerical dissipation inherent in the solver also needs to be assessed carefully. For example, the numerical dissipation must be smaller than the dissipation needed to dissipate the energy transferred from the large (resolved) scales to the subgrid scales (Nelson and Menon, 1998). It is worth noting that many commerical “LES” solvers may fail this test. The computational cost of reacting flow LES can be considerable and therefore, parallel implementation is essential. For example, a typical LES (with the G-equation) using 500,000 points requires 2 GB of memory and around 7000 single processor hours on the Cray T3E to obtain sufficient flow-through times for statistical analysis. For the LES-LEM approach us ing the same grid and around 100 LEM cells per LES cell the cost increases by a factor of four. Although this computational time appears prohibitive it is worth noting that the rapid increase in CPU speeds in recent years suggests that future simulations could be carried out effectively on PC-based clusters at a frac tion of the current cost. However, it should be kept in mind that in order to achieve rapid turnaround the grid resolution must be relatively “coarse” and this in turn, requires that the subgrid closure be more physics based and capable of incorporating the effect of missing scales accurately. 5. Results and Discussion
In this section, some representative results of LES of high-Re flows are discussed.
340
S. MENON
5.1. FLAMELET SUBGRID MODELING
To demonstrate the ability of LES-LEM in premixed cases, stagnation point flames are simulated. For these simulations a grid of 89x129x129 is used for the conventional case and a grid of 69x89x89 (with 100 LEM cells in each LES cell) is used for the LES-LEM case. More details of these cases are reported elsewhere (Chakravarthy and Menon, 2000a; Chakravarthy and Menon, 2000b). The front tracking ability of the LEM-LES method can be observed in Figure 1. As shown, LEM-LES captures the flame as a thin wrinkled front whereas the flame is captured over a broader zone by the conventional LES. These differences in the resolved flame structure directly impact the prediction of the resolved velocity field. For example, the mean and rms axial velocity profiles predicted by the LEM-LES and the conventional LES are compared to experimental data (Cho et al., 1986; Cho et al., 1988; Cheng and Shepherd, 1989) in Figure 2. The mean profiles are predicted fairly well by both models but the prediction of turbulence is much better in case of LEM-LES. Across the flame, the density decreases significantly and this causes the flow to accelerate tremendously across the flame. This results in very high (flame normal) velocities on the product side (compared to the reactants side). The unsteady oscillations of the flame at any given point in the flame brush thus, causes very high intermittency which in turn leads to an increase in This physics is captured quite accurately by the LEM-LES method. In the conventional LES, the flame structure has a finite thickness which is determined by the numerics (grid, scheme, etc.) As a consequence, the flow
SUBGRID COMBUSTION MODELING
341
acceleration (due to the density drop) is much more gradual than in the case of LES-LEM. This reduction of the flow gradient reduces flow intermittency at and hence, the conventional approach does not produce the peak in the flame brush as in the experiments and the LES-LEM simulation. This problem is also typical of LES that use finite-rate chemistry and transport equations for chemical species on the LES grid. Note that, the resolution of any real flame on a 3D grid is impossible using current computational resources. Thus, the flamelet-type burning may never be captured using conventional methods. On the other hand, the LEM-LES method appears to achieve this goal due to the combination of the features of the subgrid
342
S. MENON
LEM and the front tracking scheme. 5.2. LES OF FULL-SCALE COMBUSTOR FLOWS
Premixed combustion in a jet afterburner and in a full-scale gas turbine en gine are discussed here. The jet afterburner simulations are carried out using the flame wrinkling model (Fureby, 2000), while the gas turbine combustor flows are simulated using the dynamic flamelet model (Kim et al., 1999; Kim and Menon, 2000). Figure 3 shows the axial mean and rms profiles in the afterburner for various cases (more details are given elsewhere) using a grid of 200x80x20. The Reynolds number ranged from 44,000-88,000 (based on the afterburner height) for these simulations. It can be seen that LES predictions are in reasonable agreement with the experimental data. Further analysis of the results show that the LES captures most of the characteristics of the flame dynamics in this configuration. LES of lean premixed combustion in a General Electric LM6000 com bustor is simulated using the dynamic flamelet model (Kim et al., 1999; Kim and Menon, 2000). A highly swirling lean premixed mixture (methane air) enters the square combustor (the shape was primarily dictated by ex perimental measurement requirements) from a circular pipe at a preheated condition of 600 K. The Re (based on inlet diameter) is 330000, Swirl Number is 0.56, Karlovitz Number is 42 and Damkohler Number is 8. A relatively coarse grid of 97x65x81 is employed with the LDK closure. Inflow conditions were prescribed using normalized profiles (provided by GE) for the mean velocity field and an inflow turbulence using a prescribed energy spectrum. Due to high swirl the local turbulence is quite high and this test con dition lies in the thin reaction zone (Kim and Menon, 2000). Figures (4a) and (4b) show respectively, the mean axial velocity variation along the cen terline and a typical radial profile of the rms axial velocity. The agreement is reasonable considering the many uncertainties involved. The LES model can also be used to investigate physics of combustion dynamics since time-dependent evolution is captured in the simulation. For example, high swirl is used to stabilize premixed flame in gas turbine com bustors. Simulations using a grid of 145x65x81 and with different values of inlet Swirl number of 0.56 and 1.12 were conducted in a configuration similar to the GE LM6000 (except that now the inlet is also modelled and both inlet and combustor are circular) and compared. Note that the actual Swirl numbers at the dump plane are much lower (0.38 and 0.8, respec tively) due to decay in the inlet. Analysis of the results show that increase in swirl not only stabilizes the flame (i.e., it reduced the flame oscillation)
SUBGRID COMBUSTION MODELING
343
it also reduces the pressure oscillation in the combustor. Figure 5 shows the centerline normalized axial velocity decay. In the high swirl case, the transverse motion of the shear layer rapidly reduces the axial component of the velocity and a small region of recirculation appears near the centerline. In contrast, there is still significant axial motion in the weak swirl case and this results in coherent vortex shedding which directly interacts with the flame. Figure 6 shows a typical vortex-flame structure in the weak swirl case.
344
S. MENON
Analysis of the time-evolution of these structures (http://www.ccl.gatech. edu) show that swirling vortex rings undergo both axial and azimuthal (helical) instability that leads to their eventual breakdown. The pulsation of the swirling flame is intricately linked to the shedding of these vortices. Increase in swirl increases the radial expansion of the flow downstream of the dump plane and this leads to a quicker breakdown of the shed vortices. This in turn reduces the vortex-flame interaction leading to a more stable flame structure and much lower amplitude of pressure oscillation.
SUBGRID COMBUSTION MODELING
345
5.3. LES OF TWO-PHASE FLOWS The subgrid two-phase LES model developed earlier (Pannala and Menon, 1998; Menon and Pannala, 1997) has been used to simulate spray mixing and combustion in shear layers. In conventional two-phase LES, droplets are
346
S. MENON
tracked only up to a pre-specified cutoff size and when they become smaller than this cutoff they are assumed to instantaneously vaporize and mix. This assumption is used primarily because as the particle size decreases, the Lagrangian time-step becomes very small and the computational cost becomes exorbitant. In the LES-LEM approach, droplets below the cutoff size continue to vaporize at a rate determined by the droplet size distribu tion within the subgrid. Thus, the mixing process continues unabated till all drops (accounted in the subgrid LEM as a void fraction in the two-phase approach) have been vaporized. The advantage of this subgrid two-phase approach is demonstrated by simulating droplet vaporization (at a prescribed rate) in a temporal mix ing layer. Figure (7) shows the product density in the mixing layer as pre dicted by both methods. When the cutoff size is increased, the conventional LES predicts widely different product formation. However, since the total amount of fuel present is the same, the choice of the cutoff size should not affect the prediction of the product formation. The two-phase LES-LEM methodology predicts product formation that is relatively independent of the chosen cutoff sizes. Note that, even in the LES-LEM there is an upper limit to the cutoff size beyond which error in product formation will occur. Another advantage of the two-phase LES-LEM approach is that the overall computational cost decreases rapidly when the cutoff size is in creased due to the larger time step possible in the Lagrangian tracking of the droplets. It turns out that increasing the cutoff size by a factor of two decreases the computational cost by nearly a factor of four in-spite of the increased computational effort needed to simulate the subgrid mixing process in the LES-LEM approach. The LES-LEM two-phase model has now been implemented to study spray combustion in a full-scale gas turbine engine (e.g., DACRS configu ration of General Electric). Figure 8a shows a typical schematic and the location at which the droplets were injected. Since there is very limited data for comparison, the study so far has focused in investigating the im pact of two-way coupling on the flow dynamics. The flow conditions are similar to the weak swirl premixed study reported above and therefore, there are some similarities in the flow structures. However, the presence of the droplets (approximately, 100,000 droplet groups are in the domain when there is no vaporization) significantly changes the evolution of the flow field. Due to lack of space, only characteristic axial gas phase velocity results (to be consistent with the other cases) are discussed here. Figure 8b shows the 3D perspective of the vortex-droplet distribution for non-vaporizing case. The droplet spray spreading is strongly modulated by the shedding of the vortices from the dump plane. Figures 9a and b show respectively, the mean and rms axial velocity profiles for various spray
SUBGRID COMBUSTION MODELING
347
cases. It can be seen that the effect of two-phase coupling and infinite-rate kinetics significantly change the axial velocity fluctuation in the shear layer. Significant damping of the velocity fluctuations due to two-way interaction occurs. LES with finite-rate heat release are underway and it remains to
348
S. MENON
be seen how spray combustion differs from premixed combustion under otherwise similar flow conditions. These results will be reported in the near future. 6. Conclusions
This paper has discussed some of the modelling approaches that are poten tially applicable to high-Re reacting flows in practical systems. As noted here, the models must be able to capture the dynamics of not only the
SUBGRID COMBUSTION MODELING
349
350
S. MENON
energy transfer from the large to the small scales but also the small-scale mixing and molecular diffusion processes in order to accurately predict re acting flows. Only under special circumstances (e.g., premixed combustion in the flamelet regime) can one employ models in the resolved scale LES approach. A subgrid mixing and combustion model is also described in this paper that has the potential to deal with complex subgrid processes, including finite-rate kinetics in high-Re flows as in full-scale gas turbine engines. The LEM approach has been extended to finite-rate multi-species com bustion chemistry (Sankaran and Menon, 2000) using a In-Situ Adaptive Tabulation (ISAT) approach (Pope, 1997) and is now being implemented within the LES-LEM approach to carry out LES of reacting flows with finite-rate kinetics. Acknowledgments This work was supported in part by the Army Research Office and Gen eral Electric Power Systems Company. Computational time was provided by DOD HPC Centers at NAVO, MS, ERDC, MS and SMDC, AL under a DOD Grand Challenge Project. References Bhargava, A., Kendrick, D. W., Colket, M. B., Sowa, W. A., Casleton, K. H., and Mal oney, D. J. (2000). Pressure effect on nox and co emission from industrial gas turbines. ASME, 2000-GT-97. Cabot, W. H. and Moin, P. (1993). Large eddy simulation of scalar transport with the dynamic subgrid-scale model. In Galperin, B. and Orszag, S., editors, LES of Complex Engineering and Geophysical Flows, pages 141–158. Cambridge University Press. Chakravarthy, V. and Menon, S. (2000a). Large-eddy simulations of turbulent premixed flames in the flamelet regime. Combustion Science and Technology, 162:175–222. Chakravarthy, V. and Menon, S. (2000b). Subgrid modeling of premixed flames in the flamelet regime. Flow, Turbulance and Combustion, 65:133–161. Chakravarthy, V. and Menon, S. (2001). Linear-eddy simulations of reynolds and schmidt number effects on turbulent scalar mixing. Physics of Fluids, 13:487–499. Cheng, R. and Shepherd, I. (1989). A comparison of the velocity and scalar spectra in premixed turbulent flames. Combustion and Flame, 78:205–221. Cho, P., Law, C. K., Cheng, R. K., and Shepherd, I. G. (1988). Velocity and scalar fields of turbulent premixed flames in stagnation flow. Proceedings of the Combustion Institute, 22:739–745. Cho, P., Law, C. K., Hertzberg, J. R., and Cheng, R. K. (1986). Structure and propaga tion of turbulent premixed flames stabilized in a stagnation flow. Proceedings of the Combustion Institute, 21:1493–1499. Cook, A. W. and Riley, J. J. (1998). Subgrid-scale modeling for turbulent reacting flows. Combustion and Flame, 112:593–606. Faeth, G. M. (1983). Evaporation and combustion of sprays. Progress in Energy and Combustion Science, 9:1–76. Fureby, C. (2000). Large-eddy simulation of combustion instabilities in a jet engine afterburner model. Combustion Science and Technology, 161:213–243.
SUBGRID COMBUSTION MODELING
351
Gao, F. and O’Brien, E. (1993). A large eddy simulation scheme for turbulent reacting flows. The Physics of Fluids, 5(6):1282–1284. Germano, M., Piomelli, U., Moin, P., and Cabot, W. H. (1991). A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A, 3(11):1760–1765. Kerstein, A. R. (1989). Linear-eddy model of turbulent transport ii. Combustion and Flame, 75:397–413. Kerstein, A. R. (1991). Linear-eddy modeling of turbulent transport, part 6. microstruc ture of diffusive scalar mixing fields. Journal of Fluid Mechanics, 231:361–394. Kerstein, A. R., Ashurst, W. T., and Williams, F. A. (1988). The field equation for interface propagation in an unsteady homogeneous flow field. Physical Review A, 37:2728–2731. Kim, W.-W. and Menon, S. (1999). A new incompressible solver for large-eddy simula tions. International Journal of Numerical Fluid Mechanics, 31:983–1017. Kim, W.-W. and Menon, S. (2000). Numerical simulations of turbulent premixed flames in the thin-reaction-zones regime. Combustion Science and Technology, 160:xx–xx. Kim, W.-W., Menon, S., and Mongia, H. C. (1999). Large-eddy simulation of a gas turbine combustor flow. Combustion Science and Technology, 143:25–62. Liu, S., Meneveau, C., and Katz, J. (1994). On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. Journal of Fluid Mechanics, 275:83–119. Lund, T. S., Ghosal, S., and Moin, P. (1993). Numerical experiments with highly-variable eddy viscosity models. In Piomelli, U. and Ragab, S., editors, Engineering Applica tions of Large Eddy Simulations, volume 162 of FED, pages 7–11. ASME. Menon, S. and Calhoon, W. (1996). Subgrid mixing and molecular transport modeling for large-eddy simulations of turbulent reacting flows. Proceedings of the Combustion Institute, 26:59–66. Menon, S. and Kim, W.-W. (1996). High reynolds number flow simulations using the localized dynamic subgrid-scale model. AIAA-96-0425. Menon, S., McMurtry, P., and Kerstein, A. R. (1993). A linear eddy mixing model for large eddy simulation of turbulent combustion. In Galperin, B. and Orszag, S., editors, LES of Complex Engineering and Geophysical Flows. Cambridge University Press. Menon, S. and Pannala, S. (1997). Subgrid modeling of unsteady two-phase turbulent flows. AIAA Paper No. 97-3113. Moin, P., Squires, K., Cabot, W., and Lee, S. (1991). A dynamic subgrid-scale model for compressible turbulence and scalar transport. Physics of Fluids A, 11:2746–2754. Nelson, C. C. and Menon, S. (1998). Unsteady simulations of compressible spatial mixing layers. AIAA-98-0786. Pannala, S. and Menon, S. (1998). Large eddy simulations of two-phase turbulent flows. AIAA 98-0163, 36th AIAA Aerospace Sciences Meeting. Pocheau, A. (1994). Scale invariance in turbulent flont propagation. Physical Review E, 49:1109–1122. Pope, S. (1997). Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation. Combustion Theory Modelling, 1:41–63. Pope, S. B. (1985). Pdf methods for turbulent reactive flows. Progress in Energy and Combustion Science, pages 119–192. Sankaran, V. and Menon, S. (2000). The structure of premixed flame in the thin-reactionzones regime. Proceedings of the Combustion Institute, 28:123–129. Schumann, U. (1975). Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. Journal of Computational Physics, 18:376–404. Yakhot, V. (1988). Propagation velocity of premixed turbulent flames. Combustion Science and Technology, 60:191–214.
This page intentionally left blank
EXPERIMENTAL STUDY OF THE FILTERED PROGRESS VARIABLE APPROACH FOR LES OF PREMIXED COMBUSTION
R. KNIKKER AND D. VEYNANTE Laboratoire EM2C, CNRS - Ecole Centrale Paris 92295 Châtenay-Malabry Cedex, France Abstract. A filtered progress variable approach for large eddy simulation of premixed combustion is investigated using experimental data obtained in a V-shaped turbulent flame. The filtered flame surface density, related to the consumption rate, is extracted from visualizations of the instantaneous flame front. Modeling of the subgrid-scale contribution to the flame surface density is discussed and a new model, based on the curvature of the resolved flame, is proposed.
1. Introduction
Large-Eddy-Simulation (LES) of turbulent combustion is a promising tool to address issues where traditional Reynolds-averaged (RANS) approaches cannot be applied. Examples include flows with large-scale coherent struc tures which can couple with combustion instabilities. These structures can be captured in the resolved range of LES, whereas they have to be fully modeled within RANS. Nevertheless, a difficult problem is encountered for large eddy simula tions of premixed combustion since the flame thickness of a premixed flame is about 0.1 to 1 mm and is generally much smaller than the LES mesh size. Temperature and mass fractions are therefore very stiff vari ables and cannot be resolved on the computational grid. To overcome this difficulty, several approaches have been proposed. The filtered progress variable approach reduces the chemical mechanism defined as fresh gases to one equation for the progress variable and in burned gases, which is then spatially filtered with a filter size greater than the grid size (Boger et al., 1998). This method is closely related to the flame tracking technique or G-equation (Kerstein et al., 1988; 353 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 353-366. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
354
R. KNIKKER AND D. VEYNANTE
Menon and Kerstein, 1992; Bourlioux et al., 1996), but the filtered progress variable has the advantage of being physically well defined and can therefore easily be compared to experimental data. Another approach is to artificially thicken the flame so that it can be resolved on a coarse grid, while maintaining the laminar flame speed (Butler and O’Rourke, 1977; Angelberger et al., 1998; Colin et al., 2000). In this paper, the filtered progress variable approach will be studied using experimental data, obtained in a turbulent premixed propane/air flame. An outline of this approach is given in the next section. In section 3, the experimental setup and data analysing issues are discussed. A priori tests of flame surface density models are presented in section 4, and the conclusions are presented in section 5. 2. Filtered Progress Variable Approach 2.1. THEORY
In the classical theory of premixed flames, assuming identical mass and thermal diffusivities and adiabatic conditions, the reactive species mass fractions and the temperature are all linearly related and may be expressed in terms of a reaction progress variable in fresh gases and in burned gases), following the balance equation:
where D is the diffusivity and to this equation gives:
the reaction rate. Applying the LES filter
where denotes a density-weighted filtered quantity. The last three terms are unclosed and correspond respectively to the unresolved transport, the filtered molecular diffusion and the filtered reaction rate. 2.2. CHOICE OF LES FILTER
As already pointed out, the flame front is very thin and in order to resolve the filtered progress variable on the numerical grid the LES filter has to be chosen with care. Boger et al. (1998) proposed to use a Gaussian filter:
LES OF PREMIXED COMBUSTION
355
where the filter size is taken sufficiently large (typically where is the grid mesh size). Note that in general, this filter does not corre spond to the LES filter used in non-reactive cases. 2.3. FLAME SURFACE DENSITY CONCEPT
Instead of modeling each term separately, the terms on the right-hand side of Eq. 2 can be regrouped as a filtered flame front displacement term, describing the propagation of the iso-contours of at the speed relative to the flow field (Boger et al., 1998; Boger and Veynante, 2000):
where denotes the filtered flame surface density and the surface-averaged density-weighted displacement speed. This approach becomes particularly interesting when the chemical re action takes place in thin propagating surfaces, commonly referred to as flamelets. In this case, the filtered flame surface density can be inter preted as the flame surface area per unit volume contained within the LES filtering volume. If, in addition, the turbulence does not affect the inner structure of the reaction zone, may be estimated from the laminar and the fresh gas density as flame speed 2.4. FLAME SURFACE DENSITY MODELS
The challenge of this approach is to find an appropriate model for the fil tered flame surface density as this term represents the subgrid-scale flame dynamics. As a first step, this term may be decomposed into a resolved and an unresolved contribution (Piana et al., 1997):
where is the subgrid-scale wrinkling factor, which can be regarded as the ratio of the turbulent to the laminar flame speed. Modeling can be focussed on either the wrinkling factor or the unresolved contribution to the flame surface density. Closures for are generally based on the relation:
where represents the subgrid-scale velocity fluctuations and and are two constants of the order of unity. This relation has been investigated for example in the framework of the G-equation (Bourlioux et al., 1996; Menon
356
R. KNIKKER AND D. VEYNANTE
and Kerstein, 1992) or the artificially thickened flame model (Angelberger et al., 1998; Colin et al., 2000). Boger et al. (1998) proposed a model based on the filtered flame surface density of an infinitely thin plane flame using a Gaussian filter. The results found are approximated by a parabolic function and multiplied by for the multi-dimensional cases to include flame wrinkling effects:
Note that, when assuming an infinitely-thin flame and using the Bray-MossLibby formulation, widely used in RANS context (Bray et al., 1984), can be expressed in terms of the resolved variable as:
where is the heat release parameter defined by where and are respectively the fresh and burned gas temperatures. Based on expres sion (7), the flame surface density can again be decomposed into a resolved and an unresolved part:
where either the unresolved part or the wrinkling factor has to be mod eled. A transport equation for can also be derived (Boger et al., 1998), requiring modeling of several unclosed terms. This approach has been in vestigated in recent work (Weller et al., 1998; Hawkes and Cant, 2000), but will not be further adressed in this paper. 2.5. UNRESOLVED TRANSPORT
The unresolved transport term is often modeled using a gradient approach. However, Boger and Veynante(2000) showed that a counter-gradient trans port is found in the case of a laminar flame. For example, in the case of an infinitely-thin plane flame, we find:
Based on these findings, the author proposed to model the unresolved trans port in the more general case of a turbulent flame as:
LES OF PREMIXED COMBUSTION
357
where denotes the turbulent viscosity and the turbulent Schmidt number. Note that this formulation is in agreement with observations in RANS modeling (Veynante et al., 1997); for high values of the unresolved transport is of the counter-gradient type, whereas for high turbulence levels a gradient transport is observed. Eq. 11 is now integrated in the balance equation for thereby con veniently including the counter-gradient term in the flame surface density with In the same manner, can be term of Eq. 5 by replacing replaced by in Eq. 7. The final equation reads:
where a diffusion term has been added to ensure a correct flame front propagation when the turbulence vanishes and to control a correct resolved flame thickness (Boger and Veynante, 2000). This term may be linked to the molecular diffusion term, assuming a unity flame Reynolds number, using 3. Experimental setup
The experimental setup is shown in Fig. 1. A mixture of propane/air is injected at ambient conditions into a rectangular combustion chamber. The length, height and depth of the chamber are respectively 320, 50 and 80 mm.
The upper and lower walls are made of ceramic material for thermal iso
lation and the lateral walls are artificial quartz windows to allow optical
358
R. KNIKKER AND D. VEYNANTE
access to the entire flame. A triangular-shaped obstacle is placed in the combustion chamber to stabilize a V-shaped turbulent flame. The mean inlet velocity U and the equivalence ratio can be varied in the range of respectively 5 – 25 m/s and 0.6 – 1.2. The maximum Reynolds number based on the height of the flame holder is 41.700. 3.1. FLAME FRONT VISUALIZATION
Qualitative OH concentration measurements are obtained using planar laser induced fluorescence. A lasersheet is introduced into the combustion cham ber by two narrow windows embedded in the ceramic bricks. Images cov ering up to 12 cm downstream of the flame holder are obtained using an intensified CCD camera. A typical OH-PLIF image is shown in Fig. 2a. Burned gases are characterized by the presence of OH radicals, visible as dark regions in the image. Large concentration gradients are found at the flame front, since the reaction zone is very thin. This allows us to easily extract the flame front (Fig. 2b).
LES OF PREMIXED COMBUSTION
359
Flame front visualizations in the direction perpendicular to the streamwise direction revealed important three-dimensional effects (Knikker et al., 2000). Nevertheless, the following analysis is carried out in two dimensions.
Assuming an infinitely thin flame, fresh gases can be separated from burned gases resulting in a binarised progress variable This is then spatially filtered using the Gaussian filter defined in Eq. 3, as shown in Fig. 3a. The calculation of the filtered flame surface density is not straight-forward, since the derivatives of are not well de fined. However, according to Vervisch et al. (1995), the volume integral appearing in the LES filtering operation can be replaced by a surface inte gral:
360
R. KNIKKER AND D. VEYNANTE
where A* is the iso-surface defined by In the limit of an infinitely-thin flame, all the iso-surfaces collapse to the surface dividing fresh and burned gases and the integral over can be ignored. The filtered flame surface density is now determined by dividing the extracted flame front in small line elements and calculating the filter-weighted sum:
The subgrid-scale contribution to the flame surface density is evaluated using Eq. 9 and presented in Fig. 3b. The effect of the filtering operation is to smooth out small-scale flame structures and to thicken the flame. High unresolved contributions to the flame surface density function are found in regions of high curvature of the flame front. 4. Results 4.1. DECOMPOSITION OF THE FLAME SURFACE DENSITY
The flame surface density model of Eq. 7 is validated here using the ex perimental data. In Fig. 4a, is calculated using a filter size of and plotted as a function of for several locations in the flame. The effect of the wrinkling of the flame is to increase the flame surface density. As a consequence, the values found for lie above the solid line in the fig ure, indicating the filtered flame surface density of an infinitely-thin plane flame. The parabolic approximation of Eq. 7 is presented by the dashed
LES OF PREMIXED COMBUSTION
361
line for From this it can be seen that the flame wrinkling factor as defined by this expression is always greater or equal to one. On the other hand, the resolved flame surface density shown in Fig. 4b, is decreased by the wrinkling of the flame. The resulting unresolved contribution is, therefore, in general greater than the unresolved contribution defined by Eq. 9, provided we neglect the error of the parabolic approximation. This is illustrated in Fig. 5a.
The wrinkling factor as defined by Eq. 5 can be seen as the subgrid scale flame surface divided by the projection of this surface in the resolved propagation direction (Piana et al., 1997):
where n is the unit vector normal to the iso-contours of and N is the unit vector normal to the iso-contours of the resolved flame. They are defined as:
In regions where tends to zero, the resolved propagation direction is not well defined. This explains the unexpected high values that are found in Fig. 5b for the wrinkling factor as defined by Eq. 5. On the other hand, the wrinkling factor defined by Eq. 7 remains well bounded and is a smooth function of space.
362
R. KNIKKER AND D. VEYNANTE
4.2. UNRESOLVED FLAME SURFACE DENSITY
Some features of the unresolved flame surface density that needs to be included in the model are illustrated by Fig. 6. The upper image shows a highly wrinkled flame front, containing small-scale structures and flame pockets. The unresolved contribution to is rather distributed over the burner area and corresponds to merely an increase of the turbulent flame speed.
The lower image shows the response of the flame to acoustic oscillations, generated by two loudspeakers mounted upstream of the inlet. In the pres ence of acoustic waves, the flame features large-scale coherent structures. In contrast to the previous example, the unresolved flame surface density is now concentrated at locations where the flame front shows highest cur vature. Note, however, that three-dimensional effects are important in the first case and almost negligible in the second one. When studying the interaction between flame and acoustics, for example in the framework of combustion instabilities, it is of particular importance
LES OF PREMIXED COMBUSTION
363
to correctly predict the flame surface density as it acts as a source term in the acoustic wave equation (Candel, 1992). Previous models based on the subgrid-scale velocity fluctuations have shown to be inaccurate in the prediction of the localisation as well as the amplitude of the unresolved contribution (Nottin et al., 2000). 4.3. A RESOLVED FLAME CURVATURE MODEL
A new model is investigated, based on the absolute value of the curvature of the resolved flame:
where is a model constant. High contributions of the model can be ex pected in regions where the resolved flame is highly curved, corresponding in general to locations of high contributions of the unresolved flame surface density. Some additional spatial filtering of is necessary to smooth out large spatial variations, associated with the second order spatial deriva tives of appearing in Eq. 17. In the following examples, the Gaussian LES
364
filter is used and
R. KNIKKER AND D. VEYNANTE
is replaced by
in Eq. 17:
The modeled unresolved flame surface density is obtained by inserting the above expression for in Eq. 9. Results for the examples in the previous figures are shown in Fig. 7. An excellent correlation is found in both cases. In particular, locations of high contributions are correctly predicted. A quantitative comparison is found in Fig. 8. The unresolved contribu tion is integrated over the and compared with predicted values for two different filter sizes. An excellent reproduction of amplitude varia tions and peak locations is observed.
Values found for the model parameter remain close to one, but appear to depend on the flame conditions and the filter size. For the moment, is
LES OF PREMIXED COMBUSTION
365
obtained by an ad hoc method. In the future, may be expressed in terms of non-dimensional parameters such as or 5. Conclusions
The filtered progress variable approach is studied using experimental data obtained in a premixed turbulent flame. The flame surface density appears as a key parameter in this approach and is readily obtained from flame front visualizations using laser-induced fluorescence of the OH-radical. The flame surface density model proposed by Boger et al. (1998) is validated and compared to the more classical expression. A considerable reduction of the unresolved contribution is found because the wrinkling factor is now a well defined quantity. Modeling is still required for either the unresolved flame surface density or the flame wrinkling factor. A difficult but interesting test case is the interaction of turbulent flame with acoustic waves. Large-scale structures are observed in this case and high unresolved contributions are found in confined regions of the flame. Where models based on the subgrid-scale velocity fluctuations have proven to be inadequate (Nottin et al., 2000), improvement may be obtained using information on the resolved flame. A new model based on the curvature of the resolved flame is proposed. A priori test showed very promising results, especially for the localisation of high unresolved contributions. A model parameter is still to be specified, however. Future work will be focussed on a theoretical study of this model and its implementation in actual large eddy simulation. References Angelberger, C., D. Veynante, F. Egolfopoulos, and T. Poinsot: 1998, ‘Large eddy sim ulation of combustion instabilities in turbulent premixed flames’. In: Proceedings of the Summer Program. Stanford University - NASA Ames: Center for Turbulence Research, pp. 61–82. Boger, M. and D. Veynante: 2000, ‘Large eddy simulations of a turbulent premixed Vshape flame’. In: Eighth European Turbulence Conference. Barcelona, Spain, The Combustion Institute. Boger, M., D. Veynante, H. Boughanem, and A. Trouvé: 1998, ‘Direct numerical simula tion analysis of flame surface density concept for large eddy simulation of turbulent premixed combustion’. In: Twenty-Seventh International Symposium on Combustion. pp. 917 – 925, The Combustion Institute. Bourlioux, A., V. Moser, and R. Klein: 1996, ‘Large eddy simulations of turbulent pre mixed flames using a capturing/tracking hybrid approach’. In: Sixth International Conference on Numerical Combustion. New Orleans, Louisiana. Bray, K., P. Libby, and J. Moss: 1984, ‘Flamelet crossing frequencies and mean reaction rates in premixed turbulent combustion’. Combust. Sci. Technol. 25, 127 – 140. Butler, T. and P. O’Rourke: 1977, ‘A numerical method for two-dimensional unsteady reacting flows’. In: Sixteenth International Symposium on Combustion, pp. 1503 – 1515, The Combustion Institute.
366
R. KNIKKER AND D. VEYNANTE
Candel, S.: 1992, ‘Combustion instabilities coupled by pressure waves and their active control’. In: Twenty-Fourth International Symposium on Combustion, pp. 1277–1296, The Combustion Institute. Colin, O., F. Ducros, D. Veynante, and T. Poinsot: 2000, ‘A thickened flame model for large eddy simulations of turbulent premixed combustion’. Phys. Fluids A 12(7), 1843 – 1863. Hawkes, E. and S. Cant: 2000, ‘A flame surface density approach to large eddy simulation of premixed turbulent combustion’. Proc. Combust. Inst. 28. Kerstein, A., W. Ashurst, and F. Williams: 1988, ‘Field equation for interface propagation in an unsteady homogeneous flow field’. Phys. Rev. A 37(7), 2728–2731. Knikker, R., D. Veynante, J. Rolon, and C. Meneveau: 2000, ‘Planar laser-induced fluo rescence in a turbulent premixed flame to analyze large eddy simulation models’. In: Tenth International Symposium on Applications of Laser Techniques to Fluid Mechanics. Lisbon, Portugal, http://in3.dem.ist.utl.pt/downloads/lxlaser2000/pdf/26 - 3.pdf. Menon, S. and A. Kerstein: 1992, ‘Stochastic simulation of the structure and propagation rate of turbulent premixed flames’. In: Twenty-Fourth International Symposium on Combustion. pp. 443–450, The Combustion Institute. Nottin, C., R. Knikker, M. Boger, and D. Veynante: 2000, ‘Large eddy simulations of an acoustically excited turbulent premixed flame’. In: Twenty-Eighth International Symposium on Combustion, pp. 67–73, The Combustion Institute. Piana, J., F. Ducros, and D. Veynante: 1997, ‘Large eddy simulations of turbulent pre mixed flames based on the G-equation and a flame front wrinkling description’. In: Eleventh Symposium on Turbulent Shear Flows. Grenoble, France. Vervisch, L., E. Bidaux, K. Bray, and W. Kollmann: 1995, ‘Surface density function in premixed turbulent combustion modeling, similarities between propability density function and flame surface approach’. Phys. Fluids A 7(10), 2496–2503. Veynante, D., A. Trouvé, K. Bray, and T. Mantel: 1997, ‘Gradient and counter-gradient scalar transport in turbulent premixed flames’. J. Fluid Mech. 332, 263 – 293. Weller, H., G. Tabor, A. Gosman, and F. C.: 1998, ‘Application of a flame-wrinkling LES combustion model to a turbulent mixing layer’. In: Twenty-Seventh International Symposium on Combustion, pp. 899–907, The Combustion Institute.
THREE-DIMENSIONAL LARGE-EDDY SIMULATION OF DECAYING MAGNETOHYDRODYNAMIC TURBULENCE
W.-C. MÜLLER, B. KNAEPEN, O. AGULLO AND D. CARATI Euratom-Belgian State Association,
Physique statistique et des plasmas, CP231, Campus Plaine,
Université Libre de Bruxelles, 1050 Bruxelles, Belgium.
Abstract. The numerical large eddy simulation (LES) technique is ap plied to isotropic magnetohydrodynamic (MHD) turbulence. Two dynamic MHD subgrid-scale models of gradient diffusion type are presented for this purpose, taking into account the fundamental differences between the dis sipation mechanisms of the velocity and the magnetic field. Additional ex plicit Gaussian filtering in combination with the tensor-diffusivity mixed model approach is also considered. The LES method is successfully tested a posteriori on scalar level by comparing the obtained results to data stem ming from high-resolution direct numerical simulations of decaying threedimensional MHD turbulence.
1. Introduction
Many geo- and astrophysical systems like the earth’s inner core, large molec ular clouds or the winds and convection zones of stars, but also the heated material inside nuclear fusion experiments, consist of matter in the plasma state. These plasmas, i.e., partly or fully ionized media, can often justifiably be approximated as incompressible, Newtonian, electrically conducting flu ids in the framework of incompressible magnetohydrodynamics (MHD, see for instance (Biskamp, 1993)). The MHD equations for the fluid velocity v and the reduced magnetic induction B being the magnetic induction and the constant mass density, read
367 R. Friedrich and W. Rodi (eds.), Advances in LES of Complex Flows, 367-378. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
368
W.-C. MÜLLER ET AL.
where denotes the effective pressure, the kinematic viscosity and the magnetic diffusivity ( is rescaled to unity). The pressure is a merely passive quantity, ensuring the incompressibility of v. Due to the weak collisionality and/or large spatial dimensions of the plasmas mentioned above, the Reynolds numbers and in these systems are normally huge, making the associated flows prone to turbulence ( and represent respectively a characteristic length scale and velocity). In the following, we will concentrate on the case of unit which permits to use the magnetic magnetic Prandtl number as the sole dynamic parameter of the MHD flows Reynolds number gives a quite generic under consideration. This choice for the value of realisation of MHD turbulence, away from the two extreme cases leading to the passive convection of the magnetic field by the flow, and a situation typical for the amplification of a small-amplitude magnetic field by turbulent plasma motions (“dynamo effect”). Since direct experimental access to fully developed MHD turbulence in nature is often difficult or even impossible, one tries to circumvent this difficulty by solving equations (1)–(3) numerically. Unfortunately, the nu merical resolution necessary to render the discrete form of a turbulent MHD flow into a physically meaningful representation of the real sys tem increases rapidly with the magnetic Reynolds number This lim its direct numerical simulations (DNS) of three-dimensional MHD turbu lence on today’s supercomputers (Mac Low et al., 1998; Stone et al., 1998; Biskamp and Müller, 2000) to while typical values in the cf. e.g. (Zeldovich et al., 1983). A possible alternative sun range around to DNS is the large-eddy simulation (LES) approach, where the Reynolds number gap between simulation and reality is narrowed by restricting direct numerical computation to the large spatial scales of turbulence in combi nation with a model to incorporate the effects of the small-scale physics. The LES technique is considered a valuable tool for simulating rather large Reynolds number Navier-Stokes turbulence for more than thirty years now (Rogallo and Moin, 1984), while work on adapting the method to MHD turbulence (Yoshizawa, 1987; Passot et al., 1990; Zhou and Vahala, 1991; Theobald et al., 1994) has not received as much attention. This might in part be caused by the fact that LES small-scale models usually contain at least one free parameter, which has to be tuned to obtain optimal per formance of the simulation, turning a LES into a trial and error process. However, not long ago a dynamic procedure has been developed (Germano, 1992; Germano et al., 1991) that allows for a self-consistent calibration of the small-scale model during the simulation, leading to convincing results when used in Navier-Stokes turbulence.
LARGE-EDDY SIMULATION OF MHD TURBULENCE
369
In this article we present the extension of this procedure to isotropic three-dimensional MHD turbulence, applying two generalized gradient-diffusion models and a mixed model with an additional tensor-diffusivity term to describe the small-scale dynamics. Scalar-level tests are performed by comparing the observed temporal and spectral behavior of the kinetic and magnetic energies to the results of a high-resolution DNS of decaying MHD turbulence (Biskamp and Müller, 2000). 2. Filtering the MHD equations
The formal basis of LES is the application of a spatial gridfilter of width (Leonard, 1974) to (1)–(3). The system of filtered equations can then be solved using a coarser grid, since fluctuations on spatial scales smaller than – the subgrid scales – have been removed. In the following, quantities will be denoted by an overbar. The filtered equations for and thus read
where two unknown terms, usually referred to as subgrid-scale or filtered-
scale stress tensors, have been introduced by the filtering operation and need to be modeled: and These tensors account for effects of the subgrid-scale motions on the flow at the resolved scales larger than The filter operation can be considered as a two-level process, especially when one works in spectral space. The gridfiltering procedure is identical with the transition from a truly continous system to one of its discrete representations. For a cubic grid in Fourier space this discretization corresponds to a cubic cutoff filter setting all spatial Fourier modes with at least one wave-vector component larger than to zero, where defines the smallest resolved spatial scale of the grid. The second filteringlevel is called explicit filtering and denotes the application of an additional filter of width on top of the gridfilter. If the explicit filter ker nel in spectral space is non-zero everywhere, its effect can in principle be reversed without loss of information (de-filtering). If the filter kernel is furthermore like e.g. for the top-hat and Gaussian filters, the defilter terms take the form of an infinite series (Yeo, 1987; Leonard, 1997; Carati et al., 2001). Restricting the de-filtering series to terms of first order yields the tensor-diffusivity model. In the following we will consider both described approaches: gridfiltering with a cubic cutoff filter and explicit filtering with the combination of cubic
370
W.-C. MÜLLER ET AL.
cutoff and Gaussian filters. The Gaussian filter width filter kernel value of 0.5 for 3.
is set to yield a
MHD subgrid-scale models
The model expressions used for the subgrid-scale stress tensors and are based on the eddy-viscosity assumption widely applied in LES of Navier-Stokes turbulence (Boussinesq, 1877; Sagaut, 2001). This class of models aims at reproducing the subgrid-scale kinetic energy dissipation by assuming a linear relation between the deformation tensors at resolved and dissipative scales,
where the superscript ‘T’ denotes the transposed matrix and is the eddyviscosity. When restricting the dependence of to a characteristic mixinglength on the spatial scales under consideration and the subgrid kinetic energy dissipation dimensional analysis suggests For the magnetic energy one can proceed analogously, taking into account that the property of the magnetic field, which gives rise to resistive dissipation, is its curl rather than its deformation. This yields the model prototype
with the eddy-resistivity denoting the subgrid mag netic energy dissipation. Depending on the way of estimating the subgrid dissipation, different eddy-viscosity models can be constructed. Here we consider two variants. If lies somewhere in the inertial scale range of fully developed turbu lence and, as observed in the reference DNS, the nonlinear energy transfer between kinetic and magnetic energy is much smaller than the respective energy dissipation, and can be assumed to be independent of This yields the generalized Kolmogorov scaling subgrid model (Wong and Lilly, 1994)
where and are unknown parameter functions of time. This model is computationally very cost-effective, numerically simple to implement and is known to give satisfactory results when applied in Navier-Stokes turbulence. The classical Smagorinsky model (Smagorinsky, 1963) approximates as the product of the resolved local dissipation rate and the associated length scale leading to Together with the corresponding
LARGE-EDDY SIMULATION OF MHD TURBULENCE
extension to MHD (Theobald et al., 1994) resolved electric current density, this gives
371
being the
The MHD-extended Smagorinsky approach represents a widely used refinement of the Kolmogorov model since it additionally takes into account local gradients of the turbulent fields. In the case of additional explicit filtering with a Gaussian filter of width one can partially undo its effect on the scales larger than by including a de-filtering series (cf. section 2 of this article) in the filteredscale stress tensor. Taking into account series terms up to the first order in combination with the Kolmogorov scaling subgrid-model yields a mixed tensor-diffusivity model that, in contrast to the unconditionally dissipative and allows for an inverse energy flow from subgrid subgrid models to resolved scales:
where The Einstein summation convention applies. For a com prehensive description of this technique in the Navier–Stokes context see Ref. (Winckelmans et al., 2001). It should be noted that the inverse energy transfer capability of is based on altering the nonlinear interactions between the different scales of turbulent motion. All three models contain parameters and for which no direct relationship with the resolved large-scale quantities can be obtained without further assumptions. Therefore, one would be forced to prescribe these parameters in order to optimize the simulation results with respect to known reference characteristics. However, this problem can be circum vented by exploiting the self-similarity of fully developed turbulence. A corresponding technique, that has proven to be successful in Navier-Stokes LES, is the dynamic procedure (Germano, 1992; Lilly, 1992; Ghosal et al., 1995). 4. Dynamic procedure for MHD
The basic idea of this method is the application of a second filter – the test filter – of width to the LES equations (4)–(6). The test filtering will be indicated by the hat operation, here a cubic cutoff with symbol, In the case of additional explicit filtering (Subgrid model the width of the explicit test-filter has to be adapted accordingly to ensure scale-similarity of the resolved and the test-filtered turbulent fields.
372
W.-C. MÜLLER ET AL.
Introducing the filtered-scale stress tensors at test filter level as and leads to the Germano identities (Germano, 1992) for MHD
where the Leonard tensors and are known in terms of the resolved fields and, as such, do not require any modeling. In contrast, the filtered-scale stress tensors at grid scales and test scales have to be modeled. Since the turbulent fields are assumed to be self-similar on scales within the interval the applied models and their parameters are independent of the test filter width. Hence, for the stress tensors and at test filter level the same model and coefficients should be used as for and respectively. Due to the modeling of the filtered-scale stress tensors the identities (7) are not exactly fulfilled. The errors can be quantified as
where the model parameters are assumed to be constant in space, which is justified when dealing with spatially homogeneous turbulence. Minimizing the deviations and by variation with respect to the model coeffi cients (Lilly, 1992) yields an optimization ansatz for the filtered scale stress tensors. 5. A posteriori tests
In order to assess the LES technique and the dynamic procedure in MHD we have performed simulations of decaying isotropic turbulence using the same pseudospectral method as for the reference DNS, the same integration domain (a cube of linear extension with periodic boundary conditions) and the same set of parameters at a significantly reduced numerical resolution. The initial condition has been generated from the reference DNS data at when kinetic and magnetic energy dissipation, starting from smooth initial fields, have reached their maxima and the dissipative small-scale structures of turbulence are fully developed. The data set has been cutoff-filtered from to Fourier modes, removing about 99.8% of the originally available information. The influence of the filtered scales on the remaining large scales of motion, which still contain 90% of the total energy, has to be mimicked by the models for the filteredand scale stress tensors
LARGE-EDDY SIMULATION OF MHD TURBULENCE
373
5.1. TOTAL KINETIC AND MAGNETIC ENERGY EVOLUTION
A basic LES requirement is the reproduction of the temporal and spectral behavior of the filtered kinetic and magnetic energy. The time evolution of these quantities is shown in Figures 1 and 2, where is given, like for the reference DNS, in units of the large eddy turnover time. Di amonds represent the filtered DNS results, the solid, the dashed and the dashed-dotted lines give the energy development for LES with the models
and respectively. To evaluate the overall influence of the dy namic subgrid modeling the dotted lines show the energy curves for a LES with As expected, the results strongly deviate from the filtered DNS data, since energy dissipation is mainly due to the high wavenumber modes which have been cut off by the gridfiltering operation. A clear improvement is observed when the subgrid models and are applied. The evolution of is well reproduced in all three cases, showing that the main influence of the small-scale velocity field fluctuations on the kinetic energy is mainly dissipative. The temporal development of
374
W.-C. MÜLLER ET AL.
the magnetic energy in the LES with and also satisfactorily fits the reference data, though one observes an offset between the LES results and the DNS. Closer inspection reveals that this phenomenon is caused by an initial overshoot of the model resistivities estimated by the dynamic procedure. This effect is known from Navier-Stokes LES using dynamic models and can be reduced by allowing for a short model relaxation period at the beginning of the simulation. Due to the spatial homogeneity of the turbulent system, there is only a small systematic difference between the results obtained with the Kolmogorov scaling model which mainly depends on the spatially averaged extrapolation process inherent to the dynamic procedure, and the Smagorinsky version that additionally takes into account the local field gradients. The implicit modification of the nonlinear terms in (4) and (5) by the mixed model compared to while having only a small impact on the kinetic energy, notably slows down the decay of the magnetic energy up to However, at later times arrives at the same level for and which indicates that the tensor diffusivity term cannot counterbalance the too high energy dissipation of the Kolmogorov scaling model
LARGE-EDDY SIMULATION OF MHD TURBULENCE
375
5.2. KINETIC AND MAGNETIC ENERGY SPECTRA
The application of the virtually parameter-free dynamic subgrid models reproduces rather sensitive quantities like the angle-averaged energy spectra in good agreement with the DNS data. The kinetic energy spectra are
shown in Figure 3 at time when has decreased by a factor of about 6.5. All models yield spectra that exhibit approximately the same scaling as the filtered DNS data up to the high wavenumber range of the LES system. However, the spectra confirm the over-dissipation of the three subgrid-models as already observed in the preceding section for the global energies. The differences between the models are small, being the least dissipative. The tensor diffusivity term in obviously does not improve model. the spectral performance of the The agreement of the LES results with the filtered DNS spectrum is evidently due to the subgrid models, since the lack of such a model in the causes a large accumulation of kinetic energy over about two thirds of the available wavenumber range, which is the consequence of the miss ing filtered-scale energy dissipation in combination with the direct spectral energy cascade.
376
W.-C. MÜLLER ET AL.
The same trends are observed in Figure 4 for the angle-averaged mag netic energy spectrum However, the magnetic part of the mixed model is exhibiting less dissipation than the models and This is also in agreement with the results presented in the preceding section. In addition, one observes that the LES show a spectral magnetic energy distribution that is too low across a wide range of scales when compared to the DNS spectrum. Since the without model exhibits the same behavior, the observation cannot be explained by the subgrid models being too dissipative. This implies that the model for the filtered-scale stress tensor in equation (5) has to be more complex than the simple dissipative subgrid models and e.g., allowing for a spectrally local inflow of magnetic energy. Also the tensor-diffusivity term in model cannot outweigh this shortcoming of the underlying gradient-diffusion models. Nevertheless, the improvement in reproducing the magnetic energy reference spectrum us ing the presented subgrid-scale models compared to a is still very good.
LARGE-EDDY SIMULATION OF MHD TURBULENCE
377
6. Summary
The results presented in this article show that LES with dynamic subgrid modeling are able to reproduce the temporal and spectral energy evolution in isotropic MHD turbulence. Since decaying turbulence was considered, the LES subgrid models had to adapt to a nonstationary environment, where a continuous self-calibration of the model parameters as realized in the dynamic procedure is of particular importance. We found that simple gradient-diffusion type subgrid models perform acceptably well, while the application of computationally more costly mixed models with a tensor diffusivity term seems to be only beneficial for the magnetic field dynamics. The achieved LES performance, especially when the highly reduced nu merical resolution and the simplicity of the purely dissipative filtered-scale models are taken into account, encourages further work on dynamic subgrid modeling of MHD turbulence. Acknowledgements
D.C. and B.K. are respectively “Chercheur Qualifié” and “Chargé de recherche” of the FNRS, Belgium. This work has been supported by the contract of association EURATOM - Belgian state. The content of the publication is the sole responsibility of the authors and it does not necessarily represent the views of the Commission or its services. References Biskamp, D.: 1993, Nonlinear Magnetohydrodynamics. Cambridge: Cambridge University Press. Biskamp, D. and W.-C. Müller: 2000, ‘Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence’. Physics of Plasmas 7(12), 4889–4900. Boussinesq, J.: 1877, ‘Essai sur la théorie des eaux courantes’. Mémoires présentés par divers savants à l’Académie des Sciences, Paris 23(1), 1–680. Carati, D., G. S. Winckelmans, and H. Jeanmart: 2001, ‘On the modelling of the subgrid scale and filtered-scale stress tensors in large-eddy simulation’. Journal of Fluid Mechanics 441, 119–138. Germano, M.: 1992, ‘Turbulence: the filtering approach’. Journal of Fluid Mechanics 238, 325–336. Germano, M., U. Piomelli, P. Moin, and W. H. Cabot: 1991, ‘A dynamic subgrid-scale eddy viscosity model’. Physics of Fluids A 3(7), 1760–1765. Ghosal, S., T. Lund, P. Moin, and K. Akselvoll: 1995, ‘A dynamic localization model for large-eddy simulation of turbulent flows’. Journal of Fluid Mechanics 286, 229–255. Leonard, A.: 1974, ‘Energy cascade in large-eddy simulations of turbulent fluid flows’. Advances in Geophysics 18 A, 237–248. Leonard, A.: 1997, ‘Large-eddy simulation of chaotic convection and beyond’. In: AIAA 35th Aerospace Sciences Meeting & Exhibit, Reno, Nevada. Washington, DC, Amer ican Institute for Aeronautics and Astronautics. Lilly, D. K.: 1992, ‘A proposed modification of the Germano subgrid-scale closure method’. Physics of Fluids A 4(3), 633–635.
378
W.-C. MÜLLER ET AL.
Mac Low, M.-M., R. S. Klessen, A. Burkert, and M. D. Smith: 1998, ‘Kinetic energy decay rates of supersonic and super-alfvénic turbulence in star-forming clouds’. Physical Review Letters 80(13), 2754–2757. Passot, T., H. Politano, A. Pouquet, and P. L. Sulem: 1990, ‘Comparative study of dissipa tion modeling in two-dimensional MHD turbulence’. Theoretical and Computational Fluid Dynamics 1, 47–60. Rogallo, R. S. and P. Moin: 1984, ‘Numerical simulation of turbulent flows’. Annual Review of Fluid Mechanics 16, 99–137. Sagaut, P.: 2001, Large Eddy Simulation For Incompressible Flows. Berlin: SpringerVerlag. Smagorinsky, J.: 1963, ‘General circulation experiments with the primitive equations’. Monthly Weather Review 91(3), 99–164. Stone, J. M., E. C. Ostriker, and C. F. Gammie: 1998, ‘Dissipation in compressible magnetohydrodynamic turbulence’. Astrophysical Journal 508, L99–L102. Theobald, M. L., P. A. Fox, and S. Sofia: 1994, ‘A subgrid-scale resistivity for magneto hydrodynamics’. Physics of Plasmas 1(9), 3016–3032. Winckelmans, G. S., A. A. Wray, O. V. Vasilyev, and H. Jeanmart: 2001, ‘Explicit filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term’. Physics of Fluids 13(5), 1385–1403. Wong, V. C. and D. K. Lilly: 1994, ‘A comparison of two dynamic subgrid closure methods for turbulent thermal convection’. Physics of Fluids 6(2), 1016–1023. Yeo, W. K.: 1987, ‘A generalized high pass/low pass averaging procedure for deriving and solving turbulent flow equations’. Ph.D. thesis, Ohio State University. Yoshizawa, A.: 1987, ‘Subgrid-modeling for magnetohydrodynamic turbulent shear flows’. Physics of Fluids 30(4), 1089–1095. Zeldovich, Y. B., A. A. Ruzmaikin, and D. D. Sokoloff: 1983, Magnetic Fields In Astro physics. New York: Gordon and Breach Science Publishers. Zhou, Y. and G. Vahala: 1991, ‘Aspects of subgrid modelling and large-eddy simulation of magnetohydrodynamic turbulence’. Journal of Plasma Physics 45(2), 239–249.
