Numerical Simulation of Turbulent Flows and Noise Generation: Results of the DFG CNRS Research Groups FOR 507 and FOR 508 (Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 104)

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Author: W. Schröder | Christophe Brun | Daniel Juvé | Michael Manhart | Claus-Dieter Munz

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104

Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)

Editors W. Schröder/Aachen K. Fujii/Kanagawa W. Haase/München E.H. Hirschel/München B. van Leer/Ann Arbor M.A. Leschziner/London M. Pandolfi/Torino J. Periaux/Paris A. Rizzi/Stockholm B. Roux/Marseille Y. Shokin/Novosibirsk

Numerical Simulation of Turbulent Flows and Noise Generation Results of the DFG/CNRS Research Groups FOR 507 and FOR 508 Christophe Brun Daniel Juvé Michael Manhart Claus-Dieter Munz (Editors)

ABC

Dr. Christophe Brun Université Joseph Fourier LEGI, BP 53 38041 Grenoble CEDEX 9 France E-mail: [email protected] Prof. Daniel Juvé Ecole Centrale de Lyon Centre Acoustique, LMFA, UMR CNRS 5509 36 Avenue Guy de Collongue 69134 Ecully Cedex France E-mail: [email protected]

ISBN 978-3-540-89955-6

Prof. Dr. Michael Manhart TU München Fachgebiet für Hydromechanik Arcisstr. 21 80333 München Germany E-mail: [email protected] Prof. Dr. Claus-Dieter Munz Institut für Aerodynamik und Gasdynamik Universität Stuttgart Pfaffenwaldring 21 70569 Stuttgart Germany E-mail: [email protected]

e-ISBN 978-3-540-89956-3

DOI 10.1007/978-3-540-89956-3 Notes on Numerical Fluid Mechanics and Multidisciplinary Design

ISSN 1612-2909

Library of Congress Control Number: 2008944194 c 2009

Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 543210 springer.com

NNFM Editor Addresses

Prof. Dr. Wolfgang Schröder (General Editor) RWTH Aachen Lehrstuhl für Strömungslehre und Aerodynamisches Institut Wüllnerstr. zw. 5 u. 7 52062 Aachen Germany E-mail: offi[email protected] Prof. Dr. Kozo Fujii Space Transportation Research Division The Institute of Space and Astronautical Science 3-1-1, Yoshinodai, Sagamihara Kanagawa, 229-8510 Japan E-mail: fujii@flab.eng.isas.jaxa.jp Dr. Werner Haase Höhenkirchener Str. 19d D-85662 Hohenbrunn Germany E-mail: offi[email protected] Prof. Dr. Ernst Heinrich Hirschel (Former General Editor) Herzog-Heinrich-Weg 6 D-85604 Zorneding Germany E-mail: [email protected] Prof. Dr. Bram van Leer Department of Aerospace Engineering The University of Michigan Ann Arbor, MI 48109-2140 USA E-mail: [email protected] Prof. Dr. Michael A. Leschziner Imperial College of Science Technology and Medicine Aeronautics Department Prince Consort Road London SW7 2BY U.K. E-mail: [email protected]

Prof. Dr. Maurizio Pandolfi Politecnico di Torino Dipartimento di Ingegneria Aeronautica e Spaziale Corso Duca degli Abruzzi, 24 I-10129 Torino Italy E-mail: pandolfi@polito.it Prof. Dr. Jacques Periaux 38, Boulevard de Reuilly F-75012 Paris France E-mail: [email protected] Prof. Dr. Arthur Rizzi Department of Aeronautics KTH Royal Institute of Technology Teknikringen 8 S-10044 Stockholm Sweden E-mail: [email protected] Dr. Bernard Roux L3M – IMT La Jetée Technopole de Chateau-Gombert F-13451 Marseille Cedex 20 France E-mail: [email protected] Prof. Dr. Yurii I. Shokin Siberian Branch of the Russian Academy of Sciences Institute of Computational Technologies Ac. Lavrentyeva Ave. 6 630090 Novosibirsk Russia E-mail: [email protected]

Preface

This volume contains the reports of the research groups which participated in the French-German CNRS-DFG Program on Computational Fluid Dynamics in the period 2003-2008. This Collaborative Program was supported by the Centre Nationale de la Recherche Scientifique (CNRS) inside a Groupement de Recherche Européen and by the Deutsche Forschungsgemeinschaft (DFG) via Forschergruppen, so-called research initiatives. To be more precise, two distinct research initiatives were funded, one on ”Large Eddy Simulation (LES) of Complex Flows” and one on ”Noise Generation in Turbulent Flows”. The profound development of the LES methodology and its extension to complex geometries as well as the interaction between LES and acoustics were the general objectives of the proposals. Over the six years, the program consisted of more than 25 teams in France and Germany which did collaborate intensively within fourteen subprojects, seven on LES and seven on acoustics. The CNRS-DFG Program was initiated at the end of 2002 under the authority of the Engineering Department (ST2I) and of the Direction des Relations Européennes et Internationales (DREI) in CNRS and the Fachkollegium Str¨ omungsmechanik in DFG. The persons in charge of ST2I and DREI were Victor Sanchez, Pierre Guillon and Izo Abram, Patrick Le Quéré, Denis Veynante, and Anne-Marie Brass. From DFG, Walter Lachenmeier and Michael Meier chaired the French-German Program. The joint French-German reviewing groups were presided by Prof. Alziary de Roquefort at the meeting in Stuttgart, that took place from November 21 through 22, 2002 and by Prof. Johannes Janicka at the meeting in Giff-surYvette, that was held from November 14 through 15, 2005. The referees are renowned international scientists in the field of numerical fluid mechanics and aeroacoustics. On the reviewing team were besides the aforementioned chair persons Jean Piquet, Thierry Gallouet, Pierre Sagaut, Sébastien Candel, Hans Wengle, Wilhelm Kordulla, Gerald Warnecke, Jan Delfs, and Didier Besnard. The fourteen approved subprojects achieved excellent findings over the six-year funding period due to the strong internal collaboration within each French-German group and due to the pronounced external cooperation between the LES and acoustics teams. The Research Imitative ”Large Eddy Simulation (LES) of Complex Flows” was chaired by Michael Manhart and Christophe Brun and that on ”Noise Generation in Turbulent Flows” by Claus-Dieter Munz and Daniel Juvé, respectively. They jointly organized with their teams numerous internal meetings and several international conferences, schools, and workshops, i.e., four EUROMECH Colloquia in Darmstadt, Marseille, Dresden, and Munich and an extremely stimulating and scientifically successful summer school at CEMRACS. Special issues were published on the related works in journals and series

VIII

Preface

such as the Comptes Rendus Académie des Sciences Paris, t. 333, issue no. 1, 2005, the international journal Computers & Fluids, vol. 37, issue no. 4, 2008, the international journal Theoretical & Computational Fluid Dynamics, vol. 22, no. 3–4, 2008, the Journal of Turbulence, vol. 6, 2005, and the EDP Sciences ESAIM Proceedings, vol. 16, 2007. Approximately twenty PhD and habilitation theses were submitted in the program. A Collaborative Program of the French-German University (DFH) was supported between the Ecole Doctorale and the Graduiertenkolleg. In addition to the five special issues, approximately one hundred papers and contributions were published on the various LES and acoustics studies. Spring and summer schools were organized, e.g., Ecole’s MFN in Roscoff, CEMRACS in Marseille, ERCOFTAC SIG4 in Strasbourg, and GST in Paris, and international outstanding scientists gave invited lectures to ensure an excellent education program for the participating students. Moreover, sessions at conferences, workshops, and colloquia such as ICTIAM, GAMM, TSFP, DLES and so forth were arranged in association with the CNRS-DFG Program. The participants of the CNRS-DFG French-German Program do express their gratitude and thank to the CNRS, the Ministre de l’Enseignement Supérieur et de la Recherche (MESR) and the DFG for their continuous support of the projects and the additional grants. It is fair to recall that the idea of such a French-German Program has been developed by Egon Krause and Roger Peyret in a joint effort with Walter Lachenmeier from DFG and the Direction des Sciences de l’Ingénieur Department of CNRS in the early 90s. The present program was preceded by a two-year period 1994 through 1995 and a six-year phase 1996-2002 by an exchange program and then by a first research program, of which both were successful and therefore stimulated an extension of the French-German cooperation. The recent research topics were regularly remodelled and renewed at each stage of this initiative. The impressive results which are concisely discussed in this book are closely connected to the more than fifteen-years of scientific history of the CNRS-DFG collaboration in numerical analysis. Marseille and Aachen, November 2008

P. Bontoux W. Schröder

Table of Contents

Part I: Noise Generation in Turbulent Flows Reduced-Order Modelling of Turbulent Jets for Noise Control Michael Schlegel, Bernd R. Noack, Pierre Comte, Dmitry Kolomenskiy, Kai Schneider, Marie Farge, Dirk M. Luchtenburg, Jon E. Scouten, Gilead Tadmor . . . . . . . . . . . . . .

3

Numerical Simulation of Supersonic Jet Noise Jan Schulze, J¨ orn Sesterhenn, Peter Schmid, Christophe Bogey, Nicolas de Cacqueray, Julien Berland, Christophe Bailly . . . . . . . . . . . .

29

Fluid-Acoustic Coupling and Wave Propagation Jens Utzmann, Claus-Dieter Munz, Michael Dumbser, Eric Sonnendr¨ ucker, Stéphanie Salmon, Sébastien Jund, Emmanuel Frénod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Mechanisms and Active Control of Jet-Induced Noise Andreas Babucke, Bruno Spagnoli, Christophe Airiau, Markus Kloker, Ulrich Rist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Noise Prediction for Turbulent Coaxial Jets Seong Ryong Koh, Wolfgang Schr¨ oder, Elmar Gr¨ oschel, Matthias Meinke, Pierre Comte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Numerical Simulation of Jet Mixing Noise Associated with Engine Exhausts Dandy Eschricht, Bj¨ orn Greschner, Frank Thiele, Marc C. Jacob . . . . . 121 Part II: LES of Complex Flows Implicit Turbulence Modeling by Finite Volume Methods Stefan Hickel, Antoine Devesa, Nikolaus A. Adams . . . . . . . . . . . . . . . . . 149 Numerical Simulation of Turbulent Flows in Complex Geometries Using the Coherent Vortex Simulation Approach Based on Orthonormal Wavelet Decomposition Henning Bockhorn, Jordan A. Denev, Margarete Domingues, Carlos Falconi, Marie Farge, Jochen Fr¨ ohlich, Sˆ onia Gomes, Benjamin Kadoch, Igor Molina, Olivier Roussel, Kai Schneider . . . . . . 175

X

Table of Contents

Hybrid LES–RANS-Coupling for Complex Flows with Separation Michael Breuer, Orhan Aybay, Benoˆıt Jaffrézic, Michel Visonneau, Ganbo Deng, Emmanuel Guilmineau, Oussama Chikhaoui . . . . . . . . . . . 201 Segregated LES/RANS Coupling Conditions for the Simulation of Complex Turbulent Flows Dominic von Terzi, Ivan Mary, Jochen Fr¨ ohlich . . . . . . . . . . . . . . . . . . . 231 LES, Zonal and Seamless Hybrid LES/RANS: Rationale and Application to Free and Wall-Bounded Flows Involving Separation and Swirl ˇ c, Atabak Fadai-Ghotbi, Suad Jakirlić, Rémi Manceau, Sanjin Sari´ Bj¨ orn Kniesner, Sabrina Carpy, Gisa Kadavelil, Christophe Friess, Cameron Tropea, Jacques Borée . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Wall Scaling and Wall Models for Complex Turbulent Flows Michael Manhart, Nikolaus Peller, Margareta Petrovan Boiarciuc, Christophe Brun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 High-Order Methods for Large-Eddy Simulation in Complex Geometries Christine Baur, Patrick Bontoux, Michael Kornhaas, Mathieu Minguez, Richard Pasquetti, Michael Sch¨ afer, Eric Serre, Eric Séverac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Part I

Noise Generation in Turbulent Flows

Reduced-Order Modelling of Turbulent Jets for Noise Control Michael Schlegel, Bernd R. Noack, Pierre Comte, Dmitry Kolomenskiy, Kai Schneider, Marie Farge, Dirk M. Luchtenburg, Jon E. Scouten, and Gilead Tadmor

Abstract. A reduced-order modelling (ROM) strategy is pursued to achieve a mechanistic understanding of jet flow mechanisms targeting jet noise control. Coherent flow structures of the jet are identified by the proper orthogonal decomposition (POD) and wavelet analysis. These techniques are applied to an LES data ensemble with velocity snapshots of a three-dimensional, incompressible jet at a Reynolds number of Re = 3600. A low-dimensional Galerkin model of a three-dimensional jet is extracted and calibrated to the physical dynamics. To obtain the desired mechanistic understanding of jet noise generation, the loudest flow structures are distilled by a goal-oriented generalisation of the POD approach we term ’most observable decomposition’ (MOD). Thus, a reduction of the number of dynamically most important degrees of freedom by one order of magnitude is achieved. Capability of the presented ROM strategy for jet noise control is demonstrated by suppression of loud flow structures. Michael Schlegel · Dirk M. Luchtenburg · Jon E. Scouten · Bernd R. Noack Berlin Institute of Technology MB1, Straße des 17. Juni 135, 10623 Berlin, Germany [email protected] Pierre Comte ´ Laboratoire d’Etudes Aérodynamiques - CNRS UMR 6609 / Université de Poitiers, CEAT, 43, rue de l’aérodrome, 86036 Poitiers Cedex, France [email protected] Dmitry Kolomenskiy · Kai Schneider Centre de Mathématiques et d’Informatique and Laboratoire de Mécanique, Modélisation & Procédés Propres -CNRS UMR 6181 / Université de Provence (Aix-Marseille I), 39, rue F. JoliotCurie, 13453 Marseille - Cedex 13, France [email protected] Marie Farge Laboratoire de Météorologie Dynamique - CNRS UMR 8539 / Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris - Cedex 05 [email protected] Gilead Tadmor Northeastern University, 440 Dana Building, 360 Huntington Avenue, Boston, MA 02115-5000, USA [email protected]

C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 3–27. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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1 Introduction Jets from engine exhaust constitute the most important noise source of civil aircrafts during take-off. The noise level exceeds those of other sources like fan, combustion, and airframe noise. Hence, the suppression of jet noise has been actively pursued from the beginning of civil air traffic with jet engines, leading mainly to larger bypass ratios. As the engine diameter reaches its practical limit, further noise reductions will benefit from an intuitive understanding of the mechanisms responsible for noise generation in turbulent, subsonic jets. Yet, this understanding is still in its infancy after more than five decades of jet noise research (cmp. e.g. [52]). The complexity of this problem is ascribed to the high dimensionality and broadband spectrum of the flow state attractor. Currently, an opportunity for model-based jet noise reduction is opening up by the rapidly evolving field of reduced-order modelling (ROM) (cmp. [64]). Performance of ROM for flow control purposes is demonstrated for control of vortex shedding behind circular and D-shaped cylinders [2, 19, 31, 49, 66–68, 73] and control of cavity oscillations [36]. Reduced-order models have also enabled successful feedback control in shear flow experiments, e.g. for bluff-body drag reduction and mixing enhancements [23, 53]. Requisites for empirical analyses and modelling are data bases provided by computational aeroacoustics. Here, direct numerical simulations (e.g. [18]) and large eddy simulations (e.g. [3, 10]) contribute to the understanding of experimental data (e.g. [24]). Similar efforts are undertaken by vortex-filament models [22, 44, 70, 71]. Targeting the distillation of the desired mechanistic understanding for jet noise reduction, we pursue a ROM strategy, including the tasks of structure identification, dynamical modelling and control design (cmp. figure 1). Therefore, a velocity snapshot ensemble of an incompressible jet at Reynolds number Re = 3600 is utilised, provided by a large eddy simulation (see [7, 33] and the references therein). A main ROM challenge is represented by flow structure identification, tailored for the purposes of modelling and control. Constituting a class of flow representations, coherent flow structures are known to be a cause of the noise, as noted already in Lighthill’s classical paper in 1952 [35]. Indeed, the frequency of the local noise source scales approximately inversely with the size of the coherent structures [43]. That size increases in the streamwise direction by vortex merging, leading to a decrease of the dominant frequency. This relationship has been experimentally corroborated and employed in surprisingly effective frequency slice models [57]. Commonly, coherent structures are extracted from POD, decomposing the flow velocity most efficiently for the resolution of total kinetic energy. In present jet POD studies (e.g. [17, 21]), POD dimensions of O(100) of dynamically important degrees of freedom are revealed, which poses a serious challenge for model-based noise control. The POD method and POD analysis results for the incompressible jet are presented in section 3. Further key enablers for structure identification are represented by wavelet techniques, which were used to study turbulence in already in the early nineties [11,

Reduced-Order Modelling of Turbulent Jets for Noise Control

5

Kinematics Aeroacoustic state & filter Full−scale model

Reduced order represen− tations

Control strategy

Dynamics

Hydrodynamic state & filter

p

u

LES du/dt=F(u)

POD p= Σ a i p i +pres

POD u= Σ ai u i +u res

G. system da/dt=f(a)

Mapping p= C u

MOD * u= Σ a i* u i +u *res

FTT for ures

Spatio− temporal character− ization & filters

Wavelets u= Σ aij u ij

Actuation model

CVE u=ucoh+uinc

Control design

Identify noisy subspaces and structures in kinematics

Avoid noisy subspaces and structures in controlled dynamics

Fig. 1 Principle sketch of the reduced-order modelling (ROM) strategy, targeting jet noise control

39]. Since then different directions for wavelets and turbulence have been explored, e.g. signal processing approaches, interpretations in the multifractal community, cospectra, analysis and eduction of coherent structures using experimental data. In [13, 15] the Coherent Vortex Simulation (CVS) approach was introduced to compute and to model turbulent flows. The idea of CVS is to combine nonlinear approximation with denoising and, additionally, to exploit the properties of wavelets for numerical analysis. Wavelets yield attractive discretisations for operator equations. They allow auto–adaptive discretisations by estimating the local regularity of the solution. Furthermore, many integral and differential operators have a sparse representation in a wavelet basis and can furthermore efficiently be preconditioned using diagonal scaling. For a review we refer to the book of Cohen [6]. The idea of CVS is based on filtering turbulent flows using adaptive multiresolution techniques. Here the flow is split into two parts, a coherent flow, whose evolution is deterministically computed in an adaptive basis, and an incoherent flow, which is noise-like and whose effect on the coherent flow is modelled. Applications of this filtering, called Coherent Vortex Extraction (CVE), have been presented for homogenous isotropic turbulence [12, 14, 51],

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for mixing layers [63] and for shear and rotating turbulence [28]. The methods and results of CVE for the incompressible jet are described in section 4. To distill the desired mechanistic understanding for noise control, an acoustically optimised POD generalisation is applied for the identification of ’loud’ coherent flow structures. In previous studies of compressible jets and mixing layers, typically a dimension reduction by one order of magnitude is achieved by this method, termed ’most observable decomposition’ (MOD) (cmp. [29, 61, 62]). In these investigations, MOD modes exhibit the loudest flow structures which are interpreted in terms of known physical processes. In comparison to similar generalisations of the POD method (cmp. [4, 16, 17, 25, 37, 54], MOD is more tailored for the purposes of flow control: beside MOD’s aeroacoustical least-dimensionality MOD features additional optimal properties as a design parameter. Thus, two MOD variants are proposed for controller and observer design. The reduction of the total kinetic energy in the subspace of the ’least-energetic’ variant of MOD modes causes a reduction of the far-field fluctuation, thus enabling noise suppression by conventional energybased control. The reconstruction of flow states from given aeroacoustic data, using e.g. dynamic observers, is supplied by the second variant of MOD modes, the ’leastresidual’ MOD modes. The MOD method and its results for the incompressible jet are contained in section 5. Building on the kinematic results above, a further key enabler for flow and noise control is a portfolio of Galerkin modelling techniques addressing configurationspecific physical problems of dynamical systems. The challenges for model-based control of turbulent shear flows have been addressed by auxiliary models for the pressure term, unresolved turbulence, missing dynamically important phase space directions, compressibility effects and actuation effects [31, 34, 41, 42, 47, 48, 50, 72, 75]. Furthermore, turbulence effects are modelled by a turbulence closure generalising the ansatz of finite-time thermodynamics (FTT) for the prediction of the first and second moments of Galerkin modes [45, 46]. This FTT closure enables further system reductions and fine-scale turbulent representations respecting momentum and energy balance equations for each mode. FTT has already been successfully applied to wake flows, mixing layers and homogeneous shear turbulence, i.e. examples of simple, medium and broadband dynamics. For the first time, a 30-dimensional Galerkin model of the three-dimensional round jet is extracted in section 6 — utilising above mentioned ensemble of LES velocity snapshots. Thus, existing Galerkin models of rectangular and planar jets are supplemented (cmp. e.g. [20, 40] and references therein). In the outlook of this contribution (section 7), capability of the presented ROM strategy for jet noise control is demonstrated by first results from applications.

2 Configuration and Data Base A snapshot ensemble of a three-dimensional incompressible jet at Reynolds number Re = 3600 is provided by large eddy simulation (LES). Details of the numerical simulations are illustrated in [7, 33]. For the subsequent empirical analyses, 3000


7

Fig. 2 Sketch of the flow configuration. A three-dimensional, incompressible jet is studied at Reynolds number Re = 3600

Time

snapshots of velocity and pressure field in the post-transient regime are available in a domain extending 17D (D=jet diameter) in the streamwise x-direction and out to ±3.5D in transverse y- and spanwise z-direction. In our computations, equations are discretised by a 128 × 128 × 128 mesh of grid points, equidistantly distributed in each direction. The 3000 snapshots are equidistantly distributed over a time interval of 300 convective time units (Δ t = 0.1). Furthermore, the three-dimensional vorticity field is provided for the last 1000 snapshots. The aeroacoustic far-field is monitored by the signals of 76 far-field sensors, situated at a linear array in the zero plane of the spanwise component (cmp. figure 2). The linear array is parallel to the jet axis with a distance of 30D. The sensors are

250

0.0015 0.001

230

0.0005 0

210

-0.0005 -0.001

190

-0.0015

170 150

0

10

20

30

40

50

60

70

Sensor Fig. 3 Visualisation of the aeroacoustic far-field, computed by a Greens functions-based Lighthill solver

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equidistantly distributed from x=0D to x=75D. The aeroacoustic field is computed by a Green’s functions-based Lighthill solver described e.g. in [27]. The solver is validated against the results of a Ffowcs-Williams Hawkings solver for a Ma = 0.9 jet configuration at the same Reynolds number (cmp. [21, 38]). Thus, aeroacoustic far-field data (cmp. figure 3) are provided in an interval of 100 convective time units based on the LES data described above.

3 Coherent Structure Analysis 3.1 The Proper Orthogonal Decomposition (POD) A frequently employed method to extract coherent flow structures is represented by POD of the flow velocity u (see e.g. [26]). In the POD approximation, the velocity fluctuations u are approximated by the linear expansion into POD modes ui u (x,t) ≈

N

∑ aui (t) ui (x)

(1)

i=1

where the ai := (u , ui )Ω are defined as mode coefficients of the POD modes, latter forming an orthogonal set in the topology of the L2 Hilbert space with the inner product (g, f)Ω := Ω g · f dx. POD yields least-order modal expansions for any given resolution of total ki netic energy KΩ (u) := (u , u )Ω /2 where a time-averaging operator f := T 1 f T →∞ 2T −T

lim

dt is utilised. The mathematical statement of optimality of POD is that

the averaged projection of the hydrodynamic fluctuations onto the POD modes is maximised for a given number of POD modes. The POD modes represent the critical points of the maximum problem 2 max (2) u,φ Ω , φ ∈ H u , (φ , φ )Ω = 1

formulated in a subspace Hu of the Hilbert space L2 , satisfying regularity, incompressibility and boundary conditions. By calculus of variation, POD modes thus can be defined to be the eigenfunctions of the POD Fredholm integral equation

u (x,t) ⊗ u (x ,t) · ui (x ) dx = λiu ui (x),

(3)

Ω

where ⊗ denotes the outer product of two vectors. Hilbert-Schmidt theory assures, that there is a countable infinity of eigenvalues λiu , representing the double of total kinetic energy KΩ (u), resolved by each mode. Typically, POD modes are sorted by the size of these POD eigenvalues, starting from the largest eigenvalue λ1u . Commonly, the POD decomposition is computed by Sirovich’s POD snapshot method [69] based on a given ensemble of velocity snapshots. By application to


9

further fields (e.g. vorticity, aeroacoustic pressure), POD modes are obtained which decompose these field most efficiently for the resolution their fluctuation level (in the mentioned examples, these are the time averages of enstrophy and noise respectively). Further details of POD can be found e.g. in [26].

Fig. 4 Visualisation of coherent flow structures. Isosurfaces of the streamwise component of the first three POD modes (1 to 3 from top to bottom) are displayed for positive (light) and negative (dark) values. The Cartesian coordinates are normalised by the jet diameter

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Fig. 5 Same as figure 4, but for POD modes 20, 50 and 100 from top to bottom, respectively

3.2 POD Results Every third time step of the three-dimensional velocity snapshots is employed for POD analysis (1000 snapshots with Δ t = 0.3). First and higher POD modes of the incompressible flow are visualised in the figures 4 and 5. The most energetic structures are dominated by longitudinal streaks downstream from the breakdown of


11

%

75 − 50 −

−

0

−

0

−

25 −

250

500

750

N

Fig. 6 Residual of total kinetic energy of the POD approximation based on a velocity ensemble of 1000 snapshots over 300 convective time units. Displayed are the percentages of the non-resolved energy over the number of employed POD modes

% 75 − 50 −

1

−

0

−

25 −

10

100

N

Fig. 7 Resolution of POD of a vorticity field ensemble of 1000 snapshots in a time interval of 100 convective time units. Displayed are the percentages of the resolved average of enstrophy over the number of employed POD modes (solid line) the latter being scaled logarithmically. This resolution is compared to the POD resolution of total kinetic energy of the corresponding velocity field ensemble (dashed line)

the potential core. Helical structures become more dominant in higher modes. 90% of total kinetic energy of the incompressible jet is resolved by 284 modes (cmp. figure 6). Thus, POD reveals a large number of dynamically important degrees of freedom, which enables flow modelling but is not practical to derive the desired mechanistic understanding (cmp. figure 6). These results are consistent with wellknown previous investigations (cmp. e.g. [17, 21]).

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A similar POD analysis is employed for the last 1000 snapshots of the vorticity field (Δ t = 0.1). Here, POD reveals even higher dimensions than the POD analysis of the corresponding velocity snapshot ensemble (1000 snapshots, Δ t = 0.1). 215 POD modes are needed to resolve 90% of time averaged enstrophy while 90% of total kinetic energy of the corresponding velocity is resolved by only 101 POD modes (cmp. figure 7).

4 Wavelet Based Coherent Vortex Extraction 4.1 Coherent Vortex Extraction (CVE) Method Coherent vortices are observed in many turbulent flows which seem furthermore to be imbedded in a random background sea. Hence, a denoising procedure may allow to eliminate the noise in a given flow realization and the remaining part can then be called coherent vortices. In [14, 15], a wavelet-based method to extract coherent vortices out of both two- and three-dimensional turbulent flows was proposed, which is directly motivated by denoising. The idea is to apply the orthogonal wavelet decomposition to the vorticity field ω at a given time instant t with resolution N. In the following, we first fix the notation for the wavelet decomposition of a three dimensional vector field and summarise the main ideas of wavelet based coherent vortex extraction. For more details on the orthogonal wavelet transform, its extension to higher dimensions, we refer the reader to textbooks, e.g., [8]. For more details on the coherent vortex extraction method we refer to the original papers. A vector field u(x) is decomposed into an orthogonal wavelet series u(x) =

∑ uλ ψλ (x)

(4)

λ ∈Λ

where x ∈ Ω = [0, 17.87D] × [−3.5D, 3.5D] × [−3.5D, 3.5D], D is the diameter of the jet, and the multi–index λ = ( j, ix , iy , iz , d) denotes the scale j, the position i = (ix , iy , iz ) and the seven possible directions d = 1, ..., 7 of the wavelets.

The set Λ = λ = ( j, ix , iy , iz , d), j = 0, ..., ix , iy , iz = 0, ..., 2 j − 1 andd = 1, ..., 7 defines the corresponding index set. Due to orthogonality the coefficients are given by uλ = (u, ψλ )Ω . The coefficients measure the fluctuations of u around scale 2− j and around position i/2 j in one of the seven possible directions. The fast wavelet transform [8] yields an efficient algorithm to compute the N wavelet coefficients uλ from the N grid points values of u and has linear complexity. Here we have chosen the Coiflet 12 wavelet, which has 4 vanishing moments and is appropriate to represent the current flow simulations. The idea of the coherent vortex extraction method can be summarised in the following three step procedure: λ using the fast • Decomposition: compute the wavelet coefficients of vorticity ω wavelet transform.


13

λ , • Thresholding: apply the thresholding function ρε to the wavelet coefficients ω thus reducing the relative importance of the coefficients with small absolute value. • Reconstruction: reconstruct the coherent vorticity ωC from the thresholded wavelet coefficients using the fast inverse wavelet transform. The incoherent vorticity ωI is obtained by simple subtraction, ωI = ω − ωC . The thresholding function ρ corresponds to a if |a| > ε ρε (a) = 0 if |a| ≤ ε

(5)

where ε denotes the threshold. The thresholding parameter ε depends on the variance of the incoherent vorticity σn and on the sample size N. The threshold √ εD = σn 2 ln N (6) is motivated from denoising theory [9]. However, the variance of the incoherent vorticity is unknown, and has to be estimated from the available total vorticity ω . As a first guess we take the variance of the total vorticity, which overestimates the variance of the incoherent vorticity. Thus we split the field into coherent and incoherent parts and then take the variance of the incoherent vorticity as a new improved estimator. In [1] we have developed an iterative algorithm for this task, based on the method presented in [15]. Here we decided to perform one iteration step only which can be justified by the fast convergence of the iterative procedure and by the fact that the computational effort is minimised. Using the above algorithm, the flow is decomposed into two parts: a coherent flow, corresponding to the coherent vortices, and an incoherent flow, corresponding to the background noise [15]. This decomposition yields ω = ωC + ωI . Due to orthogonality we have (ωC , ωI )Ω = 0 and hence it follows that enstrophy is conserved, i.e., Z = ZC + ZI where Z = 12 (ω , ω )Ω . Let us mention that the complexity of the Fast Wavelet Transform (FWT) is of O(N), where N denotes the total number of grid points.

4.2 CVE Results The CVE method is applied to the vorticity of the turbulent incompressible jet at nondimensional time t = 130.1. The resolution of the computation is N = 1283. The CVE method decomposes the total vorticity ω into coherent vorticity ωC and incoherent vorticity ωI . The results of the decomposition are summarised in table 1. Figure 8 shows the enstrophy kept in the N strongest wavelet modes. The threshold is εD = 1.155, and the percentage of modes used to represent the coherent component of the vorticity field, a measure for the compression rate, is 4.75%. Therefore, only a hundred thousand modes instead of two million modes are used to represent the coherent part of the flow field, which resolves more than 91% of the

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Table 1 Statistical properties of the total, coherent and incoherent vorticity fields of one snapshot at time t=130.1 N [%] Z Z [%] Total 100 0.2972 100 Coherent 4.75 0.2715 91.35 Incoherent 95.25 0.0257 8.65

||ω ||L∞ 11.998 12.304 2.116

100 90 80 70

Z [%]

60 50 40 30 20 10 0 0.1

1 N [%]

10

100

Fig. 8 Resolved enstrophy as a function of the number of retained wavelet coefficient which have been sorted by the order of magnitude. Z and N are normalised with the reference value at full resolution and expressed in percent

Table 2 Scale-dependent compression rates Scale j Nretained /Ntotal [%] Nretained Ntotal 0 100.00 7 7 1 96.43 54 56 2 95.09 426 448 3 60.04 2151 3584 4 28.51 8175 28672 5 12.92 29632 229376 6 3.23 59255 1835008

total enstrophy. To gain further insight we now consider the percentage of retained wavelet coefficients at different scale indices j. The scale index j is related to a mean wavenumber k j = k0 2 j , where k0 is the centroid wavenumber of the chosen wavelet [11]. The wavelet representation implies that there are 7 · 23 j wavelet coefficients at a given scale j. Table 2 shows the percentage of retained wavelet coefficients at the scale index j. At large scales, i.e., for j ≤ 2, almost all coefficients are retained and correspond to the coherent part. At smaller scales, where the total number of


15

Fig. 9 Isosurfaces of the total (top), coherent (middle) and incoherent (bottom) vorticity. The nondimensional values of the isosurfaces are |ω | = 0.7

coefficients dramatically increases, the percentage of retained coefficients decreases strongly, and at the smallest scale only about 3% of the coefficients are retained. Figure 9 shows isosurfaces of vorticity at nondimensional time t = 130.1. The total vorticity shown in figure 9 (top) is characterised by well developed vortical

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M. Schlegel et al. 0

10

total coherent incoherent

−1

10

k1/3 −2

10

−3

Z(k)

10

−4

10

−5

10

k4 −6

10

−7

10

−8

10

1

2

10 k

10

Fig. 10 Enstrophy spectra of total, coherent, and incoherent vorticity. For details, see text

1

10

total coherent incoherent Gaussian fit

0

10

−1

P(ω)

10

−2

10

−3

10

−4

10

−5

10

−10

−5

0 |ω|

5

10

Fig. 11 PDF of total, coherent and incoherent vorticity, together with a Gaussian fit

structures. Those structures are retained in the coherent field, visualised in figure 9 (middle). The incoherent field, shown in figure 9 (bottom), contains no organised vortical structures and resembles noise. Its norm is about a factor 10 smaller with respect to the total vorticity. The spectral distribution of the total, coherent, and incoherent enstrophy is compared in figure 10. The spectra of the total and coherent fields coincide up to a wave number k ≈ 20, and a faster decay of the spectrum is observed for the coherent field in the dissipative range with wave-numbers k > 50. The spectrum of the incoherent field contains contributions at all wave-numbers but is significant only in the


17

dissipative range. We also observe that the total and coherent spectra exhibit a k1/3 range which is in agreement with Kolmogorov’s k−5/3 law, as the enstrophy spectrum is k2 times the energy spectrum. The spectrum of incoherent vorticity shows some k4 scaling, which corresponds to energy equipartition, i.e. k2 scaling of the energy spectrum. A study of the probability density functions (PDF) of vorticity yields information about the higher order statistics of the flow. Figure 11 shows the PDF of total, coherent, and incoherent vorticity. First, we observe that the PDF of the total and coherent vorticity almost perfectly coincide, and exhibit exponential tails. The PDF of the incoherent vorticity has a strongly reduced variance, and agrees reasonably well with the Gaussian.

5 Identification of Loud Jet Flow Structures 5.1 The Most Observable Decomposition (MOD) To obtain the desired physical understanding of jet noise generation, an aeroacoustically optimised generalisation of POD is proposed, termed most observable decomposition (MOD). This method is described in [29, 61, 62] in detail. Here, a short overview is given. In the following, hydrodynamic fluctuations and the fluctuations of the aeroacoustic far-field are considered to be prefiltered by POD decomposition via the approximations (1) and p (y,t) ≈

J

∑ aip(t) pi (y)

(7)

i=1

in the given domains Ω and Γ of near- and far-field respectively. Modes and mode coefficients of the POD of the aeroacoustic far-field are denoted by pi and aip , respectively. The latter POD modes decompose the pressure far-field most efficiently for the resolution the aeroacoustic noise level. The main idea of MOD is to impose this optimality to the optimality of the MOD approximation u (x,t) ≈

J

∑ a∗i (t) u∗i (x)

(8)

i=1

of the hydrodynamic field, where the sense of optimality is defined by the optimal resolution of the goal functional, given by the correlated level ZΓC (u) :=

p u , p u Γ /2

(9)

of fluctuations of the pressure far-field. From this perspective, the question of how to design optimality of MOD requires to know how to model the relationship of hydrodynamic field and pressure far-field.

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Table 3 Minimum principles and control perspectives of the two types of MOD MOD variant

’least-residual’ (LR-) MOD ’least-energetic’ (LE-) MOD

minimum principle of MOD approximation

minimisation of flow attractor residual

minimisation of hydrodynamic fluctuation level

control goal

flow reconstruction by a dynamic observer

reduction of total kinetic energy in loud subspace

In the most observable decomposition, a linear relationship between the hydrodynamic and aeroacoustic fluctuations is proposed after the propagation time τ p (t + τ ) = C u (t) ,

(10)

where both fluctuation variables can be considered to be represented by the vector of the respective mode coefficients, applying the POD prefilter. Thus, the linear mapping C from hydrodynamic fluctuations onto fluctuations of the pressure farfield can be identified by linear stochastic estimation utilising the POD coefficients of hydrodynamic field and pressure far-field. In MOD, it is assumed that the main events of the generation of jet noise are captured in average by the linear mapping defined in equation (10). Validity of this assumption is confirmed by a considerable body of physical evidence: the shearnoise originated from a linear source term of the velocity fluctuations been shown to dominate in free-jets in terms of the hydrodynamic, turbulent pressures [74], and to correlate better with the far-field pressure than the self-noise originated from a quadratic source term of the velocity fluctuations [30, 32, 52, 59, 60, 65]; In [5] it is furthermore demonstrated that the coherent flow structures generate noise via a linear mechanism in the region upstream of the end of the potential core. The MOD modes are obtained by the pseudoinverse images of the far-field POD modes u∗i = C− pi ,

(11)

and employing orthonormalisation. Because the definition of the pseudoinverse is typically not unique, this equation represents an ill-posed problem. Additional constraints are required to modify the problem to become well-posed. Two optimal principles are considered, addressed to flow control purposes as demonstrated in table 3 thus defining two MOD modes variants Like in the POD approach, analogues of the POD maximum problem (2) and the Fredholm integral equation (3) can be obtained for both MOD variants. The POD eigenvalues of the far-field shall coincide with the MOD eigenvalues, representing the double of the resolved portion of the correlated noise ZΓC (u) of each mode. Commonly, (LR- or LE-) MOD modes are sorted by the size of the MOD eigenvalues, starting from the largest eigenvalue.


19

%

75 − 50 −

−

0

−

0

−

25 −

25

50

75

N

Fig. 12 Residual of correlated noise of the MOD approximation. Displayed are the percentages of the non-resolved correlated noise over the number of utilised (LR- or LE-) MOD modes

5.2 MOD Results MOD results are obtained based on the POD results of subsection 3.2 and the aeroacoustic far-field data as described in section 2. Dimension reduction capability by one order of magnitude in comparison to POD dimensions is demonstrated. 90% of correlated noise ZΓC is resolved by only 33 MOD modes (cmp. Fig. 12). This result fits well with an MOD dimension of 24 for the Ma = 0.9 jet with the same Reynolds number, obtained in [29]. The first four LR-MOD modes are visualised in figure 13, showing dominant helical structures around the breakdown region of the jet potential core reminiscent to results of [16].

6 Galerkin Modelling A low-dimensional Galerkin model of the three-dimensional, incompressible jet is presented in this section. We employ 30 POD modes of the whole domain, representing 39% of total kinetic energy. The Galerkin system a˙i =

1 Re

30

∑ li j a j

j=0

30

+

∑

qi jk a j ak for i = 1, . . . , 30 ,

j,k=0

is derived from the POD Galerkin approximation (1) with a standard projection on the Navier-Stokes equations [26]. The system coefficients are defined by li j := (ui , Δ u j )Ω and qi jk := (ui , ∇ · (u j uk ))Ω , using the mean flow u0 := u. To enhance compactness of notation, the coefficient a0 ≡ 1 is introduced as well. Characteristic frequencies of the POD mode coefficients of the Navier-Stokes attractor are reproduced by the Galerkin model as illustrated in figure 14.

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Fig. 13 Visualisation of ’loud’ jet flow structures. Isosurfaces of the streamwise components of the first three (from top to bottom) least-residual MOD modes are shown for positive (bright) and negative (dark) values. The Cartesian coordinates are normalised by the jet diameter

Following Rempfer and Fasel [56], neglected fine-scale effects are modelled based on a calibration of modal turbulent eddy viscosities (cmp. [47] for further details). With this model, the coefficients amplitudes coincides as well (cmp. figure 14 again).


21

0.8 8 6

a5 4 2 0

0 2

−0.4 4 6 −0.8 8 0 0 0.6 6 5 4 a 24 3 2 1 0 0 1 2 −0.3 3 4 −0.6 5 0 0

50

100 100

150

200 t

250

300 300

50

100 100

150

200 t

250

300 300

Fig. 14 Two POD Fourier coefficients with low (top) and high (bottom) frequencies over an interval of 300 convective time units. For both coefficients, similar frequency behaviour is observed on the DNS represented Navier-Stokes attractor (dotted line) and in the Galerkin model (solid line)

7 Outlook to Applications of Jet Noise Control Two key enabler of jet noise control design are represented by the Galerkin system and the LE-MOD approach, enabled by suppression of the total kinetic energy contained in the ’loud’ LE-MOD velocity subspace, which is irreducible in respect to maintain the noise level. Thus energy-based Lyapunov design is pursued for the suppression of the energy flow into the LE-MOD subspace. As one example of control, we have modelled plasma actuation, which has reduced jet noise in flow control experiments [58] by 0.5 to over 1.0 dB. Two plasma actuators, manipulating respectively the flow in streamwise and transverse direction at the breakdown of the jet potential core, are implemented into a 100 POD mode

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400 0 (p’,p’)* 2

[%] 0

200 0 100 0

00

00

20 20

40 40

60 60

80

t

100 100

Fig. 15 Noise control via suppression of loud flow structures (LE-MOD modes). Displayed are evolutions of the level of far-field fluctuations (noise) over 100 convective time units. The former are expressed in percent of the correlated noise level ZΓC of the natural jet flow. The most noise generating flow events in the natural Galerkin model dynamics (dashed line) are suppressed by energy-based flow control (solid line). Thus a mean noise reduction by 2 dB is achieved

Galerkin model of the jet flow in thin layer parallel and symmetric to the z = 0 plane (cmp. [61]). To identify the Galerkin system, an ansatz of Rempfer [55] is pursued, according to which Galerkin projections can be effected in infinitely thin slices. On this subdomain, 100 POD modes are still required to resolve more than 80% of total kinetic energy. The actuation of the two plasma actuators is implemented employing identification of volume forces as described e.g. in [47]. For stabilisation of the controlled simulation, third order terms are implemented in the Galerkin model. Noise reduction by approximately 2 dB has been performed. Under action of the plasma actuators, the energy flow into the loud subspace is penalised based on Lyapunov control design. Thus, the most noise generating flow events are mitigated as shown in figure 15. In future application, a significant enhancement of noise reduction is expected from local actuations at the nozzle exit, implemented in the Galerkin model of the whole three-dimensional jet. Here, moreover, the modelling of the high-dimensional flow residuum via FTT opens the path to the first fully nonlinear infinite horizon control. In future applications. this form of control design is expected to overcome serious challenges of current (locally) linear design with vortex merging and multiple-scale physics.

8 Conclusions A reduced-order modelling (ROM) strategy for jet noise control is proposed pursuing the path of


(i) (ii) (iii)

23

identification and dynamic modelling of coherent structures, identification of loud structures and suppression of loud flow structures.

Coherent structures of the three-dimensional, incompressible jet are identified by proper orthogonal decomposition (POD). 284 POD modes are required to resolve 90% of the total kinetic energy. The dynamics of the most coherent structures, representing 39% of total kinetic energy, are reproduced by the 30-dimensional Galerkin model. Wavelet-based Coherent Vortex Extraction (CVE) is applied to LES simulations of a free jet. This method allows for an efficient extraction and analysis of vortical structures contained in such flows. It was found that few (4.75%) wavelet coefficients represent the coherent vortices of the flow. The results presented here motivate coherent vortex simulations (CVS) of turbulent flows for aeroacoustics. First results of CVS for three-dimensional mixing layers are presented in [63] and are promising. Further analysis of vorticity structures, enstrophy spectra and higher-order statistics suggests that the dynamical information of the total field is retained by the coherent field. It can be anticipated that a temporal integration of the coherent field will result in an evolution similar to that of the total flow field. For the incoherent flow field a viscous decay of the fluctuations can be anticipated. A mechanistic understanding for jet noise control has been extracted using a generalisation of POD, we term most observable decomposition (MOD) [29, 62]. The most sound-producing dynamics of turbulent jets are identified by this method. A reduction of dynamically relevant degrees of freedom against POD is achieved from MOD application to the incompressible jet. Two variants of MOD are tailored for the purposes of noise control design. In a first example of jet noise control, control capability of the proposed ROM strategy is demonstrated achieving a reduction of jet noise by 2 dB. From more comprehensive future investigations including the implementation of experimentally realisable actuations and finite-time thermodynamics (FTT) modelling (see [45, 46]), a significant enhancement of noise reduction is expected. It should be noted, that the presented methods are applicable to experimental data, e.g. PIV measurements, as well. Acknowledgements. We appreciate valuable stimulating discussions with Boye Ahlborn, JeanPaul Bonnet, Jacques Borée, Laurent Brizzi, Laurent Cordier, Joël Delville, Andreas Dillmann, Helmut Eckelmann, Dandy Eschricht, Jonathan Freund, William K. George, Elmar Gröschel, Hans-Christian Hege, Peter Jordan, Rudibert King, Oliver Lehmann, Claus-Dieter Munz, Mark Pastoor, Ulrich Rist, Bernd Rummler, Wolfgang Schröder, Jörn Sesterhenn, Oksana Stalnov, Frank Thiele, Charles Tinney, Maja Wänström, Mingjun Wei, and Tino Weinkauf. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grants NO 258/1-1, NO 258/2-3 and SCHL 586/1-1. Parts of the CPU time was allocated free by the CNRS supercomputing centre IDRIS under project CP2-80912. Michael Schlegel and Bernd R. Noack acknowledge visitor grants at LEA Poitiers from CNRS and AIRBUS, in particular from the EFFC and IFFC programs. Marie Farge, Dmitry Kolomenskiy and Kai Schneider thankfully acknowledge financial support from CNRS, department ST2I and the Agence Nationale de la Recherche project ’M2TFP’. Additionally, Dmitry Kolomenskiy and Kai Schneider are thankful to the Deutsch-Französische Hochschule

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in Saarbrücken, project ’S-GRK-ED-04-05’. The work of Gilead Tadmor was partially supported by the National Science Foundation (NSF) under grant number 0524070. Moreover all authors acknowledge excellent working conditions and support of the DFG-CNRS Research Group FOR 508 ”Noise Generation in Turbulent Flows” and the DFG funded Collaborative Research Center Sfb 557 ”Control of Complex Turbulent Shear Flow” at the Berlin Institute of Technology. The 3D flow visualisation has been performed with Amira Software (Zuse Institute Berlin). We are grateful for outstanding computer and software support from Martin Franke and Lars Oergel.

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40. Moreno, D., Korthapalli, A., Alkislar, M.B., Lourenco, L.M.: Low-dimensional model of a supersonic rectangular jet. Phys. Rev. E 69(026304), 1–12 (2004) 41. Morzyński, M., Stankiewicz, W., Noack, B.R., King, R., Thiele, F., Tadmor, G.: Continuous mode interpolation for control-oriented models of fluid flow. In: King, R. (ed.) Active Flow Control, Papers contributed to the Conference Active Flow Control 2006, Berlin, Germany, September 27–29, 2006. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 95, pp. 260–278. Springer, Heidelberg (2007) 42. Morzyński, M., Stankiewicz, W., Noack, B.R., Thiele, F., Tadmor, G.: Generalized mean-field model for flow control using a continuous mode interpolation. AIAA-Paper, pp. 2006–3488 (2006) 43. Narayanan, S., Barooah, P., Cohen, J.M.: Dynamics and control of an isolated jet in cross flow. AIAA J. 41, 2316–2330 (2003) 44. Narayanan, S., Noack, B.R., Meiburg, E.: Reduced-order dynamical modeling of sound generation from a jet. AIAA-Paper, 2002–0073 (2002) 45. Noack, B.R., Schlegel, M., Morzyński, M., Tadmor, G.: System reduction strategy for Galerkin models of fluid flows. Internat. J. Numer. Meth. Fluids (2008) (manuscript submitted for publication) 46. Noack, B.R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P., Tadmor, G.: A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilib. Thermodyn. 33(2), 103–148 (2008) 47. Noack, B.R.: Niederdimensionale Galerkin-Modelle für laminare und transitionelle freie Scherströmungen (transl: Low-dimensional Galerkin models of laminar and transitional free shear flows). Habilitation thesis, Fakultät V – Verkehrs- und Maschinensysteme, Berlin Institute of Technology, Germany (2005) 48. Noack, B.R., Papas, P., Monkewitz, P.A.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365 (2005) 49. Noack, B.R., Mezić, I., Tadmor, G., Banaszuk, A.: Optimal mixing in recirculation zones. Phys. Fluids 16(4), 867–888 (2004) 50. Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of lowdimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003) 51. Okamoto, N., Yoshimatsu, K., Schneider, K., Farge, M., Kaneda, Y.: Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint. Phys. Fluids 19(115109), 1–13 (2007) 52. Panda, J., Seasholtz, R.G., Elam, K.A.: Investigation of noise sources in high-speed jets via correlation measurements. J. Fluid Mech. 537, 349–385 (2005) 53. Pastoor, M., Henning, L., Noack, B.R., King, R., Tadmor, G.: Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161–196 (2008) 54. Picard, C., Delville, J.: Pressure velocity coupling in a subsonic round jet. Int. J. Heat Fluid Flow 21, 359–364 (2000) 55. Rempfer, D.: Empirische Eigenfunktionen und Galerkin-Projektionen zur Beschreibung des laminar-turbulenten Grenzschichtumschlags (transl.: Empirical eigenfunctions and Galerkin projection for the description of the laminar-turbulent boundary-layer transition). Habilitation Thesis, Universität Stuttgart (1995) 56. Rempfer, D., Fasel, F.H.: Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257–283 (1994) 57. Ribner, H.S.: Quadrupole correlations governing the pattern of jet noise. J. Fluid Mech. 38, 1–24 (1969) 58. Samimy, M., Kim, J.-H., Kastner, J., Adamovich, I., Utkin, Y.: Active Control of a Ma=0.9 Jet for Noise Mitigation Using Plasma Actuators. AIAA J. 45, 890–901 (2007) 59. Schaffar, M., Hancy, J.P.: Investigation of the noise emitting zones of a cold jet via causality correlations. J. Sound Vibr. 81(3), 377–391 (1982) 60. Scharton, T.D., White, P.H.: Simple Pressure Source Model of Jet Noise. J. Acoust. Soc. Am. 52(1), 399–412 (1972)


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61. Schlegel, M., Noack, B.R., Jordan, P., Tadmor, G.: Least-order modal flow decompositions for aerodynamic and aeroacoustic goal functionals. In: Proceedings of the 2nd Int. Conf. on Jets, Wakes and Separated Flows, Berlin Institute of Technology, Berlin, Germany (2008) 62. Schlegel, M., Noack, B.R., Lehmann, O., Gröschel, E., Schröder, W., Wei, M., Freund, J., Jordan, P.: Least-order flow representations, optimised for aeroacoustics and aerodynamics. J. Fluid Mech (2008) (manuscript in preparation for publication) 63. Schneider, K., Farge, M., Pellegrino, G., Rogers, M.: Coherent vortex simulation of threedimensional turbulent mixing layers using orthogonal wavelets. J. Fluid Mech. 534, 39–66 (2005) 64. Seiner, J.M.: A new rational approach to jet noise reduction. Theoret. Comput. Fluid Dyn. 10, 373–383 (1998) 65. Seiner, J.M., Reetoff, G.: On the distribution of source coherency in subsonic jets. AIAA-paper 1974-4, pp. 1–14 (1974) 66. Siegel, S.G., Seidel, J., Fagley, C., Luchtenburg, D.M., Cohen, K., McLaughlin, T.: Lowdimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech. 610, 1–43 (2008) 67. Siegel, S.G., Aradag, S., Seidel, J., Cohen, K., McLaughlin, T.: Low Dimensional POD Based Estimation of a 3D Turbulent Separated Flow. AIAA-Paper, 2007-0112 (2007) 68. Siegel, S., Cohen, K., McLaughlin, T.: Feedback control of a circular cylinder wake in experiment and simulation. AIAA-Paper, pp. 2003–3571 (2003) 69. Sirovich, L.: Turbulence and the dynamics of coherent structures, Part I: Coherent structures. Quart. Appl. Math. XLV, 561–571 (1987) 70. Soteriou, M.: Vortex element method — expansion about incompressible flow computation of noise generation by subsonic shear flows — the impact of external forcing. J. Turb. 4, 1–9 (2003) 71. Soteriou, M., Reba, R., Maeder, T.: Numerical study of the impact of streamwise vorticity on noise generation by jet flows. AIAA-Paper, pp. 2002–2480 (2002) 72. Tadmor, G., Noack, B.R., Morzyński, M.: Control oriented models and feedback design in fluid flow systems: A review. In: Proceedings of the 14th Mediteranean Conference on Control and Automation (2006) (invited paper) 73. Tadmor, G., Noack, B.R., Morzyński, M., Siegel, S.: Low-dimensional models for feedback flow control. Part II: Observer and controller design. AIAA-Paper, pp. 2004–2409 (2004) 74. Tinney, C.E., Jordan, P., Delville, J., Hall, A.M., Glauser, M.N.: A time-resolved estimate of the turbulence and sound source mechanism in a subsonic jet flow. J. Turbulence 8(1), 1–20 (2007) ¨ 75. Yan, P., Debiasi, M., Yuan, X., Caraballo, E., Serrani, A., Ozbay, H., Myatt, J.M., Samimy, M.: Modeling and Feedback Control for Subsonic Cavity Flows: A Collaborative Approach. In: Proc. of the 44th IEEE Conference on Decision and Control and European Control Conference CDC-ECC 2005, pp. 5492–5497 (2005)

Numerical Simulation of Supersonic Jet Noise Jan Schulze, Jörn Sesterhenn, Peter Schmid, Christophe Bogey, Nicolas de Cacqueray, Julien Berland, and Christophe Bailly

Abstract. Jets with complex shock-cell structures appear in numerous technological applications. Most supersonic jets used in aeronautics will be imperfectly expanded in flight, even those from carefully designed convergent-divergent nozzles. The adaption to the ambient pressure takes place in a sequence of oblique shocks which interact with the free shear layers and produce noise. The shock/shear-layer interaction emanates a broadband noise component. This may trigger the young shear layer at the nozzle, forming a feedback loop which results in a discrete noise component called screech . Both components are undesirable from structural and environmental (cabin noise) points of view. Screech tones are known to produce sound pressure levels of 160 dB and beyond. The focus of this research project lies in the numerical simulation of jet screech. Different numerical methods are shown with LES and DNS applications of a planar rectangular and three-dimensional jet with overset grid techniques to include complex geometries for the jet nozzle. Furthermore, a shock-capturing method is developed for high-order aeroacoustic computations. It consists in applying an adaptive second-order conservative filtering to handle discontinuities, in combination with a background selective filtering to remove grid-to-grid oscillations. The magnitude of the shock-capturing filtering is determined dynamically from the flow solutions using a procedure based on a Jameson-like shock detector. Results obtained for a shock-propagation problem are shown to validate the method, which will be now used for the simulations of supersonic jets. Jan Schulze · Jörn Sesterhenn Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, 85577 Neubiberg/München, Germany [email protected] Peter Schmid Laboratoire d’Hydrodynamique, Ecole Polytechnique, 91128 Palaiseau Cedex France, France [email protected] Christophe Bogey · Nicolas de Cacqueray · Julien Berland · Christophe Bailly Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, 69134 Ecully Cedex, France [email protected]


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1 Introduction

no

ise

Considering a supersonic jet, e.g. at the exit of a jet engine, in the over- or underexpanded case, a regular pattern of compression and expansion waves will be found within the supersonic part of the jet flow. A compression wave incident on the sonic line will be reflected as an expansion, and vice versa, cf. Fig. 1. At the location of interaction between the compression wave and the turbulent mixing layer, acoustic waves are generated. This shock-induced noise also plays an important role in what is called jet screech. This phenomenon is caused by shock-induced acoustic waves traveling upstream and forcing the ’young’ shear-layer at the nozzle exit. At this point Kelvin-Helmholtz instabilities are growing to vortices, transported downstream and interacting with the shock tips which are emanating noise again and closing a feedback loop. Experimental results indicate particularly high sound pressure levels of up to 160 dB and even beyond. Prediction and reduction of shock-induced noise, produced by modern civil aircraft with jet propulsion, traveling at high subsonic or supersonic Mach numbers is a matter of particular interest. Besides pollution of the environment by the radiated sound, the latter can also lead to high dynamic loads on parts of the aircraft causing structural fatigue and even destroy them. In the present research project, the methods of Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) will be used to compute a supersonic, threedimensional and rectangular jet that is not perfectly expanded, as it is found at the nozzle exit of jet engines for aircraft. Numerical methods of high order of accuracy are chosen for the solution of the compressible Navier-Stokes equations, which gives us the possibility to compute the sound field, generated by the supersonic jets, directly.

p_2

m co n ex pa

ns

io

p_1

sc

M>1

pr

ho ck

es sio

n

nozzle

nozzle vortices in the turbulent mixing layer

Fig. 1 Schematic view of the interaction of shock and turbulent mixing layer in a jet with emanated noise. Over-expanded ⇒ p2 > p1

Numerical Simulation of Supersonic Jet Noise

31

To properly deal with shocks in numerical simulations, a shock-capturing methodology based on selective filtering has also been developed. The need for filtering high-frequency waves is indeed a recurrent issue in high-order simulations, which has led to the design of selective filters, e.g. by Lele [14], Visbal & Gaitonde [27] and Bogey & Bailly [3]. These filters can be used for strongly non-linear problems, such as the generation of screech noise in supersonic jets as performed in [2], but it is generally recognized that they are not well suited to solutions including discontinuities. Near shocks, high-order schemes might indeed generate spurious Gibbs oscillations due to spectral truncation in the wavenumber space. Therefore, the usual approach to ensure stability for shocked flows is based on shock-capturing schemes. Such schemes might however be too dissipative for unsteady problems. In that case, their spectral properties have to be analyzed in the Fourier space [18], and their accuracy has to be checked by solving demonstrative test cases [7]. An alternative approach consists in developing an adaptive shock-capturing filtering, which is also appropriate for high-order simulations , as done for instance by Tam & Shen [26] and by Kim & Lee [12]. In the present study, a spatial filtering is thus proposed to deal with shocks in high-order non-linear computations in which a background selective filtering is implemented to remove grid-to-grid oscillations. To smooth shocks, the filtering is of second order, and its magnitude is adjusted dynamically from the flow variables so that it is negligible for linear propagation and for vortical disturbances. A procedure of shock detection is especially proposed so as to evaluate the filtering amplitude from the magnitude of the high wavenumbers of the solutions. In order to obtain a strong conservative form, the filtering procedure is also written as the difference between damping fluxes taken at the interface of two adjacent cells. In what follows, the shock-capturing method is first presented, and solutions calculated for a shock-propagation problem are then shown. More details about the method and results obtained for other test cases including linear sound propagation, vortex convection, shock-acoustic interactions in a transonic nozzle and shock-vortex interactions can be found in [6]. The organization of the paper is a follows: in the second section, the theoretical background of the used shock capturing method is presented. Recent results on DNS and LES are shown in section three using different numerical implementations.

2 Development of a Shock-Capturing Methodology 2.1 Methodology A procedure of shock detection is first derived to estimate the strength of the shockcapturing filtering from the flow variables, so that it should be significant around discontinuities but negligible everywhere else. More precisely, in order to indicate the presence of shocks, a shock detector, roughly similar to that formulated by Jameson et al. [11] making use of the second derivative of pressure, is evaluated from the

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magnitude of the high-wavenumber components of a flow variable that can be either pressure or dilatation. In some cases the use of pressure to detect shocks might however not be appropriate for distinguishing between turbulent fluctuations and shocks in an unambiguous manner. To deal with this deficiency, as also suggested previously by Ducros et al. [9], a possibility is to take into account the local property of compressibility. This led us here to perform the shock detection from dilatation Θ = ∇ · u rather than from pressure. To determine the shock sensor, the high-wavenumber components are first extracted from variable Θ using a Laplacian filter, yielding, at node i DΘ i =

1 (−Θi+1 + 2Θi − Θi−1) 4

(1)

The magnitude of the high-passed filtered dilatation is then calculated as DΘ magn = i

1 (DΘ i − DΘ i+1 )2 + (DΘ i − DΘ i−1 )2 2

(2)

and the shock sensor is defined as the ratio r expressed as ri =

DΘ magn i +ε c2i /Δ x2

(3)

where c2i = γ pi /ρi is the square of the sound speed, Δ x is the mesh spacing, and ε = 10−16 is introduced to avoid numerical divergence later in equation (4). Once the value of the shock detector r is known, the strength of the filtering has to be given. In the present approach, a threshold parameter rth is used to specify the regions where the shock-capturing filtering is employed. The filtering magnitude is evaluated by the function rth 1 rth (4) 1− σisc = + 1 − 2 ri ri For ri ≤ rth , the filtering magnitude is σisc = 0 as required. For ri > rth , that is when the level of the high-wavenumber components of dilatation are appreciable, one gets 0 < σisc < 1, and in particular σisc → 1 for ri → +∞. In this way, the second-order filter is only switched on when the dilatation gradients are strong. The threshold parameter rth is typically to be set between 10−6 and 10−4, a lower value corresponding to an application of the shock-capturing filtering on a wider region. From the results obtained for different non-linear test cases, a value of rth = 10−5 can however be recommended. The shock-capturing filtering is applied at each time step just after the background selective filtering removing grid-to-grid oscillations. Since its magnitude depending on the shock detection varies, the filtering operation is written in a conservative form as the difference between two damping fluxes taken at the interface of two adjacent cells as recommended by Kim & Lee [12] for instance. At


33

Table 1 Coefficients c j for conservative shock-capturing filtering: standard 2nd-order filter (Fo2), and optimized 2nd-order filter (Fopt), with c1− j = −c j Fo2 Fopt c1 −1/4 −0.210383 c2 0 0.039617

point i on an uniform grid, the conservative variables U are thus filtered explicitly to yield the variables sc sc sc sc Uisc = Ui − σi+ (5) 1 Di+ 1 − σi− 1 Di− 1 2

2

2

2

where the filtering strength 0 ≤ σ ≤ 1 is specified by the dynamic procedure presented previously, and the damping functions Dsc and Dsc are estimated from the i+ 1 i− 1 sc

2

variables U using the following interpolations = Dsc i+ 1 2

n

∑

c jUi+ j

2

Dsc = i− 1

and

2

j=1−n

n

∑

c jUi+ j−1

(6)

j=1−n

To determine the coefficients c j of the 2n-point interpolation defining the damping functions, one considers the non-conservative form of the filtering Uisc = Ui − σisc

n

∑

d jUi+ j

(7)

j=−n

and notes that equations (5) and (7) must be equivalent when the filtering magnitude is uniform. For a given n, the coefficients c j are then directly obtained from the coefficients d j of the corresponding non-conservative centered filter. The values found for the standard 2nd-order filter, here referred to as Fo2, are collected in Table 1. The coefficients c j calculated for another second-order filter are also reported in the Table. This filter was built up in [6] using an optimization procedure in the wavenumber space so that it displays dissipation features similar to those of the standard filter Fo2, but generates reduced phase errors. Phase errors are indeed induced by the variations of the filtering magnitude.

2.2 Application to a Shock-Propagation Problem The shock-capturing methodology is now implemented to solve the shockpropagation problem proposed by Tam [24], based on the one-dimensional Euler equations. The spatial derivatives are computed using eleven-point low-dispersion centered finite-differences, and the time integration is performed using a six-stage low-dissipation Runge-Kutta algorithm, designed in [3]. The grid spacing is uniform with Δ x = 1, and the time step is Δ t = 0.8. A background selective filtering is carried out using a eleven-point filter of order 6 optimized in the wavenumber space [6], with a uniform magnitude equal to 1, at each grid point every time step.

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J. Schulze et al. (b) 0.5

0.4

0.4

0.3

0.3

γp−1

γp−1

(a) 0.5

0.2 0.1

0.1

0 180

0.2

0 200

220 x/Δx

240

260

180

200

0.5

0.4

0.4

0.3

0.3

0.2

0.1

0

0 200

220 x/Δx

260

240

260

0.2

0.1

180

240

(d)

0.5

γp−1

γp−1

(c)

220 x/Δx

240

260

180

200

220 x/Δx

Fig. 2 Shock propagation. Pressure computed at t = 200 : (a) selective filtering of the fluxes without shock-capturing, (b) selective filtering of the variables without shock-capturing, (c) selective filtering of the variables and non-conservative shock-capturing filtering, (d) selective filtering of the variables and conservative shock-capturing filtering, using filter Fo2 and a threshold parameter rth = 10−5

The shock-capturing methodology is then applied. Note that the present discretization methods have been successfully used to perform accurate Large-Eddy Simulations of three-dimensional turbulent flows and their radiated noise [2, 4]. The pressure solution calculated at t = 200 using selective filtering of the fluxes as discussed in [5], without shock-capturing, is presented in Fig. 2(a). The pressure pulse has been dispersed, which suggests that filtering the fluxes is not appropriate when strong non-linear effects take place. The solutions obtained using selective filtering of the variables, alone or in combination with the shock-capturing filtering, are then displayed in Fig. 2(b), 2(c) and 2(d). The pressure pulse has become triangular in shape at t = 200 due to nonlinear effects. In Fig. 2(b), without using shock-capturing, a shock is indeed visible at x 249Δ x, but it is surrounded by Gibbs oscillations because of the spectral truncation of the solution. In Fig. 2(c) and 2(d), using shock-capturing, the pressure pulse however does not exhibit oscillations around the shocks which are rather sharp. The Gibbs oscillations have then been removed by the second-order filter. More precisely, as demonstrated


35

in [6], the second-order shock-capturing filtering applies near the discontinuity, but its magnitude is nil everywhere else. In addition the location of the shocks is found to differ when non-conservative or conservative filtering procedures are implemented, with shocks that are situated at x 252Δ x in Fig. 2(c) but x 249Δ x in Fig. 2(d). This result illustrates the fact that that the use of the conservative form of the filtering is required to properly calculate the speed of the shock propagation.

3 Results In the present research project several cases with different numerical methods and flow configurations are investigated. Two DNS applications with and without artificial forcing and without the presence of a nozzle (case 1 and 2). One DNS with a Joukowsky-like nozzle (case 3; still under investigation) and a LES with a rectangular nozzle. All cases are listed in Table 2.

3.1 Complex Jet Geometries The presence of a nozzle at the jet exit plays a dominant role in the screech phenomenon. Its sensitivity due to acoustic fluctuations seems to close the screech feedback loop. Recent investigations confirmed that simulations performed without the presence of a nozzle do not produce screech tones. 3.1.1 Case 1 In case 1 a fully three-dimensional direct numerical simulation (DNS) is performed solving the compressible Navier-Stokes equations, based on a characteristic-type formulation of [21] on an orthogonal grid stretched in both the stream-wise and the transverse directions. Along the span-wise direction, periodicity and statistical homogeneity are assumed. At the inlet of the domain a tanh-like velocity and pressure profile models the presence of a nozzle. In that case the computational grid contains approx. 300 million grid points with 2040 points in the stram-wise direction, 1020 points in the transverse direction and 144 points in the periodic direction. Based on the hight of the jet a Reynolds number Table 2 Investigated flow cases Case Case 1 Case 2 Case 3

method DNS DNS DNS

Grid dimensions 2040 × 1020 × 144 1024 × 512 × 16 —

used CPUs 1020 128 —

Re 30000 8000 —

nozzle no nozzle; no forcing no nozzle; forcing curvilinear nozzle

Case 4

LES

525 × 257 × 121

1

150000

Cartesian nozzle; no forcing

Cases 1, 2, 3 have been simulated using SGI-Altix 4700 with Itanium2 Madison 9M processors, Case 4 using a Nec SX-5. Case 3 is still under investigation.

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Fig. 3 Vorticity (iso-surface) and dilertation (plane) of the three-dimensional planar jet (case 1) with a Reynolds number of 30000

of 30 000 can be assumed. The spatial discretization uses a finite difference compact scheme of sixth order to capture the sound generation and propagation processes, and a spectral like method in the periodic direction. For the integration in time an explicit Krylov exponential time integration technique is implemented [20]. The code is parallelized using the Massage Passing Interface (MPI). For the current setup 1020 CPU’s are used on a SGI-Altix 4700 with Itanium2 Madison 9M processors. Approx. 12 TB of data are written to disk and 0.5 TB of main memory are used. In Fig. 3 a three-dimensional sketch with a slice of the dilatation field (div(u)) and iso-surfaces, showing the absolute vorticity, is presented. Based on the high Reynolds number, the initially laminar shear-layers undergo an early transition to turbulence about 3δω0 (shear-layer thickness) downstream of the inlet. So the oblique shocks in the jet core interact with the turbulent mixing layers and emanate shockinduced noise. For this case we do not identify any dominant frequency corresponding to a screech tone. Hence, this flow configuration is not closing the feedback loop. In the following example (case 2) we present an approach to close the feedback loop with an artificial forcing. 3.1.2 Case 2 The difference of this case to the above example is an additional forcing of the mixing layers. Two ’microphones’, placed above and below the planar jet in the subsonic co-flow measure the acoustic signal of the jet. Since these microphones


37 120

Fig. 4 The spectrum of the acoustic signal of the jet measured in the acoustic near-field close to the jet exit to capture the shock induced noise. The peak at f s ≈ 8 · 10−2 f 0 indicates the screech tone

100

SPL [dB]

80 60 40 20 0 10−2

10−1

100

101

102

f /f0

are placed close to the inlet, mainly upstream traveling noise is captured which is usually shock induced noise. This signal is then used to force the most unstable modes of the mixing layers in a tiny fringe region close to the inlet. These KelvinHelmholtz instabilities are now triggered with the acoustic signal of the jet and thus closing the feedback loop. With this approach discrete screech tones can be identified. In Fig. 4 the frequency spectrum measured in the acoustic near-field of the planar three-dimensional jet is presented. The low-frequent peak at fs ≈ 8 · 10−2 f0 corresponds to the screech tone where the reference frequency f0 is the most unstable frequency of the mixing layer: f0 = 0.132

uc δω0

(8)

Fig. 5 Pressure history on both sides of the jet in the upstream direction. Upper side ( ); Lower side ( )

p /pmax

with the mixing layer thickness δω0 and the velocity uc = (ujet core − u∞ )/2 (cf. [16]). The corresponding pressure history is depicted in Fig. 5 measured close to the nozzle exit at the upper and lower side of the planar jet. The fluctuations of the two locations seem to be 180 degrees phase shifted which is a typical phenomenon for screech tones [19]. A two-dimensional plane of the pressure field of the jet is presented in Fig. 6 where the low frequent acoustic screech noise is visible, traveling in the upstream direction. 1.5 1 0.5 0 -0.5 -1 -1.5 21

22

23 t/Ts

24

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Fig. 6 Pressure field in a two-dimensional plane of a three-dimensional jet (case 2). Screech tones traveling in the upstream direction are visible

3.1.3 Case 3 As an extension to the last two cases a nozzle will be implemented in case 3 where no forcing is needed to close the feedback loop. Although the case is still running, some details of the flow configuration are presented. 3.1.4 Nozzle Geometry In this case, again a planar and rectangular jet is used for the investigation of screech tones. As it was shown before the presence of a nozzle is important. For this reason a nozzle geometry is being implemented in the present setup based on overset grid techniques [15], [22]. The nozzle geometry is modeled by two backward facing Joukowsky profiles as shown in Fig. 11. Each profile is surrounded by a curvilinear grid which is orthogonal due to the conformal mapping of the Joukowsky transformation. Both curvilinear grids are embedded in a cartesian grid where most of the jet is simulated. These three grids are furthermore decomposed into sub-domains which allows a parallelization of the code. For the parallelization the Message Passing Interface (MPI) is used to guarantee the communication of the sub-domains. The same routines are also used to enable the communication needed for the interpolation of the curvilinear grid and the cartesian grid. This communication has to be set up carefully to define the processes involved in the interpolation process and to avoid communication deadlock. Additional communication at the boundaries of the sub-domains has to be implemented to receive ghost cells of the neighboring processes. The latter one are used to interpolate points close to the sub-domain boundaries where an interpolation stencil would extend the domain of the process. A sketch of the flow around the two backward facing Joukowsky profiles is shown in Fig. 7. Due to the


39

Fig. 7 Flow around the two backward facing Joukowsky profiles with a symmetric boundary condition in the center of the jet. The white curvilinear lines border the sub-domains. The Mach number in the jet core is M = 1.4. The pressure ratio of jet core and ambient pressure is 1.2 (under-expanded)

pressure

magnitude of the velocity

flow separation and the instabilities in the wake of the nozzle profiles, a complex shock structure appears. 3.1.5 Interpolation The interpolation is performed in an interpolation zone close to the boundary of the curvilinear grids (see bold dotted line in Fig. 11). Depending on the order of interpolation, the width of this interpolation zone has to be at lest n + 1 points, where n is the order of the interpolation. It is proposed by [10] that a fourth order Lagrange interpolation is a good compromise between accuracy and computational cost. Based on a Lagrangian interpolation in three dimensions, the values in the interpolation zone are updated within each iteration of the time integration method. Due to the closed feedback loop in our application (screech) the information of the curvilinear grid and the cartesian grid need to be interpolated in both directions. Thus, a two-way coupling is implemented. As a test case for the two-way interpolation, two overlapping grids are used. Both cartesian grids have boundary conditions like in a channel flow with two facing walls and an inlet and outlet. The coupling of the two grids takes place in an interpolation zone at the outlet of one grid and the inlet of the other grid, respectively. Although, the computational domain defines a channel, there is no mean-flow. The initial condition consists of two Gaussian-like pressure distributions, with respectively one peak on each grid. These pressure peaks produce acoustic waves propagating in the channel and finally being transported from one grid to the other and vice versa. On each two-dimensional grid, with its 65 × 32 points, the compressible NavierStokes equations are solved and integrated in time with a fourth order Runge-Kutta method. After each Runge-Kutta sub-step the values are interpolated in the interpolation zone and updated on the other domain. Both domains are shown in Fig. 8 (a). As one can see the grid points are of non-coincident nature, so interpolation is necessary. For this cartesian test case, the interpolation is based on a two-dimensional Lagrangian interpolation of third order. In Fig. 8 (b) the initial condition with the two pressure peaks is shown. After 40 time steps the acoustic waves are propagated

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(a)

Grid 2

Grid 1

Two domains (Grid 1 and Grid 2) with non-coincident grid points. Only every other point is shown. (b)

interpolation zone

Initial condition. Two pressure peaks; one on each grid. (c)

Solution after 40 time steps. The acoustics are moving in the interpolation zone. (d)

Solution after 80 time steps. The acoustics are transported from one grid to the other. Fig. 8 Interpolated domain with grid 1 and grid 2

into the interpolation zone and reflected on the channel walls (Fig. 8 (c)). Finally, after 80 iterations the waves are transported through the interpolation zone on the other grid (Fig. 8 (d)). To receive an impression of the resulting error of the used overset technique, one can solve the same initial value problem on only one grid. This reference solution is only applicable for simple cases, like this test case. Anyway, since we use non-coincident grids the reference solution will be non-coincident to the two single domains, so interpolation to compute the resulting error will be necessary again. For this case, the relative error of the pressure after 80 iterations is in the order of 10−4 . A one-dimensional slice through the center of the channel is shown in Fig. 9

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1. Transformation from Problem: Interpolate values Interpolated values on Joukowsky to unit-cube. → from the colored curvilinear → cartesian grid. 2. Perform interpolation on grid to the Cartesian one. unit-cube. Fig. 10 Inverse Joukowsky transformation of the cartesian grid to apply a Lagrangian interpolation

with the solution of the two interpolated grids and the reference solution solved on one grid. A direct interpolation of the values of the curvilinear grid on the cartesian grid is not possible since one needs a cartesian domain for the interpolation stencil. To interpolate a value from the curvilinear grid on the cartesian grid the stencil on the curvilinear grid needs to be in a cartesian domain (computational domain). These cartesian values are computed using an inverse Joukowsky transformation of the curvilinear grid. This is a different approach to [8] and [15] but fast and accurate since no system of equations has to be solved. Anyway, this approach is restricted to applications where an analytic solution to the inverse transformation of the domain exists. This is the case for a standard Joukowsky transformation. An example how the inverse transformed domain looks like is presented in Fig. 10.

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3.2 Computation of Screech Noise Using Large Eddy Simulation In the case 4, a planar supersonic jet is simulated using a compressible Large Eddy Simulation in order to compute directly its radiated sound field and to investigate noise sources. The computation aims in particular at interpreting shockinduced sound production mechanisms using the shock-leakage theory [23]. Explicit spectral-like filtering of the flow variables is applied to take account of the dissipation provided by the unresolved scales, while minimizing energy loss at the resolved scales [4]. The filtered compressible Navier-Stokes equations are therefore solved using low-dispersion and low-dissipation explicit selective filters and finite differences [3]. Time integration is carried out by a fourth-order six-stage low-storage Runge-Kutta algorithm, whose properties have been optimized in the Fourier space [1]. Periodic boundary conditions are implemented in the spanwise direction to simulate a three-dimensional planar flow. The nozzle, which is known to be necessary for screech tones emergence, is included in the computational domain, and modeled by two parallel plates. The flow inside is laminar and sonic, and no forcing is used inside the jet flow. The jet is cold and operates at underexpanded conditions so that a quasi-periodic shock cell structure is formed in the jet plume. The exit pressure is chosen so that the fully expanded jet Mach number M j is equal to 1.55. This value has been observed by Krothapalli et al. [13] to correspond to maximum screech sound radiation for a large aspect ratio rectangular jet. The Reynolds number based on M j , and on the jet height h = 3 mm, is equal to 105 . The computational domain is made of 15 × 106 points, allowing to simulate the jet up to x/h = 25 in the downstream direction. The simulation is carried out during 100 screech periods, and more than 200 CPU hours have been necessary using a Nec SX-5. The domain extension in the sideline direction is fixed to be y/h = ±8, so that the grid includes the near acoustic field generated by the jet. The near pressure field

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Fig. 12 Instantaneous snapshot of spanwise vorticity in the whole computational domain and of fluctuating pressure in a plane perpendicular to the spanwise direction. The nozzle lips are represented in gray

is finally propagated to the far-field using a wave extrapolation method based on the linearized Euler’s equations. An instantaneous snapshot of isosurfaces of the spanwise vorticity ωz in the whole computational domain is represented in Fig. 12. Shock cells can be seen inside the jet plume, and a large range of turbulence scales, including the fine scales typical of a high Reynolds number flow, are observed. Pressure isocontours in a plane perpendicular to (a)

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Fig. 14 Visualizations of the flow field and of the acoustic near field in the central (x, y)-plane, during half a period of the screech loop. The density gradient modulus |∇ρ | is plotted in gray (grayscale between 0 and 6ρ∞ /h), and pressure iso-contours are represented in the near-field (colorscale from p∞ to 1.1p∞ ). From (a) to (c): t/Ts = 0 ; 1/4 ; 1/2, where Ts is the screech period. , shock; , sound wave generated by the shock-leakage phenomenon; , rotating density gradient. , coherent structure. , low-vorticity region

the spanwise direction are also reported in Fig. 12. Upstream-propagating wavefronts associated with screech noise are clearly visible on both sides of the jet. The properties of the upstream sound field are investigated in order to characterize the screech tones. The power spectral density of the pressure measured close to the nozzle exit is given in Fig. 13(a). Three tones, corresponding to the screech tone and its first and second harmonics, are noticed in the spectrum at the Strouhal numbers St = f h/U j = 0.13, 0.26 and 0.39. Screech Strouhal numbers obtained experimentally by Panda et al. [17] for large-aspect-ratio rectangular jets, and analytically by Tam [25] for planar jets, are plotted in Fig. 13(b) as a function of the fully expanded jet Mach number M j . The value obtained from the present computation is shown to be in good agreement with values of the literature. The mechanism of screech-tone generation can thus be now discussed. Visualizations of shock/vortex interactions in the shear-layer are in particular presented in order to provide numerical evidences of the shock-leakage phenomenon [2, 23]. Shock/vortex interactions within the third shock-cell of the jet are visualized in Fig. 14(a), 14(b) and 14(c) by plotting the density gradient modulus |∇ρ | in the central (x, y)-plane and the pressure perturbations in the near-field. The compression shock is found to undergo a rotating motion, while the shock tips tend to leak through the shear layer near the regions of low vorticity. This process generates circular upstreampropagating sound waves which correspond to screech tone radiation. The shockleakage phenomenon described by Suzuki and Lele [23] through an unsteady mixing layer is shown here to contribute to the screech noise generation in 3-D jets.

4 Conclusion Different numerical approaches have been used to investigate the screech tones of planar supersonic jets. A three-dimensional DNS of a jet without the presence of a


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nozzle produced hardly screech tones (case 1). An improved implementation with an artificial closing of the feedback loop, by triggering the instabilities in the mixing layers, yield to the desired screech tones. Since the presence of the nozzle seems to be of particular importance to close the feedback loop, a complex geometry for the nozzle based on overset grid techniques is implemented and will de used for future screech investigations. A two dimensional case to test the interpolation method showed good results with a small relative error. Furthermore, a methodology based on an adaptive spatial filtering has been developed to capture shocks in non-linear problems which have to be computed using low-dissipation schemes such as aeroacoustic problems. In order to optimize the efficiency of the approach, the different components of the shock-capturing procedure, including the shock detection from the flow variables, the determination of the filtering strength from the shock detector and the second-order filter applied around the shocks, have been considered. The method has been then applied to linear test cases and to non-linear problems involving shocks, as described in detail in [6]. The results obtained show that it takes into account shocks in a proper manner, while being of easy implementation and of reasonable cost. It will be used in our future simulations of supersonic jets. Acknowledgements. The first author likes to thank the DEISA Consortium (co-funded by the EU, FP6 project 508830), for support within the DEISA Extreme Computing Initiative (www.deisa.org). The fifth author is grateful to the Centre National d’Etudes Spatiales (CNES) for financial support. Case 4 has been computed using a Nec SX-5 provided by the CNRS Computing Center (IDRIS).

References 1. Berland, J., Bogey, C., Bailly, C.: Low-dissipation and low-dispersion fourth-order RungeKutta algorithm. Comp. Fluids 35(10), 1459–1463 (2005) 2. Berland, J., Bogey, C., Bailly, C.: Numerical study of screech generation in a planar supersonic jet. Phys. Fluids 19, 075105 (2007) 3. Bogey, C., Bailly, C.: A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. Comput. Phys. 194(1), 194–214 (2004) 4. Bogey, C., Bailly, C.: Large Eddy Simulations of transitional round jets: influence of the Reynolds number on flow development and energy dissipation. Phys. Fluids 18(6), 065101 (2006) 5. Bogey, C., Bailly, C.: On the application of explicit filtering to the variables or fluxes of linear equations. J. Comput. Phys. 225, 1211–1217 (2007) 6. Bogey, C., de Cacqueray, N., Bailly, C.: A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations. J. Comput. Phys. (to appear, 2009); See also AIAA-2008-2968 (2008) 7. Daru, V., Gloerfelt, X.: Aeroacoustic computations using a high-order shock-capturing scheme. AIAA Journal 45(10), 2474–2486 (2007) 8. Desquesnes, G., et al.: On the use of a high order overlapping grid method for coupling in CFD/CAA. J. Comp. Phys. 220, 355–382 (2006) 9. Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C., Poinsot, T.: Largeeddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152(2), 517–549 (1999) 10. Guenanff, R., Sagaut, P., Manoha, E., Terracol, M., Lewandowski, R.: Theoretical aspects of a multi-domain high-order method for CAA. AIAA Paper 2003–3117 (2003)

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11. Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA-81-1259 (1981) 12. Kim, J.W., Lee, D.J.: Adaptative nonlinear artificial dissipation model for Computational Aeroacoustics. AIAA Journal 39(5), 810–818 (2001) 13. Krothapalli, A., Hsia, Y., Baganoff, D., Karamcheti, K.: The role of screech tones in mixing of an underexpanded rectangular jet. J. Sound Vib. 106, 119–143 (1986) 14. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Physics 103(1), 16–42 (1992) 15. Marsden, O., Bogey, C., Bailly, C.: High-Order Curvilinear Simulations of Flows Around Non-Cartesian Bodies. J. Comp. Aeroacoustics 13(4), 731–748 (2004) 16. Michalke, A.: On the inviscid instability of the hyperbolic-Tangent velocity profile. J. Fluid Mech. 19(4), 543–556 (1964) 17. Panda, J., Raman, G., Zaman, K.B.M.Q.: Underexpanded screeching jets from circular rectangular and elliptic nozzles. AIAA-97-1623 (1997) 18. Pirozzoli, S.: On the spectral properties of shock-capturing schemes. J. Comput. Physics 219, 489–497 (2006) 19. Raman, B.: Cessation of screech in underexpanded jets. J. Fluid Mech. 336(1), 69–90 (1997) 20. Schulze, J., Schmid, P., Sesterhenn, J.: Exponential time integration using Krylov subspaces. Int. J. Numer. Meth. Fluids (2008), doi:10.1002/fld.1902 21. Sesterhenn, J.: A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes. Computers & Fluids 30(1), 37–67 (2001) 22. Sherer, S., Scott, J.: High order compact finite-difference methods on general overset grids. J. Comp. Phys. 210, 459–496 (2005) 23. Suzuki, T., Lele, S.K.: Shock leakage through an unsteady vortex-laden mixing layer: application to jet screech. J. Fluid Mech. 490, 139–167 (2003) 24. Tam, C.K.W.: Benchmark problems and solutions. In: ICASE/LaRC Workshop on CAA, NASA CP 3300, pp. 1–13 (1995) 25. Tam, C.K.W.: The shock-cell structures and screech tone frequencies of rectangular and nonaxisymmetric supersonic jets. J. Sound Vib. 121(1), 135–147 (1988) 26. Tam, C.K.W., Shen, H.: Direct computation of nonlinear acoustic pulses using high order finite difference schemes. AIAA-93-4325 (1993) 27. Visbal, M.R., Gaitonde, D.V.: High-order-accurate methods for complex unsteady subsonic flows. AIAA Journal 37(10), 1231–1239 (1999)

Fluid-Acoustic Coupling and Wave Propagation Jens Utzmann, Claus-Dieter Munz, Michael Dumbser, Eric Sonnendrücker, Stéphanie Salmon, Sébastien Jund, and Emmanuel Frénod

Abstract. Different strategies regarding the simulation of sound generation and propagation are explored. A hydrodynamic/acoustic splitting method for computational aeroacoustics in low Mach number flows with variable density, temperature gradients and heat conduction is described. The resulting equations can be formulated as linearized Euler equations plus source terms and reduce to the linear acoustic wave equation, if convection speeds can be neglected. In contrast, a direct approach based on the coupling of different grids, time steps and equations allows a simulation of both flow and acoustics in one single calculation. For acoustic domains, the ADER discontinuous Galerkin and the conforming face finite element discretizations are efficient high order methods for unstructured grids. The proposed methods are validated and applied to complex test cases such as the simulation of an aero-engine inlet and the scattering of sound waves at a solid sphere.

1 Introduction The numerical simulation of both generation and propagation of acoustic waves in one calculation is difficult due to the multi-scale problem of aeroacoustics: While the noise producing flow features are very small but possess large amplitudes, sound Jens Utzmann · Claus-Dieter Munz · Michael Dumbser Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, 70550 Stuttgart, Germany [email protected] Eric Sonnendrücker · Stéphanie Salmon · Sébastien Jund Institut de Recherche Mathématique Avancée, Université Louis Pasteur de Strasbourg, 7, Rue René Descartes, 67084 Strasbourg Cedex, France [email protected] Emmanuel Frénod Laboratoire d’Etude et Modélisation des Environnements Littoraux, Université de Bretagne Sud, Campus de Tohhanic, 56017 Vannes Cedex, France [email protected]


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waves contain significantly less energy and have a comparatively large wavelength. A vast number of grid elements would be necessary to resolve all scales accurately in an unsteady simulation. Even then, the handling of long-distance sound propagation remains difficult due to numerical damping and dissipation. In low Mach number flows, the difference between the flow velocity and the speed of sound leads to a disparity of the local CFL numbers in the calculation domain. Consequently, the global time step of the computation becomes very small for all elements. Hybrid approaches were therefore introduced in order to overcome these difficulties. The most classical approach is the volume coupling over source terms. Here, the flow simulation is performed first, followed by the computation of acoustic source terms by an acoustic analogy. These source terms are then used in a second calculation that solves acoustic equations, for example Lighthill’s equation [16, 17] or the Ffowcs Williams-Hawkins equation [8]. More recent developments are the perturbed compressible equations (PCE) by Seo and Moon [24, 25] and the acoustic perturbation equations (APE) by Ewert and Schröder [6]. Such approaches are often referred to as ”hybrid coupling” or ”hydrodynamic/acoustic splitting” (the latter, if the flow calculation is based on incompressible equations). Another traditional approach for the simulation of acoustic problems is the Kirchhoff method (e.g., Farassat and Myers [7]). The coupling of flow and sound is treated over an integration surface around the acoustically active region, which must include every relevant source of noise. Here, the sound waves need to be resolved already in the flow simulation and the solver must be accurate enough to transport these waves to the integration surface without significant numerical damping. Then the source terms are handed over to the respective acoustic tool, for example a wave equation solver. All of these methods have in common, that there is no feedback from the acoustics to the flow. Moreover, the extraction of the sources always involves some modeling in one way or the other. Hence, along with the uncertainties in the original flow calculation, the final noise prediction underlies a lot of assumptions. As an alternative, the direct computation of both flow and noise in one calculation remains. It includes the feedback of the sound to the flow and requires the least modeling. However, due to the multi-scale problem, the computational effort can grow easily beyond any acceptable limit. In order to circumvent this problem and to facilitate simulations, the idea of heterogeneous domain decomposition has been developed. The outline of the paper is as follows. First of all, the coupling of flow and acoustics is addressed. Both a hybrid and a direct approach are proposed: The hydrodynamic/acoustic splitting for computational aeroacoustics (CAA) in low Mach number flows is investigated in the first part, then a domain decomposition method for direct simulations is described. After that, two numerical methods for the simulation of wave propagation on unstructured grids are presented, the ADER discontinuous Galerkin (ADER-DG) method and the face finite element method (Face-FEM). They are validated by convergence studies and by an examination regarding their efficiency. Last but not least, numerical examples underline the applicability of the proposed methods. Here, the propagation of acoustics in and around an aero-engine inlet is simulated with the ADER-DG scheme. Finally, the domain decomposition approach is applied to the scattering of sound waves at a solid sphere.

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2 Fluid-Acoustic Coupling 2.1 Hybrid Methods for Low Mach Number Flows Numerical methods for the compressible equations may have difficulties with the zero Mach number limit, because the speed of sound becomes infinite compared to the flow speed and thus leads to an elliptic coupling of pressure and velocity. Hence, the efficiency of all explicit numerical schemes decreases strongly in the low Mach number regime due to their stability restriction (CFL condition). Another difficulty is that the pressure in the compressible equations converges to the thermodynamic one and becomes the constant background pressure in the incompressible limit. This is the way how the equation of state for compressible flow is automatically satisfied and does not appear in the incompressible equations. There, the hydrodynamic pressure only operates as sort of a Lagrangian multiplier. Hardin and Pope [9] proposed a hydrodynamic/acoustic splitting for CAA which is called ”Expansion about Incompressible Flow” (EIF). Perturbation equations for sound propagation are derived by splitting up the compressible solution in an incompressible part and acoustic fluctuations. In the following, we briefly describe the EIF approach for low Mach number flows extended to variable density, temperature gradients, heat conduction or with compression from the boundary, see Munz et al. [21] for details. The incompressible solution, which is usually taken for the state of the expansion, is replaced by the solution at zero Mach number which takes into account the aforementioned additional physical effects (variable density, etc.). The flow equations are obtained by using the low Mach number asymptotic results as given by Klainerman and Majda [12] and by Klein [14]. The solution of the zero Mach number equations is named in the following as ρ0 , u0 , p0 , p2 , T0 . Motivated by the asymptotic results, a perturbation approach is introduced of the form

ρ (x,t) = ρ0 (x,t) + ρ2(x,t) + ρ (x,t), u(x,t) = u0 (x,t) + u (x,t), p(x,t) = p0 (t) + p2(x,t) + p (x,t).

(1) (2) (3)

The pressure is split into a thermodynamic part p0 , into the hydrodynamic part p2 and into the acoustic fluctuations p . The density is also split into the density ρ0 for M → 0, into the acoustic fluctuations ρ and into ρ2 which introduces a density change due to the hydrodynamic pressure. Note that ρ0 may depend on x, when density variations in the initial data or heat conduction are present. The corresponding temperature is obtained from the equation of state p0 = ρ0 T0 . The introduction of the hydrodynamic density variation is not inevitable at this place but favorable. These density changes are connected with pure hydrodynamic motion. In the incompressible limit, the changes do not appear, since the equation of state is satisfied by the background pressure, density and internal energy, which are all constant in space and time. For low but non-zero Mach number flows, all pressure terms have to be included into the equation of state and lead to hydrodynamic density corrections. These are associated with the flow motion and are convected

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with fluid velocity. The relations (1), (2) and (3) are inserted into the compressible Navier-Stokes equations. On the left hand side, all the derivatives in space and time of the primed variables are written in order to get an evolution equation for the perturbations. The right hand side contains all the other terms and are interpreted as the sources. Doing so and neglecting all products of primed quantities, the system

∂ ρ Dρ 2 + u0 · ∇ρ + ∇ · (ρ0u ) = − − (ρ2 + ρ )∇ · u0 , ∂t Dt ∂ u 1 1 + (u0 · ∇)u + (u · ∇)u0 + ∇p = ∇ · Te (u ), ∂t ρ0 Reρ0 Dp2 ∂ p γ + u0 · ∇p + γ p0 ∇ · u = − − (p2 + p)∇ · u0 + ∇ · λ ∇T2 ∂t Dt Pr · Re

(4) (5) (6)

is obtained. Here, the conservative formulation was dropped in favor to the simpler evolution equations for density, velocity and pressure. The left hand side of this system is usually called the linearized Euler equations (LEE). As long as smooth solutions of the acoustic equations are considered, the simpler formulation in primitive variables should be preferred. D/Dt is the abbreviation of ∂ /∂ t + u0 · ∇ and T2 is defined such that the equation of state is satisfied, which leads to the relation ρ0 T2 = p2 + p − (ρ2 + ρ )T0 . The propagation rate of the acoustic waves in this system becomes infinite, while their amplitude tends to zero when the Mach number tends to zero. The mathematical justification of the linearized acoustics was given by Klainerman and Majda [13], who showed that the solutions of these equations are uniformly bounded with respect to the Mach number in the isentropic case. If diffusion effects, compression and heat flux from the boundary may be neglected and the fluid motion is isentropic, then the main acoustic source is the term Dp2 /Dt in the pressure equation and Dρ2 /Dt in the density equations. A difficulty in the perturbation approach based on the LEE is that they do not only describe acoustic wave motion but also vorticity and entropy modes. Especially in flows where hydrodynamic instabilities are present (e.g., in shear flow), fluid instabilities may occur and increase according to the linear theory with an exponential rate and could falsify the acoustic motion. Seo and Moon [25] carefully analyzed the LEE and reformulated (u · ∇)u0 + (u0 · ∇u ) in the velocity equations into ∇(u u0 )+ two terms involving the vorticity of the acoustic propagation and the vortical hydrodynamic motion. Because the two additional terms are associated with the generation and transport of perturbed vorticity only, they are dropped in order to get the proper acoustical motion in the LPCE (linearized perturbed compressible equations). Another approach to get rid of hydrodynamic instabilities are the acoustic perturbation equations (APE) by Ewert and Schröder [6]). They reformulated the wave propagation model such that vorticity and entropy modes vanish and obtained a system of equations for the acoustic modes only. They also succeeded in extending these considerations to compressible main flow. The impact of introducing hydrodynamic density corrections could be directly seen in the isentropic case. If furthermore the convection of the acoustic perturbations


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can be neglected, then the whole set of equations simply reduces to the linear acoustic wave equation

∂ 2 p ∂ 2 p2 2 − c Δ p = − 0 ∂ t2 ∂ t2

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or, equivalently, to its formulation as a system of equations (34)-(35).

2.2 Direct Simulations with Domain Decomposition For CAA, an accurate and feasible direct simulation that considers both the generation of sound within the flow and its propagation into the far-field is hard to realize with one numerical method in a single computational domain. On the other hand, a direct approach contains automatically the interaction of the acoustic perturbations with the flow-field, a property which lacks the popular acoustic analogy models. The method proposed in the following is basically a direct simulation but it simplifies the problem for individual regions in the computational domain: The idea is to use a non-overlapping domain decomposition method where the equations, methods, grids and even time steps are adapted to meet the local physical requirements. Details can be found in Utzmann et al. [27, 28]. Inside the coupling framework, a ”zoo” of high order solvers from different classes of methods is available: On unstructured grids, a finite volume method (Dumbser et al. [2, 3]) is used for linear and nonlinear problems, as well as the ADER discontinuous Galerkin method (ADER-DG, see section 3.1 and Dumbser et al. [1, 5]). On structured grids, the space-time expansion finite volume (STE-FV, Utzmann [27]) and the ADER finite difference (ADER-FD, Lörcher et al. [18]) method are efficiently implemented for nonlinear and linear problems. These high order methods ensure excellent wave propagation capabilities throughout the entire computational domain. In the subdomains, the Navier-Stokes, Euler and the linearized Euler equations are solved. Two or more different domains Ωi are coupled at their common boundary ∂ Ω = Γ over the data in ghost elements. Depending on the discretization method, such an element can be a cell (FV and DG) or a point (FD). These ghosts are then used by the numerical methods to update the inner elements in each domain. The data are exchanged by interpolating the values from the neighbor grid onto the Gauss integration points of the ghost cells for FV and DG methods. The multi-dimensional interpolations may be performed in any order, yet the spatial order of the source domain is a reasonable choice. With a subsequent integration, the mean values (FV) or the degrees of freedom (DG) in the elements are obtained. For FD methods, the values are interpolated onto the position of the ghost points. At the coupling interface Γ , the grid configuration can be arbitrary and non-matching. Figure 1 depicts one of many possible setups: A structured FD domain Ω1 is connected to an unstructured DG domain Ω2 . The outermost domain points of Ω1 lie directly on the boundary (which is typical for FD schemes). If a fourth order ADER-FD method is used, there are two rows of ghost points, all located inside Ω2 . Last but not least, the DG domain’s single row of ghost cells extends into Ω1 . The coupling approach is also capable of handling

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Domain Ω 1

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the treatment of complex configurations with ghost cells overlapping multiple source domains (Fig. 2). Each Gauss integration point can be assigned a unique element of a neighboring grid that is the basis for the interpolation stencil. The explicit time integration methods which are used in the calculation domains Δx must follow the stability condition Δ t ∼ CFL· , where Δ x is the characteristic a element size, a is the maximum convection speed in the cell and CFL < 1 (CourantFriedrichs-Levy condition). It is obvious, that domains with very different grid spacing will also have very different maximum time steps. Usually, the domains with larger time steps would have to adopt the smallest Δ t for each iteration. However, according the domain decomposition philosophy, it would be very convenient to allow the largest possible time steps on each domain. In order to implement this form of ”local time stepping”, a subcycling method is introduced. The data between the domains are exchanged at common time levels. After the exchange, the largest possible time steps are estimated in each domain. Then the Δ t’s are sorted

Γ1

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with respect to their size and adjusted such that they are always equal or a multiple of the previous smaller one. The Δ t’s of the domains are also adjusted if a data output is imminent in order to match the desired output time. Note that time steps may be only decreased, as the stability condition must not be violated. A Taylor series in time (see equation (18) in section 3.1) is used to provide time-accurate values for the ghost elements which take part in iterations between data exchanges. The unknown time derivatives can be replaced by spatial derivatives with the CauchyKovalevskaja procedure (CK, Kowalevsky [29]). This procedure is used in LaxWendroff type time integration methods (Lax and Wendroff [15], Hirsch [10]) and is also a main ingredient of the ADER schemes. Efficient algorithms for a variety of linear and nonlinear systems of equations (LEE, Euler, Navier-Stokes, etc., Dumbser et al. [1, 3, 5]) are available in the coupling framework. It was shown on the basis of convergence studies for different constellations of grids, equations and methods, that the domain decomposition approach is capable of maintaining high order of accuracy globally (Utzmann [27]). An examination regarding high-frequency perturbations revealed a filtering process if perturbations cannot be resolved on a coarse mesh anymore. Hence, a spatial filtering operator is not a necessity. Another study showed, that the magnitude of reflections occurring at the domain boundaries are in good accordance with theoretical estimations. However, the reflections are negligible in general.

3 Numerical Methods for Wave Propagation 3.1 The ADER-DG Space and Time Discretization Dumbser et al. [1, 5] applied the ADER approach (Schwartzkopff et al. [23], Toro et al. [26]) to the discontinuous Galerkin method for linear and nonlinear hyperbolic systems. The very compact ADER-DG formulation does not need a reconstruction and thus provides the possibility to achieve high order of accuracy in space and time even on unstructured grids. This is particularly useful for accurate noise propagation in the time domain around complex obstacles or in complex geometries. The proposed implementation of the ADER-DG method is capable of solving general linear hyperbolic systems with variable coefficients and source terms on unstructured grids in 2D and 3D and the user may chose any order of accuracy in space and time he needs. In the following, the DG method is described for the linear scalar advection case in one space dimension:

∂u ∂u +a = 0. ∂t ∂x

(8)

The approximate solution in each cell is represented as a linear combination of a set of orthonormal polynomials Φi : nDOF

uh (x,t) =

∑

i=1

u(t) ˆ i Φ (x)i ,

(9)

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with uî being the degrees of freedom of the current state uh . Multiplication of the governing equation with a test function and integration over the element T (m) yields

T (m)

Φk

∂u dV + ∂t

T (m)

∂u dV = 0. ∂x

Φk a

(10)

Partial integration of the second integral on the left hand side leads to

T (m)

Φk

∂u dV + ∂t

∂ T (m)

Φk F h dS −

T (m)

∂ Φk a u dV = 0, ∂x

(11)

where F h is the numerical flux over the element surfaces. Introducing the linear combination for uh yields

T (m)

Φk

∂ uˆl Φl dV + ∂t

∂ T (m)

Φk F h dS −

T (m)

∂ Φk a uˆl Φl dV = 0. ∂x

(12)

Note that indices which appear twice stand for a summation over this index (Einstein summation). Since uî is only time-dependent, (12) can be re-written as

∂ uˆl ∂t

T (m)

Φk Φl dV +

∂ T (m)

Φk F h dS − a uˆ l

T (m)

∂ Φk Φl dV = 0. ∂x

(13)

Godunov’s exact flux is used for the numerical flux between the cell T (m) and one of its neighbors T (k j ) : Fh =

1 1 (k ) (k ) (m) (m) (a + |a|) uˆ l Φl + (a − |a|) uˆ l j Φl j . 2 2

(14)

Insertion in (13) yields

∂ uˆl ∂t

Φk Φl dV + (m) T Mkl

2

1 (m) (m) (m) ∑ 2 (a + |a|) uˆl (∂ T (m) ) Φk Φl dS j=1 j

(15)

j,0

2

+

Fkl

1 (k ) (m) (k ) ∑ 2 (a − |a|) uˆl j (∂ T (m) ) Φk Φl j dS j=1 j j,r

− a uˆl

T (m)

Fkl

∂ Φk Φl dV = 0. ∂x Kkl

Due to the special ansatz chosen for uh and for the basis functions Φi , the integrals on the left hand side can be precalculated analytically, which results in


55

2 2 ∂ uˆl 1 1 (k ) (m) Mkl + ∑ (a + |a|) Fklj,0 uˆl + ∑ (a − |a|) Fklj,r uˆl j − a Kkl uˆl = 0. (16) ∂t 2 2 j=1 j=1

Integration over one time step leads to n+1 uˆl − uˆnl Mkl +

2

1 ∑ 2 (a + |a|) Fklj,0 j=1 2

+

1

∑ 2 (a − |a|) Fklj,r

j=1

− a Kkl

Δt (m) 0

uˆl

Δt (k j ) 0

uˆl

dt

(17)

dt

Δt

uˆl dt = 0.

0

The ADER approach is now used for the time integration, for which the approximate solution uh (x,t) is expanded into a truncated Taylor series in time: tk ∂ k

N

uh (x,t) =

∑ k! ∂ t k uh (x, 0).

(18)

k=0

The time derivatives are replaced by space derivatives using the CK procedure. It makes repeatedly use of the original PDE and can be formulated for time derivatives of arbitrary order:

∂ k uh = (−1)k ∂ tk

∂ a ∂x

k uh .

(19)

Projection onto the polynomial basis yields k Δ t k+1 Φn , ∑Nk=0 (k+1)! (−1)k a ∂∂x Φi (x) uˆn (t) = uî (0) = Ini (Δ t) uî (0) Φn , Φn

(20)

Ini (Δ t)

Combining (18) and (20) results in the fully discrete ADER-DG scheme: (m) n+1 (m) n Mkl + − uˆl uˆl

1 (m) n j,0 (a + |a|) F I ( Δ t) u ˆ ∑ i kl li j=1 2 2

2

+

1

∑ 2 (a − |a|) Fklj,r Ili (Δ t)

j=1

(m) n − a Kkl Ili (Δ t) uî = 0.

(k j ) n

uî

(21)

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The ADER-DG scheme can be derived in an analogue way for systems of equations in 2D and 3D. For 2D linear hyperbolic systems

∂ up ∂ uq ∂ uq + A pq + B pq =0 ∂t ∂x ∂y

(22)

on triangular grids the fully discrete ADER-DG formulation reads: (m) n+1 (m) n uˆ pl |J| Mkl + − uˆ pl + +

3

1 2

j=1

1 2

∑ Tpqj

∑ Tpqj 3

n −1 j,0 S j F · Islin (Δ t) uˆ(m) + ATqr + ATqr Trsj in kl

n −1 j,i S j F · Islin (Δ t) uˆ(k j ) − ATqr − ATqr Trsj in kl

j=1

(m) n (m) n ξ − B∗pq |J| Kklη · Iqlin (Δ t) uîn = 0. −A∗pq |J| Kkl · Iqlin (Δ t) uîn

(23)

A pq and B pq denote the Jacobian matrices of the system. It is sufficient for the user to provide these Jacobian matrices together with their eigenvalues and left and right eigenvectors. The rest is implemented for arbitrary linear hyperbolic systems. Here, the governing equations are the linearized Euler equations, in primitive variables: ⎛ ⎛ ⎞ ⎞ u0 ρ0 0 0 v0 0 ρ 0 0 1 ⎜ 0 u0 0 ⎜ 0 v0 0 0 ⎟ ⎟ ρ0 ⎟ ⎜ ⎟ A pq = ⎜ (24) ⎝ 0 0 u0 0 ⎠ , B pq = ⎝ 0 0 v0 ρ1 ⎠ , 0 γ p0 0 u0

0

0 0 γ p 0 v0

T with the background flow ρ0 , u0 , v0 and p0 . The vector u p = ρ u v p is the vector of the fluctuations of the primitive variables density, velocity and pressure. Some considerations have to be made for the treatment of the unstructured elements in 2D: Rotational invariance of system (22) is assumed, so the flux can be written easily in a coordinate system which is aligned with the outward pointing unit normal vector on the boundary. The transformation of the vector u p from the global system to the vector uTq in an edge-aligned coordinate system is given by u p = Tpq uTq .

(25)

For the (linearized) two-dimensional Euler equations, the transformation matrix is ⎛ ⎞ 1 0 0 0 ⎜ 0 nx −ny 0 ⎟ ⎟ Tpq = ⎜ (26) ⎝ 0 ny nx 0 ⎠ . 0 0 0 1


57

Furthermore, every triangle T (m) in physical x − y space is transformed to a canonical reference triangle TE in a ξ − η reference space. Due to this, the method can be implemented much more efficiently, since all integrals can be precomputed in the reference space. The transformation is defined by x = x1 + (x2 − x1 ) ξ + (x3 − x1) η , y = y1 + (y2 − y1 ) ξ + (y3 − y1) η ,

(27)

with the inverse transformation 1 ((x3 y1 − x1 y3 ) + x (y3 − y1 ) + y (x1 − x3)) , |J| 1 ((x1 y2 − x2 y1 ) + x (y1 − y2 ) + y (x2 − x1)) , η= |J|

ξ =

(28)

where |J| = (x2 − x1 ) (y3 − y1) − (x3 − x1 ) (y2 − y1) is the determinant of the Jacobian matrix of the transformation being equal to the double of the triangle’s surface, xξ yξ J= . (29) xη yη With respect to the transformation, we have furthermore dxdy = |J| d ξ d η ,

ux uy

= J −1

uξ uη

(30)

,

(31)

A∗pq = A pq ξx + B pqξy ,

(32)

B∗pq = A pq ηx + B pqηy .

(33)

3.2 The Conforming Face Finite Element Discretization The conforming face finite element method (Face-FEM) is described in this subsection. We consider the pure wave equation (7), which is written as a first order system

∂u + ∇p = f, ∂t ∂p + ∇ · u = 0, ∂t

(34) (35)

and which corresponds to the LEE with zero background speed. A well posed variational formulation of this problem can be given: Find u ∈ H(div, Ω ) and p ∈ L2 (Ω ) such that

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d dt

Ω

d dt

u · ψ dX −

Ω

Ω

pϕ dX +

Ω

p(∇ · ψ ) dX =

f · ψ dX, ∀ψ ∈ H(div, Ω ),

Ω

(36)

(∇ · u)ϕ dX = 0, ∀ϕ ∈ L2 (Ω ).

(37)

Defining Wh ⊂ H(div, Ω ) and Vh ⊂ L2 (Ω ), two conforming finite element subspaces, respectively of dimension N and M, we consider the following approximate formulation of the variational problem: Find uh ∈ Wh and ph ∈ Vh such that

d uh · ψi dX − ph (∇ · ψi ) dX = Π (f) · ψi dX, ∀i = 1 . . . N, dt Ω Ω Ω

d ph ϕk dX + (∇ · uh )ϕk dX = 0, ∀k = 1 . . . M, dt Ω Ω

(38) (39)

where {ψ i }i=1...N (resp.{ϕk }k=1...M ) denotes the conforming finite element basis of W (resp.V ) and Π (f) denotes the canonical H(div, Ω )-conforming finite element projection of f on W . Practically we use conforming finite elements based on Raviart-Thomas spaces for Wh ⊂ H(div, Ω ) and standard discontinuous nodal finite elements for Vh ⊂ L2 (Ω ). N

∑ u j ψ j , Π (f) =

By decomposing uh =

j=1

N

∑

f j ψ j and ph =

j=1

M

∑ pl ϕl

on their

l=1

respective basis, the problem finally becomes: Find U = (u1 , . . . , uN ) and P = (p1 , . . . , pM ) such that d dt

N

∑ uj

j=1 M

d dt

∑ pl

l=1

Ω

M

l=1 N

ψj · ψi dX − ∑ pl

Ω

ϕl ϕk dX + ∑ u j j=1

Ω

Ω

ϕl (∇ · ψi ) dX =

N

∑ fj

j=1

Ω

ψj · ψi dX, ∀i = 1 . . .N,

(∇ · ψj )ϕk dX = 0,

which can be rewritten as the ordinary differential system ⎧ d ⎪ ⎪ ⎪ ⎨ dt MwU − KP = Mw F, ⎪ ⎪ d ⎪ ⎩ Mv P + K T U = 0, dt where

(Mw )1≤i, j≤N = (Mv )1≤i, j≤M = (K) 1 ≤ i ≤ N = 1≤ j≤M

Ω

Ω Ω

ψj · ψi dX, ϕi ϕ j dX, ϕ j (∇ · ψi ) dX.

(40) ∀k = 1 . . . M,

(41)


59

Denoting {σiV }i=1...M the set of linear forms associated with the finite element space V , one can prove that K T = Mv R, with (R) 1 ≤ i ≤ M = σiV (∇ · ψj ), 1≤ j≤N

so that the semi-discretized problem in space we have to solve is ⎧ d ⎪ ⎪ ⎪ ⎨ dt MwU − KP = Mw F, ⎪ ⎪ ⎪ ⎩

(42)

d P + RU = 0. dt

This can be written in a short hand notation as dV = LV, dt U 0 Mw−1 K and L = . Next, we use a Taylor expansion with V = P −R 0 V (tn+1 ) = V (tn ) + Δ t

(43)

Δ t p d pV (tn ) dV (tn ) Δ t 2 d 2V (tn ) + + · · · + + O(Δ t p+1 ). dt 2! dt 2 p! dt p

d iV = LiV , ∀i = 1, . . . , p, we can replace the time dt i derivatives and define a consistent time discretization of order p by Since equation (43) implies

Vn+1 = (I + Δ tL +

Δt2 2 Δt p p L + ···+ L )Vn = AVn . 2! p!

(44)

This corresponds directly to the Cauchy-Kovalevskaja procedure as used in (19) for a simpler test problem. Here, we use the CK procedure in a discretized form based on the semi-discrete evolution equation (43). However, as our Matrix L has only purely imaginary eigenvalues, this approach leads to unconditionally unstable discretization at certain orders. Up to order 10, only orders 3, 4, 7 and 8 are stable, whereas orders 1, 2, 5, 6, 9 and 10 are unstable. We found out that it is possible to stabilize them by adding a term Vn+1 = (I + Δ tL +

Δ t2 2 Δt p p Δ t p+1 p+1 L + ···+ L +ξ L )Vn , 2! p! (p + 1)!

(45)

which remains a consistent time discretization of order p (since ξ is in general not equal to 1). ξ can be chosen to make the stability zone as large as possible, which we can do in all cases. Another option, which proves to be better in practice, is to choose ξ such that not only the method is stable, but as little dissipative as possible under the selected CFL number. The latter is in this case a little bit smaller, but

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provides a better cost/accuracy ratio. This idea is in the same spirit as the one of Hu, Hussaini and Manthey [11].

3.3 Validation 3.3.1 Convergence The experimental order of convergence is determined for the ADER-DG and the Face-FE method. For this purpose, the setup as follows: A planar wave is initialized such that it contains only fluctuations in the right-moving characteristic wave with the eigenvalue u + c: w = w ˆ · sin(k · x), (46) with w = (w1 , w2 , w3 , w4 , w5 ) being the perturbation of the characteristic variable vector, w ˆ = (0, 0, 0, 0.001, 0) containing the amplitude of the perturbation and k = (2π , 0, 0) being the wave number vector. The primitive fluctuations are obtained by transforming with the Eigenvector matrix R belonging to the set of eigenvectors λ = (u, u, u, u + c, u − c): u = Rw , (47) ⎛

n1 ⎜ 0 ⎜ R=⎜ ⎜ n3 ⎝ −n2 0

n2 −n3 0 n1 0

n3 n2 −n1 0 0

ρ0 ρ0 2c0 2c0 n1 n1 2 −2 n2 n2 2 −2 n3 n3 2 −2 ρ0 ρ0 2c0 2c0 ,

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(48)

with n = (n1 , n2 , n2 ) = (1, 0, 0) being the normal vector of the wave. The background flow is ρ0 = 1, u0 = 0, v0 = 0, w0 = 0 and p0 = 0.714285714. The result is a wave in the primitive variables u = (ρ , u , v , w , p ), which travels in the direction of the positive x-Axis with the speed of sound c0 = 1 and with a wavelength of λ = 1. As the example is 2D, the perturbation velocity in z-direction w is set to zero. The calculation domain has the extents [−5, +5]× [−2.5, +2.5], hence exactly 10 wavelengths fit in the domain in x-direction. Periodic boundary conditions are prescribed everywhere. Figure 3 shows the regular grid. Four refinement stages (#20, #40, #60 and #80 elements in x-direction) are used to determine the experimental order of convergence of the methods for the orders O2-O6. The simulation runs until t = 20, when each wave has traveled a distance of 20 wavelengths. Finally, the error norms are calculated by taking the exact solution as a reference. Table 1 shows for both the ADER-DG and the Face-FE method that they produce the desired convergence rates. 3.3.2 Efficiency on an Irregular Grid In this test case, the ADER-DG and the Face-FE method are compared with respect to computational efficiency on an irregular unstructured grid. For this purpose, it comes in handy that the schemes follow a common methodology (arbitrary

ADER-DG P1 (O2) #20 #40 #60 #80 P2 (O3) #20 #40 #60 #80 P3 (O4) #20 #40 #60 #80 P4 (O5) #20 #40 #60 #80 P5 (O6) #20 #40 #60 #80

L1 (p ) 1.58302E-01 1.22561E-01 5.82938E-02 3.16960E-02 L1 (p ) 6.81839E-02 9.18710E-03 2.33509E-03 9.16614E-04 L1 (p ) 1.07550E-02 3.87591E-04 7.47338E-05 2.38763E-05 L1 (p ) 7.92522E-04 1.83758E-05 2.44872E-06 5.81533E-07 L1 (p ) 5.90787E-05 8.81602E-07 7.68951E-08 1.09931E-08

L2 (p ) 2.45516E-02 1.93193E-02 9.19821E-03 4.97565E-03 L2 (p ) 1.06606E-02 1.43511E-03 3.65499E-04 1.43666E-04 L2 (p ) 1.70672E-03 6.04980E-05 1.18142E-05 3.74801E-06 L2 (p ) 1.24999E-04 2.89365E-06 3.86412E-07 9.17247E-08 L2 (p ) 9.47329E-06 1.39624E-07 1.22508E-08 1.77753E-09

L∞ (p ) 5.00562E-03 3.87759E-03 1.86583E-03 1.01342E-03 L∞ (p ) 2.08229E-03 3.25075E-04 8.59878E-05 3.15096E-05 L∞ (p ) 4.01640E-04 1.56049E-05 2.89927E-06 9.26721E-07 L∞ (p ) 2.74749E-05 8.87582E-07 1.51172E-07 3.50117E-08 L∞ (p ) 3.33332E-06 4.91177E-08 5.12067E-09 9.07487E-10

OL1 OL2 OL∞ 0.37 1.83 2.12 OL1

0.35 1.83 2.14 OL2

0.37 1.80 2.12 OL∞

2.89 3.38 3.25 OL1

2.89 3.37 3.25 OL2

2.68 3.28 3.49 OL∞

4.79 4.06 3.97 OL1

4.82 4.03 3.99 OL2

4.69 4.15 3.96 OL∞

5.43 4.97 5.00 OL1

5.43 4.97 5.00 OL2

4.95 4.37 5.08 OL∞

6.07 6.08 6.08 6.02 6.00 5.58 6.76 6.71 6.01

Face-FEM P1 (O2) #20 #40 #60 #80 P2 (O3) #20 #40 #60 #80 P3 (O4) #20 #40 #60 #80 P4 (O5) #20 #40 #60 #80 P5 (O6) #20 #40 #60 #80

L1 (p ) 1.15618E-01 4.83460E-02 2.46100E-02 1.41980E-02 L1 (p ) 2.00308E-02 1.06225E-03 3.18965E-04 1.33367E-04 L1 (p ) 9.83420E-04 4.88863E-05 9.49407E-06 3.00983E-06 L1 (p ) 2.49764E-05 1.24190E-06 1.31466E-07 L1 (p ) 2.89994E-05 2.92580E-07 2.95746E-08 -

L2 (p ) 1.79022E-02 7.70888E-03 3.84259E-03 2.24103E-03 L2 (p ) 3.12586E-03 1.78868E-04 5.25784E-05 2.21841E-05 L2 (p ) 2.13116E-04 1.01645E-05 2.03491E-06 6.46171E-07 L2 (p ) 4.88336E-06 2.76073E-07 3.43140E-08 L2 (p ) 5.71709E-06 5.88622E-08 5.91868E-09 -

L∞ (p ) 3.68511E-03 1.74493E-03 8.86988E-04 5.06429E-04 L∞ (p ) 7.73576E-04 7.14012E-05 2.45934E-05 1.07211E-05 L∞ (p ) 9.85256E-05 4.87564E-06 1.27988E-06 3.68528E-07 L∞ (p ) 2.58031E-06 2.02888E-07 3.88314E-08 L∞ (p ) 2.46051E-06 3.30630E-08 4.06987E-09 -

OL1 OL2 OL∞ 1.26 1.67 1.91 OL1

1.22 1.72 1.87 OL2

1.08 1.67 1.95 OL∞

4.24 2.97 3.03 OL1

4.13 3.02 3.00 OL2

3.44 2.63 2.89 OL∞

4.33 4.04 3.99 OL1

4.39 3.97 3.99 OL2

4.34 3.30 4.33 OL∞


Table 1 Convergence rates and error norms for the ADER-DG and the Face-FE method

4.33 4.14 3.67 5.54 5.14 4.08 OL1 OL2 OL∞ 6.63 6.60 6.22 5.65 5.67 5.17 -

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2

y

1 0 -1 -2 -5

-4

-3

-2

-1

0 x

1

2

3

4

5

Fig. 3 Contour plot of p for the convergence test, #40 elements in x-direction at t = 0.0

high order FE methods on unstructured meshes) and that they are implemented in a common code environment. Hence, the conditions under which the comparison is performed involve the same • • • • • •

example grid orders of accuracy code framework compiler and compiler options (IFC 9.0) computational platform (AMD Opteron 270 Dual Core, 2 GHz, 8 GB RAM)

Both methods employ explicit ADER time integration. However, there is one drawback regarding the comparability: While the Face-FE method solves the acoustic wave equation in system form (34)-(35), the ADER-DG method solves the full set of LEE in 2D (equation (22) with Jacobians (24)) and thus one equation more. Also, the common setup does not take into account that the ADER-DG method is capable of handling local and global background flows with arbitrary Jacobians, so the full potential of the scheme is not exploited yet. It is emphasized that the degrees of

2

y

1

0

-1

-2 -5

-4

-3

-2

-1

0

x

1

2

3

4

5

Fig. 4 Grid and contour plot of p at t = 0. The polynomial in each cell is used for visualization


63

2

y

1

0

-1

-2 -5

-4

-3

-2

-1

0

x

1

2

Fig. 5 Grid and contours levels if only mean values of

3

p

4

5

plotted: The actual resolution is lost

freedom (DOF) in the efficiency factors given in Table 3 denote single components (e.g., v or p ) of the equation system’s state vector and the numbers reflect the CPU time per component. Note that the number of DOFs for the velocity components of the Face-FE method differ from the number of pressure DOFs, which is also considered in the given factors. The computational setup for the comparison is almost identical to the one from the convergence study in section 3.3.1. The only difference now is, that the grid is switched to a coarse irregular grid (Fig. 4). The coarseness of the grid turns out to be a demanding feature of the test case, which becomes clear, if only mean values are plotted for the initial condition (Fig. 5): The visualization cannot capture the actual solution on the grid and the wave vanishes. However, as the solution in each element is represented by a polynomial, this information can be used for visualization as it is done in Fig. 4. In order to test the schemes’ capability to transport acoustic waves Table 2 ADER-DG and Face-FEM: Error norms and wall-clock CPU times for different orders

ADER-DG P1 (O2) P2 (O3) P3 (O4) P4 (O5) P5 (O6) P6 (O7) P7 (O8) Face-FEM P1 (O2) P2 (O3) P3 (O4) P4 (O5) P5 (O6) P6 (O7) P7 (O8)

L1 (p ) 1.61155E-01 1.57914E-01 8.20206E-02 6.69119E-03 7.09440E-04 7.61593E-05 8.14198E-06 L1 (p ) 1.59463E-01 9.63090E-02 1.88002E-03 2.11568E-04 6.10649E-05 -

L2 (p ) 2.49445E-02 2.49953E-02 1.28810E-02 1.05309E-03 1.11974E-04 1.21084E-05 1.44290E-06 L2 (p ) 2.50092E-02 1.52586E-02 3.61499E-04 3.76543E-05 1.16797E-05 -

L∞ (p ) 5.00670E-03 5.01007E-03 2.77357E-03 2.66162E-04 3.83522E-05 7.56508E-06 8.48846E-07 L∞ (p ) 5.06886E-03 4.15299E-03 3.54854E-04 5.82064E-05 1.50615E-05 -

tCPU [s] 19.1 47.0 120.5 246.9 478.2 893.7 1618.5 tCPU [s] 4.0 31.0 88.0 668.0 951.0 -

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Table 3 ADER-DG and Face-FEM: Calculation properties and various efficiency factors

ADER-DG P1 (O2) P2 (O3) P3 (O4) P4 (O5) P5 (O6) P6 (O7) P7 (O8)

Δt 2.87E-02 1.72E-02 1.23E-02 9.56E-03 7.83E-03 6.62E-03 5.74E-03

#Iter #Elem 3486 458 5809 458 8133 458 10456 458 12780 458 15103 458 17427 458

Face-FEM P1 (O2) P2 (O3) P3 (O4) P4 (O5) P5 (O6) P6 (O7) P7 (O8)

Δt 7.11E-02 3.82E-02 4.31E-02 1.64E-02 2.31E-02 -

#Iter #Elem 1406 458 2617 458 2320 458 6084 458 4329 458 458 458

DOF(ρ +u +v +p ) Elem

tCPU [s] Elem·Iter

tCPU [s] Elem·tSim

tCPU [s] DOF·Iter

tCPU [s] DOF·tSim

12 24 40 60 84 112 144

1.19E-05 1.77E-05 3.24E-05 5.16E-05 8.17E-05 1.29E-04 2.03E-04

4.16E-04 1.03E-03 2.63E-03 5.39E-03 1.04E-02 1.95E-02 3.53E-02

9.96E-07 7.36E-07 8.09E-07 8.59E-07 9.73E-07 1.15E-06 1.41E-06

3.47E-05 4.28E-05 6.58E-05 8.98E-05 1.24E-04 1.74E-04 2.45E-04

DOF(u +v +p ) Elem

tCPU [s] Elem·Iter

tCPU [s] Elem·tSim

tCPU [s] DOF·Iter

tCPU [s] DOF·tSim

11 21 34 50 69 -

6.21E-06 2.59E-05 8.28E-05 2.40E-04 4.80E-04 -

8.73E-05 6.77E-04 1.92E-03 1.46E-02 2.08E-02 -

7.76E-07 1.57E-06 2.96E-06 5.64E-06 7.99E-06 -

1.09E-05 4.10E-05 6.86E-05 3.43E-04 3.46E-04 -

over large distances, the calculation stops at t = 100, when each wave has traveled a distance of 100 wavelengths. The error norms and the wall-clock CPU times are given in Table 2. Last but not least, the properties of the single calculations and the efficiency per element or DOF are given in Table 3. First of all, it is clear from the graphical representation of the error norms in Figs. 6, 7 and 8, that the wave can be only resolved well for higher orders of accuracy. For ADER-DG, the error norms begin to drop linearly (in log-scale) from about O4. This is the threshold when the resolution inside the elements has become sufficient for the wavelength. For lower orders, Face-FEM is very efficient: The computational costs are small, if the accepted error of the calculation may be

ADER-DG Face-FEM

100

O2 O3

O4

-2

O6

O4 O5

10-4

O2 O3 O4

O2

O3 O5

10-3

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ADER-DG Face-FEM

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10

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L2(p’)

10-1

10

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O4

-4

O6 O5

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O8

10-5 10-6 0 10

101

tCPU 10

2

103

Fig. 6 L1 -error norm of p versus tCPU

10

104

O7

O6

O6

O8

-6

10-7 0 10

101

tCPU 10

2

103

Fig. 7 L2 -error norm of p versus tCPU

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Fluid-Acoustic Coupling and Wave Propagation Fig. 8 L∞ -error norm of p versus tCPU

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10-2

O2 -3

10

-4

Linf(p’)

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ADER-DG Face-FEM

O2 O3 O4

O3

O5

O4

O5 O6 O6

10-5

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10-6

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tCPU 10

2

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104

somewhat larger. However, for rising order of accuracy, the number of DOFs rises. Hence, the costs for the inversion of the mass matrix – performed for every time step – grows. This makes the method rather expensive, if only a very small error can be accepted. In this case, the ADER-DG method closes in and even overtakes the Face-FE scheme for higher order, which is especially visible for the L∞ norm. Here, the advantage of ADER-DG lies in the fact, that the mass matrix is diagonal and can be computed and stored only once in advance. This fundamental difference between the two methods finds its representation also in the efficiency factors of Table 3: While the CPU times per DOF and iteration (or per DOF per simulation time) are smaller for Face-FEM at lower orders, this ratio inverts for higher orders in favor of the ADER-DG method.

4 Numerical Examples 4.1 Aero-Engine Inlet An application of the ADER-DG method to a realistic test problem of computational aeroacoustics (CAA) is presented in this section. The example consists in the simulation of the sound generation and propagation of a MTU JT15D aero-engine inlet. The example is taken from Panek et al. [22], where the details of the setup are described. The geometry of the engine inlet is depicted in Figure 9. The sound sources are modeled based on a large eddy simulation. The presented noise propagation results have been obtained with two different numerical methods for comparison: A fourth order block-structured FD scheme and the unstructured ADER-DG scheme presented in this paper using P3 elements (O4) with time-accurate local time stepping (LTS), see Dumbser et al. [4] for details. The Mach number distribution of the background flow has been obtained by a RANS simulation and is shown in Figure 11 (left) as a slice in the xz-plane. The mean velocities were chosen to reproduce a take-off flight condition with a fan plane Mach number of 0.4 and a flight speed of Mach 0.25 at a 3.5◦ angle of attack.

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Fig. 9 Geometry of the aero-engine inlet test case. The xy-coordinates denote reference points of the geometry

Fig. 10 Block-structured FD Chimera mesh (left) and unstructured tetrahedral DG mesh for the aero-engine inlet test case

To allow for a reasonable comparison, the size of the computational domain was equal for both the FD and DG code. The grids used for the simulation are shown in Figure 10. The curved blocks in the semi-structured FD mesh indicate the overset blocks which communicate with the underlying host grid via interpolation. In radial and axial direction every third point is shown. The unstructured DG mesh consists of tetrahedral elements and is locally refined near strongly curved boundaries. In order to compare the solution of both methods, multiple monitor points were defined. Snapshots of the fully developed perturbed pressure fields are shown in Figure 12. From visual inspection of the plots, both solutions seem to reproduce the principle lobe of the radiation equally well. The analysis of the monitored pressure signals shows a maximum difference of 2 dB in regions of significant sound pressure levels.


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Fig. 11 Mach contour lines for the aero-engine inlet test case (two different ranges depicted)

Fig. 12 Sound pressure field computed with the block-structured FD code (left) and the unstructured DG method with time-accurate local time stepping (right)

The computational time needed to simulate one millisecond of physical time for the chosen test case is listed in Table 4 for both the FD and the DG code. The CPUh/ms-value was recalculated for a single CPU core. From these results we can Table 4 Domain parameters and computational time for the FD and the ADER-DG method

Method Block-structured FD Unstructured DG with LTS

#DOF 5 469 928 2 466 080

#Elem 5 469 928 123 304

O 4 4

CPUh/ms 6.30 2.15

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deduce that thanks to the time-accurate local time stepping, the unstructured DG method performs better for this particular test case than the block-structured finite difference method.

4.2 Sphere Scattering The scattering of sound by a solid sphere shall demonstrate the capabilities of the domain decomposition approach. This example was proposed by Morris [20] for the Second CAA Workshop on Benchmark Problems (Category 1, Problems 3 and 4). He also provided details about the analytical solution, including the numerical evaluation of spherical Hankel functions [19]. The governing equations in this threedimensional example are the linearized Euler equations with the mean flow variables ρ %0 = 1, u0 = 0, v0 = 0, w0 = 0 and p0 = 0.714285714. The speed of sound is c = γ · ρp00 = 1 with γ = 1.4. A solid sphere with the radius r = 1 is located at XSphere = [0, 0, 0]. In its vicinity at XSource = [2, 0, 0], a spatially distributed source S on the right hand side of the pressure term oscillates: S = −A · exp[−B · ln2 · ((x − xSource)2 + y2 + z2 )] · cos ω t,

(49)

with A = 0.01, B = 16, xSource = 2. In the following, ω = 2π (Problem 3) is chosen as frequency, thus the expected wavelength of the generated sound is λ = cf = 1 with f = 2ωπ = 1. A sponge layer is employed at the outer boundary of the domain in order to avoid reflections. The overall computational domain is divided into one unstructured and one structured volume (Fig. 13). The region [−1.75, −1.75, −1.75] × [3.75, 1.75, 1.75] in the direct vicinity of the sphere consists of an unstructured tetrahedral mesh and includes the source region. The O4 ADER-DG method is employed and the time step in this domain is Δ t = 1.664·10−3 (CFL=0.3). The elements directly at the sphere surface are slightly refined with an interval size of Δ h = 0.2, compared to an interval size of Δ h = 0.4 at the outer boundaries. The interval size denotes the average edge length of a surface triangle. The minimum inner sphere radius of a tetrahedron is lmin = 0.0194. As it is not trivial to quantify the resolution of the ADER-DG method on tetrahedral grids, the following approximation is made: The number of degrees of freedom per element in the unstructured domain is nDOF = O·(O+1)·(O+2) = 20. By averaging the total 6 number of DOFs for the considered volume and for each spatial direction, one obtains approximately 14.62 DOFs per wavelength for the unstructured grid. A total of 1520 ghost tetrahedrons couple with the structured grid at the domain interface, which results in 97280 connecting Gauss integration points. The unstructured mesh remains fixed in the following computations. In the first calculation, the outer far field domain [−6, −6, −6] × [8, 6, 6] consists of a Cartesian mesh with an interval size of Δ h = 0.1 and thus a resolution of 10 PPW. The employed FD scheme has an order of accuracy of O8. A total of 31328 ghost points are required for the coupling with the unstructured grid.


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Fig. 13 A cut-out of the simulation domain including the structured and unstructured mesh as well as a slice of the fluctuation pressure contour plot. The structured mesh is shown in the 5 PPW configuration

In a second calculation, the interval size of the structured grid is doubled (Δ h = 0.2), hence the resolution is 5 PPW. For this configuration, the total number of structured ghost points coupling with the unstructured domain is only 6264. Last but not least, the domain decomposition results are compared to a calculation that has been performed on a single tetrahedral grid, using the O4 ADER-DG method. The calculation domain is spherical with its center at [1, 0, 0] and a radius of r = 7. In order to keep the grid similar to the one in the decomposition approach, the grid has the same size Δ h = 0.2 on the sphere surface. At the boundary, a spacing of Δ h = 0.4 is imposed, resulting in a grid for a well resolved solution. All computations were performed on a single Intel Xeon 5150 2.66GHz core. Although the solution reaches periodicity in every point of the domain very quickly (in a simulation time of less than t = 5), the calculation was run until t = 20, in order to ensure a sufficiently large sample for the analysis. Figure 14 shows the directivity plot of the root mean square fluctuation pressure. It was picked up at several radii in the xy-plane around the sphere center, r = 1 (surface of the sphere), r = 3 and r = 5. The results of the three calculations are all in good agreement with the reference solution. Tables 5 and 6 show the statistics for the three computations: Both decomposition calculations are considerably faster than the single ADER-DG calculation in terms of wall-clock time. The ta-

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0.0002

Analytical Solution Single Domain: ADER-DG O4 Domain Decomposition: ADER-DG O4, FD O8, 10 PPW Domain Decomposition: ADER-DG O4, FD O8, 5 PPW Domain Decomposition: ADER-DG O4, FD O4, 5 PPW

prms sinθ

0.0001

0

-0.0001

-0.0002 -0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

prms cosθ Fig. 14 Directivity plots of the RMS fluctuation pressure for the analytical solution, several domain decomposition calculations and the single unstructured domain calculation

bles also provide a reason why: If the DOFs per DG element are considered as the points that are necessary for the resolution of the waves, the efficiency of the DG and the FD method can be compared for this specific example. DOF denotes here Table 5 Efficiency factors for the domain decomposition and the single domain calculation tCPU [s] tCPU [s] tCPU [s] Domain O PPW Elem·Iter DOF·Iter DOF·tSim Unstr. 1 4 ≈ 15 8.37E-05 4.19E-06 2.52E-03 Str. 1 8 10 1.39E-05 1.39E-05 4.64E-04

10 PPW in the structured domain. tCPU [s] tCPU [s] tCPU [s] Domain O PPW Elem·Iter DOF·Iter DOF·tSim Unstr. 1 4 ≈ 15 8.37E-05 4.19E-06 2.36E-03 Str. 1 8 5 1.53E-05 1.53E-05 2.55E-04

5 PPW in the structured domain. tCPU [s] tCPU [s] tCPU [s] Domain O PPW Elem·Iter DOF·Iter DOF·tSim Unstr. 1 4 ≈ 15 6.94E-05 3.47E-06 2.06E-3

Single ADER-DG domain.


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Table 6 Statistics for the domain decomposition and the single unstructured domain calculation Domain Unstr. 1 Str. 1 Coupling Total

Δt tCPU [s] tCPU [%] Vol. Vol.[%] #Elem #DOF #Iter Δt Δ tmin 9941 25.5 63.2 3.1 9874 197480 12024 1.664E-03 1 18068 46.4 1948.6 96.9 1948625 1948625 668 2.995E-02 18 10958 28.1 38968 100.0 2011.8 100.0 1958499 2146105 -

10 PPW in the structured domain. Δt Domain tCPU [s] tCPU [%] Vol. Vol.[%] #Elem #DOF #Iter Δt Δ tmin Unstr. 1 9316 72.1 63.2 3.1 9874 197480 12024 1.664E-03 1 Str. 1 1301 10.1 1948.6 96.9 255362 255362 334 5.989E-02 36 Coupling 2305 17.8 Total 12922 100.0 2011.8 100.0 265236 462716 -

5 PPW in the structured domain. Domain tCPU [s] tCPU [%] Vol. Vol.[%] #Elem #DOF #Iter Δt Unstr. 1 141394 100.0 1432.6 100.0 171687 3433740 11863 1.686E-03

Δt Δ tmin

1

Single ADER-DG domain.

the entire state vector for a degree of freedom, not each component (e.g., pressure) separately. Otherwise, the total number of DOFs must be scaled up with the number 1 of variables nVar and the CPU times per DOF must be scaled down with nVar (here: tCPU nVar = 5). The CPU time per DOF and per simulation time unit DOF ·tSim shows an advantage of about one magnitude for the structured O8 FD method (Table 5). This advantage is strongly example dependent because it arises from the much bigger time step that can be made on the structured grid per iteration. The time step on the unstructured grid is limited by the DG method itself and by the finer grid that is required in order to resolve the sphere. On the other hand, the CPU time per element tCPU and iteration Elem·Iter allows an example independent (and only processor specific) statement about the methods’ cost per iteration. Here, the O4 ADER-DG method is again more expensive but becomes actually cheaper than the FD scheme, if the CPU time per DOF and iteration is considered. However, the time step Δ t per iteration is much greater for the structured grid because of a less restrictive stability condition and a more regular grid. Note that the edge spacing Δ h = 0.2 of the coarsest FD grid is the same as the interval size Δ h = 0.2 for the surface triangles on the sphere. Last but not least, a comparison between the two domain decomposition calculations (Table 6) demonstrates the benefit of using high order methods: The structured far field grid covers almost 97% of the total domain volume. By decreasing the resolution 1 from 10 PPW to 5 PPW, the calculation effort is reduced by a factor of almost 16 th! This is due to the two times bigger time step on the coarser grid, furthermore only 1 of the grid points are needed. At the same time, the sound waves are still resolved 23 well by the high order method, while a lower order method, for example O4, begins to show dissipation errors for 5 PPW at the outer regions of the domain (Fig. 14).

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5 Conclusions Novel methods for hybrid and direct aeroacoustic simulations have been developed. For low Mach number flows, a hydrodynamic/acoustic splitting method is applied. Motivated by asymptotic results for fluid flow at small Mach numbers, the expansion about incompressible flows was extended to the more general situation, when heat conduction or compression from the boundary is active and cannot be neglected. The incompressible solution is replaced by the solution of the limit equations of the compressible Navier-Stokes equations, when the Mach number tends to zero. The usually applied additional assumptions of constant temperature and density are abolished. The state of expansion is given by a subsonic approximation of the compressible Navier-Stokes equations. We introduced multiple pressure variables, a thermodynamic background pressure and a hydrodynamic pressure term, to obtain the proper limit equations. The sum of the M → 0 solution and perturbed quantities are inserted into the compressible equations to obtain evolution equations for the perturbations. A domain decomposition approach for the direct simulation of aeroacoustic problems has been developed. The basic concept is to combine different numerical methods, equations, grids and time steps for a greater efficiency. It is of secondary importance for the domain decomposition philosophy how evolved and thus complex the single solvers in the subdomains are, i.e. if they feature local time stepping algorithms, dynamic grid adaptation and so on. In fact, an increase in computational performance can already be realized by exploiting the basic properties of different schemes. The coupling approach connects different classes of methods (DG, FV, FD) on structured and unstructured grids for the solution of the Navier-Stokes, Euler and linearized Euler equations. The optimal time step can be chosen in each subdomain by employing a subcycling technique. The sphere scattering benchmark example demonstrated, that the coupling approach is especially suitable for far field computations with embedded complex geometries. For the simulation of acoustic propagation on unstructured grids, both the FaceFEM and the ADER-DG method turn out to be high order accurate and quite efficient. While being slower for lower orders of accuracy, the ADER-DG scheme closes in on the Face-FE method for higher orders and is additionally capable of handling scenarios with a complex background flow. The latter was shown for the simulation of an aero-engine inlet. Acknowledgements. This research was realized within subproject SP3 of the research group FOR508 ”Noise Generation in Turbulent Flows” and supported by the Deutsche Forschungsgemeinschaft (DFG) and the Centre national de la recherche scientifique (CNRS).

References 1. Dumbser, M.: Arbitrary High-Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. PhD thesis, Universität Stuttgart, Institut für Aerodynamik und Gasdynamik. Shaker Verlag, Aachen (2005)


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2. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics 221(2), 693–723 (2007) 3. Dumbser, M., Käser, M., Titarev, V., Toro, E.: Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics 226(2), 715–736 (2007) 4. Dumbser, M., Käser, M., Toro, E.: An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes – V. Local time stepping and p-adaptivity. Geophysical Journal International 171(2), 695–717 (2007) 5. Dumbser, M., Munz, C.-D.: Building blocks for arbitrary high order discontinuous Galerkin schemes. Journal of Scientific Computing 27(1-3), 215–230 (2006) 6. Ewert, R., Schröder, W.: Acoustic perturbation equation based on flow decomposition via source filtering. Journal of Computational Physics 188, 365–398 (2003) 7. Farassat, F., Myers, M.: Extension of Kirchoff formula to radiation from moving surfaces. Journal of Sound Vibration 123, 451–460 (1988) 8. Ffowcs Williams, J., Hawkins, D.: Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society 264, 321–342 (1969) 9. Hardin, J., Pope, D.: An acoustic/viscous splitting technique for computational aeroacoustics. Theoretical and Computational Fluid Dynamics 6, 323–340 (1994) 10. Hirsch, C.: Numerical Computation of Internal and External Flows. Fundamentals of Numerical Discretisation, vol. I. Wiley, Chichester (1988) 11. Hu, F., Hussaini, M., Manthey, J.: Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. Journal of Computational Physics 124, 177–191 (1996) 12. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Communications on Pure and Applied Mathematics 34, 481–524 (1981) 13. Klainerman, S., Majda, A.: Compressible and incompressible fluids. Communications on Pure and Applied Mathematics 35, 629–653 (1982) 14. Klein, R.: Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow. Journal of Computational Physics 121, 213–237 (1995) 15. Lax, P., Wendroff, B.: Systems of conservation laws. Communications on Pure and Applied Mathematics 13, 217–237 (1960) 16. Lighthill, M.: On sound generated aerodynamically - 1. General theory. In: Proceedings of the Royal Society of London, vol. 211(A1107), pp. 564–587 (1952) 17. Lighthill, M.: On sound generated aerodynamically - 2. Turbulence as a source of sound. In: Proceedings of the Royal Society of London, vol. 222(A1148), pp. 1–32 (1954) 18. Lörcher, F., Munz, C.-D.: Lax-Wendroff-type schemes of arbitrary order in several space dimensions. IMA Journal of Numerical Analysis 27, 593–615 (2007), doi:10.1093/imanum 19. Morris, P.: Scattering of sound from a spatially distributed, spherically symmetric source by a sphere. Journal of the Acoustical Society of America 98, 3536–3539 (1995) 20. Morris, P.: Scattering of sound by a sphere: Category 1, problems 3 and 4. In: Proceedings of Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems. NASA/CP1997-3352 (October 1997) 21. Munz, C., Dumbser, M., Roller, S.: Linearized acoustic perturbation equations for low Mach number flow with variable density and temperature. Journal of Computational Physics 224, 352–364 (2007) 22. Panek, L., Schönwald, N., Thiele, F., Dumbser, M., Munz, C.: Comparison of numerical approaches for the simulation of sound radiation from a scarfed inlet. In: Proceedings of the 14th International Conference on Sound and Vibration (ICSV 14), Cairns, Australia (2007) 23. Schwartzkopff, T., Dumbser, M., Munz, C.-D.: Fast high order ADER schemes for linear hyperbolic equations. Journal of Computational Physics 197, 532–539 (2004) 24. Seo, J., Moon, Y.: Perturbed compressible equations for aeroacoustic noise prediction at low mach numbers. AIAA Journal 43(8), 1716–1724 (2005)

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25. Seo, J., Moon, Y.: Linearized perturbed compressible equations for low mach number aeroacoustics. Journal of Computational Physics 218, 702–719 (2006) 26. Titarev, V., Toro, E.: ADER schemes for three-dimensional nonlinear hyperbolic systems. Journal of Computational Physics 204, 715–736 (2005) 27. Utzmann, J.: A Domain Decomposition Method for the Efficient Direct Simulation of Aeroacoustic Problems. PhD thesis, Universität Stuttgart, Institut für Aerodynamik und Gasdynamik. Verlag Dr. Hut, München (2008) 28. Utzmann, J., Schwartzkopff, T., Dumbser, M., Munz, C.-D.: Heterogeneous domain decomposition for computational aeroacoustics. AIAA Journal 44(10), 2231–2250 (2006) 29. Kowalevsky, S., v.: Zur Theorie der Partiellen Differentialgleichungen. Dissertation, Philosophische Facultät zu Göttingen (1874), http://www.digizeitschriften.de

Mechanisms and Active Control of Jet-Induced Noise Andreas Babucke, Bruno Spagnoli, Christophe Airiau, Markus Kloker, and Ulrich Rist

Abstract. Fundamental mechanisms of jet noise are investigated by means of direct numerical simulation. In the mixing layer, subharmonics of the respective vortex pairing are found to be responsible for the main part of the generated noise which is directed in downstream direction. By modifying the phase shift between introduced disturbances it is possible to diminish or enhance relevant portions of the emitted sound. Optimal control has been applied successfully to a plane mixing layer. In the far field, the mean noise level could be reduced. Depending on the measurement line, some distributed control or anti-noise is generated by the control. A more realistic configuration is achieved by adding a splitter plate representing the nozzle end. Rectangular serrations lead to a breakdown of the large coherent spanwise vortical structures and thus provide a noise reduction of 9dB.

1 Introduction Noise reduction is of great interest for a variety of technical applications. This is especially the case in aviation. Since most airports are located in highly populated areas, the reduction of aircraft noise can improve the quality of life for many people. Especially during take-off, jet noise is the largest aeroacoustic source of an aircraft. Recent reductions of jet noise are mainly due to an increased bypass ratio in the turbojet engine. Currently a geometric variation of the nozzle end is considered, Andreas Babucke · Markus J. Kloker · Ulrich Rist Institut für Aerodynamik und Gasdynamik (IAG), Universität Stuttgart, Pfaffenwaldring 21, 70550 Stuttgart, Germany [email protected] Bruno Spagnoli · Christophe Airiau Université de Toulouse, Paul Sabatier, Institut de Mécanique des Fluides de Toulouse (IMFT), Allée du professeur Camille Soula, 31400 Toulouse, France [email protected]


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Fig. 1 Sketch of a typical bypass jet engine with the occurring mixing layers behind the nozzles

known as chevron-nozzle. Its noise reduction is often explained by an increased mixing behind the trailing edge of the nozzle. However the underlying physical mechanisms are not yet fully understood. Within the subproject 5 Mechanisms and Active Control of Jet-induced Noise, the mechanisms of noise generation and its reduction are investigated using direct numerical simulations (DNS). Such aeroacoustic simulations face several difficulties due to largely different scales of flow field and acoustics. The hydrodynamic fluctuations are small-scale structures containing high energy compared to the acoustics with relatively large wavelengths and small amplitudes. Therefore, high resolution is required to compute the noise sources accurately. On the other hand a large computational domain is necessary to obtain the relevant portions of the acoustic far-field. Due to the small amplitudes of the emitted noise, boundary conditions have to be chosen carefully, in order not to spoil the acoustic field with reflections. Since computational resources are limited, one possibility is the simulation of low-Reynolds-number jets, e.g. done by Freund [12]. Our focus is set on the mixing layer behind the nozzle end which is responsible for the most anoying highfrequency noise. As sketched in figure 1, mixing layers occur between the primary and the bypass stream and between the bypass stream and the freestream. Having high Reynolds-number jets at an aircraft’s engine, the large diameter allows to neglect curvature. This approach is widely used [6, 9] for large-eddy or direct numerical simulations, where an S-shaped velocity profile is prescribed at the inflow. This flow field allows to investigate the fundamental mechanisms of noise generation and its control. Beyond this, the nozzle end can be included in the simulation, leading to a combination of wake and mixing layer. This provides a more realistic configuration and allows to simulate wall-mounted actuators. In this context, the engrailed nozzle end can be interpreted as a first passive ’actuator’. At the beginning of the project, a high-order, low-dispersion/low-dissipation numerical scheme for the solution of the unsteady compressible Navier-Stokes equations was available [11]. However, more complex configurations and parallelization requirements lead to the development of a new DNS code. The code is embedded in a simulation framework including initial conditions, linear stability theory (LST) and postprocessing based on EAS3 [10]. The numerical scheme has been verified for aeroacoustic simulations by comparison with the benchmark problem [9]. An alternative method of handling the multiscale problem is the coupling with an acoustic

Mechanisms and Active Control of Jet-Induced Noise

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solver. Its principle functionality could be demonstrated in cooperation with subproject 3, see [2]. IMFT has gained experience in the application of optimal control theory, see e.g. [1, 19]. The application of the adjoint equations has been extended to compressible flows [17, 18]. The open-loop control with DNS is the only approach which allows to deal with some one million of unknowns for each time step. The main drawback of the open-looped control is a poor robustness due to the absence of a feedback loop. With the real flow conditions being different from the ones used to design the optimal control, at least the control law can be inefficient or may even increase the noise emission. However, as demonstrated in [18], the sensitivity analysis which can be seen as an initial step of an optimal control approach can propose some essential information about the optimality in types and in the positions of sensors and actuators for a feedback control system. With some few runs it is then possible to have a better idea of The numerical methods used for DNS and optimal control are addressed in sections 2.1 and 2.2, respectively. The fundamental mechanisms of noise generation in a two-dimensional mixing layer are discussed in 3. Section 4 describes the optimal control applied to such a flow field. The influence of a serrated nozzle end is treated in section 5 and the emitted sound is compared to the case with a straight trailing edge.

2 Numerical Method 2.1 Direct Numerical Simulation Direct numerical simulations are performed by the DNS-code NS3D [5] solving the unsteady three-dimensional compressible Navier-Stokes equations on multiple domains. The purpose of domain decomposition is not only to increase computational performance. The combination with grid transformation and the concept of modular boundary conditions allows to compute a wide range of problems. Computation is done in non-dimensional quantities: velocities are normalized by the reference velocity u˜∞ , and all other quantities by their inflow values, marked with the subscript ˜ and the time t ∞ . Length scales are made dimensionless with a reference length L ˜ u˜∞ , where the tilde denotes dimensional values. Temperature dependence of with L/ viscosity μ is modelled using the Sutherland law:

μ˜ (T ) = μ˜ (T˜∞ ) · T 3/2 ·

1 + Ts , T + Ts

(1)

where Ts = 110.4K/T˜∞ and μ˜ (T˜∞ = 280K) = 1.735 · 10−5kg/(ms). Thermal conductivity ϑ is obtained by assuming a constant Prandtl number Pr = c p μ /ϑ . The most characteristic parameters describing a compressible viscous flow-field are the Mach number Ma = u∞ /c∞ and the Reynolds number Re = ρ∞ u∞ L/ μ∞ . We use the conservative formulation of the Navier-Stokes equations which results in the

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solution vector Q = [ρ , ρ u, ρ v, ρ w, E] containing the density, the three mass fluxes and the total energy per volume: ρ 2 · u + v2 + w2 . (2) 2 The simulation is carried out in a rectangular domain with x, y, z being the coordinates in streamwise, normal and spanwise direction, respectively. The flow is assumed to be periodic in spanwise direction. Thus a spectral discretization is used in z-direction: E = ρ · cv · T +

K

f (x, y, z,t) =

∑

Fˆk (x, y,t) · ei(kγ0 z) .

(3)

k=−K

ˆ f denotes any flow variable, √ Fk its complex Fourier coefficient, K the number of spanwise modes and i = −1. The fundamental spanwise wavenumber γ0 is given by the fundamental wavelength λz,0 representing the width of the integration domain by γ0 = 2π /λz,0. Spanwise derivatives are computed by transforming the respective variable into Fourier space, multiplying its spectral components with their wavenumbers (i · k · γ0 ) for the first derivatives or square of their wavenumbers (−k2 · γ02 ) for the second derivatives and transforming them back into physical space. Due to the non-linear terms in the Navier-Stokes equations, higher harmonic spectral modes are generated at each time step. To suppress aliasing, only 2/3 of the maximum number of modes for a specific z-resolution are used [7]. If a two-dimensional baseflow is used and introduced disturbances of u, v, ρ , T , p are symmetric and disturbances of w are antisymmetric, flow variables are symmetric/antisymmetric with respect to z = 0. Therefore only half the number of points in spanwise direction are needed (0 ≤ z ≤ λz /2), or equivalently, the Fˆk s are either purely real or imaginary. The spatial discretization in streamwise (x) and normal (y) direction is done by 6th -order compact finite differences. The tridiagonal equation systems resulting from the compact finite differences are solved using the Thomas algorithm. To reduce the aliasing error, alternating up- and downwind-biased finite differences are used for convective terms as proposed by Kloker [15]. The second derivatives are evaluated directly which distinctly better resolves the second derivatives compared to applying the first derivative twice, see [4]. Additionally, the numerical scheme is more robust since the second derivative does not vanish for the least resolved wave. Arbitrary grid transformation in the x-y plane is provided by mapping the physical grid on an equidistant computational ξ -η grid: x = x(ξ , η ) , y = y(ξ , η ) .

(4)

Time integration of the Navier-Stokes equations is done using the classical 4th -order Runge-Kutta scheme as described in [15]. At each time step and each intermediate level the biasing of the finite differences for the convective terms is changed. At the borders of each domain where no neighbor exists, specific boundary conditions can be selected. For the current investigation, we use a one-dimensional


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characteristic boundary condition [13] at the freestream. This allows straight outward-propagating acoustic waves to leave the domain. An additional damping zone drives the flow variables smoothly towards to a steady state solution, avoiding reflections due to oblique waves. Having a subsonic flow, we also use a characteristic boundary condition at the inflow. Additionally amplitude and phase distributions from linear stability theory can be prescribed to introduce defined disturbances. Their phase shift is defined with respect to the maximum amplitude of the streamwise velocity. The disturbances due to the eigenfunctions are added after the characteristic boundary condition is applied. Thus they are less affected by the characteristic treatment. The outflow is the most crucial part as one has to avoid large structures passing the boundary and contaminating the acoustic field. Therefore, a combination of grid stretching and spatial low-pass filtering is applied in the sponge region. Disturbances become increasingly badly resolved as they propagate through the sponge region. As the strength of the spatial filter depends on the step size in x-direction, perturbations are smoothly dissipated before they reach the outflow boundary. This procedure shows very low reflections and has been already applied by Colonius et al. [8]. For the splitter plate representing the nozzle end, an isothermal boundary condition is used with the wall temperature being fixed to its value from the initial condition. The pressure is obtained by extrapolation from the interior gridpoints. An extension of the wall boundary condition is the modified trailing edge, where the end of the splitter plate is no more constant along the spanwise direction. As we have grid transformation only in the x-y plane and not in z-direction, the spanwise dependency of the trailing edge is achieved by modifying the connectivity of the affected domains. Instead of regularly prescribing the wall boundary condition along the whole border of the respective subdomain, we can also define a region without wall, now. At these gridpoints, the spatial derivatives in normal direction are recomputed, now using also values from the domain on the other side of the splitter plate. The spanwise derivatives are computed in the same manner as inside the flowfield with the Fourier-transformation being applied along the whole spanwise extent of the domain. Inside the notch the full equations with the newly computed y-derivateives are solved. The concept of modular boundary conditions, chosen because of flexibility and maintainability, requires explicit boundary conditions and by that a non-compact finite-difference scheme, here. Therefore explicit finite differences have been developed with properties similar to the compact scheme used in the rest of the domain. The numerical properties of the chosen 8th -order scheme are compared with standard explicit 6th -order finite differences and the compact scheme of 6th order, regularly used in the flowfield. For the first derivative, the ∗ real and imaginary parts of the modified wavenumber kmod are shown in figure 2: the increase from order six to eight does not fully reach the good dispersion rela∗ tion of the 6th -order compact scheme but at least increases the maximum of kmod th by 10% compared with an ad hoc explicit 6 -order implementation. The imaginary part of the modified wavenumber, responsible for dissipation, shows similar

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exact Re(k*mod) FD-O8 Re(k*mod) FD-O6 compact Re(k*mod) FD-O6 Im(k*mod) FD-O8 Im(k*mod) FD-O6 compact

exact Re(k*2mod) FD O8 2 Re(k*mod) FD O6 compact 2 Re(k*mod) FD O6

8

2

6

2

k* mod

k* mod

3

4

1

2 0

0 1

0

k*

2

3

Fig. 2 Real and imaginary part of the mod∗ ified wavenumber kmod for the first derivative based on a wave with wave number k∗ = k · Δ x. Comparison of 8th -order explicit finite difference with 6th -order explicit and compact scheme

1

0

k*

3

2

Fig. 3 Square of the modified wavenumber of the second derivative for a wave with wave number k∗ = k· Δ x. Comparison of 8th order explicit finite difference with 6th -order explicit and compact scheme

characteristics as the compact scheme with the same maximum as for the rest of the domain. For the second derivative, shown by the modified wavenumber square (k∗2 )mod in figure 3, the increase of its order improves the properties of the explicit finite difference towards the compact scheme.

2.2 Optimal Control The uncontrolled mixing layer is computed with DNS. Some typical iso-vorticity lines at one time instance are shown in figure 4. The real computational domain

1.02

(Ω11 )

0.5

I (λ)

dI dλ

1

y

0

(Ω10 ) Actuation domains

0.98 −0.5 0.96

Measurement line (Ω2 )

0.92

x

Fig. 4 Actuation domain (close to the origin o the shear layer). The line in the south of the domain is the line where the reduction of noise is targeted

−1

0.94

0

0.05

λ

0.1

−1.5

Fig. 5 Actuation domain Ω11 , first global dI iteration: I(λ ) (left scaling) and (right dλ scaling)


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(Ω ) is much larger than what is shown in the figure. A line (Ω2 ) located quite far from the mixing layer instabilities defines the location where the noise emission is targeted for reduction. It is called the measurement domain, usually where sensors could be positioned. Two control domains, a smaller one (Ω11 ) and a larger one (Ω10 ) are fixed at the birth of the mixing layer. The measurement and control domains have been determined from the previous sensitivity analysis [18, 17]. There it has been demonstrated that the noise emission, far from the mixing layer, is very sensitive to any perturbation at the origin of the mixing layer. To decrease the noise emission and for the well-posedness of the control problem we target to minimize the following objective functional:

T

J(f) = 0

Ω2

(p(x,t) − p(x))2 dxdt + 20

T

0

Ω1n

fT BT Bfdxdt

(5)

where p(x) denotes the temporal mean value of the pressure at the location x in the uncontrolled case. The weighted matrix B allows to test different types of forcing (control) by keeping only the desired components of the control vector f. The 0 coefficient gives a bound to the control cost. Setting 0 = 0 leads to find the most effective control without taking into account the energy cost which could theoretically go to infinity. Several numerical experiments have shown that for our case, the cost of the energy is always negligible. Hence we will set 0 = 0 in the following for simplicity. The minimum of the functional is obtained from an iterative procedure using a conjugate gradient algorithm. The gradient of the cost functional with respect to the control variable is calculated by a small perturbation of an associated Lagrangian functional including all the constrains: the goal, the control cost, the main NavierStokes equations and the boundary conditions. It can be shown [17, 19] that the Lagrange coefficient associated to the governing equations is the adjoint vector r of the state variables (density, momentum vector and pressure) called vector q = (ρ , mx , my , p)). The adjoint variables are solutions of the adjoint compressible twodimensional Navier-Stokes equations [17]: F ∗ (q)r = (p(x,t) − p(x))2 δ (Ω2 ),

(6)

with r = (p∗ , m∗x , m∗y , ρ ∗ ) as the adjoint vector and δ (Ω2 ) a function equal to 1 over Ω2 and equal to 0 elsewhere. The source term comes from the derivative of the Lagrangian functional (eq. 5) with respect to the direct state. Finally the gradient of the Lagrangian functional with respect to the control is given by ∇Jf = 20 Bf + r.

(7)

When the control iterative algorithm is converged, this gradient is equal to zero and the solution provides the optimal control f from the adjoint state.

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3 Fundamentals of Mixing-Layer Noise 3.1 Flow Parameters The fundamental mechanisms of jet noise are investigated by considering a twodimensional mixing layer where an S-shaped velocity profile is prescribed at the inflow. The flow parameters have been closely matched to the case of Colonius et al. [9]. This allows to validate the numerical scheme for aeroacoustic simulations. The Mach numbers of the upper and lower stream are MaI = 0.5 and MaII = 0.25, respectively. As both free stream temperatures are equal (T˜I = T˜II = 280K), the ratio of the streamwise velocities is uI /uII = 2. The Reynolds number Re = ρI uI δ (x0 )/ μ = 500 is based on the vorticity thickness at the inflow which is used to normalize length scales. A cartesian grid is used with 2500x850 grid points in x- and y-direction. In streamwise direction, the grid is uniform with spacing Δ x = 0.157 up to the sponge region where the grid is highly stretched. In normal direction, the grid is continuously stretched with the smallest stepsize Δ y = 0.15 in the middle of the mixing layer and the largest spacing Δ y = 1.06 at the upper and lower boundaries. In case of the three-dimensional simulation, the z-direction is discretized with 9 grid points and Δ z = 0.491. This is equivalent to 10 spanwise modes (dealiased) with γ0 = 0.8 in the symmetric case. The initial condition, obtained from the similarity solution of the boundary-layer equations, is used for linear stability theory. The most amplified disturbance is found for the fundamental frequency ω0 = 0.6293 at the inflow. The eigenfunctions from linear stability theory are used to introduce defined disturbances at the inflow. The flow is forced with the fundamental frequency and its first three subharmonics (1/2, 1/4 and 1/8) with the maximum of |u| ˆ = 0.001 for all disturbances. The phase shift is ΔΘ = −0.028 for the first, ΔΘ = 0.141 for the second subharmonic and ΔΘ = 0.391 for the third subharmonic. In an additional simulation the phase shift of ω0 /4 is altered to ΔΘ = 3.141. Note that it is not clear how the phase shift is specified in [9] since the phase distribution varies along the normal direction. Here it is defined with respect to the maximum of |u| ˆ such that Θ = 0 at the location of |u| ˆ max in case of no phase shift. An additional steady disturbance (0, 1) with amplitude |u| ˆ = 0.01 is introduced in the three-dimensional simulation with ΔΘ = 3.141. The modes are denoted as (h, k), with h and k being the multiple of the fundamental frequency ω0 and the spanwise wavenumber γ0 , respectively.

3.2 Two-Dimensional Simulation The spanwise vorticity is shown in figure 6 being similar to the reference solution of Colonius et al. [9]. Since eigenfunctions from viscous linear stability theory instead of inviscid ones as used in [9] are used here, disturbances are introduced slightly more efficient. Accordingly the rollup of the mixing layer and the vortex pairing occur a bit earlier. The maximum amplitudes of the normal velocity v along the y-direction and the corresponding amplification rates are shown in figures 7 and 8, respectively.


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Fig. 6 Snapshot of the spanwise vorticity for ΔΘ (ω0 /4) = 0.141 after 76 fundamental periods. Contour levels range from −0.26 to 0.02 with an increment of 0.04. The reference solution of Colonius et al. [9] is shown above

In the initial region of the integration domain the amplitudes grow exponentially. Despite the spatial amplification rate (−αi ) is a very sensitive value, its values correspond well to those of linear stability theory. Further downstream, modes (1, 0), (1/2, 0), (1/4, 0) saturate at positions x = 90, x = 120 and x = 240, respectively. These streamwise locations correlate well with the positions where the respective vortices are fully developed (figure 6). At the location where (1/2, 0) and (1/4, 0) saturate (x = 120 and x = 240), the growth of the respective subharmonics (1/4, 0) and (1/8, 0) is interrupted before their amplitude increases again. The alternative phase shift ΔΘ (ω0 /4) = 3.141 does not affect the initial growth of the amplitudes as shown in figure 9. Yet the development of the second -0.2 10

-1

-0.1

^

|v|max

α i(v)

10-2

10

-3

(1,0) (1/2,0) (1/4,0) (1/8,0)

10-4 100

x 200

300

Fig. 7 Maximum amplitudes of the normal velocity v along y-direction for ΔΘ (ω0 /4) = 0.141

0 (1,0) (1/2,0) (1/4,0) LST (1,0) LST (1/2,0) LST (1/4,0)

0.1

0.2 30

40

50 x 60

70

80

Fig. 8 Amplification rates of the normal velocity v for ΔΘ (ω0 /4) = 0.141. Symbols denote results from linear stability theory

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10

-1

10

-2

^

|v|max

cph(v)

0.8

10

(1,0) (1/2,0) (1/4,0) (1/8,0) ΔΘ(ω0/4)=0.141

-3

10-4 100

x 200

300

Fig. 9 Maximum amplitudes of the normal velocity v along y-direction for ΔΘ (ω0 /4) = 3.141. Dotted lines denote the previous case with ΔΘ (ω0 /4) = 0.141

0.6 (1/2,0) (1/4,0) (1/4,0) ΔΘ(ω0)=0.141

0.4 50

100

x

150

200

Fig. 10 Phase speed c ph of the normal velocity v at y = 0 compared with the prevoius case

subharmonic differs from x ≈ 120 onwards. There, its phase is adjusted to the one of the first subharmonic. This can be seen in figure 10 showing the phase speed c ph = ω · (∂Θ /∂ x)−1 of the first two subharmonics at y = 0. Phase adjustment is found for both cases, but for ΔΘ (ω0 /4) = 3.141, the phase speed varies more and subharmonic resonance of mode (1/4, 0) is achieved with a reduced amplitude compared to ΔΘ (ω0 /4) = 0.141. Accordingly, its saturation is further downstream and faster resonance of the third subharmonic is provided. This leads to an increased amplitude of (1/8, 0) by a factor of almost four downstream of x ≥ 270.

Fig. 11 Emitted sound with frequency ω0 /4, visualized by the real part of the Fourier-transformed dilatation with contour levels ranging from −2 · 10−6 to 2 · 10−6 for ΔΘ (ω0 /4) = 0.141

Fig. 12 Real part of the Fouriertransformed dilatation field with frequency ω0 /4 for ΔΘ (ω0 /4) = 3.141. Contour levels are the same as in figure 11


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10

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10-1

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-2

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|v|max

^

^

|v|max

10-3

10

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(1,0) (1/2,0) (1/4,0) (1/8,0)

-4

10-3

10 100

x 200

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Fig. 13 Maximum amplitudes of the normal velocity v of 2-d modes (h, 0)

(1,1) (1/2,1) (1/4,1) (1/8,1) (0,1)

-4

100

x 200

300

Fig. 14 Three-dimensional disturbances (h, 1) of the maximum amplitude of v

The acoustic field is visualized by the dilatation which is the divergence of the velocity field ∇ · u. For both 2-d simulations, the sound with frequency ω0 /2 is emitted from x ≈ 120, being the position of the first vortex pairing. Both, intensity and directivity show good agreement with the reference solution of Colonius et al. [9]. In case of ΔΘ (ω0 /4) = 0.141, the sound with frequency ω0 /4 is emitted mainly perpendicular to the flow direction as shown in figure 11. With the alternative disturbance of mode (1/4, 0), the emitted sound is more intense and is directed mainly in downstream direction. The latter case corresponds to the results of [9] and [4].

3.3 Three-Dimensional Simulation The quite high amplitude of the introduced mode (0, 1) allows two-dimensional disturbances to interact with, generating unsteady oblique disturbances. The initial growth of the two-dimensional disturbances is similar up to the saturation of the first subharmonic as shown in figure 13. Non-linear interaction of the steady disturbance (0, 1) with two-dimensional instability waves rapidly genrates oblique modes, shown in figure 14. At x ≈ 130, disturbances (1, 1) and (1/2, 1) reach a level of |v| ˆ ≈ 5 · 10−5. The increased amplitudes of the oblique modes come up with a large amplitude of the steady mode (0, 1). This inhibits the growth of the second subharmonic (1/4, 0)), known from the corresponding two-dimensional case. As shown by the phase speed given in figure 15, the phase of the second subharmonic (1/4, 0) is not able to adapt to the dominant disturbance and its resonance is prevented. For this case, the spanwise vorticity at the symmetry plane z = 0 is shown in figure 16. The initial region is similar to the two-dimensional simulations: the mixing layer rolls up into vortices and the first pairing takes place at x ≈ 120. Further downstream three-dimensional effects increase and the vortical structures differ. The vortices break up into small-scale structures for x > 150. Compared to figure 6, large scales almost disappear.

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cph(v)

0.8

0.6 (1/2,0) (1/2,1) (1/4,0)

0.4 50

100

x

150

200

Fig. 15 Phase velocities of the first and second subharmonics, based on the normal velocity at y = 0 20

y

10

0

-10

-20

50

100

150

x

200

250

300

Fig. 16 Snapshot of the spanwise vorticity for the three-dimensional simulation at the symmetry plane z = 0. Contour levels correspond to those of figure 6

Fig. 17 Emitted sound with frequency ω0 /4, visualized by the real part of the Fourier-transformed dilatation with contour levels ranging from −2 · 10−6 to 2 · 10−6 for ΔΘ (ω0 /4) = 0.141

Fig. 18 Real part of the Fouriertransformed dilatation field of the undisturbed frequency 3/4 · ω0 with the same contour levels as on the left side


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The emitted sound with frequency of the second subharmonic is shown in figure 17. Compared to the corresponding two-dimensional case of figure 12, a sound reduction by roughly two is observed in the lower half of the domain. An additional source is located at x ≈ 130. Thus, the emitted sound in upstream direction is due to the suppressed resonance of mode (1/4, 0). The acoustic source at x ≈ 220 is not tonal but emits broad-band noise. This is shown exemplarily for the undisturbed frequency 3/4 · ω0 in figure 18. The above results show that subharmonics play an important role in the mechanism of sound generation. A varied phase shift can alter the process of subharmonic resonance and thus the major part of the generated noise. The resonance of twodimensional modes can be suppressed by a spanwise moculation of the flow.

4 Optimal Control of Mixing-Layer Noise 4.1 Computational Aspects A non uniform mesh with 601 × 501 nodes has been designed to improve accuracy in the mixing layer. The grid is uniform in the x-direction up to a given value x p which corresponds to the size of the physical box. Then the streamwise step size Δ x is increases successively up to the outflow boundary of the computational domain where the step size is Δ x f inal ≈ 300Δ x0 . This is a simple way for filtering undesired upstream convected structures, coming from possible numerical wave reflection from the outlet boundary. The boundary conditions in the north, south and east frontier of the computational domain are given from the characteristic theory [16]. They allow the exit of acoustic and entropy waves, limiting reflections. At the west frontier, the inlet of the domain, the flow is excited by its highest unstable eigenmode, given by linear stability theory. Its frequency is ω0 = 1.226. Forcing the flow at this location by its eigenmode decreases drastically the numerical transient behavior and increases the convergence speed. The upper flow velocity is the reference velocity with a Mach number MaI = 0.8. The Mach number of the lower stream is MaII = 0.2. The reference temperature is set to T0 = 280K. The reference length L is given from a Reynolds number of R = 2500. It corresponds to a shear layer thickness of δ given by δ /L = 0.3738 at the inlet of the computational domain. The total duration of the simulation is tend = 51.24 being 10 periods of the introduced instability, calculating over 10240 time steps. The gradient of the Lagrangian functional is determined from the direct simulation of the adjoint Navier-Stokes equation. The design of the adjoint code is very similar to the one of the direct state simulation. Non-reflecting boundary conditions based on the characteristic theory are also proposed to prevent numerical errors in the dual (adjoint) space. Note that incoming and outcoming characteristics are inverted between the direct and adjoint equations. A buffer zone is added in the dual space upstream of the computational domain used in the direct calculations, in the same way as the one existing downstream in the physical domain. Wave propagation

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is very similar in physical and in dual space. The location of the new buffer zone is explained by the fact that adjoint equations are integrated backward in time. The optimal control is obtained after approximately 5 to 7 global iterations. During each global iteration, the gradient of the cost functional is given from a conjugate-gradient algorithm which requires itself 10 iterations of the direct-adjoint simulations of the Navier-Stokes equations.

4.2 Simulation Results The adjoint simulations have been validated with a sensitivity analysis [18]. Since the adjoint variables are also gradients of the cost functional, it is possible to compare the evolution of the variation of a non-dimensional cost functional I = Jn /J0 and its gradient given by the adjoint quantity (eq. 7). It has been verified on figure 5 where the index on the horizontal axis is the index of the global iterations. λ is the parameter (or variable) usually used in the conjugate gradient algorithm which determines the steepest descent direction. The evolution of the functional and its derivatives correspond perfectly, especially the minimum is reached when the gradient (from adjoint equations) is given as null. In this case, the control is a source term in the continuity equation and it can be interpreted as some injection of mass on the larger actuation domain named (Ω11 ). The minimum found by the algorithm corresponds to a 10,4 % decrease of the functional after 5 global iterations. The mean noise reduction over the horizontal line (measurement line) is 0.5dB. It is weak compared to the previous work of Wei and Freund [20] where a reduction of 3dB was reached after 4 global iterations. Following Wei and Freund’s conclusions the actuation subtly modifies the small space and time scales of the flow in such a way that general organization is improved from an aeroacoustic viewpoint. The discrepancy between the success of the control in the two studies is explained as follows. In [20], the inlet boundary condition is the sum of the fundamental eigenmode and three harmonics. These 4 modes are perturbed randomly in the frequency domain and in amplitudes. Such an approach artificially increases the noise emission. The role of the final control is then to reduce the noise, and it happens that such artificial noise is canceled. Since we introduce just the fundamental mode without random perturbation the initial noise level is weak in our test case. Thus the actuation has less work to do, since the flow is well organized. Another point is that we have the Mach number MaI = 0.8 instead of 0.9 in [20], being initially more noisy. The difference of the sound pressure level (SPL) between the actuated and the unactuated flow is shown in figure 19. The decrease is not important on the measurement line (Ω2 ) where noise reduction was targeted, but a reduction of 3dB is reached in other locations in the shear layer. In an opposite way, some local 3 dB increase of noise level can be observed, but the noise seems not to propagate but to stay localized. Finally, noise control appears effective in a larger part of the flow, outside of the targeted line. This mean global reduction is not discussed in [20]. An analysis of the pressure spectrum has been performed. We chose to calculate the spectrum at a point on the measurement line Ω2 of coordinates (43.5, −25)


89 140

unactuated non controle controle actuated

130

SPL (dB)

120 110 100 90 80 −1 10

Fig. 19 Difference of acoustic pressure level (SPL) between the actuated case with a mass forcing and the unactuated case, in dB. The measurement line Ω2 is in the lower part of the domain

10

0

ω St

10

1

Fig. 20 Acoustic spectrum at the location (43.5, −25) in the actuated and unactuated case

Fig. 21 Difference of the acoustic level (SPL) between the actuated and the unactuated case in dB (control on the fourth component of the Navier-Stokes equations). The measurement line Ω2 is in white, at y = −24

with the largest noise reduction. The spectra in the actuated and unactuated case are presented in figure 20. A local minimum at a reduced frequency ω0 ≈ 0.61 is shown in the unactuated spectrum. This could be a subharmonic at frequency ω0 /2 of the initial fundamental mode. Even if the reduction is weak, actuation plays a positive role in a broad band of frequency, except close to ω0 ≈ 0.61 where noise emission is increased. Actuation seems to regularize the acoustic spectrum. This broadband effect should be investigated more closely in the future. A Principal Component Analysis (PCA) of the control function (actuation mode) has been used to compare with the conclusions of Wei and Freund [20]. It is a

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singular value decomposition following a Karhunen-Love transform, also known as Proper Orthogonal Decomposition (POD). The actuation is decomposed as N

f(x,t) = ∑ λi (t)Φi (x).

(8)

i=1

Details can be found in [17]. The first mode is dominant and 55.7 % of the energy is contained in the first two modes. There is a discrepancy with the previous work [20] where 10 modes are necessary to provide 50 % of the total energy. The structure of the actuation is less simple in our study than in Wei’s flow. However, similar to Wei’s conclusion, the modes are essentially spatially distributed around the mixing layer and the characteristic length scale decreases with the rank of the mode but remains in the same order of magnitude. This agreement confirms that the control essentially acts on some large scale, deeply modifying the flow structure. Simultaneous actuation on the fourth equation of the Navier-Stokes model has weakly increased the efficiency of the control, since a noise reduction of 0.6 dB has been pointed out. The conclusions given above are always valid but the area of local increase or decrease of noise emission are larger than in the previous case. The actuation on the whole Navier-Stokes equations is not so different from a unique actuation by injection of mass. It only confirms that the highest sensitivity of the noise emission is a mass source [18]. Spectrum and PCA analysis just support this conclusion. Some studies where the actuation domain was small (called Ω10 on figure 4) have demonstrated that noise reduction can be difficult or even might be impossible when the distribution of the control is too restrictive. This seems to be a limit of the open-loop control approach. If the measurement line is small (white line Ω2 on fig. 21) and the actuation domain is the larger one, a larger noise reduction of 6 dB has been reached with 6 global iterations in the control algorithm. As in the previous example, locally a reduction or an increase of 9 dB can be seen in some parts of the physical domain. However, globally and far from the mixing layer a mean reduction is conserved. The Principal Component Analysis has shown that with a small size of the targeted line which is quite far away from the noise emission, the optimal control approach provides an anti-noise control. The actuation acts as a wave which arrives at the small measurement line (approximately seen as a point) with an opposite phase and with the same amplitude as the wave which arrives at the same location but in the unactuated case. The actuation is more effective since the noise to be reduced is concentrated in a small area and the final goal is easier than the reduction of noise on a long line.

5 Serrated Nozzle End 5.1 Flow Parameters The nozzle end of a jet is modelled by a finite flat splitter plate with two-different free-stream velocities above and below. The Mach numbers MaI = 0.8 for the upper and MaII for the lower stream have been chosen according to chapter 4. Again


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the free-stream temperatures are equal (T˜I = T˜II = 280K), leading to a ratio of the streamwise velocities uI /uII = 4. The Reynolds number Re = ρ∞U1 δ1,I /μ∞ = 1000 is based on the displacement thickness δ1,I of the upper stream at the inflow. With δ1,I (x0 ) = 1, length scales are normalized with the displacement thickness of the fast stream at the inflow. The boundary layer of the lower stream corresponds to the same origin of the flat plate. The cartesian grid is made of sixteen subdomains: eight in streamwise and two in normal direction. Each subdomain contains 325 x 425 x 65 points in x-, y- and z-direction, resulting in a total number of 143.6 million gridpoints. The origin of the coordinate system (x = 0, y = 0) is located at the end of the nozzle end. Since the flow is symmetric to z = 0, the spanwise resolution corresponds to 42 spanwise modes (dealiased). The fundamental spanwise wave number is γ0 = 0.2, where Δ z = 0.2454 and λz /2 = π /γ0 = 15.708 is the spanwise extent of the domain. The mesh is uniform in streamwise direction with a step size of Δ x = 0.15 up to the sponge region, where the grid is highly stretched. The inflow is located at x = −97.5 and from x = 250 on, the mesh is smoothly stretched. In normal direction, the finest step size is Δ y = 0.15 in the middle of the domain with a continuous stretching up to a spacing of Δ y = 1.06 at the upper and lower boundaries. Due to the tiny thickness Δ y = 0.15 of the splitter plate, an isothermal boundary condition at the wall has been chosen. The temperature of the plate is T wall = 296K, being the mean value of the adiabatic wall temperatures of the two streams. The initial condition along the flat plate is obtained from similarity solutions of the boundary-layer equations, given in figure 22. Further downstream, the full boundary-layer equations are integrated downstream, providing a flow-field sufficient to serve as an initial condition and for linear stability theory. To avoid a peak of the normal velocity near the trailing edge, it is smoothed there. Since linear stability theory does not account for the wallnormal velocity, this modification does not affect its results. As shown in figure 23, the flow field keeps its wake-like shape 15

15

10

10

x = -97.5 Reδ1 = 1000

5

0

y

y

5

0

-5

-5

-10

-10

-15

x = 0 ... 96.75

-15

0

0.2

0.4

u 0.6

0.8

1

Fig. 22 Streamwise velocity profiles of the baseflow at the inflow (similarity solution of the BL equations)

0

0.2

0.4

u 0.6

0.8

1

Fig. 23 Combination of wake and mixing layer behind the thin splitter plate, obtained from downstream integration of the BL equations

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for a long range. As high amplification rates occur here, the flow is already unsteady before a pure mixing layer has developed.

5.2 Linear Stability Theory The stability diagram for two-dimensional disturbances in the upper boundary layer is shown in figure 24. In accordance to the maximum amplification, the fundamental frequency ω0 = 0.0688 is chosen. In the boundary layer, the amplification rate is only weakly dependent on the streamwise position. As the two boundary layers emerge from the same position, the lower boundary layer is stable up to the nozzle end. Behind the splitter plate, amplification rates 50 times higher than in the upper boundary layer occur due to the inflection points of the streamwise-velocity profile. Maximum amplification in the mixing layer takes place for a frequency of roughly three to four times of the fundamental frequency of the boundary layer as illustrated in figure 25.

5.3 Simulation Results The upper boundary layer is forced with the two-dimensional Tollmien-Schlichting (TS) wave (1, 0) with an amplitude of |u| ˆ = 0.005 and an additional oblique wave pair (1, 1) with |u| ˆ = 0.0005. Within this simulation, a rectangular serration with one notch per spanwise wavelength and a depth of 10 is considered. For the straight trailing edge, three-dimensional effects are less important and the reader is referred to [3]. A spectral decomposition is shown in figures 26 and 27, based on the maximum of v along y. The normal velocity has been chosen as it is less associated with upstream propagating sound. In the upper boundary layer, non-linear interaction of the introduced disturbances (1, 0) and (1, 1) generates the steady mode (0, 1) up to an amplitude of |v| ˆ = 2 · 10−5 (figure 26). From x = −25 onwards, this is exceeded by -0.01

-0.3 x = 1.35

-0.005

-0.2

0

-0.1

αi

.5

0.015 0.05

0.1

ωr

ω0 = 0.0688

-0. 15

-9 7

0

ω0 = 0.0688

0.01

x=

0.005

x=

αi

x = 14.85

0.2

0.1

Fig. 24 Spatial amplification rates from LST in the upper boundary layer

0.1

ωr 0.2

0.3

Fig. 25 Stability diagram behind the splitter plate


93 100

-2

10-2

^

^

|v|

10

|v|

100

(1,0) (1,1) (2,0) (2,1) (3,0) (3,1)

10-4

10-4 (0,1) (0,2) (0,3) (0,4)

10

10-6

-6

-50

0

x

50

0

100

Fig. 26 Maximum amplitude of the normal velocity v for steady modes (0, k)

x100

200

Fig. 27 Maximum amplitude of the normal velocity v for unsteady modes (h, 0) and (h, 1)

the upstream effect of the engrailment. The serrated trailing edge (−10 ≤ x ≤ 0) generates steady spanwise disturbances (0, k) up to |v| ˆ = 8 · 10−3 . In the notch, the combination of wake and mixing layer originates further upstream, which corresponds to the steady spanwise mode (0, 1). Its amplitude decreases behind the trailing edge up to x = 15. Higher harmonics in spanwise direction (0, 2) and (0, 4) are generated at the notch as well, staying almost constant behind the splitter plate. Figure 27 shows that the TS-wave generates higher harmonics in the upper boundary layer. With an amplitude of the driving TS-wave of |v| ˆ = 2 · 10−3, modes −4 −3 (2, 0), (3, 0) reach amplitudes of |v| ˆ = 3 · 10 and |v| ˆ = 2 · 10 , respectively. The increased steady modes at the engrailment interact with the two-dimensional waves, generating unsteady oblique modes (h, 1). Behind the splitter plate, the first two higher harmonics are growing stronger than the fundamental disturbance. Saturation is found at x ≈ 70 and x ≈ 160 for the higher harmonics and mode (1, 0), -0.04

-0.4 (1,0)

-0.02

αi(v)

αi(v)

-0.2

0

0

(1,0) (2,0) (3,0)

0.2

0.02 -80

-60 x -40

-20

0

Fig. 28 Amplification of the TS-wave in the upper boundary layer, compared with LST (symbols)

0

10

x

20

30

Fig. 29 Spatial amplification rates of 2-d disturbances behind the trailing edge. Symbols denote results from LST

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respectively. With relevant amplitudes of the oblique waves, these disturbances grow similarly to their two-dimensional counterparts and saturate at a level of |v| ˆ ≈ 4 · 10−2. Non-linear interaction of modes (h, 1) creates steady modes (0, k). As figure 26 shows, this exceeds the direct effect of the notch at x ≈ 15 and x ≈ 35 for mode (0, 1) and its higher harmonics, respectively. The spatial growth rates are compared with linear stability theory in figures 28 and 29. In the upper boundary layer, good agreement is observed. However a superimposed variation with wavelength λx ≈ 20 is visible. This is equivalent to a phase speed |c ph | = 0.22 which corresponds to the upstream propagating acoustic wave with c ph ≈ 1 − 1/MaI = −0.25. Hence the difference to LST is caused by the emitted noise of the mixing layer and not due to the three-dimensional geometry. Good agreement with LST is observed for the fundamental disturbance (1, 0) behind the

Fig. 30 Snapshot of vortical structures behind the engrailed splitter plate along two spanwise wavelengths, visualized by the isosurface Λ2 = −0.005

a)

b)

Fig. 31 Real part of the Fourier-transformed dilatation field at the symmetry plane z = 0 for ω0 : a) engrailment, b) straight trailing edges. Contour levels are in the range of ∇u = ±3 · 10−3


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splitter plate. Yet higher frequencies show a slightly lower mean amplification compared to linear stability theory. Figure 30 shows an instantaneous view of the vortical structures, visualized by the isosurface of the Λ2 vortex criterion [14]. Due to the earlier beginning of the mixing layer in the notch a spanwise deformation of the Kelvin-Helmholtz vortices occurs. At x = 50, the croissant-shaped vortex is still one coherent structure. The strong spanwise gradients lead to streamwise vortices which are twisted around the spanwise eddies. This leads to an early breakdown of the large spanwise rollers. Further downstream, the Kelvin-Helmholtz vortices known from dominantly twodimensional cases, e.g. [3], are now an accumulation of small-scale structures. The acoustic field is visualized by the real part of the Fourier-transformed dilatation field. The emission with the fundamental frequency is compared with the straight trailing edge in figure 31. The engrailed nozzle end leads to a notable reduction of the generated noise which propagates in downstream direction. In both cases an upstream emission originates from x ≈ 50 in the upper half of the domain. Its characteristic corresponds to the one due to the suppressed resonance in figure 17 for the pure mixing layer. Hence this emission is due to the phase adaptation for resonance or its suppression, respectively. The acoustic field with frequency 7/4 · ω0

a)

b)

Fig. 32 Same as figure 31 but for frequency 7/4 · ω0 : a) engrailment, b) straight trailing edges 10-2

|p|

10
18Rs and z > 35Rs . The corresponding flow solution is based on a mesh consisting of 64 blocks containing 20,142,144 cells. The minimum grid spacing in the radial direction is Δ rMIN = 0.02Rs which is chosen carefully to match the artificial inflow forcing . This minimum grid spacing is gradually increased to reach Δ r = 0.1Rs at the location of r = 16Rs. In the case of the single jet the inflow distribution is given at z = 0, r ≤ Rs and for the coaxial jet at z = 0, r ≤ R p and z = 0, R p ≤ r ≤ Rs . Note, in the present study, when an artificial velocity forcing is used to mimic the upstream turbulence intensity, the same grid resolution at the nozzle exit is chosen for numerical consistency when single and coaxial jets are investigated. That is, the various shear layers at single and coaxial jets are captured with the same grid resolution. Since the parameters involved in the artificial forcing do influence the jet development and hence the acoustics, it is essential to resolve the nozzle outflow area with the same spatial steps to obtain comparable results. A detailed discussion of the impact of the artificial forcing and the grid resolution at high Reynolds numbers on the flow and acoustic field can be found in [4]. The acoustic perturbation equations are solved on a domain which is five times larger than the LES domain, i.e., it comprises a spatial size the 55Rs in the axial and radial directions. Having a minimum spacing Δ r = 0.04Rs and a maximum spacing Δ r = 0.2Rs in the radial direction and a constant spacing in the axial direction of Δ z = 0.07Rs results in 12,359,952 cells to resolve the acoustic field.

3.3 Isothermal Single Jet In the following, the mean axial velocity profiles of the problems listed in Table 1 are compared, i.e., the development of the shear layer and the distribution of the turbulence intensities are considered. The sampling time period is Tsample = 800Rs /Us at a time step Δ tsample = 0.5Rs/Us such that 1600 LES snapshots are averaged. The findings are compared with the numerical data of the single jet configuration by Bogey and Bailly [3] and experimental findings by Zaman [36]. In Fig. 2(a) the axial profiles of the mean centerline velocity normalized by the primary nozzle exit velocity U p show the flow development on the jet centerline. The length of the potential core of the isothermal single jet is z/Rs 12. This length almost matches that of the cold coaxial jet, whereas the hot coaxial jet possesses due to the enhanced mixing process by the temperature gradient a much smaller extent of the potential core z/Rs 8. The axial profile of the jet half-width in Fig. 2(b) also evidences the hot coaxial jet to start growing slightly more upstream though the jet spreading rate of the various jets is almost alike. The axial distributions of the fluctuations of the axial velocity on the centerline are shown in Fig. 3(a). Downstream of an initial region they rise rapidly and peak slightly downstream of the end of the potential core . The hot coaxial jet reaches its maximum at z 12Rs, whereas the cold jets reach their maximum at z 15Rs . The nonlinear development is also observed in the axial distributions of the turbulence intensities of the radial velocity component in Fig. 3(b). Due to the rapid flow

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3.0

δ0.5 /Rs

UCL /Up

0.8

0.6

2.0

×

×

1.0

0.4 0

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30

0

×

×

×

10

z/Rs

20

30

z/Rs

(a) Mean velocity

(b) Jet half-width

Fig. 2 Mean velocity profiles of the single and coaxial jets; (a) mean streamwise velocity on the jet centerline and (b) evolution of jet half width δ0.5 /Rs as a function of the streamwise coordinate, (c jh ), (c jc ), · · ·· · · (s j), • (Bogey and Bailly [3], single jet), × (experiment by Zaman [36], single jet)

0.15

rms/Up

rms /Up

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0.05

0 0

0.1

0.05

10

20

z/Rs (a) Axial component

30

0 0

10

20

30

z/Rs (b) Radial component

Fig. 3 Profiles of the (a) axial and (b) radial turbulence intensities on the jet centerline as a function of the streamwise coordinate, (c jh ), (c jc ), · · · · · · (s j), • (Bogey and Bailly [3], single jet)

saturation the peak of the hot primary flow occurs at ur, rms 0.10U p and that of the cold configurations at ur, rms 0.12U p. That is, the hot primary stream enhances the nonlinear transition process. Note, if the turbulence intensities are rescaled by the single jet U p,s j , the rms values of the two coaxial jets are larger by a factor defined by the velocity ratio U p,c j /U p,s j 1.1. In brief, the turbulent distributions of the single jet and the cold coaxial jet are almost identical although the actual fluctuation level differs by the ratio of the primary velocity U p,c j /U p,s j . The rapid mixing of the hot coaxial jet leads to a steep increase of the turbulence intensities near the end of the potential core, whereas downstream of the potential core slightly larger velocity fluctuations occur in the cold jets. Since

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it is well known that there is a close link between turbulence in the core region and the sound generation [18], it is conjectured that the effective sound sources inside the hot coaxial jet generate a more powerful acoustic field. This issue will be tackled further below when the acoustic field of the hot coaxial jet will be considered. Next, the results of the statistical analysis and the impact of the turbulence structure on the overall acoustics of the single jet will be discussed. In the isothermal single jet the strong vortical motions in the turbulent shear layer generate the acoustic field. The vortex sound source is represented in the APE-4 system by the perturbed Lamb vector L which is defined such as (ω × u) in equation (4). Based on the LES solution of the core region, where the motion of the vortical structures determines the effective acoustic source, the near far-field acoustics is computed by the APE-4 system. The data processing for the acoustic simulation uses 3200 LES snapshots at a time step Δ tsample = 0.1Rs /Us . The discrete source data are quadratically interpolated. The time step of the acoustic analysis Δ tAPE,s j is 0.012Rs/a∞ to ensure a stable solution. The acoustic computations are performed over a time period TAPE,s j = 320Rs /Us . During the time interval the acoustic waves initiated in the jet core region propagate a distance seven times larger than the maximum extent of the computational domain. In Fig. 4(a) the acoustic field generated by the isothermal single jet is shown. In the core region the corresponding Lamb vector field is generated by the turbulent shear layers. The Lamb vector can be considered the vortex sound source which causes the strong acoustic radiation near the jet potential core. The time signals of the acoustic pressure at two locations are shown in Fig. 4(b). At the location ‘P1’ the short wave signals dominate, whereas at the location ‘P2’ low frequency waves are captured. The main characteristics of the stronger ‘P2’ distribution is caused by

p′

P1

0.001 ρ∞a2∞

P2

240

250

260

tUs /Ds (a) Instantaneous acoustics

(b) Acoustic signals

Fig. 4 Acoustic waves of the isothermal single jet predicted by the APE-4 system; (a) contours of the r- component of the Lamb vector fluctuations Lr (|Lr | ≤ 0.5a2∞ /Rs ) and p contour range (|p | ≤ 5 × 10−4 ρ∞ a2∞ ), (b) acoustic pressure signals at the radial location r = 15Rs , P1 (z = 12Rs ) for sideline acoustics and P2 (z = 29Rs ) for downstream acoustics

OASPL (dB)

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110

10

20

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103

PSD (Pa2 )

PSD (Pa2 )

103

102

101

(LES/APE),

Cut-Off

Cut-Off

Fig. 5 Overall sound pressure level at the circumferential locations of r = 15Rs , (LES)

102

101 0.6

fDs /Us (a) P1 (z = 12Rs )

1.2

1.8

0.6

1.2

1.8

fDs /Us (b) P2 (z = 29Rs )

Fig. 6 Sound pressure spectrum at the circumferential location r = 15Rs , the cut-off frequency is located at f Ds /Us = 1.45, (a) P1 (z = 12Rs ), (b) P2 (z = 29Rs ), (LES/APE), (LES), • (LES by Bogey and Bailly [3], single jet)

the jet development. This low-angle radiation is clearly different from the sideline acoustics in ‘P1’ where the high frequency contribution dominates. In Fig. 5 the LES solution has the peak overall sound pressure level (OASPL ) at about 12Rs downstream of the potential core, i.e., at z = 24Rs . The comparison between the direct LES and the hybrid LES/APE solutions shows the distribution to almost collapse. However, such a correspondence is only achieved if all the sources on the right-hand side of APE-4 system are taken into account. The sound spectra at the axial locations z = 12Rs and z = 29Rs at the same radial distance r = 15Rs , i.e., at ‘P1’ and ‘P2’ given in Fig. 4, are presented in Fig. 6. The sampling rate of 0.1Rs /Us suffices to resolve the maximum Strouhal number of 1.45. The pressure signal is determined to understand the characteristics of the near field


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acoustics. To be more precise, the sideline and the downstream noise propagation in Figs. 6(a) and 6(b) show a convincing agreement with the results of direct acoustic simulations by Bogey and Bailly [3]. The peak frequency of the present results occur at Strouhal number f Ds /Us 0.3 for the downstream acoustics, which is associated with the turbulence entrainment from the shear layer mixing. The sideline noise at z = 12Rs peaks at higher Strouhal number f Ds /Us 0.6, which is induced by the flow instability of the shear layer. The main concern of the present analysis is the identification of vortex sound sources, which dominate the noise generation of isothermal jets. To evidence the effective source in the subsonic stream the aerodynamic changes of eddy structures are detailed by using the spectral component of the noise source. Instantaneous data are sampled over a period of 320Rs/Us and are decomposed into 10 overlapping segments. Since the recorded signals are not periodic, the Hamming window function is used for all power spectra presented in this study. Note, a single isothermal jet configuration is analyzed and as such the dominant acoustic source is the Lamb vector fluctuation term in equation (4). The power spectra of the vortex sound source L are shown in Fig. 7. The spectra of the radial component of the Lamb vector fluctuations are illustrated for three radial locations. At each radial location the signals are determined over a circumferential surface and decomposed into the spectral density on a frequency band. Finally, the power spectral density distribution is quantified by integrating the azimuthal modes over the surface. The details of this signal processing are described in the source filtering technique by Freund [9]. 0.8

fDs /Us

fDs /Us

0.8 0.4 0 0

5

10

15

0.4 0 0

20

5

z/Rs (a) r = 0.1Rs

0.4

5

10

z/Rs (c) r = Rs

20

r ≤0.5Rs r ≥0.6Rs

0.8

fDs /Us

fDs /Us

15

(b) r = 0.5Rs

0.8

0 0

10

z/Rs

15

20

0.4 0 0

5

10

15

20

z/Rs (d) 0.1Rs ≤ r ≤ Rs

Fig. 7 Power spectral density (PSD) of the vortex sound source L in an f –z plane at several radii, (a) r = 0.1Rs , (b) r = 0.5Rs , (c) r = Rs , (d) 0.1Rs ≤ r ≤ Rs , the levels of the non-dimensional spectral density Φl R2s Us /a4∞ Ds in the interval (a) and (b) [0.25, 0.4] and (c) [0.5, 0.8], (d) peak PSD regions at 0.1 ≤ r/Rs ≤ 1

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Using this analytical procedure the power spectral density (PSD) contours are shown in a frequency( f )-space(z) plane. Near the jet centerline at r = 0.1Rs in Fig. 7(a) the spectral peak of the Lamb vector fluctuations appears at the frequency f Ds /Us 0.4, which is characteristic for downstream acoustics. The spectral density of the radial component of the Lamb vector fluctuations Φl (r, z, f ) peaks at the axial location z 15Rs . It is clear from Fig. 7(b) that at r = 0.5Rs the radial component of the power spectral density possesses a peak at an axial location z = 10Rs . In the shear layer region at r = Rs , evidenced in Fig. 7(c) the source energy quite strong in the high frequency range f Ds /Us ≥ 0.6, the axial position of the pronounced PSD regime is at z 5Rs . Performing the same analysis in the range of 0.1 ≤ r/Rs ≤ 1 identifies the radial location of the noise sources determined by the radial component of the PSD in Fig. 7(d). Near the centerline the spectral peak is determined by the downstream acoustics at f Ds /Us = 0.3 ∼ 0.4. Further off the centerline and closer to the shear layer at r = Rs , the spectral peak occurs at higher frequencies, i.e., it is characterized by sideline acoustics at f Ds /Us = 0.6 ∼ 0.8.

3.4 Heated and Unheated Coaxial Jets Analogous to the single jet analysis the computation of the acoustic field of the coaxial jets is based on the source time interval TAPE,c j = 320Rs/Us of the LES solutions. For numerical stability the time step to solve the APE system Δ tAPE,c j is 0.011Rs/a∞ . In Fig. 8 the acoustic directivities at r p = 40Rs , which are based on determining the acoustic pressure on 30 evenly distributed locations, show a unique pattern for the hot coaxial jet acoustics. The maximum OASPL occurs at θ 35 deg from the jet axis. Compared with the findings of the single and the cold coaxial jet the axial profile of the hot coaxial jet shows an approximately 5 dB higher acoustic pressure. Considering the small difference of 1 to 2 dB between the single and the cold coaxial

OASPL (dB)

130

120

110

20

40

60

θ (degree)

80

Fig. 8 Acoustic directivity of overall sound pressure level determined by the APE-4 system at r p = 40Rs : (c jh ), (c jc ), · · ·· · · (s j)


103

25° 35° 45°

10

2

55° 65° 75°

4

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10

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1

Cut-Off

10

Cut-Off

4

PSD (Pa2 )

PSD (Pa2 )

10

111

35° 45° 55° 65° 75° 85°

85°

10

1

0.6

f Ds /Us (a) Cold coaxial jet

1.2

1.8

0.6

1.2

1.8

f Ds /Us (b) Hot coaxial jet

Fig. 9 Sound spectra of acoustic radiations at angles 25◦ ≤ θ ≤ 85◦ at a distance r p = 40Rs , (a) cold coaxial jet (c jc ), (b) hot coaxial jet (c jh )

jet, it can be conjectured that the hot primary jet excites another major sound source yielding a more powerful acoustic energy over a wide frequency range. This second major source is caused by the heat content of the hot primary jet which generates a dipole-like behavior inside turbulent jets. In Fig. 9 the acoustic spectra of the cold and the hot coaxial jets are compared. The sound pressure is determined at the radial distance r p = 40Rs from the jet axis with an angular increment of 5 deg. The acoustics of the cold coaxial jet in Fig. 9(a) shows the pronounced low frequency radiation at θ ≤ 45 deg. At higher angles rather flat sound spectra are observed up to a Strouhal number of f Ds /Us 0.6. In Fig. 9(b) the acoustics of the hot coaxial jet is presented. Due to the heat impact of the temperature gradient on the noise generation the acoustic radiation at high angles is more intensified than that of the cold coaxial jet. The acoustic increment at the high angles is more confined to low frequency band. That is, the heat excited noise source corroborates the dipole sound in the sideline direction as being discussed by Morfey [21] and also predicted by the analytical solution for hot jets by Lilley [19]. In the APE-4 system the acoustic source terms of equations (3) and (4) contain the heat effects on the acoustic field. Concerning the downstream acoustics the spectral peak moves to the Strouhal number 0.5 for the hot coaxial jet against a Strouhal number 0.3 of the cold coaxial jet. This acoustic behavior is caused by the pronounced temperature gradient in the primary shear layer, which enhances the high frequency oscillations. Note, however, the present coaxial jet configurations have a mean inlet velocity distribution plus an artificial inflow forcing. This artificial excitation has to be kept in mind when the quantitative not the qualitative findings are considered. The acoustic directivity patterns of the cold and hot coaxial jets are presented in Fig. 10 for two Strouhal numbers at f Ds /Us = 0.3 and 0.5. For the hot coaxial jet the acoustic directivity shows a fundamental sound radiation at θ = 35 deg, which is higher than that of the cold coaxial jet. This result is more evident at

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130

PSD (dB)

PSD (dB)

130

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20

40

60

θ (degree)

80

120

110

(a) f Ds /Us = 0.3

20

40

60

θ (degree)

80

(b) f Ds /Us = 0.5

Fig. 10 Comparison of the acoustic directivity for two Strouhal numbers at r p = 40Rs , (a) f Ds /Us = 0.3, (b) f Ds /Us = 0.5: (c jh ), (c jc )

f Ds /Us = 0.5 than at f Ds /Us = 0.3. Due to the temperature difference between the primary and the secondary stream the waves propagate through flow fields at varying sound speed. They are refracted to form a high angle acoustic radiation. The downstream acoustics of the hot coaxial jet is dominated by the high frequency component. Besides, the intense temperature gradient on the primary shear layer excites noise which is not observed in the cold coaxial jet. The heated flow contribution to the sideline acosustics, i.e., at θ ≥ 60 deg, is evidenced in Fig. 10(a) for low frequency sound at f Ds /Us = 0.3. The maximum increase of the sound level is about 5 dB.

4 Coaxial Jets with Short Cowl Nozzle 4.1 Flow Parameters The flow parameters comprise a velocity ratio of the secondary Us and primary jet exit velocity U p of Us /U p = 0.9 for CaseA and Us /U p = 0.638 for CaseB, a Mach number 0.9 for the secondary and 0.621 and 0.877 for the primary stream of CaseA and CaseB, respectively, and a temperature ratio of Ts /Tp = 0.37. An overview of the main parameters is given in Table 2. The same flow conditions have been used in the numerical study by Andersson et al. [1] using a hybrid LES/Kirchhoff method and Yan et al. [35] using a hybrid detached-eddy simulation/Ffowcs-Williams Hawkings (DES/FWH) method for CaseB and in the experiments by Tinney et al. [32]. To reduce the computational costs the flow field inside the nozzle, i.e., upstream of the nozzle exit, was not included in the large-eddy simulation. However, the Reynoldsaveraged Navier-Stokes equations were solved to generate the inflow profiles for the LES. The grid topology and in particular the shape of the short cowl nozzle are shown in Fig. 11. The grid has about 22 × 106 number of grid points.


Z

113

X

Y

Fig. 11 The grid topology close to the nozzle tip is “bowl” shaped, i.e., grid lines from the primary nozzle exit end on the opposite side of the primary nozzle. Every second grid point is shown

4.2 Mean Flow Field Fig. 12 shows mean flow velocity profiles for CaseA (a) and CaseB (b) based on the secondary jet exit velocity of the coaxial jets at different axial cross sections compared with the experimental results. The different axial positions (x/Rs ) of the profiles are given in Table 3 relating the letters to the axial locations, respectively. A good agreement, in particular in the near nozzle region, is obtained for both cases. However, the jet CaseB breaks up earlier than predicted by the experiments resulting in a stronger mean velocity decay on the center line downstream of the potential core . It is evident that the initially three inflection points of the coaxial jet profiles merge faster to a single point for CaseA at x/Rs = 3 than for CaseB at approximately x/Rs = 5. This axial position defines the cross section from which on the coaxial jet start to develop similar to a single jet. a

b

c

d

e

f

g

3

4

a b c d e f

g h

i

k

l

EXP LES

EXP LES

3

j

r/R s

r/R s

2 2

1 1

0

0

(a) CaseA

(b) CaseB

Fig. 12 Mean flow development of coaxial jets CaseA (a) and CaseB (b) by axial velocity distribution in parallel planes perpendicular to the jet axis in comparison with experimental results [32]

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Table 2 Flow properties of the coaxial jet problems CaseA and CaseB Notation Unit Up Us Ma p Mas Maac Tp Tt p Ts Tts T∞ Re

m · s−1 m · s−1

K K K K K

CaseA

CaseB

Parameter

340.3 306.8 0.602 0.902 1.0 775.6 827.9 288.14 335.14 288.14 2 × 106

480.3 306.8 0.877 0.902 1.41 775.6 879.9 288.14 335.14 288.14 2 × 106

Primary jet velocity Secondary jet velocity Mach number (Up /a p ) of the primary jet Mach number (Us /as ) of the secondary jet Acoustic Mach number (Up /a∞ ) of the primary jet Static temperature of the primary jet Total temperature of the primary jet Static temperature of the secondary jet Total temperature of the secondary jet Ambient temperature Renolds number Re = ρ∞Us Ds /μ∞

4.3 Acoustic Field Unlike the direct acoustic approach by an LES or a DNS, the hybrid methods based on an acoustic analogy allow to separate different contributions to the noise field. These noise mechanisms are encoded in the source terms of the acoustic analogy and solutions can be superimposed exploiting the linearity of the wave operator. In Fig. 13 the near nozzle region from x/Rs = 0.5 to x/Rs = 4.5 has been extracted to analyze only the fluctuation profiles generated by the Lamb vector for CaseA and CaseB. Each profile of CaseA shows substantially lower values compared to those of CaseB. The development of the dominant noise source is not pronounced, in particular on the centerline, and only weakly in the primary shear layer. Note, both coaxial jets differ only by the primary jet exit Mach number. The low excitation of the Lamb vector fluctuations for CaseA has a profound impact on the generated acoustic field. It will be shown to be much quieter than that of CaseB. Unlike the often used integral methods, which are based on a surface integration, the computational effort of the differential equations based acoustic solver is

x/R s= 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

CaseB CaseA

r/R s

1

0.5

0

Fig. 13 Profiles of the Lamb vector fluctuations (ω × u)2 for CaseA and CaseB

Noise Prediction for Turbulent Coaxial Jets 140

Experiment LES/APE by Lamb vector LES/APE by full source terms

120

115 140 Experiment LES/APE by Lamb vector LES/APE by full source terms

SPL (dB/St)

SPL (dB/St)

120

100

100

80

80

0

0.5

St

1

(a) θ = 30 deg

1.5

0.2

0.4

0.6

0.8

1

1.2

St

(b) θ = 90 deg

Fig. 14 Sound spectra at a downstream observer point at (a) θ = 30 deg and (b) θ = 90 deg to the jet axis scaled to a 1 meter polar arc. Comparison of the sound field generated by the Lamb vector only and by the Lamb vector and the entropy source terms to experimental data [32]

determined by the total number of grid points. Therefore, only the acoustic near field has been computed. Eighteen observer points have been distributed azimuthally on circles at radius r = 15Rs and r = 20Rs from the jet axis and 40 of those circles have been placed along the jet axis from x/Rs = 0 to x/Rs = 38. To obtain the far-field noise signature, the near field results have been scaled to the far-field by using the 1/r-law assuming the center of directivity at x/Rs = 4 in the centerline. The same procedure has been applied to obtain the sound spectra on a 1 meter polar from the jet exit. The acoustic results from the circles at varying distance to the jet axis have been analyzed to identify the approximate location of the source to determine the center of directivity. The sound spectra for two selected points are shown in Fig. 14 representing the downstream noise at θ = 30 deg (a) and the sideline noise at θ = 90 deg (b) relative to the jet axis. Compared to the experiments the downstream noise is very well captures for frequencies up to St = f Ds /U js = 1.2. This Strouhal number agrees well with the maximum resolvable frequency of the acoustic mesh. Also the peak frequency at St = 0.23 is obtained by the present LES/APE approach. The additional entropy source terms enhance the noise levels at Strouhal numbers larger than 0.5. The noise radiated to the sideline at θ = 90 deg in Fig. 14 (b) agrees with the experimental measurements up to St = 0.95, whereas the spectra at higher frequencies are not captured. Since the noise to the sideline direction is mostly generated in the near nozzle region, it is conjectured that the low grid resolution near the walls and the simplified boundary conditions, that is the inflow profile from the RANS without taking the noisy unsteady structures inside the nozzle into account, generate artificial structures radiating to the near field. Furthermore, the internal damping zone is switched on directly at the nozzle exit and might affect the acoustic solution in that region. Finally, Fig. 15(a) shows the predicted far-field directivity at 60 radii from the jet axis by the Lamb vector and the entropy source terms in comparison with

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S.R. Koh et al. Experiment Andersson LES/Kirchhoff Yan, DES/FWH LES/APE Lamb & Entropy

90

CaseB Experiment CaseB LES/APE Lamb vector CaseB LES/APE Lamb & Entropy CaseA LES/APE Lamb vector

90

60

60

θ

θ 30

30

5dB

5dB

0

0

OASPL [dB]

OASPL [dB]

(a) CaseB

(b) CaseA & CaseB

Fig. 15 Acoustic directivity determined by the Lamb vector and entropy sources, (a) the acoustic field of CaseB computed by the different source contributions, (b) the acoustic fields of CaseA and CaseB computed by the Lamb vector only, OASPL comparison with numerical and experimental data [32, 1, 35] at r/Rs = 60

Table 3 Definition of the streamwise location of the cross sections, x is the axial coordinate of jets Profile No.

a

CaseA, x/Rs = CaseB, x/Rs =

3.0 4.0 6.0 8.0 10.0 13.0 17.0 20.0 0.05 0.88 2.87 3.60 5.06 6.52 7.97 8.70 9.43 10.16 12.35 14.53

b

c

d

e

f

g

h

i

j

k

l

numerical and experimental results at the same flow condition. The LES/APE results are in good agreement with the experimental results up to angles of 70 deg. This observation confirms the findings in previous studies [12] on the influence of differnet source terms on the acoustic field. To be more precise, the Lamb vector radiates dominantly in the downstream direction, whereas the entropy sources radiate at obtuse angles from the jet axis. Fig. 15(b) shows the comparison between the OASPL directivity for CaseA and CaseB generated by the Lamb vector. CaseA is substantially quieter by up to 8 dB for the downstream noise and 3 dB for the sideline noise at θ = 90 deg. Note, the main difference between the two jet problems is the reduced velocity of the primary stream in CaseA yielding a lower thrust. The outer stream does not efficiently shield the noise generated by the shear layer between the primary and the secondary jet.

5 Conclusion The flow field and the acoustic field of turbulent jets are analyzed. Concerning the pure jet configuration without nozzle effect one single jet and two coaxial jet


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configurations, one cold and one hot coaxial jet, have been exploited. The Reynolds number Re=400,000 and the Mach number M=0.9 of the single and coaxial jets were alike. For the coaxial jets the reference velocity and length scale were the velocity of the secondary jet and the diameter of the nozzle exit. The primary jet temperature of the cold coaxial configuration was the ambient temperature Tp = T∞ , whereas that of the hot coaxial jet is defined by Tp = 2.7T∞ . The main objective of the analysis was to understand the differences between single and coaxial jets and the impact of the temperature gradient on the acoustic field. The numerical method was based on a hybrid LES/APE method, i.e., first the flow field was computed by a large-eddy simulation and then the acoustic field was determined by solving the acoustic perturbation equations in the APE-4 form. The acoustic source terms consist of the Lamb vector fluctuations, the entropy source terms, and the nonlinear source terms. The overall acoustics and the corresponding sound spectra based on the hybrid LES /APE approach are in convincing agreement with the direct LES-based acoustics solution. The analysis of the single jet shows the peak locations of the power spectral density of the Lamb vector fluctuations to define the noise generation location. The sound generation in the coaxial jets is characterized by two features. The first feature is the downstream acoustics due to the large scale turbulence at the end of the jet potential core and the second one is the sideline acoustics enhanced by the temperature gradient, i.e., the low frequency sound is increased by up to 5 dB. According to the analytical solution of Lighthill’s acoustic analogy this low frequency is dipole sound due to isotopic turbulence being intensified by the temperature gradient. Furthermore, the mean temperature difference between the primary and secondary jet causes a distinct directivity pattern at θ = 35 deg from the jet axis. The flow field and the radiated sound field of realistic coaxial jets including “short cowl nozzle” has been analyzed by a hybrid LES/APE method. The high Reynolds number flows were computed in good agreement with experimental results showing the correct noise generation mechanisms to be determined. The dominant source term in the APE formulation has been shown to be the Lamb vector for angles up to 40 deg from the jet axis, while additional sources due to heating effects must be taken into account at higher angles. These source terms are generated by temperature and entropy fluctuations and by heat release effects. The present investigation shows the noise levels in the near field of the jet to be not directly connected to the turbulent statistics such as the Reynolds stresses.

References 1. Andersson, N., Eriksson, L., Davidson, L.: LES prediction of flow and acoustical field of a coaxial jet. AIAA Paper 2005-2884 (2005) 2. Bailly, C., Juvé, D.: Numerical solution of acoustic propagation problems using linearized Euler equations. AIAA Paper 98-2267 (1998) 3. Bogey, C., Bailly, C.: Direct computation of the sound radiated by a high Reynolds number, subsonic round jet. In: CEAS, CEAS Workshop from CFD to CAA (November 2002) 4. Bogey, C., Bailly, C.: Effects of inflow conditions and forcing on subsonic jet flows and noise. AIAA J. 43(5), 1000–1007 (2005)

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5. Ewert, R., Schröder, W.: Acoustic perturbation equations based on flow decomposition via source filtering. J. Comput. Phys. 188, 365–398 (2003) 6. Fischer, M., Preston, G., Bryce, W.: A modeling of the noise from simple coaxial jets, Part I: with unheated primary flow. J. Sound Vib. 209(3), 385–403 (1998) 7. Fischer, M., Preston, G., Bryce, W.: A modeling of the noise from simple coaxial jets, Part II: with heated primary flow. J. Sound Vib. 209(3), 405–417 (1998) 8. Freund, J.: A proposed inflow/outflow boundary condition for direct computation of aerodynamic sound. AIAA J. 35(4), 740–742 (1997) 9. Freund, J.: Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9. J. Fluid Mech. 438, 277–305 (2001) 10. Fureby, C., Grinstein, F.: Monotonically integrated large eddy simulation of free shear flows. AIAA J. 37(5), 544–556 (1999) 11. Gröschel, E., Meinke, M., Schröder, W.: Noise prediction for a turbulent jet using an LES/CAA method. AIAA Paper 2005-3039 (2005) 12. Gröschel, E., Meinke, M., Schröder, W.: Noise generation mechanisms in single and coaxial jets. AIAA Paper 2006-2592 (2006) 13. Gröschel, E., Schröder, W., Renze, P., Meinke, M., Comte, P.: Noise prediction for a turbulent jet using different hybrid methods. Comput. & Fluids 37, 414–426 (2008) 14. Hardin, J., Hussaini, M.: Computational aeroacoustics. Springer, Heidelberg (1993) 15. Hoch, R., Duponchel, J., Cocking, B., Bryce, W.: Studies of the influence of density jet noise. J. Sound Vib. 28, 649–668 (1973) 16. Hu, F., Hussaini, M., Manthey, J.: Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. J. Comput. Phys. 124, 177–191 (1996) 17. Koh, S., Gröschel, E., Meinke, M., Schröder, W.: Numerical analysis of sound sources in high Reynolds number single jets. AIAA Paper 2007-3591 (2007) 18. Lighthill, J.: On sound generated aeroacoustically: II. turbulence as a source of sound. Proc. R. Soc. Lond. Series A 222, 1–32 (1954) 19. Lilley, G.: The radiated noise from isotropic turbulence with application to the theory of jet noise. J. Sound Vib. 190(3), 463–476 (1996) 20. Meinke, M., Schröder, W., Krause, E., Rister, T.: A comparison of second- and sixth-order methods for large-eddy simulations. Comput. & Fluids 31, 695–718 (2002) 21. Morfey, C.: Amplification of aerodynamic noise by convected flow inhomogeneities. J. Sound Vib. 31(4), 391–397 (1973) 22. Renze, P., Schröder, W., Meinke, M.: Large-eddy simulation of film cooling flows at density gradients. Int. J. Heat& Fluid Flow 29(1), 18–34 (2008) 23. Rütten, F., Schröder, W., Meinke, M.: Large-eddy simulation of low frequency oscillations of the Dean vortices in turbulent pipe bend flows. Phys. of Fluids 17(3), 35107 (2005) 24. Schröder, W., Ewert, R.: LES-CAA coupling. In: Large-eddy simulations for acoustics. Cambridge University Press, Cambridge (2005) 25. Schröder, W., Ewert, R., Bui, T., Gröschel, E.: An LES-APE approach in computational aeroacoustics theory and applications. In: VKI Lecture Note VKI-LS 2006-05 (2006) 26. Seiner, J.M., Ponton, M.K., Jansen, B.J., Lagen, N.T.: The effects of temperature on supersonic jet noise emission. AIAA Paper 92-02-046 (1992) 27. Simonich, J., Barber, S.N.T.: High subsonic jet experiment Part I: aeroacoustical characterization, noise reduction and dimentional scaling effects. AIAA J. 39(11), 2062–2069 (2001) 28. Tam, C., Golebiowski, M., Seiner, J.: On the two components of turbulent mixing noise from supersonic jets. AIAA Paper 96-1716 (1996) 29. Tam, C., Webb, J.: Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262–281 (1993) 30. Tanna, H.: An experimental study of jet noise. Part I: turbulent mixing noise; Part II: shock associate noise. J. Sound Vib. 50, 405–444 (1977) 31. Tanna, H., Dean, P., Fisher, M.: The influence of temperature on shock-free supersonic jet noise. J. Sound Vib. 39, 429–460 (1975) 32. Tinney, C., Jordan, P., Guitton, A., Delville, J., Coiffet, F.: A study in the near pressure field of coaxial subsonic jets. AIAA Paper 2006-2589 (2006)


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33. Viswanathan, K.: Jet aeroacoustic testing: issues and implications. AIAA J. 41(9), 1674–1689 (2003) 34. Viswanathan, K.: Aeroacoustics of hot jets. J. Fluid Mech. 516, 39–82 (2004) 35. Yan, J., Tawackolian, K., Michel, U., Thiele, F.: Computation of jet noise using a hybrid approach. AIAA Paper 2005-3621 (2007) 36. Zaman, K.: Flow field and near and far sound field of a subsonic jet. J. Sound Vib. 106(1), 1–16 (1986)

Numerical Simulation of Jet Mixing Noise Associated with Engine Exhausts Dandy Eschricht, Björn Greschner, Frank Thiele, and Marc C. Jacob

Abstract. In this article the work conducted at ISTA and LMFA during the French/German research project Noise Generation in Turbulent Flows is presented. The hybrid approach applied for the noise prediction at ISTA consists of a flow simulation which is coupled to the acoustic analogy of Ffowcs-Williams& Hawkings including predictions based on volume sources. The turbulence modelling approach used in the flow simulations is the Detached-Eddy Simulation as well as the LargeEddy Simulation. Along with details on the numerical method and the turbulence modelling approach, results from compressible, three-dimensional flow simulations and their acoustic predictions are presented for the rod-airfoil configuration which acts as a simple broadband noise emitter. Additionally, results for a single-stream jet and for a coplanar, coaxial, unheated jet flow are presented. The results of the Detached-Eddy Simulation of the latter case are compared with those of a LargeEddy Simulation of the same configuration.

1 Introduction The noise emission from aircraft consists both of airframe and engine noise. Even for high bypass ratio engines, jet noise constitutes one of the major noise sources at take-off although a significant reduction has already been achieved since the first jet-engine powered aircraft. Jet noise is also important for the cabin-interior noise, whose significance may rise in the future since lighter and possibly less absorbing Dandy Eschricht · Björn Greschner · Frank Thiele Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, Sekr. MB1, Müller-Breslau Strasse 11, 10623 Berlin [email protected] Marc C. Jacob Centre Acoustique du LMFA, UMR CNRS 5509, Ecole Centrale de Lyon, Université ClaudeBernard Lyon I, F-69134 Ecully Cedex [email protected] C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 121–146. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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materials are used for the aircraft hull. It will thus have a direct influence on the passenger comfort. The jet noise reduction has been achieved in the last decades by the manufacturing of engines with increased bypass ratios that went up to values of 10 in latest designs. These higher bypass ratios make the engine larger and heavier which increases the drag and makes it more difficult to install the engine under the wing. In recent years, a new technology has been invented that is based on adding serrations (chevrons) to the nozzle lip. These chevrons induce longitudinal vortices that enhance the turbulent mixing and create a thicker shear layer with lower turbulence levels. The noise emission power could be reduced by about 3 dB with that technique, but the serrations may have the side effect of creating additional noise at higher and better audible frequencies. This unwanted effect depends on subtle changes of the nozzle geometry whose optimal design is nowadays identified with expensive and time consuming model tests. A reliable, numerical method could speed up the design process and reduce the development costs considerably. Today’s numerical noise prediction techniques can be sorted into two groups, those that rely on steady-state simulations of the flow field (RANS) and those that simulate the unsteady flow field. Since unsteady field information is not available in the RANS-based methods, they rely on semi-empirical models to reconstruct the unsteadiness of the sound field from the turbulent field information given by the RANS. While reliable predictions can be achieved with this methods for specific configurations and sometimes even for a whole class of configurations (Tam et al. [51][52]), they often require tuning of coefficients both in the RANS and the noise model to obtain the best possible result for more complex configurations. The methods in the second group are time-dependent prediction methods that rely on an unsteady simulation of the flow field, while the far-field noise is calculated either directly (Bodony et al [5][6], Bogey et al [8][13]) or by means of an acoustic analogy (Andersson et al. [1][3]). These researchers used the Large-Eddy Simulation (LES) as turbulence model in their jet flow simulations, since Direct Numerical Simulation remains infeasible for realistic Reynolds numbers, though it has been used for simulations at low Reynolds numbers [25]. LES of jet flows are mostly achieved by prescribing a measured, artificial or RANS velocity profile at the nozzle exit plane. The inner nozzle flow is therefore usually excluded from the computational domain. As today’s noise reduction techniques include modifications of the nozzle geometry (i.e serrations), more complex flows including walls have to be considered. In addition, the noise field of a realistic jet engine is affected by installation effects, i.e. pylon and wing. Furthermore, the inclusion of a part of the inner nozzle flow into the simulation avoids the specification of inflow conditions at a non-planar nozzle exit. Such a specification would be inconvenient especially for a serrated nozzle. In order to include the inner nozzle flow, reliable wall closures must be applied as long as the representation of the wall boundary layer turbulence by highly resolved grids remains un-feasible for high Reynolds number flows.

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The Detached-Eddy Simulation first proposed by Spalart et al. [49] is one of the numerical methods that promises to offer the required features that are an effective modelling of wall boundary layers and a reproduction of resolved structures that are important broadband noise sources. Even as the first applications of the method were of a pure aerodynamic type, the resolved turbulent structures visible already suggested an LES-like behaviour in resolved regions. Some intrinsic limitations of the original DES approach, such as modelled-stress depletion and log-layer mismatch, have been overcome in the past years by introducing the concepts of Delayed Detached-Eddy Simulation (DDES) and Improved Delayed Detached-Eddy Simulation (IDDES) [53][46]. Since the DES seemed promising, the work of our group during the DFG/CNRS project was focused on the application of that method for noise predictions of various flow configurations. In the beginning of the project, the general prediction procedure was established: it consists of a compressible flow simulation which is coupled to an acoustic analogy for the far-field prediction. This work was conducted on basis of the rod-airfoil configuration which even as a relative simple configuration includes tonal and broadband noise. The flow simulations of this configuration could be carried out on relative small computational meshes, thus reducing the required computational effort. Moreover, as the configuration includes a bluff body flow, it is better suited for the DES as it was originally intended. A further reason for choosing this configuration was that the French partner could directly provide the experimental data [33] and a numerical study [14][15]. An overview of the configuration and a description of the work accomplished is given in section 3.1. To gain some experience with general numerical handling of jet flow configurations, a simulation of an unheated, single-stream jet was carried out using LES. Some details and results are given in section 3.2.1. Subsequently, the configuration of a coplanar, coaxial jet was examined numerically. For this configuration a comparison between a LES and a DES was made that showed a fair agreement of the two methods when applied under similar conditions. An overview of the work on this configuration is given in section 3.2.2. This article is organised as follows. An overview over the hybrid noise prediction method is given in section 2. The configurations to which this prediction methodology has been applied are described in section 3. The articles finishes with a conclusion in section 4 that additionally contains an outlook into future work considered.

2 Numerical Method This section provides an overview over specific details of the flow solver code used for the numerical simulation in section 2.1. Further included in this section are details concerning the turbulence modelling applied in section 2.1.1 even as it might vary in different applications. The method used for the acoustic far-field prediction is described in 2.2.

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2.1 Flow Solver The flow solver elaN3D solves the Reynolds-Averaged Navier-Stokes equations using the finite volume method. Though the simplicity of the method has drawbacks in terms of numerical accuracy (second order in space and time), the implemented turbulence modelling features and the lack of a time step upper limit in terms of the local CFL number, make it still an efficient tool for unsteady flow field predictions, and it is commonly used for simulation purposes by the CFD group at ISTA. The flow solver solves the unsteady Reynolds-averaged continuity, momentum and energy equations on structured, multi-block grids. The continuity equation is solved via a pressure correction algorithm (SIMPLE), where a close coupling between the convective velocity and the pressure is achieved by the Rhie and Chow interpolation method [41].

∂ ρ ∂ ρuj + =0 ∂t ∂xj ∂ ρ ui ∂ ρ ui u j ∂ p ∂ σi j ∂ τi j + =− + + ∂t ∂xj ∂ xi ∂ x j ∂xj ∂ ρ T ∂ ρ u jT ∂p ∂ p ∂ q˙ j ∂ ui + + uj = − + (τi j + σi j ) cp ∂t ∂xj ∂t ∂xj ∂xj ∂xj

(1) (2) (3)

Here, σi j and τi j denote the molecular and the Reynolds stress tensor, respectively, while c p is the isobaric, specific heat capacity. The temperature diffusive flux q j is modelled similar to Fick’s law containing a molecular and turbulent part q = c p (μPr+Tμt ) ∂ T /∂ x j . The two stress tensors are defined as 1 1 2 σi j = 2μ Si j − Skk and τi j = 2μt Si j − Skk − ρ kδi j 3 3 3 1 ∂ u j ∂ ui with Si j = + 2 ∂ xi ∂ x j

(4)

where the definition of the Reynolds stresses τi j follows the ansatz of Boussinesq [16]. The turbulent viscosity μt that appears in the Reynolds stress tensor is to be determined by a turbulence model. Inside the flow solver elaN3D a variety of models including models for Detached-Eddy Simulation and Large-Eddy Simulation can be chosen. 2.1.1 Turbulence Modelling The Detached-Eddy Simulation was developed [49] to combine the strength of RANS models with that of Large-Eddy Simulation. Unlike Unsteady-RANS, DES allows for small scales in regions where the grid is fine enough similar to LES. The reproduction of wall-boundary layers is possible even if the grid at the walls does not support the necessary, fine scale turbulence as the DES model acts as a RANS model in this case. Because of the simple implementation and the impressive results


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in its first applications [45][54], DES became a modelling approach commonly used for aerodynamic applications and was later also used in simulations for aeroacoustic predictions [43]. The model formulation of the DES is based on traditional RANS models. These are adjusted to include a dependence on the local grid spacing. The turbulent length scale LRANS , which is inherent to the RANS model, is replaced by a length scale L which depends on the local grid spacing. DES has been used on basis of different one-equation and two-equation turbulence models by a number of researchers. Results shown here were obtained with DES based on different RANS models. To give an insight into the DES, the changes made to the RANS model to obtain the DES model shall be explained in the following using the Wilcox k-ω model which was used to obtain the results presented in section 3.2. & ∂ ρk ∂ ρu jk ∂ μt ∂ k + + Pk − ρ Dk = μ+ (5) ∂t ∂xj ∂xj σk ∂ x j & ∂ ρω ∂ ρ u j ω ∂ μt ∂ ω ω + + (cω 1 Pk − ρ cω 2 kω ) = μ+ (6) ∂t ∂xj ∂xj σω ∂ x j k with further relations for the turbulent production rate Pk , the dissipation rate Dk and the turbulent viscosity μt : Pk = μt

' 2Si j Si j ,

μt = ρ

k , ω

Dk = c μ k ω

(7)

As mentioned before, the modifications that make the DES model from the RANS model are based on the a length scale replacement in the RANS model. The turbulent length scale LRANS associated with this k-ω model is LRANS =

k1/2 cμ ω

→

Dk =

k3/2 LRANS

(8)

which is then inserted into the dissipation term of the k-equation. To allow the model to discern between resolved and unresolved regions in the flow field, LRANS is replaced by LDES which is a function of the local grid spacing. k3/2 LDES with Δ = max(Δ x, Δ y, Δ z)

LDES = min(CDES Δ , LRANS )

→

Dk =

(9) (10)

As LDES is always smaller than LRANS , the new dissipation term is always larger in regions of finer grid spacing. The turbulent kinetic energy is therefore reduced and along with it, the turbulent viscosity. One common problem of the DES is the so called ”grey area” problem. It appears in a flow region where the LES-mode of the model becomes activated but where true LES content in form of resolved turbulent fluctuations is still absent. This results in

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an undefined modelling state as it can not be named true LES nor RANS. The problem is further enhanced by eddy viscosity that is convected from a RANS-region into the LES-region via the convection of k and ω . These higher levels of eddy viscosity damp the development of fine scale turbulent structures. Yan et al. [55] tried different length scale substitutions in the DES on a bluff body flow and found only a weak influence as the Kelvin-Helmholtz instability is strong for this separated flow and the recirculating wake turbulence stimulates the development of LES content. For jet flows, however, recirculating wake disturbances are missing and the instability of the shear layer is much weaker. Therefore, additional substitutions are adopted, i.e the length scale is inserted into the equation for the eddy viscosity. √ √ μt = cμ LRANS k → μt = cμ LDES k (11) The eddy viscosity is directly reduced in fine grid regions. This reduction is independent of the reduction of k via the increased dissipation as the viscosity is now directly proportional to the length scale. The effects were studied by Yan et al [56] for a short-cowl nozzle configuration. An enhanced development of turbulent structures was found especially in the initial shear layer. Hence, the additional substitutions are seen as a viable means to reduce the grey area problem. 2.1.2 Non-reflecting Boundary Condition The boundary conditions are of importance especially for compressible flow simulations aiming at acoustic predictions. Reflected waves from the boundaries can disturb the simulated sound field seriously and corrupt the predicted far-field noise. To circumvent this in the flow simulation, a non-reflective boundary condition has been implemented into the flow solver. This boundary condition is based on the one published by Bogey et al. [7] with small modifications. Bogey et al. [7] define a radial derivative in the direction from a virtual source position inside the computational domain where rî is the unit vector in direction from the source to the position on the boundary. The boundary condition handles all incoming waves as if they have their origin in the virtual source position. In the implemented, modified version this is replaced by the derivative in the direction normal to the boundary.

∂ ∂ 1 1 + = rî + ∂r r ∂ xi r

→

∂ ∂ ni = ∂ xi ∂n

(12)

The modified version assumes all incoming waves as normal to the boundary. Both versions do not work optimally if the directional assumption is not correct. The reason for the modification are problems with the stability of Bogey’s version, especially at intersections of inflow and wall boundaries where the radial direction can have a wall-normal component that creates a velocity component in wall direction.


The implemented equations for the the inflow boundaries are ⎞ ⎛ ⎞ ⎛ ρ ρ − ρ¯ ∂ ⎝ ⎠ ∂ ⎝ u i + vg · ui − u¯i ⎠ = 0 ∂t p ∂ n p − p¯

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(13)

where vg is the group velocity of the waves where vg = c + ni u¯i , and c and ni are the speed of sound and the outer normal unit vector at the boundary. The outflow boundary condition is described by the following equations ∂ρ ∂ ∂ 1 ∂p + u¯ j + u¯ j (ρ − ρ¯ ) = 2 (p − p) ¯ ∂t ∂xj c¯ ∂t ∂xj 1 ∂ ∂ ui ∂ (14) + u¯ j (ui − u¯i ) = − (p − p) ¯ ∂t ∂xj ρ¯ ∂ xi ∂p ∂ + vg (p − p) ¯ =0 ∂t ∂n This set of equations is in principle designed to allow convected structures in the flow field (vortices and entropy modes) to leave the domain. But in practice small errors do exist and because of the large energy contained in the convective modes, even a small error in the boundary condition leads to an acoustically large pressure perturbation visible as a reflection at the outflow boundary. Therefore, Bogey et al. [7] make use of a buffer zone near the downstream end of the computational domain. Inside the buffer zone, a damping term is added to the generic, transport equation ∂ ρΨ ∂ ρ u jΨ + (15) = −δ˜ (x ) Ψ − Ψ¯ . . . ∂t ∂xj where the damping function δ˜ controls the damping strength and can be defined space dependent as the configuration demands. The damping function typically takes the form x − xa 2 δ˜ (x ) = δ (16) xe − xa with the damping constant δ , and the position dependent part that contains the start and end position of the damping zone xa and xe , respectively.

2.2 Far-Field Prediction As the direct simulation of the sound propagation is computationally demanding, especially for broadband and high frequency noise configurations, the flow field results have been extended to the far field either by applying ISTA’s acoustic analogy solver C3NOISE or the advanced time, acoustic prediction code FoxHawk which is provided by LMFA. Both solve the acoustic analogy of FfowcsWilliams&Hawkings [24] while ISTA’s solver is only capable of noise predictions

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based on surfaces, FoxHawk additionally allows the evaluation of volume sources in the prediction. The far-field solution can in principle also be obtained from a solution including volume sources after Lighthill [36] or by the Kirchhoff-integral (KH). Lighthill’s acoustic analogy requires time-resolved volume data which requires a huge amount of storage capacity for three-dimensional configurations. The surface based methods, FWH and KH, are more efficient as only surface, i.e 2D, data is required. However, they do not include the sound radiated by eddies outside the surface. The surface must therfore be chosen to enclose all sound sources in the flow field. Comparing the two types of surface based approaches, FWH and KH, Brentner et al. [17] have shown that their prediction results are not equivalent. In this study, the far-field predictions of the FWH and the KH method were investigated for a moving airfoil configuration, and a strong dependence on flowfield non-linearities for the KH method was found. This dependence is due to the fact that the derivation of KH-integral is based on linear assumptions. The FWH method was found to be much more independent of non-linearities and of the surface position. It was therefore chosen as the preferred far-field prediction method. The solver C3NOISE solves the FWH-equation in the time domain using the far-field solution commonly referred to as formulation 1A by Farassat [21][22][23] which is given in equation (17).

ρo (U˙ n + Un˙ ) ρoUn (rM˙ r + coMr − coM 2 ) 4π p(x,t) = + dS r(1 − Mr ) r2 (1 − Mr )3 S ret (17)

˙ 1 Lr + co r(Lr − LM ) Lr (rM˙ r + co Mr − co M 2 ) + dS + co S r(1 − Mr )2 r2 (1 − Mr )3 ret using the notations Ui = vi + (ρ /ρo)(ui − vi ) in the thickness noise term and Li = (pδi j + ρ ui (u j − v j ))n j in the loading noise term. Here, ρo denotes the reference density, vi is the velocity of the surface and Mi = co vi the surface Mach number based on the reference speed of sound co . A subscript denotes a scalar product with either the surface outer normal, unit vector ni or the unit radiation direction vector rî . To be independent of the system architecture on which the simulation is run, the data for the far-field prediction is written in an architecture-independent data format ([31]). All processes of the parallel running simulation write their surface data into separate files, which is later collected into single files per surface that can be processed with the analogy solver C3NOISE. The surface integration is performed using a source-time based algorithm that interpolates the contribution of each surface element at the observer time t. This allows to sequentially proceed through the available source time steps without the need to keep a possibly large number of source time steps in memory as would be required for observer-time based algorithms. The algorithm used is quite similar to the one used in FoxHawk as proposed by Casalino [18].


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3 Applications 3.1 Rod-Airfoil Interaction The configuration is that of a symmetric airfoil located one chord downstream of a rod, whose wake contains both periodic and broadband vortical fluctuations. In particular, a significant broadening of the main Strouhal peak has been observed at subcritical vortex shedding conditions. Recent studies have shown that the rod-airfoil configuration is particularly suitable for the assessment of CFD codes in modelling broadband noise sources. An experimental investigation of the rod-airfoil configuration was carried out in the high-speed, subsonic, anechoic wind tunnel [33] of the Ecole Centrale de Lyon. A symmetric NACA0012 airfoil (chord c = 0.1 m) and a circular rod (d/c = 0.1) were placed in the potential core of a jet. The airfoil was located one chord-length downstream of the rod. Both bodies extended 30d in the span-wise direction and were supported by rigid smooth plates. The incoming velocity was 72 m/s with a turbulence intensity Tu = 0.8 %. The corresponding rod diameter based Reynolds number Red was about 48000, that based on the chord length was about 480000 and the Mach number approximately 0.2. The computational grid extends over 180d in the stream-wise direction, 120d in the cross-stream direction and 3d in the span-wise direction. The two-dimensional grid shown in figure 1 is repeated at 30 span-wise locations, with a constant spacing Δ z= 0.1d. The three-dimensional grid consists of about 2.3 million cells. The first cell at the wall extends y+ < 1.5 wall units in wall-normal direction. The grid is locally refined in the vicinity of airfoil and rod to reduce the cell number in outer regions while concentrating cells in the region where they are required. The refinement ratio between inner and outer grid used is 1 : 2. The time step used in the aerodynamic simulation is Δ t = 10−6 s representing a sample Strouhal number of Std = 139 based on the rod diameter. The results shown in the following are based on a DES using the Cubic-ExplicitAlgebraic-Stress LL-k-ε Model (CEASM). This CEASM model is a two-layer model which was developed for complex wall bounded flows. For a two-equation model, the turbulence length scale LRANS can appear in more than one term of the model equations (compare section 2.1.1). Although the standard approach consists in substituting this length scale in the dissipation term of the k-equation, both a wall normal distance and a locally determined length scale are used in the CEASM

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model. For an exhaustive representation of the model equations, tensor representation and model constants, the reader is referred to Lübcke et al. [37]. Typical instantaneous flow patterns emerging from the CEASM-DES are depicted in figure 2 in terms of iso-lines of the span-wise velocity component in the center plane of the computational domain. Strong span-wise effects are visible as the fluctuations of the span-wise velocity are of the same magnitude as the incoming velocity. A vortex-shedding pattern can be observed downstream of the rod, although the strong span-wise effects prevent the formation of a regular Karman vortex street. The non-dimensional peak shedding frequency found in the CEASM-DES is St = 0.183 which is very close to the value St = 0.19 found experimentally. A far-field result obtained from the acoustic-analogy solver is given in figure 3. The far-field pressure time series used for the power-spectral density in the figure is based on a virtual surface enclosing the rod and the airfoil. Since the simulated span of three diameters is 10 times smaller than the span of the experimental setup, a scaling correction has been applied as suggested by Kato [35] (also see Greschner et al. [27]). As mentioned before, the peak frequency in the simulation is slightly underpredicted. This can also be observed in the far-field prediction. The level at the peak frequency is underpredicted by about 4 dB but it must be noted that there is a certain level of uncertainty arising from the 2D–3D correction and that the powerspectral density is not an adequate means for comparison of harmonical, spectral peaks. The decay rate of the high frequency noise however, is in good agreement with the measurements. It can be noted in general that noise predictions for this configuration are in fair agreement with the measurements, though a dependence on the underlying turbulence model of the DES has been noted by Greschner et al. [28] that is especially pronounced in the main shedding frequency predicted. Discrepancies have been observed between far-field computations using different integration surfaces. These far-field differences could be due either to volume sources that are implicitly taken into account in the volume surrounded by the integration surfaces (and which consequently differ from one surface to another), or to the crossing of the surfaces by volume sources during the computation. This problem is a major issue when applying analogies to jet-type flows since they usually extend out of the computational domain at least in the stream-wise direction and the integration surfaces thus


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Fig. 3 Far-field prediction from enclosing surface at θ = 90o compared to the measurement from [27]

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necessarily cut the flow domain. In order to sort out the first (physical) discrepancy from the second (spurious) one, volume source contributions following the work of Lighthill [36] have to be computed inside and around the domains contained in the control surfaces. To achieve this, volume source terms had to be included into the FWH-solver FoxHawk that was provided by the LMFA. These were validated on a simple 3D flow configuration involving only quadrupoles sources as now walls were present. The configuration was that of the leapfrogging of two ring vortices that was computed according to Möhring [39] and Schram [44]. The acoustic farfield was obtained in two ways, an analytical evaluation based on Möhring’s analogy and a numerical Lighthill-based volume source formulation where the flow data was created using the analytical flow field and fed into FoxHawk. The comparison of the two far-field estimates validated the numerical implementation of the volume terms in FoxHawk. Further a DNS of the vortex leapfrogging was carried out using ISTA’s flow solver elaN3D without any turbulent closure model (Jacob et al. [32]) during the summer school CEMRACS 2005 in Luminy which was organised by the DFG/CNRS group. The coupling of the flow solver with the volume source computations of FoxHawk was established in order to give a realistic picture of vortex leapfrogging noise as it might occur in circular jets. The successful noise prediction of the leapfrogging test case allowed to gain confidence in the volume term computations that needed to be carried out. Once the volume-source prediction was validated, noise predictions for the rodairfoil configuration based on volume sources and various surfaces were carried out. Additionally, the volume based predictions were compared to predictions based on wall and free surfaces. As expected from theory, it was found that the prediction based on an appropriately chosen free surface contains the contribution of the volume sources. The study showed further that volume sources from the rod wake, although they are not significant sources of sound at low Mach numbers in theory, can play a non-negligible role at high frequencies (as shown in figure 4). Further information is given in Greschner et al. [29][26].

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Fig. 4 Far-field prediction at θ = 90o based on wall and volume sources compared to the measurement from [26]

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3.2 Jet-Flow Configurations The LES is the methodology most commonly used for the simulation of jet flows as the proper capturing of broadband noise sources in a jet requires the resolution of the larger scales which is inherent in the LES [6][8][13][19]. Though mostly good results were achieved, questions concerning details of the turbulence model were raised. Andersson et al. [1][4][2][3] studied single-stream and later coaxial jets with LES. It can be argued that the azimuthal resolution is too coarse in most computational meshes especially at larger Reynolds number. The size of the local grid spacing Δ = max(Δ x, Δ y, Δ z) in the shear layer is therefore dominated by the azimuthal resolution. An artificial reduction of that grid spacing leads to a reduction of turbulent viscosity. The use of Δ = (Δ x Δ y Δ z)1/3 or even Δ = min(Δ x, Δ y, Δ z) was found to decrease the suppression of the initial shear layer instability and to improve the results. Bogey&Bailly [8][12] studied the effects of the Reynolds number on the LES results. In additional work, the effects of the subgrid-scale model used in the LES is discussed [9][11]. It was argued that the effective Reynolds number of the simulation is based on the modelled viscosity as Newtonian and Reynolds shear stresses have a similar form (compare eq. (4)). Because of the strong Reynolds-number dependence of jet flows, a strong effect on the simulation results is found. Bogey et al. therefore omit turbulence modelling and obtain the diffusion of smallest scales via explicit, spatial filters of high order. A similar approach is used by Spalart et al. [50], who also omit any turbulence modelling but do not use spatial filtering. A good agreement with measurements is found even for large Strouhal numbers. Measures to reduce the modelled viscosity are therefore taken in the simulations of the coplanar jet (see section 3.2.2). Furthermore, the issue of the transition to turbulence in the initial shear layer is of importance. Bogey&Bailly [10] apply forcing in the initial shear layer in their LES and study the effects of forcing parameter and the initial shear layer thickness on the flow and sound field. A dependence of the results on the forcing parameter and the thickness of the shear layer is observed. McMullan et al. [38] also describes a dependence of the transition to turbulence on the initial shear layer thickness in their LES. Andersson et al. [1][4][2][3] and Spalart et al. [50] do not apply artificial forcing and do not report problems with the transition. Additionally,


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Andersson includes the inner nozzle in his simulation and uses a wall model for the LES. Especially with regard to future work concerning modified nozzle lips, we adopt the methodology of including the inner nozzle into the simulation and work without unsteady forcing, as the specification of nozzle exit conditions would become difficult at non-planar nozzle exits. 3.2.1 Single-Stream Jet The first jet flow configuration studied during the DFG/CNRS project is that of an unheated, single-stream jet at a nozzle exit Mach number of Ma = 0.6 and a Reynolds number of Re = 1.1 · 106 , based on nozzle diameter and velocity at the nozzle exit. Experimental investigations of this configurations are described in Simonich et al. [47] and Narayanan et al. [40]. The grid used in the simulation consisted of 5.7 million cells with an azimuthal resolution of 80 cells. The time step size chosen for the simulation corresponds to a sampling Strouhal number of StΔ t = 140. The total number of 350 convective time units D/U j has been calculated. Far-field data was written for the last 200 convective time units and used in the acoustic prediction. Turbulence modelling was done using the unmodified Smagorinsky-model [48] as the subgrid-scale model of the LES , using the definition of grid length scale Δ = max(Δ x, Δ y, Δ z). The instantaneous flow field is depicted in figure 5 in terms of an iso-surface of the λ2 -criterion which is used to detect vortices in the flow field. It can be observed that the initial shear layer region up x ≤ 4D is dominated by vortex rings that are purely convected downstream. An unsteady analysis showed that these vortex rings start a leapfrogging-cycle near the end of the potential core which is located at x = 5D both in the experiment and in the simulation. The leapfrogging-cycle is never completed, because turbulent, three-dimensional structures are developed quickly by the simulation in the region around the end of the potential core. The missing turbulent structures in the initial shear layer lead to an decreased turbulent mixing. The effects of this can clearly be observed in the time averaged, axial velocity given in figure 6. The shear layer grows only slowly before x ≤ 3.5D due to the missing mixing. Further downstream, the mixing enhances suddenly as three-dimensional structures develop. This leads to the rounded shape of the potential core.

Fig. 5 Instantaneous isosurface of λ2 = −2 · 106

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Fig. 6 Iso-lines of the time averaged, axial velocity r/D

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For a jet of this Reynolds number, one would expect a more triangle shaped potential core which is sketched in figure 6 as the dash-dotted lines. If turbulent structures were present in the shear layer, their turbulent mixing would lead to a faster and more constant growing of the shear layer which would produce the triangle shaped potential core. The far-field predictions calculated from the simulated flow field are shown in the left-hand side of figure 7 as directivity of the overall sound pressure level. The levels calculated by the LES/FWH approach are compared to far-field measurements of Jordan et al. [34] that were conducted during the EU-funded project JEAN. It must be noted that the Mach number of the single-stream jet in the experiments was Ma = 0.75 instead of Ma = 0.6, though the Reynolds number is of the same order of magnitude. The results of the LES/FWH approach have therefore been scaled by the squared velocity ratio of the two jets (U0.75 /U0.6 )2 . The lower Mach number of the numerical simulation should yield a smaller convective amplification in the rear arc in comparison to the experiment. Despite the facts that the Mach number differs from the experiment and that turbulent structures inside the shear layer are missing in the simulation, a fair agreement between experiment and LES/FWH prediction is found. The error of the numerical prediction is generally less than 3 dB, expect for the very low angles to the jet axis where the overprediction is about 5 dB. This overprediction is even larger if the smaller convective amplification of the numerical simulation is considered. A similar tendency to overpredict the low and the high angles can also be found in the LES of Andersson [1] who used the Mach number of the experiments, but did the simulation at a Reynolds number of 50000. Though the overprediction at low angles 120

120 Exp., Jordan et al, 2002 LES/Kirchhoff., Andersson, 2003; Ma=0.75, cold Exp. +/- 5dB, Jordan et al, 2002 LES/FWH, rescaled

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is much less pronounced in the work of Andersson. The FWH prediction from our LES shows an additional overprediction at θ = 90o. To examine this further a decomposition into azimuthal modes was carried out for the time series of the predicted, far-field pressure. The modal content of this decomposition is depicted in the right-hand side of figure 7. The modal content of the predicted far-field pressure is dominated by the ring mode m = 0, especially at low angles and around θ = 90o. The overprediction at these angles is therefore attributed to the vortex-ring structures inside the shear layer which were not present in the LES of Andersson. 3.2.2 Coaxial, Coplanar Jet The aim of this work is to compare the results of a LES with that of a DES carried out under similar conditions. While there are a number of flow configurations that have been studied with both LES and DES, though mostly for airframe noise configurations different flow solvers, far-field prediction methods and grids were used which make a straightforward interpretation of the results difficult. Comprehensive experiments were conducted by Guitton et al [30] concerning unheated, coplanar, coaxial jets of moderate Mach number during the EU-funded project CoJeN. The experimental data was obtained in an anechoic chamber at the LEA in Poitiers and included LDA, PIV data as well as near-field and far-field pressure measurements. The coplanar nozzle from which the flow streams emerge is depicted in figure 8. The nozzle itself and the nozzle inflow parameters have been designed to let the fluid emerge at ambient temperature for both the primary and the secondary stream. The velocity ratio between secondary and primary stream used in the simulation is γ = 0.5, where the Mach number of the primary stream is M p = 0.5. Both are similar to the conditions of the measurements. The Reynolds number of the configuration is Re=1.1 million based on the outer nozzle diameter D=0.1m and the exit velocity of the primary stream. In this work, the approach of Andersson et al. is adopted, which is the definition of the local grid spacing based on the cubic root of the cell volume. This modification is used for both LES and DES to achieve similar modelling conditions. More details can be found in Eschricht et al [20]. To achieve a stable solution especially in the near wall region, flux blending is used in the complete computational domain. In Fig. 8 Sketch of the nozzle geometry (Guitton et al [30])

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the near wall region cell face values are interpolated nearly completely by the TVD scheme. Away from the wall, the blending function is quickly transformed to lead to 95% CDS and only 5% TVD scheme everywhere away from the walls to ensure accuracy. The grid used for the numerical simulation consists of 8.1 million cells and extends over −3<x/D |w|

Fig. 1 Flowchart of CVE decomposition

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which yields the same result. A flowchart of the CVE algorithm is depicted in Fig. 1. The induced coherent and incoherent velocity fields are computed using Biot–Savat’s kernel (BS), v = ∇ × (∇−2 ω ), from the coherent and incoherent vorticity fields, respectively.

2.3 CVE of Turbulent Flows at Different Reynolds Numbers In the case of N = 2563 we study the vorticity field, for the N = 5123 and 20483 dataset, as they are are too large, we will only consider subcubes of 2563 . Table 1 shows the statistical analysis comparing the total, coherent and incoherent flows for Reynolds number Re = 20000 and Table 2 shows the same for 4 different 2563 subcubes extracted from the N = 5123 cube data set which correspond to a flow with Reynolds number Re = 60000, and for 4 other different 2563 subcubes of the N = 20483 cube which correspond to a flow with Reynolds number Re = 400000. We found that the coherent flow corresponds to 3.6% of the wavelet coefficients and retains from 92.5% to 90.7% of the total enstrophy and more than 99% of the total energy. The remaining incoherent flow represents about 95% of the wavelet coefficients but retains from 7.5% to 9.2% of the enstrophy and less than 1% of the energy. A visualization of the modulus of vorticity and their coherent and incoherent contributions are shown in Figure 2. We observe that almost all Table 1 CVE decomposition of a flow field computed at resolution N = 2563 for Reynolds number Re = 20000

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coherent incoherent 3.59 96.41

Energy 2.955 2.946 0.004 % of Energy 100 99.7 0.14 min value -5.92 -5.93 -0.31 max value 6.16 6.17 0.33 velocity skewness 0.1432 0.1439 0.0016 skewness 0.14317 0.14385 0.15956 velocity flatness 2.7818 2.7802 3.48 flatness 2.7818 2.7802 3.4816

Enstrophy 212.80 % of enstrophy 100 min value -230.26 max value 245.86 vorticity skewness -0.013 vorticity flatness 8.08

196.93 15.87 92.54 7.46 -243.29 -34.18 259.24 38.19 -0.016 0.68 ·10−3 8.31 4.92

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Table 2 Statistical analysis of the CVE filtering of several vorticity fields computed at resolution N = 2563 , 5123 , 20483 , which correspond to Reynolds number Re = 20000, 60000, 400000 respectively

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5755.875 5224.8975 580.9725 100 90.105 9.895 -0.1125 0.020925 -0.225755333 14.4425 15.6775 5.6975

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structures are preserved in the coherent vorticity while they are no organized structures left in the incoherent vorticity. Indeed, the total and coherent vorticity fields present vortex tubes, while the incoherent vorticity looks structureless and noiselike. The vorticity PDFs and spectra for the total flow and for the coherent flow are similar, i.e., non-Gaussian and long-range correlated, while the incoherent flow is decorrelated and has a much reduced variance (Fig. 3). We observe that all along the inertial range the coherent flow presents a similar energy and enstrophy spectra

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compared to the spectra of the total flow, whereas they differ only in the dissipative range for k ≥ 30. The reconstruction of the coherent part of these fields requires only, for Re = 60000, 3.6% of the wavelet coefficients and retains ≈ 92% of the


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enstrophy , and only ≈ 3.7% of the wavelet coefficients maintains about 91% of the enstrophy for Re = 40000. Additionally, there are nonsignificant changes in the skewness and flatness between the total field and the coherent field for the different Re . Moreover we do not observe significant variations between the statistical properties of the four different subcubes for each for Re . The total, coherent and incoherent vorticity fields of the vorticity field is visualized in Figure 2 for Re = 20000, 60000, 400000. As observed in the Re = 20000, almost all structures are preserved in the coherent contribution and none remain in the incoherent contribution. However with the increase of Re , we observe the presence of more structures in the analyzed subcubes. In [26] the extraction of coherent structures is done for homogenous isotropic turbulent flows. A similar CVE is used for different resolutions 2563 , 5123, 10243 , 20483 which correspond to Re = 30000, 70000, 200000, 600000 respectively. The main difference is that the CVE is applied on all the fields and not on subcubes as we do here. The authors find that the compression rate increases with Re . In our analysis on the subcubes, the compression rates are nearly constant ∼ 3.6% for the different Re (20000, 60000, 400000), so that, the Donoho threshold increases with Re . This could suggest that the compression rate is limited by the resolution, i.e., the number of octaves available. Indeed the resolution of all cubes we analysed is constant at N = 2563. Moreover the decay of wavelet coefficients is smaller for the subcubes, but the PDFs and spectra are very similar to the ones obtained in our study. From the present results we conjecture that modelling the effect of the discarded modes on the resolved modes is easier to perform using the Coherent Vortex Simulation (CVS) approach introduced in [14]. CVS computes all degrees of freedom which contribute to the flow nonlinearity, i.e., the coherent modes, whatever their scale, while the remaining degrees of freedom, i.e., the incoherent modes, are discarded to model turbulent dissipation. The method actually combines an Eulerian projection of the solution with a Lagrangian procedure for the adaption of the computational basis: for more details we refer the reader to [14]. The next step to demonstrate the potential of CVS is to develop an adaptive wavelet solver for the 3D Navier–Stokes equations.

3 Coherent Vortex Simulation of Turbulent Mixing Layers In this section, we present an extension of the Coherent Vortex Simulation (CVS) [14] method to compressible flows. The CVS method is based on the observation that turbulent flows contain both an organized part, the coherent vortices, and a random part, the incoherent background flow. The separation into coherent and incoherent contributions is done using a non-linear wavelet filtering. The evolution of the coherent part is computed in physical space using a finite volume scheme on a locally refined grid, while the incoherent part is discarded during the flow evolution, which models turbulent dissipation. To discretize the convective terms, we use a 2–4 McCormack scheme [17], while the diffusive terms are discretized using a second–order centered scheme in space. The time integration is done by an

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Table 3 Comparison between DNS and CVS of a 3D compressible mixing layer, Ma = 0.3, Re = 200. CPU time required on a Pentium IV 2.5 GHz, percentages of CPU time, required memory, total energy E and total enstrophy Z in comparison with the DNS computation Method CPU time % CPU % Mem %E %Z DNS 7 day 09 h 40 min 08 s 100.0 % 100.0 % 100.0 % 100.0 % CVS 2 day 07 h 00 min 03 s 29.0 % 30.2 % 98.3 % 93.4 %

Fig. 4 CVS of a 3D compressible mixing layer, Ma = 0.3, Re = 200. Top: isosurfaces of vorticity ||ω || = 0.6 (black), ||ω || = 0.4 (dark gray), and ||ω || = 0.2 (light gray) at t = 18, (left) t = 37 (center), and t = 80 (right). Bottom: corresponding adaptive grids

explicit second-order Runge-Kutta scheme. The wavelet basis used for the filtering relies on the cell-average multiresolution analysis developed by Harten [19]. After the filtering, the discarded coefficients are removed from memory, so that both CPU time and memory requirements are significantly reduced in comparison with the DNS computation. The data structure is organized into a graded-tree form to be able to navigate through it. To perform the CVS computations, a three-dimensional adaptive multiresolution algorithm [30], originally developed for reaction-diffusion equations, has been extended to the compressible Navier-Stokes equations. Extensions of the adaptive multiresolution scheme to the compressible Euler equations can be found in [6]. As example, we apply the CVS method here to compute a time-developing threedimensional turbulent mixing layer in the weakly compressible regime. CVS computations of incompressible turbulent mixing layers have been presented in [32]. In this test-case, both layers have the same initial velocity norm, but opposite directions. An initial three-dimensional sinusoidal perturbation is added to the basic profile. The Reynolds and Mach numbers, based on the initial velocity norm and


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half the initial layer thickness, are set to 200 and 0.3, respectively. The results are compared with the one obtained by DNS performed on the regular finest grid with the same numerical scheme. We find that only 17.9 % wavelet coefficients contain around 98.3 % of the energy and 93.4 of the enstrophy. Taking into account all the nodes of the tree data structure, these wavelet coefficients represent 30.2% of the 1283 = 2097152 cells that the fine-grid computation requires. Concerning the CPU time, it only represents 29.0% of the one required by the DNS, i.e. CVS is in the present case three times faster than DNS. These results show that the CVS method yields accurate results in comparison with DNS, while significantly reducing the CPU time and memory requirements. Further work will focus on the CVS of compressible mixing layers with larger values of both Mach and Reynolds numbers, i.e. in a more turbulent and more compressible regime, for which higher compression of memory and CPU time are expected, cf. [31].

4 Adaptive Multiresolution Methods for Evolutionary PDEs Up to now, different approaches have been investigated to define adaptive space discretizations, some emerge from ad hoc criteria, others are based on more elaborated a posteriori error estimators using control strategies by solving computational expensive adjoint problems. In the framework of the current project we focused on multiresolution based schemes (MR) for evolutionary PDEs. The multiresolution data representation is the main idea of the MR method. The decay of the MR coefficients gives information on local regularity of the solution. Therewith the truncation error can be estimated and coarser grids can be used in regions where this error is small and the solution is smooth. An adaptive grid can be introduced by suitable thresholding of the multiresolution representation where only significant coefficients are retained. Hence a given discretization on a uniform mesh can be accelerated as the number of costly flux evaluations is significantly reduced, while maintaining the accuracy of the discretization. The memory requirement could also be reduced, for example using a dynamic tree data structure.

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A main bottleneck of most of these space–adaptive methods, which typically employ explicit or semi–explicit time discretizations, is that the finest spatial grid size imposes a small time step in order to fulfill the stability criterion of the time scheme. Hence, for extensive grid refinement with a huge number of refinement levels, a very small size of the time step is implied. The aim in the present project was to develop local scale dependent time stepping for the space adaptive multiresolution scheme introduced in [30]. The idea is to obtain additional speed up of this efficient space adaptive scheme by introducing at large scales larger time steps without violating the stability condition of the explicit time scheme and less flux evaluation due to larger time steps. The originality of our work is to combine, in the framework of the cell-average multiresolution analysis, a local time stepping together with multi-stage Runge-Kutta methods. The


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synchronization we propose differs from the one introduced in [25], where singlestage methods were used. Starting point of our work is the fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry which based on a second order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The number of costly flux evaluations is significantly reduced. For details we refer the reader to [30]. For time discretization we use an explicit Runge-Kutta scheme of second order with a scale dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On large scales the time step can be increased without severe violation of the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure which allows memory compression and CPU time reduction. In [6] we present numerical validations for test problems in one space dimension which demonstrate the efficiency and accuracy of the local time stepping

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scheme with respect to both the multiresolution scheme with global time stepping and the finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrated the memory reduction and the CPU speed-up for a flame instability. For details we again refer the interested reader to [6]. In [6, 7] we extended the above scheme and developed an adaptive time stepping scheme with automatic error control. The adaptive time integration method is based on a Runge–Kutta–Fehlberg method, which allows an estimation of the local error in time. An original limiting technique is also proposed to avoid non admissible choices for the time step. The adaptivity in space is done through a multiresolution method, which automatically detects the local regularity of the solution and hence guarantees automatic grid adaption in space. The costly numerical fluxes are evaluated on this locally refined, while ensuring strict conservativity. The implementation uses graded tree data structures which allows an efficient representation of the solution on adaptive grids with reduced memory requirements. Applications are shown for the compressible Euler equations [6].

4.1 Advection Equation in 1D As example (see [7]) we consider the linear advection equation

∂u ∂u +c = 0, ∂t ∂x where u = u(x,t), t ≥ 0, (x, y) ∈ [− 12 , 12 ], and c > 0 is the constant velocity. The boundary conditions are periodic and the initial condition is u(x, 0) = exp −50 x2 . The elapsed time is set to t = 1, so that the final solution is equal to the initial one. In Figure 6, we show the time evolution of the time step for different initial CFL values, i.e., for different initial time step sizes, using the finite volume scheme on a regular grid with global time adaptivity and the RKF 2(3) method. We observe that the time steps tend to converge in all cases to a time step around 5.2 × 10−3. To avoid a “bad choice” of the initial CFL values, the code allows a bigger limiter in the beginning time steps. These “bad choices” of initial CFL could however increase the global error, as presented in Table 4. The automatic step size control of the solution reduces the number of time steps and hence the computational cost (cf. Table 4). We can also observe that the final error with respect to the analytical solution is reduced for the time adaptive schemes, compared to the finite volume scheme with fixed time stepping, except for the initial CFL = 1 using the L∞ norm. In Figure 8, we compare L2 , L1 , L∞ norms for the RK 3 method with CFL = 0.5 and the RKF 2(3) method. The results show that the choice of the initial time step does not influence the final error, since all computations yield a similar result. Figure 7 shows the CPU time spent for different choices of the initial CFL. The CPU


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Table 4 Initial CFL, number of time steps, CPU time, initial and final time steps, and errors for the advection equation. Mesh containing 256 points, δ0 = 10−3 , Smin = 0.01, S0 = 0.10. The first line corresponds to the constant time step with CFL=0.5 Init. CFL 0.50 1.00 0.75 0.50 0.25 0.10

# steps 513 367 381 388 452 587

% CPU compression 100 90 96 93 105 133

Initial time step 3.91e-03 7.64e-03 5.73e-03 3.90e-03 2.00e-03 7.98e-04

Final time step 3.91e-03 5.27e-03 5.18e-03 5.15e-03 4.88e-03 4.94e-03

L∞ -error L2 -error L1 -error 8.61e-02 8.80e-02 8.52e-02 8.52e-02 8.58e-02 8.56e-02

1.68e-02 1.65e-02 1.58e-02 1.58e-02 1.64e-02 1.60e-02

6.84e-03 6.27e-03 5.97e-03 5.97e-03 6.57e-03 6.23e-03

time decreases as the initial CFL increases. This is directly related to the number of time steps needed to compute the solution at t = 1, as could be observed in Fig. 9. For this test case, we can thus conclude that RKF 2(3) is more efficient than the conventional RK 3 method with a fixed time step.

5 Wavelet Decomposition for LES - Implementation in the LESOCC2 Code All simulations of the test cases presented further have been performed with the block-structured finite volume code LESOCC2 [20]. In this code the wavelet decomposition was implemented and used to derive subgrid-scale models for LES. Details are given in [4, 5]. Performing this work in the code LESOCC2 had the advantage of employing an already parallelized code which runs efficiently on a large number of processors with disrtibuted memory access [3].

5.1 Wavelet Subgrid-Scale Modelling The wavelet decomposition used here is based on Hartens biorthogonal approach [18, 29]. It is applied here to generate a two-level decomposition to obtain the wavelet details . The details from this wavelet decomposition are then used to derive an eddy-viscosity subgrid-scale model in the framework of LES according to the following equation: % νt = C(Vol p )(1/3) [detail(u)]2 + [detail(v)]2 + [detail(w)]2 . (3) Here, C denotes a model constant which was set equal to 0.02 based on calibration for turbulent channel flow. The quantity Vol p is the volume of the computational cell on the finer grid, while u, v and w denote the instantaneous values of the velocity components. The above equation was derived based on dimensional reasoning. Alternatively, this model can be obtained in analogy to the structure-function model [23] by substituting the structure functions with the wavelet-details. Further

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exploitation of the idea of wavelet decompositions and how they can be used in the framework of LES with mixing of passive scalars is given in [4].

5.2 Numerical Details for the Simulated Cases The finite-volume code LESOCC2 developed at the University of Karlsruhe for Large Eddy Simulations of incompressible flows uses second-order central schemes for the spatial discretization of all terms. The time-marching is explicit and is based on a second-order Runge-Kutta scheme. The time step is variable computed with a CFL number equal to 0.6. The grid is block-structured, hexahedral, collocated and curvilinear. Whenever possible a wall-resolving grid was applied for the flow regions close to rigid walls. The Smagorinsky model was used with model constant Cs = 0.1 and a van Driest damping function. With the above wavelet model, no wall damping was used. For the cases where the grid near the wall was too coarse to be wall-resolving, the Werner-Wengle wall function was employed for the Smagorinsky model, while for the wavelet-based model the no-slip boundary condition was applied.

6 Complex Flows Simulation: Presentation of Test Case Results In the framework of the DFG-CNRS cooperation of FOR 507, several test cases were defined to assess LES model development for complex flows. The following cases were computed in the present project and will be discussed below: • Plane turbulent channel flow with Reynolds numbers Reτ = 180, 395 and 590; • Periodic hill flow, Re = 10 600 and Re = 37 000; • Flow in a model combustor with swirl, Re = 50 500, swirl number S = 0.6. In all subsequent Figures the wavelet model (3) is labeled as ”present”. Computations with the Smagorinsky model were undertaken with all other parameters unchanged in order to provide sound reference data for the evaluation of the performance of the new model. For the hill flow at the lower Reynolds number, the fine-grid LES solution of Breuer [1] obtained on a grid with 12Mio cells is used as an additional reference data set (REF-MB).

6.1 Plane Channel Flow The numerical grid consists of about 250, 000 control volumes, and is the same for all three Reynolds numbers. As a result, the first node (cell center) in the wall-normal direction is positioned at y+ = 1.34 for Reτ = 180, at y+ = 2.94 for Reτ = 395, and at y+ = 4.40 for Reτ = 590. For the two higher Reynolds numbers, computations with the present model were carried out also with a finer, wallresolving grid with about 400, 000 control volumes. This finer grid resulted in


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Fig. 10 Resolved turbulent stresses, Reτ = 180. Left: Present subgrid-scale model compared to the DNS data from [22]. Right: The Smagorinsky model compared to the DNS data from [22]

only very minor improvement so that they are not reproduced here for the sake of brevity. Results for the resolved Reynolds stresses from the present model and from the Smagorinsky model are compared with DNS data from [22, 24] and presented in Figures 10 to 12. For the lowest Reynolds number the present model is clearly superior to the Smagorinsky model for all turbulence quantities. For Reτ = 395 and Reτ = 590 the present model overpredicts the streamwise stresses u u , while the Smagorinsky model predicts them with a good accuracy. However, the position of the peak with respect to the y/h coordinate is well predicted by both models. For all other quantities (and especially for the shear stresses u v ) the present model is superior to the Smagorinsky model. DNS-MKM DNS-MKM <w´w´>DNS-MKM -DNS-MKM PRESENT PRESENT <w´w´>PRESENT -PRESENT

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6.2 Periodic Hill Flow The periodic hill flow test case was calculated for two different Reynolds numbers. Comparison is performed with data from other groups of the DFG − CNRS collaboration, both numerical [1] and experimental [28]. Fig. 13 shows the geometry and computed streamlines. Configuration and physical issues of this test case are discussed in [16]. In [34], SGS and wall-modelling issues were investigated. The numerical grid employed consists of about 1Mio control volumes. Details are provided in Table 5. Figure 14 shows the dimensionless streamwise velocity for all models and the experiments at the lower Reynolds number 10595. Compared to the experiments of [28], all numerical simulations overpredict the maximal streamwise velocity by a small amount with the deviation being largest for the present model. In the same Figure data for the resolved turbulent stresses are compared. Here, the new simulations show good agreement with the experimental and numerical reference data. The observed deviations are substantially smaller than those between different SGS models in [34].

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Table 5 Summary of the numerical simulations of the periodic hill flow. Index ”s” - separation point, index ”r” - reattachment point, ta is averaging time and tx is flow-trough time, LR stands for Re=10595 and HR - for Re=37000 Case

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281 × 222 × 200 160 × 100 × 60 160 × 100 × 60 160 × 100 × 60 160 × 100 × 60

0.0018 0.0084 0.0085 0.0083 0.0079

141 72 72 73 73

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We now turn to the position x/h = 0.5 located at the beginning of the recirculation zone, with corresponding data shown in Fig. 15. As above, the numerical solutions slightly underpredict the mean streamwise velocity in the lower part of the domain (y/h < 1.8), while yielding somewhat larger values in the upper part. Concerning

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the vertical velocity component v, it is to be observed that its value is substantially smaller than u, so that differences should not be over-interpreted. Closest to the experiments are the data obtained with the Smagorinsky model, while the present model yields larger values. The resolved Reynolds stresses in the lower pictures of the figure agree very well with the reference data. Here, v v exhibits the largest visual differences, with the present model and the Smagorinsky model being similar in the upper part and the present model better in the lower part. But again absolute value of v v is small. In the following Fig. 16 we present results for the hill flow with Re = 37000. It is the first time that this case is computed since the experimental data have been obtained only recently. Hence, only the experimental data of [28] are available for comparison. Since the Reynolds number is higher than for the previous case it is expected that the contribution of the SGS-modelling is larger. In order to be able to assess this, a solution without a SGS-model was carried out (referenced as PDNS with P standing for ”Pseudo”). For the mean streamwise velocity, this PDNS solution yields results which are closer to the experimental data than obtained with the


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other simulations. For the resolved Reynolds stresses, however, and in particular for u u , the results are worse than those from the other cases. Comparing the results with the new model and the Smagorinsky model shows that the Reynolds stresses are largely the same in both cases. This is even more observed for the mean flow. The overall analysis carried out so far shows that the agreement between the numerical simulations and the experiments is better for the lower Reynolds number than for the higher one. In order to investigate the effect of the Reynolds number, results for both Re = 10595 and Re = 37000 are presented in Fig. 17 at x/h = 4.0 - a position at which the recirculation zone approximately ends. The mean streamwise velocity u in the experiment becomes more uniform with increasing Reynolds number. The same trend is observed with the present model, while for the Smagorinsky model the curves are almost identical between the two cases. Concerning the Reynolds stresses it is observed that these, when scaled with Ub2 , decay with increasing Reynolds number in the experiment. The results of the simulations, all obtained on the same grid, are very much alike. Only for u u slight deviations are observed showing a small trend in opposite direction, i.e. increasing scaled fluctuations with

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EXP-RAPP Re=10595 EXP-RAPP Re=37000 PRESENT Re=10595 PRESENT Re=37000 SMAG. Re=10595 SMAG. Re=37000

2

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2 b

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Fig. 17 Comparison of results for the two Reynolds numbers considered, x/h = 4. For additional explanations see caption of Fig. 14

Re. In particular, the results of the new model are close to those of the Smagorinsky model. These observations can be related to the mean eddy viscosity displayed in Fig. 13. The change of this quantity with Re (not displayed here) is very small. The difference between the two models can be appreciated in that figure for Re = 37000 and is about 10-15%. One time step with the present wavelet model was about 5% faster than the Smagorinsky model with respect to the CPU-time for the cases with Reynolds number 37000. Additional information about the computations performed as well as for the separation and reattachment points is given in Table 5.

6.3 Flow in a Model Combustor with Swirl The annular swirl combustor experimentally investigated at the Technical University of Darmstadt [27, 33] was another testcase in the DFG-CNRS collaboration. In the following we focus on the burner geometry and compute the flow prior to the outlet into the combustion chamber, as it was done in [27].


197

Z 0.08

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Fig. 18 Numerical blocks with the section x = −0.04, where comparison with measurements are presented (left). Inflow section at x = −0.165 with the averaged streamlines and instantaneous pressure fluctuation (right)

This annular swirler as depicted in Fig. 18 has an inner wall and an outer wall and no blades. Its dimensions are the following: length 0.180[m], radius at the inflow section 0.085[m], inner radius of the annulus ri = 0.020[m] and outer radius ro = 0.050[m]. The Reynolds number based on the hydraulic diameter of the annulus is Re = 50500 with a bulk velocity Ub = 10.94[m/s] used as characteristic velocity. The swirl number in the plane of comparison was to S = 0.6 and is visualized by the streamlines of the mean flow in the right graph of Fig. 18. This was achieved with the following values at the inflow boundary of the domain: radial component 4.51[m/s] and tangential component 3.01[m/s]. The computational grid employed contains 1.9 Mio computational nodes decomposed into 40 blocks as depicted in Fig. 18. The computation was performed over 12.5 flow-through times and averaging was accumulated over 10.5 flow-trough times. Using 20 processors on an HP XC4000 this simulation took approximately 72 hours wall-clock time. The contour plot in Fig. 18 provides instantaneous data of the pressure fluctuations convey an impression of the structure and the size of turbulent fluctuations at this position. Figure 19 presents a comparison between numerical and experimental results in the annular swirler, at x = −0.040[m]. This section had been selected for measurements in the experiments [27]. From the different simulations carried out in [33], the one with the Dynamic Smagorinsky model and a boundary condition set denoted as ”simplified swirler” has been used here for comparison as it corresponds most closely to the present set of boundary conditions and computational domain. In the figures these results are denoted ”DSM” and the corresponding experimental data from [33] have been used for comparison, too. The present simulation were undertaken using the Smagorinsky model as well as the new wavelet model.

H. Bockhorn et al. <w>/Ub - tangential velocity

/Ub - axial velocity

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Exp. (S=0.6) DSM SMAG. PRESENT

2 1.5 1 0.5 0 0.2

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urms/Ub

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1

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1


0

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0.4 0.6 (r-ri)/(ro-ri)

Fig. 19 Comparison of different numerical simulations with experiments at x = −0.04[m]. DSM ˇ c and Jakirlić [33] and experimental data courtesy to Sari´

Figure 19 shows that the DSM simulation of [33] agrees well with the experimental data. The simulations performed in the present paper with the two SGS models considered produced virtually identical data. They yield a mean streamwise velocity profile which is close to the reference data. The deviation for the mean circumferential velocity w between the present models and the DSM is larger, in particular near the inner wall. The reason for this behavior is the substantially lower level of fluctuations compared to the reference simulation and the experiment. This, in turn, may be due to the use of stationary inflow conditions (in [33] this issue is not discussed). Computations with different boundary conditions and numerical parameters are under way to elucidate this issue.

7 Conclusions The present paper gives examples of analysis and simulation of complex turbulent flows with a particular emphasis on the utilization of the orthogonal and biorthogonal wavelet decomposition for their study and computation. Theoretical background and applications of the wavelet based coherent vortex extraction method are presented for homogeneous isotropic turbulence at different Reynolds numbers. A summary of the developed adaptive multiresolution method for evolutionary PDEs is presented. First fully adaptive computations of 3d mixing layers using Coherent Vortex Simulation are shown. Features like local scale dependent time stepping are also illustrated and examples for one dimensional problems are given.


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Computations with the developed wavelet models have been performed on complex geometries and flows with high Reynolds numbers applicable to engineering problems. For this purpose three test cases selected in the present DFG-CNRS cooperation were simulated comprising non-orthogonal grids and swirl. The simulations with the finite-volume code LESOCC2 show that the wavelet-based models are robust and efficient for the purpose of parallel computations, showing a good accuracy when compared to classical subgrid-scale models like the one of Smagorinsky. Acknowledgements. Computations have been performed on the national super computer HP XC4000 at the High Performance Computing Centrer Stuttgart (HLRS) under the grant with ˇ c for acronym DNS-jet. The authors would like to thank also Dominic von Terzi and Sanjin Sari´ the support in setting up the test-case simulations and to P. K. Yeung and D. Donzis from Georgia Tech for the dataset related to [35] they provided.

References 1. Breuer, M.: New reference data for the Hill Flow Test Case: DFG - CNRS Research Unit 507, A French-German Research Group (2005), http://www.hy.bv.tum.de/DFG-CNRS 2. Daubechies, I.: Ten Lectures on wavelets. In: CBMS-NSF Conferences in Applied Mathematics, vol. 61. SIAM, Philadelphia (1992) 3. Denev, J.A., Fröhlich, J., Bockhorn, H.: Direct Numerical Simulation of a Round Jet into a Crossflow - Analysis and Required Resources. In: Nagel, W.E., Kröner, D., Resch, M. (eds.) Transactions of the High Performance Computing Center, Stuttgart (HLRS), pp. 339–350. Springer, Heidelberg (2007) 4. Denev, J.A., Fröhlich, J., Falconi, C., Bockhorn, H.: Direct Numerical Simulation and Large Eddy Simulation of transport processes in a jet in crossflow. In: Mewes, D., et al. (eds.) Springer series on Heat and Mass Transfer (to appear, 2008) 5. Denev, J.A., Falconi, C., Fröhlich, J., Bockhorn, H.: Wavelet-adapted sub-grid scale models for LES. In: Second International Conference on Turbulence and Interaction, Sainte-Luce, Martinique, 31 May - 5 June (to appear, 2009) (submitted) 6. Domingues, M., Gomes, S., Roussel, O., Schneider, K.: Space-Time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations. Appl. Num. Math. (April 2008) (accepted) 7. Domingues, M., Roussel, O., Schneider, K.: An adaptive multiresolution method for parabolic PDEs with time-step control. Int. J. Num. Meth. Eng. (October 2008) (accepted) 8. Domingues, M., Gomes, S., Roussel, O., Schneider, K.: An adaptive multiresolution scheme with local timestepping for evolutionary PDEs. J. Comput. Phys. 227, 3758–3780 (2008) 9. Donoho, D.: Unconditional bases are optimal bases for data compression and statistical estimation. Appl. Comput. Harmon. Anal. 1, 100–115 (1993) 10. Donoho, D., Johnstone, I.: Ideal spatial adaption via wavelet shrinkage. Biometrica 81, 425– 455 (1994) 11. Farge, M.: Wavelet transforms and their applications to turbulence. Ann. Rev. of Fluid Mech. 24, 395–457 (1992) 12. Farge, M., Schneider, K., Kevlahan, N.: Non-gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthonormal wavelet basis. Phys. Fluids 11(8), 2187–2201 (1999) 13. Farge, M., Pellegrino, G., Schneider, K.: Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett. 87(5), 054501 (2001)

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14. Farge, M., Schneider, K.: Coherent vortex simulation (CVS), a semi-deterministic turbulence model using wavelets. Flow, Turbulence and Combustion 66(4), 393–426 (2001) 15. Farge, M., Schneider, K., Pellegrino, G., Wray, A.A., Rogallo, R.S.: Coherent vortex extraction in three-dimensional homogeneous turbulence: comparision between CVS-wavelet and POD-Fourier decompositions. Phys. Fluids 15(10), 2886–2896 (2003) 16. Fröhlich, J., Mellen, C.P., Rodi, W., Temmerman, L., Leschziner, M.A.: Highly resolved largeeddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 526, 19–66 (2005) 17. Gottlieb, D., Turkel, E.: Dissipative two-four methods for time-dependent problems. J. Comput. Phys. 30, 703–723 (1976) 18. Harten, A.: Discrete multi-resolution analysis and generalized wavelets. J. Appl. Num. Math. 12, 153–193 (1993) 19. Harten, A.: Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 1305–1342 (1995) 20. Hinterberger, C.: Three-dimensional and depth-averaged Large-Eddy-Simulation of flat water flows. PhD thesis, Inst. Hydromechanics, Univ. of Karlsruhe, 296 p. (2004) 21. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) 22. Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1987) 23. Métais, O., Lesieur, M.: Spectral large-eddy simulations of isotropic and stably-stratified turbulence. J. Fluid Mech. 239, 157–194 (1992) 24. Moser, R., Kim, J., Mansour, N.: Direct numerical simulation of turbulent channel flow up to Retau = 590. Phys. Fluids 11, 943–945 (1999) 25. Müller, S., Stiriba, Y.: Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comput. 30(3), 493–531 (2007) 26. Okamoto, N., Yoshimatsu, K., Schneider, K., Farge, M., Kaneda, Y.: Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint. Phys. of Fluids 19, 115109 (2007) ˇ c, S., Jakirlić, S., Tropea, C.: Experimental 27. Palm, R., Grundmann, S., Weismüller, M., Sari´ characterisation and modelling of inflow conditions for a gas turbine swirl combustor. Int. J. Heat Fluid Flow 27, 924–936 (2006) 28. Rapp, C.: Experimentelle Studie der turbulenten Strömung u¨ ber periodische Hügel’, Dissertation, Technische Universität München (2008) 29. Roussel, O.: Development of an multiresolution adaptive three-dimensional algorithm for the solution of parabolic partial differential equations. Application to thermo-diffusive flame instabilities. Ph.D. Thesis, University of Aix-Marseille II, 147 p. (2003) 30. Roussel, O., Schneider, K., Tsigulin, A., Bockhorn, H.: A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J. Comput. Phys. 188(2), 493–523 (2003) 31. Roussel, O., Schneider, K.: Coherent Vortex Simulation of compressible turbulent mixing layers using adaptive multiresolution methods. J. Comput. Phys. (submitted, 2008) 32. Schneider, K., Farge, M., Pellegrino, G., Rogers, M.: Coherent vortex simulation of three– dimensional turbulent mixing layers using orthogonal wavelets. J. Fluid Mech. 534, 39–66 (2005) ˇ c, S., Jakirlić, S., Cavar, ˇ 33. Sari´ D., Kniesner, B., Altenhöfer, P., Tropea, C.: Computational study of mean flow and turbulence strucutre in inflow system of a swirl combustor. In: Friedrich, R., et al. (eds.) Procs. of 5th Int. Symp. on Turbulence and Shear Flow Phenomena, TU-Munich, Garching, Germany, August 27-29, vol. 2, pp. 911–918 (2007) 34. Temmerman, L., Leschziner, M.A., Mellen, C.P., Fröhlich, J.: Investigation of wall-function approximations and subgrid-scale models in LES of separated flow in a channel with streamwise periodic constrictions. Int. J. Heat Fluid Flow 24, 157–180 (2005) 35. Yeung, P.K., Donzis, D.A., Sreenivasan, K.R.: High-Reynolds-number simulation of turbulent mixing. Phys. of Fluids 17, 081703 (2005)

Hybrid LES–RANS-Coupling for Complex Flows with Separation Michael Breuer, Orhan Aybay, Benoˆıt Jaffrézic, Michel Visonneau, Ganbo Deng, Emmanuel Guilmineau, and Oussama Chikhaoui

Abstract. The investigations carried out within the French–German DFG–CNRS Research Initiative on ‘LES of Complex Flows (FOR 507)’ aimed at the development and application of hybrid LES–RANS methods for turbulent flows with separation. The objectives were twofold. On the one hand, the performance of a well-established hybrid approach, namely the detached-eddy simulation, was studied based on really challenging test cases such as the flow over the 3D hill and around the Willy car model. On the other hand, a new hybrid LES–RANS methodology was set up which aims at overcoming certain drawbacks associated with DES. It relies on the coupling of a near-wall RANS model with LES for the outer flow which allows an appropriate representation of large-scale flow phenomena. In order to take the anisotropies in the near-wall region rigorously into account, an explicit algebraic Reynolds stress model was chosen and its performance in comparison with linear eddy-viscosity models was analyzed for different test cases such as the flow over 2D hills and in a 3D diffuser. A further advantage of the new hybrid method is that no interface predefinition is required. Instead the interface is dynamically determined on-the-fly based on instantaneous physical quantities which guarantees that local changes in the flow are accounted for.

1 Introduction The simulation of complex turbulent flows at high Reynolds numbers encountered in technical applications is still a challenge. In principle, industry is willing to Michael Breuer · Orhan Aybay Fluid Mechanics at Institute of Mechanics, University of the Federal Armed Forces Hamburg, Holstenhofweg 85, Postbox 70 08 22, D–22043 Hamburg, Germany [email protected] Benoˆıt Jaffrézic Institute of Fluid Mechanics, University of Erlangen–Nürnberg, Cauerstr. 4, D–91058 Erlangen, Germany (formerly also address of M.B. and O.A.) Michel Visonneau · Ganbo Deng · Emmanuel Guilmineau · Oussama Chikhaoui Laboratoire de Mécanique des Fluides CNRS UMR 6598, Division Modélisation Numérique, Ecole Centrale de Nantes, France [email protected] C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 201–229. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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apply simulation methodologies which go beyond the capabilities of the well-known Reynolds-averaged Navier-Stokes (RANS) equation approach. That is especially true for complex flow phenomena for which RANS evidently suffers from reliability, e.g., large-scale separation and reattachment or vortex shedding. For such flows large-eddy simulations (LES) are obviously the better choice. However, engineering applications are in most cases wall-bounded flows, whereby wall-resolved LES requires very fine near-wall grid resolutions. According to Piomelli and Chasnov [34] the limits are set to Δ x+ = O(50–150), Δ z+ = O(15–40) and y+ 1st pt < 2, for the streamwise, spanwise and wall-normal direction, respectively (see also [52]). In order to avoid those resolution requirements at least in both wall-tangential directions, the idea of combining LES with RANS applied in the near-wall region was borne some years back. Needless to mention that in the meantime, beside this class of methods a variety of other hybrid LES–RANS concepts were proposed with partially different objectives but all aiming at a reduction of CPU costs compared to pure LES. Providing an overview of the different methodologies developed so far is beyond the scope of the present paper and we refer to the vastly growing literature, see, e.g., [4, 6–10, 13–15, 28, 29, 32, 33, 38–40, 43, 46–48, 50, 51]. To find a generally accepted taxonomy (e.g. segregated, seamless, zonal, nonzonal, unified, . . . ) even seems to be difficult and has led to more confusion rather than clearness. The most widely known hybrid approach is the detached-eddy simulation (DES) [33, 43, 46, 47] based on a simple modification of the one-equation Spalart-Allmaras RANS model [45]. The target of the DES designers was mainly external aerodynamic flows at high Re. Therefore, the main application area is unsteady turbulent flows with large separation regions for which RANS predictions do not work properly. Here, DES has been successful (see, e.g., [43, 50]). In general, the application of DES seems to be reasonable and reliable to such turbulent flows, whose dynamics are defined by large-scale separated vortices playing a dominant role in the energy balance. One main disadvantage of DES is that the LES–RANS interface position is not determined by the physics of the flow but instead predefined by the user during the grid generation process. Lately, Spalart et al. [48] proposed a new version of DES denoted Delayed DES (DDES) which ought to overcome the inaccurate behavior of DES observed for thick boundary layers and shallow separation regions. Basically, a new length scale is defined which now also depends on the eddy viscosity. First tests with DDES show an improvement of the results compared to DES regarding the ”grid-induced separation” problem [32]. Moreover, Travin et al. [51] proposed a further approach (IDDES) based on the recent DDES model which should solve the well-known log-law mismatch problem. However, that includes a bunch of rather complicated and incomprehensible functions and thus has not widely been used yet. A second drawback of DES and its variants is the Spalart-Allmaras model itself which is not especially adapted to the challenges of modeling the near-wall region appropriately. Close to a wall, large anisotropies of the Reynolds stresses are observed. Thus, a full Reynolds stress model (RSM) is obviously the best choice. However, RSM requires the solution of at least seven additional partial differential

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equations. Furthermore, these equations often turn out to be stiff. Hence the computational effort for RSM is too expensive to be used within a hybrid technique. A much cheaper alternative is found by an explicit algebraic RSM which avoids the solution of transport equations for each component. Instead algebraic relations are applied to account for the anisotropy emerging especially in the near-wall region. Such an anisotropy-resolving closure is implemented and evaluated in the present hybrid LES–URANS approach. Thus the paper includes two main parts. On the one hand the performance of DES for complex, practically relevant flow configurations is assessed. On the other hand an alternative hybrid method is suggested and evaluated.

2 Detached-Eddy Simulation The idea behind DES is to switch from the standard RANS model to an LES model in those regions where the turbulent length scale, Lt , predicted by the RANS model is larger than the local grid spacing. Several models of DES exist according to the RANS model used.

2.1 Spalart-Allmaras DES The Spalart-Allmaras DES, SA-DES, is derived from the Spalart-Allmaras oneequation model [45] (SA model) which solves a transport equation for a modified eddy viscosity, ν˜ . The modification of the SA model in order to obtain the DES formulation is a redefinition of the minimum wall distance d with a new length scale, d˜ = min(d,CDES Δ ) where Δ is the local maximum grid spacing in the three directions. The model constant CDES was calibrated in isotropic turbulence [43], yielding CDES = 0.65.

2.2 SST-DES The SST-DES [49] uses the SST turbulence model as RANS model. In this case, the length scale used in the computation of the dissipation rate in the equation for the turbulent kinetic energy is replaced by the local grid spacing Δ . The DES modification in the SST model is applied to the dissipation term in the k-equation as follows: Lt ∗ ∗ ε = β kω → β kω FDES with FDES = ,1 , (1) CDES Δ where ε is the dissipation rate, Δ is the maximum local √ grid spacing (Δ = max(Δ x, Δ y, Δ z), β ∗ is a constant of the SST model, Lt = k/β ∗ ω and CDES = 0.78 is a calibration constant of the DES formulation.

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3 Hybrid LES-URANS Method The hybrid method proposed relies on the idea to apply RANS or more specifically URANS in those regions, where statistical turbulence models in general perform properly. On the other hand, where large unsteady vortical structures are present in those regions LES is used in order to resolve them directly. Thus URANS is used for the near-wall region, whereas LES is performed in the remaining computational domain. Overall this approach strongly reduces the resolution requirements since the application of URANS for the prediction of attached boundary layers allows to dramatically decrease the near-wall resolution with the exception of the wall-normal direction [33]. This raises the hope that hybrid LES–URANS techniques can be used with acceptable effort even for high-Re flows encountered in technical applications. When setting up a hybrid approach, several questions have to be answered. The main ones are: Which models should be used in the URANS and LES regions? How should the LES–URANS interface be defined and how should both regions be coupled?

3.1 Choice of RANS and SGS Models In order to take the anisotropy of the Reynolds stresses in the near-wall region reasonably into account, the explicit algebraic Reynolds stress model (EARSM) proposed by Wallin and Johansson [53] is used here. It represents a compromise between the too expensive full RSMs and classical linear eddy-viscosity models (LEVM) based on the Boussinesq approximation relating the shear stress component to the mean velocity gradient. Such a hybrid method relying on a LEVM for the viscosity-affected near-wall RANS mode (i.e., the one-equation model proposed by Rodi et al. [37]) was previously developed [13] within the project and will be used here for comparison purposes. However, it is well known that a LEVM is not capable to account for the stress anisotropy. That will be demonstrated below on sample applications. For the implementation into a CFD code, the EARSM can be formally expressed in terms of a non-linear eddy-viscosity model (NLEVM). Its extra computational effort is small, but still requiring solely the solution of one additional transport equation for the turbulent kinetic energy. The non-closed stress tensor in the Reynolds-averaged momentum equations is written as [53]: 2 (ex) ef f (2) ui u j mod = kmod δi j − 2Cμ Si j + ai j 3 (ex)

Here, ai j = f (S2 , Ω 2 , Sn Ω m , f1 , ...) with S2 = Sik Sk j and Ω 2 = Ω ik Ω k j represents an extra tensor which takes the anisotropy of the stresses into account and is computed explicitly based on the mean strain Si j and rotation tensors Ω i j defined as - τ ∂U i ∂U j τ ∂U i ∂U j ν k , Ωij = , τ = max (, 3) Si j = + − ,Cτ 2 ∂xj ∂ xi 2 ∂xj ∂ xi ε ε


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normalized by the turbulent time scale τ suggested by Durbin [23]. As shown in [53] the model obtains the correct near-wall behavior. The parameter Cμ within the relation for the eddy viscosity νt = Cμ · vc · lc , where vc denotes a characteristic velocity and lc stands for a characteristic length scale, is not a constant but dynamically calculated, thus denoted Cμef f . For three-dimensional mean flows it is defined as 1 Cμef f = − f1 (β1 + IIΩ β6 ) , 2

(4)

where the damping function f1 ensures the correct near-wall behavior. The invariants (e.g., IIΩ ) and the various β -coefficients appearing in the EARSM can be found in [28, 53]. However, the EARSM itself is not complete since the length scale lc and the velocity scale vc are not defined. These have to be supplied by an additional scale-determining part. In principle, that can be achieved by a two-equation model such as a classical k-ε or k-ω model, which in the context of RANS is a natural choice since one transport equation is solved for the velocity scale and one for the length scale. However, for the present hybrid methodology a more or less unique modeling strategy in both regions has several advantages. Since in LES the length scale is naturally given by the filter width Δ , a one-equation model with a transport equation for the velocity scale is preferred. Thus the length scale in the URANS region has to be defined by an algebraic relation. The resulting strategy consists of a single transport equation for the modeled turbulent kinetic energy kmod = kRANS = v2c in RANS mode and the subgrid-scale (SGS) turbulent kinetic energy kmod = kSGS in LES [28]. It reads

∂ kmod ∂ kmod +U j =P + D − ε ∂t ∂xj

(5)

where P = −(ui uj )mod ∂ U i /∂ x j represents the production term closed by relation (2). Thus, in the present non-linear model for the RANS zone the production term P is determined on the basis of the consistent Reynolds stress formulation including the anisotropy term which compared to the originally applied linear model [13] improves the production term and subsequently the prediction of kmod . Note that the extension to EARSM is actually not used in the LES mode. The unknown diffusion term D can be closed by a classical gradient-diffusion approach as done for the LES zone and previously also for the RANS zone [13]. However, for EARSM the enhanced representation of the Reynolds stresses can be introduced into D by applying the diffusion model of Daly and Harlow [19] which is preferred here. Finally, the dissipation rate ε needs to be defined. For the LES zone 3/2 the dissipation rate is set to ε = Cd kmod /Δ yielding the one-equation SGS model of Schumann [41]. Nearly identical one-equation SGS models were proposed by Sagaut [38], Yoshizawa and Horiuti [54], and Pope [35] which differ from Schumann’s model purely by the constants. In a recent study [13] the model constants by Schumann (Cd = 1.0,Cμ = 0.048) were found to be slightly favorable and thus are

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Table 1 Overview on modeling assumption in the k-equation for the hybrid versions LEVM and NLEVM; σk = 1.0, cs = 0.11 [25]; the missing relations and constants can be found in [13, 28] URANS

P

D

LEVM ∂U νt ∂∂Ux ji + ∂ xij ∂∂Ux ji ∂ ∂xj

(ν + σνt ) ∂ ∂kmod x k

j

LES

EARSM / NLEVM (ex) ∂ U i −kmod 23 δi j − 2Cμef f Si j + ai j ∂xj

νt

∂U i ∂xj

+

∂U j ∂ xi

∂U i ∂xj

Use extra anisotropy same gradient-diffusion model or

same grad.-diff. model

∂ kmod ∂ kmod ∂ + ∂∂x j cs kmod ε (ui u j )mod ∂ xi ∂xj ν ∂xj Daly & Harlow [19] (v2 )1/2 · kmod /lε ,v

ε

same model (Rodi et al. [37]; y∗ ≤ 60) or

3/2

Cd · kmod /Δ

∗ with lε = Cl · y · 1 − exp (− Ay ε ) Chen & Patel [16]; y∗ ≤ 250 3/2

kmod lε

νt

1/2

2 /ε Cμef f · kmod

(v2 )1/2 · lμ ,v

Cμ · kmod · Δ

used here. For the URANS zone the formulation of ε can be overtaken from the nearwall one-equation model by Rodi et al. [37] as it was done for the hybrid method relying on the LEVM [13]. However, the hybrid method based on the EARSM also allows to take the improved model by Chen and Patel [16] into account which reads 3/2

k ε = mod lε

where

∗ y lε = Cl · y · 1 − exp − Aε

(6)

−3/4

with Aε = 2Cl , Cl = κ Cμ , κ = 0.418, and Cμ = 0.09. Both formulations rely on algebraic relations for the length scales, where Rodi et al.’s modeling assumption has the disadvantage that it is only applicable for y∗ ≤ 60, whereas Chen and Patel’s expression allows to apply the near-wall model up to y∗ ≈ 250. Thus the latter is generally preferred here. Table 1 summarizes the most important features of the present hybrid approach relying on the EARSM but formulated as a NLEVM. For comparison the formulation of the previously developed hybrid method relying on the LEVM [13] is also added.

3.2 Interface Definition and Treatment Based on previous experiences [13, 28] the predefinition of LES and URANS regions is avoided in the present approach and a gradual transition between both methods is assured. The dynamic interface criterion [13] relies on the instantaneous


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distribution of the modeled turbulent kinetic energy kmod and the wall distance y leading to & ≤ Cswitch,y∗ =⇒ URANS mode 1/2 (7) y∗ = kmod · y/ν > Cswitch,y∗ =⇒ LES mode For y∗ ≤ Cswitch,y∗ the method works in URANS mode, otherwise the LES mode is applied. It is worth to mention that the instantaneous interface position strongly varies in space and time depending on the local flow field close to the wall. Thus it relies upon physical quantities accounting for characteristic flow properties and automatically reacts on dynamic flow field variations (see Fig. 1(d)). For example, if kmod increases near the wall, the thickness of the URANS zone will automatically decrease. Contrarily, the URANS zone will become thicker, if kmod decreases in the vicinity of the wall. In principle, the constant Cswitch,y∗ allows the user to vary the average interface position and to study its effect on the results as will be shown below for the periodic hill flow test case. Typically, however, it should be set so that the interface is located in the logarithmic wall layer, e.g., y∗ = 50 − 250. Hence in practice Cswitch,y∗ is fixed by the physics and is not an arbitrary parameter. This interface criterion denoted version A partially provides no sharply delimited LES–URANS regions (see Fig. 1(d)). Therefore, an enhanced version guaranteeing a sharp interface without RANS islands (denoted version Asi) was additionally taken into account, see, e.g., [13, 28]. This special interface treatment which converts all undesired RANS islands back to LES, yields useful information whether the RANS islands found for version A influence the results or not. As will be shown below, the irregular appearance of RANS islands is not at all critical for the quality of the results so that the simpler criterion A is generally favored. Finally, it should be mentioned that presently an interface treatment without synthetic or reconstructed turbulence is preferred in order to assess the method in its simplest form. Nevertheless, an enrichment procedure could be easily added if required. For the test cases investigated below, however, it was not necessary.

4 Numerical Methods 4.1 Solver LESOCC Used for the Hybrid LES–URANS Approach The LES code LESOCC used for the solution of the filtered Navier-Stokes equations at LSTM Erlangen, is a 3D finite-volume solver for arbitrary non-orthogonal blockstructured grids [2, 3]. The spatial discretization of all fluxes is based on a central scheme of second-order accuracy. Time advancement is performed by a predictorcorrector scheme. A low-storage multi-stage Runge-Kutta method is applied for integrating the momentum equations in the predictor step. Within the corrector step the Poisson equation for the pressure correction is solved implicitly by an incomplete LU decomposition method. In order to ensure the coupling of pressure and velocity fields on non-staggered grids, the momentum interpolation technique by Rhie

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and Chow [36] is used. The additional transport equation for kmod is discretized with the same scheme. LESOCC is highly vectorized and additionally parallelized by domain decomposition with explicit message passing based on MPI. Beside the hybrid approach different SGS models for LES are implemented (Smagorinsky model [44], dynamic Smagorinsky model [24] and one-equation SGS models [35, 38, 41, 54]) as well as DES (Spalart-Allmaras model [47] and an improved variant [4]). These features are used to provide reference data for comparison with the new hybrid method.

4.2 Solver ISIS-CFD Used for DES To handle complex geometries efficiently, an unstructured code ISIS-CFD [20–22] was developed at EC Nantes. It is a finite-volume code solving the Navier-Stokes equations for an incompressible fluid using arbitrary control volumes. The convective fluxes are discretized with a second-order central scheme or an upwind-blended scheme. A second-order central scheme is applied to the diffusion fluxes. A secondorder backward differences implicit scheme is employed for time discretization. Pseudo-time stepping is used for steady flow computations. The Rhie and Chow approach [36] is employed to evaluate the mass flux in a non-staggered layout. The velocity-pressure coupling is achieved by a SIMPLE-like algorithm. The pressure equation is solved with a preconditioned conjugate gradient solver. The code was initially designed for RANS simulation. Several statistical models ranging from one-equation Spalart-Allmaras, two-equation linear and non-linear models such as k-ω SST model and an explicit algebraic stress model, to seven-equations differential Reynolds stress model are implemented and extensively validated [20–22]. ISIS-CFD was also validated successfully in the context of DNS, LES and hybrid LES/RANS (mainly DES) upon different classical test cases [18].

5 Test Cases and Details of Numerical Simulations The emphasis here is on separated flows including large-scale structures at which hybrid LES–URANS methods are mainly aiming at. In the course of this project a variety of test cases with different degrees of complexity were considered.

5.1 Flow over Periodic 2D Hills The flow in a channel with streamwise periodic constrictions at Reb = 10, 595 (based on the bulk velocity Ub and the hill height h) [31] presents a geometrically simple but still challenging test case. It allows to perform basic investigations based on a complex flow including pressure-induced separation on a curved surface and subsequent reattachment. That is the main reason why this test case was extensively used within the entire research unit FOR 507. The dimensions of the domain are: Lx = 9.0 h (streamwise), Ly = 3.036 h (wall-normal) and Lz = 4.5h (spanwise) [31]. The flow is assumed to be periodic in the streamwise direction.


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The first task within the French–German cooperation was to provide a kind of reference solution which could be used for evaluating the hybrid simulations. This was fulfilled by a highly wall-resolved LES [5, 11, 12] consisting of a block-structured curvilinear grid with about 12.4 × 106 CVs using the finite-volume code LESOCC (Sect. 4.1). Some details about this solution denoted in the following WR–LES should be provided here. The grid points are clustered in the vicinity of the lower and upper walls as well as in the region of the free shear layer. The resolution of the near-wall region is described by the distribution of non-dimensional y+ values defined as y+ = Δ ycc · uτ /ν where Δ ycc denotes the distance of the cell center from the wall. On the lower wall, y+ is below 0.45 with a mean value of about 0.2 except at the windward side of the hill. Here the largest values of the wall shear stress are observed and y+ reaches its maximum of about 1.2. Regarding the wall-normal resolution the grid satisfies the requirements of a wall-resolved LES [34, 52]. Compared to previous studies especially the number of grid points in the wall-normal direction is increased in WR–LES. Furthermore, the upper wall is also resolved by a DNSlike representation (y+ ≤ 0.95). Thus, the application of a wall function is avoided. Owing to the increased resolution in streamwise and spanwise directions the cell sizes expressed in wall units are below Δ x+ = 20 and Δ z+ = 9 and thus substantially lower than the recommendations for wall-resolved LES [34]. Additionally, a complementary numerical/experimental investigation on the flow over periodic hills was carried out for a wide Reynolds number range of 100 ≤ Re ≤ 10, 595 [11, 12] which studied all physical aspects of this flow in detail. Based on this wall-resolved LES a huge amount of common studies of the entire research group were carried out (see, e.g., [6, 7, 39, 40]) aiming at detailed investigations of all issues and concerns related to LES, DES and other hybrid LES–RANS approaches. Furthermore, the test case was taken into account to develop and assess the present hybrid method, see, e.g., [8–10, 14, 15, 18, 28, 29]. In this report, hybrid simulations performed on two different grids are considered; the standard grid A consists of 160 × 100 × 60 ≈ 1.0 × 106 CVs and grid B consists of 80 × 100 × 30 ≈ 0.25 × 106 CVs in streamwise, wall-normal (Δ ycrest /h = 5. × 10−3 , 1st CV height) and spanwise directions. Grid B is extracted from grid A by retaining every second grid point in streamwise and spanwise directions. The ratios of grid points in relation to the reference case WR–LES are about 13 and 52, respectively.

5.2 Flow through a 3D Diffuser The turbulent flow through a 3D diffuser is a quite new test case which was recently proposed for the 13th ERCOFTAC Workshop on Refined Turbulence Modeling, Sept. 25–26, 2008, in Graz [1]. It is based on the experimental measurements by Cherry et al. [17] who investigated two different configurations by the magnetic resonance velocimetry. Here diffuser # 1 is considered. In contrast to most of the diffuser flows studied before which were nominally 2D, a geometrically 3D setup was chosen which additionally avoided any symmetries to eliminate the problem of swaping separated flow regions between the two diverging walls.

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A long development channel with a rectangular cross-section of height h = 1 cm and an aspect ratio of 1:3.33 upstream of the diffuser guarantees a fully developed channel flow and thus clearly defined boundary conditions at the inlet. The diffuser with a length of 15 cm is attached directly to the exit of the development channel and possesses a 4 cm square outlet, giving an area expansion ratio of 4.8. All walls were straight, except one side and the top wall which expanded at an angle of 2.56◦ and 11.3◦, respectively. The corners between the straight and inclined walls were smoothed with a fillet of radius 6 cm. The Reynolds number based on the height of the inlet channel and the bulk velocity was approximately 10,000. Additionally, an outlet transition device was mounted behind the diffuser, see [17]. As expected, a 3D boundary layer separation is observed in the diffuser. The strong adverse pressure gradient leads to a rapid boundary layer growth and finally to flow separation starting at the corner between the two diverging walls and developing across the top expanding wall. The associated unsteadiness plays an important role yielding a turbulent flow with highly non-equilibrium characteristics which renders a challenge for turbulence modeling. This test case was considered to further evaluate the performance of the hybrid LES–RANS method proposed. A block-structured grid with about 3.25 million CVs was generated which covers a computational domain including an inlet duct (−5 ≤ x/h ≤ 0, aspect ratio 1:3.33), the diffuser #1 itself (0 ≤ x/h ≤ 15) and an outlet duct with a 4 cm square cross-section (15 ≤ x/h ≤ 27.5). In order to avoid a recirculation at the outlet a contracting duct (27.5 ≤ x/h ≤ 37.5) with a 3 cm square cross-section at the tail was attached. The near-wall resolutions at the diffuser inlet and outlet are (Δ y; Δ z)min,CV /h = 2.33 × 10−3 and (Δ y; Δ z)min,CV /h = 5. × 10−3, respectively. The corresponding estimated y+ values of the wall-nearest grid point is about 0.5. A precursor simulation of a 1:3.33 duct flow with periodic boundary conditions was used to provide reliable inflow data at the inlet located x/h = −5. No-slip boundary conditions were imposed at all walls and a convective outflow boundary condition was applied at the outlet. The hybrid approach relying on the EARSM with the diffusion model by Daly and Harlow [19] and the dissipation model by Chen & Patel [16] were considered (see Table 1). Owing to the low Reynolds number of the diffuser flow, the interface criterion was set to Cswitch,y∗ = 60 using version A.

5.3 Flow over Axisymmetric 3D Hill The axisymmetric 3D hill was a benchmark test case proposed by the 11th and 12th ERCOFTAC Workshop on Refined Turbulence Modeling to understand the formation and structure of vortical 3D separations of a turbulent boundary layer over axisymmetric hills or bumps. The measurements were conducted by Simpson et al. [42] at Virginia Polytechnic Institute and State University. The geometry is an axisymmetric hill with a height h = 78 mm and a radius of the circular base a = 2h. The shape is defined by Bessel functions. The Reynolds number, based on the height h and the nominal speed Ure f = 27.5 m/s, is Reh = 130,000. The thickness of the boundary layer at 2 hill-heights upstream of the hill, is approximatively 0.5h. The mean flow is closely symmetric about the centerline. Complex vortical separations


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occur on the leeside and merge into two large streamwise vortices downstream. The flow along the streamwise centerline at x/h = 0.39 is a downwashing reattachment flow and only one mean vortex exists on each side of the centerline. Unfortunately, no upstream conditions are available, so that the inlet flow cannot be specified directly. Instead, profiles of velocity and Reynolds stresses have been measured in the duct with the hill removed at the location corresponding to the hill centerline. To generate inlet conditions, pre-cursor calibrations were performed over a length of 20 hills heights, and the reference hill-top conditions, as returned by the computation, were determined by matching the solution to the measured duct-flow profiles. Two block-structured grids have been generated: a coarse mesh with (81 × 41 × 65) CVs and a fine mesh with (161 × 81 ×130) CVs. For the DES predictions, the RANS region is represented by the 11 to 13 first CVs in wall-normal direction for the coarse grid and 3 to 5 CVs for the fine mesh. Additionally, a RANS simulation has been carried out with the EARSM turbulence model on the coarse grid.

5.4 Flow around Willy Car Model The Willy body is a simplified car model which is a square-back model. It was used to understand the physics of separated flows around bluff bodies with application to side gust wind effects on automobiles. A generic model with no-sharp corners on the fore body and a square base was more convenient for analyzing separations limited to its leeward side and base. Thus, a yaw angle dependency can be studied. Experiments were carried out in the semi-open test section at the Conservatoire National des Arts et Métiers (CNAM) [26]. The length of the model is L = 675 mm whereas the maximum height and width are 192 mm and 240 mm, respectively. The Reynolds number based on the length of the body and the bulk velocity (20 m/s) was 900,000. The value of the yaw angle β is positive when the right side of the car model is windward. Tomographies of total pressure are measured using a Kiel probe. Wall pressures are measured along a generatrix, noted (A), and on the base of the model. The forces measured on the Willy model as a function of the yawing angle β show the same typical characteristics as can be observed on real cars [27]: a quasi-linear evolution of the side force and a decrease of the yawing moment slope at high yaw angles. At the lower leeward side, a vortex takes birth in an open separation and grows as the yaw angle goes up. This vortex is close to the body at the small yaw angles and detaches at the yaw angle between 15◦ and 20◦ . For larger yaw angles (β ≥ 20◦ ), a vortex exists on the upper leeward side of the model and it grows in size with increasing yaw angle and spreads out toward the upstream model. The body is placed in the channel with a cross-section of 3 L× 1.6 L. The front face of the model is located at a distance of 4 L from the channel inlet and the downstream length between the base of the body and the channel outlet is 5.4 L. Numerical accuracy is established by performing three DES predictions at the yaw angle β = 10◦ on different computational grids containing 1.6, 6.6 and

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10.8 million nodes. For all grids, the wall-normal resolution is y+ = 0.5. The time step is 3 × 10−4 s. For each time step, 20 non-linear iterations are carried out to reduce the non-linear residuals by two orders of magnitude.

6 Results of the Hybrid Approach for the 2D Hill Flow 6.1 Interface and Anisotropy First, the functioning of the hybrid method will be demonstrated based on two examples. Figs. 1(a)–1(c) show the averaged (time plus spanwise direction) distributions of the modeled, resolved and total turbulent kinetic energy for the periodic hill at Reb = 10, 595. As required, the modeled contribution is high in the near-wall RANS region for both the upper and the lower walls. On the other hand, in the region far away from the walls, where the method works in LES mode, the contribution of the subgrid scales and thus kmod is small1 . Large-scale flow phenomena such as the Kelvin-Helmholtz instability in the free shear layer are aspired to be handled in LES mode. As visible in Fig. 1(b) this goal is achieved which is evident from the high level of resolved turbulent kinetic energy kres in this region. An additional proof is provided by Fig. 1(d) which depicts a snapshot of the LES–RANS interface located in the logarithmic layer at y∗ = Cswitch,y∗ = 200 at an arbitrarily chosen temporal and spatial instant. The black areas in the vicinity of the walls define the RANS

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region, whereas the white space represents the LES region. Additionally, the grey line denotes the average interface position. Two observations can be made. First, the interface position strongly varies in space and time which is fully intended in order to automatically react and adapt to the local flow conditions. Second, no sharply delimited LES–RANS regions are obtained and thus irregularly RANS islands appear. The reason for this behavior is discussed in detail in [13, 28]. In order to prove that the irregular appearance of these RANS islands is of no relevance for the quality of the results, the auxiliary interface criterion version Asi was introduced. The outcome is presented in Sect. 6.4. In order to show that the anisotropy is indeed realized by the EARSM, this issue is depicted in Fig. 2 for a channel flow simulation at Reτ = 2003. Based on the total Reynolds stresses (resolved + modeled) the scalar invariants II = ai j a ji and III = ai j aik a jk were determined and plotted in the anisotropy-invariant map by Lumley and Newman [30]. From that diagram it is clearly visible that based on the LEVM the state of two-component turbulence is not reached close to the wall. Contrarily, the NLEVM is able to reach the state of two-component turbulence in the vicinity of the wall. Therefore, a clear improvement resulting from the application of the EARSM in the near-wall RANS region is obvious. As shown in [28] the log-law mismatch is also improved by the EARSM compared to the LEVM. For the hill flow case, a variety of investigations was carried out comparing the linear and the non-linear model (EARSM) and investigating several model variants for the production, dissipation and diffusion terms in the k-equation of the EARSM. Furthermore, a series of different grid resolutions was evaluated and the influence of the interface position and interface criterion were studied. For the sake of brevity in the following, four different issues are addressed and the rest can be found in [8–10, 14, 15, 18, 28, 29].

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6.2 Comparison of the Hybrid Technique Based on LEVM vs. EARSM The evaluation relies on grid A , the interface criterion A with Cswitch,y∗ = 60, and the formulation of ε by Rodi et al. [37]. Fig. 3 depicts results of the linear (denoted L) and the non-linear model (denoted nL). Regarding the profiles of the mean streamwise velocity U (Fig. 3(a)) a better agreement with the reference data WR–LES is visible for nL, especially in the free shear layer and in the vicinity of the upper wall. Otherwise the results of both L and nL are in good accordance with WR– LES. Concerning the normal Reynolds stress u u (not shown here) a very good agreement between nL and WR–LES is found. Deviations between L and WR–LES are visible close to the upper wall and in the shear layer. The differences between L and nL are even more obvious for the second normal Reynolds stress v v (Fig. 3(b)), where L yields strange peaks in the vicinity of the upper wall. In the recirculation region both hybrid variants deliver an underprediction of v v with respect to WR–LES. The shear stress component u v (Fig. 3(c)) predicted by nL shows a very good agreement with WR–LES, whereas large deviations are observed for L in the shear layer. Finally, Fig. 3(d) depicts the wall shear stress distribution τw along the lower wall. For L a strong non-physical peak is found at the hill crest. This behavior was also noticed in pure RANS simulations and is caused by inaccurate predictions of the modeled Reynolds stresses. This disagreement with WR–LES completely disappears for nL. Moreover, nL offers a better prediction of τw along the

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entire wall. The separation and reattachment regions are reasonably predicted. Thus, the overall accuracy of the hybrid method is strongly enhanced by nL (EARSM) compared to L.

6.3 Influence of the Interface Position The evaluation relies on grid A and the non-linear model based on the diffusion model by Daly and Harlow [19] and the dissipation formulation by Chen and Patel [16]. The interface criterion A is applied and the position is varied between Cswitch,y∗ = 60, 160, and 200. Fig. 4(a) depicts the mean streamwise velocity U. Overall a good agreement between nL and WR–LES is observed independent of the interface position. That in general also applies for the normal Reynolds stresses (not shown here), where the results slightly improve for the larger Cswitch,y∗ values. The satisfactory agreement for all interface positions is also visible for the Reynolds shear stress component depicted in Fig. 4(b). The same trend is found for the wall shear stress distribution (not shown here). Thus as desired, the results do not strongly depend on the interface position as long as the interface is located in the log-layer.

6.4 Influence of the Interface Criterion As discussed above, the interface criterion A sometimes leads to RANS islands within the LES region. These islands appear irregularly and only survive a certain short time interval, see, e.g., Fig. 1(d). Therefore, the question arises whether these islands are really critical or not. For that purpose the interface criterion Asi guaranteeing a sharp interface without RANS islands was implemented [13, 28]. The evaluation carried out on grid A using nL and the interface position Cswitch,y∗ = 200 yields nearly no differences between the results using the interface criteria A and Asi. That is depicted in Fig. 5 for the mean streamwise velocity U, the Reynolds shear stress u v , the total turbulent kinetic energy, and the wall shear stress τw .

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Hence it can be concluded that the RANS islands do not disturb the overall solution significantly. Therefore, it is not necessary to get rid of them by the time-consuming sorting algorithm applied within Asi.

6.5 Influence of the Grid Resolution Finally, simulations based on nL (interface criterion A, Cswitch,y∗ = 200, D by Daly and Harlow [19], ε by Chen and Patel [16]) are carried out on grid A and B. The comparison with WR–LES shows that the results are nearly independent regarding the streamwise and spanwise resolution, which both differ between grid A and B by a factor of 2. The fine-grid results are marginally better than the coarse-grid predictions. That is obvious from Fig. 6 which depicts the mean streamwise velocity U, the Reynolds shear stress u v , the total turbulent kinetic energy, and the wall shear stress τw for both grids. Overall a very good agreement with WR–LES (e.g. for the separation and reattachment length, see Fig. 6(d)) is found also on the coarse grid using only a quarter of a million grid points. Compared to the wall-resolved reference simulation the ratio of required grid points is larger than 1:50. Taking additionally the larger time step size into account, a tremendous saving of CPU time is achieved by the hybrid approach.

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7 Results of the Hybrid Approach for the 3D Diffuser Flow Fig. 7 depicts the time-averaged streamwise velocity at five cross-sections downstream of the inlet of the diffuser located at x/h = 0. The bold line indicates zerostreamwise-velocity and thus incloses the recirculation region. The upper figures show the experimental measurements by Cherry et al. [17] whereas the lower ones are the predictions based on the hybrid method using the EARSM. As visible from both rows the recirculation starts at the upper-right corner, which is the corner between the two diverging walls. At x/h = 5 (see Figs. 7(b) and 7(g)) the separation bubbles remain in the corner. At the next cross-section shown at x/h = 8 it is obvious that the recirculation regions has started to spread across the top of the diffuser. Further downstream at x/h = 12 and 15 a massive separation region is visible covering the entire top wall of the diffuser. Overall an excellent agreement between the hybrid LES–RANS prediction and the measurements is found. This circumstance is not self-evident for all simulations carried out at the ERCOFTAC Workshop [1]. All RANS predictions based on eddy-viscosity models were found to predict a qualitatively wrong results showing a separation region at the side wall and not at the top wall as observed in the experiment and the hybrid simulation. RANS predictions based on RSM partially do a better job but still do not reach a similar level of agreement as the hybrid simulation.

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Fig. 8 displays streamwise turbulent velocity fluctuation data at the same crosssections as before. Except in the first cross-section at x/h = 2 a satisfactory agreement between measurement and prediction is found. For the front part of the diffuser a high turbulence intensity is observed in the vicinity of the upper-right corner, i.e., in the shear layer bounding the separation bubble [17]. Further downstream the region of high turbulence starts to spread into the horizontal region parallel to the top wall which is closely related to the spreading of the recirculation region observed. A quantitative comparison between the measurements [17] and the predictions is shown in Fig. 9 based on profiles of the mean streamwise velocity and RMS values of the streamwise velocity fluctuations at z/B = 0.25 (B = 3.33 cm). An

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excellent agreement is found for the mean velocity including the recirculation and recovery region. Some deviations are visible for the RMS values especially close to the straight wall. Overall, however, the prediction and the measurement are still in good agreement strengthening the reliability of the hybrid method proposed.

8 Results of the SA-DES for the 3D Hill Flow The result of the matching process for the inlet conditions (see Sect. 5.3) is indicated in Fig. 10 for two turbulence models. The conditions returned by the solution 4 hill heights upstream of the matching location obtained with the EARSM turbulence model were taken as inlet conditions for the hill flow prediction. Fig. 11 compares the predicted skin-friction distributions on the hill surface with a topological map extracted from the oil-film experiments. As is evident, there are major differences between both model predictions and the experimental results. The experiment shows three distinct vortex pairs while the RANS model (EARSM) predicts a single pair. The SA-DES gives an intermediate result. A considerable difference between both simulations is obvious concerning the position of the vortex. With the RANS model, these vortices extend on the base of the hill and do not permit fluid to be drawn into the base from above to feed the wake. With SA-DES, the

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vortices are located higher on the hill and are more separated from the symmetry plane than those obtained with the RANS simulation. So, this position induces to the lateral spread of the wake.

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To examine the wake statistics in detail, streamwise and spanwise velocity profiles at various spanwise locations on the plane x/h = 3.63 are shown in Fig. 12. The differences between the RANS predictions and the experiments reflect the much more intense reverse flow predicted in the leeward central portion on the hill’s surface. The DES predictions on both, the coarse and the fine grid, yield a better agreement. However, various differences exist between both DES simulations. With the fine grid, the agreement is less good. This is probably due to the RANS region that is only limited to 3 to 5 cells while, with the coarse mesh, the RANS region is described by 11 to 13 cells. Consistently with the differences in the mean flow, the computed Reynolds stresses, depicted in Fig. 13, decrease rapidly in the spanwise direction. Downstream of the hill, in the wake region, both simulations present some similarities in the region of the symmetry plane where a vortex exists. In addition, the DES simulation predicts another vortex toward z/h ≈ 1.5 (see Fig. 14).

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9 Results of SST-DES for the Willy Car Model A comparison of the pressure coefficient along a generatrix of the model for the three different grid resolutions using SST-DES can be seen in Fig. 15. Although the results are nearly identical for all meshes, small discrepancies exist in the vicinity of the model base. Table 2 summarizes the evolution of the drag force versus the mesh resolution. One notices that the drag coefficient is similar for the medium and fine meshes. Consequently, for the other yaw angles, the medium grid is used. Figs. 16, 17, 18, and 19 present a comparison of the total pressure coefficient between the EARSM simulation results, the results obtained by the SST-DES prediction and the experiment. For the EARSM simulations the grid consists of approximatively 20 million nodes. The boundary conditions are the same as those used for the DES. For the yaw angle β = 0◦ , the flow is fully attached to the body. With the EARSM turbulence model, the wake of the feet is more intense. For the yaw angle β = 10◦, the boundary layer does not separate on the leeward side of the body. One just observes a thickening of the boundary layer on the leeward side. For the yaw angle β = 20◦ , a vortex is separated from the body and is convected in the direction of the free stream. This vortex is the confluence of the wake of the front feet and the central cylinder. As a result, on the leeward side of the model a longitudinal vortex can be seen clearly. For the yaw angle β = 30◦, this vortex remains attached to the wall. For all yaw angles, the wake of the feet is more intense for the EARSM

1.0

Table 2 Drag force versus the mesh

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turbulence model than for the SST-DES. Overall, the SST-DES predictions show a better agreement with the experiment than the RANS predictions. The evolution of the pressure coefficient along a generatrix is depicted in Fig. 20 for different yaw angles β . The results show that the wall pressure is correctly simulated on the windward and leeward sides up to β = 20◦ . For these yaw angles, the simulations based on the EARSM turbulence model and the SST-DES produce similar results. For the yaw angle β = 30◦ , the windward side is well predicted whereas for the leeward side, some discrepancies appear for both simulations, particularly for the pressure gradient. The flow separation appears earlier in the EARSM simulation than in the SST-DES prediction which is in better agreement with the experiments. The difference between the simulations are more clearly visible in Fig. 21 which presents the pressure coefficient on the base of the model for a horizontal plane Z = 50 mm. One can see that, for all yaw angles, the EARSM simulation predicts a pressure distribution which is very uniform except for the yaw angle β = 30◦ . For this yaw angle, the pressure value reaches a maximum around Y = 0 mm. For all yaw angles, the simulations based on the EARSM are not in good agreement with the experiments. Applying the SST-DES model, a maximum pressure close to the side of the model is predicted which agrees with the experimental data. Moreover, in the SST-DES

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prediction this maximum increases with increasing yaw angle, and the same trend is found in the experiment. The drag coefficient, the side force coefficient, and the yawing moment coefficient are given in Fig. 22. The drag increases up to the yaw angle β = 20◦ . Above this angle, the drag decreases. This behavior is typical for a square-back model [27]. For small yaw angles up to β = 20◦ , the numerical simulation based on the EARSM turbulence model underpredicts the drag coefficient whereas the computation based on the SST-DES slightly overpredicts this force coefficient. For larger yaw angles, both simulations overpredict the drag coefficient. The difference in the drag between both simulations is due to the position and the size of the vortex on the leeward side of the model. However, the numerical results obtained using the SST-DES model are in better agreement with the experimental data than the RANS predictions. The side force shows a linear evolution. Predictions with the SST-DES yield a clear improvement compared to RANS. The yawing moment coefficient rises linearly with the yaw angle up to β = 25◦ . At large yaw angles, the slope of the yawing moment decreases. One can see that the yawing moment obtained at yaw angle β = 30◦ based on the EARSM turbulence model decreases compared to the previous angle (β = 25◦ ) whereas with the SST-DES, solely the slope of the curve decreases as in the experiments.


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Fig. 22 Force coefficients and yawing moment coefficient versus the yaw angle β

10 Conclusions and Outlook A hybrid LES–RANS method relying on an EARSM for the viscosity-affected nearwall region has been developed. The approach has several advantages. The model solely solves one additional transport equation for the modeled turbulent kinetic

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energy supplying the scale-determining part. Nevertheless, the high anisotropy in the vicinity of the wall is accounted for. The computational costs increase only marginally and thus are not comparable to the effort required for a differential RSM. Furthermore, the method possesses a high numerical stability which is important for practical applications. Another key feature is the dynamically predicted interface location avoiding the predefinition by the user and adjusting itself to the flow condition. Beside the classical channel flow case the hybrid method has been extensively investigated based on two test cases for separated flows. A third one, i.e., the flow around the Ahmed car model, is in progress. The results confirm that the new methodology works reliably and yields highly satisfactory results even on relatively coarse grids. Furthermore, DES predictions are compared against RANS results obtained for steady-state flows. For this purpose, challenging test cases such as the 3D hill flow and the flow around the Willy car model at different yaw angles are taken into considerations. The results are highly encouraging in that for mean quantities DES predictions are clearly superior to RANS. Acknowledgements. The project was financially supported by the Deutsche Forschungsgemeinschaft and CNRS within the French-German programme ’LES for Complex Flows’(FOR 507), i.e. BR 1847/8. The authors gratefully acknowledge many fruitful discussions and an active exchange with members of this research programme. The computations were partially carried out on the German Federal Top-Level Computers Hitachi SR 8000-F1 and SGI ALTIX at LRZ Munich. All computations performed by the French partner were carried out on the IDRIS/CINES supercomputers through research projects 1308 and 80129. All kinds of support are gratefully acknowledged.

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29. Jaffrézic, B., Breuer, M., Chikhaoui, O., Deng, G., Visonneau, M.: Towards LES-RANSCoupling for Complex Flows with Separation. In: Cancès, E., Gerbeau, J.F. (eds.) ESAIM Proc., CEMRACS 2005, CAA and CFD in Turbulent Flows, Marseille, France, July 18– August 26, 2005, vol. 16, pp. 89–113 (2007) 30. Lumley, J.L., Newman, G.: The Return to Isotropy of Homogeneous Turbulence. J. Fluid Mech. 82, 161–178 (1977) 31. Mellen, C.P., Fröhlich, J., Rodi, W.: Large-Eddy Simulation of the Flow over Periodic Hills. In: Deville, M., Owens, R. (eds.) 16th IMACS World Cong., Lausanne, Switzerland (2000) 32. Menter, F.R., Kuntz, M.: Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow behind Vehicles. In: McCallen, R., Browand, F., Ross, J. (eds.) Symp. on The aerodynamics of heavy vehicles: trucks, buses and trains, Monterey, USA, December 2–6, 2002. Springer, Heidelberg (2004) 33. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D., Spalart, P.R.: An Approach to Wall Modeling in Large-Eddy Simulations. Phys. Fluids 12(7), 1629–1632 (2000) 34. Piomelli, U., Chasnov, J.R.: Large-Eddy Simulations: Theory and Applications. In: Hallbäck, M., Henningson, D.S., Johansson, A.V., Alfredson, P.H. (eds.) Turbulence and Transition Modeling, pp. 269–331. Kluwer, Dordrecht (1996) 35. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000) 36. Rhie, C.M., Chow, W.L.: Numerical Study of the Turbulent Flow past an Airfoil with TrailingEdge Separation. AIAA Journal 21, 1525–1532 (1983) 37. Rodi, W., Mansour, N.N., Michelassi, V.: One-Equation Near-Wall Turbulence Modeling with the Aid of Direct Simulation Data. J. Fluids Eng. 115, 196–205 (1993) 38. Sagaut, P.: Large-Eddy Simulation for Incompressible Flows – An Introduction. Springer, Heidelberg (2001) ˇ c, S., Jakirlić, S., Breuer, M., Jaffrézic, B., Deng, G., Chikhaoui, O., Fröhlich, J., von 39. Sari´ Terzi, D., Manhart, M., Peller, N.: Evaluation of Detached-Eddy Simulations for Predicting the Flow Over Periodic Hills. In: Cancès, E., Gerbeau, J.F. (eds.) ESAIM Proc., CEMRACS 2005, CAA and CFD in Turbulent Flows, Marseille, France, July 18–August 26, 2005, vol. 16, pp. 133–145 (2007) ˇ c, S., Jakirlić, S., Breuer, M., Jaffrézic, B., Deng, G., Chikhaoui, O., Fröhlich, J., von Terzi, 40. Sari´ D., Manhart, M., Peller, N.: Collaborative Assessment of Eddy-Resolving Strategies for the Flow Over Periodic Hills. AIAA Journal (2009) (subm.) 41. Schumann, U.: Subgrid-Scale Model for Finite-Difference Simulations of Turb. Flows in Plane Channels and Annuli. J. Comput. Phys. 18, 376–404 (1975) 42. Simpson, R.L., Long, C.H., Byun, G.: Study of Vortical Separation from an Axisymmetric Hill. Int. J. Heat Fluid Flow 23, 582–591 (2002) 43. Shur, M., Spalart, P.R., Strelets, M., Travin, A.: Detached-Eddy Simulation of an Airfoil at High Angle of Attack. In: Rodi, W., Laurence, D. (eds.) Fourth Int. Symp. on Engineering Turbulence Modeling and Measurements, Corsica, France. Engineering Turbulence Modeling and Experiments, vol. 4, pp. 669–678. Elsevier, Amsterdam (1999) 44. Smagorinsky, J.: General Circulation Experiments with the Primitive Equations, I, The Basic Experiment. Mon. Weather Rev. 91, 99–165 (1963) 45. Spalart, P.R., Allmaras, S.R.: A One-Equation Turbulence Model for Aerodynamic Flows. La Recherche Aérospatiale 1, 5–21 (1994) 46. Spalart, P.R., Jou, W.-H., Strelets, M., Allmaras, S.R.: Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach. In: Liu, C., Liu, Z. (eds.) Advances in DNS/LES, 1st AFOSR Int. Conf. on DNS/LES. Greyden Press, Columbus (1997) 47. Spalart, P.R.: Strategies for Turbulence Modeling and Simulations. Int. J. Heat Fluid Flow 21, 252–263 (2000) 48. Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M., Travin, A.: A New Version of Detached-Eddy Simulation Resistant to Ambiguous Grid Densities. Theor. Comput. Fluid Dyn. 20, 181–195 (2006)


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Segregated LES/RANS Coupling Conditions for the Simulation of Complex Turbulent Flows Dominic von Terzi, Ivan Mary, and Jochen Fröhlich

Abstract. The paper presents hybrid LES/RANS computations of turbulent flows with a segregated approach. This approach employs strict steady RANS and strict LES in pre-defined regions of the computational domain coupling the solution between them by specifically designed interfaces. The latter can be obtained by enhancement of the interfaces needed for domain decomposition in any blockstructured code. The paper covers inflow and outflow conditions of the LES subdomain when linked to a RANS domain as well as tangential coupling. Compressible as well as incompressible simulations are reported.

1 Introduction In recent years Large Eddy Simulation (LES) has been applied more and more for the simulation of complex flows, since computer power has increased and made a certain range of non-trivial problems accessible for simulation with this approach [22]. LES has clear superiority over Reynolds-Averaged Navier–Stokes (RANS) methods for the simulation of complex flows, especially when large-scale structures dominate the turbulent transport, and when dynamic forces and noise generation must be evaluated [7]. However, LES is still too costly to be applied routinely in flows of practical interest. This is due to the high resolution requirements, particularly in Dominic von Terzi Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Kaiserstraße 12, 76128 Karlsruhe, Germany [email protected] Ivan Mary ONERA, Châtillon, 29 av. Division Leclerc, BP 72, 92322, Châtillon Cedex, France [email protected] Jochen Fröhlich Institut of Fluid Mechanics, Technical University of Dresden, George-Bähr-Straße 3c, 01062 Dresden, Germany [email protected] C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 231–252. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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D. von Terzi, I. Mary, and J. Fröhlich

Fig. 1 Classification of eddy-resolving strategies for the simulation of turbulent flows

wall-bounded flows, and the need to determine time-averages by executing a very large number of time steps for flows with no homogeneous directions [11]. RANS methods, on the other hand, are able to determine the mean flow at a cost typically about two orders of magnitude lower than required for LES [5]. In many cases, especially for attached flows, jets and wakes, the resulting accuracy can be sufficient for engineering purposes. Hence, the construction of hybrid methods combining LES and RANS approaches in the same simulation has become an important area of research in recent years. The underlying philosophy is ”to perform LES only where it is needed while using RANS in regions where it is reliable and efficient” [10]. In the following, all approaches aiming to resolve in some part of the computational domain a sizable amount of turbulent kinetic energy (TKE) instead of modelling it completely are termed ”eddy-resolving strategies.” This comprises Direct Numerical Simulation (DNS), LES, all hybrid LES/RANS models and Unsteady RANS (URANS). During the last years, numerous hybrid LES/RANS models were proposed based on different concepts. In a recent article [10], the first two authors reviewed the most important ones and developed a classification which is represented in condensed form in Fig. 1. DNS employs a grid fine enough to resolve even the smallest scales of velocity fluctuations so that no model apart from boundary conditions is needed. The term ”Second Generation URANS” (2G-URANS) was coined to designate URANS models particularly conceived for unsteady simulations such as SAS [16]. Unified models compute a continuous velocity field by blending or switching of LES and RANS models or model quantities in the momentum equation and/or in auxiliary transport equations for turbulence quantities. During the course of the project ”LES/RANS coupling for the simulation of complex flows,” on which is reported here, the present authors also contributed to the assessment of the capabilities and limitations of unified and 2G-URANS models [2–4, 31, 32]. In addition, the problems arising for LES with sudden or strong changes in resolution [8, 24] were studied. Nevertheless, the focus of the work remained at the development of segregated modelling strategies. The outcome of this work is reported in the following.

2 Segregated LES/RANS Coupling A segregated model uses LES and (steady) RANS in pre-defined separated subdomains which are connected to each other by specialized coupling conditions at

Segregated LES/RANS Coupling Conditions for the Simulation

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their interfaces. These interfaces are required to account for the presence and absence of turbulent fluctuations on the LES and RANS sides of the modelling boundary, respectively. The advantage of this approach is that each of the methods, LES as well as RANS, is used in its own regime of validity. In particular, no grey zone is present in such simulations contrary to unified models. In such a grey zone, the continuous velocity field experiences a shift between both approaches. Here, the grid is neither coarse enough to turn the model into a true RANS model since still substantial TKE is resolved, nor is it fine enough to warrant a suitable resolution for LES. Furthermore, turbulent fluctuations are usually not created at a sufficient rate in this grey zone to provide resolved turbulence to the LES domain. Segregated models avoid this problem by allowing for a discontinuity of the flow quantities at the interface. As a consequence, the jump has to be specified by explicit coupling conditions that connect the LES and RANS domains. In the following, we name ΩLES the LES domain, ΩRANS the RANS domain and Ω the union of both, the total computaitonal domain. Regardless which approach to hybrid LES/RANS coupling is employed, unified, 2G-URANS, or segregated, one has to distinguish between the situations depicted in Fig. 2, here sketched for a segregated approach. Three different situations can be distinguished by the direction of the mean flow with respect to the position of the interface: a RANS zone upstream of the LES (”inflow-type”), RANS downstream of LES (”outflow-type”) and a tangential alignment of the regions with respect to the mean flow. Of course, in a particular hybrid LES/RANS simulation not necessarily all of these situations need to appear. For segregated LES/RANS hybrid methods, at each interface, coupling conditions for all independent variables of the simulation need to be prescribed, e.g. the (resolved) velocities, pressure or density, and turbulence quantities of any auxiliary transport equation. For all types of interfaces, mean quantities are provided by the RANS calculation and can be coupled directly to the explicitly averaged LES velocities. Such an averaging procedure simultaneous to the computation of the unsteady LES solution is routinely applied for diverse reasons anyway. For the LES-side, the mean values are readily available from the RANS computation, but fluctuations need to be prescribed. The key to a successful coupling is that turbulent fluctuations must be removed or created in a way consistent to the physical processes occurring

Fig. 2 Possible types of interfaces between LES and RANS; the arrow indicates the principal direction of the mean flow

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Fig. 3 Implementation of coupling conditions for the streamwise velocity in a finite volume flow solver using ghost cells at the block boundary: RANS on coarser grid provides mean flow data for LES ghost cells on a finer grid, fluctuations if needed are generated according to the type of interface; explicitly averaged LES data are assigned to the RANS ghost cells; the thick line indicates the block boundary

at the interface without violating the (different) turbulence modelling assumptions on both sides of this boundary. The coupling can be implemented based on interfaces common to flows solvers with block structure, i.e. the already required transfer of information between the blocks of the grid can be exploited to ”add on” the coupling condition for cases where the modelling strategy changes as well. This is illustrated in Fig. 3 for the concept of ghost cells used in the finite volume solvers employed in the present study and the (resolved) streamwise velocity as the flow quantity to be coupled. A RANS model is typically employed on a coarser grid in the block below the interface, marked by a horizontal line in the figure. Its ghost cells, shown as the large black dashed squares above this line are filled by an appropriate explicit averaging process of the LES data that was obtained at the corresponding location. The LES, performed on a finer grid in the block above the interface, is to be exposed to unsteady data in its ghost cells, the small cells in the line indicated by the index j − 1 below the interface. This data is constructed by adding suitable fluctuations to the mean value provided by the RANS domain. How these fluctuations are generated depends on the type of the interface and the sophistication level of the segregated model and will be discussed in more detail below. In the remainder of this paper, selected coupling conditions developed by the present authors are presented that have been successful or promising. They were constructed based on physical arguments and were thoroughly tested and compared to alternative methods, such that the authors feel confident enough to propose these techniques for the use in segregated LES/RANS coupling for the simulation of complex flows. As a bare minimum (in case of the tangential coupling), they constitute a


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very promising starting point for further enhancement. The developments reported here benefitted from the mutually different backgrounds of the project partners and were undertaken in a complementary way so as to cover the entire picture: all types of interfaces, inflow, outflow and tangential, as well as compressible and incompressible flows. Finally, a simulation with segregated LES/RANS coupling of the separated flow in an asymmetric diffuser is presented as an example for a typical application.

3 Outflow Coupling In the course of the present study, the outflow-type situation was addressed first, since it served as an intermediate step toward tangential coupling and because it seemed easy at first sight. While this is true for some quantities and several canonical flow configurations, the authors experienced that for incompressible solvers and only slightly more challenging flows, it is definitely not trivial. This led to the coupling condition described below. The underlying idea for the outflow-type interface condition proposed in [27, 29] is that, for any RANS zone downstream of an LES zone, the primary task of a hybrid LES/RANS coupling is to propagate mean flow information upstream. At the same time, for flows with stationary statistics, the LES should provide only mean flow data to the RANS domain. Since the LES delivers unsteady data, the interface has to allow for the fluctuations to leave the LES domain without reflections. To this end, the proposed velocity interface condition couples the explicitly Reynolds-averaged velocity at the LES outflow directly to the RANS inflow boundary, whereas fluctuations are convected out of the LES domain using a one-dimensional, linear convection equation with a given convection speed Uc . The inherent assumption for such a coupling is that the downstream transport of fluctuations across the interface is dominated by convection. For this to be true, Uc needs to be directed towards the RANS domain and its magnitude should be considerably larger than the amplitude of the fluctuations, hence restricting this method to outflow-type interfaces. In addition, laminar and modeled turbulent diffusion across the interface must be negligible which, however, is uncritical for turbulent flows and adequately resolved LES. This proposed ”convective condition” for the velocity coupling is general and contains the enrichment strategy of Quéméré and Sagaut [23] as the limiting case of an infinite convection speed of the fluctuations [29]. Contrary to enrichment, no constant needs to be calibrated if the local mean velocity at the interface is used as the convection velocity for the fluctuations. The convective coupling condition for a fluctuation φ is implemented in its discrete form using a first-order upwind difference in the direction normal to the interface (index j in Fig. 3) and a so-called θ –scheme with 0 ≤ θ ≤ 1 in time (t = m Δ t): m+1 m m+1 φ j−1 = C1 φ m j−1 + C2 φ j + C3 φ j

.

(1)

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Here, C1 , C2 and C3 are coefficients depending on the convection speed Uc , the spatial mesh size, the time step and the choice of θ . For θ = 0.5 and θ = 1, as chosen for the simulations presented here, this results in the implicit second-order accurate trapezoid rule and the implicit first-order Euler method, respectively. In the following . . .LES designates an explicit averaging procedure in time and homogeneous directions applied in the LES domain. A double prime denotes fluctuations with respect to this average and the average inherent to the Reynolds–averaging procedure. Suppose the interface is located at the face between the cells with index j − 1 (RANS–side) and j (LES–side). The resulting coupling conditions then read U jm+1 = um+1 LES j

(2)

for the streamwise velocity U at the inflow boundary of the RANS calculation and m+1 m+1 um+1 j−1 = U j−1 + u j−1

(3)

for the resolved streamwise velocity u at the LES outflow boundary, with the fluctuation u obtained from (1). All other velocity components are computed accordingly. Note that setting C1 = C2 = 0 and C3 = CE in (1), recovers the ad hoc formula for enrichment with its calibration constant CE . For incompressible flows with well-posed boundary conditions, mass conservation inside the fluid domain is implicitly enforced through the pressure field or an equivalent constraint variable, e.g. the streamfunction.These variables are governed by a Poissontype equation such that a convective condition or a scaled copying cannot be applied to their fluctuations and a different way of coupling the LES and RANS domains needs to be devised. In the following we restrict ourselves to formulations involving the pressure.Two distinctpossibilitiesof handling thisvariableattheinterfacewerescrutinized in [27]. One possibility is to solve the instantaneous pressure globally in Ω , the union of the LES and the RANS domains. The overall velocity boundary conditions for Ω are adjusted as usual with a projection method for incompressible flow such that the integral of mass flow over the boundary ∂ Ω is zero to machine precision as described below. This treatment of the pressure introduces a strong coupling that enforces instant mass conservation in the complete fluid domain. If the algorithm for the Poisson solver employed uses a domain decomposition technique no adjustment to the algorithm is necessary making this a very attractive approach. However, complications may arise due to the so-called modified pressure caused by many turbulence models [29]. In these cases the pressure gradient is altered by an additional gradient resulting from the trace of the Reynolds stress tensor in ΩRANS and the corresponding quantity of the subgrid-scale (SGS) model in ΩLES . both additional terms are different in the respective sub-domains so that a jump accross the interface is induced. Additional to this problem this method turned out to be unstable in the presence of large fluctuations [27]. The alternative approach is to decouple the pressure fields in ΩLES and ΩRANS completely. In this case, both the pressure field and the velocity field are discontinuous at the interface and mass conservation across this boundary is not guaranteed.


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When instantaneous mass conservation over the sub-domains is violated, the boundary conditions for the Poisson solver, applied in the individual sub-domain, become ill-posed and the solver converges poorly or not at all. In other words, the right hand side of the Poisson equation must fulfill a compatibility condition for the solution to exist, and this is ensured by a minute correction of the velocity field, if needed. As mentioned above, this issue occurs routinely with projection methods for incompressible flows. A remedy implemented in many flow solvers is a global mass flux correction applied to the velocities at the outflow boundaries. The mass flux over all inflow boundaries m˙ in is computed as well as the actual mass flux over all exit boundaries m˙ ∗out resulting from the uncorrected exit velocities u∗i . The velocity components at all outflow cells can then be scaled with the same factor ui = fm u∗i

with

fm =

|m˙ in | m˙ ∗out

.

(4)

In practical simulations, the mass flux ratio fm in (4) is usually very close to one. This “global correction” can also be applied to the velocities constructed with (2) and (3) at a single outflow-type interface. For the LES boundary cells, one needs to replace m˙ ∗out in (4) with the mass flux m˙ LES leaving the LES domain. Conversely, the velocities in the RANS boundary cell are then scaled using the magnitude of the mass flux entering the RANS domain m˙ RANS . For coupling of LES and RANS in case of complex geometries, one might have multiple embedded LES domains or tangential interfaces making the global correction cumbersome, if not impossible. Hence, a local approximation to the global flux correction above was proposed in [27] with |m˙ in | in (4) being replaced, at each interface, by 1 m˙ interface = m˙ LES + m˙ RANS (5) 2 with m˙ LES and m˙ RANS determined by integration over the corresponding interface. This method was applied in the example given below. In the course of the present study, several test cases have been investigated, e.g. swirl flow, turbulent channel flow and the flow over periodic hills [19–21, 25– 27, 29, 30]. Various locations of the LES-to-RANS interface, alternative coupling conditions from the literature or own modifications thereof and a host of possible combinations and parameters for the coupling conditions outlined above were scrutinized. This included different handling of turbulence quantities, velocities and pressure as well as the use of incompressible and compressible flow solvers. The handling of the turbulence quantities turned out to be uncritical [21]. For incompressible flow, however, the issue of mass conservation at the interface and the related pressure coupling conditions could, in some cases, decide upon success or failure of a simulation [27, 29]. In general, the convective coupling condition for the velocities was superior to classical enrichment. Only the most challenging setup is shown below, i.e. the incompressible hill flow with the interface placed upstream of the crest of the second hill as illustrated in Fig. 4. For this setup, a simulation without the RANS zone using a standard

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Fig. 4 Convective outflow coupling for the flow over periodic hills with the LES-to-RANS interface at x ≈ 7; instantaneous streamwise velocity (top) and mean streamwise velocity (bottom)

convective outflow condition diverges and classical enrichment with any kind of pressure coupling failed to obtain a result at all [30]. The simulation is divided into three distinct zones: The first zone is computed with LES using wall functions and periodic boundary conditions in the downstream direction serving as inflow generator for the second zone. 200 × 64 × 92 interior cells are used in the downstream, wall-normal and lateral direction, respectively. For the second zone, also LES is performed using the same resolution and wall-function as in Zone 1, however, before the crest of the next hill is reached, the simulation switches from LES to RANS (using the Spalart–Allmaras model). At the outflow of the RANS domain, Neumann boundary conditions are applied. Typical results obtained with convective coupling, decoupled pressure fields and explicit mass flux correction are displayed in Fig. 4. The instantaneous streamwise velocity contours show that the RANS flow field is completely steady. No reflections can be seen in the LES domain. The mean streamlines reveal that for the twodimensional RANS solution reattachment occurs far too late, consistent with RANS results in the literature [13]. On the other hand, the LES in Zone 2 delivers results similar to the reference solution of Zone 1, albeit with a slightly longer recirculation region. Both reattachment lengths of 4.1 h and 4.3 h for Zone 1 and 2, respectively, are sufficiently close to the reference values of 4.6 to 4.7 h in [9] obtained with a substantially finer grid. This example demonstrates the crucial importance of pressure decoupling in conjunction with mass conservation for complex situations.


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4 Tangential Coupling For tangential interfaces, sub-domain boundaries are more or less aligned with streamlines of the mean flow. If these interfaces are close to walls with the RANS region between the LES domain and the wall, the problem is analogous to near-wall modeling of LES using a two-layer approach. With segregated modeling, however, the velocities are discontinuous across the interface, since only mean values are directly coupled. Fluctuations need to be provided separately. Tangential coupling with segregated modeling has so far been proposed only in [23] using the enrichment strategy discussed in Sect. 3. In this reference, the method was applied to turbulent channel flow and the flow over a bluff body. Here, we follow the same premise, namely that copying fluctuations from inside the LES zones to ghost cells on the other side of the interface provides physically realistic structures. But instead of scaling the structures with an ad hoc constant we use the information from the RANS model to appropriately scale the amplitude of the fluctuations. Hence the solution in the ghost cells should have the desired statistical properties determined by the RANS zone and be statistically consistent. Here, a first step of the investigation is presented where the RANS solution is frozen. Suppose the interface is an x−z plane located between the indices j −1 and j in the y–direction and that the RANS zone is below the LES region, i.e. the scenario depicted in Fig. 3 above. In the present method, the values of the u−component of the instantaneous solution in the LES ghost cells are then determined as RANS LES uLES j−1 = u j−1 + f u u j

with

uLES = uLES − uLES j j j LES

.

(6)

The other velocity components are determined analogously. Note the different indices ( j − 1 and j) in the left part of the equation representing the copying of fluctuations accross the interface. The notation uLES instead of u, etc., is preferred here for clarity. The other velocity components are determined analogously. When employing a turbulence model which is based on a K–equation, it is natural to scale the velocity fluctuations to √ LES this yields f = f = f = the desired value of K. With fK = K RANS /K f . For u v w K j j−1 other models that do not provide a value K RANS , an alternative scaling was designed to yield matching of the total turbulent shear stress which is the central component of most eddy-viscosity based statistical turbulence models [28]. Rescaling the amplitudes of the copied LES-fluctuations is meant to enforce chosen statistics provided by the RANS calculation, but it can not account for a phase shift in the fluctuation. A phase shift may occur due to the retardation of the faster moving fluid from above the interface when it is moved to the slower moving fluid below. In order to mimic this retardation effect, a relaxation was introduced to the fluctuation in (6) LES m+1 m LES uLES , (7) = ε uLES − uLES j j | − u j LES + (1 − ε ) u j | j LES where m represents the time-level of the fluctuation and ε is the relaxation factor.

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Table 1 Simulations reported and resulting Reynolds numbers Reτ based on friction velocity for the top and the bottom wall, DNS reference data from [17] and [6] for the lower and higher Reynolds numbers, respectively method

Re = 7000 Re = 20580 method top bottom top bottom

Re = 7000 Re = 20580 top bottom top bottom

DNS LES RANS-KO RANS-SA N-KO

395 382 410 406 410

410 410 406 – –

395 382 410 406 383

934 934 E-KO 890 890 K-KO 1084 1084 S-SA 1120 1120 K-KO-R (0.5) 1084 889 E-KO-R (0.5)

404 396 404 – –

1084 1084 – 1084 1084

952 910 – 917 950

Turbulent channel flow at two bulk Reynolds numbers (Re = 7000 and 20580) was used to assess the tangential coupling conditions for a RANS zone placed between LES and the wall. The investigated cases are compiled in Tab. 1. Here, only results for the highest Re are presented, for more details see [28]. Apart from the obvious naming for DNS, LES and RANS, the first letter in the case abbreviation identifies the method generating fluctuations, i.e. no fluctuations (N), classical enrichment with scaling constant 0.95 (E), scaling based on the turbulent kinetic energy (K) and the total turbulent shear stress (S). Letters behind the hyphen identify the RANS model used: KO or SA for the K-ω and Spalart– Allmaras models, respectively. For cases with relaxation, the letter R is appended after a second hyphen. Since the bulk velocity was enforced, the Reynolds number based on friction velocity is a calculation result and reported in Tab. 1. For the hybrid simulations, the RANS zone next to the upper wall was frozen such that Reτ of the RANS calculation is maintained whereas the corresponding value at the lower wall adjusts such that the total mass flux is maintained. The values obtained for Reτ on both walls are included in Tab. 1. All simulations were computed with decoupled pressure fields and no explicit mass flux correction at the interface. The domain size is 2 π × 2 × π in the streamwise (x), wall-normal (y), and spanwise (z) direction, respectively, with periodic boundary conditions in x and z. The bulk Reynolds number was imposed by a volume force in the x−momentum equation and an appropriate controller. A grid of 64 × 64 × 64 cells with stretching in the wall-normal direction was used throughout. For the low Reynolds number + + case this results in a near-wall scaling of y+ 1 ≈ 0.5, Δ x ≈ 40, and Δ z ≈ 20 and for + + + the high Re case in y1 ≈ 1, Δ x ≈ 90, and Δ z ≈ 50. The RANS domain extends over the last 16 points in the wall-normal direction (1.885 ≤ y ≤ 2). Here, the grid is two-dimensional with only one cell in the spanwise direction. The time step was Δ t = 0.01 and statistics were sampled over taver ≥ 1500 δ /Ubulk. All averages were taken in time and over wall-parallel planes. As initial condition for the hybrid simulations, the results of the RANS computation were used with the addition of random noise in the LES part of the domain. A coarse grid was chosen on purpose in


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Fig. 5 Wall-normal profiles for turbulent channel flow simulation at Reτ ≈ 1000 with tangential LES/RANS coupling: mean streamwise velocity (top left), resolved turbulent kinetic energy (modeled for RANS, top right) resolved Reynolds shear stress (bottom left) and resolved wall-normal Reynolds stress (bottom right); DNS reference data from [6]

order to more clearly see whether the RANS layer yields any improvements. For the results shown, the interface is placed within the logarithmic region of the U + –profile (y+ ≈ 120). Fig. 5 demonstrates that simulation K-KO delivered the best results with respect to the streamwise mean velocity improving over pure LES, pure RANS and enrichment in a smooth fashion. For K-KO, also the U + –profile for the lower (LES) wall is shown. A slight shift in the constant of the log-profile can be discerned. This shift is due to the adjustment of the wall friction to the enforced mass flux, but otherwise the structure of the flow seems unperturbed. The resolved u−fluctuations however exhibit an overprediction close to the interface. Although the K–scaling represents a clear improvement over the original enrichment technique, this artificial increase of resolved fluctuations in the vicinity of the interface is still bothersome. It indicates that at least some of the flow structures are not represented accurately. A possible reason for this may be the strict vertical displacement of the fluctuations whereas, in particular for coarse grids, real flow structures might be transported to the location of the ghost cells at an angle leading to a time shift of the fluctuation. To assess whether such a mechanism might indeed be at work here, both the K–scaled and the classical enrichment technique were repeated with the relaxation of (7). Results are only shown for a relaxation factor of ε = 0.5 here, but other factors were tested as well and very similar results were observed. The relaxation has indeed a drastic

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effect on the flow structures as can be seen in the resolved Reynolds stresses plotted in Fig. 5. The shear stress profile is improved and, in particular for case K-KO-R, excellent agreement with the DNS data is obtained. This seems to be mainly due to a better prediction of the wall-normal velocity fluctuations, since the resolved wallnormal Reynolds stress also exhibits very good agreement with the DNS data near the interface. Also a considerable improvement can be seen in the lateral stresses (not shown here). The longitudinal Reynolds stress, however, is now overpredicted even more. Since it contributes most to the resolved turbulent kinetic energy this also attains higher values near the interface. A more careful look at flow structures in the vicinity of the interface (see Fig. 6) reveals that, for pure LES, predominately longitudinal structures of a velocity deficit are present. Enrichment leads indeed to unsteady flow structures but with shorter streamwise extension. The K–scaling improves on the resemblance and arguably realistic flow structures are generated, but this method still falls short, since the longitudinal structures appear to be shorter than in the LES. For the instant in time shown here, the relaxation yields a picture similar to the LES but with a lack in structures with a velocity deficit. However, at another instant (shown in [28]) the high-frequency content of the fluctuations was missing and only the large-scale structures survived. The reason for this intermittent behaviour is still unclear and needs further scrutiny.

Fig. 6 Illustration of flow structures for turbulent channel flow simulation at Reτ ≈ 1000 with tangential LES/RANS coupling: Shown are contours of the streamwise velocity fluctuation in the LES cells adjacent to the LES/RANS interface. Top left: pure LES; top right: enrichment; bottom right: K–scaled; bottom left: K–scaled with relaxation


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5 Inflow Coupling At an inflow interface within a turbulent flow field realistic turbulent fluctuations need to be generated in order for the LES to provide a realistic answer. The generation of fluctuations at this boundary can be performed with higher or lower perfection, depending on the flow investigated and the purpose of the simulation conducted. The method described in the following is a combination of a database technique and rescaling as used in the recycling technique for the generation of unsteady inflow data [15]. More details can be found in [18]. At the upstream end of the LES domain, values of the mean conservative fields are known from the upstream RANS domain, so that only the determination of the fluctuating part of these fields is needed. These fluctuations are extracted from a database, then scaled to fit the desired time and length scales and to have the proper magnitude and finally added to the average field at the inflow plane. These inflow conditions were developed for (compressible) wall-bounded turbulent boundary layers with spanwise homogeneity. In the wall-normal direction (y), the boundary layer is divided into an inner and an outer region where the viscosity and the turbulence dominate, respectively. The same division is applied to the fluctuations, hence, the velocity, density, and pressure fluctuations are decomposed as (ui )reg (x, y, z,t) = φireg (x) fi (y/lyreg , z/lzreg ,t/t reg ) (T )reg (x, y, z,t) = ψ reg (x) h(y/lyreg , z/lzreg ,t/t reg)

(8)

(ρ )reg (x, y, z,t) = χ reg (x) g(y/lyreg , z/lzreg ,t/t reg ) . As above, the double prime indicates fluctuations with respect to the Reynolds average. The upper index reg stands for the region, either inner or outer region, while φi , χ and ψ have the same dimension as their respective variable and determine the magnitude of the fluctuations. The functions fi , g and h are homogeneous in the spanwise direction. These formula are used to rescale the inner and outer fluctuations of the database. First, the scales and magnitudes of the fluctuating velocity fields are determine. For the outer region, the study on the vortex organization in a boundary layer of Adrian et al. [1] provides appropriate information on these scales. The size of the “hairpin packets” in the outer layer are of the order of the boundary layer thickness inner = 0.8U . So δ in each direction. Moreover their convective velocity is about Uconv ∞ outer outer outer the length scales lx , ly and lz are equal to the boundary layer thickness. The outer = δ /(0.8U ) ∼ δ /U . As the outer retime scale is defined by t outer = lxouter /Uconv ∞ ∞ gion is more dependent on the free-stream flow, several scales are relevant for the magnitude φiouter of the fluctuations: the friction velocity uτ , the free-stream velocity U∞ and the square root of the maximum TKE. Two velocity scalings were applied for the study undertaken: uτ and U∞ . In the inner region, where the viscosity dominates, the natural length-scales lxinner , lyinner and lzinner are determined by the viscous length lv . The convective velocity Uconv is of the order of 0.6U∞ . Therefore the time inner = l /(0.6U ). However the same time scale as in scale could be t inner = lxinner /Uconv v ∞

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the outer region was retained to maintain the temporal coherence between the inner and the outer region at the inlet plane. The magnitude φiinner is set by the friction velocity uτ . Second, the scales and magnitude of the density and temperature fields are needed. The same length and time scales as for the outer and inner velocity fields are retained. For boundary layers with adiabatic wall at moderate Mach number, an approximation of the stagnation enthalpy yields: T u − (γ − 1)M 2 u T

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where u denotes the streamwise component of velocity and M the Mach number and the tilde Favre filtering. Therefore, assuming the average and fluctuating velocity to have the same magnitude, the magnitude of inner and outer temperature fields are ψ inner = ψ outer = − (γ − 1)M∞2 T∞ . The equation derived from the approximation the equation of state at moderate Mach number

ρ T − ρ¯ T

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shows that the magnitude of the density fluctuations also varies with the square of the Mach number: χ inner = χ outer = (γ − 1)M∞2 ρ¯ ∞ . According to the above construction the fluctuating fields are expressed as: y z t (ui )inner (x, y, z,t) = uτ (x) fi ( lv (x) , lv (x) , δ (x)/U ) ∞ (x) y z t (T )inner (x, y, z,t) = − (γ − 1)M∞2 T∞ (x) h( lv (x) , lv (x) , δ (x)/U ) ∞ (x) t ) (ρ )inner (x, y, z,t) = (γ − 1)M∞2 ρ¯ ∞ (x) g( lyv , lzv , δ (x)/U ∞ (x) y z t , δ (x) , δ (x)/U ) (ui )outer (x, y, z,t) = (uτ (x) or U∞ (x)) fi ( δ (x) ∞ (x)

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y z t (T )outer (x, y, z,t) = − (γ − 1)M∞2 T∞ (x) h( δ (x) , δ (x) , δ (x)/U ) ∞ (x) y z t , δ (x) , δ (x)/U ) (ρ )outer (x, y, z,t) = (γ − 1)M∞2 ρ¯ ∞ (x) g( δ (x) ∞ (x)

Slight modifications must be made to take into account, on one hand, that the duration of most computations exceeds the duration of the database, and on the other hand, that in the spanwise direction periodic boundary conditions are applied. The inner and outer profiles of fluctuations are combined by the weighting function proposed by Lund et al. [15]. The inflow conditions were assessed using a simple test case, the flat plate with zero pressure gradient. First, a reference computation (R1) at Mach 0.14 was performed. For this computation, the inlet the boundary layer is laminar and random perturbations were added to the velocity to trigger the transition towards turbulence


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Fig. 7 Computational domain for generating the database (left) and for the subsequent validation (right)

(see Fig. 7(a)). As shown in the figure, a database is created storing samples at a given plane for this computation R1. Subsequently, simulations were carried out to study the influence of parameters of the database such as duration and subsampling of the solution. Computations were also done to assess the scalings when the Mach number and the Reynolds number based on momentum thickness Reθ at the inlet plane are different from those of the database. These computations were set up as follows. LES was employed to solve the boundary layer while resolving the rest of the computational domain with a RANS simulation (see Fig. 7). In the region where the original LES domain is replaced with RANS, statistics of the corresponding reference simulation were prescribed so that the mean profile at the inlet is alwasy the same as for the reference computation. For all simulations adiabatic wall were prescribed at the bottom of the boundary layer, subsonic inlet and outlet for RANS domains were used, based on characteristics and at the top of the RANS domain a Neumann condition was employed. At the dowmstream end and the top of the LES domain, conditions defined in section 3 and 4 were applied using enrichment downstream end and no enrichment at the top. The LES meshes employed follow the classical requirements: with x, y and z being the coordinates in streamwise, wall-normal and spanwise direction, respectively, Δ x+ = 50, Δ z+ = 20 and Δ y+ = 1 in the first cell were used. The database was created from the reference simulation R1 at Mach 0.14. The plane where the samples were stored is located where the flow is turned into a fully turbulent one. To locate the plane the reduced variable x∗ = x/δre f is introduced where δre f refers to the boundary layer thickness at the plane of the database, equal to 3 ∗ 10−3 meters. At this plane, x∗ = 50 and Reθ = 878. A number of 104 samples were stored every Δ t = 3 ∗ 10−7s, leading to the normalized time step Δ t ∗ = Δ t/(δre f /U∞ ) = 4.78 ∗ 10−3 and the normalized duration Tdb = 47.8. This duration corresponds to the convection of around thirty streaks. The database needs about 1GB storage. The subsequently conducted hybrid simulations used only the fluctuating part of the database solution. To evaluate the sensitivity of the proposed method to changes in the Reynolds number, the database generated for Reθ = 878 was used as an inflow condition at a different position, namely at x∗ = 80 with Reθ = 1306 (see

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Fig. 7(right)). Here at the LES inflow plane, the reference length for normalization, the boundary layer thickness, is 5 ∗ 10−3m. Fig. 8 displays the streamwise evolution of the profiles of the mean streamwise velocity, as well as the streamwise and spanwise Reynolds stresses for such a simulation, named S6. Whatever the location downstream of the inflow plane, the profiles of mean streamwise velovity for this simulation closely agree with the reference R1. For the Reynolds stress components discrepancies appear, especially for the spanwise Reynolds stress at the inlet plane and one boundary layer thickness downstream where these are underestimated. Five boundary layers downstream of the inlet plane all profiles match the the reference data. This suggests that the correct structures in the inner region are quicker regenerated than in the outer region. The underestimation of the spanwise Reynolds stresses may be due to the lack of rescaling in this direction since this is technically difficult due to the periodic boundary conditions. The influence of the time step between two successive samples of the database on the solution was also studied. To reduce the computational cost, the configuration of simulation S6 was retained with the rescaling. Three computations were done subsampling the database, retaining every 10th sample (S7), every 50th sample (S8),


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and every 100th sample (S9). The normalized time steps Δ t ∗ of the databases employed for S7, S8 and S9 are 4.78 ∗ 10−2, 2.39 ∗ 10−2 and 4.78 ∗ 10−1, respectively. The influence of subsampling on the velocity profiles is displayed in Fig. 8. The velocity and rms profiles of S6 and S7 agree. By subsampling every 50th sample slight differences between the Reynolds stress profiles of S6 and S8 appear at the inlet plane. Subsampling by a factor of 100 these are larger, but five boundary layer thicknesses downstream of the inlet plane all profiles are very close to the reference solution R1. So it seems that only very large-scale structures need to be well described. To assess the Mach number rescaling of the database, a flate plate at M=0.7 has been studied. In this case, the quality of agreement between the computed solution and the reference is similar to that obtained with the Reynolds number rescaling presented above in Fig. 8 [18]. The conclusions of these computations are that the duration of the database should be as long as possible with at least a duration corresponding to the convection of around 20 boundary layer thicknesses. The discontinuity related to the ’rewind’ of the database perturbes mainly the pressure field. On the other hand the database time step can be increased significantly without influencing the flow. These results confirm the observations of Keating et al. [14] that the large scales (those whose size is larger than the integral length scale) play the dominant role in the generation of turbulence downstream of the inflow plane.

6 Combined Application to a Complex Flow The prediction of internal flow over curved surfaces provides a challenge to any simulation approach. An example considered here is an air intake configuration displayed in Fig. 9. Indeed the boundary layers are subjected to curvature and pressure gradient effects, which often lead to boundary layer separation. Experiments on this

Fig. 9 Air intake simulation: RANS zone (dark grey) and LES zone (light grey). Flow is from left to right in x−direction. The steady RANS solution in the inflow part is symmetric so that only half the span was computed and then mirrored

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configuration were undertaken by Geffroy and Sipp [12] with an inflow duct of 15cm height and a bulk velocity of 30m/s leading to a Reynolds number of 300000. Despite the continuous improvement of modelisation, RANS can only predict the low frequency part of an unsteady signal, whereas it suffers from a lack of accuracy in case of separation. LES is relatively free of these problems, but it leads to high computational cost. Therefore, a segregated RANS/LES approach is very interesting for the simulation of this flow. In the simulation presented here, attached boundary layers are modelled with the Spalart-Allmaras model, whereas the region of separated flow is taken into account by an LES zone (see Fig. 9).

Fig. 10 Schlieren view of pressure fluctuation in the LES zone: general view (top), Zoom at the upstream corner near the RANS/LES interface (bottom)


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At the interfaces between RANS and LES zones, the specific conditions presented above are used to generate or filter turbulence. A slight modification of the LES inflow turbulence condition is introduced here to take into account the corners formed by the lower wall and the side walls. Different simulations were performed to assess the influence of mesh resolution and side walls effects. All these simulation were conducted with 12 to 30 million cells. The correct function of the proposed inflow and outflow coupling methods can be appreciated by means of Fig. 10. Realistic turbulence is present even immediately after the inflow plane. At the outflow of the LES domain, the treatment is also correct, as obviously no damping of the turbulent fluctuations is introduced by the boundary condition. On the other hand, a failure of the LES tangential coupling can be observed in the right graph of Fig. 10 inside the side wall boundary layer. The dark colour indicates that at this location hardly any unsteady perturbation is present in the boundary layer near the inflow. This is due to the fact that the streamlines enter from the RANS zone at the upper face of this block. As the enrichment procedure, and not the inflow boundary condition, is used at this interface, the fluctuations are not generated here. Despite the problem at the interface between the RANS and the LES domain in the side wall boundary layer mentioned above, the accuracy of the present 3D simulation is very encouraging. Indeed, the comparisons with the experimental data for both, the mean velocity and the turbulent kinetic energy (TKE) is favourable over a substantial distance from the inlet as displayed in Fig. 11. There is however still room for improvement since further downstream the deviations increase and all simulations predict too long a recirculation zone. The importance of accounting for the sidewalls is revealed by the fact that the simulation employing a periodic condition in spanwise direction with the smallest spanwise extent (M1) performs not as well as the others.

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7 Conclusions Techniques for “coupling LES where needed with RANS where sufficient” are interesting from a practical point of view in order to reduce the simulation cost for complex turbulent flows of engineering interest. The simulation of an air intake flow at high Reynolds number presented here illustrates this statement by the fact that a full simulation of this flow by LES would not be possible to date. The RANS-to-LES inflow boundary is very similar in nature to the pure LES inflow condition. The database method proposed and validated here is a convenient approach to cope with the task of providing realistic turbulent fluctuations. In the present situation, the repetition of the data did not have an impact on the result. For simulations concerned with aeroacoustics this might be different. In such cases the database should be supplemented with an additional scrambling or forcing mechanism to avoid exact reproduction of the inflow signal after some laps of time. In the simulations for compressible flow, tangential and outflow coupling were applied as well employing the enrichment procedure of [23] and the overall performance of these conditions was good. The latter two conditions were further enhanced and generalized, so that in particular for the outflow condition any need for choosing a stabilizing parameter is removed. This was performed in the incompressible framework and issues related to the somewhat more envolved mass conservation in this case were discussed. The extension of the tangential coupling condition performed better than the enrichment procedure, but still leaves room for improvement. This is subject of ongoing work. Acknowledgements. The authors like to thank Deutsche Forschungsgemeinschaft (DFG) for its financial support through project SP4 in the DFG/CNRS research group FOR 507 ”LES of Complex Flows”. ONERA, Châtillon, supported the project through funding of a PhD thesis. We also like to thank the French and German colleagues in FOR 507 for many fruitful discussions and a productive and collaborative atmosphere. The provision of computing time by SCC Karlsruhe, ZIH Dresden, and ONERA is gratefully acknowledged.

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5. Breuer, M., Lakehal, D., Rodi, W.: Flow around a surface mounted cubical obstacle: Comparison of LES and RANS–results. In: Deville, M., Gavrilakis, S., Ryming, I. (eds.) Computation of 3D Complex flows. Notes on Numerical Fluid Mechanics, vol. 53, pp. 22–30. Vieweg Verlag (1996) 6. Del Alamo, J., Jimenez, J.: Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144 (2004) 7. Fröhlich, J.: Large Eddy Simulation turbulenter Strömungen. Teubner Verlag (2006) 8. Fröhlich, J., Denev, J.A., Hinterberger, C., Bockhorn, H.: On the impact of tangential grid refinement on subgrid-scale modelling in large eddy simulation. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds.) NMA 2006. LNCS, vol. 4310, pp. 550–557. Springer, Heidelberg (2007) 9. Fröhlich, J., Mellen, C.P., Rodi, W., Temmerman, L., Leschziner, M.A.: Highly–resolved large eddy simulations of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 526, 19–66 (2005g) 10. Fröhlich, J., von Terzi, D.: Hybrid LES/RANS methods for the simulation of turbulent flows. Prog. Aerospace Sci. 44, 349–377 (2008) 11. Garc´ıa-Villalba, M., Stoesser, T., von Terzi, D., Wissink, J., Fröhlich, J., Rodi, W.: Large eddy simulation of turbulent separated flow over a three-dimensional hill. In: Proc. 11th European turbulence Conference, Porto, Portugal. CD-ROM (2007) 12. Geffroy, P., Sipp, D.: Contrôle des décollements - etude expérimentale fondamentale a` d’une manche coudée. RTS PEA CACV, Sous-tache 2.1.3.1, internal report ONERA (2006) 13. Jakirlić, S., Jester-Zürker, R., Tropea, C. (eds.): 9th ERCOFTAC / IAHR / COST Workshop on Refined Turbulence Modelling. Darmstadt University of Technology (2001) 14. Keating, A., Piomelli, U., Balaras, E., Kaltenbach, H.-J.: A priori and a posteriori tests of inflow conditions for large-eddy simulation 16, 4696–4712 (2004) 15. Lund, T., Wu, X., Squires, K.D.: Generation of inflow data for spatially developing boundary layer simulations. J. Comp. Phys. 140, 233–254 (1998) 16. Menter, F., Kuntz, M., Bender, R.: A scale-adaptive simulation model for turbulent flow predictions. AIAA Paper 2003-0767 (2003) 17. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids 11, 943–945 (1999) 18. Nolin, G., Mary, I., Ta, P.L.: LES inflow conditions for turbulent boundary layers of complex flows using database. AIAA paper 2006-2888 (2006) 19. Nolin, G., Mary, I., Ta-Phuoc, L.: RANS eddy viscosity reconstruction from LES flow field. AIAA paper 2005-4998 (2005) 20. Nolin, G., Mary, I., Ta-Phuoc, L.: RANS eddy viscosity reconstruction from LES flow field for zonal RANS/LES simulations of transitional and turbulent flows. In: 40th Colloque d’Aérodynamique Appliquée, Toulouse (2005) 21. Nolin, G., Mary, I., Ta-Phuoc, L., Hinterberger, C., Fröhlich, J.: Coupling from LES to RANS using eddy-viscosity models. In: Lemballais, E., Friedrich, R., Geurts, B., Metais, O. (eds.) Direct and Large Eddy Simulation VI, pp. 679–686. Springer, Heidelberg (2006) 22. Pope, S.: Ten questions concerning the large-eddy simulation of turbulent flows. New J. Physics 6, 35 (2004) 23. Quéméré, P., Sagaut, P.: Zonal multi–domain RANS/LES simulations of turbulent flows. Int. J. Numer. Meth. Fluids 40, 903–925 (2002) 24. Tomm, S., von Terzi, D., Fröhlich, J.: Interaction of the filter and spatial discretisation operators for large-eddy simulation using the approximate deconvolution model. In: Proc. Appl. Math. Mech. (to appear, 2008) 25. von Terzi, D., Fröhlich, J., Rodi, W.: Hybrid techniques for large-eddy simulations of complex turbulent flows. High Performance Computing in Science and Engineering 2, 195–213 (2008) 26. von Terzi, D., Hinterberger, C., Garcia-Villalba, M., Fröhlich, J., Rodi, W., Mary, I.: LES with downstream RANS for the flow over periodic hills and a model combustor flow. In: Proc. of EUROMECH Colloquium 469, Dresden, pp. 6–8 (December 2005)

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LES, Zonal and Seamless Hybrid LES/RANS: Rationale and Application to Free and Wall-Bounded Flows Involving Separation and Swirl ˇ c, Atabak Fadai-Ghotbi, Suad Jakirlić, Rémi Manceau, Sanjin Sari´ Björn Kniesner, Sabrina Carpy, Gisa Kadavelil, Christophe Friess, Cameron Tropea, and Jacques Borée

Abstract. An overview is given of the activities in the framework of the GermanFrench Research Group on ”LES of Complex Flows” (DFG-CNRS FOR 507) with respect to the development of zonal and seamless hybrid LES/RANS computational methods based on a near-wall Eddy-Viscosity Model (EVM) and a near-wall SecondMoment Closure (SMC) respectively. The zonal scheme represents a two layer model with a two-equation EVM-RANS model covering the near-wall layer and the true LES employing the zero-equation subgrid-scale (SGS) model of Smagorinsky resolving the core flow. Due attention was payed to the exchange of the variables between the ensemble-averaged RANS field and the spatially-filtered LES field across the discrete interface separating the two sub-regions. A procedure for controlling the interface position in the flow domain was also in focus of the present investigations. After considering a few introductory test cases (fully-developed channel flow, flows separating from sharp-edged surfaces) the feasibility of the method was validated against the available experiments in a single tubo-annular, swirl combustor configuration (Exp.: Palm et al., [39]) and in the separated flows in a 3-D diffuser (Exp. Cherry et al., [10]) and over a 2-D hump including the case with the separation control by steady suction (Exp. Greenblatt et al., [23]). The seamless LES/RANS method employs the so-called Elliptic-Blending Reynolds-Stress Model (EB-RSM, Manceau, [33]; Manceau and Hanjalic, [34]) being active in the entire flow field. This RANS-based SGS model represents a near-wall Second-Moment Closure model relying on the elliptic relaxation method. The model coefficient multiplying the destruction term in the transport equation for the scale-supplying variable ε (dissipation rate of the turbulence kinetic energy) was made filter-width (corresponding ˇ c · Björn Kniesner · Gisa Kadavelil · Cameron Tropea Suad Jakirlić · Sanjin Sari´ Fachgebiet Strömungslehre und Aerodynamik, Technische Universität Darmstadt, Petersenstr. 30, 64287 Darmstadt, Germany [email protected] Rémi Manceau · Atabak Fadai-Ghotbi · Sabrina Carpy · Christophe Friess · Jacques Borée Laboratoire d’études aérodynamiques (LEA), Université de Poitiers, ENSMA, CNRS, Bd Marie et Pierre Curie, 86962 Futuroscope-Chasseneuil, France [email protected] C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 253–282. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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to the grid spacing) dependent, i.e. dependent on the location of the spectral cutoff, by applying a multiscale modelling procedure originating from spectral splitting of filtered turbulence in line with the Partially Integrated Transport Model (PITM) proposed by Dejoan and Schiestel, [48] and Chaouat and Schiestel, [8]. Herewith, the dissipation rate level was obtained, which suppresses the turbulence intensity towards the subgrid (i.e. subscale) level in the regions where large coherent structures dominate the flow. The resulting model was validated by computing some free flows (a temporal mixing layer) and wall-bounded flows (a fully-developed channel flow). Finally, the PITM method applied to the high-Reynolds number RSM model due to Speziale et al., [53] was used to compute the flow separated from a 2-D hill (with reference LES by Fröhlich et al., [19] and Breuer, [6]). In addition, all relevant cases were computed by the conventional LES method aiming at mutual comparison of the predictive capabilities of the afore-mentioned methods with respect to the quality of results and space-time resolution issues.

1 Introduction The computational method for solving turbulent flows, known as Large-Eddy Simulation (LES), has increasingly become a tool of choice for engineering use. Rapid advances in computer hardware and developments of numerical algorithms and subgrid-scale models have enabled a broader use of the LES method. However, in spite of significant progress, the opinion is frequently expressed that LES will not reach the standard of an industrially relevant numerical tool in the near future. The main reason for such a statement, beside the uncertainties with the definition of inflow conditions and other numerically relevant problems (see e.g. the work of Moin, [37]), is the treatment of the near-wall regions. The wall-resolved LES becomes progressively important with the Reynolds number (number of grid cells in the viscous near-wall region is proportional to Re1.76 compared to N ∝ Re0.4 in the off-wall region). This implies that almost 50 % of the total number of the numerical nodes should be situated in the viscous sublayer and the buffer layer, Pope, [41], Temmerman et al., [55]. Furthermore, the stresses residing in subgrid scales depend strongly on the filter type and filter width, the latter corresponding directly to the grid spacing in the case of the Smagorinsky-related subgrid-scale models. This feature comes into focus especially in the immediate wall vicinity, the free mixing i.e. shear layer regions and recirculation zones, all flow phenomena being massively encountered in a combustor configuration. These regions are traditionally meshed by an irregular, highly non-uniform, anisotropic grids implying a lower cell aspect ratio. Application of such an anisotropic grid invalidates the isotropy assumptions, implicitly implemented in most SGS models. Keeping in mind the fact that the anisotropies in the turbulent flow not resolved by the grid, represent the anisotropies of the SGS motion, it becomes clear that Smagorynski-like models are regarded as inappropriate to deal with anisotropic meshes and the associated turbulence. On the other hand, the RANS (Reynolds-Averaged Navier Stokes) models deal satisfactorily with the

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grid anisotropy issues. These introductory remarks emphasize clearly the objectives of the present project: a) development of a zonal procedure merging an appropriate, transport-equation-based RANS model being active in the wall vicinity with the conventional LES covering the core flow (meshed by grid cells close to isotropic) and b) development of a seamless computational method relying on a RANS-based SGS transport model on the Second-Moment Closure level being capable of dealing with mesh and turbulence anisotropies present in complex flows when employing coarser grids complying with the spectral cutoff position not situated in the inertial sub-region, the Smagorinsky-like models are based upon.

2 Zonal Hybrid LES/RANS Method The work reported here aims at developing a hybrid LES/RANS (HLR) model for turbulent flows by accounting for the near-wall effects with a RANS model and capturing the large-scale structures and bulk unsteadiness in the reminder of the flow field using a coarse LES. The goal is to achieve the accuracy comparable to that of a conventional LES, but at much lower computational costs. The method development is outlined in the next section followed by the results obtained by its application to selected flow configurations affected by separation and swirl. In order to illustrate some important issues with respect to coupling the LES and RANS regions the results of the channel flow simulations at a moderate Reynolds number Rem = 24000 (DNS: Abe et al., [1]) are shown. The dimensions of computational domain were (Lx , Ly , Lz ) = (2π h, 2h, π h) with 2h being the channel height. The computations with the hybrid model have been performed on several grids. Results shown in Figs. 2-3 were obtained on the grids (Nx , Ny , Nz ) = (48, 64, 72) and (24, 64, 36). These were coarsened by the factor of 8 and 32 respectively compared to the fine LES (96, 128, 144) .

2.1 Method Rationale The functional principle of the present two-layer hybrid LES/RANS scheme coupling a low-Reynolds-number k − ε RANS model with Large-Eddy Simulation (LES) is illustrated in Fig. 1. The RANS model covers the near-wall region and the LES model the remainder of the flow domain. Both methods share the same temporal resolution. The equations governing the velocity field operate as the Reynolds-averaged Navier-Stokes equations in the near-wall layer (U i represents the ensemble-averaged velocity field Ui ) or as the filtered Navier-Stokes equations in the outer layer (U i represents the spatially filtered velocity field). The eddyviscosity νm originates from the model formulation either for the subgrid-stress tensor τi j or for the Reynolds-stress tensor ui u j . The latter tensors are expressed in terms of the mean strain tensor Si j via the Boussinesq’s relationship. Two different subgrid-scale (SGS) models in LES were considered, the Smagorinsky model

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(Δ = (Δx × Δy × Δz )1/3 , |S| = (Si j Si j )1/2 , Cs = 0.1) and the one-equation model for the residual kinetic energy (Yoshizawa and Horiuti, [59]; Ck = 0.07, Cε = σk = 1.0), Fig. 1. Various low-Reynolds number k − ε models differing basically in the transport equation for the scale-determining variable are applied in the near-wall RANS region. The model equations, with ε representing either the ”isotropic” (ε˜ = ε − 2ν (∂ k1/2 /∂ xn )2 , Launder and Sharma, [31]; Chien, [11]) or the ”homogeneous” part of the total viscous dissipation rate (ε h = ε − 0.5Dkν ; inhomogeneous part follows exactly by adding one half of the molecular diffusion of the turbulent kinetic energy 0.5Dkν , Jakirlic and Hanjalic, [28]). The equations for ε h and ε are related by Dε h /Dt = Dε /Dt − 0.5∂ (ν∂ ε /∂ x j )/∂ x j . The latter concept offers certain advantages with respect to the asymptotic behaviour of turbulence quantities when approaching the solid wall. Interested readers are referred to the original publications for details about the model coefficients Cε ,1 and Cε ,2 , near-wall and viscous damping functions f2 and f μ and the production term Pε ,3 . The coefficients and functions in the k − ε h model are adopted in line with the Launder and Sharma’s proposal. Independently of the models used, a major issue in a zonal hybrid LES/RANS strategy is the coupling of the instantaneous LES field and the ensemble-averaged RANS field at the interface. In the present method the coupling is realized via the turbulent viscosity, which makes it possible to obtain solutions using one system of equations. This means practically that the equations of motion are solved in the entire solution domain irrespective of the flow sub-region (LES or RANS).

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Depending on the flow zone, the hybrid model implies the determination of the turbulent viscosity νm either from the RANS or from the LES formulation, Fig. 1. Because ε and also k in the case of the Yoshizawa-Horiuti model, are not provided within the LES sub-domain, their SGS values are estimated using the proposal of Mason and Callen, [35] (see also Sagaut, [44]), Fig. 1. The RANS equations for k and ε are solved in the entire flow field, but with the discretization coefficients taking zero values in the LES sub-region. By manipulating appropriately the source terms, the numerical solution of these equations in the framework of the finite volume method provides the interface values of the kRANS and εRANS being equal to the corresponding SGS values. By doing so, the boundary condition at the LES/RANS interface (ifce) implying the equality of the modelled turbulence viscosities (by assuming the continuity of their resolved contributions across the interface, Temmerman et al., [55]) at both sides of the interface:

νt,i f ce |RANS−side = νSGS,i f ce |LES−side is implicitly imposed without any further adjustment, see Fig. 2 for illustration. In such a way a smooth transition of the turbulence viscosity is ensured. Due to the relatively high turbulence viscosity in the RANS region, the fluctuations are strongly damped and do not recover up to some distance behind the interface. The typical outcome of such a circumstance is the so-called velocity mismatch in the region of interface, Fig. 3-left. In order to reduce or to eliminate this velocity bump, a so-called forcing method was applied. The idea of the forcing method is to generate fluctuations at or around the interface in order to accelerate their recovery. A method originating from a digital-filter-based generation of inflow data for spatially developing DNS and LES due to Klein et al., [30] was adopted in the present work. Its use increases the computational costs to some extent. However, it should be emphasized that the fluctuations are computed only at the interface. The steps to be performed are summarized as follows: (a) creation of random fluctuations, (b) filtering of fluctuations in space and time, (c) adjustment to local Reynolds stresses and (d) introduction into momentum equations through a source term. The strength of the forcing depends on the interface position. The closer the interface to the wall,

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the less forcing is needed. One important observation emerged. It seems to be sufficient to introduce the forcing only in the direction normal to the interface. Fig. 3-left displays also the result obtained applying the method of smoothing the turbulence viscosity by adjusting the RANS model parameters, such as Cμ (see the work of Temmerman et al., [55]; the form of the blending function used here is given by Cμ = 0.09 + (Cμ ,i f ce − 0.09)y+/y+ i f ce ). The results of the reference LES are also shown. Fig. 3-right displays the turbulent kinetic energy profile obtained by the HLR method using the forcing at the interface and the pure LES using the same (coarse) grid. It is clearly visible that the HLR method with forcing performs better than the corresponding coarse LES. The results obtained are even very close to the fine LES. The computational costs of HLR exceed that of the corresponding coarse LES by about 30 % due to the additionally solved equations for k and ε . Fig. 3-right shows the profile of the total (tot) k for the case with forcing, which was obtained by summing up the resolved (res) and modelled (mod) parts. Due to unsteady treatment of the RANS layer and the RANS field excitation by the adjacent instantaneous LES field through the interface, one can observe a resolved part also in this region. The modelled part clearly diminishes when crossing the interface at about y+ = 100. The HLR results shown in Figs. 2-3 are all obtained by using the Chien k − ε˜ model in the RANS layer and the Smagorinsky model in the LES region. Other model combinations resulted in similar results (not shown here). One of the advantages of a zonal approach is the possibility to predefine the LESRANS interface. However, in unknown flow configurations, this could be a difficult issue. Therefore, a certain criteria expressed in terms of a control parameter should be introduced. Presently, a following control parameter kmod ∗ k = kmod + kres is adopted, representing the ratio of the modelled (SGS) to the total turbulent kinetic energy in the LES region, averaged over all grid cells in homogeneous direction at the interface on the LES side. As soon as this value exceeds about 20 %, the interface is moved farther from the wall and in opposite direction when the values


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go below 20 %. This additionally ensures that in the limit of a very fine grid (very low level of the residual turbulence) LES is performed in the most of the solution domain. Contrary, in the case of a coarse grid, RANS prevails. As the interface separates the near wall region from the reminder of the flow, it would be suitable to choose a wall-defined parameter for denoting the interface location. In the present study, the dimensionless wall distance y+ was adopted. Despite possible difficulties in respect to the definition of y+ in the flow domains where the wall shear stress approaches zero, as e.g. in separation and reattachment regions, no problems in the course of the computations have arisen (let us recall that the same non-dimensional wall distance y+ is regularly used in the Van Driest’s wall-damping of νt also in LES of separating and reattaching flows). It is noted, that the interface y+ is not active in the computational procedure. It only denotes the computational nodes, at which the prescribed value of k∗ is obtained. Fig. 4-left displays the time evolution of the interface position in the high-Re number backward-facing step flow (Exp.: Vogel and Eaton, [58]), averaged over all cells at the interface. Fig. 4-right shows a snapshot of the instantaneous velocity field with the corresponding evolution of the RANS/LES interface along the upper wall and the lower (step) wall. Numerical Code Description. The computational results presented were obtained by using the in-house computer code FASTEST-3D (Flow Analysis by Solving Transport Equations Simulating Turbulence), [18] based on a finite volume numerical method for solving both three-dimensional filtered and Reynolds-averaged Navier-Stokes equations on block structured, body fitted, non-orthogonal meshes. Block interfaces are treated in a conservative manner, consistent with the treatment of inner cell faces. A cell centered (collocated) variable arrangement and Cartesian vector and tensor components are used. The equations are linearised and solved sequentially using an iterative ILU method. The velocity-pressure coupling is ensured by the pressure-correction method based on the SIMPLE algorithm which is embedded in a geometric multi-grid scheme with standard restriction and prolongation. The code is parallelized applying the Message Passing Interface (MPI) technique for communication between the processors. The convective transport of all variables was discretized by a second-order, central differencing scheme for LES. In the case

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of the HLR method for k− and ε − equation some upwinding is used by applying the so called ”flux blending” technique. The time discretization is accomplished applying the second order (implicit) Crank-Nicolson method.

2.2 Application to Separating and Swirling Flows As the main target was to model flows involving swirl and separation, the experiments of turbulent flow within the inlet section (consisting of the swirl generator and appropriate annular and central pipes) and the flue of a single tubo-annular combustor (Palm et al. [39]), a turbulent flow over a hump mounted on the channel wall at a high-Re number Rec = 9.36 · 105 (Greenblat et al., [23]) and a separated flow in a 3-D plane diffuser (Cherry et al., [10]) were chosen as the reference. 2.2.1 Single Tubo-Annular Gas Turbine Combustor Model One of the topic of the present study was numerical analysis of the variable swirl intensity influence on the flow characteristics in a tubo-annular combustion chamber, which was experimentally investigated by Palm et al., [39]. The measurements were performed over a range of swirl intensities and Reynolds numbers related to the main stream and swirling annular flow. In addition to the LES and HLR methods, an eddy-viscosity-based turbulence model, denoted by ζ − f (Hanjalic et al., [25]), was applied in the RANS framework. Three different cases corresponding to the swirl intensities of S = 0.0, 0.6 and 1.0 were simulated. Schematic of the flow considered is depicted in Fig. 5 together with the operating flow parameters. The inner diameter of the main flow Dm is 36 mm, whereas the inner and outer (Dc ) diameters of the annular section are 40 mm and 100 mm, respectively. The diameter of the flue D f is 200 mm. The swirl generator is based on the ’movable block’ design, Leuckel (1969). By rotating an inner and an outer annular block relative to each other, varying positions of tangential and radial channels are created. With a pure radial inlet, a non-swirling flow is obtained, and with a pure tangential inlet, the maximum swirl

Fig. 5 Schematic of the combustion chamber model and operating flow conditions


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Fig. 6 Mean streamline patterns within the swirl generator (left) and the flue (right)

is generated. To give a first impression about the flow topology in a swirl combustor, a sketch of the mean flow pattern obtained by the ζ − f model is shown in Fig. 6. Two sets of combustor simulations were performed: one accounting for the entire inlet section including (simplified) swirl generator system (Fig. 7 and one with the swirling inflow generated computationally (not accounting for the swirl generator) at the experimental locations within the annular pipe (cross-section located 40 mm upstream of the expansion. In case of the latter simulations the swirling inflow was generated using the method proposed by Pierce and Moin, [40]. This method assumed fully developed flow conditions, whereby the (equilibrium) swirling motion was created by introducing a fictitious pressure gradient into the momentum equation governing the circumferential velocity. The magnitude of the pressure gradient (with constant value over the cross-section) was iteratively adjusted until the computed U and W velocity fields satisfied the prescribed swirl intensity S. The solution domain with the length of Lx = 2.67π (Rc − Rm ) was meshed by Cartesian grid with Nx × Nr × NΘ = 64 × 49 × 128 cells. The maximum value of the CFL number was 0.85. The dynamic Smagorinsky model was used for this purpose. The obtained instantaneous velocity fields across the annular pipe were prescribed for the consequent flue simulations. In order to partially avoid the complex geometry of the swirler two geometrical simplifications were followed in the present study: one excluding the annular-like inflow plenum and second excluding the channels between movable blocks (i.e. guide vanes). The tangential (azimuthal) velocity component was adjusted in time until the desired swirl intensity was obtained. The solution domain consisting of the flue, simplified swirler with annular and central pipes was discretized by a Cartesian grid (Nx × Nr × Nθ ) comprising about 4.6 Mio. cells in total (192 × 161 × 128 cells in the flue, 96 × 49 × 128 cells in the swirler and 19 × 5 × (32 × 32) cells in the end part of the central pipe). The time step chosen (Δ t = 6 · 10−5s, i.e. 7.7 · 10−4D f /U f ) corresponds to CFL ≤ 0.5 in the largest part of solution domain. For the highest swirl intensity case (S = 1.0), the grid was refined in axial direction by reducing the computational domain size Lx = 4D f (S = 0 and 0.6) to Lx = 2D f , preserving the number of computational nodes. Thus, streamwise resolution was effectively doubled. This is a compromise regarding computational effort and resolution

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Fig. 7 A grid slice of a simplified swirl generator (left) including the combustor flue and contours of mean velocity in the x − y plane obtained by LES for S = 1.0

requirements of LES. The LES computations were running two to three flow through times before taking the flow statistics which represent the average over two flow through times. Separate precursor LES of a fully developed, non-swirling (central) pipe flow corresponding to the experiment (Rem = 23500) had to be conducted to generate the pipe inflow data for both sets of combustor simulations. More details ˇ c et al., [45, 46]. about grid resolution assessment can be found in Sari´ Prior to considering the flow within the flue, an intensive study of the flow structure within the swirl generator system was conducted. Fig. 7-right displays the contours of the mean axial velocity field for the case with S = 1.0 in the radial and 90o -turned part of the inlet section and the annular pipe indicating very complex flow pattern deviating substantially from the equilibrium (fully-developed) conditions. Strong acceleration of the flow occurs when the flow in the radial part of the swirler transforms into the axial motion within the annular pipe (red areas). In this case with strong swirl a swirl-induced recirculation zone bounded at the inner wall (thin blue area) was detected corresponding to the region with strongest acceleration of the circumferential flow. The comparison of the experiments and the computationally obtained axial velocity and shear stress profiles within the annular pipe of the inlet section for non-swirling case and the case featuring maximum swirl intensity S = 1.0 are presented in Fig. 8. Figs. 9 show the influence of the swirl intensity on the mean flow structure illustrated by the evolution of the axial velocity profiles at selected locations in the near field of the flow within the flue immediately after expansion featured by most intensive mixing of the swirling annular jet and non-swirling inner jet. Swirl intensity influence on the flow in terms of enhanced tendency towards the free recirculation zone generation is obvious. The non-swirling flow configuration (Fig. 9-upper) is characterized by a long annular (wall-mounted) corner bubble. Besides a corner bubble (which is of substantially shorter length compared to the non-swirling case), the most important feature of the swirling flow configuration is a swirl-induced, free flow reversal in the core region, Figs. 9-lower. Figs. 9 illustrate the intensification of the velocity magnitude (with respect to both the shear layer and the back-flow

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region) and the strengthening of the curvature of the shear layer by increasing the swirl intensity. The radial flow becomes more intensive, hence promoting the mixing. This swirl-induced separation represents actually a transition from the supercritical flow state related to the non-swirling central stream to the sub-critical situation with respect to the flow reversal in the core (vortex breakdown phenomena; Escudier and Keller, [16]). It contributes significantly to the corner bubble shortening. The mean velocity results exhibit very good agreement with experimental data in the regions of the shear and outer layers. However, both simulations (LES and HLR) result in a separation region, whose onset is not situated at the symmetry axis. The computationally obtained separation zone is lifted in the radial direction exhibiting an annular form, with a central jet going through the middle (note the continuously positive value of the centerline velocity, Fig. 9-lower). A similar outcome was obtained experimentally but for the higher velocities of the central jet relative to the velocity of the annular jet (lower mass flow ratios, not considered in the present work). The origin of such anomaly could lay, besides the reasons pertinent to the grid resolution, in the poor representation of the swirling outflow through the boundary condition prescribed. The importance of the outlet boundary conditions has been recognized in some previous works, Escudier et al., [17]. Flow visualization by isosurfaces of pressure fluctuations are shown in Figs. 10 for both non-swirling (S = 0.0) and swirling cases (1.0). With increase in the swirl intensity, the coherent structures appear to become dominant and clearly illustrate an enhanced spreding of the annular and central jets discharging into a sudden expansion, i.e. enhancement of turbulent mixing.

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Fig. 10 Visualization of coherent flow structures in LES of swirling flow in the combustor - isosurface of the instantaneous pressure fluctuation p for various swirl intensities

2.2.2 Flow over a 2-D, Wall-Mounted Hump: Separation Control by Steady Suction The next flow configuration considered is the turbulent flow over a smoothlycontoured, wall-mounted hump (Fig. 11) at high chord-based Reynolds number Rec = 936000 situated in a plane channel (height 0.909c; c = 0.42m). This flow


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was experimentally examined at the NASA Langley Research Center (Greenblat et al., [23]). In addition to the LES and present HLR method this flow is also simulated by using the DES method (Detached Eddy Simulation; Spalart et al., [50]), another hybrid LES/RANS model. The solution domain (dimensions: 6.14c × 0.909c × 0.152c) was meshed with a grid comprising approximately 1.72.0 Mio. cells (426 × 145 × 28 and 426 × 145 × 32 respectively) when applying the DES and coarse LES/present hybrid LES/RANS. The solution domain employed for DES with a somewhat larger spanwise dimension (0.2c), was meshed by the grid with interface being situated between y+ i f ce ≈ 10 − 100 (recirculation zone) and y+ ≈ 250 (flow region around the hump). The reference LES was performed on i f ce the grid with almost 4 Mio. cells (426 × 145 × 64). In all the LES and DES computations available experimental velocity profiles were imposed at the inlet plane placed at 2.14c upstream of the hump leading edge. The instantaneous inflow corresponding to a boundary layer developing at the bottom wall used for the HLR simulations was generated by using the method proposed by Klein et al., [30]. The necessary RANS profiles of all turbulence quantities were obtained by applying the near-wall Reynolds-stress model by Jakirlic and Hanjalic, [28]. The pressure coefficient evolution and the mean streamwise velocity profiles and shear stress profiles are shown in Figs. 12 and 13 for the baseline case and the case with separated boundary layer control by spatially uniform steady suction through a thin opening situated at the natural separation point. Figs. 13 show the streamwise evolution of the axial velocity and shear stress profiles in all characteristic flows regions: separation zone, reattachment and recovery regions in the flow control case. The LES method, despite a relatively coarse mesh

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for such a high Reynolds number, wall-bounded flow, was capable of capturing important effects of the flow control by steady suction qualitatively and quantitatively. The DES results are almost identical to those obtained by using the conventional LES in the baseline case (Fig. 12-left). It is especially encouraging knowing that a fairly coarse grid was applied. However, poor performance in the suction case (see also Fig. 12-right) indicates the importance of the DES grid design (the same grid as in the baseline case was used) with respect to the interface position. It is determined in advance by the grid used, not depending on the flow investigated. Unlike in the DES method, the position of the LES/RANS interface is variable in the proposed hybrid LES/RANS method, depending on the prescribed (20 %) ratio of the modelled (SGS) to total kinetic energy of turbulence. Fig. 11 displays the development of the interface related to the lower wall. The distance of the interface to the wall expressed in non-dimensional units corresponds to y+ ≈ 200, which represents the value averaged over the interface-closest grid cells on the LES side. Unlike a somewhat larger recirculation zone and consequently slower recovery in the baseline case (not shown here) the results for the steady suction case exhibit excellent agreement with the reference data. 2.2.3 Separated Flow in a 3-D Plane Diffuser An incompressible flow in a 3-D diffuser with deflected upper wall (with an angle of expansion α = 11.3o) and one side wall (with an angle of expansion α = 2.56o) at the inlet-duct-height-based Reynolds number of Reh = 10000 (Fig. 14), for which


267

Fig. 14 Time-averaged axial velocity field in the 3-D diffuser obtained by the present HLR method

the experimentally obtained reference database was provided by Cherry et al., [10], was studied computationally by using the LES (Large Eddy Simulation) and DES (Detached Eddy Simulation) methods as well as the present HLR (Hybrid LES/RANS) method. The focus of the investigation was on the adverse pressure gradient effects evoked by the duct expansion on the size and shape of the threedimensional flow separation pattern. The inflow data were generated by a precursor simulation of the fully developed duct flow using the respective models. The solution domain for the DES and HLR simulations comprised a part of the development duct (5h), the diffuser section (15h) and the straight outlet duct (12.5h). At its outlet cross-section the convective outflow conditions were applied. No-slip boundary conditions were applied at the walls. The grid applied in both simulations contained 224 × 62 × 134 cells (approximatelly 1.86 Mio. grid cells in total). The dimensionless time step Δ t = 0.028 (normalized by the inlet channel parameters Ubulk = 1m/s and h = 1cm) was used in the computations providing the CFL number less than unity throughout the solution domain (CFLmax ≈ 0.76). The wall-adjacent cell size in wall units pertinent to the HLR simulations along bottom and upper walls corresponded to Δ x+ = 10 − 100 and Δ z+ = 5 − 20 in the central plane (z/B = 1/2). The solution domain adopted for the LES simulation, extended to account for the part of the convergent transitioning duct (≈ 9h), was meshed with almost 4 Mio. grid cells (Kadavelil et al., [29]). The wall boundary layers are resolved with y+ values of approximately O(1). The corresponding inlet data are generated by a simultaneously running the periodic simulation of a fully-developed channel flow with the same cross section as the diffuser inlet. To allow the flow through the diffusor to influence the flowfield in the development channel, a part (5 channel hights) of this channel has been modelled in front of the diffuser. The final position of the LES/RANS interface determined in accordance with the criterion explained earlier corresponded to y+ ≈ 50 in the HLR simulation. Mean axial velocity and streamwise stress component profiles at selected streamwise locations in all characteristic flow regions in the vertical plane corresponding to z/B = 1/2 (B = 3.33cm) are shown in the following figures

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Fig. 15 Evolution of the axial velocity and streamwise stress component profiles in the flow in a 3-D diffuser

confirming superiority of the HLR-results over the DES and LES results obtained using the grid with the same (DES), i.e. finer (LES) resolution.

3 Partially Integrated Transport Modelling (PITM) for Free and Wall-Bounded Flows Schiestel & Dejoan (2005) [48] and Chaouat & Schiestel (2005) [8] have recently shown that seamless hybrid LES/RANS methods can provide a good compromise between RANS and LES, in particular when they are based on differential subgridstress models. The objective of such approaches is twofold: first, it provides a methodology to continuously switch from a LES, in regions where it is of interest, to RANS in other regions, in order to reduce the computational cost; secondly, the approach provides the possibility to perform a Very-Large Eddy Simulation (VLES). Indeed, with a differential subgrid-stress model based on a refined physical modelling (similar to Reynolds-stress or at least algebraic Reynolds-stress model), the possibility of defining a cutoff wavenumber in the productive region of the spectrum (contrary to LES, which requires a cutoff in the inertial region) can be at the


269

origin of a significant reduction of the CPU cost. The accuracy of the method will thus strongly depend on the way the subgrid stresses are modelled, and in particular on the way the transition from a LES-behaviour to a RANS-behaviour is imposed. Moreover, the method will rely on the RANS model in the near-wall regions, where an accurate LES is almost at the price of a direct numerical simulation. In order to obtain a hybrid model, a spectral splitting is introduced, as for LES. However, the main difference with LES lies in the fact that the cutoff wavenumber can be located everywhere between the Kolmogorov scale (DNS) and zero (RANS). Therefore, the subgrid stresses cannot be simply represented by a Smagorinsky-like model, but must be accounted for in a more sophisticated way. Different recent studies (e.g., Hadˇzić et al., 2001 [24], Tardu and Da Costa, 2001 [54], Revel et al., 2005 [42], Carpy and Manceau, 2005 [7]), have shown that in an unsteady context, a correct representation of the time lag between stresses and strains, entering the turbulence production mechanisms, is necessary to obtain realistic predictions. Therefore, in hybrid LES/RANS models, in which the cutoff wavenumber can be located right in the energetic scales, the complexity of the production mechanisms cannot be represented by Smagorinsky-like models. Hadˇzić et al., 2001 [24] and Carpy and Manceau, 2005 [7] have shown that second-moment closures are able to account for the response of the turbulence field to large-scale varying strains. They are moreover attractive with respect to their ability to reproduce complex effects such as curvature, rotation, stratification, etc.

3.1 The PITM Approach In the PITM method (Partially Integrated Transport Model) proposed by Schiestel & Dejoan [48], the turbulent spectrum is divided into 3 zones, delimited by the wavenumbers κc and κd , as shown in figure 16. κd is to be located sufficiently far from the energetic range for assuming that the energy contained between κd and ∞ is negligible. Introducing the spectral cutoff filter (κc ) into the Navier–Stokes equations leads to the following equations, for the filtered velocity and the subfilter stress ∂ τi j SGS 1 ∂ P˜ ∂ 2U˜ i ∂ U˜ i ˜ ∂ U˜ i +Uj =− +ν − , (1) ∂t ∂xj ρ ∂ xi ∂ x j∂ x j ∂xj . / 0 1 ∂ ui uj uk ∂ 2 τi j SGS ∂ τi j SGS ∂ τi j SGS ∂ ui ∂ uj ˜ + Uk =− +ν − 2ν ∂t ∂x ∂x ∂ xk ∂ xk ∂ xk ∂ xk k k ν T Ci j SGS Di j SGS Di j SGS εi j SGS ∂ U˜ j ∂ U˜ i 1 1 ∂ p ∂ p ui − uj − −τikSGS − τ jk SGS . ρ ∂xj ρ ∂ xi ∂ xk ∂ xk Pi j SGS φi∗j SGS

(2)

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Fig. 16 Spectral partitioning used in the PITM method

4

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Fig. 17 Second-moment closure URANS computation of a temporal mixing layer using a 4-eddy-containing computational box. Vorticity contours

Moreover, in homogeneous turbulence, the energy-spectrum function E(t, κ ) satisfies ∂E (t, κ ) = −λi j Ai j (t, κ ) + B(t, κ ) − 2νκ 2E(t, κ ) (3) ∂t where λi j is the mean velocity gradient, Ai j (t, κ ) the Fourier transform of the twopoint correlation tensor and B(t, κ ) the spectral transfer term. Integrating equation (3) over the three intervals defined in figure 16, it can be shown [48] that the usual dissipation equation can be used CS ∂ε ∂ε P ε ∂ ∂ε + U˜ j = Cε1 − Cε∗2 + νδlm + τlmSGS T (4) ∂t ∂xj T T ∂ xl σε ∂ xm if the coefficient Cε∗2 is made a function of the ratio modelled energy/total energy fk = kSGS /k such that Cε∗2 = Cε1 + fk (Cε2 − Cε1 ) , (5) where Cε1 and Cε2 are the standard values of the coefficients in a RANS model. Since the filtered equations (1) and (2) are formally identical to standard RANS equations, it is possible to impose a continuous transition from a RANS behaviour to a LES behaviour. In the model, this behaviour has to be linked to the turbulent scales, and this can be done simply by prescribing a relation between the ratio modelled energy/total energy fk and κc . By prescribing a κ −5/3 -shape turbulent spectrum and by integrating between κc and ∞, the relation fk = 1/(β0 ηc 2/3 ) can be obtained,


271

where ηc is the ratio between the filter width and the integral scale

ηc = κc

k3/2 . ε

(6)

In order to ensure the correct RANS limit (limηc →0 fk ), the following empirical modification was proposed by [48] fk (x,t) =

1 1 + β0ηc 2/3

(7)

Similarly to conventional LES, κc can be related to the local cell size π /(Δx Δy Δz )1/3 . In that case, the transition from LES to RANS is only driven by the ratio ηc between the integral length scale k3/2 /ε and the cell size. This theoretical derivation shows that a second moment closure (RSM) can be used as a hybrid LES/RANS model, as soon as the transport equations are made sensitive to the non-dimensional cutoff wavenumber ηc = κc k3/2 /ε . The variation of Cε∗2 , which drives the destruction mechanism in the dissipation equation, is thus the cornerstone of the method: it ensures that, whatever the cutoff wavenumber, the dissipation rate remains the same (the dissipation rate being active at Kolmogorov’s scale, well beyond the cutoff wavenumber), while the resolved energy is a function of the cutoff wavenumber. Presentation of the Code. All the computations are performed with the opensource Software Code Saturne, a parallel, finite volume solver on unstructured grids, developed at EDF [2], distributed [3] under Gnu GPL license. Space discretization is based on a collocation of all the variables at the centre of gravity of the cells. Velocity/pressure coupling is ensured by the SIMPLEC algorithm, with a Rhie & Chow interpolation in the pressure-correction step. The Poisson equation is solved with a conjugate gradient method. Time advancement is based on a CrankNicolson scheme. Spatial derivatives are approximated by a second-order centraldifference scheme (CDS) for the resolved velocity field and a first-order upwinddifference scheme (UDS) for the subgrid turbulence field.

3.2 Temporal Mixing Layer This methodology has been tested in the case of a temporal mixing layer, using a standard second moment closure RANS model (SMC) [32] as the limiting form when κc → 0, and compared with a URANS model, based on the same transport equations, but with a constant Cε2 coefficient, and with a LES. All the computations were performed in comparison with the DNS of Roger & Moser [43] at Re = δm Δ U/ν = 800, where δm is the momentum thickness and Δ U the velocity difference. In the DNS, the mixing layer is strongly forced, with initial perturbation of the order of 90% of the mean kinetic energy. Similarly, in order to accelerate the

272


appearance of large-scale structures, all our computations have been forced with a white noise and a perturbation at the most amplified mode, with an amplitude of 1 % of the kinetic energy. The computations show very realistic structures and a correct growing of the momentum thickness, provided that a sufficient modelling level is used: URANS computation using a linear eddy-viscosity model does not exhibit any large-scale structure, but rather a diffusive thickening of the mixing layer. On the contrary, URANS and PITM computations using second-moment closure either as Reynoldsstress model or as subgrid-stress model, respectively, show large-scale structures that grow until pairing occurs. Figure 17 shows the appearance of four KelvinHelmholtz vortices, the first pairing and the second pairing, in a 2D URANS computation. Similarly, figure 18 shows the pairing that occurs in the 3D PITM computation. Figure 19 shows a comparison of the tridimensional character of the solutions of 3D URANS, PITM and LES computations at the same time, on the same mesh, with the same perturbation introduced at the initial time step. It appears that the URANS model is able to sustain some oscillations in the spanwise direction generated by the initial perturbation, but does not develop 3D structures, while PITM is very similar to LES. Actually, these computations clearly show that PITM experiences a transition from a URANS behaviour to a LES behaviour, which is driven by the ratio ηc = κc k3/2 /ε . This can be seen in particular on figure 20, which shows the time history of the momentum thickness: during the initial phase, before pairing, PITM follows URANS, while after the pairing that occurs around T = 80Δ U/δω0 , it progressively tends towards LES. This can be explained by the time-evolution of the parameter ηc : as time evolves, the integral length scale of turbulence grows, while the mesh remains constant, such that ηc increases. The PITM switches to URANS when the mesh is coarse compared to the integral length scale, considering that it is too coarse for computing in LES mode, and progressively goes to LES when the separation between the (fixed) filter width and the integral length scale becomes sufficient.

Fig. 18 PITM computation of the temporal mixing layer using a subgrid-stress transport model. Vorticity contours


URANS

273

LES

PITM Fig. 19 Comparison at T = 65Δ U/δω0 of 3D URANS, LES and PITM computations on the same mesh. Isocontours of a passive scalar coloured by Q

This progressive switch to LES explains why the PITM computation is able to reproduce the tridimensional structure of the flow. This result is very encouraging since the aim of the method is precisely to offer the possibility of a progressive transition from URANS to LES depending on the local (in space or in time) refinement of the mesh.

3.3 Derivation of a Near-Wall Model Based on Elliptic Blending In order to extend the model presented in the previous section to the near-wall region, the next step of the present work consists in the application of the elliptic blending strategy to reproduce the non-viscous, non-local blocking effect of the wall. The new model is thus an adaptation to the hybrid LES/RANS approach of the Elliptic-Blending Reynolds Stress Model (EB-RSM) [34, 33], which was successfully applied to different configurations in a RANS methodology [33, 56, 5, 57, 49, 12]. Modelling issues are presented and discussed. The new model is derived and calibrated in a channel flow at Reτ = 395, in comparison against DNS data [38]. 3.3.1 The Elliptic Blending Model The term of Eq. (2) that deserves a particular attention in the near-wall region is the velocity–pressure gradient correlation (hereafter the pressure term) φi∗j . An adaptation of the EB-RSM model [33], usually applied in the RANS context, is used. The model blends the ”homogeneous” (away from the wall) and the near-wall models of the pressure term φi∗j and the dissipation tensor εi j using

274


Fig. 20 Time evolution of the momentum thickness δm /δm0

3 URANS PITM LES

0

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φi∗j = (1 − α 2)φiwj + α 2 φihj , τi j 2 εi j = (1 − α 2) SGS ε + α 2 εδi j , km 3

(8) (9)

where km = 12 τii is the modelled fluctuating kinetic energy and α is a blending coefficient which goes from zero at the wall, to unity far from the wall. Similarly to the elliptic relaxation method of Durbin [14, 15], the nonlocal character of the pressure is reproduced by solving an elliptic equation for the blending function

α − L2 ∇2 α = 1. SGS

(10)

The near-wall form of the model φiwj can be shown [33] to be consistent with the asymptotic behaviours of all the variables at the wall when it is taken as ε 1 φiwj = −5 τik n j nk + τ jk ni nk − τkl nk nl (ni n j + δi j ) . (11) km 2 A generalization of the concept of wall-normal vector is used here: n = ∇α /∇α , which is applicable everywhere in the domain. It is worth noting that, even though the RANS region is subject to unsteadiness coming from the LES region, the asymptotic analysis is still valid, since the unsteady term remains negligible compared to other terms of the budgets [34]. The Speziale, Sarkar and Gatski (SSG) model [53] is used for φihj . The association of the elliptic blending model and the PITM approach leads to some modelling issues. Firstly, it will be shown that the parameter fk , which drives the transition from a RANS to a LES behaviour, varies too rapidly in the near-wall region when the formulation Eq. (7) is used. In order to enforce a RANS behaviour in the region strongly influenced by the wall, the elliptic blending function will be introduced in fk . Secondly, the length scale of the elliptic equation Eq. (10) must be made a function of the cutoff wave-number. Indeed, the non-local blocking effect of the wall on the subgrid stresses is a consequence of the incompressibility constraint for the non-resolved scales. In a hybrid context, this blocking effect must be imposed

LES, Zonal and Seamless Hybrid LES/RANS 4

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Fig. 21 Influence of the form of f k . Mesh 1, component τ11 . Profile of resolved (RES), modelled (SGS) and total stress. Left: f k given by Eq. (12) with Cg = 6.6. Middle: f k given by Eq. (14), Cg = 6.6. Right: f k given by Eq. (14), Cg = 2

only on the modelled scales, which implies that the correlation length scale LSGS , entering Eq. (10), must be decreased compared to the RANS context, where all the scales of motion are modelled. In order to investigate the previously mentioned issues, the equations of the model are solved in the case of a channel flow at Reτ = uτ H/(2ν ) = 395. The first mesh used contains 32 × 54 × 32 cells in the x, y and z directions, respectively. The centre of the first cell is located at y+ = 1.5, and the elongations in the streamwise and spanwise directions are Δ x+ = 100 and Δ z+ = 50. This mesh is clearly not sufficiently fine for a LES, and therefore well suited for testing the hybrid LES/RANS methodology. Since hybrid LES/RANS results are often very sensitive to grid refinement, and in particular to Δ x+ and Δ z+ , a second mesh was designed using Δ x+ = 50 and Δ z+ = 25, and containing 64 × 70 × 64 cells. 3.3.2 Modelling of the Parameter fk The formulation fk =

1 2/3

1 + β0ηc

,

(12)

with ηc given by Eq. (6), is an empirical modification [8, 48] of the analytical value fk =

1 2/3

β0 ηc

,

(13)

obtained by integration of a Kolmogorov spectrum, in order to ensure the compatibility with the RANS limit (limκc →0 fk = 1). However, using formulation (12) with 2/3 mesh 1, it is noticed that the condition β0 ηc 1 is not satisfied at the centre of the channel, such that the theoretical formulation (13), is not recovered in the LES region. Moreover, as shown in Fig. 21 (left), the resolved part of the Reynolds stress increases very rapidly as a function of the distance to the wall and is strongly overestimated. In order to avoid these two undesirable effects, it is proposed to use fk = (1 − α 2) + α 2

1 2/3

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,

(14)

S. Jakirlić et al. 1

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Fig. 22 Influence of LSGS . Mesh 1, component τ22 . Profiles of resolved (RES), modelled (SGS) and total stress. Left: using the RANS length scale. Right: using the RANS length scale arbitrarily divided by a factor 1.5

which enables a better control of the transition RANS–LES, as shown in Fig. 21 (middle), because fk is not only a function of the local cell size, but also of the distance to the wall, implicitly contained in α . Moreover, [21] showed that the evaluation of the cutoff wavenumber from the grid size by κc = 2π /2Δ is too optimistic when using a second order accurate central difference scheme, and recommended the use of κc = 2π /CgΔ with Cg = 6. A range of values between 2 and 20 have been tested for the Cg parameter, which enters the evaluation of β0 . For large values of Cg , fk = 1 is obtained all across the channel, thus leading to a RANS solution. Cg < 4.5 leads to a too weak subgridscale dissipation and a strong overestimation of the total Reynolds stresses, Fig. 21 (right). In the range [4.5, 10], the turbulence statistics are weakly dependent on the value of Cg . The optimal value was found to be Cg 6.6 (β0 = 0.20), which is very close to the value suggested by [21], Fig. 21 (middle). 3.3.3 Length Scale for the Wall-Blocking Effect

0

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Fig. 23 Profile of resolved (RES), modelled (SGS) and total shear stress. Left: mesh 1. Right: mesh 2


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Fig. 25 Profiles of total Reynolds stresses

) L = CL max

+

k3/2 ,Cη η ε

, (15)

where η is the Kolmogorov scale. This length scale characterizes the nonlocal influence felt by the non-resolved motion. In the hybrid context, since the length scale of the non-resolved fluctuations is dependent on the width of the filter, the length scale must be consequently modified. Fig. 22 shows the influence of arbitrary reducing the length scale by a factor 1.5. It is seen that the decrease of LSGS modifies the anisotropy, by reducing the blocking effect, i.e., the inhibition of the redistribution from τ11 to τ22 . As expected, the blocking effect only affects the subgrid scales, leaving the resolved scales almost unchanged. Such a reduction of the length scale is quite naturally obtained for the first term in Eq. (15) by replacing the integral length scale k3/2 /ε by the length scale characterizing the largest subgrid eddies 3/2 3/2 km /ε . However, the second term must be reduced by the same factor, i.e., fk , which yields , ) 3/2 km 3/2 (16) LSGS = CL max , fk Cη η ε The results given by this formulation, are shown in Figs. 23, 24 and 25. The partition of turbulent energy among resolved and subgrid scales is shown in Fig. 24. It can be seen that near the wall, the SGS part is dominant and decreases toward the centre of the channel, where the resolved part in turn becomes dominant. The same trend is observed for the shear stress on Fig. 23. When the mesh is refined, the cutoff wave number is increased, and the balance resolved/modelled stress is modified as expected.

3.4 PITM-Based High-Re RSM: Application to Separated Flows Starting from the Differential Reynolds-Stress Model (DSM) proposed by Speziale, Sarkar and Gatski - SSG, [53] with the quadratic formulation for the pressure strain model term in the RANS framework modifications of the destruction term,

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Fig. 26 Instantaneous velocity field and associated streamlines obtained by the PITM-DSM model (left) and corresponding profiles of the Cε∗2 -coefficient at several streamwise locations (right)

i.e. its model coefficient Cε∗2 in the transport equation for the corresponding scalesupplying variable were made in line with the PITM method, Eq. (5), with the coefficients fk = (1 + βη ηC3 )2/9 , βη = (2CK /3)2/9 and CK = 1, Chaouat and Schiestel, [9]. The model coefficients in the production and destruction terms take their standard values Cε1 = 1.44 and Cε2 = 1.83. Such a modification (see Fig. 26-right for the behaviour of the Cε∗2 model coefficient) provides a dissipation rate level which suppresses the turbulence intensity towards the subgrid (i.e. subscale) level in the region where large coherent structures (e.g., separated shear layer) with a broader spectrum dominate the flow, allowing in such a way evolution of structural features of the associated turbulence. Herewith, a seamless coupling, i.e. a smooth transition from LES to RANS and opposite is enabled. Some selected results (Figs. 26 and 27) obtained by computing the periodic flow over a 2-D hill (Fröhlich et al., [19]; Breuer, [6]), exhibiting a number of features typically associated with the highly-unsteady shear layer that separates the main stream from the recirculation, demonstrate that the model presented (denoted by PITM-DSM) is capable of representing the subgrid-stress transport in the framework of LES in the flows dominated by the organized, large-scale coherent structures, influencing to a large extent the overall flow behaviour. In addition to the PITM-DSM, the results of the 3-D computations using the same model in the RANS

3

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Fig. 27 Evolution of the mean axial velocity profiles in the periodic flow over a 2-D hill


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framework (RANS-DSM) are also displayed for the sake of their comparative assessment. Unlike in some flows separated from the sharp-edged surfaces (with fixed separation point; e.g., backward-facing step geometries), the RANS models can neither qualitatively nor quantitatively reproduce the flows separated from continuous surfaces, where the separation point oscillates over a larger portion of the wall, causing an intensive spreading of the separated shear layer. The outcome is a much longer recirculation zone reflecting a fairly poor reproduction of the mean flow and turbulence evolution (Fig. 27). The PITM-DSM predictions obtained employing a significantly coarser grid (338.000 cells) are comparable with the reference LES (e.g., Breuer - 13 Mio. cells) within all characteristic flow regions: separation, recirculation zone and recovery and are clearly superior over the RANS results.

4 Conclusions The potential of the proposed zonal hybrid LES/RANS model scheme was illustrated by computing a series of wall-bounded flow configurations featured by separation, reattachment and swirl effects in a broad range of Reynolds numbers. Very promising results with respect to the structural characteristics of the instantaneous flow field, the mean velocity field and associated integral parameters (e.g., pressure coefficient, flow spreading rate) as well as the turbulence quantities demonstrate the model feasibility and applicability in a broad range of complex, wall-bounded turbulent flows. The employment of the present hybrid model enables a substantial reduction of the number of grid points compared to conventional LES method. Concerning spatial and temporal resolution a coarsening by a factor of 4 to 8 and 2 to 4 respectively is shown to be possible by the results quality being comparable to the LES method. It is particularly valid for the integral and wall-related characteristics, which, at marginal increase of the costs (among others, more equations are solved in the hybrid framework), show a considerable improvement compared to the LES on the same grid Furthermore, the present work have confirmed the ability of transport models to fulfil the necessary triple role to be played in a seamless hybrid RANS/LES approach: the role of a RANS model when the cutoff wavenumber goes to zero, the role of a LES subgrid-scale model when the cutoff lies in the inertial region, and the role of a VLES subgrid-scale model for a cutoff in the productive region of the spectrum. The PITM methodology enables to control the transition from RANS to LES by comparing the integral scale to the local mesh refinement. In a channel flow, the model, based on the elliptic blending strategy for the representation of the wall effects, continuously transitions from LES to RANS when the wall is approached, providing the basis for a continuous (or global) zonal approach. Moreover, in a temporal mixing layer, the integral length scale grows with time, while the mesh is fixed, such that the model undergoes a temporal transition from URANS to LES, and, for instance, the momentum thickness clearly switches from one solution to the other. This is now encouraging for the VLES objective of this kind of approaches, i.e., the possibility of obtaining a LES-type solution on a coarse mesh.

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Wall Scaling and Wall Models for Complex Turbulent Flows Michael Manhart, Nikolaus Peller, Margareta Petrovan Boiarciuc, and Christophe Brun

Abstract. The near wall behaviour of complex turbulent flows is investigated by dimensional considerations and a priori investigations. We consider separated incompressible and compressible flows. In incompressible turbulent flows, it is found that the flow in the immediate vicinity of the wall is dependent on two parameters, based on wall friction and local pressure gradient, respectively. In compressible turbulent flows, a third parameter based on the wall heat flux is relevant for the flow profiles in vicinity of the wall. While in the viscous sub-layer, an exact law of the wall can be formulated, the buffer and log-layer is dependent on empirical formulations. The significance of the wall scaling is shown by carefully carried out Direct Numerical Simulations and Large-Eddy Simulations of incompressible and compressible separated flows.

1 Introduction The near wall region of turbulent flows is in the focus of intense research activities for more than half a century. One of the main reason for these research activities is the fact that the inner region, i.e. the region that is dominated by wall effects, gets thinner and thinner with increasing Reynolds number and is therefore increasingly difficult to resolve by numerical methods. For the accurate prediction of high Reynolds number flows, it is therefore crucial to be able to model the inner region of the boundary layer. Michael Manhart · Nikolaus Peller Technische Universität München, Fachgebiet Hydromechanik, Arcisstrasse 21, 80333 Munich, Germany [email protected] Magareta Petrovan Boiarciuc Polytech’Orléans/PRISME, 8, rue Léonard de Vinci, 45072 Orleans CEDEX 2, France [email protected] Christophe Brun Université Joseph Fourier, LEGI, BP 53 38041 Grenoble CEDEX 9, France [email protected] C. Brun et al. (Eds.): Num. Sim. of Turbulent Flows & Noise Generation, NNFM 104, pp. 283–308. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com

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It has been noted that in incompressible turbulent boundary layer flows turbulent quantities close to the wall can be scaled by the so called inner coordinates that are based on wall shear stress. The region for which these inner coordinates can be applied can be divided into viscous and turbulent region in which viscous stresses are dominant or negligible, respectively, with respect to turbulent stresses. In the socalled buffer layer both stresses are of comparable intensity. In the viscous region, the velocity profile is linear with respect to wall distance, in the turbulent region, it has been found that the velocity profile depends logarithmicly on wall distance. This law of the wall can be considered as the back-bone of wall models implemented in current simulation codes. Unfortunately, the law of the wall fails in some flow situations, namely turbulent flows with separation/reattachment, with strong pressure gradients and compressible flows. Alternative solutions have been sought for. For separated incompressible flows, it has been shown by e.g. Simpson [27] that the standard law of the wall fails and that a quadratic dependence of the velocity from the wall distance is observed. The failure of the law of the wall is a consequence of the wall shear stress going to zero at the points of separation and reattachment. Therefore, another quantity has to control the near wall profiles. In several works a scaling based on the local pressure gradient has been proposed [31, 28, 29]. Manhart et al. [18] have proposed a coupled scaling based on both, wall shear stress and local pressure gradient, and shown that this scaling is universal for incompressible turbulent flows with or without pressure gradients. For compressible flows, even for attached flows, strong density and viscosity changes have to be taken into account when formulating velocity or temperature scales. It was expressed for long by Morkovin [22] in the following terms: ’The large scale motion should be statistically coupled to the thermal field almost exclusively through mean values of ρ , μ , λ , and the generalized law of the wall so that with a variable lateral stretching factor, it may resemble the incompressible motion.’ While the so called van Driest transformation was sufficient for adiabatic boundary layers to account for density changes at the wall, extensions to non adiabatic cases needs to reflect fluid properties gradients at the wall. This is the case for e.g. in a fully developed compressible channel flow with constant wall temperature. When flow distorsion acts on the top of a compressible wall layer, a complex relation exists between wall friction, wall pressure gradient, and wall heat flux. It is one of the objectives of the present paper to develop scaling laws valid up to such complex compressible flow configuration. In the next section, we present the analysis of near wall flow quantities for compressible and incompressible flow that undergo separation and reattachment. In section 2.2, we introduce scaling parameters and a suitable non-dimensionalization. During the following sections, we introduce wall laws for the viscous regions in compressible (sec. 2.3) and incompressible (sec. 2.5) flows and present formulations for the turbulent layer. The numerical codes and the analyzed test cases are presented in section 3. The results for incompressible flow are discussed in section 4, the ones for compressible flow are discussed in section5. Section 6 concludes the paper.

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2 Near Wall Scaling in Turbulent Flows We analyze the near-wall behavior of equilibrium velocity profiles in the proximity of a wall. The central ingredient of our analysis is the assumption that close to the wall the flow quantities are governed by wall shear stress, streamwise pressure gradient and wall heat flux in the general compressible case. As a consequence of this assumption, we formulate velocity, temperature and length scales by these quantities. If there was a universal behaviour of the near wall flow quantities, their profiles should collapse after being made dimensionless by these scales. We start our analysis by the governing equations.

2.1 Governing Equations The compressible Navier-Stokes equations write in a conservative form for the mass the momentum and the total energy, respectively

∂ (ρ u j ) ∂ρ + =0 ∂t ∂xj ∂ (ρ ui ) ∂ + (ρ ui u j + pδi j − μ Si j ) = 0 ∂t ∂xj ∂ (ρ e) ∂ ∂T + (ρ e + p)u j − μ Si j ui − λ =0 ∂t ∂xj ∂xj

(1) (2) (3)

The total energy, the strain rain tensor and the equation of state are defined as, respectively 1 ρ e = ρ cv T + ρ u2i 2 1 ∂ ui ∂ u j 2 ∂ uk + − δi j Si j = 2 ∂ x j ∂ xi 3 ∂ xk p = ρ RT

(4) (5) (6)

The fluid is assumed to be an ideal gas (air) with constant ratio of specific heats c γ = cvp = 1.4, R = c p − cv = 287JKg−1K −1 and constant Prandtl number Pr = 0.7. A power law μ = μ (T ) = μw (T /Tw )0.7 is considered for the molecular dynamic viscosity, as a relative good approximation of the Sutherland law, and thermal conc μ (T ) ductivity λ (T ) is given by λ (T ) = p Pr . We consider equilibrium velocity profiles in two-dimensional flows for which the mean streamwise momentum balance comprises contributions of convection, pressure gradient, diffusion and Reynolds stresses. We apply ensemble averaging on equations (2) - (3) and neglect convection contribution to derive the following equations for mean velocity and mean temperature,

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∂ p ∂ ∂ u + μ − ρ u v − ρ uv 0=− ∂x ∂y ∂y γ ∂ p ∂ ∂ Ti u + λ − ρ C p T v − ρ uu v 0=− γ −1 ∂x ∂y ∂y

(7) (8)

where the total temperature [5, 21] is defined as Ti = T +

Prm 2 u 2c p

(9)

with the mixing Prandtl number Prm given by [20]. In the viscous region, the convective and Reynolds terms can be neglected. If, like it is done in the standard boundary layer assumptions, we neglect the second derivative in streamwise direction and assume that the streamwise pressure gradient is constant with respect to the wall normal direction y, then the solution is completely governed by the wall shear stress τw , the local pressure gradient ∂∂p x , and the wall heat flux qw , including as well the fluid properties at the wall (viscosity μw , conductivity λw and density ρw at the wall).

∂P , μw , ρ w ) ∂x ∂P T (y) = f (y, τw , , q w , μw , ρ w , λ w ) ∂x ∂P , q w , μw , ρ w , λ w ) Ti (y) = f (y, τw , ∂x

U(y) = f (y, τw ,

(10) (11) (12)

The main issue concerning a possible formulation of a wall model is the range of validity, i.e. how far away from the wall the underlying assumptions can be made.

2.2 Scaling Parameters Provided, there is a universal behavior of the near-wall velocity profile, we seek for a proper scaling, i.e. proper velocity, length, temperature and time scales to nondimensionalize the momentum equation. Since, we seek these reference values for the near-wall region, we refer to them as inner scales. The standard reference velocity for inner scaling is the friction velocity uτ , based on wall shear stress. 1/2 τw uτ = ρw

(13)

Following Simpson [26], we define an additional velocity scale u p , which is based on the streamwise pressure gradient at the wall. We have to take wall quantities because of compressibility effects.


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μw ∂ p 1/3 u p = 2 ρw ∂ x

(14)

According to Simpson, the former is relevant when the wall shear stress dominates the pressure gradient, the latter is relevant when the pressure gradient is dominating (e.g. close to separation or reattachment). Here, we propose instead to use a combined scale uτ p that is taking both into account. % (15) uτ p = u2τ + u2p We refer to this velocity scale as extended velocity scale or extended inner scaling, alternatively. In order to describe which of the two effects, friction and pressure gradient, is preponderant, the ratio α ∈ [0, 1] is defined as [18]

α=

u2τ u2τ = u2τ + u2p u2τ p

(16)

We extend similar dimensional arguments to define a friction temperature Tτ =

−qw , ρw c p uτ

(17)

u2p , 2c p

(18)

a pressure gradient based temperature Tp = and a combined temperature Tτ p = Tτ + Tp. The ratio β ∈ [0, 1] defined by

β=

Tτ Tτ + Tp

(19)

(20)

quantifies the relative importance of each of the two involved inner temperature scales, Tτ and Tp . Based on the extended inner scale for the velocity, uτ p , and for the temperature, Tτ p , the non-dimensional velocity, u∗ , the non-dimensional total temperature Ti ∗ and the non-dimensional length y∗ are defined as u uτ p Ti Ti ∗ = Tτ p ρ w yuτ p y∗ = μw u∗ =

(21) (22) (23)

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With this scaling, the velocity and total temperature profiles in the viscous region including pressure gradient effects write in non-dimensional form as a function of only three non-dimensional parameters u∗ = f (y∗ , α ) ∗

(24)

∗

Ti = f (y , α , β ) The ratio between classical (u+ ; Ti + ; y+ ) and extended (u∗ ; Ti ∗ ; y∗ ) inner coordinates can be expressed in terms of α and β , √ u∗ = α + u Ti ∗ =β Ti + y∗ 1 = √ y+ α

(25) (26) (27)

2.3 Viscous Region of Compressible Flows After integration from the wall, equations (7) and (8) write: ∂ p μ (T ) ∂ u∗ (1 − α )3/2y∗ + sign(τw )α = sign μw ∂ y∗ ∂x

y∗ √ 2γ μ (T ) ∂ Ti ∗ ∂ p sign (1 − = β ) 1 − α Pr u∗ dy∗ μw ∂ y∗ γ −1 ∂x 0 √ + sign(−qw)β α Pr

(28) (29)

A second integration from the wall of equation (28) leads to a parabolic velocity profile [31, 30]: ∂ p (1 − α )3/2 c∗ 2 ∗ ∗ c∗ u (y ) = sign(τw )α y + sign y2 (30) ∂x 2 where yc∗ =

y∗ 0

2

yc∗ 2 =

y∗2 0

μw μw ∗ dy∗ = y μ (T ) μ1

(31)

μw 2 μw ∗ 2 dy∗ = y μ (T ) μ2

(32)

are integral lengthscales. A second integration of Equation (29) from the wall leads to a polynomial total temperature profile:


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√ Ti ∗ (y∗ ) = Tw∗ + sign(−qw)β α Pryc∗ (33) √ 2 γ ∂ p α 1 − α μw c∗3 (1 − α ) μw c∗4 sign (1 − β )Pr + y + y γ −1 ∂x 3 μ1 3 12 μ2 4 with 3

yc∗ 3 = 4

yc∗ 4 =

y∗3

(34)

0

μw 3 μw ∗ 3 dy∗ = y μ (T ) μ3 4 μw μw ∗ 4 dy∗ = y μ (T ) μ4

(35)

0

y∗4

For flows with zero pressure gradient (α = β = 1.0), we obtain the classical law of the wall for compressible boundary layer equations with u∗ = u+ , Ti ∗ = Ti + and yc∗ = yc+ , u+ = yc+ γ −1 Pr Mτc 2 u+2 = Pr yc+ Ti + − Tw+ = T + − Tw+ + 2

y+ μw dy+ yc+ = μ (T ) 0

(36) (37) (38)

which is consistent with the expected linear law at the wall for both the velocity and the total temperature, and accounts for near wall temperature variations in an alternative way to the existing compressible scaling [1, 11, 14]. Note that the integral lengthscale yc+ is equivalent to the standard y+ unit in the asymptotic incompressible case. For flows with zero wall friction (α = 0.0) and zero wall heat flux (β = 0.0), we obtain a quadratic law accounting for the pressure gradient effect only: 1 c∗ 2 y 2 2

(39)

Ti ∗ (y) = Tw∗ +

(40)

u∗ (y) =

1 γ μw 4 Pr yc∗ 4 12 γ − 1 μ2

2.4 Turbulent Region of Compressible Flows In a compressible channel flow, the resulting integral Reynolds number based on the integral lengthscale defined above Recτ = hc+ =

h+ 0

μw dy+ μ (T )

(41)

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is a realistic measure of the turbulent state of the flow which accounts for local fluid properties in the near wall region. This is of particular interest for flows with non-adiabatic wall conditions which are affected by strong temperature gradients at the wall. For compressible wall layers, the van Driest transformation [1, 10] consists in correcting the velocity and temperature variations at the wall based on changes in density. Changes in viscosity are accounted for to generalize the van Driest transformation to wall bounded flows with strong wall temperature gradients. For an attached boundary layer (α = 1.), we write 2

ρ 1 du+ = ln yc+ + Cuc ρ κ 0 w 2

Ti + + y μw ρ Prt == + dTi + = ln yc+ + Ctc yc+ μ ρw κ Tw c γ −1 + c2 +2 Prm Mτ u = T + 2

uc+ = Ti c+ − Tw+

u+ + y μw

yc+ μ

(42)

(43)

For non zero presure gradient situations (α = 1), the development of wall models in the turbulent region is not trivial because of coupling between distorsion and compressibility effects. Extension of the incompressible case (see following sections) should be considered for the mean profiles. For the RMS profiles, the situation is more obvious. Based on the α and β parameters previously defined, the velocity and the temperature fluctuations might be also affected by both distorsion and compressibility effects, respectively, 2 + μ √ y ρ + w uc∗ α c+ u (44) rms = y μ ρw rms 2 √ y+ μw ρ + c∗ v (45) vrms = α c+ y μ ρw rms 2 + μ √ y ρ + w α c+ w (46) wc∗ rms = y μ ρw rms 2 y+ μw ρ + c∗ T (47) Trms = β c+ y μ ρw rms

2.5 Viscous Region of Incompressible Flows The functional relationship for equation (25) is obtained by integrating the momentum equation (7) in wall normal direction under the above mentioned conditions.


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Dividing this equation by u3τ p /ν yields its dimensionless form, which is given here without convection and Reynolds terms. ∂ p ∂ 2 u∗ (1 − α )3/2 + 0 = sign (48) ∂x ∂ y∗ 2 P1

D1

The signs of the pressure gradient and wall friction are necessary to define the direction of the flow and the direction of the pressure forces, in complement to the amplitude of α . We integrate equations (48) twice in order to obtain an explicit formulation for the dimensionless velocity u∗ (or alternatively we consider equation 30 in the limit of incompressibility). ∂ p 1 ∗ ∗ ∗ u (y ) = sign(τw )α y + sign (1 − α )3/2y∗ 2 (49) ∂x 2 The range of validity of equation (49) is of high importance for the construction of explicit wall models. It has to be defined in terms of the dimensionless wall distance y∗ and will certainly be a function of α . It is the purpose of this paper to investigate the range of validity for various α under various flow conditions. The range of validity will strongly depend on how fast the Reynolds terms and the convective terms gain weight in the momentum balance, when moving away from the wall. If, however the increase of the Reynolds term with respect to wall distance follows a universal form (determined by wall conditions), then we would observe a universal behavior of the velocity profiles even when equation (49) is no longer valid. This is the case e.g. in the buffer layer of a zero pressure gradient boundary layer which renders a universal velocity profile and is described by empirical means only up to date. The classical law of the wall u+ = y+ can be written in terms of the extended inner coordinates u∗ = α y∗ (50) It has to be noted that in case of vanishing streamwise pressure gradient (α = 1.0), the classical law of the wall for the viscous region u∗ = y∗

(or alternatively: u+ = y+ )

(51)

is recovered. It is generally accepted now, that this equation is valid below y+ = 5. For larger wall distances, the influence of the Reynolds and convective terms in the momentum balance (7) gives rise to a deviation from this law (in the so called buffer layer region). When the wall shear stress is zero, a quadratic velocity profile is obtained in the viscous region 1 u∗ = y∗2 2

(52)

For realistic flow configurations, both contributions are blended in equation (49).

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2.6 Turbulent Region of Incompressible Flows The turbulent region is defined as the region in which turbulent stresses are dominant and viscous stresses can be neglected. For the case of zero pressure gradient an asymptotic solution for attached flows at high Reynolds numbers can be written in standard inner coordinates [33] u+ =

1 ln(y+ ) + C. κ

(53)

Where κ ≈ 0.41 and C ≈ 5.0. For the case of zero wall shear stress and an adverse pressure gradient, Tennekes and Lumley [33] derived another asymptotic solution u y u p + b, = a ln up ν

(54)

in which a ≈ 5 and b ≈ 8 according to experimental data of Stratford [30]. It is evident and has been shown by many authors (e.g. [27]), that equation (53) is not applicable for flows near separation or reattachment and that equation (54) is not valid in flows without strong pressure gradients. The main point is the fact that, according to asymptotic theory, there exists a log-layer in both limits. This asymptotic behaviour will be obtained in the high Reynolds number limit for wall distances that are small compared to the boundary layer thickness and large compared to the thickness of the viscous sublayer. Whether it is also obervable in mixed regions, i.e. in regions of intermediate α , will be checked with DNS data throughout this paper. Based on the two length scales uτ and u p , Shih et al. [25] have done an asymptotic analysis of the near wall flow for intermediate situations, i.e. α -values. Similarly to Manhart et al. [18], they defined a combined velocity scale uc = uτ + u p . According to Tennekes [32], they decompose the momentum equation into two parts, in which the pressure gradient or the wall shear stress are set to zero, respectively. This leads to two pairs u, u v of which the first obeys a momentum equation with zero pressure gradient and the second one with zero wall stress. Thus they have two asymptotic solutions for the inertial sublayer, one for the wall stress part and one for the pressure gradient part. They match the coefficients in such a way that they fit the two limiting solutions, equations (53) and (54), with their respective coefficients. This gives a smooth variation of the log law between the two limiting solutions. With the scaling uτ p instead of uτ , equation (53) gives the following for the wall stress contribution 1 uτ ln(y∗ ) + C1 κ uτ p uτ uτ 1 ln( ) +C C1 = uτ p κ uτ p

u∗1 =

or in terms of the parameter α :

(55) (56)


1√ α ln(y∗ ) + C1 κ √ 1 √ ln( α ) + C C1 = α κ

u∗1 =

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(57) (58)

The same can be done for the pressure gradient part of the solution, equation (54) √ u∗2 = a 1 − α ln(y∗ ) + C2 (59) √ √ (60) C2 = 1 − α ln( 1 − α ) + b , with a and b fitted to the zero wall stress solution. In summation, one can write u∗ = Aln(y∗ ) + B (61) ∂ p √ 1√ a 1−α α + sign A = sign(τw ) κ ∂x √ √ ∂P √ 1 √ B = sign(τw ) α ln( α ) + C + sign 1 − α ln( 1 − α ) + b κ ∂x with the constants a and b fitted to the zero wall shear stress solution. We have two branches of solution due to the fact that the pressure gradient can be favourable with the wall shear stress or adverse with it, respectively. Note that this solution is obtained under the assumption of an asymptotic solution at high Reynolds numbers in which the boundary layer thickness is sufficiently large compared to the thickness of the viscous sublayer. This is equivalent with the assumption that the outer layer effects will be sufficiently separated from the viscous sublayer. Whether a logarithmic layer can indeed be observed, is one of the question being adressed in this paper.

3 Test Cases We investigate several types of complex flows, compressible turbulent channel flow with an adverse force that leads to separation on the lower wall of the channel (CCH), incompressible turbulent boundary layer along a flat plate with separation and reattachment (BL) ([17]) and incompressible flow over a periodic arrangement of two-dimensional hills (PH) ([2]). The incompressible flows have been computed by means of Direct Numerical Simulation (DNS), the compressible one by wall resolved Large Eddy Simulation (LES). In the following, we shortly describe the numerical schemes used for the simulations.

3.1 Incompressible Flow Solver We solve the Navier-Stokes equations for incompressible flows, by a Finite Volume solver on a staggered, non-equidistant Cartesian grid. The spatial approximations

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are second order accurate and use centered interpolations and differentiations. Time integration is performed via a fractional step method using a leapfrog scheme with time lagged diffusion [24] for the momentum equation (1). The Poisson equation for the pressure correction is solved by Stone’s strongly implicit procedure (SIP). See e.g. Ferziger and Peric [7] for discussion of these standard methods. The zonal grid implementation used for the simulation of the separated flat plate boundary layer is based on overlapping grids with a refinement factor of two. We used full two-way coupling and conservative second order interpolation at the fine/coarse interface. The algorithm is documented in detail in [16]. In the periodic hill case, the curved geometry of the hills is represented by an Immersed Boundary method. A third order least squares interpolation at the body surface allows a stable and highly accurate interpolation at the surface [23].

3.2 Compressible Flow Solver We take advantage of the conservative form of the compressible Navier-Stokes equations (1-3) [6], which are further discretised on a cartesian collocated grid at each time step, with a fourth-order extension of the Mc Cormack scheme proposed by Gottlieb & Turkel [9]. The time advancement is split in two steps, a predictor and a corrector step, which results in a globally centered scheme of second-order accuracy in time. The computing time step is determined based on both a CFL and a viscous stability criterion which must both be considered for the present wallbounded compressible flow in a channel. MPI parallelisation is performed to allow for solving a spatially developing compressible flow in a channel.

3.3 Separating Turbulent Boundary Layer along a Flat Plate (BL) The turbulent boundary layer simulation is designed according to an experiment performed by [13]. In this experiment, a turbulent boundary layer developing along a circular cylinder is subjected to a streamwise adverse pressure gradient until separation occurs. In comparison to the experiment the Reynolds number has been lowered to Reθ = 870 (based on inlet free-stream velocity and inlet momentum thickness) because of computational expenses. The simulation has been performed in a rectangular domain selected in a way to cover the separation bubble (see Figure 1). Based on wall shear stress of the oncoming boundary layer, the grid spacings were Δ x+ = 11.7, Δ y+ = 1.6 and Δ z+ = 7.2 in streamwise, wall normal and spanwise direction, respectively. It was shown in a grid study and by comparison with the experiment of [13] that the resolution and the size of the computational domain were sufficient to get a highly accurate data base of this flow. We used a fully turbulent inflow condition, generated by fluctuations taken from a downstream position which are superposed to a time averaged boundary layer profile corresponding to the desired boundary layer thickness and Reynolds number. In spanwise direction, we used periodic conditions, at the outflow a zero-gradient for the velocities. The bottom wall is described by a no-slip condition and at the top, the desired pressure profile is


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y/δ0 TD ITD ID

Fig. 1 Streamlines for BL case

x/δ0

Fig. 2 Instantaneous streamwise velocity component at the wall and in a streamwise/wall-normal plane

prescribed in combination with a zero gradient for the velocities. The separation of the boundary layer (Reθ = 870) is driven by an adverse pressure gradient which is adjusted to yield a marginal separation with a maximal backflow coefficient of 70%. The separation bubble starts at x/δo = 44 and is thin compared to the inlet boundary layer thickness. The dynamics in the separation zone are strongly affected by large scale structures reaching from the shear layer above the separation bubble to the wall. Figure 2 gives an idea of the complexity of the instantaneous flow. The separation line is not fixed, but strongly fluctuates in space and time. The labels in Figure 1 mark the regions of incipient detachment (ID, 1% backflow), intermittent transitory detachment (ITD, 20% backflow) and transitory detachment (TD, 50% backflow) according to the definitions of [27]. TD is at the same position as τw = 0.0. The point of incipient detachment is identifiable with the first occurrence of these strong large scale structures. According to Simpson, this is also the point at which the mean velocity profile starts to deviate from the logarithmic law of the wall. From this point on, one would expect that standard wall models based on the law of the wall start to fail. From x/δ0 = 34 (ITD) on both, linear and logarithmic laws start to fail.

3.4 Channel Flow with Constrictions (PH) Our second test case is the turbulent flow in a channel with periodically arranged hills at the bottom wall. The numerical setup was introduced by [19]. A detailed investigation of this flow at Re = 10595 has been undertaken by [8] on the basis of a

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y/h

x/h

Fig. 3 Streamlines for PH case

LES. Based on hill height h, the channel extends 9.0h in streamwise, 4.5h in spanwise and 3.035h in wall-normal direction. The hills are arranged at the distance 9.0h. The Reynolds number based on the bulk velocity and the hill height is Re = 5600. In streamwise and spanwise direction periodic boundary conditions are used. On the hill surface a no-slip condition is applied as well as on the top of the channel wall. The configuration can be seen in Figure 3 which shows streamlines of the average velocity. The flow separates at the hill crest at x/h = 0.17 and forms a large separation bubble reaching x/h = 5.0. The separation in this case is not only due to an adverse pressure gradient but also the result of the strong streamwise curvature of the lower wall. Thus it establishes a completely different test bed than the flat plate boundary layer for our investigations. The computational grid has 233 · 106 grid cells. The resulting near-wall grid spacings based on maximum wall shear stress are Δ x+ = 8, Δ y+ = 6.3 and Δ z+ = 1.2 in streamwise, spanwise and wall normal direction, respectively. A considerable negative wall shear stress exists at x/h ≈ 3.0 within the recirculation bubble which shows that a strong backflow develops in the recirculation zone. A second tiny recirculation zone is present at x/h ≈ 7.2 just before the hill. On the windward side of the hill, a sharp peak of wall shear stress evolves that is the result of the strong acceleration of the flow due to the convergence of the channel in this region. This acceleration is driven by a pressure drop, i.e. a negative streamwise pressure gradient. Shortly after the main separation, the roll-up process in the shear layer bounding the separation zone leads to large scale fluctuations of high turbulence intensity (Figure 4). Nevertheless, there is a relatively calm region in the separation bubble. The fluctuations of the separation line are comparably smaller than in the flat plate case, since it is also determined by the point of maximum curvature at the hill crest. The reattachment line however fluctuates strongly and instants in time occur, during which the whole bottom wall shows reverse flow. y/h

x/h Fig. 4 Flow over a periodic arrangement of hills: instantaneous streamwise velocity component. The black line indicates u = 0


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Tw Fig. 5 Geometry of the spatially developing channel flow with distorsion y/h

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Fig. 6 Mean velocity (left) and temperature (right) isovalues for CCH case; Re=4880 and M=0.7

3.5 Compressible Channel Flow with Adverse Force (CCH) LES is performed with the Filtered Structure Function model [15] to compute a turbulent 3-dimensional compressible channel flow (CCH) for Re = 4880 and M = 0.7. An artificial distributed adverse body force is used to yield flow distorsion at the bottom wall (figure 5). Both the streamwise velocity field and the temperature field are perturbed by the source term and the deviation from a symetric profile of the channel will be further modeled. The computational domain is split in 7 subdomain of size Lx × Ly × Lz = 4π h × 2h × 43 π h in the streamwise x, wall-normal y and spanwise z directions, respectively, h being the half-height of the channel. The first subdomain is used to generate realistic turbulent inflow conditions for the remaining part of the channel. The number of grid points for each subdomain is nx × ny × nz = 128 × 65 × 81 that is to say about 5 millions points in a whole. A hyperbolic-tangent type stretching is applied in the wall-normal direction to capture the main physical aspects in the near wall region down to y+ w = 0.2. In this context, any side effects due to the SGS model might be reduced at the wall.

4 Incompressible Flow We now turn to the near-wall behaviour of the terms in the momentum balance and the behaviour of the velocity profiles. In what follows, we investigate first the velocity profiles normalized by uτ p . We will check (i) the validity of the viscous law of the wall and (ii) whether there exists an inertial region in which the profiles follow a log law. Then, we investigate the terms of the momentum balance. The

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main question being addressed here is whether they behave in a universal way under the normalization by uτ p . For this, we select positions in the flow cases in which specific values of α are obtained. Here, we restrict ourselves to three values of α values, namely α ≈ 0.0 being representative for a situation with zero wall shear stress, α ≈ 0.3 being representative for similar strengths of pressure gradient and wall shear stress in the momentum equation and and α ≈ 0.6, which stands for an intermediate situation. For higher values, the wall shear stress is dominant and the profiles can be approximated in a sufficiently accurate way by the standard laws. One has to take care of the signs of the wall shear stress and the streamwise pressure gradient. When both have the same sign, their effect on the velocity profile is in the same direction. When they have opposite signs, they act in opposite direction and it can be possible that the flow changes direction at a certain wall distance. Therefore, we distinguish between these two situations.

4.1 Velocity Profiles It is well known that the standard law of the wall (equations 51 and 53) can not be applied in flows with strong pressure gradients. We demonstrate this by plotting in figure 7 the velocity profiles used in the plots hereafter in standard wall units, u+ and y+ . The great scatter of the velocity profiles in standard inner coordinates and their poor accordance with the standard law of the wall is obvious. As a result, wall shear stress predictions can be wrong even at moderate wall distances. Some profiles have a thin recirculation zone - below 20 wall units. It is evident that such change in sign of the velocity profile is not captured by the standard law of the wall. in what follows, we sort the profiles according to their α value. We start our evaluation of the velocity profiles for α ≈ 0.6 which marks an intermediate situation in which the profiles in the viscous region are mainly determined by the wall stress with already considerable influence of the pressure gradient. For the case when wall stress and pressure gradient act in the same direction (figure 8), the viscous region is not strongly affected by the quadratic term in equation (49) and the classical law of the wall is still a good approximation. There is a logarithmic

25 x 20 2.44*log(x) + 5 15 u^+

Fig. 7 Velocity profiles at various streamwise positions in separating boundary layer flow and flow over periodic hills in standard inner coordinates. Lines: law of the wall, equations (51) and (53)

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Fig. 8 Velocity profiles in extended inner coordinates for α ≈ 0.6. Left: viscous region; right: turbulent region. (τw ) · (∂ P/∂ x) > 0

2


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Fig. 9 Velocity profiles in extended inner coordinates for α ≈ 0.6. Left: viscous region; right: turbulent region. (τw ) · (∂ P/∂ x) < 0

region in the range 5 < y∗ < 20, which is not matched by the classical log law. The asymptotic theory is closer to the profiles but has a too high gradient. The best fit is obtained for A = 2.5 and B = −0.5 in equation (62). Next, we take a look at the situation in which the pressure gradient acts opposite as the wall shear stress (see figure 9). The profiles show a thin recirculation zone below y∗ ≈ 20. In this situation, the standard law of the wall fails even in the viscous sublayer. The extended law of the wall (49) gives a good approximation up to y∗ ≈ 2.0. It also would predict the reversal of the flow but too early. A log layer is only present in one of the profiles shown between 10 < y∗ < 40. It is not clear whether a log layer would come up at higher Reynolds numbers. If this region was interpreted as a log layer, neither the standard log law nor the asymptotic theory would be sufficient to predict it. When the pressure gradient is increased, at α ≈ 0.3, there is still a log law, when the pressure gradient acts in the same direction as the wall stress (Figure 11). The log region extends between 5 < y∗ < 30. But, its coefficients are different from both, standard law of the wall and asymptotic theory. The same can be said for the case when the pressure gradient and wall stress act in opposite direction (Figure 10). There is a log law between 5 < y∗ < 30, but its coefficients are different

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6


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classical asymptotic theory 3.5*log(x) - 2.5

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Fig. 10 Velocity profiles in extended inner coordinates for α ≈ 0.3. Left: viscous region; right: turbulent region. (τw ) · (∂ P/∂ x) > 0

from both, standard log law and asymptotic theory. The thin reverse flow region is a consequence of the pressure gradient acting in opposite direction than the wall stress. Under both signs of pressure gradient, the extended law of the wall performs well in the viscous region which extends to y∗ ≈ 2. The standard law of the wall is not valid at the points of zero wall shear stress, i.e. the points of separation and reattachment. From Figure 12 it becomes evident that the extended law of the wall gives a good approximation in the viscous region that reaches y∗ ≈ 2. As in the other cases, there exists a log layer, for which A = 4 and B = −4 are accurate values of the coefficients. Note, that these are not in accordance with the ones proposed by Tennekes and Lumley [33]. The inspection of velocity profiles under the extended scaling has revealed the following observations: The viscous region extends up to at least about y∗ = 2. Below this, the extended law of the wall gives good approximations of the velocity profiles. It is even able to predict flow reversal in thin layers at the wall. A log region is present in most cases, except for α ≈ 0.6 and flow reversal close to the wall. For this situation, it is not clear whether a log region would come up at higher

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Fig. 11 Velocity profiles in extended inner coordinates for α ≈ 0.3. Left: viscous region; right: turbulent region. (τw ) · (∂ P/∂ x) < 0


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Fig. 12 Velocity profiles in extended inner coordinates for α ≈ 0.0. Left: viscous region; right: turbulent region

0

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Fig. 13 Momentum balances in extended inner coordinates for α ≈ 0.6. Left: (τw ) · (∂ P/∂ x) > 0 Right:(τw ) · (∂ P/∂ x) < 0. viscous term D1; Reynolds and convective term (R1+C1); + pressure term P1

Reynolds numbers. The log region extends in the most cases from 5 < y∗ < 30. The coefficients of the log law do not coincide with the ones obtained from asymptotic theory. The important point is that velocity profiles taken from various flow situations and normalized by the extended inner units coincide up to about 30 wall units. This means that up to this wall distance the flow is mainly governed by wall shear stress and streamwise pressure gradient. The outer flow affects the velocity profiles only at larger wall distances. Equation (49) is a good approximation for the profiles to about 2 wall units. At first glance the range of validity seems to be small compared to the 5 wall units being the limit of validity for the linear law of the wall in zero pressure gradient boundary layers. However, one has to bear in mind that in outer coordinates this is given in terms of δ0 and h, respectively, which is defined by the ratio (ν /uτ p )/δ0 and (ν /uτ p )/h. For the profiles considered here, 2y∗ > 0.025δ for the separating boundary layer and 2y∗ > 0.01δ . For comparison, 5 classical wall units are about 0.014δ0 in the oncoming flow of the separating turbulent boundary layer.

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4.2 Momentum Balances We consider the mean streamwise momentum balance (7). It comprises contributions of convection (C1), pressure gradient (P1), diffusion (D1) and Reynolds stresses (R1). We follow [12] and add the convective (C1) and Reynolds (R1) terms. The main issue related to these balances is the relative importance of each term and the range where they can be neglected. In addition to this the question arises if universality can be found for separate terms or their respective sums under a certain scaling. For the wall shear dominated situation, we selected three profiles for the case when shear stress and pressure gradient act in the same direction and three for the other, in total four profiles from (BL) and two from (PH). In Figure 13 terms of the momentum balance (equation 7) are plotted: viscous term (D1), pressure term (P1) and the sum of Reynolds and convective terms (R1 + C1). The profiles are plotted on top of each other in extended inner coordinates. One can observe that the assumption of a constant pressure gradient is valid in good approximation up to a wall distance of y∗ = 50. Due to the normalisation, the pressure gradient assumes −(1 − α )(3/2) = −0.25 at the wall. The absolute value of the viscous term is equal to the pressure term at the wall since the Reynolds terms go to zero. The most important observation is that despite deviations close to the wall the Reynolds and diffusive terms of all cases have similar slopes and reach the same maximum amplitudes. The Reynolds terms reach their maxima at about y∗ = 4. The viscous law of the wall extends up to about the half of that wall distance. At larger wall distances, all Reynolds terms fall together indicating universal behaviour under the extended normalization for the wall distances shown here. Now we turn to the situation where wall friction and pressure gradient act in opposite directions. The terms of the momentum balance are plotted in Figure 13 (right). The agreement among different cases is even better than in the latter case where pressure gradient and wall friction have the same sign. This time all Reynolds terms and all convective terms, respectively, lie on top of each other. Only for the periodic hill (PH) case deviations in the pressure gradient can be observed. The assumption of a constant pressure gradient starts to fail at y∗ = 10.

1

0.5 D1*,(R1*+C1*),P1*

Fig. 14 Momentum balance in extended inner coordinates for α ≈ 0.0. τw ≈ 0. ——— BL at x/δ0 = 42; – · – · – · BL at x/δ0 = 73; · · · · · · PH at x/h = 7; viscous term D1; Reynolds and convective term (R1+C1). + pressure term P1;

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Fig. 15 Left: Inner velocity scales based on friction and wall heat flux (dashed line), pressure gradient at the wall (full line) and combined quantities (symbols). right: α ratio in the distorted compressible channel at M = 0.7 and Re = 4880

When the wall shear stress is zero (α ≈ 0), we do not have to distinguish between the two cases depending on the sign of pressure gradient. The wall friction is zero and the value of the pressure gradient in the non-dimensional momentum balance yields one. The limiting behaviour can be observed in Figure 14. As in the previous cases, differences among the cases can be mainly observed close to the wall. Apart from the wall all profiles behave similarly. As in the preceeding case the profile for the periodic hill (PH) shows a pressure gradient which does not remain constant over y∗ ≈ 15. The sum of Reynolds and convective terms (R1+C1) reaches a maximum at about 3 extended inner units. They are equally strong as the viscous terms at y∗ = 1. Consequently, the viscous region reaches only about 1 or two wall units. For the boundary layer case, this means at least 0.04δ0 and for the periodic hill case about 0.01h. Note, that this scale in independent of Reynolds number as it is completely determined by the streamwise pressure gradient at the corresponding positions.

5 Compressible Flow 5.1 Situations with Pressure Gradient Effect (α ≈ 0.8) The spatially developing compressible flow in a channel is driven by a favorable pressure gradient (pressure drop). The artificial source term applied at the lower wall of the channel yieds a pressure gradient changed of sign in the streamwise direction. The resulting pressure based velocity u p is about twice larger in amplitude (Fig. 15). The flow distorsion produced by this local pressure gradient leads to a reduction of the friction velocity uτ by 10%. The global uτ p has at last a lower contribution than in non distorted region of the channel flow. A similar trend is obtained for the temperature scales at the wall since the temperature field is also subjected to a distorsion.

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Fig. 17 Mean velocity profiles (left) and mean total temperature profiles (right) for a fully developped compressible channel flow at Re = 4880, M = 0.7. Periodical channel flow (full line) and spatially developed channel flow (symbols)

But, since the Mach number of the present configuration is relatively low (M = 0.7) the effect on Tp , Tτ and Tτ p is weak (not shown). The α parameter of the present flow is shown on figure 15 and confirms the above discussion with an extremum (α ≈ 0.8) reached at the 4πx h 2.55 location. The extended scaling derived in section 2.3 for compressible viscous wall layer is applied to the velocity and temperature flow fields obtained from the present simulation (Fig. 16). The α correction is very efficient to move the validity of the extended law of the wall to y∗ = 5 at least for the mean velocity profile. The correction for the mean temperature profile is much weaker due to to low Mach number and thus not necessary (not shown). RMS profiles for the streamwise velocity are plotted on Figure 16. The α correction consists of a restriction of the scatter obtained on the maximum RMS value and the final plot based on extended scaling fit the classical profiles for incompressible non distorted channel flow, which is a tribute towards universality of the present scaling analysis.


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5.2 Wall Shear Dominated Situations (α ≈ 1.0) For the flow cases without distorsion, wall layers develop for both the velocity and the temperature field and the proper scaling has been shown in section 2.3 and 2.4 to be strongly dependent on compressibility effects in terms of viscosity and density gradients at the wall. The extension of van Driest transformation to the present flow configuration involving non-adiabatic wall conditions is efficient for both the velocity and total temperature profiles which show, once properly rescaled, a trend very close to the incompressible logarithmic law of the wall with a constant slope of about ccu = 6.9 for the velocity and about ctc = 3.1 for the temperature [21, 4, 5]. Once corrected with the local changes in density and viscosity at the wall, the RMS profiles at Recτ = 277 are also fairly close to the incompressible case (for same friction Reynolds number). This is again a strong argument towards universality of compressible turbulent wall bounded flows [3]. uc+ rms

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6 Conclusions We introduced a novel approach to scale near wall profiles of turbulent quantities in complex turbulent flows. One of the main ingredients for this approach is the assumption that in sufficiently high Reynolds number flows, there is a layer close to the wall in which the flow is dominated by wall effects. We introduce an extended scaling that remedies one of the problems that arise from the classical wall stress scaling. The extended scaling defines a combined velocity and a combined temperature based on wall shear stress and streamwise pressure gradient as determining parameters. From this, a non-dimensional velocity, temperature and length can be obtained. Near wall quantities depend of three parameters, non-dimensional wall distance, ratio of friction to combined velocity and ratio of friction to combined temperature. The near wall region is divided in two layers, the viscous and the turbulent layer. In the viscous layer, the law of the wall is extended by the proposed scaling. For compressible flows, the present scaling is applied to both the velocity and the total temperature profiles. First, at temperature scale based on the streamwise pressure gradient at the wall is proposed to better represent situations with low wall heat flux (nearly local adiabaticity) for which the friction temperature is not a relevant quantity. Second, the strong changes in both density and viscosity in the near wall region are accounted for in an integral formulation of the law of the wall. A universal scaling is proposed which is valid for both compressible and incompressible wall shear flows. We assess the extended scaling by direct or highly resolved large eddy simulations of three different flows. These flows include incompressible turbulent boundary layer flows with separation and reattachment along a flat plate, separation and reattachment along a curved wall and a distorted turbulent compressible flow. We demonstrate that in the incompressible flows, the extended law of the wall for the viscous region is valid up to approximately two extended inner units. We find a distinct turbulent region in which the velocity profile follows a log law between about 5 and 30 extended inner units with one exception. At a an intermediate ratio between wall stress to combined velocity and opposite effect of wall stress and pressure gradient, a thin flow reversal at the wall appears and a distinct log layer is not present. This could be a consequence of the low Reynolds number. One of the important results is that the velocity and the stress profiles are universal under the extended scaling up to about 30 wall units.

References 1. Bradshaw, P.: Compressible turbulent shear layers. Annu. Rev. Fluid. Mech. 9, 33–54 (1977) 2. Breuer, M., Peller, N., Rapp, C., Manhart, M.: Flow over periodic hills - numerical and experimental study over a wide range of Reynolds numbers. Computers and Fluids 38, 433–457 (2009) 3. Brun, C., Petrovan Boiarciuc, M., Haberkorn, M., Comte, P.: Large eddy simulation of compressible channel flow. arguments in favour of universality of compressible turbulent wall bounded flows. Theor. Comput. Fluid Dyn. 22(3), 189–212 (2008)


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4. Carvin, C., Debiève, J.F., Smits, A.J.: The near wall temperature profile of turbulent boundary layers. In: AIAA Meeting (1988) 5. Debieve, J.F., Dupont, P., Smith, D.R., Smits, A.J.: Supersonic turbulent boundary layer subjected to step changes in wall temperature. AIAA Journal 35(1), 51–57 (1997) 6. Ducros, F., Comte, P., Lesieur, M.: Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech. 326, 1–36 (1996) 7. Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics, 2nd edn. Springer, Berlin (1997) 8. Fröhlich, J., Mellen, C.P., Rodi, W., Temmerman, L., Leschziner, M.: Highly resolved largeeddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 526, 19–66 (2005) 9. Gottlieb, D., Turkel, E.: Dissipative two-four methods for time-dependent problems. Math. Comp. 30, 703–723 (1976) 10. Huang, P.G., Coleman, G.N.: Van driest transformation and compressible wall-bounded flows. AIAA J. 305, 185–218 (1994) 11. Huang, P.G., Coleman, G.N., Bradshaw, P.: Compressible turbulent channel flows: Dns results and modelling. J. Fluid Mech. 305, 185–218 (1995) 12. Kaltenbach, H.-J.: A priori testing of wall models for separated flows. Phys. Fluids. 15(10), 3048–3068 (2003) 13. Kalter, M., Fernholz, H.H.: The influence of free-stream turbulence on an axisymmetric turbulent boundary layer in, and relaxing from, an adverse pressure gradient. In: 5th European Turbulence Conference, Siena 1994 (1994) 14. Lechner, R., Sesterhenn, J., Friedrich, R.: Turbulent supersonic channel flow. Journal of Turbulence 2, 1–25 (2001) 15. Lesieur, M., Métais, O.: New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid. Mech. 28, 45–82 (1996) 16. Manhart, M.: A zonal grid algorithm for DNS of turbulent boundary layers. Computers and Fluids 33(3), 435–461 (2004) 17. Manhart, M., Friedrich, R.: DNS of a turbulent boundary layer with separation. Int. J. Heat and Fluid Flow 23(5), 572–581 (2002) 18. Manhart, M., Peller, N., Brun, C.: Near-wall scaling for turbulent boundary layers with adverse pressure gradient. Theoretical and Computational Fluid Dynamics 22(3-4), 243–260 (2008) 19. Mellen, C.P., Frhlich, J., Rodi, W.: Large-eddy simulation of the flow over periodic hills. In: 16th IMACS World Congress, Lausanne, Switzerland (2000) 20. Michel, M.: Aérodynamique expérimentale. Rebuffet (1962) 21. Michel, M., Quemard, C., Durand, R.: ONERA NT Technical Report (1969) 22. Morkovin, M.V.: Effects of compressibility on turbulent flows. In: Favre, A.J. (ed.) Mécanique de la turbulence, pp. 367–380. CNRS (1961) 23. Peller, N., Le Duc, A., Tremblay, F., Manhart, M.: High-order stable interpolations for immersed boundary methods. International Journal for Numerical Methods in Fluids 52, 1175– 1193 (2006) 24. Schumann, U.: Linear stability of Finite Difference equations for three-dimensional flow problems. J. Comp. Phys. 18, 465–470 (1975) 25. Shih, T., Povinelli, L.A., Liu, N., Potapczuk, M.G., Lumley, J.L.: A generalized wall function. NASA/TM-1999-209398 (1999) 26. Simpson, R.-L.: A model for the backflow mean velocity profile. AIAA J. 21, 142 (1983) 27. Simpson, R.L.: Turbulent boundary-layer separation. Ann. Rev. Fluid Mech. 21, 205–234 (1989) 28. Simpson, R.L., Chew, Y.T., Shivaprasad, B.G.: The structure of a separating turbulent boundary layer. Part 1. mean flow and Reynolds stresses. J. Fluid Mech. 113, 23–51 (1981) 29. Skote, M., Henningson, D.S., Henkes, R.A.W.M.: Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients. Flow, Turbulence and Combustion 60, 47–85 (1998)

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30. Stratford, B.S.: An experimental flow with zero skin friction throughout its region of pressure rise. J. Fluid Mech. 5, 17–35 (1959) 31. Stratford, B.S.: The prediction of separation of the turbulent boundary layer. J. Fluid Mech. 5, 1–16 (1959) 32. Tennekes, H.: Outline of a second order theory for turbulent pipe flow. AIAA Journal 6, 1735 (1968) 33. Tennekes, H., Lumley, J.L.: A first course in turbulence. MIT Press, Cambridge (1972)

High-Order Methods for Large-Eddy Simulation in Complex Geometries Christine Baur, Patrick Bontoux, Michael Kornhaas, Mathieu Minguez, Richard Pasquetti, Michael Schäfer, Eric Serre, and Eric Séverac

Abstract. Developing high-order methods for large-eddy simulation (LES) is of interest to avoid mixing between subgrid scale modeling contributions and approximation errors of the numerical method. Two different approaches are investigated. The first one focuses on the so-called Spectral Vanishing Viscosity LES (SVV-LES) approach, which allows to extend the well known capabilities of spectral methods from laminar to turbulent flows, while the second one rather investigates the possibility of extending a second order finite volume code to higher order approximations. For the SVV-LES approach, a volume penalization like technique is used to address complex geometries.

1 Introduction Spectral methods are known to be well suited for laminar or transitional flows in simple geometries (Cartesian, cylindrical, spherical...). Our goal is here to describe some routes allowing to address more complex flows, especially turbulent flows in complex geometries. In the frame of spectral / spectral element methods the spectral vanishing viscosity (SVV) technique is especially interesting, because allowing to dissipate the energy which accumulates in the high frequency range of the spectral approximation without losing the so-called spectral accuracy, i.e. the exponential decrease of the approximation error with the discretization parameter as soon as Patrick Bontoux · Mathieu Minguez · Eric Serre · Eric Séverac Lab. M2P2, Université Paul Cézanne / Aix-Marseille III, IMT, La Jetée´ , Tech. Château-Gombert, 38 rue Frédéric Joliot-Curie 13451 Marseille, France [email protected] Mathieu Minguez · Richard Pasquetti UMR CNRS 6621, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice, France [email protected] Christine Baur · Michael Kornhaas · Michael Schäfer Technische Universität Darmstadt [email protected]


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the solution is smooth. Initially developed for 1D hyperbolic problems [28, 56], typically the Burgers equation, the SVV technique has been extended to the 3D incompressible Navier-Stokes (NS) equations, both in Cartesian and cylindrical geometries, with very satisfactory results for turbulent rotor-stator flows. To handle complex geometries, one may think to use a domain decomposition technique. Such an approach was previously developed for 2D geometries resulting from gathering rectangles, see [9], and so it was natural to extend it to more complex situations. This method, similar to the well known spectral element method, but in the frame of collocation approximations, was unfortunately not successful because yielding to very ill posed problems [48]. On the contrary, the so-called pseudo-penalization method [40], i.e. a volume penalization method with implicit formulation of the penalization term, has allowed to address complex geometries, especially the square cylinder wake and the flow over a simplified car model (the “Ahmed body” [1]). However, for very high Reynolds number flows the boundary layers cannot be resolved with such an embedding technique and a near wall correction appeared necessary to obtain satisfactory results. At the same time, a high order (4 th order) finite volume method is currently developed, the LES capability being presently based on a classical subgrid scale (SGS) model, i.e. the dynamic Smagorinsky model. The paper provides a synthesis of works carried out these last four years and so partially relies on papers already published, especially [19, 23, 31, 32, 34, 35, 39, 43]. It is organized as follows : In Section 2 we present the SVV-LES modeling and give results obtained for a rotor-stator flow, i.e. a complex flow in a simple geometry. In Section 3, we describe the pseudo-penalization technique together with the near wall correction and give results obtained for the square cylinder and for the Ahmed body. In Section 4 we describe the high order finite volume method presently developed and give some preliminary results for efficiency investigations and benchmark cases.

2 SVV-LES Method - Application to a Rotor-Stator Flow In the frame of collocation methods, our SVV-LES methodology, see e.g. [37], yields to consider the following stabilized dimensionless NS equations : Dt u = −∇p + νΔSVV u ∇·u = 0

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with usual notations (t: time, u : velocity, p: pressure, Dt : material derivative, ν : inverse of the Reynolds number) and where ΔSVV if a SVV modified Laplacian, which depends on the discretization parameter, say N, and such that ΔSVV → Δ if N → +∞. Similar developments can be carried out if a weak approximation of the NS equations is considered, see [37, 59]. The ΔSVV operator is hereafter defined and the SVV-LES approach is then applied to the computation of a turbulent rotor-stator flow.

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2.1 SVV Stabilization Basically, the SVV method relies on the introduction of a discretization dependent artificial diffusive term only active in the high frequency range of the numerical approximation. For 1D partial differential equations, with N for the discretization parameter and uN (x) for the numerical solution, such an additional term reads: VN ≡ εN ∂x (QN (∂x uN ))

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with QiNi the 1D viscosity operator acting in direction i. Thus, we actually use a diagonal matrix form of the operator εN QN , just as one introduces a non-scalar diffusivity when anisotropic media are considered. Coming back to the NS equation, the SVV stabilization can be simply implemented by combining the diffusion and (vectorial) SVV term to obtain,

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The determination of the ”right values” of the SVV parameters, εN and mN , is discussed in [36]. In this paper we suggest, on the basis of numerical experiments, to simply minimize the magnitude of the SVV stabilization term. A similar formulation may be established for the cylindrical coordinate system used for rotor-stator flows, the main complication coming from the fact that the vector Laplacian operator is no-longer diagonal. We refer to [39, 50] for a complete description. The algorithm was first introduced in [39].

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The main features of the Cartesian and cylindrical codes that have been developed, using this SVV-LES methodology, are mentioned hereafter. More details may be found in [6] for the Cartesian code and in [44, 46] for the cylindrical code. • The time-scheme is second order accurate: Second order backward differences are used for the approximation of the time derivative of the velocity. • The SVV modified diffusion terms are treated implicitly. • The convective terms are treated explicitly. A second order Adams-Bashforth extrapolation is used in the cylindrical code whereas an “Operator Integration Factor” Semi-Lagrangian method is used in the Cartesian code. • Projection methods are used to get a divergence-free velocity field. It relies on the so-called unique grid PN − PN−2 approximation in the Cartesian code whereas one uses a preliminary solve for the pressure in the cylindrical code. • Fourier-Galerkin approximations are used in the homogeneous direction, i.e. in the spanwise and azimuthal directions for the Cartesian and cylindrical codes, respectively. Chebyshev collocation approximations are used in the nonhomogeneous directions. • Specific to the Cartesian code: (i) A domain decomposition technique is used in the streamwise direction and (ii) the bluff body is modeled by using a pseudopenalization technique.

2.2 Application to a Rotor-Stator Flow We investigate a rotor-stator flow within a cavity made of two discs enclosing an annular domain of radial extent Δ R = R1 − R0 , where R0 and R1 are the internal and the external radii, respectively. Two stationary cylinders of height H, termed the shaft and the shroud, bound the domain, see Fig. 1. Two parameters define the shape of the system : These may be taken as the curvature parameter Rm = (R0 + R1 )/Δ R and the aspect ratio L = Δ R/H. Here, Rm = L = 5. The reference time and velocity are Ω −1 and Ω R1 , respectively, and the Reynolds number is defined as Re = Ω R21 /ν , where ν is the kinematic viscosity.

Fig. 1 Schematic of the rotor-stator cavity


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The normalized dimensionless coordinates in any meridian plane are (r = rdim /Δ R − Rm , z = 2zdim /H − 1) ∈ [−1, 1]2 (rdim and zdim are the corresponding “dimensioned” variables). No-slip boundary conditions are applied to all walls : ur = uθ = uz = 0, except on the rotating disk (z = 1) where uθ = (Rm + r)/(Rm + 1). The junction of the stationary cylinders with the rotor is regularized employing a boundary layer function, uθ = exp(−(z − 1)/ μ ), where the value of the shape parameter μ = 6. 10−3 provides a reasonable representation of experimental conditions (there is a thin gap between the edge of the rotating disc and the stationary sidewall), while retaining spectral accuracy. Rotor-stator flows are very challenging for numerical modeling, particularly in turbulent regimes, see [47, 49, 50] and references herein. A characteristic feature of such flows is indeed the coexistence of adjacent coupled flow regions involving laminar, transitional and turbulent regions which are completely different in terms of the flow characteristics. Moreover, the turbulence is strongly inhomogeneous and anisotropic because of confinement, flow curvature and rotational effects. A high Reynolds number flow, Re = 106 , is computed to show the capability of the SVV technique to investigate the turbulence features of rotor-stator flows. The computational grid has been chosen relatively coarse for this Reynolds number with 81 × 160 × 81 points in (r, θ , z) directions, respectively, and a time step Δ t = 10−4 was used. As a comparison, this grid is twice coarser than the one used by Andersson and Lygren [2] in their LES of ”case D” study of a ”wide gap” cavity. It permits however to correctly solve the boundary layers : Values of z+ = hzVτ /ν around 0.65 and less than 0.6 are found for the Ekman and Bödewadt layers, respectively. Concerning the SVV parameters we use εN = 2/N and mN = N/2 in all directions, with here N = (80, 80, 80). At Re = 106 , both layers are turbulent as shown by the isolines of the mean turbulent kinetic energy in Fig. 2. The figure provides qualitative evidence that the turbulence is mainly concentrated near the two discs. The maximum of turbulent kinetic energy is located at the rotor-shroud junction where the flow, accelerated by the rotor, impinges the stationary outer cylinder. The flow consists of rather tangled co-rotating vortices which originate near the shroud and move radially inward on the stator and outward on the rotor, following the main flow direction. The growth

Fig. 2 Isolines of the instantaneous fields of Q-criterion for θ = π /4 (top) and θ -averaged turbulent kinetic energy (bottom), Re = 106

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of these vortices strongly affects not only the structure of the two disc boundary layers but also the geostrophic core. This is revealed by plotting the isolines of the Q-criterion in the meridian plane (see Fig. 2). In the vicinity of the inner and outer cylinders there is now a strong mixing between both boundary layers, involving a large number of vortices of different scales. This underlines the important effects that the inner and outer cylinders have on the characteristics of the turbulence. The axial profiles at mid radius of the mean radial and tangential component of the velocity show a more sharpened profiles and a larger core region than at Re = 7. 104 (see Fig. 3). The core rotates at Kuθ /(Ω rdim ) = 0.3919 at mid-radius, which corre2 /ν = 694444. This value of K is sponds to a local Reynolds number Reloc = Ω rdim in close agreement both with the numerical work of [2] (K = 0.40 at Reloc = 640000 in ”case D”) and experimental results of Itoh et al. [17] (K = 0.41). The small underestimation of K is mainly due to differences in the problem modeling (a cylindrical cavity for [17], an homogeneous cavity in radial direction for [2], and a shrouded rotor-stator annular cavity for the SVV-LES). Indeed, the stationary inner cylinder

Fig. 3 Axial profiles at mid radius of the mean radial velocity (top left), mean tangential velocity (top right), square root of the radial (bottom left) and azimuthal (bottom right) normal Reynolds Stresses in the Ekman (solid line) and in the Bödewadt (dashed line) layers, Re = 106 . Mean velocities are normalized by the local velocity of the rotor. Square root of Reynolds stresses are normalized by the local friction velocity, Vτ , on the corresponding disc


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greatly stabilizes the flow and the shroud tends to slow down the flow by dissipating a lot of energy in the impinging jet and in the Stewardson layer, that follows the Ekman layer at the upper part of the shroud. As no reference profile is available at this Reynolds number, we have compared two local parameters introduced by Daily and Nece [8] and based on the wall friction velocity: The skin-friction drag coefficient Cθ = 2(Vτ /rdim Ω )2 that is derived from the torque exerted by the fluid on the two discs and the local stress Reynolds number Reτ = rdimVτ /ν . SVV-LES calculations give values at mid-radius equal to Cθ = 1.6 10−3 in the Ekman layer and Cθ = 8.74 10−4 in the Bödewadt layer. Such values are slightly smaller than those found in [2], where Cθ = 2.05 10−3 and Cθ = 1.15 10−3, respectively. The relative dissipation rate between both layers due to the skin-friction, Cθ (rotor)/Cθ (stator), is equal to 1.83 for the SVV-LES. This value is in close agreement with the value 1.79 of [2] for the ”case D” and with the value 2 measured experimentally in [17]. The local stress Reynolds numbers at mid-radius are found equal to Reτ = 525 at the rotor and Reτ = 288 at the stator. These values, which show that the level of turbulence at mid-radius is larger in the Ekman layer than in the Bödewadt layer, are very close to those obtained by Andersson and Lygren [2], Reτ = 525 and Reτ = 284, respectively.

3 Pseudo-Penalization Method and Bluff Body Wake Flows To address bluff body wake flows with the SVV-LES spectral (Cartesian) solver one uses a volume penalization like method. It is described hereafter together with the corrections that have been implemented to improve the results in near wall region. The computation of the square cylinder wake allows to validate this Near Wall (NW) treatment. The approach is then applied to a challenging benchmark, the Ahmed body [1, 26], slant angle 25o. Till now RANS approaches fail to describe this flow and no LES results are completely satisfactory, see e.g. [7, 14, 29] and [10, 11, 15, 16, 24], respectively.

3.1 Modeling of the Bluff Body A volume penalization method is used to model the obstacle, but no penalization term is however explicitly introduced in the momentum equation. As suggested in [40], this is implicitly done through the time discretized NS equations. With χ for the characteristic function of the obstacle and assuming that the linear diffusive term is treated implicitly and the non-linear convective term explicitly, the pseudo-penalization method consists of solving at each time-step:

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α n+1 u − ∇pn+1 = (1 − χ¯ )fn+1 τ ∇ · un+1 = 0

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where n is the time index, τ (≡ Δ t) the time-step and α a scheme dependent coefficient (α = 3/2 for a second-order backward finite difference scheme). The pair (un , pn ) is the numerical approximation of (u, p) at time tn and fn+1 is an easily identifiable source term, which also depends on the time scheme. Finally, χ¯ is a regularized characteristic function, in practice obtained from local averages of the function χ . Clearly, inside the obstacle (un+1 , pn+1 ) solves the steady Stokes equations with a O(1/τ ) penalization term. Then it appears that inside the obstacle the velocity approximately vanishes, i.e. is essentially O(τ ) 1. As soon as the Reynolds number is really high, as it is e.g. for the Ahmed body flow at Re = 768000, the boundary layers cannot be resolved properly by the mesh. This has motivated a lot of researches during the three last decades on Near Wall Modeling (NWM), see e.g. [42]. The problem is especially difficult when the flow shows large detachments. Several approaches have been suggested, based on the boundary layer equations or requiring to resolve joined equations, see [58], patching techniques, DES (Detached Eddy Simulation) methods, see [30]... Such approaches have essentially been developed for finite volume approximations and it is not straightforward to implement them in a spectral solver and when using a penalization type method. The results presented in this paper have been obtained with a cruder approach but which has allowed significant improvements : • The characteristic function of the obstacle is not smoothed, • the control parameters of the SVV technique, εN and mN , are relaxed in NW region. Point 2 may be formulated by completing the momentum equation (1) with the body force term NW fNW = χNW ν (ΔSVV − ΔSVV )u (8) with χNW a second characteristic function used to localize the NW adjustment and NW is defined like Δ where the operator ΔSVV SVV but makes use of a smaller value of εN and / or a greater value of mN . In practice, we have only increased the value of mN within the NW region.

3.2 Validation for the Square Cylinder The geometry corresponds to the one proposed by the ERFOCTAC test case [57]. A square cylinder of unitary side is placed at the origin of the coordinate system, within the computational domain (−4.5, 15) × (−7, 7) × (−2, 2). The entire domain is decomposed in the x-streamwise direction in eight sub-domains with the interfaces located at x = {−4.5, −2, −0.55, 0.55, 2, 4.5, 7.5, 10.5, 15}. In each sub-domain we use polynomials of degree Nx = 30, Ny = 350 and Nz = 16, so that one has about 2, 7.106 mesh points. Using a mapping in the y-vertical direction, the first grid point is located at a distance of Δ y = 0.015 that does not allow a fine description of the boundary layer, as already mentioned in the conclusions of the ERCOFTAC test case [57], for Δ y = 0.01.


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Fig. 4 Instantaneous isosurfaces of the pressure without (left) and with (right) near wall treatment

At the inlet an uniform velocity profile Ue is imposed, whereas the outlet is treated with a convective condition at velocity Ue . The characteristic Reynolds number of the flow, based on Ue and D, is equal to Re=21400. According to stability tests, ' the SVV parameters have been chosen as ' √ (mNx , mNy , mNz ) = (3 Nx , 4 Ny , 2 Nz /2) and εN = 1/N in NW regions whereas √ in the complete domain (mN , εN ) = ( N, 1/N). The simulations have been performed on the NEC-SX8 of IDRIS, and required 5 GB of memory and 73 CPU hours to get convergence. Instantaneous isosurfaces of the pressure, as obtained without and with near wall treatment are provided in Fig. 4. Clearly, the filtering of the characteristic function induces a shift of the detachment line. Statistics above the cylinder are presented in Fig. 5, for the mean streamwise velocity and for the streamwise Reynolds stress. Results are given for the standard SVV-LES, without filtering of the characteristic function and with the near wall relaxation of the SVV parameters. Comparisons are provided with the experimental results of [27].

3.3 Simulations of the Ahmed Body Flow Computations were carried out in the domain (−7.25, 7.25) × (0, 3.47) × (−2.35, 2.35), using the height of the car model as reference length and the rear part being located at x = 0. The computational domain is partitioned in 8 subdomains. In each one the discretization is 41 × 191 × 340, obtained with Chebyshev polynomials of degrees Nx = 40 and Ny = 190 and trigonometric polynomials of degree Nz = 170. Since Gauss-Lobatto-Chebyshev points naturally accumulate at the end-points, subdomains interfaces have been located at the front and rear part of the Ahmed body. In the vertical direction, non-linear mappings are used to accumulate

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grid-points on the upper part of the bluff body. The mesh makes use of about 21 millions of grid-points. Starting from the fluid at rest, a computation was first carried out on a rough mesh (discretization is roughly divided by 2 in each direction) till a turbulent flow is well established, say at the dimensionless time t = 100. Then, the solution was interpolated on a fine mesh and the simulation was continued till t = 160. Statistics were only computed during the last 40 time units of the simulation, in order to avoid any pollution coming from the coarse mesh computation. The dimensionless time-step was equal to τ = 2.10−3 and the CPU time to 9 s for one time-step, i.e. approximately 9.510−8s per iteration and degree of freedom on the NEC SX8 computer of the IDRIS center. The SVV stabilization is governed by the threshold frequency mN and the SVV amplitude εN . Independently √ of the spatial (x, y, z) direction, outside the boundary layer we have used mN = N and εN√= 1/N. ' Within √ the boundary layer, we have used anisotropic values, i.e. mN = {2 Nx , 5 Ny , 4 Nz } and again εN = 1/N. Results obtained at the reference Reynolds number Re = 768000 and comparisons with results obtained at Re = 8322, as in the experiments of [13, 51], are provided. Same meshes and time-steps have been used in the two cases and the values


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of the SVV control parameters are the same, both inside and outside the boundary layers. However, for computational cost reasons we only used 12 time units to get the statistical results at the lower Reynolds number. As described hereafter, the topology of the two flows are in fact close, showing that the flow is not very sensitive to the Reynolds number. More details may be found in [31, 34]. In Fig. 6 we compare isolines of the mean streamfunction in the vertical median plane z = 0. One clearly observes three recirculation zones, on the upper part of the body, over the slant and behind the obstacle. Moreover, recirculation bubbles similar to the one in the upper part occur at the lateral walls. Despite similar, these recirculation zones appear larger, i.e. longer and thicker, at the lower Reynolds number. One should mention that experimentally such recirculations may or may not be observed, depending on the laminar or turbulent feature of the upstream flow. Behind the obstacle the topology is slightly different : The recirculation zone shows two contra-rotating recirculation bubbles and one observes that the lower one is less pronounced at Re = 8322, which probably results from a thicker boundary layer developing under the body.

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In Fig. 7 are shown velocity fields in the plane x = 1.34 for the two values of the Reynolds number. Clearly, the cone like trailing vortices which escape from the edge of the slant are similar. However, at the low Re the vortices appear (i) slightly more distant and (ii) are located lower. We compare in Fig. 8 mean streamwise velocity and turbulent kinetic energy profiles. Under the body one observes that the flow is nearly laminar for the low Re simulation : a parabolic like profile is obtained and the turbulent kinetic energy is close to 0. This is not very surprising, since the local Reynolds number based on the distance from the body to the ground equals 1445. Over the slant, one recovers the conclusion inferred by the mean streamlines that the reattachment is delayed. For the higher Re detailed comparisons with the experiments of Lienhart et al. [26] are provided in [33], but to be self contained we present in Fig. 9 such comparisons, for the mean streamwise velocity and for the turbulent kinetic energy. We also point out the influence of the NW correction. Even if the SVV-LES profiles


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compare rather well with the experimental data, one observes at the beginning of the slant a deficit of streamwise velocity associated to an overestimation of the turbulent kinetic. The phenomenon is in fact confined in the median region, say for −0.5 < z < 0.5 and develops from the front part of the body, downstream of the recirculation bubble localized on the upper part, see Fig. 6. Of course, the results obtained here at the lower value of the Reynolds number are certainly more reliable, since the SVV stabilization plays here a less important role. This may be visualized by computing the dissipation rate of the turbulent kinetic energy, see [31, 34, 38] for details.

4 High-Order Finite Volume Method and Applications To increase the geometrical flexibility of the concerned Chebyshev collocation pseudo–spectral method, a combination of a mapping technique together with a multi-block decomposition algorithm has been investigated [48]. The mapping and the ’generalized’ domain decomposition technique increase the condition numbers of the discrete systems which turned out to be a crucial issue with respect to an efficient solution [43]. Due to this fact that the extension of the spectral methods to more complex geometries leads not only to ill-conditioned and full system matrices but also to high memory requirement. An efficient computation of an accurately resolved turbulent flow seems not to be possible with such an approach. A compromise between the mentioned drawbacks of the spectral methods and the required high accuracy for the computation of turbulent flows with LES is seen in the finitevolume method in conjunction with compact finite difference methods.

4.1 Numerical Method and LES Modeling The basis is the finite volume solver FASTEST [52]. It solves the filtered NavierStokes equations on boundary fitted, block structured hexahedral grids. The convective and diffusive fluxes are approximated with a second-order central difference scheme. In order to minimize discretization errors a fourth order compact scheme is developed for flux approximations (section 4.2). Subgrid stresses are computed using the Smagorinsky model with the dynamic approach of Germano [12]. The implicit second order Crank-Nicolson scheme is applied for time discretization. Pressure velocity coupling is realized with a SIMPLE type algorithm which is embedded in a geometric multigrid scheme with standard restriction and prolongation [4]. The resulting linear systems of equations are solved with an ILU method.

4.2 Fourth-Order Compact Finite Volume Scheme A method to achieve a higher order spatial accuracy is the compact finite difference method (e.g. [41]). Compared to conventional finite difference methods the finite

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Fig. 10 Notation of a control volume

compact difference schemes reflect the high-frequency, short-wave parts of the solution in a better way and have a better fine-scale resolution [25]. For the discretization of the averaged convective and diffusive cell face fluxes a fourth-order compact finite difference scheme suggested by [41] is employed. To show the procedure of the interpolation the discretization of an east cell face of a control volume for an orthogonal 2D grid (Fig. 10) is taken exemplarily. Unknowns of convective fluxes are interpolated as 1 1 3 uw + ue + uee = (uP + uE ). 4 4 4

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4.3 Efficiency Investigations of the Finite Volume Solver For the efficiency investigations we chose the well known test case “Periodic flow over a 2D hill” [18] which is a periodic segment of a channel constricted by “2D hills” at the lower wall. The Reynolds number Rh = 11600 is based on the hill height h. The computational domain is periodic in streamwise as well as in spanwise directions what avoids uncertainties due to unknown boundary conditions. Although the simple geometry, the flow shows different features like separation on curved surfaces, strong recirculation and reattachment. The test case geometry is shown in Fig. 11. The computational domain is of the size 9h × 3.03h × 4.5h. The flow is driven by a pressure gradient that is adjusted to the desired Reynolds Number of Rh = 11600. On the top and bottom walls no slip boundary conditions are applied. Two grid resolutions were considered both with a fully resolved wall boundary layer and y+ values at the wall below unity. A reference solution was computed utilizing a highly resolved LES on a 11.8 × 106 control volume (CV) grid. The below mentioned parameter studies are carried out on a coarser 1.47 × 106 CV grid. In many cases, especially when structured or block structured grids are used for wall bounded flows, the highest Courant (CFL) numbers occur only in few control volumes whereas for the most part of the computational domain they are much below unity. Also the locations with the highest Courant number vary with time. Fig. 12

Fig. 12 Spatial distribution of the CFL number for CFLmax ≈ 1. CFL > 0.3

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Fig. 13 Influence of the time step size. Profiles of the mean velocity component /ub and the mean streamwise fluctuations /u2b at x/h = 0, 05 and x/h = 2

shows the control volumes with CFL > 0.3 for a single time step with a maximum Courant number of CFLmax ≈ 1. With this background the use of implicit time discretization schemes and their ability to handle time step sizes with corresponding CFL > 1 seems to be a promising way to increase the efficiency of the simulations since with the same number of time steps a greater time interval can be covered. If the quality of the results is not affected, the computational time - especially when turbulence statistics are of interest - can be reduced significantly. To gain information how greater time step sizes affect the quality of the computational results and the efficiency of the simulations, computations with corresponding maximum CFL numbers of approximately 1, 2, and 5 were carried out. Averaging was performed for 40 flow-through times. Fig. 13 shows the profiles of the mean streamwise velocity component /ub and the mean streamwise fluctuations /u2b for the three considered time step sizes as well as the reference solution obtained on the fine 11.8 × 106 CV grid at the positions x/h = 0.05 and x/h = 2. At both considered locations the obtained velocity profiles are in very good agreement with the reference solution (LES - fine grid). There are no significant differences between the results for the different time step sizes. Also the fluctuations in streamwise direction are in good agreement to each other for all three time step sizes and the reference solution. The deviation to the reference solution at the position x/h = 2 is in the expected range since only the resolved part is plotted and this part is greater on the finer grid. It can be stated that the time step size - at least in the considered range - has no negative influence on the quality of the computational results. But it has great influence on the efficiency of the computations. In table 1 the computational times for a given time interval of 0.05s (equivalent to 1000 time steps with CFL ≈ 1) are summarized. For the time step size with CFL ≈ 2 almost half of the computational time could be saved whereas for the greatest considered time step size (CFL ≈ 5) the convergence behavior became worse and therefore it was less efficient as the simulation with CFL ≈ 2.


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Table 1 Computational time for a given time interval of 0.05s (2 IBM Power 5 Processors with 1.9 GHz)

Δ t (CFL) 5 × 10−5 s (≈ 1) 10−4 s (≈ 2) 2.5 × 10−4 s (≈ 5) Computational time 76.4 × 103 s 38, 3 × 103 s 46, 8 × 103 s

Further numerical investigations like the influence of the convergence criterion on quality and efficiency and more detailed analysis of the flow features haven been published in [23].

4.4 Investigation of Efficiency Improvements by an Algebraic Multigrid Method The finite volume method leads to a large sparse system of equations. To solve this equation system efficiently multigrid methods are used. In this section a geometric and an algebraic multigrid method are investigated. For the simulations performed in this study the SAMG package by Fraunhofer SCAI [55] was used as algebraic multigrid (AMG) solver. While the coarser grid levels of the geometric multigrid method are dependent on the mesh and the geometry the coarser grid levels of the AMG solver are constructed based just on the system matrix. A schematic comparison between the two multigrid solvers is shown in Fig. 14. For the efficiency investigations a laminar and a turbulent test series are considered. The geometry corresponds to a channel flow through a labyrinth (see Fig. 15-17). The idea is achieving a more and more

Fig. 14 Comparison of geometric and algebraic multigrid method

Fig. 15 Test case LAB1, velocity profile at y = 0.1 m

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complex geometry and investigating the computational effort for the different test cases with the two methods. The number of control volumes is almost the same for each test case (around 575 × 103 CV). In the laminar case the inlet velocity of the fluid is u = 0.45 m/s in x-direction with a density ρ = 1 kg/m3 and kinematic viscosity μ = 10−3 kg/ms yielding the Reynolds number Re = 77. The resulting linear sets of equations are solved with a full multigrid strategy for the geometric multigrid method and V-cycles for the algebraic method. Results of the calculations are shown in Fig. 15-17. In Fig. 18 the computing times for the different test cases are shown. It can be seen that the more complex the geometry the more advantageous the AMG solver becomes.

Fig. 18 Computing time for laminar flow test series


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Fig. 21 Computing time for LAB3

For the turbulent test series the inlet velocity is u = 16 m/s and thus the Reynolds number is Re = 2743. For the LES simulation a dynamic Smagorinsky model is chosen with time step Δ t = 5 · 10−5s. The results are shown in Fig. 19 and 20. The geometric and algebraic multigrid methods are compared with single grid calculations. Due to the fact that in the residuals of the geometric multigrid calculation oscillations appeared [53] a method where the calculations start as a geometric multigrid method and switch to the single grid method [53] are also considered (GMGSG). The results are summarized in Fig. 21 (computational time corresponds

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Fig. 22 Geometry of the diffuser (diffuser 1) taken from [5]

to 20 time steps). Even the geometric multigrid method without switching is faster than the AMG solver. The best results are obtained with the GMGSG approach. Thus for laminar test cases the AMG solver appears to be a promising method, but for the considered turbulent test cases it seems to be not.

4.5 Benchmark Computations for Flow in a 3D Diffuser An incompressible flow through a 3D diffuser (ERCOFTAC Test Case 13.2: Flow in a 3D diffuser - Diffuser 1) with deflected upper wall and one deflected side wall is studied by means of LES at a bulk Reynolds number Rh = 10000 based on the height h of the inlet duct. Experimental data are available from the work of Cherrye et al. [5]. The flow shows an adverse pressure gradient caused by the duct expansion. The focus of the study is on the size and shape of the three dimensional flow separation pattern. The adverse pressure gradient and the high sensitivity of the flow to small changes of the geometry makes it a challenging test case for turbulence modeling. The test case geometry is depicted in Fig. 22.

Fig. 23 Computational domain. Only each second grid line is shown


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Fig. 24 Contours of mean streamwise velocity on cross sections x/h = 2, 5, 8, 12 and 15 in comparison to experimental data

According to the experimental setup of Cherrye et al. [5] a fully developed turbulent channel flow is used as inlet boundary condition. This inlet data is generated by a simultaneously running periodic simulation of a channel with the same cross section as the diffuser inlet. To allow the flow through the diffuser to influence the flow field in the development channel a part (5 channel heights) of the development channel is modeled in front of the diffuser. Furthermore the convergent part behind the diffuser (compare Fig. 22) is modeled to avoid back flow at the outlet plane. The

Fig. 25 Mean streamwise velocity profiles /ubulk on the slice z/h = 1 in comparison to experimental data

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Fig. 26 Mean streamwise velocity profiles /ubulk on the slice z/h = 2.625 in comparison to experimental data

computational domain including the development channel and the convergent part is depicted in Fig. 23. Two cases are considered. One with dynamic Smagorinsky model and the second one without subgrid model. The grid size was approximately 4 million CV including the development channel and the convergent part behind the diffuser. The wall boundary layers are fully resolved with y+ values below unity at all walls. The time step size is chosen in a way that the corresponding Courant number is around one. This leads to a time step size of 1.1 × 10−4 s. Averaging is performed for approximately 80000 time steps. The simulations were carried out on 16 IBM Power 5 CPUs with a load balancing efficiency of 95%. This leads to a computational times per time step of approximately 14s. Fig. 24 shows contours of the mean streamwise velocity component in comparison to experimental data taken from [5] at the cross sections x/h = 2, 5, 8, 12 and 15. The computations show a good agreement with experimental data for all considered locations. The overall flow features are well captured. Also the size and shape of the recirculation bubble is well reproduced (= 0 thicker line). Fig. 25 shows mean streamwise velocity profiles on the slice z/h = 1 in comparison to experimental data from Cherrye et al. [5]. Both simulations show a relatively good agreement with the experiment. But at streamwise position x/h = 2 the velocity close to the upper wall differs from experimental values. Since this position is close to the separation point, the deviation of the computational data leads to misprediction of the recirculation at streamwise positions x/h = 6 and x/h = 8. At the positions further downstream the agreement with the experiment is better. There are no significant differences between the two simulations with and without subgrid model.


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In Fig. 26 mean streamwise velocity profiles on the slice z/h = 2.625 are shown. At this spanwise position recirculation is present at the greatest part of the domain (compare Fig. 24 and 22). Again at the position x/h = 2 the velocity at the upper wall differs from experimental data what again leads to misprediction of the recirculation at positions further downstream. At this slice the two simulations with and without model differ from each other. The simulation without subgrid model shows a better agreement with experiments than the one with dynamic Smagorinsky model, especially recirculation at streamwise positions x/h = 6 is predicted much better. The differences to the experiments are most probably caused by a to coarse grid resolution in streamwise direction in combination with too big aspect rations of the control volumes (compare Fig. 23). The influence of the grid resolution and the subgrid model are investigated in detail in present studies. Detailed simulation data of the LES with dynamic Smagorinsky model, with profiles of all velocity components and Reynolds stresses, at various streamwise and spanwise positions can be found in [19]. Acknowledgements. The SVV-LES calculations were carried out on the NEC SX8 vector computer of the CNRS computational center IDRIS (project 074055 and 084055). Computational resources of the mesocentre SIGAMM of the OCA were also used. The authors gratefully acknowledge support from the DFG and the CNRS trough the DFG-CNRS program ”LES of complex flows”.

References 1. Ahmed, S.R., Ramm, G.: Salient Features of the Time-Averaged Ground Vehicle Wake, SAEPaper 840300 (1984) 2. Andersson, H.I., Lygren, M.: LES of open rotor-stator flow. J. Heat Fluid Flow 27(4), 551–557 (2006) 3. Baur, C., Sternel, D.C., Schäfer, M.: Efficiency investigation of an algebraic multigrid solver for complex geometries (to appear) 4. Briggs, W.L., Van Emden, H., McCormick, S.F.: A Multigrid Tutorial. SIAM, Philadelphia (2000) 5. Cherrye, E.M., Elkins, C.J., Eaton, J.K.: Geometric sensitivity of three-dimensional separated flows. J. Heat Fluid Flow 29(3), 803–811 (2008) 6. Cousin, L., Pasquetti, R.: High-order methods for the simulation of transitional to turbulent wakes. In: Lu, Y., Sun, W., Tang, T. (eds.) Advances in Scientific Computing and applications, pp. 133–143. Science Press, Beijing (2004) 7. Craft, T.J., Gant, S.E., Iacovides, H., Launder, B.E., Robinson, C.M.E.: Computational study of flow around the Ahmed car body (case 9.4). In: 9th ERCOFTAC/IAHR Workshop on Refined Turbulence on Modelling Darmstadt University of Technology, Germany, October 4-5 (2001) 8. Daily, J.W., Nece, R.E.: Chamber dimension effects on induced flow and frictional resistance of enclosed rotating discs. ASME J. Basic Eng. 82, 217–232 (1960) 9. Droll, P.: Numerical simulation of fluid flow with pseudo-spectral methods, Phd Dissertation, TUD (2002) 10. Durand, L., Kuntz, M., Menter, F.: Validation of CFX-5 for the Ahmed Car Body, CFX Validation Report, CFX-Val 13/1002 (2002)

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Index

Comte-Bellot and Corrsin test Kolmogorov function 152 1/r-law 115 2-D hump, 265 3-D axisymmetric hill, 201, 210 diffuser, 201, 209, 266, 328

153

acoustic analogy, 127, 130, 140 directivity, 111 field, 100 perturbation equations, 100, 102 spectrum, 89, 95 adaptive local deconvolution method, 151 shock-capturing, 31 ADER 53 adjoint 81, 87, 88 adverse pressure gradient 163 aero-engine inlet 65 Ahmed body 310, 315, 317 air intake 247 ALDM 151 amplification rate 83, 93 anisotropy 203, 205 annular swirl combustor 196 APE 50, 100, 102 B¨ odewadt layer 313 backward-facing step flow Bluff body wake 315 boundary condition, 78, 87, 126 layer, 163 Burgers equation 151

259

Cauchy-Kovalevskaja (CK) procedure 53 channel flow 273, 275 coaxial jet 103 code Code Saturne, 271

FASTEST-3D, 259, 321 Coherent Vortex Extraction (CVE) 12 coherent vortices 12 compact finite volume scheme 321 compatibility condition 237 computational aeroacoustics (CAA) 99, 100 convective condition 235 cost functional 81, 88 Courant number 323 curvilinear grid 39 cut-cell approach, 158 technique, 160 database technique 243 DDES 202 deconvolution 150 degree of freedom 51 delayed detached-eddy simulation 202 DES 123, 124, 135, 201–203 differential subgrid-stress model 268 diffusion 205 dilatation 85, 86, 95 dipole 100, 111 direct simulation 51 discontinuous Galerkin (DG) 53 dissipation 205 DNS 30, 77, 80 domain actuation, 80, 88 decomposition, 38, 51, 77 measurement, 81, 89 EARSM 204 eddy-damped quasi-normal Markovian approximation 151 eddy-resolving strategies for turbulent flows 232 eddy-viscosity subgrid model 189 EDQNM 151 Ekman layer 313

336

Index

elliptic blending model, 274 relaxation method, 274 enrichment 235 error norms 60 evolutionary optimization 151 explicit algebraic Reynolds stress model 204 Face-FEM 57 far-field prediction 127, 130, 134, 138, 140 filtering 151 finite differences, 78, 79 volumes, 150 forcing method 257 free flows 254 FWH 127, 128, 130, 134, 138, 140 Galerkin model, 19 system, 19 Gibbs oscillations 31 gradient-diffusion approach 205 grey zone 233 grid 129, 133, 136 grid-induced separation 202 grids structured, 51 unstructured, 60 heat effects, 111 transfer, 157 high-order methods, 53, 309 simulations, 31 homogeneous isotropic turbulence 151 hybrid LES/RANS, 201, 231, 269, 271, 275 LES/URANS, 204 methods, 48 hydrodynamic/acoustic splitting 49 ILES 149 immersed boundary implicit LES 149 incoherent vorticity

158, 159 13

inflow condition for LES, 233, 243 forcing, 105 interface criterion 206 interpolation Lagrange, 39 zone, 39 jet noise 100 Joukowsky profile

38

K-ω model 241 Kirchhoff method

48

Lamb vector 107 leapfrogging 131, 133 LES 30, 102, 123, 133, 135, 202, 231 LES/RANS classification of hybrid approaches, 232 interface, 257 pressure coupling in segregated LES/RANS, 236 seamless hybrid, 268 segregated modelling, 231, 232 tangential interface, 239 unified models, 232 LEVM 204 linear eddy-viscosity model 204 linearized Euler equations (LEE), 56, 100 perturbed compressible equations (LPCE), 50 log-law mismatch 202 mass matrix 64 mean value 51 Message Passing Interface (MPI) 38 mixing layer 76, 82, 87 MOD approximation, 17 eigenvalues, 18 modes, 18 mode variants, 18 modified differential equation analysis 151 most observable decomposition (MOD) 17 multi-scale problem 47 multigrid

Index algebraic, 325 geometric, 325 Navier-Stokes 30 NLEVM 204 non-equilibrium 171 non-linear eddy-viscosity model numerical flux, 151 viscosity, 151 OASPL 108 optimal control 80–81, 87 order of convergence 60 orthogonal wavelet decomposition outflow condition for LES 233 overall sound pressure level 108 overset grid techniques 38

204

12

parallelization 38 passive scalar 157 periodic hill flow 159, 192, 201, 208, 237, 278 phase shift, 82, 84 speed, 84, 94 PITM method 269, 278 POD approximation, 8 eigenvalues, 8 Fredholm integral equation, 8 modes, 8 potential core 105, 113 power spectra 109 production 205 proper orthogonal decomposition (POD) 8 pseudo-penalization method 315 RANS 231 reattachment 202 regularization 151 resonance 84, 85 reverse-flow parameter 167 Reynolds-stress model 268, 272 rod-airfoil 129 Rotor-stator flow 310, 312 round jet 103

337

scale similarity 151 scaling for compressible boundary layer 244 screech 29, 30 second generation URANS 232 separation 159, 163, 202 separation control 264 shock-capturing 31 shock-cell 29 shock-induced noise 30 short cowl nozzle 112 single tubo-annular combustor 260 Smagorinsky model 151, 330 source terms 48 Spalart-Allmaras model 202, 248 spectral cutoff, 269 methods, 309 splitting, 269 vanishing viscosity (SVV), 309–311 splitter plate 79, 91, 94 square cylinder wake 315, 316 STE-FV 51 subgrid-scale modeling 149, 205 subgrid-stress transport model 272 subsampling 247 swirling flows 260 Taylor expansion 151 temporal mixing layer 271, 272 tensor dissipation 151 truncation error 149 turbulent boundary layer, 243 channel flow, 154, 240 two-way coupling 39 volume coupling, 48 penalization method, 310 vortex 82, 85, 95, 130, 131, 137 vortex sound 107 wall-bounded flows 253 wavelet details 189 WENO 150, 151 Willy car model 201, 211

Author Index

Adams, Nikolaus A. 149 Airiau, Christophe 75 Aybay, Orhan 201

Jakirlić, Suad 253 Jund, Sébastien 47

Babucke, Andreas 75 Bailly, Christophe 29 Baur, Christine 309 Berland, Julien 29 Bockhorn, Henning 175 Bogey, Christophe 29 Boiarciuc, Margareta Petrovan Bontoux, Patrick 309 Borée, Jacques 253 Breuer, Michael 201 Brun, Christophe 283 Carpy, Sabrina 253 Chikhaoui, Oussama 201 Comte, Pierre 3, 99 de Cacqueray, Nicolas 29 Denev, Jordan A. 175 Deng, Ganbo 201 Devesa, Antoine 149 Domingues, Margarete 175 Dumbser, Michael 47 Eschricht, Dandy

121

Fadai-Ghotbi, Atabak 253 Falconi, Carlos 175 Farge, Marie 3, 175 Frénod, Emmanuel 47 Friess, Christophe 253 Fr¨ ohlich, Jochen 175, 231 Gomes, Sˆ onia 175 Greschner, Bj¨ orn 121 Gr¨ oschel, Elmar 99 Guilmineau, Emmanuel Hickel, Stefan Jacob, Marc C. Jaffrézic, Benoˆıt

149 121 201

201

Kadavelil, Gisa 253 Kadoch, Benjamin 175 Kloker, Markus 75 Kniesner, Bj¨ orn 253 Koh, Seong Ryong 99 Kolomenskiy, Dmitry 3 Kornhaas, Michael 309 283 Luchtenburg, Dirk M.

3

Manceau, Rémi 253 Manhart, Michael 283 Mary, Ivan 231 Meinke, Matthias 99 Minguez, Mathieu 309 Molina, Igor 175 Munz, Claus-Dieter 47 Noack, Bernd R.

3

Pasquetti, Richard 309 Peller, Nikolaus 283 Rist, Ulrich 75 Roussel, Olivier 175 Salmon, Stéphanie 47 ˇ c, Sanjin 253 Sari´ Sch¨ afer, Michael 309 Schlegel, Michael 3 Schmid, Peter 29 Schneider, Kai 3, 175 Schr¨ oder, Wolfgang 99 Schulze, Jan 29 Scouten, Jon E. 3 Serre, Eric 309 Sesterhenn, J¨ orn 29 Séverac, Eric 309 Sonnendr¨ ucker, Eric 47 Spagnoli, Bruno 75

340

Author Index

Tadmor, Gilead Thiele, Frank

3

Utzmann, Jens

121

Tropea, Cameron

253

47

Visonneau, Michel von Terzi, Dominic

201 231

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