Large-Eddy Simulation for Acoustics (2007)(en)(441s)

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LARGE-EDDY SIMULATION FOR ACOUSTICS Noise pollution around airports, trains, and industries increasingly attracts environmental concern and regulation. Designers and researchers have intensified the use of large-eddy simulation (LES) for noise-reduced industrial design and acoustical research. This book, written by thirty experts, presents the theoretical background of acoustics and LES followed by details about numerical methods such as discretization schemes, boundary conditions, and coupling aspects. Industrially relevant, hybrid Reynolds-averaged Navier–Stokes/LES techniques for acoustic source predictions are discussed in detail. Many applications are featured ranging from simple geometries for mixing layers and jet flows to complex wing and car geometries. Selected applications include recent scientific investigations at industrial and university research institutions. Presently perfect solution methodologies that address all relevant applications do not exist; however, the book presents a state-of-the-art collection of methods, tools, and evaluation methodologies. The advantages and weaknesses of both the commercial and research methodologies are carefully presented. Claus Wagner received his Ph.D. in Fluid Dynamics in 1995 at the Technical University of Munich, Germany. He is Honorary Professor for Industrial Aerodynamics, Ilmenau University of Technology, Germany. Since 1998, he has been a scientist in and head of the Section of Numerical Simulations for Technical Flows of the Institute for Aerodynamics and Flow Technology, German Aerospace Center, G¨ottingen, Germany. Dr. Wagner’s research includes experimental investigations of the resonant control of nonlinear dynamic systems, theoretical and numerical investigations of thermal convection in cylindrical containers, and direct numerical simulation and large-eddy simulations of turbulent flows in different configurations. He has held visiting positions in Gainsville, Florida, USA as well as in Bremen, Germany. Thomas H¨uttl received his Ph.D. in Fluid Dynamics in 1999 at the Technical University of Munich, Germany. His academic research included work on the direct numerical simulation of turbulent flows in curved and coiled pipes, and direct and large-eddy simulations of boundary layer flows with and without separation, in the framework of a French–German DFG-CNRSCooperation project. He has held visiting positions in Nantes, France and Bangalore, India. Between 2000 and 2003, he was a senior engineer for aeroacoustics at MTU Aero Engines GmbH, Germany’s leading manufacturer of engine modules and components and of complete aero engines. Dr. H¨uttl led MTU’s contribution for the European research project TurboNoiseCFD and contributed to the European research project SILENCER. After some years working as IT quality manager and internal auditor, he is now Chief Privacy Officer for the entire MTU Aero Engines concern. Among his many honors, Dr. H¨uttl was elected a member of the Senate of the DGLR, Deutsche Gesellschaft f¨ur Luft- und Raumfahrt - Lilienthal - Oberth e.V. in 2003 and 2006. Pierre Sagaut received his DEA in Mechanics in 1991 and his Ph.D. in Fluid Mechanics in 1995 at Universit´e Pierre et Marie Curie – Paris 6 (Paris, France). He worked as a research engineer at ONERA (French National Aerospace Research Center) from 1995 to 2002. He has been a professor in mechanics at University Pierre et Marie Curie – Paris 6 since 2002. He is ´ also part-time Professor at Ecole Polytechnique (France) and scientific consultant at ONERA, IFP, and CERFACS (France). His main research interests are fluid mechanics, aeroacoustics, numerical simulation of turbulent flows (both direct and large-eddy simulation), and numerical methods. He is also involved in uncertainty modeling for computational fluid dynamics (CFD). He has authored and coauthored more than sixty papers in peer-reviewed international journals and 130 proceedings papers. He is the author of several books dealing with turbulence modeling and simulation. He is member of several editorial boards: Theoretical and Computational Fluid Dynamics, Journal of Scientific Computing, and Progress in CFD. He received the ONERA award three times for the best publication and the John Green Prize (delivered by ICAS, 2002).

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Cambridge Aerospace Series Editors Wei Shyy and Michael J. Rycroft 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

J. M. Rolfe and K. J. Staples (eds.): Flight Simulation P. Berlin: The Geostationary Applications Satellite M. J. T. Smith: Aircraft Noise N. X. Vinh: Flight Mechanics of High-Performance Aircraft W. A. Mair and D. L. Birdsall: Aircraft Performance M. J. Abzug and E. E. Larrabee: Airplane Stability and Control M. J. Sidi: Spacecraft Dynamics and Control J. D. Anderson: A History of Aerodynamics A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principles of Space Instrument Design G. A. Khoury and J. D. Gillett (eds.): Airship Technology J. Fielding: Introduction to Aircraft Design J. G. Leishman: Principles of Helicopter Aerodynamics, 2nd Edition J. Katz and A. Plotkin: Low Speed Aerodynamics, 2nd Edition M. J. Abzug and E. E. Larrabee: Airplane Stability and Control: A History of the Technologies that Made Aviation Possible, 2nd Edition D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamics and Aeroelasticity W. Fehse: Automatic Rendezvous and Docking of Spacecraft R. D. Flack: Fundamentals of Jet Propulsion with Applications E. A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines Doyle D. Knight: Elements of Numerical Methods for High-Speed Flows C. Wagner, T. H¨uttl, and P. Sagaut: Large-Eddy Simulation for Acoustics

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Large-Eddy Simulation for Acoustics Edited by CLAUS WAGNER German Aerospace Center

¨ THOMAS H UTTL MTU Aero Engines GmbH

PIERRE SAGAUT Universite´ Pierre et Marie Curie

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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521871440 © Cambridge University Press 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 eBook (EBL) ISBN-13 978-0-511-29466-2 ISBN-10 0-511-29466-2 eBook (EBL) hardback ISBN-13 978-0-521-87144-0 hardback ISBN-10 0-521-87144-1

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

List of Figures and Tables

page xiii

Contributors

xxi

Preface

xxv

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The importance of acoustic research Thomas Huttl ¨ 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.1.7

Health effects Activity effects Annoyance Technical noise sources Political and social reactions to noise Reactions of industry Research on acoustics by LES

1.2 Introduction to computational aeroacoustics Manuel Keßler 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5

Definition History Aeroacoustics Conceptual approaches Remaining problem areas for sound computation

1.3 State of the art: LES for acoustics Claus Wagner, Oliver Fleig, and Thomas Huttl ¨ 1.3.1 Broadband noise prediction in general 1.3.2 Broadband noise prediction based on LES 2

1 1 2 2 3 4 5 5 7 7 7 8 9 13 15 15 15 17

Theoretical Background: Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Avraham Hirschberg and Sjoerd Rienstra 2.1 Introduction to aeroacoustics 2.2 Fluid dynamics 2.2.1 Mass, momentum, and energy equations 2.2.2 Constitutive equations

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CONTENTS

2.2.3

Approximations and alternative forms of the basic equations 2.3 Free-space acoustics of a quiescent fluid 2.3.1 Orders of magnitude 2.3.2 Wave equation and sources of sound 2.3.3 Green’s function and integral formulation 2.3.4 Inverse problem and uniqueness of source 2.3.5 Elementary solutions of the wave equation 2.3.6 Acoustic energy and impedance 2.3.7 Free-space Green’s function 2.3.8 Multipole expansion 2.3.9 Doppler effect 2.3.10 Uniform mean flow, plane waves, and edge diffraction 2.4 Aeroacoustic analogies 2.4.1 Lighthill’s analogy 2.4.2 Curle’s formulation 2.4.3 Ffowcs Williams–Hawkings formulation 2.4.4 Choice of aeroacoustic variable 2.4.5 Vortex sound 2.5 Confined flows 2.5.1 Wave propagation in a duct 2.5.2 Low-frequency Green’s function in an infinitely long uniform duct 2.5.3 Low-frequency Green’s function in a duct with a discontinuity 2.5.4 Aeroacoustics of an open pipe termination 3

30 33 33 35 36 38 39 44 47 47 49 52 55 55 59 60 62 65 68 68 72 74 76

Theoretical Background: Large-Eddy Simulation . . . . . . . . . . . . . . . . . . . . 89 Pierre Sagaut 3.1 Introduction to large-eddy simulation 3.1.1 General issues 3.1.2 Large-eddy simulation: Underlying assumptions 3.2 Mathematical models and governing equations 3.2.1 The Navier–Stokes equations 3.2.2 The filtering procedure 3.2.3 Governing equations for LES 3.2.4 Extension for compressible flows 3.2.5 Filtering on real-life computational grids 3.3 Basic numerical issues in large-eddy simulation 3.3.1 Grid resolution requirements 3.3.2 Numerical error: Analysis and consequences 3.3.3 Time advancement 3.4 Subgrid-scale modeling for the incompressible case 3.4.1 The closure problem 3.4.2 Functional modeling 3.4.3 Structural modeling 3.4.4 Linear combination models, full deconvolution, and Leray’s regularization 3.4.5 Extended deconvolution approach for arbitrary nonlinear terms 3.4.6 Multilevel closures 3.4.7 The dynamic procedure

89 89 90 91 91 92 95 98 100 105 105 109 112 112 112 113 118 119 120 121 121

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3.5 Extension of subgrid models for the compressible case 3.5.1 Background 3.5.2 Extension of functional models 3.5.3 Extension of structural models 3.5.4 The MILES concept for compressible flows 4

125 125 125 126 126

Use of Hybrid RANS–LES for Acoustic Source Predictions . . . . . . . . . . . . 128 Paul Batten, Philippe Spalart, and Marc Terracol 4.1 Introduction to hybrid RANS–LES methods 4.2 Global hybrid approaches 4.2.1 The approach of Speziale 4.2.2 Detached-eddy simulation 4.2.3 LNS 4.2.4 The approach of Menter, Kunz, and Bender 4.2.5 Defining the filter width 4.2.6 Modeling the noise from unresolved scales 4.2.7 Synthetic reconstruction of turbulence 4.2.8 The NLAS approach of Batten, Goldberg, and Chakravarthy 4.3 Zonal hybrid approaches 4.3.1 The approach of Quem ´ er ´ e´ and Sagaut 4.3.2 The approach of Labourasse and Sagaut 4.3.3 Zonal-interface boundary coupling 4.4 Examples using hybrid RANS–LES formulations 4.4.1 Flow in the wake of a car wing mirror 4.4.2 Unsteady flow in the slat cove of a high-lift airfoil 4.5 Summary of hybrid RANS–LES methods

5

128 130 130 131 133 138 140 142 143 145 148 150 150 152 154 154 158 163

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1 Spatial and temporal discretization schemes Tim Broeckhoven, Jan Ramboer, Sergey Smirnov, and Chris Lacor 5.1.1 5.1.2 5.1.3 5.1.4

Introduction to discretization schemes Dispersion and dissipation errors Spatial discretization schemes Temporal discretization schemes

5.2 Boundary conditions for LES Michael Breuer 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5

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Outflow boundary conditions Inflow boundary conditions Boundary conditions for solid walls Far-field boundary conditions for compressible flows Final remark for discretization schemes

5.3 Boundary conditions: Acoustics Fang Q. Hu 5.3.1 Characteristic nonreflecting boundary condition 5.3.2 Radiation boundary condition 5.3.3 Absorbing-zone techniques

167 167 168 170 197 201 203 204 208 214 215 216 217 218 218

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CONTENTS

5.3.4 Perfectly matched layers 5.3.5 Summary of boundary conditions for acoustics

6

220 222

5.4 Some concepts of LES–CAA coupling Wolfgang Schroder ¨ and Roland Ewert

222

5.4.1 LES inflow boundary 5.4.2 Silent embedded boundaries

225 232

Applications and Results of Large-Eddy Simulations for Acoustics . . . . . 238 6.1 Plane and axisymmetric mixing layers Christophe Bogey and Christophe Bailly 6.1.1 Plane mixing layer 6.1.2 Axisymmetric mixing layers – jets 6.1.3 Concluding remarks for mixing layer simulations 6.2 Far-field jet acoustics Daniel J. Bodony and Sanjiva K. Lele 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6

Introduction to jet acoustics Numerics of jet simulations Results for jet simulations Future directions of jet acoustics Conclusions for far-field jet acoustics Acknowledgments

6.3 Cavity noise Xavier Gloerfelt, Christophe Bogey, and Christophe Bailly 6.3.1 6.3.2 6.3.3 6.3.4

Introduction to cavity noise Overview of cavity-flow simulations Recent achievements using LES Concluding remarks for cavity noise

6.4 Aeroelastic noise Sandrine Vergne, Jean-Marc Auger, Fred Peri ´ e, ´ Andre´ Jacques, and Dimitri Nicolopoulos 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.4.6 6.4.7 6.4.8 6.4.9

Introduction to aeroelastic noise Fluid–structure interaction Numerical simulation Application Simulation model Numerical results Mesh influence Conclusions for aeroelastic noise prediction Acknowledgment

6.5 Trailing-edge noise Roland Ewert and Eric Manoha 6.5.1 Introduction to trailing-edge noise simulation using LES 6.5.2 Trailing-edge noise simulation using LES and APE 6.5.3 Trailing-edge noise simulation using LES, Euler equations, and the Kirchhoff integral 6.5.4 Unsteady pressure-field analysis

238 239 241 244 245 245 247 252 259 262 262 262 262 263 266 272 272

272 274 276 278 279 283 290 293 293 293 293 296 315 320

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6.6 Blunt bodies (cylinder, cars) Franco Magagnato 6.6.1 Overview of blunt-body simulations 6.6.2 Circular cylinder 6.6.3 Car 6.7 Internal flows Philippe Lafon, Fabien Crouzet, and Jean Paul Devos 6.7.1 Introduction to internal flows 6.7.2 Computation of acoustic fluctuations due to turbulence-generated noise at low Mach number 6.7.3 Computation of flow acoustic coupling in low-Mach-number ducted flows 6.7.4 Computation of aeroacoustic instabilities in high-Machnumber ducted flow 6.7.5 Conclusions for internal flow prediction 6.8 Industrial aeroacoustics analyses Fred Mendonca ¸ 6.8.1 6.8.2 6.8.3 6.8.4 6.8.5 6.8.6 7

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Introduction to industrial aeroacoustics analyses Preliminary considerations A two-step CFD modeling process (steady-state and transient) Postprocessing through acoustic coupling Conclusions for industrial aeroacoustics analyses Acknowledgments

333 333 335 345 349 349 349 351 354 355 356 356 357 358 370 376 376

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Claus Wagner, Pierre Sagaut, and Thomas Huttl ¨ 7.1 Governing equations and acoustic analogies

378

7.2 Numerical errors

384

7.3 Initial and boundary conditions

385

7.4 Examples

386

Appendix A. Nomenclature

389

Appendix B. Abbreviations

391

References

395

Index

429

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List of Figures and Tables

Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5

2.6 2.7 2.8

2.9 3.1 3.2 3.3 3.4 4.1 4.2

Magnetic levitation hover train project Transrapid Noise prediction methods Insulator of a high-speed train’s pantograph Sketch of the insulator of a high-speed train’s pantograph Far-field sound-pressure spectra Monopole, dipole, and quadrupole generating waves on the surface of the water around a boat Sketch of scattered plane wave with mean flow A potential flow through the vocal folds is silent Straight duct of arbitrary cross section (a) Method of images applied to a source at y = (y1 , y2 , y3 ) at a distance y3 from a hard wall x3 = 0 has a Green’s function: G(x ,t|y ,τ ) = G0 (x,t|y,τ ) + G0 (x,t|y∗ ,τ ) with y∗ = (y1 , y2 , −y3 ). (b) A source between two parallel hard walls generates an infinite row of images. (c) A source in a rectangular duct generates an array of sources The end correction for no flow (M j = 0) and a little flow (M j = 0.01) Plane-wave reflection coefficient |R| and end correction δ at jet exhaust without coflow for M j = 0.01, 0.1, . . . , 0.6 Plane-wave reflection coefficient |R| and end correction δ at jet exhaust with M j = 0.3 and coflow velocities Mo/M j = 0, 0.25, 0.5, 0.75, 1 Acoustic flow at a pipe outlet (a) for an unflanged pipe termination and (b) for a horn Schematic kinetic energy spectra of resolved and subgrid scales with spectral overlap (Gaussian filter) Streaks in the inner layer of the boundary layer Schematic of kinetic energy transfer in isotropic turbulence Schematic of the two-level filtering procedure and the Germano identity Turbulence energy spectrum partitioned into resolvable and unresolvable frequencies Required near-wall mesh resolutions with DNS, traditional LES, global hybrid RANS–LES, and nonlinear acoustics solvers (NLAS) based on disturbance equations

4 12 21 22 22 44 54 66 68

73 84 85

86 87 95 107 114 122 135

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LIST OF FIGURES AND TABLES

4.3

4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 5.1 5.2 5.3 5.4

5.5 5.6

5.7 5.8 5.9 5.10 5.11 5.12

Initial startup transient at probe 111 (cylinder rear face) predicted by unsteady RANS and hybrid RANS–LES (LNS model) using identical base model, mesh, and time step Instantaneous streamwise vorticity contours (with streamwise velocity shading) predicted by hybrid RANS–LES model Resolved and unresolved (synthetically generated) signals for probe 111 (cylinder rear face) Sound-pressure levels determined by resolved, unresolved, and composite signals for probe 111 (cylinder rear face) Instantaneous streamwise vorticity contours (with streamwise velocity shading) predicted by the nonlinear acoustics solver (NLAS) Sound-pressure levels determined by NLAS method at probe 111 (cylinder rear face) Location of the LES subdomain (displayed mesh shows every eighth grid line) Mean flow streamlines. Left: RANS computation, Right: Hybrid RANS–LES computation Instantaneous Schlieren-like view Isovalue contours of the dilatation field  Acoustic pressure spectrum at location S2 Acoustic pressure spectrum in the recirculation bubble (location S3) Isovalue contours of the dilatation field  Acoustic pressure spectrum at the slat’s trailing edge (location S4) Acoustic pressure spectrum in the slat’s wake Resolution of different explicit and compact (implicit) schemes Comparison of the resolution of Taylor and Fourier difference schemes Effect of α f on a second- and an eighth-order filter Perturbation pressure of an acoustic wave initiated by a Gaussian pulse: comparison of solution with smooth and randomly perturbed mesh Example for a reasonable choice of the integration domain (inflow and outflow boundaries) for the flow past a wing in a wind tunnel Von Karm ´ an ´ vortex street past an inclined wing (NACA–4415) at Re = 20,000 and α = 12◦ visualized by streaklines; four different time instants of a shedding cycle in the vicinity of the outflow boundary are shown Sketch of Lund et al.’s (1998) procedure for generating appropriate inflow conditions for a boundary layer flow Example for the generation of inflow data for a 90◦ bend using a second simulation for a straight duct flow with periodic b.c. Law of the wall u+ (y+ ) in a turbulent boundary layer without or with only a weak pressure gradient (half-logarithmic plot) Sketch of the computational domains to determine, for example, trailing-edge noise with the hybrid approach Sketch of the rescaling concept Sketch of the flat-plate boundary layer domain (left) and the trailing-edge domain (right). The procedure to provide the inlet distribution to simulate trailing-edge flow is visualized

155 155 156 157 157 158 159 159 160 160 161 162 162 163 164 178 180 188

197 202

205 206 207 211 224 226

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LIST OF FIGURES AND TABLES

5.13

5.14 5.15 5.16 6.1

6.2

6.3

6.4 6.5 6.6

6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21

LES of a turbulent boundary layer for Reθ0 = 1400 and M∞ = 0.4. Skin-friction coefficient c f versus Reθ for different rescaling formulations Transfer function | F˜ | as a function of wave number α scaled by damping zone thickness d Spurious sound waves and velocity field generated at an artificial boundary at x = 0 Pressure distribution on y = 35 in Figure 5.15 for several thickness values d and Biot–Savart’s law (denoted as compensation) Simulation of a 2D mixing layer. (a) Snapshot of the dilatation field  = ∇.u on the whole calculation domain, levels in s−1 . (b) View of the pairing zone with the vorticity field in the mixing layer and the dilatation field outside LES of a ReD = 6.5 × 104 subsonic jet. Time evolution of the fluctuating pressure p in Pa as a function of t ∗ = tu j /D, at x = 16r 0 , y = 8r 0 , and z = 0 LES of a ReD = 6.5 × 104 subsonic jet. Snapshot of the vorticity norm in the flow field and of the fluctuating pressure outside in the plane z = 0 at t ∗ = 7.5 LES of a ReD = 6.5 × 104 subsonic jet. Snapshots of the vorticity norm in the plane z = 0 at times: (a) t ∗ = 2.2, (b) t ∗ = 3.5 LES of a ReD = 4 × 105 subsonic jet. Snapshot of the vorticity norm in the flow field and of the fluctuating pressure outside in the plane z = 0 LES of a ReD = 4 × 105 subsonic jet. (a) Pressure spectra at  (x = 11r 0 , r = 15r 0 ). (b) Profiles of vrms /u j in the shear layer for r = r 0 . Different simulations: LESac (- – -), LESampl (. . . . . . ), LESshear (- - -), LESmode (− · −·) Schematic of a turbulent jet issuing into a still fluid OASPL directivity at a distance of 30D j from the unheated, Mach 0.9 jet exit Centerline distribution of streamwise root-mean-square fluctuations Centerline distribution of density root-mean-square fluctuations normalized by the difference (ρ j − ρ∞ ) Far-field OASPL taken at a distance of 30D j from the nozzle exit Integral Lagrangian time scale of streamwise fluctuations near the end of the potential core One-third–octave spectral comparisons of an unheated, Mach 0.5 jet of Bodony and Lele (2004) with the 195 m/s data of Lush (1971) Azimuthal correlation of the far-field sound field of LES data from Bogey, Bailly, and Juve´ (2003) (− · −) and Bodony and Lele (2004) (—) Far-field OASPL taken at a distance of 30D j from the nozzle exit Far-field acoustic spectra taken at a distance of 30D j from the nozzle exit Transition toward a wake mode for large L/δθ ratio (2D DNS of the flow over an L/D = 4 cavity, at M = 0.5, and ReD = 4800) LES of a deep cavity. Influence of the boundary layer turbulence on cavity noise Mode switching The plate model

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240

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244 247 249 254 255 256 257 258 258 260 260 265 268 270 271 275

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LIST OF FIGURES AND TABLES

6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 6.35 6.36 6.37 6.38 6.39 6.40 6.41 6.42 6.43 6.44 6.45 6.46 6.47

6.48 6.49 6.50 6.51

Sketch of the experimental setup Wall-pressure transducer positions on the elastic steel plate Vibroacoustics measurements: (+) accelerometers and (•) microphones positions General mesh dimensions General sizes of the mesh Detailed mesh around the obstacle Contours of the streamwise mean velocity U1 /U0 near the ruler in the median plane Streamwise velocity fluctuations urms /U0 near the ruler in the median plane Velocity profiles at the origin of the reference: (- — -) measurement; (+ + +) simulation Pressure coefficient Wall-pressure fluctuation rms levels calculated at various positions along the median plane Comparison of calculated (· · · ) and measured (+ + +) wall-pressure fluctuation rms levels on the plate PSD of wall-pressure fluctuations measured on the flat plate Coherence γ 2 of wall-pressure fluctuations measured on the flat plate for streamwise points using the same notation as in Figure 6.34 Coherence γ 2 of wall-pressure fluctuations measured on the flat plate for spanwise points using the same notation as in Figure 6.34 Phase velocity Up of wall-pressure fluctuations measured on the flat plate for streamwise points using the same notation as in Figure 6.34 Acceleration PSD aa – channel 26 Acoustic pressure PSD pp – channel 30 (Pref = 2 × 10−5 Pa) using the same notation as in Figure 6.38 First mesh: detailed mesh around the obstacle Comparison of the two simulations – wall-pressure fluctuations DSP on the plate Comparison of the two simulations – acoustic pressure in cavity using the same notation as in Figure 6.41 Numerical simulation of airfoil aerodynamic noise: possible hybrid strategies Sketch of the computational domains to determine trailing-edge noise with the hybrid approach Coordinate system and nomenclature used to determine corrections for a 2D acoustic simulation LES subdomain at the trailing edge (left) and horizontal weighting function (right) Damping function |F (αd )| over vortical wavenumber in x1 -direction α = 2π/λv scaled with the damping zone on- and offset width d = (l 2 − l 1 )/2 LES grid with partially resolved plate and every second grid point shown Acoustic grid scaled with the plate length l Enlargement of the leading- (left) and trailing-edge region (right) Visualization of the instantaneous flow field

278 279 280 280 281 281 283 283 284 285 285 286 287 288 289 290 291 291 292 292 293 295 297 301 304

305 305 306 307 307

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LIST OF FIGURES AND TABLES

6.52 6.53

6.54 6.55

6.56

6.57 6.58 6.59 6.60 6.61 6.62 6.63 6.64 6.65 6.66 6.67 6.68 6.69 6.70 6.71

6.72

6.73 6.74 6.75 6.76

APE source terms L  = [ω  × u] = (L x , L y )T , L x (left), L y (right), and CAA grid Pressure contours of the trailing-edge problem M = 0.15, Re = 7 × 105 , at time level T = 3.0 with APE solution equations (6.32, 6.33) Pressure contours of the trailing-edge problem M = 0.15, Re = 7 × 105 at time level T = 3.0 Directivity 1/2 (θ, r, f ) with ∼ = nondimensional power spectral ∞   2 2 2  , i.e., 0 (θ, r, f )df = p 2 (θ, r )/ density (PSD) of p (θ, r )/ ρ∞ c∞   2 2 ρ∞ c∞ Comparison of the trailing-edge noise directivities 1/2 (θ, r, f ) for ¨ r = 1.5 applying Equations (6.32, 6.33) ( = ˆ PSD of p 2 ) and Mohring’s 2 (1999) acoustic analogy ( = ˆ PSD of B ), Equations (6.43. 6.44) Sound-pressure level (SPL) versus frequency for a receiving point in r = 1.5 above the trailing edge for various directions θ (see Figure 6.45) Generation of a periodical source term via window weighting CAA grid A harmonic test source over the trailing edge (left) and directivities obtained for M = 0, 0.088 with LEE and APE (right) applied APE source term (ω  × u) y (left) and sound radiation from the trailing edge (right) Computational grid Contours of instantaneous Mach-number isovalues Flow streamlines at the trailing edge: instantaneous (above) and time-averaged (below) Wall-pressure fluctuations on pressure and suction sides near the TE Evolution of the wall-pressure PSD along the chord Evolution of the wall-pressure PSD along the chord Wave-number–frequency spectrum of wall-pressure fluctuations on the airfoil suction side at x/C = 0.9 Wave-number–frequency spectrum of wall-pressure fluctuations on the airfoil suction side at x/C = 0.5 Instantaneous isovalues of pressure fluctuations obtained from LES data Evolution of the wall-pressure spectra along the vertical grid line x = C (starting from the TE upper corner) with respect to the vertical distance z Spanwise evolution of the coherence of the surface-pressure field on the suction side near the TE (x/C = 0.9958). The frequency bandwidths are integrated Spanwise evolution of the coherence of the pressure field at distance z0 = 37.9 mm from the TE at x/C = 1 Final problem-adapted acoustic grid Final problem-adapted acoustic grid (closer view) Isovalue contours of instantaneous pressure fluctuation field (range ±2 Pa, black and white) computed from (i) LES inside the injection interface and (ii) E3P (from LES data injection) outside the injection interface

308

309 310

310

311 311 313 313 314 315 319 320 321 322 323 324 324 325 326

326

327 328 328 330

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LIST OF FIGURES AND TABLES

6.77

6.78

6.79 6.80 6.81 6.82 6.83 6.84 6.85 6.86 6.87 6.88 6.89 6.90 6.91 6.92 6.93 6.94 6.95 6.96 6.97 6.98 6.99 6.100 6.101 6.102 6.103 6.104

6.105 6.106 6.107 6.108 6.109 6.110

Isovalue contours of instantaneous pressure fluctuation field (range ±2 Pa, black and white) computed from (i) LES inside the injection interface and (ii) E3P (from LES data injection) outside the injection interface (closer view) Isovalue contours (range ±3 Pa black and white) of instantaneous pressure fluctuation field computed from (i) LES data inside the injection interface, (ii) Euler data (from LES data injection) between the injection interface and the Kirchhoff control surface, and (iii) from Kirchhoff integration data beyond the Kirchhoff control surface Numerical mesh (2D plane) Sound-pressure level of LES Sound-pressure level of 3D URANS Experimental setup Instantaneous streamlines in 2D URANS simulation Instantaneous streamlines in LES with adaptive model Lift and drag coefficients of the cylinder in the third-finest grid Lift and drag coefficients of the cylinder in the second-finest grid Lift and drag coefficients of the cylinder in the finest grid Acoustic density fluctuations of 2D URANS simulation in the finest grid Sound-pressure level of 2D URANS simulation in finest grid Acoustic density fluctuations of LES with adaptive model in second-finest grid Sound-pressure level of LES with adaptive model in second-finest grid Acoustic density fluctuations of 3D LES simulation in finest grid Sound pressure level of LES with adaptive model in finest grid Sound pressure level of LES in finest grid with Smagorinsky and Lilley model The Ahmed body from Ahmed et al. (1984) (isosurface of zero streamwise velocity from Kapadia et al. 2003) Mesh for the CFD model from Volkswagen Streamlines in the wake of the CFD model Vorticity in the wake of the CFD model Pressure coefficient in the symmetry plane of the CFD model Sound-pressure level at an observer point of x = 10 m, y = 10 m, and z = 1 m Aerodynamic computational domain Acoustic computational domain LES velocity field; longitudinal component (U = 14 ms−1 , t = 6.6 × 10−2 s) Acoustic results: acoustic power radiated by the diaphragm with respect to the mean velocity (top) and acoustic power spectrum for U = 14 ms−1 (bottom) Geometry of the cavity duct system Snapshots of the pressure in the duct (left) and the vorticity in the cavity (right) during a period of the oscillation Geometry of the sudden enlargment Snapshots of the Mach number for τ1 = 5.5 Snapshots of the Mach number for τ2 = 2.65 Opel 2004 Astra

331

333 335 337 337 338 339 340 340 341 341 341 342 342 343 343 343 344 345 346 347 347 348 348 351 351 352

352 353 354 355 355 356 360

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LIST OF FIGURES AND TABLES

6.111 Lilley turbulence shear-source distribution illustrated by isosurfaces for the idealized wing-mirror example of Siegert et al. (1999) 6.112 Mesh frequency cutoff (MFC) estimate; idealized wing-mirror example of Siegert et al. (1999) 6.113 Predicted versus measured pressure spectra; idealized wing-mirror example of Siegert et al. (1999) 6.114 Mach 0.85 cavity: symmetry-plane snapshot at t = 0.3 s; DES/k–ε (top) and URANS/k–ε (bottom) 6.115 Overall (a) and band-limited (b) Prms along cavity ceiling centerline 6.116 PSD (kPa2 /Hz) at location x/L = 0.45 6.117 Diesel injector primary liquid spray breakup; liquid-free surface with synthetic inlet perturbation (top) and without inlet perturbation (bottom) 6.118 Audi A2 full-vehicle geometry with localized domain shown in dark 6.119 Instantaneous velocity magnitude field 6.120 SPL against frequency at Microphone 4 6.121 Experimental (top) and simulated (bottom) pressure trace to 0.3 s at x/L = 0.95 6.122 Sampling effects on overall Prms (kPa) along the cavity ceiling for M219 experimental data 6.123 Sampling effects on overall Prms (kPa) along the cavity ceiling for CFD data 6.124 Resonator geometry: application challenge from BEHR GmbH in the DESTINY-AAC project 6.125 Velocity contours (top), pressure–time traces (bottom left), and spectral magnitude (bottom right) at three bulk velocities (4, 8, and 12 m/s) taken at the neck of the resonator 6.126 Acoustic response for 8 m/s case at the microphone 6.127 Experimental prototype with inlet cylinder and outlet filter removed and locations of far-field monitors 6.128 Comparison of steady-state RANS and snapshots from the DES calculation 6.129 Surface acoustic pressure (Pa) on the exterior model (a) and acoustic pressure in the far-field (b) 6.130 Computed and measured dB(A) levels at the nine microphone locations at the blade-passing frequency (BPF)

361 362 363 365 366 367 367 368 368 369 369 370 370 372

373 374 374 375 375 376

Tables 1.1 3.1 3.2 3.3

3.4 3.5 3.6

Sound-pressure levels for common sounds Examples of usual spatial convolution filters Various decompositions for the nonlinear terms Resolution requirements referred to Kolmogorov length scale η used in DNS based on spectral methods of some incompressible homogeneous and wall-bounded flows Typical mesh size (expressed in wall units) for DNS and LES of boundary layer flow Definition of simulation types for compressible flows Modified wave-number analysis of some classical centered finite difference schemes

2 94 97

105 107 109 109

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LIST OF FIGURES AND TABLES

5.1 5.2 5.3 5.4 6.1

Coefficients for the DRP scheme of Equation (5.27) Coefficients for filter formula 5.72 Optimized coefficients of the amplification factor for the LDDRK schemes Typical mesh sizes (expressed in wall units) for a boundary layer flow using DNS, wall-resolved LES, and LES with an appropriate wall model LES of a ReD = 4 × 105 subsonic jet. Sideline sound levels and  -maxima in the shear layer for the different simulations vrms

174 188 200 209 244

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Contributors

Jean-Marc Auger PSA Peugeot Citro¨en 1 route de Gisy F-78943 V´elizy-Villacoublay Cedex FRANCE e-mail: [email protected] Dr. Christophe Bailly Centre Acoustique LMFA & UMR CNRS 5509 Ecole Centrale de Lyon 36, avenue Guy de Collongue F-69134 Ecully FRANCE e-mail: [email protected] Dr. Paul Batten Metacomp Technologies, Inc. 28632-B Roadside Drive Suite 255 Agoura Hills, CA 91301 USA e-mail: [email protected] Dr. Daniel J. Bodony Stanford University Department of Aeronautics and Astronautics Stanford, CA 94305-4035 USA e-mail: [email protected]

Dr. Christophe Bogey Centre Acoustique LMFA & UMR CNRS 5509 Ecole Centrale de Lyon 36, avenue Guy de Collongue F-69134 Ecully FRANCE e-mail: [email protected] Priv.-Doz. Dr.-Ing. Michael Breuer Lehrstuhl f¨ur Str¨omungsmechanik (LSTM) Universit¨at Erlangen-N¨urnberg Cauerstr. 4 D-91058 Erlangen GERMANY e-mail: [email protected] Ir. Tim Broeckhoven Vrije Universiteit Brussel Department of Mechanical Engineering Fluid Mechanics and Thermodynamics Research Group Pleinlaan 2 B-1050 Brussels BELGIUM e-mail: [email protected]

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CONTRIBUTORS

Fabien Crouzet Electricite de France Analysis in Mechanics and Acoustics Department 1 avenue du General de Gaulle F-92141 Clamart Cedex FRANCE e-mail: [email protected] Jean Paul Devos Electricite de France Analysis in Mechanics and Acoustics Department 1 avenue du General de Gaulle F-92141 Clamart Cedex FRANCE e-mail: [email protected] Dr.-Ing. Roland Ewert DLR Institute of Aerodynamics and Flow Technology Technical Acoustics Lilienthalplatz 7 D-38108 Braunschweig GERMANY e-mail: [email protected] Oliver Fleig The University of Tokyo Graduate School of Engineering Department of Mechanical Engineering 7-3-1 Hongo, Bunkyo-Ku Tokyo 113-8656 JAPAN e-mail: [email protected] Xavier Gloerfelt Laboratoire SINUMEF ENSAM (Ecole Nationale Sup´erieure d’Arts et M´etiers) 151, boulevard de l’Hopital F-75013 Paris FRANCE e-mail: [email protected] Prof. Dr. Ir. Avraham Hirschberg Faculteit Technische Natuurkunde Technische Universiteit Eindhoven CC 2.24, Postbus 513 NL-5600 MB Eindhoven THE NETHERLANDS e-mail: [email protected]

Prof. Fang Q. Hu Department of Mathematics and Statistics Old Dominion University Norfolk, VA 23529 USA e-mail: [email protected] Dr. Thomas Huttl ¨ MTU Aero Engines GmbH Dachauer Str. 665 D-80995 M¨unchen GERMANY e-mail: [email protected] Andr´e Jacques Mcube 54, Rue Montgrand - BP 232 F-13178 Marseille Cedex 20 FRANCE e-mail: [email protected] Dr. Manuel Keßler, Dipl.-Phys. Institut f¨ur Aerodynamik und Gasdynamik Universit¨at Stuttgart Pfaffenwaldring 21 D-70550 Stuttgart GERMANY e-mail: [email protected] Prof. Dr. Ir. Chris Lacor Vrije Universiteit Brussel Department of Mechanical Engineering Fluid Mechanics and Thermodynamics Research Group Pleinlaan 2 B-1050 Brussels BELGIUM e-mail: [email protected] Philippe Lafon Laboratoire de Mecanique des Structures Industrielles Durables (LaMSID) UMR CNRS EDF 2832 1 avenue du General de Gaulle F-92141 Clamart Cedex FRANCE e-mail: [email protected]

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CONTRIBUTORS

Sanjiva K. Lele Stanford University Department of Aeronautics and Astronautics & Department of Mechanical Engineering Stanford, CA 94305-4035 USA e-mail: [email protected]

Ir. Jan Ramboer Vrije Universiteit Brussel Department of Mechanical Engineering Fluid Mechanics and Thermodynamics Research Group Pleinlaan 2 B-1050 Brussels BELGIUM e-mail: [email protected]

Dr.-Ing. Franco Magagnato Fachgebiet Str¨omungsmaschinen Universit¨at Karlsruhe (TH) Kaiserstrasse 12 D-76128 Karlsruhe GERMANY e-mail: [email protected]

Dr. Sjoerd W. Rienstra Department of Mathematics and Computer Science Eindhoven University of Technology P.O. Box 513 NL-5600 MB Eindhoven THE NETHERLANDS e-mail: [email protected]

Eric Manoha ONERA/DSNA/BREC BP 72 F-92322 Chatillon Cedex FRANCE e-mail: [email protected] Fred G. Mendon¸ca CDadapco CFD Engineering Services Manager, London CD adapco Group, UK 200 Shepherds Bush Road London W6 7NY ENGLAND e-mail: [email protected] Dimitri Nicolopoulos Mcube 54, Rue Montgrand - BP 232 F-13178 Marseille Cedex 20 FRANCE e-mail: [email protected] Fred P´eri´e Mcube 54, Rue Montgrand - BP 232 F-13178 Marseille Cedex 20 FRANCE e-mail: [email protected]

Prof. Pierre Sagaut LMM - UPMC/CNRS Laboratoire de mod´elisation en m´ecanique Universit´e Pierre et Marie Curie Boite 162, 4 place Jussieu F-75252 Paris Cedex 05 FRANCE e-mail: [email protected] Univ.-Prof. Dr.-Ing. Wolfgang Schr¨oder Lehrstuhl f¨ur Str¨omungslehre und Aerodynamisches Institut RWTH Aachen W¨ullnerstr. zw. 5 u. 7 D-52062 Aachen GERMANY e-mail: [email protected] Ir. Sergey Smirnov Vrije Universiteit Brussel Department of Mechanical Engineering Fluid Mechanics and Thermodynamics Research Group Pleinlaan 2 B-1050 Brussels BELGIUM e-mail: [email protected] Philippe Spalart Boeing Commercial Airplanes P.O. Box 3707 Seattle, WA 98124-2207 USA e-mail: [email protected]

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CONTRIBUTORS

Marc Terracol ONERA 29 avenue de la Division Leclerc F-92320 Chatillon FRANCE e-mail: [email protected] Sandrine Vergne PSA Peugeot Citro¨en 2 route de Gisy F-78943 V´elizy-Villacoublay Cedex FRANCE e-mail: [email protected]

Dr. Claus Wagner Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V., DLR Institut f¨ur Aerodynamik und Str¨omungstechnik Abt. Technische Str¨omungen Bunsenstraße 10 D-37073 G¨ottingen GERMANY e-mail: [email protected]

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Preface

Two branches of the same tree are growing together: Acoustics and the large-eddy simulation (LES) technique are based on the same fundamental equations of fluid dynamics. In the past, both scientific disciplines developed independently from each other. Acoustics is one of the classical disciplines of mechanics, having its roots in Greek and Roman times. LES is a comparatively young field of research that has benefited from the exponential growth in computational possibilities over the last few decades. Each scientific community has developed its own methods, definitions, and conventions, and it sometimes seems that experts and scientists in acoustics and LES techniques speak different languages. During the last few years, the LES and the acoustics communities realized that LES can be a comprehensive tool for acoustical research and design and intensified its use. This book presents the current state of the art for LES used in acoustical investigations and comprises 19 contributions from 30 authors, each an expert in his field of research. A general introduction to the subject is followed by descriptions of the theoretical background of acoustics and of LES. A chapter on hybrid RANS–LES for acoustic source predictions follows. More details are given for numerical methods, such as discretization schemes, boundary conditions, and coupling aspects. Numerous applications are discussed ranging from simple geometries for mixing layers and jet flows to complex wing or car geometries. The selected applications deal with recent scientific investigations at universities and research institutes as well as applied studies at industrial companies. Side areas of LES for acoustics are addressed in a contribution on vibroacoustics. The book is a collection of different methods, tools, and evaluation methodologies. Currently it is not possible to offer a perfect solution methodology that generally covers all possible applications. Although interesting results of several commercial codes are presented, a recommendation for any specific solver cannot be made because a benchmark of the codes has not been established and several other codes have not been considered yet. Each method, both scientific and commercial, has its individual advantages and weaknesses. It was also not our intention to harmonize the definitions xxv

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PREFACE

and conventions in acoustics and LES computing. Therefore, the same nomenclature is not used by all authors. The book is intended to be used by students, researchers, engineers, and code developers willing to become more familiar with the use of the LES technique for acoustical studies. The limitations of the method have been outlined as well as its requirements. The reader should acquire an impression of possible and appropriate applications for this methodology. The editors would welcome any initiatives motivated by this book for international cooperation in the development or application of LES for acoustics. The idea for this book came from Eric Willner, the former commissioning editor for engineering at Cambridge University Press, when he read the first call for papers for the International Workshop on LES for Acoustics organized by Thomas H¨uttl, Claus Wagner, and Jan Delfs in G¨ottingen, 2002.∗ At this time, Cambridge University Press was actively seeking a book on LES for acoustics for its aerospace series. Thomas H¨uttl and Claus Wagner agreed to edit a scientific book based on the contributions of the workshop in G¨ottingen. Several speakers and participants of the workshop and other experts promised to contribute to the book, which was conceived as more of a scientific handbook than a simple workshop proceedings. Pierre Sagaut separately developed the idea of a book on LES for acoustics and joined the team of editors. The book would not exist without the contributions from each of the authors. The editors are not only grateful for these contributions but also for valuable review comments from several authors during two book reviews as well as interesting scientific discussions of review comments and proposals. We would also like to thank Peter Gordon, Senior Editor of Engineering at Cambridge University Press, for his help in preparing the book but also for enthusiasm, patience, and confidence during the last 2 years when the progress of the book was sometimes slow but never stopped. Thomas H¨uttl gratefully acknowledges the advice and comments of MTU Aero Engines aeroacoustics specialist Fritz Kennepohl, who introduced him to the secrets of acoustics during the TurboNoiseCFD research project. Pierre Sagaut, Thomas H¨uttl, and Claus Wagner Europe, May 2006 ∗

International ERCOFTAC-DGLR-DLR-Workshop on LES for Acoustics organized by T. H¨uttl, C. Wagner and J. Delfs, German Aerospace Center (DLR), G¨ottingen, Germany, 7–8 October 2002.

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Introduction

1.1 The importance of acoustic research Thomas Huttl ¨ There are some aspects of acoustics that significantly affect the quality of our daily lives. By speaking, we transfer information and knowledge from one person to another. The sound of rain, of wind or anything else gives us orientation and aids optical perceptions. Music can fascinate us and stimulate our emotions and moods. Pleasant sounds and music positively affect health by their calming character. The negative side of acoustics is noise. Noise is the most commonly cited form of environmental pollution. Noise is easily detected by the human hearing system. Its effects can be cumulative, and it influences our work environment as well as our leisure. Even the quality of our sleep is reduced if we are exposed to noise. In recent decades, the effects of noise on people have been studied intensively. 1.1.1 Health effects

There is no doubt that noise has an impact on health. Very loud sounds are clearly highly injurious to people as well as animals. Table 1.1 shows sound-pressure levels (dB(A)) for common sounds. At sound-pressure levels of 160–165 dB(20 kHz) flies die when exposed only for a short time. With these exposures levels, human beings become tired, may experience facial pain, and may develop burned skin. When the sound pressure is lowered, reactions to the sound decrease. Long-term exposure to high noise levels of about 90 dB(A) can result in permanent hearing loss. Even for a steady daily noise level of 75 dB(A) for 8 hours per day, there is a risk of permanent hearing damage after 40 years of exposure. Apart from the auditory effects of noise, nonauditory effects on health are well known. Noise can induce a range of physiological response reactions such as increases in blood pressure, heart rate, and breathing, and these reactions are not confined to high noise levels and sudden noise events but are also true for noise levels commonly experienced in noisy environments such as busy streets (Nelson 1987). 1

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INTRODUCTION

Table 1.1. Sound-pressure levels for common sounds (dB(A))

Common sounds

30 50 60 70 80 90 100 110 120 130 140 180

whisper rainfall, quiet office, refrigerator dishwasher, normal conversation traffic, vacuum cleaner, restaurant alarm clock, subway, factory noise electric razor, lawnmower, heavy truck or road drill at 7 m garbage truck, chain saw, stereo system set above halfway mark rock concert, power saw jet takeoff, nightclub, thunder jack hammer shotgun, air raid system rocket-launching pad

Exposure to excessive noise during pregnancy may result in high-frequency hearing loss in newborns and may be associated with prematurity and intrauterine growth retardation (American Academy of Pediatrics 1997). 1.1.2 Activity effects

Noise influences human activities such as sleep, communication, and people’s general performance. Studies have shown that noise can affect sleep in many ways. Noise may shorten the length of the sleep period and increase the number or frequency of awakenings, and it may affect the duration of the various stages of sleep. Given the importance of sleep for individual health, a perturbation of sleep is usually not tolerated by people. First- and second-grade school children chronically exposed to aircraft noise have significant deficits in reading as indexed by standardized reading tests administered under quiet conditions. Chronic noise may also lead to deficits in children’s speech acquisition (Evans and Maxwell 1997). Other studies show the impact of noise from aircraft, road traffic, and trains on long-term recall and recognition (Hygge 1993; Groll-Knapp and Stidl 1999). Mood and behavioral abnormalities can also be related to noise exposure. 1.1.3 Annoyance

In addition to the direct effects of noise on sleep, communication, and performance there are also indirect consequences of annoyance or disturbance that are related to the way a person feels about noise. Annoyance is a very complex psychological phenomenon, and it is scarcely possible to define or measure it here. For physically loud sounds the perceived annoyance is often equivalent to the perceived loudness. For physically soft sounds (rustling papers at the movies, people talking while watching television), the perceived annoyance deviates greatly from perceived loudness. Thus, for physically soft sounds, more attention is paid to other aspects of the sound than to the loudness-related

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1.1 THE IMPORTANCE OF ACOUSTIC RESEARCH

sound-pressure level (Berglund, Preis, and Rankin 1990). Nevertheless, annoyance effects can also influence mood and behavior and can ultimately cause even severe health problems. 1.1.4 Technical noise sources

Many noise sources are man-made – especially transportation noise from road traffic, aircraft, and trains. Other technical noise sources can also be annoying such as wind turbines or cooling and climate systems. Noise can be produced by several physical interaction mechanisms: r Solid-body friction noise (e.g., gearbox), r Solid-body vibration (e.g., unbalanced rotating machines), r Combustion noise (e.g., piston engines), r Shocks (e.g., explosions or pneumatic hammer), and r Aerodynamic noise (e.g., vortex–structure interaction)

In this book we concentrate on aerodynamic, flow-induced noise. Aerodynamic noise is one of the major contributors to external vehicle noise emission as well as of internal vehicle noise due to the transmission of the externally generated noise through structure and window surfaces into the cabin. Aerodynamic noise becomes dominant at driving speeds exceeding 100 km/h when compared with structure-borne, power train, and tire noise, for which substantial noise reduction has been achieved. The interaction of the flow with the geometrical singularities of the vehicle body produces unsteady turbulent flows – often detached – resulting in an increased aerodynamic noise radiation (Vergne et al. 2002). Aircraft noise is dominant for residents near airports when planes fly at low altitudes such as during departure and landing. The engines, especially their free-jet flow – but also flaps, wings, airbrakes, landing gear, or openings – contribute significantly to the total sound emission. When an aircraft is flying at cruising altitude, the aerodynamic noise of the aircraft body and propeller, or engine noise, can be annoying for passengers and crew. For trains, the bow collector and the wheel–rail interaction are dominant noise sources. For high-speed trains, especially unconventional concepts like the magnetic levitation hover train project Transrapid (Figure 1.1), the relevance of aerodynamic noise is increased. Wind turbines in operation emit aerodynamic noise that can be perceived by humans. With respect to large wind turbines with low rotational speed, the main contribution to aerodynamic noise is the narrow-band noise caused by blade–tower interaction. Smaller, fast-rotating wind turbines, on the other hand, emit broadband noise, which is mostly caused by vortex–blade interaction (Fleig, Iida, and Arakawa 2002). Cooling and climate systems are designed to regulate temperature and filter out pollution. Their power units often emit tonal noise at constant frequency that is transported through the duct system. Furthermore, tonal noise due to aeroacoustic resonance can

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INTRODUCTION

Figure 1.1. Magnetic levitation hover train project Transrapid.

be activated (Hein, Hohage, and Koch 2004). Although the noise level of cooling and climate systems is usually very low, it is annoying and should be considered by the designer.

1.1.5 Political and social reactions to noise

Noise is becoming generally accepted as an environmental and even health hazard to the population. Public reaction to noise problems is causing governments to adopt laws, regulations, and guidelines for the certification of noise-emitting vehicles and machines as well as for temporal or spatial limitations on their use. Aircraft and jet engine manufacturers face increasingly stringent noise requirements for near-airport operations worldwide (Bodony and Lele 2002a). Old-fashioned and noisy aircraft may not be operated from an increasing number of airports or their operators must pay additional fees for noise emission. Some aircraft may not be operated during the night if their noise emission is too high. Airlines have to consider the noise-related airport fees in their operating costs or reduce the noise emissions by adding new or additional noise-reducing devices to their planes and jet engines. A sociocultural aspect of traffic noise is that the prices for houses and apartments are lower if they are close to highways and roads, railways, or flying routes near the airports. Costly noise-reduction measures like fences, walls, and special windows are required to compensate for the increased traffic noise.

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1.1 THE IMPORTANCE OF ACOUSTIC RESEARCH

Butterworth-Hayes (2004) describes the European Union (EU) plans to overtake the United States as the largest aerospace power in the world by 2020. The 2001 report “European Aeronautics: A Vision for 2020,” which was drawn up by a group of “wise men” to advise the European Commission (EC) on the future of the continent’s aerospace industry, laid out a series of research objectives that Europe’s civil aircraft manufacturers must achieve to ensure dominance of the market. One of the key objectives of “Vision 2020” research is the halving of perceived aircraft noise. This means, in particular, reducing external noise by 4–5 dB and 10 dB per operation in the short and long terms, respectively. For rotorcraft, the objective is to reduce the noise footprint area by 50% and external noise by 6 dB and 10 dB over the short and long terms, respectively.

1.1.6 Reactions of industry

The aircraft industry projects a growth in passenger kilometers of 100% or more in the next 15 years. Satisfying the resulting demand for larger or faster airplanes, or both, requires that new airplanes be designed. The same is true for future high-speed trains, which, in order to be able to compete with air traffic, have to become faster as well. This would boost the emitted noise levels tremendously if these trains were built based on today’s technology because the emitted aerodynamic noise increases approximately with the fifth to sixth power of the vehicle speed (Schreiber 1995). After having achieved significant progress in reducing the level of other primary noise sources such as piston engine noise or road-contact noise, the automotive industry is now contending with the major problem of interior and exterior noise related to aerodynamic effects. Aeroacoustics is becomimg increasingly important in many other fields such as the energy industry or personal computer manufacturing industry. Sensitivity to noise is increasing all over the world. Therefore, significant noise reduction in new airplanes, trains, and cars is mandatory if this growth in the transportation system is to be accepted by the population and their political representatives. This noise reduction can only be realized if the design process is guided by robust and fast computational aeroacoustic methods. Owing to the lack of commercial software, many companies and research institutes are using their specialized, self-developed, in-house codes for solving engineering problems. Remarkable progress has also been made by designers of commercial codes for aeroacoustic applications, although these codes are currently still under development (Wagner and H¨uttl 2002).

1.1.7 Research on acoustics by LES

Aeroacoustics is the scientific discipline between fluid mechanics and classical acoustics. It considers sound generated by aerodynamic forces or motions originating from (turbulent) flows (Ihme and Breuer 2002). Initially, experimental investigations were

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INTRODUCTION

used to derive some empirical relations for estimating the noise emission of new technical products. However, owing to strongly increased computer performance, the numerical simulation of acoustic fields generated by fluid flow, called computational aeroacoustics (CAA), has become very attractive. At this time no unique solution procedure exists for all acoustic problems. Instead, various strategies have been developed, each with individual advantages and disadvantages (Ihme and Breuer 2002). For applications with complex, inhomogeneous flows and flow-induced noise radiation, the most promising and commonly used numerical technique is to adopt a hybrid approach.∗ In such an approach, the sound-generation and sound-propagation processes are considered separately. A nonlinear aerodynamic near field in which the aerodynamic perturbations are generating the sound is matched to a linear acoustic far field in which no flow or homogeneous flow exists and the sound waves are only propagating. The underlying assumption is that there is no feedback of the acoustic waves on the flow. Coupled or hybrid approaches are currently being created by several research groups and code developers. Such efforts have also been the focus of two recently finished European research projects: Application of Large-Eddy Simulation to the Solution of Industrial Problems (ALESSIA) and TurboNoiseCFD.† The primary aim of the EU ALESSIA project‡ has been to develop software tools for the simulation of fluctuating flows by large-eddy simulation (LES), with a particular focus on flow-induced acoustics (Montavon 2002). Commercial computational fluid dynamics (CFD) and CAA codes have been interfaced within ALESSIA. The aim of the TurboNoiseCFD project was to contribute to the objective of a 10-dB reduction in 10 years in aircraft external perceived noise through new design technologies. To achieve this objective, the aircraft engine manufacturing industry significantly enforces to reduce engine noise levels at the source. In response to this challenge, new methods have been created that will aid in the design of low-noise turbomachinery components based on the adaptation of existing CFD software and its integration with propagation and radiation models. Besides other techniques, an LES methodology has been tested for aeroacoustical evaluations of broadband noise (Boudet, Grosjean, and Jacob 2003; Jacob et al. 2005). After the successful funding of noise reserach programs under the Fifth Research Framework Program (1997–2001), the EC continues to invest money in research projects aimed at reducing the environmental impact of aviation (fuel consumption, noise pollution, and emissions of carbon dioxide (CO2 ), nitrous oxides (NOX ), and other chemical pollutants) in its Sixth Research Framework Program (2002–2006).

∗

† ‡

Other coupling methods are also called hybrid methods such as the coupling of the Reynolds-averaged Navier–Stokes (RANS) approach and large-eddy simulation (LES) for detached-eddy simulation (DES); see Chapter 4. Turbomachinery noise-source CFD models for low-noise aircraft designs. This project was funded by the EC within the GROWTH Fifth Framework Program 1998–2002. ALESSIA is an ESPRIT Project funded at 50% by the EC.

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1.2 INTRODUCTION TO COMPUTATIONAL AEROACOUSTICS

1.2 Introduction to computational aeroacoustics Manuel Keßler Computational fluid dynamics (CFD) has reached a level of maturity that permits many industrially relevant problems to be solved routinely using commercially available tools, although some difficult problems are still out of reach. Consequently, the research interest in the fluid mechanics community has shifted slightly, and there is now a large and still growing group of experts engaged in the field of acoustics. For a long time their work was mostly based on analytical and experimental studies, but the astonishing advances in computer technology have made a numerical approach feasible. That approach is called computational aeroacoustics (CAA). 1.2.1 Definition

Numerous definitions exist for CAA reflecting the many people who have been attached to this subject. We understand CAA here in the broadest possible sense – that is, as a process using some kind of numerical computation to produce acoustical information for aerodynamic phenomena. That obviously includes all flavors of acoustical transport techniques (Lighthill’s acoustic analogy, the Kirchhoff method, the Ffowcs Williams–Hawkings equation), linearized Euler approaches, combined procedures with CFD, semiempirical treatments like stochastic noise generation (SNGR), and even compressible direct numerical simulation (DNS) – admittedly very rarely used for acoustic analysis these days. Experimentors evidently use numerical computations for data processing and evaluation as well. However, although experimental studies are of substantial significance in the efforts to find appropriate models, tune constants, and validate computations, they do not constitute CAA. 1.2.2 History

Even though sufficient computing power has become available only recently, CAA has quite a long tradition in the fluid mechanics community. Among the earliest efforts was the paper by Gutin (1948) published first in Russia in 1936. However, modern CAA rests mostly on the shoulders of Sir James Lighthill (1952, 1954), who published what are certainly the most influential and therefore the most cited papers in aeroacoustics. He introduced the idea of representing sound as the difference between the actual flow and a reference flow – usually a quiescent medium at rest. Because sound pressure and velocities are in general small perturbations around a background flow, approximations are possible to simplify the problem. In the late 1960s, his acoustic analogy approach was extended by Ffowcs Williams and Hawkings (1969) to the case of moving surfaces immersed in the flow (and acoustic) field, which is second in citations only to the work of Lighthill. Another classic of its time was Goldstein’s (1976) book.

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Nevertheless, owing to the highly abstract and mathematical presentation of the subject and the lack of appropriate aerodynamic simulation data, progress was slow during the next two decades. Only sound emitted from jets received considerable interest in the early days of CAA – mostly for their simple geometry (because there are no solid walls) as well as the importance of this sound source for the noise levels of aircraft developed at that time. Many discoveries of basic sound-generation mechanisms and scaling laws date from this time. Computers have only recently become powerful enough to tackle other, more difficult aeroacoustical problems one way or another with sufficient accuracy to provide results valuable for industry in the design process. Associated with increasing societal interest in noise reduction, another golden age in aeroacoustics dawned in the late 1980s and early 1990s. CAA gained momentum since then, and there is still no slowdown in sight. 1.2.3 Aeroacoustics

Aerodynamic noise occurs because of two basically different phenomena. The first one is impulsive noise, which is a result of moving surfaces or surfaces in nonuniform flow conditions. The displacement effect of an immersed body in motion and the nonstationary aerodynamic loads on the body surface generate pressure fluctuations that are radiated as sound. This kind of noise is deterministic and relatively easy to extract from aerodynamic simulations because the required resolution in space and time to predict the acoustics is similar to the demands from the aerodynamic computation. Aerodynamic noise arises primarily from rotating systems (e.g., helicopter rotors, wind turbines, turbine engine fans, and ventilators). If the surfaces move at speeds comparable to the speed of sound or there is an interaction between a rotor and a stator wake, these tonal noise components can be dominant. The other noise mechanism is the result of turbulence and therefore arises in nearly every engineering application. Turbulence is by its very nature stochastic and therefore has a broad frequency spectrum. Interestingly enough, turbulent energy is converted into acoustic energy most efficiently in the vicinity of sharp edges (e.g., at the trailing edge of an aircraft wing). In this case the uncorrelated turbulent eddies flowing over the upper and lower sides of the edge have to relax with each other, generating locally very strong equalizing flows that result in highly nonstationary pressure spikes. Another major source of turbulence sound is jet flows, in which the shear layer in the mixing zone again radiates into the far field. A third – but here neglected – phenomenon is the case of combustion noise, which is a result of the chemical reactions and the subsequent introduction of energy into the flow. As previously stated, turbulence noise almost always exists, and as a consequence aerodynamic noise is usually a broadband noise sometimes augmented by narrow-band tonal components coming from impulsive noise sources. Impulsive noise can usually be derived from nonstationary aerodynamic calculations. Like CFD, turbulence noise is

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much more difficult to simulate because the turbulence has to be either simulated fully, as in DNS, or modeled as in the Reynolds-averaged Navier–Stokes (RANS) approach, or something in between, as in large-eddy simulation (LES) or hybrid computations. 1.2.4 Conceptual approaches

In the CFD area, several tools have been developed to a very high level of maturity that makes them not only useful as a scientific research instrument but also as an industrial design tool. Some of these tools have reached sufficient reliability to make them helpful for users not considered experts in their field. However, pushing tools to their limits can lead to disaster – and often to nonobvious paths. Because CAA is a more recent domain of activity, the situation is much less favorable. There is not yet a clear path to follow for reliable acoustical information for each and every application, which may consist of such different things as a cooling fan for a personal computer or a supersonic jet driving an airplane. Consequently, many different techniques exist nowadays, each working well in one area and failing totally in another. We try to classify a few of them in the following paragraphs. Direct methods can be considered the most exact technology for CAA and are comparable to the DNS in the CFD field. The complete, fully coupled compressible Euler or Navier–Stokes equations are solved in the domain of interest for the unsteady combined flow and acoustic field from the aerodynamic effective area down to the far-field observer. They do not include any modeling of the sound (besides, possibly, a turbulence model) and thus do not suffer from modeling or approximation errors. Of course, they require tremendous computational resources because – especially in the case of small-Mach-number flows – flow and acoustics represent a multiscale problem with its inherent difficulties. The difficulty here is that the small acoustic perturbations are not drowned out by numerical errors of the much larger aerodynamic forces. Space and time resolution requirements for the aerodynamic data combined with the large distances up to an observer in the far field give rise to ridiculously high numbers of cells and time steps. Even if the necessary computer power were available, the discretization schemes well known from CFD do not work very well in CAA applications because they have dispersion and diffusion errors that are much too high. A plane wave is usually severely distorted and dampened after being transported for just a few wavelengths, which is clearly too short for the common case of an observer in the far field. Direct methods are deceptively attractive because well-known and well-understood CFD packages promise to provide aerodynamic and aeroacoustic data at the same time. Sometimes they do work surprisingly well – mostly in those cases in which the differences between aerodynamics and aeroacoustics are negligible – as in transonic problems. However, for many or most other problems they do not because the basic requirements of CFD and CAA are just too different. Several CFD schemes are tuned specifically to suppress spurious acoustic waves, which of course is a bad idea when one is interested in the acoustic properties of a problem at hand. CFD usually is designed

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to solve a near-field problem because the perturbations from the mean flow vanish quickly. Furthermore, the flow in this region is usually highly nonlinear but basically stationary, or at least changing only slowly (aside from turbulent motions). Acoustics, on the other hand, is clearly a far-field problem in which sound is generated locally in the aerodynamic active area and passively radiated outside to an observer with a smaller exponent of decrease. Outside the aerodynamic active area where the sound is generated the perturbations are small, and a linear description is usually sufficient. However, noise is inherently unsteady with time scales quite comparable to turbulent eddies even if the spatial wavelengths are large compared with aerodynamic ones by an order of the reciprocal of the Mach number. These different properties (linear versus nonlinear, far-field versus near-field, timedependent versus (quasi) stationary, large versus small spatial scale) obviously necessitate different tradeoffs regarding computational schemes. No currently known schemes score best on all possible requirements of CFD and CAA; therefore, methods especially adapted for the specific demands of an application will always be superior to more general ones that necessarily have to balance all their characteristics carefully. Most of the computational aeroacoustic tools in successful use nowadays are therefore of the hybrid type in which sound generation due to aerodynamics is more or less decoupled from the acoustic transport process to the far field, making it possible for tailored algorithms to be used for both tasks. This decoupling leads straightforwardly to an arbitrary combination of a sound generation method with another sound transport method. CFD Sources: On the sound-generation side some kind of CFD tool is primarily in place. If a direct coupling mechanism to the transport method is used it has to provide sound data in the coupling region (surface or volume). In this case aeroacoustic applicability is very important; that is, dispersion and diffusion errors must be at the lowest possible levels. However, the demands are not as high as for direct methods because an undistorted transport has to be sustained only up to the coupling regions, which are seldom farther away than a few wavelengths. A small error in phase and amplitude is therefore acceptable. Semiempirical Sources: Alternatively, the sound sources can be reconstructed semiempirically from CFD data derived mostly from turbulence quantities. A straightforward, steady RANS computation provides information about turbulent length and time scales that translate by empirical relations into sound-source spectra. These spectra are then radiated by one of the transport methods described in the next paragraph. Of course, this process depends heavily on the soundness of the empirics and the validation data used to calibrate them. However, the methods can be very fast and reliable for obtaining a judgment between two close configurations (e.g., in the acoustic optimization of an airfoil shape). In such situations, with a small and well-defined application domain close to the calibration data, they are fairly useful.

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After identification of the sound sources in the aerodynamic active area, the noise generated in this near field has to be transported outside to an observer. Again we may choose between two alternatives: Computational transport methods are similar to a CFD computation, in the sense that they solve some partial differential equations(s) in the entire field up to the observer. In contrast to direct methods, they do not simulate the aerodynamic area as well but concentrate on the solely acoustic domain. Therefore, several assumptions and, consequently, approximations are allowable to achieve accurate solutions efficiently. Usually computational transport methods solve simple equations such as the linearized Euler equations (LEE) or simply the wave equation. Because these are both linear, many problems encountered with the full set of equations do not occur, and consequently the discretization schemes can be highly tuned to reach the desired low level of dispersion and diffusion errors. On the boundary between the CAA domain and the CFD computation, the CFD solution is used as a boundary condition for the CAA simulation. Owing to the change in discretization, resolution, and even equations the definition of proper transmission conditions is a highly nontrivial task. The computational cost can grow tremendously if the observer is far away because all the space in between has to be solved for. However, if an entire noise carpet computation is required in contrast to a single-observer spectrum, this approach may be beneficial inasmuch as the cost is the same regardless of the number of observer points. Analytical transport techniques employ an integrated form of the relevant acoustic propagation equation – either Kirchhoff’s surface integral or the Ffowcs Williams–Hawkings (FW–H) equation. In this case the sound pressure at an observer at a specific point in time is computed by an integration-of-source term along a surface – either a physical one or surrounding the aerodynamic area – and possibly additional volume integrals outside the surface in the case of the FW–H equation. Owing to the finite speed of sound and the deterministic relationship between emission and observer time of a signal, there has to be some kind of interpolation of the data at least on one side. In the case of parts of the integration surface or volume moving at transonic speeds, the integrals become highly singular because of the Doppler effect, which leads to difficulties regarding the numerical stability of the procedure. Taking all these different techniques together, we end up with a general map of noise prediction methods shown in Figure 1.2. Given all the different numerical formulations and specific implementations of each method, there is a plethora of variants to choose from. In this book we want to delve in more detail into the promising part of this tree emphasized in bold.

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12 Wave equation

Computational transport

Acoustic analogy (FW–H)

Analytical transport

Kirchhoff integral

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Figure 1.2. Noise prediction methods.

Linearised Euler equation

Acoustic perturbation equation (APE)

Transport method

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Sound generation

Hybrid methods

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CFD analysis

Direct methods

CAA

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1.2.5 Remaining problem areas for sound computation

Despite the progress made in recent years and the substantial number of researchers in the field, there are still several unsolved issues involving the numerical prediction of noise. Because CFD still has to struggle with (at least) turbulence modeling, transition to turbulence, and separation (even more so with relaminarization and reattachment), the CAA community has not yet reached consensus about the proper treatment of sound computations in most application areas. Consequently, some remarks concerning these issues seem to be in order. Sound is a fundamental nonstationary process. In physical time, the sound waves generated by some aerodynamic processes in the active region have to be transported for at least some distance. So that the sound generated (e.g., at a trailing edge) will not be totally destroyed, it is of paramount importance to perform this transport process as accurately as possible. This holds especially for two basic features of sound: the wave speed and the amplitude. Only specifically tuned methods can guarantee that waves will be supported for a time and distance long enough to be of practical interest, and even fewer methods can do this for the correct phase speed. Short wavelengths at the edge of resolvability by the spatial discretization are particularly prone to numerical errors. They usually encounter totally nonphysical propagation properties – possibly up to the point that the acoustic solution is totally spoiled. 1.2.5.1 Discretization

The main problem of CAA is the disparity of energy, length, and time scales between the aerodynamics and the aeroacoustics – especially at smaller Mach numbers. The ratio of noise energy to mechanical energy is on the order of Pnoise /Pmech ≈ 10−4 M5 . For a low-speed case of M = 0.1 we obtain a ratio of 10−9 , and even for an airliner at M = 0.7 the number is only 10−5 . In an aerodynamic simulation we of course introduce numerical errors. To be at the same level as the acoustics we are interested in, these errors have to be five to nine orders of magnitude smaller than the intended physical values. To obtain acceptable signal-to-noise ratios for sound levels, we have to add at least one more order of magnitude. In this sense, basically every CFD simulation is very loud, counting just the numerical errors that introduce numerical noise. Common CFD schemes are adapted to stationary simulations and therefore just suppress acoustic waves, and so for the aerodynamic community the problem seems to be solved. For the purpose of aeroacoustics, on the other hand, the discretization schemes in both space and time must be specifically tuned. Diffusion and dispersion errors have to be reduced to the lowest possible level. This noble goal is achieved only partly, but progress has been made to develop highly accurate discretization techniques to reach the desired levels. Another problem is the introduction of artifical noise sources caused by numerical errors that may overwhelm the physical sound sources and thus generate entirely unreliable noise information. Finally, if we use dimensional splitting

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techniques, discontinuities in the metrics of grid lines (e.g., at corners and edges) may cause spurious noise reflections.

1.2.5.2 Boundary conditions

At the very heart of CAA lies the proper handling of boundary conditions, which is a situation not very different from that of CFD. Because acoustics is a radiation problem, basically all the sound energy will sooner or later try to leave the computational domain. On solid walls, we obviously get reflections, which we can handle straightforwardly just as in CFD. However, on artificial far-field boundaries, the physics dictates a straight pass-through without any spurious reflections. Although this requirement seems to be obvious, it is indeed very hard to fulfill sufficiently. Several concepts have been developed for this specific problem and optimized for one application or another, but the downside is that a generally accepted solution does not yet exist. Interior boundaries can be problematic as well. If we couple a CFD tool to a computational transport method, there is a difference between the models, and we may therefore obtain spurious reflections at the interface of the boundaries. The same holds for other interior interfaces, even in the same code (e.g., in the interpolation region between Chimera-type overlapping grids). Any difference in modeling, discretization, or resolution is a potential cause for most unwanted reflections.

1.2.5.3 Coupling

Owing to the fundamental differences between aerodynamics and aeroacoustics in many applications, a hybrid approach seems to be a sensible way to tackle the sound prediction problem, as stated before. However, in this case the problem of coupling between the two different algorithms inevitably arises. There is no theory of backscattering (i.e., a feedback from the acoustics to the aerodynamics) because this is not relevant in any engineering application. Even the sole direction from a flow solution to a sound computation, however, is difficult enough to handle properly in order to guarantee continuity, conservation, and, again, the avoidance of artificial reflections at the coupling while admitting physical acoustic waves. This is especially true when the two schemes operate in different spaces (e.g., the flow solver in physical time and the acoustic prediction in the frequency domain).

1.2.5.4 Conclusions

After having looked at some of the issues and – partly unsolved – problems in computational aeroacoustics, we can legitimately claim that CAA will be a demanding and exciting field for surely some time to come. So let us step forward to the collective wisdom of the experts in the hope of overcoming some of the obstacles in order to help the researcher and later on the engineer with his or her daily work on the subject of “design to noise.”

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1.3 STATE OF THE ART: LES FOR ACOUSTICS

1.3 State of the art: LES for acoustics Claus Wagner, Oliver Fleig, and Thomas Huttl ¨ 1.3.1 Broadband noise prediction in general

People’s perception of noise is changing and their sensitivity toward it is increasing all over the world. Significant noise reduction to accommodate this development has been realized by controlling or damping tonal noise sources, but a further increase in noise pollution is expected. The aircraft industry, for example, expects a growth in passenger kilometers of 100% or more in the next 15 years. At the same time new high-speed trains are under development that will allow railways to compete with air traffic on short-distance routes. Further noise reduction is mandatory if this growth in the transportation system is to be accepted by people and their political representatives. To achieve this, the design process needs support from computer-based noise prediction tools. To provide the needed robust and efficient methods, research on reliable CFD and CAA methods has increased tremendously in the last several years. The currently applied numerical approaches range from extremely costly DNS to hybrid approaches that solve timeaveraged governing equations to obtain the near flow field and a wave propagation equation fed by a synthetic turbulent flow field. During the last 20 years, CFD has made impressive progress. There is no doubt that CFD methods will play an indispensable role in the industrial design process, but for several reasons the use of CFD methods for aeroacoustic predictions faces many difficulties. One of the reasons is that the methods solve the Reynolds-averaged (time-averaged) Navier–Stokes (RANS) equations, which are generally time independent. Aeroacoustics problems though, are, by definition, highly time dependent. This is the reason more sophisticated methods must be developed and investigated to predict broadband noise generation and propagation reliably. The straightforward method for predicting broadband noise is to solve the compressible Navier–Stokes equations using direct numerical simulations. Initial work on this approach by Mitchell, Lele, and Moin (1997) and Mankbadi et al. (1998) focused on supersonic jets. Resolving all scales of a turbulent flow in a DNS properly requires that the discretized equations be solved on extremely fine grids because the size of the smallest turbulent scales decreases with increasing Reynolds numbers. For threedimensional flows, this results in computing times that scale with the third power of the Reynolds number. Because most technically relevant flows are characterized by high Reynolds numbers, it can easily be shown that it will be impossible to use DNS for applied turbulent flow problems in the near future. Additionally, a huge computational domain has to be chosen to simulate the far-field acoustics. To fulfill these two conditions at the same time will be a challenge for some generations of researchers to come. Nevertheless, direct numerical calculations of noise have been performed for some simple geometries. Examples are the two-dimensional driven cavity flow, for

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which direct simulations of the flow and the noise have been conducted by Gloerfelt, Bailly, and Juv´e (2001); Colonius, Basu, and Rowley (1999); and Shieh and Morris (1999b). A more promising approach for technically relevant aeroacoustic problems is to apply hybrid methods. With this approach, the near-field aerodynamics are computed to obtain velocity and pressure fluctuations that form the acoustic source terms for a separate computation of the far-field acoustics. The reason for splitting off aerodynamics from aeroacoustics is the great disparity of levels and length scales between the flow and aeroacoustic fields. For both the simulation of the flow and the computation of the sound waves, a variety of methods exist that differ in accuracy and in their demand for computational resources. As mentioned before, flow simulations can be conducted in DNS with the restriction to flows with low Reynolds numbers and rather simple geometries. On the other hand RANS simulations in CFD are applicable for complex geometries and high Reynolds numbers. To make predictions of the radiated noise it is necessary to describe the properties of the turbulence. Because a complete simulation of turbulence and its generated and radiated noise are computationally too expensive for high-Reynolds-number flows, noise predictions are often based on the solution of RANS equations using a two-equation turbulence model like the (k–ε) turbulence model. Such solutions provide an estimate of the amplitude of turbulent velocity fluctuations, but because this approach provides only time-averaged properties it is necessary to make assumptions about the statistical characteristics of the turbulence. In particular, two-point cross correlation of the turbulent noise sources must be approximated. One of the earliest attempts to couple estimates of the statistical properties from steady-flow predictions with a noise model based on acoustic analogy was the approach developed by Balsa and Gliebe (1997). More recent extensions have used RANS solutions for the flow field based on a (k–ε) turbulence model. Tam and Auriault (1999), Page et al. (2001), and Bailly, Candel, and Lafon (1996), for example, performed RANS computations of jet flows to predict the radiated noise based on such a coupled method. The perfectly suited method to compute the large-scale fluctuations, which are known to contribute most to the noise generated in many problems, is the LES technique. To investigate this, Avital, Sandham, and Luo (1996) conducted DNS of temporally evolving mixing layers with transonic and supersonic velocities and used these to estimate the sound emitted by corresponding spatially evolving mixing layers. The sound predictions were obtained applying Lighthill’s analogy, and a large-scale model for the implementation of Phillips’ analogy (Phillips 1960). For high Reynolds numbers, this method allows prediction of the dynamics of the large turbulent scales, whereas the effect of the fine scales is modeled using a subgrid-scale model. When one considers turbulent flows in the absence of walls like jet flows or isotropic turbulence, it is easy to estimate the spatial resolution required to capture a wave number at the lower limit of the inertial range in a LES. Assuming a Reynolds number of Re = 8.7 × 106 for a supersonic turbulent jet, Tam (1998b) estimated that a mesh with 2.1 × 109 grid points is required to accurately perform a LES of the noise of a supersonic jet. Although currently very few research institutions have the computer capability to conduct such

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simulations, the use of LES as a tool for noise research appears to be feasible. However, practical use of LES for noise prediction and design is still some years away. The most general methodology for the prediction of far-field jet noise is to compute the near-field unsteady flow field using a DNS or LES technique in conjunction with an acoustic analogy. Following the flow computation, an acoustic source term is determined from the large-scale fluctuating flow field. The far-field acoustics are then calculated in a larger domain on grids, which are generally characterized by their coarser spatial resolution. As determined by the aeroacoustic problem considered, different methods can be applied. The most common approaches for noise predictions are discussed in the next section.

1.3.2 Broadband noise prediction based on LES

In recent years a variety of large-eddy simulations were performed with the objective of generating the source terms for wave propagation equations. In spite of this, the investigated configurations are limited. We want to give some examples for the application of LES to acoustics and classify them with respect to the investigated configuration. Note that the work listed below is not complete. Our main objective is to demonstrate that the hybrid approach has been applied for noise predictions in different configurations with a variety of combinations that used subgrid-scale models and noise prediction methods. 1.3.2.1 Decaying and forced isotropic turbulence

Seror et al. (2000) addressed the problem of evaluating and modeling the contribution of the unresolved scales to the radiated noise production when Lighthill’s analogy is employed together with LES. To achieve this, they split the Lighthill tensor into three parts: a high-frequency part that is not resolved in LES, the filtered Lighthill tensor computed from filtered LES variables, and a subgrid-scale tensor. They performed DNS and LES of decaying isotropic turbulence to determine the three parts of the Lighthill tensor for both approaches and evaluated the subgrid and high-frequency contributions to the complete Lighthill tensor. This work showed that the high-frequency part of the Lighthill tensor does not contribute significantly to noise production if cutoff wave numbers of the usual values are employed, but this work also indicated that the subgridscale contribution cannot be neglected. With a similar objective Seror and Sagaut (2000) and Seror et al. (2001) computed forced isotropic turbulence by means of DNS and LES together with a noise prediction based on Lighthill’s analogy. DNS was performed on a grid with 1923 points, whereas large-eddy simulations with the structure function model were conducted on two grids with 323 and 643 points, respectively. The authors investigated whether noise radiation generated in forced isotropic turbulence can be estimated using a hybrid LES–Lighthill analogy approach. They confirmed that the high-frequency part of Lighthill’s tensor does not contribute significantly to the overall noise production and showed that the subgridscale intensity and the subgrid-scale fluctuating pressure can not be neglected. Again

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they addressed the parametrization of subgrid-scale effects based on a scale-similarity model and demonstrated the efficiency of the latter in a priori and a posteriori tests.

1.3.2.2 Jet flows

The prediction of high-speed jet noise is one important problem that largely benefits from recent developments in LES. Jet noise is an important component of noise emission from civilian aircraft. For engines having low-to-medium bypass ratios, jet noise reduction can be achieved by internal mixing of the core and the bypass streams; however, for the high-bypass streams of engines having high bypass ratios, the noise reduction benefit from internal mixing is small. Therefore, jet noise is the most prominent source at full power takeoff conditions. To design nozzles for which a significant jet noise reduction can be achieved for high bypass ratios, many research groups have conducted applied and fundamental research based on numerical simulations of the turbulent flow field and the radiated noise. In the initial work on jet noise, Mankbadi et al. (1994, 1998) and Mitchell et al. (1997) concentrated on supersonic jet noise. Mankbadi et al. (1994) and Morris, Wang et al. (1997) performed large-eddy simulations of supersonic jet flow and noise using the Smagorinsky subgrid-scale eddy viscosity model. Mankbadi et al. (1994, 1998) restricted themselves to two-dimensional simulations, and Morris, Wang et al. (1997) conducted three-dimensional computations for the case of a Mach 2.0 jet and presented numerical results in qualitative agreement with measurements. Kolbe et al. (1996) and Chyczewski and Long (1996) carried out LES of rectangular jets, but they relied on numerical and artificial damping to dissipate the turbulence energy. Mitchell, Lele, and Moin (1999) performed direct numerical simulations for both the near-field flow and the far-field sound radiated from subsonic and supersonic twodimensional, axisymmetric jets. The predicted acoustic far field was found to agree with predictions from Lighthill’s acoustic analogy. Freund (1999) performed threedimensional DNS of a randomly forced round jet at Re = 3.6 × 103 and M = 0.9 using 25 million grid points. Predicting the far-field acoustic pressure by solving a wave equation within the near-field pressure data from the DNS, they also obtained good agreement with experimental data. More work on jet flow and its noise generation was performed by Boersma and Lele (1999). They conducted three-dimensional LES of a round jet under the same conditions as Freund (1999) using a dynamic Smagorinsky model, but no sound radiation predictions were reported. Additionally, Mankbadi, Shih et al. (2000) studied supersonic jet noise using two-dimensional LES with a nondynamic Smagorinsky model, but the computational domain did not include part of the acoustic far field. Bogey, Bailly, and Juv´e (2000a) conducted LES of a jet for a Reynolds number Re = 6.5 × 104 and Mach number M = 0.9 with the standard Smagorinsky model. On the basis of the obtained unsteady results they directly computed the aerodynamic noise. The mean flow and turbulence intensities, as well as sound radiation directivity and sound levels, were in good agreement with experimental data.

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1.3 STATE OF THE ART: LES FOR ACOUSTICS

Direct numerical simulations of the near field of a two-dimensional, axisymmetric, supersonic hot jet were performed by Gamet and Estivalezes (1998) for Reynolds number Re = 1 × 103 and Mach number M = 2. They also conducted a LES on the near field of a three-dimensional round jet at the same Mach number but for Re = 3 × 104 . Kirchhoff’s (1877) method has been used to predict the directivity of the radiated sound field. The comparison of the results with experimental data from the literature showed that three-dimensional LES predictions yield better agreement than two-dimensional axisymmetric DNS predictions. The more complex case of simulating the near-field flow and far-field sound radiation of subsonic jets is the subject of Zhao, Frankel, and Mongeau (2001a). They performed LES for a Reynolds number, based on the jet nozzle diameter, of Re = 5 × 103 with a dynamic Smagorinsky and a dynamic mixed subgrid-scale turbulence model. LES predictions obtained with these two turbulence models were similar, although the mixed model resulted in higher turbulence and sound levels. Both the near-field velocity statistics and the far-field sound directivity, as determined with Kirchhoff ’s (1877) method, were found to be in good agreement with experimental data and DNS results from the literature.

1.3.2.3 Flow around airfoils and cylinders

Wang and Moin (2000) performed LES computations of the flow past an asymmetrically beveled trailing edge of a flat strut at a chord Reynolds number of Re = 2.15 × 106 . Their computed mean and fluctuating velocity profiles compare reasonably well with experimental measurements. The far-field acoustic calculation was facilitated by an integral-form solution to the Lighthill equation according to Ffowcs Williams and Hawkings (1969). The authors concluded that, for an accurate predicted noise radiation using the partial source field included in the LES domain, the size of the spanwise domain must be larger than the coherence length of the source field in this direction. The incompressible flow around the blunt trailing edge of a thick plate was investigated by Manoha, Troff, and Sagaut (2000), who performed LES using a gradient diffusion eddy viscosity model. The simulations were conducted for a Reynolds number based on the far-field flow velocity and the plate thickness H of Re H = 1000 on a grid with 161 points in streamwise, 51 points in spanwise, and 201 points in lateral directions, respectively. To determine the far-field noise, they applied the theory of Ffowcs Williams and Hawkings (1969). Although they compared their results with experimental data obtained for a different Reynolds number, they reported good agreement of computed and measured radiated acoustic fields. The unsteady incompressible flow around an blunt trailing edge has also been the subject of LES of the turbulent flow around a NACA0012 airfoil for a Reynolds number – based on the free-stream velocity and the plate thickness – of Re = 1000 by Troff, Manoha, and Sagaut (1997). They solved the filtered Navier–Stokes equation using an eddy viscosity subgrid-scale model on a nonstaggered grid with a secondorder accurate hybrid finite difference–finite volume method. Comparing wall-pressure

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INTRODUCTION

fluctuations and the far-field noise applying Ffowcs Williams–Hawkings theory, they observed good agreement with airfoil noise measurements. More work on trailing-edge flow was reported by Schr¨oder et al. (2001), who conducted LES of a trailing-edge flow in conjunction with noise prediction based on solutions of the linearized Euler’s equation. The unsteady flow around a high-lift configuration was calculated via LES by Manoha, Delahay, Redonnet et al. (2001) using the selective mixed-scale model. The integral methods for the far-field noise prediction of Kirchhoff (1877) and Ffowcs Williams and Hawkings (1969) were used. In Japan, first attempts to apply LES for the prediction of aerodynamic sound were made by Kobayashi and Satake (1991) and Kato et al. (1991) for a flat plate and a circular cylinder, respectively. Kobayashi and Satake (1991) numerically predicted the aerodynamic sound generated by a flat plate in a low-Mach-number turbulent flow. The flow field was first calculated by LES on staggered grids by assuming incompressibility and making use of a wall model and the Smagorinsky subgrid-scale model. A total of 195,000 grid points were used to simulate the flow for a Reynolds number based on the length of the flat plate and the average flow speed of Re = 89,290. Using the pressure fluctuations obtained by LES, the generated aerodynamic sound was predicted by Lighthill’s equation. Kato et al. (1991) computed the aerodynamic sound radiated from the low-Machnumber turbulent wake of a circular cylinder for a Reynolds number based on the cylinder diameter of Re = 10,000. The unsteady flow field around the circular cylinder was solved by LES using an incompressible streamline upwind finite-element method. The sound pressure was computed by Lighthill–Curle’s equation using the fluctuating surface pressure obtained from LES. The numerical results were compared with the measured data obtained in a low-noise wind tunnel. Lift and drag coefficients as well as vortex shedding frequency agreed well with experimental measurements. Simulation results were compared against velocity fluctuation spectra obtained by hotwire measurements. The computational frequency spectra reached an inertial −5/3 range slope. In the low-frequency domain, good agreement with experimental measurements was observed. However, in the high-frequency domain, the power levels were underestimated in the simulation most likely because of insufficient grid resolution. Further, the sound-pressure levels were overestimated by the LES simulation in the high-frequency domain. This was attributed to the fact that the span of the simulated cylinder was 2 diameters whereas the experimental cylinder had a span of 50 cylinder diameters. Miyake, Bando, and Hori (1993) performed a numerical sound source analysis for aerodynamic sound generated by uniform flow over a square cylinder at a Reynolds number of Re = 10,000. The incompressible, time-dependent turbulent flow was first calculated by LES, and the sound generation in the far field was then obtained by a surface integral on the square cylinder using Curle’s equation modified for low-Machnumber flows. The sound-source intensity maps on the square cylinder thus obtained could be used to detect dominant noise origins.

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Figure 1.3. Insulator of a high-speed train’s pantograph.

1.3.2.4 Internal flows

Coupling LES with a linearized Euler equation, Crouzet et al. (2002) investigated the flow and the sound radiation in a duct with a diaphragm. They performed two computations for different upstream velocities and project source terms which were calculated on the LES grid and on the acoustic grid. The results are in good agreement with experimental ones. A correction of the Lighthill tensor by adding a subgrid-scale tensor based on properties simulated in LES of channel flow was proposed by Piomelli, Streett, and Sarkar (1997).

1.3.2.5 Applied flow problems

The aerodynamic and acoustic behavior of an exhaust diffuser was investigated by Jayatunga et al. (2001) by means of LES and the Ffowcs Williams and Hawkings version of the acoustic analogy. Considerable research on LES and acoustics has been also carried out in Japan – especially with respect to practical applications of LES to acoustics for very complex geometries at high Reynolds numbers. In this respect Kato et al. (2000) used LES to analyze aerodynamic noise emitted from the insulator of a high-speed train’s pantograph (see Figures 1.3 and 1.4) and to identify the sound-generating vortical structures with the aim of finding noise-reducing concepts. The insulator is composed of a main circular cylinder and several circular disks. Its unique geometry is responsible for the occurrence of large separation regions

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Figure 1.4. Sketch of the insulator of a high-speed train’s pantograph.

and thus aerodynamic noise, preventing a further speedup of high-speed trains. The authors simulated the turbulent near wake and resulting far-field sound by decomposing the flow field and the resulting acoustic field with low-Mach-number flow assumption. The Reynolds number based on the uniform wind velocity and the representative diameter of the insulator was 140,000. They solved the unsteady flow field using LES with the Smagorinsky model. The complex shape was treated using overset grids to allow enhanced grid resolution in the vicinity of the wall. The resulting far-field sound pressure was calculated by Lighthill–Curle’s equation using the surface-pressure fluctuations obtained by LES. As seen in Figure 1.5, the predicted sound-pressure levels quantitatively agree up to around 2.5 kHz with the ones measured in a low-noise-level wind tunnel for the fine mesh using 6 million elements.

Figure 1.5. Far-field sound-pressure spectra. From Kato et al. (2000); reprinted by permission of the American Institute of Aeronatics and Astronautics, Inc.

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The authors found that the predicted sound-pressure levels were strongly affected by the grid resolution near the object. They investigated the origin of the sound sources and found that the longitudinal vortices generated by the circular disks are primarily responsible for the generation of sound from this flow. Through the same numerical methods, but with moving overset grids, Kato et al. (2002) performed a numerical simulation of the aerodynamic noise radiated from an engine-cooling propeller fan using a total of 4 million finite elements. The far-field sound was predicted by feeding the surface pressure fluctuations obtained by LES into the Ffowcs Williams–Hawkings equation. The computed static pressure rise of the fan quantitatively agreed with the experimentally measured values, indicating that the flow around the fan blades was accurately computed by LES. The predicted trend of the overall sound-pressure levels was found to be consistent with those measured for a propeller fan in general.

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Theoretical Background: Aeroacoustics Avraham Hirschberg and Sjoerd Rienstra

2.1 Introduction to aeroacoustics Owing to the nonlinearity of the governing equations it is very difficult to predict the sound production of fluid flows. This sound production occurs typically at high-speed flows, for which nonlinear inertial terms in the equation of motion are much larger than the viscous terms (high Reynolds numbers). Because sound production represents only a minute fraction of the energy in the flow, the direct prediction of sound generation is very difficult. This is particularly dramatic in free space and at low subsonic speeds. The fact that the sound field is in some sense a small perturbation of the flow can, however, be used to obtain approximate solutions. Aeroacoustics provides such approximations and at the same time a definition of the acoustical field as an extrapolation of an ideal reference flow. The difference between the actual flow and the reference flow is identified as a source of sound. This idea was introduced by Lighthill (1952, 1954), who called this an analogy. A second key idea of Lighthill’s (1954) is the use of integral equations as a formal solution. The sound field is obtained as a convolution of the Green’s function and the sound source. The Green’s function is the linear response of the reference flow, used to define the acoustical field, to an impulsive point source. A great advantage of this formulation is that random errors in the sound source are averaged out by the integration. Because the source also depends on the sound field, this expression is not yet a solution of the problem. However, under free-field conditions one can often neglect this feedback from the acoustical field to the flow. In that case the integral formulation provides a solution. When the flow is confined, the acoustical energy can accumulate into resonant modes. Because the acoustical particle displacement velocity can attain the same order of magnitude as the main flow velocity, the feedback from the acoustical field to the sound sources can be very significant. This leads to self-sustained oscillations, which we call whistling. Despite the back reaction, the ideas of the analogy appear to remain useful. Because linear acoustics is used to determine a suitable Green’s function, it is important to obtain basic insight into properties of elementary wave equation solutions. 24

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2.2 FLUID DYNAMICS

We will focus here on the wave equation describing the propagation of pressure perturbations in a uniform stagnant (quiescent) fluid. Although in acoustics of quiescent media it is rather indifferent whether we consider a wave equation for the pressure or the density, we will see that in aeroacoustics the choice of a different variable corresponds to a different choice of the reference flow and hence to another analogy. It seems paradoxical that analogies are not equivalent inasmuch as they are all reformulations of the basic equations of fluid dynamics. The reason is that the analogy is used as an approximation. Such an approximation is based on some intuition and usually empirical observations. An example of such an approximation was already mentioned above. In free-field conditions we often neglect the influence of the acoustical feedback on the sound sources. Even though Lighthill’s analogy is very general and useful for order-of-magnitude estimates, it is less convenient when used to predict sound production by numerical simulations. One of the problems is that the sound source deduced from Lighthill’s analogy is spatially rather extended, leading to slowly converging integrals. For lowMach-number isothermal flow we will see that aerodynamic sound production is entirely due to mean flow velocity fluctuations, which may be described directly in terms of the underlying vortex dynamics. This is more convenient because vorticity is in general limited to a much smaller region in space than the corresponding velocity field (Lighthill’s sound sources). This leads to the idea of using an irrotational flow as reference flow. The result is called vortex sound theory. Vortex sound theory is not only numerically efficient but also allows us to translate the very efficient vortex-dynamical description of elementary flows directly into sound production properties of these flows. We present here only a short summary of the elements of acoustics and aeroacoustics. The structure of this chapter is inspired by the books of Dowling and Ffowcs Williams (1983) and Crighton et al. (1992). A more advanced discussion is provided in textbooks by Pierce (1981), Temkin (2001), Morse and Ingard (1968), Goldstein (1976), Blake (1986), Crighton et al. (1992), Hubbard (1995), Howe (1998, 2002), and Rienstra and Hirschberg (2001). The influence of wall vibration is discussed in, among others, Cremer and Heckl (1988), Junger and Feit (1986), and Norton (1989). In the sections of this chapter we will consider the following: r Some fluid dynamics (Section 2.2), r Free-space acoustics (Section 2.3), r Aeroacoustic analogies (Section 2.4), and r Aeroacoustics of confined flows (Section 2.5),

2.2 Fluid dynamics 2.2.1 Mass, momentum, and energy equations

We consider the motion of fluids in the continuum approximation. This means that quantities such as the velocity v and the density ρ are smooth functions of space and

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time coordinates (x, t) as considered in Prandtl and Tietjens (1957), Batchelor (1976), Landau and Lifshitz (1987), Schlichting (1968), Kundu (1990), Paterson (1983), and Durand (1963). We consider the fundamental equations of mass, momentum, and energy applied to an infinitesimally small fluid particle of volume V . We call this a material element. We define the density of the material element equal to ρ, and the mass is therefore simply ρV . Because the mass is conserved, as denoted by d(ρV ) = ρdV + V dρ = 0,

(2.1)

the rate of change of the density observed while moving with the fluid velocity v is equal to minus the dilatation rate 1 DV 1 Dρ =− = −∇ · v, (2.2) ρ Dt V Dt where ∇ denotes a symbolic vector of partial derivatives used to describe gradient, divergence, curl, and convective derivative. In Cartesian coordinates it is given by ∇ = ( ∂∂x1 , ∂∂x2 , ∂∂x3 ). The Lagrangian time derivative Dρ/Dt is related to the Eulerian time derivative ∂ρ/∂t by ∂ρ Dρ = + (v · ∇)ρ. (2.3) Dt ∂t For a Cartesian coordinate system x = (x1 , x2 , x3 ), we can write this in the index notation as follows: ∂ρ Dρ ∂ρ , = + vi (2.4) Dt ∂t ∂ xi ∂ρ ∂ρ ∂ρ ∂ρ where vi = v1 + v2 + v3 . ∂ xi ∂ x1 ∂ x2 ∂ x3 According to the convention of Einstein, the repetition of the index i implies a summation over this dead index. Substitution of definition (2.4) into Equation (2.3) yields the mass conservation law applied to a fixed infinitesimal volume element: ∂ρ ∂ρvi ∂ρ + ∇ · (ρv) = 0, or + = 0. ∂t ∂t ∂ xi

(2.5)

We call this the conservation form of the mass equation. For convenience one can introduce a mass source term Q m in this equation: ∂ρ ∂ρvi = Qm . + ∂t ∂ xi

(2.6)

In a nonrelativistic approximation such a mass source term is of course zero and is only introduced to represent the influence on the flow of a complex phenomenon (such as combustion) within the framework of a model that ignores the details of this process. Therefore, there is some ambiguity in the definition of Q m . We should actually specify whether the injected mass has momentum and whether it has a different thermodynamic state than the surrounding fluid. In agreement with the nonrelativistic approximation, we apply the second law of Newton to a fluid particle: ρ

Dv = −∇ · P + f, Dt

(2.7)

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where f is the density of the force field acting on the bulk of the fluid, and −∇ · P is the net force acting on the surface of the infinitesimal volume element. This force is expressed in terms of a stress tensor P. Using the mass conservation law (2.5) without mass source term (Q m = 0), we obtain the momentum equation in conservation form: ∂Pi j ∂ρvi v j ∂ρv ∂ρvi =− + fi . + ∇ · (P + ρvv) = f, or + ∂t ∂t ∂x j ∂x j

(2.8)

The isotropic part pδi j of this tensor corresponds to the effect of the hydrodynamic pressure p = Pii /3: Pi j = pδi j − σi j ,

(2.9)

where δi j = 0 for i = j and δi j = 1 for i = j. The deviation σi j from the hydrostatic behavior corresponds in a simple fluid to the effect of viscosity. We define a simple fluid as a fluid for which σi j is symmetrical (Batchelor 1976). The energy equation applied to a material element is ρ

 D  e + 12 v 2 = −∇ · q − ∇ · (P · v) + f · v + Q w , Dt

(2.10)

where e is the internal energy per unit of mass, v = v, q the heat flux, and Q w the heat production per unit of volume. In conservation form this equation becomes    ∂  ρ e + 12 v 2 + ∇ · ρv(e + 12 v 2 ) = − ∇ · q − ∇ · (P · v) + f · v + Q w , ∂t (2.11a) or, in index notation,    ∂Pi j v j ∂  ∂qi ∂  − + f i vi + Q w . ρ e + 12 v 2 + ρvi e + 12 v 2 = − ∂t ∂ xi ∂ xi ∂ xi

(2.11b)

The mass, momentum, and energy conservation laws in differential form are only valid when the derivatives of the flow variables are defined. When those laws are applied to a finite volume V , one obtains integral formulations that are also valid in the presence of discontinuities such as shock waves. For an arbitrary volume V enclosed by a surface S with outer normal n, we have   d ρ dV + ρ(v − b) · n dS = 0, (2.12a) dt V S     d ρv dV + ρv(v − b) · n dS = − P · n dS + f dV (2.12b) dt V S S V       d 1 2 ρ e + 2 v dV + ρ e + 12 v 2 (v − b) · n dS dt V S    f · v dV, (2.12c) = − q · n dS − (P ·v) · n dS + S

S

V

where b is the velocity of the control surface S. For a material control volume we have v·n = b·n. For a fixed control volume we have b = 0.

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2.2.2 Constitutive equations

The mass, momentum, and energy equations (2.5), (2.8), and (2.11) involve many more unknowns than equations. The additional information needed to obtain a complete set of equations is provided by empirical information in the form of constitutive equations. An excellent approximation is obtained by assuming the fluid to be locally in thermodynamic equilibrium – that is, within a material element (van Kuiken 1995). This implies, for a homogeneous fluid, that two intrinsic state variables fully determine the state of the fluid. For acoustics it is convenient to choose the density of mass ρ and the specific entropy (i.e., per unit of mass) s as variables. All other intrinsic state variables are a function of ρ and s. Hence, the specific energy e is completely defined by a relation e = e(ρ, s).

(2.13)

This is what we call a thermal equation of state. This equation is determined empirically. Variations of e may therefore be written as     ∂e ∂e dρ + ds. (2.14) de = ∂ρ s ∂s ρ Comparison with the fundamental equation of thermodynamics, de = T ds − pdρ −1 , provides thermodynamic equations for the temperature T and the pressure p:   ∂e T = , ∂s ρ and

 p=ρ

2

∂e ∂ρ

(2.15)

(2.16)

 .

(2.17)

s

Because p is also a function of ρ and s, we have     ∂p ∂p dρ + ds. dp = ∂ρ s ∂s ρ

(2.18)

Because sound is defined as isentropic (ds = 0) pressure-density perturbations, the speed of sound c = c(ρ, s) is defined by   ∂p . (2.19) c= ∂ρ s An extensive discussion of the speed of sound in air and water is provided by Pierce (1981). In many applications the fluid considered is air at ambient pressure and temperature. Under such conditions we can assume the ideal gas law to be valid: p = ρ RT,

(2.20)

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where R is the specific gas constant, which is the ratio R = kB /m w of the constant of Boltzmann, kB , and of the mass of a molecule, m w . By definition, for such an ideal gas the energy density only depends on T , e = e(T ), and we have

p  (2.21) c = γ = γ RT , ρ where γ = cp /cv is the Poisson ratio of the specific (i.e., per unit of mass) heat capacities at, respectively, constant volume,   ∂e , (2.22) cv = ∂T ρ and constant pressure,

 cp =

∂i ∂T

 ,

(2.23)

p

where i is the enthalpy per unit of mass defined by i =e+

p . ρ

(2.24)

For an ideal gas, we have cp − cv = R. An ideal gas with constant specific heats is called a perfect gas. As we consider local thermodynamic equilibrium, it is reasonable (Schlichting 1968; van Kuiken 1995) to assume that transport processes are determined by linear functions of the gradients of the flow-state variables. This corresponds to a Newtonian fluid behavior   (2.25) σi j = 2η Di j − 13 Dkk δi j + μv Dkk δi j , where the rate-of-strain tensor Di j is defined by   ∂v j 1 ∂vi . + Di j = 2 ∂x j ∂ xi

(2.26)

Note that Dkk = ∇ · v takes into account the effect of dilatation. In thermodynamic equilibrium, according to the hypothesis of Stokes, one assumes that the bulk viscosity μv vanishes. The dynamic viscosity η is a function of the thermodynamic state of the fluid. For an ideal gas, η is a function of the temperature only. Although the assumption of vanishing bulk viscosity μv is initially an excellent approximation, one observes significant effects of the bulk viscosity in acoustical applications such as propagation over large distances (Pierce 1981). This deviation from local thermodynamic equilibrium is due, in air, to the finite relaxation time of rotational degrees of freedom of molecules. The corresponding approximation for the heat flux q is the law of Fourier, qi = −K

∂T , ∂ xi

(2.27)

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where K is the heat conductivity. For an ideal gas, K is a function of the temperature only. It is convenient to introduce the kinematic viscosity ν and the heat diffusivity a: η ν= (2.28) ρ and a=

K . ρcp

(2.29)

The kinematic viscosity and the heat diffusivity are diffusion coefficients for momentum and heat transfer, respectively. For an ideal gas, both transfer processes are determined by the same molecular velocities and similar free molecular path. This explains why the Prandtl number Pr = ν/a is of order unity. For air at ambient pressure and temperature, Pr = 0.72. 2.2.3 Approximations and alternative forms of the basic equations

Starting from the energy equation (2.11) and using the thermodynamic law (2.15), one can derive an equation for the entropy: ρT

Ds = −∇ · q + σ :∇v + Q w , Dt

(2.30)

where the scalar product “:” of two tensors (or the double contraction of their product) is defined in Cartesian coordinates by σ :τ =

3 3

σi j τ ji .

i=1 j=1

If heat transfer and viscous dissipation are negligible and there are no heat sources, the entropy equation reduces to Ds = 0. Dt

(2.31)

Hence, the entropy of a material element remains constant and the flow is isentropic. When the entropy is uniform, we call the flow homentropic, and thus ∇s = 0. An isentropic flow originating from a reservoir with uniform state is homentropic. When there is no source of entropy, the sound generation is dominated by the fluctuations of the Reynolds stress∗ ρvi v j . Therefore, sound generation often corresponds to conditions for which the term |∂ρvi v j /∂ x j | in the momentum equation (2.8) is large compared with |∂σi j /∂ x j |. Assuming that both gradients scale with the same length D while the velocity scales with U0 (a “main flow velocity”), we find Re = U0 D/ν  1, ∗

In this chapter, the tensor ρvi v j includes the linear and the steady component of the full Reynolds stress and not only the turbulent component as commonly used in the computational fluid dynamics community.

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where Re is the Reynolds number. In such a case one can also show that the dissipation is limited to thin boundary layers near the wall and that, for time scales of the order of U0 /D, the bulk of the flow can be considered as isentropic. Note that the demonstration of this statement in aeroacoustics has been the subject of research for a long time (Morfey 1976, 2003; Obermeier 1985; Verzicco et al. 1997). It is not a trivial statement. Actually, a turbulent flow is essentially dissipative. On the time scales relevant to sound production dissipation is negligible outside the viscous boundary layers at walls (Morfey 2003). Similarly, we often assume that heat transfer is limited to thin boundary layers at the wall and that the bulk of the flow is essentially isothermal. We will see in Section 2.4, that, when the entropy of the flow is not uniform, the convection of those inhomogeneities is an important source of sound. We have now discussed the problem of dissipation and heat transfer in the source region. We will later consider the effect of friction and heat transfer on wave propagation. In a frictionless flow, the momentum equation (2.8) reduces to the equation of Euler as follows: ρ

Dv = −∇ p + f. Dt

(2.32)

Using the definition of enthalpy (2.24), i = e + p/ρ, combined with the fundamental equation (2.15), T ds = de + pd(1/ρ), we find Dv f = −∇i + T ∇s + . Dt ρ

(2.33)

The acceleration Dv/Dt can be split up into an effect of the time dependence of the flow, ∂v/∂t, an acceleration in the direction of the streamlines, ∇( 12 v 2 ), and a Coriolis acceleration due to the rotation, ω = ∇×v, of the fluid as follows:   ∂v Dv = + ∇ 12 v 2 + ω × v. Dt ∂t

(2.34)

Substitution of Equations (2.34) and (2.33) in Euler’s equation (2.32) yields ∂v f + ∇B = −ω × v + T ∇s + , ∂t ρ

(2.35)

where the total enthalpy or Bernoulli constant B is defined by B = i + 12 v 2 .

(2.36)

In general, the flow velocity field v can be expressed in terms of a scalar potential φ and a vector stream function ψ: v = ∇φ + ∇×ψ.

(2.37)

There is an ambiguity in this definition that may be removed by some additional condition. One can, for example, impose ∇· ψ = 0. In most of the problems considered,

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the ambiguity is removed by boundary conditions imposed on φ and ψ. Although the scalar potential φ is related to the dilatation rate ∇ · v = ∇2 φ,

(2.38)

because ∇ · (∇×ψ) = 0, the vector stream function ψ is related to the vorticity

ω = ∇×v = ∇×(∇×ψ)

(2.39)

because ∇×∇φ = 0. This will be used as an argument to introduce the unsteady component of the potential velocity field as a definition for the acoustical field within the framework of vortex sound theory (see Section 2.4.5). For a homentropic (∇s = 0) potential flow (v = ∇φ) without external forces (f = 0), the momentum equation (2.32) can be integrated to obtain the equation of Bernoulli: ∂φ + B = g(t), ∂t

(2.40)

where the function g(t) can be absorbed into the definition of the potential φ without any loss of generality. In the vortex sound theory (Section 2.4.5), where the reference flow (acoustic field) corresponds to the unsteady component of the potential flow ∇φ, the source is the difference between the actual flow and the potential flow. Therefore, the sources are directly related to the vorticity ω. In a homentropic flow the density is a function ρ = ρ( p) of the pressure p only (barotropic fluid). In such a case we can eliminate the pressure from the equation of Euler by taking the curl of this equation (2.32). We obtain an equation for the vorticity (Saffman 1992) as follows:   f Dω = ω · ∇v − ω∇ · v + ∇× . (2.41) Dt ρ In the absence of external forces, the equation reduces to a purely kinematic equation. Solving this equation yields, with ω = ∇×v, the velocity field. This approach is most effective for two-dimensional plane flows v = (v1 (x1 , x2 ), v2 (x1 , x2 ), 0). In that case the vorticity equation reduces to Dω3 /Dt = 0. The study of such flows provides much insight into the behavior of vorticity near sharp edges. Mathematically, the assumed absent viscosity yields a set of equations and boundary conditions that have no unique solution. By adding the empirically observed condition that no vorticity is produced anywhere, we have again a unique solution. This, however, is not exactly true near sharp edges. As determined by the Reynolds number and the (dimensionless) frequency and amplitude, a certain amount of vorticity is shed from a sharp edge. For high enough Reynolds number and low enough frequency and amplitude, the amount of shed vorticity is just enough to remove the singularity of the potential flow around the edge. This is the so-called Kutta condition (Prandtl and Tietjens 1957; Landau and Lifshitz 1987; Rienstra 1981a; Paterson 1983; Crighton 1985).

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When the flow is nearly incompressible (such as in acoustical waves), we can approximate the enthalpy by  p dp  . (2.42) i= ρ ρ0 Under these circumstances, the equation of Bernoulli (2.40) reduces to p ∂φ + 12 v 2 + = 0. ∂t ρ0

(2.43)

When considering acoustical waves propagating in a uniform stagnant medium, we may neglect the quadratically small term 12 v 2 , which yields the linearized equation of Bernoulli: ∂φ p + = 0. ∂t ρ0

(2.44)

2.3 Free-space acoustics of a quiescent fluid 2.3.1 Orders of magnitude

In acoustics one considers small perturbations of a flow. This will allow us to linearize the conservation laws and constitutive equations described in the previous section (2.2). We will focus here on acoustic perturbations of a uniform stagnant (quiescent) fluid. For that particular case we will now discuss orders of magnitude of various effects. This will justify the approximations that we use further on. We will focus on the pressure perturbations p  that propagate as waves and that can be detected by the human ear. For harmonic pressure fluctuations, the audio range is 20 Hz ≤ f ≤ 20 kHz. The sound-pressure level (SPL) measured in decibels (dB) is defined by    prms , SPL = 20 log10 pref

(2.45)

(2.46)

where pref = 2 × 10−5 Pa for sound propagating in gases and pref = 10−6 Pa for propagation in other media. The sound intensity I = I·n is defined as the time-averaged energy flux associated to the acoustic wave, propagating in direction n. The intensity level (IL) measured in decibels (dB) is given by   I

, (2.47) IL = 10 log10 Iref where in air Iref = 10−12 Wm−2 . The reference intensity level Iref is related to the reference pressure pref by the relationship valid for propagating plane waves, I =

p 2 , ρ0 c0

(2.48)

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because in air at ambient conditions ρ0 c0  400 kg m−2 s−1 . The time-averaged power P generated by a sound source is the flux integral of the intensity I over a surface enclosing the source. The sound power level (PWL) measured in decibels (dB) is defined by   P

, (2.49) PWL = 10 log10 Pref where Pref = 10−12 W corresponds to the power flowing through a surface of 1 m2 surface area with an intensity I = Iref . The threshold of hearing (for good ears) at 1 kHz is typically around SPL = 0 dB. This corresponds physically to the thermal fluctuations in the flux of molecules colliding with our eardrum. In order to detect 1 kHz, we can at most integrate the signal over about 0.5 ms. At ambient conditions, this corresponds to the collision of N  1020 molecules with our eardrum.√The thermal fluctuations in the measured pressure are therefore of the order of p0 / N = 10−5 Pa, with p0 the atmospheric pressure. The maximum sensitivity of the ear is around 3 kHz (pitch of a policeman’s whistle), which is due to the quarter-wave-length resonance of our outer ear – a channel of about 2.5-cm depth. The threshold of pain is around SPL = 140 dB. Even at such high levels we have pressure fluctuations only of the order p  / p0 = O(10−3 ). The corresponding density fluctuations are p ρ = , (2.50) ρ0 ρ0 c02 which are also of the order of 10−3 because in air ρ0 c02 / p0 = γ = c p /cv  1.4. This justifies the linearization of the equations. Note that in a liquid the condition for linearization ρ  /ρ0 1 does not imply a small value of the pressure fluctuations because p  / p0 = (ρ0 c02 / p0 )(ρ  /ρ0 ) whereas ρ0 c02  p0 . In water ρ0 c02 = 2 × 109 Pa. We should note that, when considering wave propagation over large distances, nonlinear wave steepening will play a significant role. In a pipe this can easily result in the formation of shock waves. This explains the occurrence of brassy sound in trombones at fortissimo levels (Hirschberg 1995; Hirschberg et al. 1996). Also, in sound generated by aircraft, nonlinear wave distortion significantly contributes to the spectral distribution (Crighton et al. 1992). For a propagating acoustic plane wave the pressure fluctuations p  are associated to the velocity u  of fluid particles in the direction of propagation. We will see later that u =

p . ρ 0 c0

(2.51)

The amplitude δ of the fluid particle displacement is for a harmonic wave with circular frequency ω given by δ = |u  |/ω. At f = 1 kHz, the threshold of hearing (0 dB) corresponds to δ = 10−11 m. At the threshold of pain we find δ = 10−4 m. Such small displacements also justify the use of a linearized theory. When the acoustical displacement δ attains the same order of magnitude as the radius of curvature of a wall, one will observe acoustical flow separation and the formation of vortices. In a pipe, when

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δ approaches the pipe cross-sectional radius, one will observe acoustical streaming. At the pipe outlet this will result in periodic vortex shedding (Ingard and Labate 1950; Ingard and Ising 1967; Disselhorst and van Wijngaarden 1980; Peters and Hirschberg 1993; Peters et al. 1993; Dalmont et al. 2002). In woodwind musical instruments and bass-reflex ports of loudspeaker boxes this is a common phenomenon (Hirschberg 1995; Roozen et al. 1998; Dalmont et al. 2002). When deriving a wave equation in the next section, we will not only linearize the basic equations but will also neglect friction and heat transfer. This corresponds to the assumption that, in an acoustical wave with wavelength λ = c/ f , the unsteady Reynolds number 

|ρ ∂u | λ2 f ∂t (2.52) =O Re unst = 2 ν |η ∂∂ xu2 | is very large. For air ν = 1.5 × 10−5 m2 s−1 , and thus for f = 1 kHz we find Re unst = O(107 ). We therefore expect that viscosity only plays a role on very large distances. Because the Prandtl number is of order unity Pr = O(1) in a gas, we also expect heat transfer to be negligible. At high frequencies, however, we observe a much stronger attenuation due to nonequilibrium effects (bulk viscosity). This results in a strong absorption of these high frequencies when we listen to aircraft at large distances. Furthermore, in the presence of walls, viscothermal dissipation will also be much larger. The amplitude of a plane wave traveling along a tube of cross-sectional radius R will attenuate exponentially exp(−αx) with the distance x. The attenuation coefficient is given for typical audio conditions by Pierce (1981) and Scheichl (2004): √   πfν γ −1 . (2.53) 1+ √ α= Rc0 Pr In most woodwind musical instruments at low pitches the viscothermal dissipation losses are larger than the sound radiation power (Fletcher and Rossing 1998).

2.3.2 Wave equation and sources of sound

We consider the propagation of pressure perturbations p  in an otherwise quiescent fluid. The perturbations of the uniform constant reference state p0 , ρ0 , s0 , v0 are defined by p  = p − p0 ,

ρ  = ρ − ρ0 ,

s  = s − s0 ,

v = v − v0 ,

(2.54)

where, for a quiescent fluid v0 = 0. We assume that f, Q w and the perturbations p  / p0 , ρ  /ρ0 , . . . are small so that we can linearize the basic equations. Furthermore we neglect heat transfer and viscous effects. The equations of motion (2.5, 2.8, and 2.30) reduce to ∂ρ  + ρ0 ∇·v = 0, ∂t

ρ0

∂v + ∇ p  = f, ∂t

ρ0 T0

∂s  = Qw, ∂t

(2.55)

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and the constitutive equation (2.19) becomes   ∂p s. p  = c02 ρ  + ∂s ρ

(2.56)

Subtracting the divergence of the linearized momentum equation from the time derivative of the linearized mass conservation law yields ∂ 2ρ − ∇2 p  = −∇ · f. ∂t 2

(2.57)

Combining the entropy equation with the constitutive equation yields 2  (∂ p/∂s)ρ ∂ Q w ∂ 2 p 2∂ ρ . = c + 0 ∂t 2 ∂t 2 ρ0 T0 ∂t

(2.58)

Elimination of the density fluctuations from Equations (2.57) and (2.58) yields a nonhomogeneous wave equation: 1 ∂ 2 p − ∇2 p  = q, c02 ∂t 2 q=

(2.59)

(∂ p/∂s)ρ ∂ Q w − ∇ · f. ρ0 c02 T0 ∂t

The first source term corresponds to the dilatation of the fluid as a result of heat production in processes such as unsteady combustion or condensation. This type of sound-generation mechanism has been discussed in detail by Morfey (1973) and Dowling and Ffowcs Williams (1983). The second term describes the sound production by a nonuniform, unsteady external force field. When one considers a moving body, the reaction of the body to the force exerted by the fluid can be represented by such a force field. An example of this is a model of the sound radiated by a rotor calculated by concentrating the lift force of each wing into a point force. This model will be discussed in Section 2.4.2 and corresponds to the first theory of propeller sound generation as formulated by Gutin (1948) and commonly used in many applications (Brouwer 1992; Roger 2004). We introduced q(x, t) as shorthand notation for the source term in the wave equation. In the absence of a source term, q = 0, the sound field is due to initial perturbations or boundary conditions. In the next section we present a general solution of the wave equation.

2.3.3 Green’s function and integral formulation

Using Green’s theorem (Morse and Feshbach 1953), we can obtain an integral equation that includes the effects of the sources, the boundary conditions, and the initial conditions on the acoustic field. The Green’s function G(x, t|y, τ ) is defined as the

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response of the flow to an impulsive point source represented by delta functions of space and time: 1 ∂2G − ∇2 G = δ(x − y)δ(t − τ ), c02 ∂t 2

(2.60)

where δ(x − y) = δ(x1 − y1 )δ(x2 − y2 )δ(x3 − y3 ). The delta function δ(t) is not a common function with a pointwise meaning but a generalized function (Crighton et al. 1992) formally defined by its filter property:  ∞ F(t)δ(t) dt = F(0) (2.61) −∞

for any well-behaving function F(t). The definition of the Green’s function G is completed by specifying boundary conditions at a surface S with outer normal n that encloses both the source placed at position y and the observer placed at position x. Because this is an acoustical phenomenon, we follow Crighton’s (1992) suggestion to call the observer a listener. A quite general linear boundary condition is a linear relationship between the value of the Green’s function G at the surface S and the (history of the) gradient n · ∇G at the same point. If this relationship is a property of the surface and independent of G, we describe the surface as locally reacting. Such a boundary condition is usually expressed in the Fourier domain in terms of an impedance Z (ω) of the surface S – that is, the ratio between pressure and normal velocity component – as follows: Gˆ i Z (ω) = , ωρ0 n · ∇x Gˆ where Gˆ is the Fourier-transformed Green’s function defined by  ∞ 1 ˆ G(x, t|y, τ ) e−iωt dt G(x, ω|y, τ ) = 2π −∞ and its inverse

 G(x, t|y, τ ) =

∞ −∞

ˆ G(x, ω|y, τ ) e iωt dω.

(2.62)

(2.63)

(2.64)

(Always check the sign convention in the exponential! Here, we used exp(+iωt). This is not essential as long as the same convention is used throughout!) A problem when using Fourier analysis is that the causality of the solution is not self-evident. We need to impose restrictions on the functional dependence of Z and 1/Z on the frequency ω (Rienstra 1988; Rienstra and Hirschberg 2001). Causality implies that there is no response before the pulse δ(x − y)δ(t − τ ) has been released, and so G(x, t|y, τ ) = 0

for t < τ.

(2.65)

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Consider a Green’s function G not necessarily satisfying the actual boundary condition prevailing on S and a source q not necessarily vanishing before some time t0 . For the wave equation (2.59) we then find the formal solution  t  q(y, τ )G(x, t|y, τ )dVy dτ p (x, t) = t0

+

V

 t t0

+

1 c02



 G(x, t|y, τ )∇y p  − p  (y, t)∇y G ·n dSy dτ

S

  G(x, t|y, τ ) V

∂G ∂ p − p  (y, τ ) ∂τ ∂τ

 τ =t0

dVy ,

(2.66)

where dVy = dy1 dy2 dy3 . The first integral is the convolution of the source q with the pulse response G, the Green’s function. The second integral represents the effect of differences between the actual physical boundary conditions on the surface S and the conditions applied to the Green’s function. When the Green’s function satisfies the same locally reacting linear boundary conditions as the actual field, this surface integral vanishes. In that case we say that the Green’s function is “tailored.” The last integral represents the contribution of the initial conditions at t0 to the acoustic field. If q = 0 and p  = 0 before some time, we can choose t0 = −∞ and leave this term out. Note that in the derivation of the integral equation (2.66) we have made use of the reciprocity relation for the Green’s function (Morse and Feshbach 1953): G(x, t|y, τ ) = G(y, −τ |x, −t).

(2.67)

Owing to the symmetry of the wave operator considered, the acoustical response measured in x at time t of a source placed in y fired at time τ is equal to the response measured in y at time −τ of a source placed in x fired at time −t. The change of sign of the time t → −τ and τ → −t is necessary to respect causality. The reciprocity relation will be used later to determine the low-frequency approximation of a tailored Green’s function. This method is extensively used by Howe (1998, 2002). It is a particularly powerful method for flow near a discontinuity at a wall. In many cases, however, it is more convenient to use a very simple Green’s function such as the free-space Green’s function G 0 . We will introduce this Green’s function after we have obtained some elementary solutions of the homogeneous wave equation in free space.

2.3.4 Inverse problem and uniqueness of source

It can be shown that for given boundary conditions and sources q(x, t) the wave equation has a unique solution (Morse and Feshbach 1953). However, different sources can produce the same acoustical field. A good audio system is able to produce a music performance that is just as realistic as the original. Mathematically the nonuniqueness of the source is demonstrated by the following enlightening example of Dowling and

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Ffowcs Williams (1983). Let us assume that p  (x, t) is a solution of the nonhomogeneous wave equation 1 ∂ 2 p − ∇2 p  = q(x, t) c02 ∂t 2

(2.68)

in which q(x, t) = 0 in a limited volume V . Outside V the source vanishes, and thus, / V . However, p  + q satisfies the q(x, t) = 0. As a result, p  + q = p  for any x ∈ equation 1 ∂ 2q 1 ∂ 2 ( p  + q) 2  − ∇ ( p + q) = q(x, t) + − ∇2 q, 2 ∂t 2 c0 c02 ∂t 2

(2.69)

which has in general a different source term than Equation (2.68). To determine the source from any measured acoustical field outside the source region, we need a physical model of the source. This is typical of any inverse problem in which the solution is not unique. When using microphone arrays to determine the sound sources responsible for aircraft noise, one usually assumes that the sound field is built up of so-called monopole sound sources (Sijtsma, Oerlemans, and Holthusen, 2001). We will see later that the sound sources are more accurately described in terms of dipoles or quadrupoles (Section 2.4.1). Under such circumstances it is hazardous to extrapolate such a monopole model to angles outside the measuring range of the microphone array or to the field from flow Mach numbers other than used in the experiments.

2.3.5 Elementary solutions of the wave equation

We consider two elementary solutions of the homogeneous wave equation (q = 0) 1 ∂ 2 p − ∇2 p  = 0 c02 ∂t 2

(2.70)

that will be used as building blocks to obtain more complex solutions: r the plane wave and r the spherical symmetric wave.

We assume in both cases that these waves have been generated by some boundary condition or initial condition. We consider their propagation through an in-all-directions infinitely large quiescent fluid we call “free space.” We first consider plane waves. These are uniform in any plane normal to the direction of propagation. Let us assume that the waves propagate in the x1 direction, in which case p  = p  (x1 , t) and the wave equation reduces to 1 ∂ 2 p ∂ 2 p − = 0. c02 ∂t 2 ∂ x12

(2.71)

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This one-dimensional wave equation has the solution of d’Alembert,     x1 x1  p =F t− +G t + , c0 c0

(2.72)

where F represents a wave traveling in the positive x1 direction and G travels in the opposite direction. This result is easily verified by applying the chain rule for differentiation. The functions F and G are determined by the initial and boundary conditions. Consider for example the acoustic field generated by an infinite plane wall oscillating around x1 = 0 with a velocity u 0 (t) in the x1 direction. In linear approximation v1 (0, t) = u 0 (t); that is, the acoustical velocity at x1 = 0 is assumed to be equal to the wall velocity. It is furthermore implicitly assumed in the definition of “free space” that no waves are generated at infinity. Therefore, we have for x1 > 0 that G = 0. Using the linearized equation of motion (2.55) in the absence of external force field f = 0, ρ0

∂v1 ∂ p = − 1, ∂t ∂ x1

(2.73)

p  = ρ0 c0 v1 .

(2.74)

we find

We call ρ0 c0 the specific acoustical impedance of the fluid. Using the boundary condition v1 (0, t) = u 0 (t) and p  (x1 , t) = F(t − x1 /c0 ), we find p  = ρ0 c0 u 0 (t − x1 /c0 )

(2.75)

as a solution for x1 > 0. This equation states that perturbations, observed at time t at position x1 , are generated at the wall x1 = 0 at time t − x1 /c0 . The time te = t − x1 /c0 is called the emission time or retarded time. In a similar way, we find p  = −ρ0 c0 u 0 (t + x1 /c0 )

(2.76)

for x1 < 0 if the wall is of zero thickness and perturbs the fluid at either side. By analogy of Equation (2.72), we easily find for a plane-wave solution propagating in a direction given by the unit vector n the most general form   n·x  . (2.77) p =F t− c0 For the particular case of harmonic waves the plane-wave solution is written in complex notation as p  = pˆ e iωt−ik· x ,

(2.78)

where k = kn is the wave vector, k = ω/c0 is the wave number, and pˆ is the amplitude. The complex notation is a shorthand notation for ˆ cos(ωt − k · x) − ( p) ˆ sin(ωt − k · x). p  = ( pˆ e iωt−ik· x ) = ( p)

(2.79)

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By means of Fourier analysis in time, an arbitrary time dependence can be represented by a sum or integral of harmonics functions. In a similar way, general spatial distributions can be developed in terms of plane waves. Another important elementary solution of the homogeneous wave equation (2.70) is the spherically symmetric wave. In that case the pressure is only a function p  (r, t) of time and the distance r to the origin.  2 ∂ F  1 ∂ 2r F ∂ r ∂r = r ∂r 2 , we find that the wave equaBy identifying ∇2 F(r ) = r12 ∂r tion (2.70) reduces for r > 0 to ∂ 2 pr 1 ∂ 2 pr − = 0. ∂r 2 c02 ∂t 2

(2.80)

Note that at r = 0 the equation is singular. As we will see this will correspond to a possible point source. Equation (2.80) implies that the product p r of the pressure p  and the radius r satisfies the 1D wave equation and may be expressed as a solution of d’Alembert, p =

1 [F(t − r/c0 ) + G(t + r/c0 )] , r

(2.81)

in which F represents outgoing waves and G denotes incoming waves. In many applications we will assume that there are no incoming waves G = 0. We call these free-field conditions. We now focus on the behavior of outgoing harmonic waves: p =

A iωt−ikr e , r

(2.82)

where A is the amplitude and k = ω/c0 the wave number. The radial fluid particle velocity vr associated with the wave can be calculated by using the radial component of the momentum equation (2.55): ρ0

∂ p ∂vr =− . ∂t ∂r

We find the following: vr

p = ρ0 c0



i 1− kr

(2.83)  .

(2.84)

At distances r large compared with the wavelength λ = 2π/k (kr = 2πr/λ  1) we find the same behavior as for a plane wave (2.74). The spatial variation due to the harmonic wave motion dominates over the effect of the radial expansion. We call this the far-field behavior. In contrast to this, we have for kr 1 the near-field behavior in which the velocity vr is inversely proportional to the square of the distance r . This is indeed the expected incompressible flow behavior. Over small distances the speed of sound is effectively infinite because any perturbation arrives without delay in time. As a result, the mass flux is conserved and vr r 2 is constant. All this can be understood by the observation that |(∂ 2 p  /∂t 2 )/[c02 (∂ 2 p  /∂ 2r 2 )] ∼ (kr )2 , and thus the wave equation reduces to the equation of Laplace ∇2 p  = 0 for kr → 0.

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Outgoing spherical symmetric waves correspond to what is commonly called a monopole sound field. Such a field can be generated by a harmonically pulsating rigid sphere with radius a: a = a0 + aˆ e iωt .

(2.85)

ˆ 0 ) we have In linear approximation (in a/a ˆ vˆr (a0 ) = iωa.

(2.86)

Combining this boundary condition with Equations (2.82) and (2.84), we obtain the amplitude A of the wave: p = −

ρ0 ω2 a0 aˆ a0 iωt−ik(r −a0 ) e . 1 + ika0 r

(2.87)

In the low-frequency limit ka0 1 we see that the amplitude of the radiated sound field decreases with the frequency. If the volume flux V = 4πa02 vr (a0 ) = 4π ia02 ωaˆ generated at the surface of the sphere is kept fixed, the sound pressure p  decreases linearly with decreasing frequency: p =

iωρ0 V iωt−ik(r −a0 ) e . 4πr

(2.88)

A monopole field can, for example, be generated by unsteady combustion, which corresponds to the entropy source term in the wave equation. This will occur in particular for a spherically symmetric combustion. In general the monopole field will be dominant when the source region is small compared with the acoustic wavelength ka0 1. We call a region that is small compared with the wavelength a compact region. We have seen that a compact pulsating sphere is a rather inefficient source of sound under freefield conditions. More formally, a monopole source corresponds to a localized volume source or point source placed at position y: q(x, t) =

∂V δ(x − y). ∂t

(2.89)

We will discuss this approach more in detail later. Note the time derivative in the source term of Equation (2.89). It reflects the fact that a steady flow does not produce any sound. Using the monopole solution (2.82), we can build more complex solutions. If p0 is a solution of the wave equation (2.70), any spatial derivatives ∂ p0 /∂ xi are also solutions because the wave equation has constant coefficients and the derivatives may be interchanged. A first-order spatial derivative of the monopole field is called a dipole field. Second-order spatial derivatives correspond to quadrupole fields. An example of a dipole field is the acoustic field generated by a rigid sphere translating harmonically in a certain direction x1 with a velocity vs = vˆs e iωt . The radial velocity vr (a0 , θ) on the surface of the sphere is given by vˆr (a0 , θ) = vˆs cos θ,

(2.90)

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where θ is the angle between the position vector on the sphere and the translation direction x1 . Because we have the identity  ∂r ∂ x1 = x12 + x22 + x32 = = cos θ, (2.91) ∂ x1 ∂ x1 r we can write for the dipole field ∂ pˆ = A ∂ x1



e−ikr r



∂ = A cos θ ∂r



e−ikr r

 .

Substitution of Equation (2.92) into the momentum equation (2.83) yields   ∂ 2 e−ikr iωρ0 vˆr = −A cos θ 2 . ∂r r We apply this equation at r = a0 . Comparison with Equation (2.90) yields  a 2 iωρ0 vˆs a0 cos θ 0 e−ik(r −a0 ) . (1 + ikr ) pˆ = 2 + 2ika0 − (ka0 )2 r

(2.92)

(2.93)

(2.94)

Another example is the calculation of the field p  generated by an unsteady, nonuniform force field f = ( f 1 , f 2 , f 3 ). Following Equation (2.59), we have 1 ∂ 2 p − ∇2 p  = −∇·f. c02 ∂t 2

(2.95)

Let us assume that we have obtained a solution F1 of the wave equation in free space, thus satisfying 1 ∂ 2 F1 − ∇2 F1 = − f 1 . c02 ∂t 2

(2.96)

Then we may find the solution p  of Equation (2.95) in free space by taking the space derivative of F1 as follows: p =

∂ F1 . ∂ x1

(2.97)

This indicates that the dipole field is related to forces exerted on the flow. Another way to deduce the relationship between forces and dipole fields is to consider the dipole as the field obtained by placing two opposite monopole sources of amplitude V at a distance y1 from each other. Taking the limit of y1 → 0 while we keep V y1 constant yields a dipole field. Because in free space changes in source position y are equivalent to changes in listener position x, it is obvious that this limit relates to the spatial derivative of the monopole field. If we consider now the two oscillating volume sources forming the dipole, there will be a mass flow V from one source to the other. Such an unsteady mass flow is associated with an unsteady momentum flux. This unsteady momentum flux must, following Newton, be produced by an external force acting on the flow (Prandtl and Tietjens 1957; Durand 1963). Hence we see that a dipole is not possible without the action of a force. This idea is illustrated in Figure 2.1 in which we consider waves generated by a boat on the water surface.

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Figure 2.1. Monopole, dipole, and quadrupole generating waves on the surface of the water around a boat.

When a person jumps up and down in the boat, he or she produces an unsteady volume injection, and this generates a monopole wave field around the boat. When two persons on the boat play with a ball, they will exert a force on the boat each time they throw or catch the ball. Exchanging the ball results in an oscillating force on the boat. This will make the boat translate, and this in turn generates a dipole wave field. We could say that two individuals fighting with each other is a reasonable model for a quadrupole. This indicates that quadrupoles are in general much less efficient in producing waves then monopoles or dipoles. This indeed appears to be the case. It is often stated that Lighthill (1954) has demonstrated that the sound produced by a free, turbulent isentropic flow has the character of a quadrupole. A better way of putting it is that, because there is no net volume injection due to entropy production nor any external force field, in such flows the sound field can at most be a quadrupole field (Hirschberg and Schram 1995). Therefore, Lighthill’s statement is actually that we should ignore any monopole or dipole emerging from a poor description of the flow. We will consider this in more detail in Section 2.4.1. 2.3.6 Acoustic energy and impedance

The definition of acoustical energy is not obvious when we define the acoustic field on the basis of linearized equations. The energy is essentially quadratic in the perturbations. We may anticipate therefore that there is some arbitrariness in the definition of acoustical energy. This problem has been the subject of many discussions in the literature (Goldstein 1976; Pierce 1981; Myers 1986a, 1986b, 1991; Landau and Lifshitz 1987; Jenvey 1989; M¨ohring 2001). In the particular case of the acoustics of a quiescent fluid, the approach proposed by Kirchhoff (Landau and Lifshitz 1987), starting from the linearized equations (2.55), appears to be equivalent to the result obtained by expanding the energy equation (2.11) up to the second order (Landau and Lifshitz 1987).

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After elimination of the density by using the constitutive equation, we can write the linearized mass conservation in the form   1 ∂ p ∂s  1 ∂p ∇·v = + ρ (2.98) 0 c02 ∂t c02 ∂s ρ ∂t and the momentum equation in the form ∂v + ∇ p  = f. ∂t

ρ0

(2.99)

We multiply the first equation (2.98) by p  /ρ0 and add the result to the scalar product of the second equation (2.99) with v to obtain the acoustic energy equation ∂E + ∇ · I = −D, ∂t

(2.100)

where we have defined the acoustic energy E by E=

1 1 p 2 2 . ρ0 v  + 2 2 ρ0 c02

(2.101)

The intensity I, defined as I = p  v ,

(2.102)

is identified as the flux of acoustic energy. The dissipation D is the power per unit volume delivered by the acoustical field to the sources    ∂p 1  ∂s D=− − f · v . p (2.103) ρ0 c02 ∂s ρ ∂t From the mass conservation law (2.98), we see that the source term (∂ p/∂s)ρ /(ρ0 c02 )(∂s  /∂t) in the dissipation corresponds to the dilatation rate induced by the source. This allows us to relate the first term in the dissipation to the work of the acoustical field due to the change in volume (dW = p  dV ). For harmonically oscillating fields p  = pˆ e iωt , v = vˆ e iωt , the time-averaged E

of the acoustic energy is (of course) independent of time:  2π/ω ω E = E dt; (2.104) 2π 0 hence, the energy equation (2.100) reduces to ∇ · I = − D .

(2.105)

By integration of this equation over a volume enclosing the sources, we find the source power   (2.106) P = − D dV = I · n dS, V

S

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where n is the outer normal to the control surface S. If we assume an impedance boundary condition on the surface S, Z (ω) =

pˆ , vˆ ·n

(2.107)

we have I ·n = 12 (Z )|ˆv ·n|2 .

(2.108)

We see that the real part (Z ) of the impedance Z is associated with the transport of acoustic energy through the surface S. The imaginary part is associated with pressure differences induced by the inertia of the flow. We can now easily verify, by using Equation (2.106), that the spherically symmetric wave solution (2.82) satisfies the acoustic energy conservation law. The r −1 dependence of the pressure (Equation (2.81)) in a simple outgoing wave results in a conserved value of 4πr 2 I ·n. To illustrate this we consider the impedance of a pulsating sphere of radius a0 . From Equation (2.87) we find for the impedance Z of the surface of the sphere that   ρ 0 c0 i pˆ Z= = 1 + . (2.109) vˆr ka0 1 + (ka10 )2 The real part is given by (Z ) = ρ0 c0

(ka0 )2 . 1 + (ka0 )2

(2.110)

We see that for a large sphere, ka0  1, the impedance is equal to ρ0 c0 , which is the impedance experienced by a plane wave (Equation (2.74)) of any plane control surface. For a compact sphere ka0 1, we see that (Z )  ρ0 c0 (ka0 )2 , which implies very little energy transfer and thus a very inefficient sound source. The imaginary part (Z ) of the impedance of the sphere, given by (Z ) = ρ0 c0

ka0 , 1 + (ka0 )2

(2.111)

vanishes for ka0 → ∞. For a compact sphere, ka0 1, it corresponds to the pressure calculated by means of the linearized equation of Bernoulli (2.44) if we assume ˆ 0 /r ) around the sphere. (Note that φ∞ − φ(a0 ) = an incompressible flow vˆr = iωa(a ∞ ˆ a0 vr dr = iωaa0 .) Furthermore, we note that, in order to deliver acoustical energy, a volume source needs to be surrounded by a field of high pressure. This occurs when it is surrounded by a surface of which the real part of the impedance is large. A force field needs a large velocity fluctuation to produce acoustical energy efficiently. This corresponds to a large real part of the acoustical admittance Y = 1/Z .

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2.3.7 Free-space Green’s function

The free-space Green’s function G 0 is the acoustical field generated at the observer’s position x at time t by a pulse δ(x − y)δ(t − τ ) released in y at time τ . To calculate the free-space Green’s function G 0 , we will make use of the Fourier transform (2.63, 2.64). We seek a spherically symmetric wave solution (2.82) of the form A (2.112) Gˆ 0 = e−ikr , r where r = x − y. To determine the amplitude A, we integrate the wave equation (2.60) over a compact sphere of radius a0 around y. Making use of the properties of the delta function we find that  e−iωτ = (k 2 Gˆ 0 + ∇2 Gˆ 0 ) dV − 2π V

   ˆ ∂ Gˆ 0 2 ˆ 2 ∂ G0  ∇ G 0 dV = . (2.113) dS = 4πa0 ∂r V S ∂r r =a0

Using the near-field approximation (∂ Gˆ 0 /∂r )r =a0  −A/a02 , we can calculate the amplitude A. We find 1 e−iωτ −ir/c0 , 8π 2r which leads by (generalized) inverse Fourier transformation to Gˆ 0 =

(2.114)

1 (2.115) δ(t − τ − r/c0 ). 4πr We observe at time t at a distance r from the source a pulse corresponding to the impulsion delivered at the emission time: r (2.116) te = t − . c0 G0 =

Because G 0 depends only on r = x − y rather than on the individual values of x and y, the free-space Green’s function not only satisfies the reciprocity relation (2.67) but also the symmetry relation: ∂G 0 ∂r ∂G 0 ∂r ∂G 0 ∂G 0 = =− =− . ∂ xi ∂r ∂ xi ∂r ∂ yi ∂ yi

(2.117)

Approaching the source by the listener has the same effect as approaching the listener by the source ∂r/∂ xi = −∂r/∂ yi . 2.3.8 Multipole expansion

We can use the free-space Green’s function G 0 to obtain a more formal definition of monopoles, dipoles, quadrupoles, and so on. As we will see, this corresponds to the use of a Taylor expansion of the free-space Green’s function. We will consider the far-field

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p  in free space of a compact source distribution q(x, t). To derive the general multipole expansion, we will first consider the field at a single frequency. By using the free-field Green’s function Gˆ 0 (x|y) =

e−ikr , 4πr

(2.118)

ˆ e iωt in a we find the acoustic field for a given time-harmonic source distribution q(x) finite volume V to be given by   e−ikr  ˆ ˆ G 0 (x|y) dVy = ˆ dVy . (2.119) q(y) q(y) pˆ = 4πr V V Suppose the origin is chosen inside V . We are interested in the far field (i.e., x is large) and a compact source (i.e., k L is small, where L is the typical diameter of V ). This double limit can be taken in several ways. Because we are interested in the radiation properties of the source, which corresponds to kx ≥ O(1), we will keep kx fixed. In that case, the limit of small k is the same as small y, and we can expand in a Taylor series around y = 0:     2 x·y y2 (x·y)2 2 1/2 = x 1 − + − + ... r = x − 2(x·y) + y x2 2x2 2x4   1 y2 2 y sin θ + . . . , cos θ + = x 1 − x 2 x2 where θ is the angle between x and y, and   3   e−ikx e−ikr 1 = 1 + (1 + ikx) x j yj + . . . r x x2 j=1  l+m+n  ∞ e−ikr ∂ y1l y2m y3n . = l! m! n! ∂ y1l ∂ y2m ∂ y3n r y1 =y2 =y3 =0 l,m,n=0

(2.120)

Utilizing the symmetry of r as a function of x and y, we find this is equivalent to  −ikx  ∞ e e−ikr (−1)l+m+n l m n ∂ l+m+n = y1 y2 y3 l m n . (2.121) r l! m! n! x ∂ x1 ∂ x2 ∂ x3 l,m,n=0 The acoustic field is then given by  −ikx   ∞ e 1 ∂ l+m+n (−1)l+m+n  l m n ˆ dy l m n . pˆ = y1 y2 y3 q(y) 4π l,m,n=0 l! m! n! V x ∂ x1 ∂ x2 ∂ x3

(2.122)

Because each term in the expansion is by itself a solution of the reduced wave equation, this series yields a representation in which the source is replaced by a sum of elementary sources (monopole, dipoles, quadrupoles – in other words, multipoles) placed at the origin (y = 0). Expression (2.122) is the multipole expansion of a field from a finite source in the Fourier domain. From this result, we can obtain the corresponding expansion in the time domain.

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From the integral formulation (2.66), we have the acoustic field from a source q(x, t):   ∞ q(y, t − r/c0 ) δ(t − τ − r/c0 ) p = dydτ = dy. (2.123) q(y, τ ) 4πr 4πr V −∞ V If the dominating frequencies in the spectrum of q(x, t) are low, such that ωL/c0 is small, we obtain by Fourier synthesis of Equation (2.122) the multipole expansion in the time domain (see Goldstein 1976):    ∞ 1 1 (−1)l+m+n ∂ l+m+n l m n y y y q(y, t ) dy p = e 4π l,m,n=0 l! m! n! ∂ x1l ∂ x2m ∂ x3n x V 1 2 3   ∞ ∂ l+m+n (−1)l+m+n (2.124) μlmn (te ) , = l m n 4π x l,m,n=0 ∂ x 1 ∂ x 2 ∂ x 3 where te = t − x/c0 is the emission time and μlmn (t) is defined by  l m n y1 y2 y3 μlmn (t) = q(y, t) dy. V l! m! n!

(2.125)

The (lmn)-th term of the expansion (2.124) is called a multipole of order 2l+m+n . The 20 -order term corresponds to a monopole, a concentrated volume source at y = 0 with  source strength μ000 = V q(y, t)dVy , which is called the monopole strength. Because each term is a function of x only, the partial derivatives to xi can be rewritten into expressions containing derivatives to x. In general, these expressions are rather complicated, and so we will not try to give the general formulas here. For very large x, each multipole further simplifies because     μ (te ) μ(te ) xl 1 ∂ μ(te ) = − − ∂ xl x c0 x x2 x −

xl ∂ μ (te ) xl =− μ(te ). c0 x x c0 x2 ∂t

(2.126)

This leads to p 

∞

x1l x2m x3n ∂ l+m+n μlmn (te ), 4π (c0 x)l+m+n x ∂t l+m+n l,m,n=0

(x → ∞).

(2.127)

Most results in Section 2.3.9 will be presented in this far-field approximation.

2.3.9 Doppler effect

We can use the Green’s function formalism to determine the effect of the movement of a source on the radiated sound field. We consider a point source localized at the point xs (t): q(x, t) = Q(t)δ(x − xs (t)).

(2.128)

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For free-field conditions, using Equation (2.115), we find    ∞ y − x Q(τ )δ(y − xs (τ ))  δ t −τ − dVy dτ. p (x, t) = 4π y − x c0 −∞ V

(2.129)

After integration over space, using the property (2.61) of the delta function, we obtain    ∞ R Q(τ )  p (x, t) = δ t −τ − dτ, (2.130) c0 −∞ 4π R where R = R.

R(τ, x) = x − xs (τ ),

(2.131)

The contributions of this integral are limited to the zeros of the argument of the δfunction. In other words, this is an integral of the type  ∞  tn +ε d F(τ )δ(g(τ )) dτ = F(τ )δ((τ − tn ) dτ g(tn )) dτ −∞

n

=

n

tn −ε

F(tn ) d | dτ

g(tn )|

,

(2.132)

where τ = tn corresponds to the roots of g(τ ). In the present application, we have g(τ ) = t − τ −

R(τ, x) , c0

(2.133)

and so dg R·vs = −1 + = −1 + Mr , dτ Rc0

where vs =

dxs dτ

(2.134)

and Mr is the component of the source velocity vs in the direction of the listener scaled by the sound speed c0 . We call this the relative Mach number of the source. It is positive for a source approaching the observer and negative for a source receding from the observer. It can be shown that, for subsonic source velocities |Mr | < 1, the equation g(te ) = 0 or c0 (t − te ) = R(te , x)

(2.135)

has a single root, which is to be identified as the emission time te . Hence, we find for the acoustic field the Li´enard–Wiechert potential (Jackson 1999) p  (x, t) =

Q(te ) . 4π R(1 − Mr )

(2.136)

When the source moves supersonically along a curve, multiple solution te can occur. This may lead to a focusing of the sound into certain regions of space, leading to the so-called super bang phenomenon. The increase (when approaching) or decrease (when receding) of the amplitude is called the Doppler amplification, and the factor (1 − Mr )−1 is called the Doppler factor. This Doppler factor is best known from its occurrence in the increased or decreased pitch of the sound experienced by the listener. For a sound source harmonically oscillating

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with frequency ω that is high compared with the typical sound source velocity variations, the listener experiences at time t a frequency ω d(ωte ) . = dt 1 − Mr

(2.137)

The right-hand side is obtained by implicit differentiation of Equation (2.135). Hence, the observed frequency is the emitted one multiplied by the Doppler factor. In this discussion we ignored the physical character of the source. If, for example, we consider a monopole source with a volume injection rate V (t), the source is given by q(x, t) = ρ0

∂ (V (t)δ(x − xs (t))) . ∂t

The corresponding sound field is   ρ0 V (te ) ∂ p  (x, t) = ∂t 4π R(te , x)(1 − Mr (te ))   ρ0 V (te ) 1 ∂ . = 1 − Mr (te ) ∂te 4π R(te , x)(1 − Mr (te ))

(2.138)

(2.139)

Although this is for an arbitrary source V and path xs an extremely complex solution, it is interesting to note that, even for a constant volume flux V , there is sound production when the source velocity vs is nonuniform. In a similar way, we may consider the sound field generated by a point force F(t): q(x, t) = −∇·[F(t) δ(x − xs (t))]. The produced sound field is given in a far-field approximation by   R ∂ F · . p  (x, t) = c0 R(1 − Mr ) ∂te 4π R(1 − Mr )

(2.140)

(2.141)

Even when the source flies at constant velocity dtd xs this solution involves high powers of the Doppler factor. An interesting application of this theory is the sound production by a rotating blade. The blade can be represented by a point force (mainly the lift force concentrated in a point) and a compact moving body of constant volume Vb (the blade). The lift noise (the contribution of the lift force) can be calculated by means of Equation (2.141). Note that in practice this lift is not only the steady thrust of the rotating blade but also contains the unsteady component due to interaction of the blades with obstacles like supports or with a turbulent or nonuniform inflow. The effect of the volume of the blade can be shown to be given by the second time derivative   ρ0 Vb ∂2  . (2.142) p (x, t) = 2 ∂t 4π R(1 − Mr ) Even though the blade volume remains constant, the displacement of air by the rotating blade induces a sound production. This so-called thickness noise depends on a higher

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power of the Doppler factor than the lift noise. This implies that, at low Mach numbers such as prevail for a ventilation fan, the lift noise will be dominant. At high-source Mach numbers, such as aircraft propellers, the lift noise will still dominate at takeoff when the required thrust is high, but at cruise conditions the thickness noise becomes comparable because the tip Mach number is close to unity (or even higher). We will see in the next section that the sound produced by turbulence in a free jet has the character of the field of a quadrupole distribution. Because the vortices that produce the sound are convected with the main flow, there will be a very significant Doppler effect at high Mach numbers. This results in a radiation field mainly directed about the flow direction. Owing to convective effects on the wave propagation, the sound is, however, deflected in the shear layers of the jet. This explains why, along the jet axis, there is a so-called cone of silence (Goldstein 1976; Hubbard 1995).

2.3.10 Uniform mean flow, plane waves, and edge diffraction

The problem of a source, observer, and scattering objects moving together steadily in a uniform stagnant medium is the same as the problem of a fixed source, observer, and objects in a uniform mean flow. If the mean flow is in the x direction and the perturbations are small and irrotational, we have for potential φ, pressure p, density ρ, and velocity v the problem given by   ∂ 2 ∂ 2φ ∂ 2φ 1 ∂ ∂ 2φ + 2 + 2 − 2 φ = 0, + U0 ∂x2 ∂y ∂z ∂x c0 ∂t   ∂ ∂ + U0 φ, p = c02 ρ, v = ∇φ, p = −ρ0 ∂t ∂x

(2.143)

where U0 , ρ0 , and c0 denote the mean flow velocity, density, and sound speed, respectively. We assume in the following that |U0 | < c0 . The equation for φ is known as the convected wave equation. 2.3.10.1 Lorentz or Prandtl–Glauert transformation

By the transformation (in an aerodynamic context named after Prandtl and Glauert but qua form originally due to Lorentz) X=

x , β

T = βt +

M X, c0

M=

U0 , c0

β=

 1 − M 2,

(2.144)

the convected wave equation may be associated with a stationary problem with solution φ(x, y, z, t) = ψ(X, y, z, T ) satisfying ∂ 2ψ ∂ 2ψ 1 ∂ 2ψ ∂ 2ψ + + − = 0, ∂ X2 ∂ y2 ∂z 2 c02 ∂ T 2

ρ0 p=− β



∂ ∂ + U0 ∂T ∂X

 ψ.

(2.145)

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For a time-harmonic field e iωt φ(x, y, z) = e iT ψ(X, y, z) or φ(x, y, z) = e iK M X ψ(X, y, z), where  = ω/β, k = ω/c0 , and K = k/β, we have ∂ 2ψ ∂ 2ψ ∂ 2ψ + + + K 2 ψ = 0. ∂ X2 ∂ y2 ∂z 2

(2.146)

The pressure may be obtained from ψ, but because p satisfies the convected wave equation too, we may also associate the pressure field directly by the same transformation with a corresponding stationary pressure field. The results are not equivalent, however, and especially when the field contains singularities some care is in order. The pressure obtained directly is no more singular than the pressure of the stationary problem, but the pressure obtained via the potential is one order more singular owing to the convected derivative. See the example in Sections 2.3.10.2 and 2.3.10.3. 2.3.10.2 Plane waves

A plane wave (in x, y plane) may be given by 

x cos θn + y sin θn φi = exp −ik 1 + M cos θn





 cos(θ − θn ) = exp −ikr , 1 + M cos θn

(2.147)

where θn is the direction of the normal to the phase plane and x = r cos θ, y = r sin θ. This is physically not the most natural form, however, because θn is due to the mean flow, not the direction of propagation. By comparison with a point source field far away or from the intensity vector   ∂ ∂ φe y I = (ρ0 v + ρv0 )(c02 ρ/ρ0 + v0 ·v) ∼ β 2 φ − ik Mφ e x + ∂x ∂y ∼ (M + cos θn )e x + sin θn e y ,

(2.148)

we can learn that θs , the direction of propagation (the direction of any shadows), is given by M + cos θn , cos θs =  1 + 2M cos θn + M 2

sin θs = 

sin θn 1 + 2M cos θn + M 2

.

(2.149)

By introducing the transformed angle s cos θs M + cos θn cos s =  = , 1 + M cos θn 1 − M 2 sin2 θs

(2.150)

β sin θs β sin θn = sin s =  2 2 1 + M cos θn 1 − M sin θs

(2.151)

and the transformed polar coordinates X = R cos , y = R sin , we obtain the plane wave φi = exp (i K M X − i K R cos( − s )) .

(2.152)

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Figure 2.2. Sketch of scattered plane wave with mean flow.

2.3.10.3 Half-plane diffraction problem

By using the foregoing transformation, we obtain from the classical Sommerfeld solution for the half-plane diffraction problem (see Jones 1964) of a plane wave (Equation (2.152)) incident on a solid half-plane along y = 0, x < 0 (Figure 2.2), the following solution (see Rienstra 1981a):   (2.153) φ(x, y) = exp (i K M X − i K R) F(s ) + F(¯ s ) , where eiπ/4 2 F(z) = √ e iz π



∞

e−it dt 2

(2.154)

z

and s , ¯ s = (2K R)1/2 sin 12 ( ∓ s ).

(2.155)

An interesting feature of this solution is the following. When we derive the corresponding pressure   p(x, y) = exp (i K M X − i K R) F(s ) + F(¯ s ) e−iπ/4 M cos 12 s exp (i K M X − i K R) sin 12  + √ π 1 − M cos s



2 KR

1/2 ,

(2.156)

we see immediately that the first part is just a multiple of the solution of the potential, and so the second part has to be a solution too. Furthermore, the first part is regular like φ, whereas this second part is singular at the scattering edge. Because the second part decays for any R → ∞, it does not describe the incident plane wave, and so it may be dropped if we do not accept the singularity in p at the edge. By considering this solution, sin 1  pv (x, y) = exp (i K M X − i K R) √ 2 , KR

(2.157)

a bit deeper, we find that it has no continuous potential that decays to zero for large |y| (see Jones 1972; Rienstra 1981a). This solution corresponds to the field of vorticity (in

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the form of a vortex sheet) that is being shed from the edge. This may be more clear if we construct the corresponding potential φv for large x, which is 

 ω ω φv ∼ sign(y) exp − |y| − i x , U0 U0

pv ≈ 0.

(2.158)

In conclusion, we obtain the continuous-potential, singular solution by transforming the no-flow solution in potential form and the discontinuous-potential, regular solution from the no-flow solution in pressure form. The difference between both is the field of the shed vortex sheet. The assumption that just enough vorticity is shed that the pressure field is no longer singular is known as the unsteady Kutta condition. Physically, the amount of vortex shedding is controlled by the viscous boundary layer thickness compared with the acoustic wavelength and the amplitude (and the Mach number for high-speed flow). These effects are not included in the present acoustic model; therefore, they have to be included by an additional edge condition (e.g., the Kutta condition). Because vorticity can only be shed from a trailing edge, a regular solution is only possible if M > 0. If M < 0, the edge is a leading edge and we have to leave the singular behavior as it is. Note that the same physical phenomenon seems to occur at a transition from a hard to a soft wall in the presence of mean flow. Normally, at the edge there will be a singularity. When the soft wall allows a surface wave of particular type accounting for a modulated vortex sheet along the line surface to form, a Kutta condition can be applied that removes the singularity – possibly at the expense of a (spatial) instability (Quinn and Howe 1984; Rienstra 1986, 2002). It should be emphasized that this results from a linear model and that an instability that is too severe may not be acceptable in a fully nonlinear model.

2.4 Aeroacoustic analogies 2.4.1 Lighthill’s analogy

Until now we have considered the acoustic field generated in a quiescent fluid by an imposed external force field f or by heat production Q w . We have further assumed that the sources induce linear perturbations of the reference quiescent fluid state. Lighthill (1952) proposed a generalization of this approach in the case of an arbitrary source region surrounded by a quiescent fluid. Hence, we no longer assume that the flow in the source region is a linear perturbation of the reference state. We assume only that the listener is surrounded by a quiescent reference fluid ( p0 , ρ0 , s0 , c0 uniformly constant and v0 = 0) in which the small acoustic perturbations are accurately described by the homogeneous linear wave equation (2.70). Lighthill’s key idea was to derive from the exact equations of mass conservation (2.6) with Q m = 0 and momentum conservation (2.8) a nonhomogeneous wave equation that reduces to the homogeneous wave equation (2.70) in a region surrounding the listener.

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By taking the time derivative of the mass conservation law (2.6) and subtracting the divergence of the momentum equation (2.8), we obtain ∂2 ∂ fi ∂ 2ρ = (Pi j + ρvi v j ) − . ∂t 2 ∂ xi ∂ x j ∂ xi

(2.159)

∂  By adding the term c0−2 ∂t 2 p to both sides and by making use of definition (2.9) (i.e., σi j = pδi j − Pi j ), we can write Equation (2.159) as    1 ∂ 2 p p ∂ 2 p ∂2 ∂ fi ∂2  (2.160) − = (ρv v − σ ) − + − ρ i j ij ∂ xi ∂ x j ∂ xi ∂t 2 c02 c02 ∂t 2 ∂ xi2 2

in which the perturbations p  and ρ  are defined by p  = p − p0

and

ρ  = ρ − ρ0 .

(2.161)

This equation is called the analogy of Lighthill. Note that neither ρ  /ρ0 nor p  / p0 is necessarily small in the so-called source region. In fact, Equation (2.160) is exact. ∂2  Furthermore, because we obtained this equation by adding the term c0−2 ∂t 2 p to both sides, this equation is valid for any value of the velocity c0 . In fact, we could have chosen c0 = 3 × 108 ms−1 (the speed of light in vacuum) or c0 = 1 mm/century. Of course, the equation would be quite meaningless then. By choosing for c0 and p0 the values of the reference quiescent fluid surrounding the listener, we recover the homogeneous wave equation (2.70) whenever the right-hand side of Equation (2.160) is negligible. Hence, Equation (2.160) is a generalization of Equation (2.59), which was derived for linear perturbations of a quiescent fluid. We should now realize that we did not introduce any approximation to Equation (2.160), and so this is exact. Therefore, this equation does not provide any new information that was not already contained in the equations of mass conservation (2.6) and of momentum (2.8). In fact, we have lost some information. We started with four exact Equations (2.6, 2.8) and eleven unknowns (vi , p, ρ, σi j ). We are now left with one Equation (2.160) and still eleven unknowns. Obviously, without additional information and approximations we have not gotten any closer to a solution for the acoustic flow. The first step in making Lighthill’s analogy useful was already described above. We have identified a listener around which the flow behaves like linear acoustic perturbations and is described by the homogenous wave equation (2.70). This assumption is valid in many applications. When we listen, under normal circumstances, to a flute player we have conditions that are quite reasonably close to these assumptions. At this stage the most important contribution of Lighthill’s analogy is that it generalizes (2.161) the equations for the fluctuations ρ  and p  to the entire space even in a highly nonlinear source region. The next steps are the introduction of approximations to estimate the source terms – that is, the right-hand side of Equation (2.160). We recognize in the ∂ 2 p  right-hand side of Equation (2.160) the term ∂t 2 ( 2 − ρ ), which is a generalization of c0 the entropy production term in Equation (2.59). As shown by Morfey (1973) (see Crighton et al. 1992), this term includes complex effects due to the convection of entropy nonuniformities. The effect of external forces f

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is the same in both Equation (2.70) and Lighthill’s analogy (2.160). We have, however, now removed the condition that the force should only induce a small perturbation to the reference state. An arbitrary force is allowed as long as we take into account any additional effects it may have on the other terms at the right-hand side of Equation (2.160). We observe additional terms due to the viscous stress σi j and the Reynolds stress ρvi v j . The viscous stress σi j is induced by molecular transport of momentum, whereas ρvi v j takes into account the nonlinear convection of momentum. One of Lighthill’s (1952) key ideas is that, when the entropy term and the external forces are negligible, the flow will only produce sound at high velocities, corresponding to high Reynolds numbers. He therefore assumed that viscous effects are negligible and reduce the sound source to the nonlinear convective effects (∂ 2 ρvi v j /∂ xi ∂ x j ). It is worth noting that a confirmation of this assumption’s reasonableness was provided qualitatively by Morfey (1976, 2003) and Obermeier (1985) only about 30 years after Lighthill’s original publication. A quantitative discussion for noise produced by vortex pairing was provided by Verzicco et al. (1997). An additional assumption, commonly used, is that feedback from the acoustic field to the source is negligible. Hence, we can calculate the source term from a numerical simulation that ignores any acoustic wave propagation and subsequently predict the sound production outside the flow. In extreme cases of low-Mach-number flow, a locally incompressible flow simulation of the source region can be used to predict the (essentially compressible!) sound field. Equation (2.160) can formally be solved by an integral formulation of the type (2.66). This will have the additional benefit of reducing the effect of random errors in the source flow on the predicted acoustic field. One can state that such an integral formulation combined with Lighthill’s analogy allows to be obtained a maximum of information concerning the sound production for a given information about the flow field. A spectacular example of this is Lighthill’s prediction that the power radiated to free space by a free turbulent isothermal jet scales as the eighth power, U08 , of the jet velocity. This result is obtained from the formal solution for free-space conditions:    ∞ r 2 δ t − τ − ∂ c 0 dVy dτ ρvi v j p  (x, t) = ∂ xi ∂ x j −∞ V 4πr   ρvi v j  ∂2 = dVy , (2.162) ∂ xi ∂ x j V 4πr τ =te where r = x − y and te = t − r/c0 . Assuming that the sound is produced mainly by the large turbulent structures with a typical length scale of the width D of the jet, we estimate the dominating frequency to be f = U0 /D, where U0 is the jet velocity at the exit of the nozzle. Hence, the ratio of jet diameter to the acoustical wavelength D f /c0 = U0 /c0 = M. This implies that, at low Mach numbers, we can neglect variations of the retarded time te if the source region is limited to a few pipe diameters. Because the acoustic power decreases very quickly with decreasing flow velocity, we have indeed

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a source volume of the order of D 3 . Inasmuch as v ≈ U0 and ρ ≈ ρ0 , we may assume that ρvi v j ∼ ρ0 U02 . From the far-field approximation ∂/∂ xi = −∂/(c0 ∂t) ∼ 2π f /c0 , we find p  (x, t) ∼ ρ0 U02 M 2

D , x

(2.163)

where we ignore the effect of convection on the sound production (Goldstein 1976; Hubbard 1995). The radiated power is thus found to be I  4πx2 p  vr ∼ ρ0 U03 M 5 D 2 ∼ U08 .

(2.164)

This scaling law appears indeed to be quite accurate for subsonic isothermal free jets. Note that this law implies that we can achieve a dramatic reduction of aircraft jet noise production by decreasing the flow Mach number. In order to retain the necessary thrust, the jet cross section has to be increased. This is exactly what happened in the 1960s and 1970s when the high-bypass turbofan aeroengines replaced the older turbofan engines without or with little bypass flow. This is known as the “turbofan revolution” (Smith 1989; Hubbard 1995). As stated by Crighton et al. (1992) and Powell (1990), Lighthill’s theory is rather unique in its prediction of a physical phenomenon before experiments were accurate enough to verify it. This made Lighthill’s analogy famous. Note that we have actually discarded any contribution from entropy fluctuations or external forces. This means that, if we use as input for the analogy data obtained from a numerical simulation that include significant viscous dissipation and spurious forces, we still would predict the same scaling law. Because the amplitude of the acoustic pressure generated by a compact monopole and dipole would scale in free-field conditions as M 2 and M 3 , respectively, these spurious sources easily dominate the predicted sound amplitude. This is one reason why most of the direct numerical simulations of sound production by a subsonic flow in free-field conditions are carried out at high Mach numbers (M  0.9). In the presence of walls, the sound radiation by turbulence can be dramatically enhanced. In the next section, we will see that compact bodies will radiate a dipole sound field associated with the force they exert on the flow as a reaction to the hydrodynamic force of the flow applied to them. Sharp edges are particularly efficient radiators. This corresponds to our common experience with the production of sounds like the consonant /s/. The interaction of the sharp edges of our teeth is essential. In free-field conditions, much attention has been devoted to the so-called trailing-edge noise of aircraft wings (Goldstein 1976; Blake 1986; Hubbard 1995; Howe 1998). The scaling rule for this sound field is p  ∼ M 5/2 , which is just between a monopole and a dipole. For confined subsonic flows at low frequencies, the scaling rules are quite different. A compact turbulent flow in an infinitely long straight duct will produce an acoustic field that scales according to p  ∼ M 6 (Ffowcs Williams 1969; Michalke

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1989; Rienstra and Hirschberg 2001). The sound field from a monopole and an axially aligned dipole will both scale with M 2 , whereas transversal dipoles will not radiate any sound. We will consider elements of the acoustics and aeroacoustics of confined flows in Section 2.5. The scaling rules for supersonic free jets are obscured by the temperature difference between the flow and the environment (Tam 2001). Globally, however, one expects a power that is limited by a U03 law because an extrapolation of the scaling rule ∼ U08 would imply that the acoustical power generated by the flow would soon become larger than the kinetic energy flux of the flow, which scales as 12 ρ0 U03 . At high Mach numbers, the theory has to be modified to take intrinsic temperature differences into account.

2.4.2 Curle’s formulation

The integral formulation of Lighthill’s analogy can be generalized for flows in the presence of walls. We will discuss later the use of tailored Green’s functions (Section 2.5.3). We now consider Curle’s (1955) approach. We use the free-space Green’s function G 0 (2.116). Instead of the pressure p  as the aeroacoustical variable we use the density ρ  . Subtracting from both sides of Equation (2.159) the term c02 (∂ 2 ρ  /∂ xi2 ), we obtain the analogy of Lighthill for the density, ∂ 2 Ti j ∂ 2ρ ∂ 2ρ ∂ fi − c02 2 = − , 2 ∂t ∂ xi ∂ x j ∂ xi ∂ xi

(2.165)

in which the stress tensor of Lighthill, Ti j , is defined by Ti j = Pi j + ρvi v j − c02 ρδi j .

(2.166)

We further assume that f = 0 and focus on the other sound sources. We have selected here the density as the dependent variable because this was Lighthill’s (1952) original choice. We will later discuss the implications of this choice (Section 2.4.4). We consider a fixed surface S with outer normal n and apply Green’s theorem (2.66) to the volume V outside of S. Note that n is chosen toward the interior of V , and so the sign convention of n is opposite to the sign convention used in Equation (2.66). By means of partial integration and by utilizing the symmetry properties ∂G 0 /∂ xi = −∂G 0 /∂ yi (2.123) and ∂G 0 /∂τ = −∂G 0 /∂t of the Green’s function G 0 , we obtain (Goldstein 1976; Rienstra and Hirschberg 2001) p  (x, t) = c02 ρ  (x, t) =

∂2 ∂ xi ∂ x j ∂ − ∂x j

  V

  S

Ti j 4πr

 τ =te

dVy +

Pi j + ρvi v j 4πr

 τ =te

∂ ∂t

  ρvi  n i dS S 4πr τ =te

n i dS,

(2.167)

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which is clearly a generalization of Equation (2.162). In this equation we used the assumption that, at the listener’s position, p  = c02 ρ  . In the far-field approximation (2.126), we find  



  1 ∂ ρvi  dVy + n i dS 4π ∂t S r τ =te V τ =te    Pi j + ρvi v j xj ∂ n i dS. (2.168) + 4π xc0 ∂t S r τ =te

xi x j ∂2 p (x, t)  4π x2 c02 ∂t 2 

Ti j r

For a compact body we can neglect the variations of te over the surface and can write r = x if we choose the origin y = 0 inside or near the body. Assuming that Ti j decays fast enough, we have in that case xi x j ∂2 p (x, t)  4π x3 c02 ∂t 2 

xj ∂ + 2 4π x ∂t

 

V

S

1 ∂ [Ti j ]τ =tedVy + 4π x ∂t

[Pi j + ρvi v j ]τ =ten i dS,

 S

[ρvi ]τ =ten i dS (2.169)

and te = t − x/c0 . The second integral corresponds to the monopole sound field generated by the mass flux through the surface S. The third integral corresponds to the dipole sound field generated by the instantaneous force −F j of the surface to  the surrounding fluid. This is the reaction of the surface to the force F j = − S (Pi j + ρvi v j )n i dS of the flow on the surface. This result is a generalization of Gutin’s principle (Gutin 1948; Goldstein 1976). Using this theory, we easily understand that a rotor blade moving in a nonuniform flow field will generate sound owing to the unsteady hydrodynamic forces on the blade. At low Mach number, this will easily dominate the Doppler effect due to the rotation. Wind rotors placed downwind of the supporting mast are cheap because they are hydrodynamically stable. There is no need for a feedback system to keep them in the wind. However, the interaction of the wake of the mast with the rotor blades causes dramatic noise problems (Hubbard and Shepherd 1991; Wagner, Bareiss, and Guidati 1996).

2.4.3 Ffowcs Williams–Hawkings formulation

Although Curle’s formulation discussed in the previous section assumes a fixed control surface S, the formulation of Ffowcs Williams and Hawkings allows the use of a moving control surface S(t). The key idea is to include the effect of the surface in the differential equation (2.165) as described by Ffowcs Williams and Hawkings (1969), Goldstein (1976), and Crighton et al. (1992). This is achieved by a most elegant and efficient utilization of generalized functions (viz. so-called surface distributions).

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We assume that the volume B(t) enclosed by the surface S(t) and this surface are sufficiently smooth to allow the definition of a smooth function h(x, t) such that ⎫ h(x, t) < 0 if x ∈ B(t), ⎪ ⎪ ⎬ (2.170) h(x, t) = 0 if x ∈ S(t), ⎪ ⎪ ⎭ h(x, t) > 0 outside B(t). Consider any physical quantity, like ρ  , defined outside B(t) and extend its definition to all space by giving it a value equal to zero inside B(t). This is efficiently done by assuming ρ  smoothly defined everywhere. Then by multiplying it by the Heaviside function H (h) we create a new variable ρ  H (h) that vanishes within the volume B(t) (where H (h) ≡ 0) and is equal to ρ  outside B(t) (where H (h) ≡ 1). The next step will be to extend the prevailing equations to the whole space by adding suitable surface sources. To achieve this we need the normal n to the surface S(t) given by   ∇h . (2.171) n= ∇h h=0 We assume that the surface S(t) is parameterized∗ in time and space by the coordinates (t; λ, μ). A point xs (t) ∈ S(t) with fixed parameters λ and μ is moving with the velocity b. Hence, we have h(xs , t) = 0 and ∂h = −b ·∇h = −(b·n)∇h. (2.172) ∂t After multiplying the mass conservation equation (2.6) and the momentum equation (2.8) by H (h) and reordering terms, we obtain the following equations valid everywhere: ∂  [ρ H ] + ∇·[ρvH ] = [ρ0 b + ρ(v − b)] ·∇H, ∂t

(2.173a)

∂ [ρvH ] + ∇· [(P + ρvv)H ] = [P + ρv(v − b)] ·∇H, (2.173b) ∂t where H stands for H (h). Because ∇H = δ(h)∇h, the equations can be interpreted as generalizations of the mass and momentum equations with surface sources at S. Using the relations above and following Lighthill’s procedure for acoustic variable p  = p − p0 , we find 1 ∂2  [ p H ] − ∇2 [ p  H ] = ∇· [∇·[(ρvv − σ)H ]] − ∇·[fH ] c02 ∂t 2     ∂2 p ∂  [(ρ0 b + ρ(v − b)) ·∇H ] + 2 −ρ H + 2 ∂t ∂t c0 − ∇·[( p  I − σ + ρv(v − b)) ·∇H ], ∗

(2.174)

When S(t) is the surface of a solid and undeformable body, it is natural to assume a spatial parametrization that is materially attached to the surface. Like the auxiliary function h, this parametrization is not unique.

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where (I)i j = δi j and p0 is the uniform reference value of the pressure. Note that ∇·(∇·( p0 I H )) = ∇·( p0 I ·∇H ). For a solid surface, v·n = b·n. In that case, by applying Green’s theorem and using the free-space Green’s function, we find p  (x, t) =

   (ρvi v j − σi j )H ∂2 dVy ∂ x i ∂ x j R3 4πr τ =te        2 fH ( p /c0 − ρ  )H ∂ ∂2 − dV + dVy y ∂ xi R3 4πr τ =te ∂t 2 R3 4πr τ =te    ∂ ρ0 b·n + dS ∂t S(te ) 4πr (1 − Mr ) τ =te     p n i − σi j n j ∂ dS, (2.175) − ∂ xi S(te ) 4πr (1 − Mr ) τ =te

where r = x − y and Mr = b·(x − y)/r c0 , and we use the following generalizations of Equation (2.132):   g(x) dS, (2.176a) g(x)δ(h(x)) dx = 3 R S |∇h|   g(x) ·∇H (h(x)) dx = (g ·∇h)δ(h) dx 3 3 R R   g ·∇h = (2.176b) dS = g(x)·n(x) dS. S |∇h| S The first three integrals correspond to the contribution of the flow around the surface S(t), and the last two integrals represent generalizations of the thickness noise and sound generated by the surface forces we have discussed earlier. A reduced form, widely used for subsonic propeller and fan noise when volume sources and surface stresses are negligible, is thus (Farassat 1981)    1 ∂ ρ0 b·n  dS p (x, t) = 4π ∂t S(te ) r (1 − Mr ) τ =te    p ni 1 ∂ dS. (2.177) − 4π ∂ xi S(te ) r (1 − Mr ) τ =te 2.4.4 Choice of aeroacoustic variable

In the previous discussion we used p  as the dependent aeroacoustical variable to introduce the analogy of Lighthill (Section 2.4.1). We then used ρ  to introduce the formulation of Curle describing the effects of stationary boundaries (Section 2.4.2). Finally, for the formulation of Ffowcs Williams and Hawkings describing the effect of moving boundaries, we retuned to p  . In acoustics this would have been indifferent

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because the two variables are related by the equation of state p  = c02 ρ  . Inasmuch as we assume linear acoustics of quiescent fluids to be valid around the listener, we made use of this relationship in Equation (2.167). Actually, in aeroacoustics there is a subtle difference that appears when we compare the source terms of the two wave equations    p ∂ 2 p ∂2 ∂ fi ∂2 1 ∂ 2 p  (2.160) − = (ρv v − σ ) − + − ρ i j ij ∂ xi ∂ x j ∂ xi ∂t 2 c02 c02 ∂t 2 ∂ xi2 and 2  ∂2 ∂ fi ∂2 ∂ 2ρ 2∂ ρ − c = (ρv v − σ ) − + ( p  − c02 ρ  ). i j ij 0 ∂t 2 ∂ xi ∂ x j ∂ xi ∂ xi2 ∂ xi2

(2.178)

Without further approximation, these two forms are both equivalent. However, considered as an analogy, the right-hand sides are assumed to be known and act as given source distribution. In that case we see that when p  is used as the aeroacoustical variable, the effect of entropy fluctuations ∂ 2 (( p  /c02 ) − ρ  )/∂t 2 has the character of a monopole sound source. On the other hand, when ρ  is used the apparently same effects produce a quadrupole distribution ∂ 2 ( p  − c02 ρ  )/∂ xi2 that is qualitatively different. Of course, there is no difference if we consider the exact equations, but if we do not introduce any approximation the analogy is just a reformulation of basic equations without much use. Clearly, we have to be careful in selecting the aeroacoustic variable. When considering sound production by subsonic flames, we should choose p  as the aeroacoustical variable because most of the sound is produced by the volume changes associated with the combustion. When we neglect convection effects, we have exactly the source term ∂ 2 (( p  /c02 ) − ρ  )/∂t 2 . As shown by Morfey (1973) and Crighton et al. (1992), this term includes complex effects due to the convection of entropy nonuniformities. They become explicit when we use the equation of state (2.17) as applied to a material element,   Ds ∂p Dp 2 Dρ =c + , (2.179) Dt Dt ∂s ρ Dt in combination with Equation (2.14): Ds De D T = +p Dt Dt Dt

  1 . ρ

Morfey (1973) obtains the following result:   2    ∂ ρe Dρ  ∂ 2 ρe c ρ2 ∂ T Ds  −1+ − 2 = + 2 + ∇·(ρe v) , ∂t ∂t ρ Dt c02 c0 ∂ρ s Dt

(2.180)

(2.181)

where the excess density ρe is defined by ρe = ρ  − ( p  /c02 ). The first term vanishes in a subsonic free jet of an ideal gas with constant heat capacity. The second term is the entropy production (combustion) term corrected for convective effects. The last term corresponds to the force exerted by a patch of fluid with a different density on its surroundings in an accelerating flow. This may be compared with a buoyancy (Archimedes)

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effect. It is induced by the fact that, owing to the difference of density between the particle and its surroundings, the pressure gradient imposed by the surroundings of a particle does not match its acceleration. The question arises whether ρ  could be a useful choice too. If we consider a turbulent isentropic flow in a region with a speed of sound c that differs strongly from the speed of sound c0 at the listener position, the source term can be rewritten as ∂ 2 p  (1 − (c0 /c)2 )/∂ xi2 . In a subsonic turbulent flow we expect the local pressure fluctuations p  to scale with 12 ρU02 . Hence, the analogy indicates that, when the speed of sound c0 at the listener position is much higher than the speed of sound c in the source region, there will be a strong enhancement of the sound production compared with a similar flow in a uniform fluid. Such a spectacular effect does indeed occur when we consider a flow in a water–air mixture such as obtained by directing the nozzle of the shower toward the surface of the water. As the result of air entrainment there will be a volume fraction β of air in the water flow. The density of the mixture will be ρ = (1 − β)ρwater + βρair .

(2.182)

If we assume a quasi-static response of the bubbles and neglect dissolution of air in the water, the compressibility 1/(ρc2 ) of the mixture will be the sum of the compressibilities of both phases (van Wijngaarden 1972; Crighton et al. 1992): 1−β β 1 = + . 2 2 2 ρc ρwater cwater ρair cair

(2.183)

In the case of water–air mixtures for not-too-small nor too-large values of β, the density is mainly determined by the water phase, whereas the air determines the compressibility. Hence, we find for the speed of sound that c2 

2 ρair cair . β(1 − β)ρwater

(2.184)

2 Typical values are β = 0.5, ρair cair = 1.4 × 105 Pa , ρwater = 103 kg m−3 , and cwater = 3 −1 1.5 × 10 ms . We find c  2 × 101 ms−1 , and, if we have our head underwater, (c0 /c)2  5 × 103 . This should result in an enhancement in SPL on the order of 60 dB. Indeed the flow is much noisier than when we avoid air entrainment by putting the shower nozzle underwater. Other choices of aeroacoustic variables lead to different analogies. In many cases such analogies tend to avoid the problem induced by the fact that the analogy of Lighthill does not distinguish between propagation and production of sound waves in a strongly nonuniform flow, which induces refraction. This becomes very important in supersonic flow. In such cases the source is not compact. Simple results like Equation (2.166) are not valid anymore. Alternative analogies that attempt to overcome such problems are described in the literature (Goldstein 1976; Hubbard 1995; Lilley 2004).

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2.4 AEROACOUSTIC ANALOGIES

2.4.5 Vortex sound

One of the drawbacks of the analogy of Lighthill (2.160) is that the source term is spatially quite extended. As observed by Powell (1964), the sound production in subsonic homentropic flows is associated to the dynamics of vortices. Vorticity ω = ∇×v appears to be spatially less extended than the Reynolds stress ρvi v j . The reason for this is that around vortices there is a large region of potential flow that actually does not produce any sound. In its original form, the vortex sound theory (Powell 1964) was applied to free-field conditions at low Mach numbers. It was only a special form of Lighthill’s analogy stressing the role of vorticity. In the case of free-field conditions, the vortex sound theory greatly enhances the predicted sound field. Various modifications of the theory of Powell have been proposed that more explicitly impose the conservation of momentum and energy on the flow in the source region (M¨ohring 1978; Hirschberg and Schram 1995; Schram and Hirschberg 2003). This also improves the performance of the theory. Howe (1975, 1998, 2002) has generalized the theory of Powell (1964) to allow its application to confined flows and conditions in which the listener is placed into a potential flow rather than a quiescent fluid. In this theory the fluctuations of the total enthalpy B  = ( p  /ρ0 ) + v  v0 appear as a natural aeroacoustical variable (Howe 1975; Doak 1989, 1995; Musafir 1997). In general form the vortex sound theory can be applied to arbitrary Mach numbers. A similar analogy was derived by M¨ohring (2001) on the basis of acoustical energy considerations. However, such analogies become quite obscure. They do not provide many intuitive insights and can only be used numerically as proposed by Ewert and Schr¨oder (2004). We consider now the case of low subsonic flows in that Howe (1980) proposed a very nice energy corollary that provides much insight into the role of vorticity in sound production. We propose here an intuitive approach to this theory. The key idea of the theory is that a potential flow is silent. This is illustrated in Figure 2.3 in which we consider a sketch of human vocal folds. The oscillation of the vocal folds results in a variable volume flow from our lungs into the vocal tract. This variable volume flux is the source of sound. Seeking a simplified model for the flow through the glottis, we consider the Reynolds number Re and the Strouhal number St. For a typical flow velocity U0 = 30 ms−1 , a length scale on the glottis on the order of L = 2 mm, a frequency f = 102 Hz, and a kinematic viscosity ν = 1.5 × 10−5 m2 s−1 , we have Re = U0 L/ν = O(103 ) and St = f L/U0 = O(10−2 ). A quasi-steady, frictionless approximation seems promising. Because the Mach number is low, M = O(10−1 ), and the flow is compact, L/λ = MSt = O(10−3 ), we assume that the flow is locally incompressible. Under such circumstances, the equation of Bernoulli in the form p + 12 ρ0 v 2 = pt with pt a constant should be valid. About one diameter upstream and downstream of the vocal folds we expect a uniform flow velocity. Assuming the flow channel upstream of the vocal folds (trachea) to have the same cross section as the channel downstream (vocal tract), we find that the velocities should be equal. As a consequence we conclude that there is no pressure difference across the vocal folds. The volume flux is therefore not controlled

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Figure 2.3. A potential flow through the vocal folds is silent. Sound is produced by the volume-flows-control associated with flow separation and formation of a free jet. This implies vorticity injection in the main flow.

by the opening of the folds! This implies that they cannot produce any sound. This corresponds to the paradox of d’Alembert, which is solved by realizing that we can never neglect the effect of friction at the walls. Even at high Reynolds numbers there are always thin viscous boundary layers near the walls. In these boundary layers the pressure is essentially equal to that of the main flow, but the velocity decreases to zero at the wall. This implies that the stagnation pressure pt is lower than its value in the main flow. Hence, the fluid in the boundary layers cannot flow against the strong adverse pressure gradient in the diverging part of the glottis. This results in separation of the boundary layers and the formation of a free jet. Turbulent dissipation in the free jet explains the flow control by the oscillating vocal folds and the pressure difference across the glottis. Of course, the fluid from the boundary layers injected into the main flow has vorticity. Hence, we see that vorticity injection into the main flow is associated with the production of sound. This illustrates the statement of M¨uller and Obermeier (1988) that “vortices are the voice of the flow.” As explained in Section 2.2.3, the velocity field can be separated into a potential and a vortical flow. The potential part ∇φ of the flow is associated with the dilatation rate ∇·v = ∇2 φ of fluid particles in the flow. Because the acoustical flow is essentially compressible and unsteady, Howe (1980) proposed to define the acoustic field as the unsteady component of the potential flow uac , uac = ∇φ  ,

(2.185)

in which φ  = φ − φ0 is the time-dependent part of the potential φ. For a homentropic flow, we can write the equation of Euler (2.35) in the form ∂v fc + ∇B = ∂t ρ

(2.186)

in which fc = −ρ(ω × v) is the density of the Coriolis force associated with the vorticity of the flow. When ω = 0, we have a potential flow. Hence, we identify fc /ρ as the source of sound. At low Mach number for compact flows, we can neglect the variation

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of the density so that fc is the source of sound. Under such circumstances we can, in first approximation, apply the energy equation (2.107) in the form  (2.187) P = fc ·uac dV, V

where P is the time-averaged acoustical power generated by the vortices and V is the volume in which the vorticity ω is confined. The success of this equation is due to the fact that, even when using a highly simplified vortex model, it provides a fair prediction of the sound production. A first explanation for this is that it is an integral formulation that smooths out random errors. A second aspect is that, when considering models of isolated vortices with time-dependent ∂  induced circulation , it ignores the effect of the spurious pressure difference −ρ ∂t by the model along the “feeding line” of the vortex. This corresponds to the force necessary for the transport of vorticity toward the “free” vortex. Hence, a model that does not satisfy the momentum equation still gives reasonable predictions because we ignore the effect of spurious forces. The theory of Howe provides much insight. In particular, Equation (2.187) clarifies the essential role of sharp edges at which the potential flow is singular and vortex shedding occurs. In Section 2.5.4 we consider examples of the use of this theory. For the sake of completeness we now provide a more formal form of the analogy of Howe (2002). Starting from the divergence of the momentum conservation equation (2.186), ∂∇·v + ∇2 B = −∇·(ω × v), ∂t

(2.188)

and eliminating ∇·v by means of the mass conservation law (2.5) ∇·v = −

1 Dρ , ρ Dt

(2.189)

we obtain for an isentropic flow 1 D20 B  1 D20 B  ∂ 1 Di 2  , − ∇ B = ∇·(ω × v) + − c2 Dt 2 c2 Dt 2 ∂t c2 Dt

(2.190)

∂ B D0 B  = + (∇φ0 ) ·∇B  , Dt ∂t

(2.191)

where

and we make use of the equation of state for isentropic flows: 1 Dp 1 Di 1 Dρ = 2 = 2 . ρ Dt c ρ Dt c Dt

(2.192)

As in Howe (1998), the first source term ∇·(ω × v) in Equation (2.190) is dominant at low Mach numbers. Although the resulting wave equation (2.190) looks relatively simple, for an arbitrary steady potential flow ∇φ0 it can only be solved numerically.

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Figure 2.4. Straight duct of arbitrary cross section.

2.5 Confined flows 2.5.1 Wave propagation in a duct

For a narrow, hard-walled duct, the only waves that propagate are of the form given by Equation (2.72), which is for a time-harmonic sound field in complex notation the pair of plane waves p  (x, t) = A e iωt−ikx +B e iωt+ikx ,

(2.193)

where k = ω/c0 and the medium is uniform and stagnant. Note that each plane wave is self-similar in x apart from a phase change. This solution can be generalized for wider ducts or higher frequencies as follows. The time-harmonic sound field in a duct of constant cross section with linear boundary conditions that are independent of the axial coordinate may be described by an infinite sum of special solutions – modes – that retain their shape when traveling down the duct. They consist of an exponential term multiplied by an eigenfunction of the Laplace eigenvalue problem on a duct cross section. Consider the two-dimensional area A in the (y, z) plane with a smooth boundary ∂A and an externally directed unit normal n. By shifting A in the x direction, we obtain the duct D given by D = {(x, y, z)|(0, y, z) ∈ A}

(2.194)

with axial cross sections being copies of A and for which the normal vectors n are the same for all x (Figure 2.4). In the usual complex notation (with +iωt–sign convention), the acoustic field p  (x, t) ≡ p  (x, ω) e iωt ,

v (x, t) ≡ v (x, ω) e iωt

(2.195)

satisfies, in the duct (x ∈ D), the equations ∇2 p  + k 2 p  = 0,

iωρ0 v + ∇ p  = 0.

(2.196)

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At the duct wall we assume the impedance boundary condition p  = Z (v ·n) for x ∈ ∂D.

(2.197)

Hard walls correspond to Z = ∞. The solution of this problem may be given by p  (x, y, z) =

∞

Cn ψn (y, z) e−iκn x ,

(2.198)

n=1

where ψn are the eigenfunctions of the Laplace operator reduced to A, that is, solutions of   2 ∂ ∂2 + 2 ψ = α 2 ψ for (y, z) ∈ A, − ∂ y2 ∂z (2.199) −iωρ0 ψ = (n ·∇ψ)Z for (y, z) ∈ ∂A, where α 2 is the corresponding  eigenvalue. The axial wave number κn is given by one of the square roots κn = ± k 2 − αn2 (+ for right and − for left running), whereas the branch of the square root is to be taken such that (κn ) ≥ 0 and (κn ) ≤ 0. Therefore, the left-running and right-running modes are usually explicitly given as p  (x, y, z) =

∞

  ψn (y, z) An e−iκn x +Bn e iκn x .

(2.200)

n=1

Each term in the series expansion – that is, ψn (y, z) e−iκn x – is called a duct mode. For hard walls, the eigenvalues αn2 are real and positive, except for the first one, which is α1 = 0. In the hard-wall case we have the important distinction between k > αn , where κn is real and the mode is propagating, and k < αn , where κn is imaginary and the mode is evanescent (i.e., exponentially) decaying. Propagating modes are described as being “cut on,” and evanescent modes are “cut off.” The frequency ω = c0 αn is called the “cutoff frequency” of the mode. For low frequencies, the only cut-on mode is the plane wave (with cutoff frequency zero): ψ1 = 1,

α1 = 0,

κ1 = ±k.

(2.201)

The next eigenvalue α2 is typically of the order of a number between 3 and 4 divided by the√duct diameter D. As a result, any higher modes of a sound field with frequency ω < 10 c0 /D decay faster than exp(−x/D). At any distance more than, say, two diameters away from a source, the field is well described by just plane waves. If the duct cross section is circular or rectangular and the boundary condition is uniform everywhere, the solutions of the eigenvalue problem are relatively simple and may be found by separation of variables. These eigensolutions consist of combinations of trigonometric and Bessel functions in the circular case or combinations of trigonometric functions in the rectangular case. In particular, for a cylindrical duct we have in polar coordinates (r, θ) the spiraling modes ψ = Jm (αmμr ) e−imθ ,

m ∈ Z,

(2.202)

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where Jm is a Bessel function of integer order m. Note that J−m = (−)m Jm . Positive m correspond to counterclockwise-rotating modes (phase ωt − mθ = constant) and negative m with clockwise-rotating modes. With a fixed source it is sometimes clearer to distinguish between symmetric and antisymmetric modes by using sin(mθ) and cos(mθ ) rather than e−imθ . For a rectangular duct we have    sin(βμ y) sin(γν y) ψ= , (2.203) cos(βμ y) cos(γν y) 2 where αμν = βμ2 + γν2 . Note that, owing to the symmetry, in both geometries the eigenvalue has multiplicity 2 – that is, for each eigenvalue α there are two eigenfunctions. This is not the case for an arbitrary cross section. Some other geometries, like ellipses, do also allow explicit solutions, but only in special cases such as with hard walls. For other geometries one has to fall back on numerical methods for the eigenvalue problem. Without mean flow, the problem is symmetric, and to each eigenvalue there corresponds a right-running and a left-running mode because both κn and −κn can occur. The modes form a complete set of basis functions for the solutions to the wave equation. These modes are not exactly orthogonal to each other, but the complex conjugated modes (more precisely, the solutions of the adjoint – which is here the complex conjugated – problem) are mutually orthogonal or “biorthogonal” with ψn . It follows that, if we specify ψˆ n = ψn∗ (the asterisk denotes the complex conjugate), we have   def ∗ ˆ ˆ ψn ψm dσ = δnm ψn2 dσ = δnm (ψn , ψˆ n ), (2.204) (ψn , ψm ) = A

A

δnm = 1 if n = m and is zero otherwise. Note that hard-walled modes are real, and so ψˆ n = ψn ; thus, hard-walled modes are orthogonal. The biorthogonality can be used to determine the coefficients of an expansion. Assume we have the pressure given along cross section x = 0 at the right side of a source such that only right-running modes occur. We have p  (0, y, z) = F(y, z) =

∞

An ψn (y, z).

(2.205)

n=1

After multiplication of the left- and right-hand sides by ψˆ m and integration across A, we obtain Am =

(F, ψˆ m ) . (ψm , ψˆ m )

(2.206)

In a similar way we can determine an expansion of the Green’s function Gˆ for a duct (Morse and Feshbach 1953) as follows: ˆ G(x, y, z|ξ, η, ζ ) = 12 i

∞ ψn (y, z)ψn (η, ζ ) n=1

κn (ψn , ψˆ n )

e−iκn |x−ξ | .

(2.207)

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This result can also be obtained, albeit in a more laborious way, by means of Fourier transformation to x. Solutions within more complex geometries consisting of piecewise constant pipe elements can be obtained by matching two series of expansions in modes of adjacent pipe segments. This procedure involves an approximation by truncation of the mode expansion. This is quite successfully applied to stepwise changes in pipe cross section (Morse and Feshbach 1953; Kergomard, Garcia, and Taguy 1989; Rienstra and Hirschberg 2001), the diaphragm (Morse and Feshbach 1953; Rienstra and Hirschberg 2001), the elbow (Miles 1947), the bend (Cabelli 1980), the T-joint (Redmore and Mulholland 1982), and the tone-hole of woodwinds (Dubos et al. 1999). A subtle point is that the physical singularity of the sound field at the sharp edges is related to the convergence rate of the series expansion. It is therefore important, when using such approximations, to tune the number of modes carefully in the two series expansions (Rienstra and Hirschberg 2001). With mean flow, the problem is not symmetric anymore, but for uniform mean flow in a hard-walled duct this is only a minor obstacle because the problem can be transformed to an equivalent no-flow problem as we described in Section 2.3.10.1. This is not possible in the combination of mean flow with soft walls – at least when we use the Ingard–Myers boundary condition (Ingard 1959; Myers 1980) for inviscid flow along an impedance wall, where the eigenvalue appears in the boundary condition. We have different eigenvalues for the left- and right-running modes, and we now write the following instead of Equation (2.200): p  (x, y, z) =

∞

+

−

An ψn+ (y, z) e−iκn x +Bn ψn− (y, z) e−iκn x .

(2.208)

n=1

Orthogonality cannot be used to determine the amplitudes, but a Green’s function can still be determined by Fourier transformation to the axial coordinate. Furthermore, there are indications that the system may be unstable for certain impedances with the result that one seemingly upstream-running mode is to be interpreted as really a downstreamrunning instability (Rienstra 2002). Modes can also be obtained in more complex situations such as pipes with a main flow (Eversman 1993; Pagneux and Froelich 2001; Rienstra 2002) or nonrigid walls (Rienstra 1985). In many applications the prediction of mode propagation allows a significant insight into sound radiation problems. An example of such an application is the Tyler and Sofrin (1962) rules for the design of aircraft turbines, minimizing the number of excited cut-on modes and thus the radiation of sound generated by rotor stator interaction. When the duct and its possible mean flow vary in the axial direction, modes that are strictly self-similar in x are in general not possible. However, if the duct varies slowly as, for example, in an aeroengine flow duct, we can still identify approximate, “almost” modes that retain their shape but have a slowly varying amplitude, eigenvalue, and axial wave number (Rienstra 1999, 2003). (The approximation is known as a variant of the

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WKB approximation.) If the duct and mean flow variations scale on a so-called slow variable X = εx, where ε is small, we can describe a slowly varying mode by p  (x, y, z) = Nn (X )ψn (y, z; X ) e−i

x κn (εξ )dξ

,

(2.209)

where ψn with κn is a mode with wave number at cross section A(X ). The variable X acts as no more than a parameter. An adiabatic invariant can be identified yielding the varying amplitude. If the modes are normalized such that (ψn , ψˆ n ) = 1 and the mean flow is nearly uniform (i.e., modulated by the duct), the amplitude Nn is given for hard-walled ducts by  1/2 = constant, ρ0 Nn2 k 2 − (1 − M 2 )αn2

(2.210)

where ρ0 , k = ω/c0 , and M = U0 /c0 depend on X and eigenvalue αn corresponds to the mode ψn at position X . For soft walls the expression is similar. When the mode passes a turning point – that is, where k = (1 − M 2 )1/2 αn – the solution breaks down because the incident mode couples with its back-running counterpart (the mode “reflects”). A local analysis is possible to describe this effect (Ovenden 2002, 2005; Rienstra 2003; Ovenden et al. 2004).

2.5.2 Low-frequency Green’s function in an infinitely long uniform duct

At frequencies below the cutoff frequency for higher-order modes, the acoustical field in a duct is, at some distance from the source, dominated by the plane-wave mode. We expect, therefore, the Green’s function to be independent of the transverse coordinate of the listener’s position. If we use the principle of reciprocity (2.67), this implies also that the Green’s function G should not depend on the transverse coordinate of the source. Hence, we can use a one-dimensional Green’s function g defined by 1 ∂2g ∂2g − 2 = δ(x3 − y3 )δ(t − τ ). c02 ∂t 2 ∂ x3

(2.211)

A solution of this equation will be nonzero only for t > τ and of the form f (t − x3 /c0 ) for x3 > y3 and g(t + x3 /c0 ) for x3 < y3 . So we try (or see Kanwal 1998) the auxiliary function   x x + H (−x)g t + , (2.212) ϕ(x, t) = H (x) f t − c c where H denotes the Heaviside step function, and note that it has the properties   x x ∂ 2ϕ   t − t + + H (−x)g , = H (x) f ∂t 2 c c   1 x 1 x ∂ϕ = δ(x) f (t) − δ(x)g(t) − H (x) f  t − + H (−x)g  t + , ∂x c c c c

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Figure 2.5. (a) A method of images applied to a source at y = (y1 , y2 , y3 ) at a distance y3 from a hard wall x3 = 0 has a Green’s function: G(x ,t|y ,τ ) = G0 (x,t|y,τ ) + G0 (x,t|y∗ ,τ ) with y∗ = (y1 ,y2 ,−y3 ). (b) A source between two parallel hard walls generates an infinite row of images. (c) A source in a rectangular duct generates an array of sources.

where we introduced δ(x) f (t − x/c) = δ(x) f (t) and similarly for g, ∂ 2ϕ 1 = δ  (x)( f (t) − g(t)) − δ(x)( f  (t) + g  (t)) 2 ∂x c    1 x x 1 + 2 H (−x)g  t + , + 2 H (x) f  t − c c c c and hence satisfies ∂ 2ϕ 1 1 ∂ 2ϕ − = −δ  (x)( f (t) − g(t)) + δ(x)( f  (t) + g  (t)) c2 ∂t 2 ∂x2 c = δ(t)δ(x)

(2.213)

if f (t) = g(t) = 12 cH (t). By a simple coordinate transformation we find henceforth that   |x3 − y3 | 1 g(x3 , t|y3 , τ ) = 2 c0 H t − τ − . (2.214) c0 A more generic approach may be the use of Fourier transformation in t and x (see Rienstra and Hirschberg 2001). This result can also be understood by using the method of images (Pierce 1981). The acoustical field of a point source placed at a distance d from an infinite plane hard wall will generate the same acoustical field as the original source and its image placed in free space (Figure 2.5a). The image’s position is along the normal to the plane at the distance d from the plane opposite to the source. The image takes into account the effect of the waves’ reflection on the plane. A source placed between two hard plane walls will generate an infinite row of images (Figure 2.5b). A source placed in a rectangular duct at y = (y1 , y2 , y3 ) has the same effect as an array of sources in the (y1 , y2 ) plane, as shown in Figure 2.5c. Each source of the array generates a pulse G 0 = δ(t − te )/(4πr ) with te = t − r/c0 . For a given listener coordinate (0, 0, x3 ) in the duct, we have r 2 = y12 + y22 + (x3 − y3 )2 and dte = −dr/c0 .

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The number of images corresponding to an emission time between te and te + dte scales for fixed t, x, and y3 because π d(y12 + y22 ) = π dr 2 = 2πr dr = 2π c0r dte . Because G 0 scales as r −1 , the increase in number of sources with r exactly compensates the decrease of G 0 . This results in Heaviside step function behavior for a delta function source behavior. Note that, for a source placed between two rigid planes, we obtain a two-dimensional response intermediate between the delta pulse and the Heaviside step. One observes a long decay scaling with t −1/2 . Thunder acts qualitatively as a line source with a two-dimensional acoustical field. This phenomenon explains the long decay time of thunder sound. This makes acoustics in two dimensions quite complex and different from three- and one-dimensional acoustics (Dowling and Ffowcs Williams 1983). A two-dimensional acoustical field does not have a simple near-field behavior as a three-dimensional acoustical field. This essential difference between three- and twodimensional acoustics is a major problem when one considers the sound production by a two-dimensional model of a three-dimensional flow. Extending the two-dimensional model to the acoustical far field will dramatically exaggerate radiation losses. Placing a radiation condition at a finite distance will provide results that depend on the distance between the boundary and the flow. In an analogous way, we may find that, in the presence of a uniform subsonic main flow U0 in the duct, the Green’s function, satisfying   ∂ 2 1 ∂ ∂2g + U g − 2 = δ(x3 − y3 )δ(t − τ ), (2.215) 0 2 ∂x ∂x c0 ∂t is given by

 x3 − y3 g(x3 , t|y3 , τ ) = − y3 )H t − τ − c0 + U 0   x3 − y3 1 . + 2 c0 H (y3 − x3 )H t − τ + c0 − U 0 

1 c H (x3 2 0

(2.216)

Note that this Green’s function satisfies the reverse-flow reciprocity principle (Howe 1975) rather than the reciprocity principle (2.67). When we exchange the source and listener’s positions, we should also reverse the main flow in order to keep the travel time r/(c0 ± U0 ) of the waves between the source and the listener constant. 2.5.3 Low-frequency Green’s function in a duct with a discontinuity

We want to introduce the concept of low-frequency Green’s function by considering the effect of a discontinuity in an infinitely long duct. We will use the reciprocity principle (2.67) to determine the Green’s function. This implies that, for a source placed at the discontinuity, we can deduce the Green’s function by considering the sound field generated by a source placed at the position of the listener. For a source placed far from the discontinuity and a listener placed at the same side of the discontinuity as the source, the Green’s function will be built up of the waves generated at the source plus the reflection of one of the waves at the discontinuity. A listener placed beyond the

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discontinuity will only be reached by the waves transmitted through the discontinuity. In other words the problem reduces to the determination of the reflected and transmitted waves at the discontinuity. We limit ourselves to the case of a compact transition region x I ≤ x3 ≤ x I I of cross-sectional area S(x3 ) between two semi-infinite ducts of cross section S I at the side of the source and S I I at the opposite side. In the transition region we will now for simplicity assume a quasi–one-dimensional, incompressible potential flow v3 (x3 , t). Applying the integral (2.12a) mass conservation law across the discontinuity, we have for x I ≤ x3 ≤ x I I S(x3 )v3 (x3 , t) = S I v3 (x I , t) = S I I v3 (x I I , t).

(2.217)

The equation of Bernoulli (2.44) applied in x I ≤ x3 ≤ x I I between x I and x3 yields  x3  ∂ ∂ p I − p  (x3 , t) = ρ0 [φ(x3 , t) − φ I ] = ρ0 v3 dx3 ∂t ∂t x I  x3   SI ∂v3 (x I , t) = ρ0 dx3 . (2.218) S(x ) ∂t 3 xI −ikx3 ikx3 −ikx3 For harmonic waves pˆ I = p + + p− and pˆ I I = p + we have I e I e II e   −ikx I ikx I −ikx I I SI p+ − p− = SI I p+ I e I e II e

and

  −ikx I ikx I −ikx I I −ikx I ikx I p+ + p− − p+ = ik0 L eff p + − p− I e I e II e I e I e

with the effective length L eff defined by  L eff =

xI I xI

SI dx3 . S(x3 )

(2.219)

(2.220)

We find for the reflection coefficient R and the transmission coefficient T that R=

ikx I p− S I − S I I (1 − ik L eff ) I e + −ikx I = S I + S I I (1 + ik L eff ) PI e

T =

−ikx I I p+ II e + −ikx I . pI e

and (2.221)

These results reduce to the well-known result R = [(S I − S I I )/(S I + S I I )] and T = 2S I /(S I + S I I ) in the limit of k L eff → 0. For an ideal open pipe end at x I = 0 we find R = −1 by taking the limit S I /S I I → 0 and x I = 0 (Morse and Ingard 1968; Dowling and Ffowcs Williams 1983). For a closed pipe at x I = 0, we find R = 1 by taking the limit S I /S I I → ∞. When S I = S I I we recover the result for a diaphragm in a pipe R = ik L eff /(2 + ik L eff ) and T = 2/(2 + ik L eff ). In the limit k L eff → 0 the Green’s function g we are looking for is, for x I > y3 ,   c0 |x3 − y3 | g = H t −τ + 2 c0   x3 + y3 − 2x I c0 for x3 < x I , + R H t −τ + 2 c0

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and   x3 − y3 c0 g = T H t −τ − 2 c0

for

x I < x3 .

(2.222)

Considering the case x I > y3 while we neglect the effect of reflections R = 0 and T = 1, we can write for k L eff 1 along x I < x3 < x I I   x I + xeff − y3 c0 , (2.223) g = H t −τ − 2 c0 where

 xeff =

x3 xI

SI dx3 . S(x3 )

(2.224)

The effective coordinate xeff corresponds to the potential difference induced between x3 and x I by a flow having at x I a unit velocity v I (x I ) = 1. We can apply the same equations to the problem of an arbitrarily shaped discontinuity in a pipe if we replace the definition (2.224) of the effective position xeff by the more general definition xeff =

φ(x3 ) − φ I , v I

(2.225)

where we do not assume a quasi–one-dimensional flow at the discontinuity. This corresponds to the low-frequency Green’s function proposed by Howe (1975) for a discontinuity in a pipe. This generalization allows one to calculate the effect of a thin diaphragm in a pipe (Morse and Ingard 1968; Pierce 1981) or the reflection at an elbow (Bruggeman 1987) by using potential theory for incompressible flows. This approach is limited to compact regions but has the great advantage of providing a detailed model of the acoustical flow at such a discontinuity even in the neighborhood of sharp edges. Note that near a sharp edge an expansion of the solution in pipe modes fails to converge. Such theories are first-order approximations in a matched asymptotic expansion procedure (Lesser and Crighton 1975). The matched asymptotic expansion procedure permits a systematic approximation that can be extended to higher order in the small parameter k L eff . In the present case we have neglected terms of order (k L eff )2 . We take the inertia of the flow at the discontinuity into account but neglect the effect of compressibility. The same kind of approximation leads to the low-frequency Green’s function for a compact rigid body in free space proposed by Howe (1998, 2002). 2.5.4 Aeroacoustics of an open pipe termination 2.5.4.1 Introduction to open pipe termination acoustics

Modeling of the aeroacoustical behavior of internal flow involves in many cases an open pipe termination. This is not only a boundary condition for the calculations of the internal acoustical field but also a source of sound for the external acoustical field. In the presence of a main flow or at high amplitudes this involves a complex, unsteady flow at the pipe termination. This problem is often oversimplified and underestimated. At low

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frequencies, for example, the boundary condition p  = 0 is often used. In the presence of a main flow this corresponds to a quasi-steady response of the free jet outside the pipe. This model is quite reasonable to describe the reflection of acoustical waves. When, however, we consider the convection of vorticity, such a model induces spurious sounds and flow phenomena because vortices cannot be transported through such a boundary. Also such a model does not predict the spectacular effect of the shape of the nozzle edges on the aeroacoustical behavior of the pipe termination. Under particular flow conditions and for specific nozzle shapes, the pipe termination can be a source of sound. Coupling with acoustical resonances of the pipe system results in whistling. We give here a short survey of the aeroacoustical behavior of open pipe terminations. 2.5.4.2 Low-frequency linear behavior without main flow

We consider plane harmonic waves propagating in the x1 direction along a duct of uniform cross-sectional area A: p  = pˆ e iωt = ( p + e−ikx1 + p − e ikx1 ) e iωt .

(2.226)

These waves induce a volume flow at the open pipe termination in x1 with an amplitude φV = Avˆ1 = A

p+ − p− . ρ 0 c0

(2.227)

The boundary condition at the open pipe termination x1 = 0 for the internal acoustical field can be expressed in terms of a radiation impedance as follows:   p+ + p− pˆ = ρ0 c 0 + . (2.228) Zp = vˆ1 x1 =0 p − p− The acoustical power P radiated at the pipe termination is given by P = ( p  v1 )x1 =0 A = 12 vˆ1 vˆ1∗ A(Z p ),

(2.229)

where the asterisk denotes the complex conjugate. The real part of this impedance can be determined by applying the conservation of acoustical energy (Equation (2.101)) between the pipe exit and the far field outside the pipe. We consider first a thin-walled, unflanged pipe emerging into free space. In that case the far field will be dominated by the monopole radiation (Equation (2.84)): iωρ0 φV iωt−ikr iωρ0 Avˆ1 iωt−ikr . e e = (2.230) 4πr 4πr The corresponding radial velocity becomes, for kr  1, equal to vr  p  /(ρ0 c0 ). Together with Equation (2.230) this gives, for the power P radiated in the far field, p 

P = p  vr 4πr 2 

1 1 2 ρ0 c0 k 2 A2 vˆ1 vˆ1∗ . p  4πr 2 = ρ 0 c0 8π

(2.231)

Combining Equations (2.229) and (2.231) with the conservation of acoustical energy yields (Z p ) = ρ0 c0

k2A . 4π

(2.232)

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When we consider a pipe emerging from a hard wall into a half-infinite free space (flanged pipe), the radiation energy is distributed over a surface 2πr 2 rather than 4πr 2 . Furthermore, the acoustical amplitude is increased by a factor of two owing to reflection of waves at the wall. The combination of both effects results in an increase of (Z p ) by a factor of two compared with the unflanged pipe termination case (Equation (2.232)). Further confinement of the pipe outlet will further increase the radiation impedance. The imaginary part (Z p ) of the radiation impedance takes the inertia of the acoustical flow outside the pipe into account. This effect is often expressed in terms of an end correction δ: (Z p ) = ρ0 c0 kδ.

(2.233)

This is the length of a pipe segment that has the same inertia as the outer flow. In first√ order approximation in the small parameter k A/π , the reflected wave p − seems to be generated from a reflection of the incoming wave p + , without phase shift, from an ideal open pipe termination with p  = 0 placed at x1 = δ. The pipe behaves acoustically as if it were longer by a length δ if we assume that it is terminated by ideal open ends. In contrast to (Z p ), which is determined by global flow properties, the end correction δ of an open pipe termination is sensitive to the details of the local flow around the pipe outlet. For a pipe with circular cross section A = πa 2 and for infinitely thin walls, Levine and Schwinger (1948) found in the low-frequency limit δ  0.61a. For thick walls the end correction increases gradually from δ = 0.61a up to the flanged pipe limit δ = 0.82 (Ando 1969; Peters et al. 1993; Nederveen 1998; Dalmont, Nederveen, and Joly 2001). Please note that the order of magnitude of the radiation impedance of an unflanged pipe is found by considering the low-frequency limit of the radiation impedance Z s of a compact sphere of radius a. Following Equations (2.82) and (2.83), we have Z s = ρ 0 c0

ika + (ka)2  ρ0 c0 (ika + (ka)2 + . . . ). 1 + (ka)2

(2.234)

The simple behavior just discussed is typical for a radiation into a three-dimensional free field. Confinement will drastically affect the radiation impedance. If we consider, for example, the radiation of a pipe termination emerging between two closely spaced parallel hard walls, the radiation field will be two-dimensional. The radiation impedance has a complex behavior in such a case (Lesser and Lewis 1972). One can not, for example, identify a locally incompressible near field of the pipe termination. In general the radiation impedance will be much larger than in a three-dimensional case. It is quite tempting to use a two-dimensional flow simulation to describe the oscillating grazing flow along a wall cavity. It is, however, essential to realize that, with this twodimensional calculation, we will dramatically overestimate the radiation losses of the flow. The results of the calculations will actually depend on the size of the calculation domain.

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2.5.4.3 High-frequency linear behavior without main flow

For high frequencies, any incident mode of amplitude A, frequency ω, circumferential order m, and radial order μ is reflected in several propagating radial modes. Owing to cylindrical symmetry, no mode is reflected in another circumferential order. Outside the pipe we have in the far field pmμ (x, r )  ρ0 c02 ADmμ (ξ )

e−ik k

(k → ∞),

(2.235)

where x =  cos ξ , r =  sin ξ , and Dmμ (ξ ) is called the directivity function, and |Dmμ (ξ )| is the radiation pattern. Because each mode has its own spiraling phaseplane orientation, the radiated pattern consists of lobes interlaced by zeros. Each zero is found at the propagation direction of a reflected mode (that is, at ξ = arc sin(αmν /k)), except for the direction of the incident mode (that is, at ξmμ = arc sin(αmμ /k)). As may be expected, here we find the radiation maximum or main lobe. The field inside may be written as   ∞ 2 iωt−imθ −iκmμ x iκmν x + Rmμν Jm (αmν r ) e Jm (αmμr ) e , (2.236) p(x, r, θ, t) = ρ0 c0 A e ν=1

where the branch of the square root 2 κmμ =

 2 k 2 − αmμ

(2.237)

is chosen such that κmμ is either positive real or negative imaginary. The matrix Rm = {Rmμν } is called the reflection matrix. Explicit analytical expressions (in the form of complex contour integrals) are found by application of the Wiener–Hopf method (Levine and Schwinger 1948; Crighton et al. 1992). A subtlety in the solution of the scattering problem is the duct edge. Here, the boundary condition is, technically speaking, not applicable, and without further condition we have no unique solution owing to possible point or line sources “hiding” at the edge. So we have to add the so-called edge condition of finite energy in any neighborhood of the edge. This amounts to an integrable squared velocity field ∼ |∇ p|2 . It transpires that p varies near the edge like the square root of the distance. 2.5.4.4 Influence of main flow on linear behavior at low frequencies

Flow has a dramatic effect on the radiation impedance Z p of an open pipe termination. We consider the effect of a uniform subsonic flow velocity U0 in the pipe. In this section we limit our discussion to low frequencies. By low frequencies we do not only mean that we limit ourselves to plane-wave propagation in the pipe, but we also assume that the flow at the pipe termination is locally quasi-steady. This corresponds to a low Strouhal number limit St = ωa/U0 1. Because the Strouhal number is related to the Mach number M = U0 /c0 by St = ka/M, the combination of St 1 with M < 1 implies that we consider very long wavelength ka 1.

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The acoustic field in the pipe is described by the convective wave equation   ∂ ∂ 2  ∂ 2 p + U0 p − c02 2 = 0. (2.238) ∂t ∂ x1 ∂ x1 The solution of this wave equation is p  = p + e iωt−ik

+

x1

+ p − e iωt+ik

−

x1

(2.239)

with k± =

ω . c0 ± U0

(2.240)

The fluctuations v1 in the flow velocity are obtained by applying the linearized momen∂ + U0 ∂∂x1 )v1 = − ∂∂x1 p  : tum equation ρ0 ( ∂t v1 =

1  + iωt−ik + x1 −  − p − e iωt+ik x1 . p e ρ 0 c0

(2.241)

Following the vortex sound theory (Howe 2002), the acoustic field is most efficiently described in terms of fluctuation B  = U0 v1 + p  /ρ0 of the total enthalpy B = i + v12 /2: B  = B + e iωt−ik with ±

B =p

±

+

x1

+ B − e iωt+ik



U0 1± c0

−

x1

(2.242)

 .

(2.243)

We consider a flow leaving the pipe. As the result of flow separation at the pipe outlet, a free jet will be formed. It can be demonstrated that the only possible subsonic jet flow is one in which the pressure is uniform and equal to the pressure in the surroundings: p = patm (Shapiro 1953). Following this model, we have the boundary condition p  = 0 at the open pipe termination x1 = 0. This implies a pressure reflection coefficient p− = −1 p+

(2.244)

and a total enthalpy reflection coefficient p − c0 − U 0 c0 − U0 B− = =− . + + B p c0 + U 0 c0 + U 0

(2.245)

This result implies a loss of acoustical energy. In the free jet the kinetic energy of the flow will be dissipated without any pressure recovery. Because an acoustical modulation of the flow at the pipe termination implies a modulation of the kinetic energy in the jet, one can expect an absorption of acoustical energy. The vorticity in the shear layers of the free jet acts as a source of sound. The end correction appears to be determined by details of the flow just outside the pipe. As we will see in the next section, in the limit of very low Strouhal numbers for an unflanged circular pipe, δ  0.2a (Rienstra 1981b, 1983; Peters et al. 1993) which is a rather unexpected result.

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The preceding model corresponds to the very simple boundary condition p  = 0 at the pipe outlet x1 = 0. Although this model appears to be quite accurate in the limit St 1, it should be used with care. A vortex with a dimension of the order of the pipe diameter corresponds to a perturbation with a Strouhal number of order unity. A boundary condition of uniform pressure at a pipe termination will not allow such a vortex to flow out of the pipe! This is of course in contradiction to the experimental observations that vortices do leave the pipe through an open pipe termination. In the case of inflow, we can have a potential flow. In such a case the radiation impedance for low frequencies will not be strongly affected by the flow. If flow separation occurs, one can consider the use of a quasi-steady model (van Wijngaarden 1968). It is interesting to note that a diaphragm placed at the pipe outlet can be used as a nonreflecting pipe termination at a critical Mach number M = U0 /c0 . This anechoic pipe termination behavior was predicted by Bechert (1980) using a quasi-steady flow model. Applying the quasi-steady equation of Bernoulli for an incompressible flow between the pipe termination x1 = 0 with cross section A and the free jet of cross section A j formed at the diaphragm, we have 1 ρ (U0 2 0

+ v1 )2 + p(0, t) = 12 ρ0 (U0 + v1 )2

A , Aj

(2.246)

where we assume that in the jet the pressure remains constant, p j = 0. Substitution of p  = p + + p − and v1 = ( p + − p − )/(ρ0 c0 ) yields after linearization   U 0 A2 − 1 1 − + 2 c0 A p ,  2j =− (2.247) − A p 1 + Uc00 A 2 − 1 j

which vanishes for U0 /c0 = ((A/A j )2 − 1)−1 . Generalization of this result to arbitrary subsonic Mach numbers is discussed by Hofmans et al. (2000). Application to the design of silencers is discussed by Durrieu et al. (2001). When the Mach number is increased, we will eventually reach a critical flow at the diaphragm (choked flow, unit Mach number at the diaphragm). In this case the Mach number of a steady flow in the pipe is imposed by the ratio A/A j of pipe and diaphragm cross sections. It will in a quasi-steady theory remain constant, independent of any acoustical perturbation. The condition c v1 = U0 c0

(2.248)

combined with the assumption of an ideal gas behavior ( p  / p0 = γρ  /ρ0 and c02 = γ p0 /ρ0 ) yields (Anthoine, Buchlin, and Hirschberg 2002) p 2 .  = ρ 0 c0 v1 M(γ − 1)

(2.249)

A more elaborate discussion of the acoustical response of a choked nozzle is provided by Marble and Candel (1977).

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2.5.4.5 Influence of main flow on linear behavior at high frequencies

The most complete model available of sound radiation from a flow duct with jet that is analytically tractable is the following. It is a semi-infinite hard-walled duct with walls of zero thickness containing inviscid uniform mean flow, density, and sound speed, the values of which may differ inside and outside the duct or jet. The interface between jet and outer medium is modeled as a vortex sheet. Its analytically exact but mathematically rather involved solution of the Wiener–Hopf type was given by Munt (1977, 1990). The solution contains all the elements of the no-flow solution such as reflection into radial modes and a radiation pattern with mode-related lobes, but the presence of flow adds several particular features. Of course, there are geometrical effects like a redirecting of the radiation pattern and refraction across the jet interface, but there are also two effects due to coupling with the mean flow. First, for a velocity different across the vortex sheet, the jet is unstable. This instability is mathematically recognizable by means of a causality analysis in the complex frequency plane. Second, there is the edge condition, which is even more subtle than without flow (see Section 2.3.10.3). The condition of finite energy is still necessary to select physically possible solutions but is now not enough for a unique solution. We still have a choice corresponding to the amount of acoustic vorticity shed from the trailing edge. The possibility of vortex shedding is included in the model, but its amplitude is not yet fixed because it is determined by viscous and nonlinear processes that are not included. We have to add an extra condition. One such condition is full regularity of the field at the trailing edge, the Kutta condition, which corresponds physically to the maximum amount of vortex shedding possible. Any other non-Kutta–condition solution will be singular at the edge, but only one corresponds to no vortex shedding. Because it is the shed vorticity that excites the jet instability, the strict absence of the instability is a way to apply the condition of no vortex shedding for jets. If the mean flow is uniform everywhere, the absence of vortex shedding is most easily typified by a continuous potential. An alternative way to characterize the Kutta condition is via the streamline emanating from the edge given by r = a + (h(x) exp(iωt − imθ)). Without the Kutta condition, its shape for x ↓ 0 is like h(x) = O(x 1/2 ). With the Kutta condition, it is like h(x) = O(x 3/2 ). The Kutta condition seems to be the relevant condition for relatively low frequencies and acoustic amplitudes and high Reynolds numbers (Crighton 1985). Therefore, it will be the condition that is supposed here throughout. As governed by the mean flow Reynolds number, dimensionless frequency, and amplitude, other conditions (generalized Kutta conditions) are also possible (Rienstra 1981a, 1984). Physically, the shed vorticity, while moving near the edge of the solid duct wall, produces some additional sound and thus adds some acoustic energy to the sound field. At the same time a certain amount of acoustic energy and a certain amount of mean flow energy are needed for the creation of vorticity. Usually the net profit of energy is negative (acoustic energy is lost to the hydrodynamic vorticity), but this is not necessary (Rienstra 1981a).

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Very interesting effects of vortex shedding occur for low frequencies, and thus we will consider results of Munt’s solution in more detail in the next section. 2.5.4.6 Frequency dependence of the effect of flow on the radiation impedance

The experiments by Bechert, Michel, and Pfizenmaier (1978) showed for the first time a dramatic loss of acoustic energy when a long plane wave from upstream of a jet exhaust was partially reflected and partially transmitted at the exit. Although only a part was reflected, practically nothing was recovered in the radiated field outside. This was explained by Howe (1979) for low Mach numbers by showing that, owing to the presence of mean flow, the sound field sheds vorticity from the edge in such way that acoustic energy is converted into the acoustically undetected hydrodynamic energy. Cargill (1982) then showed that the Munt model (Munt 1977) includes all the reported effects for any Mach number. This makes the Munt model extremely interesting for studying various aspects of sound field mean flow coupling. Let us consider the predictions of the Munt model for plane waves of relatively low wave number k in a duct of radius a with Mach jet number M j . The feature that is directly related to the energy absorption is the modulus of the pressure reflection coefficient |R|. With the Kutta condition, it tends to unity for low Helmholtz number ka, which yields for the transmitted power in the duct PT ∼ (1 + M j )2 − |R|2 (1 − M j )2 – a finite value in contrast to the vanishing radiated value of O((ka)2 ). The difference is the energy lost in the vortices. Apart from the modulus reflection coefficient (Munt 1990), few results are reported from the Munt solution pertaining to the low-frequency range. Therefore, we present here some recent results obtained by in’t Panhuis (2003). A rather remarkable feature is the behavior of the end correction for a jet without coflow. We define the end correction as the virtual point just outside the duct exit against which the plane wave appears to reflect with a free-field boundary condition: the modulus of the plane-wave pressure | p| attains a minimum. If we write R = −r e−iθ , then the plane-wave pressure is given by −i

p(x) ∼ e and so

kx 1+M j

−r e

kx i 1+M −iθ j

,

 2k M x j −θ , | p(x)|2 ∼ 1 + r 2 − 2r cos 1 − M 2j

(2.250)

which attains its minimum for x = δ, where   1 − M 2j θ δ = . a 2ka

(2.251)

(2.252)

A marked difference is the nonuniform limit of the end correction for ka → 0 and M j → 0. As long as the Strouhal number ka/M j is large, it converges to the no-flow

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δ ⁄a

ka Figure 2.6. The end correction for no flow (M j = 0) and a little flow (M j = 0.01). Note the nonuniform behavior for ka → 0, M j → 0.

value ∼ 0.6127, but once ka M j , the Strouhal number becomes small and the behavior changes completely: the final value will be ∼ 0.2554. Therefore, results for low Mach number and low Helmholtz number are usually better presented on the basis of the Strouhal number. See Figure 2.6, where Munt’s solution of very low Mach number is compared with the Levine and Schwinger (1948) results for no flow. This nonuniform limit was predicted by Rienstra (1981b, 1983) and actually experimentally confirmed more than 10 years later by Peters et al. (1993). A pressure reflection coefficient larger than 1 for ka between 0 and, say, 1 was already reported by Munt (1990). In Figure 2.7 the reflection coefficients together with the corresponding end corrections are given for various Mach numbers (no flow outside, no density or sound speed difference). Note that the limiting values for ka → 0 of |R| are unity for any Mach number; the end corrections converge to 0.2554(1 − M 2j )1/2 . Finally, in Figure 2.8 the effect of coflow is shown. When the outer Mach number Mo varies from zero to the jet Mach number M j , both reflection coefficient and end correction return to a behavior similar to the no-flow case. 2.5.4.7 Whistling

As determined by the shape of the nozzle, a pipe termination can become a source of sound. This is a familiar phenomenon. By adjusting our lips and blowing at a critical velocity, we can whistle. Sound production by such a smoothly curved pipe outlet was first studied in laboratory experiments by Blake and Powell (1986) and by Wilson et al. (1971). We propose a qualitative model of this phenomenon in terms of vortex sound theory. The same model has been used to explain the behavior of a whistler-nozzle (Hill and Greene 1977; Hirschberg et al. 1989; Howe 1998). The model has also been applied to other configurations in which self-sustained oscillations of the flow can be described in terms of discrete vortex shedding. Examples of such phenomena are

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1.2 1.1 1

0.1

0.01

0.2

0.3

0.4

0.5

0.6

0.9 0.8 0.7 0.6 0.5 0.4

0

0.5

1

1.5

0.01 0.6 0.1 0.5 0.2 0.3 0.4

0.4 0.5 0.6

0.3

0.2

0.1

0

0.5

1

1.5

Figure 2.7. Plane-wave reflection coefficient |R| and end correction δ at jet exhaust without coflow for M j = 0.01, 0.1, . . . , 0.6.

the edge-tone (Holger, Wilson, and Beavers 1977), the Helmholtz resonator (Nelson, Halliwell, and Doak 1983; Mongeau et al. 1997; Ricot et al. 2001; Dequand, Luo et al. 2003), the splitter plate (Welsh and Stokes 1984), the closed side branches (Bruggeman et al. 1991; Ziada and B¨uhlmann 1992), the flute (Verge et al. 1997a, 1997b; Dequand, Willems et al., 2003), the wall cavities (Thompson, Hourigan, and Welsh 1992; Howe 1998), the diffusors (van Lier et al. 2001), and the solid-propellant rocket engine (Anthoine et al. 2002). Other examples of related self-sustained flow oscillations are described in many review papers and textbooks: Rockwell (1983), Rockwell and Naudascher (1978), Blake (1986), and Blevins (1990). Let us start our discussion by reconsidering the acoustical response of an unflanged pipe termination in the case of a uniform main flow U0 directed toward the pipe outlet. The pipe cross section has a radius a. Flow separation at the sharp edges of the pipe

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1.2 1.1 0.5 0.75

1

0 0.25

1

0.9 0.8 0.7 0.6 0.5 0.4

0

0.5

1

0.5

1

1.5

0.65 0.6 1

0.55 0.75 0.5 0.50 0.45 0.25

0.4 0.35

0 0.3 0.25 0.2

0

1.5

Figure 2.8. Plane-wave reflection coefficient |R| and end correction δ at jet exhaust with M j = 0.3 and coflow velocities Mo/M j = 0, 0.25, 0.5,0.75, 1.

outlet results in the formation of a free jet. We consider the reflection of harmonic plane waves with a frequency f traveling down the pipe in the direction of the main flow. The acoustical flow uac is defined following the vortex sound theory (Howe 1980) as the unsteady part of the potential flow component of the flow field. This potential flow bends around the sharp edges of the pipe outlet, as illustrated in Figure 2.9. In a potential flow, the centripetal force on the fluid particles, allowing a curved stream line, is provided by the pressure gradient. The corresponding acceleration is −|uac |2 /R, where R is the radius of curvature of the stream line. This implies that the pressure decreases toward the interior of the bend. As a consequence the velocity should increase. This is easily verified in a quasi-steady approximation by applying the equation of Bernoulli. At a sharp edge, the radius of curvature of the streamlines

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Figure 2.9. Acoustic flow at a pipe outlet (a) for an unflanged pipe termination and (b) for a horn.

in the potential flow vanishes. This implies a singular behavior. The pressure becomes infinitely low and the velocity infinitely large. The singular behavior of the flow at a sharp edge implies that viscous forces can never be neglected at such an edge. The result is that flow separation occurs, which induces the shedding of vorticity. The vorticity is such that it compensates the potential flow singularity. Within the framework of a frictionless theory, this is the so-called Kutta condition (Crighton 1985). At moderate acoustical amplitudes |uac |/U0 1, the amount of vorticity generated by the acoustic field will be negligible compared with the vorticity shedding induced by the main flow. The acoustic field does, however, trigger an instability of the shear layers of the free jet at the pipe exit. This instability results in the concentration of the vorticity into coherent structures that can be described as vortex rings. From experiments it appears that a new vortex starts to be formed at the pipe outlet at the beginning of each acoustical oscillation period, when the acoustical velocity turns from pipe inwards to pipe outwards. We call this t = 0. The vortices travel along the pipe axis with a convective velocity Uc , which is about half the main flow velocity U0 . The circulation of the vortex ring increases almost linearly in time as it accumulates the vorticity shed at the edge (Nelson et al. 1983; Bruggeman 1987; Dequand, Willems et al. 2003). After one oscillation period t = T , a new vortex is shed and the vortex ring travels farther downstream. Following the theory of Howe (1980), the acoustical power generated by the vorticity is given by fc ·uac = −ρ(ω×v)·uac .

(2.253)

This theory predicts that the vorticity field will initially absorb energy from the acoustic field. This seems logical inasmuch as the acoustic field is perturbing the shear layer. This initial absorption is strong because, near the edge of the pipe exit, the acoustical velocity is normal to the convective velocity v = (Uc , 0, 0). Furthermore, the acoustical velocity is large because of the singular behavior near the sharp edge. The theory of Howe (1980) predicts that, after half an oscillation period T /2 < t < T , the vortex will produce sound because the acoustic flow is reversed whereas the signs of the convective velocity and the vorticity do not change. In spite of the growth of the vortex, the sound production will be weaker than the initial absorption that occurred for 0 < t < T /2.

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This occurs because the vortex moves away from the edge singularity. Both the magnitude of the acoustical velocity and the angle between the vortex path and the acoustical streamlines decrease. This discussion has provided a qualitative understanding of the sound absorption predicted by linear theory in the previous sections. An important feature of this discussion is that it has stressed that the sound absorption is a balance between the initial absorption for 0 < t < T /2 and the sound production for T /2 < t < T . This implies that it should be possible to obtain a net sound production by reducing the initial absorption. This is done by using a pipe with rounded edges like our lips or a horn. The singular behavior at the flow separation point is suppressed. Furthermore the convective velocity near the flow separation point is almost parallel to the potential flow lines of the acoustic field. This reduces the initial absorption so much that at critical flow velocities sound production is observed. Typically one finds an optimal sound production when the travel time of the vortices along the lips corresponds to an oscillation period of the acoustic field. Experimentally, one observes this for a Strouhal number based on the radius of curvature R of the lips given by St R = f R/U0 = 0.2. When acoustical energy can accumulate in a resonant mode of the pipe system upstream of the pipe termination, one can observe self-sustained oscillation, which is referred to as whistling. This occurs when the sound production is large enough to compensate for viscothermal losses and radiation losses of the resonator. It is important to realize that in such a case the oscillation amplitude is limited by nonlinear effects. This can be an increase of losses at high amplitudes or a saturation of the source. In the particular case of whistling, the main nonlinear-amplitude-limiting effect is the saturation of the circulation of the vortex rings. Once the vortex has accumulated all the vorticity available in the shear layer, its circulation has reached a maximum value. In the case of a flute, the main nonlinear-amplitude-limiting mechanism is additional shedding of vorticity at sharp edges such as the labium (Verge et al. 1997a). The present example also illustrates that sound production does not necessarily involve impingement of a vortex or shear layer on a sharp edge. This qualitative discussion indicates that, for Strouhal number of order unity, a numerical flow simulation of a pipe system should include a model of the dynamical response of the free jet formed at a pipe outlet. Simple boundary conditions such as assuming a constant outlet pressure are only reasonable at low Strouhal numbers.

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Theoretical Background: Large-Eddy Simulation Pierre Sagaut

3.1 Introduction to large-eddy simulation 3.1.1 General issues

This chapter is devoted to large-eddy simulation (LES) of turbulent flows. The framework is restricted to Newtonian, single-phase, nonreactive fluids without external forcing or coupling as in magnetohydrodynamics. The primary approximations for unsteady simulations of turbulent flows are the following: r Direct numerical simulation (DNS), which is based on the direct resolution of the

full, unsteady Navier–Stokes equations without any additional physical assumptions or models. To get reliable results, one must represent all the dynamically active scales of motion in the simulation. This means that the grid spacing x and the time step t must be fine enough to capture the dynamics of the smallest scales of the flow down to the Kolmogorov scale, referred to as η, and that the computational domain must be large enough to represent the largest scales. These criteria lead to a high computational cost, which is responsible for the fact that DNS is nowadays almost only used for theoretical analysis and accurate understanding of flow dynamics and is not a “brute force” engineering tool. r Averaged or filtered simulations: To reduce the complexity of the simulation (and then lower the computational effort), a classical technique is to apply an averaging or filtering procedure to the Navier–Stokes equations, yielding new equations for a variable that is smoother than the original solution of the Navier–Stokes equations because the averaging or filtering procedure removes the small scales or high frequencies of the solution. Because it is smoother, the smallest scales are no longer of the order of the Kolmogorov scale but are comparable with a cutoff length scale  > η (to be discussed in Section 3.2.2), resulting in a lower complexity. As a consequence, the averaged or filtered simulation can be interpreted as a DNS of a new fluid flow with a modified constitutive law. The high frequencies are no longer captured by the computation, but their action on the resolved scales (through the nonlinear terms appearing in the equations) must be taken into account via the use of 89

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a statistical model. The most popular averaging- and filtering-related methods are the following: – Reynolds-averaged Navier–Stokes (RANS), which relies on a statistical average. The ergodic theorem says that procedure can be asymptotically interpreted as a time-averaging procedure (if the solution fulfills some given requirements), leading to steady computations in the general case. Unsteady RANS can also be obtained when the statistical average is related to a conditional or phaseaveraging procedure, or both. Note that the RANS approach does not permit explicit control of the complexity of the simulation because the cutoff frequency can not be specified during the averaging procedure. – Large-eddy simulation (LES), which is based on a filtering operator, leading to unsteady 3D computations. The filtering procedure (to be detailed in Section 3.2) can be explicit (i.e., associated with the application of a convolution filter to the DNS solution) or implicit (i.e., imposed by numerical errors, the computational mesh, or modeling errors) or even a blending of these two possibilities. 3.1.2 Large-eddy simulation: Underlying assumptions

Large-eddy simulation (Rogallo and Moin 1984; Sagaut 2002) is based on a scale separation, the smallest scales of the exact solution being parametrized via the use of a statistical model referred to as a subgrid-scale model. The removal of the highest frequencies is carried out considering the following assumptions, which are derived from the local isotropy hypothesis formulated by Kolmogorov (1941): r Large scales of the flow

– characterize the flow (i.e., drive its dynamics). In particular, the driving physical mechanisms, which are responsible for transition to turbulence and production of the turbulent kinetic energy, are carried by the large scales. – are sensitive to boundary conditions and so are anisotropic. – contain the main part of the total fluctuating kinetic energy (a minimum threshold of 80 to 90% is commonly agreed). r Small (subgrid) scales of the flow – have a universal character and are isotropic. This corresponds to the local isotropy hypothesis of Kolmogorov. – are only responsible for the viscous dissipation. No driving mechanism should be associated to that range of scales. – are weak. They contain only a few percent of the total kinetic energy. If these hypotheses are verified, one can expect that retaining only the large scales of the flow will make it possible to reduce the cost of the computation while still capturing the desired features of the flow. It is important to note that these features of large and small scales can be seen as a physical definition of these two families of scales of motion. This definition is to be compared with the previous one, which is based on the definition of an arbitrary cutoff length . These two definitions are not contradictory

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in practice because experience shows that LES yields satisfactory results when  is chosen so that it is compatible with the physical criteria.

3.2 Mathematical models and governing equations 3.2.1 The Navier–Stokes equations

The discussion here is restricted to the flows of compressible and incompressible Newtonian fluids governed by the Navier–Stokes equations. Effects of stratification, buoyancy, combustion, or magnetic forces introduce new physical phenomena and characteristic scales, which will not be considered here. It is worth noting, however, that they increase the simulation complexity in degree rather than kind. For compressible flows, the resulting mathematical model is ∂ρ + ∇ · (ρu) = 0 ∂t

(3.1)

∂ρu + ∇ · (ρu ⊗ u + p) = ∇ · τv (3.2) ∂t ∂E (3.3) + ∇ · (u(E + p)) = ∇ · (uτv ) − ∇ · qT , ∂t where ρ, ρu, and E are the density, the momentum, and the total energy, respectively. The ⊗ symbol refers to the tensorial product – that is, (u ⊗ v)i j ≡ u i v j . The viscous stress tensor and the heat flux are referred to as τv and qT , respectively. The system is closed assuming that the fluid is a perfect gas: p = ρ RT , T being the temperature and R the perfect gas constant (R = 287.6 J kg−1 K−1 for air). If temperature-dependent viscosity and diffusivity are considered – that is, μ = μ(T ) and κ = κ(T ) – then the viscous stresses are nonlinear functions of the variables. Viscous stresses and heat flux are expressed as follows using the Stokes assumption:   2 (3.4) τv = μ ∇u + ∇T u − (∇ · u)Id 3 qT = −κ∇T,

(3.5)

where Id is the identity tensor. Temperature-dependent viscosity and diffusivity are usually computed using the Sutherland empirical law:  T 1 + S/273.15 (3.6) μ(T ) = μ(273.15) 273.15 1 + S/T with μ(273.15) = 1.711 10−5 kg m−1 s−1 and S = 110.4 K (for air). For temperatures lower than 120 K, the Sutherland law is extended as T . (3.7) 120 Diffusivity κ(T ) is obtained assuming the molecular Prandtl number Pr constant. Its value is 0.7 for air at ambient temperature. μ(T ) = μ(120)

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In the incompressible case, the system (3.1)–(3.3) simplifies as ∇·u =0 ∂u + ∇ · (u ⊗ u) = −∇ p + ν∇2 u. ∂t

(3.8) (3.9)

The energy is totally decoupled from the continuity and momentum equation, and thus the temperature can be treated as a passive scalar if stratification and dilatational effects are neglected: ∂T + ∇ · (uT ) = κ∇2 T. ∂t

(3.10)

3.2.2 The filtering procedure 3.2.2.1 Definition

As previously mentioned in Section 3.1.1, the effective filter observed in practical LES computations originates from very different sources. This effective filter appears as a mixture of the following elements: r The grid filter: the choice of a computational grid with mesh size x induces the

definition of the maximum resolution of the computation. No scale smaller than the mesh size can be captured. According to the Nyquist theorem, the corresponding cutoff wave number is kc = π/x on uniform grids. r The theoretical filter, which is the filter to be applied to the exact solution of the Navier–Stokes equations to smooth it. This filter has a characteristic cutoff length . r The numerical filter: the numerical error, which is not uniformly distributed over the resolved frequencies, can also be interpreted as a filter. When local numerical methods such as finite element, finite volume, or finite difference methods are used to solve the governing equations, the numerical error is observed to be an increasing function of the wave number. Consequently, the dynamics of the highest frequencies resolved on the computational grid are only poorly captured, and these scales can be considered as being filtered. r The subgrid model filter (Magnient, Sagaut, and Deville 2001; Mason and Brown 1994; Mason and Callen 1986; Muchinsky 1996): During the computation, the only term that contains information related to the convolution filter is the subgrid model (through the use of the filter length scale  to define the subgrid viscosity, for example). But the exact filtered velocity is the solution of the filtered equations including the exact subgrid tensor, and we have to account for the fact that subgrid models are not exact, and introduce new errors, which modify the original filter. The theoretical filter is then transformed into a subgrid model filter. Very little is known at the present time about the numerical filter and the subgrid model filter. The emphasis is put here on the theoretical filter applied to the continuous Navier–Stokes equations. Leonard (1974) proposes to model it as the application of a

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convolution filter to the exact solution. The filtered part u of the variable u is defined by the following convolution operator (denoted by the symbol hereafter):  t  +∞ u(x, t) = G(, θ , |x − x  |, t − t  )u(x  , t  )d x  dt  −∞

−∞

= G(, θ) u(x, t),

(3.11)

where G(, θ, |x − x  |, t − t  ) is the kernel of the filter. The two arbitrary parameters  and θ are the cutoff length and the cutoff time, respectively. It can be proven that G must depend on the distance |x − x  | to preserve certain symmetries of the Navier– Stokes equations, such as the Galilean invariance (Fureby and Tabor 1997; Ghosal 1999; Oberlack 1997). The small-scale subgrid part u  is then defined as u  (x, t) = u(x, t) − u(x, t).

(3.12)

From the definition of u, one can see that u  will be different for each filter kernel. For the sake of simplicity, we restrict ourselves here to the case of an isotropic filter. We recall that a filter is said to be isotropic if both its mathematical expression and cutoff scales do not depend on the orientation of the Cartesian axes and the location in space. This implies that the domain is not bounded and that  and θ are constant in space and time. The extension for space-dependent cutoff scales and bounded domains is presented in Section 3.2.5. Almost all authors have considered spatial filtering only, resulting in G = G(, |x −  x |). Eulerian time-domain filtering has recently been revisited by Pruett (2000), who considered causal filters of the form G = G(θ , t − t  ). 3.2.2.2 Properties

To obtain tractable governing equations for LES, we assume that the filtering operator satisfies the following conditions: r Linearity:

u + v = u + v.

(3.13)

This property is automatically satisfied because we are considering the convolution filter (see Equation (3.11)). r Preservation of the constant:  +∞ a = a ⇐⇒ G(, |x − x  |)d x  = 1, ∀x. (3.14) −∞

r Commutation with derivatives:



 ∂ , G = 0, ∂s

s = x, t,

where the commutator is defined as [a, b](u) = a ◦ b(u) − b ◦ a(u).

(3.15)

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Table 3.1. Examples of usual spatial convolution filters Name Gaussian filter Sharp cutoff filter Box/top-hat filter

G(, |x − x  |) 2 2 6/π exp(−6|x − x  |2 / ) sin(|x − x  |kc )/(|x − x  |kc )

1/ if |x − x  | ≤ /2 0 otherwise

 G(k) 2

exp(−k 2  /24)

1 if k ≤ kc 0 otherwise sin(k/2)/(k/2)

kc = π/

A difference with the RANS∗ approach appears here – such filters, in the general case, are not Reynolds operators (i.e., they do not belong to the family of the projector operators): u = u,

u  = 0.

(3.16)

Note that projective filters can not be inverted because they are associated with an irreversible loss of information, whereas some filters verifying relation (3.16) can be inverted. In the latter case, the filtering procedure is nothing but a change of variables. The associated filtering operator in Fourier space can be written as  u (k, t),  u (k, t) = G(k)

(3.17)

 are the Fourier transforms of u and G (i.e., the transfer function of where  u and G(k) the filter), respectively. Expressions for three filters of common use are given in Table 3.1. The Gaussian filter possesses the two interesting properties of being quickly decaying and positive in both physical and Fourier space. The Gaussian and the box filter are called smooth filters because they do not lead to a sharp scale separation between resolved and subgrid scales in the Fourier space, resulting in a spectral overlap. This property will be of great importance for certain modeling strategies. The sharp cutoff filter is a projective filter, whereas the Gaussian and the box filters satisfy Equation (3.16). Typical spectra of the resolved and subgrid scales are displayed in Figure 3.1. 3.2.2.3 Other filtering procedures

Other filtering procedures, based on mixed spatial and time-domain filters (Dakhoul and Bedford 1986a, 1986b) or the projection of the exact solution on a finite basis of continuous functions have also been proposed (Hughes et al. 1998) but will not be detailed here because their use is less common. We will only mention the case of the scale selection through projection of the exact continuous solution onto a finite basis {φi (x, t)}, i = 1, . . . , N because such a projection is the cornerstone of all discretization methods based on Galerkin-type procedures. If we restrict ourselves to the common ∗

It is worth noticing that Osborne Reynolds, in his original 1884 paper, used a volume average that is equivalent to the so-called box filter in the LES framework.

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Figure 3.1. Schematic kinetic energy spectra of resolved and subgrid scales with spectral overlap (Gaussian filter).

case of time-independent basis functions, the exact solution u is transformed into  u i (t)φi (x), (3.18) u d (x, t) = d (u)(x) = i=1,N

where u i (t) are the coefficients of the decomposition and functions φi (x) may be chosen among a wide class of functions, ranging from piecewise constant functions (finite volume approach, finite element) to trigonometric functions (Fourier method) and highorder polynomials (pseudospectral methods) and proper orthogonal disposition (POD) modes (POD–Galerkin methods). The filtering procedure here is associated with the choice of the projection basis. The transfer function of this implicit filter is formally written as   =

φl (x), e2πikx  , (3.19) G(k) l=1,N

where the ·, · is related to an ad hoc scalar product in the domain . The dimension of the projection basis being finite, we have |G(k)| −→ 0 when |k| −→ ∞, enlightening the existence of a filter.

3.2.3 Governing equations for LES 3.2.3.1 First example: Generic conservation law

We first present the case of a generic nonlinear conservation law to show how LES governing equations are derived from their original unfiltered counterpart. Let us consider the conservation law ∂u + ∇ · F(u, u) = 0, ∂t

(3.20)

where the nonlinear flux function F is assumed to exhibit a quadratic behavior with respect to u. Applying the convolution filter to Equation (3.20), one obtains ∂u + ∇ · F(u, u) = 0. ∂t

(3.21)

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Using the commutation property (3.15), we find that this relation simplifies as ∂u + ∇ · F(u, u) = 0. ∂t

(3.22)

The last step consists in rewriting the filtered nonlinear flux F(u, u) as a function of the new filtered unknown, u. This is achieved writing u = u + u  and inserting this decomposition into the nonlinear term, yielding

∂u (3.23) + ∇ · F(u, u) = −∇ · F(u, u  ) + F(u  , u) + F(u  , u  ) , ∂t where all the terms that can not be directly computed from the known quantity u have been grouped in the right-hand side. These terms can not be exactly computed because they explicitely depend on u  , which is not contained in the LES problem inasmuch as it corresponds to scales that have been eliminated. Thus, the terms will be approximated through the use of a subgrid model, which is a function of u. 3.2.3.2 Incompressible flows

Applying a convolution filter to the Navier–Stokes equations for incompressible flows, and taking into account all the properties of the filter, one obtains the governing equations for LES: ∂u + ∇ · (u ⊗ u) = −∇ p + ν∇2 u ∂t ∇ · u = 0.

(3.24) (3.25)

This set of equations can not be directly used because the nonlinear term u ⊗ u must first be decomposed as a function of the only acceptable variables, which are u and u  . Two decompositions of the nonlinear term are now presented. 3.2.3.3 Leonard’s decomposition

A first decomposition was proposed by Leonard (1974). It is obtained by inserting the decomposition u = u + u  into the nonlinear term, yielding u ⊗ u = (u + u  ) ⊗ (u + u  ) = u  ⊗ u + u ⊗ u   + u  ⊗ u + resolved

C:Cross terms

(3.26) u   ⊗ u.

(3.27)

R:Reynolds stresses

The resolved term can be further decomposed as u ⊗ u = u  ⊗ u + (u ⊗ u − u ⊗ u) .    new resolved

(3.28)

L:Leonard stress tensor

The decomposition procedure makes three tensors appear as follows: r The Leonard stress tensor L, which corresponds to the fluctuations of the interac-

tions between resolved scales (zero for RANS).

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Table 3.2. Various decompositions for the nonlinear terms Form of the resolved convection term

Expression for τ

u⊗u u⊗u

C+R L +C + R

r The cross-stress tensor C, which accounts for direct interactions between resolved

and unresolved scales (zero for RANS).

r The subgrid Reynolds stress tensor R, which is associated to the action of subgrid

scales on the resolved field (Reynolds tensor for RANS). Two possibilities arise for the definition of the subgrid-scale tensor τ that depend on the choice of the formulation of the resolved convection term (see Table 3.2). These two decompositions can be used, but they introduce some interesting conceptual problems. Consider the basic philosophy of LES: filtered equations are derived, and classical wisdom tells us that all the terms appearing in the equations must be filtered terms (i.e., appear as the filtered part of something). Only the first decomposition τ = R + C satisfies that condition. This is especially true when the filtering operator is associated with the definition of a computational grid (and overbar just means “defined on the grid”): the convection term is computed on the same grid as the filtered variables and then should be written as u ⊗ u. 3.2.3.4 Germano’s consistent decomposition

Another possibility is to build a decomposition based on the generalized central moments of Germano (1986, 1987, 1992). The generalized central moment of two quantities u and v, for the filter kernel G, is defined as {u, v}G = G (uv) − (G u)(G v) = u v − u v.

(3.29)

Using that definition, we can write generalized subgrid tensors as L = {u, u}G ,

    C = u, u  G + u  , u G ,

  R = u, u G .

(3.30)

It can be easily verified that L + C + R ≡ L + C + R.

(3.31)

Germano named this decomposition the consistent decomposition because all the terms can be recast into the generalized central moments formalism. One can verify that each term of that decomposition is Galilean invariant.

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3.2.4 Extension for compressible flows 3.2.4.1 Definition of the filtered variables

We now discuss the extension of the filtering procedure, as previously defined for incompressible isochoric flows, to variable density and compressible flows. The definition of the filtered set of equations results from three preliminary choices: r The original set of unfiltered variables or equations. r The filter (same low-pass as in the incompressible case). r The set of filtered variables.

The first problem arises because a very large number of formulations of the compressible Navier–Stokes equations can be found that rely on the choice of basic variables for describing the flow. Velocity is usually described using velocity or momentum, but mass conservation and energy equations can be expressed using two independent variables among a large set: density, entropy, total energy, internal energy, enthalpy, and so on. The last point, which is the most difficult one when dealing with compressible flows, will be detailed now. Let us consider the set of the conservative variables (ρ, ρu, E). A straightforward use of the filtering procedure for incompressible flows leads to the definition of the filtered variables (ρ, ρu, E). The term ρu can be rewritten as ρu ≡ ρ u,

(3.32)

where  u = ρu/ρ is the mass-weighted filtered velocity. It is worthwhile noting that the  operator can not be interpreted as a convolution-like filtering operator, and it does not commute with derivatives. It corresponds only to a change of variable.∗ Here appears the problem of the choice or definition of the basic set of variables for the LES of compressible flows. It arises because ρu, E, and p are nonlinear functions of the other variables and then can be decomposed. On the basis of that choice, different subgrid terms will arise from the filtered equations, inducing a need for specific subgrid modeling. There is an infinite number of possibilities for the definition of the variables of the filtered problem, starting from (ρ, ρu, E). Two examples are as follows: r System I: as proposed by Comte and Lesieur (Ducros, Comte, and Lesieur 1996)

u , E) supplemented by the and Vreman (1995), the set of filtered variables is (ρ, ρ macropressure π and the macrotemperature θ . These two quantities are defined as 1 π = p + τkk , 3 ∗

2  + γ (γ − 1)M τkk ρ, θ=T 2

(3.33)

It is worth noting that this mass-weighting approach was proposed by Osborne Reynolds in his 1884 paper!

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where M is the Mach number. The associated filtered equation of state is 3γ − 5 (3.34) τkk . 6 The uncomputable term τkk requires a priori an adequate model. But some authors 2 p, where Msgs is propose to neglect it, arguing that it can be recast as τkk = γ Msgs the Mach number based on the characteristic velocity scale of the subgrid motion, which is generally very small compared with the mean flow Mach number M. Comte and Lesieur also argue that γ = 5/3 for monoatomic gases like argon or helium, leading to a vanishing contribution of the subgrid term in the filtered equation of state. r System II: as proposed in references (Sreedhar and Ragab 1994; Vreman 1995;  suppleu , E) Vreman, Geurts, and Kuerten 1997), the selected variables are (ρ,  . mented by the filtered pressure p and the mass-weighted filtered temperature T  is defined as The synthetic total energy E π = ρ Rθ −

= E

p u2 + ρ γ −1

(3.35)

and corresponds to the computable part of the total energy. Using this set of variables, we find that no subgrid contribution appears in the filtered equation of state. The choice of a system of governing equations is seen to change the closure problem by modifying the subgrid terms. The total complexity of the system being conserved, the choice depends essentially on the preference of the practioners. In the previous examples, system I will not give direct access to the pressure and the temperature, whereas system II does. Consequently, system II will be preferred for applications dealing with aeroacoustics and aero-optics. But system I is known to yield simpler energy equations and is more easily implemented. 3.2.4.2 Example of filtered equations

We now illustrate the problem by writing the set of filtered equations corresponding to system II. Simple algebra yields ∂ρ u ) = 0, + ∇ · (ρ ∂t

(3.36)

∂ρ u + ∇ · (ρ u u) + ∇ p − ∇ ·  τv = A 1 + A 2 , ∂t

(3.37)

 ∂E  + p)) − ∇ · ( u τv ) + ∇ ·  qT + ∇ · ( u( E ∂t = −B1 − B2 − B3 + B4 + B5 + B6 − B7 ,

(3.38)

where  τv and  qT are the viscous stresses and the heat flux computed from the resolved variables and are equal to their respective filtered counterparts if the viscosity

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and the diffusivity are assumed to be constant. The subgrid terms A1 , A2 , B1 ,. . . , B7 are defined as ⊗ u − u ⊗ u ), τ ≡ ρ(u B1 =

1 ∇ γ −1

A1 = −∇ · τ,

· ( pu − p u ),

B3 = ∇ · (τ u ),

(3.39)

B2 = p∇ · u − p∇ ·  u,

B4 = τ ∇ ·  u,

u − τv u ), B6 = ∇ · (τv 

A2 = ∇ · (τv −  τv ),

B5 = τv ∇u − τv ∇ u,

B7 = ∇ · (qT −  qT ).

(3.40)

Terms A2 , B6 , and B7 arise from the possible dependency of molecular viscosity and diffusivity on the temperature and correspond to a new nonconvective source of nonlinearity. The use of the mass-weighthed filtered variable  u instead of the filtered velocity u prevents the appearance of subgrid terms in the continuity equation.

3.2.5 Filtering on real-life computational grids

Only isotropic filters have been considered in Section 3.2.2. These filters can be defined, only on unbounded domains; on bounded domains, the filter kernel must be modified near the boundaries in order to remain definite, resulting in a violation of the homogeneity property. Another problem is that isotropic filters have a constant cutoff length scale and are not appropriate to describe what is occuring on general curvilinear grids. This lack of homogeneity will result in the loss of the property of commutation with the spatial derivatives, and the filtered Navier–Stokes equations, as written at the beginning of this chapter, are no longer valid. The general form of the commutation error for space-dependent cutoff length filters in a bounded fluid domain  can be written in the following compact form (Fureby and Tabor 1997):      ∂ ∂G ∂

φ , G (φ) = + G((x), x − x  )φ(x  )n(x  )ds, (3.41) ∂x ∂x ∂ ∂ where φ(x) is a dummy function and n(x) is the outward unit normal vector to the boundary of . The first term on the right-hand side arises from the fact that  varies in space, and the second term comes from the boundedness of the domain. Two possibilities arise to derive the filtered Navier–Stokes equations on general meshes: (i) Apply the filter to the basic equations written in Cartesian coordinates. (ii) Rewrite the filtered equations in general coordinates to solve them on general grids. or (i) Write the Navier–Stokes equations in general coordinates. (ii) Apply the filter.

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These two possibilities yield very different sets of governing equations for LES. They are described in the following section. 3.2.5.1 Filtering the equations in Cartesian coordinates

This solution was first proposed by Ghosal and Moin (1995), who addressed the problem of the definition of a filtering procedure well suited for the case of a variable cutoff length scale filter on a bounded domain [a, b]. In this section, Greek symbols are related to coordinates in an unbounded reference domain [−∞, +∞] (with uniform mesh spacing ), whereas Roman symbols refer to coordinates in the physical (bounded) domain. Let f be the coordinate transformation ξ = f (x).

(3.42)

This function is monotone and differentiable (with f  ≡ d f /d x), and f (a) = −∞,

f (b) = +∞.

(3.43)

Using that change of variable, one can associate a variable cutoff length scale δ(x) (defined on the physical domain) to the constant length scale  (defined in the reference space):  . (3.44) f  (x) Let ψ(x), x ∈ [a, b] be the function to be filtered. The next step is the definition of the filtering procedure on the physical domain. This is done using the following four-step algorithm: δ(x) =

(i) Redefine the convolution filter (in the isotropic case, i.e., on the reference space) as    ξ −η 1 +∞ φ(ξ ) = G φ(η)dη. (3.45)  −∞  (ii) Operate the inverse change of variable x = f −1 (ξ ) and define φ(ξ ) = ψ( f −1 (ξ )). (iii) Apply the isotropic filtering procedure (3.45) to φ(ξ ):    1 +∞ ξ −η ψ(x) ≡ φ(ξ ) = G φ(η)dη. (3.46)  −∞  (iv) Come back to the physical space (operate the inverse change of variable):    1 b f (x) − f (y) ψ(y) f  (y)dy. ψ(x) = G (3.47)  a  This procedure leads to the definition of second-order commuting filters (SOCF), 2 whose commutation error with spatial derivatives is O( ). It means that the classical, filtered Navier–Stokes equations, which were given in the previous sections, obtained using the commutation property, are only a second-order accurate approximation of the true filtered problem. One can demonstrate that a straightforward extension of the classical filtering procedure yields a zeroth-order accurate method.

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Vasilyev, Lund, and Moin (1998) have recently demonstrated that this method can be extended to arbitrary orders of accuracy by choosing a filter kernel G with an ad hoc number of vanishing moments. More precisely, the commutation error will be of order n O(δ ) if α(G, 0) = 1, α(G, k) = 0,

(3.48) 1 ≤ k ≤ n,

(3.49)

α(G, k) < ∞, k ≥ n with

(3.50)

 α(G, p) =

ξ p G(ξ )dξ.

(3.51)

3.2.5.2 Filtering the equations in general coordinates

The incompressible Navier–Stokes equations written in generalized coordinates read as follows: ∂ (J −1 ξik u i ) = 0, ∂ξ k ∂ −1 ∂ ∂ 1 ∂ (J u i ) + k (U k u i ) = − k (J −1 ξik p) + ∂t ∂ξ ∂ξ Re ∂ξ k

(3.52) 

J −1 G kl

 ∂ (u ) , i ∂ξ l

(3.53)

where ξ k are the coordinate directions in the transformed space, ξik = ∂ξ k /∂ xi , J −1 is the Jacobian of the transformation, G kl = ξik ξil denotes the contravariant metric tensor, u i the Cartesian components of the velocity field, and U k = J −1 ξik u i the contravariant flux across the ξ k = constant plane. Jordan (1999) proposed to operate the filtering in the transformed plane, leading to the following set of governing equations for LES: ∂ (J −1 ξik u i ) = 0, ∂ξ k ∂ −1 ∂ ∂ 1 ∂ (J u i ) + k (U k u i ) = − k (J −1 ξik p) + ∂t ∂ξ ∂ξ Re ∂ξ k −

(3.54) 

 ∂ J −1 G kl l (u i ) , ∂ξ

∂ (σ k ) ∂ξ k i

(3.55)

where the contravariant counterpart of the subgrid tensor is defined as σik = J −1 ξ kj u i u j − J −1 ξ kj u j u i = U k u i − U k u i .

(3.56)

The metrics being computed by a discrete approximation in practice, they can be considered as filtered quantities, yielding U k = J −1 ξ kj u j  J −1 ξ kj u j .

(3.57)

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Jordan (1999) proved that the commutation error vanishes if the variables are first transformed into computational space before being filtered because a uniform kernel can be defined in this space. For compressible flows, one obtains the following set of equations (it is assumed that metrics are filtered quantities):

∂ ∂ −1 J ρ + k J −1 ξ kj ρ u j = 0, (3.58) ∂t ∂ξ

  ∂ −1 ∂ u j + pδi j −  J ρ u i + k J −1 ξ kj ρ ui τv,i j = A1 + A2 , (3.59) ∂t ∂ξ   ∂ −1  ∂  + p) −  J E + k J −1 ξ kj  ui τv,i j +  qT, j u j (E ∂t ∂ξ = −B1 − B2 − B3 + B4 + B5 + B6 − B7 . The subgrid terms A1 , A2 , B1 , . . . , B7 are now defined as

  ∂ ∂ τv,i j , A1 = − k J −1 ξ kj τi j , A2 = k J −1 ξ kj τv,i j −  ∂ξ ∂ξ    ∂u ∂ u 1 ∂ −1 k  j j B1 = J ξ j pu j − p u j , B2 = J −1 ξ kj p k − p k , γ − 1 ∂ξ k ∂ξ ∂ξ   ∂u ∂ u ∂ −1 k j j u i k J ξ j τi j , B5 = J −1 ξ kj τv,i j k − τv,i j k , B3 + B4 =  ∂ξ ∂ξ ∂ξ     ∂ ∂ ui −  u i , B7 = k J −1 ξ kj qT, j −  B6 = k J −1 ξ kj τv,i j  τv,i j  qT, j . ∂ξ ∂ξ

(3.60)

(3.61) (3.62)

(3.63) (3.64)

We observe that the set of filtered equations in generalized coordinates is formally similar to the previous one, the main difference being that metrics are now intrinsically included in the definition of the problem. As a consequence, the derivation of physical subgrid models is more difficult (What is the physics of contravariant velocity components?), and the subgrid models used in practice are derived from subgrid models built using the set of filtered Navier–Stokes equations in Cartesian coordinates. 3.2.5.3 What is the filtering length scale on general meshes?

Many subgrid models explicitly account for the filtering length scale . In most of the computations,  is computed as a function of the mesh topology. For Cartesian grids, with “relatively” isotropic grids, the most popular formula is  = (xyz)1/3 . For more anisotropic grids, other empirical values are   = Max(x, y, z) or  = (x 2 + y 2 + z 2 )/3.

(3.65)

(3.66)

For curvilinear meshes, a commonly used rough estimate is  = (V )1/3

(3.67)

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for finite volume methods (where V is the volume of the control volume) and  = (J ξ ηζ )1/3

(3.68)

for finite difference methods (where J is the Jacobian, and ξ, η, ζ the grid spacing in the reference space). There is no general theory for defining the best approximation, and the choice is very often made to render the implementation as simple as possible. The optimization of the LES solution is then carried out by changing the value of the constant in the subgrid model and redesigning the computational grid (i.e., making it as isotropic as possible).

3.2.5.4 Discrete test filters

Many techniques and models related to LES involve the use of a second filtering level, which is applied to the resolved field. During the computation, that filter, referred to as the test filter, is approximated using some weighted linear combinations of the values of the resolved field. The two most popular one-dimensional discrete test filters are the following: 1 (φk+1 + 2φk + φk−1 ), 4 1 φ k = (φk+1 + 4φk + φk−1 ). 6 φk =

(3.69) (3.70)

Two- and three-dimensional filters are then obtained using a one-dimensional test filter in each√direction. Note that the characteristic length scales of these filters are not equal:  = 6x for the first one, and  = 2x for the second one. This difference must be accounted for when using the dynamic procedure, the best results being obtained  are the cutoff lengths of the LES  = 2 (Najjar and Tafti 1996), where  and  with  filter and the discrete test filter, respectively. The numerical experiments have shown that the subgrid models that rely on a test filter are almost all very sensitive to its discrete features. Another point when one deals with the dynamic procedure is that the best results are generally obtained when using test filters restricted to the homogeneous directions of the flow (but if at least two periodic directions are available). The use of three-dimensional test filters may reduce the efficiency of the dynamic procedure. The two filters presented above can be used on structured meshes. Similar filters can be designed on unstructured meshes by discretizing the differential operator associated with the targeted filter or by solving a linear system to compute the weighting coefficients of the neighboring points. Optimized discrete filters can also be derived, which yield the best least-squares approximation of a required transfer function over a frequency band (Sagaut and Grohens 1999; Vasilyev et al. 1998).

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Table 3.3. Resolution requirements referred to Kolmogorov length scale η used in DNS based on spectral methods of some incompressible homogeneous and wall-bounded flows Flow

Resolution in ηa

Boundary layer Homogeneous shear Isotropic turbulence

x  15, y  0.33, z  5b x  8, y  4, z  4 x  4.5, y  4.5, z  4.5

η at the wall is used for the wall-bounded flows. x, y, and z refer to the streamwise, wall-normal, and spanwise direction, respectively. Source: Adapetd from Moin and Mahesh (1998).

a

b

3.3 Basic numerical issues in large-eddy simulation Both DNS and LES have been implemented using all known numerical approaches, including vortex methods and Lattice Boltzmann methods. This section is devoted to basic issues that apply for all simulations based on finite difference, finite element, finite volume, and spectral (including spectral element) methods. This section is devoted to the presentation of two very important issues: the required grid resolution for obtaining reliable results, and the effects of numerical errors. In both cases, the results dealing with DNS will be recalled.

3.3.1 Grid resolution requirements 3.3.1.1 Homogeneous and free-shear flows

As mentioned in Section 3.1.1, the main constraint dealing with resolution in DNStype computations is that all relevant scales of the flow must be directly captured. This means that the computational domain must be large enough to accommodate the largest scales of the flow and that the mesh size must be small enough to represent the smallest dynamically active scales. In practice, the former condition means that the two-point correlation of the fluctuations must vanish within the domain in each direction of space. The latter corresponds to the fact that the viscous dissipation of kinetic energy must be captured. The viscous dissipation takes place in the wave-number band0.1 ≤ kη ≤ 1, corresponding to a length-scale band of about 6η to 60η, where η = ν 3 /ε is the Kolmogorov scale. Here, ε is the viscous dissipation rate of kinetic energy. As a consequence, the resulting criterion for DNS of turbulent flows is x ∝ η. This is also equivalent to requiring that the “wormlike” structures observed in isotropic turbulence, whose typical diameter is 4–10η, must be correctly represented on the grid. Examples of grid resolution for incompressible flows are displayed in Table 3.3. The computational effort required for DNS computations can be estimated from a priori knowledge about the physics of the flow. For isotropic turbulence, the ratio between the integral scale (characteristic of large 3/4 scales), L, and the Kolmogorov scale evolves as L/η ∝ Re L , where Re L = LU/ν is

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the Reynolds number based on L and U = 12 u i u i . Thus, the number of grid points, N , required to perform DNS of isotropic turbulence in a box of volume L 3 scales as √ 9/4 Re L . Let TL be the integral time scale of turbulence and Tη = ν/ε the Kolmogorov time scale. The resolution requirement is t ∝ Tη . Thus, the minimal number of time 1/2 steps is proportional to TL /Tη ∝ Re L . The minimum complexity of the simulation – that is, the number of times the Navier–Stokes equations should be solved – is then of 11/4 the order of Re L . Similar scaling laws can be derived for free-shear flows without shocks. The basic philosophy of LES is to use coarser grids than DNS while still correctly capturing the large-scale dynamics. As mentioned in Section 3.1.2, large scales can be physically defined as scales responsible for turbulent fluctuation mechanisms. Most subgrid models are efficient in representing the turbulent energy cascade only. Thus, the LES computational grid must be fine enough to resolve production of turbulence and transition to turbulence directly. For free-shear flows and separated-shear layers, this constraint is weakened because turbulence production is associated with scales whose size is of the order of the shear layer thickness δ. Consequently, a typical mesh size is δ/100 ≤ x ≤ δ/10, irrespective of the Reynolds-number value. 3.3.1.2 Wall-bounded flows

Wall-bounded flows induce more drastic constraints (Moin and Mahesh 1998; Piomelli and Balaras 2002; Rogallo and Moin 1984). Large scales are of the order of the boundary layer thickness in the outer layer of the boundary layer, and the relevant scaling for the turbulent motion responsible for the turbulence production in the inner layer is the √ viscous length ν/u τ , where u τ = τw /ρ is the friction velocity. In incompressible  boundary layers, channel flows, and pipe flows we have u τ ∝ U C f and C f ∝ Reα , where C f = 2τw /ρU 2 and U are the skin friction coefficient and the outer velocity scale of the flow, respectively. Experimental data yield α  0.2 − 0.25. The streaky structures (Figure 3.2) observed in the inner layer have constant spatial characteristics in wall units, and therefore the grid spacing must be of constant size in this coordinate system. Therefore, the number of grid points in each direction scales as −α/2

N=

L LU Re L L = ∝ + x x ν/u τ ν

1−α/2

∝ Re L

3(1−α/2)

.

(3.71)

∼ Re2.625 − Re2.7 The total number of grid points is then proportional to Re L L L . The size of the time steps t and the characteristic time of large events scale as −1/2 and L/U , respectively. The estimated number of time steps is then (L/u τ )Re L α+1/2 3.5−α/2 . The complexity of the simulation is then Re L ∼ Re3.4 of the order Re L L . Recommended values for the mesh size (expressed in wall units) are displayed in the first column of Table 3.4.

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Table 3.4. Typical mesh size (expressed in wall units) for DNS and LES of boundary layer flow

+

x (streamwise) y + (spanwise) Min(z + ) (wall-normal) Number of points in 0 < z + < 10

DNS

Wall-resolving LES

10–15 5 1 3

50–100 10–20 1 3

For wall-bounded flows, the size of the events responsible for the turbulent production is a function of the Reynolds number, and the mesh size for LES obeys the same scaling law as for DNS. Such LES simulations with a direct resolution of the dynamics of the inner layer are referred to as wall-resolving LES. Note, however, that LES makes it possible to achieve considerable savings by coarsening the mesh size: Values like x + ≈ 50 (streamwise direction) and y + ≈ 15 (spanwise direction) are usual and must be compared with their DNS counterparts. A more significant cost reduction might be achieved if a specific model for the inner layer, referred to as a wall stress model, were used. The wall-modeling LES approach is discussed in Section 5.2.3. 3.3.1.3 Sound waves

Compressible flows introduce new characteristic scales. The typical wavelength of acoustic waves in turbulent flows of interest is generally larger than the Kolmogorov scale, and thus the aerodynamic mesh is well adapted. This result is quickly recovered by considering a fluctuation with frequency ω f in time (the associated period is noted as T f = 2π/ω f ). If it is a turbulent hydrodynamic fluctuation, we have T f = L t /u t , where L t and u t are the corresponding length and velocity scale, respectively. Assuming that the considered turbulent fluctuation is located within the inertial range

Figure 3.2. Streaks in the inner layer of the boundary layer.

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of thekinetic energy spectrum and that the Kolmogorov hypotheses hold, we have u t = 2π E(2π/L t )/L t , yielding 

2/3 L t = T f u t = T f 2π E(2π/L t ) . (3.72) Now, considering an acoustic fluctuation, we have L a = T f (U + c), where U and c are the local mean flow velocity and the sound speed, respectively. The ratio of these two lengths is U +c U 1 La = = + , Lt uf uf Mf

(3.73)

where M f = u f /c is the Mach number associated with the fluctuation. In practical flows, we have M f  1 and |U |  |u f |, showing that the acoustic waves will be directly resolved on the LES grid. 3.3.1.4 Shock waves

Another problem associated with compressible flows is the presence of shocks. In this case, even the definition of DNS must be revised. The shock introduces a new length scale associated with its thickness referred to as δshock . The problem induced by the presence of a shock is twofold. First, the Navier–Stokes equations are relevant to describe the internal dynamics of the shock in the low-supersonic regime only (for Mach numbers M ≤ 1.2−1.3). Second, the thickness of the shock is usually much smaller than the Kolmogorov scale η. Moin and Mahesh (1998) estimate their ratio as M − 1 η ∼ 0.13 Reλ , (3.74) δshock Mt where Mt is the turbulent Mach number based on the square root of the turbulent  kinetic energy and Reλ the Reynolds number based on the Taylor microscale λ = 2ν u  2 /ε (brackets stand for the Reynolds average). For most realistic applications, one has η/δshock ≥ 10. As determined by the respective size of these length scales, several approximations can be defined, which are summarized in Table 3.5. It is observed that both quasi-DNS and LES are based on shock capturing. Here again, new problems arise: the control of the numerical dissipation (to be discussed in Section 3.2.2) and the required mesh size to capture shock–turbulence interaction. For shock–isotropic turbulence interaction, DNS performed by Lee, Lele, and Moin (1993, 1997) has shown that the mesh must be fine enough to capture the shock corrugation caused by turbulence to recover good results. This is in agreement with results of linear interaction approximation, which shows that the shock deformation plays an important role (Mahesh et al. 1995, 1997). An empirical criterion is x/y ∼ u rms /U , where x and y ≈ η are the mesh size in the shock-normal and shock-parallel directions, respectively. Velocity scales U and u rms are taken in the shock-normal direction. Published DNS results correspond to u rms /U = 0.1−0.2.

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Table 3.5. Definition of simulation types for compressible flows Casea

Shock treatment

Turbulence treatment

Simulation type

x < δshock < η δshock < x < η δshock < η < x

shock-resolving shock-capturing shock-capturing

direct simulation direct simulation subgrid modeling

DNS quasi-DNS LES

a η,

δshock , and x are the Kolmogorov scale, the shock thickness, and the mesh size, respectively.

The same restriction applies for shock–turbulence interaction (Ducros et al. 1999; Garnier, Sagaut, and Deville 2001, 2002): shock corrugation must be directly represented, and the LES mesh is very similar to the DNS one in the vicinity of the shock.

3.3.2 Numerical error: Analysis and consequences 3.3.2.1 Direct numerical simulation

The preceding discussion deals with scaling laws, and the remaining question is, How many grid points are required to represent an event (eddy, wave) of characteristic size l? The answer is obviously a function of the numerical method. Several error types are present that contribute to the global efficiency of the method. The differentiation error is measured using the Fourier analysis and the modified wave-number approach. Considering a test solution in one dimension f (x) = eikx , we find its exact derivative is ik f (x), whereas a discrete scheme yields an approximate solution of the form ik  f (x), where k  is the modified wave number. Some algebra shows that k  is a function of k and x. The Fourier spectral method leads to exact differentiation and k  = k. Using the modified wave-number analysis, one can evaluate the minimal value of the ratio l/x required to obtain an arbitrary level of accuracy. This value is also referred to as the required number of points per wavelength (PPW). Typical results are displayed in Table 3.6. The second main source of error is aliasing: when functions are represented using a finite number of basis functions, nonlinear terms require more degrees of freedom and thus can not be exactly represented using the same set of basis function. The Table 3.6. Modified wave-number analysis of some classical centered finite difference schemes Scheme

PPW (β = 0.1)a

PPW (β = 0.01)

PPW (β = 0.001)

Second-order, explicit Fourth-order, explicit Sixth-order, explicit Fourth-order, implicit Sixth-order, implicit

8 4.54 3.70 3.38 2.85

25 8.69 5.71 5.71 4

100 15.38 8.69 10 5.71

Points per wavelength (PPW) required for an arbitrary level of accuracy; parameter β = |k  − k|/k measures the error. Source:Adapted From Table III of Lele (1992).

a

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resulting error will pollute all resolved frequencies, the highest ones being the most corrupted. This error is reported to result either in numerical instability or excessive damping. The differentiation error is known to lower the effect of the aliasing error, and so the higher the accuracy of the scheme, the larger the aliasing error. Aliasing error is also a function of the form of the convection terms: although conservative quasilinear forms exhibit similar properties, the skew-symmetric form 12 [u∇u + ∇ · (u ⊗ u)] significantly reduces the aliasing error in both incompressible and compressible flows (Blaisdell, Spyropoulos and Qin 1996). Ducros et al. (2000) proposed discrete schemes for the conservative form that mimic properties of the skew-symmetric form. Compressible flow computations introduce a new nonlinearity through the equation of state, whose influence is less known. For spectral methods, dealiasing is performed using the so-called 2/3 rule: for a total number of Fourier modes equal to N , all modes k such that k > 2N /3 are set to zero at each time steps. High-order upwind biaised schemes, because they damp the highest resolved frequencies, are also observed to partly cure the aliasing problem. Quasi-DNS relies on the use of shock-capturing schemes. These schemes being very dissipative, most authors used hybrid schemes, which consist of a blending of the usual DNS scheme far from the shock and high-order shock-capturing scheme in the vicinity of the shock. 3.3.2.2 Large-eddy simulation

The requirement of controlling the numerical error appears more stringent for LES than for DNS because the LES cutoff is supposed to occur within scales that are much more energetic than for DNS, leading to a much higher level of numerical error. This may become very problematic if the numerical scheme introduces some artificial dissipation (artificial viscosity, upwind scheme, filter, etc.) because the amount of numerical nonphysical dissipation may overwhelm the physical drain of resolved kinetic energy associated to the energy cascade. The possible situations depending on the relative values of the numerical dissipation εnum , and the subgrid dissipation, εsgs are summarized as follows: r Controlled LES: εnum  εsgs . The physical model is dominant, and explicit subgrid-

scale modeling is required.

r Intermediary case: εnum  εsgs . Some specific stabilized numerical schemes may

serve as subgrid model (MILES approach), and the use of explicit subgrid models is not justified. r Uncontrolled LES: εnum  εsgs . No subgrid model is needed, and results are strongly case-dependent. Practical experience reveals that the best results are obtained using centered, nondissipative schemes to make it possible to capture a much broader resolved band of scales than dissipative methods. However, because they do not provide stabilization, these methods require the use of much finer grids to ensure numerical stability.

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A particular case is the monotonically integrated large-eddy simulation (MILES) approach (Boris et al. 1992), which relies on the use of dissipative schemes that mimic the energy cascade. This approach is detailed in Section 3.5.4. Various shock-capturing schemes and stabilized methods have been analyzed, leading to possibly satisfactory results – especially for free-shear flows or massively separated flows. Garnier et al. (1999) showed that higher-order statistics of turbulence may be affected when the MILES approach is used. The two main reasons for this are that numerical dissipation is most of the time much higher than the physical drain of resolved kinetic energy and that the spectral and spatial distributions of these two dissipations are not the same. The use of stabilized methods is sometimes necessary to enforce physical behavior of the solution. Examples are flows with shocks, flows at very high Reynolds number on coarse grids, and flows with additional advected variables such as passive scalar or chemical species. In these cases, the usual subgrid models are known to fail to enforce entropic or realizability constraints, and numerical stabilization must be used. The problem is then to keep the level of numerical dissipation as low as possible. For the shock problem, an efficient solution proposed by Ducros et al. (1999) is to introduce the following shock sensor in front of the dissipative part of the scheme: |∇ · u| . |∇ · u| + |∇ × u|

(3.75)

To reduce the numerical dissipation provided by usual stabilized methods for compressible flows without shocks, new methods based on entropy splitting (Sandham, Li, and Yee 2002) or wiggle detection (Mary and Sagaut 2002) have recently been used successfully. Mary and Sagaut (2002) defined the wiggle detector i as (in one dimension at grid point number i) iφ = sign [(φi+2 − φi+1 )(φi+1 − φi )] ,

Wψk =

1 if 0 otherwise

i−1 i (iψk + i+1 ψk ) < 0 or (ψk + ψk ) < 0

i = Maxk (Wψk ),

(3.76) ,

(3.77) (3.78)

where ψk is the vector of conservative variables. Centered schemes introduce dispersive errors instead of dissipative errors. They do not appear directly in the budget equation for the mean resolved kinetic energy and thus are not explicit competitors of subgrid models for the energy cascade. The level of error was shown by Ghosal (1996) to be much larger than the amplitude of the true subgrid terms – even for eighth-order accurate schemes – but, owing to their dispersive character, centered schemes are known to yield good results when they are applicable. A possible way to minimize the numerical error with regard to the subgrid model is to employ the prefiltering technique (Ghosal 1996). That technique consists of choosing a cutoff length scale  for the LES filter larger than the mesh spacing x. Then, while keeping the same value for , the dominance of the subgrid terms can be recovered by decreasing x (i.e., refining the discretization grid). Theoretical works have shown

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that when a centered, fourth-order accurate scheme having x < /2 is used, the subgrid contribution is expected to be dominant. The use of this technique, however, induces a large increase in the cost of the computation (by a factor of (/x)4 if the computation is performed using the same Courant number; i.e., U t/x = const, where U is a characteristic convection speed and t is the time step). This is the reason why it is not popular among LES practitioners. 3.3.3 Time advancement

The time advancement scheme and time step are chosen so as to enforce numerical stability and accurate description of the resolved scales of motion. The Courant number ut/x based on the velocity of the fluid, u, is usually taken of the order of unity. In compressible flows, the fluid velocity must be replaced by the highest velocity of acoustic waves, |u| + c. These constraints make the use of explicit schemes very attractive. Implicit methods can be used for convective terms in the subsonic regime if acoustic modes are not important. In this case, the relevant velocity scale is still the fluid velocity. Diffusive terms are also often treated implicitly – especially in wall-bounded flows – in which the use of very small meshes in the vicinity of the wall may yield a very stringent time-step restriction if explicit time integration is carried out.

3.4 Subgrid-scale modeling for the incompressible case 3.4.1 The closure problem

To obtain a tractable set of governing equations, one must close the problem (i.e., the subgrid tensor τ must be expressed as a function of the unknowns of the filtered problem u). As mentioned in Section 3.2.2 the effective filter in an LES computation appears to be a blending of theoretical and numerical filters. Thus, subgrid scales can be grouped into scales smaller than the effective cutoff length  but larger than the Nyquist length 2x, and scales smaller than 2x. The former are sometimes called subfilter scales, whereas the latter are referred to as subgrid scales. The former can be reconstructed on the grid if the filter is invertible, but the latter can not. Two philosophies can be identified for subgrid model derivation, which lead to very different solutions (Sagaut 2002): r Functional modeling: the idea is to introduce a new term in the filtered equations

that has the same effect (dispersive, dissipative) on the resolved scales as the subgrid scales but that can not be considered as a true model for the tensor τ . It should rather be interpreted as a model for the term ∇ · τ . r Structural modeling: on the opposite side, structural modeling will aim at building a model for the subgrid tensor τ itself, whatever the nature of the interaction between resolved and subgrid scales may be. Multilevel simulations, in which subgrid scales are evaluated on an auxiliary grid, are a special case of structural models.

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Reconstruction of interactions between scales that can be captured on the computational grid is generally carried out using structural models, whereas representation of interaction between resolved scales and scales smaller than the Nyquist frequency kc = π/x is performed using dissipative functional models. This point is more precisely discussed in Section 3.4.4. As guidelines for the derivation of subgrid models, we can think of the following constraints: r Physical constraints:

– Subgrid models should have the same symmetry properties as the true subgrid terms. It is interesting, however, to note that most of the time the discretized Navier–Stokes equations do not have the same properties. – Subgrid models should be consistent (i.e., return a zero contribution when the flow is fully resolved), leading to an automatic switch to DNS. – Subgrid models should have the same effect on the resolved scales as the true subgrid scales (dispersion, dissipation, diffusion). r Numerical constraints (it is important to remember that subgrid models are developed to perform simulations on computers): – Subgrid models must be realistic from the computational point of view: The extra cost for their evaluation must be acceptable (if you really can afford a large extra cost, then it is sometimes better to use a finer grid with a simpler subgrid model!). As a consequence, subgrid models should be local in space and time. – Subgrid models should not induce numerical instability. – Subgrid models should be “numerically robust” – that is, once discretized, the effect of the model should remain the same as in the continuous case (no spurious effects). These constraints seem to be very simple, but a careful analysis shows that many existing subgrid models do not satisfy all of them. A large number of models have been proposed but will not be described here. See the monograph by Sagaut for a more exhaustive presentation (Sagaut 2002).

3.4.2 Functional modeling

Functional modeling uses the dynamics of the isotropic turbulence as a basis. Both theoretical and numerical studies show that the net effect of the subgrid scales is a drain of kinetic energy from the resolved scales (see Chollet and Lesieur 1981; Domaradzki, Liu, and Brachet 1993; Kraichnan 1971, 1976; Lesieur 1997; Leslie and Quarini 1979; Zhou 1993). That process is directly linked to the energy cascade from large to small scales (see Figure 3.3). A weaker inverse energy cascade is also detected (i.e., transfer of kinetic energy from small subgrid scales to large resolved scales), but that secondary process, commonly referred to as backscatter, is neglected by almost all the LES users. There are two reasons for this: (1) Several models have been proposed, but none of them has been found to be satisfactory. (2) When accounting for inverse cascade, energy is

113

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Figure 3.3. Schematic of kinetic energy transfer in isotropic turbulence.

injected into the resolved field, yielding possible numerical instabilities. See, however, Carati, Ghosal, and Moin (1995) and Leith (1990) for backscatter models in physical space. A possible use for this backscatter model could be to cure the overdissipation provided in most MILES approaches. A simple way to account for that net drain of energy is to parameterize it as an additional dissipation. This is done by defining an eddy viscosity in the same spirit that is done in RANS modeling. One can then write τ D = −2νt S, S =

1 (∇u + ∇t u). 2

(3.79)

It is important to note that only the deviatoric part τ D = τ − (τkk /3)Id is taken into account because S is a traceless tensor in the incompressible case. The isotropic part (τkk /3)Id is added to the filtered pressure, leading to the definition of the pseudopressure

= p + τkk /3. The closure problem now consists in the definition of the eddy viscosity νt . Before describing the most popular eddy viscosity models, let us note that the use of a scalar viscosity is a simplification – a simple tensorial analysis shows that νt should be a fourth-order tensor: τi j = νt i jkl S kl .

(3.80)

This kind of eddy viscosity model is local in space and time (and is then very simple to use), but theoretical analysis demonstrates that it should be nonlocal in space and time (Yoshizawa 1979, 1984). The locality is recovered only when the existence of a spectral gap between resolved and subgrid scales is assumed. An equivalent statement is that the characteristic scales of the subgrid modes are much smaller than those of the resolved field. With L 0 and l0 being the characteristic length scale of the

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resolved and subgrid fields, and T0 and t0 the associated time scales, this condition reads l0  L 0 , t0  T0 .

(3.81)

It is worth noting that this condition is fulfilled for RANS modeling, but not for LES, because the kinetic energy spectrum is a continuous function and the cutoff is assumed to occur in the inertial range; there is no spectral gap. 3.4.2.1 Smagorinsky model

The most popular subgrid model is certainly the Smagorinsky model, which is also the oldest one (Smagorinsky 1963). It is obtained by performing a simple dimensional analysis. Considering that νt ∝ l02 t0−1 ,

(3.82)

we find that the problem is now to evaluate the two characteristic scales l0 and t0 . Assuming that the cutoff length scale is representative of the subgrid modes, one can write l0 = Cs ,

(3.83)

where Cs is a constant to be evaluated. The evaluation of the time scale t0 is a bit more complex and requires new assumptions on the dynamics. We first assume that the local equilibrium hypothesis is satisfied – that is, the production rate of kinetic energy is equal to the transfer rate across the cutoff, which is equal to the dissipation rate by the viscous effect, resulting in an automatic adaptation of subgrid scales to the resolved ones (i.e., the information propagates at an infinite speed along the spectrum). The characteristic time scale of the subgrid modes is then equal to that of the resolved scales, which is assumed to be the turnover time defined as (3.84) T0 = 1/ 2S i j S i j = t0 , leading to νt = (C S ) 2S i j S i j . 2

(3.85)

The constant of the model remains to be evaluated. A theoretical value of the constant can be derived under the assumptions that the spectrum is a Kolmogorov spectrum – that is, E(k) = K 0  2/3 k −5/3 , with K 0  1.4 – and that filter is a sharp cutoff filter, yielding C S  0.18. Note that this spectrum is a nonphysical one because it corresponds to an ∞ infinite value of the turbulent kinetic energy 0 E(k)dk. That value is not a “universal” one and depends a priori on the energy spectrum. Lower values should be used for shear flows (typically 0.1 for channel flow; Deardorff 1970) in order to account for the contribution of the mean shear into the evaluation

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of the time scale T0 . Another possibility when dealing with shear flows is to use the splitting technique of Schumann (1975): (1)

(2)

τiDj = −2νt (S i j − S i j ) − 2νt S i j  with (1)

νt

= (C1 )2 2(S i j − S i j )(S i j − S i j ), (2) νt = (C2 )2 2 S i j  S i j ,

(3.86)

(3.87) (3.88)

where  is the statistical average operator (here, average over time/homogeneous directions). That splitting yields an improvement of the results but is restricted to configurations in which the statistical average can be performed. 3.4.2.2 WALE model

The wall-adapting local eddy viscosity (WALE) model was designed by Nicoud and Ducros (1993) to recover the correct asymptotic behavior of the subgrid viscosity at the wall in zero-pressure gradient incompressible boundary layers:

3/2 Sidj Sidj (3.89) νt = (Cw )2  5/2 d d 5/4 Si j Si j + Si j Si j with  1  Sidj = S ik S k j + ik k j − δi j S mn S mn − mn mn , 3 =

1 (∇u − ∇t u). 2

(3.90) (3.91)

3.4.2.3 Mixed-scale model

Another popular eddy viscosity subgrid model is the mixed-scale model (Sagaut 2002), which reads νt = Cm 

3/2 1/4 qc (2S i j S i j )1/2 ,

(3.92)

where the constant Cm is equal to 0.06. The quantity qc is the kinetic energy of the highest resolved frequencies, which is assumed to be a good surrogate for the subgrid kinetic energy. It is evaluated during the simulation by applying a second filter with  > , referred to as the test filter (symbol ), to the resolved field u: cutoff length  1 u)2 . (3.93) (u −  2 In practice, a three-point discrete test filter is used to compute qc (see Section 3.2.5.4 for a description of discrete filters). This model appears to have different properties when compared with the two previous ones: although the Smagorinsky and the structure function models are only sensitive to the gradient of the solution and then return nonzero qc =

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values for the laminar uniform shear flow, the mixed model is also sensitive to the 2 u ∝  ∇2 u. second-order derivatives of the solution; a three-point test filter yields u −  This is a good illustration of the problem of defining subgrid models by looking only at the resolved scale features. 3.4.2.4 Estimation of the subgrid-scale kinetic energy

As mentioned in Section 3.4.2 subgrid viscosity models for incompressible flows are relevant models for the deviatoric part of the subgrid tensor only. As a consequence, the subgrid kinetic energy, which is tied to the trace of the subgrid tensor, is not a direct output of these models and must be modeled separately when it is required for some physical analysis (but it is important to note that its knowledge is not necessary for integrating the closed, filtered equations). First, we emphasize that the trace of the subgrid tensor is not equal to the filtered subgrid kinetic energy q sgs = 12 u i u i . The exact relation is (Knaepen, Debliquy, and Carati 2002; Sagaut 2002)  1 1 u l u l − u l u l = q sgs + u l u l = k. τll = 2 2

(3.94)

It is seen that the equality is recovered only if the filter is a Reynolds operator. Some models relying on an evolution equation for q sgs have been proposed but will not be described here. We will restrict ourselves to algebraic models. Assuming that the Kolmogorov spectrum holds for all wave numbers and introduc = α, α > 1, one obtains ing a secondary filter (noted by a tilde) with cutoff length  the simple expression  +∞ 1 L ii q sgs = E(k)dk = u i ui . (3.95) , L ii = u i u i −  2 α 2/3 − 1 π/ The two filters are assumed here to be sharp spectral cutoff filters. The idea of introducing an auxiliary filter can also be used to obtain a self-adaptive version of the Yoshizawa model, leading to 2

q sgs = Cq  |S|2 ,

Cq =

L ii



2

(α 2 | S|

 − |S|)

.

(3.96)

A last, improved expression was proposed by Knaepen et al. (2002) to account for viscous effects at very high wave numbers: q sgs =

1 L ii (1 − β 2/3 ), 2 α 2/3 − 1

(3.97)

where  3/4 2τi j S i j π kc 4/3 , β = , kc = , k = kc − k 3ν A 

A=

π 2/3 L ii 3

2/3

(1 − α 2/3 )

.

(3.98)

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3.4.3 Structural modeling

Many ways of reconstructing and approximating subgrid scales have been proposed that rely on very different approaches: approximate deconvolution, scale-similaritybased models, deterministic vortex models, and so on. The emphasis is put here on the first two classes of models because they are by far the most popular ones and can be interpreted as different implementations of the same fundamental idea. The key idea is to approximate the subgrid tensor as τi j ≡ u i u j − u i u j ∼ u i u j − u i u j ,

(3.99)

where u is a synthetic velocity field reconstructed from u, which approximates the exact unfiltered solution u. Computing u is equivalent to searching for inverting the filtering procedure. This is possible, at least theoretically, for nonprojective filters. Because exact deconvolution is impossible in practice, an approximate estimate is used. A first way to compute u is proposed by Stolz and Adams (1999), Adams and Stolz (2001), and Stolz, Adams, and Kleiser (2001a, 2001b). If the filter kernel G has an inverse G −1 , the latter can be obtained using the van Cittert iterative procedure:  (Id − G)l = lim G −1 (3.100) G −1 = (Id − (Id − G))−1 = lim p . p→∞

p→∞

l=0, p

An approximate defiltered field is then obtained using a truncated series expansion: u = G −1 p u

(3.101)

with p = 5 in practice. The scale-similarity model proposed by Bardina (Bardina, Ferziger, and Reynolds 1983; Meneveau and Katz 2000) is recovered by taking p = 0, yielding u = u and τi j = u i u j − u i u j .

(3.102)

Another way to compute u is to approximate the filtering operator as a differential operator. Writing the Taylor series expansion of the velocity field at point x as u(y) = u(x) +

 (y − x)l ∂ l u (x) l! ∂ xl l=1,∞

(3.103)

and inserting it into Equation (3.11), we obtain u(x) =

 (−1)k k ∂ku  Mk (x) k (x), k! ∂x k=0,∞

(3.104)

where Mk is the kth-order moment of the kernel G (the flow field is now assumed to be 2π -periodic in the three directions of space):  Mk (x) = (−1)

k

(π+x)/

(π −x)/

G(, x, y)y k dy.

(3.105)

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The approximate inverse operator is computed as the inverse of the approximate filter operator obtained considering a finite Taylor series expansion, leading to  −1  (−1)k k ∂k

 Mk (x) k u(x, t). (3.106) u (x, t) = k! ∂x k=0, p Several practical implementations have been proposed: direct inversion of the matrix associated to the implicit discretized problem associated with Equation (3.106) (Fischer and Iliescu 2001) or explicit approximation of its solution. The latter is obtained by considering  as a small parameter and expanding relation (3.106) using the well-known relation (1 − )−1 = 1 −  +  2 − . . . . Carati, Winckelmans, and Jeanmart (1999) derived a general expression for the resulting model, which is valid for a large family of filter kernels, including all symmetric kernels: τi j =

 l,m=0,∞;(l,m) =(0,0)

Clm

∂ l ui ∂ m u j , ∂ xl ∂ xm

(3.107)

where Clm are filter-dependent coefficients. This general expression shows that models like Clark’s gradient model (Clark, Ferziger, and Reynolds 1979) and the tensor eddy diffusivity model (Winckelmans et al. 2001) are nothing but particular members of a more general family. These models can be further improved by adjusting the constant that appears in front of them. This makes it possible to account for the fact that the best low-order approximation coefficients are not those of the infinite Taylor series expansions (e.g., Wagner 2001, 2003). 3.4.4 Linear combination models, full deconvolution, and Leray’s regularization

The deconvolution models presented in Section 3.4.3 are not able to reconstruct scales smaller than the grid spacing x and so are not able to account for nonlocal energy transfer across the cutoff. As a consequence, they must be supplemented by another model specially designed for this purpose. A simple way to do this is to use functional models of the eddy viscosity type based on the description of the kinetic energy transfers associated with nonlocal interactions. Thus, a full deconvolution model is obtained by operating a linear combination of a deconvolution-type structural model with a functional model of the eddy viscosity type. The use of these linear combination models is encouraged because many theoretical arguments and practical observation prove it is beneficial. The main reasons are as follows: r The structural models presented in Section 3.4.3 are known to dissipate no or very

small amounts of resolved kinetic energy, but they are very good at predicting the anisotropy of the subgrid scales. They exhibit very good correlation coefficients with

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exact subgrid stresses. Subgrid viscosity models provide much more dissipation but do not properly represent the subgrid tensor. r Shao, Sarkar, and Pantano (1999) did show that structural models represent the rapid part of the subgrid tensor very well (i.e., the part associated to the mean flow gradient), whereas functional models are adequate to account for the slow part. The rapid part is important when production of kinetic energy is much larger than dissipation or when filter length is of the same order as the integral scale of turbulence. It is responsible for the anisotropy production of the subgrid stresses. It is worth noting that eddy viscosity can be replaced by other resolved turbulent kinetic energy sinks. Possible ways are the use of numerical dissipation (MILES approach) or the implementation of a penalization term as proposed by Adams and Stolz (2001): 1 (Id − G −1 p G) u, τξ

(3.108)

where τξ is a relaxation time. Another approach for deriving stable deconvolution models, based on mathematical results dealing with functional analysis of Navier–Stokes-like equations, was proposed by Geurts and Holm (2003). Considering Leray’s analysis on regularized Navier–Stokes equations as a starting point, these authors suggest using the following asymmetric model: τi j ≡ u i u j − u i u j ∼ u i u j − u i u j .

(3.109)

3.4.5 Extended deconvolution approach for arbitrary nonlinear terms

A very interesting feature of the deconvolution approach is that it can be used to close subgrid terms arising from any kind of nonlinearity because it is a general approach that relies not on any assumption on the nonlinear term but on the filter. A general closure procedure for arbitrary nonlinear functions of a scalar Z was proposed by Pantano and Sarkar (2001). Given a nonlinear function f , the associated subgrid term arising through the filtering procedure is τ f (Z ) = f (Z ) − f (Z ).

(3.110)

The key idea, as in classical deconvolution models, is to approximate f (Z ) by f (Z ), with Z = G −1 p Z , yielding τ f (Z )  f (Z ) − f (Z ).

(3.111)

Expanding relation (3.100), one can write the approximate defiltered variable as Z = Z + c0 (Z − Z ) + c1 (Z − 2Z + Z ) + . . . ,

(3.112)

where all scalar coefficients ci are equal to 1. The new idea is to optimize values of these coefficients to enforce new constraints on Z . Pantano and Sarkar (2001) considered

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the first-order reconstruction (i.e., ci = 0, i ≥ 2) and searched for a value of c0 such that the exact and modeled fields have averaged filtered moments of order 2 that are equal: 

2



(Z 2 (x) − Z (x))d x =

 

2 Z 2 (x) − Z (x) d x.

(3.113)

Given the analytical expression for both the transfer function of G and the spectrum of Z , an exact expression for c0 can be found.

3.4.6 Multilevel closures

Another way to evaluate the subgrid tensor is to compute the subgrid scales explicitly using a multilevel approach. This approach can be seen as an extension of the deconvolution procedure based on direct evaluation instead of filter inversion. With u on a mesh of size x, the idea is known to evaluate u  on an auxiliary mesh of size x  < x (typically, x  = x/2). The approximate unfiltered field on mesh x  is u = u + u  . The nonlinear term (u ⊗ u) is computed on mesh x  and then restricted on mesh x. In order to save time with respect to DNS, the evaluation of the fluctuating field  u on the secondary grid must involve some simplifications. A first possibility is to compute it using random functions, low-order chaotic dynamic systems or kinematic models rather than solving partial differential equations fractal interpolation procedure (Scotti and Meneveau 1997), subgrid-scale estimation procedure (Domaradzki and Liu 1995), . . . . Another approach is to use simplified models derived from the Navier– Stokes equations instead of the full Navier–Stokes equations (modified subgrid-scale estimation procedure (Domaradzki and Yee 2000), local Galerkin approximation (McDonough and Bywater 1986). A last way of reducing the complexity is to use the Navier–Stokes equations but on a grid such that all subgrid scales are not captured (variational multiscale method (Hughes, Mazzei, and Jansen 2000; Hughes, Mazzei et al. 2001; Hughes, Oberai, and Mazzei 2001), Terracol’s multilevel algorithm (Terracol, Sagaut, and Basdevant 2001)). Here, u  is solution of a new LES problem. Further saving can be obtained defining a cycling strategy between the two grids and eventually using more than two resolution levels (Terracol et al. 2001).

3.4.7 The dynamic procedure

All the eddy viscosity models presented in Section 3.4 exhibit a constant that was set looking at the isotropic turbulence case. An idea to minimize modeling errors is to adjust that constant at each point and at each time step in order to get the best possible adaptation of the selected subgrid model to the local state of the resolved field. This can be done using the dynamic procedure, which relies on the Germano identity.

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Figure 3.4. Schematic of the two-level filtering procedure and the Germano identity.

3.4.7.1 Germano identity

Germano derived the following exact identity (Germano et al. 1991) that establishes a . link between the subgrid tensors obtained at the two different filtering levels . and   ≥ : with respective filtering length scales  and      (u i u j − u i u j ) = (u i u j − u i u j ) − (u i u j − u i u j ) ,          L imj

Ti j

(3.114)

 τi j

where L m is a resolved tensor at the first filtering level (bar symbol), which can be directly computed, and T and τ are, respectively, the subgrid tensors obtained at the second and first filtering level (see Figure 3.4). 3.4.7.2 Computation of the subgrid model constant

The Germano identity can be used to compute the optimal value of the constant of subgrid models (Germano et al. 1991). Formally rewriting the closure as τi j = C f i j (u, ),

(3.115)

where C is the constant to be computed and f i j (u, ) the subgrid model itself (either of functional or structural type), and assuming that the same model with the same constant can be employed to close the problem at the second filtering level, we obtain  u, ). Ti j = C f i j (

(3.116)

Inserting Equations (3.115) and (3.116) into Equation (3.114), we can define the residual Ri j :  + C u, ) f i j (u, ). Ri j = L imj − C f i j (

(3.117)

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To enable a simple evaluation of C, we rewrite the residual as (Ronchi, Ypma, and Canuto 1992)  − Ri j = L imj − C( f i j ( u, ) f i j (u, )).   

(3.118)

Mi j

The constant is now computed to minimize a given norm of the residual. The most used solution is to operate a least-squares minimization of the residual (Lilly 1992) – that is, find C such that ∂ Ri j Ri j = 0, ∂C

(3.119)

yielding C=

L imj Mi j Mi j Mi j

.

(3.120)

The use of that relation at each grid point and each time step allows us to adapt the subgrid model to the local state of the flow. Subgrid models whose constant(s) are computed using this procedure are referred to as dynamic models. The main interesting properties of dynamic models are as follows: r They vanish automatically in the near-wall region (so there is no need for a wall-

damping function).

r They vanish in fully resolved regions of the flow (automatic switch to DNS). r They are able to capture the transition process.

Numerical experiments show that the dynamic procedure also has some drawbacks. First, the constant C may happen to have some infinite values. Second, when used together with an eddy viscosity model, the dynamic constant happens to take negative values corresponding to a negative subgrid viscosity. The physical meaning of this phenomenon is not proven, and it has deleterious effects on the numerical stability. This is why several stabilization procedures have been proposed. The first one, referred to as the clipping procedure (Zang, Street, and Koseff 1993), is to bound the dynamic constant so that the total viscosity (ν + νt ) remains nonnegative. The second one is to average the numerator and denominator of Equation (3.120). Averaging can be performed over homogeneous direction(s) (Germano et al. 1991; Zhao and Voke 1996), over time (Zang et al. 1993), along streamlines (Meneveau, Lund, and Cabot 1996), or over neighboring grid points (Zang et al. 1993). Another possibility is to remove the highest resolved frequency from the velocity field before computing the constant. 3.4.7.3 Extension to multiparameter models

The dynamic procedure can also be used to compute the constants of multiparameter models (such as the linear combination models). We now consider the following model: τi j = C1 f i j (u, ) + C2 gi j (u, ).

(3.121)

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Following the same procedure, one can evaluate the two constants by solving the two problems ∂ Ri j Ri j = 0, ∂C2

∂ Ri j Ri j = 0, ∂C1

(3.122)

which leads to the inversion of a 2 × 2 matrix. The residual is now defined as  

  − g u,  . (3.123) G   Ri j = L i j − C1 f i j (u, ) − f i j (u, ) − C2 gi j  u,  ij Numerical experiments have shown that direct use of this two-parameter procedure can lead to numerical instability when it is applied to the linear combination of a subgrid viscosity and a soft deconvolution model. This instability occurs because the subgrid viscosity constant appears as the scaling factor of a residual term (instead of a firstorder contribution to the modeling of the subgrid tensor), the soft deconvolution models exhibiting a very high correlation level with the true subgrid viscosity tensor. It was also proven by Kobayashi and Shimomura (2003) that the soft deconvolution part may induce a strong antidissipative effect in boundary layers. To cure this problem, it is recommended to first evaluate the constant of the subgrid viscosity part (without taking into account the soft deconvolution part) and then dynamically adjust the constant of the second part of the composite model. 3.4.7.4 Accounting for numerical errors

Previous developments dealing with the dynamic procedure were all devoted to the adjustment of the subgrid model constant with the purpose of best approximating the subgrid tensor. This procedure does not take into account the discretization errors that occur when computing the divergence of the subgrid tensor (i.e., the subgrid force that is the true term appearing in the momentum equation). The dynamic procedure can also be used to account for the numerical errors by using it at the vector level and not at the tensor level (Morinishi and Vasilyev 2002; Sagaut 2002). A solution is to write ∂ ∂ τi j = C f i j (u, ), ∂x j ∂x j

∂ ∂  Ti j = C f i j ( u, ), ∂x j ∂x j

(3.124)

leading to the following definition of the vectorial residual: Ri =

∂ ∂ m ∂  −  L i j − C( f i j ( u, ) f i j (u, )). ∂x j ∂x j ∂x j   

(3.125)

Ni

The least-squares minimization procedure yields the new expression for the constant C: C=

Ni ∂ L imj /∂ x j Ni Ni

.

(3.126)

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3.5 EXTENSION OF SUBGRID MODELS FOR THE COMPRESSIBLE CASE

3.5 Extension of subgrid models for the compressible case 3.5.1 Background

The extension of the subgrid models to the compressible case can mostly be interpreted as a variable-density extension of the incompressible models rather than the development of true compressible subgrid models relying on the physics of compressible turbulence. The main reasons are as follows: r The subgrid Mach number Msgs is generally small, and thus compressible effects

at the subgrid motion level are expected to be small. Erlerbacher et al. (1992) indicate that subgrid compressiblity effects can be neglected for subgrid Mach number less than or equal to 0.4, and, because the subgrid Mach number is smaller than the turbulent Mach number Mt , they propose to neglect these effects in flows with a turbulent Mach number smaller than 0.6. An example is the boundary layer flow without strong cooling or heating effects in which the Morkovin hypothesis applies up to Mach number M  5 and subgrid modes are nearly incompressible. Another point is that a priori evaluation of subgrid terms using DNS data reveals that some subgrid terms can be neglected for a large class of flows. Examination of DNS data shows that subgrid term A2 (see Equation (3.39)) in the momentum equations can be neglected in all cases. The most significant terms in the energy equation (as defined in Equations (3.39)–(3.40)) are B1 through B5 , whereas B6 and B7 are about one to two orders of magnitude smaller. r The main compressible effects are expected to be taken into account by the resolved modes. Thus, functional models will be extended with variable-density extensions of incompressible subgrid models for the momentum equations, and models for the passive scalar case will be used as a basis for the energy equation. Structural models will also be extended but are not so restrictive because they do not rely on assumptions about the dynamics. 3.5.2 Extension of functional models

Functional models are extended in a very simple way to compressible flows. Still considering the splitting of the subgrid tensor into two parts 1 (3.127) τ = τ D + τkk Id , 3 we find that the deviatoric part is parameterized using the following variable-density extension of the incompressible eddy viscosity model:   1 D  (3.128) τ = −2ρνt S − Skk Id , 3 where νt is evaluated using one of the previously discussed subgrid models. A few subgrid models for the isotropic part τkk have been proposed (Erlerbacher et al. 1990,

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1992; Salvetti and Banerjee 1994), but it appears that in most of the existing computations it is very small in front of the filtered pressure and then is often neglected (see also Section 3.4.2.4). In the filtered energy equation discussed in Section 3.2.4.2, terms B1 and B2 still need to be parameterized. Vreman et al. (1997) and many other authors who have been dealing with similar formulations have employed a simple Boussinesq-like model: B1 + B2 = −∇ · (κt ∇T ),

(3.129)

where the eddy diffusivity is linked to the eddy viscosity through a turbulent Prandtl number Prt = νt /κt . Usual values for this Prandtl number are Prt ∈ [0.3, 0.6]. It can also be computed using a dynamic procedure. 3.5.3 Extension of structural models

The extension of the structural subgrid models is straightforward and will not be discussed here. The procedure is the same as in compressible flows considering massweighted filtered variables. The subgrid tensor is approximated as     (ρu)i (ρu) j (ρu i )(ρu j ) 1 1 − ρu i ρu j ≈ − (ρu) i (ρu) j . (3.130) τi j =

ρ ρ ρ ρ Subgrid models for compressible flows are derived applying the deconvolution procedure (iterative or explicit) to the filtered variables. An interesting point is that these extensions do not rely on any assumption on the compressible interscale dynamics, and numerical experiments have shown that they were able to account for compressible effects on the subgrid dynamics. Although explicit deconvolution based on the Taylor series expansion technique may yield some problem near shocks or interfaces (Garnier et al. 2002), Adams recently proved that the van Cittert iterative procedure has some shock-capturing capability (Adams 2002). 3.5.4 The MILES concept for compressible flows

It was shown in Section 3.3.2 that numerical errors may occur to overwhelm the subgrid model, leading to uncontrolled simulations. However, some amount of numerical dissipation is necessary to guarantee that some realizability constraints will be fulfilled: positive of specific mass fraction, entropic solution near shock waves, temperature physical bound preservation, and so on. A huge amount of work is devoted to the development of numerical methods that are able to enforce these realizability constraints. A common feature of all these methods is that they rely on the introduction of numerical dissipation. Thus, the problem of producing realiable LES with stabilized numerical schemes arises. To solve this problem, Boris et al. (1992) proposed a new concept referred to as monotonically integrated large-eddy simulation (MILES), which is based on the idea that numerical schemes can be designed that are capable of enforcing the realizability

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constraints while providing a subgrid–viscosity-like dissipation in turbulent flows. With such a scheme, usual explicit subgrid models are no longer needed, and LES-like results can be obtained by just solving the Navier–Stokes equations. The MILES approach was further investigated by Fureby and Grinstein (1999, 2002), who considered the case of flux-limited finite volume schemes. If it is assumed that the total numerical convective flux function v f is evaluated as   v f = v Hf − (1 − ) v Hf − v Lf , (3.131) where v Hf , v Lf , and  are a high-order flux function, a low-order (dissipative) flux function, and the limiter function, respectively, the Navier–Stokes equations supplemented by the leading numerical error terms are ∂ρ + ∇ · (ρu) = 0 ∂t ∂ρu + ∇ · (ρu ⊗ u + p) − ∇ · τv = ∇ · (E 1M + E 2M ) ∂t ∂E + ∇ · (u(E + p)) − ∇ · (uτv ) + ∇ · qT = ∇ · (E 1e + E 2e ) ∂t E 1M = ρ(C∇T u + ∇uC T ), E 1e = ρC∇e,

E 2M = ρχ 2 (∇ud ⊗ ∇ud), E 2e = ρχ 2 (∇e · d)∇ud,

(3.132) (3.133) (3.134) (3.135) (3.136)

where d is the topology vector connecting the neigboring cells, e = E/ρ, χ is a scalar function that depends on the flux limiter , and C = χ (u ⊗ d). Looking at these error terms, we see that E 1M and E 1e can be interpreted as tensorial subgrid viscosity models, whereas E 2M and E 2e are close to scale-similarity models. As a consequence, the MILES approach can yield relevant results if  is adequatly defined.

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Use of Hybrid RANS–LES for Acoustic Source Predictions Paul Batten, Philippe Spalart, and Marc Terracol

4.1 Introduction to hybrid RANS–LES methods The title of this book implies the current or impending feasibility of large-eddy simulation (LES) for use in noise prediction problems. The potential accuracy of LES is generally well regarded; however, the cost of such calculations requires a specifically focused algorithmic and computational effort in order to make LES affordable for practical engineering problems. At high Reynolds numbers, such problems can defeat LES by raising the range of scales beyond affordability owing to the extremely small size of the eddies in the viscous sublayer or even the size of the dominant eddies in the bulk of the boundary layer (Spalart et al. 1997). This chapter outlines a variety of recently developed hybrid methodologies that specifically address the issue of computational cost and make the simulation of large-scale, sound-generating flow structures tractable with existing computer resources. The hybrid nature of the methods discussed here involves the simultaneous use of (or blending between) statistical Reynolds-averaged Navier–Stokes (RANS) and traditional LES within the noise-source region. This hybrid character is distinct from the methods used in traditional acoustic analogies for far-field propagation of sound. Those issues remain here; however, this chapter specifically considers methods offering an affordable prediction of both the mean flow and pressure fluctuations in the near field. A common premise throughout the methods described in this chapter is that routine engineering predictions of acoustic disturbances (or simply turbulence) arising from realistic Reynolds-number flows must lead to a cost that is substantially less than that of traditional, full-domain LES. Thus, the methods discussed herein consider the application of LES to only those regions containing the most significant and geometrydependent noise-generating structures. The emphasis is on methods that are able to contain the cost of establishing both coherent and broadband noise sources without losing all accuracy. Additionally, because by nature hybrid RANS–LES methods resolve a narrower fraction of the frequency spectrum than traditional (full-domain) LES, this chapter also considers the possibility of extracting the missing noise that would have originated from the unresolved scales in the flow. 128

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The use of full-domain LES (involving a single subgrid model) may appear to be a natural choice for aeroacoustics problems because of the weak modeling dependency and its inherently time-dependent nature. Historically, however, cost has been a decisive factor favoring the use of semianalytic models capable of extracting noise from either unsteady RANS predictions or directly from statistical data. Cost will continue to be a major consideration in future use of LES for acoustics problems, but its application is made practical here by carefully confining the role of LES to the most critical noisesource regions. The finest scale structures are still expected to be modeled statistically, and far-field propagation is expected to be handled by some alternative approach. In general, the intention is to limit the use of LES or hybrid RANS–LES to the (nonlinear) noise-source generation region with subsequent propagation handled via some linear propagation method, which would ideally be analytic and not even involve a computational mesh. The statistical modeling employed in the hybrid methods discussed here has more impact than simple subgrid-scale (SGS) modeling in the sense that it may model even the dominant eddies, but then only within some specific “RANS regions.” The recent developments in hybrid RANS–LES are natural candidates for engineering aeroacoustic calculations because the statistical modeling employed is restricted to the finest scales with the larger, more energetic and more significant scales of motion (determined via a choice of mesh and time step) being simulated directly. The weaker reliance on empirical modeling, relative to unsteady RANS, is particularly welcome for problems involving fluid resonance or discrete tones arising from geometry-related scales. Although empirical modeling of tonal noise has had some success for very specific geometries (such as the square cavity analysis of Rossiter (1964) and Heller, Holmes, and Covert (1971), in general, the prediction of fluid-dynamic or fluid-resonant instabilities associated with the pockets of separated flow over, for example, base regions, wings, and cavities, requires accurate numerical modeling of the dominant, large-scale, coherent structures. General experience with unsteady RANS has been mixed because the outcome of such predictions has been dependent on the particular flow and type of RANS model employed. Some progress has been reported using unsteady RANS to predict the largest scales of motion (e.g., Zhang, Bachman, and Fasel 2000b; and Iaccarino and Durbin 2000) even though the RANS equations do not achieve the correct limit (direct numerical simulation) as the mesh spacing  → 0. Although unsteady RANS may give useful predictions for certain flows, its use in areas related to aeroacoustics is regarded by many as an unsafe practice because RANS imposes hard empirical limits on the representation of the flow physics – limits that cannot be shifted by increased temporal and spatial resolution, as would be the case in LES. There is not even a consensus over whether such unsteady RANS simulations should be strictly three-dimensional (i.e., even when the geometry is only two-dimensional). This creates a need for an intermediate strategy that is richer than RANS but avoids the cost of a full-domain LES. One of the earliest mentions of hybrid RANS–LES was in a paper by Schumann (1975). However, Schumann’s proposal applied only to the lower section of the boundary

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layer; more recent methods typically envision treating the entire boundary layer with a RANS model. Little attention was paid to the subject until the more recent papers of Speziale (1996) and Spalart et al. (1997). These hybrid RANS–LES methods and more recent developments inspired by them have shown that statistical modeling can be successfully blended with LES, yielding enhanced predictions of both turbulence statistics and unsteady flow dynamics at a fraction of the cost of traditional LES (e.g., Spalart et al. 1997; Germano 1999; Batten, Goldberg, and Chakravarthy 2000, 2002b; Strelets 2001; Menter, Kuntz, and Bender 2003; Labourasse 2002; Labourasse and Sagaut 2002; Spalart and Squires 2003). A brief background on these hybrid methods is presented in Sections 4.2 and 4.3, and examples are considered in Section 4.4. An emerging strategy that is considered to show promise for acoustics predictions is the use of nonlinear disturbance equations (Morris, Long et al. 1997) in which statistical modeling is used to determine an initial mean field about which fluctuations are synthesized or allowed to develop naturally in resolvable scales. Recent developments in this area, which we refer to as nonlinear acoustics solvers, are also discussed along with their connection to the existing range of hybrid RANS–LES methods. Many of the methods discussed in this chapter are relatively new and the associated terminology is still evolving, but these techniques can be broadly categorized into two major classes (discussed in Sections 4.2 and 4.3) corresponding to global and zonal hybrid methods.

4.2 Global hybrid approaches Global (or nonzonal) hybrid RANS–LES methods rely on a single set of model equations and a continuous treatment that blends between RANS and LES approaches. Such methods can be considered as a form of very large eddy simulation with subgrid stresses that are designed to reach RANS levels in certain limits of coarse or highly stretched meshes. As in traditional LES, the subgrid model is designed to account for the shorter-wavelength, higher-frequency velocity fluctuations in a statistical sense; hence, full-spectrum predictions require an extended hybrid strategy in which resolved scales are simulated directly with additional modeling employed to extract the missing components of the noise from the remaining statistics. 4.2.1 The approach of Speziale M

Speziale (1996) presented a hybrid framework in which the stress tensor u i u j provided by a (conventional RANS) Reynolds-stress transport model was damped via u i u j = α u i u j

M

(4.1)

in which α = [1 − exp(−β L  /L k )]n ,

(4.2)

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where β and n are some (unspecified) parameters, L  is some representative mesh spacing, and L k is the Kolmogorov length scale, 3

1

4. L k = ν 4 /

(4.3)

In Speziale’s approach, a regular RANS closure is recovered when L  is much larger than L k , whereas the subgrid stresses vanish completely as L  → 0. However, several issues were never completely specified by Speziale (1996) such as the definition of the parameters β, n in Equation (4.2), and the definition of the model equations used to derive the undamped Reynolds stresses. Further, more complete, developments of Speziale’s Kolmogorov length-scale-dependent blending have since been pursued at the University of Arizona (e.g., von Terzi and Fasel 2002) and CRAFT Technologies (Arunajatesan and Sinha 2001). Knowledge of the Kolmogorov length scale is, however, not strictly needed to ensure that the correct DNS behavior is reached in the limit of vanishing mesh spacing, and there are many possible choices of such blending functions (partly indicated by the free parameters in Equation (4.2)). In addition, properly reaching both the DNS and RANS limits is no guarantee that a concept provides a viable LES mode in between. Speziale’s precise intention for the use of Equation (4.1) may never be known, but there are (at least) two valid interpretations, both of which are represented in subsequent methods reviewed in this chapter. The first interpretation is that the modified M (undamped) stresses u i u j be computed via a coupled set of time-dependent transport equations evolving with the equations for the resolved scales. This interpretation is consistent with the DES and LNS methods described in Sections 4.2.2 and 4.2.3, M respectively. The second interpretation is that u i u j be computed in advance (or in a separate, time-dependent computation), with an independent set of equations solved subsequently (or concomitantly) for the resolved-scale fluctuations. This interpretation is closer to the NLAS and Labourasse–Sagaut methods described in Sections 4.2.8 and Labourasse and Sagaut (2002), respectively.

4.2.2 Detached-eddy simulation

The detached-eddy simulation (DES) method was the most widely used hybrid method in the 2000–2004 time frame thanks to clear objectives, open publications, resources for detailed testing (Strelets 2001), commonality with a known RANS model, and a simple and complete formulation (Spalart et al. 1997). Its impetus came from estimates that, for a wing and similar flow problems, LES will have to wait until the year 2045, even assuming that “wall modeling” has been achieved. It is then obvious that for now any practical method will treat the regions of thin, unchallenging boundary layer with RANS. Driven toward the simplest, nonzonal concept, the reasoning arrived at the use of a one-equation RANS model sensitized to the filter width  in the LES region, which creates a plausible SGS model. From the onset, the “gray area” in which the turbulence

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needs to convert from fully modeled (in the boundary layer) to mostly resolved (in massive separation) was recognized as potentially delicate. The Spalart–Allmaras (S–A) model does not need to be introduced here. The specific DES modification concerns its destruction term. In RANS, it represents blocking by a nearby wall and scales with the wall distance. In LES (within DES), it represents a limit on the mixing length and scales with the filter width. DES takes the classical view that the filter width is best tied to the grid spacing in order to make the best use of the available resolution. This is discussed further in Section 4.2.5. Therefore, the length scale injected into the model is d˜ ≡ min(d, CDES ),

(4.4)

where d is the traditional wall distance. In a prototypical DES, the entire boundary layer uses the d˜ = d branch and the S–A RANS model is fully active. The separated region, given an adequate grid, uses the d˜ = CDES  branch, and the SGS model has many similarities with the Smagorinsky model – certainly its scaling with grid spacing and dissipation rate. The effect of Equation (4.4) is very close to that of computing a RANS model and an SGS model separately and injecting the weaker of the two effective viscosities into the momentum equations; this describes LNS quite well (see Section 4.2.3). However, the DES transport equation for eddy viscosity involves a finite rate of dissipation, resulting in an eddy viscosity field that tends to be smoother than that obtained via a pure Smagorinsky model (equivalent levels being reached only when the DES production and dissipation terms become balanced). Whether this smoothness has a beneficial effect on predictions within the LES region has yet to be established. The objectives of DES overlap with those of Speziale (1996), but the focus on treating the entire boundary layer with RANS appears to represent a clear difference. In addition, DES practice has benefited from the routine control of transition in the S–A model for which there is no better demonstration than the circular cylinder case (Travin et al. 2000). LES is many years away from a serious attempt at its “drag crisis,” and yet DES reproduced the major quantitative features of this flow in addition to nontrivial qualitative features such as three-dimensional chaos and intense modulations of the vortex shedding. The frontier for this flow resides in the intermediate Reynolds-number range (which is unfortunately very wide: roughly 105 to 107 !). The physics mingles separation and transition, which S–A is not capable of, and thus, the DES misses both the very low drag values and the spontaneous lifting cases. As of 2004, DES enjoys an international user base and a publication stream that is often successful and sometimes critical (some representative calculations are presented in Section 6.8). The principal cause for concern has been in cases in which the grid is progressively refined until the d˜ = CDES  branch of the minimum in Equation (4.4) intrudes inside the boundary layer. The result is a weakened eddy viscosity but one that is not weak enough to allow LES eddies to form; as a result, the separation line moves too far forward. This phenomenon needs careful monitoring, and a permanent solution is yet to be found. DES is now appearing in vendor computational fluid

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dynamics (CFD) codes in which it has been demonstrated with very uneven levels of understanding. Educating users to the complexities of designing a DES or even an LES is a serious challenge. The statement that DES can function with a coarser grid than LES is too often made to sound general, but in fact it is correct only in the boundary layer. The coupling of DES with sound prediction methods has not progressed far, but the confidence in its capabilities to reproduce the dominant eddies of a separated flow is good enough to be optimistic. Most probably, the coupling will be through Kirchhoff or Ffowcs Williams–Hawkings (FW–H) approaches for several reasons. Such methods allow most of the grid points to be placed in the turbulent or nonlinear flow region. Surface integrals can then be used to determine far-field propagation by exploiting the pressure disturbances generated directly on a near-field-containing surface via the DES model. The instantaneous effects of the Lighthill stress tensor (which would need to be included in any volume-integral solution to Lighthill’s equation) are negligible if the surfaces surrounding the turbulent region are designed well. A notable challenge for most flows arises when turbulence is transported downstream and will cross the Kirchhoff or FW–H surface before it has become negligible; as a result, careful corrections and tests are indispensable. 4.2.3 LNS

The limited numerical scales (LNS) concept was presented by Batten et al. (2000) as a means of closure for Speziale’s (1996) approach and (in 1999) was the first implementation of a hybrid RANS–LES method in a commercial CFD solver. From its inception, the LNS method has been used exclusively within a nonlinear eddy viscosity framework that is more general, but also slightly more complex, than that of existing DES formulations. In its basic mode of operation, LNS exhibits hybrid RANS–LES characteristics similar to DES, which will involve a RANS treatment of the entire boundary layer if the near-wall region is meshed with similarly high-aspect-ratio cells. However, LNS achieves this behavior through an instantaneous limit on the Reynolds stresses (and hence production term) rather than through an elevated rate of dissipation, as in DES. The two-equation, nonlinear eddy viscosity framework is not an essential ingredient for LNS, although its use has some theoretical advantages in situations in which the underlying RANS model is used to predict the primary separation location. The use of a nonlinear framework is more significant in recent LNS developments that require a realizability property of the local stress tensor. This is needed to allow a reconstruction and transfer of data between different modes (statistical and directly resolved representations) in a manner that aims to preserve the total shear stress. That process becomes considerably more challenging with a single-equation framework that provides no description of the turbulence kinetic energy. However, recent extensions of DES to a two-equation framework (e.g., Strelets 2001) might form suitable candidates for use with the data transfer procedures outlined in Section 4.2.7.

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The primary aim of LNS is to identify resolvable and unresolvable fractions of the turbulence kinetic energy using an unambiguous expression for Speziale’s latency parameter (4.2). A key distinguishing feature of LNS is that it contains no empirical constants beyond those appearing in the baseline RANS and LES models. Whereas in traditional turbulence closures models are calibrated as a complete entity, in LNS the key parameter – the turbulence shear stress in an axis aligned with the mean shear strain – is the only factor in determining the model blending. Thus, the LNS framework is immediately able to exploit improvements in either RANS or LES modeling by directly inserting any enhanced component model. No model recalibration is required. LNS achieves this goal by redefining the latency factor in Speziale’s method via the ratio of effective-viscosity norms   min ( f μ L .V )LES , ( f μ L .V )RANS (4.5) α= ( f μ L .V )RANS in which L and V denote length and velocity scales, respectively. The ( f μ L .V )LES and ( f μ L .V )RANS products denote norms for the effective viscosity arising from the component LES and RANS models, respectively. If consistent models are assumed for both LES and RANS stress tensors, then the latency factor above simply selects the shear stress of minimum magnitude (in an isotropic, Boussinesq model, such as the Smagorinsky, this is the only relevant parameter). The use of the weaker of the two effective viscosities is closely reminiscent of DES and likely to be a common feature of RANS–LES hybrids. In current LNS computations, the same low-Reynolds-number damping function f μ is used in both the RANS and LES component models. The conventional Smagorinsky model has an eddy viscosity defined as  (4.6) μt = Cs (L i )2 Skl∗ Skl∗ in which Cs is the Smagorinsky coefficient (here taken as 0.05) and L  is a length-scale parameter defined for an arbitrary, unstructured control volume such as rc − rk |), L i = 4 max (| k=0...n

(4.7)

where n is the number of faces forming cell i, rc is the centroid of cell i, and rk is the midpoint of face k. For certain cell types (prismatic, tetrahedral), a preferred definition may be the recent (unpublished) recommendation of Spalart to use twice the cell diameter. This guarantees that wavelengths of L  would be resolvable at any orientation on the mesh. (The subject of defining this filter width L  is discussed further in Section 4.2.5.) The Smagorinsky SGS model requires some form of nearwall damping because, for any finite mesh spacing L  , the nonvanishing strain rate would otherwise imply a nonvanishing turbulent shear stress at the wall. Commonly employed forms of near-wall damping for the Smagorinsky model are of the van Driest type; for example,  b (4.8) f μ = 1 − exp(−y + /25)a .

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Energy production

Inertial range

α Figure 4.1. Turbulence energy spectrum partitioned into resolvable and unresolvable frequencies.

Dissipation

αk

Resolvable kinetic energy

2 π /Δ

Unresolvable kinetic energy

Wave number

The ad hoc nature of this damping is felt to justify the use of a single f μ function for both RANS and LES component models. The use of a single low-Reynolds-number damping function for both RANS and LES components is, in fact, a common feature of both LNS and DES hybrid formulations. If the near wall were resolved with a fine isotropic mesh, the use of a separate f μ for the SGS model might be justified – particularly if the Smagorinsky SGS model is to be replaced by a self-tuning or dynamic SGS model that might require less damping. However, the reader should note that the low-Reynolds-number damping of the LES subgrid stresses is not expected to be significant in any of the hybrid calculations presented here because the near-wall layer is intended to be treated via conventional RANS modeling, which is forced by the use of high-aspect-ratio near-wall cells. For the purpose of hybridization with the Smagorinsky model, the use of a single f μ function simplifies the definition of α as follows: α=

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min [(L .V )LES , (L .V )RANS ] . (L .V )RANS

(4.9)

The exact formulation for the current component RANS and LES models is given later in Equation (4.19). The resulting equations then behave as RANS if α = 1, or LES if α < 1. Thus, in LNS, the “gray zone” between pure RANS and pure LES relates only to the data on which the equations operate. In LNS, the energy fraction αk is interpreted as unresolvable subgrid turbulence kinetic energy that can only be modeled. The (1 − α)k component is interpreted as resolvable turbulence kinetic energy, which, given the local grid resolution, could be represented directly (see Figure 4.1). The sum total of k in LNS represents the current level of statistically represented turbulence energy, which is not the same as the k that would be achieved using a conventional RANS model. Similarly, the quantity α is interpreted as the dissipation that applies to the unresolvable scales, and the quantity (1 − α) is interpreted as the dissipation or transfer that applies to the resolvable scales. For a linear Boussinesq closure, the preceding definitions imply μt = αμtM (i.e., the eddy viscosity simply gets multiplied by a number

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between 0 and 1). Although there is no specific requirement on the baseline RANS model, in the present work the LNS model equations are based on a nonlinear k− model in which the Reynolds stress tensor is defined via a tensorial expansion cubic in the mean strain and vorticity tensors: ∂ ∂ρ  k k u i ∂ρ = + ∂t ∂ xi ∂ xi ∂ ∂ρ  ∂ρ   u i  = + ∂t ∂ xi ∂ xi





μt μ+ σ

μt μ+ σk



 ∂k  + Pk − ρ  ∂ xi

∂   + E)Tt−1 . + (C1 Pk − C2 ρ  ∂ xi

(4.10)

(4.11)

In the preceding equation, ui   ∂ , Pk = −ρ u i uj ∂x j

(4.12)

and

2 1 ∗ ∗ k μt ∗ ∗ ∗    Sik Sk j − Skl Skl δi j ρ u i u j = αρ kδi j − μt Si j + c1 3   3  k μt ik Sk∗j +  jk Ski∗  

k μt 1 ik  jk − lk lk δi j + c3   3

+ c2

 ∗ k2  ∗ μt ∗  + S  S Skl l j li ki k j  2

2 ∗ k2 μt + c5 2 mn nl δi j il lm Sm∗ j + Sil∗ lm m j − Slm   3

+ c4

+ c6

k2 ∗ ∗ ∗ k2 ∗ μt μt S S S + c S kl kl 7 i j kl kl  2  2 i j

with Si∗j

=

∂ uj ∂ ui + ∂x j ∂ xi

−

∂ uj ∂ ui , − i j = ∂x j ∂ xi   k 1 ∗ ∗ S= S S ,   2 ij ij   k 1 = i j i j .   2

uk 2 ∂ δi j , 3 ∂ xk

(4.13)

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Note that the expression above for the anisotropy tensor is currently used throughout (i.e., in both RANS and LES components models). A realizable turbulence time scale is defined as    (4.14) Tt = k max 1, ξ −1 , √ √ k 2 /(ν ) and Cτ = 2. The eddy viscosity is defined as where ξ = Rt /Cτ with Rt =  k 2 / , μt = αCμ f μ ρ

(4.15)

with 2/3 , A1 + S + 0.9 3/4 15/4 c1 = , c2 = , 3 (1000 + S )Cμ (1000 + S 3 )Cμ

Cμ =

c3 =

−19/4 , (1000 + S 3 )Cμ

c5 = 0, and

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c4 = −10Cμ2 ,

c6 = −2Cμ2 , c7 = −c6 ,

  1 1 − e−Aμ Rt √ max 1, , fμ = ξ 1 − e − Rt  1  1 k 2 , (ν ) 4  Tt τ , E = A Eτ ρ max     k ∂ k ∂τ ,0 , τ = . τ = max ∂x j ∂x j  

(4.16) (4.17) (4.18)

From Equation (4.9), the latency parameter α is defined for the current choice of RANS and LES component models as    2  Skl∗ Skl∗ /2 Cs L i ,1 (4.19) α = min k 2 / Cμ +δ with δ some small parameter O(10−20 ) to allow α → 1 without singularities in lowReynolds-number regions. The remaining model constants are A1 = 1.25, C1 = 1.44, C2 = 1.92, σk = 1.0, σ = 1.3, Aμ = 0.01, A Eτ = 0.15, Cs = 0.05. When fine isotropic grid regions are encountered by the LNS method, the scaling of the predicted Reynolds stress tensor by α causes the effective viscosity to be reduced instantly to the levels implied by the underlying LES subgrid model, and the local flow also experiences a decreased rate of production owing to the reduced magnitude of the stress tensor components. In the basic version of LNS, nothing is done

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with the resolvable-scale (1 − α)k energy fraction, which continues to decay according to the local dissipation rate. More recent efforts aim to convert this statistical data into resolved-scale fluctuations (Batten, Goldberg, and Chakravarthy 2004). The use of realizable stress tensors to generate a synthetic reconstruction of RANS data is discussed separately in Section 4.2.7. 4.2.4 The approach of Menter, Kunz, and Bender

A potential problem can arise with the previously discussed hybrid RANS–LES methods when flow is transported (either through mean convection or turbulent mixing) across regions of varying mesh resolution. Because the predicted subgrid stresses in these methods are intentionally sensitized to the local mesh resolution, the model equations will change with L  . However, a switch from RANS to LES mode (or vice versa) does not imply an instantaneous transformation of the data (such as the velocity field) on which those equations operate. This can be problematic if the grid experiences a significant clustering (with decreasing aspect ratio) within the boundary layer (or indeed in any region of significant turbulent and mean-flow interaction) because a sudden switch to LES would reduce the unresolved stresses and the approaching statistically steady flow would have no opportunity to initiate or sustain the fluctuations that would be required to compensate by an increase in the resolved stresses. For separated flows around sufficiently simple geometries, suitable mesh design can eliminate this problem by maintaining uniformly high-aspect-ratio cells in the near-wall layer. However, for realistic geometries involving abrupt surface curvatures, it can become difficult to satisfy such stringent meshing constraints, which would also preclude or limit the use of local adaptive mesh refinement. Although these problems are now reasonably well understood, there have been few proposed solutions. A recent extension of the LNS method attempts to address this issue by automatically extracting statistically represented kinetic energy and injecting this into the resolvable scales (Batten, Goldberg, and Chakravarthy 2004). This approach relies on a tensorial extension of Kraichnan’s (1969) method described in Section 4.2.7. However, calculations by Batten, Goldberg, and Chakravarthy (2004) on a plane-channel flow with streamwise grid clustering showed that several channel half-heights were required to switch to a fully developed hybrid RANS–LES boundary layer – a distance that would not be of any practical help in regions containing abrupt clusterings owing to surface curvature. For sufficiently simple geometries, the recycling approach of Lund, Wu, and Squires (1998) can be used to maintain unsteady inlet data; however, its generalization presents some challenges, and, again, its use is not practical for localized mesh clusterings. Furthermore, rescaling only accounts for the effects of data transported through mean convection and does not address the problem in hybrid RANS–LES of turbulent transport of kinetic energy through the boundary layer. If the near-wall layer is described only in terms of statistical data, this presents the unsteady outer flow with a nonphysical quasi-steady layer that can cause a (resolved-scale) component laminarization and, as a result, a nonphysical description of the total shear stress. Rescaling can always be

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applied to match statistics at a certain inlet plane, but this does not guarantee that those statistics will be preserved downstream even as far as the end of the recycling zone. A recent proposal by Menter et al. (2003) offers an intriguing alternative. The Menter et al. (2003) approach is based on a single transport equation for the eddy viscosity:



νt 2 ∂ μt ∂νt ∂(ρνt ) ρu j νt = c1 μt S − c2 ρ + + . (4.20) ∂t ∂x j LvK ∂x j σ ∂x j For a simple boundary layer, the von K´arm´an length scale L vK is defined as    ∂u/∂ y   . L vK =  2 ∂ u/∂ y 2 

(4.21)

For general multidimensional calculations, Menter et al. (2003) suggest two alternative invariant formulations for L vK (note the missing square roots in the original paper of Menter et al. 2003):   (S.S)   L vK =   (4.22) ∂S ∂S . ∂x j ∂x j or L vK

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   =

∂u i ∂u i ∂x j ∂x j ∂ 2 ul ∂ 2 ul ∂ xm2 ∂ xn2

.

(4.23)

In an earlier paper on the derivation of a single-equation, eddy-viscosity transport model, Menter (1997) modified the dissipation based on the preceding length scale in order to avoid the singularity caused by a vanishing mean-strain rate:  !



νt 2 νt 2 ∂νt ∂νt ⇒ c2 ρ min , . (4.24) c2 ρ LvK LvK ∂x j ∂x j By removing the second argument on the right-hand side of Equation (4.24) and instead using some grid-related tolerance on L vK , the dissipation in the resulting hybrid RANS– LES model becomes sensitized to the local flow structures. It is curious to note that this expression for a dissipation length scale arose naturally in the derivation of a single-equation model for the k− eddy viscosity (Menter 1997). The claim of Menter et al. (2003) that their approach contains no explicit grid dependency is not strictly correct, however, because some fraction of  is still explicitly required as a tolerance on L vK in order to allow reasonable agreement with isotropic decaying turbulence (Menter et al. 2003). This suggests that the numerically determined L vK underestimates the required dissipation length scale – an effect that is probably required if the largerscale eddies are to be allowed to follow their physical breakdown to the Kolmogorov scale but one that defeats the objective of entirely removing the  dependency from the model. Nevertheless, variants of this method may help to preserve the near-wall RANS layer in situations in which mesh refinement is used (locally) to help represent complex surface topography. This issue is considered further in the following section.

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4.2.5 Defining the filter width

The definition of a characteristic length scale is a key factor in the design of nonzonal hybrid RANS–LES models (just as in pure LES) because this allows the approach to distinguish between unresolvable and resolvable (or at least potentially resolvable) scales of motion. Because the orientation of local flow structures cannot be known in advance (except perhaps in the vicinity of a wall), a safe requirement is to choose the smallest wavelength that can be supported at any orientation to the local mesh. This leads us to a measure for the local Nyquist grid wavelength, given by expression (4.7), which, for a Cartesian mesh, reduces to L  = 2 max(x, y, z)

(4.25)

with the additional factor of 2 (often absorbed into the Smagorinsky constant) accounting for the wavelength corresponding to the grid Nyquist frequency. A similar measure of filter length is used in both LNS and DES methods. However, there is no universal agreement on this definition, and it is still common for many practitioners to employ measures such as the cube root of the volume (as in the approach of Labourasse and Sagaut 2002), or even to replace the maximum in Equation (4.25) with a minimum. Such definitions reach the wrong limit for extreme cell shapes such as a “chopstick,” but this has not been recognized in typical LES of free-shear flows, with their nearly cubic cells. Hybrid methods do make the issue more critical in that they lead to cells that are far from cubic. Additional filter-length controls are currently under investigation within the LNS method. For example, the insistence that the time step be small enough to resolve the convective transport of any grid-supportable structures accurately leads to a Courant condition based on the local fluid velocity (which defines a basic condition for accuracy in nonlinear flow problems). This can be used to provide an additional safety factor in the local length-scale definition:   L  = 2 max x, y, z, u i2 t ,

(4.26)

where u i is the local fluid velocity relative to the mesh. This ensures that, irrespective of the local spatial resolution, the RANS solution is recovered when t becomes large. This additional constraint can help in regions of sharp surface curvature (such as corners) where the mesh resolution may become uniformly fine but the temporal resolution would not permit those very-fine-scale structures to be predicted accurately (these structures would be on the order of the smallest boundary layer spacing in all directions whereas the global time step is typically set to match the larger cell spacings away from the immediate near-wall region). However, the modified length scale (4.26) is recognized as being incomplete. In stagnation and reattachment regions, Equation (4.26) will not augment the local filter width because of the vanishing mean kinetic

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energy. Additional length-scale constraints have been proposed such as the following suggestion of Mani and Paynter (2002):

  √ (4.27) L  = 2 max x, y, z, u i2 t, kt . The approach above has not been widely tested but is also considered potentially problematic because large initial values for k (such as those from a RANS solution) could perpetuate local RANS behavior. Another alternative would be to introduce a solution-dependent length scale such as the von K´arm´an length scale (4.22) used by Menter et al. (2003) in their recently proposed hybrid approach. A disadvantage of using the Menter et al. (2003) method directly is that velocity profile curvature in any one direction can force adoption of a small length scale (analogous to choosing min[x, y, z] in LNS or DES), and thus it becomes impossible to guarantee a near-wall RANS layer (note that the computations presented in Menter et al. (2003) were performed with wall functions). A safer approach would be to introduce the von K´arm´an length scale into Equation (4.25), giving L  = 2 max[x, y, z, L vK ].

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(4.28)

The preceding approach ensures that near-wall RANS behavior will be maintained on stretched, high-aspect cells irrespective of the normal-to-wall component of velocity curvature. The constraint above is being tested as a modification to the LNS method, denoted LNSSAS , but could equally be employed in a modified DESSAS method by using Equation (4.28) in place of the usual grid scale (4.25). Current experience suggests that modification (4.28) can be unpredictable – in some cases allowing the length scale to diminish in situations in which a preserved RANS layer was desired, and in other cases preventing the length scale from diminishing in flows displaying only weak instabilities. In the latter case, LNSSAS or DESSAS methods may never breed any unsteady motions, and hence such modifications make a priori identification of resolved and unresolved noise sources difficult. Furthermore, this approach puts the interface region between RANS and LES at the most critical location in the flow – the very point at which the primary separations are initiated. A simpler approach, which has proven effective in certain LNS calculations, is to explicitly enforce a minimum filter width   (4.29) L  = 2 max x, y, z, L  min in which L  min is a user-supplied length scale corresponding to the size of the largeeddy resolving cells (typically located away from the near wall). If suitably chosen, this can prevent the rapid switching from RANS → LES → RANS in regions of abrupt near-surface mesh clustering, thereby preserving the near-wall RANS layer. For many problems, however, a more durable solution would be to initiate the transfer of data to the LES mode some distance upstream of the interaction region, which would require the use of recycling techniques, LES databases, or synthetic reconstruction or

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forcing using available RANS statistics. Research in the area of synthetic turbulence is growing rapidly, and a brief review is provided in Section 4.2.7. 4.2.6 Modeling the noise from unresolved scales

The aim of time-dependent techniques such as LES or hybrid RANS–LES is to account only for the largest scales of motion and hence, by implication, the most significant noise-generating scales. There remains, inevitably, an upper bound on the frequency that can be predicted directly via such a time-dependent approach as a result of finite spatial and temporal resolution. The objective of the global hybrid RANS–LES methods is to reduce the overwhelming computer cost of traditional LES by resorting to RANS in certain locations (such as the near-wall region). Thus, the range of scales predicted by a global hybrid RANS–LES technique is likely to be less than that of a full-domain or zonal LES (if it is assumed these were computationally affordable); hence, we may expect the problem of missing noise from unresolved scales to be exacerbated by the use of global hybrid RANS–LES. The coarse mesh limit of hybrid RANS–LES may still adequately reproduce turbulence statistics but will leave the corresponding scales of motion unresolved. It is therefore necessary to provide some mechanism to extract noise from the unresolved fraction of hybrid RANS–LES simulations. To facilitate this, we consider the noise sources to be split into resolvable and unresolvable components with all resolvable components assumed to be accounted for via the hybrid RANS–LES approach. This step is trivial with a constant filter length but can be complicated by the use of ambiguous or solution-dependent (and hence time-varying) filter lengths, as in the approach of Menter et al. (2003). Because analytic methods are needed to generate and propagate unresolved noise sources, Lighthill’s equation is considered written in tensor form as follows: 2 ∂ 2 Ti j ∂ 2ρ 2 ∂ ρ − c∞ = , 2 2 ∂t ∂ xi ∂ x j ∂ xi

in which

  2 Ti j = ρu i u j − τi j + p − c∞ ρ δi j .

(4.30)

(4.31)

The usual assumption (in what is otherwise an exact mathematical manipulation of the Navier–Stokes equations) is that the right-hand-side terms are known and independent of the left-hand side, which then simply represents a wave-propagation operator. A solution to this equation was proposed by Curle (1955) and a more general extension to moving surfaces by Ffowcs Williams and Hawkings (1969). These two approaches are essentially equivalent for fixed (nonmoving surfaces). Curle’s approach is based on a solution to the inhomogeneous wave equation (4.30) on a finite domain: """ 1 1 ∂ 2 Ti j  ρ(x j , t) = dV 2 4π c∞ r ∂ yi ∂ y j

"" 1 ∂ρ 1 ∂r 1 ∂r ∂ρ 1 + 2 ρ+ d S, (4.32) − 4π r ∂n r ∂n c∞r ∂n ∂τ

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where n defines the local surface normal. A solution to Curle’s equation, which is valid for arbitrary r , can be written as (e.g., Larsson 2002)

""" li l j ∂ 2 Ti j 3li l j − δi j 3li l j − δi j ∂ Ti j 1  + + Ti j d V p (xi , t) = 2 r ∂t 2 4π c∞ c∞ r 2 ∂t r3



"" pδi j − τi j ∂τi j 1 ∂p 1 + δi j − + d S. (4.33) li n j 4π c∞r ∂t ∂t r2 Given a suitable mathematical model for the subgrid velocity fluctuations, Equation (4.33) can be used to completely determine the acoustic pressure generated and transmitted from the unresolved scales. Representative results presented later in this chapter were generated using the modeled, synthetic reconstruction of Batten, Goldberg, and Chakravarthy (2004), which is presented in detail in Section 4.2.7.

4.2.7 Synthetic reconstruction of turbulence

This section considers some recent proposals for generating a field of synthetic, or artificial, velocity fluctuations from a given set of turbulence statistics typical of those obtained from a classical RANS method. Synthetic turbulence is required in several areas critical to LES and acoustics: (i) The generation of unsteady, turbulent inlet velocity fields – particularly for inhomogeneous (e.g., spatially developing) flows. (ii) The stimulation of large-scale (resolvable) eddy structures at coarse–fine interfaces or mesh clusterings in hybrid RANS–LES. (iii) The generation of full- or part-spectrum synthetic noise sources for use with numerical or analytic acoustic wave propagation methods. The prescription of an unsteady inlet condition for an LES is probably the most pressing of the issues above. Obtaining quality unsteady inlet data (consistent with the local numerical resolution and Navier–Stokes equations) is a problem that is becoming increasingly important because of recent emphasis on hybrid RANS–LES methods. For LES or hybrid RANS–LES, the imposition of a steady or quasi-steady inlet velocity field is always incorrect for a fully turbulent flow; however, the resulting errors may not be significant in situations in which a strong, inherent instability exists downstream and is able to overwhelm any upstream disturbances. Much of the early work on hybrid RANS– LES concentrated on exactly this type of flow, but, because the range of applications has widened, the issue of unsteady boundary conditions has become more acute. A synthetic reconstruction of turbulence is attractive because it offers an inexpensive method of generating an unsteady flow field from any localized set of RANS statistics. For inlet boundary conditions, the main alternative is the use of a separate, auxiliary section of LES, which may contain more physics than a synthetic model but adds a significant overhead to the calculation and creates a large burden for the end user in specifying the separate LES zone, the positioning of any overlapping or recycled inlet

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planes, and the interpolation and rescaling of mean (and possibly fluctuating) quantities to account for developing boundary layers or other inhomogeneities. The synthetic method outlined in Equation (4.34) was developed by Batten, Goldberg, and Chakravarthy (2004) as an anisotropic extension to Kraichnan’s (1969) method. This method allows a reconstruction of spatially and temporally coherent turbulent velocity fluctuations from a given dissipation rate and set of second moments. This formulation follows from a simplification of the tensor scaling proposed by Smirnov, Shi, and Celik (2001) in which the need to compute similarity transformations is avoided:  u i (x j , t) = aik

N      2 # pkn cos dˆ nj xˆ j + ωn tˆ + qkn sin dˆ nj xˆ j + ωn tˆ N n=1

(4.34)

V xˆ j = 2π x j /L , tˆ = 2π t/τ, dˆnj = d nj n , V = L/τ, c  3   n n n n cn = u u d d /d d , pin = i jk ηnj dkn , qin = i jk ξ nj dkn , 2 l m l m k k

1 . ηin , ξin = N (0, 1), ωn = N (1, 1), din = N 0, 2 In the preceding equation, ai j is the Cholesky decomposition of u i u j . The Cholesky decomposition is the lower triangular matrix ai j , with aik akTj = u i u j . For a symmetric, positive, definite Reynolds stress tensor, u i u j , ai j can be determined as ⎛ ⎞   u1u1 0 0 ⎜ ⎟  ⎜   ⎟   2 (4.35) ai j = ⎜ u 1 u 2 /a11 ⎟. u 2 u 2 − a21 0 ⎝ ⎠  2 2 u 1 u 3 /a11 (u 2 u 3 − a21 a31 )/a22 u 3 u 3 − a31 − a32 Note that all elements of the Cholesky decomposition are real if the target stress tensor is realizable. In the synthetic model Equation (4.34) above, the time and space correlations are represented by scaling the local time and distance coordinates in the Fourier model by the local turbulence time scale τ and (anisotropic) velocity scale cn , which is a tensorially invariant measure in the direction of the modal wave vector din . This feature tends to elongate the synthetic eddies preferentially in the direction of the strongest correlations, producing spherical-shaped eddies in which the turbulence is closer to isotropy (typically, away from walls) and flatter eddies near solid surfaces where the normal-to-wall velocity tends to be preferentially damped by inviscid blocking effects. In the long-time average, the synthesized time-dependent flow field will reproduce the given length and time correlations and all second moments. The storage requirements for this method are dictated only by the number of modes and are independent of the grid size. Because each mode in this Fourier reconstruction has an associated wavelength (the wave-vector modulus), it is possible to reconstruct fluctuations over the full spectrum, or, by selectively summing over each mode (e.g., if L/|d n | < L  ), it is possible

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to reconstruct only that portion of the spectrum that would remain unresolved in an LES or hybrid RANS–LES. It is, of course, equally possible to reconstruct only those components that are resolvable on the local mesh – for example, in hybrid RANS–LES methods in which the finer scales are treated via a subgrid model. The latter situation is particularly relevant to hybrid RANS–LES calculations in regions in which the filter width suddenly (or gradually) diminishes, implying a potential switch from RANS to LES mode. Another useful feature of this mathematical description is that the modeled function is continuously (and easily) differentiable, making it simple to compute the temporal or spatial derivatives, or both, needed for analytic acoustic propagation methods such as Equation (4.33). An important practical consideration for inflow conditions is the subsequent spatial evolution of the synthetic turbulence before its arrival at the interaction region and, in particular, the preservation and transmission of the imposed target statistics. Synthetic inlet data can be expected to undergo a transient period during which physical structures develop. Experience suggests that the turbulence energy and shear stresses can sometimes undergo an initial decay during this transient. This decay may be partly due to the presence of nonphysical structures but also results from numerical and (modeled) subgrid viscosity, which may damp signals at wavelengths that the reconstruction criteria in the synthetic turbulence expected to be resolvable (as determined by the SGS model and numerical scheme employed, the smallest supported structures may be significantly larger than the filter width L  ). Recent work by Spille-Kohoff and Kaltenbach (2001) and Keating et al. (2004) suggests that an imposed forcing at a sequence of control planes downstream of the inlet can help to mitigate unwanted decays in the shear stress from synthetically reconstructed inlet data. The controlled forcing approach currently lacks some generality and burdens the user with the task of imposing appropriate data at the various control planes; however, the method is much less expensive than using an auxiliary LES. Future research is expected to improve the quality of synthetic turbulence data, which should reduce or eliminate the need for additional forcing terms.

4.2.8 The NLAS approach of Batten, Goldberg, and Chakravarthy

An alternative form of hybrid method was presented by Batten et al. (2002c) and Batten, Ribaldone et al. (2004) as a means of reducing the diffusive effect of the subgrid model and directly accounting for the generation of acoustic waves from SGS structures. The derivation of this NLAS begins with the Navier–Stokes equations: ∂ Fiv ∂ Fi ∂Q − = 0, + ∂t ∂ xi ∂ xi ⎤ ρ Q = ⎣ ρu j ⎦ , e ⎡

⎡

⎤ ρu i Fi = ⎣ ρu i u j + pδi j ⎦ , u i (e + p)

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(4.36) ⎡

⎤ 0 ⎦. Fiv = ⎣ τi j −θi + u k τki

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In aeroacoustics, manipulations of these equations are usually made in order to generate simplified wave equations that are amenable to analytic solutions (given a separate hypothesis and model for the noise sources). Here, we consider a system of equations that is formally identical to the Navier–Stokes equations (i.e., containing no simplifying assumptions or elimination of variables) but is derived by considering each primitive variable as being split into mean and fluctuating quantities, φ = φ + φ  . Substituting into the Navier–Stokes equations and rearranging for fluctuation and mean quantities gives a system of perturbation equations referred to as nonlinear disturbance equations (NLDE):   v ∂ Fiv ∂ Fi ∂ Q ∂Q ∂ Fi ∂ Fi − =− + , (4.37) + − ∂t ∂ xi ∂ xi ∂t ∂ xi ∂ xi ⎡

⎤ ρ Q = ⎣ρ uj ⎦, e

⎡

⎤ ρ ui F i = ⎣ ρ u i u j + pδi j ⎦ , u i (e + p)

⎤ ρ Q  = ⎣ ρu j + ρ  u j + ρ  u j ⎦ , e ⎡



 v 

Fi

⎡

v Fi

⎤ 0 ⎦, =⎣ τij −θ i + u k τ ki

⎤ 0 ⎦, τij =⎣    −θi + u k τ ki + u k τki ⎡

⎤ ⎡ ⎤ ρ  u i ρu i + ρ  u i Fi = ⎣ρ  u i u j + ρu i u j + ρu i u j + p  δi j⎦ + ⎣ρu i u j + ρ  u i u j + ρ  u i u j + ρ  u i u j⎦ . u i (e + p) + u i (e + p  ) u i (e + p  ) ⎡

Neglecting density fluctuations and taking time averages of the preceding system of equations cause the evolution terms and all flux terms linear in the perturbations to vanish, resulting in LHS = RHS = in which

∂ Ri ∂ xi

(4.38)

⎡

⎤ 0 ⎢ ⎥ ρu i u j Ri = ⎣ ⎦. 1         c p ρT u i + ρu i u k u k + 2 ρu k u k u i + u k τki

The equations above correspond to the standard Reynolds stress tensor and heat flux terms. The key step is to obtain these unknown terms in advance from a classical RANS method. The unresolvable (short-wavelength) contribution to these terms can then be generated using a synthetic model of turbulence such as that described in Section 4.2.7. With these mean quantities and statistics established, time-dependent computations can then be made to determine the perturbations about this mean using Equation (4.37). Naturally, the time-dependent computation may reveal more physics than was available in the precursor (steady-state) RANS calculation; hence, the time-averaged fluctuations

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DNS

LES

Hybrid RANS–LES

NLAS– NLDE

Figure 4.2. Required near-wall mesh resolutions with DNS, traditional LES, global hybrid RANS–LES, and nonlinear acoustics solvers (NLAS) based on disturbance equations.

from the second, time-dependent calculation are not expected to agree fully with the initial RANS mean data.∗ The corresponding inviscid system of disturbance equations was first proposed by Morris, Long et al. (1997), and there may be many situations in which inviscid disturbance equations are sufficient – particularly when using separate acoustics meshes in which cell sizes are too large to properly resolve viscous layers. A grid-converged Reynolds stress tensor can still be imposed through the background (RANS) field, but the resulting stresses (in addition to cell-size-related numerical viscosity), are likely to dominate the effects of the physical viscosity on such meshes, making the retention of viscous terms unnecessary. However, more recent computations involving disturbance equations (e.g., Batten et al. 2002c; Batten, Ribaldone et al. 2004; Labourasse and Sagaut 2002) have favored the more general approach of retaining viscous terms. If the local mesh is unable to support any resolved-scale fluctuations (e.g., in coarse or high-aspect-ratio cells), the NLAS method of Batten et al. (2002c) can still access a set of (grid-converged) RANS statistics to provide a description of the subgrid. In the NLAS method, these data are provided in the form of a synthetic reconstruction of the RANS statistics (see Section 4.2.7). This NLAS method offers several interesting advantages over both traditional LES and more conventional hybrid RANS–LES methods such as DES (Section 4.2.2) or LNS (Section 4.2.3). NLAS provides a more sophisticated subgrid treatment that allows the extraction of acoustic sources from the temporal variation within the (modeled) subgrid structures; grid requirements can be relaxed in the near-wall region during the NLAS transient calculation (the quasi-steady near-wall RANS solution being obtained a priori; see Figure 4.2); the dissipative effects of a subgrid eddy viscosity model are avoided; thus, on coarser meshes, NLAS proves less diffusive. In terms of cost, NLAS is competitive with linearized Euler equation (LEE) and acoustic-perturbation equation (APE) methods (e.g., Ewert, Meinke, and Schr¨oder 2001b) having very similar mesh resolution and flux evaluation requirements, but the ∗

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If the only interest were the noise emission from the initial RANS solution, the NLDE mean could be forced to agree with the RANS mean by subtracting out an averaged source term at each time step. In general, however, the objective is also to use the available information on the unsteady flow physics to improve the mean-flow predictions.

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synthetic source reconstruction is less costly in NLAS owing to the imposed filtering. In addition, NLAS has the important advantages over LEE- and APE-based methods in that it can account both for broadband, turbulence-related noise and discrete tones arising from coherent structures or resonance, and (like traditional LES) its empirical modeling content decreases with increasing spatial resolution. The use of background RANS statistics also has the potential to reduce the far-field mesh requirements. In the commercial CFD++ code (see Peroomian et al. 1998), NLAS can operate on a subdomain with absorbing far-field boundaries defined by a truncated subset of the original RANS calculation, which provides not only turbulence statistics but also spatially varying mean-field data that can be used to place the outer boundary conditions much closer to the regions of interest. Experience with methods based on nonlinear disturbance equations and a priori RANS calculations is less extensive than with DES (Section 4.2.2) or LNS (Section 4.2.3), and some issues have yet to be resolved. For example, the near-wall treatment might be brought into question if RANS and NLDE mean flows were substantially different in separation or reattachment locations. It may also prove difficult to produce a statistically steady RANS solution for flows that have an inherent, violent unsteadiness (although conventional hybrid RANS–LES methods are likely to be well suited to this category of flow). Nonlinear disturbance equations are, however, expected to outperform more traditional hybrid RANS–LES methods on flows with weak shear instabilities.

4.3 Zonal hybrid approaches Zonal hybrid methods are based on a purely discontinuous treatment between RANS and LES approaches. This is achieved by means of a domain decomposition, with each domain being solved in either pure RANS or pure LES mode, as dictated in advance by the user. The previous global approaches assume a continuous treatment between the RANS and LES modes and thus introduce an acknowledged “gray area” in which the approach may act as neither RANS nor LES. The use of “pure” LES and “pure” RANS zones may remove the issue of these gray areas from within the zones; however, it also creates a clear requirement to account for the discontinuous boundary condition between the two distinct descriptions of the flow given by RANS and LES. This required two-way interaction typically satisfies conservation only in the long-time average because the RANS zones require boundary conditions comprising averaged (statistical) data, whereas the LES zones require boundary condition data containing the full range of unsteady motion, which must ultimately be consistent with the RANStransported statistics, including local scales of length and time. Although such internal boundary conditions are a clear requirement for zonal methods, similar treatments are also urgently needed in certain global hybrid RANS–LES applications. For example, in the case of an abrupt mesh refinement (i.e., if the global hybrid approaches were deployed on the same zonal meshes) the two approaches would exactly coincide because

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the finer mesh in the LES zones would force a switch to LES behavior within the global RANS–LES hybrids. Pure zonal methods, if sufficiently well resolved, allow the possibility of directly generating a locally accurate description of the unsteady noise sources in the LES zones with less (or perhaps no) modeling being required to extract the full spectrum of near-field noise (far-field radiation will still require additional procedures). However, because of the differing near-wall resolution requirements,  and therefore t are expected to be much smaller in zonal calculations than in calculations using global hybrid RANS–LES approaches. Finer resolution in the LES zones would mitigate the requirement for additional modeling to extract noise from unresolved scales but would again require much larger (possibly prohibitive) computing resources. The difficult (and, as yet, not fully solved) problem of the coupling between RANS and LES regions currently reduces the range of applicability for both global and zonal methods. The simplest possible coupling relies on a direct injection of the RANS values at the boundaries of the LES zone and an average (or simply a direct injection; Georgiadis, Alexander, and Reshotko 2003) of the LES field to create a boundary condition for the RANS region. However, such a simple coupling would be of limited practical use. Failing to take into account the intrinsically different natures of the RANS and LES descriptions of the flow field would prevent the transmission of turbulence data (in either resolved or statistical form) from one region to the other. An LES data representation will generally contain much more information than the corresponding RANS data, and thus the RANS field cannot be used to generate data for the LES region without additional modeling, which must include further assumptions on local length scales, time scales, and energy distributions. This topic is discussed further in Section 4.3.3. Recent work carried out by Qu´em´er´e and Sagaut (2002) and Labourasse and Sagaut (2002) involves zonal approaches that consider discontinuous boundaries between RANS and LES regions. Both these methods also adopt the use of NLDE, which can be viewed as an extension of the multilevel–multiresolution formalism of Harten (1994, 1996) and Terracol et al. (2001) in which two levels of solution resolution are introduced by the respective RANS and LES data representations. Each LES-filtered (resolved-scale) flow quantity φ is decomposed into φfiltered = φ + φ  ,

(4.39)

where the overbar denotes the Reynolds average and φ  is the frequency complement between the RANS and the LES representation of φ. This last quantity is often referred to as the detail in the multilevel representation. The complete field φ can be considered as being decomposed using a triple decomposition: φcomplete = φ + φ  + φSGS ,

(4.40)

where φSGS refers to the additional (unknown) detail modeled by the subgrid terms.

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4.3.1 The approach of Quem ´ er ´ e´ and Sagaut

The approach proposed by Qu´em´er´e and Sagaut (2002) was one of the first attempts to derive a consistent discontinuous coupling between the two (RANS and LES) approaches. The authors consider a global RANS domain in which a small overlapping region is devoted to LES. In this region, a two-way coupling exists as follows: r The information provided by the RANS solver is used to derive some boundary

conditions for the LES domain. To take into account the discontinuity between the two fields, the authors have proposed two procedures to synthesize turbulent fluctuations. The first one relies on an extrapolation of the details (φ  ) computed in the LES region in the ghost cells of the LES domain. In these cells, a reconstruction of φ is then performed with the averaged field φ given by RANS. The authors indicate that such a treatment is only valuable in the case of “lateral” or “outflow” boundaries. For inflow boundaries, the authors have proposed to use a predictor simulation and to apply the rescaling suggested by Lund et al. (1998) (see the description at end of this section). r A feedback from the LES region to the steady RANS or unsteady Reynolds-averaged Navier–Stokes (URANS) region also exists. This is achieved by averaging the LES field to provide data for the RANS field boundary conditions in the overlap region. In this region, traditional RANS transport equations are used to determine the quantities k, ε, and μt (Qu´em´er´e and Sagaut 2002). The potential of this method was first demonstrated on plane-channel flow calculations in cases in which the LES regions corresponded to the near-wall layer or to the core region of the channel. In the latter case, the underlying idea of the method is very close to DES (in which RANS–LES regions are effectively fixed a priori by the wall-distance function and choices of local cell sizes). Finally, the authors have applied their approach to the simulation of a base flow with a small LES region located at the trailing edge of a flat plate computed by URANS. 4.3.2 The approach of Labourasse and Sagaut

The approach proposed by Labourasse and Sagaut (2002), is again based on the solution of the NLDE in certain zones, and an initial (or separate) RANS simulation is used to provide the global mean-field and boundary conditions for the LES domains. In the terminology used by Labourasse and Sagaut (2002), the NLDE read  ∂ Fiv    ∂ Fi   ∂ Q ∂ Fi   ∂ Fiv   + Q +Q − Q + Q − τSGS = Q − Q − τRANS . ∂t ∂ xi ∂ xi ∂ xi ∂ xi (4.41) The two terms τSGS and τRANS denote, respectively, unknown terms in the LES and RANS equations that both require modeling. The system of equations (4.41),

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which describes the temporal evolution of the details or difference between the LES and RANS solutions, is then solved in the LES region instead of the classical LES equations. This minimizes the sensitivity of the solution to numerical errors or to the use of relatively coarse discretization grids, or both. In addition, the perturbation field Q  can sometimes display less sensitivity to the numerical treatment at boundaries than the total field values Q + Q  (because any errors are then committed on a smaller fraction of the total field value). Indeed, the correlation lengths and times associated with the perturbations are smaller than those of the full-field variables, which can lead to a more rapid damping of the numerical errors at boundaries. However, the authors have shown (Sagaut et al. 2003) that a careful treatment is still needed to prevent strong reflections at the interfaces of the LES-like (viscous NLDE) domain for both the unsteady turbulent structures and acoustic waves. The authors have thus proposed to use an extension of existing characteristic theory applied to the perturbation equations. In the nonconservative form, the Navier–Stokes equations can be written ∂V ∂V ∂V ∂V +A +B +C = VIS ∂t ∂x ∂y ∂z

(4.42)

where V = (ρ, u 1 , u 2 , u 3 , p)T is the primitive variable vector, VIS accounts for the viscosity, and SGS terms and A, B, and C are the classical convective-flux Jacobian matrices. V can be divided into three parts according to the NLDE decomposition: V = V + V  + VSGS .

(4.43)

The first part deals with the mean field, whereas the second part is connected with the fluctuating field V  = (ρ  , u 1 , u 2 , u 3 , p  )T . As in all other works dealing with the subject, the contribution of the SGS term VSGS is neglected in this characteristic analysis. This leads to the following equation for the fluctuating field: ∂V  ∂t

VIS −









+ A ∂∂Vx + B ∂∂Vy + C ∂∂zV =

∂V ∂t

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 + A ∂∂Vx + B ∂∂Vy + C ∂∂zV − VIS ,

(4.44)

where VIS and VIS account, respectively, for the fluctuating and mean part of the viscosity terms. By analogy with the extension of the characteristic theory to viscous flows (e.g., Poinsot and Lele 1992), only the convection terms of the fluctuations are considered, which leads us to neglect the right-hand side of Equation (4.44). The resulting equation is ∂V  ∂V  ∂V  ∂V  +A +B +C = 0. ∂t ∂x ∂y ∂z

(4.45)

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The matrix E = An x + Bn y + Cn z (where n = (nx , ny , nz )T is the vector normal to the considered boundary) has to be diagonalized to set conditions on the characteristic variables of the equation:  = L E L −1 = diag(λ1 , λ2 , λ3 , λ4 , λ5 ) = diag(u n , u n , u n , u n + c, u n − c),

(4.46)

where u n is the projection of the velocity vector on the normal to the boundary, c is the sound velocity, and L is the matrix of the left eigenvectors. The temporal evolutions of the primitive variables (δV  ) and those of the characteristic variables are then linked by δW  = LδV  or δV  = L −1 δW  ,

(4.47)

where W  is the vector of characteristic variables. A nonreflecting boundary condition for the fluctuating field can then finally be derived following Thomson (1987) as ⎧ ⎨0 if λi ≤ 0 , ∂W (4.48) ≡ δW  = W  n+1 − W  n ⎩ ∂t if λi > 0 . t The approach proposed by Labourasse and Sagaut (2002) has been applied successfully in both academic and more realistic configurations. Initial computations were performed on a plane-channel flow configuration and revealed the ability of the method to handle coarser meshes than regular LES and to work with reduced LES domains (Labourasse and Sagaut 2002). Several computations have also been performed in the case of a low-pressure T106 turbine blade (Labourasse and Sagaut 2002; Sagaut et al. 2003), where a small LES–NLDE computational domain was localized at the blade’s trailing edge while the overall configuration was treated by the RANS approach. These computations have allowed the authors to perform a critical analysis of the boundary conditions used for the LES–NLDE domain. The result of this analysis is that a nonreflective boundary treatment, such as the one described in Equation (4.48), is necessary to allow the correct transmission through the boundary of both turbulent vortical structures and acoustic waves. The approach has also been applied more recently to the case of the unsteady flow in the slat cove of a high-lift airfoil (Terracol et al. 2003). The computations performed in this case have enabled an analysis of the various flow phenomena responsible for acoustic wave emission. Some of the results obtained thus are reported in Section 4.4. 4.3.3 Zonal-interface boundary coupling

Despite the reported success of the zonal RANS–LES approaches described in this chapter, the coupling between the two (RANS and LES) approaches at the domain interfaces remains an open problem. Indeed, the numerical treatment proposed by Labourasse and Sagaut (2002) can only be applied to lateral or outflow interfaces (where the mean flow is parallel to the interface or passes from the LES to the RANS region). However, the

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numerical treatment of inflow sections (where fully turbulent flow enters the LES region) remains a more challenging problem. The zonal approach proposed by Labourasse and Sagaut (2002) has been successfully applied to application cases in which the flow at the inflow boundary of the LES domain was quasi-laminar but would fail to accurately reproduce flows with turbulent inflow conditions in the LES region∗ . To handle turbulent flows at this interface, the method needs to incorporate a more detailed data transfer procedure such as a synthetic reconstruction from the given RANS statistics. Several approaches are possible for generating these inflow conditions for the LES region. The simplest approach involves a superposition of white noise on the mean velocity profile. A further elaboration would be a white noise signal that would reproduce the single-point statistics. However, most previously reported attempts in this direction have failed owing to the rapid dissipation of Nyquist-frequency signals by the subgrid treatment. Important requirements in this synthetic reconstruction are realistic spatial and temporal correlations that allow the maintenance, regeneration, or both of physical turbulent structures (Chung and Sung 1997). Section 4.2.7 presents a method for synthesizing turbulent (RANS) data that reproduces the second moments as well as the two-point correlations. This method is already being applied to RANS–LES interfaces (e.g., Batten, Goldberg, and Chakravarthy 2004). For sufficiently simple geometries, an alternative strategy relies on the use of a periodic recycling and rescaling such as that proposed by Lund et al. (1998) to generate simulation inflow data for incompressible turbulent boundary layers. This strategy has been extended to compressible flows by several authors (Sagaut et al. 2003; Urbin and Knight 2001; Stolz and Adams 2001; Schr¨oder et al. 2001) and has also been retained by Qu´em´er´e in his zonal RANS–LES approach. The idea of this treatment is to extract the turbulent fluctuations in a plane downstream of the inflow interface location† and to inject them at the inflow after a rescaling based on the ratio of the skin-friction velocity and thickness at the two different locations. The rescaling approach is harder to implement in general (i.e., in complex geometries) and, as determined by the scaling, may not exactly reproduce the intended second moments, length scales, and time scales but has the advantage (relative to most existing synthetic models) of requiring a shorter transient period to reestablish self-sustaining physical disturbances. The methods discussed in this section for imposing inflow LES boundary conditions are relatively new and have had little application in the context of acoustic source prediction. However, there is every justification to expect reasonable results. A good synthetic model of the turbulent fluctuations at the inflow interface of the LES domain should allow equivalently good predictions both of the flow downstream and of the direct noise emission from that region of synthetic turbulence. It is imperative, however, ∗

†

Treatment of LES inflow sections is also a critical issue for global hybrid RANS–LES methods. The hybrid RANS–LES predictions of Fan et al. (2002, 2003) failed dramatically without an appropriate imposition of the unsteady approach boundary layer. Another solution is to use a second simulation with a periodicity condition and a forcing term in the streamwise direction to provide the fluctuations. See Qu´em´er´e and Sagaut (2002) for example.

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to ensure that these fluctuations are not additive – that is, the underlying data must be represented either as RANS (with no fluctuations) or as LES with synthesized fluctuations (and appropriately diminished levels of statistical turbulence energy k) – and that any synthesized disturbances are then allowed to develop naturally as part of the solution to the time-dependent Navier–Stokes equations. Reconstruction of unsteady, synthetic turbulence fields is discussed further in Section 4.2.7.

4.4 Examples using hybrid RANS–LES formulations This section presents selected examples of calculations using some of the described hybrid RANS–LES formulations. The first example considers the application of global hybrid RANS–LES approaches to the wake of a car wing mirror together with the use of a synthetic reconstruction to account for the noise from unresolved scales. The second example presents results from a zonal NLDE approach to the simulation of the flow in a slat cove of a high-lift wing device.

4.4.1 Flow in the wake of a car wing mirror

This example illustrates the use of hybrid RANS–LES in predicting resolved-scale noise sources with a synthetic reconstruction being used in conjunction with an analytic wave propagation approach to account for noise from the unresolved scales. A second set of results demonstrates the application of Batten, Ribaldone et al.’s (2004) NLAS. The problem consists of a low-speed (55.5 m/s) air flow over a hemispherical half-cylinder mounted on a flat plate representative of a generic car wing mirror. Predictions are compared with experimental measurements taken on the downstream surface of the mirror. The hybrid approaches used here are the LNS and NLAS methods described in Sections 4.2.3 and 4.2.8, respectively. The mesh used for this case was intentionally coarse, consisting of approximately 400,000 hexahedral elements with wall functions employed on all solid surfaces. Three layers of absorbing cells were used adjacent to all far-field boundaries by the introduction of a source term of the form K (U − U∞ ), where K is some damping coefficient and U∞ corresponds to the far-field data used to define the boundary condition. For the computation of the unresolved scales (used in conjunction with LNS), the synthetic reconstruction described in Section 4.2.7 was carried out on a reduced mesh generated by cutting away cells outside an isosurface containing 99% of the total mean turbulence energy, as predicted by an a priori steady-state RANS calculation (refer to Section 6.8 for alternative methods of identifying dominant sound-source regions). Figure 4.3 shows the initial startup transient computed by the LNS hybrid method compared with its baseline (cubic) RANS model at a probe location (111) on the downstream face of the cylinder. Both hybrid and conventional RANS calculations used the same mesh, the same time step, and the same spatial and temporal integration

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250 Resolved signal (hybrid RANS–LES) Resolved signal (URANS)

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Figure 4.3. Initial startup transient at probe 111 (cylinder rear face) predicted by unsteady RANS and hybrid RANS–LES (LNS model) using identical base model, mesh, and time step.

procedures. Nevertheless, Figure 4.3 shows a large difference in the predicted amplitudes at this probe location. Specifically, the unsteady RANS solution predicts very regular oscillations but of such low amplitude they are not visible to the eye when plotted on a scale comparable with the LNS output. Figure 4.4 shows an instantaneous isosurface of the streamwise vorticity shaded with the streamwise velocity. The coarseness of the mesh is evident in the exclusively

Figure 4.4. Instantaneous streamwise vorticity contours (with streamwise velocity shading) predicted by hybrid RANS–LES model.

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300 Unresolved (synthesized) signal Resolved (hybrid RANS–LES) signal 200

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Time (s) Figure 4.5. Resolved and unresolved (synthetically generated) signals for probe 111 (cylinder rear face).

large-scale structures predicted. As a result, higher-frequency components will be missing from the LNS predicted data; hence, a reduced domain was used to synthesize the missing high-frequency noise components as discussed in Section 4.2.6. Figure 4.5 shows the resolved and synthesized noise components at probe 111. Their respective Fourier transforms are shown in terms of sound-pressure levels in Figure 4.6. This figure also illustrates the signal-processed output from the RANS and LNS simulations in isolation. The isolated LNS predictions fall slightly under the data at higher frequencies as expected. The unsteady RANS predictions show a single discrete tone with a significant underestimation of the broadband noise levels. The composite signal shows the closest agreement with data, and the discrete nature of the energy distribution in the synthetic reconstruction model is evident in the higher-frequency components. Figure 4.7 shows an instantaneous isosurface of the streamwise vorticity computed using the NLAS method. This calculation was performed on the same mesh (but with the outer, absorbing layer boundaries slightly truncated) and using the same choice of time step as the hybrid RANS–LES calculation. A comparison of Figures 4.7 and 4.4 shows the increased resolution obtained with the acoustics solver relative to the hybrid RANS–LES. For any finite mesh resolution, one can expect a high-frequency cutoff for both traditional hybrid RANS–LES and NLAS methods; however, at these low flow speeds (approximately Mach 0.1), that limit is an order of magnitude less

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150 Composite signal Resolved signal (hybrid RANS–LES) Resolved signal (URANS) Exp. - Probe 111

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Figure 4.6. Sound-pressure levels determined by resolved, unresolved, and composite signals for probe 111 (cylinder rear face).

restrictive with NLAS, which contains a model for the time-dependent variation of the noise-generating flow structures at subgrid scales. Figure 4.8 shows the NLAS-predicted sound-pressure levels at probe 111. The levels compare quite favorably with the basic hybrid RANS–LES predictions, showing less suppression of the higher frequencies.

Figure 4.7. Instantaneous streamwise vorticity contours (with streamwise velocity shading) predicted by the nonlinear acoustics solver (NLAS).

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150 140

Exp. - probe 111 NLAS - probe 111

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Figure 4.8. Sound-pressure levels determined by NLAS method at probe 111 (cylinder rear face).

4.4.2 Unsteady flow in the slat cove of a high-lift airfoil

This section describes the unsteady numerical simulation in the slat cove of a high-lift wing. A three-element high-lift wing with deployed slat and flap is considered with the following physical parameters: the wing chord (in cruise configuration) was equal to 0.61 m, and the upstream velocity was 65.5 m/s, leading to a chord Reynolds number of 2.5 million. A steady, two-dimensional RANS computation was first carried out over the full high-lift configuration using the S–A model on a 411,664-point grid. The instantaneous three-dimensional turbulent fluctuation field was then computed using the zonal RANS–LES method of Labourasse and Sagaut (2002) (described in Section 4.3.2) in a small region encompassing the slat cove (see Figure 4.9). It is worth noting that the LES subdomain is, itself, decomposed into several computational grid blocks. The hybrid RANS–LES simulations were carried out on three-dimensional meshes. The grid distribution was the same as for the RANS simulation in the (x, z) plane (with ≈110,000 points per plane). Several simulations were performed corresponding to different spanwise extents (2.7 and 27% of the slat chord), spanwise mesh sizes (32 or 52 points), or both. The fluctuating field was assumed to be periodic in the spanwise direction. As mentioned previously, both explicit and implicit simulations were performed to get a broadband description of the flow physics. In this chapter, only the results with the 27% chord extent with 52 mesh points in the spanwise direction are presented. Simulations with the reduced value of a chord length were found to remain quasi-two-dimensional, and thus the span was clearly not large enough to represent the transverse unstable modes.

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Figure 4.9. Location of the LES subdomain (displayed mesh shows every eighth grid line).

Firstly, one can note that the hybrid RANS–LES computation adjusts the mean field in comparison with the quasi-steady RANS approach, as can be seen in Figure 4.10, which illustrates mean-flow streamlines inside the slat cove. Two additional separation zones are present in the hybrid RANS–LES result: the first is just behind the slat’s trailing edge; the second, more significant separation zone is found at the leading edge of the main wing body. This highlights the difficulties of accurately simulating this type of flow using traditional RANS methods. Indeed, Khorrami, Singer, and David (2002)

Figure 4.10. Mean flow streamlines. Left: RANS computation, Right: Hybrid RANS–LES computation. Taken from Terracol et al. (2005) (Fig. 27, p. 221) with kind permission of Springer Science and Business Media.

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Figure 4.11. Instantaneous Schlieren-like view. Taken from Terracol et al. (2005) (Fig. 28, p. 221) with kind permission of Springer Science and Business Media.

found it necessary to switch off the model effects in the leading-edge region in order to restore this second separation zone in their URANS computations. An instantaneous, Schlieren-like view of the field in an (x, z) plane is shown in Figure 4.11, and the dilatation field  = ∇ · u is displayed in Figure 4.12. This last

Figure 4.12. Isovalue contours of the dilatation field . Taken from Terracol et al. (2005) (Fig. 29, p. 221) with kind permission of Springer Science and Business Media.

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p (db)

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Figure 4.13. Acoustic pressure spectrum at location S2. Taken from Terracol et al. (2005) (Fig. 31, p. 222) with kind permission of Springer Science and Business Media.

quantity is proportional to the acoustic pressure time derivative and permits observation of very-low-frequency oscillations. These two figures make it possible to investigate the flow structure and the resulting acoustic wave propagation. Looking at these figures, one can observe several basic flow features associated with noise emission related to specific frequencies in the pressure spectra: r A mixing layer develops in the shear region that bounds the main recirculation

bubble on the suction side of the slat (S2). Owing to strong streamline curvature effects, eddies of opposing spanwise vorticity are not symmetric. Pairing is also observed. Figure 4.13 shows the acoustic spectrum obtained just behind the slat’s trailing edge with an associated main frequency of 1.5 kHz, having subharmonics (due to vortex pairing) and higher-order harmonics. r A secondary recirculation bubble is observed on the slat suction side (S1) that interacts with the coherent vortices swept along the slat surface by the main recirculation. The frequency spectrum observed at this location (figure not shown) was extremely similar to that obtained in the mixing layer S2, suggesting a strong interaction between the two flow features S1 and S2. r An important secondary turbulent recirculation bubble is observed in the gap on the main body surface (S3). Two different phenomena occur here: (1) a mixing layer subjected to a Kelvin–Helmholtz instability with a main vortex shedding frequency of about 10 kHz, and (2) an erratic, lower-frequency vortex shedding is detected that could be explained by the interaction of the classic “breathing mode” of the recirculation bubble with the chaotic advection of coherent vortices coming from

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Figure 4.14. Acoustic pressure spectrum in the recirculation bubble (location S3). Taken from Terracol et al. (2005) (Fig. 33, p. 223) with kind permission of Springer Science and Business Media.

the mixing layer through the gap. Figure 4.14 presents the sound-pressure spectrum obtained in the center of the recirculation bubble, which appears to be a full spectrum owing to the high level of turbulence observed in the gap. r The last phenomenon noted occurs in the slat wake (S4). Recalling that the slat has a blunt trailing edge, we note that a coherent vortex shedding with a main frequency of about 30 kHz is detected. A more precise analysis of this isolated flow

Figure 4.15. Isovalue contours of the dilatation field . Taken from Terracol et al. (2005) (Fig. 34, p. 224) with kind permission of Springer Science and Business Media.

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Figure 4.16. Acoustic pressure spectrum at the slat’s trailing edge (location S4). Taken from Terracol et al. (2005) (Fig. 35, p. 224) with kind permission of Springer Science and Business Media.

feature has also been carried out by means of a numerical simulation using a smaller computational domain localized around the slat’s trailing edge. This simulation took into account the boundary layer on the upper side of the slat, and the influence of the turbulent bubble in the gap was removed. A global view of the dilatation field obtained in this case on the full computational domain considered is shown in Figure 4.15. An acoustic wave pattern is clearly emitted at the trailing edge with a frequency of about 30 kHz, as mentioned previously. This frequency is associated with the vortex shedding and recovered in the acoustic spectrum at the trailing edge (Figure 4.16). In the wake, the main frequency quickly becomes about 15 kHz (Figure 4.17), which is explained by the vortex pairing in the wake.

4.5 Summary of hybrid RANS–LES methods In this chapter we have attempted to summarize several hybrid RANS–LES approaches that show promise for computational aeroacoustics problems. Various new approaches have been described, ranging from global approaches that solve a single set of equations throughout the entire domain to zonal methods that explicitly impose pure RANS or pure LES in individual zones according to some initial domain decomposition. For the task of simulating the turbulence and near-field noise, all methods presented here are considered good potential candidates, with accuracy that is expected to be superior to unsteady RANS and a cost anticipated to be significantly less than that of

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80 70 60 50

p (db)

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3

10

4

F (Hz)

10

5

Figure 4.17. Acoustic pressure spectrum in the slat’s wake. Taken from Terracol et al. (2005) (Fig. 35, p. 224) with kind permission of Springer Science and Business Media.

a traditional, full-domain LES. The question of accuracy relative to full-domain LES remains to be assessed. In comparison with traditional LES, hybrid RANS–LES methods are intended to allow larger mesh spacings (at least in certain regions) and hence larger time steps; thus, they can result in direct resolution of a smaller portion of the total frequency spectrum. This is not a weakness of the hybrid models themselves but simply an expected result of exploiting the coarser spatial and temporal resolutions with which hybrid methods are able to operate. The issue of unresolvable noise sources has been discussed, and two possible approaches to help extend the range of predicted frequencies have been described. Both approaches involve synthetic reconstruction of an unsteady velocity field that reproduces key properties of the underlying (statistically represented) turbulence. One approach considered was a nonlinear acoustics solver that uses a stochastic subgrid model; the other involves augmenting the frequency spectra of traditional LES or hybrid RANS–LES in order to separately model the generation and transmission of the noise from unresolvable scales. Of the hybrid methods discussed, the DES approach currently has the widest experience base. The simplicity of the DES formulation is clearly appealing and is likely to make this the first method of choice for hybrid RANS–LES implementations in new or existing CFD codes. The LNS method has an increasing user base in the commercial CFD++ code, where the slight increase in model complexity is not an issue for the end user. Differences between these two methods are expected to be small when used as explicitly intended by the DES. The current challenge is to extend the range of applicability of these models to handle impinging flow, thin separation, and automatic

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initiation of LES fluctuations as RANS data are transported from coarse-mesh into fine-mesh regions. This data exchange between statistical (RANS) and resolved (LES) region is likely to be a pivotal issue in the near future because users will increasingly expect CFD codes to allow the use of embedded LES regions, akin to the use of zonal hybrid RANS–LES but determined automatically through mesh refinement in regions where detailed knowledge of the local flow structures and sound sources is required. With the major challenge already facing companies to train CFD engineers adequately in this increasingly multidisciplinary field, it is unlikely that manual intervention at complex internal boundaries would be practical or even possible. Therefore, the CFD code itself will be expected to automatically handle the (abrupt or gradual) interface between the RANS and LES regions. The possibility of treating these zonal interfaces automatically is something we expect to see emerging soon from commercial CFD vendors. The transfer from LES to RANS data requires nothing more than an averaging process. The transfer from RANS to LES data is slightly more complex, but procedures are described in this chapter that can create a synthetic field of velocity fluctuations that achieves the desired second moments as well as the scales of time and length given a set of RANS statistics. Such a process requires knowledge of realizable covariances of the velocity fluctuations – information that is directly available within the LNS model. Similar developments are also under way within the DES framework. The concept of using local structure size estimates for the dissipation rate in hybrid RANS–LES models has been introduced with reference to the recent Menter et al. (2003) proposal. This approach has not yet been widely tested, but its aim and philosophy seem well grounded and may lead (for certain cases) to a simpler alternative than the use of synthetic reconstruction. Further developments are required, but the ideas presented by Menter et al. (2003) are expected to find their way into other hybrid RANS–LES approaches as the problems associated with handling increasingly complex geometries and general meshes become more prevalent. The use of nonlinear perturbation or disturbance equations, such as the system proposed by Morris, Long et al. (1997) and its viscous extension, has been discussed. Although such methods are not yet very popular, they have the potential to be more accurate in the computational treatment of disturbances that are small relative to their solution mean. Several advantages of using these equations are highlighted in the work of Batten et al. (2002c), Batten, Ribaldone et al. (2004), and Labourasse and Sagaut (2002). The use of background (statistically steady) RANS mean fields is an area that shows potential for reducing the resolution requirements for a partially resolved LES. Existing hybrid RANS–LES formulations that solve a single set of equations require sufficient near-wall resolution in the direction normal to the wall, whereas a priori RANS calculations with subsequent interpolation to a more uniform acoustics mesh can mitigate this requirement because a set of grid-converged background statistics can always be obtained from the a priori RANS solution. In the hybrid RANS–LES approach of Labourasse and Sagaut (2002) and the NLAS approach of Batten et al. (2002c), a grid-converged set of background statistics (synthetically reconstructed in the latter

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case) are always available wherever the mesh resolution is unable to support the local flow structures. Finally, we note that the choice of both numerical method and SGS closure is expected to continue to play a crucial role in LES and hybrid RANS–LES. In most (if not all) existing LES or hybrid RANS–LES codes, the SGS model and numerical flux treatments are oblivious to one another, typically resulting in a mixing that is too strong. Some developments in viscous limiter functions were presented by Toro (1992) for simplified model equations, but further developments would be welcome progress for use with LES of the full Navier–Stokes equations.

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5.1 Spatial and temporal discretization schemes Tim Broeckhoven, Jan Ramboer, Sergey Smirnov, and Chris Lacor 5.1.1 Introduction to discretization schemes

In contrast to standard computational fluid dynamics (CFD) applications, for which second-order accuracy in space is sufficient for engineering purposes, the requirements on the schemes are much more stringent in the computation of aeroacoustical applications. There are several reasons for this. First, the amplitudes of acoustic waves are several orders of magnitude smaller than the average aerodynamic field amplitudes. In addition, the length scales of acoustic waves, typically the principal acoustic wavelengths, are some orders of magnitude larger than the dimensions of the sound-generating perturbations (vortices and turbulent eddies). Thirdly, sound generated by turbulence is broadband noise with often three orders of magnitude difference between the largest and the smallest acoustic wavelengths. Finally, acoustic waves propagate at the speed of sound (which is not necessarily comparable to the mean flow velocity) over large distances in all spatial directions, whereas aerodynamic perturbations are only convected by the mean flow. Moreover, one is usually interested in the noise level at the far field, implying that the waves have to be traced accurately over long distances. This requires numerical methods with higher accuracy than routinely applied in CFD codes. In particular, the discretization of the convective operator (i.e., the first-order derivative) requires special attention. In this respect, the actual order of the scheme, which can be determined based on a Taylor expansion analysis, is not the primary concern. Although this order gives an indication of the error reduction when one goes to finer grids, it gives no information about the dispersive and dissipative behavior of the scheme. This information, which is related to the resolution of the scheme as opposed to the accuracy (Adams and Shariff 1996) can be obtained from a Fourier mode analysis. The resolution is then the ability of the scheme to represent Fourier modes 167

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of increasing wave numbers accurately. The dispersive errors of the scheme cause an error in the phase of the Fourier waves as compared with the exact solution from the differential equation, whereas the dissipative errors cause an error in the amplitude of the Fourier waves (e.g., Hirsch 1988). Similar arguments apply to the temporal discretization. The acoustic waves have to be tracked accurately in time. Again, more important than the temporal order of accuracy of the time integration method are its dispersive and dissipative behavior. For the solution to be accurate, these errors have to be minimized as much as possible. This chapter is organized as follows. First, the notion of dispersive and dissipative errors is briefly reviewed, both for spatial and temporal discretization schemes. Next an overview of spatial discretization strategies for use in acoustical simulations is given. The final section is devoted to temporal discretization schemes. 5.1.2 Dispersion and dissipation errors

Consider the scalar convection equation ∂u ∂u +a =0 ∂t ∂x

(5.1)

uˆ = F(t)e I kx .

(5.2)

and a Fourier wave

Substitution of Equation (5.2) in Equation (5.1) gives the following equation for F: dF = −I akt. F

(5.3)

F = e−I akt ,

(5.4)

Solution of this equation gives

and the Fourier wave of type (5.2) satisfying Equation (5.1) is given exactly by uˆ ex = e I k(x−at) .

(5.5)

5.1.2.1 Dispersion and dissipation errors of the spatial scheme

We again consider Equation (5.1), but semidiscretized in the sense that a spatial discretization scheme is used for the spatial derivative. Equation (5.1), written in point i, becomes ∂u i = R(u i , u i−1 , u i+1 , u i−2 , u i+2 , . . .) (5.6) ∂t i . The letter R is used to indicate that this is with R the discretization stencil of −a ∂u ∂x the so-called residual. As dictated by the scheme used, the value of u in point i, u i , as well as in surrounding nodes (u i−1 , u i+1 , u i−2 , u i+2 , ...) may be used. Substitution of the Fourier wave (5.2) leads to the following equation for F:

dF = Rt, F

(5.7)

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where R is the so-called Fourier footprint of the residual, which can be decomposed in a real part, Rr , and an imaginary part, Ri : R = Rr + I Ri .

(5.8)

F = eRr t e I Ri t .

(5.9)

The solution of Equation (5.7) is

Combining Equations (5.2) and (5.9), we find that the solution of the semidiscretized equation is therefore uˆ = eRr t e I k(x+

Ri k

t)

.

(5.10)

Comparing Equation (5.5) with Equation (5.10), one notes that two errors arise because of the discretization: r An error on the amplitude of the wave: The exact amplitude is 1, whereas, with

spatial discretization, an amplitude e−Rr t is obtained. This is the dissipation error.

r An error on the propagation speed of the wave: The exact speed is a (for all wave

numbers k), whereas, with spatial discretization, the wave speed depends on the wave number and is given by − Rk i . This is the dispersion error. Some authors define a numerical wave number k ∗ as R ≡ −I ak ∗ .

(5.11)

The solution of the semidiscretized Equation (5.9) becomes k∗

uˆ = e I k(x− k at) .

(5.12)

Hence, if the difference between numerical and actual wave speed describes the error the real part of k ∗ corresponds to dispersive errors, whereas the imaginary part corresponds to a dissipative error. As an example, consider the second-order central scheme u i+1 − u i−1 ∂u i +a = 0. ∂t 2x

(5.13)

The Fourier footprint is given by R = −I a

sin(kx) . x

(5.14)

The Fourier footprint is purely imaginary, and hence the central scheme does not introduce a dissipation error. The dispersion error disp , defined as the ratio of numerical wave speed and exact wave speed, is given by disp =

sin(kx) . kx

(5.15)

Note that disp is also the ratio of numerical and actual wave speed with the numerical wave speed as defined in Equation (5.11).

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5.1.2.2 Dispersion and dissipation errors of the temporal scheme

We consider the semidiscretized Equation (5.6) and introduce a temporal discretization. To fix thoughts, consider the Euler explicit scheme u in+1 − u in n n n n = R(u in , u i−1 , u i+1 , u i−2 , u i+2 , . . .), t

(5.16)

where the subscript indicates the time level of the solution. Substitution of the Fourier wave (5.2) gives F n+1 − F n = RF n t

(5.17)

F n+1 = 1 + tR. Fn

(5.18)

and

 n+1    To study the stability of the temporal discretization it suffices to check that  FF n  ≤ 1. For a study of dispersion and dissipation properties, the correct evolution of F in time has to be considered. This is given by Equation (5.9), and hence  n+1  F = eRt . (5.19) F n ex Comparison of Equations (5.18) and (5.19) shows that, for a given spatial discretization scheme, the temporal discretization in general introduces both an error in the module of F (the dissipation error) and in the phase of F (the dispersion error). This analysis is easily extendable to other discretization schemes. For example, for Runge–Kutta schemes, Equation (5.18) becomes F n+1 = G(tR), Fn

(5.20)

where G is a polynomial of degree q with q the number of Runge–Kutta stages. Expressed in terms of the numerical wave speed, Equation (5.11), one has to compare ∗ G(−I tak ∗ ) with e−I tak . 5.1.3 Spatial discretization schemes

The equations to be solved in computational aeroacoustics (CAA) applications are the Navier–Stokes equations, either in their full form or in a simplified form, by omitting viscous terms (Euler equations) or by linearizing (e.g., linearized Euler equations (LEE)). This system can compactly be written as ∂F ∂G 1 ∂V ∂W ∂U + + = + ∂t ∂x ∂y Re ∂ x ∂y

(5.21)

with U the vector of unknowns; F, G the inviscid fluxes that depend on U; and V, W the viscous fluxes depending both on U and its derivatives ∇U . The inviscid fluxes describe

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the convective transport, and, as such, their discretization requires special attention and accurate schemes. In the following sections we will at first consider a simplified scalar convection equation of the form ∂u ∂u + =0 ∂t ∂x

(5.22)

and focus on the discretization of the ∂∂ux term. Where needed, the extension to the system above will be made and further explained. 5.1.3.1 Classical central- and upwind-type schemes Central schemes

The most straigthforward approach is to use central-type schemes – that is, the derivative of u is discretized as   M ∂u 1  ≈ a j u i+ j , (5.23) ∂x i x j=−M where M defines the stencil width. To determine the coefficients a j and the order of the scheme, one can decompose the u i+ j in a Taylor expansion around i and equate leftand right-hand sides of Equation (5.23) to the maximum possible order. One finds that a0 = 0 and a j = −a− j and that the order is 2M. The simplest scheme is the well-known second-order central scheme (M = 1):   1 ∂u (u i+1 − u i−1 ) . (5.24) ≈ ∂x i 2x Although this scheme is routinely used in standard CFD applications, where it is usually supplemented by some artificial dissipation terms (Jameson, Schmidt, and Turkel 1981), its accuracy seems insufficient for CAA applications and large-eddy simulation (LES) – especially for the accurate simulation of sound wave propagation. In the context of a hybrid approach, however, this scheme has been used by several authors in the near field for the determination of the sound sources (e.g., Biedron et al. 2001; Manoha et al. 2000; Boersma 2002). Also note that this scheme is often used in LES. One of the reasons is that, in its finite volume form, it can be shown (Ducros et al. 2000) to be in the so-called skewsymmetric form. As recalled by Kravchenko and Moin (1997), such schemes have a built-in dealiasing property that makes them attractive for use in LES. However, in most CAA applications higher-order variants of the central scheme are ¨ or¨uk and Long 1996a, 1996b, 1997; Ahuja, Ozy¨ ¨ or¨uk, and Long 2000), where used (Ozy¨ a fourth-order accurate scheme is used for applications of ducted turbofan noise. This latter scheme corresponds to M = 2 and is given by     1 1 1 ∂u ≈ − u i+2 + 2u i+1 − 2u i−1 + u i−2 . (5.25) ∂x i 3x 4 4 This fourth-order accurate scheme is also popular in LES simulations. Ducros et al. (2000) show that it can also be put in skew-symmetric form.

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Chyczewski, Morris, and Long (2000b) use a sixth-order finite difference scheme in combination with the nonlinear disturbance equation (NLD) approach of Morris, Long et al. (1997). Applications, however, are restricted to a fully developed turbulent boundary layer and the channel flow without noise predictions. In the NLD approach, the unknowns are split in a mean value, which is assumed independent of time, and a perturbation. The mean solution is obtained as a Reynolds-averaged Navier–Stokes solution with usually a simple algebraic turbulence model. Substitution of the full solution in the Navier–Stokes system leads then to the NLD equations, which contain both linear and nonlinear fluctuation terms, and a mean-flow source term independent of the fluctuations. To simplify the system, we assume that the disturbances are essentially inviscid in nature so that viscous perturbation terms can be neglected. Note that the preceding schemes only guarantee high-order accuracy on Cartesian and uniform grids. This poses a problem because almost all applications require nonuniform grids (i.e., grids with clustering and, in most cases, also non-Cartesian grids). In the finite difference context, this problem is circumvented by applying the discretization above in computational space, where the grid is uniform and Cartesian. However, one has to make sure then that the errors of the Jacobian transformation do not reduce the accuracy of the scheme (cf. Gamet et al. 1999; see also Section 5.1.3.3). Upwind schemes

In the central schemes of the previous section, the discretization stencil is symmetric with respect to the point where the derivative is evaluated. In upwind-type schemes, the discretization stencil is upwind (or downwind) biased – that is, upwind (downwind) points have a higher contribution weight. For fully upwind (downwind) schemes only upwind (downwind) points contribute to the discretization stencil. In the present context “upwind” is used as a generic term indicating both upwind and downwind schemes. Several authors use similar upwind-type schemes as developed for CFD (and not CAA) applications. In Smith et al. (2000) a Roe-type scheme with monotone upstreamcentered schemes for conservation laws (MUSCL) extrapolation and different limiters is proposed. The Roe-type scheme, or flux difference splitting scheme, was developed in the 1980s by Roe (1981) and is a Godunov-type scheme in the sense that the numerical flux formulation is based on the solution of an approximate Riemann problem. Hence, the numerical flux needs information from the cells left and right of the interface. In the first-order accurate version, only the cells sharing the interface are used. With the MUSCL approach, the accuracy of the scheme is increased by also involving cells farther away from the interface. This allows an increase in the accuracy of the scheme to a second-order or even third-order one with the so-called κ-schemes (Van Leer 1979). Limiters are used to restore the monotonicity of the scheme by incorporating some nonlinearity in accordance with the Godunov theorem that states that monotone linear schemes for convection problems can at most be first-order accurate. In Smith et al. (2000), the Roe scheme is applied for cavity acoustics at supersonic freestream in the framework of LES simulations.

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Ewert et al. (2002) and El-Askary, Ewert, and Schr¨oder (2002) use the secondorder advection upstream splitting method (AUSM) (Liou and Steffen 1993) for the LES simulation of the near field of the flow around a sharp leading edge and around a cylinder. The radiation to the far field is calculated via a linearized Euler approach using a dispersion-relation-preserving (DRP) scheme (cf. Section 5.1.3.2) with artifical selective damping (ASD) (cf. Section 5.1.3.2). Ekaterinaris (1999b) extends the stencil used in the determination of the dissipative term of the Roe scheme to obtain, respectively, a third- and fifth-order accurate scheme. He shows results for the computation of the sound generation from a corotating vortex pair using the Hardin–Pope two-step approach (Hardin and Pope 1994). In this two-step approach the Euler equations are linearized around the mean flow of an incompressible computation. Because pressure and density variations obtained by this means are no longer isentropically related, a further decomposition of the density is used. Nance, Viswanathan, and Sankar (1997) improve the extrapolation, typically used in MUSCL-type schemes by incorporating ideas of the DRP (cf. Section 5.1.3.2) L , the following formula is schemes. To obtain the left value on cell face i + 1/2, u i+1/2 used: L = a−2 u i−2 + a−1 u i−1 + a0 u i + a1 u i+1 + a2 u i+2 . u i+1/2

(5.26)

Four of the five coefficients a j are determined by Taylor expansions of the righthand side. This fixes the order of the extrapolation. The remaining degree of freedom is used to improve the resolution. This is achieved by taking the Fourier transform of Equation (5.26) and minimizing the dispersion error. As in the DRP schemes, the error is expressed as an integral of the resolution error (cf. Section 5.1.3.2), and optimization is only achieved for wave numbers with wavelength larger than 4x (i.e., the integral ranges in between −π/2 and π/2). Casper and Meadows (1995) propose the use of essentially nonoscillatory (ENO) schemes for CAA problems involving shockwaves. Weighted essentially nonoscillatory (WENO) schemes are used by Seror and Sagaut (2002) in aeroacoustic simulations of a supersonic rectangular jet with a hybrid LES–Kirchhoff or Lighthill method. Several authors have also proposed upwind-type schemes in the context of finite elements (e.g., Baggag et al. 1999; Lemaire, Marquez, and Jansen 1999; Kato et al. 2000). 5.1.3.2 DRP schemes Symmetric DRP schemes

The DRP schemes have been developed by Tam and Webb (1993). The basic idea is to use explicit, high-bandwidth, finite difference stencils for the first-order derivatives: 

∂f ∂x

 ≈ i

M 1  a j f i+ j . x j=−N

(5.27)

In contrast to the central schemes of Section 5.1.3.1, only part of the available coefficients a j of the stencil are used to fix the order of the scheme. The remaining coefficients

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Table 5.1. Coefficients for the DRP scheme of Equation (5.27) Source

a0

a1 = a−1

a2 = a−2

a3 = a−3

Tam and Webb Tam and Shen

0 0

0.79926643 0.77088238

−0.18941314 −0.16670590

0.02651995 0.20843142

Source: Modified from Tam and Shen (1993).

are tuned so as to minimize the resolution error. This is the difference between the physical wave number k and the numerical wave number k¯ which is given by M I  a j e I jkx k¯ = − x j=−N

(5.28)

with I 2 = −1. In Tam and Webb (1993) only symmetric schemes are considered (i.e., N = M). This implies that k¯ is real and the errors are only dispersive. The error to be optimized is expressed as  β 2 ¯ E= |kx − kx| d(kx). (5.29) −β

The value of β determining the range of the integral was originally chosen as π/2 by Tam and Webb (1993). This means that optimization is performed only for waves with a wavelength larger than 4x. The reason for not optimizing over the complete wave-number range (−π, +π ) is that, irrespective of the scheme used, the numerical wave number goes to zero for kx = π . It therefore makes no sense to try to optimize up to the highest wave numbers (see also Kim and Lee (1996) for optimization in the framework of compact schemes). According to Lockard, Brentner, and Atkins (1994), the choice β = π/2 already places a very stringent requirement on the minimization that can only be met by schemes with some significant overshoots (the numerical wave number is larger than the actual one) in the lower wave-number range of the wavenumber diagram. The better resolution for high wave numbers is then at the cost of a reduced resolution for lower wave numbers. This was also recognized by Tam and Shen (1993) in later work. Table 5.1 shows the coefficients a j of Equation (5.27) as obtained by Tam and Webb (1993) and by Tam and Shen (1993). In both cases, a seven-point stencil was used (i.e., N = M = 3 in Equation (5.27)). Fourth-order accuracy in space was imposed, leaving one degree of freedom for the optimization. In Lockard et al. (1994) the integral therefore extends only up to waves with wavelengths of 7x or larger (i.e., β = 2π/7). The DRP scheme is often used in combination with the LEE approach (e.g., Agarwal and Morris 2000) for acoustic scattering by a fuselage and Fleig et al. (2002) for the aeroacoustics of a wind turbine blade. The central-type DRP schemes do not have any inherent dissipation mechanism to eliminate spurious short-wavelength numerical waves. As a result, these schemes allow a pileup of energy in the high-wave-number range that might render the scheme unstable. In order to dissipate this energy, some dissipative terms can be added.

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One of the first schemes to incorporate such dissipation terms was the Jameson– Schmidt–Turkel scheme (Jameson et al. 1981), which nowadays is widely used in CFD codes. For CAA applications this type of dissipation is not well adapted as shown by Kim and Lee (2000), who combine it with compact schemes (cf. Section 5.1.3.3). Morris, Long, and Scheidegger (1999), Morris, Scheidegger, and Long (2000), and Agarwal and Morris (2000) use a standard sixth-order artificial dissipation. Originally they also include a second-order dissipation in some applications (Morris et al. 1999), but in a later paper this term is omitted (Morris et al. 2000). In Morris et al. (1999, 2000), applications are noise generation of both circular and rectangular supersonic jets. The NLD approach (cf. Section 5.1.3.1) is based on an LES formulation (Morris, Long et al. 1997). Both a direct as well as a hybrid approach are used. In the latter case, a Kirchhoff or a Ffowcs Williams–Hawkings method is used for the far-field radiation. Tam, Webb, and Dong (1993) propose ASD. A damping term of the following form is added to the right-hand side of the equation: D=−

3 νa  d j f i+ j . (x)2 j=−3

(5.30)

For a Fourier wave F(k) ≡ e I kx with k the wave number, the resulting damping is νa D(kx) = − (x)2



3 

 dje

I jkx

F(k).

(5.31)

j=−3

The coefficients d j can now be optimized in order to tune this damping so that it is essentially active in the high-wave-number range (Tam et al. 1993). Note that the original ASD proposal is not in conservative form. Extension to conservation form is presented by Kim and Lee (2000); see also Section 5.1.3.3. Bogey, Bailly, and Juv´e (1999, 2000a) use the DRP scheme with ASD in their LES solver and show results for mixing layer noise (Bogey et al. 1999) and the sound radiated by a circular subsonic jet (Bogey et al. 2000a). Asymmetric DRP schemes

Dissipation terms can also be directly incorporated in the scheme by deviating from a central-type formulation. Lockard et al. (1994) therefore relax the requirement of symmetric schemes. This implies that k¯ now contains an imaginery part, leading to dissipative errors. The integral to be optimized contains, then, a weighted contribution from both types of errors:  E=

β

−β



2 2 ¯ ¯ + (σ − 1) Im(kx) σ Re(kx − kx) d(kx)

(5.32)

with σ a weighting function that allows the emphasis to be put on either the real or ¯ ≤ 0, in order to avoid the imaginary part. Because one has to be careful that I m(k)

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growing modes, a modified integral is also proposed to have more flexibility to force the imaginary part to be negative:  β

2

2 ¯ ¯ + (σ − 1) Im(kx) − γ sinμ (kx/2) d(kx) σ Re(kx − kx) E= −β

(5.33) with parameter γ always negative and μ a positive integer. Lockard et al. (1994) derive, then, optimized third-order accurate schemes with different stencils to be used in their ENO code. A similar optimization in the context of compact-upwind schemes (cf. Section 5.1.3.4) is formulated by Adams and Shariff (1996). 5.1.3.3 Compact central schemes

In recent years, so-called compact (or Pad´e-type) schemes (Lele 1992; Kim and Lee 1996) have gained increasing popularity in applications such as DNS (Gamet et al. 1999), LES (Boersma and Lele 1999; Meinke et al. 2002), and CAA (Gamet et al. 1999; Koutsavdis, Blaisdell, and Lyrintzis 2000; Povitsky and Morris 2000) as an alternative to spectral methods. The main advantage of these schemes is that, while providing a better representation of the shorter length scales of the solution as compared with classical finite difference and finite volume schemes, they allow the use more complex mesh geometries than the spectral methods, which are limited to applications in simple domains and with simple boundary conditions. Here, it should be mentioned that combinations of spectral methods with h-type methods, using local polynomial expansions in the cells of the mesh, do not suffer from the preceding limitations. This is the case for the so-called spectral/hp element methods (Karniadakis and Sherwin 1999) and the discontinuous Galerkin method, which is further described in Section 5.1.3.6. Originally, compact schemes were defined in the finite difference context, in one dimension and on uniform grids. One of the first applications of compact finite differencing to solve differential problems can be found in the work of Collatz (1966, 1972). However, it is only recently, in view of the applications mentioned above, that there has been a renewed interest in these types of schemes. 1D formulation

Finite difference formulations

The main idea of a Pad´e-type finite difference scheme is to construct the approximation of the differential equation to be solved with not only node values being unknowns but also the derivatives. For the first-order derivative of any scalar quantity u, the Pad´e-type approximation can be written as follows (Lele 1992): 









γ u i+2 + κu i+1 + u i + κu i−1 + γ u i−2 =C

u i+3 − u i−3 u i+2 − u i−2 u i+1 − u i−1 + B +A , 6h 4h 2h

(5.34)

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where u  denotes the first-order derivative, h is the grid spacing (in the assumption of uniform mesh), and γ , κ, A, B, and C are unknown coefficients that determine the accuracy of the approximation. These coefficients can be obtained by developing all the variables in a Taylor series about point i and requiring the coefficients of the resulting expansion to vanish up to some definite term. The first nonzero coefficient will determine the formal truncation error. Note that the scheme of Equation (5.34) is purely central and therefore will only introduce dispersive errors. It is also possible to construct compact approximations for the higher-order derivatives (e.g., Lele 1992), where also second-order derivatives are considered. However, in the present context, the focus is on the convective terms, and schemes for higher-order derivatives will therefore not be discussed. If γ and κ do not both equal zero, Equation (5.34) defines the derivative in an implicit way. Note that if γ = κ = 0, the standard central schemes, with different stencil widths, as determined by the coefficients A, B, and C, are retrieved. Using the notation of Adams and Shariff (1996), one can write Equation (5.34) symbolically as

 (5.35) Lu i = (Ru)i , where L and R are compact notations for the stencils on the left- and right-hand sides, respectively. Note that in theory these stencils can even be wider than the ones appearing in Equation (5.34). In case γ = 0, the resulting system is tridiagonal. Lele (1992) shows that the maximum order of accuracy is then 8. If, in addition, the choice C = 0 is made, a oneparameter family of fourth-order schemes results. The most compact scheme within this family corresponds to the choice B = 0 and is given by 1  3 u i+1 − u i−1 1  u i+1 + u i + u i−1 . = 4 4 2 2h

(5.36)

For γ = 0, the system becomes pentadiagonal and the order of accuracy can be increased to 10. Lele investigated the resolution of the compact schemes (of implicit type; i.e., γ and κ not both equal to zero) and found that they have much better resolution properties as compared with the explicit schemes (γ =κ=0) with a comparable order of accuracy. Figure 5.1 illustrates this; it shows the numerical wave number k ∗ x (cf. Section 5.1.2.1) as a function of the exact wave number for different schemes. The diagonal line corresponds to no error. It is observed that the bandwidth of wave numbers for which dispersion errors are small grows with the order of the scheme. The fourth-order explicit scheme, which is basically scheme (5.25), gives an important improvement compared with the second-order scheme of Equation (5.24). However, the implicit fourth-order scheme is clearly superior to its explicit counterpart. This is also confirmed by Wilson, Demuren, and Carpenter (1998). They introduce a maximal numerical wave number k¯c that defines the region of acceptable accuracy; that is, ¯ |kx − kx| < 0.01

for

0 < k < k¯c .

(5.37)

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3.14

exact second-order fourth-order explicit fourth-order implicit sixth-order implicit eighth-order implicit

2.09

Modified Wave number 1.05

0.00

0.00

1.05 2.09 Wave number

3.14

Figure 5.1. Resolution of different explicit and compact (implicit) schemes.

The number of grid points per wavelength (PPW) for this maximal wavelength k¯c is given by P PW =

2π . k¯c x

(5.38)

Wilson et al. (1998) found that PPW decreases from 28.6 for the second-order accurate scheme to 4.1 for the sixth-order compact scheme. Lele (1992) also proposesd to further optimize the schemes by imposing that, for certain wave numbers, the numerical wave number equals the exact wave number. For example, if one uses scheme (5.34) and imposes only fourth-order accuracy, three degrees of freedom result, which can be used to equal numerical and exact wave number in three points. The choice of these wave numbers is somewhat empirical, although Lele argues that the result is quite insensitive to the choice made. For this fourth-order scheme, Lele chooses kx = 2.2, 2.3, 2.4

(5.39)

as points for optimization. The resulting scheme clearly outperforms the tenth-order accurate scheme in terms of dispersive behavior. Note that the DRP scheme of Tam and Webb (1993) is based on a more refined idea of optimization. Their approach was adapted to compact schemes by Kim and Lee (1996). In the latter paper a weighting function is introduced to (1) make the integral analytically integrable and (2) weight the integrated error more in the highwave-number range. Note that the first reason for using a weighting function does not apply to explicit-type schemes (κ = γ = 0) such as the DRP schemes. Concerning the choice of β (cf. Equation (5.29)), Kim and Lee (1996) mention they obtain good

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schemes for values up to β = 0.9π . They adapt the range, as determined by the scheme, to obtain maximum resolution properties. Yet, some of their schemes also produce an overshoot (i.e., the numerical wave number exceeds the physical one), which, according to Lockard et al. (1994), is not a desirable feature (cf. also Section 5.1.3.2). A result of this optimization procedure is that better resolution properties can sometimes be obtained for lower-order accurate schemes because more degrees of freedom are available. For instance, for tridiagonal schemes of type (5.34) (γ = 0), a fourthorder accurate scheme offers two degrees of freedom (the variables κ and C can be tuned to minimize the integral), but a second-order accurate scheme has three degrees of freedom (in addition B can be tuned). This results in a second-order accurate scheme with better resolution properties than the fourth-order accurate one (Kim and Lee 1996). An alternative way for optimization is proposed by Tang and Baeder (1997). Their optimization is based on the description of the compact schemes via Hermitian interpolation. Their approach is illustrated below for a three-point scheme involving u i−1 , u i , u i+1 . First, consider a Lagrangian interpolation, which will lead to standard central schemes. The Lagrangian interpolation function q(x) for a three-point scheme is a polynomial of power 2: q(x) = a0 + a1 (x − xi ) + a2 (x − xi )2 .

(5.40)

Equating q(x) in the nodal points i − 1, i, and i + 1 to, u i−1 , u i , and u i+1 , respectively, allows us to determine the coefficients ai . The discretization formula for u i is then obtained because u i ≈ q  (xi ) = a1 . One obtains the standard central scheme u i ≈

u i+1 − u i+1 . 2x

(5.41)

The compact scheme (5.36) is obtained via Hermitian interpolation, and q(x) is now a polynomial of order 4: q(x) = a0 + a1 (x − xi ) + a2 (x − xi )2 + a3 (x − xi )3 + a4 (x − xi )4 .

(5.42)

Two additional conditions are needed to determine ai as follows:  , q  (xi−1 ) = u i−1

(5.43)

 . q  (xi+1 ) = u i+1

(5.44)

The discretization formula for u i is again obtained because u i ≈ q  (xi ) = a1 . It can easily be checked that Equation (5.36) is retrieved. Tang and Baeder (1997) propose the use of trigonometric functions instead of power polynomials for the interpolation, leading to so-called Fourier difference schemes as opposed to Taylor difference schemes. For the same example as above, the interpolation function becomes     π (x − xi ) π (x − xi ) + a2 sin , (5.45) q(x) = a0 + a1 cos 2x 2x

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Figure 5.2. Comparison of the resolution of Taylor and Fourier difference schemes.

where the half-wavelength of the cosine and sine functions corresponds to the stencil width of 2x. Equating q(x) in the nodal points i − 1, i, and i + 1 to u i−1 , u i , and, respectively, u i+1 allows us to determine the coefficients ai . The discretization formula π a2 . One obtains for u i is then obtained because u i ≈ q  (xi ) = 2x u i ≈

π u i+1 − u i+1 . 2 2x

(5.46)

Comparing the numerical wave number of this scheme with the one of Equation (5.41), one sees that the numerical wave number is multiplied with π2 . As a result the dispersion errors are more uniformly distributed in the frequency domain, but there is also a large accuracy contamination in the low-frequency range. This is illustrated in Figure 5.2, where Fourier difference schemes of second and sixth order are compared with their Taylor counterparts. As a result, this low-order trigonometric interpolation does not lead to a reasonable scheme. Although the sixth-order Fourier difference scheme seems superior to the sixthorder Taylor scheme, it has to be noted that the Fourier difference scheme always has a zero-order truncation error, which is of course not acceptable from a mesh refinement point of view. Tang and Baeder (1997) therefore propose to combine the Fourier difference approach with the Taylor difference approach by replacing only the higher-order power

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polynomials of the Hermitian interpolation with higher-order trigonometric functions. For the compact scheme above, the interpolation function (5.42) is replaced by     x − xi x − xi 2 + a4 sin lπ q(x) = a0 + a1 (x − xi ) + a2 (x − xi ) + a3 cos lπ L L (5.47) with L the stencil width (i.e., 2x) and l a free parameter. The expression for u i is obtained in exactly the same way as for the compact scheme (5.36). By varying l, the dispersive behavior of the scheme can be improved. Note that l = 2 corresponds to the maximum wave number visible on the present stencil (wavelength of 2x). Experience shows that l must be well below this maximum wave-number value in order to produce useful optimized schemes. Tang and Baeder (1997) also further refine this strategy by including trigonometric functions of two different orders, l1 and l2 , in the interpolation. Results shown in Tang and Baeder (1997) are limited, however, to 1D convection of a Gaussian wave. Finally, note that the compactness of the schemes can further be improved by use of some prefactorization as proposed by Hixon (2000). This allows us to replace the tridiagonal (respectively, pentadiagonal) systems by products of two bidiagonal (respectively, tridiagonal) systems. Hixon shows results for some CAA benchmark problems. If the grid is nonuniform, the formulations above can be applied in computational space (where the grid is uniform). However, this may lead to some loss of accuracy related to the Jacobian transformation; see Section 5.1.3.3. Finite volume formulations

The finite volume (FV) method is inherently conservative, which is not the case for the finite difference (FD) approach in which special attention must be paid to the conservation properties of the schemes. However, because the FV method is formulated in physical space the construction of compact schemes is less straightforward (especially on nonuniform grids and in multidimensions), and only few papers on the subject can be found in the literature. To our knowledge, one of the first papers dealing with a formulation of compact schemes within the FV context is by Gaitonde and Shang (1997). Their scheme is based on an implicit reconstruction step relating cell face values to cell averaged values. Gaitonde and Shang (1997) discuss a 1D formulation and consider the extension into multidimensions only for linear equations and uniform grids. Applications are shown for the 1D propagation of a transverse electromagnetic wave and a 2D spherical dipole field. A similar approach, but restricted to 1D equations, was proposed more recently by Kobayashi (1999). This approach was later extended to the multidimensional Navier– Stokes system by Pereira, Kobayashi, and Pereira (2000). Although results are only shown on Cartesian grids, the proposed formulation can also be used for non-Cartesian grids by working in computational rather than physical space. In that case, errors in the calculation of the derivatives of the transformation might reduce the global

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accuracy of the scheme (cf. Section 5.1.3.3). Lacor, Smirnov, and Baelmans (2000, 2004), and Smirnov, Lacor, and Baelmans (2001a) independently developed a similar approach that is also applicable to Navier–Stokes on arbitrary, structured, non-Cartesian grids. In 1D all the different approaches are equivalent. Consider the following generic 1D advection equation: ∂ f (u) ∂u + = 0. ∂t ∂x

(5.48)

Integration over mesh cell i, with cell faces i + 1/2 and i + 1/2, leads to ∂ u¯ i x + f i+1/2 − f i−1/2 = 0 ∂t with u¯ i the cell-averaged value of cell i: u¯ i =

1 x



(5.49)

xi+1/2

u(x)d x.

(5.50)

xi−1/2

Note that relation (5.49) is still exact, and the order of the scheme will be determined by the order with which the fluxes f i+1/2 and f i−1/2 are evaluated. Because the relation between f and u is known, this reduces to the problem of evaluating u i+1/2 and u i−1/2 (i.e., the value of the unknowns at the cell faces). An approach based on an implicit interpolation formula is proposed: βu i−3/2 + αu i−1/2 + u i+1/2 + αu i+3/2 + βu i+5/2 =a

u¯ i+1 + u¯ i u¯ i+2 + u¯ i−1 u¯ i+3 + u¯ i−2 +b +c . 2 2 2

(5.51)

This implicit step is equivalent to the implicit procedure used in the FD schemes (cf. Equation (5.34)) and allows us to achieve high orders of accuracy with relatively small stencils. Taylor expansions can be used to determine the interpolation coefficients to fix the order of the interpolation and hence the order of the scheme. If the mesh is nonuniform, the grid metrics will appear in the expressions for the coefficients, which can then be stored. Extension to multidimensions and Navier–Stokes

Finite difference formulations

Because compact schemes define the derivatives in an implicit way, the extension to arbitrary grids and the Navier–Stokes system is not always straightforward. In the FD context however, the formulations of Lele (1992) can be used directly in Navier– Stokes codes. The idea is to apply the 1D formulation of Equation (5.34) in each of the three grid directions in computational (as opposed to physical) space. This approach is used by many authors (e.g., Visbal and Gaitonde 1998, 2002; Gaitonde and Visbal 1999; Ekaterinaris 1999b). Visbal and Gaitonde (1998, 2002) use basically a tridiagonal fourth-order accurate and a pentadiagonal sixth-order accurate scheme. Near the boundaries they use a one-sided formulation for the point on the boundary

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(usually fourth-order accurate) and the tridiagonal fourth-order scheme for the point next to the boundary. Ekaterinaris (1999b) uses tridiagonal fourth- and sixth-order accurate schemes. He mentions that the cost of using these schemes is comparable to that of standard, explicit schemes provided the inverses of the implicit matrices are stored. Ekaterinaris (1999b) also uses the fourth-order compact scheme in the implicit operator of the Beam–Warming scheme (Beam and Warming 1976) to replace the standard central scheme for approximating the Jacobian derivatives. This reduces the need for many subiterations to eliminate linearization and factorization errors. Also, the diagonalized version of Pulliam and Chaussee (1981) is extended using this approach. Ekaterinaris (1999a) also proposes a compact formulation when an upwind implicit operator is used. The standard first-order upwind discretization of the derivatives of the split Jacobians is replaced by compact upwind formulations. When using a fully upwind formulation such as the Cockburn and Shu scheme (Equations (5.73) through (5.75)), this leads to pentadiagonal systems. An alternative is to use upwind-biased central stencils as in the schemes proposed by Adams and Shariff (1996) (cf. Section 5.1.3.4). By keeping only three points (instead of five in the Adams and Shariff schemes) in the left- and right-hand sides, such as in the schemes proposed by Zhong (1998), the tridiagonal structure can be kept. The authors above all make use of the Jacobian transformation to define the compact schemes in computational space. This is a straightforward procedure that is briefly explained below. Consider the nondimensionalized Navier–Stokes system (below the 2D system is presented for compactness; extension to 3D is straightforward): ∂F ∂G 1 ∂V ∂W ∂U + + = + ∂t ∂x ∂y Re ∂ x ∂y

(5.52)

with F, G the inviscid flux vectors and V , W the viscous flux vectors. If the partial derivatives are denoted as a subscript, the preceding equation becomes Ut + Fx + G y =

 1

Vx + W y . Re

(5.53)

We consider a transformation from physical time-space (x, y, t) to computational timespace (ξ, η, τ ) with τ = t, ξ = ξ (x, y, t), η = η(x, y, t).

(5.54)

Application of the chain rule to Equation (5.53) gives Uτ + ξt Uξ + ηt Uη + ξx Fξ + ηx Fη + ξ y G ξ + η y G η =

 1

ξx Vξ + ηx Vη + ξ y Wξ + η y Wη . Re

(5.55)

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This equation is in weak conservation form because the metrics terms appear as coefficients in the differential equation. It can be put in so-called strong conservation form by multiplying first with J −1 ≡ xξ yη − xη yξ , the inverse Jacobian of the transformation, and using the chain rule on all terms such as J

−1

 ξx Fξ ≡

ξx F J



 ξ

−F

ξx J

 ξ

.

(5.56)

If terms are regrouped, this leads to 

   ξt U + ξ x F + ξ y G ηt U + η x F + η y G + + J J τ ξ η

     ηx V + η y W ξx V + ξ y W 1 = + + T. Re J J ξ η

U J





(5.57)

The extra term T is given by 

  

    η  η  ξt ξx 1 t x + + + +F T =U J τ J ξ J η J ξ J η

   

  η  η  ξy ξx 1 y x + + V − +G J ξ J η Re J ξ J η

   η  ξy 1 y W + . − Re J ξ J η

(5.58)

Equation (5.58) contains the so-called invariants of the transformation (between brackets). It can be shown analytically that these invariants are identically zero. However, when evaluated numerically the invariants might be nonzero because the numerical derivatives do not necessarily commute. This implies that, for a uniform flow, Equation (5.57) is not satisfied: all terms will disappear numerically, except T . Hence, special care must be taken to make the invariants also numerically equal to zero. Once this is satisfied, the resulting equation, now in strong conservation form, reads 

U J

 τ

∂ Gˆ 1 ∂ Fˆ + = + ∂ξ ∂η Re



∂ Vˆ ∂ Wˆ + ∂ξ ∂η

 (5.59)

with ⎤ ρ uˆ 1 ⎢ ρu uˆ + ξx p ⎥ ⎥, Fˆ = ⎢ J ⎣ ρv uˆ + ξ y p ⎦ ˆ E + p) − ξt p u(ρ ⎡

(5.60)

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⎤ ρ vˆ 1 ⎢ ρu vˆ + ηx p ⎥ ⎥, Gˆ = ⎢ J ⎣ ρv vˆ + η y p ⎦ v(ρ ˆ E + p) − ηt p

(5.61)

 1

ξx V + ξ y W , Vˆ = J

(5.62)

⎡

 1

ηx V + η y W , Wˆ = J ˆ vˆ the contravariant velocities, defined as and u,

(5.63)

uˆ = ξt + ξx u + ξ y v,

(5.64)

vˆ = ηt + ηx u + η y v.

(5.65)

Because in most cases the transformation from physical to computational space is not known analytically, the metrics (ξt , ξx , ξ y , ηt , ηx , η y ) are calculated numerically on a cell-by-cell basis. A convenient way of establishing the coordinate transformation is by making use of isoparametric elements. The transformation of the element in physical space to an element in the curvilinear coordinates of computational space is then described via the following relation: x =

4 

x i Ni (ξ, η)

(5.66)

i=1

with x i as the coordinates of the nodes of the physical elements and Ni the shape functions, function of the curvilinear coordinates ξ, η. The preceding relation allows us to calculate xξ , xη , yξ , yη immediately. These are related to ξx , ξ y , ηx , η y via the inverse relations: ξx = J yη ; ξ y = −J xη ; ηx = −J yξ ; η y = J xξ .

(5.67)

As mentioned by Gamet et al. (1999), the Jacobian transformation can lead to large errors in case of nonsmoothly varying mesh spacings. In the general case, the theoretical accuracy of the compact schemes (as derived in 1D on uniform grids) can be maintained only for FD schemes that take into account the stretching of the grid. The critical point of the Jacobian transformation is whether the Jacobians, which contain first and second derivatives of the transformation, can be defined and calculated without appreciable loss in overall accuracy. The loss of accuracy on curvilinear grids is also discussed by Gaitonde and Visbal (1999). Gamet et al. (1999) therefore take the nonuniformity of the mesh directly into account by adapting the coefficients of the compact stencils. The coefficients in Equation (5.34) now depend on the index i. The conditions for a given order of accuracy (resulting from Taylor expansions) will now involve also the mesh spacing and lead to appropriate expressions for the coefficients. The important gain in accuracy using such

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an approach, as compared with a Jacobian transformation, is clearly quantified in Li, Ma, and Fu (2000). However, the method of Gamet et al. (1999) only deals with nonuniform but Cartesian grids. If the grid is non-Cartesian, a Jacobian tranformation must still be used. Freund (2001) and Freund, Lele, and Moin (1998) use a sixth-order compact scheme in DNS simulations of the noise of a circular subsonic and supersonic jet, respectively. LES results of this subsonic jet using the same compact scheme but supplemented with a sixth-order filter to remove energy from the poorly resolved wave numbers, where the subgrid-scale model is ineffectual, are shown by Bodony and Lele (2002a). Mitchell et al. (1999) use the fourth-order scheme in their DNS simulations of an axisymmetric jet. The same scheme is used by Povitsky (2001) for aeroacoustics of spherical pulse propagation in 3D stagnation flows and the flow over a circular cylinder. Colonius, Lele, and Moin (1997) use both fourth- and sixth-order accurate schemes in the DNS simulation of sound generation in a mixing layer. Mankbadi, Hixon, and Povinelli (2000) apply the prefactored compact scheme of Hixon to a very large eddy simulation (VLES) simulation of the noise generation of a supersonic, heated round jet. Finite volume formulations

Below, we restrict ourselves to 2D for the sake of clarity. Consider the 2D Euler equations u t + f x + g y = 0. Integrating over a cell, one obtains ∂ i j u¯ i, j + ∂t

(5.68)

 (−gd x + f dy) = 0

(5.69)

with i j the volume (area) of the cell “i j” (with i and j the mesh directions) and u¯ i, j the cell-averaged value; that is,   1 u(x, y)d xd y. (5.70) u¯ i, j ≡ i, j i, j The accuracy of the scheme is determined by the accuracy with which the line integrals  (second term of Equation (5.69)) are determined. Consider the flux f dy, which is over the cell faces transverse to the i-direction; if the flux is linear (e.g., f ≡ u), as is the  case for a linear 2D convection problem, the integral udy can be determined with a formula similar to Equation (5.51). In the left-hand side the line integrals on consecutive cell faces in the i-direction appear, and in the right-hand side the cell-averaged values in the same direction, keeping the j-index constant. However, on arbitrary grids this 1D-like approach, as proposed in Pereira et al. (2000) and Pereira, Kobayashi, and Pereira (2001), cannot guarantee a high order of accuracy anymore. Pereira et al. (2000, 2001) therefore use a transformation to a Cartesian grid as in FD methods. An alternative, avoiding transformations, is to use a multidimensional stencil. In Lacor et al. (2000) this is achieved by extending the stencil in the right-hand side by including cell-averaged values with a different j-index.

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For Euler and Navier–Stokes applications, the cell-face integrals are calculated  for all the primitive variables. The nonlinear parts of the fluxes (e.g., uvdy) can be obtained by combining the integrals of the primitive variables 1 y



1 uvdy = y



1 udy y

 vdy + O(h 2 ),

(5.71)

where y is the length of the cell face. Note that the approximation above is only secondorder accurate, automatically reducing the order of the scheme to 2. Pereira et al. (2000) propose a correction to the relation above valid for Cartesian grids, making it fourthorder accurate. In Lacor et al. (2000) and Smirnov, Lacor, and Baelmans (2001) this relation is extended for use on arbitrary grids. Applications with these FV compact schemes are shown for convection-based model problems (Pereira et al. 2000; Lacor et al. 2000); a laminar flat-plate calculation and LES of channel flow (Lacor et al. 2004; Smirnov, Lacor, and Baelmans 2001). Stabilization using filters or artificial dissipation

Because the compact schemes formulated in the previous sections are purely central, they do not guarantee monotonicity. Also, because Fourier waves are not damped, the nonlinear character of the Navier–Stokes system may lead to a pileup of energy in the high-wave-number range, which can lead to numerical instability unless there is some dissipation or filter to remove this energy. Different filters, both explicit and implicit, have been investigated by Lele (1992). The order of the filter should of course be at least of the same order of accuracy as the compact scheme. The simplest filters are those of explicit type. Their use in the present context is, however, limited (e.g., Pruett et al. 1995). Compact, implicit filters are more accurate than the equivalent explicit filters and therefore more popular. They also allow us to achieve a high-order accuracy on a narrower stencil than the explicit filters. The two main disadvantages are that (1) a matrix must be inverted and (2) the boundary stencil has a large effect on the interior accuracy of the filter (Hixon 1999). Visbal and Gaitonde (1998, 1999, 2002), Gaitonde and Visbal (1999), Koutsavdis, Blaisdell, and Lyrintzis (1999), and Ekaterinaris (1999a, 1999b) use such implicit, lowpass filters (based on the suggestions of Lele (1992)) to remove unwanted oscillations and waves from the solutions and to suppress numerical instabilities arising from all sources, including those associated with mesh nonuniformities, boundary conditions, or low local grid resolution. ˆ one Denoting a component of the solution vector as φ and its filtered value as φ, can describe the filter as (cf. Visbal and Gaitonde 2002) α f φˆ i−1 + φˆ i + α f φˆ i+1 =

N  an n=0

2

(φi+n + φi−n ) .

(5.72)

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Table 5.2. Coefficients for filter formula (5.72) Order

a0

a1

a2

2

10

1 + αf 2 3α 5 + 4f 8 5α 11 + 8f 16 70α 93 + 128f 128 126α 193 + 256 f 256

1 + αf 2 1 + αf 2 17α 15 + 16 f 32 18α 7 + 16 f 16 302α 105 + 256 f 256

α −1 + 4f 8 3α −3 + 8f 16 14α −7 + 32 f 32 30α −15 + 64 f 64

Order

a3

a4

a5

α 1 − 16f 32 α 1 − 8f 16 90α 45 − 512f 512

−1 128 −5 256

4 6 8

2 4 6 8 10

+ +

αf 64 10α f 256

1 256

−

2α f 256

Source: After Visbal and Gaitonde (2002).

The coefficients a j are derived in terms of α f with Taylor- and Fourier-series analyses, and α f is an adjustable parameter in the range −0.5 < α f ≤ 0.5 with higher values corresponding to a less dissipative filter. Table 5.2 gives the values of a j for filters of orders varying from 2 to 10. Figure 5.3 shows the effect of α f on a second- and an eighth-order filter. The bottleneck with filtering is the procedure near the boundaries where the same stencil as for inner points cannot be used. One possibility is to use a filter of lower

Figure 5.3. Effect of α f on a second- and an eighth-order filter.

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order (with reduced stencil); see Visbal and Gaitonde (1998). In a later paper the same authors propose an improved approach based on one-sided, but still high-order, filters (Gaitonde and Visbal 2000). According to Visbal and Gaitonde (1998), filtering seems superior to artificial dissipation. In Koutsavdis et al. (1999), the application for CAA is investigated in the framework of LEE on Cartesian, nonuniform grids. The conclusion is that filtering is very useful because it removes unwanted q-waves; thus, grid refinement to suppress these waves is not needed. This allows us to obtain solutions with an accuracy that could only be realized on much finer grids if the filtering is not used. An alternative way to stabilize the scheme is through the use of some artificial dissipation – either standard Jameson dissipation (Jameson et al. 1981) or, in view of CAA applications, an ASD procedure, as proposed by Tam (1995) (cf. Section 5.1.3.2). Shim, Kim, and Lee (1999) and Kim and Lee (2000) recently proposed a conservative implementation of this approach and combined it with their optimized compact FD schemes of Kim and Lee (1996). They also added a second-order dissipation term to be able to deal with nonlinear discontinuities. Comparison on a bell-shaped linear pulse propagation shows the same accuracy as with the standard ASD scheme but a significant improvement compared with the result obtained with Jameson-type dissipation. In Shim et al. (1999) this scheme is applied to the study of radiation of multiple pure tone noise from an aircraft engine inlet. The use of ASD in combination with compact schemes is also discussed in Broeckhoven et al. (2003). 5.1.3.4 Compact upwind schemes

In contrast to the central compact schemes, upwind compact schemes have an inherent dissipation. As for the dispersive errors, it is important that the dissipative errors mainly act on the higher-frequency waves. This ensures that the resolution of the lowerfrequency waves will not be spoiled by dissipation. On the other hand, energy aliasing back from high wave numbers will be damped efficiently. 1D formulation

Finite difference formulations

The general notation used in Equation (5.35) means that the operators L and R are not of the central type anymore. Different basic schemes can be attributed to different authors. A third-order upwind scheme is proposed by Cockburn and Shu (1994) in which the implicit operator L is of the central type but the explicit operator R is upwind biased:

Lu 

 i

1  1  ≡ − u i−1 + u i − u i+1 5 5

(5.73)

and (Ru)i ≡

3 (3u i − 4u i−1 + u i−2 ) . 10x

(5.74)

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In the downwind version, the R operator becomes 3 (−3u i + 4u i+1 − u i+2 ) . (5.75) 10x Upwind schemes with both the left- and right-hand-side operators upwind biased have been formulated by Tolstykh and Shirobokov (1995) (third-order) and Fu and Ma (1997) (third- and fifth-order accurate). The Tolstykh scheme is defined as (Ru)i ≡



Lu 

 i

≡

5  1  , u + u i − u i+1 8 i−1 8

3 (u i − u i−1 ) , 2x and the schemes of Fu and Ma (1997) as (Ru)i ≡



(Ru)i ≡

(Ru)i ≡



(5.76) (5.77)

1  u + u i , 2 i−1

(5.78)

1 (u i+1 + 4u i − 5u i−1 ) , 4x

(5.79)

Lu 

i



≡

2  u + u i , 3 i−1

(5.80)

1 (−u i+2 + 12u i+1 + u i − 44u i−1 − 3u i−2 ) . 36x

(5.81)

Lu 

i

≡

In Li et al. (2000), the Fu and Ma schemes are extended for nonuniform grids using an approach similar to that of Gamet et al. (1999) (cf. Section 5.1.3.3). Adams and Shariff (1996) use an optimization procedure (see also Section 5.1.3.2) to derive two upwind-type schemes based on a five-point stencil both in left- and right-hand-side form. The first scheme is described as being compact upwind with high dissipation (CUHD) and is designed to be about as dissipative at nonresolved wave numbers as a noncompact upwind scheme while giving a much better representation of the dispersion relation. It is given by (numbers are only approximate; see appendix of Adams and Shariff 1996 for the exact numbers)

     + 0.734u i−1 + u i − 0.169u i+1 − 0.061u i+2 , (5.82) Lu i ≡ 0.036u i−2 1 (−0.176u i−2 − 1.137u i−1 + 1.063u i + 0.452u i+1 − 0.201u i+2 ) . (5.83) x The second scheme, denoted as compact upwind with low dissipation (CULD), is designed to be less dissipative (about one order of magnitude lower dissipation than CUHD). It is defined as (numbers are only approximate; see appendix of Adams and Shariff 1996 for the exact numbers)

     + 0.456u i−1 + u i + 0.477u i+1 + 0.039u i+2 , (5.84) Lu i ≡ 0.028u i−2 (Ru)i ≡

(Ru)i ≡

1 (−0.119u i−2 − 0.748u i−1 + 0.0005u i + 0.718u i+1 + 0.148u i+2 ). (5.85) x

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The schemes above can easily be used for 1D linear convection. However, for 1D Euler and Navier–Stokes applications, one needs a mechanism to choose between the upwind- or downwind-biased versions. Such mechanisms were already developed for standard upwind-type schemes and can be combined with the compact upwind schemes. Cockburn and Shu (1994) combine the 1D Euler equation  ∂u  + f i = 0 (5.86) ∂t i in different points to bring the implicit operator in front of the time derivative. Using a compact scheme of the form (5.35) leads to ∂ u¯ i + (R f )i = 0 ∂t

(5.87)

u¯ i ≡ (Lu)i .

(5.88)

with

This can be put in conservation form by noticing that R f can be rewritten as (R f )i = (N f )i+1/2 − (N f )i−1/2 .

(5.89)

For example, for the fourth-order central compact scheme one has (N f )i+1/2 ≡

3 ( f i + f i+1 ) . 4x

(5.90)

The numerical flux fˆ ≡ N f can be split into positive and negative parts for easier upwinding as follows: fˆ = fˆ+ + fˆ− = N f + + N f − .

(5.91)

Different splittings are possible; Cockburn and Shu (1994) propose a Lax–Friedrichs splitting. Note that the calculation of the (splitted) fluxes requires the u i values, whereas only u¯ i is known from solving the equations. This requires a reconstruction step. The +/− numerical fluxes on the cell faces i + 1/2 are compared with the +/− fluxes in neighboring points i for purposes of limiting. According to the theory of upwind schemes, the f + ( f − ) fluxes should be obtained, respectively, by upwind- or downwind-biased formulas. This calls for a compact formula (5.35) where the operator R is, respectively, upwind or downwind. Cockburn and Shu (1994) also use the central scheme of Equation (5.36). This works well for smooth flows. On flows with discontinuities, however, the results are inaccurate, although the scheme remains stable as a result of the limiter’s action. A related approach is proposed in Ravichandran (1997). In contrast to Cockburn and Shu (1994), the flux is split before applying the compact scheme:  ∂u  + − + f  i + f  i = 0. (5.92) ∂t i

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Ravichandran (1997) uses kinetic flux vector splitting (Mandal and Deshpande 1994) for its robustness as compared with Van Leer or Steger–Warming flux vector splitting. Upwind- (downwind-)biased compact formulas of type (5.35) are used for f  i+ ( f  i− ). The corresponding operators are denoted with a + or − sign, that is,   + (5.93) L + f   = R+ f + i , i

  − L − f   = R− f − i . i

(5.94)

Formally one can therefore write  + + f  i = L −1 , + R+ f i

(5.95)

 − − . f  i = L −1 − R− f i

(5.96)

This can be substituted in Equation (5.92) but is first put in conservation form similar to that of Equation (5.89); for example, 



  (5.97) R+ f + i = N+ f + i+1/2 − N+ f + i−1/2 and similar for R− f − . This leads to  

∂u  + −1 − + L −1 + N+ f + L − N− f  i+1/2 ∂t i 

−1 − − L + N+ f + + L −1 = 0. − N− f i+1/2

(5.98)

The numerical flux fˆ is thus given by + −1 − fˆ = L −1 + N+ f + L − N− f .

(5.99)

The +/− numerical fluxes are compared with the +/− fluxes in neighboring points for purposes of limiting. The compact schemes used for the split fluxes are those of Cockburn and Shu (1994; cf. Equations (5.73)–(5.75)), Tolstykh and Shirobokov (1995), and Fu and Ma (1997). Deng and Maekawa (1997) use a cell-centered version of Equation (5.34), which, according to Lele (1992), has a better dispersive behavior than the unstaggered formula (5.34). Equation (5.34) is, for example, appropriate for FD schemes, where the solution is stored in the mesh modes; the cell-centered version uses the derivatives in the cell centers in the left-hand side whereas in the right-hand side the fluxes on the cell interfaces appear. The expression for the flux derivative of the 1D Euler equation becomes   + f i + κ f i+1 = κ f i−1

 b

 a

f i+1/2 − f i−1/2 + f i+3/2 − f i−3/2 . h h

(5.100)

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Deng and Maekawa (1997) replace the flux in the right-hand side with expressions for the numerical flux fˆ from upwind-type schemes such as Roe’s flux difference splitting scheme or flux vctor splitting. For example, for the Roe scheme one has

L  1 R  1 R  L f u i+1/2 + f u i+1/2 − |a| u i+1/2 − u i+1/2 . (5.101) fˆi+1/2 = 2 2 The u L and u R values are obtained via implicit tridiagonal, fifth-order interpolation upwind and downwind biased, respectively. In case discontinuities are present, interpolation across the discontinuity is avoided (for reasons of accuracy) by switching locally to bidiagonal interpolation, either upwind or downwind, as determined by the location of the discontinuity. The choice of the appropriate interpolation is based on ideas similar to those used in ENO schemes by measuring the “smoothness” of the different stencils using first- and second-order differences. In a subsequent paper, Deng and Mao (1997) combine the three interpolations in a weighted formulation. The eight coefficients are chosen so that, in smooth regions, the interpolation is fifth-order accurate and third order near discontinuities. In the latter case, the weights are such that no interpolation across the discontinuity is guaranteed. The determination of the weights also uses the first- and second-order differences as in Deng and Maekawa (1997). In a recent paper by Deng and Mao (2001), extension to multidimensional Navier–Stokes is proposed. Also, dissipative terms are added in the right-hand side of Equation (5.100), leading to so-called dissipative, weighted compact schemes. The dissipative terms are based on split fluxes and are therefore only used in combination with flux vector splitting schemes. In Deng and Mao (2001), only results with the explicit version of the schemes are shown; that is, κ ≡ 0 in Equation (5.100). Finite volume formulations

A systematic analysis of upwind-type compact schemes was done by Ramboer, Smirnov, and Lacor (2002) and Ramboer and Lacor (2002). The approach is similar to that of the 1D central compact scheme, but Equation (5.51) is replaced with an upwind-biased formulation. Three types of schemes can be considered: those with both L and R operator upwind biased (LURU); schemes with L operator upwind biased and R operator central; and schemes with L operator central and R operator upwind biased. Through symbolic manipulation, the properties (order of accuracy, dispersive and dissipative error) of schemes of each category have been investigated, thereby restricting them to at most a five-point implicit operator and a six-point explicit operator. The conclusion is that the LURU schemes are preferable. Two promising schemes are a bidiagonal thirdorder accurate scheme and a tridiagonal fifth-order scheme. The following interpolation (cf. Equation (5.51)) is used: 5 1 1 u i−1/2 + u i+1/2 = u¯ i + u¯ i+1 2 4 4 for the third-order scheme and 1 1 19 10 1 u i−1/2 + u i+1/2 + u i+3/2 = u¯ i−1 + u¯ i + u¯ i+1 2 6 18 18 18

(5.102)

(5.103)

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for the fifth-order scheme. The expressions above correspond to the upwind-biased formulation. The downwind-biased formulation follows immediately from analogy. On nonuniform grids, the coefficients will depend on the grid metrics as is the case for the central-type schemes (cf. Section 5.1.3.3). When applied to the Euler/Navier–Stokes system, Roe’s flux difference splitting scheme is used to distinguish between positive and negative eigenvalues. The cell face flux is given by f i+1/2 =

  1 L f + f R − |A|(u R − u L ), 2 2

(5.104)

where the R (L) values are obtained with, respectively, downwind (upwind) interpolation. The parameter  (which equals 1 in the original Roe scheme) allows to reduce the dissipation; see also Lin, Yu, and Shih (1997) and Bui (1999). Extension to multidimensions and Navier–Stokes

Finite difference formulations

Cockburn and Shu (1994) also extend their method to multidimensions for Cartesian grids. The operators A and B now become L x , Rx for x-derivatives and L y , R y for the y-derivatives. Starting from the 2D Euler system ∂u ∂f ∂g + + = 0, ∂t ∂x ∂y

(5.105)

we find that the resulting equation is



 ∂ u¯ i + L y Rx f i + L x R y g i = 0 ∂t

(5.106)

 u¯ i ≡ L y L x u i .

(5.107)

with

The numerical fluxes are now defined as

 fˆi+1/2, j = L y N x f + + N x f − i+1/2, j ,

(5.108)

 fˆi, j+1/2 = L x N y f + + N y f − i, j+1/2 .

(5.109)

The approach of Ravichandran (1997) is easily extended to multidimensions by applying the same strategy in each direction of the computational space. Finite volume formulations

The upwind schemes defined in Section 5.1.3.4 can be extended in much the same way as in the central compact schemes of Section 5.1.3.3. Instead of determining the    flux udy, two fluxes are determined – namely, u L dy and u R dy. The formulas for these fluxes are again implicit but respectively upwind and downwind biased. In Ramboer et al. (2002) and Ramboer, Smirnov, and Lacor (2003) results are shown for, respectively, a vortex and a pressure pulse superposed on a uniform 2D Euler flow. By

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tuning the dissipation with the parameter  of Equation (5.104), one can obtain very accurate results – especially with the fifth-order scheme. 5.1.3.5 Predictor-corrector schemes

Other high-order accurate schemes have been formulated based on a predictor-corrector approach. Gottlieb and Turkel (1976) modified the McCormack scheme (McCormack 1969) to increase the accuracy to fourth order. Bayliss (1985) proposed a sixth-order accurate scheme. In Snyder and Scott (1999) different predictor-corrector schemes are tested for CAA applications, using category 2 benchmark problems. The Gottlieb and Turkel scheme outperforms the DRP scheme of Tam and Webb (1993) and is competitive with the Bayliss predictor-corrector scheme. In Hixon (2000), compact McCormack-type schemes are proposed. A prefactorization method is used to reduce a nondissipative central difference stencil to two lower-order biased stencils that have easily solved reduced matrices. However, these high-accuracy McCormack schemes have some drawbacks. If inherent damping in the biased stencil is too large, it can damp out waves the original central difference scheme could have propagated accurately; conversely, if the inherent damping is too small, nonlinear waves may generate unresolved high-frequency waves that will destroy the solution accuracy (Hixon 1998). 5.1.3.6 Discontinuous Galerkin methods

The discontinuous Galerkin (DG) method is a compact finite element projection method that provides a practical framework for the development of high-order methods using unstructured grids. The higher-order accuracy is obtained by representing the solution in each cell as a high-degree polynomial whose time evolution is governed by a local Galerkin projection. In contrast to standard finite element methods, the DG methods enforce the conservation law only locally. This allows them to have a mass matrix that can easily be made to be the identity and that therefore does not necessitate “lumping” or matrix inversion while being highly accurate and nonlinearly stable. Because of the local character, the solution may be discontinuous across different elements. As a result, the basis for the polynomials is essentially unrestricted. However, for computational efficiency, an orthogonal basis is chosen relative to the inner product. Note, however, that, even if the polynomials are not orthogonal, one still only needs to invert a small mass matrix and there is never a global mass matrix as in a typical finite element method. The DG method can also be interpreted as an extension of a FV method, incorporating notions such as approximate Riemann solvers, numerical fluxes, and slope limiters into a finite element framework. However, instead of only one degree of freedom per cell as in a finite volume method (viz., the cell average of the solution), there are, in 2D, (n + 1)(n + 2)/2 degrees of freedom for a polynomial of order n. These degrees of freedom are chosen as the coefficients of the polynomial when expanded in a local basis. Consider, for instance, the Euler system ∂U + ∇F = 0. ∂t

(5.110)

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In each cell, U is expanded in terms of the polynomials bk , k = 1, . . . , N : U=

N 

Uk bk .

(5.111)

k=1

The discretized equation is obtained by multiplying Equation (5.110) by b j and integrating over the cell:   ∂U ∇Fb j d V = 0. (5.112) bjdV + V ∂t V Using the expansion (5.111) and applying Green’s theorem, one obtains    N  ∂Uk bk b j d V − F∇b j d V + Fb j d S = 0. ∂t V V k=1

(5.113)

This equation shows that the derivative of Uk is retrieved by inverting a matrix M with  components Mk j ≡ V bk b j d V . The surface integral requires an expression for the flux on the cell faces. Because the solution is discontinuous across cell faces (as in FV methods), numerical fluxes based on Riemann solvers and incorporating limiters can be used. One can show that, for polynomials of degree p, the order of accuracy is at least p + 12 (Johnson and Pitk¨arata 1986). Equation (5.113) is usually discretized by evaluating the integrals using quadrature formulas of the required order, which is 2 p for the volume integral and 2 p + 1 for the edge integral (Cockburn, Hou, and Shu 1990). This limits the usefulness of the method because the number of terms in the quadrature summations significantly exceeds the number of unknowns, making the method computationally expensive. However, Atkins and Shu (1996, 1998) described a quadrature-free formulation. The application of the DG method to hyperbolic systems in combination with Runge–Kutta methods, the so-called Runge–Kutta discontinuous Galerkin (RKDG) method, was thoroughly investigated by Cockburn (1999) and Cockburn, Karniadakis, and Shu (2000) for a detailed survey. Hu, Hussaini, and Rasetarinera (1999) studied the wave propagation properties of the DG method. Based on an analysis of scalar convection, they found that in a formulation with upwind fluxes the dissipative errors are dominant, whereas the dispersive error is negligible for wave numbers up to a value equal to the order of the method. In a centered flux formulation, there is no dissipative error, but the range of wave numbers for which the dispersion error is negligible is smaller than with the upwind flux. Because DG methods are especially useful on unstructured grids, the influence of the mesh was also investigated. From an analysis of the 2D wave equation, it is found that an unstructuredlike triangular mesh has better dispersion and dissipation properties than a structured quadrilateral grid or than triangular grids derived from such a grid. Also, the properties are less prone to anisotropy (i.e., vary less with the orientation of the Fourier waves). Atkins and Lockard (1999) used the RKDG method to solve the linearized Euler equations (LEE) to study acoustic scattering from a two-dimensional slat and a three-dimensional blended-wing-body combination. In addition, they showed, for a

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Figure 5.4. Perturbation pressure of an acoustic wave initiated by a Gaussian pulse: comparison of solution with smooth and randomly perturbed mesh.

three-dimensional wave propagation in a cube, that the RKDG method is insensitive to the mesh smoothness: on a smooth tetrahedral mesh and the same mesh with 20% random perturbation of each grid point, the two solutions are almost indistinguishable as shown in Figure 5.4. Stanescu, Hussaini, and Farassat (2002) use the spectral element implementation of DG of Kopriva, Woodruff, and Hussaini (2000) for the computation of sound radiation from aircraft engine sources to the far field. The nonlinear Euler equations are solved, and only the radiation of inlet noise into a quiescent fluid is modeled. Both nacelle alone and fuselage-nacelle and fuselage-wing-nacelle configurations are considered. In a later paper by Xu et al. (2003), the same methodology is applied to an actual two-engine jet aircraft and compared with a spectral element solution in the frequency domain. The results show that trends of the noise field are well predicted by both methods. In Baggag et al. (1999) the DG method is applied to a benchmark problem of the 1997 CAA workshop.

5.1.4 Temporal discretization schemes 5.1.4.1 Runge–Kutta schemes

By far the most popular schemes in CAA are the Runge–Kutta schemes. They are explicit and can be formulated up to an arbitrary order of accuracy. Many authors use the classical Runge–Kutta scheme, whether or not in a low-storage form to reduce the usage of memory. Recently optimized Runge–Kutta schemes have been developed with better dissipative and dispersive behavior.

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Classical Runge–Kutta schemes

The Navier–Stokes equations, Euler equations, or linearized Euler equations, which are usually solved in CAA, consist of a set of coupled partial differential equations. By spatially discretizing these equations, one obtains a set of coupled ordinary differential equations. Therefore, the time integration is concerned with solving the initial value problem dU dt

= F (t, U (t)) ; U (t = 0) = U0

(5.114)

with U the vector of unknowns. The most general q-stage explicit Runge–Kutta scheme (Kennedy, Carpenter, and Lewis 1999) to integrate from time level t n to time level t n+1 can be written as U(i) = U(n) + t U(n+1) = U(n) + t

i−1  j=1 q 

ai j F( j) ,

(5.115)

b j F( j) ,

(5.116)

j=1

where 

F(i) = F t (i) , U(i) , t

(i)

=t

ci =

(n)

s 

(5.117)

+ ci t,

(5.118)

ai j

(5.119)

j=1

and the stage number i runs from 1 to q. The formulation above requires considerable memory because, at stage i, all F( j) on the previous stages need to be known. The Runge–Kutta schemes can be implemented in a low-storage form, given by U(i) = U(n) + tαi F(i−1) , q  U(n+1) = U(n) + t βi F(i) .

(5.120) (5.121)

i=1

A classical fourth-order Runge–Kutta scheme corresponds to the choice α1 = 12 , α2 = 12 , α3 = 1,

(5.122)

β1 = 16 , β2 = 13 , β3 = 13 , β6 = 16 .

(5.123)

The classical Runge–Kutta scheme given by Equations (5.122) and (5.123) is often used in CAA in combination with DNS or LES for the calculation of the near-field sources. Examples are LES simulations of jet flows by Zhao, Frankel, and Mongeau (2000), Morris et al. (1999), and Bogey et al. (2000a) and DNS simulations of jet flows by Freund (2001) and Mitchell et al. (1999) as well as DNS of sound generation in a mixing layer by Colonius et al. (1997).

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The same scheme has also been used for solving the LEE by, for example Bailly, Bogey, and Juv´e (2000), Agarwal and Morris (2000), Povitsky (2001), and Ekaterinaris (1999c). In Yu, Hsieh, and Tsai (1995) the dissipation and dispersion errors for this Runge–Kutta scheme are studied in combination with several higher-order spatial discretizations and results of vortex calculations based on the nonlinear Euler equations are shown. Kim and Lee (1996) compared classical Runge–Kutta schemes of different orders in combination with a fourth-order spatial discretization. Going from a second-order Runge–Kutta scheme to a fourth-order Runge–Kutta scheme significantly improved their results. However, going to even higher-order Runge–Kutta schemes did not bring much improvement. This might explain why the classical forth-order Runge–Kutta scheme is so popular. As a last remark, it is important to mention that another four-stage Runge–Kutta scheme commonly used in CFD applications U(i) = U(n) + tαi F(i−1) , U(n+1) = U(n) + tF(q) ,

(5.124) (5.125)

is only fourth-order accurate in time for linear advection problems with constant speed. In case of nonlinearity, this scheme has a maximum second-order accuracy in time. 5.1.4.2 Optimized schemes

Time integration schemes can be optimized for different properties such as linear and nonlinear stability, accuracy efficiency, error control reliability, and dissipation and dispersion accuracy (Kennedy et al. 1999). In CAA applications, where one is especially concerned about the dissipative and dispersive behavior of discretized equations, optimization of dissipation and dispersion properties of the time integration scheme is of paramount importance. However, until now only a few authors have considered optimizing time integration schemes. A very commonly used optimized time integration scheme is the one by Hu, Hussaini, and Manthey (1996). Several Runge–Kutta schemes are optimized for their dispersion and dissipation behavior and are baptized as low-dissipation and lowdispersion Runge–Kutta schemes (LDDRK). The optimization is achieved by minimizing the difference between the numerical amplification factor, G(−I tak ∗ ), and ∗ the exact amplification factor, e−I tak , as explained in Section 5.1.2.2. The following integral is therefore minimized:   |G(−I σ ) − e−I σ |2 dσ, (5.126) 0 ∗

where σ = tak and  specifies the range of σ in the optimization. In Hu et al. (1996) the coefficients for a four-, five-, and six-stage optimized Runge–Kutta scheme (LDDRK4, LDDRK5, LDDRK6) are given.

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Table 5.3. Optimized coefficients of the amplification factor for the LDDRK schemes Stages

c3

c4

c4

c4

4 5 6

0.162997 0.166558 1/3!

0.0407574 0.0395041 1/4!

– 0.00781071 0.00781005

– – 0.00132141

Source: Hu et al. (1996).

The coefficients of the amplification factor for these schemes are given in Table 5.3. All of these are kept second-order accurate in time (c1 = 1, c2 = 12 ) while minimizing the dispersion and dissipation errors. A second method in Hu et al. (1996) to optimize time integration schemes is the use of different coefficients for the Runge–Kutta schemes in two alternating time steps. This leads to a further reduction of the dispersion and dissipation errors while maintaining a higher order of accuracy. To do this, the product of the amplification factors of the two different schemes is compared with the square of the exact amplification factor. The coefficients are determined such that the following integral is minimized: 



|G 1 (−I σ )G 2 (−I σ ) − e−I 2σ |2 dσ

(5.127)

0

with G 1 , G 2 the polynomials corresponding to the two Runge–Kutta schemes. Two fourth-order optimized schemes are given, a four–six alternating scheme (LDDRK4–6), and a five–six alternating scheme (LDDRK5–6). Note that Hu et al. (1996) only consider central spatial schemes, ensuring that the numerical wave number and hence σ are real. In his extensive study on different numerical methods used in CAA, Goodrich (1999) concludes that the six-stage optimized LDDRK scheme by Hu et al. (1996) in combination with at least a sixth-order spatial differencing can provide between one and two orders of magnitude decrease in error at a given grid density. However, when the fourth-order central scheme or the DRP scheme of Tam and Webb (1993) is applied, no significant improved is found by using an optimized Runge–Kutta scheme. Other applications of the schemes formulated by Hu et al. (1996) can be found in Ewert et al. (2001a), who use LDDRK5–6 in combination with the fourth-order spatial DRP scheme, as well as Bogey et al. (1999), Bogey, Bailly, and Juv´e (2000b), and Morris, Long et al. (1997), who both use the LDDRK–4 scheme. A nonlinear extension of the LDDRK5–6 is used by Mankbadi, Hixon, and Povinelli (2000) for a VLES application. Further details of this nonlinear extension are described in Stanescu and Habashi (1998). Recently, Bogey and Bailly (2004) presented an optimization of Runge–Kutta schemes based on ideas similar to those of Hu et al. (1996). The integrals to be optimized are defined somewhat differently, leading to different schemes. Optimized versions of two five-stage and six-stage Runge–Kutta schemes are presented. The stability of these

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5.2 BOUNDARY CONDITIONS FOR LES

optimized schemes is better than that of the five- and six-stage algorithms proposed by Hu et al. (1996). Tam and Webb (1993) are among the few authors who optimized a four-level timeintegration scheme. The procedure is comparable to the one followed by Hu et al. (1996). By taking the Laplace transform of the four-level time-integration scheme given by   3  dU (n− j) bj , (5.128) U(n+1) − U(n) t dt j=0 it is easy to find the effective angular frequency of the time discretization:

 i e−iωt − 1 ω= .  t 3j=0 b j e−i jωt

(5.129)

The scheme is optimized by minimizing the integral error E 1  0.5   σ [ (ωt − ωt)]2 + (1 − σ ) × [ (ωt − ωt)]2 d (ωt) (5.130) E1 = −0.5

with , denoting the real and the imaginary part of their argument, respectively. The real part of ω describes the dispersive behavior and the imaginary part the dissipative behavior. For σ = 0.36, the following coefficients are obtained: b0 = 2.30255809, b2 = 1.57434093,

b1 = −2.49100760, b3 = −0.38589142.

(5.131)

5.1.4.3 Other time-integration schemes

Most of the predictor-corrector schemes used in CAA, like the modified McCormack scheme by Gottlieb and Turkel (1976) discussed in Section 5.1.3.5, are second-order accurate in time. This scheme is used in Gamet and Estivalezes (1998), Mankbadi et al. (1994), Mankbadi, Hixon et al. (1995), and Mankbadi, Shih et al. (1995) for jet noise applications. Another predictor-corrector scheme, proposed by Bayliss (1985), is also second-order in time. Manoha et al. (2000) use a second-order accurate semiimplicit Adams–Bashforth/Crank–Nicolson scheme for LES calculation of trailingedge noise.

5.2 Boundary conditions for LES Michael Breuer The filtered Navier–Stokes equations used for LES predictions are in general elliptic in space and parabolic in time. Hence, to solve these equations for a specific flow configuration under consideration, boundary conditions at all borders of the integration domain and initial conditions for all flow variables in the entire field are required.

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Nozzle

Tu = 0 Airfoil / Wing

x1

feasible inflow boundary ui , p, T = f (x1, y, z, t)

feasible outflow boundary

Figure 5.5. Example for a reasonable choice of the integration domain (inflow and outflow boundaries) for the flow past a wing in a wind tunnel.

With a few exceptions, such as predictions of the transition process or fundamental turbulence investigations, in most cases the initial conditions play a subsidiary role because the statistically steady-state flow status is of major concern, which should be reached independently from the initial conditions. An example of an exception is the decay of a homogeneous isotropic flow often used for basic investigations in turbulence research. For this purpose a cubical integration domain with periodic boundary conditions in all directions is chosen, requiring the initialization of the flow field. Based on the scalar energy spectrum of von K´arm´an and Pao (see Hinze 1975), a homogeneous isotropic and divergence-free vector field is generated in spectral space and then transformed back to the physical space by fast Fourier transformation. This procedure leads to reasonable initial conditions with prescribed values for the turbulent kinetic energy k0 , the dissipation rate 0 , and the smallest and largest wave numbers of the flow field. On these initial conditions the basis of basic investigations on the decaying turbulence can be carried out using LES or DNS predictions (e.g., Domaradzki and Rogallo 1990; Rogallo 1981). In most practical LES applications, however, initial conditions are not of prior importance. Here appropriate initial conditions can be chosen to shorten the simulation time until statistically steady state is achieved. For that purpose a variety of tricks are available such as symmetry breaking, superposition of perturbations, and successive grid refinement (Breuer 2002). Compared with initial conditions, boundary conditions (b.c.) play a dominant role for LES – especially for spatially inhomogeneous flows – and will therefore be discussed in detail. On the basis of their mathematical character, b.c. are typically classified into Neumann b.c., Dirichlet b.c., combinations of Neumann and Dirichlet b.c., and periodic b.c. Physically, we can distinguish between two kinds of situations. Either a physical boundary such as a solid wall or an artificial boundary of the integration domain has to be modeled. The latter appears if the solution domain constitutes only a part of the total flow field typically encountered for predictions of external flows. A typical example is shown in Figure 5.5, which depicts an experimental investigation of

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the flow around an inclined wing. To simulate the corresponding flow numerically, a reasonable integration domain has to be chosen not encompassing the entire wind tunnel. This leads to artificial inflow and outflow boundaries that require physically meaningful approximations of the flow situation. These are often difficult to formulate. The quality of these b.c., however, determines the extension of the computational domain required for a high-quality LES and hence strongly influences the efficiency of the prediction. Unlike at physical boundaries, natural b.c. of the physical model can be applied in most cases. In the following sections, the most important kinds of b.c. for LES are discussed (i.e., outflow b.c., inflow b.c., and b.c. for solid walls). For the sake of simplicity, first the flow field without additional scalars is considered solely. At the end some hints for the extension toward nonisothermal flows and far-field b.c. for compressible flows are given. 5.2.1 Outflow boundary conditions

The integration of the (filtered) Navier–Stokes equations requires a finite solution domain. Hence, a chosen outflow boundary can always be interpreted as an artificial cut through the flow field (see Figure 5.5). Survey, for example, the flow around a wing, where the flow in the vicinity of the airfoil and the resulting forces and noise sources are of interest whereas the far wake is of less importance. Consequently, the grid points will be clustered in the vicinity of the wing. The outflow margin of the computational domain will be chosen as close as possible to the body in order to restrict the total size of the domain. The flow variables at this artificial boundary have to be approximated in a physically meaningful manner in order not to influence the solution of the conservation equations within the internal field by upstream-traveling perturbations. In order to guarantee this condition, on the one hand the outflow boundary has to be shifted downstream as far as possible which conflicts with the condition above. Furthermore, this downstream shift is strongly restricted by the number of grid points acceptable and the grid stretching allowed. It is very important for the location of the boundary that no reverse flow exist on the entire plane and that the streamlines be more or less parallel. Consequently, a compromise for the location of the outflow margin has to be found taking all these aspects seriously into account. On the other hand, the formulation of the boundary conditions itself plays a major role. For stationary laminar or Reynolds-averaged flow predictions very often simple extrapolations based on zero-, first-, or second-order polynomials along streamlines or grid lines are applied. Especially, the condition of a fully developed flow in the main flow direction χ (e.g., x, y, or z) is most often used. It leads to the extremely simple Neumann boundary condition ∂u i /∂χ|outflow = 0, which does a good job for steadystate flows at high Reynolds numbers. The reason is the weak upstream propagation of perturbations under these flow conditions. LES predictions are, however, inherently unsteady and dominated by vortical structures. Under these conditions, the reflecting boundary conditions of zero or any higher order mentioned above lead to unsatisfactory results. This is not surprising because they

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are physically not assured. The problem of upstream-traveling perturbations triggered by outflow boundary conditions is a long-lasting topic of general interest for CFD. Thus, a variety of so-called nonreflecting boundary conditions (see Section 5.2.4) have been proposed in the literature – especially for compressible subsonic and supersonic flows (e.g., Hirsch 1990 or Engquist and Majda 1977). Appropriate outflow boundary conditions for LES have to ensure that vortices can approach and pass the outflow boundary without significant disturbances or reflections into the inner domain. For this purpose, a convective boundary condition (Orlanski 1976) of the type  ∂u i  ∂u i = 0 (5.132) + Uconv ∂t ∂χ outflow has proven to work extremely well (cf. Breuer 1998, 2000, 2002). Equation (5.132) represents nothing other than a simplified and linearized one-dimensional transport equation in the main flow direction χ – namely the momentum equation for the filtered velocity component u i , and Uconv denotes a constant mean convection velocity in the χ-direction. Typically, Uconv is independent of the position at the outflow plane and has to be adjusted with respect to the flow simulated. One criterion for the appropriate choice of Uconv may be the mass flow rate at the outlet. For incompressible flows this mass flow rate has to balance the inflow mass flux (i.e., satisfy the global mass conservation). The gradient ∂u i /∂χ |outflow substitutionally denotes the gradient in the x-, y-, or z-direction as determined by the direction of the mean flow. Numerically, the gradient in Equation (5.132) is approximated by one-sided differences. This convective boundary condition has proven to minimize the problem of pressure waves traveling upstream from the outlet and hence to avoid the propagation of errors from the outflow boundary into the computational domain. Consequently, it is widely used for LES predictions of external as well as internal flows. In Figure 5.6, the application of this boundary condition is demonstrated for an LES prediction of the flow past an inclined wing at Re = 20,000 and α = 12◦ (Breuer and Joviˇci´c 2001; Breuer 2002). Only a part of the entire three-dimensional integration domain is shown. However, the right boundary depicted in each subfigure is in fact a part of the actual outflow boundary. Hence, the figure clearly reveals that the large-scale vortices are passing the outflow boundary and are leaving the computational domain without large perturbations reflected back to the internal flow domain.

5.2.2 Inflow boundary conditions

Like the outlet boundary the inflow boundary typically represents an artificial cut through the flow field. If, for example, the sample flow configuration installed in a wind tunnel is numerically simulated from above, it is impossible and impractical to take the flow in the entire experimental setup into account. Instead a reasonably chosen subdomain composed of the wing is typically considered. As a direct consequence LES predictions require appropriate inflow data of the Dirichlet type that adequately represent the flow field at or near the nozzle of the wind tunnel. For the prediction of laminar

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Figure 5.6. Von Karm ´ an ´ vortex street past an inclined wing (NACA–4415) at Re = 20,000 and α = 12◦ visualized by streaklines; four different time instants of a shedding cycle in the vicinity of the outflow boundary are shown.

flows the specification of a steady velocity (or pressure) field at the inlet is sufficient in most cases. For turbulent flow predictions based on the Reynolds-averaged Navier– Stokes (RANS) equations using, for example, two-equation closure models such as the standard k– model, additional specifications for the level of the turbulent kinetic energy k and its dissipation rate  are required. Typically, k and  are expressed in terms of the turbulence intensity Tu and the length scales l of the energy-carrying vortices. For LES partially resolving the spectrum of turbulent length scales in the flow, appropriate b.c. that mimic the unsteady vortical flow at the inlet are essential. These have to be physically meaningful instantaneous data for the entire inflow plane satisfying characteristic autocorrelations and cross-correlations of the velocity components between each other. Because the inflow b.c. can have a strong influence not only in the vicinity of the inlet but also for the entire flow development, the specification has to be made with care. Examples can be found in Klein, Sadiki, and Janicka (2003), Lund et al. (1998), and Stanley and Sarkar (2000) for DNS of spatially developing flows and where in principle no differences exist between LES and DNS concerning the challenge of appropriate b.c. Three special cases leading to simple resorts should be discussed first as follows: (i) Laminar inflow: In the first case, transition to turbulence appears far away from the inflow plane and consequently takes place within the computational domain as a part of the solution. Hence, the inflow data can be represented by laminar steady-state flow. An example is the subcritical flow past a cylinder where transition takes place in the freeshear layers. This case is reasonably simulated by LES (Breuer 1998, 2000, 2002) using a constant velocity profile (without any perturbations at the inlet) mimicking an ideal wind tunnel or free-flight experiment with zero turbulence intensity.

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rescaling

arbitrary position

Auxiliary Simulation

inflow

or

outflow

Main Simulation

inflow

outflow

Figure 5.7. Sketch of Lund et al.’s (1998) procedure for generating appropriate inflow conditions for a boundary layer flow.

(ii) Periodic b.c. with or without rescaling: The second way out of the dilemma is the use of periodic b.c. that avoid inflow and outflow b.c. completely. However, the applicability of this most elegant boundary condition is restricted to flow configurations that are indeed periodic owing to their geometry, such as turbine blades, or statistically homogeneous in one or more coordinate directions such as channel, pipe, or duct flows. In the second case the length of the integration domain L has to be chosen such that the two-point correlations of the fluctuating quantities drop to zero along the homogeneous direction within L/2. Spalart and Leonard (1987) and Spalart (1988) extended the area of application of periodic b.c. to turbulent boundary layers by using a coordinate transformation. Because of the complexity of Spalart and Leonard’s approach and its restriction to flows with small mean streamwise variation compared with the transverse variation, Lund et al. (1998) proposed an enhanced version. Their method is based on an auxiliary simulation applying a much simpler rescaling procedure than Spalart and Leonard’s for the velocity field from a downstream location in order to generate its own inflow conditions. For this simplified transformation the periodicity of the b.c. is no longer valid. However, this is not a critical issue because, in contrast to Spalart and Leonard’s method (Spalart and Leonard 1987; Spalart 1988), the rescaled b.c. are applied to the additional simulation only, which is solely carried out in order to produce appropriate inflow data for the main simulation. This main LES prediction directly employs one plane of the auxiliary simulation as inflow data. The entire procedure is depicted in Figure 5.7. It is worth noting that the rescaling method of Lund et al. (1998) can be extended to compressible flows in order to take Mach number and temperature effects into account. The corresponding procedure was proposed by El-Askary, Meinke, and

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o

Storage of data or on-the-fly simulation

Inflow plane

Periodic Duct

90 bend

Outflow plane Δp Periodic boundary conditions

Figure 5.8. Example for the generation of inflow data for a 90◦ bend using a second simulation for a straight duct flow with periodic b.c. (Breuer 2002).

Schr¨oder (2001). Following Spalart and Leonard (1987) and Spalart (1988), all flow variables are first decomposed into a mean and a fluctuating part. Then the temperature as well as the velocity components are rescaled using appropriate scaling laws for the inner and the outer layer separately. For a detailed description of this method, refer to Section 5.4.1 or El-Askary et al. (2001). (iii) Reusage of data from auxiliary simulations: The technique of Lund et al. (1998) also fits to the third special resort – that is, the reusage of data obtained by another LES or DNS prediction applying periodic b.c. Examples are fully inhomogeneous flows such as the flow through a 90◦ bend (see Figure 5.8) or around an obstacle mounted at the bottom wall of a plane channel (Breuer 2002). To generate appropriate inflow conditions for these cases, additional simulations for a straight duct or a plane channel with the same cross sections have to be carried out applying periodic b.c. in the homogeneous flow directions. The instantaneous data from one plane of these auxiliary simulations can then be applied as inflow b.c. for the inhomogeneous flows. The inflow data from the additional simulation can either be stored in a preprocessing step to the intrinsic simulation, leading to an enormous amount of data to be stored, or generated on the fly, running both simulations in parallel. The second option is favorably used on parallel computers, directly exchanging the data between the processors by a communications library such as message-passing interface. In order to reduce the additional computational efforts with respect to CPU time or disk space, the successive reusage of a previously generated time series or the application of a frozen spatial turbulence field making use of Taylor’s hypothesis can be taken into account under certain conditions. All three special cases discussed above lead to physically reasonable inflow conditions of high quality. However, their application is restricted to specific flows and hence is not practicable in general. An alternative is the generation of artificial inflow data purely based on the knowledge of the flow geometry and eventually some experimental data. The first step of this procedure is to split up the (filtered) velocity field u i into a

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steady, constant mean value u i  and a fluctuating component u i according to Reynolds’ approach:    u i (xi , t)|inflow = ui (xi )|inflow + u i (xi , t) . (5.133) inflow

Typically the mean values are known from experiments or theory. As a consequence, the generation of artificial data can be restricted to the fluctuating components. A variety of methods have been proposed for that purpose in the literature. Here only some important techniques can be mentioned. The simplest but also the worst technique to generate time series for the fluctuating velocities is the application of random-number generators producing white noise. The only free parameter to adjust is the root-mean-square (rms) values of the fluctuations. A serious drawback of this method is that there is absolutely no way to take any spatial or temporal correlations into account. Consequently, the inflow data produced have nothing in common with the physical situation. Furthermore, time series produced by randomnumber generators mainly consist of high-frequency components that are immediately damped out by the numerical methods used to solve the filtered conservation equations – especially when coarse grids are used in the vicinity of the inflow plane as is often the case. Thus, the results of these inflow conditions are more or less identical to laminar inflow data. In order to improve the situation by taking low wave numbers into account, Lee, Lele, and Moin (1992) suggested a method based on stochastic fluctuations with a prescribed energy spectrum. In general, the idea of all methods proposed is to produce inflow data that satisfy certain statistical properties such as rms values, cross-correlations, higherorder moments, length and time scales, or energy spectra. In a recent publication, Klein et al. (2003) provided a good overview of these techniques, typically consisting of two steps. The first is the generation of a provisional three-dimensional signal for each velocity component which possesses a prescribed two-point statistic. In the second step the cross-correlations between different velocity components are taken into account by using a method proposed by Lund et al. (1998). For the first step two methods are applicable. The first was provided by Lee et al. (1992) and is based on an inverse Fourier transform. The second was recently suggested by Klein et al. (2003) and is based on digital filtering of random data. This method allows a prescribed second-order (one-point) statistic as well as autocorrelation functions to be reproduced. Compared with inverse Fourier transforms (Lund et al. 1998), the new procedure possesses several advantages such as simplicity, flexibility, and accuracy. In Klein et al. (2003) these were impressively demonstrated based on two different test cases; however, the technique is generally applicable.

5.2.3 Boundary conditions for solid walls 5.2.3.1 Introduction to solid-wall boundary conditions

A key technology for the application of LES to high-Reynolds-number flows is an appropriate wall-modeling strategy. Of course, Stokes’s no-slip boundary condition and

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Table 5.4. Typical mesh sizes (expressed in wall units) for a boundary layer flow using DNS, wall-resolved LES, and LES with an appropriate wall model

Streamwise Spanwise Wall-normal

x + z + min(y + )

Number of points in

0 < y + < 10

DNS

Wall-resolved LES

LES with wall model

10–15 5 1

50–150 10–40 1

100–600 100–300 30–150

3–5

3–5

–

Source: Sagaut (2004).

the impermeability condition are also valid for turbulent flows at solid walls. However, extremely large velocity gradients are found in the near-wall region that are mainly responsible for the production of turbulent kinetic energy. A prediction based on the (filtered) Navier–Stokes equations has to resolve these velocity gradients adequately in order to determine the level of turbulence production and hence the Reynolds stresses and wall friction reasonably. This solely requires several grid points in the thin viscous sublayer (0 ≤ y + ≤ 5; y + denotes the wall-normal coordinate in wall units). In contrast to a RANS prediction, an extremely fine grid is required not only in the wall-normal direction but in all spatial directions to resolve the near-wall turbulence, including coherent structures such as the well-known high- and low-speed streaks. Typical mesh sizes for DNS and LES prediction of an attached boundary layer are displayed in Table 5.4. These values are common, but the quality of the results exhibits a strong dependence on the size of the mesh – especially in LES. As mentioned above and validated by numerical experiments, the quality of a wall-resolved LES prediction strongly depends on a good resolution of the inner layer (see Table 5.4). Furthermore, the resolutions in the streamwise (x + ) and the spanwise (z + ) directions are very important parameters that also govern the quality of the solution (Sagaut 2004): r High-resolution LES (i.e., x + ≤ 50 and z + ≤ 12):

→ good agreement of the predicted skin friction in plane channel compared with DNS or experiments when nondissipative numerical methods are used. r Medium-resolution LES (i.e., x + ≤ 100 and z + ≤ 30): → thicker and shorter streaks, error on the skin friction. r Poor-resolution LES (i.e., x + ≥ 100 and z + ≥ 30): → unphysical streaks, large error on the skin friction. Physically these considerations reflect the fact that, in an attached turbulent boundary layer, the smallest eddies approximately scale with the distance from the wall, leading to a DNS-like resolution requirement for wall-resolved LES (Spalart et al. 1997). Piomelli and Balaras (2002) estimated the computational costs for this case to be proportional to Re2.4 . For practically relevant high-Re flows such an extremely fine resolution is not achievable and workable. For a flow around an airfoil at cruise flight condition [Re = O(107 )] Spalart et al. (1997) estimated the number of required grid points and

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time steps to 1011 and 5 · 106 , respectively. Thus, such computations will not be feasible during the next decades. 5.2.3.2 Classical wall models

As a remedy for this dilemma, wall models (sometimes called wall functions) bridging the near-wall region and allowing placement of the first grid point in the logarithmic part of the velocity profile (30 ≤ y + ≤ 500) have been developed. Consequently, the near-wall behavior is not resolved in detail, which also leads to reduced requirements with respect to the grid characteristics in streamwise and spanwise directions. In total, the resolution requirements are drastically reduced and the computational costs scale only with Re0.5 , allowing high-Reynolds-number flows to be tackled. One of the first wall models developed and applied to flows in plane channels and annuli was the pioneering one by Schumann (1975). It is based on the phase coincidence of the instantaneous wall shear stress τ12,w and the tangential velocity component u at the grid point nearest the wall. The wall-normal velocity component v is defined by the impermeability condition. For a flat wall parallel to the x–z plane, the wall model for a colocated grid in nondimensional form reads τ12,w (x, z, t) =

τw  u(x, y2 , z, t) , u(y2 )

τ22,w (x, z, t) = 0 τ32,w (x, z, t) =

with

(5.134)

v w (x, z, t) = 0 ,

(5.135)

1 w(x, y2 , z, t) . Re y2

(5.136)

The notation ... represents statistical Reynolds averaging. If homogeneous flow directions are present, additional averaging along lines or planes is applicable. The expression y2 denotes the wall-normal distance of the first off-the-wall grid point and w the wall-normal velocity difference of the spanwise component. The coincidence assumed between the wall shear stress τ12,w and the velocity component u has been experimentally verified. The relationship between these two quantities is established based on the law of the wall for the averaged flow (e.g., the relation between τ12,w  and u). As determined by the wall distance y2 , a corresponding law for the viscous sublayer (VS), the logarithmic buffer layer (BL), and the logarithmic outer layer (OL) is assumed (see Figure 5.9): u +  = y +

for:

0 ≤ y+ < 5

(VS)

(5.137)

u +  = a2 ln (y + ) + b2

for:

5 ≤ y + < 30

(BL)

(5.138)

u +  = a3 ln (y + ) + b3

for:

30 ≤ y + ≤ 500 (OL), (5.139) √ where y + = y u τ  Re , u +  = u/u τ , and u τ  = τw /ρ. The constants are a2 = 5.0, b2 = −3.05, a3 = 2.5 = 1/κ, and b3 = 5.0−5.2. Based on Schumann’s approach, Piomelli et al. (1989) developed a slightly enhanced version that takes the inclination of the near-wall structures and the resulting temporal delay between the tangential velocity and the wall shear stress into account (shifted

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30

u+ u+= y +

20 Figure 5.9. Law of the wall u+ (y+ ) in a turbulent boundary layer without or with only a weak pressure gradient (half-logarithmic plot).

+ 1 + u = __ κ ln y + C

10

0 1

10

100 + 1000 y

logarithmic buffer viscous layer (BL) sublayer (VS)

10000

logarithmic outer layer (OL)

boundary condition). For the most important wall shear-stress component τ12,w , this model reads τ12,w (x, z, t) =

τw  u(x + s, y2 , z, t) , u(y2 )

(5.140)

where the spatial delay is determined by s = |y2 | cot (θ ) with θopt ≈ 8–13◦ . Another wall model suggested by Piomelli et al. (1989) is the ejection boundary condition, which is based on the observation that the near-wall dynamic is dominated by sweeps and ejections. Both models lead to slightly improved results for the plane channel flow compared with Schumann’s original formulation. A major drawback of all these models is that they are difficult to assign to complex, statistically three-dimensional flows because they require the determination of the averaged wall-shear-stress and velocity. Additionally, the use of the customarily applied laws of the wall is highly questionable for flows with large pressure gradients or even local separation and recirculation regions. A further extension of Schumann’s and Piomelli’s models was provided by Bagwell et al. (1993). On the basis of a linear stochastic estimation approach, they use the entire velocity field in a plane parallel to the wall in order to determine the wall shear stress, which a priori requires a two-point correlation tensor generally not available. Following Schumann’s concept, Werner and Wengle (1993) suggested a wall model that is also based on the phase coincidence mentioned above but that applies the laws of the wall (viscous sublayer and power law) directly to the instantaneous velocity field. This simplifies the determination of the wall shear stress and allows the use of their model in flows with separations. However, the application of the laws of the wall to instantaneous velocities and in separated flows is theoretically not justified. Hoffmann and Benocci (1995) derived an analytical expression for the local wall shear stress. They analytically integrated the boundary layer equations coupled with an

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algebraic eddy-viscosity model. Neglecting both advective terms and approximating the unsteady term, they finally ended up with an expression for the wall shear stress leading to satisfactory results for plane-channel flow and rotating-channel flow. An improved model was also derived by Manhart (2001) that takes the local instantaneous pressure gradient in the streamwise direction into account. Actually, the search for appropriate wall models is the ongoing. Abel, Stojkovi´c, and Breuer (2003, 2004) applied advanced statistical techniques such as nonlinear stochastic estimation to instantaneous DNS and wall-resolved LES data to identify the most important model terms and to derive appropriate wall models for LES. In a first attempt this technique was successfully applied to attached flows but the extension to separated flows is actually carried out. 5.2.3.3 Zonal and nonzonal approaches

Another possibility for wall modeling is to use so-called zonal or nonzonal approaches based on the explicit solution of a different set of equations in the inner and outer layers (Balaras and Benocci 1994; Balaras, Benocci, and Piomelli 1996; Cabot and Moin 1999; Piomelli and Balaras 2002). The basic assumption is that the interaction between the near-wall region and the outer region is weak. For modeling the inner near-wall region, simplified governing equations such as the two- and three-dimensional thin boundary layer equations or the RANS equations with a statistical turbulence model are taken into account. These equations are solved on an embedded inner grid in the direct vicinity of the wall, whereas the original LES prediction is carried out on an outer grid not resolving the near-wall region. This leads to a two-layer model as proposed by Balaras and Benocci (1994) and extensively tested by Balaras et al. (1996). The boundary layer equations for the inner layer read   ∂ ∂p ∂ ∂u i ∂u i + (ν + νt ) , (5.141) (u n u i ) = − + ∂t ∂ xi ∂ xi ∂ xn ∂ xn where n denotes the wall-normal direction y and i = 1 and 3 if we assume a wall parallel to the x–z plane. The wall-normal velocity component u n is predicted based on the mass conservation equation for the inner layer: y un = − 0

∂u i  dy . ∂ xi

(5.142)

The system is closed by setting the wall boundary condition for the inner layer to the no-slip condition u i = 0, and at the upper boundary the velocity distribution is obtained from the outer-flow LES prediction as a “freestream” condition (Piomelli and Balaras 2002). Furthermore, the pressure gradient ∂ p/∂ xi in Equations (5.141) is assumed to be independent of y in the inner layer and thus taken from the outer-flow prediction. Consequently, no Poisson equation for the pressure has to be solved and the costs for solving the two momentum equations in the inner layer are only marginally higher than

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using equilibrium boundary conditions such as in the wall models of Schumann (1975) and Piomelli et al. (1989). The quality of the results depends on the choice of the model for the eddy viscosity νt . The most commonly used model for νt , originally applied by Balaras and Benocci (1994) and Balaras et al. (1996), is a simple mixing-length model with near-wall damping. It reads

 3  y+ 2 + + , (5.143) νt = (κ y) |S| D(y ) with D(y ) = 1 − exp − + A where κ denotes the von K´arm´an constant, |S| the magnitude of the strain rate, y + the distance from the wall in wall units, and A+ = 25, a constant. Finally, the wall-stress components obtained from the integration of Equations (5.141) in the inner layer are used as boundary conditions for the outer LES prediction. The two-layer model was successfully applied to channel flow with and without extra rotation and backward-facing step flow. In the latter case, however, the boundary layer equations used within that model are no longer valid in the vicinity of the separation region. If the full RANS equations are applied instead, a hybrid LES–RANS approach is achieved. Because the regions for RANS and LES are defined in advance, this method is called a zonal approach (see also Chapter 4 for more details). The counterparts to the zonal techniques discussed above are nonzonal hybrid LES– RANS approaches. Here a gradual transition between both methods takes place based on an automated switch, ideally removing the need for user-defined information. Speziale’s (1998) formulation belongs to this latter class. Continuing the idea of Speziale, Zhang, Bachman, and Fasel (2000a) have demonstrated the first successful applications of this hybrid concept for a flat-plate boundary layer with and without separation. Conceptually, their approach called flow simulation methodology is very similar to the detached-eddy simulation (DES) proposed by Spalart et al. (1997) and Spalart (2000), which is more widely known. The DES approach can still be considered a zonal method because the two domains (LES–RANS) are fully determined by the grid topology and the segmentation is independent of the flow solution. In DES, attached flow regions (attached eddies) are distinguished from separated flow regions (detached eddies). The former are properly predicted based on RANS with statistical turbulence models, whereas the latter, including large-scale unsteady vortices, are computed more reasonably by LES. Thus, DES could be represented as a natural hybrid method combining RANS and LES. This means that, near solid boundaries, the governing equations work in the RANS mode (i.e., all turbulent stresses should be modeled). Furthermore, pressure and velocity fields are time averaged, and unsteady vortical structures should not be resolved directly. Far away from solid boundaries, the method switches to the LES mode. Hence, the basic concept is to combine the advantages of both methods, yielding an optimal solution at least for a special class of flows, and to afford predictions of high-Reynolds-number flows with reasonable computational effort. For further details, refer to Chapter 4. However, a variety of open issues need to be addressed before one can rely on hybrid methods. These include, in particular, the demand for appropriate coupling

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techniques between LES and RANS, adaptive control mechanisms, and proper SGS– RANS models. Because approximate wall-boundary conditions partially rely on RANS modeling and the classical wall functions discussed above are a limiting case of a RANS model in the near-wall region, both topics are strongly interconnected. For a complete review of wall-modeling strategies, refer to Piomelli and Balaras (2002). 5.2.3.4 Pressure and temperature wall-boundary conditions

The previous section solely concentrated on the velocity field. Appropriate boundary conditions for the pressure field (if required at all) are not so critical. They depend on the flow problem considered and on the numerical methodology applied. For example, the boundary layer over a flat plate at rest exhibits a zero wall-normal pressure gradient at the wall, which can be discretized by a Neumann boundary condition. For a more general case of a flow over a curved, moving, or rotating surface or when external forces such as buoyancy or centrifugal forces are present, large pressure gradients may appear in the vicinity of the surface. In that case it is advisable to determine the pressure gradient based on a simplified momentum equation in the wall-normal direction using one-sided finite differences or to extrapolate the pressure at the wall from the internal region to the boundary. Because these techniques used in the context of LES do not deviate from standard applications for laminar or RANS flow predictions, we refer to the basic literature for CFD. One important issue remains. In the case of nonisothermal flows in which an additional energy equation (e.g., for the temperature T or the enthalpy h) has to be solved, a situation similar to that of the velocity field arises. In principle, a wall heat flux q˙w or wall temperature Tw can be prescribed and discretized without further approximations. However, like the viscous wall layer for the velocity field, this measure requires the conductive wall layer to be resolved. As determined by the molecular Prandtl number Pr = μc p /λ = ν/α of the fluid, which describes the ratio of diffusivities for momentum and heat, this layer might be even thinner than the viscous layer (e.g., for water, < < < 1.7 < ∼ Pr ∼ 13.7, and for oil, 50 ∼ Pr ∼ 100,000). If a fine near-wall resolution is not possible or desired, suitable time-dependent formulations of wall models for the temperature equation have to be applied. These models relate the instantaneous local heat fluxes to the temperature fluctuations at the grid point nearest the wall by using time-averaged wall laws (Gr¨otzbach 1981, 1987; Gr¨otzbach and W¨orner 1999). Consequently, these models are basically analogous to the wall models for the velocity field such as the models of Schumann (1975) and Piomelli et al. (1989) described above. Furthermore, any additional scalar transport equation (e.g., for species) can be treated in much the same way as the temperature equation. 5.2.4 Far-field boundary conditions for compressible flows

Most of the boundary conditions discussed above in the context of incompressible flows can easily be extended to compressible flows (e.g., the physical boundary conditions at solid walls). However, at artificial boundaries arising owing to finite integration domains

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for the prediction of external compressible flows, appropriate far-field boundary conditions are required. This issue is directly related to the boundary conditions applied to the acoustic simulation explained in detail in Section 5.3. Hence, we only address this topic briefly and refer to this section and the cited references therein for further details. For the derivation, all viscous effects are neglected in the far field, leading to the Euler equations. Although this assumption is in general questionable for vortical flows, it is a reasonable condition under practical aspects such as a large extension of the domain and a highly stretched grid with a coarse resolution in the vicinity of the farfield boundaries. The three-dimensional Euler equations for a compressible fluid are a hyperbolic system of five equations with five real eigenvalues. These eigenvalues define the directions along which information is transported. Owing to different signs of the eigenvalues, subsonic flows (Mach number M = u/a < 1) and supersonic flows (M > 1) have to be distinguished. If a positive mean flow direction (u > 0) for a supersonic flow is assumed, all eigenvalues are positive. This is the most uncritical case because the transport of information takes place only in a positive coordinate direction. For an inflow boundary, this means ρbc = ρ∞ , u bc = u ∞ , vbc = v∞ , wbc = w∞ , pbc = p∞ ,

(5.144)

that is, the values of all variables at the boundary (bc) are prescribed by the undisturbed inflow (∞). For an outflow boundary, this means ρbc = ρint , u bc = u int , vbc = vint , wbc = wint , pbc = pint ,

(5.145)

that is, the values of all variables at the boundary are set equal to the values of the first grid point of the internal domain (int). For a subsonic flow (M < 1), four eigenvalues are positive and only one eigenvalue is negative. In contrast to the supersonic case, this means that four variables have to be prescribed by the undisturbed flow at the inlet and one variable has to be extrapolated from the internal region to the boundary (e.g., by a Neumann boundary condition). Correspondingly, at the outlet, four variables at the boundary are given by the internal field, whereas one variable is defined from outside. The latter may lead to reflections. As shown in Section 5.3, the Euler equations can be transformed to a characteristic form that builds the base for nonreflecting boundary conditions. For that purpose the Jacobian matrix A in the Euler equations is reconstructed to the new matrix A+ , using only positive eigenvalues. On the basis of this modified system of equations nonreflecting boundary conditions were derived (Hirsch 1990; Engquist and Majda 1977). More details, including the corresponding reflection coefficients, are presented in Section 5.3.

5.2.5 Final remark for discretization schemes

In conclusion, boundary conditions play a dominant role for LES predictions of high quality, and the derivation of appropriate b.c. is not a trivial task. Especially for inflow

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boundaries and solid walls, intensive research activities are ongoing to widen the applicability of LES, partially leading to hybrid LES–RANS approaches. This will result in exciting new techniques and prospects for LES.

5.3 Boundary conditions: Acoustics Fang Q. Hu With current computing capabilities, it remains difficult to compute the turbulence and its acoustic radiation at the same time in most practical problems. One apparent compromise is to compute the turbulence and the acoustic field separately. An LES or unsteady RANS calculation, which captures the main physics of the underlying noise-generating flow, is conducted first. Then a second calculation is carried out for the acoustic radiation of the turbulent flow to the far field, using the result of the turbulence calculation as the input. The acoustic calculation is often done by solving the Ffowcs Williams–Hawkings equation or the linearized Euler equations. This section deals with the implementation of far-field nonreflecting boundary conditions for the linearized Euler equations. Issues related to the boundary conditions for the LES calculations have been discussed in the previous section. At the far-field boundary, it is possible to use the linearized Euler equation because the viscous and nonlinear effects are often negligible. For convenience, we will use the following linearized Euler equation as the model equation: ∂u ∂u ∂u ∂u +A +B +C + Du = 0, (5.146) ∂t ∂x ∂y ∂z where ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ 0 0 ρ¯ 0 0 U¯ ρ¯ 0 0 0 ρ 1⎟ ⎜0 0 ⎜0 ⎜ u ⎟ 0 0 0⎟ U¯ 0 0 ⎜ ⎟ ⎜ ⎜ ⎟ ρ¯ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 0 0 ρ1¯ ⎟ , u = ⎜ v ⎟,A = ⎜0 0 U¯ 0 0 ⎟ , B = ⎜0 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎝0 0 ⎝0 ⎝w ⎠ 0 0 0⎠ 0 0 U¯ 0 ⎠ p 0 0 γ P¯0 0 0 0 γ P¯0 0 0 U¯ ⎛

0 ⎜0 ⎜ ⎜ C = ⎜0 ⎜ ⎝0 0

0 0 0 0 0

0 0 0 0 0

ρ¯ 0 0 0 γ P¯0

⎞ ⎛ 0 0 ⎟ ⎜ 0⎟ ⎜0 ⎟ ⎜ 0 ⎟ , D = ⎜0 ⎜ 1⎟ ⎝0 ρ¯ ⎠ 0 0

0 0 0 0 0

d ρ¯ dy d U¯ dy

0 0 0

0 0 0 0 0

⎞ 0 0⎟ ⎟ ⎟ , 0⎟ ⎟ 0⎠ 0

and where U¯ , ρ, ¯ and P¯0 are the mean velocity, density, and pressure, respectively, and γ is the specific heats ratio. We have assumed a parallel mean flow that varies only in the y-direction. In many practical situations, the far field has a uniform mean flow.

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We will review the most commonly used nonreflecting boundary conditions, including the characteristic method, radiation conditions, and absorbing zones as well as the recently emerged perfectly matched layer technique. 5.3.1 Characteristic nonreflecting boundary condition

The characteristic nonreflecting boundary condition is based on the characteristic splitting of the Jacobian matrices A, B, or C in Equation (5.146) at a boundary where the x-, y- , or z-coordinate is constant, respectively (Giles 1990; Thompson 1990). Extensions to viscous Navier–Stokes equations can be found in Poinsot and Lele (1992). For example, consider a nonreflecting boundary at an outflow boundary x = x0 . Let the characteristic decomposition of A be A = EE−1 ,

(5.147)

where  is the eigenvalue diagonal matrix and E is the eigenvector matrix. Then, for boundary grids at the nonreflecting boundary, the differential equation is modified to be ∂u ∂u ∂u ∂u + A+ +B +C + Du = 0, (5.148) ∂t ∂x ∂y ∂z where A+ is reconstructed from Equation (5.147) using only the positive eigenvalues, namely, A+ = E+ E−1 . The spatial derivative in the direction normal to the boundary, the x derivative in this example, can now be evaluated using backward differences. Note that the characteristic nonreflecting boundary condition is not exact because matrices A, B, and C in Equation (5.146) are not simultaneously diagonalizable. In fact, for a uniform mean flow of Mach number M, the reflection coefficient for an outgoing acoustic wave is found to be 1 − cos φi , (5.149) Racoustic = 1 − cos φr where φi and φr are the angles of the outgoing and reflected waves, respectively (Hu and Atkins 2003). Here the reflected angle φr is related to φi as sin φi sin φr = . 1 + M cos φr 1 + M cos φi

(5.150)

The reflection coefficient for an outgoing vorticity wave is found to be Rvorticity =

sin φi , 1 − cos φr

(5.151)

where the incident and reflected angles are related by sin φr tan φi = . 1 + M cos φr M

(5.152)

Equations (5.149) and (5.151) indicate that the characteristic nonreflecting boundary condition is most effective for outgoing waves with a nearly normal incident angle.

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Several techniques are available in the literature that can be used to circumvent this limitation partially (Atkins and Casper 1994; Higdon 1987). 5.3.2 Radiation boundary condition

The radiation boundary condition is based on the assumption that certain asymptotic forms can be developed for the solutions at the far field (Bayliss and Turkel 1982a). The asymptotic expansion is usually written in ascending powers of 1/r , where r is the distance from the boundary point to the source of sound. Once the asymptotic expansion is known, differential equations are derived such that they can be satisfied by the asymptotic expansion up to a certain order of 1/r . These differential equations constitute the radiation boundary condition. In this way, the radiation boundary condition is approximately matched to the Euler equation. The radiation boundary conditions have only outward characteristics; thus, they can be easily discretized with backward differences or without any information from the domain exterior to the computational domain. A far-field radiation boundary condition that lets out acoustic waves in two dimensions has been given by Tam (1998a): u 1 ∂u ∂u + + = 0, V (θ) ∂t ∂r 2r

(5.153)

√ where V (θ) = M cos θ + 1 − M 2 cos2 θ . Here M is the mean flow Mach number and θ the angle of the boundary grid point measured from the direction of the flow. For 3D acoustic waves, Bayliss and Turkel (1982a) give a series of radiation conditions with increasing order of accuracy. The leading approximation yields the radiation condition p ∂p ∂p + + =0 (5.154) ∂t ∂r r for the pressure component. Because the radiation boundary condition is based on the asymptotic expansion of the solution, it works best if the nonreflecting boundary is far away from the source of the sound. 5.3.3 Absorbing-zone techniques

These methods are variously referred to as “absorbing zones,” “sponge layers,” “exit zones,” or “buffer zones” in the literature. The basic strategy of these methods is to introduce additional zones of grid points, or layers, to surround the physical domain so that outgoing disturbances are attenuated in the added zones and thus minimize the reflections. The numerical solutions inside absorbing zones need not be physical as long as the use of the zone does not cause significant reflection back into the physical domain and the zone is numerically stable. Various ways of constructing the absorbing

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zones have been proposed in the literature. Some of these approaches will be reviewed here. We note that these approaches are often implemented in a combined fashion. In some cases, the absorbing zone is terminated with a characteristics-based, nonreflecting boundary condition at the exit end of the zone. 5.3.3.1 Artificial dissipation and damping

In this method, an absorbing zone is created and appended to the physical computational domain in which the governing equations are modified to mimic a physical dissipation mechanism (Israeli and Orszag 1981). For the Euler and Navier–Stokes equations, the artificial damping term can easily be introduced into the governing equations as follows: ∂u = L(u) − ν(u − u0 ), ∂t

(5.155)

where u is the solution vector and L(u) denotes the spatial operators of the equations. The damping coefficient ν assumes a positive value and should be increased slowly inside the zone. Here, u0 is the time-independent mean value in the absorbing zone (Freund 1997). Kosloff and Kosloff (1986) analyzed a system similar to Equation (5.155) for the two-dimensional wave equation in which, in particular, a reflection coefficient of a multilayer absorbing zone was calculated. 5.3.3.2 Grid stretching and numerical filtering

Attenuation of the solution may also be achieved by using purely numerical means. One such approach, used by Rai and Moin (1991) and Colonius, Lele, and Moin (1993), is to create a “sponge layer,” or “exit zone,” in which the grids are stretched and coarsened. When an outgoing wave enters the sponge layer, it becomes underresolved in the coarsened grid. Because most numerical schemes have a built-in mechanism of dissipating the disturbances in unresolved scales, the numerical solutions inside the sponge layer are attenuated through numerical dissipation. Computationally, grid stretching is equivalent to modifying spatial derivative terms. For example, if the grids in the x-direction are to be stretched, one can replace the x derivative by 1 ∂ ∂ −→ , ∂x α(x) ∂ x

(5.156)

where α(x) ≥ 1 is an increasing function with α = 1 at the start of the added zone. Then, inside the sponge, spatial derivative terms can be discretized in the same way as those in the interior region. The stretching has to be done very gradually, and α(x) should be a slowly varying function. A sudden increase in grid spacing can cause numerical reflection (Hu and Atkins 2003; Vichnevetsky 1981). A model of smoothly stretching the grid has been discussed by Colonius et al. (1993).

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The attenuation of the numerical solution can be enhanced by applying low-pass numerical filters inside the added zone, thus reducing the length of the exit zone. Colonius et al. (1993) enforced a five-point explicit filter operation in the sponge layer. Visbal and Gaitonde (2001) used implicit high-order filters. Visbal and Gaitonde (2001) noted that, because grid stretching can cause high-frequency grid-to-grid oscillations, high-order filters should be applied to the solutions in the physical domain as well. Examples were presented by Visbal and Gaitonde (2001) in which grid stretching could even be done nonsmoothly when combined with high-order filters. Liu and Liu (1993) combined a grid stretching in the streamwise direction with an increase in viscous dissipation. The performance of the boundary condition depends on the gradualness with which the mean flow is increased.

5.3.3.3 Modification of convective mean velocity

Streett and Macaraeg (1989) and Ta’asan and Nark (1995) modified the mean flow inside the buffer zone for the compressible Euler equations so that it increased gradually and eventually became supersonic at the end of the buffer domain. At that point, because the flow is supersonic and outward, termination of the grid will not cause any reflection. Freund (1997) combined this approach with additional damping terms similar to those used in a sponge layer. The formulation and implementation of the buffer zone appears to be relatively simple. It essentially involves an addition to the governing equations of artificial convective terms of the form U0 (x, y)

∂u ∂u + V0 (x, y) , ∂x ∂y

(5.157)

where U0 and V0 are the artificial velocities that are zero at the start of the buffer zone and gradually increase to become supersonic in the outward direction.

5.3.4 Perfectly matched layers

The perfectly matched layer (PML) technique employs the same strategy as that used in the absorbing-zone techniques reviewed in the previous section. The main difference is that the PML equations are constructed in such a way that they match perfectly to the governing equations of the physical domain while being absorbing for all disturbances that enter the PML zone. The match is perfect when the interface of the physical and PML domains is reflectionless for waves of any frequency and angle. As such, PML zones are usually much shorter compared with other absorbing zones and are less sensitive to parametric variations of the zone. The first PML formulation was introduced by Berenger (1994) for the Maxwell equation in computational electromagnetics. A numerically stable PML formulation for the linearized Euler equation was recently given by Hu (2001).

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The PML technique can be viewed as a complex change of variable in the frequency domain (Collino and Monk 1998; Gedney 1996; Petropoulos 2000; Zhao and Cangellaris 1996). For instance, a change of variable  x i σx d x, (5.158) x → x + σˆ x , where σˆ x = ω x0 would be used for an x-layer starting at x = x0 , where σx is the absorption coefficient and can be any positive function of x or a constant. For the linearized Euler equation with a convective mean flow, it has been pointed out by Hu (2001) that a Prandtl– Glauert-type, or a Lorentz-type, transformation should be applied to the Euler equation before employing the complex change of variable (5.158) to avoid causing instability in the PML domain. In three-space dimensions, the PML equation for Equation (5.146) with a uniform mean flow can be expressed as follows: ∂u yz ∂ux z ∂ux y ∂u +A +B +C + u∗ + σx βAu yz = 0, (5.159) ∂t ∂x ∂y ∂z where U¯ (5.160) β= 2 a¯ − U¯ 2 and u yz , ux z , ux y , ux , and u∗ denote u yz = u + (σ y + σz )q1 + σ y σz q2 ,

(5.161)

ux z = u + (σx + σz )q1 + σx σz q2 ,

(5.162)

ux y = u + (σx + σ y )q1 + σx σ y q2 , ux = u + σx q1 ,

(5.163)

u∗ = (σx + σ y + σz )u + (σ y σz + σx σz + σx σ y )q1 + σx σ y σz q2 .

(5.164)

Here q1 and q2 are auxiliary variables and are computed as ∂q1 = u, ∂t

(5.165)

∂q2 = q1 . (5.166) ∂t The absorption coefficients σx , σ y , and σz are functions of x, y, and z, respectively, with the Euler equation as the special case of σx = σ y = σz = 0. The auxiliary variables q1 and q2 are needed only inside the PML zones. Equation (5.159) can be further simplified wherever any of the absorption coefficients is zero. In particular, q2 is not needed in a region where any two of the absorption coefficients are zero. In two dimensions, we have a simpler version, ∂u ∂u y ∂ux +A +B + u∗ + σx βAu y = 0, ∂t ∂x ∂y

(5.167)

where ux = u + σx q1 , u y = u + σ y q1 , u∗ = (σx + σ y )u + σx σ y q1 .

(5.168)

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Further details on the numerical implementation of PML equations are available in Hu (1996a, 1996b, 2001, 2002). PMLs for nonuniform mean flows are given by Hu (2004).

5.3.5 Summary of boundary conditions for acoustics

As far as accuracy is concerned, the PML technique appears to be the most accurate among all the methods reviewed here. However, the PML equations are not yet available for every type of nonreflecting boundaries to be encountered in practical situations. So far, this technique has only been developed for a parallel mean flow in a direction aligned with one of the coordinates. The technique is still very much under active development (Abarbanel, Gottlieb, and Hesthaven 1999; Hagstrom and Nazarov 2003; Hu and Atkins 2003; Hu 2004, 2005). In this regard, the absorbing-zone techniques can be applied to a wider class of problems and are often coupled with the characteristic boundary conditions. For problems with a centralized and compact noise source, the radiation condition offers an effective alternative.

5.4 Some concepts of LES–CAA coupling Wolfgang Schroder ¨ and Roland Ewert It has already been stated in Chapter 1 that community noise is one of the major problems to be addressed by the aircraft industry, to improve the quality of life in the neighborhood of airports, and to ensure the current growth of passenger numbers. Theoretically, radiated jet noise as well as airframe noise can be determined by solving the unsteady, compressible Navier–Stokes equations. However, this straightforward method to predict the acoustic field of technically relevant subsonic turbulent flow problems significantly exceeds today’s computational capabilities. To substantiate this statement, let us briefly approximate the computational effort for the direct approach of the sound field. If the characteristic length is denoted by L and η is the Kolmogorov length scale to describe the smallest eddies, the total number of grid points of a uniform mesh is  3 L . (5.169) Nuni ≈ η Considering L to be a multiple of the integral length scale of turbulence  (i.e., 3/4 L = c f ) and isotropic turbulence with /η ≈ O (Ret ), we obtain 9/4

N f ≈ c3f Ret ,

(5.170)

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% where the turbulent Reynolds number Ret = u 2  / ν is used and c f > 1 represents a constant. To take into account the acoustic length scale, we consider the Strouhal number Sr = L ω/u that can be interpreted to be proportional to the ratio of the characteristic length L to the length scale of the vortices λω ≈ u/ω (i.e., Sr ≈ L/λω ). The Helmholtz number H e = L ω/c relates L to the acoustic wavelength λa ≈ c/ω such that the vortical and the acoustic structures are connected via H e = Sr M, where M is the Mach number characteristic to the flow problem. More precisely, in the case of low-speed flows the acoustic length is M −1 larger than the vortical length λa ≈ λω M −1 . Using the relation between L and λa via the Helmholtz number L ≈ ca λa and expressing the product λω M −1 by the integral length scale and the turbulent Mach number λω M −1 ≈  Mt−1 such that L ≈ ca  Mt−1 with ca > 1 holds, we obtain for the grid resolution on a uniform mesh N f a ≈ ca3 Mt−3 Ret . 9/4

(5.171)

Equation (5.170) approximates the grid resolution of the turbulent flow field, and Equation (5.171) estimates the number of cells to resolve the flow field plus the acoustic field. If we drop the dependence on the constants c f and ca , it is evident that N f a scales with Mt−3 compared with N f . The turbulent Mach number Mt is in the range of O(10−2 ) for subsonic jets and boundary layers. Therefore, the increased requirement in storage resources is extremely severe in low-speed flows. Moreover, besides this memory problem there is the issue of accuracy. Because acoustic pressures can be almost neglected compared with hydrodynamic pressures, the numerical approximations in CAA have to be more accurate than in classical CFD methods. Given the strong discrepancy in the characteristic length of the flow field and that of the sound field, it is more or less natural to separate the computation of both problems – that is, to apply the hybrid approach to the analysis of low-speed flow and noise problems. In the first step, the flow field is determined by solving the Navier–Stokes equations via a traditional CFD method, and in the second step, the sound field is computed by a numerical solution of an appropriate system of acoustic equations. Rather than using approximate models to describe the sound sources occurring in the acoustic equations, we base our discussion on the assumption that large-eddy simulations are performed to predict the acoustic source terms accurately. That is, an LES is conducted just in the region where the noise is generated, as indicated by the inner domain in Figure 5.10 for the trailing-edge noise problem, and the LES data are used as acoustic input to determine the sound propagation over the complete outer domain in Figure 5.10. Numerous sound propagation equations exist – among other formulations various acoustic analogies, the most famous of which is Lighthill’s acoustic analogy (Lighthill 1952), and some equations that are based, generally speaking, on some form of the linearized Euler equations (Hardin and Pope 1994; Bailly et al. 2000). In our analysis

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l x

e

Figure 5.10. Sketch of the computational domains to determine, for example, trailing-edge noise with the hybrid approach. The LES domain encompasses the vicinity of the trailing edge, whereas the computational aeroacoustics domain includes the whole airfoil owing to the less stringent demands concerning the grid resolution (xacoustic ∼ xLES /M). Thus, scattering at the leading edge captured and directivities predicted.

we use the acoustic perturbation equations that were derived in Ewert and Schr¨oder (2003): ∂ρ  + ∇ · (ρ  u¯ + ρ u ) = 0, ∂t   p ∂u + ∇ u · u + ∇ = qm , ∂t ρ ∂ p ¯  ) = qe + c¯2 ∇ · (ρ  u¯ + ρu ∂t

(5.172)

(5.173)

(5.174)

with the source terms determined by the LES qm = − (ω  × u¯ + ω¯ × u ) + T  ∇ s − s  ∇ T¯ , qe =

γ p¯ ∂s  , c p ∂t

(5.175)

(5.176)

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where the customary notation has been used (i.e., u is the velocity, ω the vorticity, ρ the density, T the temperature, p the pressure, s the entropy, and γ = c p /cv the ratio of specific heats). These equations describe mean flow convection effects, but unlike other systems for the acoustic field they do not possess instabilities for any nonuniform mean flow field with arbitrary density gradients (Ewert and Schr¨oder 2003). The coupling between the LES and the CAA solutions is based on the source terms in Equations (5.175) and (5.176), which are cyclically fed into the acoustic simulation. Over the time interval, the windowing concept, which is used in signal processing to provide a smooth periodic distribution, is applied to the source data to avoid artificial discontinuities between the data samples. When the formulations for the acoustical field (like those presented in Hardin and Pope 1994; Bailly et al. 2000; Shen and Sørensen 2001) are used, the coupling procedure is more or less the same. Only different source terms occur that might require various input data from the solution. Because a detailed comparison of those source formulations is beyond the scope of this section, the interested reader is referred to Ewert, Meinke, and Schr¨oder (2001a) for a more in-depth discussion of the differences. Furthermore, to make this contribution concise and yet thorough, we will discuss neither the impact of different interpolation strategies between the LES and the CAA domain, nor the influence of different windowing formulations of the source terms, nor the pros and cons of various weak or strong coupling concepts, nor the bearings of the interaction on the acoustical and the flow field with respect to phenomena such as receptivity. However, because the efficiency of the hybrid approach, which is the basis for the coupling discussion, strongly depends on the locality of the LES solution and on an acoustically accurate formulation of the boundary conditions on the boundary between the flow domain and the acoustic domain (Figure 5.10), the following discussion of the LES–CAA coupling focuses on two issues that in general do not receive sufficient attention in the context of the two-step approach. That is, we consider a concept to prescribe inflow distributions for an LES locally (El-Askary et al. 2001, 2003) and a silent condition on the LES–CAA boundary (Ewert, Meinke, and Schr¨oder 2002). The fundamentals of large-eddy simulations, sound-propagation equations, and appropriate discretizations are described at length in Chapters 2, 3, and 4 and Section 5.1, respectively. 5.4.1 LES inflow boundary

Although the boundary conditions of local large-eddy simulations were already discussed in Section 5.2, we will address once again in the following section the issue of inflow formulations with emphasis on compressible flows with and without weak pressure gradients. The incorporation of varying pressure distributions is of major importance. For example, in the vicinity of the trailing edge of an airfoil, which is one of the most significant noise sources during takeoff and landing, the academic condition of a constant pressure distribution is not fulfilled. Because the impact of the upstream condition persists for large distances downstream, it defines the size of the LES domain. The better the inflow condition, the smaller the streamwise extension of

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the LES region and as such the more efficient the hybrid approach no matter what kind of description is used for the acoustic step. In this sense the turbulent inflow boundary conditions in spatially developing compressible flows are extremely significant for the hybrid concept. A simple inlet formulation is based on periodic boundary conditions, which unfortunately are restricted to a few simple geometries like channel flow. Spalart and Leonard (1987) were able to extend the application of periodic formulations by introducing a transformation, which is defined via the minimum streamwise inhomogeneity of the boundary layer. Thus, their approach is restricted to spatially evolving boundary layers, the mean streamwise variation of which is small compared with the transverse alteration. It goes without saying that periodic boundary conditions are very appealing, but the formulation of Spalart and Leonard (1987) suffers from several additional intricate terms whose evaluation is very costly. Therefore, Lund et al. (1998) reformulated the concept discussed in Spalart and Leonard (1987). Unlike Spalart and Leonard (1987), they transform only the boundary conditions by using standard scaling laws for an equilibrium turbulent boundary layer. Their formulation is much simpler; however, the periodicity of the boundary conditions is no longer valid. An auxiliary simulation, which produces its own inflow distribution by rescaling the velocity field from a downstream location and reintroducing it at the inlet, is used to extract instantaneous distributions of velocity. A schematic of the procedure, which yields so-called semiperiodic or scaled boundary conditions, is shown in Figure 5.11. In the following, the rescaling method is described. First, the discussion focuses on compressible flows at zero pressure gradient; then, a formulation for a weakly variable pressure gradient in the mainstream direction is introduced, which was already successfully used in Meinke et al. (2004) to simulate airfoil trailing-edge flow. To be able to account for Mach number and temperature effects across the boundary layer, El-Askary et al. (2001, 2003) extended the rescaling method of Lund et al. (1998)

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to compressible fluids. Following Spalart and Leonard (1987), all flow components are decomposed into a mean and fluctuating part, and then an appropriate scaling law is applied to each quantity separately for the inner and outer layer. Consequently, the velocity fluctuation u reads u = u(x, y, z, t) − U (x, y),

(5.177)

where u = (u, v, w)T is the instantaneous velocity and U = (U, V, W )T the mean velocity. Note, however, that the spanwise component vanishes such that no rescaling in the spanwise direction is necessary. The mean streamwise velocity component at the downstream station is rescaled (index “re”) and linked with that at the inlet (index “in”) inner + (yin ) = βs Uvd,re (yin+ ), Uvd,in

(5.178)

outer Uvd,in (ηin ) = βs Uvd,re (ηin ) + (1 − βs )Uvd,∞ .

(5.179)

In these equations the inner law coordinate y + = u τ y/ν is defined via the friction √ velocity u τ = τw /ρ, η represents the outer coordinate, and the subscript “vd” refers to the van Driest transformation, which is applied only to the streamwise component   U (x, y) 1 Uvd (x, y) −1 sin B (5.180) = U∞ B U∞ with & ' ' B=(

1

γ −1 2 r M∞ 2 2 + γ −1 M∞ 2

r

,

(5.181)

where r represents the recovery factor and M∞ is the freestream Mach number. The velocity fluctuations are analogously formulated:



 

 inner + yin , z, t = βs u re yin+ , z, t , u in

(5.182)

 inner

 u in (ηin , z, t) = βs u re (ηin , z, t) ,

(5.183)

where the rescaling factor βs is defined as   u τ,in βs = > 1. u τ,re

(5.184)

The superscripts “inner” and “outer” denote the inner and outer layer components of the variables, respectively. To account for compressibility effects the temperature is rescaled, too. For the mean static temperature T , we locally use the equation of Walz (1969),   T |U |2 =1+ A 1− 2 , (5.185) T∞ U∞

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where 2 A = 0.5 (γ − 1) r M∞

(5.186)

with the ratio of specific heats, γ. Furthermore, the temperature fluctuation T  can be calculated if one assumes a negligible total temperature fluctuation Tt compared with the static temperature fluctuation. This assumption is valid at Mach numbers in the range M∞ ≤ 3.0 (Bradshaw 1977). Then, by focusing on the streamwise velocity component and by neglecting the higher-order velocity fluctuations, we find that the static temperature fluctuation reads T  = − U u  /c p or, in rewritten form, T  (y, z, t) u  (y, z, t) = − (γ − 1) M 2 , T (y, z) U (x, z)

(5.187)

where M is the local Mach number determined by the mean velocity and the mean temperature. Using Equations (5.185) and (5.187) and following the rescaling process, we find that the equations for the rescaled temperature and its fluctuations read Tininner (yin+ ) = βs2 Tre (yin+ ) + C1 T∞ ,

(5.188)

 Tin,inner yin+ , z, t = βs2 Tre (yin+ , z, t),

(5.189)

Tinouter (ηin ) = βs2 Tre (ηin ) − C2 T  in

outer

Ure (ηin ) T∞ + C3 T∞ , U∞

(5.190)

(u  )re (ηin , z, t) T∞ U∞

(5.191)

(ηin , z, t) = βs2 Tre (ηin , z, t) − C2

with C1 = (1 + A) (1 − βs2 ), C2 = 2 A βs (1 − βs ), C3 = (1 − βs ) [1 + βs + 2 A βs ]. We now discuss the impact of an adverse pressure gradient. The equilibrium-type similarity analysis for boundary layers with pressure gradient is explained in Tennekes and Lumley (1972) and Castillo and Walker (2002). For the characterization of these equilibrium turbulent boundary layers, it is sufficient that a single pressure gradient parameter is constant. In the similarity solution discussed in Castillo and Walker (2002), the pressure gradient parameter m m=

dUδ δ dx dδ Uδ d x

= constant

(5.192)

is constant, where Uδ is the velocity at the edge of the boundary layer. The self-similar turbulent boundary layer is obtained if Uδ ∼ x m . This relation can be inferred from keeping u τ /Uδ = constant. For nonzero values of m, integration of the m-equation

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yields Uδ 1/m . In other words, δ ≈ x and dδ/d x = constant. Castillo and Walker (2002) use in their similarity solution for equilibrium boundary layers the relation u τ 2 ≈ Uδ 2

dδ . dx

(5.193)

If δ ≈ x is inserted in this relation, u τ ≈ Uδ can be concluded. This means, for the inner layer of an equilibrium turbulent boundary layer, that the pressure gradient has hardly any effect on its similarity. Thus, we can apply the same inflow condition as derived for the inner layer of the zero-pressure-gradient boundary layer. In the outer layer, the experimental results by Castillo and Walker (2002) show, for a wide range of Reynolds numbers, that the equation Uδ − U = f (y/δ) Uδ

(5.194)

gives similar velocity profiles in the outer part of the boundary layer. If this relation is applied in the inlet and the rescaling cross section, we obtain Uinouter = βapg Ure (ηin ),

(5.195)

where βapg = Uδ in /Uδ re . In Equations (5.196) and (5.197) below, the velocity fluctuations are assumed to obey the same relation: + +  (u )inner in (yin , z, t) = βapg (u )re (yin , z, t),

(5.196)

 (u )outer in (ηin , z, t) = βapg (u )re (ηin , z, t).

(5.197)

The temperature fluctuations in the inner and outer layer are conjectured to satisfy Morkovin’s hypothesis, which is the basis for the Bradshaw (1977) relation (5.187), such that they can be expressed by T  in

2 = βapg T  re (yin+ , z, t),

(5.198)

T  in

2 = βapg T  re (ηin , z, t).

(5.199)

inner

outer

The mean temperature distribution is determined by Equation (5.185) of Walz (1969). A smooth distribution of the profile over the entire boundary layer with and without pressure gradient is obtained by forming a weighted average of the inner and outer profile W f (ηin ), (ψ)in = ψininner [1 − W f (ηin )] + (ψ)outer in

(5.200)

¯ inner = () + (ψ  )inner (ψ)inner in in in ,

(5.201)

¯ outer (ψ)outer = () + (ψ  )outer in in in ,

(5.202)

where

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Turbulent Streamwise Velocity Profile

L

L1

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Figure 5.12. Sketch of the flat-plate boundary layer domain (left) and the trailing-edge domain (right). The procedure to provide the inlet distribution to simulate trailing-edge flow is visualized.

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¯ is the rescaled time-averaged part of the variable and its rescaled fluctuation in which   is ψ . The weighting function W f (η) ensures a smooth transition between the inner and outer layer: W f (η) =

1 2

 1+

tanh [αω (η − b)/(1 − 2b) η + b] tanh(αω )

 (5.203)

with αw = 4 and b = 0.2. The parameter αw controls the width of the region, in which the function varies from W f (η = 0) = 0 to W f (η = 1) = 1. A linear interpolation is used to determine the corresponding locations in the inlet and rescaling planes. The rescaling operation requires the friction velocity u τ and the boundary layer thickness δ at the rescaling station and the inlet. At the rescaling station these quantities are determined by the solution of the mean velocity profile, whereas at the inlet they must be specified. It is sufficient to fix δ at the inlet, but u τ is evaluated via )   1 ρwall,re δ1re [2(n−1)] u τ,in = u τ,re , (5.204) ρwall,in δ1in where δ1 is the displacement thickness and ρwall is the local density at the wall; the exponent is n = 5. Equation (5.204) can be derived using the customary power law −1/n −1/n approximations, c f ∼ Rex , δ1 /x ∼ Rex , where c f is the skin-friction coefficient. Unlike Lund et al. (1998), who use the momentum thickness, the displacement thickness is incorporated in the rescaling formulation to reduce the nonlinearity effects of the momentum thickness. The integrand of the displacement thickness is a linear function of the velocity, whereas that of the momentum thickness is a quadratic function. The latter accumulates more inaccuracies from the spanwise average such that a slightly less benign behavior at the boundaries results. We return now to the inflow boundary condition of the LES domain in Figure 5.10; the boundary layer solution is coupled with the LES that provides the acoustic sources by using the vector of solution of the boundary layer computation at time steps t = tn = n t in the cross section x = xct with xin < xct < xre as inflow distribution at t = tn , as shown in Figure 5.12. That is, the flow computation in the first step of the hybrid approach is based on m + 1 large-eddy simulations performed simultaneously, where m corresponds to the number of inlet boundaries of the acoustic source region above a solid surface (i.e., for the trailing-edge noise problem of a flat plate sketched in Figure 5.12, m = 2 holds). The importance of the thorough formulation of the rescaling method is shown by the skin-friction distribution in Figure 5.13 for the flow over an adiabatic flat plate at M∞ = 0.4 and Reo = 1400, where o is the momentum thickness at the inlet. The comparison of the distributions computed using a simple temperature rescaling, which assumes the mean and fluctuation temperature to be determined by the corresponding velocity quantities, and the aforementioned rescaling method evidence the superiority of the latter as far as the agreement with data from the literature (Murlis, Tsai, and Bradshaw 1982; Falkner 1943; Fernholz 1971) is concerned.

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0.0045 0.004 0.0035 0.003 cf

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0.0015 0.001 1400 1500 1600 1700 1800 1900 2000 2100 Reθ Figure 5.13. LES of a turbulent boundary layer for Reθ0 = 1400 and M∞ = 0.4. Skin-friction coefficient c f versus Reθ for different rescaling formulations.

5.4.2 Silent embedded boundaries

Aeroacoustics deals with the minutest energy levels compared with the entire flow. Even when the computational analysis is based on a smooth one-block mesh, an approximation especially designed to simulate the acoustic field has to be used to ensure that the acoustics is not determined by the numerical procedure but by the acoustic source mechanisms contained in the flow field. It goes without saying that particular attention has to be paid to the formulation of boundary conditions because their accurate definition may have unexpectedly large effects on the sound field. This is, for example, known from the broadband amplification phenomenon (Bechert and Pfizenmaier 1975; Moore 1977). This extreme sensitivity of the sound field against perturbations at the boundaries holds not only at outer but also at inner boundaries such as those encountered in the hybrid approach in which different matching blocks covering the LES and the acoustic domain possess artificial boundaries (Figure 5.10). Owing to the discontinuity in the vorticity distribution, numerical noise is generated at these embedded boundaries that can, remarkably, falsify the acoustic field. This issue has been briefly discussed by various authors (Crighton 1993; Mitchell, Lele, and Moin 1995b; Wang and Moin 2000; Kalitzin and Wilde 2000) in the context of airframe noise and jet noise. To remedy this problem Mitchell et al. (1995b) and Kalitzin and Wilde (2000) implement a controlled decay of the acoustic source term at the boundary between the flow domain and the acoustic domain. To reach a better insight into spurious sound that is generated by the mismatch of the vorticity distribution on the artificial boundary between the inner LES subdomain and the outer acoustic domain and to get an idea of how to find formulations to avoid this phenomenon, we start by considering the following formulation for the source term q m in Equation (5.175) of the momentum equation of the acoustic perturbation equations (5.172, 5.173, 5.174): qm = − H ( f ) L.

(5.205)

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¯ × u is a perturbed form of the Lamb vector. This The quantity L = ω × u¯ + ω vortex term represents the major acoustic source when, for example, trailing-edge noise is considered. The function f (x) is f (x) < 0 in the acoustic domain and f (x) > 0 in the LES domain such that the Heaviside function is H ( f ) = 1 within the source region and H ( f ) = 0 outside. The gradient of H is nonzero only on the boundary between the LES and the acoustic domain on which the gradient of f is in the direction of the normal vector on the boundary n pointing into the interior of the acoustic region ∇ f =n. Equation (5.205) limits the vorticity distribution just to the acoustic source region. This truncation leads to a discontinuity on the boundary, which generates a spurious velocity field with vorticity confined to the boundary that encloses the acoustic region. In the following the discussion of the generation of vortical perturbations due to the modified source defined in Equation (5.205) will be based on the momentum equation (5.173). Using ∇H ( f ) = δ( f )∇ f , where δ is the Dirac delta function, and taking the curl of the velocity field ω  = ∇ × u with the truncated source in Equation (5.205) result in ∂ω  = − ∇ × H ( f )L, ∂t

(5.206)

∂ω  = − [H ( f )(∇ × L) + δ( f )(∇ f × L)], ∂t

(5.207)

∂ω  = − [H ( f )(∇ × L) + δ( f )(n × L)]. (5.208) ∂t Note that only the second term on the right-hand side represents a vorticity source on the boundary. That is, it is exactly this expression that excites spurious perturbations on the boundary. In the following, it will be shown that at a perpendicular flow over the boundary only rotational velocities will be generated by the modified source. For this reason, a potential ϕ and a vector potential ψ are defined to decompose the source qm into an irrotational ∇ ϕ and a solenoidal part ∇ × ψ: qm = ∇ ϕ + ∇ × ψ.

(5.209)

Taking div qm eliminates the ψ-term and leads to the following formulation for the irrotational velocities: ∇2 ϕ = − H ( f ) (∇ · L) + δ( f )(n · L).

(5.210)

The source term on the boundary δ( f )(n · L) vanishes because, for an orthogonal flow across the boundary, the Lamb vector L and the normal vector n are perpendicular to each other. That is, no spurious irrotational velocities are generated. Considering curl qm removes the potential ϕ such that the right-hand side of Equation (5.208) is completely determined by the vector potential ψ. Hence, it can be concluded from the analysis that, on the boundary, only spurious vortical perturbations are excited by δ( f )(n × L); that is, ∂ω  svp = − δ( f ) (n × L), ∂t where the subscript svp denotes spurious vortical perturbations.

(5.211)

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Having derived an equation for spurious perturbations generated at embedded boundaries, we turn now to the discussion of how to suppress this spurious vorticity. Two methods will be described; the first is based on the Biot–Savart formula. We consider a two-dimensional formulation. Let S be the boundary, where the source is located, (x  , y  )T the source point, (x, y)T a point in the domain, and r the distance between (x  , y  )T and (x, y)T ; we will use the basic properties of the Dirac delta function. Then, the term containing the perturbation velocities can be computed by the following integral:    ∂u svp (n × L)z y − y x − x  T = − , d S = −qcorr , (5.212) ∂t 2π r r r S where the subscript z denotes the component of the source term orthogonal to the x–y plane. This correction term qcorr , when added to the acoustic source −L, will prevent the occurrence of spurious vorticity, and thus a silent artificial boundary without any spurious sound is determined. The computation of the compensation expression hardly impairs the efficiency of the hybrid approach because qcorr can be simultaneously calculated with the acoustic source L. In the second method, to avoid spurious sound generated by artificial boundaries, a damping zone of finite thickness d is imposed at the boundary x = 0 – that is, | x | < d/2. As far as the mathematical formulation of the modified boundary source in Equation (5.205) is concerned, this means a smooth filter function h(x); for instance, ⎧ 0 ⎪ ⎪ ⎨

 h(x) = 12 1 + sin πdx ⎪ ⎪ ⎩ 1

for x ≤ − d2 , for −

d 2

for x ≥

< x < − d2 ,

(5.213)

d 2

is substituted for the Heaviside function qm = −h (x) L, yielding for the spurious perturbation on the boundary ∂ω svp ∂t

=−

∂h (n × L). ∂x

(5.214)

Under the assumption of passively convected vorticity in the positive x-direction – ˆ y) – that is, the vorticity is simply shifted by xˆ = u ∞ t such that L(x, y, t) = L(x − x, the boundary equation reads ∂ω svp ∂t

=−

∂h q (x − x) ˆ ∂x

(5.215)

with q = n × L. Note that the y-dependence has been dropped for convenience. This equation expresses that a spurious vorticity source occurs at any x within the zone between the acoustic and the LES domain, which is the origin of a numerically induced acoustic wave. To be able to determine the overall impact of all sources via a simple integration over the thickness d of the damping zone, we assume compactness (i.e.,

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Figure 5.14. Transfer function | F˜ | as a function of wave number α scaled by damping zone thickness d.

d is much smaller than the acoustic wavelength). Taking into account the vanishing gradient of h(x) outside the zone |x| ≥ d/2, we obtain the total influence via  ∞ ∂ω svp, tot ∂h ∂h ˆ dx = − ≈− q (x − x) ∗ q; (5.216) ∂t ∂x −∞ ∂ x that is, the entire impact corresponds to the spurious sound source convoluted by the gradient of the filter function h. Because the Fourier transform of the convolution is equal to the product of the Fourier transforms of the functions, the preceding equation reads in Fourier space . svp ∂ω

. ∂h q, ˜ (5.217) ∂t ∂x where the equal sign has been substituted for the approximate sign. From Equation (5.217) it is evident that in wave-number space the spurious vorticity source q˜ is filtered x. To analyze the effect of wave-number-dependent filtering, consider the filter by ∂h/∂ function defined in Equation (5.213). The gradient is given by /π cos ( πdx ) for | x | < d2 ∂h 2d (5.218) = F(x) = ∂x 0 otherwise =−

and the Fourier transform of F(x) reads    α d 1 1 π ˜ F(α) = cos 2 2 π −αd π +αd

(5.219)

˜ with the wave number of the vortical perturbation α = 2π/λ. The graph of | F(α)| in Figure 5.14 evidences a low-pass distribution (i.e., perturbations with wavelength λ  d or α d  1 are hardly modified, whereas small disturbances satisfying λ < d or α d  1 are effectively filtered). To show the efficiency of both methods to suppress spurious sound generated at the flow-acoustic boundary, the Biot–Savart and the damping zone approach, we consider the artificial noise caused at a boundary located at x = 0 in Figure 5.15. The boundary represents the case encountered in a hybrid computational aeroacoustics approach,

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where the vorticity distribution of the LES subdomain, assumed to be located at x > 0, does not exactly correspond to that of the outer domain, which is described by x < 0. That is, the outer velocity field does not coincide with the inner velocity field. The impact of such an unbalanced distribution is evidenced when the effect of a convecting point source being turned on within an interval located in the immediate vicinity of x = 0 in Figure 5.15 is simulated. In general, the point source acts like a force in the y-direction (Figure 5.15), which produces a doublet-like velocity field, the axis of which points normal to the mainstream direction. The convection of the source causes the velocity field of a downstream-generated doublet to balance the velocity distribution of the upstream doublet. In other words, the remaining velocity field describes a silent vortex convecting with the mean velocity. If, however, the initial velocity field is not properly prescribed (e.g., by suddenly turning on the point source), the generated vortex is not balanced by the velocity distribution. Therefore, a steady counterrotating vortex compared to the convecting vortex occurs at the initial source position. Because, as was shown in Ewert and Schr¨oder (2003), the acoustic perturbation equations do not possess the convection property of vorticity, the vortex dynamics is completely determined by the source term. From the Kutta–Zhukhovski theorem it is known that, in a mean flow field, a steady vortex is related to a pressure field. Hence, under zero initial conditions for the pressure field, the steady-state pressure distributions of the initial vortex are compensated by opposite-pressure fluctuations that propagate as acoustic waves. The behavior of the convecting point source is visualized in Figure 5.15. A vortex convects to the right, and a steady counterclockwise rotating vortex, whose origin is at x = 0, y = 0, produces propagating acoustic waves leaving the computational domain.

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5.4 SOME CONCEPTS OF LES–CAA COUPLING

Figure 5.16. Pressure distribution on y = 35 in Figure 5.15 for several thickness values d and Biot–Savart’s law (denoted as compensation).

The impact of the suppression concept is evidenced in Figure 5.16 by the distribution of the spurious pressure on the line y = 35. Without any compensation, a strong overshoot occurs that is drastically reduced when the filter function (5.213) is turned on within the damping zone | x | ≤ d/2. The larger the thickness of this zone (i.e., a wider range of wave numbers is filtered), the smaller the amplitudes of the spurious pressure waves. In fact, at d = 20 the suppression is as effective as it is when the Biot–Savart law (5.212) is used. It can be concluded from the findings of the numerical analysis of the convecting vortex problem that both methods can be applied to avoid spurious sound waves induced by boundaries embedded in the overall computational domain. However, the more straightforward implementation makes the damping zone approach the method of choice when the acoustic field of technically relevant problems is to be computed.

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Applications and Results of Large-Eddy Simulations for Acoustics

6.1 Plane and axisymmetric mixing layers Christophe Bogey and Christophe Bailly Velocity-gradient regions between two streams are found in numerous flows such as plane or axisymmetric jets. These simple flow configurations, usually referred to as shear layers or mixing layers, have been extensively investigated as reported in the review of Ho and Huerre (1984). These studies have allowed better understanding of the transition to turbulence occurring in initially laminar or transitional shear layers thanks notably to the development of instability theories (Michalke 1984) and to the observations of coherent structures. Among the experiments supporting the latter observations, the famous one conducted by Brown and Roshko (1974) shows clearly that large-scale coherent structures are intrinsic features of mixing layers even at high Reynolds numbers. These structures, through their interactions such as vortex mergings, have been recognized to be appreciably responsible for the spreading of the shear layers or for noise generation, as stated by Winant and Browand (1974). From the preceding considerations, the large-eddy simulation (LES) approach appears especially well suited to mixing-layer computations because with LES all the scales larger than the grid size, and consequently a large part of the coherent structures, are calculated. The earlier large-eddy simulations of practical flows have thus often involved mixing layers. They have permitted testing of the LES methodology for simple transitional flows. For instance, Vreman et al. (1997) investigated the effects of several subgrid modelings for a temporal mixing layer, and Doris, Tenaud, and Ta Phuoc (2000) studied the influence of different numerical schemes and inlet conditions for a spatially developing mixing layer. LES has also been applied with the aim of improving the description of the turbulent structures in mixing layers. An example is given by Comte, Silvestrini, and B´egou (1998), who, using LES, investigated the formation of streamwise vortices in mixing layers developing spatially. As a further step, LES seems appropriate for determining the noise radiated by these large structures, which are likely 238

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to be dominant in transitional shear layers (Winant and Browand 1974), although their contribution in fully turbulent shear layers where no pairing occurs is questionable, as claimed, for instance, by Bridges and Hussain (1987). The first simulations of free-shear flows to compute their radiated noise from the unsteady compressible flow equations were, however, performed using direct numerical simulation (DNS). These simulations were DNS of bidimensional and axisymmetric mixing layers and were carried out in the Stanford group by Colonius et al. (1997) and by Mitchell et al. (1999), respectively. They showed the feasibility of the direct noise computation (DNC) by directly calculating the noise generated by vortex pairings in mixing layers. Afterward, DNC has been naturally applied using DNS for threedimensional mixing layers. A clear illustration is provided by Fortun´e, Lamballais, and Gervais (2001), who studied by DNS the effects of temperature on the sound generated by temporally evolving mixing layers. Because DNS is limited to low-Reynolds-number flows, LES must be used to perform DNC at high Reynolds numbers (i.e., for flows with realistic turbulence). In what follows, some recent works of the authors are presented to illustrate the use of LES to investigate the sound generated in mixing layers. Three configurations are shown: the first one involves a bidimensional mixing layer, and the two others involve circular subsonic jets at different Reynolds numbers with initially transional shear layers. In the three cases, the noise radiated by the turbulence is obtained directly from LES, and the sound-generation mechanisms are discussed. 6.1.1 Plane mixing layer

As in DNS, with the work of Colonius et al. (1997), the first DNC using LES has often been performed for simple bidimensional mixing layers. The first illustration in this chapter therefore deals with the computation of the sound generated by vortex pairing in a 2D mixing layer using large-scale simulation and the Smagorinsky model for the subgrid modeling. All the details of the simulation can be found in Bogey et al. (2000b). The inflow of the mixing layer is defined by an hyperbolic-tangent velocity profile as    2y , (6.1) u 1 (y) = Um 1 + Ru tanh δω (0) where Um = 0.3 c0 and Ru = U/(2Um ) with U = U2 − U1 = 0.48 c0 − 0.12 c0 (c0 is the mean speed of sound) and δω (0) is the initial vorticity thickness of the shear layer. The Reynolds number based on the initial vorticity thickness is equal to Reω = δω (0)U/ν = 12, 800 (ν is the molecular viscosity). To control the first vortex pairing, the mixing layer is classically forced at its fundamental frequency f 0 and its subharmonic f 0 /2. The fundamental frequency f 0  0.132 Um /δω (0) corresponds to the most amplified instability in the initial shear layer predicted by the linear instability theory (Michalke 1984). This kind of forcing

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allows fixing the location of the vortex pairings around x = 70δω (0) in the downstream direction and their frequency as f p = f 0 /2. The dilatation field  = ∇.u obtained directly from the simulation on the whole computational domain is displayed in Figure 6.1(a). Dilatation is indeed related to the acoustic pressure in the uniform streams by    ∂p ∂ p 1 + Ui with i ∈ [1, 2]. (6.2)  = ∇.u = − ∂x ρ0 c02 ∂t Wave fronts originating clearly from the zone of vortex pairings are observed. As expected, the acoustic wavelength corresponds exactly to the pairing frequency. This wavelength appears to be modulated by the effects of the uniform streams on sound propagation. The wave fronts have typical oval forms – especially in the upper high-velocity stream. The noise directivities in the high- and low-velocity streams are also appreciably altered by the mean flow effects on sound waves. Note that these interactions between the flow and the acoustic waves have been investigated using

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two-step approaches with the DNC solution as reference solution. Both Lighthill’s analogy in Bogey, Gloerfelt, and Bailly (2003) and the approach based on the linearized Euler equations in Bogey, Bailly, and Juv´e (2002) have thus been applied. Finally, to study the sound source, a view of the pairing zone is presented in Figure 6.1(b) with the vorticity field in the shear layer and the dilatation field outside. A double spiral structure with four lobes corresponding to a rotating quadrupole is observed. This kind of acoustic source was described for a corotating vortex pair in the analytical works of Powell (1964) and later computed through DNS (Mitchell, Lele, and Moin 1995a). The present simulation has thus allowed to reveal the presence of such a source in a mixing layer. 6.1.2 Axisymmetric mixing layers – jets

This section reports some recent investigations of the noise generated in the axisymmetric shear layers of circular jets computed by LES. Links between the shear layer turbulence and the noise radiated in the sideline direction are shown for two jets at different Reynolds numbers. 6.1.2.1 Moderate-Reynolds-number jet

The first three-dimensional flow simulated by LES in our group to compute its radiated noise was a circular jet at Mach number M = u j /c0 = 0.9 and Reynolds number Re D = u j D/ν = 6.5 × 104 , where u j is the jet inflow velocity and D its initial diameter. In this simulation, the inflow shear layer was forced with random velocity disturbances to seed the turbulence, and the Smagorinsky model was used. Please refer to Bogey, Bailly, and Juv´e (2003) for all the simulation parameters. The flow and acoustic results have been successfully compared with corresponding measurements to show the feasibility of directly computing jet noise using LES. Connections between the dynamics of the turbulent structures in the jet and the radiated waves have then been tracked, as was done in experiments by Hileman and Samimy (2001) for instance. In this way, a strong link has been exhibited between the intrusion of turbulent structures in the jet core and the noise radiated in the downstream direction. To detect possible noise-generation mechanisms occurring in the shear layer, we have recorded the pressure at a point defining a wide angle from the flow direction. Experiments such as those of Zaman (1986) have indeed shown that jet noise in the sideline direction can be related to the turbulence in the shear layer. The observation point has been chosen to be at x = 16r0 , y = 8r0 , and z = 0. This point is at 9.4r0 from the end of the potential core at x = 11r0 and y = z = 0, where the sound sources are classically assumed to be located, at an angle of about 60o with respect to the downstream direction. The signal of pressure at this point is displayed in Figure 6.2 during a nondimensional time period T ∗ = T u j /D = 40. An attempt has then been made to connect the peak of highest amplitude observed for t ∗ = 6.9 with the interactions between the turbulent structures in the jet. The sound wave corresponding to the pressure peak is shown in Figure 6.3. It is located at about

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Figure 6.2. LES of an ReD = 6.5 × 104 subsonic jet. Time evolution of the fluctuating pressure p in Pa as a function of t ∗ = tu j /D, at x = 16r 0 , y = 8r 0 , and z = 0. Reprinted from Bogey, Bailly, and Juve´ (2003) with the kind permission of Springer Science and Business Media.

x = 18r0 and y = 9r0 . Its directivity is marked for wide angles between 30o and 70o , which supports that it be generated in the shear layer. To find the turbulent event likely to generate the present acoustic wave, we evaluated the time delay between its emission and its arrival at the observation point by assuming a propagation at the mean sound speed. A delay of t ∗ = 4.2 has been obtained, which implies that the generation occurs for t ∗  2.7. The vorticity in the plane z = 0 has therefore been depicted at t ∗ = 2.2 and t ∗ = 3.5 in Figure 6.4. An isolated large structure is observed in the upper shear layer in Figure 6.4(a) at x  11r0 and y  r0 . It interacts with other structures to merge with the vortical field located downstream in Figure 6.4(b). This kind of vortex pairing is conjectured to be responsible for the acoustic wave of Figure 6.3. More generally, this observation supports that the interactions between coherent structures in the shear layer contribute to jet noise at moderate Reynolds numbers. 6.1.2.2 High-Reynolds-number jet

Subsequent to the simulation reported just above, the LES of a circular jet at the same Mach number M = 0.9 but at higher Reynolds number Re D = 4 × 105 was carried out.

Figure 6.3. LES of a ReD = 6.5 × 104 subsonic jet. Snapshot of the vorticity norm in the flow field and of the fluctuating pressure outside in the plane z = 0 at t ∗ = 7.5. Reprinted from Bogey, Bailly, and Juve´ (2003) with the kind permission of Springer Science and Business Media.

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Figure 6.4. LES of a ReD = 6.5 × 104 subsonic jet. Snapshots of the vorticity norm in the plane z = 0 at times (a) t ∗ = 2.2, (b) t ∗ = 3.5. Reprinted from Bogey, Bailly, and Juve´ (2003) with kind permission of Springer Science and Business Media.

This simulation, referred to as LESac, has been performed using a selective filtering alone for modeling the dissipative effects of the unresolved scales. Flow and sound features have been shown in Bogey and Bailly (2006) to correspond well to what is observed for high-Reynolds-number jets – particularly regarding the changes in the acoustic field according to the observation angle. Further, large-eddy simulations have then permitted investigations of the effects of the inflow conditions and of the subgrid modelings on the jet properties in Bogey and Bailly (2005b) and Bogey and Bailly (2005a), respectively. In the latter study, the results with the dynamic Smagorinsky model and with the filtering alone designed not to affect the resolved scales (Bogey and Bailly 2004) have been discussed. A view of the vorticity field in the jet and of the pressure field outside is represented in Figure 6.5 for the simulation LESac. Two kinds of acoustic waves are visible: a lowfrequency wave with high amplitude located near x = 20r0 and y = 6r0 predominant in the downstream direction, and waves characterized by higher frequencies, for wider angles, appearing to come mainly from the turbulent axisymmetric shear layer around x = 8r0 . 16

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Figure 6.5. LES of a ReD = 4 × 105 subsonic jet. Snapshot of the vorticity norm in the flow field and of the fluctuating pressure outside in the plane z = 0.

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Table 6.1. LES of a Re D = 4 × 105 subsonic jet. Sideline  -maxima in the shear layer for the sound levels and vrms different simulations OASPL in dB at

LESshear LESampl LESac LESmode



(x = 11r0 , r = 15r0 )

max

125.2 124.7 124.1 121.8

0.196 0.190 0.186 0.168

 (r =r ) vrms 0 uj



To exhibit links between the shear-layer turbulence and the sideline noise, we now report some flow and sound results provided by four simulations using various inflow conditions for the jet in Bogey and Bailly (2005b). The sound spectra at x = 11r0 and r = 15r0 obtained from the different large-eddy simulations are thus provided in Figure 6.6(a). Following Zaman (1986), who demonstrated that the noise sources can be approximated by the maxima of turbulence intensities, the profiles at r = r0 of the root-mean-square (rms) values of the radial fluctuating velocity v  are also plotted in Figure 6.6(b). From the four simulations, it appears clearly that the sideline noise levels  peak values. are arranged in the same order as the vrms This correspondence is emphasized in Table 6.1: the amplitudes of the sideline  peak values in the shear layer. The present sound levels vary accurately with the vrms LES results thus provide a new indication that sideline jet noise is predominantly generated by the turbulence developing in the axisymmetric shear layer. Further studies are, however, required to clearly identify the generation mechanisms involved at high Reynolds numbers. 6.1.3 Concluding remarks for mixing-layer simulations

Direct noise computation is an outstanding approach for studying the sound generated by free-shear flows because it permits us to correlate turbulence events with the sound

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Figure 6.6. LES of a ReD = 4 × 105 subsonic jet. (a) Pressure spectra at x = 11r 0 , r = 15r 0 ; (b) Profiles  /u in the shear layer for r =r . Different simulations: LESac (——), LESampl (.......), LESshear of vrms j 0 (- - -), LESmode (-. -. -. ).

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6.2 FAR-FIELD JET ACOUSTICS

far field. In this context, LES seems the better tool to clarify Reynolds-number effects – especially for subsonic jet noise. The dependence on Reynolds number is particularly expected for the noise-generation mechanisms observed in mixing layers whose initial state, transitional or turbulent, is fixed by the Reynolds-number value. In this section, only the noise radiated in the sideline direction from the jet axis has been discussed. It has been shown that its generation likely takes place in the developing mixing layer. Note also that a similar analysis has been conducted for the downstream noise. This jet noise component was correlated in Bogey, Bailly, and Juv´e (2003) with the periodic intrusion of vortical structures into the potential core. Its level was recently demonstrated in Bogey and Bailly (2005b) to vary as the maximum of turbulence intensities on the jet centerline.

6.2 Far-field jet acoustics Daniel J. Bodony and Sanjiva K. Lele 6.2.1 Introduction to jet acoustics

It can be said that most of the current research on the generation of sound by turbulent flows has had its origin in the prediction of jet noise (in particular the noise from jet exhaust plumes) by Lighthill (1952, 1954) in his two papers “On Sound Generated Aerodynamically”.∗ Lighthill’s U 8j scaling of the radiated noise intensity, where U j is the jet exit velocity, gave engine manufacturers an initial tool by which to design quieter engines and led, in part, to the development of internal and external mixers and the turbofan engine. In the early 1970s, it became clear that a more detailed knowledge of the noise sources and the resulting radiation field would be needed to further reduce jet noise. Lilley’s (1974) inhomogeneous convective wave equation, based on the work of Phillips (1960), gave further insight into the noise sources and attempted to separate out the sound-generation and sound-propagation processes that are present in the jet.† The numerical prediction of jet noise for design applications came with the introduction of the MGB‡ jet prediction tool (Balsa et al. 1978). The MGB code uses a Reynolds-averaged Navier–Stokes (RANS) mean flow solution to define local length scale, time scale, and source strength parameters for a semiempirical source model. The source model is based on a simplified Lighthill quadrupole source term with an ∗

†

‡

It should, however, be noted that the importance of reducing jet noise for commercial air transport use had been recognized prior to Lighthill’s work in the experiments of Morley (1939), Westley and Lilley (1952), and others. It is beyond the scope of this section to discuss the acoustic analogies of Lighthill (1952, 1954), Lilley (1974), M¨ohring (1999), Fedorchenko (2000), and others. Many of these analogies have been formulated to directly address the prediction of jet noise. The name MGB stems from its primary authors’ last names: Mani–Gliebe–Balsa.

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approximate high-frequency solution of Lilley’s equation for propagation to the far field. The noise source modeling of the original MGB formulation continues to be refined (e.g., see Khavaran, Bridges, and Freund 2002; Khavaran and Bridges 2004 and the references within), and the more recent versions of the code are referred to as MGBK. Tam and Auriault (1998, 1999) have proposed a RANS-based prediction scheme for the “fine-scale mixing noise” component of jet noise. In their formulation, a modified k– turbulence model provides parameters for a semiempirically based space–time correlation function of the fluctuation turbulence kinetic energy which, in contrast to the source term used in the MGBK code, is not based on Lilley’s (1974) form of Lighthill’s (1952, 1954) analogy. Instead, in the spirit of kinetic theory, they postulate a relationship between the turbulence kinetic energy and fluctuating pressure. An adjoint field is used to project the near-field source onto the far field. Tam’s reformulation, like MGBK, retains the strong dependence on a RANS solution with a calibrated, semiempirical noise-source model. The fundamental limitation of the aforementioned RANS-based approaches is that the flow unsteadiness is modeled semiempirically. Observations of the two-point, timeseparated statistical correlation u i (x, t)u j (x + ξ, t + τ ) as a function of ξ and τ suggest using the functional form of the noise model (Goldstein 1976). However, statistics of the noise-source terms themselves have not yet been measured in the laboratory owing primarily to the difficulty in experimentally measuring the necessary products. Freund (2003) has recently detailed statistics of the Lighthill source term using DNS data in a low-Reynolds-number jet. In a real flow it is the temporal evolution of a particular turbulent eddy within its immediate environment that generates sound, which suggests that source models solely using the RANS mean flow to provide values for model constants will be limited in their ability to predict the radiated noise accurately. These methods perform reasonably well for generic axisymmetric or rectangular nozzles, but they are often unable (Lord and Feng 2000) to account for subtle nozzle design changes such as the addition of nozzle chevrons or tabs. Moreover, they have difficulties at the low- and high-frequency ends of the acoustic spectrum (NASA Glenn Research Center 2001). It is believed that noise-prediction tools that do not model the flow unsteadiness and involve fewer a priori assumptions will better be able to represent the noise sources. In this regard, LES of jet flows is seen to represent the next step in jet noise predictions because some of the unsteadiness naturally present in the exhaust plume is retained. The remainder of this section discusses the use of LES in jet noise prediction with an emphasis on the far-field sound. We further confine our discussion to those simulations and associated techniques that directly compute the turbulence-generated sound. That is, the simulations of incompressible jets from which the sound is extracted via a sound source assumption (usually that of Lighthill 1952, 1954) will not be addressed here, but the reader is encouraged to visit the work of Zhao, Frankel, and Mongeau (2001b), Boersma (2002, 2003), or Rembold and Kleiser (2003), for example, for analogy-based jet noise predictions.

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Figure 6.7. Schematic of a turbulent jet issuing into a still fluid. The jet near field, where the LES equations are solved, is situated within an exterior domain into which the sound is propagated. Not to scale.

6.2.2 Numerics of jet simulations

A schematic of the computational domain used in the simulations of Gamet and Estivalezes (1998), Zhao et al. (2001a), Bodony and Lele (2002a, 2003b, 2004), Uzun, Blaisdell, and Lyrintzis (2002, 2003), and Uzun (2003) into a quiescent environment is shown in Figure 6.7 along with the defining coordinate system. In this setup, two distinct regions are used: the first is the LES domain, where the filtered equations of motion are solved using a subgrid-scale flux closure. It is in this portion of the domain that the inflow boundary conditions are specified, the jet develops, and, ultimately, the sound is generated. Outside this area is the wave equation domain, where, with approximation, the sound is propagated to locations far removed from the jet plume. There have also been several studies (Bogey, Bailly, and Juv´e 2000a; Bogey and Bailly 2005a, 2005b, 2006; Morris et al. 2002) that retain a portion of the far acoustic field in the near-field solution in lieu of a separate wave-propagation zone. 6.2.2.1 Near-field discretization

Many of the previous discussions in this book on numerical discretization apply equally well to the simulation of jet flows and to the radiated noise. One point, however, that should be emphasized is the correspondence between the dispersion relation of the

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numerical scheme (including the temporal discretization) and that of the radiated sound. Because air is an approximately nondispersive medium for low-amplitude sound waves at the conditions of interest in jet noise, the numerical method must reflect this by not introducing any artificial dispersion. Often it is necessary to correctly capture the sound propagation over a distance of many wavelengths for a range of frequencies so that numerics-induced dispersion or dissipation will be exaggerated. The most popular numerical methods of this sort are the dispersion-relationpreserving (DRP) scheme of Tam and Webb (1993) (see also Bogey and Bailly 2004) and the combination of a Pad´e derivative scheme (Lele 1992) and a high-order time advancement scheme, usually a Runge–Kutta variant. The mixing layer calculations of Lui and Lele (2001, 2002a, 2002b, 2003) used an optimized compact scheme coupled with the fourth-order, low-dispersion Runge–Kutta scheme of Hu et al. (1996) (using the low-storage version of Stanescu and Habashi 1998) and found the method to be highly accurate and numerically efficient. The resolving characteristics of finite difference methods have not been completely translated into finite element or finite volume schemes. Such schemes are quite convenient when complex geometries are involved, as is true for most engineering applications, and their efficient use for aeroacoustic problems is needed. Initial attempts at using finite elements to include the nozzle in jet LES calculations by Al-Qadi and Scott (2002), DeBonis and Scott (2002), and Jansen, Maeder, and Reba (2002), for example, show promise but, to date, have had mixed success. The DRP scheme has also been applied to jet flows with the nozzle present by Dong and Mankbadi (1999). 6.2.2.2 Boundary conditions

For simulations of jet exhaust noise and other flow-generated noise situations the boundary conditions are crucial in establishing the environment into which the flow develops. Aside from the need for nonreflecting boundary treatments, jet noise predictions have been found to be sensitive to the inlet conditions associated with the quasi-laminar annular shear layer (Morris et al. 2002; Bodony and Lele 2002a; Bogey, Bailly, and Juv´e 2003). In real jet flows, the annular shear layers exiting the engine nozzle are extremely thin (0 /D j ≈ O(10−3 ), where 0 is the shear-layer momentum thickness and D j the jet diameter) and turbulent. In an LES of the corresponding flow, the shear layers are thicker (0 /D j ∼ O(10−2 )) with imposed quasi-laminar disturbances both due to the limited available resolution. (For reference, the DNS of a low-Reynolds-number jet by Freund (2001) had an initial momentum thickness of 0 /D j = 0.02.) The thicker initial shear layer implies that, for the same velocity difference across the shear layer, the rate of energy extraction from the mean to the shear-layer disturbances is lower than in the real flow. This appears to increase the sensitivity of the jet-radiated sound to the inlet conditions. In particular, the degree of azimuthal correlation of the inlet disturbances is found to substantially influence the radiated noise levels with more strongly correlated disturbances leading to increased radiated sound.

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90 60

Figure 6.8. OASPL directivity at a distance of 30D j from the unheated, Mach 0.9 jet exit. Nearly axisymmetric disturbances, —; azimuthally decorrelated disturbances, − −; data of Stromberg McLaughlin, and Troutt (1980), ◦; MolloChristensen, Kolpin, and Martucelli (1964), ; Freund (2001), −  −; Bogey, Bailly, and Juve´ (2000a), − −.

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This relationship between the azimuthal correlation of inlet disturbances and the radiated sound was established experimentally by Zaman (1985), who found that the noise from axisymmetric disturbances was approximately 5 dB higher than from lesscorrelated disturbances. Similar changes in sound levels have been found in the available LES studies, and Figure 6.8 exhibits the change found by Bodony and Lele (2002a). Note that the inlet disturbances used by Bodony and Lele (2002a) in the “nearly axisymmetric disturbances” (solid line) case of Figure 6.8 were identical in form to those used in the DNS of Freund (2001). Bogey, Bailly, and Juv´e (2003) further studied the effect of the inlet conditions on the radiated noise by varying the initial shear-layer thickness, the azimuthal mode composition, and overall forcing amplitude of the quasi-laminar disturbances. They also found that more strongly correlated inlet disturbances increase the jet noise. In addition they noted that a thinner initial shear-layer thickness reduces the downstream-radiated noise while increasing the noise levels at observers situated near 90◦ from the jet axis. The introduction of disturbances into the annular shear layer must be such that, in addition to inducing natural turbulence development, no spurious noise is generated in the vicinity of the inlet plane. That boundary noise generation can be a problem is easily seen in the simulations of Constantinescu and Lele (2001). There the inlet conditions radiated sound at levels similar to the jet itself (see their Figure 2) and have marked peaks in the sound spectrum. For the avoidance of such difficulties, two approaches have been used successfully to date. The first method attempts to construct disturbance profiles that are divergence free, or nearly so. Bogey, Bailly, and Juv´e (2003) and Uzun (2003), for example, use disturbances of the form u x = f 1 (θ )g1 (t)h 1 (x, r ),

(6.3)

u r = f 2 (θ )g1 (t)h 2 (x, r ),

(6.4)

u θ = 0,

(6.5)

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where the functions h 1 and h 2 are chosen such that ∇ · u  = 0 (see also Glaze and Frankel 2003). This technique works well for cold low- and moderate-Mach-number jets where the density is nearly uniform. One could generalize this somewhat by specifying that, for example, u x = f 1 (θ )g1 (t)h 1 (x)i 1 (r )

(6.6)

with u r , u θ used similarly and requiring that the functions { f k , h k , i k }3k=1 be such that u  is divergence free. This yields a set of differential equations relating the f, h, i that may be solved with additional constraints. Similarly, one may also vary the inlet momentum thickness 0 as a function of time and azimuthal location; that is, u x Uj

0 (θ, t) =

1 4B

= (1 − tanh [0 (θ, t) (r/ro − ro /r )]) ,

 St U + 1m,n Amn cos Dmnj t j + mn cos (mθ + mn ) ,

(6.7) (6.8)

where B is the time-averaged momentum thickness and the functions , , and St specify the fluctuation randomness. Used originally by Freund (2001), this method has been used subsequently by Zhao et al. (2001a) and Bodony and Lele (2002a, 2003b, 2004). The second approach uses the nonradiating eigenfunction solutions to the linearized spatial instability problem associated with the inlet mean profile as the disturbance profiles. Assuming a modal decomposition of a disturbance flow variable q  (x, r, θ, t) as ˆ ) exp{i(ωt − kx − nθ)}, for integer n, we find that the pressure disturbance ampliq(r tude pˆ for an axisymmetric mean flow is governed by the cylindrical Rayleigh equation     1 dρ 2k (ω − ku)2 n2 1 du d d2 2 ˆ ˆ − + p + − − k pˆ = 0 p + dr 2 r ρ dr (ω − ku) dr dr r2 a2 (6.9) subject to the boundary conditions ˆ <∞ | p| pˆ → 0

as r → 0,

(6.10)

as r → ∞.

(6.11)

The spatial stability problem is characterized by fixing ω to be real and solving for the complex-valued eigenvalue k. The remaining flow variables are given as linear combinations of pˆ and its radial derivative. This method is guaranteed to yield nonradiating disturbances provided the phase speed ω/kr , with k = kr + iki , of each (ω, n, k) triplet is subsonic relative to the ambient fluid. This condition is satisfied in unheated jets with M j < 1.5 and is a function of the temperature ratio T j /T∞ for heated jets. When the disturbance phase speed is supersonic, the instability waves radiate directly to the far field (Tam and Morris 1980). Application of this method to rectangular jets has been discussed by Rembold and Kleiser (2003) and by Zhao et al. (2001a) and Bodony and Lele (2003b, 2004) for circular jets.

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6.2.2.3 Far-field sound extrapolation

In experimental investigations of jet noise, sound measurements are generally taken in the acoustic far field, usually at distances of 30D j and beyond from the nozzle exit. This region, where the nearly incompressible pressure fluctuations directly associated with the turbulence of the jet are negligible, is generally uniform; thus, the propagation of sound is greatly simplified relative to the environment nearer to the jet. It is therefore not necessary to include the acoustic far field in the LES but instead to restrict the computational domain laterally to a few jet diameters away from the jet centerline. Continuation of the sound field to the far field is done either analytically or with a simplified numerical method. One such technique, using a Kirchhoff surface, is described below; alternatives may be found in the literature. For the Ffowcs Williams–Hawking surface, see Brentner and Farassat (1998) and the references therein as well as Uzun (2003); for a time-marching numerical solution of the wave equation, see Freund (2000). The Kirchhoff surface method is based on an integral solution of the wave equation  −2 2 2 (6.12) a∞ ∂ /∂t − ∂ 2 /∂ x j ∂ x j ( p −  p) = 0 for an inviscid, quiescent medium with constant sound speed a∞ and has been reviewed by Lyrintzis (1994). The presence of a uniformly moving fluid outside the jet, as there would be for a jet in flight, can be accommodated by a change of coordinates. This simple linear model of sound propagation is valid for noise of most high-Reynoldsnumber applications and for the frequency ranges and propagation distances of interest. Extremely loud jets (with sound-pressure intensity levels in excess of 150 dB) or jets with “crackle” (Ffowcs Williams, Simson, and Virchis 1975) may require a more complicated description of the propagation. The application to jet noise is straightforward: pressure fluctuations about the average p  = ( p −  p) are collected on a cylindrical shell of radius Rs surrounding the jet. The shell is chosen to be sufficiently far away from the hydrodynamic region of the jet so that the sound propagation is sufficiently described by Equation (6.12) and extends from the computational inlet to the exit. The “ends” of the shell, which are perpendicular to the jet axis, are generally not included because they do not lie in linear regions of the flow; their absence may be accounted for (Freund, Lele, and Moin 1996) but is often ignored with the restriction that the far-field sound predictions are valid over a limited range of angles from the jet axis. Using the pressure fluctuations specified on the shell, we can solve the wave equation (6.12) in the partially transformed form 

ω2 n2 d2 −1 d 2 + 2 − k − 2 pˆ = 0, +r (6.13) dr 2 dr a∞ r which may also be obtained directly from Equation (6.9). Equation (6.13) is Bessel’s ˆ ; k, n, ω) = equation and is subject to the boundary conditions that, for r = Rs , p(r ˆ s ; k, n, ω), and that as r → ∞ the solutions represent outgoing traveling waves. p(R 2 − k 2 > 0 radiate as sound and contribute to the Only those waves that satisfy ω2 /a∞ far field. Near r = Rs , however, the pressure field may be dominated by the nonradiating components. The far-field pressure fluctuations are found by integrating Equation (6.13)

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to the desired value of r and inverse transforming. The overall solution at a given field point (x, r, θ, t) is then seen to be 1 ( p −  p)(x, r, θ, t) = (2π )2

+∞ 

−∞

N θ /2−1

ˆ s ; k, n, ω) p(R

n=−Nθ /2



2 − k2 r ω2 /a∞ 

ei(ωt−kx−nθ) dkdω, (6.14) × (m) 2 2 2 Hn Rs ω /a∞ − k (m)

Hn

(m)

where Hn is the nth-order Hankel function with m = 1 for ω > 0 and m = 2 for ω < 0. The radial location of the Kirchhoff surface is important to ensure that the nonlinearity, nonuniformities, or both of the jet near field are sufficiently low to justify the assumptions made in the method’s derivation. If the surface is placed too close to the near field, the sound levels can be overpredicted by 5–7 dB (Brentner and Farassat 1998; Uzun 2003).

6.2.3 Results for jet simulations

When examined in detail, the structure of a turbulent jet is complex with a wide range of spatial and temporal scales involved in the dynamics. The far-field noise is generally broadband with spectral characteristics that depend on the angle from the jet axis and jet operating conditions. Experimental investigations have shown that higher-frequency sources are typically located closer to the nozzle exit while lower-frequency sources can be found farther downstream (NASA Glenn Research Center 2001). Moreover, the turbulent structure and corresponding sound sources are functions of the jet exit Mach number M j (Freund, Lele, and Moin 2000) and jet temperature ratio T j /T∞ . To date, most of the LES studies of far-field noise have been of moderate-to-high subsonic jets owing, in part, to the existence of quality experimental data and the desire to avoid cases with shocks present. Notable exceptions to this are the incompressible jet studies of Boersma (2002, 2003) and the perfectly expanded Mach 2.1 unheated jet of Morris et al. (2002). Because the simulation of moderate-to-high subsonic jets has received the most attention, it will be the main topic in the remainder of this discussion. A brief survey of the relevant studies will precede a more detailed review of quantitative results. 6.2.3.1 Brief literature survey

Initial attempts at using LES to study compressible jets began in the late 1990s. Estivalezes and Gamet (1996) and Gamet and Estivalezes (1998) used MacCormack’s scheme to investigate the near-jet region of a hot, Mach 2 jet in two and three dimensions with a Kirchhoff surface to extract the far-field sound. Choi et al. (1999) compared MGBK predictions with LES predictions of an off-design supersonic nozzle using a Kirchhoff surface. This was followed by the simulation of Dong and Mankbadi (1999) of nozzle-ejector mixer operated at a jet Mach number of 1.5 with Mach 0.4 coflow.

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Boersma and Lele (1999) performed an exploratory study investigating the suitability of LES for the jet noise problem, focusing on the subgrid-scale modeling. They found that the compressible subgrid-scale model of Moin et al. (1991) performed reasonably well in capturing the mean and rms fields but noted some dependence on the boundary conditions. Constantinescu and Lele (2001) furthered the work of Boersma and Lele (1999) by applying a highly accurate centerline treatment (Constantinescu and Lele 2002) and focusing on the boundary conditions for a Mach 0.9 jet. The resulting nearfield data compared favorably with the experimental data, and they found that using non-Favre-weighted variables improved the robustness of the solver. It is difficult to extract far-field acoustic data from their investigation because their inlet condition appears to be strongly radiating, corrupting the sound field but apparently having little effect on the hydrodynamic field. Bodony and Lele (2002a, 2003b, 2004) refined the work of Boersma and Lele (1999) and Constantinescu and Lele (2001, 2002) by implementing improved boundary conditions, allowing the necessary entrainment through the lateral boundaries and a physically realizable inlet disturbance specification. The Mach 0.9 jet of Bogey et al. (2000a), Bogey, Bailly, and Juv´e (2003), and Bogey and Bailly (2005a, 2005b, 2006), using a rectangular grid solver and various subgrid-scale closures, compared favorably with the available turbulence experimental data in the self-similar region of the jet plume. In addition, their computational domain included a portion of the far field in which they collected pressure fluctuation data. The spectra of pressure fluctuations had anomalous behavior at lower frequencies, which they attributed to the boundary conditions. When the low-frequency portion of the acoustic spectra was neglected, Bogey et al. (2000a) found overall sound-pressure levels (OASPL) and directivity patterns to be in reasonable agreement with the experimental data. Zhao et al. (2001a) found similar acoustic field characteristics using a dynamic subgrid-scale model and Kirchhoff surface. More recently, Bogey, Bailly, and Juv´e (2003) refined their initial results by, through boundary condition improvement, removing the excess low-frequency fluctuations in their acoustic spectra and performing a parameter study on the inlet conditions. Uzun, Blaisdell, and Lyrintzis (2002, 2003), also using a rectangular grid solver, computed the sound of a series of isothermal jets at varying Reynolds numbers with particular attention to the location of the Kirchhoff and Ffowcs Williams–Hawkings surfaces. Bodony and Lele (2004) have examined the ability of LES to capture heating effects in moderate-to-high subsonic jets. A unique study by Andersson (2003) of a Mach 0.75 jet (heated and unheated) was the computational counterpart to a series of measurements at the same conditions using the nozzle geometry of the experiment. When a finite volume method is used the unheated and heated jets slightly underpredict the streamwise rms fluctuations along the centerline with a peak location farther upstream than measured experimentally. OASPL directivity comparisons show ±3 dB agreement over a wide range of angles for the unheated and hot jets. In working toward the development of a general jet noise prediction methodology Shur, Spalart, and Strelets (2003) investigated the use of a multiblock solver with an implicit LES model, in which the subgrid-scale dissipation is accomplished through

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0.2

0.15

0.1

0.05

0 0

10

20

30

40

50

Figure 6.9. Centerline distribution of streamwise root-mean-square fluctuations. Legend: —, Bodony and Lele (2004); − −, Bogey, Bailly, and Juve´ (2003); •, Lau, Morris, and Fisher (1979); , Zaman (1986); , Arakeri et al. (2003).

numerical upwinding, on an impressive array of jet conditions and effective nozzle geometries, including nozzles with chevrons. The nozzles are not directly included, however, and the influence of the nozzle geometries is modeled through a specified inflow velocity profile. A Ffowcs Williams–Hawking surface provides extrapolation of the sound to the far field. Without explicitly forcing unsteadiness into the incoming fluid they found, similar to Andersson (2003), OASPL agreement of ±3 dB over a range of conditions. 6.2.3.2 Discussion of LES acoustic results

There are believed to be two main types of sound generation in subsonic turbulent jet plumes: the turbulence-generated sound concentrated in the vigorous mixing region near the end of the potential core and the lower-frequency, large-scale instability wave-generated noise. (We exclude in this discussion other important noise sources – especially the high-frequency noise generated in the thin, near-nozzle shear layers. See Section 6.2.4.2.) Each noise component is argued to have a unique acoustic spectrum (Tam, Golebiowski, and Seiner 1996; Goldstein 2003), and the overall sound produced by a jet contains significant contributions from both. It is thus necessary for an LES noise prediction to capture both phenomena. For the turbulence-generated sound associated with the end of the potential core, simple reasoning suggests that a necessary, but by no means sufficient, statistical quantity to be captured accurately is the centerline distribution of the turbulence fluctuation levels. Several data are shown in Figure 6.9 of the streamwise rms velocity fluctuations. (Note: In this and other figures, the data have been shifted along the x-axis to yield a common potential core collapse location.) The rapid rise of the fluctuation levels is due to the merging of the annular shear layers and the subsequent induced

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6.2 FAR-FIELD JET ACOUSTICS

0.5

0.4

0.3

0.2

0.1

0 0

10

20

30

40

50

Figure 6.10. Centerline distribution of density root-mean-square fluctuations normalized by the difference (ρ j − ρ∞ ). Legend: —, LES data of an unheated, Mach 0.9 jet by Bodony and Lele (2004); , Panda and Seasholtz (2002).

mixing. In the data shown, the fluctuations peak near 12–15% of the jet exit velocity. The LES predictions appear to be consistently higher than the experimental measurements, which is probably a result of the weakly turbulent shear layers of the simulations relative to the fully turbulent experimental shear layers. In all cases the width of the peak is roughly the same with a uniform rate-of-decrease fluctuation level with increasing axial position. In those studies with excessive azimuthal correlation of inlet disturbances (Bodony and Lele 2002a; Morris et al. 2002), the increased axisymmetric coherence manifests as a large overshoot of the centerline u x,rms levels. A 20% increase in the peak rms value was typical. Downstream of the peak, however, there was a rapid decay of the fluctuation intensities, and thus beyond x/ro = 25, roughly, there was agreement with experimental data in the self-similar region. It should be emphasized, in light of these data, that validation of jet LES data in the self-similar range is not sufficient and may yield inaccurate conclusions. The role of the density fluctuations in sound generation has not yet been well established owing, in part, to the difficulty is measuring ρ  within the jet. Lighthill (1952) argued that the velocity fluctuations are primarily responsible for the quadrupole stress and thus that the density may be taken as uniform with a value equal to that of the density in the ambient fluid. Crow (1970) put the neglect of density in the sound source on firm ground for incompressible jets. For higher-speed or hot jets, or both, this is not expected to be the case, but there are not yet sufficient data from which to draw a conclusion. The density fluctuations in unheated jets have recently been measured by Panda and Seasholtz (2002) using the Rayleigh scattering technique. Their results further support the neglect of the density fluctuation considerations in sound generation. Figure 6.10 shows the density rms levels along the jet centerline for

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Figure 6.11. Far-field OASPL taken at a distance of 30D j from the nozzle exit. Legend: —, Bodony and Lele (2004); − −, Bogey, Bailly, and Juve´ (2003); , Freund (2001); ◦, Stromberg et al. (1980); , Mollo-Christensen et al. (1964).

an unheated, Mach 0.9 jet normalized by the difference ρ j − ρ∞ . The LES-predicted density fluctuations are roughly twice those measured by Panda and Seasholtz (2002). That the LES-predicted sound fields are in agreement with those of laboratory-measured jets (see Figure 6.11) further suggests the insensitivity of the sound to density fluctuations for unheated, subsonic jets. The role of the density fluctuations for hot jets is just beginning to be established (Panda et al. 2004; Bodony and Lele 2004). Beyond the fluctuation levels, it is the space–time structure of the turbulence that plays a central role in the statistical properties of the sound field (Goldstein 2003). More precisely it is the space–time character of the sound source that is important, but, through approximation, sound field may be related to the velocity correlation u i (x, t)u j (x + ξ, t + τ ). Two parameters that characterize the correlation are the integral length scale and the moving-axis, Lagrangian integral time scale T . Measurements of these quantities is difficult, and limited data are thus available. Davies, Fisher, and Barratt (1963) found that, for a low-Mach-number jet (M j near 0.45), the Lagrangian time scale could be correlated to the local inverse shear,    4.5  ,  T = ∂U /∂r 

(6.15)

x

for locations near the lip line at the end of the potential core. This argument implicitly states that the turbulence in the shear layer and potential core region is essentially equivalent to homogeneously sheared turbulence (Blaisdell, Mansour, and Reynolds 1993). That such a relation exists in the potential core region of a turbulent jet is unclear. However, Figure 6.12 shows that, for the limited data available, the correlation is approximately retained in the LES for those points near the lip line (r/ro = 1). The filled-in circles of Figure 6.12 correspond to measurements taken just downstream of the potential core, beginning with r/ro = 0.5 and increasing in increments of 0.1. The

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4 3.5 3 Figure 6.12. Integral Lagrangian time scale of streamwise fluctuations near the end of the potential core. Legend: ◦, LES data of Bodony and Lele (2004) with r /r o > 1; •, LES data of Bodony and Lele (2004) with r /r o < 1; —, correlation of Davies et al. (1963).

2.5 2 1.5 1 5

10

open circles are for r/ro > 1 with similar increments. For those locations near the lip line, the correlation of T with the local inverse shear holds in the manner of Davies et al. (1963). There is a rapid departure for locations r/ro > 1, which is believed due to the quasi-laminar nature of the incoming, annular shear layers. These results suggest that, at least for r/ro < 1, the jet space–time structure of the turbulence is relatively well captured in the LES. Similar agreement was found in the Mach 0.75 jets of Andersson (2003). With the possible exception of the density fluctuations, the agreement between the LES data and experiment would suggest that the radiated noise for the turbulencegenerated noise should be properly predicted. This is borne out in Figure 6.11 of the OASPL directivity where, to a large extent, the LES-predicted levels are within the scatter of the experimental data. Near the maximum intensity angle of 30◦ , the LES data have little scatter relative to the other angles and are consistently below the DNS data of Freund (2001), which may be due to low-Reynolds-number effects in the DNS. Over the downstream angles, the LES–experimental data agreement is encouraging. The spectral content of the sound as a function of polar angle  is given in Figure 6.13, where the simulation data of Bodony and Lele (2004) have been scaled to the physical dimensions of apparatus used by Lush (1971) for his 195-m/s jet. Note that the narrow-band spectra of the LES have been synthesized into 1/3-octave band spectra for the comparison. At lowest polar angle, both the peak frequency and amplitude are well captured by the simulation. For the intermediate angle of 45◦ the simulation again captures the peak frequency and amplitude but with little energy in frequency components much beyond the peak frequency. By 90◦ , the peak frequency is missed and the simulation spectra are evidently a low-pass filtered version of the experimental data. The energy at higher frequencies that is missing in the simulation but present in the measurements is generated in two regions: (i) near the nozzle exit

15

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60 40 20 10

2

3

4

10

10

5

10

60 40 20 10

2

10

3

4

10

5

10

60 40 20 10

2

10

3

4

10

5

10

Figure 6.13. One-third–octave spectral comparisons of an unheated, Mach 0.5 jet of Bodony and Lele (2004) with the 195-m/s data of Lush (1971). Legend: —, simulation; •, Lush (1971).

and (ii) within the jet plume very near the potential core with the first location being the more dominant. The effect of missing the near-nozzle shear layers is quite evident, and yet the predictions are quite good for those frequencies that are present in the simulations. Bogey and Bailly (2006) and Bodony and Lele (2004) have found (see Figure 6.14) that the azimuthal correlation of the sound in the far field is in relatively good agreement with the experimental measurements of Maestrello (1976). At small angles to the jet axis, where the large-scale instability waves are dominant, the sound field shows a large degree of azimuthal correlation around the jet, indicating the presence of strong, axisymmetric disturbances. Michalke (1984) notes that the jet is a highly selective amplifier with a strong preference for axisymmetric motions. Thus, the creation of axisymmetric

1 0.75 Figure 6.14. Azimuthal correlation of the far-field sound field of LES data from Bogey, Bailly, and Juve´ (2003) (− · −) and Bodony and Lele (2004) (—) compared with experimental data by Maestrello (1976) () at θ = 30◦ and θ = 90◦ .

0.5 0.25 0 − 0.25 0

30

60

90

120

150

180

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modes through convection is manifested in the sound field at these low angles. For larger angles to the jet axis, where the large-scale instability waves are less important, a more rapid decorrelation with increasing separation angle is found. The question of the robustness of these results to changes in the subgrid model is an important one that has only recently been investigated. By surveying the literature cited throughout this article, one can see that, for those studies that use high-fidelity numerics,∗ the variety of methods taken to compute the jet near field, the acoustic far field, or both do not have a significant effect on the turbulence intensities or on the radiated sound field. Bogey and Bailly (2005a) examined the sound-field changes due to different subgrid-scale models (dynamic coefficient Smagorinsky with and without a turbulence kinetic energy correction) and to a simulation using selective filtering alone to remove high-frequency energy. The kinetic energy correction appears to have little effect. In changing from the usual LES procedure of including a subgrid-scale model to the use of selective filtering, Bogey, Bailly, and Juv´e (2003) reported a downstream shift in the jet development and small, but statistically meaningful, changes in the turbulence field. The acoustic fields do differ in this case; in particular, the spectra show a shift toward lower frequencies for a fixed point caused by the lateral shift of the jet as a whole when the selective filtering procedure is used. However, the two models have roughly the same rate of energy decrease at larger Strouhal numbers, suggesting that the high-frequency noise is not significantly affected. Likewise, Zhao et al. (2001a) found that a dynamic mixed model produced a jet with slightly higher turbulence level and a louder sound field relative to the dynamic Smagorinsky alone but without significant overall change.

6.2.4 Future directions of jet acoustics

On the basis of the results presented in the previous section, it appears that the use of LES to predict the noise radiated by high subsonic jets is justified. There is evidence that the predicted noise-generation-relevant turbulence and radiated acoustic field characteristics are in agreement with existing experimental and numerical data. Before LES as a noise prediction tool becomes widespread, however, several issues need to be addressed. What follows is a brief discussion of the LES-specific issues that are deemed important; it is by no means exhaustive. 6.2.4.1 Resolution effects on the near and far fields

As discussed in Section 6.2.3.2, the peaked spectrum at 30◦ is believed to be related to the large-scale jet motions with length scales on the order of the jet diameter. It is of interest to know the degree to which a coarse-grid LES, which is not able to predict the near-field turbulence levels, accurately can reproduce the low-frequency noise. One ∗

That is, those studies with an implementation that did not introduce numerical diffusion, dispersion, or both.

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Figure 6.15. Far-field OASPL taken at a distance of 30D j from the nozzle exit. Legend: —, 100,000-point simulation; − −, 1,000,000-point simulation of Bodony and Lele (2004); , Freund (2001); ◦, Stromberg et al. (1980); , Mollo-Christensen et al. (1964).

such comparison for the OASPL is given in Figure 6.15 and for the spectra in Figure 6.16 (dashed curves in (a) and (b)). The coarse LES, with one-tenth the number of grid points but with the same inflow and lateral boundary conditions, is able to reproduce the OASPL to within 2 dB and retains the spectral peak location and amplitude near St = 0.2. The higher-frequency noise is clearly missed. That the poorly resolved LES, in terms of the turbulence, is capable of estimating the low-frequency noise at shallow angles may be of interest because the MGBK and Tam–Auriault formulations (Tam and Auriault 1998, 1999) are unable to do so in their current form (cf. Choi et al. 1999). Further work is needed to characterize the resolution dependence of the radiated sound field to determine the usefulness of a hybrid approach using analogy-based methods for the high-frequency noise and a coarse-grid LES for the low-frequency noise.

SPL

100 80

(a)

60 40 -2

10

10

-1

St

10

0

SPL

100 80

(b)

60 40 -2

10

10−1

St

10

0

Figure 6.16. Far-field acoustic spectra taken at a distance of 30D j from the nozzle exit. Legend: —, 1,000,000-point LES simulation of Bodony and Lele (2004); − −, 100,000-point LES simulation; − · −, similarity spectra of Tam et al. (1996). In the upper (lower) figure, the low-frequency (high-frequency) spectra of Tam et al. (1996) are used.

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6.2.4.2 Subgrid-scale noise model

As a corollary to Section 6.2.4.1, one issue of the truncation of scales in an LES is the effect on the sound generated by the missing scales of motion. Because only the largest scales, as determined by the grid resolution, are represented in an LES, the influence of the subgrid scales on the resolved scales is modeled in the closures used in the mass, momentum, and energy equations. The role of the subgrid-scale motions in the sound-generation process is, as yet, not understood. At the Reynolds numbers of realistic flows, the subgrid-scale noise may well account for a significant fraction of the relevant frequencies in the far-field spectrum. Bodony and Lele (2003a) recently estimated that, for an LES of a Mach 0.9 jet to capture the St > 0.5 far-field noise at 90◦ from the jet axis to within 1 dB, a grid spacing of 0.0008D j –0.0016D j would be necessary for axial locations of 1 ≤ x/D j ≤ 2. Current simulations have a minimum grid spacing of ≈ 0.01D j ; recovering the missing noise with better-resolution simulations, including in the near-nozzle regions with thin, turbulent, annular shear layers, is a long way off. To illustrate the missing noise, consider the spectra of Figure 6.16. Here the difference in the two simulations, as discussed previously, is apparent: the finer-resolution computation has a larger frequency content. Compared with the empirical data parameterized by Tam et al. (1996), both simulations fall far below for St > 1. Because the experimental data were measured under realistic conditions, the elevated spectral levels are due primarily to the lack of the thin, turbulent, annular shear layers present near the nozzle exit. It is these shear layers, where the inherent time scales are much shorter than those found in the plume, in which the high-frequency sources lie. Using the estimate of the previous paragraph with that of Freund and Lele (2004) shows that the number of points needed to include both the near-nozzle shear layers and the jet plume is on the order of 1–10 billion. This number of points is beyond current computational capacity and thus an alternative approach is needed. One such approach is the statistical subgrid noise model recently proposed by Bodony and Lele (2002b, 2003a) using Goldstein’s (2003) “generalized acoustic analogy.” In their formulation, a field variable, for instance the velocity u i , is decom(L) (M) posed into a large-scale component u i and a “missing-scale” component u i . The large-scale variable would be that computed in an LES. On joining this decomposition with Goldstein’s (2003) source term Bodony and Lele (2002b, 2003a) proposed (L) (M) a subgrid-scale noise model composed of terms of the form u i u j that represents the interaction of the large- and missing-scale components. Further work is needed to establish the validity of this formulation. Another approach is to develop numerical techniques that allow one to compute the near-nozzle region along with the turbulent plume at reasonable cost. The work of Jansen et al. (2002), for example, illustrates the difficulty in including the nozzle directly but clearly defines the gains that may be had in considering small sections of the nozzle when the nozzle geometry warrants such a simplification. The introduction of the discontinuous Galerkin (Hu et al. 1999) and compact finite volume (Piller and Stalio 2004), with their better resolution properties, will be necessary.

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6.2.5 Conclusions for far-field jet acoustics

The suitability of LES to the prediction of jet noise has been well established for unheated, high-subsonic jets. The introduction of high-bandwidth, high-accuracy numerical techniques has allowed for the computation of both the unsteady, near-field turbulence found in the jet exhaust plume and in its radiated sound. Numerous research groups have reported agreement between their LES results and experimental data for both the near-field fluctuations and the far-field sound characteristics for high-subsonic jets at moderate Reynolds numbers. The most recent investigations have exhibited the correct acoustic spectral characteristics over a range of frequencies. Ongoing investigations are exploring the influnce of the subgrid-scale model and the grid resolution on the radiated sound. Future work is focused on extending the LES results to higher frequencies through the development of subgrid-scale noise models. 6.2.6 Acknowledgments

The authors gratefully acknowledge support from the Aeroacoustics Research Consortium, a government and industry consortium managed by the Ohio Aerospace Institute, Cleveland, Ohio, U.S.A.

6.3 Cavity noise Xavier Gloerfelt, Christophe Bogey, and Christophe Bailly 6.3.1 Introduction to cavity noise

Impinging shear layers give rise to intense, self-sustained oscillations as well as noise radiation. Flow past a cavity belongs to this class of flows and has been extensively studied for the last 50 years because of its practical interest, its geometrical simplicity, and the diversity of the theoretical questions that it raises. The generic features are assessed in detail in the reviews of Rockwell and Naudascher (1978); Tam and Block (1978); Rockwell (1983); Komerath, Ahuja, and Chambers (1987); Howe (1997); or Colonius (2001). The self-sustained oscillations arise from a feedback loop consisting of the following chain of events. The growth and convection of instability waves in the shear layer induce large-amplitude pressure disturbances as the vortical perturbations impinge on the downstream corner of the cavity. The upstream influence of the generated pressure fluctuations provides further excitation of the instabilities in the shear layer – especially in its most receptive region near the upstream edge. A stable phase criterion is then installed between the downstream and the upstream edges of the cavity. This complex phenomenon is often greatly simplified to build lumped models such as the Rossiter formula (Rossiter 1964) in which the free-shear layer is viewed as two-dimensional and

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the recirculating flow is neglected. At high Mach numbers, this simple formula succeeds in predicting the admissible Strouhal numbers, although it provides no information on the amplitude of the self-sustained oscillations and fails to indicate which of the multiple modes will be predominant. At low Mach numbers, for frequencies close to those of the acoustic resonances, the acoustic forcing by the cavity resonance can overwhelm that provided by the feedback loop. Besides, this kind of semiempirical model is independent of the Reynolds number and cannot describe the changes between an initially laminar or turbulent incoming boundary layer. The ways to study cavity-flow phenomena are thus complete experiments or numerical simulations. An overview of previous simulations and recent advances is provided in Section 6.3.2. The 2D limitations are explained in view of previous works. Recent achievements by LES are described in Section 6.3.3. We attempt to identify important issues such as the influence of the turbulence or the coexistence of multiple tones, which will require further studies. 6.3.2 Overview of cavity-flow simulations 6.3.2.1 CFD simulations of cavity flow Use of URANS approaches

The first CFD computations of unsteady cavity flows used the two-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) equations with a turbulence model. The early work of Hankey and Shang (1980), using 78 × 52 grid points, was promising, showing fair agreement with frequencies of oscillations and with distributions of the pressure coefficient along the cavity walls from experiments. With the development of CFD codes in the 1990s, numerous URANS simulations were performed from the subsonic to hypersonic regimes. Various turbulence models were coupled: one-equation models based on the eddy viscosity concept of Boussinesq, such as the Cebeci–Smith (see Hankey and Shang 1980) or the Baldwin–Lomax models (see Baysal and Stallings 1987), or two-equation models such as the k–ε (see Suhs 1987; Shih, Hamed, and Yenan 1994) or the k–ω models (see Zhang 1995; Henderson, Badcock, and Richards 2000). The effectiveness of such models for separated flows remains an open question. For instance Slimon, Davis, and Wagner (1998) found that simulations display a strong sensitivity to the choice of the turbulence model. Tam, Orkwis, and Disimile (1995) also showed that the results are affected by high values taken by the turbulent viscosity. They even noticed better results for the estimation of the time-averaged surface pressure field with a zero-equation turbulence model, which led Rona and Dieudonn´e (2000) to prefer to study laminar flow motion. Orkwis et al. (1998) also showed how the results can be affected by the resolution algorithm. The first 3D applications were carried out in the late 1980s by Suhs (1987), Rizzetta (1988), and Srinivasan and Baysal (1991) and captured additional features of the 3D shear layer and recirculation inside the cavity but yielded limited improvements owing to the relatively coarse-mesh grid used. The development of industrial RANS codes allowed the study of complex geometrical configurations, including passive or active devices for the control of cavity oscillations (Kim and Chokani 1990; Suhs 1993;

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Baysal, Ten, and Fouladi 1994; Ota et al. 1994; Lamp and Chokani 1997; Zhang et al. 1998; Soemarwoto and Kok 2001; Bortz et al. 2002). Use of hybrid RANS–LES approaches

To enhance the description of the unsteady features of the separated shear layer at high Reynolds numbers, Shieh and Morris (2000) or Paliath and Morris (2003) tested in 2D the use of a hybrid RANS–LES approach based on the detached-eddy simulation (DES) concept. Extensions to 3D flows were carried out recently by Sinha, Arunajatesan, and Ukeiley (2000), Shieh and Morris (2001), and Hamed, Basu, and Das (2003). This method aims at exploiting the best features of both approaches by solving RANS equations in the turbulent boundary layer and performing LES in the separated shear layer. 6.3.2.2 CAA simulations of cavity noise Two-step methods

The two-step approaches separate the flow calculation and the noise propagation problem in order to apply the most appropriate method at each step. One of the most famous is the acoustic analogy of Lighthill (1952), extended by Curle (1955) and Ffowcs Williams and Hawkings (1969) (FW–H) to take into account the effects of solid boundaries. Zhang, Rona, and Lilley (1995) used URANS simulations and Curle’s spatial formulation to obtain far-field spectra of cavity noise, but no validation was proposed. In Gloerfelt, Bailly, and Juv´e (2003), the results of a direct computation of the noise by DNS were successfully compared with three integral methods: the FW–H analogy, the Kirchhoff method, and a wave extrapolation method based on the FW–H equation. A wave extrapolation method based on the FW–H equation was also applied by Ashcroft and Zhang (2001) to extend a compressible URANS solution to the acoustic far field. Note that these extrapolation methods require a compressible simulation below the extrapolation surface, whereas the source terms for the acoustic analogy can be deduced from an incompressible simulation. Hardin and Pope proposed another two-step method: the viscous flow is obtained from an incompressible simulation, and a correction to the constant density is defined. The acoustic radiation is then obtained from the numerical solution of perturbed, compressible equations. This technique was applied to the computation of flowinduced cavity noise by Hardin and Pope (1995), Slimon et al. (1998), and Moon et al. (1999). These methods suffer, however, from two difficulties: the modeling of the source terms from aerodynamic fluctuations and the ability of the wave operator to include complex acoustic flow interactions. Direct noise computations

Because flow and acoustic fluctuations are solutions of the compressible Navier–Stokes equations, it is possible to obtain both fields in the same calculation. However, owing

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Figure 6.17. Transition toward a wake mode for large L/δθ ratio (2D DNS of the flow over an L/D = 4 cavity, at M = 0.5, and ReD = 4800). Snapshots of the vorticity field (levels in s−1 ) for two successive times T/2 and T during one oscillation period T . On the left, a shear-layer mode is observed for L/δθ1  63, whereas, on the right, a wake mode is visible for L/δθ 2  280 (Gloerfelt, Bailly, and Juve´ 2000).

to the great disparities between these two quantities, in classical computational fluid dynamics (CFD), acoustical phenomena are not resolved accurately because of the numerical schemes used and inadequate grid-cell size or time step. Moreover, reflections due to the boundary conditions can shade the physical acoustic wave field. High-order algorithms minimizing dispersion and dissipation and nonreflecting boundary conditions have therefore been developed that permit DNC. In the case of cavity noise, initial attempts of DNC were made for supersonic flows by Zhang (1995) and Rona and Dieudonn´e (2000) for which the amplitude of the shock waves is very strong and the acoustic waves can not travel upstream. The first computations of the noise induced by a cavity with a subsonic grazing flow were carried out by Colonius et al. (1999), Rowley, Colonius, and Baasu (2002), and Shieh and Morris (1999b) using 2D DNS. These simulations showed a transition toward a new flow regime when the ratio L/δθ of the cavity length over the momentum thickness became large, as already observed by Najm and Ghoniem (1991) by using a vortex method. This regime is characterized by the shedding of a single vortex that occupies the entire cavity and overshadows the role of the smaller-scale vortices of the separated shear layer. The periodic ejection of this structure is associated with an increase of the cavity drag. The same numerical bifurcation was also noted by Gloerfelt et al. (2000), as illustrated in Figure 6.17. Rowley et al. (2002) called the new regime a wake mode because of similarities with the transition observed in the experiments of Gharib and Roshko (1987) in a water tunnel. However, the presence of a wake mode is not seen in experiments of compressible cavity flows. Did it result from the very low Reynolds numbers imposed by DNS or from the 2D approach? To investigate higher Reynolds numbers, Shieh and Morris (2000) applied computational aeroacoustics (CAA) tools to solve hybrid URANS–LES. The transition to a wake mode was still observed, indicating that it could be related to the 2D behavior rather than to the Reynolds number. A subsequent 3D study by Shieh and Morris (2001) did not show a wake mode transition for the same configurations,

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confirming the 2D nature of the bifurcation. In a 3D vortex, stretching significantly modifies the turbulent mixing between counterrotating vortices in the recirculation region and prevents untimely transition to the wake mode. Another 2D DNC using DNS of the cavity flow was performed in the framework of the third CAA workshop by Kurbatskii and Tam (1999), Shieh and Morris (1999a), and Heo and Lee (2001) for a door cavity. Heo and Lee (2001) studied the effects of cover plates obstructing the cavity opening. Koh and Moon (2003) simulated 2D compressible turbulent flows and compared a URANS approach using a second-moment turbulence closure and a limiter for the energy balance with the DES model. Bidimensional behavior of the recirculation zone was visible as well in the computations of Heo, Kim, and Lee (2003). The latter used 2D DNS with high-order numerical schemes and probed the feedback loop through correlations. 6.3.3 Recent achievements using LES 6.3.3.1 A challenging test case

The previous simulations indicate that a 3D approach is required to describe the turbulent mixing inside the cavity and, if necessary, the turbulent character of the incoming flow. A DNS solving all the scales (down to the Kolmogorov scale) would be impractical. To achieve reasonable computational cost, and given the dominance of coherent structures in the shear layer, cavity flows seem a good candidate for LES. This configuration is, however, a challenging test case insofar as the flow can be viewed as a synthesis of a simple shear flow in the mixing layer above the cavity, a complex shear flow inside the cavity, and multiscale wall boundary layers. Basic difficulties in cavity-flow LES are thus the modeling of the unresolved subgrid scales (SGS), the description of the near-wall region, and the intensive computational resources required. The first point is still subject to discussion. The most popular solution is the use of an eddy viscosity model linearly related to the resolved stress tensor such as Smagorinsky or dynamic Smagorinsky models, which are referred to as SM and DSM, respectively. These models are often overdissipative in high-Reynolds-number configurations (Fureby and Grinstein 2002), and it can be inferred that the eddy viscosity may have a nonnegligible effect on the shear-layer dynamics and that a less dissipative model would be helpful to preserve high-Reynolds-number features. Alternative approaches have been proposed. Zang et al. (1993) used a mixed model that combined the DSM and the similarity model of Bardina for turbulent recirculating flows in driven cavities, reducing the magnitude of the dynamically computed coefficient. Boris et al. (1992) have suggested damping the turbulent energy by the numerical procedure. In their monotone integrated LES (MILES), the numerical dissipation is provided by the use of low-order upwind schemes. The use of high-order explicit filtering appears more appropriate in order to control the numerical dissipation and the cutoff between resolved and unresolved scales. For instance, Visbal and Rizzetta (2002) obtained better results using compact filtering alone than with the DSM for isotropic turbulence. In the same manner, Bogey and Bailly (2005c) compared the DSM and the use of selective

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filtering alone for a high-Reynolds-number high-subsonic jet. Their results support the fact that the eddy viscosity artificially decreases the Reynolds number. The second difficulty for wall-bounded flows concerns the description of the near-wall structures. The resolution of the inner layer exceeds present computational capabilities already at moderate Reynolds numbers. Methods to bypass the wall layer are required to perform high-Reynolds-number LES. One alternative is proposed by the hybrid RANS–LES approaches, which switch to RANS solutions in the boundary layers. Another possibility is the use of wall models for the inner layer (Piomelli and Balaras 2002) even if their extension to complex configurations is not simple (see Shih, Povinelli, and Liu 2003). The application of the SGS model to wall-bounded flows is also questionable. The simplest approach is the use of damping functions to account for the scale reduction near the walls such as the van Driest function. Several modifications reviewed by Piomelli et al. (1989) or Sagaut, Montreuil, and Labb´e (1999) have been proposed to model the near-wall effects. This kind of model is also hardly extendable to more complex configurations. A dynamic procedure, such as that of DSM, may overcome this difficulty by using the information from the smaller resolved scales. Note that the latter problem is automatically avoided in alternative approaches based on implicit SGS contributions such as MILES or the compact or selective filtering approaches. Owing to the preceding considerations and the computational ressources required, very few large-eddy simulations have been carried out until now. The first applications concerned transonic or supersonic flows over shallow cavities, representing weapon bays of military aircrafts. Large-eddy simulations by Dubief and Delcayre (2000), Smith (2001), or Lillberg and Fureby (2000) reproduced high-Reynolds-number configurations with fairly coarse resolutions. Rizzetta and Visbal (2002) carried out LES of weapon bays with L/D = 5, M = 1.19, and Re L = 2 × 105 . The computations involved 20.6 million grid points and used high-order compact schemes and DSM to describe the fine-scale structures. Two computations were performed with or without suppression devices based on high-frequency mass injection. Another interesting weapons bay LES without the acoustic field was performed by Larchevˆeque, Sagaut, and Lˆe (2003). The existence of strong transverse acoustic modes in the cavity was highlighted. Preliminary results with the DSM by Oh and Colonius (2002) can also be quoted. Three recent aeroacoustic contributions by Larchevˆeque, Sagaut, Mary et al. (2003), Gloerfelt, Bogey, Bailly, and Juv´e (2002), and Gloerfelt et al. (2003a) are presented below. 6.3.3.2 LES of a deep cavity in a channel

Larchevˆeque, Sagaut, Mary et al. (2003) performed LES of the flow over a deep cavity with a length-to-depth ratio of L/D = 0.42, a freestream Mach number of 0.8, and a Reynolds number Re L = 8.6 × 105 , reproducing exactly the experiment of Forestier, Jacquin, and Geffroy (2003). Two LES strategies were tested: a traditional LES with an eddy viscosity model, where νt was determined by the selective mixed model, using a second-order centered scheme, and a MILES approach in which the intrinsic dissipation of an upwind scheme mimicked the dissipative behavior of the unresolved

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Figure 6.18. LES of a deep cavity.

scales. In both simulations, a third-order compact Runge–Kutta algorithm was used together with a wall model based on the logarithmic law. The turbulence level inside the boundary layer was provided by the superimposition of random fluctuations in the inlet plane. The two LES strategies gave a very good agreement with measured spectral peak levels. Reynolds and phase averages were also remarkably well reproduced, even if the turbulent levels were slightly higher with MILES. This is probably related to the more dissipative discretization scheme in the MILES approach. The acoustic field was obtained thanks to a 2D–3D domain coupling. Figure 6.18 shows a view of the computational domain. Structures are visualized using isosurface of the Q-criterion equal to 10(U∞ /L)2 combined with a Schlieren-like picture in the background (Larchevˆeque, Sagaut, Mary et al. 2003). The Q-criterion defines as vortex tubes the regions in which the second invariant of velocity gradient tensor Q = 1/2 (i2j − si2j ) is positive. This view reveals the existence of a characteristic lattice of multiply reflected waves in the channel similar to that observed experimentally by Forestier et al. (2003). The presence of upper wall reflections can trigger the shedding of new vortices. 6.3.3.3 Influence of the incoming-flow turbulence

Three-dimensional large-eddy simulations were performed by the authors to describe the influence of the small scales and of the intermittency of the turbulence. In his pioneering work on cavity oscillations, Karamcheti (1955) noticed a different behavior whether a laminar or a turbulent boundary layer interacted with the cavity. The frequency of the oscillations for a turbulent inflow was slightly lower than the one measured with a laminar inflow. Moreover, a reduction in the amplitude of the pressure fluctuations was observed together with the emergence of a low-frequency component in the spectra for a turbulent inflow. The origin of this low-frequency component, with roughly half the frequency of the fundamental, remains unexplained. One of Karamcheti’s configurations was reproduced by Gloerfelt, Bogey, Bailly, and Juv´e (2002) using LES with two different incoming flows: a laminar Blasius boundary layer with no forcing, and a turbulent boundary layer generated by superimposing random Fourier

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modes on a turbulent mean profile. The length-to-depth ratio was L/D = 3, with D = 2.54 mm as in the experiment, and the freestream Mach number was M = 0.8, giving a Reynolds number Re L = 1.46 × 105 . The governing equations were integrated in time using an explicit low-storage, six-step Runge–Kutta scheme optimized in the frequency space. The derivatives were calculated by using optimized finite differences with an eleven-point stencil for the convective fluxes, and fourth-order finite differences for the viscous and heat fluxes. As part of the algorithm, a selective filtering using an eleven-point stencil was incorporated in each direction to eliminate grid-to-grid oscillations. The coefficients of the Runge–Kutta, of the finite differences, and of the filtering are given in Bogey and Bailly (2002). The computations lasted 40 hours for 5.4 million grid points (CPU time of 0.6 μs per grid point and per iteration). The effects of the unresolved SGS were modeled via the Smagorinsky eddy viscosity with a self-adaptive van Driest damping function near the solid boundaries for the turbulent inflow simulation. No model was applied for the initially laminar case because SM cannot describe the turbulent transition. For examining the role of the turbulence model, incorporated solely in the initially turbulent simulation, the turbulent case was also performed with the use of selective filtering alone (i.e., without an explicitely added SGS model) in a subsequent study by Gloerfelt, Bogey, and Bailly (2002). The results are displayed in Figure 6.19 for the three large-eddy simulations. On the left, snapshots of the instantaneous vorticity modulus ω (top view at x2 = 0.06 D + sideview at x3 = 0) are shown. The levels are between 0 and 2 × 106 s−1 for the three cases. The vorticity snapshots indicate that the coherent structures are actually made up of several small-scale vortices more concentrated with the laminar inflow than with the turbulent inflow. The vorticity concentration is also enhanced more with SM than with the explicit filtering. The features of the shear layer, with finer scales without SM, support the fact that the effective Reynolds number is reduced by the use of an added eddy viscosity. In the right column of Figure 6.19, the pressure field in the computational domain and OASPL is shown at the point (−D, 4.6D, 0) in decibels (ref 2 × 10−5 Pa). The pressure grayscale is the same for the three cases, with levels between −3000 and 3000 Pa (Gloerfelt, Bogey, and Bailly 2002; Gloerfelt, Bogey, Bailly, and Juv´e 2002). The structure of the radiated field showed a strong downstream directivity owing to the convection of acoustic wave fronts by the mean flow combined with the reflections on the cavity walls. The acoustic emission was directly linked to the impingement of the coherent structures, and the pressure levels seemed correlated with the vorticity concentration of the structure. The highest amplitude is found for the laminar configuration in agreement with Karamcheti’s results (Karamcheti 1955). The comparison of acoustic spectra for the turbulent configurations with or without SM indicated that the levels are lower without the model. A part of the energy is dumped into the more numerous small scales, which induces a less coherent interaction with the downstream edge, and, consequently, reduces the acoustic emission. The results also showed the emergence of a low-frequency component for the turbulent case without SGS model, as in Karamcheti’s measurements.

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Figure 6.19. Influence of the boundary layer turbulence on cavity noise. LES of the L/D = 3, M = 0.8 configuration of Karamcheti’s experiments with laminar inflow (top); turbulent inflow with SM (CS = 0.18, van Driest damping) (middle); and turbulent inflow with selective filtering alone (bottom).

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Figure 6.20. Mode switching. Three isocontours of instantaneous modulus ω of vorticity (1, 2, and 3 × 106 s−1 ) are superimposed on the instantaneous vorticity averaged over the spanwise direction (grayscale between −1.2 and 1.2 × 106 s−1 ) for two instants during one cycle. A mode I, with one big roll in the shear layer, is visible on the top, and a mode II, with two coherent rolls, is identifiable on the bottom (Gloerfelt et al. 2003a).

6.3.3.4 Switching between competitive modes

The previous configuration (L/D = 3, M = 0.8) was simulated with periodic boundary conditions in the spanwise direction by Gloerfelt et al. (2003a), allowing a finer resolution in this direction. The same algorithms were used, and the transition toward a realistic turbulent boundary layer ahead of the cavity was successfully validated. The SGS contribution used SM with a Smagorinsky constant CS = 0.1 and an adaptive van Driest function near the walls. Attention was primarily focused on the intermittence of the turbulence, leading to the coexistence of two peaks in the spectra for turbulent conditions (Karamcheti 1955). The switching between modes I and II of the cavity oscillations was demonstrated (see Figure 6.20). This switching was not random in time but followed a cyclical pattern with a successive alternation between the two modes even if a certain level of intermittency was observed in this succession. In turbulent conditions, the coherent structures display a shape of clusters of small scales, and the alternation of different sizes of the dominant structures can proceed by a reorganization of these clusters. The examination of the early shear-layer growth revealed that the formation of large-scale structures was possible through a collective interaction after the upstream corner, corresponding to the fusion of several smaller vortices shedded at the most unstable frequency of the shear layer. This phenomenon appears to be of fundamental interest in the selection process of the main oscillation frequencies. Another interesting finding is the strong unsteadiness of the recirculating flow within the cavity, which seems severely coupled to the shear-layer modulations. However, the role of the recirculating flow on the switching phenomenon is not clear because the coexistence of different modes has been noticed in the self-sustained oscillations for configurations without adjacent recirculating flows. The acoustic field clearly indicates the predominance of the lower component, whereas the Schlieren visualizations of

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Karamcheti indicate the predominance of the high component. The development of the lower mode in the LES was enhanced by the numerics – especially by the eddy viscosity – and thus a more violent impingement on the downstream corner of the cavity was induced and dominated the acoustic radiation.

6.3.4 Concluding remarks for cavity noise

For high-Reynolds-number flows, the dynamics of the small scales and the intermittence of the turbulence may lead to the coexistence of different modes with the flow switching between them. This phenomenon can hardly be described by RANS methods and is better suited to LES. The first complete aeroacoustic computations are very recent. The large-eddy simulations described here reveal the dependence of the results on the resolution and on the SGS model. The radiated field induced by the cavity flow is greatly sensitive to small flow modifications such as the concentration of the coherent structures. The computation of the correct acoustic levels in the far field is thus a challenging problem. It requires subsequent investigations on the role of the SGS models and on the effects of the numerical resolution. Significant progress relies on the direct comparisons with experimental databases. Such databases with aerodynamic and acoustic information have been lacking and are just beginning to emerge (Forestier et al. 2003). The simultaneous obtainment of the aerodynamic and acoustic fields from the compressible Navier–Stokes equations is made possible by the use of very accurate algorithms and of quiet boundary conditions. This can provide further insights into the noise-generation mechanisms. In particular, the investigation of the nonlinear interactions between modes by Gloerfelt et al. (2003b), the characterization of the phase relationship between the two corners of the cavity – especially the identification of the acoustic feedback path – or the study of the growth of the shear-layer instabilities may help toward a better understanding of the selection process in order to design reduced models and efficient control strategies.

6.4 Aeroelastic noise Sandrine Vergne, Jean-Marc Auger, Fred Peri ´ e, ´ Andre´ Jacques, and Dimitri Nicolopoulos 6.4.1 Introduction to aeroelastic noise

Aerodynamic noise is one of the major contributors to internal vehicle noise; it becomes dominant above 400 Hz at driving speeds exceeding 100 km/h when compared with structure-borne, power train, and tire noise for which substantial noise reduction has

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been achieved. The interaction of the flow with the geometrical singularities of the vehicle body produces transient turbulent flows, often detached, resulting in an increased aerodynamic noise generation. Two main phenomena are involved: r Aeroacoustic noise: The production and decay of turbulent structures within the

flow yields the generation of acoustic sources. That is propagative pressure waves by opposition to the mostly convective nature of the flow around the car. The pressure waves produce a loading of the vehicle body and side windows which do transmit part of the noise inside the vehicle. r Aeroelastic noise: The convected pressure waves fluctuations (pseudo noise), although not propagative, do also load the vehicle structures and therefore induce vibrations that will radiate noise inside the passenger compartment. This section focuses mainly on this second kind of noise, which is also named “aerovibroacoustics.” Flow-induced vibration phenomena are a ubiquitous feature in numerous engineering applications from dynamic stability (e.g., airplane stability, deformations of building structures, and bridges under wind loads) to structural radiation (e.g., the trailing-edge noise described in Howe (2000) or the flow-induced sound inside a plane or a car due to the pressure fluctuations of the boundary layer on the exterior of the vehicle). In the first case, the vibrations are undesirable, causing material fatigue at best and catastrophic failure at worst. The interactions between flow and structure are strong. In the second case, flow-induced forces on bodies and structures induce vibration. This vibration, in turn, induces additional noise and may even modify the unsteady fluid flow. In many cases, the structural surfaces do vibrate, but these vibrations cannot alter the flow motion. Turbulent structures remain unaffected by the acoustic effects. This assumption corresponds to the implicit hypothesis, made in acoustic analogies, that the acoustic pressures are small in comparison with the turbulence pressures. These flow–structure interactions may be considered weak. Several research tasks are dedicated to the improvement of numerical and experimental methods of building hybrid structural–acoustic models based on a combination of experimental and numerical data. The global purpose is to combine equivalent sources or forces derived from experiments and vibroacoustic, transfer path models in order to calculate the pressure at the passengers’ ears within the vehicle compartment (Mein, Dupuy, and Bohineust 2002; Bohineust, Bardot, and Dupuy 1996; Bohineust et al. 1999). These hybrid models are now available for several stages of the design and development process for power train and tire noise but are still under development for aerodynamic noise calculation. Such hybrid models have been applied to test cases like a flat plate, either coupled or not coupled to a cavity, and submitted to a turbulent boundary layer flow. The structure problem could be solved using the normal-mode analysis technique: the structure is assumed to respond in its normal modes of vibration, which describe a condition of resonance. Therefore, given a known description of the fluid characteristics on the

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surface, the vibroacoustics response of the system could be evaluated. For a turbulent boundary layer, the excitation is described by a semiempirical wave-number frequency spectrum model such as the well-known Corcos model (Corcos 1964), the Smol’yakov and Tkachenko model (Smol’yakov and Tkachenko 1991), theoretical ones such as the Chase model (Chase 1987), or the Ffowcs Williams model (Ffowcs Williams 1982). In most cases (Graham 1997; Leclercq 1999), computed acoustic responses radiated by the plate strongly overestimate the experimental results above a threshold frequency, and the accuracy of the predicted response levels is very sensitive to the space–time characteristics of the turbulent flow (i.e., the coherence and phase velocity of the wall-pressure fluctuations). Moreover, research efforts are aimed at introducing numerical approaches to calculate aerodynamic noise. Many studies (P´erot et al. 2004; Lokhande, Sovani, and Xu 2003; Reichl et al. 2003) deal with the estimation of the aerodynamic noise by using LES computation, but the acoustics receivers are put outside the vehicle. Theses numerical approaches can help toward an understanding of the physical mechanisms responsible for the noise generation. To better understand the coupling between turbulence and vibroacoustics, we consider a numerical method based on unsteady, fully coupled resolution of 3D compressible Navier–Stokes equations and structural conservation laws. This method has been applied to several types of problems such as aeroacoustic simulation of a rear side-mirror vehicle (Obrist, Nicolopoulose, and Jacques 2002), flow fan noise (Roy and Cho 1999), and automotive exhaust to resolve coupled problems featuring noise transmission through a vibrating structural part (Nicolopoulos, P´eri´e, and Jacques 2004; Sakurai, Endo, and P´eri´e 2002). In this section, we will focus on the ability of the Radioss CFD–CAA code to calculate a typical case of many industrial applications: the vibroacoustic response of an elastic plate–cavity system excited by a turbulent flow. The behavior of the model is studied in terms of its three components: hydrodynamic (flow field and wall pressure fluctuations), structural (plate accelerations), and acoustic (cavity response). These different components are compared with an existing experimental database obtained by Peugeot Societe Anonyme (PSA) (Leclercq 1999). 6.4.2 Fluid–structure interaction

We consider the plate to be driven by the wall-pressure fluctuations induced by the turbulent flow and by the acoustics fluctuations due to plate motions (Figure 6.21). This model is given by the equations of motion, ms

∂w ∂ ∂ 2w + Ds w = − ( pt + p2 − p1 ), +β 2 ∂t ∂t ∂t

(6.16)

and acoustics (Helmholtz equation), p0 −

1 ∂ 2 pi = 0, c02 ∂t 2

(6.17)

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Figure 6.21. The plate model.

where w(x1 , x2 , ω) is the plate normal velocity, Ds is the plate bending stiffness, m s is the plate mass per unit area, and β is a mechanical damping coefficient of the panel. The plate bending stiffness Ds is related to the plate Young’s modulus E, thickness h, and Poisson’s ratio ν, by Ds = Eh 3 /12(1 − ν 2 ). The plate vibrations are excited by turbulent wall-pressure fluctuations pt (xi , ω) on the upper surface of the plate. The terms p1 (xi , ω) and p2 (xi , ω) are the internal and external acoustic pressure generated by the motion of the panel, respectively. This system of equations is completed by the boundary conditions of the plate (simply supported, clamped) and by the condition linking pi and w at the plate surface: ∂w ∂ pi = −ρ0 . ∂ x3 ∂t

(6.18)

The external flow is governed by the Navier–Stokes equations ∂(ρu i ) ∂ρ = 0, + ∂t ∂ xi ∂( pδi j − τi j ) ∂(ρu i ) ∂(ρu i u j ) =− , + ∂t ∂x j ∂x j

(6.19) (6.20)

where u i are the flow velocity components, ρ the volumic mass, c0 the sound velocity, and τi j the viscosity shear stress. An unsteady slipping boundary is applied on the moving panel as u · n(t) = w · n(t),

(6.21)

where u is the flow velocity, w the panel velocity, and n(t) the normal to the surface that is dependent on time t. In most literature (Blake 1970a, 1970b; Kraichnan 1956) for boundary layer excitations, authors have assumed that whatever surface motion occurs does not influence the fluid dynamics (i.e., u · n(t) = 0). This weak-coupling assumption is physically valid for most fluid–structure interaction. The weak-coupling assumption is not used in the considerations presented in this section.

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6.4.3 Numerical simulation

The approach taken in this section for computational aeroacoustics attempts to take several physics into account in a single simulation: r Unsteadiness of the flow involving macroscopic scales; r Dissipation due to microscopic turbulent scales; r Acoustic contribution of macroscopic turbulence; r Acoustic wave propagation, including impedance of boundaries; r Noise transmission to structures; and r Structural response. The CAA methodology is grounded on the following numerical choices: r The transient solution of the full, compressible Navier–Stokes equations that al-

low for the propagation of pressure waves as well as fluid flow. Its double precision implementation guarantees enough accuracy for both average values (flow) and fluctuations of the calculated variables (macroscale turbulence and acoustic waves). In these classes of aeroacoustic problems, the dynamic range of the pressure fluctuation can be from 2 × 10−5 Pa up to several thousand Pa. r An explicit time integrator (first-order) that provides enough time resolution to capture bifurcation phenomena. This method yields a fairly cheap computational cost per element per time step (at least one magnitude order cheaper than an implicit solution) at the expense of a high number of time steps. The simulations are performed with the explicit module of Radioss CFD–CAA developed by Mcube and Mecalog. The implementation of Navier–Stokes equations is based on the arbitrary Lagrangian Eulerian (ALE) formulation, as presented in Donea → (1982): each point in space has the usual material velocity − u as well as a grid velocity − → v describing the a priori arbitrary movement of nodal points. Conservation laws may be written as  ∂ρ → → → + [(− u −− v ) · ∇]− u − ∇ · σ = 0, ∂t

(6.22)

∂ρ → → → + [(− u −− v ) · ∇]ρ + ρ∇ · − u = 0, ∂t

(6.23)

∂ρe → → → + [(− u −− v ) · ∇]ρe + (ρe + p)∇ · − u = 0. ∂t

(6.24)

 ρ

→ → Note that, for − u =− v , Equations (6.22)–(6.24) reduce to the Lagrangian descrip− → tion, whereas for v = 0, they describe the Eulerian case. As a consequence, a single formulation is able to describe the evolution of physical variables in the laboratory reference frame and in a grid with any arbitrary movement (e.g., owing to the grid deformation required to accommodate deformation and movement of structures).

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The spatial integration is achieved with a second-order finite element formulation with a streamline upwind Petrov–Galerkin (SUPG) integration scheme (Brooks and Hughes 1982). The time integration is performed with an explicit time-centered integration scheme whose stability is governed by Courant’s condition,   h , (6.25) t = min → c + − u where h is the minimum element dimension and c is the speed of sound. These result in very small time steps that, together with SUPG, provide minimal numerical diffusion and enough accuracy to capture and to convey the resolved vortices. These schemes allow turbulent structures greater than ≈10 elements to be resolved by the grid. Smaller scales are taken into account by the Smagorinsky SGS model with Cs = 0.1. The proposed method does not pretend to resolve turbulence near walls. A special treatment (Benyahia 2003) assumes a logarithmic profile and computes the turbulent viscosity at the boundary layer as if all turbulent structures were unresolved there. Special care is also taken for sound absorption due to unresolved turbulence by adding a viscous bulk behavior: Pvisc = ρνsgs V˙ /V,

(6.26)

where Pvisc is the corrective pressure dependent of the density ρ, the subgrid viscosity νsgs for the LES, and the flow velocity fluctuation rate V˙ /V . This correction has enhanced the sound-pressure wave reflections on structures through boundary layers. Note that the compressible nature of Equations (6.22)–(6.24) takes into account acoustic wave propagation within the computational domain. This allows direct capture and propagation of the aeroacoustic noise sources, avoiding the tedious data storage and propagation calculations needed by the volume terms in Lighthill’s approach. However, it requires modeling the radiation at the computational domain boundaries. The linearized Euler equation of Bayliss and Turkel (1982b) provides an efficient treatment of this problem: ∂u n c ∂p = ρc + (P∞ − p), ∂t ∂t 2lc

(6.27)

where u n is the velocity normal at the boundary, and P∞ is the stagnation pressure. The first term on the right-hand side is the Sommerfeld impedance, and the last one is a relaxation term toward the average value either imposed or derived by assuming continuity: ∂p = 0. → ∂− n

(6.28)

This equation is equivalent to a high-pass filter. Moreover, a Fourier analysis of Equation (6.27) shows that these two terms match the radiation impedance of a monopole at distance 2lc . This means that the computational domain boundaries should be constructed so that the simulated acoustic sources are far enough apart to be seen as monopoles.

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Figure 6.22. Sketch of the experimental setup: small ruler of size L = 0.010 m and H = 0.009 m and elastic plate.

The finite element implementation allows a simple nodal approach to connect different parts – either by simply sharing nodal connectivity or through the use of interfaces. Several nonconforming interfaces are available for CAA applications: r Fluid–fluid, r Fluid wall–structure, r Tied structures, and r Periodic.

By its nature the ALE formulation allows a two-way coupling of the fluid and the Lagrangian domains. This means that noise can be transmitted to structures and vice versa. It also means that structures can be modeled through the same formulation using a wide variety of finite elements and kinematics conditions, namely, r thin or thick shell, quads, or triangles; r beams; r springs and dampers; r rigid bodies; and so on.

6.4.4 Application 6.4.4.1 Test case description

An existing test case is used to evaluate model results as defined and fully investigated by PSA (Leclercq 1999). It was built in order to analyze and to improve the comprehension of the mechanisms of energy transfers between a turbulent flow and the vibroacoustics. In the experimental setup (Figure 6.22), a flat plate is excited by a turbulent flow and coupled to a cavity. This system should represent a very simplified vehicle structure for which the flat plate and the cavity represent, respectively, a vehicle panel and the cabin; the flow velocity is of the order of 140 km/h, for which the aerodynamic noise is dominant. As a way of approaching the complex flows observed around the vehicles, the flow is perturbed by a small ruler, which could represent a seal or a windshield wiper. Upstream, a turbulent boundary layer fully develops on a thick, rectangular, flat table, 2.4 m long and 0.9 m wide. At a distance of 0.05 m before the fence, the turbulent boundary layer thickness is estimated at 0.02 m. The 40 m/s flow and its turbulent

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Figure 6.23. Wall-pressure transducer positions on the elastic steel plate.

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25 0

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boundary layer are then perturbed by a small ruler, 0.009 m high and 0.01 m wide, mounted on the floor (Figure 6.23). Downstream, the resulting turbulence excites a 1-mm-thickness elastic steel plate (0.366 m × 0.306 m) coupled to a cavity (0.370 m long, 0.310 m wide, and 0.280 m high). 6.4.4.2 Experimental setup

An experimental study has been conducted to characterize this test case accurately (Leclercq 1999). Measurements involve flow spatiotemporal characteristics and vibroacoustics response. Velocity profiles (particle image velocimetry, hot-wire anemometry) are measured to characterize the flow field. Because the flexible plate cannot be equipped, wall-pressure measurements are made over a rigid plate in a first step. The area is equipped with 42 wall-pressure transducers with their positions as shown in Figure 6.23. The transducers are developed by PSA to characterize the wall pressure field, accurately (Leclercq and Bohineust 2002). Note that this approach implicitly assumes a weak coupling (i.e., that pressure fluctuations are dominated by the turbulence and not modified by plate vibrations). In a second step, the vibroacoustics response of the flexible plate coupled to the acoustic cavity is then studied by fitting the experimental setup with transducers to measure the plate vibratory field (six accelerometers) and the cavity acoustic field (five microphones) (Figure 6.24). 6.4.5 Simulation model 6.4.5.1 Mesh description Mesh sizes and general dimensions

The computational domain begins 50 mm upstream of the obstacle and is 960 mm long, as shown in Figure 6.25. As conditions for accurately performing aeroacoustic simulation, it has been established by P´eri´e (2002) that r The Strouhal number associated with the element dimension must be less than 1/12

in the acoustic sources region. This gives in our case a smallest mesh size length of 3 mm (the frequency range of interest being 40–3000 Hz). r To capture the structures generated by the obstacle (whose dimensions are 10 mm × 9 mm) a mesh requires smaller size. Twenty elements per side of the

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x3 (m)

24

26 32

Figure 6.24. Vibroacoustics measurements: (+) accelerometers and (•) microphone positions.

30 0.1

0

0

x2 (m)

0.1

x1 (m)

obstacle give a size of 0.5 mm. As the minimal size of the whole mesh, this 0.5 mm induces the critical time step (Courant’s condition), 1.3 ms, which should be compared with the 0.6 s of physical duration necessary for simulation. r Twelve elements per smallest wavelength are required in the propagation zones, in order to have minimal attenuation and dispersion when acoustic waves travel across the mesh. This gives 25 mm in our case. r Mesh consists of hexahedral elements. The mesh size in the transverse direction (y) has the constant value of 5.2 mm. The plate and the cavity

The plate is meshed with shell elements so that its flexion is represented by at least six elements of size h per smallest wavelength. Considering that the celerity of the flexion waves is given by C f = (B/Ms )1/4 ω1/2 , B=

Figure 6.25. General mesh dimensions.

Eh 3 , 12(1 − ν 2 )

(6.29) (6.30)

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Figure 6.26. General sizes of the mesh.

where B is the bulk modulus, Ms the surface mass, ω the pulsation, E the Young modulus, ν the Poisson coefficient, and h the plate thickness, we find that this gives a size of 8 mm for the plate elements size. Because only acoustic propagation is simulated in the cavity, the following values were used: 6.2 mm in x , 10.6 mm in y , and 15 mm in z . They all widely satisfy the propagation criterion of 25 mm. Mesh description

First, an automatic triangle mesh of the x–z plane is generated. The triangular elements are then recombined to create quadrangular elements. This allows a greater flexibility in the mesh size repartition. Then, the 2D mesh is extruded along the spanwise direction. This mesh is particularly well refined around the obstacle, where the cells keep a structured topology. The mesh density areas can be seen in Figures 6.26 and 6.27. The average 0.5-mm-density region follows the high shear layer whose location was identified in the first model. Near the plate, the mesh dimension is 1.0 × 5.2 × 1.0 mm3 . The total size of this mesh is 1,098,310 hexahedral elements, 1,710 shell elements, and 1,141,433 nodes.

0.1 0.09 0.08 0.07

Z

0.06 0.05 0.04 0.03 0.02 0.01 0

0.05

0.1

Figure 6.27. Detailed mesh around the obstacle.

X

0.15

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6.4.5.2 Numerical model

Air is modeled using a linearized compressibility assumption equivalent to an isothermal equation of state. Connection between the fluid main part, the cavity, and the plate

The fluid, the cavity, and the plate meshes are connected through a nonconforming Lagrange–Lagrange interface. The nodes of the fluid and of the cavity in contact with the plate are set Lagrangian. Initial and boundary conditions

r For initializing the fluid flow, a velocity of 40 m/s, oriented in the x-direction, is

used as an initial condition for the first time step.

r Continuous fluid motion is forced through an imposed velocity condition at the inlet

of the system; the velocity profile Vx (z) close to the wall describes the upstream boundary layer as estimated in the experiment. The y and z components of the velocity are set to zero. r A constant pressure of 0 Pa is applied at the outlet of the computational domain, whereas the upper boundary has a zero pressure gradient. r A silent boundary treatment is applied at both outlet and at the upper boundary: the frequency components of the pressure waves greater than the cutting frequency of 273 Hz are absorbed. The acoustic impedance at boundaries is in fact equivalent to the radiated impedance of a monopole located at 100 mm away. r On the cavity sides and on the floor (including the obstacle), the velocity is set to zero (no slip). r Symmetry conditions are imposed on the lateral planes (Vy = Vnormal = 0). r For modeling the sealing of the plate, a series of springs are used and their stiffnesses are adjusted to approach the experimental resonance frequencies. These resonance frequencies are the damping coefficient are obtained by classical vibration measurements (a simple freedom degrees extraction method is used to posttreat the frequencies response of a punctual excitation). The translation stiffness is set to 1.5 × 107 N/m, and the rotation stiffness to 15 N/m. The damping coefficients have been input as 50% of the critical damping, which gives in translation 85 kg/s and in rotation 0.278 × 10−3 kg/s. Those values were calibrated earlier during the evaluation of the hybrid method described in Section 6.4.3. r The plate is set Lagrangian. r The two-way fluid–structure coupling is achieved by the ALE formulation applied to the fluid regions, whose meshes actually deform near the surface of the plate. Analysis

Two phases are required to perform the aeroacoustic simulation: r During a highly transient phase, the flow and the plate vibrations converge toward

a stabilized regime until an energy and acoustic convergence is reached. This phase lasts 0.1 s of physical time.

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Figure 6.28. Contours of the streamwise mean velocity U1 /U0 near the ruler in the median plane (levels between −0.4 and 1.2; white line: isocontour of 0).

x3 /h

10 5 0

0

10

x1/ h

20

30

r Once convergence is obtained, the simulation is continued in order to record pressure

signals at selected locations. The sampling frequency is 10 kHz. A duration of 0.5 s is necessary to have 10 consecutive blocks of 512 records ready to perform averaged spectra. The total computing time is 70 elapsed hours and 550 CPU hours on a CRAY SX6 with eight processors. 6.4.6 Numerical results 6.4.6.1 Flow-field analysis Mean flow

To validate the simulation, one must first evaluate the CAA code’s ability to simulate the flow accurately. Figure 6.28 presents the modulus of the mean velocity scaled on U0 near the ruler in the vertical median plane. The figure clearly shows that the flow separates on the ruler, the single recirculating region encompassing both the top and the rear faces. The flow reattaches on the horizontal wall at approximately 20 h. This flow separation generates turbulence, as shown by the nondimensional streamwise fluctuations u rms /U0 in Figure 6.29. The more significant turbulent kinetic energy is observed near the reattachment point (about 15–20 h). The simulation leads to an overprediction of the reattachment length from those observed in the literature. With a block of length 2 h, Moss and Baker (1980) measured the reattaching point at a distance of 12 h downstream from the front face. Mohsen (1967) measured similar separation length downstream from a fence (around 11–12 h). The simulated profiles for the mean streamwise velocity U1 and the root-meansquare (rms) velocities u 1 and u 3 , respectively, in the streamwise direction and in the direction normal to the wall are compared with experimental data at about 42 h downstream from the ruler (Figure 6.30). The experimental mean velocity profile is always strongly perturbed by the fence, which induced a mean velocity drop and a strong turbulent layer increase. The numerical mean velocity profile is less disturbed 10 Figure 6.29. Streamwise velocity fluctuations urms /U0 near the ruler in the median plane (levels between 0 and 0.3; black line: isocontour of 0.25).

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Figure 6.30. Velocity profiles at the origin of the reference: (—-) measurement; (+ + +) simulation. (a) Mean streamwise velocity U1 , (b) rms streamwise velocity u1 , (c) rms normal velocity u3 .

by the fence. The nondimensional streamwise mean velocity U1 /U0 and rms velocity u 1 /U0 are overestimated by the computation in the turbulent boundary layer. Thus, the boundary layer thickness calculated is underestimated: the momentum thickness is around 0.26 h for 0.65 h measured. Mean pressure and rms levels

The mean pressure distribution downstream from the ruler is similar to the observation of Mohsen (1967) and downstream fences in terms of absolute levels and trends (Figure 6.31). The minimum level appears in the separation bubble region, and the maximum just downstream from the reattachment point. The extent of the region with rapid pressure rise is longer for this computation (about 10 h) than in Mohsen’s (1967) observation (8 fence heights). At about x1 = 18 h, the maximum rms pressure level is observed about 0.13 times the dynamic pressure q (Figure 6.32). This value is greater than the observation of Mohsen (1967), who found a factor of 0.06 for a 51.8 m/s velocity downstream from a fence. The comparisons with our experimental data show a good agreement for the

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Cp

0.25 0

Figure 6.31. Pressure coefficient.

0

20

40 60 x /h

80

100

1

points situated in the middle of the flat plate (Figure 6.33). For the two upstream points, the level is overestimated, and this is consistent with the overpredicted reattachment length already observed. 6.4.6.2 Wall-pressure fluctuations

In the classical random vibration theory, the turbulent loading function of structures subject to low-speed flow is described by the space–time cross correlation between two points separated by ξ = (ξ1 , ξ2 ) or its time and spatial Fourier transform, the wavevector frequency spectrum. Rather than study directly the cross spectrum, we can use the following expression: S pp (x, ξ, f) = pp (x, f)γ (x, ξ, f)e(−i2πfξ/Up (x,ξ,f)) ,

(6.31)

where r pp (x, f) is the autospectrum of the fluctuating wall pressures, r U p (x, ξ, f) is the turbulent eddy convection velocity in the nominal flow direction,

and

r γ 2 (x, ξ, f) is the coherence function (Bendat and Piersol 1980), which is sometimes

called the coherency-squared function. The shape and properties of these three functions of space and frequency are important for the computation of the dynamic response of a panel subjected to a similar 0.15 0.125 Figure 6.32. Wall-pressure-fluctuation rms levels calculated at various positions along the median plane.

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prms / q

35 31

27

25

Figure 6.33. Comparison of calculated (· · · ) and measured (+ + +) wall-pressure fluctuation rms levels on the plate.

0.05

0 20

39 30

40 x /h

50

60

1

excitation. In this study, we use the same formalism to validate our computations: the autospectrum is calculated at different positions in the flow, the coherence functions are evaluated for streamwise and spanwise point pairs, and the phase velocity is computed along the main flow direction. Power spectral density (PSD)

As shown in Figures 6.34(a) and 6.34(b), the wall-pressure fluctuation PSD simulated on the flat plate are in good agreement with experimental results obtained for points located downstream far enough from the ruler (points 25 and 40 in Figure 6.23). The discrepancies appear for frequencies higher than 500 Hz; levels become underpredicted (about 15 dB at 1000 Hz), corresponding to a cutoff effect of the model. Some oscillations on computated spectra are mainly caused by the limited number of available blocks (only 21 blocks of 512 points) used to perform spectrum averaging. For points closer to the ruler (points 31 and 35 in Figure 6.23), results show important discrepancies as long as the location goes up (Figures 6.34(c) and (d)): the power spectral density (PSD) intensities are overvalued for low frequencies ( f < 500 Hz). In this region close to the reattachment point, the rms pressure level is still decreasing quickly, as shown in Figure 6.32. Therefore, the result comparison is quite sensitive to the prediction of the recirculation length. In conclusion, far enough from the reattachment zone, the simulation correctly the predicts the low-frequency structures, and thus the wall-pressure power spectral densities, up to a frequency around 500–700 Hz. Cross-spectral densities (CSD)

The coherence γ 2 is compared at different positions in the flow for streamwise and spanwise transducer pairs with ξ separation. The phase velocity Up is only measured along the main flow direction. For all experimental and numerical comparisons, the computed coherence in the streamwise direction is overrated for high frequencies (Figure 6.35) for which as already observed in the section on PSD, the limits of computation are reached: in fact, the wall-pressure PSD showed that there is no excitation at high frequencies (predicted pressure levels are 20 dB below for f > 700 Hz). For lower frequencies, the computed coherence is also overvalued for the smaller separation distance (Figure 6.35(a): ξ = 0.011 m) but underestimated when the distance increases.

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r ef

(f) / P2 ]

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600 f (Hz) Channel 31

ref

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ref

(f) / P2 ]

10 log10 [

pp

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Figure 6.34. PSD of wall-pressure fluctuations measured on the flat plate (point locations in Figure 6.23): (—-) measurement; (- - -) simulation (Pref = 2 × 10−5 Pa).

(f) / P2 ]

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Figure 6.35. Coherence γ 2 of wall-pressure fluctuations measured on the flat plate for streamwise points using the same notation as in Figure 6.34. (a) ξ1 = 0.011 m, (b) ξ1 = 0.040 m, (c) ξ1 = 0.109 m.

As mentioned in Section 6.4.4.2, wall-pressure fluctuations are measured on a rigid test plate. Noise due to plate vibration is not taken into account; an experimental analysis done by comparing a rigid and an elastic plate case at downstream locations shows more important pressure fluctuations and a coherence drop for frequencies below 200 Hz (Leclercq 1999) for the elastic case as against the rigid one. This is in good agreement with the numerical observations, which undervalued the experimental coherence for the greatest separation distance (Figure 6.35(c)). The comparisons of spanwise coherence are similar with those in the streamwise direction (Figure 6.36). Furthermore, coherence variation versus separation distance is correctly reproduced by the model. The computed phase velocity (established from cross-spectrum phase of Bendat and Piersol 1980) is in good agreement with the experimental results (Figure 6.37); the

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Figure 6.36. Coherence γ 2 of wall-pressure fluctuations measured on the flat plate for spanwise points using the same notation as in Figure 6.34. (a) ξ2 = 0.010 m, (b) ξ2 = 0.030 m, (c) ξ2 = 0.078 m.

error is lower than 10%. The convection velocity is calculated correctly by the explicit integration scheme together with the nondiffusive SUPG spatial numerical scheme. 6.4.6.3 Vibroacoustics response

The plate response to aeroelastic loading is owed, on the one hand, to the intensity and to the spatial distribution of the stress, and, on the other hand, to the intrinsic characteristics of the structure – that is, as represented by its modal base. For the lower frequencies ( f < 1000 Hz), the modal density is weak and the spatial coupling between modal shapes and the wall-pressure loading (coherence, convection speed) determines the plate response. The comparisons of numerical and experimental acceleration PSD

aa show a good agreement (Figure 6.38). The frequency shift is estimated to be due to the parametric adjustment of the plate eigenfrequencies, which are controlled by

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Figure 6.37. Phase velocity Up of wall-pressure fluctuations measured on the flat plate for streamwise points using the same notation as in Figure 6.34. (a) ξ1 = 0.011 m, (b) ξ1 = 0.011 m, (c) ξ1 = 0.040 m.

structural boundary conditions (i.e., spring parameters). For frequencies greater than 700 Hz, the computed PSD level is underestimated (about 10 dB); again the limits of the simulation are reached. For acoustic pressure p in the cavity, results are quite similar to those found for the plate acceleration a (Figure 6.39). In conclusion, vibroacoustic quantities are correctly modeled in the frequency range below 700 Hz. For frequencies above 700 Hz, the underprediction of numerical results increases with increasing frequency.

6.4.7 Mesh influence

Unfortunately, present computer performance limitations do not allow use of a refined mesh everywhere in the computational domain. Despite ever-increasing computational performance, it is unlikely that this situation will significantly change in the near future.

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Channel 26 2 10log10[Φaa(f) / aref]

140 120

Figure 6.38. Acceleration PSD aa – channel 26 (aref = 1. × 10−6 m·s−2 ): (—–) measurement; (- - - -) simulation.

100 80 60

0

250

500 f (Hz)

750

1000

Models able to compute acoustic frequencies up to 5000 Hz around a side mirror and window with simulations long enough to record 20 samples will feature over 10 million nodes and need to be computed for about 1 million time steps. Conducting such a simulation in a reasonable engineering turnaround time requires access to highly powerful supercomputers featuring hundreds of processors. For the case studied here, the numerical model initially ran for 19 days on Fujitsu VPP5000 (one processor). The time was reduced to 55 hours by using a CRAY SX6 (eight processors), and more recently to 37 hours on thirty-two Intel IA64 processors. The CPU time is getting reasonable, but still requires large computing servers. Therefore, a careful mesh design phase must be carried out to make sure that the CPU time is kept under control; meanwhile the quality of the results is satisfactory. The general idea is to alternate as fast as possible between a “noise generation” fine mesh criteria to a “noise propagation” coarser one. This is achieved by making sure that the highly turbulent zones are meshed with the fine criteria. Several techniques can be used to do this, among which is the examination of experimental or existing numerical RANS visualizations of the flow. In this case, an initial mesh has been built and a careful examination of the flow results led to the development of a second mesh, with the same number of elements but a better spatial repartition: r The first one was achieved by using a block-structured technique, which allows

refinement close to the obstacle (where the mesh size is 0.5 × 5.2 × 0.5 mm3 ) Channel 30 100

Figure 6.39. Acoustic pressure PSD pp – channel 30 (Pref = 2. × 10−5 Pa) using the same notation as in Figure 6.38.

2 10log10[Φpp(f) / pref]

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0.1 0.09 0.08 0.07

Z

0.06 0.05 0.04 0.03 0.02 0.01 0

0.05

0.1

0.15

X

0.2

Figure 6.40. First mesh: detailed mesh around the obstacle.

and coarsening in the far field to reach the value of 10.0 × 5.2 × 6.6 mm3 (Figure 6.40). The total number of elements for this mesh is 974,752 hexahedral elements, 1710 quad shells, and 1,017,424 nodes. This simulation was designed to be accurate up to a frequency of about 750 Hz. r The second one (see Section 6.4.5.1) was, in particular, refined near the flat plate in order to better capture the smaller structures generated by the obstacle, which propagate above the plate. This mesh was supposed to solve turbulent structures up to 3200 Hz in the streamwise direction but only up to around 615 Hz in the spanwise direction.

10 log 0 [ φ (f) / P2 ] 1 pp ref

The comparison of wall-pressure fluctuation spectra shows some gain above 500 Hz (Figures 6.41, 6.42) when the mesh is refined in the streamwise direction (second model), but the improvement is not as good as expected: the spanwise mesh size probably also needs to be refined to really improve the upper frequency limit of the model. Channel 25

100 90

m

80 s2

70 60 50

s1 200

400 600 f (Hz)

800

1000

Figure 6.41. Comparison of the two simulations – wall-pressure fluctuations DSP on the plate: m = measurement (—), s2 (– – –), s1 (–. –. –).

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Channel 30 100

2

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75 50 25 0

0

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500 f (Hz)

750

6.4.8 Conclusions for aeroelastic noise prediction

In this section, we have demonstrated the capability of the Radioss CFD–CAA code to predict aeroelastic and vibroacoustic phenomena due to turbulence in the wake of a ruler. This approach exhibits accurate predictions below approximately 700 Hz. For higher frequencies, the levels are underestimated. A refined mesh in the streamwise flow provides somewhat better results, but the poor spanwise resolution of the model is believed to limit the development of structures smaller than 0.05 m (corresponding to frequencies higher than 500 Hz). Another research track is to evaluate the dissipation of wall functions. The fully coupled numerical approach presented here provides better results than those obtained by a hybrid method combining modal analysis for the plate and the cavity with a semiempirical model of the wall-pressure loading, which overestimated the acceleration and acoustic radiation of the plate above 700 Hz. The main advantage of a compressible ALE–CFD code like Radioss CFD–CAA is its ability to solve flow field; noise, source generation; propagation, including model behaviors; and radiation and transmission to structures in a single simulation. 6.4.9 Acknowledgment

The authors would like to acknowledge Dominik Obrist of CRAY Computer Inc. and Ian Godfrey of FUJITSU Systems for their active support throughout the study.

6.5 Trailing-edge noise Roland Ewert and Eric Manoha 6.5.1 Introduction to trailing-edge noise simulation using LES

The applications described in Sections 6.5.2 and 6.5.3 address the general context of the numerical prediction of the aerodynamic noise generated by aircraft in the vicinity

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of urban airports. It is known that, for the largest existing and future aircraft, the noise radiated in the approach phase is evenly generated by the engines (working at reduced thrust) and the airframe. Airframe noise sources are known to be mainly located in landing gears and in high-lift devices, slats, and flaps, which are deployed during the approach. Among all the complex noise-source mechanisms associated with high-lift devices, trailing-edge noise has probably received the most extensive attention (mostly theoretical and experimental) and is a perfect candidate for the evaluation of the emerging CAA techniques of numerical simulation based on LES. The trailing-edge noise mechanism is associated with the acoustic scattering of the airfoil turbulent boundary layer convected at the trailing edge. In this process, the quadrupolar (acoustically inefficient) nature of the convected eddies is transformed into dipolar sources, which are much more acoustically efficient. The numerical simulation of the complete problem, including the local noise-source simulation and the far-field noise radiation, faces the difficulty of handling fluctuations covering a very extended range of length scales and amplitudes: turbulent eddies that generate noise have small lengh scales but high energy, whereas the acoustic waves radiated away have comparatively very long wavelengths but small amplitude. Thus, trailing-edge noise simulation is still beyond the capabilities of complete DNS, and hybrid methods are used in most practical cases. Figure 6.43 sketches the possible numerical strategies, showing how the near-field turbulent flow and the far-field noise are computed separately. The idea is to divide the physical space into several domains in which specific physical mechanisms are simulated by using the most adequate set of equations with the most economic discretization strategy. Computational fluid dynamic techniques are used to simulate the near-field flow, which contains the aerodynamic noise sources. Available techniques include steady RANS computations, in conjunction with stochastic models of the wavenumber–frequency spectrum of the turbulence (B´echara et al. 1994; Bailly, Lafon, and Candel 1995; Kalitzin, Kalitzin, and Wilde 2000; Billson, Eriksson, and Davidson 2003), unsteady RANS methods (Singer et al. 1999; Khorrami, Singer, and Berkman 2002), LES (Seror et al. 1999; Wang and Moin 2000; Piomelli 2001; Boudet, Casalino, Jacob, and Ferrand 2003; Boudet, Grosjean, and Jacob 2003; Manoha, Delahay, Sagaut et al. 2001; Manoha et al. 2002; Ewert, Meinke, and Schr¨oder 2002; Ewert, Schr¨oder, Meinke, and El-Askary 2002; Ewert et al. 2003; Sorg¨uven, Magagnato, and Gabi 2003), and nonlinear disturbance equations (Chyczewski, Morris, and Long 2000a; Terracol et al. 2003). This local flow solution has to be coupled to an acoustic numerical technique for the prediction of far-field noise. The most practical formulations are the integral methods such as the Ffowcs Williams–Hawkings equation (Ffowcs Williams and Hawkings 1969), the boundary element method (BEM) (Manoha et al. 1999), and the Kirchhoff integral (Prieur and Rahier 1998; Rahier and Prieur 1997). Integral methods assume that, beyond a given distance from the noise sources and body surfaces, the sound propagates in a medium at rest or moving with uniform velocity. This assumption becomes a strong limitation when the radiated noise results from a surface integration or a control interface located near solid walls, where

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Local turbulence prediction: Unsteady CFD (DNS, LES, unsteady RANS) Steady CFD (RANS) + stochastic model

Flow

Airfoil Turbulent flow Low-turbulence Nonuniform mean flow

Outer region with uniform flow Integral methods: - Acoustic analogy - Kirchhoff integral - BEM

Acoustic propagation through inhomogeneous flow using any modified formulation of linearized Euler equations

Far-field observation point Figure 6.43. Numerical simulation of airfoil aerodynamic noise: possible hybrid strategies.

velocity gradients are significant. In that case, only the (linearized or nonlinear) Euler equations in perturbation form (and derivatives of these equations) governing the acoustic propagation may account for the propagation in nonhomogeneous flows. This is obtained at the price of a significant computational effort because the propagation domain must be meshed separately with an adequate resolution with respect to the smallest acoustic wavelength and also because finite difference, high-order schemes are needed to ensure numerical accuracy and low dispersion of acoustic wave propagation (Tam and Webb 1993). Moreover, applications to realistic geometries, including airfoils, need three-dimensional curvilinear grids (Grogger et al. 2000) on which the use of high-order schemes is not straightforward. However, the domain in which Euler equations is used can be strictly limited to regions in which velocity gradients are significant: in most practical airframe noise problems, an external boundary can be found beyond which the flow can be assumed uniform, and thus integral methods can be used for the noise prediction at very long distances from the airfoil. The critical point is the coupling technique between the LES and the Euler domain. Two different techniques are described in this section, and both are applied to the simulation of trailing-edge noise. The first coupling technique, developed in Section 6.5.2, is based on a volume right-hand-side source term for linear disturbance equations determined from the unsteady flow data provided by the LES. In this case, the “acoustic” grid

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used for the acoustic simulation can be completely dissociated from the “aerodynamic grid” used for the LES, the results of which are used as a local-volume source term for acoustic propagation. This method has been followed by several authors (B´echara et al. 1994; Bailly et al. 1995; Billson, Eriksson, and Davidson 2002; Ewert, Meinke, and Schr¨oder 2000, 2001a), often using different formulations of the source term. Section 6.5.3 decribes a more straightforward technique based on direct injection, without additional filter or source-term reconstruction, of LES perturbation data (obtained by subtracting the time-averaged mean flow) at the border of the Euler domain. This process relies on the compressibility of LES and its ability to simulate the local acoustic field correctly (Manoha, Delahay, Sagaut et al. 2001; Manoha et al. 2002). 6.5.2 Trailing-edge noise simulation using LES and APE 6.5.2.1 Hybrid simulation approach

This section concerns a hybrid acoustic simulation approach to airframe noise utilizing LES to resolve the unsteady compressible flow problem in the hydrodynamic near field and subsequently applying an acoustic analogy based on linear acoustic perturbation equations (APE) to determine the related sound radiation toward the far field. Computational results of the considered hybrid simulation concept are presented for trailing-edge noise generated at a flat plate at Mach number M = 0.15, Reynolds number Re = 7 × 105 , and an airfoil at M = 0.088 and Re = 8 × 105 . Figure 6.44 sketches the approach for the plate problem. Separating the analyses of the flow field and the acoustic field in general makes it possible to take advantage of the disparity of the turbulent and acoustic length scales at low Mach numbers, where the latter scale with M −1 compared with the former. For the trailing-edge noise problem, the LES is limited to a subdomain close to the trailing edge. Hence, 2 × 106 mesh points are sufficient for the resolution of the Re = 7 × 105 problem. The acoustic domain has a substantially larger extension compared with the LES domain owing to the increased grid spacing allowed for the acoustic simulation and comprises, apart from the LES region, the remaining geometry not resolved in the unsteady flow simulation. The acoustic simulation based on APE considers mean-flow convection and refraction effects of a nonuniform mean flow such that the computational domain of the LES has to comprise only the significant acoustic source region in the immediate vicinity of the trailing edge. The nonuniform mean flow in the acoustic domain in general is provided by a steady-state RANS solution. The linearized acoustic perturbation equations can be considered modifications of the linearized Euler equations (LEE), which completely prevent the unbounded growth of hydrodynamic instabilities in critical mean flows. Because the LEE describe the propagation of acoustic, vorticity, and entropy waves, unstable solutions of the LEE in a globally unstable mean flow occur. If the LEE are excited by additional sources, the instabilities will cause divergent solutions that are limited neither by nonlinear saturation nor by viscous damping (Ewert et al. 2001a). On the basis of APE that encode a convected wave equation for nonuniform mean flows, an acoustic analogy is derived from the full governing-fluid equations

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Figure 6.44. Sketch of the computational domains to determine trailing-edge noise with the hybrid approach. The LES domain encompasses the vicinity of the trailing edge, whereas the computational aeroacoustics (CAA) domain includes the whole airfoil owing to the less stringent demands concerning the grid resolution (xacoustic ∼ xLES /M, M = O(10−1 )). Thus, scattering at the leading edge can be captured and directivities can be predicted.

that permits the coupling of the flow and acoustic simulation via appropriate source terms. The source reconstruction from the compressible flow simulation involves only simple numerical operations that allow a straightforward computation. For the discretization of the acoustic perturbation equations, numerical methods developed in the framework of CAA are applied. Section 6.5.2.2 discusses the methods used in the acoustic simulation step in their application to the trailing-edge noise problems. Section 6.5.2.3 presents the simulation results for the flat plate at M = 0.15 and the transonic airfoil at M = 0.088. 6.5.2.2 Acoustic simulation step Acoustic perturbation equations (APE)

The acoustic perturbation equations (APE), which have been proposed in Ewert and Schr¨oder (2003) – see also Section 5.4 – constitute an equation system for the pressure and velocity perturbations ( p  , u  )T :   p ∂ p 2  (6.32) + c0 ∇ · ρ0 u + 2 u 0 = c02 qc , ∂t c0    p (6.33) = q m . ∂ u  t + ∇ u 0 · u  + ∇ ρ0 The subscript 0 denotes mean-flow quantities (i.e., ρ0 , p0 , and u 0 are the density, pressure, and the velocity, respectively, of the time-averaged flow). Furthermore,

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√ c0 = γ p0 /ρ0 is the sonic speed with γ the isentropic exponent. The right-hand-side sources qc and q m will be defined in the discussion of the APE-based acoustic analogy. The homogeneous equation system with the sources qc = 0, q m = 0 can be shown to be equivalent to the convective wave equation for irrotational mean flows. To obtain the wave operator the perturbation velocity u  is split into an irrotational plus a remaining part that contains all the vorticity: u  = ∇ϕ + u r .

(6.34)

Because u r is not solenoidal, the decomposition becomes uniquely defined after the additional condition is imposed that the unsteady pressure be expressed only in terms of the unsteady acoustic potential ϕ by p  = −ρ0

D0 ϕ , Dt

(6.35)

with the substantial time derivative D0 /Dt = ∂/∂t + u 0 · ∇. Introduction of Equations (6.34) and (6.35) into Equations (6.32) and (6.33) yields the equivalent system for the variables (ϕ, u r ):     1 ∇ · (ρ0 u r ) D0 1 D0 (ρ − ∇ · ∇) ϕ = , (6.36) Lϕ = 0 Dt c02 Dt ρ0 ρ0

∂ u r t + ∇ (u 0 · u r ) = 0.

(6.37)

The left-hand side of Equation (6.36) is the convective wave operator for an acoustic potential ϕ. It is Pierce’s approximate wave equation (Pierce 1990), which was also derived by Goldstein (1978) and recently used in Golubev and Atassi (1998) and Cooper and Peake (2001). The wave operator is equivalent to the linearized wave operator of M¨ohring’s acoustic analogy (M¨ohring 1979, 1999), which reads    qtot D0 1 D0 B  1  , (6.38) − ∇ · ρ0 ∇B  = − LB = 2 Dt c0 Dt ρ0 ρ0 where B  denotes the perturbations of the total enthalpy, which is linked, with the pressure, to linear order via D0 B  /Dt = 1/ρ0 ∂ p  /∂t with thermal conductivity and viscous effects neglected. As discussed by Howe (1998), for homentropic, high-Reynoldsnumber flows, Howe’s acoustic analogy (Howe 1975) agrees with Equation (6.38). M¨ohring (1999) showed that the wave operator L is stable for arbitrary mean flows. Unlike the LEE, the entropy mode is a priori completely removed in the APE system and the behavior of the vortical perturbations is governed by Equation (6.37). Taking the curl of Equation (6.37) yields the vorticity equation ∂ω

r t = 0;

(6.39)

that is, the vorticity convection property of the LEE is also removed. Because vortical Equation (6.37) does not support growing instabilities and Equation (6.36) is stable, the equivalent APE system does not possess instabilities. This is an important feature if linear perturbations are forced with sources, because growing instabilities will cause

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divergent solutions that are limited by neither nonlinear saturation nor viscous effects. Instabilities appear in the LEE for unstable mean flows. Although the homogeneous APE system describes wave propagation properly in irrotational mean flows, even in mean flows with mild levels of mean vorticity the homogeneous perturbation equation can resolve convection effects with good accuracy, as was demonstrated in Ewert and Schr¨oder (2003), who, to quantify these convection errors, considered the propagation of acoustic waves of frequency f through a shear layer of thickness 2δ and velocity jump ±u 0 . Hence, in the shear layer a mean vorticity of magnitude ω0  u 0 /δ is apparent. Then the convection error is a function of a Strouhal number defined by the mean vorticity ω0 and acoustic frequency f (i.e., St = f /ω0 ). The irrotational case is defined by the high-frequency limit of the Strouhal number tending to infinity (i.e., St = f /ω0 → ∞). However, it was demonstrated that the wave operator still resolves refraction effects for Strouhal numbers of order O(1) properly (see also the discussion in Section 6.5.2.3 and Figure 6.60).

APE-based acoustic analogy

An acoustic analogy with dependent variables p  and u  can be found that recasts the Navier–Stokes equations in nonlinear disturbance formulation such that the left-hand side equals the APE system. The remaining terms lumped together on the right-hand side constitute the sources (Ewert and Schr¨oder 2003, 2004). The right-hand side sources become   ρ 0 D 0 s  , qc = −∇ · ρ  u  + c p Dt

 + T  ∇s0 − s  ∇T0 q m = − (ω × u)    (u  )2 ∇·τ − ∇ + , 2 ρ

(6.40)

(6.41)

where (. . .) := (. . .) − (. . .) denotes perturbations of terms obtained by subtracting from an actual term its time average, indicated by an overline. A major vortex source term is the perturbed Lamb vector L = ω × u , that is,  

 = −ω 0 × u  − ω  × u 0 − ω  × u  . q m = − L  = − (ω × u)

(6.42)

A similar vortex source term appears in the acoustic analogies of Powell (1964), Howe (1975), and M¨ohring (1979). It can also be easily computed from a compressible unsteady flow simulation. The extension of the acoustic analogy concept from a scalar wave equation to a set of linearized inhomogeneous equations was recently also discussed by Goldstein (2003).

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Ehrenfried, Meyer, and Dillmann (2003) integrated M¨ohring’s (1999) acoustic analogy equation (6.38), numerically using the system ∂ B  t + u 0 ∇B  +

c02 c2 ∇·w

 = 0 qs , ρ0 ρ0

(6.43)

∂w



 + ρ0 ∇B  = −(ρ L) + u 0 ∇ · w ∂t

(6.44)

with w

 = (ρ u ) . For more detailed information about the right-hand-side sources, refer to Ehrenfried et al. (2003). Therefore, Equations (6.32) and (6.33) and Equations (6.43) and (6.44) constitute two formally different sets of perturbation equations for the same wave operator equation (6.38). As was discussed by M¨ohring (1999), the wave operator L of Equation (6.38) is self-adjoint. However, the APE can be shown not to have the self-adjoint property. Hence, the adjoint equations differ formally from the APE but must also encode the self-adjoint wave operator L of Equation (6.38). In what follows

 to the it is demonstrated that, by appropriately renaming the variables from B  and w adjoint variables of the APE, for instance ( pa , u a )T , the formal system equations (6.43) and (6.44) give the adjoint equations to the APE, and vice versa. Furthermore, as was discussed in Ewert, Schr¨oder et al. (2002), the system equations (6.32) and (6.33) can also be used to integrate M¨ohring’s (1999) acoustic analogy equation (6.38). However, unlike Equations (6.43) and (6.44), the system equations (6.32) and (6.33) are based on the usual primitive variables ( p  , u  )T and hence immediately give exact solution to the most important acoustic variable in the complete acoustic domain p  . Computational methods

The computational problem to solve for perturbation variables U = (ρ  , u  , p  )T over a mean flow field (ρ0 , u 0 , p0 )T looks formally like ∂ U ∂ U

+ Ai + H U = S. ∂t ∂ xi

(6.45)

In Equation (6.45), Einstein’s summation condition holds (i.e., products of terms with equal indices are summed over all spatial directions), and a source vector S appears on the right-hand side. For the acoustic perturbation equations, Equations (6.32) and (6.33), the matrices and source terms become, in 2D, ⎛ ⎞ ⎛ ⎞ u 0i qc ρ0 δ1i ρ0 δ2i 0 ⎜0 ⎜ ⎟ u 01 δ1i δ1i /ρ0 ⎟ u 01 δ1i ⎟ , S = ⎜ qm1 ⎟ , Ai = ⎜ (6.46) ⎝0 ⎠ ⎝ u 01 δ2i u 02 δ2i δ2i /ρ0 qm2 ⎠ 0

γ p0 δ1i ⎛ ∂u 0i

∂ xi

⎜ ⎜ 0 ⎜ H =⎜ ⎜ 0 ⎜ ⎝ 0

γ p0 δ2i

u 0i

∂ρ0 ∂ x1 ∂u 01 ∂ x1

∂ρ0 ∂ x2 ∂u 02 ∂ x1

∂u 01 ∂ x2

∂u 02 ∂ x2

0 c02 ∂ρ ∂ x1

0 c02 ∂ρ ∂ x2

⎞

0

c02 qc

⎟ 0 ⎟ − ρ12 ∂ρ ∂ x1 ⎟ 0 ⎟, 0 ⎟ − ρ12 ∂ρ ∂ x ⎟ 2 0 ⎠

c02 ∂∂xi

u 0i c02

(6.47)

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Figure 6.45. Coordinate system and nomenclature used to determine corrections for a 2D acoustic simulation; R denotes a 2D polar radius vector in the x1 –x2 plane, ζ R = x3 the spanwise coordinate of the 3D radius vector r = | r |, and r = (x1 , x2 , x3 ) = (R cos θ, R sin θ, ζ R)T .

where u 0i denotes the ith component of the mean flow u 0 and δi j is the Kronecker symbol with δi j = 0 for i = j and δi j = 1 otherwise. The components of the source vector S are defined by Equations (6.40) and (6.41), where qmi denotes the ith component of q m . Note that the governing equation for the density is decoupled from the remaining part such that the first equation could be dropped. The spatial discretization is accomplished with the fourth-order DRP scheme of Tam and Webb (1993). For use on curvilinear multiblock grids, Equation (6.45) is extended with a metric, where the metric terms are also computed consistently with the DRP scheme. At far-field boundaries just the asymptotic radiation boundary condition of Tam and Webb (1993) is used because, even at outflow boundaries, no nonacoustic convection modes are apparent. The solid-wall boundary condition is implemented based on a ghost point beneath the surface that yields the proper surface-wall normal pressure gradient (Tam and Dong 1994). For the suppression of high-frequency spurious waves, artificial selective damping (according to Tam and Dong 1993) has been used. The time integration is carried out with the fourth-order, alternating, two-step, low-dissipation, and low-dispersion Runge–Kutta scheme (LDDRK5–6) proposed by Hu et al. (1996).

Determination of volume sources

The volume sources are computed from Equation (6.42) expressing the velocity fluctuations and the fluctuating vorticity through the LES solution. The acoustic simulations are carried out in 2D using only the x1 and x2 components of the source term. As was discussed in Manoha et al. (1999), Oberai, Roknaldin, and Hughes (2002), Casper and Farassat (2003), and Ewert et al. (2003), for cylindrical geometries of constant 2D cross sections and if one considers observer positions just in the spanwise, symmetric x1 –x2 plane (see Figure 6.45), the 3D problem can be reduced to an equivalent 2D problem. According to the findings of Manoha et al. (1999), Casper and Farassat (2003), and

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Oberai et al. (2002), only the zeroth spanwise wave number of the spanwise, Fouriertransformed source needs to be considered. Oberai et al. (2002) and Ewert et al. (2003) used a reduced 2D source in a pure 2D simulation to correct the sound-pressure levels later from 2D (decay ∝ R −1/2 ) to 3D (decay ∝ R −1 ). Casper and Farassat (2003) utilized only the zeroth spanwise wave-number component of the source for a 3D acoustic simulation. The latter approach allows the spanwise acoustic grid resolution to be determined by the acoustic rather than the turbulent length scale; hence, the full wing span L is resolvable in the acoustic simulation step. This avoids, furthermore, the necessity of an additional 2D-to-3D sound-pressure-level correction. However, note that, in general, for the flow and acoustic problem, respectively, different demands concerning the spanwise grid resolution and extension exist that are naturally taken into account with a hybrid prediction method. In particular, consider the trailing-edge problem resolved with a direct method that combines both the LES and the acoustic simulations. In the LES the spanwise extension of the computational domain is typically only on the order of the turbulent length scale, and periodic boundary conditions are applied to reduce the computational cost to an acceptable level. Because in low-Mach-number flows the turbulent length scale is significantly smaller than the acoustical length scale, applying periodic boundary conditions for the compressible problem means to simulate an unphysically correlated acoustic source in the spanwise direction, which yields radiated acoustic waves that decay ∝ R −1/2 . Physically, the waves generated by one LES slice of spanwise width  decay ∝ R −1 . The spanwise, unphysically correlated acoustic signal was observed by Manoha, Delahay, Sagaut et al. (2001) by analyzing the acoustic signal contained in the LES of a trailing-edge flow. The correlated source causes an overprediction of the sound-pressure level (SPL). Kato et al. (1993) stated the SPL to be about 14 dB too high. This issue was also observed by Wang and Moin (2000). To recover the proper physical decay, one could use an acoustic computation with absorbing boundaries in the spanwise direction. However, a computational acoustic domain of small spanwise width yields acoustic waves that leave the sidewise boundary surfaces at very shallow angles. Most nonreflecting boundary conditions will not work properly under these conditions (i.e., the reflection causes a false decay law of the sound waves). On the other hand, if the full wingspan L is considered, the spanwise resolution limited by the flow length scale restricts the computational problem because of high memory requirements. Because an integral length scale of the acoustic source in the spanwise direction is small compared with the acoustic wavelength for small Mach numbers, the source is compact in the spanwise direction; hence, an acoustic simulation with a spanwise extension  and periodic boundary conditions is equivalent to a 2D acoustic simulation with a spanwise-averaged acoustic source  1 /2  q d x3 . (6.48) qˆ =  −/2 Here, qˆ indicates a component of the spanwise-averaged source, and  q is a component of the source from the full 3D LES simulation. If the source equation (6.48) is used

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in a 2D acoustic simulation, the sound-pressure correction from 2D to 3D becomes, according to Ewert et al. (2003),  1 − i k2  ˆ . (6.49) p (0, R, θ, )  p(R, θ, ) 2 πR ˆ In Equation (6.49),  p (ζ R, R, θ, ) denotes the 3D, and p(R, θ, ) the 2D, frequency-related Fourier transform of the sound pressure. Furthermore,  = 2π f √ is the angular frequency, k = /c0 the wave number, and i = −1. Hence, the correction is frequency dependent and only applies in the x1 –x2 plane, where ζ R = 0 (see Figure 6.45, which also defines all other appearing quantities). The result agrees with the findings of Oberai et al. (2002) that were derived for a vanishing mean flow. In addition, the derivation of Equation (6.49) also incorporates the convection effect of a uniform low-Mach-number mean flow. The correction does affect the final spectral SPL distribution; however, it has no impact on the θ -dependent directivity, obtained from a 2D simulation. Furthermore, the mean-flow amplification remains the same. In other words, the 2D simulation delivers directivities and Doppler mean-flow amplifications that are also valid in the x1 –x2 plane for the 3D case. From Equation (6.49) the SPL correction can be deduced to be   f 2 S P L 3D, = S P L 2D + 10 log . (6.50) Rc0 Apparently the 2D sound-pressure spectrum is shifted 3 dB/octave toward the higher frequencies by the correction. The 3D-corrected pressure S P L 3D, of Equation (6.50) is the sound radiated by one slice of width . For a finite spanwise extension L (Figure 6.45), an additional correction   L (6.51) S P L 3D,L = S P L 3D, + 10 log  has to be used. This correction was given by Kato et al. (1993) and is based on the assumption that all slices of width  along the wingspan are uncorrelated, whereas the spanwise extension L is small compared with R. The correction also only affects the final spectral SPL distribution but leaves the directivity unchanged. On the basis of Taylor’s hypothesis of convecting vorticity, the nondimensionalizing of Equation (6.49) yields, with k = /c0 ,  ∝ u 0 /λv , where λv is the length scale of a vortical disturbance, the scaling  ˆ p ∝ M 1/2 p. In other words, the correction Equation (6.49) serves also as the correction of the powerscaling law (i.e., the sound intensities in 3D scale with the power of the Mach number increased by one compared with the 2D case). In general the computational domain of the LES is a small subdomain inside the computational domain of the acoustic simulation (see Figure 6.44). Because the vanishing and generation of vorticity are always related to a strong noise-source mechanism,

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1

x2 u0

W 1(x1)

0.75

LES subdomain

0.5

l1

x1c

l2

0.25 0

x1

l1

Plate

x1

l2

Figure 6.46. LES subdomain at the trailing edge (left) and horizontal weighting function (right).

the simple truncation of the vorticity-source-term Equation (6.42) at the boundaries of the LES domain in the mean-flow direction will cause spurious noise sources in the acoustic simulation. The spurious sound generated owing to the termination of the source of Lighthill’s (1952) acoustic analogy was discussed by Wang, Lele, and Moin (1996) for a vortex-shedding airfoil. The problem was also addressed by Crighton (1991, 1993). Furthermore, the same difficulty was observed by Mankbadi et al. (1994) and Mitchell et al. (1999) in jet noise computations. Mitchell et al. (1999) used slowly decaying Lighthill source terms at the outflow boundaries to damp the spurious sound source. For the APE source-term equation (6.42) the spurious noise-source mechanism was discussed by Ewert, Meinke, and Schr¨oder (2002). It was shown that the spurious noise sources can be avoided by smoothly switching the sources on and off inside the resolved source region in the mean-flow convection direction. Because the vortical structures are floating for the flat-plate problem mainly in the x1 -direction parallel to the plate, this is accomplished applying a spatial weighting function W1 (x1 ) in that direction according to Figure 6.46. Ewert, Meinke, and Schr¨oder (2002) demonstrated that a low-pass behavior for spurious sound is obtained, which is a function of the nondimensional wave number αd = 2π d/λv , where, according to Figure 6.46, d = (l2 − l1 )/2 is the width of the on- and offset zone and λv is the wavelength of a vortical disturbance (related to an acoustic wavelength of same frequency via λv = M∞ λ). Figure 6.47 depicts the lowpass behavior for a Hann window-weighting function. As a rule of thumb, d should be chosen larger than the largest vortical structure λv that is related to the lowest resolved frequency via f = u c /λv with the convection velocity u c ≈ 0.66u 0 . Altogether, the nonzero components of the vortex source vector S = (0, qm1 , qm2 , 0)T of Equation (6.45) are determined for the 2D acoustic simulation by the first two components of 1 q m (x1 , x2 ) = − 



/2

−/2

[ω(

x ) × u ( x )] · W1 (x1 )d x3 .

(6.52)

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Figure 6.47. Damping function |F (αd)| over vortical wave number in x1 -direction α = 2π/λv scaled with the damping zone on- and offset width d = (l 2 − l 1 )/2.

Damping |F(αd)|

6.5 TRAILING-EDGE NOISE

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

αd

6.5.2.3 Simulation of trailing-edge noise Large-eddy simulation I: Flat plate

An LES is performed for the compressible flow over the trailing edge of the flat plate. The discretization of the convective fluxes of the Navier–Stokes equations is based on a second-order accurate advective upstream-splitting method, described in Liou and Steffen (1993), with a central approximation for the pressure derivative. The viscous stresses are discretized using central differences of second-order accuracy. Furthermore, an explicit five-step Runge–Kutta time-stepping scheme of second-order accuracy is used for the temporal integration. The coefficients are chosen to maximize the stability of a central scheme. The scheme is described in more detail in Meinke et al. (2002). The Reynolds number based on the length of the plate is Re = 7 × 105 , and the freestream Mach number is M = 0.15. The flat plate possesses a thickness d = 0.175%. To reduce the computational effort while capturing the essential physics, numerical simulations are conducted in a domain that contains only 25% of the total plate length. Figure 6.48 shows a plane of the LES grid. Seventeen points in the

y

5

0

-5

-10

0

x

10

Figure 6.48. LES grid with partially resolved plate and every second grid point shown; dimensions are related to the inflow boundary layer δ0 , which scales with 1/52.228 related to the plate length l (i.e., the complete plate extends from x = −52.228 to 0.0).

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2 1.5 1 0.5

y

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-0.5 -1 -1.5 -2 -2

-1

x

0

1

Figure 6.49. Acoustic grid scaled with the plate length l (i.e., plate extends from x = −1.0 to 0.0). Every fourth grid point is shown.

spanwise direction have been used to yield a grid with 2.22 × 106 points. The spanwise extension is 0.64δ0 . The boundary layer thickness at the inlet is δ0 /l = 1/52.228. The instantaneous inflow data are generated via an auxiliary LES of an adiabatic flat-plate boundary layer using a compressible rescaling method. More details can be found in El-Askary et al. (2001). Large-eddy simulation II: Airfoil

An LES for the flow over a complete 3D airfoil is performed using a block-structured mesh with a C-type inner block and an O-type outer block. The free-flow Mach number is M = 0.088. The Reynolds number based on the chord length is Re = 8 × 105 ; therefore, a spanwise extension of the mesh of 0.3 percent chord suffices. Experience has shown that an LES for the flow over an airfoil requires a resolution of the nearwall cells in inner law scaling in the range of x + ≈ 100, y + ≈ 2, and z + ≈ 20, where x, y, z denote the streamwise, normal, and spanwise coordinates, respectively. This requirement is approximately satisfied over the whole airfoil. The complete mesh contains 7,323,651 grid points, 1686 of which are distributed on the airfoil – 373 in the wake, 197 in the normal, and 17 in the spanwise direction. Capturing the laminar– turbulent transition, which is known from experiments to occur just downstream of the nose region, was accomplished by clustering the mesh in the area 0 ≤ x ≤ 0.2 (Figure 6.50). Furthermore, a high resolution was generated near the trailing edge to ensure that the vortical structures in the boundary layers on the upper and lower surface and in the wake were well captured because these structures determine the acoustic source

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Figure 6.50. Enlargement of the leading- (left) and trailing-edge region (right).

terms of the acoustic perturbation equations and as such the overall acoustical field. On the right of Figure 6.51 the vortical structures in the aft region of the airfoil and in the wake are visualized. The streaky structures in the near-wall region are elongated near the trailing edge, and spanwise vortices form and grow in the wake region. The left of Figure 6.51 shows the vortical structures in the suction-side boundary layer. The occurrence of multiple streaky structures in the spanwise direction indicates the LES to have a sufficiently large spatial extension in the spanwise direction. More details can be found in Ewert et al. (2003). Acoustic simulation I: Flat plate

The acoustic grid is shown in Figure 6.49. The flat plate extends from −1 to 0, and the finite thickness is resolved. The grid is clustered in the wake of the plate to resolve the source term properly. Computing the acoustic source term equation (6.52) entails sampling 503 time levels of the LES with a time increment of t = 4 × 10−3 (reference time tref = l/c∞ , plate length l). Using 10 points per period is sufficient to resolve

Y

X Z

Figure 6.51. Visualization of the instantaneous flow field: Coherent structures on the airfoil surface (left) and vortex structures in the wake (right).

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-1.0E-03 -6.0E-04 -2.0E-04 2.0E-04

6.0E-04

1.0E-03

0.03 0.02 0.01

y

0

– 0.01 – 0.02 – 0.03 – 0.04

– 0.02

0

x

-2.0E-02 -1.2E-02 -4.0E-03 4.0E-03

0.02

0.04

Figure 6.52. APE source terms L  = [ω

× u ] = (L x , L y )T , L x (top), L y (bottom), and CAA grid. 1.2E-02

2.0E-02

0.03 0.02

y

0.01 0

– 0.01 – 0.02 – 0.03 – 0.04

– 0.02

0

x

0.02

0

frequencies up to 42 kHz, whereas the lowest resolution is, according to the number of time levels, on the order of 800 Hz. Owing to the fine grid resolution of the LES close to the plate surface, the temporal resolution of the LES is much finer such that only every 160th time step is used. Figure 6.52 shows snapshots of the source terms that follow from Equation (6.52). The quantitative comparison evidences that the main contribution to the source term comes from the y-component. The visible vortical structures mainly convect in the downstream direction, where the temporal resolution of the acoustic simulation is sufficient to resolve the source term properly. Note that the source based on Lamb’s vector is small close to the wall owing to the no-slip boundary condition where the LES has its finest resolution. The upper-half planes in Figure 6.52 show the acoustic grid in the vicinity of the plate. The acoustic grid possesses 2 × 105 points, 17,585 of which are located in the LES domain. The source term is computed on the LES grid and then

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p’ -1E-06

0

1E-06

2 1.5 1 0.5

y

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-0.5 -1 -1.5 -2 -2.5

-2

-1.5

-1

-0.5

x

0

0.5

1

1.5

Figure 6.53. Pressure contours of the trailing-edge problem M = 0.15, Re = 7 × 105 , at time level T = 3.0 using APE solution equations (6.32, 6.33).

bilinearly interpolated onto the acoustic grid. Entropy fluctuations are neglected in the source computation. The source term is ramped up and down at the interior LES boundaries by applying a weighting function to avoid spurious sound sources. According to the discussion of Section 6.5.2.2 and Figure 6.47, the spatial extension of the ramping function must be on the order of the wavelength of the vortical disturbances. Because the LES domain is enclosed by a C-shaped sponge layer (Figure 6.48), a natural damping zone is obtained at the outflow boundary where the source will decay to zero. At the inflow boundary the computed source is weighted over a width d = 0.15. Figure 6.53 shows a snapshot of the pressure field that evidences acoustic waves generated at the trailing edge. No spurious sound sources (e.g., from the LES inflow boundary located at x = −0.25) are visible. Figure 6.54 shows pressure contours for the same time level computed on a finer grid with a grid refinement factor of 1.5 for each coordinate direction. The contours of the coarse and the fine grid simulation evidence a grid-independent solution. Figure 6.55 shows the directivity for four different frequencies and a circle centered at the trailing edge with radius r = 1.5. The directivities agree qualitatively very well with the shapes of the approximate harmonic, noncompact edge-noise Green’s function according to Howe (2001). In the compact limit, the product of wave number and chord length tends to zero (i.e., kl → 0). In this limit, the directivity recovers a dipole eight-characteristic ∝ sin (θ). For the opposite case with

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p’ -1E-06

0

1E-06

1.5

1

0.5

0

Y

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-0.5

-1

-1.5 -2

-1.5

-1

-0.5

X

0

0.5

1

Figure 6.54. Pressure contours of the trailing-edge problem M = 0.15, Re = 7 × 105 at time level T = 3.0 with APE solution equations (6.32, 6.33) on acoustic grid refined by a factor of 1.5 in each coordinate direction.

kl → ∞, the ratio of chord length to wave length tends to infinity (i.e., the cardioid directivity of the semi-infinite flat plate ∝ sin(θ/2)) follows. With increasing wave numbers, a multilobe pattern appears whose envelopes converge against the cardioid shape. Figure 6.56 compares the directivities from the APE system equations (6.32) and (6.33)

2×10

−7

1.5×10

−7

1×10

5×10

kl=3.072 kl=9.215 kl=15.358 kl=21.502

−7

−7

0

−2×10

−7

−1×10

−7

0

 2 2 – that is, Figure 6.55. Directivity 1/2 (θ, r, f ) with = ˆ nondimensional PSD of p 2 (θ, r )/ ρ∞ c∞ ∞  2 2 2 0 (θ, r, f )d f = p (θ, r )/ ρ∞ c∞ ; APE simulation for r = 1.5, origin at the trailing edge, for various nondimensional wave numbers.

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2×10

−7

1.5×10

−7

1×10

−7

5×10

−7

0

p’, kl=9.215 p’, kl=15.358 B’, kl=9.215 B’, kl=15.358

−2×10

−7

−1×10

−7

0

Figure 6.56. Comparison of the trailing-edge noise directivities 1/2 (θ, r, f ) for r = 1.5 applying Equations (6.32) and (6.33) ( =  PSD of p 2 ) and Mohring’s ¨ (1999) acoustic analogy ( =  PSD of B 2 ), Equations (6.43) and (6.44).

with those based on the perturbation system equations (6.43) and (6.44), Section 6.5.2.2, for the acoustic analogy of M¨ohring (1999). Note that the acoustic variable of the acoustic analogy of M¨ohring (1999) – that is, the total perturbation enthalpy B  – is related in the far field to the perturbation pressure via p  ≈ ρ0 B  . The directivities of both methods agree qualitatively and quantitatively well. Figure 6.57 depicts the PSD in three receiving points with a distance r ≈ 1.5 to the trailing edge. The three receiving points are related to the polar angles θ = 45◦ , 90◦ , and 135◦ , respectively, where all angles are defined similarly to those of Figure 6.45. The SPL and dimensional frequencies are computed based on the chord length of 0.2 m 70 o

60

θ = 90

θ = 135

S P L [dB]

50

θ = 45

o

o

40

30

20

10

0

5

10

f [kHz]

15

20

Figure 6.57. Sound-pressure level (SPL) versus frequency for a receiving point in r = 1.5 above the trailing edge for various directions θ (see Figure 6.45).

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and ambient thermodynamic properties. The dimensional pressure levels are related to a reference pressure of pref = 2 × 10−5 Pa such that the SPL follows from   peff (6.53) , p0dB = 2 × 10−5 Pa, S P L[dB] = 20 log p0dB with peff := ( ∗  f ∗ )1/2 , where ∗ (θ, r, f ) is the dimensional PSD of p  2 , and  f ∗ = 1 Hz the dimensional unit frequency. The definition of the PSD used here is based on the dimensional single-sided PSD  ∞  ∞ ∗ ∗ ∗ 2 3 2  p (θ, r ) =

(θ, r, f ) d f = ρ∞ c∞l (θ, r, f ) d f ∗ . (6.54) 0

0

In Equation (6.54), denotes the nondimensional, single-sided PSD that follows 2 and f ref = c∞ /l as reference pressure from an acoustic simulation using pref = ρ∞ c∞ and frequency, respectively, for nondimensionalizing with the subscripts ∞ indicating undisturbed flow properties. Hence, 2 ( f )1/2 peff = ρ∞ c∞

(6.55)

in Equation (6.53) with  f =  f ∗ / f ref . The spectral distributions evidence the decline of the SPL over the frequency band. Acoustic simulation II: Airfoil

The LES data sample for the airfoil in a subsonic flow contains 350 discrete time levels that cover a nondimensional time interval T = 2.975, which is nondimensionalized with the sonic speed and the chord length as references. With a temporal resolution of 20 points per time period, the data sample is sufficient to resolve physically frequencies up to 5 kHz based on a chord length of 0.4 m, which was chosen as the high-frequency limit. Furthermore, the time interval yields a frequency increment of T −1 (i.e., physically  f = 350 Hz). However, because it needs about three time units to propagate the initial disturbances out of the CAA domain plus at least one data sample period to sample results (e.g., for directivities), the source data sample has to be fed periodically into the acoustic simulation. To avoid the strong spurious sound sources that appear because the sample data ends do not match smoothly, the samples need to be manipulated to prevent spurious sound sources without losing essential physical information. Because spurious sound sources can be avoided by smoothly ramping up and down the timedependent sources, the source data sample has been weighted with a window function. The -window depicted in Figure 6.58 evidences the least-steep ramp function to appear if the whole half-data period is used to switch on the source. However, for the )/2 is used. In addition airfoil source computation a Hann window N (t) = (1 − cos 2πt T to the temporal data, windowing is added to one weighted data sample with the same weighted sample time shifted by T /2 to yield finally the weighted source (e.g., for the first half-period),     T T ·N t+ , 0 ≤ t ≤ T /2. q(t) = q(t) · N (t) + q t + 2 2

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0.4 0.3

1

N(t) 0

(ω × u)y

0.2 0.1 0 -

0.1

-

0.2

-

0.3

-

0.4 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

t, T=2.97518 0.4 0.3

t T/2

T/2

I

(ω × u)y

0.2

f(t)

0.1 0 -

0.1

-

0.2

-

0.3

-0.4 0

1

2

3

4

5

t, T=2.97518

Figure 6.58. Generation of a periodical source term via window weighting.

Figure 6.58 depicts on the right a comparison between unweighted and weighted sources with the Hann window applied (the left-hand-side sketch shows, for illustration, a Bartlett window). One can easily show that all source components that are 2n/T periodical remain unchanged by the filtering based on the Hann window. Figure 6.58 also sketches the 2/T periodical source obtained from the filtering; hence, the filtering doubles the lowest resolved frequency as well as the frequency increment. With the periodic source, after a transient period in any receiving point a correspondingly periodical signal is obtained that can be exploited to compute Fourier transforms (i.e., no additional windowing is necessary to compute sound pressure spectra). Spurious noise at interior boundaries is prevented by switching the source term of the airfoil simulation on and off over a region of 0.4 chord length in accordance with the lowest resolved frequencies. Figure 6.59 shows details of the curvilinear multiblock mesh used for the acoustic simulation of the transonic airfoil case. The computational domain is resolved with a

Figure 6.59. CAA grid.

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2 1.5

p 0.0010 0.0009 0.0008 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 -0.0006 -0.0008 -0.0009 -0.0010

1

y

0.5 0 -0.5 -1 -1.5 -2 -1.5

-1

-0.5

0

0.5

x

1

1.5

2

2.5

0.00035 0.0003

Figure 6.60. A harmonic test source over the trailing edge (top) and directivities obtained for M = 0, 0.088 with LEE and APE (bottom) applied.

LEE, M=0 LEE, M=0.088 APE, M=0.088

0.00025 0.0002 0.00015 1E-04 5E-05 0 -5E-05 -0.0001

-0.0002

-0.0001

0

1E-04

0.0002

10-block mesh that contains 86,902 grid points. Because, for acoustic simulations, an equidistant Cartesian grid is optimal, an H-topology in the far field has been chosen in combination with a C-grid topology around the wing section. The topology yields two singular points where the continuation of the grid lines is not unique. It was found that those points only cause strong spurious signals in combination with convecting vorticity passing through them. However, vorticity waves are a priori avoided when applying the APEs. The grid was chosen to have a resolution of 10 points per wavelength for the highest resolved frequency of 5 kHz based on a chord length of 0.4 m. The acoustic simulation was carried out with the time-averaged mean flow from the LES simulation that in this case covers the whole acoustic domain. Figure 6.60 shows the sound field generated by a pointlike harmonic test source with an equivalent frequency of 5 kHz that was placed above the trailing edge. The simulations were carried out with and without viscous mean flow applying the LEE and the APE. The directivities depicted on the top of Figure 6.60 evidence that the APE with mean flow delivers the same directivities as the LEE simulation for nonuniform mean flows. Interestingly, the simulations with a mean flow with a Mach number of just M = 0.088 show clear

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(

uy)

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0.2000

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– 0.1 0.9

Figure 6.61. APE source term (ω

× u ) y (top) and sound radiation from the trailing edge (bottom).

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differences compared with the case without mean flow. That is, even at very small Mach numbers convection effects seem to be important that are not described by methods based on the simple wave equation. Figure 6.61 (top) presents snapshots of the APE source term close to the trailing edge. The sound radiated from the trailing edge is depicted on the bottom panel of Figure 6.61. Analogous to the flat plate case, no waves are radiated into the up- and downstream direction. Owing to the nonuniform mean flow, the acoustic lobes on the airfoil pressure side are more pronounced than that on the suction side.

6.5.3 Trailing-edge noise simulation using LES, Euler equations, and the Kirchhoff integral 6.5.3.1 Objective

The objective is to simulate the noise radiated by an NACA-0012 airfoil with a blunted trailing edge. The proposed strategy combines three different techniques.

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A near-field compressible LES accurately simulates the local aerodynamic noise sources and is able to provide the acoustic field generated at a very short distance, including refraction effects through the inhomogeneous unsteady flow and scattering effects on solid surfaces. Then this LES local solution is coupled to a Euler solver in a midfield domain. LEScomputed perturbations are directly injected into the Euler domain, in which they propagate outward through the nonuniform mean flow, naturally accounting for refraction effects in regions where mean velocity gradients are significant. The radial extent of this Euler domain is set such that the mean flow is rather uniform beyond this limit. Finally, the acoustic field radiated at the external boundary of the Euler domain becomes the entry data of a Kirchhoff integration, which provides the noise radiated in the far field. 6.5.3.2 Large-eddy simulation

This section is devoted to the unsteady flow simulation via LES. First, the main features of the LES are recalled. Then the airfoil geometry and the computational grid are presented as well as the numerical procedure and storage method. Several basic aerodynamic results are given at the end of this section. Code implementation

The LES method described below has been implemented in the FLU3M solver, an industrial CFD software that has been developed at the Office National d’Etudes et de Recherches Aerospatiales (ONERA) for several years. This solver is based on the discretization of the compressible Navier–Stokes equations on multiblock, structured meshes by a finite volume technique. A second-order accurate, implicit temporal integration is achieved thanks to an approximate Newton method. Filtered Navier–Stokes equations

These equations are detailed in Chapter 5. A dimensionless form of the Navier–Stokes → equations is retained, which means that the local instantaneous velocity − u , tempera− → ture T , density ρ, abscissa x , dynamic viscosity μ, and thermal conductivity κ are normalized, respectively, by U0 , T0 , ρ0 , C, μ0 , and κ0 . The three-dimensional, unsteady, filtered Navier–Stokes equations are used in conservative form for a viscous, compressible Newtonian fluid. Any flow variable φ can be written as φ = φ + φ  , where φ represents the large-scale part of the variable and φ  its small-scale part. The Favre filtering operator φ˜ = ρφ/ρ, classicaly defined as a convolution product on the computational domain (Sagaut 2001), is assumed to commute with time and spatial derivatives. The eddy viscosity must be expressed by an SGS model. The equations are supplemented by a filtered equation of state. Selective mixed-scale model

The selective mixed-scale model, developed by Sagaut and Lenormand, has been retained, because it realizes a good compromise between accuracy, stability, and

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computational cost (Lenormand, Sagaut, and Ta Phuoc 2000; Lenormand, Sagaut, Ta Phuoc, and Comte 2000). Moreover, the use of a selective function allows this model to handle transitional flows, which is one of the key points of the present application. This selective mixed-scale model is detailed in Chapter 5. Numerical method

The Navier–Stokes equations are discretized using a cell-centered finite volume technique and structured, multiblock meshes. The viscous fluxes are discretized by a secondorder accurate centered scheme. For efficiency reasons, an implicit time integration is employed to deal with the very small grid size encountered near the wall. Indeed the large disparity between the acoustic wave speed and the advection velocity at low Mach numbers renders explicit time integration inefficient. This occurs because numerical stability considerations impose small time steps on the acoustic waves, whereas the physics is mainly driven by the solenoidal part of the flow, whose time scale, being associated with advection, is larger. Then a three-level backward differentiation formula is used to approximate the temporal derivative of Q c , leading to second-order accuracy. An approximate Newton method is employed to solve the nonlinear problem. At each iteration of this inner process, the inversion of the linear system relies on the lower-upper symmetric Gauss–Seidel (LU-SGS) implicit method originally proposed in Yoon and Jameson (1987). More details about these numerical points are available in Mary and Sagaut (2001) and Weber and Ducros (2000). Usually LES requires high-order centered schemes for the Euler fluxes discretization (with spectral-like resolution) to minimize dispersive and dissipative numerical errors. However, such schemes cannot be applied easily in complex geometry. Indeed, most of the aerodynamic codes able to deal with such a geometry are based on a finite volume technique in order to handle degenerated cells. Thus, getting a high-order method becomes very time consuming owing to the to high-order quadrature needed to compute the fluxes along the cell boundaries. Because several works, for instance Wu et al. (1999), have shown that LES can be carried out with a low-order centered scheme in case of sufficient mesh resolution, only a secondorder accurate scheme is employed in this study. However, a special effort has been carried out to minimize the intrinsic dissipation of the scheme. Thus, the second-order accurate hybrid upwind-centered discretization developed in Mary and Sagaut (2001) has been used in this study to achieve a good compromise between robustness and accuracy. The key point of the scheme is the use of a sensor that allows some numerical dissipation to be introduced locally when a numerical wiggle is detected on one of the primitive variables. Therefore, the effect of the SGS model is not influenced by the numerical dissipation as long as odd–even decoupling is not detected in the flow solution. Geometry, computational grid, and numerical procedure

A two-dimensional (constant section in the spanwise direction) NACA-0012 airfoil with a C = 0.6096 m chord and a blunted trailing edge (TE) of thickness H = 2.5 mm

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(0.4% of the chord) has been retained for this study with reference to the airfoil noise experiment conducted at NASA in 1980 by Brooks and Hodgson (1981). In this experiment, the airfoil had a span S = 0.46 m (or 75% of the chord), whereas in the present simulation the LES computational domain has a spanwise extent representing only 3.3% of the chord. The upstream flow velocity is 69.45 m/s (maximal velocity in Brooks and Hodgson 1981), the Mach number is 0.205, and the Reynolds number based on upstream velocity and chord is 2.86 million. The airfoil angle of attack with respect to the flow direction is 5◦ . The 3D curvilinear computational grid is obtained by replication in the spanwise direction ( y) of a 2D curvilinear structured grid made of two domains. Domain 1 is located upstream from the TE (C-shape), with 309 points along the airfoil body and 97 points in the radial direction. Domain 2 is located downstream from the TE with 227 points in the z direction (including 35 points on the TE bluntness) and 103 points in the x direction. The influence of boundary conditions is minimized by using large computational domain. As shown in Figure 6.62, this 2D grid extends nearly 10 chords above and below the profile as well as upstream and downstream. Regions in the vicinity of solid walls are highly refined. The whole mesh is made of 1.76 million points. The horizontal plane z = 0 is a plane of symmetry. In any (x, z) plane, the smallest cells are located at the TE corners with dimensions (in wall units) z + = 1.5 in the direction normal to the wall and x + = 15 along the chord. From the corners, the grid is streched in the streamwise direction (x) with a stretching coefficient equal to 1.06 (upstream) and 1.09 (downstream). In the z direction, the stretching coefficients are equal to 1.23 toward the z = 0 plane and 1.12 toward the grid periphery. This 2D grid is replicated 32 times in the spanwise direction (y) with a constant step y = 10−3 C. A no-slip condition is applied at the airfoil surface and a periodic conditon is imposed in the spanwise direction. Nonreflecting characteristic boundary conditions are applied for the far field. Moreover, a steady RANS computation using Baldwin–Lomax models provides an initial flow solution. The time accuracy of the results is ensured by taking the physical time step equal to 0.5 s, meaning a sampling frequency of 2 MHz. An initial phase of 100,000 time steps was achieved; then the useful computation was performed over 130,000 time steps with a total duration of 65 ms, representing the convection of the flow over 7.5 airfoil chords. The requirements of acoustic computations led to a storage of one time sample every 100th LES time step, meaning a storage sampling of 20 kHz, with a useful frequency band of 10 kHz. The total computation cost was 360 CPU hours on an NEC-SX5 computer. Aerodynamic results

In this section, the aerodynamic field computed via LES near the airfoil is analyzed in detail. This field is dominated by turbulent structures in the airfoil’s boundary layers and wake. At small distance of the airfoil, the pressure field is mostly acoustic and is generated by the acoustic scattering at the airfoil TE of (i) the turbulent boundary layers convected on both sides (broadband noise) and (ii) the alternated vortex shedding generated by the TE bluntness (narrow-band component). Owing to the stretching of

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Figure 6.62. Computational grid.

the LES computational grid, which acts as an acoustic low-pass frequency filter, this acoustic field cannot radiate farther than a half-chord from the body. Figure 6.63 displays contours of instantaneous Mach-number isovalues showing the development of the turbulent boundary layers and the wake. The transition between laminar and turbulent flows occurs at x/C = 0.15 on the suction side and at x/C = 0.9 on the pressure side. Mean velocity profiles near the TE were obtained by time-averaging instantaneous data. The integration of these velocity profiles provided turbulent boundary layer (TBL) momentum thicknesses of 0.8 mm on the pressure side and 1.5 mm on the suction side. Brooks and Hodgson’s (1981) measurement on a similar airfoil was slightly higher (3.9 mm), which could be explained by the roughness strips of random carbon they used to trip the boundary layer artificially at x/C = 0.15.

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Figure 6.63. Contours of instantaneous Mach-number isovalues.

It is thought that the excessive slenderness of the computed TBL may also be explained by an insufficient grid refinement in the chord direction x near the walls in the transition region. However, it will be shown in the next section that the computed flow presents general qualitative and quantitative features that make it suitable for aeroacoustic applications. Figure 6.64 shows details of the instantaneous (above) and time-averaged (below) flow streamlines around the blunted trailing edge. Time-averaged results have been favorably compared with the results of the mean flow computed separately using a RANS code.

6.5.4 Unsteady pressure-field analysis 6.5.4.1 Surface-pressure fluctuations Time histories

Figure 6.65 (top) shows time histories of the wall-pressure fluctuations near the TE at symmetrical points located on the pressure and suction sides. This plot exhibits a time period of 2 ms, meaning a dominant mechanism at a frequency near 5 kHz, and a clear phase opposition between both sides. This opposition is confirmed by the crossspectrum phase between both signals shown in Figure 6.65 (bottom). These features are typical of a mechanism of alternated vortex shedding at the TE. Power spectral densities

Figure 6.66 shows the evolution along the chord (from the leading edge up to the TE) of the wall-pressure PSD (0–10 kHz) on the suction side (above) and the pressure side

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Figure 6.64. Flow streamlines at the trailing edge: instantaneous (above) and time-averaged (below).

(below). Spectra are characterized, on both airfoil sides, by a narrow-band component emerging out of a wideband continuum. Located near 4.7 kHz, the narrowband bump is due to the vortex-shedding mechanism. It is perceived on the whole chord length with identical levels on both airfoil sides. It will be shown later that this component corresponds to acoustic waves generated at the TE and propagating in the upstream direction. The wideband continuum is generated by the convected turbulent boundary layers. Because the airfoil has a positive 5◦ angle of attack, the laminar-to-turbulent transition occurs at 15% of the chord on the suction side and at 90% of the chord on the pressure side. From this transition point toward the TE, the boundary layer thickness grows, enriching spectra with higher-frequency components. Notice that, on both airfoil sides, the vortex-shedding component emerges from the continuum near the TE (x/C > 0.98), then is masked by the near-field boundary layer, and finally emerges again at x/C < 0.85.

Comparison with experimental data

In their airfoil noise experiment (performed with the same velocity, tripped TBL, but zero incidence), Brooks and Hodgson (1981) measured surface pressure with arrays

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Figure 6.65. Wall-pressure fluctuations on pressure and suction sides near the TE. Above: time histories; below: cross-spectrum phase.

of pinhole sensors of 0.34 mm diameter. The closest distance between sensors and the TE was at 2.54 mm. Figure 6.67 compares the spectrum measured by this sensor with computed data at the same position (x/C = 0.9958). The comparison is qualitatively satisfying: the same narrow-band component emerges by ≈ 7 dB from a wide-band continuum. However, the quantitative comparison shows that predicted levels are overestimated by nearly 3 dB and that the predicted vortex-shedding frequency (4.7 kHz) is also overestimated (3.5 kHz in the experiment). Because this frequency is governed by a Strouhal number based on the wake thickness (sum of the TE thickness and TBL displacement thicknesses), this overestimation may be partly explained by the excessive slenderness of the computed TBL with respect to the experimental one. However, this cannot be explained solely by the absence of transition triggering in the simulation.

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Figure 6.66. Evolution of the wall-pressure PSD along the chord.

Wave-number–frequency spectra

The spectral densities presented earlier in this section are single-point data that do not provide any information on the propagative features of the wall-pressure field. This information can be revealed by processing a discretized, space-time, Fourier transform of any function p(x, t), providing the so-called wave-number–frequency spectrum (k, ω). For example, a monodimensional wave with frequency ω0 and velocity V0 propagating in the x-direction, for instance p(x, t) = eiω0 (t−x/V0 ) , will appear in the (k, ω) domain as a Delta function centered at k = ω0 /V0 and ω = ω0 . With the same process, a white noise propagating at celerity c0 in the x-direction will appear as a ridge along the line k = ω/c0 or along the line k = −ω/c0 if the sound travels in the opposite (−x) direction. Figure 6.68 presents the results of this process applied to the wall-pressure field computed via LES on the airfoil pressure side near the TE at x/C = 0.9. The wave-number

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Figure 6.67. Evolution of the wall-pressure PSD along the chord.

domain extent [−π/x, π/x] depends on the local space sampling x of the pressure field or the size of the grid cells. A wide-band “convective ridge” can be observed along the k = ω/Uc line generated by the convection in the flow direction, of eddies inside the TBL at an average velocity Uc slightly inferior to the local mean flow velocity

Figure 6.68. Wave-number–frequency spectrum of wall-presssure fluctuations on the airfoil suction side at x/C = 0.9.

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Figure 6.69. Wave-number–frequency spectrum of wall-pressure fluctuations on the airfoil suction side at x/C = 0.5.

Ue . Another wide-band component can be observed along the k = −ω/(c0 − Ue ) corresponding to an acoustic propagation in the direction opposite to the flow with a peak at 4.7 kHz, the vortex-shedding frequency. Figure 6.69 presents the same results obtained at midchord (x/C = 0.5). The convective ridge and the upstream acoustic component are again present. One interesting result is a new component k = ω/(c0 + Ue ), which corresponds to acoustic waves (also with a peak at 4.7 kHz) propagating in the flow direction (downstream), which may be generated by the diffraction, at the airfoil leading edge, of the acoustic waves propagating upstream.

6.5.4.2 Flow-pressure fluctuations Instantaneous maps of pressure fluctuations

The existence of a sound field generated at the TE, strongly suggested by the surfacepressure results above, is confirmed in Figure 6.70, which shows instantaneous isovalues of the pressure fluctuations inside the flow. At every point of the LES grid, pressure fluctuations are computed by subtracting the time-averaged pressure to the instantaneous pressure. The grayscale table is adapted to the low values of the acoustic fluctuations (±6 Pa), although there are much larger pressure fluctuations (up to 500 Pa) in the TBL and in the wake. Concentric waves are clearly observed near the TE, with a wavelength corresponding to the vortex-shedding frequency 4.7 kHz which, as shown earlier in this section, dominates the source spectrum by a few decibels. An animation was generated from 300 identical maps plotted for successive time steps (t = 5.10−5 s), allowing the acoustic radiation in the fluid and the turbulence convection along the airfoil chord to be visualized simultaneously with their respective celerity.

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Figure 6.70. Instantaneous isovalues of pressure fluctuations obtained from LES data.

Low-pass filtering by the CFD grid

It is interesting to note that the wave pattern corresponding to the vortex-shedding noise vanishes at a half-chord from the airfoil, while larger wavelengths are observed much farther away. This is explained by the radial stretching of the LES grid, which acts on the noise field as a low-pass filter. It is known that the propagation of an acoustic wave will not be correctly simulated if its wavelength is discretized with less than four or six cells. To illustrate this, Figure 6.71 shows the evolution of the wall-pressure spectra

Figure 6.71. Evolution of the wall-pressure spectra along the vertical grid line x = C (starting from the TE upper corner) with respect to the vertical distance z.

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Figure 6.72. Spanwise evolution of the coherence of the surface-pressure field on the suction side near the TE (x/C = 0.9958). The frequency bandwidths are integrated.

along the vertical grid line x = C (starting from the TE upper corner) with respect to the distance from the TE (in logarithmic scale). This plot clearly shows that low-frequency waves propagate much farther than high-frequency waves.

6.5.4.3 Spanwise coherences

Inside the TBL and the wake, turbulent structures or eddies are convected at low velocities, which means that typical spanwise length scales may be smaller than the spanwise extent of the LES domain and correctly simulated. To illustrate this, Figure 6.72 shows the spanwise evolution of the coherence of the surface-pressure field near the TE in several frequency bandwidths. It is defined as γ 2 (y, f ) =

|S pp (0, y, f )|2 , |S p (0, f )||Sp(y, f )|

(6.56)

where S p (y, f ) denotes the surface-pressure spectrum at the spanwise position y, and S pp (y1 , y2 , f ) denotes the surface-pressure cross spectrum between spanwise positions y1 and y2 . The coherence rapidly vanishes at any frequency bandwidth, meaning that integral length scales are smaller than the spanwise extent of the LES domain. It can be seen that the largest length scales correspond to the vortex-shedding frequency, which is a mechanism significantly correlated in the spanwise direction. Figure 6.73 shows spanwise coherence of the pressure field above the TE on the grid line j0 = 54 located at distance z 0 = 37.9 mm from the TE, where pressure fluctuations are expected to be mostly acoustic. As explained in the next paragraph, this particular grid line will be the location of the LES–Euler coupling. Figure 6.74 shows that typical length scales seem to be much longer than the spanwise extent of the LES grid – especially at low frequency and the frequency of vortex shedding. In other words, the very-near-field turbulent flow is strongly three-dimensional, whereas the acoustic field simulated at some distance from the airfoil is mostly two-dimensional, which confirms that a two-dimensional LES–Euler coupling is reliable.

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Figure 6.73. Spanwise evolution of the coherence of the pressure field at distance z0 = 37.9 mm from the TE at x/C = 1. The integrated frequency bandwidths are the same those used in as Figure 6.72.

6.5.4.4 LES–Euler coupling process

This section describes how the acoustic field generated at a very short distance by LES propagates through the nonuniform mean flow by resolution of the Euler equations. The main steps of this LES–Euler coupling are (i) the derivation of an acoustic grid and (ii) the splitting of LES data into a mean flow and a perturbation field. Propagation code

A code named E3P (propagation via Euler equations under a small perturbation hypothesis) has been developed at ONERA by Redonnet, Manoha, and Sagaut (2001) relying on the discretized Euler equations in a perturbation formulation based on the splitting of the total field in a mean flow (subscript “o”) and a perturbation field (subscript “p”):   [∂t uo + ∇ · Fo ] + ∂t up + ∇ · Fp = qo + qp + qs .

(6.57)

Here, the vectorial quantities u, F, and q, respectively, denote the unknowns, fluxes, and source terms. These equations are discretized in conservative form and can either

Figure 6.74. Final problem-adapted acoustic grid.

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be linearized (only first-order terms are kept) or nonlinearized (all linear and nonlinear terms are conserved). Cartesian or curvilinear 2D–3D (monodomain) structured grids can be handled. In the space domain, the discretization uses high-order, finite difference explicit or implicit schemes. In the time domain, multistep schemes (Adams–Bashforth or Runge–Kutta) are implemented. Specific boundary conditions are used for solid surfaces, nonreflexive borders, periodicity, or perturbation injections. Nonuniform mean flows are taken into account. The characteristics of the code, as well as several test cases of acoustic scattering on solid objects within nonuniform 2D and 3D flows, have been described in detail in Redonnet et al. (2001). The numerical tools developed in E3P are now implemented in a multipurpose CFD–CAA solver named sAbrinA (solver for aeroacoustics broadband interactions from aerodynamics) built from the general multiblock solver FLU3M, which already houses the LES–DNS solvers. This new solver is another step toward an integrated tool for simultaneously coupling CFD and propagation calculations. Acoustic grid derivation

The simulation of viscous turbulent flows via RANS or LES methods requires a high level of grid refinement near the solid walls and otherwise uses strong grid stretching in the far field to benefit from the almost uniform flow conditions. Consequently, CFD grids are notoriously unadapted to the simulation of acoustic propagation via the Euler equations, which must be discretized on homogeneous grids with cells of rather constant size all over the domain. In the present case, the coupling of LES and Euler equations is required to create a new grid for the acoustic domain. An acoustic grid was derived from the LES grid following specific constraints: (i) the homogeneity of grid refinement; (ii) the average cell size (with respect to the smallest wavelengths); (iii) intrinsic limitations of the E3P code, which, for example, cannot handle multiblock structured grid; and (iv) the coupling interface does not intercept the wake. This acoustic grid is shown in Figure 6.74. The interior border, on which LES data will be injected in the Euler domain, follows the airfoil’s surface at an average distance of 1% of the chord length. The outer border of the acoustic domain is approximately one chord away from the airfoil because it was confirmed that the mean flow is quasi-uniform beyond this distance. All cells have a quasi-constant size close to λmin /6, for the smallest considered wavelength λmin (corresponding to the highest frequency of interest – 10 kHz, for instance). The grid involves 183 C-lines and 866 radial lines, or 158,478 points. Figure 6.75 presents a closer view of this grid near the airfoil. Data injection process

The E3P code is linked with unsteady CFD computations via an interface: a 2D (respectively, 1D) interface is required for a 3D (respectively, 2D) Euler calculation. The principle of the data injection process is to reinitialize the solution vector u p at each time step on a ghost-point distribution outside the Euler domain with the injected perturbation values. The number of ghost points depends on the maximum order of the

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Figure 6.75. Final problem-adapted acoustic grid (closer view).

used spatial scheme and spatial filter. In the present application, a sixth-order scheme (seven-point stencil) is used along with a tenth-order filter (eleven-point stencil), and thus the filter stencil requires five ghost points outside the domain, the data being finally injected on six points (five ghost points plus the interface point). Mean flow–perturbation splitting

The E3P code handles perturbations that propagate on a steady inhomogeneous flow. In our case, the mean flow was obtained at any point of the LES domain by time-averaging the unsteady data. This method assumes that the LES computation was long enough to reach a stabilized mean flow, which was actually untrue: its average magnitude is subject to a slow drift. This problem was minimized by limiting time averaging to the last quarter (300 time steps) of the whole stored LES available duration (1300 time steps) to ensure that the mean flow would be as stabilized as possible, but this problem may have induced a bias in the coupling process. LES–Euler coupling result

Figure 6.76 shows isovalue contours of an instantaneous pressure fluctuation field computed from (i) LES inside the injection interface and (ii) E3P (from LES data injection) outside the injection interface. Figure 6.77 shows a closer view of the contours in Figure 6.76 centered on the airfoil. This view shows that there is no discontinuity at the injection interface between the LES wave fronts and the E3P wave fronts.

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Figure 6.76. Isovalue contours of instantaneous pressure fluctuation field (range ±2 Pa, black and white) computed from (i) LES inside the injection interface and (ii) E3P (from LES data injection) outside the injection interface.

6.5.4.5 Full simulation, including LES, Euler, and Kirchhoff integration

Finally, this section describes the full three-step CAA process: the acoustic field radiated at the external boundary of the Euler domain (where the mean flow can be considered as uniform) becomes the entry data of a Kirchhoff integration, which provides the noise radiated in the far field. Kirchhoff formulation

The KIM (Kirchhoff integration method) code provides the noise radiated by any threedimensional surface in a flow with uniform velocity in the direction x given the density (or pressure with isentropy assumed or p  = c02 ρ  ) and its normal gradient along this surface. The formulation is (Prieur and Rahier 1998)    1 DXY 1 → − → − →  − d Sdτ, (6.58) f K ( X , Y , τ )δ t − τ + ρ ( X , t) = 4π d c0

Figure 6.77. Isovalue contours of instantaneous pressure fluctuation field (range ±2 Pa, black and white) computed from (i) LES inside the injection interface and (ii) E3P (from LES data injection) outside the injection interface (closer view).

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where D X Y is the distance, corrected from convection effects, between the source point, located on the integration surface, and the observer point (both defined in a reference framework in which the surface S is at rest): 

DXY = d=

d−M0 (X 1 −Y1 ) , β2



(1 − β 2 )δi j + β 2 (X i − Yi )2 .

(6.59)

Here, M0 is the mean-flow Mach number and β is a Prandtl–Glauert factor (1 − M 2 ). − → − → The expression f K ( X , Y , τ ) is given by ρ ∂ρ  ∂ρ  − → − → − → − → → n · ( X − Y ) + M02 n 1 − f K ( X , Y , τ ) = 2 (1 − M02 )− d ∂Y1 ∂n " ! − → − → − → n · ( X − Y ) ∂ρ  1 + . M0 n 1 + c0 d ∂τ

(6.60)

This formulation assumes that the pressure field on S satisfies the convected wave equation: 

 D2 2 − a0 ρ  = 0 Dτ 2

with

∂ D ∂ , = + U0 Dτ ∂τ ∂ X1

(6.61)

which governs the propagation of acoustic waves in a medium with uniform velocity U0 in the direction X 1 . The space–time discretization of the Kirchhoff integration on a given closed control surface assumes that temporal fluctuations of fluid pressure are time sampled (with time step t and duration N t) at any cell (with maximal dimension σ ) of a two-layer (with maximal normal separation n between both layers) surface grid. If λa and Ta = λa /c0 denote the acoustic wavelength and period, the crucial parameters of the process are σ/λa , n/λa , and t/Ta . The parameter σ/λa must be as small as possible to ensure that the acoustic field will be correctly discretized on the control surface – especially if the code is not implemented with a “noncompact cell” treatment. The parameters n/λa and t/Ta must also be minimized to ensure that the normal gradients and time derivative of the pressure fluctuations will be accurately computed. Finally, a long duration N t is necessary for (i) statistical accuracy of the spectral processes applied to random signals and (ii) limiting the effects of truncation due to the delay between maximum and minimum propagation times D X Y /c0 . Final result

The whole three-step CAA process is finally applied to the NACA-0012 case, including a far-field noise prediction performed by use of a Kirchhoff integration based on the pressure field and its normal derivative on the external boundary of the Euler domain. Figure 6.78 shows the superimposition of a pressure fluctuation field obtained from LES (below the LES–Euler interface), a pressure fluctuation field obtained from E3P by injection of LES data on the new acoustic grid (between the two interfaces), and a

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Figure 6.78. Isovalue contours (range ±3 Pa black and white) of instantaneous pressure fluctuation field computed from (i) LES data inside the injection interface, (ii) Euler data (from LES data injection) between the injection interface and the Kirchhoff control surface, and (iii) Kirchhoff integration data beyond the Kirchhoff control surface.

pressure fluctuation field obtained from Kirchhoff by integration of Euler data on the surface indicated by the highest black line (above the E3P–Kirchhoff interface). The wave fronts are continuous from one domain to another. An interesting point is that acoustic waves emitted by the airfoil’s leading edge are hardly visible on this figure. This phenomenon has been already observed on wall-pressure wave-number–frequency spectra computed at midchord on the suction side (Manoha, Delahay, Sagaut et al. 2001) and confirms that the leading edge acts as a geometrical singularity on which incident acoustic waves coming from the trailing edge are scattered.

6.6 Blunt bodies (cylinder, cars) Franco Magagnato 6.6.1 Overview of blunt-body simulations

The flow-induced noise around blunt bodies such as cylinders and cars is described in this section. The numerical simulations of this flow field have been under investigation for many years. Owing to the high computational requirements (especially for the flow around a car), the first calculations were made with the help of RANS (e.g., Fritz, Magagnato, and Rieger 1991, 1992). Beaudan and Moin (1994) performed the first LES of the flow past a circular cylinder at a relatively low Reynolds number of Re = 3900.

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Later, others performed LES calculations of the flow past circular cylinders but at higher Reynolds numbers (Breuer 1997; Fr¨ohlich et al. 1998; Magagnato and Gabi 2000). All these calculations were done without aeroacoustical predictions. It appears that only a few investigations of the aeroacoustical aspects of the flow past a cylinder have been performed up to now. Batten et al. (2002a) have computed the flow past a square cylinder using their new method called limited numerical scales (LNS) and a conventional unsteady RANS approach based on a cubic eddy-viscosity turbulence model. They found that the predicted SPL of the LNS approach was some 20 dB higher compared with predictions made through URANS. They attributed this to the reduced levels of numerical dissipation in the LNS calculations. Because the computational requirements of an LES in conjunction with the LEE of the far-field noise are too demanding, most calculations on cylinders and cars have been done so far with the acoustic analogy (mainly with Ffowcs Williams–Hawkings). Experience in computational aeroacoustics shows that, in the far field, the accuracy of acoustic signals is determined by the accuracy of the flow-field prediction rather than by acoustic analogy. Although it is well known that URANS equations cannot explore unsteady flow accurately, this approach is still a very popular numerical procedure in computational aeroacoustics because of the relatively low CPU time and memory requirements. URANS equations resolve only a narrow band of frequencies present in the flow and average the rest of the frequency domain. Directional numerical solution has to be carried out to resolve all of the fluctuations. However, because DNS requires very fine grid resolution and therefore very high CPU time and memory, today DNS is only affordable for very low Reynolds numbers and simple geometries. A good compromise is LES, which resolves a fairly large range of frequencies and models only a small part of the flow. It is commonly assumed that the unresolved part of the flow consists of very small eddies that exhibit a homogeneous and isotropic structure. As long as this assumption is valid, even an algebraic model, like the SGS model from Smagorinsky, provides reasonable results. To keep this assumption valid requires that all important scales be resolved, which is a manageable task for small Reynolds numbers and simple geometries. However, in industrially relevant flows, for example in turbomachines, very complicated turbulence structures prevail. Resolving all relevant turbulence scales is challenging work even with supercomputers using high processing power and the memory access rate available today. Unless the computational mesh is fine enough to capture all relevant scales of turbulence, unresolved fluctuations cannot be considered isotropic or homogenous. Better turbulence models than algebraic ones have to be employed to model the unresolved part fairly. For this purpose a two-equation turbulence model has been proposed by Magagnato and Gabi (2000). This so-called adaptive k–τ model can be used for all cell Reynolds numbers in the unsteady case. It has the property of reducing to a DNS if the temporal and spatial resolution of the flow field is on the order of the Kolmogorov microscale, but if the fluctuations are not resolved the model reduces to a standard two-equation model. Another feature of the model is that the backward scattering of energy transfer is taken into account.

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Figure 6.79. Numerical mesh (2D plane).

6.6.2 Circular cylinder 6.6.2.1 Etkin test case

A comparison between 2D and 3D URANS, LES, and the experiment from Etkin, Korbacher, and Keefe (1957) of the flow past a circular cylinder has been made by Pantle (2002) and Magagnato, Pantle, and Gabi (2002). Etkin et al. (1957) performed experiments in the turbulent flow region of the flow past a circular cylinder. The acoustical frequency spectrum of the computation was expected to show some noise owing to the turbulent sources entering the acoustical computation. For the comparison, an observer at a distance of 0.6 m perpendicular to the main flow direction above the cylinder was considered. The main flow velocity was U = 68.6 m/s, giving a Mach number of M = 0.2; the cylinder diameter was d = 0.0125 m. The flow medium was air at ambient temperature, and the Reynolds number based on the cylinder diameter was Re = 60,000. The unsteady computation was performed using the k–τ model of Speziale, Abid, and Anderson (1990). The turbulence level was set according to the results of Etkin et al. (1957) to 0.3%. For the turbulent computation some information about turbulent length scale or eddy viscosity was also required that could not be found in the experimental description. Thus, as a starting value, a reasonable eddy viscosity ratio was chosen. Owing to previous computing experiences it was set to μt /μl = 1.0. The computational schemes were of second-order accuracy both in space and time (Magagnato 1998). The numerical mesh consisted of about 50,000 cells (Figure 6.79) for the 2D case, whereas about 3,200,000 cells were used for the 3D case. The meshes showed a distance between the cylinder and the mesh boundary of about 17 diameters in the upstream direction, 20 diameters in the directions perpendicular to the flow, and 43 diameters downstream. At the boundaries upstream and parallel to the flow, a far-field condition was applied; downstream, the static pressure was set. In the third direction (along the cylinder) there was a symmetry condition for the 2D case, whereas a periodic

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condition was used in 3D. For the LES and the 3D URANS calculation the spanwise extension of the cylinder was chosen three times the diameter of the cylinder. This is more than necessary to capture the correlations of the turbulent length scales (approximately one diameter is sufficient) but may be insufficient for the acoustical length scales of this low-Mach-number flow, which are much larger compared with the turbulent ones (see Manoha, Delahay, Sagaut et al. 2001). For the LES, the Smagorinsky–Lilley SGS model was used. The numerical scheme was a central differencing scheme in combination with a classical artificial dissipation scheme based on Jameson et al. (1981) but with a strongly reduced numerical dissipation compared with a URANS calculation. In many previous calculations it was found that the numerical dissipation of a compressible solver is absolutely necessary for the stability of the calculation but must be reduced in the range of 3 to 5% compared with URANS calculations to avoid overdamping of the turbulent small-scale fluctuations. The time step used for the acoustical computation was t = 2 × 10−5 s. Pantle (2002) found that the 2D URANS calculation gave an SPL that was too high owing to the two-dimensionality of the flow field and consequently larger lift and drag fluctuations compared with the LES results. The well-known, strong, threedimensional flow field observed in experiments obviously could not be predicted by the 2D simulation, and consequently an overprediction of the SPL of about 8 dB was found. The same conclusions have been drawn by Boudet et al. (2002). They investigated the flow past a cylinder measured by Michard, Jacob, and Grosjean (2002). Their prediction overestimated the SPL at the vortex-shedding frequency by 25%. The question was now, could a 3D URANS calculation resolve the three-dimensional structure of this flow, resulting in a reduction of the SPL? Boudet, Casalino et al. (2003) as well as Pantle (2002) found that a 3D URANS was not able to predict a 3D flow field owing to the high level of eddy viscosity distribution in the wake of the cylinder. Therefore the 3D URANS gave essentially a 2D flow-field solution with the same SPL. Only in the LES, a three-dimensional flow field was predicted and the lift and drag oscillations were considerably reduced as well because the SPL was predicted about 15 dB lower compared with the URANS calculations. Pantle (2002) found in a calculation that, in comparison with the experiment of Etkin et al. (1957), the main peak of the SPL was predicted 5.5 dB lower than the experimental value of SPL = 117 dB. The fundamental frequency predicted by the 2D, 3D URANS calculations as well as LES was at 1260 Hz, whereas in the experiment it was predicted at 1000 Hz. Because this was compared in the one-third-octave band spectrum, an uncertainty remains about the real discrepancy between calculations and experiment. In Figures 6.80 and 6.81 the SPL obtained by the computation and the comparison with the noise-reduced experimental findings is shown. Because these measurements were taken 45 years ago it is not certain how reliable they are. 6.6.2.2 ECL test case

A more recent experiment was carried out in the test facilities of the Ecole Centrale de Lyon by Michard et al. (2002) and Jacob et al. (2005). Again a circular cylinder and an airfoil in the wake of a cylinder were investigated. Because this experiment is

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Exp. (Etkin) LES

130

SPL [dB]

120 110 100 90 80

103

104 f [Hz]

Figure 6.80. Sound-pressure level of LES.

considered a benchmark validation case for CFD codes, both aerodynamic and acoustic data were measured. Another reason besides precise and purposive measurements is that, although all solid bodies are stationary, the flow in the experiment imitates the flow at rotating blades. The experimental setup is shown in Figure 6.82. The airfoil chord is aligned with the center of the cylinder cross section, and it is placed one chord length after the cylinder to avoid significant feedback of the airfoil onto the shedding. The experiments were carried out with different cylinder diameters and inlet velocities. In the numerical simulations of Magagnato, Sorg¨uven, and Gabi (2003), one configuration is chosen. In this reference configuration the diameter of the cylinder is 0.01 m and 0.3 m long in the spanwise direction. An anechoic wind tunnel provides air with 70 m/s inlet velocity and 1% turbulent intensity. The Reynolds number based on the cylinder diameter was Re = 4.8 × 104 . Exp. (Etkin) URANS

130

SPL [dB]

120 110 100 90 80

102

103

104 f[Hz]

Figure 6.81. Sound-pressure level of 3D URANS.

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Figure 6.82. Experimental setup.

The observer point for the acoustic measurements was placed at a distance of R = 1.38 m from the airfoil midpoint at an observation angle of θ = 90◦ in the midspan plane. The resolution of measurements is given as 2 Hz. Because the detailed experimental data are not yet freely available, the experimental data presented here are taken from the published papers and are therefore subject to rounding errors. The computational grid for the cylinder-only configuration was similar to the one used for the Etkin et al. (1957) case. The grid for the cylinder extended 20 times the cylinder diameter upstream and in the directions parallel to the main flow and 40 times the downstream. It consisted of 72 blocks and 3.2 × 106 control volumes in the finest mesh. All simulations were carried out with up-to-the-wall integration, which means that y + is kept less than or equal to 1 in the finest mesh levels. As determined by the space correlation considerations, the spanwise length was set to two times the cylinder diameter in both grids and resolved with 65 control volumes in the finest grid levels. This was considered necessary to resolve the flow field better in this direction. The total number of control volumes was limited by increasing the expansion ratio beyond the vertical region. The effects of this are discussed in the next section by comparison of two configurations. The boundary conditions used for the acoustical calculations are extraordinarily important because they can affect the acoustic data easily. In the simulations far field, static pressure and periodic boundary conditions are used. The far-field boundary condition can be considered nonreflecting or at least weakly reflecting because Riemann’s invariants have been used. At the outlet, the static pressure is kept constant. These boundary conditions are indeed reflective, but two factors allow us to neglect the eventual reflections. First, the outlet boundary is set far away from the cylinder. Hence, the variations of the pressure near the boundary are small. Secondly, the smallest wavelength the control volumes near the boundary can resolve is much higher than for the wavelengths relevant to the acoustic analysis. The eventually reflected

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Figure 6.83. Instantaneous streamlines in 2D URANS simulation.

waves would therefore not affect the acoustic computation. The fluid medium was air at room temperature, and the thermodynamic data were set accordingly. Velocity in the far field was 70 m/s and had a block profile. The outlet static pressure was set to p = 104,326.23 Pa. Turbulence intensity was set to 1% in the far field as it was measured in the experiment. Regarding that, the eddy viscosity ratio was set equal to 5. The computed acoustic signal needs a correlation owing to the differences in numerical and experimental configurations. Because the spanwise length of the cylinder and the airfoil is much smaller in the computations than in the experiments, the numerical acoustic signal has to be correlated. For this purpose the following empirical formula according to Etkin et al. (1957) was used: I =

ρ0 2   2 6 L c L St V , 32a03

(6.62)

where ρ0 is the reference density, a0 is the reference speed of sound, St the Strouhal number, V the inlet velocity, and cL the fluctuating force coefficient. By applying Equation (6.62) to two different lengths L exp and L sim and rearranging, we get   L exp S P L = 20 . (6.63) L sim Because the length in the spanwise direction is 0.3 m in the experiment and 0.02 m in the simulation, Equation (6.63) yields S P L of 23.52 dB. All computational acoustic results were corrected with this value. It should be mentioned that other correlations have been proposed. Kato et al. (1994) proposed a correlation that gives a smaller correction than Equation (6.63). As an overview of the flow around the cylinder, the streamlines and the contours of the velocity component parallel to the main flow v are shown in Figures 6.83 and 6.84. These are instantaneous snapshots of the 2D URANS and LES (adaptive k–τ ) calculations, respectively. Both are results at the finest grid levels. In 2D simulations only a few vortices are resolved; they represent a pure periodic structure. After about four times the cylinder diameter in the wake, the streamlines become completely parallel to the main flow again. The results of LES simulations exhibit a much larger von K´arm´an vortex street. The frequency of the vortex structure is still dominated by the same shedding frequency, but a spectral broadening around

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Figure 6.84. Instantaneous streamlines in LES with adaptive model.

this frequency is observed. The vortices have lost their uniformity and exhibit a more chaotic structure. Figures 6.85 through 6.87 show the lift and drag coefficients history of the cylinder in different grid levels of the LES. Here the lift fluctuates around cL = 0 with an amplitude around cL = 1. The perfect periodicity of lift and drag coefficients in the third-finest grid level shows that the flow is still two-dimensional, although the grid is 3D. This signifies that the grid is not fine enough to resolve the three-dimensional structures in the flow. In a finer grid level (Figure 6.86), three-dimensionality is captured. Fluctuations with different frequencies are generated, but the amplitudes are of the same extent. In the finest grid level, the shedding frequencies cover a larger range, and the amplitudes of the lift and drag coefficients sink dramatically. This shows that the turbulent energy of the flow is distributed in all three dimensions and is totally threedimensional. It has to be mentioned here that the velocity component in the spanwise direction is ±10 m/s in the second-finest grid, whereas it is ±50 m/s in the finest one. The 2D simulations result in a pure, periodic acoustic signal at the observation point, as seen in Figure 6.88. The acoustic signal ρ  fluctuates between ±3 × 10−6 kg/m3 . Correspondingly, fast Fourier transform analysis shows a clear peak at the dominant frequency f = 1612 Hz, S P L = 106 dB and its harmonics at 2

CL, CD

1

0

Figure 6.85. Lift and drag coefficients of the cylinder in the third-finest grid.

−1

−2

0.0375

t [G]

0.0725

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Figure 6.86. Lift and drag coefficients of the cylinder in the second-finest grid.

CL, CD

1

0

−1

−2

0.1025

t [G]

0.1075

2

1

Figure 6.87. Lift and drag coefficients of the cylinder in the finest grid.

CL, CD

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−1

−2

Figure 6.88. Acoustic density fluctuations of 2D URANS simulation in the finest grid.

0.1325

t [G]

0.1375

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Figure 6.89. Sound-pressure level of 2D URANS simulation in finest grid.

f = 3274 Hz, S P L = 71 dB, and f = 4887 Hz, S P L = 71 dB (Figure 6.89). Although the tendency of the sound-pressure spectrum is predicted correctly, both frequencies and amplitudes of the peaks are miscalculated. The second-finest mesh in 3D simulations results in a more chaotic acoustic signal (Figure 6.90). The corresponding sound-pressure spectra indicate that fluctuations with different frequencies besides the dominating one exist (Figure 6.91). The amplitudes are on average about ±3 × 10−6 kg/m3 . Because the amplitudes in acoustic signals are comparable with the 2D case, the SPL is also on the same level. Therefore frequency is predicted better ( f = 1466 Hz), but the amplitude is similar to the 2D case (S P L = 109 dB). Figures 6.92 and 6.93 give the acoustic signal and the sound-pressure spectra of the finest grid in 3D simulations. Here, the amplitudes of the acoustic signal sink dramatically in comparison with the coarser grids. This is a clear result of the threedimensionality of the calculated flow field. Figure 6.93 shows that the SPL is damped because of these phenomena. Also, the frequency of the dominating fluctuations is predicted better. Here, the main peak is at 1466 Hz and 84 dB, which corresponds

Figure 6.90. Acoustic density fluctuations of LES with adaptive model in second-finest grid.

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Num Exp

100

Figure 6.91. Sound-pressure level of LES with adaptive model in the second-finest grid.

SPL [dB]

80

60

40 2000 f [Hz]

4000

6000

Figure 6.92. Acoustic density fluctuations of 3D LES simulation in finest grid.

Num. Exp.

80

Figure 6.93. Sound-pressure level of LES with adaptive model in finest grid.

SPL [dB]

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2000 f [Hz]

4000

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Num. Exp.

SPL [dB]

80

Figure 6.94. Sound-pressure level of LES in finest grid with the Smagorinsky and Lilley model.

60

40 2000 f [Hz]

4000

6000

to about 7% and 0.5% error in frequency and SPL, respectively. One cannot see the positions of harmonics clearly because of the many fluctuations in the computational SPL curve. It was assumed that these would vanish with more acoustic data. Figures 6.93 and 6.94 show the comparison between two different LES models, the Smagorinsky and Lilley model and the adaptive k–τ model. Both provide similar results with respect to the position of the peak, whereas the adaptive k–τ model is slightly better. At lower and higher frequencies, both models achieve higher SPL curves than in the experiment, but the adaptive k–τ model is about 50% better than the Smagorinsky and Lilley model in these regions. Montavon et al. (2002) have made calculations as well as measurements on a cylinder in cross flow at two different Reynolds numbers. The low Reynolds number of Re = 3900 was calculated as an LES using CFX-5 commercial software. The aerodynamic results were in very good agreement compared with their own experiments as well as with published work in the literature (Lourenco and Shih 1993). The higher Reynolds number of Re = 140,000 gave less satisfactory aerodynamics results compared with the experiments by Cantwell and Coles (1983). Although the distribution of the fluctuating pressure coefficient agrees fairly well with the experiment of Norberg (1992), the back pressure coefficient was predicted to be about −1, which underestimates the measurements by about 20%. This could be due to a low resolution of the flow field, for only 225,000 elements were used for both Reynolds numbers. The acoustical results were obtained with SYSNOISE (LMS SYSNOISE 2002), which solves the acoustic analogy equation of Ffowcs Williams–Hawkings with the direct boundary element method. At the higher Reynolds number, the prediction of the overall SPL was found to be about 5 dB higher than in the experiment, whereas at the lower Reynolds number the overestimation was on the order of 10 dB. The reason for this error was not clearly understood, but Montavon et al. (2002) assumed that it had to be connected with the overprediction of the drag coefficient by 30%, which could mean that the dipole sources were generally overestimated.

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Figure 6.95. The Ahmed body from Ahmed et al. (1984) (isosurface of zero streamwise velocc ity from Kapadia et al., 2003). Reprinted with permission from SAE Paper # 840300 1984 SAE International.

6.6.3 Car

The prediction of the flow field around a car is without any doubt of great practical importance. In the early days of the application of CFD in car aerodynamics, the drag and lift generated by a car were the main driving forces. Nowadays car designers are interested additionally in the pressure distribution around a specific shape in order to get correct boundary conditions for internal flow-field predictions (e.g., the engine cooling device or the air conditioning inlet and outlet position). Recently the noise generated by the flow around a car became increasingly important for the car industry mainly because of comfort factors but also in response to more stringent noise regulations. It appears that only very few URANS, DES, or LES results have been reported in the open literature so far. Perzon and Davidson (2000) have calculated the flow around the Association for Structural and Multidisciplinary Organization (ASMO) model with URANS. The geometry of the ASMO model is mainly used for testing CFD codes. Perzon and Davidson (2000) have found the pressure distribution around the car in the symmetry plane to be in good agreement with the measurement whereas the total drag deviates significantly. Hinterberger, Garcia-Villalba, and Rodi (2003) and Howard and Pourquie (2002) have made large-eddy simulations around the Ahmed body (Ahmed, Ramm, and Faltin 1984). The Ahmed body is a generic car shape with a relatively simple geometry and extensive measurement data are available (see Figure 6.95). Kapadia, Roy, and Wurtzler (2003) have performed a DES about the same geometry. The resolution of these calculations ranged from 1,700,000 points (Kapadia et al. 2003) up to 18,500,000 (Hinterberger et al. 2003). The comparison of the predicted flow structure with the experiment was well captured, but some discrepancy, especially in the lower part of the slant back, exists for all calculations. None of them have predicted the flow-induced noise.

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Figure 6.96. Mesh for the CFD model from Volkswagen.

Another experiment around a generic car shape has been made by the Volkswagen AG (see W¨ustenberg and Hupertz 1995). This geometry (see Figure 6.96) is a bit closer to realistic cars and is as well very detailed. An LES of this CFD model has been made by Magagnato, Pritz, and Gabi (2005) using a 3D grid provided by Volkswagen AG. The full mesh has been obtained by mirroring the original grid at the symmetry plane. A total of 5,200,000 points covers the flow field, which extends four lengths in the upstream direction, three lengths in the lateral direction, five lengths in the downstream direction, and three lengths of the car perpendicular to the car. The freestream velocity was u = 63 m/s with a turbulence level of T u = 1%; the density was ρ = 0.62, and the temperature was T = 292 K. This results in a Reynolds number of about Re = 7,350,000 based on the length of the car. It is clear that the number of grid points above mentioned can not adequately resolve the flow field in such a way that a large part of it lies well inside the inertial subrange of the energy spectrum. It is probably more a very large eddy simulation. The simulation has been made by using the Smagorinsky and Lilley SGS model, a second-order central differencing scheme in space, and a second-order accurate implicit scheme in time (dual time stepping). The calculations were done on an IBM SP2 parallel computer using 32 processors. The global computational time was about 80 h. In Figures 6.97 and 6.98 the streamlines and the vorticity magnitude at an instantaneous time are shown. The strong three-dimensional character of the flow field – especially in the wake – is clearly visible. The comparison of the pressure distribution in the symmetry plane of the CFD model between experiment and calculation shows a very good agreement over the roof and in the major part of the car underbody (see Figure 6.99). Except close to the front of the underbody, a considerable deviation can be observed. This is very surprising

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Figure 6.97. Streamlines in the wake of the CFD model.

because the flow in this region is mainly a potentially dominated flow that is very easily predictable by CFD. One possible explanation could be the different application of the wall-boundary condition in front of the car. In the experiment, a boundary layer sucking in front of the car is used, whereas in the calculation a symmetry condition is applied in the front part ending with a wall boundary condition just below the front part of the car. Unfortunately, no acoustical data have been measured in the experiment. Thus, the calculated SPL with the acoustic analogy at an observer point of x = 10 m, y = 10 m, and z = 1 m is only for demonstration purposes. The SPL collected at 512 samples is shown in Figure 6.100. The main peak in the SPL is about 57.9 dB at a frequency of 41 Hz. Over a range of about 100 Hz the

Figure 6.98. Vorticity in the wake of the CFD model.

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Figure 6.99. Pressure coefficient in the symmetry plane of the CFD model.

amplitude of the SPL remains on the order of 50 dB, whereas at higher frequencies it drops very quickly. Another approach of predicting aeroacoustics noise generated by a car is used by Gr¨unewald and Basel (see Isensee and Frenzel 2003). They have been using the BEM in conjunction with unsteady calculation methods for the prediction of noise in a recent study of the German Ministry of Education and Research (Leiser Verkehr). There are also some calculations published with commercial codes (Correa, Massa, and Zajas 2003; Mendon¸ca 2002) calculating the flow around a generic mirror with LES and acoustic analogy. The corresponding experiment has been set up to be used 60

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as a simplified mirror producing noise in the passenger compartment. Many car manufacturers are concerned about passively reducing the noise emitted to the driver and the passengers.

6.7 Internal flows Philippe Lafon, Fabien Crouzet, and Jean Paul Devos 6.7.1 Introduction to internal flows

Among the many aeroacoustics phenomena that may occur in internal flows, the experience of the industrial context leads us to consider the following first: r Acoustic fluctuations due to turbulence-generated noise at low Mach number, r Self-sustained oscillations and flow acoustic coupling at low Mach number, and r Aeroacoustic instabilities at high Mach number.

These phenomena are similar to the ones observed in free flows, but in the context of confined flows, many features of these phenomena are enhanced; for example, flow– acoustic interactions are stronger. The numerical approaches based on LES or very large eddy simulation (VLES) we present in this chapter were developed in an industrial context. Thus, direct acoustic simulations by compressible DNS are out of our scope. Section 6.7.2 presents a hybrid approach based on the computation of incompressible LES and LEE.∗ This approach is able to take into account the generation and the propagation of acoustics disturbances due to the presence of control flow devices (diaphragms, valves, etc.) in industrial ducts. In Section 6.7.3, a “direct” approach based on the computation of nonlinear Euler equations is able to model the flow–acoustics interactions at low Mach numbers when there are feedback phenomena and acoustic resonances. In Section 6.7.4, the same set of equations is solved in order to model transient aeroacoustic phenomena in high-Mach-number flows. All these cases were defined and computed to obtain information about real problems occurring in industrial configurations. 6.7.2 Computation of acoustic fluctuations due to turbulence-generated noise at low Mach number 6.7.2.1 Source modeling

The modeling approach used here is a hybrid one. Aeroacoustic methods that separately consider aerodynamic and acoustic modeling are called hybrid approaches. ∗

This work was carried out within the framework of the PREDIT research program supported by the French government.

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Approaches based on Lighthill’s analogy are of this type, but because they rely on an integral solution they are not well suited for confined problems. That is why Euler equations have been chosen since the first developments (B´echara et al. 1994; Bailly, Lafon, and Candel 1996; Bogey et al. 2002). A source term has been defined on the momentum equation of the Euler system. This source was built to match Lilley’s equation for a one-dimensional sheared flow. This source term is expressed by  # ∂u it ∂u it − u jt . (6.64) Si = − u jt ∂x j ∂x j The subscript t in the source term expression denotes turbulent fluctuating quantities. This source term may be calculated by several methods. An approximate approach is possible by using stochastic modeling for turbulent fields (B´echara et al. 1994; Lafon 1997; Longatte, Lafon, and Candel 1998; Bailly and Juv´e 1999). Here we use LES data to calculate the source term. The source term can also be formulated in terms of vorticity (Howe 1975). In this case, it is more appropriate to use a vortex method for computing the flow (Hofmans 1998). Note that in supersonic cases the knowledge of a broadband turbulent source term is not necessary because, in such situations, the noise radiation is mainly due to the excitation of instability waves. Thus, a compressible LES with suitable inflow excitations can be used (Dong and Mankbadi 1996). 6.7.2.2 Flow and acoustic computations LES computation

LES computations are carried out with the ESTET code developed by the R&D Division of Electricit´e de France. It solves LES equations for incompressible flow using a conservative finite volume method on a Cartesian grid. The classical Smagorinski model is used, and periodic boundary conditions are imposed in the transverse direction. Acoustic computation

The source terms need to be calculated from the transient LES results. It was shown by Crouzet et al. (2002) that these results have to be very clean because any nonphysical disturbances in the LES results lead to large errors in the source terms. Acoustic computations are carried out with the EOLE3D code developed by the R&D Division of Electricit´e de France. It solves the LEE on a structured Cartesian grid using a DRP scheme proposed by Tam and Webb (1993). 6.7.2.3 Results in the case of a diaphragm in a duct

The configuration of a 3D duct obstructed by a 2D diaphragm is chosen because some experimental data are available. This simple case is considered to behave, at least qualitatively, like many industrial cases when turbulent acoustic sources are generated downstream of flow control devices (e.g., diaphragm, valves, etc.).

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U

80 mm

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3 mm

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Figure 6.101. Aerodynamic computational domain. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

The computational domain for the LES computations is presented in Figure 6.101. The computational domain for the acoustic that is presented in Figure 6.102 is much longer, and thus the acoustic far field can be obtained in the duct. The LES grid is refined close to the diaphragm, whereas the acoustic one is uniform. Figure 6.103 displays a snapshot of the instantaneous longitudinal velocity for U = 14 ms−1 . Figure 6.104 displays the acoustic results. The upper graph shows the evolution of the acoustic power radiated by the diaphragm sources with respect to the average velocity in the duct. The results are compared with the classical U 4 law, and the agreement is good. The lower graph compares the computed and the experimental spectra of the acoustic power. 6.7.3 Computation of flow acoustic coupling in low-Mach-number ducted flows Fluid–acoustic coupling

In confined flow, self-sustained oscillations due to feedback phenomena or acoustic resonances are very common. The basic physics and analysis may be found in Rockwell and Naudascher (1978). More recently, several computational studies analyzed these kinds of flows in free or confined configurations (e.g., Dequand, Hulshoff et al. 2003; Gloerfelt, Bailly, and Juv´e 2003; Lafon et al. 2003). In such flows, large coherent eddies drive the whole dynamics. So, computing LES or VLES in these cases seems to be relevant. Furthermore, such flow dynamics and associated acoustics are nonviscous phenomena. Thus, one might expect to calculate these configurations by means of nonlinear Euler equations, provided some suitable wall treatment is introduced if boundary layers have a great influence on flow development.

80 mm

35 mm 400 mm

3 mm

1200 mm

40 mm

Figure 6.102. Acoustic computational domain. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

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Figure 6.103. LES velocity field; longitudinal component (U = 14 ms−1 , t = 6.6 × 10−2 s). Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc. 120 Experiments Computation 4 U law

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Figure 6.104. Acoustic results: acoustic power radiated by the diaphragm with respect to the mean velocity (top) and acoustic power spectrum for U = 14 ms−1 (bottom). Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

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Figure 6.105. Geometry of the cavity duct system.

Fluid–acoustic computation using nonlinear Euler equations

The Euler equations are solved on a 2D curvilinear grid using a conservative finitevolume method (Lafon and Devos 1993). The total variation diminishing algorithm developed by Harten (1983) is applied. For inlet and outlet boundary conditions, characteristic variables, and compatibility equations are used. As is proposed in Poinsot and Lele (1992), additional terms are introduced in the equations in order, first, to drive the solutions toward a target state at infinity and second to ensure absorbing conditions for acoustic waves. Results in the case of a ducted cavity

The geometry of the confined cavity we have studied is shown in Figure 6.105. This configuration is a 2D case modeling a real industrial problem that has been also treated with 3D numerical methods and 2D and 3D experiments on a scale model. The values of the parameters defining the case are as follows: r d = 0.05 m r e = 0.008 m r h = 0.02 m r H = 0.137 m r L = 0.073 m;

the sound speed is 343 m s−1 . The U0 mean velocity in the duct is 62.8 m s−1 . This value of the velocity is the one that produced the strongest fluctuations measured in the experimental models. Figure 6.106 shows snapshots of the time evolution during an oscillation period of the pressure in the whole duct and the vorticity in the cavity. Note that the evolution of the flow in the cavity and of the acoustics in the duct are coupled: it is shown that, when the eddies interact with the downstream corner of the cavity (t0 , t0 + T ), the acoustic response in the duct is clearly organized in accordance with the first transverse mode pattern. Experimental results and more detailed numerical analysis showed that

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Figure 6.106. Snapshots of the pressure in the duct (left) and the vorticity in the cavity (right) during a period of the oscillation. From top to botton: t0 , t0 + T/4, t0 + T/2, t0 + 3T/4, t0 + T .

the fluctuations in the duct are at a maximum when this coupling occurs (Devos and Lafon 2003; Lafon et al. 2003).

6.7.4 Computation of aeroacoustic instabilities in high-Mach-number ducted flow 6.7.4.1 Presentation of the test case

Industrial applications involving high-pressure steam flows are controlled by systems like pressure-reducing valves or diffusers. Downstream of these sudden expansions, the flow may become unstable by strong aeroacoustic coupling. To determine the fundamental mechanisms of these instabilities, researchers have carried out experimental studies. Numerous results are given and analyzed in the paper of Meier et al. (1978). Several different cases are presented, but the case of a rectangular duct with a sudden enlargment is very typical of many industrial configurations. The geometry of the case is presented in Figure 6.107. The parameter that controls the evolution of the flow is the ratio of the upstream pressure on the downstream pressure, τ = Pupstream /Pdownstream . At high values of τ the flow is entirely supersonic. When τ is decreased, the flow becomes unstable and a cyclic oscillation appears. Complex flow patterns can be characterized

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L

H2

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H1 = 0.3 H2

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Figure 6.107. Geometry of the sudden enlargment. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

involving normal and oblique shock waves, formation of supersonic jets, reentry of fluid at the downstream boundary, and so on. 6.7.4.2 Results in the case of a sudden enlargment in a 2D duct

Results for two pressure ratios τ are presented in Figures 6.108 and 6.109: τ1 = 5.5 and τ2 = 2.65. For the highest pressure ratio the flow is not fully detached from the walls and the flow oscillates between two states: one involving a strong normal shock associated with two supersonic jets on the sides and the other one in which the normal shock disappears. For the lowest pressure ratios, the flow is detached from one wall and oscillates from a state in which shock patterns are still present to another one in which the structures are fully destroyed. 6.7.5 Conclusions for internal flow prediction

We have presented methods able to deal with aeroacoustic phenomena in internal flows. These methods, based on transient flow computations (LES and VLES), give data for defining source terms in hybrid approaches for subsonic flow. When applied to

Figure 6.108. Snapshots of the Mach number for τ1 = 5.5. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

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Figure 6.109. Snapshots of the Mach number for τ2 = 2.65. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.

compressible computations, these methods are able to catch flow and acoustic feedback and aeroacoustic coupling directly. In comparison with free-space situations, internal configurations have the following characteristics: r There is no need to have a very large grid in order to deal with the acoustic far field. r The influence of solid boundaries is of course greater both for flow computation

(boundary layers) and for acoustics computation (resonances).

6.8 Industrial aeroacoustics analyses Fred Mendonca ¸ 6.8.1 Introduction to industrial aeroacoustics analyses

It is true of aeroacoustics, as with most simulation-based engineering, that the engineer is pulled between two extremes: whether to undertake fully rigorous analyses with all the implied resource demands or, instead, to perform analyses of limited scope with quantified risk because the time scales in the industrial design cycles require as much. When the analyses involve large-eddy simulation (LES), this dilemma is brought sharply into focus. In this section we attempt to qualify important questions relating to the aeroacoustics simulation process to help the analyst make value judgements before, during, and after LES calculations, concentrating on data analysis and interpretation, because the processing of information ultimately defines the true worth of the outcome. This section should be read with an industry perspective. It aims to extract maximum value from the modeling process by gathering useful information from computational fluid dynamics (CFD) analyses to maximize knowledge of the acoustics properties of simulated systems. In industry, these systems are diverse, covering many sectors – most notably transportation. Some current topics of interest are listed in the following paragraphs. The commercial aerospace operator’s main noise-associated concerns are well documented: high-lift devices (Agarwal and Morris 2004), landing gears (Souliez

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et al. 2002), jet nozzles (Morris and Farassat 2002), and cabin and cockpit climate control devices (Mendon¸ca et al. 2005). Military operations focus on the structural and aerodynamic instabilities of weapons and weapons bays (Henshaw 2000; Allen, Mendon¸ca, and Kirkham 2005). Automobile manufacturers and component suppliers are keen to demonstrate expertise in applied technologies for noise minimization in several areas. External components such as sideview mirrors (Siegert, Schwarz, and Reichenberger 1999), A-pillars, and wiperblades directly excite the vehicle’s panels, which transmit noise to the driver’s ears. Under the hood, noise from turbomachines such as the cooling fan (Algermissen, Siegert, and Spindler 2001) and turbocharger can be heard above the idling engine. Ducting and climate control system components such as the blower fan (Barone et al. 2003; Read et al. 2004; Dubief et al. 2005) and flaps introduce noise directly into the passenger compartment. Open apertures such as sunroofs and windows suffer buffeting caused by oscillation of the separated shear layer at the aperture’s leading edge. In what follows, we exploit a variety of CFD and postprocessing approaches, covering steady-state and transient calculations pertinent to all of the preceding topics of interest. The exploitation leads toward an expert-system analysis template for industrial aeroacoustics. Postprocessing of steady-state CFD to resynthesization noise or to gain an approximate measure of the grid frequency cutoff are useful precursors to fully time-accurate transient calculations. Later, we turn our attention to noise propagation. Some examples of the “hybrid” approach coupling transient CFD to a separate noise propagation analysis are presented, but we start with a reminder of some basic building blocks in the process. 6.8.2 Preliminary considerations 6.8.2.1 Hybrid nature of flow-noise generation and propagation mechanisms

We begin with the premise that general purpose, commercially available CFD codes, widely used in industry and capable of producing quality flow solutions for complexgeometry mainstream applications using second-order accurate discretization methods in space and time, are sufficiently accurate to predict flow-induced noise sources through direct capture of the dynamic flow structures but are insufficient to propagate the associated noise to the far field (see also Section 6.6). This limitation arises because the spatial discretization is only of second-order accuracy, but this is not necessarily a major impediment. In fact, the two mechanisms, noise-source generation and propagation, are conveniently separated. Acoustic pressures are typically orders of magnitude smaller than the flow-fluctuating pressures (pseudo-sound). The flow generates noise and influences its propagation via, for example, convection and diffusion. Conversely, the sound field generally does not generate flow. This useful dissection allows for the flow analysis and the propagation analysis to be performed separately and consecutively in that order and is referred to as the “hybrid method” for aeroacoustic simulation. One notable exception is aeroacoustic resonance, in which feedback occurs between the fluctuating flow and propagating pressure waves. This circumstance only occurs when the geometry contains duct or cavity length scales that are equivalent to the

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acoustic wavelengths excited by the flow. Examples are resonators in automotive intake and exhaust systems (in which these devices are purposely introduced), passenger cabins with open windows or sunroofs (in which the resonance is unintended and distinctly unpleasant), and aerospace cavities such as those housing the landing gear and weapons stores. Aeroacoustics resonance has been successfully simulated by Inagaki et al. (2002) and is further illustrated here in Section 6.8.4.1 to stress that compressibility is the mechanism by which acoustic-flow feedback is facilitated. 6.8.2.2 Domain size, meshing strategy, and temporal resolution

The engineer should include in the CFD analysis sufficient volume around the subject geometry to ensure r all the flow noise mechanisms of interest are contained within, r the boundary values are known, and r that the numerical treatment at boundaries allows for wave transmission, or r that the boundaries are placed far enough away for artificial reflections to be dissi-

pated before they influence the noise-generation mechanisms. Meshes should as far as possible be orthogonal and isotropic (unit aspect ratio) – the former to preserve the spatial accuracy, and the latter to ensure that the shape of the noisegenerating flow structures captured by the simulation is not distorted. Some readers will take the assertion of cell isotropy to be excessive, imposing an unreasonable burden on resolution especially in the boundary layer: instead, the choice of cell structure should be consistent with resolution of target-flow features. For example, vortex streaks are stretched features confined to the boundary layer; therefore, streamwise cell stretching is a natural and consistent choice. Isotropic meshes, aimed at resolving authentically three-dimensional structures in the bulk flow, do not preclude the use of high-aspectratio cells in the boundary layer aimed at resolving streaks. Time-accurate (transient) simulations will be able to resolve frequencies to a maximum of the inverse of twice the time-step increment, corresponding to the Nyquist requirement of two points in a wave form. The author prefers to recommend a more conservative ten points per wave form to give a better chance for the amplitude, and hence the noise intensity, to be resolved. 6.8.3 A two-step CFD modeling process (steady-state and transient)

We define two steps in the modeling process commencing with CFD in steady state, after which useful postprocessing is performed as a precursor to calculations in the transient mode. 6.8.3.1 Steady-state calculations

Synthetic reconstruction of the turbulent fluctuations from steady-state calculations can lead to useful approximations of relative noise-source magnitudes and locations. One such method is described in Section 6.8.3.2. Similar synthetic reconstructions may be used to account for the broadband contributions lost in the SGS (see Section 4.2.7),

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or to introduce perturbations at the domain inlet with time-varying coherent structures representative of turbulence statistics (Benhamadouche, Jarrin, and Laurence 2003). Through postprocessing of steady-state RANS solutions, a mesh frequency cutoff (MFC) measure is calculable. This simple method of Mendon¸ca et al. (2005) gives a reasonable approximation of the limiting frequency the mesh is locally able to resolve. The validity of the method is verified through direct comparison with the predicted spectra from a transient solution and measurements. 6.8.3.2 Steady-state postprocessing Approximation of noise sources through synthetic reconstruction

Standard turbulence closures include transport equations for the mean kinetic energy k of the mean turbulent structures and their dissipation rate ε, or other turbulence quantities from which k and ε may be derived. Together with the mean flow, this information is used to resynthesize the fluctuating velocity (Debatin 1999; Kraichnan 1970) vi (x, t) based on a Fourier decomposition of the von K´arm´an energy spectrum E(α, t) across wave numbers α as follows:  ∞ 3 2 E(α, t)dα = vi = k, (6.65) 2 0  ∞ α 2 E(α, t)dα = ε. (6.66) 2ν 0

Reconstruction of the fluctuations, sometimes referred to as stochastic noise generation and radiation (SNGR), arises from the wave-number decomposition vi (x, t) =

N √  2 vn cos [αn (x − V t) + n + ωn t] en .

(6.67)

n=0

This equation describes the turbulent fluctuation as the sum of N harmonic waves αn with a specific orientation en , amplitude νn , turbulent frequency ωn , convective speed V , and phase shift n that have been selected from a set of appropriate probability density functions. The noise characteristics associated with the fluctuations can be inferred through the Lilley (1969) acoustic analogy. If we neglect viscous and entropy effects, the Lilleyequation source term (left-hand side of Equation (6.68) below depicting gradients of the instantaneous velocity vi,∗ j,k ) can be separated into various contributions from the turbulent fluctuations (B), the mean flow distortion (A), and mixed mean–turbulent gradients (C). From dimensional analysis, the largest contribution comes from (B): B

A %& ' %& '$ ∂vk ∂v j ∂vi ∂vk ∂v j ∂vi = −2 −2 −2 ∂ xi ∂ xk ∂ x j ∂ xi ∂ xk ∂ x j ∂ xi ∂ xk ∂ x j

∂vk∗

∂v ∗j

∂vi∗

$

$

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Figure 6.110. Opel 2004 Astra (courtesy of Opel AG).

Figure 6.110 illustrates an isosurface of the power of the synthetic source represented by term (B), which henceforth will be referred to herein as the Lilley source, arising from the steady solution flow over a full automobile. There are two obvious benefits to using this methodology as a postprocessing tool in steady-state calculations. First, the relative magnitudes of sources are identifiable; this is useful either to compare one design against another or to rate the relative source magnitudes from different components on the same vehicle. We see in Figure 6.110 that the largest shear-noise sources originate from the sideview mirror, wiperblade cavity, A-pillar, wheelwell, and wake. Secondly, the method provides information about the location and expanse of the sources. Thus, in the preparation of a transient analysis on more detailed or localized analyses, it is useful in deciding where to refine the mesh, where to place domain boundaries, and where to locate monitoring points. The often-cited idealized wing-mirror study of Siegert et al. (1999) demonstrates the suitability of this method well. From Figure 6.111, the analyst is able to identify clearly those regions in which the Lilley source indicator dominates and is therefore able to plan a concentration of mesh density there. The inset in Figure 6.111 shows four levels of successive refinement, the finest level adjacent to the mirror surface, with refinements extending into the separated wake region The analyst should also be aware of the limitations of this technique. As a steady-state method, it is incapable of representing noise sources that arise from large-scale transient features (large-scale here means larger than the grid scale) such as vortex shedding or moving boundaries.

Mesh frequency cutoff (MFC) measure of Mendonca ¸

The main value in performing an MFC analysis as postprocessed from a steady-state solution is to be able to decipher in advance of a transient calculation whether the chosen grid has sufficient resolution to capture the turbulence flow structures in the frequency range of interest. Given a cell dimension  and local turbulent kinetic energy k, the smallest length scale of a turbulent eddy structure captured by the mesh is 2; its associated isotropic

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Figure 6.111. Lilley turbulence shear-source distribution illustrated by isosurfaces for the idealized wing-mirror example of Siegert et al. (1999).

( 2 fluctuation velocity is k . Therefore, the maximum frequency f MC reasonably 3 resolved by the local grid spacing  is ( 2 k 3 . (6.69) f MC = 2 The choice of k, the mean turbulent kinetic energy contained by the mean turbulent structures, to represent the turbulence velocity fluctuation is easily justified. Most widely used RANS turbulence models solve transport equations for it directly; for those that do not, it is easily derivable. The transport equation for k is derived directly from the unsteady Navier–Stokes equations and therefore contains contributions from all the important mechanisms: convection, diffusion, production, and dissipation. So long as the mean flow features are well captured in the modeling, the k solution also tends to be reasonably grid independent. This measure is used for aeroacoustics in preference to one derived from the turbulence time scale (i.e., the ratio of kinetic energy to its dissipation rate k/ε, which is widely used to estimate the time scales for fully resolved LES). This is more representative of the dissipative scales, which, in frequency and energy content, are, it is hoped, orders removed from the human hearing range or peak sensitivity. Because this measure is derived from a steady-state solution, some limitations are inherent. The frequencies associated with time-varying, large-scale motions such as vortex shedding, which convect through the mesh, will not be accounted for. Instead, its usefulness is to approximate the frequencies of the turbulence scales modeled in RANS that become resolved in LES. In other words, this measure is more valid for the broadband and less so for narrow-band excitations (Mendon¸ca et al. 2005).

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Figure 6.112. Mesh frequency cutoff (MFC) estimate; idealized wing-mirror example of Siegert et al. (1999).

The maximum frequency f MC represents an important step in the analysis process. For this value to be correctly applied and interpreted, certain conditions should be met. First, the mesh should be isotropic (unit aspect ratio) as recommended earlier; in this case, the cube root of the volume becomes an appropriate measure of the hexahedral cell dimension  (appropriate modifications are required for perfect tetrahedra, polyhedra, etc.). Secondly, the boundary conditions applied to k, especially at the inflow, should be realistic. Thirdly, the mesh must be fine enough to capture the important mean-flow features. Most of the cases chosen in this section to support the MFC measure, although quite typical of industrial applications, are lowMach-number flows. Further validations are necessary for transonic and higher-Machnumber flows. With respect to the idealized wing mirror of Siegert et al. (1999), Figure 6.112 shows the local mesh frequency cutoff distribution f MC . As expected, we observe discontinuities in the measure across the mesh refinement interface. Close to the mirror surface, where the grid is most densely packed, the measure suggests that frequencies well in excess of 1 kHz will be resolved. In the separated shear layer, the cutoff frequency is lower (somewhere in the region 800–900 Hz). Practical justification of this measure through comparison with a fully time-accurate transient calculation is given in Section 6.8.3.3 (see Figure 6.113).

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Figure 6.113. Predicted versus measured pressure spectra; idealized wing-mirror example of Siegert et al. (1999).

6.8.3.3 Transient calculations

Transient calculations permit the direct computation of the noise sources (within the limits of the modeling used), and are useful in determining the spectra in the noise-source region directly (Figure 6.113 is typical) or accumulating flow data for compilation of equivalent noise sources for noise propagation to the far field. Figure 6.113 shows the computed versus measured pressure spectra at a point in the wake of the idealized wing mirror. We observe a good qualitative and quantitative (within 2 dB) prediction over a wide frequency range up to about 900 Hz, beyond which the prediction trails off from the measured spectrum. This is indicative of a lack of spatial resolution and is usually manifested by an underprediction of soundpressure level. The energy contained by (unresolved) eddy structures smaller than the grid dimensions is lost in the subgrid scale. Section 4.2.7 demonstrates that there are possibilities for reconstructing the spectral contribution from the subgrid scales using SNGR, for example, but note that such methods also have inherent limitations. It is useful to note that, with the ability to identify the cutoff value from a steadystate calculation, the analyst is afforded a useful additional step in the process. He or she is free to refine the mesh before the transient simulation by direct extrapolation of the MFC measure. For example, if the measure indicates 500 Hz when 1000 Hz is targeted, the user is obliged to increase the spatial resolution locally by a factor of 2 in all directions. With respect to the choice of time-step size, though the Nyquist criterion suggests the minimum frequency resolvable in a time-based simulation is half the inverse of

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the time step, a more conservative recommendation is made here: that is to use onetenth of the inverse of the time step to allow for a better capture of the waveform and amplitude of any oscillating signal. The recommendation is supported by Mendon¸ca et al. (2005), who have shown that a cutoff proportional to the time-step size is observed in the predicted spectra. In contrast to the underpredictions that result from inadequate spatial resolution, the temporal cutoff is manifested in the simulated spectrum by a leveling off in the sound-pressure level. This can be an indication that the solution is unable to distinguish between, or separate out, the energy content of the resolved structures across wave numbers that are higher than the applied temporal resolution allows. The following sections provide an overview of some additional key issues: turbulence modeling in unsteady flow, inlet perturbations, discretization practices, and global-to-local domain boundary mapping. We discuss the practicability of turbulence modeling for industrial applications. Turbulence modeling

We now offer some limited examples of turbulence modeling for transient flows. Despite the title of this book, we start with RANS models applied to unsteady flows, so-called URANS, to provide a suitable comparison with LES-based models. We see that the former have inherent limitations in their ability to predict broadband flow excitations. RANS–LES combinations are capable of overcoming these limitations by invoking LES where it is necessary while retaining the economy of RANS where LES would otherwise be unaffordable. “Necessary,” for aeroacoustics, is taken to mean the bulk flow – particularly separated flow regions rather than boundary layers. The studies of Mendon¸ca et al. (2003), Allen and Mendon¸ca (2004), and Allen et al. (2005) into the M219 high-speed cavity flows note that URANS performs well where flow excitations result in a dominant narrow-band tone but potentially fails where multiple modes of roughly the same order of magnitude exist, or when the flow excitations are broadband or both. In this example, the performance of one particular hybrid approach, DES (Spalart et al. 1997), is compared with a URANS model for a well-documented high-speed cavity flow (Henshaw 2000). A range of mesh sizes has been assessed ranging from 1 to 2.8 million cells, containing 0.8 to 2.5 million cells in the cavity and shear layer, respectively, and showing minimal mesh dependency in the results. Figure 6.114 illustrates the instantaneous flow fields from DES and URANS simulations in a section of the three-dimensional, 4-in.-deep cavity configured without bay doors (L/D = 5, W/D = 1) at M = 0.85. Both calculations were performed on the same mesh, which was designed for URANS with wall functions, using the same timestep size, 2 × 10−5 s. The computational times for the DES and URANS calculations are virtually identical. Many more eddy structures are prevalent in the DES solution, implying greater fidelity in capturing broadband noise. This is confirmed by the measures of rms pressure along the cavity ceiling, illustrated in Figure 6.115, and spectra at the midpoint along the ceiling shown in Figure 6.116. URANS fails to pick up the second and fourth (Rossiter) acoustic modes observed at

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Figure 6.114. Mach 0.85 cavity: symmetry-plane snapshot at t = 0.3 s; DES/k−ε (top), and URANS/k−ε (bottom).

approximately 380 Hz and 800 Hz, respectively, demonstrating a major failing of the model. DES, by contrast, predicts the background levels and modal behavior extremely well. Inlet perturbations

At the inlet of the computational domain, mean flow parameters are usually well known; these are a necessary requirement for RANS simulations, and we note happily that experimentalists are well attuned to this need and usually provide high-quality data needed for simulation. In LES, by contrast, correct modeling can depend strongly on the perturbations about the mean flow. For example, the primary breakup of the liquid core discharging from a circular duct into the atmosphere, as is typical in a spray nozzle, requires instabilities from the duct flow to initiate breakup of the free surface (Buonfiglioli and Mendon¸ca 2005), as shown in Figure 6.117. Instabilities may be accounted for by a full, transient LES precursor calculation in the duct, but this procedure is inordinately expensive and, for the moment, impractical in any industrial application. Cheaper solutions are becoming better understood and more widely applied. These are based on synthetic excitations, which, unlike “white” noise, are correlated in space and time. One such method, described by Benhamadouche et al. (2003), uses knowledge of the local mean-flow turbulence, k and ε, to impose synthetic vortical structures in the plane of inlet boundary and streamwise perturbations normal to it. The synthetic structures are shown to be self-sustaining and also to generate fully developed turbulence within a few duct dimensions downstream of the perturbed inlet.

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Figure 6.115. Overall (a) and band-limited (b) Prms along cavity ceiling centerline.

Boundary mapping – global to local domain interfacing

Irrespective of the speed of computing, it will always be desirable with LES to focus on localized domains. Localized domain analysis depends crucially on the provision of well-defined boundary conditions – primarily at the inlet with respect to mean profiles. In the following example (Read and Mendon¸ca 2004), mapping of boundary conditions from a steady-state calculation in a global domain to a transient simulation in a localized domain is described. The focus of this case is the aeroacoustic behavior of the Audi A2’s wing mirror. A steady-state calculation was first made around the full vehicle. A localized domain mesh was then generated (seen in dark color in Figure 6.118) containing approximately 3.8 million cells; many more would be required for LES on

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Figure 6.116. PSD (kPa2 /Hz) at location x/L = 0.45.

the full vehicle. The steady-field values were mapped onto the boundaries of the local domain – that is, velocities at the upstream and four adjacent side planes and pressure at the localized domain outlet; all were held at the steady values for the duration of the transient calculation. Figure 6.119 shows a snapshot of the resulting transient DES calculation, which is dominated by large-eddy structures in the wake of the wing mirror. The pressure spectrum at a point on the mirror face is shown in Figure 6.120 and, when compared with the measurement, gives immediate credence to this methodology. Additionally, mapping of the turbulence levels together with synthetic perturbations allows for even nonsteady inlet conditions.

Figure 6.117. Diesel injector primary liquid spray breakup; liquid-free surface with synthetic inlet perturbation (top) and without inlet perturbation (bottom).

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Figure 6.118. Audi A2 full-vehicle geometry with localized domain shown in dark (courtesy of Audi AG).

6.8.3.4 Transient postprocessing

For all transient flow calculation aimed at aeroacoustic sources prediction, we recommend that some basic sanity checks be performed. By exemplification on industrial cases, these sanity checks also justify the recommended processes defined earlier in this section. We address data sampling in an attempt to highlight pitfalls and necessary minimum requirements. Transient data sampling and spectral postprocessing

To illustrate some key features and pitfalls in the processing of transient data, we return to the high-speed cavity case. The unsteady pressures (measured and simulated) are

Figure 6.119. Instantaneous velocity magnitude field (courtesy of Audi AG).

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Figure 6.120. SPL against frequency at Microphone 4 (courtesy of Audi AG).

inherently noisy, as is apparent from Figure 6.121 corresponding to the samples at the downstream end of the cavity at x/L = 0.95. The experimental data (Henshaw 2000) are compiled using 34 windows (rectangular boxcar with zero overlap), each of 0.1 s duration and corresponding to approximately 50 freestream passes over the length of the cavity. Overlapping of the data, using for instance Hanning windowing with 50% overlap, has negligible effect. From the rms pressure curves of the experimental data in Figure 6.122, it is evident that the minimum sample of 1.0 s must be processed to obtain a result comparable to the original 3.4-s sample. From these assessments, we can begin to make the following recommendations for this case. LES simulations should provide data equivalent to an elapsed time of 0.5 s, allowing the initial 0.1 s to be discarded and the remaining 0.4 s to be processed. This translates into the first 50 freestream passes being discarded and a minimum of 200 further passes required for statistically steady data to be produced. Shorter samples should be interpreted with care. Figure 6.123 shows the Prms curves from the simulation (DES) over a range of sample periods averaged in the same manner as the experimental data but with the first 0.1 s of the sample discarded to eliminate startup effects. The curves begin to converge from the 0.2-s sample onward, and a sample with a total duration of 0.4 s is required to obtain a negligible level of deviation between windows.

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Figure 6.122. Sampling effects on overall Prms (kPa) along the cavity ceiling for M219 experimental data.

6.8.4 Postprocessing through acoustic coupling

As stated in Section 6.8.2.1, the assumption of the hybrid nature of aeroacoustics allows the separation of these two physical mechanisms (source generation and propagation) during simulation. Two classes of simulation method are enumerated. The first involves integrals (volume, surface, line) using information that comes directly from transient or even steady-state CFD calculations that are enumerated here

Figure 6.123. Sampling effects on overall Prms (kPa) along the cavity ceiling for CFD data.

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mainly through reference for brevity. Some limitations are inherent; that is, they do not account for multiple surface reflections; they do not allow for surface or volume attenuation (damping) and therefore are not generally advisable for internal flows; and they do not generally allow for refraction effects owing to interaction of the acoustic waves with nonuniform mean flow (particularly shear layers). Morris and Farassat (2002) described the use of acoustic analogies and alternative methods for jet noise predictions, all of which are based on steady-state (RANS) k−ε-based prediction, and concentrate on directionality in the prediction of the propagated jet noise spectrum. Also based on RANS calculations, Agarwal and Morris (2004) demonstrate propagation of noise generated in the slat-cove region of a high-lift device through the mean flow into the far field using only three empirical constants. Greschner et al. (2004) have made a study of different RANS-based DES calculations of the noise generated and propagated in a complex transient interaction between a cylinder wake and downstream aerofoil, illustrating the use of “permeable” Ffowcs Williams–Hawkings surface integrals. The second class involves frequency-domain-boundary or finite element computations formulated around acoustic analogies, that compute the propagation, including reflections and surface-volume damping, of aeroacoustic sources that have been derived from transient CFD calculations. Recently, efforts have been made to incorporate interaction of the acoustics with mean-flow variations, but as yet these have not been validated fully. The next section presents some key applications using this class of source-propagation coupling. 6.8.4.1 Frequency-domain methods

Frequency-domain methods offer an efficient methodology for noise propagation simulation using commercially available software. They require input of equivalent sources such as monopoles, surface or rotating dipoles, and quadrupoles. It is typical for the latter two types to be compiled from synthesized sources or at best from transient LES simulations that contain modeled contributions from both narrow-band and broadband sources. Surface and volume sources

In the following example describing one-way coupling between a precursor CFD calculation and frequency-domain acoustics postcalculation, important features are demonstrated with the primary objective of simulating resonance in a cavity. r First, the CFD calculation has the following characteristics:

– It captures broadband and narrow-band excitations. – The frequency of the narrow-band oscillation (due to the shear-layer fluctuations at the neck of the cavity) is linearly proportional to the cavity bypass velocity. – The amplitude of the narrow-band oscillations maximizes at the cavity resonance frequency.

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Figure 6.124. Resonator geometry: application challenge from BEHR GmbH in the DESTINY-AAC (Detached Eddy Simulation for the Transportation Industry – Aero Acoustics) project (courtesy of BEHR GmbH).

– The underlying second-order numerics are insufficient to amplify any excitation at the cavity resonance frequency. Virtual microphones in the cavity do not detect any amplification of power at the cavity resonance frequency except when the shear-layer oscillation corresponds with it. r Second, the characteristics of the acoustics calculation are as follows: – Volume sources from the CFD containing both broadband and narrow-band power are exported onto the acoustics mesh. – Subsequent solution of the propagation of the sources reproduces an amplification of the acoustic power at the cavity resonance frequency even though the shear layer oscillates at a different frequency. Figure 6.124 shows a schematic of the rig and typical flow conditions. On the basis of the connecting rod length and diameter, box volume, and orifice correction L = πr/2, a resonance frequency of 358 Hz found analytically using ) π · r2 a · (6.70) f0 = 2π V · (l + l) agrees closely with the measured resonance. In the CFD simulation, transient effects are dominated by the oscillation of the shear layer at the neck of the resonator and subsequent downstream shedding (see velocity contours in Figure 6.125). The system responds to three upstream bulk velocities (4, 8, and 12 m/s) with a near-linear increase in the frequency of the shear-layer oscillation (183, 305, and 538 Hz, respectively), but nonlinear influence on magnitude of the fluctuations. The maximum of the three amplitudes occurs at the intermediate velocity, wherein the frequency is closest to the resonator’s resonant frequency, clearly demonstrating the effects of compressibility in the interplay between velocity and pressure in the CFD solution.

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Figure 6.125. Velocity contours (top), pressure–time traces (bottom left), and spectral magnitude (bottom right) at three bulk velocities (4, 8, and 12 m/s) taken at the neck of the resonator (courtesy of BEHR GmbH).

Coupling between a CFD solution (here obtained by STAR-CD) and a commercial acoustics code (here ACTRAN) requires interpolated values of the Lighthill tensor at the nodes of a volume acoustics mesh (Caro et al. 2005). The subsequent frequency-domain calculation for the 8 m/s inlet bulk-velocity case, computed here with a resolution of 25 Hz, shows, in Figure 6.126, a magnification of the acoustic pressure at the resonant frequency. This result requires that excitation from the CFD contain broadband power spectral distribution with nonzero power at the resonant frequency. The only mechanism through which this is possible is LES turbulence modeling, which is here achieved using DES. Fan sources

In this final example, the noise source and subsequent radiation from a subsonic compressor fan is modeled. The noise from the blower fan, containing 47 blades rotating at 3770 rpm in a typical automotive heating, ventilation, and air-conditioning system was assessed in isolation (see Figure 6.127). Experimental measurements were taken in the semianechoic chamber at Denso Thermal Systems at the nine far-field points illustrated in Figure 6.127 around the rig comprising the blower housing (scroll), electric motor, cylindrical inlet duct, and a rectangular outlet duct with a filter at the outlet.

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Figure 6.126. Acoustic response for 8 m/s case at the microphone (courtesy of BEHR GmbH and Free Field Technology, Caro et al. 2005).

A mesh consisting of approximately 1.5 million cells was initially run steady-state using the multiple-rotating frames of reference method and then restarted transiently using the k–ε variant of DES with the mesh around the fan blades rotating relative to the static volumes across a sliding interface. The flow field evolved to a pseudosteady condition after approximately eight full rotations of the fan. Data were then output for propagation in the commercial acoustic code SYSNOISE to the far-field observer locations. A generalized Ffowcs Williams–Hawkings method was used. It requires the storing of forces on an individual blade of the fan (taken to be representative) over one complete rotation of the fan. Fuller details can be found in Barone et al. (2003). Sources from the housing surface and the volume attenuation effects of the filter were

Figure 6.127. Experimental prototype with inlet cylinder and outlet filter removed and locations of far-field monitors (courtesy of Denso Thermal Systems).

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Figure 6.128. Comparison of steady-state RANS and snapshots from the DES calculation (courtesy of Denso Thermal Systems; Barone et al. 2003).

neglected. Figure 6.128 shows a velocity magnitude in a section through the blower from both the RANS and DES calculations at different times during one rotation of the fan. The capture and subsequent convection of large eddies can clearly be seen in the DES snapshots. Figure 6.129 shows the predicted sound fields from the acoustics code on the exterior of the blower housing and noise propagated to far-field locations. Figure 6.130 gives a comparison of the computed result and experiment for the blade-passing frequency (BPF). Agreement within 5 dB is observed. More importantly, the directional variation of noise magnitude is correctly modeled. (a)

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Figure 6.129. Surface acoustic pressure (Pa) on the exterior model (a) and acoustic pressure in the far field (b) (courtesy of Denso Thermal Systems and LMS International; Barone et al. 2003).

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Figure 6.130. Computed and measured dB(A) levels at the nine microphone locations at the bladepassing frequency (BPF) (courtesy of Denso Thermal Systems and LMS International; Barone et al. 2003).

6.8.5 Conclusions for industrial aeroacoustics analyses

Various techniques for adding value to the aeroacoustics calculations through CFD and acoustics postprocessing have been assessed. These include steady-state analyses of noise source locations, their relative magnitudes, and synthesized velocity fluctuations. For transient calculations, the importance of using appropriate turbulence modeling for tonal and broadband flow excitations has been stressed as has the need for thoughtful meshing and calibration of discretization practices on the subgrid model. Justifications have been offered for the use of the approximate steady-state methods through equivalent transient validations and, in particular, the use of a steady-state MFC analysis in the assessment of mesh suitability for capturing acoustic-generation mechanisms in the desired frequency range. Even though the theories and methodologies applied to LES and aeroacoustics are constantly evolving, the processes described herein (especially Section 6.8.3) offer a framework in which to exploit CFD calculations on industrial applications in a controlled and thoughtful manner, each step adding a little more knowledge toward the understanding of the aeroacoustics signature of complete systems or their components. The usefulness of coupling CFD to acoustic propagation has been demonstrated, showing that one-way interaction between the flow and acoustics for a variety of industrial applications can be sufficient.

6.8.6 Acknowledgments

The author acknowledges the involvement of Airbus, BAE SYSTEMS, QinetiQ, Bombardier Transportation, Audi, BMW, DaimlerChrysler, Opel AG, Air International, BEHR, Denso, and Valeo, all of which have contributed significantly in the form of experimental facilities, measurement data, people time, engineering insight, and project

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funding. Willingness on the part of CFD code vendors to pursue detailed studies bridging research methods and commercial utilization on industrial cases and investing in increased accuracy and robust validated modeling practices has been essential. The benefits are embodied in CFD results from the commercial code STAR-CD (2004) reported extensively here. In particular, the author is grateful to colleagues A. Read, R. Allen, M. Buonfiglioli, and collaborators T. Rung, K. Debatin, V. Joshi, F. Brotz, M. Schrumpf, M. Islam, F. Klimetzek, F. Barone, P. Durello, F. Dubief, F. Werner, and A. Senf.

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Conclusions Claus Wagner, Pierre Sagaut, and Thomas Huttl ¨

Noise is becoming generally accepted as an environmental and even health hazard to the population. Many noise sources are man-made – especially transportation noise from road traffic, aircraft, and trains. Other technical noise sources can also be annoying such as wind turbines or cooling and climate systems. Governmental reactions to noise problems and loss of public acceptance are laws, regulations, decrees, and guidelines for the certification of noise-emitting vehicles and machines as well as temporal or spatial limitations of their use. Aircraft and jet engine manufacturers in particular face increasingly stringent noise requirements for near-airport operations worldwide. Aerodynamic noise is one of the major contributors to external vehicle noise emission. It also contributes to internal vehicle noise owing to the transmission of the externally generated noise through structure and window surfaces into the cabin. Aerodynamic noise becomes dominant at driving speeds exceeding 100 km/h when compared with structure-borne, power train, and tire noise for which substantial noise reduction has been achieved. The interaction of the flow with the geometrical singularities of the vehicle body produces unsteady turbulent flows, often detached, resulting in an increased aerodynamic noise radiation. To achieve these noise reductions, the European Commission, for example, has laid out a series of research objectives. In order to meet the challenging goals proclaimed, the design process needs to be supported by computer-based noise prediction tools.

7.1 Governing equations and acoustic analogies In principle, the way to predict aerodynamic noise generation and propagation is straightforward. The governing equations are those of mass, momentum, and energy that involve more unknowns than equations. The additional information needed to obtain a complete set of equations is provided by empirical information in the form of constitutive equations. An excellent approximation can be obtained by assuming the fluid to be locally in thermodynamic equilibrium. This implies, for a homogeneous fluid, that two intrinsic-state variables fully determine the state of the fluid. For acoustics it is 378

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convenient to choose the mass density ρ and the specific entropy s as variables. When there is no source of entropy, the sound generation is dominated by the fluctuations of the Reynolds stress. With respect to local thermodynamic equilibrium, it is reasonable to assume that transport processes are determined by linear functions of the gradients of the flow-state variables. This leads to a Newtonian fluid behavior and to the fully compressible and unsteady Navier–Stokes equations, which can be numerically solved by means of direct numerical simulation (DNS) without any additional physical assumptions or models. All the dynamically active scales of motion must be represented in the simulation to ensure reliable results. This means that the grid spacing x and the time step t must be fine enough to capture the dynamics of the smallest scales of the flow down to the Kolmogorov scale and that the computational domain must be large enough to represent the largest scales. These criteria lead to a high computational cost, which is responsible for the fact that DNS is nowadays almost only used for theoretical analysis and accurate understanding of flow dynamics and is not a “brute force” engineering tool. A classical technique for reducing the complexity of the simulation (and then lowering the computational effort) is to apply an averaging–filtering procedure to the Navier– Stokes equations, yielding new equations for a variable that is smoother than the original solution of the Navier–Stokes equations because the averaging–filtering procedure removes the small scales and high frequencies of the solution. Although the high frequencies are no longer captured by the computation, their action on the resolved scales can be taken into account via the use of a statistical model. The most popular averaging and filtering operations lead to the Reynolds-averaged Navier–Stokes (RANS) equation, which relies on a statistical average leading to steady computations in the general case. Unsteady RANS (URANS) can also be obtained by applying the statistical average of a conditional or phase-averaging procedure, or both. Note that the RANS approach does not allow an explicit control of the complexity of the simulation because the cutoff frequency can not be specified during the averaging procedure. The second popular filtering approach leads to the large-eddy simulation (LES) technique, which is based on a filtering operator that results in unsteady 3D computations. The filtering procedure can be explicitly associated with the application of a convolution filter to the DNS solution or implicitly imposed by numerical errors, the computational mesh, or modeling errors, or even by the blending of these two possibilities. The requirement of controlling the numerical error appears more stringent for LES than for DNS because the LES cutoff is supposed to occur within scales that are much more energetic than for DNS, leading to a much higher level of numerical error. This may become very problematic if the numerical scheme introduces some artificial dissipation (artificial viscosity, upwind scheme, filter, etc.) because the amount of numerical nonphysical dissipation may happen to overwhelm the physical drain of resolved kinetic energy associated with the energy cascade. Practical experience reveals that the best results are obtained using centered, nondissipative schemes, which make it possible to capture a much broader resolved band of scales than dissipative methods; however, because nondissipative schemes do not provide stabilization, they require much finer grids

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to ensure numerical stability. Centered schemes introduce dispersive errors instead of dissipative errors. Although statistical modeling is in general not suitable for aeroacoustic prediction owing to the removal of the unsteadiness, LES provides the unsteady flow with the use of a subgrid-scale model. Several subgrid-scale models are available that can be classified according to the assumption made during their derivation. Functional modeling uses the dynamics of the isotropic turbulence as a basis. Both theoretical and numerical studies show that the net effect of the subgrid scales is a drain of kinetic energy from the resolved scales. A simple way to account for that net drain of energy is to parameterize it as an additional dissipation. This is done by defining an eddy viscosity in the same spirit as in RANS modeling. Note that subgrid viscosity models for incompressible flows are relevant models for the deviatoric part of the subgrid tensor only. As a consequence, the subgrid kinetic energy, which is tied to the trace of the subgrid tensor, is not a direct output of these models and must be modeled separately if it is required for some physical analysis. The second class of subgrid-scale models use structural modeling. Many ways of reconstructing or approximating subgrid scales have been proposed that rely on either approximate deconvolution, scale similarity, or deterministic vortex models. Deconvolution models are not able to reconstruct scales smaller than the grid spacing x and so are not able to account for nonlocal energy transfer across the cutoff. As a consequence, they must be supplemented by another model specially designed for this purpose. A simple way to do this is to use functional models of the eddy-viscosity type, which are based on the description of the kinetic energy transfers associated with nonlocal interactions. Thus, a full deconvolution model is obtained by operating a linear combination of a deconvolution-type structural model with a functional model of the eddy-viscosity type. All the eddy-viscosity models exhibit a constant that was set when the isotropic turbulence case was considered. An idea to minimize modeling errors is to adjust that constant at each point and at each time step to obtain the best possible adaptation of the selected subgrid model to the local state of the resolved field. This can be done using the dynamic procedure, which relies on the Germano identity. For more complex applications and industrial problems, coupling of acoustic and aerodynamic prediction has been attempted to several hybrid RANS–LES approaches that show promise for computational aeroacoustics problems. Various new ones have been described, ranging from global ones that solve a single set of equations throughout the entire domain to zonal approaches that explicitly impose pure RANS or pure LES in individual zones according to some initial domain decomposition. In comparison with traditional LES, hybrid RANS–LES methods are intended to allow larger mesh spacings and hence larger time steps, and thus they can result in a smaller portion of the total frequency spectrum being directly resolved. This is not a weakness of the hybrid models themselves but simply an expected result of exploiting the coarser spatial and temporal resolutions with which hybrid methods are able to operate. The issue of unresolvable noise sources has been discussed, and two possible approaches to help extend the range of predicted frequencies have been described. Both approaches involve synthetic reconstruction of an unsteady velocity field that reproduces

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key properties of the underlying (statistically represented) turbulence. One approach considered was a nonlinear acoustics solver using a stochastic subgrid model; the other involves augmenting the frequency spectra of traditional LES or hybrid RANS–LES to separately model the generation and transmission of the noise from unresolvable scales. Finally, we note that the choice of both numerical method and subgrid-scale closure is expected to continue to play a crucial role in LES, hybrid RANS–LES, and acoustics. In most existing LES or hybrid RANS–LES codes, the subgrid-scale model and numerical flux treatments are independent of one another, typically resulting in a mixing that is too strong. Aerodynamic noise occurs because of two basically different phenomena. The first one is impulsive noise – a result of moving surfaces or surfaces in nonuniform flow conditions. The displacement effect of an immersed body in motion as well as the nonstationary aerodynamic loads on the body surface generate pressure fluctuations that are radiated as sound. This kind of noise is deterministic and relatively easy to extract from aerodynamic simulations because the required resolution in space and time to predict the acoustics is similar to the demands from the aerodynamic computation. This noise appears primarily with rotating systems such as helicopter rotors, wind turbines, turbine engine fans, and ventilators. In particular, if the surfaces move at speeds comparable to the speed of sound or there is an interaction between a rotor and a stator wake, these tonal noise components can be dominant. The other noise mechanism is the result of turbulence and therefore arises more or less powerfully in nearly every engineering application because turbulence is by its very nature stochastic and therefore has a broad frequency spectrum. Interestingly enough, turbulent energy is converted to acoustic energy most efficiently in the vicinity of sharp edges as, for example, at the trailing edge of an aircraft wing. In this case the uncorrelated turbulent eddies flowing over the upper and lower sides of the edge have to relax with each other, generating very locally strong equalizing flows that result in highly nonstationary pressure spikes. Another major source of turbulence sound is jet flows, in which the shear layer in the mixing zone again radiates into the far field. Given the nonlinearity of the governing equations, it is very difficult to predict the sound production of fluid flows. This sound production occurs typically at high-speed flows for which nonlinear inertial terms in the equation of motion are much larger than the viscous terms (high Reynolds numbers). Because sound production represents only a very minute fraction of the energy in the flow, the direct prediction of sound generation is very difficult. This difficulty is particularly dramatic in free space and at low subsonic speeds. Solving the complete, fully coupled, compressible Navier– Stokes equations requires tremendous computational resources because – especially on small-Mach-number flows – flow and acoustics represent a multiscale problem with its inherent difficulties. The problem is that the small acoustic perturbations are not drowned out by numerical errors of the much larger aerodynamic forces. Space and time resolution for the aerodynamic data combined with the large regions up to an observer in the far field give rise to ridiculously high numbers of cells and time steps; however, even if the computer power were available, the discretization schemes well known

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CONCLUSIONS

from numerical fluid dynamics do not work very well in computational aeroacoustic applications because they usually have higher dispersion. Typically a plane wave is severely distorted and sometimes dampened after being transported along a distance of just a few wavelengths. Such a distance is clearly too short for the common case of an observer in the far field. A more promising approach for technically relevant aeroacoustic problems is to apply hybrid methods. With this approach the near-field aerodynamics are computed to obtain velocity and pressure fluctuations, which form the acoustic source terms for a separate computation of the far-field acoustics. The reason for splitting off aerodynamics from aeroacoustics is the great disparity of levels and length scales between the flow and aeroacoustic fields. In acoustics, one considers small perturbations of a flow. This allows linearizing the conservation laws and constitutive equations. To derive a wave equation for the prediction of noise propagation, one not only linearizes the basic equations but friction and heat transfer are also neglected. Using Green’s theorem we can obtain an integral equation that includes the effects of the sources, the boundary conditions, and the initial conditions on the acoustic field. Green’s function is defined as the response of the flow to an impulsive point source. To determine the source from any measured acoustical field outside the source region, we need a physical model of the source. This is typical of any inverse problem in which the solution is not unique. When using microphone arrays to determine the sound sources responsible for aircraft noise, one usually assumes that the sound field is built up of so-called monopole sound sources; however, sound sources are more accurately described in terms of dipoles or quadrupoles. Under such circumstances it is hazardous to extrapolate such a monopole model to angles outside the measuring range of the microphone array or to the range of flow Mach numbers other than used in the experiments. It is often stated that Lighthill has demonstrated that the sound produced by a free, turbulent, isentropic flow has the character of a quadrupole. The reason is that, because in many flows there is no net volume injection owing to entropy production nor any external force field, the sound field can at most be a quadrupole field. Therefore, Lighthill’s statement is actually that we should ignore any monopole or dipole emerging from a poor description of the flow. Sound, for example, produced by turbulence in a free jet has the character of a quadrupole distribution field. Because the vortices that produce the sound are convected with the main flow, there will be a significant Doppler effect at high Mach numbers. This results in a radiation field mainly directed about the flow direction. Owing to convective effects on the wave propagation, the sound is, however, deflected in the shear layers of the jet. This explains that, along the jet axis, there is a so-called cone of silence. We can use the Green’s function formalism to determine the effect of the movement of a source on the radiated sound field. The problem of a source, observer, and scattering objects moving together steadily in a uniform stagnant medium is the same as the problem of a fixed source, observer, and objects in a uniform mean flow.

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Lighthill proposed an approach to the case of an arbitrary source region surrounded by a quiescent fluid. The key idea of Lighthill was to derive from the exact equations of mass conservation and momentum conservation a nonhomogeneous wave equation that reduces to the homogeneous wave equation in a region surrounding the listener. When the entropy term and the external forces are negligible, the flow will only produce sound at high velocities corresponding to high Reynolds numbers. He therefore assumed that viscous effects are negligible and reduce the sound source to the nonlinear convective effects. Hence, we can calculate the source term from a numerical simulation that ignores any acoustic wave propagation and subsequently predicts the sound production outside the flow. In extreme cases of low-Mach-number flow, a locally incompressible flow simulation of the source region can be used to predict the sound field. One can state that such an integral formulation combined with Lighthill’s analogy allows us to obtain a maximum of information concerning the sound production for a given fact flow field. A spectacular example of this is Lighthill’s prediction that the power radiated to free space by a free, turbulent, isothermal jet scales as the eighth power U08 of the jet velocity. The first step in making Lighthill’s analogy useful is to identify a listener around which the flow behaves like linear acoustic perturbations and is described by the homogenous wave equation. This is an assumption valid in many applications. When we listen, under normal circumstances, to a flute player we have conditions that are quite reasonably close to these assumptions. At this stage, the most important contribution of Lighthill’s analogy is that it generalizes the equations for the fluctuations ρ  and p  to the entire space even in a highly nonlinear source region. Then, the next step is to introduce approximations to estimate the source terms. The integral formulation of Lighthill’s analogy can be generalized for flows in the presence of walls. To do so, Curle used the free-space Green’s function and, instead of the pressure p  as aeroacoustical variable, the density ρ  . Using his theory, we understand easily that a rotor blade moving in a nonuniform flow field will generate sound owing to the unsteady hydrodynamic forces on the blade. At low Mach number this will easily dominate the Doppler effect because of the rotation. Wind rotors placed downwind of the supporting mast are cheap because they are hydrodynamically stable. There is no need for a feedback system to keep them in the wind. However, the interaction of the wake of the mast with the rotor blades causes dramatic noise problems. Although the formulation of Curle just discussed assumes a fixed control surface S, the formulation of Ffowcs Williams and Hawkings allows the use of a moving control surface S(t). The key idea is to include the effect of the surface in the differential equation. For the formulation of Ffowcs Williams and Hawkings describing the effect of moving boundaries, p  is used again as the aeroacoustical variable. In acoustics this would have been immaterial because the two variables are related by the equation of state. Actually, in aeroacoustics there is a subtle difference that appears when we compare the source terms of the two wave equations. In that case, we see that when p  is used as the aeroacoustical variable, the effect of entropy fluctuations has the character of a monopole sound source. On the other hand, when ρ  is used, the apparently same effects produce a quadrupole distribution that is qualitatively different. Of course, there

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CONCLUSIONS

is no difference if we consider the exact equations, but if we do not introduce any approximation the analogy is just a reformulation of basic equations without much use. Clearly, we have to be careful in selecting the aeroacoustic variable. Other choices of aeroacoustic variables lead to different analogies; however, such analogies become quite obscure. They do not, moreover, provide much intuitive insight and can only be used numerically. In many cases such analogies tend to avoid the problem induced by the inability of the Lighthill analogy to distinguish between propagation and production of sound waves in a strongly nonuniform flow, which induces refraction. This becomes very important in supersonic flow. In such cases the source is not compact. One of the problems is that the sound source deduced from Lighthill’s analogy is spatially rather extended, leading to slowly converging integrals. For low-Mach-number isothermal flow the aerodynamic sound production is entirely due to mean-flow velocity fluctuations, which may be described directly in terms of the underlying vortex dynamics. This is more convenient because vorticity is in general limited to a much smaller region in space than the corresponding velocity field (Lighthill’s sound sources). This leads to the idea of using an irrotational flow as reference flow. The result is called vortex sound theory, which is not only numerically efficient but also allows us to translate the very efficient vortex-dynamical description of elementary flows directly into sound production properties of these flows. There are techniques that employ an integrated form of the relevant acoustic propagation equation – that is, either Kirchhoff’s surface integral or the Ffowcs Williams– Hawkings (FW–H) equation. In this case the sound pressure at an observer at a specific point in time is computed by an integration-of-source term along a surface – either a physical one or surrounding the aerodynamic area – and possibly additional volume integrals outside the surface in the case of the FW–H equation. Owing to the finite speed of sound and the deterministic relationship between emission and observer time of a signal, there has to be some kind of interpolation of the data at least on one side. In the case of parts of the integration surface or volume moving at transonic speeds, the integrals become highly singular because of the Doppler effect, which leads to difficulties regarding the numerical stability of the procedure. When the flow is confined, the acoustical energy can accumulate in resonant modes. Because the acoustical particle displacement velocity can reach the same order of magnitude as the main flow velocity, the feedback from the acoustical field to the sound sources can be very significant. This leads to the occurrence of self-sustained oscillations we call whistling. In spite of the backreaction, the ideas of the analogy appear to remain useful.

7.2 Numerical errors The main problem of computational aeroacoustics (CAA) is the disparity of energy, length, and time scales between the aerodynamics and the aeroacoustics – especially at smaller Mach numbers. In an aerodynamic simulation, we of course introduce

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7.3 INITIAL AND BOUNDARY CONDITIONS

numerical errors. To be at the same level as the acoustics we are interested in, these errors have to be five to nine orders of magnitude smaller than the intended physical values. To obtain acceptable signal-to-noise ratios for sound levels, we have to add at least one more order of magnitude. In this sense, basically every flow simulation is very loud, counting just the numerical errors that introduce numerical noise. Common computational fluid dynamics (CFD) schemes are adapted to stationary simulations and therefore just suppress acoustic waves; thus, for the aerodynamic community the problem seems to be solved. This means that, for CAA, diffusion and dispersion errors have to be reduced to the lowest possible level. This is of major importance because the amplitudes of acoustic waves are several orders of magnitude smaller than the average aerodynamic field amplitudes. In addition, their length scales, typically the principal acoustic wavelengths, are some orders of magnitude larger than the dimensions of the sound-generating perturbations (vortices and turbulent eddies). Moreover, sound generated by turbulence is broadband noise with often three orders of magnitude difference between the largest and the smallest acoustic wavelengths. Finally, acoustic waves propagate at the speed of sound (which is not necessarily comparable to the mean flow velocity) over large distances in all spatial directions, whereas aerodynamic perturbations are only convected by the mean flow. Furthermore, one is usually interested in the noise level at the far field, implying that the waves have to be traced accurately over long distances. Similar arguments apply to the temporal discretization. The acoustic waves have to be tracked accurately in time. Again, more important than the temporal order of accuracy of the time integration method are its dispersive and dissipative behavior. For the solution to be accurate, these errors have to be minimized as much as possible.

7.3 Initial and boundary conditions At the very heart of CAA and CFD lies the proper handling of boundary conditions. Because acoustics is a radiation problem, basically all the sound energy will sooner or later try to leave the computational domain. On solid walls, reflections obviously arise, which we can handle in a straightforward manner just as in CFD. However, on artificial far-field boundaries the physics dictates a straight pass-through without any spurious reflections. Although this requirement seems to be obvious, it is indeed very hard to fulfill sufficiently. Several concepts have been developed for this specific problem optimized for one application or another, but the downside is that a generally accepted solution does not yet exist. As far as accuracy is concerned, the perfectly matched layer (PML) technique appears to be the most accurate among all the methods reviewed above. However, the PML equations are not yet available for every type of nonreflecting boundaries likely to be encountered in practical situations. So far, they have only been developed for a parallel mean flow in a direction aligned with one of the coordinates. The technique is still very much under active development. In this regard, the alternatives are the

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CONCLUSIONS

absorbing-zone techniques that can be applied to a wider class of problems and are often coupled with the characteristic boundary conditions. For problems with a centralized and compact noise source, the radiation condition offers an effective alternative. With a few exceptions such as predictions of the transition process or fundamental turbulence investigations, in most cases the initial conditions play a subsidiary role because the statistically steady-state flow status, which should be reached independently of the initial conditions, is of major concern. An example of an exception is the decay of a homogeneous isotropic flow often used for basic investigations in turbulence research. For this purpose a cubical integration domain with periodic boundary conditions in all directions is chosen requiring the initialization of the flow field. On the basis of the scalar energy spectrum a homogeneous isotropic and divergence-free vector field is generated in spectral space and then transformed back to the physical space by fast Fourier transformation. This procedure leads to reasonable initial conditions with prescribed values for the turbulent kinetic energy, the dissipation rate, and the smallest and largest wave numbers of the flow field.

7.4 Examples From the presented examples we conclude that direct noise computation is an outstanding approach for the sound generated by free-shear flows because it permits turbulence events to be correlated with the sound far field. In this context, LES seems the better tool to clarify Reynolds-number effects – especially for subsonic jet noise. The dependence on Reynolds number is particularly expected for the noise-generation mechanisms observed in mixing layers whose initial state may be either transitional or turbulent. The suitability of LES to the prediction of jet noise has been well established for cold, high-subsonic jets. The introduction of high-bandwidth, high-accuracy numerical techniques has allowed for the computation of both the unsteady, near-field turbulence found in a jet exhaust plume and its radiated sound. Numerous research groups have reported agreement between their LES results and experimental data for both the near-field fluctuations and the far-field sound characteristics for high-subsonic jets at moderate Reynolds numbers. The most recent investigations have exhibited the correct acoustic spectral characteristics over a range of frequencies. Ongoing investigations are exploring the role the subgrid-scale (SGS) model and the grid resolution has on the radiated sound. Future work is focused on extending the LES results to higher frequencies through the development of SGS noise models. For high-Reynolds-number flows, the dynamics of the small scales and the intermittence of the turbulence may lead to the coexistence of different modes, the flow switching between them. This phenomenon can hardly be described by RANS methods and is better suited to LES. The first complete aeroacoustic computations are very recent. The large-eddy simulations described here reveal the dependence of the results on the resolution and on the SGS model. The radiated field induced by the cavity

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7.4 EXAMPLES

flow is greatly sensitive to small flow modifications such as the concentration of the coherent structures. The computation of the correct acoustic levels in the far field is thus a challenging problem. It requires subsequent investigations of the role of the SGS models and on the effects of the numerical resolution. Significant progress has relied on direct comparisons with experimental databases. Such databases with aerodynamic and acoustic information have been lacking and are just beginning to emerge. As an example, it was also demonstrated that the Radioss CFD–CAA code is capable of predicting aeroelastic and vibroacoustic phenomena due to turbulence in the wake of a ruler. The presented approach exhibits accurate predictions below about 700 Hz. For higher frequencies, the levels are underestimated. A refined mesh in the streamwise flow provides somewhat better results, but the poor spanwise resolution of the model is believed to limit the development of structures smaller than 0.05 m (corresponding to frequencies higher than 500 Hz). Another research track is to evaluate the dissipation of wall functions. The fully coupled numerical approach presented here provides results opposite to those obtained by a hybrid method combining modal analysis for the plate and the cavity with a semiempirical model of the wall-pressure loading, which overestimated the acceleration and acoustic radiation of the plate above 700 Hz. The main advantage of a compressible arbitrary Lagrangian–Eulerian–CFD code like Radioss CFD– CAA is its ability to solve the following in a single simulation: flow field; noise; source generation; propagation, including model behaviors; and radiation and transmission to structures. The whole three-step CAA process was finally applied to the NACA-0012 case, including a far-field noise prediction performed by use of a Kirchhoff integration based on the pressure field and its normal derivative on the external boundary of the Euler domain. It was shown that the wave fronts are continuous from one domain to another. An interesting point is that acoustic waves emitted by the airfoil’s leading edge are hardly visible. This confirmed that the leading edge acts as a geometrical singularity on which incident acoustic waves coming from the trailing edge are scattered. Because the computational requirements of an LES in conjunction with the linearized Euler equations of the far-field noise are too high, most calculations on cylinders and cars have been done so far with the acoustic analogy (mainly with FW–H). Experience in computational aeroacoustics shows that, in the far field, the accuracy of acoustic signal is determined by the accuracy of the flow-field prediction rather than by acoustic analogy. Although it is well known that URANS equations cannot explore unsteady flow accurately, they are still very popular in computational aeroacoustics because of the relatively low CPU time and memory requirements. URANS equations resolve only a narrow band of present frequencies in the flow and average the rest of the frequency domain. Direct numerical simulation has to be carried out to resolve all of the fluctuations. However, because DNS requires very fine grid resolution and therefore very high CPU time and memory, today it is only affordable for very low Reynolds numbers and simple geometries. A good compromise is LES, which resolves a fairly large range of frequencies and models only a small part of the flow. It is commonly

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CONCLUSIONS

assumed that the unresolved part of the flow consists of very small eddies that exhibit a homogeneous and isotropic structure. As long as this assumption is valid, even an algebraic model, like the SGS model from Smagorinsky, provides reasonable results. To keep this assumption valid, one must resolve all important scales, which is an affordable task for small Reynolds numbers and simple geometries.

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APPENDIX A

Nomenclature

A.1 Symbols A1 , A2 B1 , . . . , B7 c C Cf E G J −1 k kc = π/ L M Msgs Mt p qT R Re T Tη u Uk uτ

subgrid stress tensors in the compressible momentum equation subgrid stress tensors in the compressible energy equation speed of sound cross-stress tensor skin friction coefficient total energy convolution filter kernel Jacobian of the curvilinear grid transform wave number LES cutoff wave number Leonard tensor Mach number subgrid-scale Mach number turbulent Mach number pressure heat flux subgrid Reynolds stress tensor Reynolds number temperature Kolmogorov time scale velocity vector kth component of the contravariant flux friction velocity

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A.2 Greek symbols   

LES cutoff length

test filter cutoff length x, y, z mesh size in direction x, y, z t time step of the computation ε kinetic energy dissipation rate numerical dissipation rate εnum subgrid dissipation rate εsgs η Kolmogorov length scale κ molecular diffusivity μ dynamic viscosity ν kinematic viscosity ω frequency ρ density τ subgrid stress tensor viscous stress tensor τv θ LES cutoff time

A.3 Mathematical operators ui φ φ  φ  φ

φ+  E τv q˜T ⊗

ith component of the velocity vector filtered part of φ subgrid part of φ mass-weighted filtered part of φ Fourier transform of φ φ expressed in boundary layer wall units synthetic filtered total energy synthetic filtered viscous stress tensor synthetic filtered heat flux tensorial product convolution product

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APPENDIX B

Abbreviations

AIAA ALE ALESSIA APE ASD ASME ASMO AUSM b.c. BC BEM BL BMBF CAA CEAS CFD CFL CNRS CPU CSD CUHD CULD DES DESTINY DFG DG DGLR

American Institute of Aeronautics and Astronautics arbitrary Lagrangian–Eulerian formulation application of large-eddy simulation to the solution of industrial problems acoustic perturbation equations artificial selective damping American Society of Mechanical Engineers Association for Structural and Multidisciplinary Optimization advective upstream-splitting method boundary conditions boundary conditions boundary element method logarithmic buffer layer Bundesministerium f¨ur Bildung und Forschung computational aeroacoustics Confederation of European Aerospace Societies computational fluid dynamics Courant–Friedrichs–Lewy Centre National de la Recherche Scientifique central processing unit cross-spectral densities compact upwind with high dissipation compact upwind with low dissipation detached-eddy simulation Detached Eddy Simulation for the Transportation Industry Deutsche Forschungsgemeinschaft discontinuous Galerkin Deutsche Gesellschaft f¨ur Luft- und Raumfahrt - Lilienthal - Oberth e.V. 391

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ABBREVIATIONS

DLR DNC DNS DRP DSM ECL ENO ENSAM ERCOFTAC EU FD FDS FFT FSM FV FW–H IBM IL KFVS KIM LCRU LDDRK LDDRK4 LDDRK5 LDDRK5-6 LDDRK6 LEE LES LHS LMFA LNS LSTM LURC LURU LU-SGS MGB MGBK

Deutsches Zentrum f¨ur Luft- und Raumfahrt direct noise computation direct numerical simulation dispersion-relation-preserving dynamic Smagorinsky model Ecole Centrale de Lyon essentially nonoscillatory Ecole Nationale Sup´erieure d’Arts et M´etiers European Research Community on Flow, Turbulence and Combustion European Union finite difference flux-difference splitting fast Fourier transformation flow simulation methodology finite volume Ffowcs Williams–Hawkings International Business Machines intensity level kinetic flux vector splitting Kirchhoff integration method L-operator central and R-operator upwind-biased low-dissipation and low-dispersion Runge–Kutta scheme four-stage optimized low-dissipation and low-dispersion Runge–Kutta scheme five-stage optimized low-dissipation and low-dispersion Runge–Kutta scheme alternating two-step low-dissipation and low-dispersion Runge–Kutta scheme six-stage optimized low-dissipation and low-dispersion Runge–Kutta scheme linearized Euler equations large-eddy simulation left-hand side Laboratoire de M´ecanique des Fluides et Acoustique limited numerical scales Lehrstuhl f¨ur Str¨omungsmechanik L-operator upwind-biased and R-operator central L-operator and R-operator upwind-biased lower-upper symmetric Gauss–Seidel Mani–Gliebe–Balsa Mani–Gliebe–Balsa–Khavaran

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ABBREVIATIONS

MILES MPI MUSCL NACA NASA NLAS NLD NLDE NRBC OASPL OL ONERA PC PIV PML POD PPW PREDIT PSA PSD PWL R&D RANS RHS RK4 RKDG rms RNP S–A sAbrinA SATIN sgs; SGS SM SNGR SOCF SPL SUPG svp SWING

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monotonically integrated large-eddy simulation message-passing interface monotone upstream-centered schemes for conservation laws National Advisory Committee for Aeronautics National Aeronautics and Space Administration nonlinear acoustics solver nonlinear disturbance nonlinear disturbance equations nonreflecting boundary conditions overall sound-pressure level logarithmic outer layer Office National d’Etudes et de Recherches Aerospatiales personal computer particle image velocimetry perfectly matched layer proper orthogonal decomposition points per wavelength Programme de Recherche et d’Innovation dans les Transports Terrestres Peugeot Societe Anonyme power spectral density sound power level research and development Reynolds-averaged Navier–Stokes right-hand side fourth-order Runge–Kutta Runge–Kutta discontinuous Galerkin root mean square Reynolds-number-preserving Spalart–Allmaras (model) solver for aeroacoustics broadband interactions from aerodynamics statistical approach to turbulence-induced noise subgrid scale Smagorinsky model stochastic noise generation and radiation second-order commuting filters sound-pressure level streamline-upwind Petrov–Galerkin (integration scheme) spurious vortical perturbations simulation of wing-flow noise generation

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TBL TBLE TE TEM TurboNoiseCFD

turbulent boundary layer thin boundary layer equations trailing edge transverse electromagnetic turbomachinery noise-source CFD models for low-noise aircraft designs TVD total variation diminishing URANS, (U)RANS unsteady Reynolds-averaged Navier–Stokes VLES very large eddy simulation VS viscous sublayer WALE wall-adapting local eddy viscosity WENO weighted essentially nonoscillatory WKB Wentzel–Kramers–Brillouin

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References

Abarbanel, S., Gottlieb, D. and Hesthaven, J.S. (1999): Well-posed perfectly matched layers for advective acoustics, Journal of Computational Physics, 154:266–283. Abel, M., Stojkovi´c, D. and Breuer, M. (2003): Development of wall models for LES based on nonlinear stochastic estimation, Fifth Workshop on DNS and LES: DLES-5, Munich University of Technology, Germany, Aug. 27–29. Abel, M., Stojkovi´c, D. and Breuer, M. (2004): Nonlinear stochastic estimation of wall models for LES, International Journal of Heat and Fluid Flow, 27(2):267–278. Adams, N. (2002): A subgrid-scale deconvolution approach for shock capturing, Journal of Computational Physics, 178:391–426. Adams, N.A. and Shariff, K. (1996): A high-resolution hybrid compact-ENO scheme for shockturbulence interaction problems, Journal of Computational Physics, 127:27–51. Adams, N.A. and Stolz, S. (2001): Deconvolution methods for subgrid-scale approximation in LES, Modern Simulation Strategies for Turbulent Flow, B.J. Geurts (ed.), Edwards, Flourtown, PA, pp. 21–44. Agarwal, A. and Morris, P.J. (2000): Direct Simulation of Acoustic Scattering by a Rotorcraft Fuselage, AIAA Paper 2000-2030. Agarwal, A. and Morris, P.J. (2004): Broadband Noise from the Unsteady Flow in a Slat Cove, AIAA Paper 2004-854. Ahmed, S.R., Ramm, G. and Faltin, G. (1984): Some Salient Features of the Time-Averaged Ground Vehicle Wake, SAE Technical Paper Series 840 300. ¨ or¨uk, Y. and Long, L.N. (2000): Computational Simulations of Fore and Aft Radiation Ahuja, V., Ozy¨ from Ducted Fans, AIAA Paper 2000-1943. Algermissen, G., Siegert, R. and Spindler, T. (2001): Numerical Simulation of Aeroacoustic Sound Generated by Fans under Installation Conditions, AIAA Paper 2001-2174. Allen, R. and Mendon¸ca, F. (2004): DES Validations of Cavity Acoustics over the Subsonic to Supersonic Range, AIAA Paper 2004-2862. Allen, R., Mendon¸ca, F. and Kirkham, D. (2005): RANS and DES turbulence model predictions of noise on the M219 cavity at M = 0.85, International Journal of Aeroacoustics, 4(1,2):135–152. Al-Qadi, I.M.A. and Scott, J.N. (2002): High Reynolds Number Simulation of Under-expanded Supersonic Rectangular Jet, AIAA Paper 2002-2474. American Academy of Pediatrics (1997): Policy Statement: Noise: A hazard for the fetus and newborn (RE9728), Pediatrics, 100(4):724–727. Andersson, N. (2003): A Study of Mach 0.75 Jets and Their Radiated Sound Using Large-Eddy Simulation. Ph.D. thesis, Chalmers University of Technology, G¨oteborg, Sweden. Ando, Y. (1969): On the sound radiation from semi-infinite circular pipe of certain wall thickness, Acustica, 22:219–225.

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November 17, 2006

REFERENCES

Anthoine, J., Buchlin, J.-M. and Hirschberg, A. (2002): Effect of nozzle on resonance in large srm: theoretical modeling, Journal of Propulsion and Power, 18:304–311. Arakeri, V.H., Krothapalli, A., Siddavaram, V., Alkislar, M. and Lourenco, L. (2003): On the use of microjets to suppress turbulence in a Mach 0.9 axisymmetric jet. Journal of Fluid Mechanics, 490:75–98. Arunajatesan, S. and Sinha, N. (2001): Unified Unsteady RANS-LES Simulations of Cavity Flowfields, AIAA Paper 2001-0516. Ashcroft, G. and Zhang, X. (2001): A Computational Investigation of the Noise Radiated by FlowInduced Cavity Oscillations, AIAA Paper 2001-0512. Atkins, H. and Casper, J. (1994): Nonreflective boundary conditions for high-order methods, AIAA Journal, 32:512–518. Atkins, H.L. and Lockard, D.P. (1999): A High-Order Method Using Unstructured Grids for the Aeroacoustic Analysis of Realistic Aircraft Configurations, AIAA Paper 99-1945. Atkins, H.L. and Shu, C.-W. (1996): Quadrature-Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations, ICASE Report No. 96-51. Atkins, H.L. and Shu, C.-W. (1998): Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations, AIAA Journal, 36:775–782. Avital, E.J., Sandham, N.D. and Luo, K.H. (1996): Sound Generation Using Data from Direct Numerical Simulations of Mixing Layers, AIAA Paper 96-1778. ¨ Baggag, A., Atkins, H., Ozturan, C. and Keyes, D. (1999): Parallelization of an Object-Oriented Unstructured Aeroacoustics Solver, ICASE Report 99-11. Bagwell, T.G., Adrian, R.J., Moser, R.D. and Kim, J. (1993): Improved approximation of wall shear stress boundary conditions for large-eddy simulation, Near-Wall Turbulent Flows, R.M.C. So, C.G. Speziale and B.E. Launder (eds.), Elsevier Science Publ., New York, pp. 265–275. Bailly, C., Bogey, C. and Juv´e, D. (2000): Computation of Flow Noise Using Source Terms in Linearized Euler’s Equations, AIAA Paper 2000-2047. Bailly, C., Candel, S. and Lafon, P. (1996): Computation of jet mixing noise from statistical source models and a compressible turbulence closure, Journal of Sound and Vibration, 194(2):219–242. Bailly, C. and Juv´e, D. (1999): A Stochastic Approach to Compute Subsonic Noise Using Linearized Euler’s Equations, AIAA Paper 99-1872. Bailly, C., Lafon, P. and Candel, S. (1995): A Stochastic Approach to Compute Noise Generation and Radiation of Free Turbulent Flows, AIAA Paper 95-092. Bailly, C., Lafon, P. and Candel, S. (1996): Computation of Noise Generation and Propagation for Free and Confined Turbulent Flows, AIAA Paper 96-1732. Balaras, E. and Benocci, C. (1994): Subgrid scale models in finite-difference simulations of complex wall bounded flows, AGARD CP 551, Dec. 1994, pp. 2.1–2.6, 74th Fluid Dynamics Symp., Chania, Crete, Greece, April 2004. Balaras, E., Benocci, C. and Piomelli, U. (1996): Two-layer approximate boundary conditions for large-eddy simulations, AIAA Journal, 34(6):1111–1119. Balsa, T.F. and Gliebe, P.R. (1997): Aerodynamics and noise of coaxial jets, AIAA Journal, 15(11):1550–1558. Balsa, T.F., Gliebe, P.R., Kantola, R.A., Mani, R., Strings, E.J. and Wong II, J.C.F. (1978): High Velocity Jet Noise Source Location and Reduction, FAA Technical Report No. FAA-RD-76-79. Bardina, J., Ferziger, J.H. and Reynolds, W.C. (1983): Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows, Report TF-19, Thermosciences Division, Department of Mechanical Engineering, Stanford University. Barone, F., Durello, P., Read, A., Mendon¸ca, F., Tournour, M. and El Hachemi, Z. (2003): Investigation of the tonal noise radiated by subsonic fans using the aero-acoustic analogy, Fan Noise 2003, 2nd International Symposium, Senlis, France, Sept. 23–25, 2003. Batchelor, G. (1976): An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, UK. Batten, P., Goldberg, U. and Chakravarthy, S. (2000): Sub-grid Turbulence Modeling for Unsteady Flow with Acoustic Resonance, AIAA Paper 00-0473.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Batten, P., Goldberg, U. and Chakravarthy, S. (2002a): Acoustics Predictions Using Less than LES, Proceedings of the LES for Acoustics Workshop, G¨ottingen. Batten, P., Goldberg, U. and Chakravarthy, S. (2002b): LNS – An Approach Towards Embedded LES, AIAA Paper 2002-0427. Batten, P., Goldberg, U. and Chakravarthy, S. (2002c): Reconstructed Sub-grid Methods for Acoustics Predictions at all Reynolds Numbers, AIAA Paper 2002-2516. Batten, P., Goldberg, U. and Chakravarthy, S. (2004): Interfacing statistical turbulence closures with large eddy simulation, AIAA Journal, 42(3):485–492. Batten, P., Ribaldone, E., Casella, M. and Chakravarthy, S. (2004): Towards a Generalized Non-linear Acoustics Solver, AIAA Paper 2004-3001. Bayliss, A. (1985): A Fourth-Order Scheme for the Unsteady Compressible Navier–Stokes Equations, ICASE Report 85-44. Bayliss, A. and Turkel, E. (1982a): Far-field boundary condition for compressible flows, Journal of Computational Physics, 48:182–199. Bayliss, A. and Turkel, E. (1982b): Outflow boundary conditions for fluid dynamics, SIAM Journal on Scientific and Statistical Computing, 3:250–253. Baysal, O. and Stallings, R.L. (1987): Computational and experimental investigation of cavity flowfields, AIAA Journal, 26(1):6–7. Baysal, O., Ten, G.-W. and Fouladi, K. (1994): Navier–Stokes computations of cavity aeroacoustics with suppression devices, ASME Journal of Vibration and Acoustics, 116:105–112. Beam, R.M. and Warming, R.F. (1976): An implicit finite-difference algorithm for hyperbolic systems in conservation-law form, Journal of Computational Physics, 22:87–110. Beaudan, P. and Moin, P. (1994): Numerical Experiments on the Flow Past a Circular Cylinder at Sub-critical Reynolds Number. Technical Report TF-62, Stanford University. B´echara, W., Bailly, C., Lafon, P. and Candel, S. (1994): Stochastic approach to noise modeling for free turbulent flows, AIAA Journal, 32(3):455–463. Bechert, D.W. (1980): Sound absorption caused by vorticity shedding, demonstrated with a jet flow. Journal of Sound and Vibration, 70:389–405. Bechert, D.W., Michel, U. and Pfizenmaier, E. (1978): Experiments on the transmission on sound through jets. AIAA Journal, 16:873–874; also AIAA Paper 77-1278. Bechert, D.W. and Pfizenmaier, E. (1975): On the amplification of broadband jet noise by a pure tone excitation. Journal of Sound Vibration, 43:581–587. Bendat, J.S. and Piersol, A.G. (1980): Engineering Applications of Correlation and Spectral Analysis, Wiley-Interscience Publication. Benhamadouche, S., Jarrin, N. and Laurence, D. (2003): Synthetic turbulent inflow conditions for large eddy simulation, Journal of Turbulence Heat and Mass Transfer, 4:467–474. Benyahia, A. (2003): Contribution a` la mod´elisation num´erique du comportement de la couche limite instationnaire: validation par comparaison aux donn´ees exp´erimentales sur un profil oscillant. Ph.D. thesis, Universit´e de la M´editerran´e, France. Berenger, J.-P. (1994): A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114:185–200. Berglund, B., Preis, A. and Rankin, K. (1990): Relationship between loudness and annoyance for ten community sounds. Environment-International, 16:523–531. Biedron, R.T., Rumsey, C.L., Podboy, G.G. and Dunn, M.H. (2001): Predicting the Rotor-Stator Interaction Acoustics of a Ducted Fan Engine, AIAA Paper 2001-0664. Billson, M., Eriksson, L.E. and Davidson, L. (2002): Acoustic Source Terms for the Linear Euler Equations in Conservative Form, AIAA Paper 2002-2582. Billson, M., Eriksson, L.E. and Davidson, L. (2003): Jet Noise Prediction Using Stochastic Turbulence Modeling, AIAA Paper 2003-3282. Blaisdell, G.A., Mansour, N.N. and Reynolds, W.C. (1993): Compressibility effects on the growth and structure of homogeneous turbulent shear flow, Journal of Fluid Mechanics, 256:443–485. Blaisdell, G.A., Spyropoulos, E.T. and Qin J.H. (1996): The effect of the formulation of nonlinear terms on aliasing errors in spectral methods, Applied Numerical Mathematics, 21:207–219.

12:29

397

P1: JZZ references

CUFX063/Wagner

398

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Blake, W.K. (1970a): Mechanics of Flow-Induced Sound and Vibration. Volume I. General Concepts and Elementary Sources, Academic Press. Blake, W.K. (1970b): Mechanics of Flow-Induced Sound and Vibration. Volume II. Complex FlowStructure Interaction, Academic Press. Blake, W.K. (1986): Mechanics of Flow-induced Sound and Vibration, Volume I. Academic Press, Orlando. Blake, W.K. and Powell, A. (1986): The development of contemporary views of flow-tone generation, Recent Advances in Aeroacoustics, A. Krothapalli, C.A. Smith, Springer, Berlin, New York, pp. 247–345. Blevins, R.D. (1990): Flow-Induced Vibration, 2nd ed., Van Nostrand Reinhold, New York. Bodony, D.J. and Lele, S.K. (2002a): Large-eddy simulation of turbulent jets and progress towards a subgrid scale noise model, Proceedings of the International Workshop on “LES for Acoustics,” October 7–8, German Aerospace Center, DLR, G¨ottingen, Germany, DGLR-Report 200203. Bodony, D.J. and Lele, S.K. (2002b): Spatial Scale Decomposition of Shear Layer Turbulence and the Sound Sources Associated with the Missing Scales in a Large-Eddy Simulation, AIAA Paper 2002-2454. Bodony, D.J. and Lele, S.K. (2003a): A Stochastic Subgrid Scale Noise Model for Noise Predictions of Subsonic Jets, AIAA Paper 2003-3252. Bodony, D.J. and Lele, S.K. (2003b): Large-Eddy Simulation of Jet Engine Exhaust Noise, ERCOFTAC Bulletin 58, Theme Issue: Aeroacoustics: 21–24. Bodony, D.J. and Lele, S.K. (2004): Jet Noise Prediction of Cold and Hot Subsonic Jets Using LargeEddy Simulation, AIAA Paper 2004-3022. Boersma, B.J. (2002): Large-eddy simulation of the sound field of a round turbulent jet, Proceedings of the International Workshop on “LES for Acoustics,” October 7–8, German Aerospace Center, DLR, G¨ottingen, Germany, DGLR-Report 2002-03. Boersma, B.J. (2003): Large-Eddy Simulation of the Sound Field of a Round Turbulent Jet, ERCOFTAC Bulletin 58, Theme Issue: Aeroacoustics: 51–54. Boersma, B.J. and Lele, S.K. (1999): Large Eddy Simulation of a Mach 0.9 Turbulent Jet, AIAA Paper 1999-1874. Bogey, C. and Bailly, C. (2002): A Family of Low Dispersive and Low Dissipative Explicit Schemes for Computing Aerodynamic Noise, AIAA Paper 2002-2509. Bogey, C. and Bailly, C. (2004): A family of low dispersive and low dissipative explicit schemes for flow and noise computations, Journal of Computational Physics, 194(1):194–214. Bogey, C. and Bailly, C. (2006): Computation of a high Reynolds number jet and its radiated noise using LES based on explicit filtering, Computers and Fluids, 35(10):1344–1358. See also: Direct computation of the sound radiated by a high-Reynolds number, subsonic round jet, Presented at the CEAS Workshop From CFD to CAA, November 7–8, 2002, NTUA, Athens, Greece. Bogey, C. and Bailly, C. (2005a): Decrease of the effective Reynolds number with eddy-viscosity subgrid-scale modeling, AIAA Journal, 43(2):437–439. See also LES of a High Reynolds, High Subsonic Jet: Effects of the Subgrid Modellings on Flow and Noise, AIAA Paper 20033557. Bogey, C. and Bailly, C. (2005b): Effects of inflow conditions and forcing on subsonic jet flows and noise, AIAA Journal, 43(5):1000–1007. See also LES of a High Reynolds, High Subsonic Jet: Effects of the Inflow Conditions on Flow and Noise, AIAA Paper 2003-3170. Bogey, C., Bailly, C. and Juv´e, D. (1999): Computation of Mixing Layer Noise Using Large-Eddy Simulation, AIAA Paper 99-1871. Bogey, C., Bailly, C. and Juv´e, D. (2000a): Computation of the Sound Radiated by a 3-D Jet Using Large-Eddy Simulation, AIAA Paper 2000-2009. Bogey, C., Bailly, C. and Juv´e, D. (2000b): Numerical simulation of the sound generated by vortex pairing in a mixing layer, AIAA Journal, 38(12):2210–2218. Bogey, C., Bailly, C. and Juv´e, D. (2002): Computation of flow noise using source terms in linearized Euler’s equations, AIAA Journal, 40(2):235–243.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Bogey, C., Bailly, C. and Juv´e, D. (2003): Noise investigation of a high subsonic, moderate Reynolds number jet using a compressible LES, Theoretical Computational and Fluid Dynamics, 16(4):273– 297. Bogey, C., Gloerfelt, X. and Bailly, C. (2003): An illustration of the inclusion of sound–flow interactions in Lighthill’s equation, AIAA Journal, 41(8):1604–1606. Bohineust, X., Bardot, A. and Dupuy, F. (1996): Vehicle noise design modifications analysis using truncated transfer path techniques, ISMA 21, Leuven, Belgium, Sept. 18–20, 1996. Bohineust, X., Floc’h, C., Sol, A. and Giardi, H. (1999): Hybrid structure acoustic vehicle models, Proc. 3 Stuttgarter Symoposium, Krastsahrwesen und Verbrennungsmotoren, Feb. 23–25, 1999. Boris, J.P., Grinstein, F.F., Oran, E.S. and Kolbe, R.L. (1992): New insights into large-eddy simulation, Fluid Dynamics Research, 10:199–228. Bortz, D.M., Rubio, A.D., Banks, H.T., Cain, A.B. and Smith, R.C. (2002): Control of open bay acoustics by harmonic mass injection, International Journal of Aeroacoustics, 1(1):65–81. Boudet, J., Casalino, D., Jacob, M.C. and Ferrand, P. (2002): Unsteady RANS computations of the flow past an airfoil in the wake of a rod, ASME Fluid Engineering Division Summer Meeting, Montreal, Canada, FEDSM2002-31343. Boudet, J., Casalino, D., Jacob, M.C. and Ferrand, P. (2003): Prediction of Sound Radiated by a Rod using Large Eddy Simulation. In Proceedings 8th AIAA/CEAS Aeroacoustics Conference, 2003. AIAA Paper 2003-3217. Boudet, J., Grosjean, N. and Jacob, M. (2003): Wake–airfoil interaction as broadband noise source, Computational Aeroacoustics: From Acoustic Sources Modeling to Far-Field Radiated Noise Prediction, Colloquium EUROMECH 449, December 9–12, Chamonix, France. Bradshaw, P. (1977): Compressible turbulent shear layers, Annual Review of Fluid Mechanics, 9:35– 54. Brentner, K.S. and Farassat, F. (1998): An analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces, AIAA Journal, 36(8):1379–1386. Breuer, M. (1997): Numerical and modelling influences on large-eddy simulations for the flow past a circular cylinder, Proceedings of the 11th Turbulent Shear Flow Conference, Sept. 1997, Grenoble, France. Breuer, M. (1998): Large-eddy simulation of the sub-critical flow past a circular cylinder: Numerical and modeling aspects, International Journal for Numerical Methods in Fluids, 28:1281–1302. Breuer, M. (2000): A challenging test case for large-eddy simulation: High Reynolds number circular cylinder flow, International Journal of Heat and Fluid Flow, 21(5):648–654. Breuer, M. (2002): Direkte numerische Simulation und Large–Eddy Simulation turbulenter Str¨omungen auf Hochleistungsrechnern, Habilitationsschrift, Universit¨at Erlangen–N¨urnberg, Berichte aus der Str¨omungstechnik, Shaker Verlag, Aachen, Germany. Breuer, M. and Joviˇci´c, N. (2001): An LES investigation of the separated flow past an airfoil at high angle of attack, 4th Workshop on DNS & LES, Enschede, The Netherlands, July 18–20, 2001, ERCOFTAC Series, 8, pp. 165–172, Direct and Large–Eddy Simulation IV, B.J. Geurts, R. Friedrich, and O. M´etais (eds.), Kluwer Academic Publishers, Dordrecht. Bridges, J.E. and Hussain, A.K.M.F. (1987): Roles of initial condition and vortex pairing in jet noise, Journal of Sound Vibration, 117(2):289–311. Broeckhoven, T., Smirnov, S., Ramboer, J. and Lacor, C. (2003): Finite volume formulation of compact upwind and central schemes with artificial selective damping, CD-ROM Proc. Second M.I.T. Conference on Computational Fluid and Solid Mechanics., Boston, June 17–20, 2003, SIAM Journal on Scientific and Statistical Computing 21(3):441–467. Brooks, A.N. and Hughes, T.J.R. (1982): Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32:199–259. Brooks, T.F. and Hodgson, T.H. (1981): Prediction and comparison of trailing edge noise using measured surface pressures, Journal of Sound and Vibration, 78(1):69–117. Brouwer, H.H. (1992): On the use of the method of matched asymptotic expansions in propeller aerodynamics and acoustics, Journal of Fluid Mechanics, 242:117–143.

12:29

399

P1: JZZ references

CUFX063/Wagner

400

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Brown, G.L. and Roshko, A. (1974): On density effects and large structure in turbulent mixing layers, Journal of Fluid Mechanics, 64(4):775–816. Bruggeman, J.C. (1987): The propagation of low-frequency sound in a two-dimensional duct system with T joints and right angle bends: T joints theory and experiment, Journal of the Acoustical Society of America, 82:1045–1051. Bruggeman, J.C., Hirschberg, A., van Dongen, M.E.H., Wijnands, A.P.J. and Gorter, J. (1991): Selfsustained aero-acoustic pulsations in gas transport systems: Experimental study of the influence of closed side branches, Journal of Sound and Vibration, 150:371–393. Bui, T.T. (1999): A Parallel, Finite-Volume Algorithm for Large-Eddy Simulation of Turbulent Flows. NASA TM-206570. Buonfiglioli, M. and Mendon¸ca, F. (2005): LES-VOF simulation of primary diesel break-up with synthetic inlet perturbations, ILASS Americas, 18th Annual Conference on Liquid Atomisation and Spray Systems, Irvine, CA, May 2005. Butterworth-Hayes, P. (2004): Europe reaches for aerospace dominance, Aerospace America, August 2004, American Institute of Aeronautics and Astronautics, pp. 36–40. Cabelli, A. (1980): The acoustic characteristics of duct bends, Journal of Sound and Vibration, 68:369–388. Cabot, W.H. and Moin, P. (1999): Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flow, Journal of Flow, Turbulence and Combustion, 63(1,4):269–291. Cantwell, B. and Coles, D. (1983): An experiment study of entrainement and transport in the turbulent near wake of a circular cylinder, Journal of Fluid Mechanics, 136:321–374. Carati, D., Ghosal, S. and Moin, P. (1995): On the representation of backscatter in dynamic localization models, Physics of Fluids, 7(3):606–616. Carati, D., Winckelmans, G.S. and Jeanmart, H. (1999): Exact expansions for filtered-scales modelling with a wide class of LES filters, Direct and Large Eddy Simulation III, Voke, Sandham and Kleiser (eds.), Kluwer Academic Publishers, Dordrecht, pp. 213–224. Cargill, A.M. (1982): Low-frequency sound radiation and generation due to the interaction of unsteady flow with a jet pipe, Journal of Fluid Mechanics, 121:59–105. Caro, S., Ploumhans, P., Gallez, X., Brotz, F., Schrumpf, M., Read, A. and Mendon¸ca, F. (2005): Aeroacoustic Simulation of the Noise Radiated by a Helmholz Resonator in a Duct, AIAA Paper 2005-3067. Casper, J. and Farassat, F. (2003): Trailing Edge Noise Prediction Based on a New Acoustic Formulation, AIAA Paper 2002-2477. Casper, J. and Meadows, K.R. (1995): Using High-Order Accurate Essentially Non-oscillatory Schemes for Aeroacoustic Applications, AIAA Paper 95-0163. Castillo, L. and Walker, J. (2002): Effect of upstream conditions on the outer flow of turbulent bounday layers, AIAA Journal, 40:1292–1299. Chase, D.M. (1987): The character of the turbulent wall pressure spectrum at subconvective wavenumbers and a suggested comprehensive model, Journal of Sound and Vibration, 112:125–147. Choi, D., Barber, T.J., Chiappetta, L.M. and Nishimura, M. (1999): LES Simulation of High Reynolds Number Jet Flows, AIAA Paper 99-0230. Chollet, J.P. and Lesieur, M. (1981): Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures, Journal of Atmospheric Sciences, 38:2747–2757. Chung, Y.M. and Sung, H.J. (1997): Comparative study of inflow conditions for spatially evolving simulation, AIAA Journal, 35(2):269–274. Chyczewski, T.S. and Long, L.N. (1996): A Direct Numerical Study of a Supersonic Rectangular Jet Using the Full Navier–Stokes Equations, AIAA Paper 96-1730. Chyczewski, T., Morris, P. and Long, L. (2000a): Trailing Edge Noise Prediction: Large-Eddy Simulation of Wall Bounded Shear Flow Using the Nonlinear Disturbance Equations, Applied Research Laboratory, The Pennsylvania State University, State College, PA, Final Report, Contract No. N-00014-99-0290, Sept. 26, 2000. Chyczewski, T., Morris, P. and Long, L. (2000b): Large-Eddy Simulation of Wall Bounded Shear Flow Using the Nonlinear Disturbance Equations, AIAA Paper 2000-2007.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Clark, R.A., Ferziger, J.H. and Reynolds, W.C. (1979): Evaluation of subgrid-scale models using an accurately simulated turbulent flow, Journal of Fluid Mechanics, 91(1):1–16. Cockburn, B. (1999): Discontinuous Galerkin methods for convection dominated problems. In H. Deconinck and T.J. Barth (eds.), Lecture Notes in Computational Science and Engineering, Springer-Verlag, 9:69–224. Cockburn, B., Hou, S. and Shu, C.-W. (1990): TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Mathematics of Computation, 54(190):545–581. Cockburn, B., Karniadakis, G.E. and Shu, C.-W. (2000): Discontinuous Galerkin methods. Theory, computation and applications. In Lecture Notes in Computational Science and Engineering, Springer-Verlag, 11:3–50. Cockburn, B. and Shu, C.-W. (1994): Nonlinearly stable compact schemes for shock calculations, SIAM Journal on Numerical Analysis, 31:607–627. Collatz, L. (1966): The Numerical Treatment of Differential Equations, Springer-Verlag. Collatz, L. (1972): Hermitian Methods for Initial-Value Problems in Partial Differential Equations, Proceedings of the Royal Irish Academy Conference on Numerical Analysis, Topics in Numerical Analysis, J.J.H. Miller (ed.), pp. 41–61. Collino, F. and Monk, P. (1998): The perfectly matched layer in curvilinear coordinates, SIAM Journal on Scientific and Statistical Computing, 19(6):2016. Colonius, T. (2001): An Overview of Simulation, Modeling and Active Control of Flow/Acoustic Resonance in Open Cavities, AIAA Paper 2001-0076. Colonius, T., Basu, A.J. and Rowley, C.W. (1999): Numerical Investigation of the Flow Past a Cavity, AIAA Paper 99-1912. Colonius, T., Lele, S.K. and Moin, P. (1993): Boundary conditions for direct computation of aerodynamic sound generation, AIAA Journal, 31(9):1574–1582. Colonius, T., Lele, S.K. and Moin, P. (1997): Sound generation in a mixing layer, Journal of Fluid Mechanics, 330:375–409. Comte, P., Silvestrini, J.H. and B´egou, P. (1998): Streamwise vortices in large-eddy simulations of mixing layers, European Journal of Mechanics B – Fluids, 17(4):615–637. Constantinescu, G.S. and Lele, S.K. (2001): Large-Eddy Simulation of a Nearly Sonic Turbulent Jet and Its Radiated Noise, AIAA Paper 2001-0376. Constantinescu, G.S. and Lele, S.K. (2002): A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions, Journal of Computational Physics, 183:165–186. Cooper, A.J. and Peake, N. (2001): Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow, Journal of Fuid Mechanics, 445:207–234. Corcos, G.M. (1964): The structure of the turbulent pressure field in boundary layer flows, Journal of Fluid Mechanics, 18(3):353–378. Correa, J., Massa, B. and Zajas, S. (2003): Fluent Announces Coupling with LMS SYSNOISE for Acoustics Modeling, press release PR61, Fluent Inc., August 6, 2003, Lebanon, NH, USA and Leuven, Belgium, URL: http//www.fluent.com/about/news/pr/pr61.htm. Cremer, L. and Heckl, M. (1988): Structure-Borne Sound, 2nd ed., translated and revised by E.E. Ungar, Springer-Verlag, Berlin. Crighton, D.G. (1985): The Kutta condition in unsteady flow, Annual Review of Fluid Mechanics, 17:411–445. Crighton, D.G. (1991): Airframe noise. In Aeroacoustics of Flight Vehicles, H.H. Hubbard (ed.), NASA TR 90-3052. Crighton, D.G. (1992): private communication. Crighton, D.G. (1993): Computational aeroacoustics for low Mach number flows. Computational Aeroacoustics, J.C. Hardin and M.Y. Hussaini (eds.), Springer-Verlag, New York, pp. 50– 68. Crighton, D.G., Dowling, A.P., Ffowcs Williams, J.E., Heckl, M. and Leppington, F.G. (1992): Modern Methods in Analytical Acoustics, Lecture Notes, Springer-Verlag, London.

12:29

401

P1: JZZ references

CUFX063/Wagner

402

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Crouzet, F., Lafon, P., Buchal, T. and Laurence, D. (2002): Aerodynamic Noise Prediction in Internal Flows Using LES and Linear Euler Equations, AIAA Paper 2002-2568. Crow, S.C. (1970): Aerodynamic sound emission as a singular perturbation problem, Studies in Applied Mathematics, 49(1):21–44. Curle, N. (1955): The influence of solid boundaries upon aerodynamic sound, Procedings of the Royal Society of London, Series A, 231:505–514. Dakhoul, Y.M. and Bedford, K.W. (1986a): Improved averaging method for turbulent flow simulation. Part 1: Theoretical development and application to Burger’s transport equation, International Journal for Numerical Methods in Fluids, 6:49–64. Dakhoul, Y.M. and Bedford, K.W. (1986b): Improved averaging method for turbulent flow simulation. Part 2: Calculations and verification. International Journal for Numerical Methods in Fluids, 6: 65–82. Dalmont, J.P., Nederveen, C.J., Dubos, V., Ollivier, S., M´eserette, V. and te Sligte, E. (2002): Experimental determination of the equivalent circuit of an open side hole: Linear and non-linear behaviour, EAA, 46:567–575. Dalmont, J.P., Nederveen, C.J. and Joly, N. (2001): Radiation impedance of tubes with different flanges: Numerical and experimental investigations. Journal of Sound and Vibration, 244:505–534. Davies, P.O.A.L., Fisher, M.J. and Barratt, M.J. (1963): The characteristics of the turbulence in the mixing region of a round jet, Journal of Fluid Mechanics, 15:337–367. Deardorff, J.W. (1970): A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, Journal of Fluid Mechanics, 41:453–465. Debatin, K. (1999): “Lokalisation aeroakustischer Quellen in Scherschichtdominanten Str¨omungen,” Universit¨at Karlsruhe. DeBonis, J.R. and Scott, J.N. (2002): Large-eddy simulation of a turbulent compressible round jet, AIAA Journal, 40(7):1346–1354. Deng, X. and Maekawa, H. (1997): Compact high-order accurate nonlinear schemes, Journal of Computational Physics, 130:77–91. Deng, X. and Mao, M. (1997): Weighted Compact High-Order Nonlinear Schemes for the Euler Equations, AIAA Paper 97-1941. Deng, X. and Mao, M. (2001): High-Order Dissipative Weighted Compact Nonlinear Schemes for Euler and Navier–Stokes Equations, AIAA Paper 2001-2626. Dequand, S., Hulshoff, S., Van Kuijk, H., Willems, J. and Hirschberg, A. (2003): Helmholtz-like resonator self-sustained oscillations, Part 2: Detailed flow measurements and numerical simulations, AIAA Journal, 41(3):416–423. Dequand, S., Luo, X., Willems, J. and Hirschberg, A. (2003): Helmholtz-like resonator self-sustained oscillations, Part 1: Acoustical measurements and analytical models, AIAA Journal, 41:408–415. Dequand, S., Willems, J.F.H., Leroux, M., Vullings, R., van Weert, M., Thieulot, C. and Hirschberg, A. (2003): Simplified models of flue instruments: Influence of mouth geometry on the sound source, Journal of the Acoustical Society of America, 113:1724–1735. Devos, J.P. and Lafon, P. (2003): Computation of the Noise Generated by a Shallow Cavity in a Duct and Test of a Solution for Lowering the Noise, AIAA Paper 03-3105. Disselhorst, J.H.M. and van Wijngaarden, L. (1980): Flow in the exit of open pipes during acoustic resonance, Journal of Fluid Mechanics, 99:293–319. Doak, P.E. (1989): Momentum potential theory of energy flux carried by momentum fluctuations, Journal of Sound and Vibration, 131:67–90. Doak, P.E. (1995): Fluctuating total enthalpy as a generalized field, Acoustical Physics, 44:677–685. Domaradzki, J.A. and Liu, W. (1995): Approximation of subgrid-scale energy transfer based on the dynamics of the resolved scales of turbulence, Physics of Fluids, 7(8):2025–2035. Domaradzki, J.A., Liu, W. and Brachet, M.E. (1993): An analysis of subgrid-scale interactions in numerically simulated isotropic turbulence, Physics of Fluids, 5(7):1747–1759. Domaradzki, J.A. and Rogallo, R.S. (1990): Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence, Physics of Fluids A, 2:412–426. Domaradzki, J.A. and Yee, P.P. (2000): The subgrid-scale estimation model for high Reynolds number turbulence, Physics of Fluids, 12(1):193–196.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Donea, J. (1982): An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid-structure interaction, Computer Methods in Applied Mechanical Engineering, 33:689–723. Dong, T.Z. and Mankbadi, R.R. (1996): Large-Eddy Simulations of Sound due to Turbulence Mixing inside an Ejector, AIAA Paper 96-1731. Dong, T.Z. and Mankbadi, R.R. (1999): Simulation of unsteady flow in nozzle-ejector mixer, Journal of Propulsion and Power, 15(4):539–543. Doris, L., Tenaud, C. and Ta Phuoc, L. (2000): LES of spatially developing 3D compressible mixing layer, C. R. Acad. Sci. Paris, t. 328, S´erie IIb:567–573. Dowling, A.P. and Ffowcs Williams, J.E. (1983): Sound and Sources of Sound, Ellis Horwood Publishers, Chichester, UK. Dubief, F., Naji, S., Lewis, M. and Mendon¸ca, F. (2005): Prediction of Aeroacoustic Noise in Automotive Centrifugal Fans: Validation of CFD and Acoustic Simulation Techniques, AIAA Paper 2005-2921. Dubief, Y. and Delcayre, F. (2000): On coherent-vortex identification in turbulence, Journal of Turbulence, 1:011. Dubos, V., Keefe, D., Kergomard, J., Dalmont, J.P. and Nederveen, C.J. (1999): Theory of sound propagation in a duct with a branched tube using modal decomposition, Acustica-Acta Acustica, 85:153–169. Ducros, F., Comte, P. and Lesieur, M. (1996): Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate, Journal of Fluid Mechanics, 326:1–36. Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C. and Poinsot, T. (1999): Large-eddy simulation of shock/homogeneous turbulence interaction, Journal of Computational Physics, 152:517–549. Ducros, F., Laporte, F., Soul`eres, T., Guinot, V., Moinat, P. and Caruelle, B. (2000): High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: Application to compressible flows, Journal of Computational Physics, 161:114–139. Durand, W.F. (1963): Aerodynamic Theory, Dover, New York. Durrieu, P.P.J.M., Hofmans, G.C.J., Ajello, G., Boot, R.J.J., Auregan, Y., Hirschberg, A. and Peters, M.C.A.M. (2001): Quasi-steady aero-acoustic response of orifices, Journal of the Acoustical Society of America, 110:1859–1872. Ehrenfried, K., Meyer, C. and Dillmann, A. (2003): Simulation of Sound Propagation in Boundary Layers Based on M¨ohring’s Acoustic Analogy, AIAA Paper 2003-3272. Ekaterinaris, J.A. (1999a): Implicit, High-Order Accurate in Space Algorithms for Navier–Stokes Equations, AIAA Paper 99-3257. Ekaterinaris, J.A. (1999b): Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics, Journal of Computational Physics, 156:272–299. Ekaterinaris, J.A. (1999c): New formulation of Hardin–Pope equations for aeroacoustics, AIAA Journal, 37:1033–1039. El-Askary, W.A., Ewert, R. and Schr¨oder, W. (2002): On large eddy simulation as a tool to predict acoustical fields, Proc. Workshop LES for Acoustics, G¨ottingen. El-Askary, W.A., Meinke, M. and Schr¨oder, W. (2001): Towards the numerical analysis of trailing-edge noise, Proceedings of the German Aerospace Conference, Hamburg, DGLR–JT, 2001–176. El-Askary, W.A., Schr¨oder, W. and Meinke, M. (2003): LES of Compressible Wall-Bounded Flows, AIAA Paper 2003-3554. Engquist, B. and Majda, A. (1977): Absorbing boundary conditions for the numerical solution of waves, Mathematics of Computation, 31:629–651. Erlerbacher, G., Hussaini, M.Y., Kreiss, M.O. and Sarkar, S. (1990): The analysis and simulation of compressible turbulence, Theoretical and Computational Fluid Dynamics, 2:73–95. Erlerbacher, G., Hussaini, M.Y., Speziale, C.G. and Zang, T.A. (1992): Toward the large-eddy simulation of compressible turbulent flows, Journal of Fluid Mechanics, 238:155–185. Estivalezes, J.L. and Gamet, L. (1996): From jet flow computations to far-field noise prediction. FED-Vol. 238, 1996 Fluids Engineering Division Conference, Volume 3. Etkin, B., Korbacher, G.K. and Keefe, R.T. (1957): Acoustic radiation from a stationary cylinder in a fluid stream (aeolian tones), Journal of the Acoustical Society of America, 29:30–36.

12:29

403

P1: JZZ references

CUFX063/Wagner

404

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Evans, G.W. and Maxwell, L. (1997): Chronic noise exposure and reading deficits: The mediating effects of language acquisition, Environment and Behaviour, 29:638–656. Eversman, W. (1993): Theoretical models for duct acoustic propagation and radiation, Aeroacoustics of Flight Vehicles, Theory and Practice – Volume 2: Noise Control, H.H. Hubbard (ed.), The Acoustical Society of America. Ewert, R., Meinke, M. and Schr¨oder, W. (2000): Aeroacoustic Source Terms for the Linearized EulerEquations, AIAA Paper 2000-2046. Ewert, R., Meinke, M. and Schr¨oder, W. (2001a): Comparison of Source Term Formulations for a Hybrid CFD/CAA method, AIAA Paper 2001-2200, 7th AIAA/CEAS Aeroacoustic Conference. Ewert, R., Meinke, M. and Schr¨oder, W. (2001b): Computation of aeroacoustic sound via hybrid CFD/CAA methods, NATO RTO/AVT Symposium on Development in Computational Aero- and Hydro-Acoustics, Manchester, UK, Oct. 8–11. Ewert, R., Meinke, M. and Schr¨oder, W. (2002): Computation of Trailing Edge Noise via LES and Acoustic Perturbation Equations, AIAA Paper 2002-2467. Ewert, R. and Schr¨oder, W. (2003): Acoustic perturbation equations based on flow decomposition via source filtering, Journal of Computational Physics, 188(2):365–398. Ewert, R. and Schr¨oder, W. (2004): On the simulation of trailing edge noise with a hybrid LES/APE method, Journal of Sound and Vibration, 270:509–524. Ewert, R., Schr¨oder, W., Meinke, M. and El-Askary, W.A. (2002): LES as a Basis to Determine Sound Emission, AIAA Paper 2002-0568. Ewert, R., Zhang, Q., Schr¨oder, W. and Delfs, J. (2003): Computation of Trailing Edge Noise of a 3D Lifting Airfoil in Turbulent Subsonic Flow, AIAA Paper 2003-3114. Falkner, V.M. (1943): The resistance of a smooth flat plate with turbulent boundary layer, Aircraft Engineering, 15:65–58. Fan, T.C., Xiao, X., Edwards, J.R., Hassan, H.A. and Baurle, R.A. (2002): Hybrid LES/RANS Simulation of a Shock Wave/Boundary Layer Interaction, AIAA Paper 2002-0431. Fan, T.C., Xiao, X., Edwards, J.R., Hassan, H.A. and Baurle, R.A. (2003): Hybrid LES/RANS Simulation of a Mach 3 Shock Wave/Boundary Layer Interaction, AIAA Paper 2003-0080. Farassat, F. (1981): Linear acoustic formulas for calculation of rotating blade noise, AIAA Journal, 19(9):1122–1130. Fedorchenko, A.T. (2000): On some fundamental flaws in present aeroacoustic theory, Journal of Sound Vibration, 232(4):719–782. Fernholz, H.H. (1971): Ein halbempirisches Gesetz f¨ur die Wandreibung in kompressiblen turbulenten Grenzschichten bei isothermer und adiabater Wand, ZAMM, 51:T146–T147. Ffowcs Williams, J.E. (1969): Hydrodynamic noise, Annual Review of Fluid Mechanics, 1:197–222. Ffowcs Williams, J.E. (1982): Boundary layer pressures and the Corcos model: A development to incorporate low wavenumber constraints, Journal of Fluid Mechanics, 125:9–25. Ffowcs Williams, J.E. and Hawkings, D.L. (1969): Sound generated by turbulence and surfaces in arbitrary motion, Philosophical Transactions of the Royal Society of London Series A, 264(1151):321– 342. Ffowcs Williams, J.E., Simson, J. and Virchis, V.J. (1975): “Crackle”: An annoying component of jet noise, Journal of Fluid Mechanics, 71:251–271. Fischer, P. and Iliescu, T. (2001): A 3D channel flow simulation at Reτ = 180 using a rational LES model. DNS/LES Progress and Challenges, Liu , Sakell and Beutner (eds.), Greyden Press, Columbus, OH, pp. 283–290. Fleig, O., Iida, M. and Arakawa, C. (2002): Aeroacoustics simulation around a wind turbine blade using compressible LES and linearized Euler equations, Proceedings of the International Workshop on “LES for Acoustics”, 7–8 October, German Aerospace Center, DLR, G¨ottingen, Germany, DGLRReport 2002-03. Fletcher, N.H. and Rossing, T.D. (1998): The Physics of Musical Instruments, Springer-Verlag, New York, 2nd ed. Forestier, N., Jacquin, L. and Geffroy, P. (2003): The mixing layer over a deep cavity at high-subsonic speed, Journal of Fluid Mechanics, 475:101–145.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Fortun´e, V., Lamballais, E. and Gervais, Y. (2001): Study of Temperature Effects on Radiated Noise from Mixing Layers Using DNS, AIAA Paper 2001-2257. Freund, J.B. (1997): Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound, AIAA Journal, 35(4):740–743. Freund, J.B. (1999): Acoustic Sources in a Turbulent Jet: A Direct Numerical Simulation Study, AIAA Paper 99-1858. Freund, J.B. (2000): A simple method for computing far-field sound in aeroacoustic computations. Journal of Computational Physics, 157:796–800. Freund, J.B. (2001): Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9, Journal of Fluid Mechanics, 438:277–305. Freund, J.B. (2003): Noise-source turbulence statistics and the noise from a Mach 0.9 jet, Physics of Fluids 15(6):1788–1799. Freund, J.B. and Lele, S.K. (2004): Computer simulation and prediction of jet noise, High Speed Jet Flows, G. Raman, D. Mclaughlin, and P. Morris (eds.), Taylor & Francis, London. Freund, J.B., Lele, S.K. and Moin, P. (1996): Calculation of the radiated sound field using an open Kirchhoff surface, AIAA Journal, 34(5):909–915. Freund, J.B., Lele, S.K. and Moin, P. (1998): Direct Simulation of a Mach 1.92 Jet and Its Sound Field, AIAA Paper 98-2291. Freund, J.B., Lele, S.K. and Moin, P. (2000): Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate, Journal of Fluid Mechanics, 421:229–267. Fritz, W., Magagnato, F. and Rieger, H. (1991): Computational Methods for Prediction of Turbulent Flows around Automobiles, VDI-Kongress, Fellbach. Fritz, W., Magagnato, F. and Rieger, H. (1992): Computational methods for prediction of turbulent flows around automobiles, Proceedings of the 3rd International Conference of Innovation and Reliability in Automotive Design and Testing, Florence. Fr¨ohlich, J., Rodi, W., Kessler, P., Parpais, S., Bertoglio, J.P. and Laurence, D. (1998): Large-eddy simulations of flow around circular cylinders on structured and unstructured grids, in E.H. Hirschel (ed.), Notes on Numerical Fluid Mechanics, 66:319–338. Fu, D. and Ma, Y. (1997): A high order accurate difference scheme for complex flow fields, Journal of Computational Physics, 134:1–15. Fureby, C. and Grinstein, F.F. (1999): Monotonically large-eddy simulation of free shear flows, AIAA Journal, 37(5):544–556. Fureby, C. and Grinstein, F.F. (2002): Large-eddy simulation of high-Reynolds number free and wall-bounded flows, Journal of Computational Physics, 181:68–97. Fureby, C. and Tabor, G. (1997): Mathematical and physical constraints on large-eddy simulations, Theoretical and Computational Fluid Dynamics, 9:85–102. Gaitonde, D. and Shang, J.S. (1997): Optimized compact-difference-based finite-volume schemes for linear wave phenomena, Journal of Computational Physics, 138:617–643. Gaitonde, D. and Visbal, M. (1999): Further Development of a Navier–Stokes Solution Procedure Based on Higher-Order Formulas, AIAA Paper 99-0557. Gaitonde, D. and Visbal, M. (2000): Pade-Type Higher-Order Boundary Filters for the Navier–Stokes Equations, AIAA Journal, 38:2103. Gamet, L., Ducros, F., Nicoud, F. and Poinsot, T. (1999): Compact finite difference schemes on nonuniform meshes. Application to direct numerical simulations of compressible flows, International Journal for Numerical Methods in Fluids, 29:159–191. Gamet, L. and Estivalezes, J.L. (1998): Application of large-eddy simulations and Kirchhoff method to jet noise prediction, AIAA Journal, 36(12):2170–2178. Garnier, E., Mossi, M., Sagaut, P., Deville, M. and Comte, P. (1999): On the use of shock-capturing schemes for large-eddy simulation, Journal of Computational Physics, 153:273–311. Garnier, E., Sagaut, P. and Deville, M. (2001): A class of explicit ENO filters with application to unsteady flows, Journal of Computational Physics, 170:184–204. Garnier, E., Sagaut, P. and Deville, M. (2002): Large-eddy simulation of shock/homogeneous turbulence interaction, Computers and Fluids, 31(2):245–268.

12:29

405

P1: JZZ references

CUFX063/Wagner

406

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Gedney, S.D. (1996): An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices, IEEE Transactions on Antennas Propagation, 44:1630. Georgiadis, N.J., Alexander, J.I.D. and Reshotko, E. (2003): Hybrid Reynolds-averaged Navier–Stokes/large-eddy simulations of supersonic turbulent mixing, AIAA Journal, 41(2): 218–229. Germano, M. (1986): A proposal for a redefinition of the turbulent stresses in the filtered Navier– Stokes equations, Physics of Fluids, 29(7):2323–2324. Germano, M. (1987): On the Non-Reynolds Averages in Turbulence, AIAA Paper 87-1297. Germano, M. (1992): Turbulence: The filtering approach, Journal of Fluid Mechanics, 238:325–336. Germano, M. (1999): From RANS to DNS: Towards a bridging model, Direct and Large-Eddy Simulation III, Kluwer Academic Publishers, Dordrecht, pp. 225–236. Germano, M., Piomelli, U., Moin, P. and Cabot, W.H. (1991): A dynamic subgrid-scale eddy viscosity model, Physics of Fluids A, 3(7):1760–1765. Geurts, B.J. and Holm, D.D. (2003): Regularization modeling for large-eddy simulation. Physics of Fluids, 15(1):L13–L16. Gharib, M. and Roshko, A. (1987): The effect of flow oscillations on cavity drag, Journal of Fluid Mechanics, 177:501–530. Ghosal, S. (1996): An analysis of numerical errors in large-eddy simulations of turbulence, Journal of Computational Physics, 125:187–206. Ghosal, S. (1999): Mathematical and physical constraints on large-eddy simulation of turbulence, AIAA Journal, 37(4):425–433. Ghosal, S. and Moin, P. (1995): The basic equations for the large-eddy simulation of turbulent flows in complex geometry, Journal of Computational Physics, 118:24–37. Giles, M.B. (1990): Non-reflecting boundary conditions for Euler equation calculations, AIAA Journal, 28:2050–2058. Glaze, D.J. and Frankel, S.H. (2003): Stochastic inlet conditions for large-eddy simulation of a fully turbulent jet, AIAA Journal, 41(6):1064–1073. Gloerfelt, X., Bailly, C. and Juv´e, D. (2000): Calcul direct du rayonnement acoustique d’un e´ coulement affleurant une cavit´e, C. R. Acad. Sci., t. 328, S´erie IIb:625–631. Gloerfelt, X., Bailly, C. and Juv´e, D. (2001): Direct calculation of cavity noise and validation of acoustic analogies, Proc. of NATO/RTO Applied Vehicle Technology Panel, Symposium on Aging Mechanisms and Control, Development in Computational Aero- and Hydro-Acoustics, Manchester, UK, 8–11 October. Gloerfelt, X., Bailly, C. and Juv´e, D. (2003): Direct computation of the noise radiated by a subsonic cavity flow and application of integral methods, Journal of Sound and Vibration, 226(1):119–146. Gloerfelt, X., Bogey, C. and Bailly, C. (2002): LES of the noise radiated by a flow over a rectangular cavity, ERCOFTAC Workshop on LES for Acoustics, 7–8 October, DLR G¨ottingen, Germany. Gloerfelt, X., Bogey, C. and Bailly, C. (2003a): Numerical evidence of mode switching in the flowinduced oscillations by a cavity, International Journal of Aeroacoustics, 2(2):99–124. Gloerfelt, X., Bogey, C. and Bailly, C. (2003b): Numerical Investigation of the Coexistence of Multiple Tones in Flow-Induced Cavity Noise, AIAA Paper 2003-3234. Gloerfelt, X., Bogey, C., Bailly, C. and Juv´e, D. (2002): Aerodynamic Noise Induced by Laminar and Turbulent Boundary Layers over Rectangular Cavities, AIAA Paper 2002-2476. Goldstein, M.E. (1976): Aeroacoustics, McGraw-Hill Book Company, New York. Goldstein, M.E. (1978): Unsteady vortical and entropic disturbances of potential flows round arbitrary obstacles, Journal of Fluid Mechanics, 891:433–468. Goldstein, M.E. (2003): A generalized acoustic analogy, Journal of Fluid Mechanics, 488:315– 333. Golubev, V.V. and Atassi, H.M. (1998): Acoustic-vorticity waves in swirling flows, Journal of Sound and Vibration, 209(2):203–222. Goodrich, J.W. (1999): A Comparison of Numerical Methods for Computational Aeroacoustics, AIAA Paper 99-1943. Gottlieb, D. and Turkel, E. (1976): Dissipative two-four methods for time dependent problems, Mathematics of Computation, 30:703–723.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Graham, W.R. (1997): A comparison of models for the wavenumber-frequency spectrum of turbulent boundary layer pressures, Journal of Sound and Vibration, 206(4):541–565. Greschner, B., Thiele, F., Casalino, D. and Jacob, M.C. (2004): Influence of Turbulence Modelling on the Broadband Noise Simulation for Complex Flows, AIAA Paper 2004-2926. Grogger, H.A., Delfs, J.W., Lauke, T.G., Lummer, M. and Yin, J. (2000): Simulation of leading-edge noise of airfoils using CAA based on body-fitted grids, 7th International Congress on Sound and Vibration, Garmisch-Partenkirchen, Germany, 2000. Groll-Knapp, E. and Stidl, H.-G. (1999): Literaturzusammenstellung zum Thema Auswirkungen von Flugl¨arm auf den Menschen, Institut f¨ur Umwelthygiene der Universit¨at Wien, Vienna, Dezember. Gr¨otzbach, G. (1981): Numerical simulation of turbulent temperature fluctuations in liquid metals, International Journal of Heat and Mass Transfer, 24:475–490. Gr¨otzbach, G. (1987): Direct numerical and large-eddy simulation of turbulent channel flows, Encyclopedia of Fluid Mechanics, Cheremisinoff, N.P. (ed.), 6:1337–1391, Gulf Publ., Houston. Gr¨otzbach, G. and W¨orner, M. (1999): Direct numerical and large-eddy simulations in nuclear applications, International Journal of Heat and Mass Transfer, 20:222–240. Gutin, L. (1948): On the Sound Field of a Rotating Propeller, NACA, No. TM 1195 (Original in Russian: Zhurnal Tekhnicheskoi Fiziki 12:76–83, 1936). Hagstrom, T. and Nazarov, I. (2003): Perfectly Matched Layers and Radiation Boundary Conditions for Shear Flow Calculations, AIAA Paper 2003-3298. Hamed, A., Basu, D. and Das, K. (2003): Detached Eddy Simulation of Supersonic Flow over a Cavity, AIAA Paper 2003-0549. Hankey, W.L. and Shang, J.S. (1980): Analyses of pressure oscillations in an open cavity, AIAA Journal, 18(8):892–898. Hardin, J.C. and Pope, D.S. (1994): An acoustic/viscous splitting technique for computational aeroacoustics, Theoretical and Computational Fluid Dynamics, 6:323–340. Hardin, J.C. and Pope, D.S. (1995): Sound generation by flow over a two-dimensional cavity, AIAA Journal, 33(3):407–412. Harten, A. (1983): High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics, 49:357–393. Harten, A. (1994): Multiresolution Representation and Numerical Algorithms: A Brief Review, ICASE Report No 94-59. Harten, A. (1996): Multiresolution representation of data: A general framework, SIAM Journal of Numerical Analysis, 33(3):1205–1256. Hein, S., Hohage, T. and Koch, W. (2004): On resonances in open systems, Journal of Fluid Mechanics, 506:255–284. Heller, H.H., Holmes, D.G. and Covert, E.E. (1971): Flow induced pressure oscillations in shallow cavities, Journal of Sound and Vibration, 18(4):545–555. Henderson, J., Badcock, K. and Richards, B.E. (2000): Subsonic and Transonic Transitional Cavity Flows, AIAA Paper 2000-1966. Henshaw, M.J. de C. (2000): M219 Cavity Case in Verification and Validation Data for Computational Unsteady Aerodynamics, RTO-TR-26, AC/323(AVT)TP/19. Heo, D.N., Kim, J.W. and Lee, D.J. (2003): Study on Noise Characteristics of an Open Cavity with Cross-Correlation Analysis, AIAA Paper 2003-3104. Heo, D.N. and Lee, D.J. (2001): Numerical Investigation of the Cover-Plates Effects on the Rectangular Open Cavity, AIAA Paper 2001-2127. Higdon, R.L. (1987): Numerical absorbing boundary conditions for the wave equation, Mathematical Computation, 49:65–90. Hileman, J. and Samimy, M. (2001): Turbulence structures and the acoustic far field of a Mach 1.3 jet, AIAA Journal, 39(9):1716–1727. Hill, W.G. and Greene, P.R. (1977): Increased turbulent jet mixing rates by self-excited acoustic oscillations, Transactions of the ASME, Journal of Fluids Engineering, 99:520–525. Hinterberger, C., Garcia-Villalba, M. and Rodi, W. (2003): LES of flow around the Ahmed body: Lecture notes in applied mechanics, Proceedings of the UEF Conference on “The Aerodynamics of Heavy Vehicles: Trucks, Busses and Trains,” Monterey, CA, 2002, Springer-Verlag.

12:29

407

P1: JZZ references

CUFX063/Wagner

408

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Hinze, J.O. (1975): Turbulence, McGraw-Hill, New York. Hirsch, C. (1988): Numerical Computation of Internal and External Flows, Volume 1, John Wiley & Sons, New York. Hirsch, C. (1990): Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley Series in Numerical Methods in Engineering, John Wiley & Sons, New York. Hirschberg, A. (1995): Aero-acoustics of wind instruments. Mechanics of Musical Instruments, A. Hirschberg, J. Kergomard, and G. Weinreich (eds.), pp. 291–369, Vienna New York. International Centre for Mechanical Sciences, Springer-Verlag, Udine, Italy. Hirschberg, A., Bruggeman, J.C., Wijnands, A.P.J. and Smits, N. (1989): The whistler-nozzle and horn as aero-acoustic sound sources in pipe systems, Acustica, 68:157–160. Hirschberg, A., Gilbert, J., Msallam, R. and Wijnands, A.P.J. (1996): Shock waves in trombones. Journal of the Acoustical Society of America, 99(3):1754–1758. Hirschberg, A. and Schram, C. (1995): A primitive approach to aeroacoustics, Waves in Fluids, Y. Auregan, A. Maurel, and V. Pagneux (eds.), Springer-Verlag, New York, pp. 291–369. Hixon, R. (1998): A New Class of Compact Schemes, AIAA Paper 99-0367. Hixon, R. (1999): Prefactored Compact Filters for Computational Aeroacoustics, 37th AIAA Aerospace Sciences Meeting, AIAA Paper 99-0358. Hixon, R. (2000): Prefactored small-stencil compact schemes, Journal of Computational Physics, 165:522–541. Ho, C.-M. and Huerre, P. (1984): Perturbed free shear layers, Annual Review of Fluid Mechanics, 16:365–424. Hoffmann, G. and Benocci, C. (1995): Approximate wall boundary conditions for large-eddy simulations, Advances in Turbulence V, R. Benzi (ed.), Kluwer Academic Publishers, pp. 222–228. Hofmans, G.C.J. (1998): Vortex Sound in Confined Flows, Ph.D. thesis, Technical University of Eindhoven. Hofmans, G.C.J., Boot, R.J.J., Durrieu, P.P.J., Auregan, Y. and Hirschberg, A. (2000): Aeroacoustic response of a slit-shaped diaphragm in a pipe at low Helmholtz number, Part 1: Quasi-steady results, Journal of Sound and Vibration, 244:35–56. Holger, D.K., Wilson, T.A. and Beavers, G.S. (1977): Fluid mechanics of the edge tone, Journal of the Acoustical Society of America, 62:1116–1128. Howard, R.J.A. and Pourquie, M. (2002): Large-eddy simulation of the Ahmed reference model, Journal of Turbulence, 42(3):1–18. Howe, M.S. (1975): Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute, Journal of Fluid Mechanics, 71:625–673. Howe, M.S. (1979): Attenuation of sound in a low Mach number nozzle flow, Journal of Fluid Mechanics, 91:209–229. Howe, M.S. (1980): The dissipation of sound at an edge, Journal of Sound and Vibration, 70:407–411. Howe, M.S. (1997): Edge, cavity and aperture tones at very low Mach numbers, Journal of Fluid Mechanics, 330:61–84. Howe, M.S. (1998): Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, UK. Howe, M.S. (2000): Analytical Representations of Fluid-Structure Interaction Noise for Large Eddy Simulations, Report AM-00-003, Boston University, College of Engineering. Howe, M.S. (2001): Edge-source acoustic Green’s function for an airfoil of arbitrary chord, with application to trailing edge noise. Quarterly Journal of Mechanics and Applied Mathematics, 54(1):139–155. Howe, M.S. (2002): Theory of Vortex Sound, Cambridge University Press, Cambridge, UK. Hu, F.Q. (1996a): On absorbing boundary conditions of linearized Euler equations by a perfectly matched layer, Journal of Computational Physics, 129:201–219. Hu, F.Q. (1996b): On Perfectly Matched Layer as an Absorbing Boundary Condition, AIAA Paper 96-1664. Hu, F.Q. (2001): A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, Journal of Computational Physics, 173:455–480.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Hu, F.Q. (2002): On Constructing Stable Perfectly Matched Layers as an Absorbing Boundary Condition for Euler Equations, AIAA Paper 2002-0227. Hu, F.Q. (2004): On Using Perfectly Matched Layer for the Euler Equations with a Non-uniform Mean Flow, AIAA Paper 2004-2966. Hu, F.Q. (2005): A perfectly matched layer absorbing boundary condition for linearized Euler equations with a non-uniform mean flow, Journal of Computational Physics, 208:469–492. Hu, F.Q. and Atkins, H.L. (2003): A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method, AIAA Paper 2003-3301. Hu, F.Q., Hussaini, M.Y. and Manthey, J.L. (1996): Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics, Journal of Computational Physics, 124:177–191. Hu, F.Q., Hussaini, M.Y. and Rasetarinera, P. (1999): An analysis of the discontinuous galerkin method for wave propagation problems, Journal of Computational Physics, 151:921–946. Hubbard, H.H. (1995): Aeroacoustics of Flight Vehicles: Theory and Practice. Volume 1 Noise Sources; Volume 2 Noise Control (NASA Reference Publication 1258), Acoustical Society of America. Hubbard, H.H. and Shepherd, K.P. (1991): Aeroacoustics of large wind turbines, Journal of the Acoustical Society of America, 89:2495–2508. Hughes, T.J.R., Feijoo, G.R., Mazzei, L. and Quincy, J.B. (1998): The variational multiscale method – A paradigm for computational mechanics, Computational Methods in Applied Mechanical Engineering, 166:2–24. Hughes, T.J.R., Mazzei, L. and Jansen, K.E. (2000): Large-eddy simulation and the variational multiscale method, Computers in Visual Sciences, 3:47–59. Hughes, T.J.R., Mazzei, L., Oberai, A.A. and Wray, A.A. (2001): The multiscale formulation of largeeddy simulation: Decay of homogeneous isotropic turbulence, Physics of Fluids, 13(2):505–512. Hughes, T.J.R., Oberai, A.A. and Mazzei, L. (2001): Large-eddy simulation of turbulent channel flows by the variational multiscale method, Physics of Fluids, 13(6):1784–1799. Hygge, S (1993): A comparison between the impact of noise from aircraft, road traffic and trains on long-term recall and recognition of a text in childred aged 12–14 years. Schriftenreihe des Vereins f u¨ r Wasser-, Boden- und Lufthygiene 88:416–427. Iaccarino, G. and Durbin, P. (2000): Unsteady 3D RANS Simulations Using the v2-f Model, Center for Turbulence Research, Annual Research Briefs. Ihme, M. and Breuer, M. (2002): Sound prediction for a turbulent co-flowing jet based on LES, Proceedings of the International Workshop on “LES for Acoustics,” 7–8 October, German Aerospace Center, DLR, G¨ottingen, Germany, DGLR-Report 2002-03. Inagaki, M., Murata, O., Kondoh, T. and Abe, K. (2002): Numerical prediction of fluid-resonant oscillation at low mach number, AIAA Journal, 40(9):1823–1908. Ingard, K.U. (1959): Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission, Journal of the Acoustical Society of America, 31(7):1035–1036. Ingard, K.U. and Labate, S. (1950): Acoustic circulation effects and the non-linear impedance of orifices, Journal of the Acoustical Society of America, 22:211–218. Ingard, U. and Ising, H. (1967): Acoustic non-linearity of an orifice, Journal of the Acoustical Society of America, 42:6–17. in’t Panhuis, P. (2003): Calculations of the Sound Pressure Reflection Coefficient and the Acoustic End Correction of a Semi-Infinite Circular Pipe Issuing a Subsonic Cold or Hot Jet with Co-Flow, Internal report, MWL, Aeronautical and Vehicle Engineering, Project EU-ARTEMIS GRD1-200025507, KTH, Stockholm. Supervisors H. Bod´en, S.W. Rienstra. Isensee, S. and Frenzel, J. (2003): Leiser Verkehr, L¨armforschung im Forschungsprogramm Mobilit¨at und Verkehr, BMBF Publik, Bundesministerium f¨ur Bildung und Forschung (BMBF) (ed.), Bonn, June 2003, URL: http://www.bmbf.de/pub/leiser verkehr.pdf. Israeli, M. and Orszag, S.A. (1981): Approximation of radiation boundary conditions, Journal of Computational Physics, 41:115–135. Jackson, J.D. (1999): Classical Electrodynamics, John Wiley & Sons, New York, 3rd edition. Jacob, M.C., Boudet, J., Casalino, D. and Michard, M. (2005): A rod-airfoil experiment as benchmark for broadband noise modelling, Theoretical and Computational Fluid Dynamics, 19(3):171–196. See also Proc. of the 3rd SWING Aeroacoustics Workshop, Stuttgart, 2002.

12:29

409

P1: JZZ references

CUFX063/Wagner

410

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Jameson, A., Schmidt, W. and Turkel, E. (1981): Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge–Kutta Time-Stepping Schemes, AIAA Paper 81-1259. Jansen, K., Maeder, T. and Reba, R. (2002): Finite-Element Based Large-Eddy Simulation of the Near-Nozzle Region of a Compressible Round Jet, AIAA Paper 2002-2358. Jayatunga, C., Kroeff, G., Carotte, J.F., McGuick, J.J. and Petersson, A.T. (2001): Computational aero-acoustic studies of an exhaust diffuser, Proc. of NATO/RTO Applied Vehicle Technology Panel, Symposium on Aging Mechanisms and Control, Development in Computational Aero- and HydroAcoustics, Manchester, UK, 8–11 October. Jenvey, P.L. (1989): The sound power from turbulence: A theory of the exchange of energy between the acoustic and non-acoustic fields, Journal of Sound and Vibration, 131:37–66. Johnson, C. and Pitk¨arata, J. (1986): An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Mathematics of Computation, 46(176):1–26. Jones, D.S. (1964): The Theory of Electromagnetism, Pergamon Press, Oxford, UK. Jones, D.S. (1972): Aerodynamic sound due to a source near a half-plane, Journal of the Institute for Mathematics and Applications, 9:114–122. Jordan, S.A. (1999): A large-eddy simulation methodology in generalized curvilinear coordinates, Journal of Computational Physics, 148:322–340. Junger, M.C. and Feit, D. (1986): Sound, Structures and their Interaction, MIT Press, Cambridge, Massachusetts, 2nd edition. Kalitzin, G., Kalitzin, N. and Wilde, A. (2000): A Factorization Scheme for RANS Turbulence Models and SNGR Predictions of Trailing Edge Noise, AIAA Paper 2000-1982, 6th AIAA-CEAS Aeroacoustics Conference, Lahaina (Hawaii), 12–14 June, 2000. Kalitzin, N. and Wilde, A. (2000): Stochastic modeling of turbulence in CAA simulations of airframe noise, Second Aeroacoustic Workshop, H. K¨orner and J.W. Delfs (eds.), TU Dresden Press, Braunschweig. Kanwal, R.P. (1998): Generalized Functions: Theory and Technique, Birkh¨auser, Boston, 2nd edition. Kapadia, S., Roy, S. and Wurtzler, J. (2003): Detached Eddy Simulation over a Reference Ahmed Car Model, AIAA Paper 2003-0857, 41st Aerospace Sciences Meeting and Exhibit, 6–9 January, Reno, NV. Karamcheti, K. (1955): Acoustic Radiation from Two-Dimensional Rectangular Cutouts in Aerodynamic Surfaces, Technical Note 3487, NACA. Karniadakis, G.E. and Sherwin, S.J. (1999): Spectral/hp Element Methods for CFD, Numerical Mathematics and Scientific Computing, Oxford University Press, New York. Kato, C., Iida, A., Hattori, M. and Inadama, S. (2000): Numerical Simulation of Aerodynamic Sound Source in the Wake of a Complex Object, 6th AIAA/CEAS Aeroacoustics Conference, Lahaina, Hawaii, AIAA Paper 2000-1942. Kato, C., Iida, A., Takano, Y., Fujita, H. and Ikegawa, M. (1993): Numerical Prediction of Aerodynamic Noise Radiated from Low Mach Number Turbulent Wake, AIAA Paper 93-0145. Kato, C., Shishido, S., Miyazawa, M., Yoshiki, H., Ito, H. and Tsubota, H. (2002): Numerical prediction of aerodynamic noise radiated from a propeller fan, The Fifth JSME-KSME Fluids Engineering Conference, Nagoya, Japan. Kato, C., Takano, Y., Iida, A., Fujita, H. and Ikakawa, M. (1994): Numerical prediction of aerodynamic sound by large-eddy simulation, Transaction of the Japan Society of Mechanical Engineers, Part B, 60(569):126–132. Kato, C., Takano, Y., Iida, A. and Ikegawa, M. (1991): Numerical computation of aerodynamic noise radiation by the large-eddy simulation (Japanese), 5th Numerical Fluid Dynamics Symposium, Tokyo, Japan, 195–198. Keating, A., Piomelli, U., Balaras, E. and Kaltenbach, H.-J. (2004): Inflow conditions for large-eddy simulation, Submitted to Physics of Fluids. Kennedy, C.A., Carpenter, M.H. and Lewis, R.M. (1999): Low-Storage Explicit Runge–Kutta Schemes for the Compressible Navier–Stokes Equations, ICASE Report No. 99-22. Kergomard, J., Garcia, A. and Taguy, G. (1989): Analysis of higher order mode effects in an expansion chamber using modal theory and equivalent electrical circuits, Journal of Sound and Vibration, 129:457–475.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Khavaran, A. and Bridges, J. (2004): Modelling of Turbulence Generated Noise in Jets, AIAA Paper 2004-2983. Khavaran, A., Bridges, J. and Freund, J.B. (2002): A Parametric Study of Fine-Scale Turbulence Mixing Noise, AIAA Paper 2002-2419. Khorrami, M.R., Singer, B.A. and Berkman, M.E. (2002): Time-accurate simulations and acoustic analysis of slat free shear layer, AIAA Journal, 40(7):1284–1291. Khorrami, M., Singer, B. and David, P.L. (2002): Time-Accurate Simulations and Acoustic Analysis of Slat Free Shear Layers: Part II, AIAA Paper 2002-2579. Kim, I. and Chokani, N. (1990): Navier–Stokes Simulation of Unsteady Supersonic Cavity Flowfield with Passive Control, AIAA Paper 90-3101-CP. Kim, J.W. and Lee, D.J. (1996): Optimized compact finite difference schemes with maximum resolution, AIAA Journal, 34(5):887–893. Kim, J.W. and Lee, D.J. (2000): Adaptive Nonlinear Artificial Dissipation Model for Computational Aeroacoustics, AIAA Paper 2000-1978. Kirchhoff, G. (1877): Vorlesungen u¨ ber Mathematische Physik. Bd. 1: Mechanik. 2. Aufl. Teubner, Leipzig. Klein, M., Sadiki, A. and Janicka, J. (2003): A digital filter based generation of inflow data for spatially developing direct numerical or large-eddy simulations, Journal of Computational Physics, 186(2):652–665. Knaepen, B., Debliquy, O. and Carati, D. (2002): Subgrid-scale energy and pseudo pressure in largeeddy simulation, Physics of Fluids, 14(12):4235–4241. Kobayashi, H. and Shimomura, Y. (2003): Inapplicability of the dynamic Clark model to the large eddy simulation of incompressible turbulent channel flows, Physics of Fluids, 15(3):L29–L32. Kobayashi, M.H. (1999): On a class of pad´e finite volume methods, Journal of Computational Physics, 156:137–180. Kobayashi, T. and Satake, M. (1991): Numerical prediction of aerodynamic sound using large-eddy simulation (Japanese), 5th Numerical Fluid Dynamics Symposium, Tokyo, Japan, pp. 191–194. Koh, S.R. and Moon, Y.J. (2003): Aeroacoustic Computations of Turbulent Flows over an Open Cavity, AIAA Paper 2003-3102. Kolbe, R., Kailasanath, K., Young, T., Boris, J. and Landsberg, A. (1996): Numerical simulations of flow modification of supersonic rectangular jets, AIAA Journal, 34:902–908. Kolmogorov, A.N. (1941): Dissipation of energy in the locally isotropic turbulence, Comptes Rendus (Doklady) de l’Acad´emie des Sciences de l’URSS, XXXII(1):16–18. Komerath, N.M., Ahuja, K.K. and Chambers, F.W. (1987): Prediction and Measurement of Flows over Cavities – A Survey, AIAA Paper 87-0166. Kopriva, D.A., Woodruff, S.L. and Hussaini, M.Y. (2000): Discontinuous spectral element approximation of Maxwell’s equations. Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G.E. Karniadakis, and C.W. Shu (eds.), Lecture Notes in Computational Science and Engineering, 11:355–361. Springer Verlag, New York. Kosloff, R. and Kosloff, D. (1986): Absorbing boundary conditions for wave propagation problems, Journal of Computational Physics, 63:363–376. Koutsavdis, E.K., Blaisdell, G.A. and Lyrintzis, A.S. (1999): On the Use of Compact Schemes with Spatial Siltering in Computational Aeroacoustics, AIAA Paper 99-0360. Koutsavdis, E.K., Blaisdell, G.A. and Lyrintzis, A.S. (2000): Compact schemes with spatial filtering in computational aeroacoustics, AIAA Journal, 38:713. Kraichnan, R.H. (1956): Pressure fluctuation in turbulent flow over a flat plate, Journal of the Acoustical Society of America, 38(3):378–380. Kraichnan, R.H. (1969): Diffusion by a random velocity field, Physics of Fluids, 13(1):22–31. Kraichnan, R.H. (1970): Diffusion by a random velocity field, Journal Computational Physics, 13(1):22–31. Kraichnan, R.H. (1971): Inertial-range transfer in two- and three-dimensional turbulence, Journal of Fluid Mechanics, 47(3):525–535. Kraichnan, R.H. (1976): Eddy viscosity in two and three dimensions, Journal of Atmospheric Sciences, 33:1521–1536.

12:29

411

P1: JZZ references

CUFX063/Wagner

412

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Kravchenko, A.V. and Moin, P. (1997): On the effect of numerical errors in large-eddy simulations of turbulent flows, Journal Computational Physics, 131:310–322. Kundu, P.K. (1990): Fluid Mechanics, Academic Press, San Diego. Kurbatskii, K.K. and Tam, C.K.W. (1999): Direct numerical simulation of automobile cavity tones, 3rd Computational Aeroacoustics Workshop, pp. 371–383, Ohio Aerospace Institute, Cleveland, OH, Nov. 8–10. Labourasse, E. (2002): Reconstruction des fluctuations turbulentes par une approche hybride RANS/LES. Ph. D. thesis (11 December 2002), Onera, Universite Pierre et Marie Curie – Paris VI. Labourasse, E. and Sagaut, P. (2002): Reconstruction of turbulent fluctuations using a hybrid RANS/LES approach, Journal of Computational Physics, 182(1):301–336. Lacor, C., Smirnov, S. and Baelmans, M. (2000): A finite volume formulation for compact schemes on arbitrary meshes with applications to LES, CFD 2000, N. Satofuka (ed.), pp. 479–484. SpringerVerlag 1st Int. Conf. on CFD, Kyoto, 11–14 July 2000. Lacor, C., Smirnov, S. and Baelmans, M. (2004): A finite volume formulation of compact central schemes on arbitrary, structured grids, Journal of Computational Physics, 198:535–566. Lafon, P. (1997): Noise of confined flows, Von Karman Institute for Fluid Dynamics, Lecture Series 1997-07, Sept. 15–18, 1997. Lafon, P., Caillaud, S., Devos, J.P. and Lambert, C. (2003): Shallow cavity noise aeroacoustical coupling and its structural effects on a piping steam line, Journal of Fluids and Structures, 18:695– 713. Lafon, P. and Devos, J.P. (1993): Numerical Prediction of Instabilities in Transonic Internal Flows Using an Euler TVD Code, AIAA Paper 93-0072. Lamp, A.M. and Chokani, N. (1997): Computation of cavity flows with suppression using jet blowing, Journal of Aircraft, 34(4):545–551. Landau, L.D. and Lifshitz, E.M. (1987): Fluid Mechanics, Pergamon Press, Oxford, UK, 2nd edition. Larchevˆeque, L., Sagaut, P. and Lˆe, T.-H. (2003): Large-Eddy Simulations of Flows in Weapon Bays, AIAA Paper 2003-0778. Larchevˆeque, L., Sagaut, P., Mary, I., Labb´e, O. and Comte, P. (2003): Large-eddy simulation of a compressible flow past a deep cavity, Physics of Fluids, 15(1):193–200. Larsson, J. (2002): Computational Aero Acoustics for Vehicle Applications, Licentiate, Chalmers University of Technology, Dept. of Thermo and Fluid Dynamics. Lau, J.C., Morris, P.J. and Fisher, M.J. (1979): Measurements in subsonic and supersonic free jets using a laser velocimeter, Journal of Fluid Mechanics, 93(1):1–27. Leclercq, D.J.J. (1999): Mod´elisation de la r´eponse vibro-acoustique d’une structure coupl´ee a` une cavit´e en pr´esence d’un e´ coulement turbulent, Ph.D. thesis, Departement G´enie M´ecanique, Universit´e de Technologie de Compi`egne, France. Leclercq, D.J.J. and Bohineust X. (2002): Investigation and modelling of the wall pressure field beneath a turbulent boundary layer at low and medium frequencies, Journal of Sound and Vibration, 257(3):477–501. Lee, S., Lele, S.K. and Moin, P. (1992): Simulation of spatially evolving turbulence and the applicability of Taylor’s hypothesis in compressible flow, Physics of Fluids A, 4(7):1521–1530. Lee, S., Lele, S. and Moin, P. (1993): Direct numerical simulation of isotropic turbulence interacting with a weak shock wave, Journal of Fluid Mechanics, 251:533–562. Corrigendum in 264:373– 374. Lee, S., Lele, S. and Moin, P. (1997): Interaction of isotropic turbulence with shock waves: Effect of shock strength, Journal of Fluid Mechanics, 340:225–247. Leith, C.E. (1990): Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer, Physics of Fluids A, 2(3):297–299. Lele, S.K. (1992): Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 103:16–42. Lemaire, S., Marquez, B. and Jansen, K.E. (1999): A Comparison between Lighthill Analogy and the Method of Kirchhoff Surfaces for Aeroacoustic Predictions in Finite Elements, AIAA Paper 99-0233, 37th AIAA Aerospace Sciences Meeting.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Lenormand, E., Sagaut, P. and Ta Phuoc, L. (2000): Large-eddy simulations of subsonic and supersonic channel flow at moderate Reynolds number, International Journal for Numerical Methods in Fluids, 32:369–406. Lenormand, E., Sagaut, P., Ta Phuoc, L. and Comte, P. (2000): Subgrid-scale models for large-eddy simulations of compressible wall bounded flows, AIAA Journal, 38:1340–1350. Leonard, A. (1974): Energy cascade in large-eddy simulations of turbulent fluid flows, Advances in Geophysics A, 18:237–248. Lesieur, M. (1997): Turbulence in Fluids (3rd edition), Kluwer, Dordrecht. Leslie, D.C. and Quarini, G.L. (1979): The application of turbulence theory to the formulation of subgrid modelling procedures, Journal of Fluid Mechanics, 91(1):65–91. Lesser, M.B. and Crighton, D.G. (1975): Physical acoustics and the method of matched asymptotic expansions, Physical Acoustics, Volume XI, W.P. Mason and R.N. Thurston (eds.), Academic Press, New York. Lesser, M.B. and Lewis, J.A. (1972): Applications of matched asymptotic expansions methods to acoustics I. The Webster–Horn equation and the stepped duct, Journal of the Acoustical Society of America, 51:1664–1669. Levine, H. and Schwinger, J. (1948): On the radiation of sound from an unflanged circular pipe, Physical Review, 73:383–406. Li, X., Ma, Y. and Fu, D. (2000): DNS of incompressible turbulent channel flow with upwind compact scheme on non-uniform meshes, CFD Journal, 8(4):536–543. Lighthill, M.J. (1952): On sound generated aerodynamically. Part I: General theory, Proceedings of the Royal Society of London, A211:564–587. Lighthill, M.J. (1954): On sound generated aerodynamically. Part II: Turbulence as a source of sound, Proceedings of the Royal Society of London, A222:1–32. Lillberg, E. and Fureby, C. (2000): Large-Eddy Simulations of Supersonic Cavity Flows, AIAA Paper 2000-2411. Lilley, G.M. (1969): Generation of Sound in a Mining Region, Lockheed Aircraft Company, F-3361571-C-1663. Lilley, G.M. (1974): On the Noise from Jets, Technical Report AGARD CP-131. Lilley, M.G. (2004): The source of aerodynamic noise, International Journal of Aeroacoustics, 2:241– 253. Lilly, D.K. (1992): A proposed modification of the Germano subgrid-scale closure method, Physics of Fluids A, 4(3):633–635. Lin, S.Y., Yu, F. and Shih, S.-C. (1997): Numerical Study of MUSCL Schemes for Computational Aeroacoustics, AIAA Paper 97-0023. Liou, M.S. and Steffen, C.J. (1993): A new flux splitting scheme, Journal of Computational Physics, 107:23–39. Liu, C. and Liu, Z. (1993): High order finite difference and multigrid methods for spatially evolving instability in a planar channel, Journal of Computational Physics, 106:92–100. LMS SYSNOISE. (2002): LMS SYSNOISE Rev 5.6, Computational Vibro-Acoustics, On-Line User’s manual, LMS International NV, Interleuvenlaan 68, Leuven, Belgium. Lockard, D.P., Brentner, K.S. and Atkins, H.L. (1994): High-Accuracy Algorithms for Computational Aeroacoustics, AIAA Paper 94-0460. Lokhande, B., Sovani, S. and Xu, J. (2003): Computational Aeroacoustics Analysis of a Generic Side View Mirror, SAE Paper 2003-01-1698. Longatte, E., Lafon, P. and Candel, S. (1998): Computation of Noise Generation by Turbulence in Internal Flows, AIAA Paper 98-2332. Lord, W. and Feng, J. (2000): Physics of jet noise suppression, Proceedings of the Jet Noise Workshop, NASA/CP–2001-211152, pp. 617–630, Cleveland, OH, NASA Glenn Research Center. Lourenco, L. and Shih, C. (1993): Characteristics of the Plane Turbulent Near Wake of a Circular Cylinder, A Particle Image Velocimetry Study. Adrian, Springer-Verlag. Lui, C. and Lele, S.K. (2001): Direct Numerical Simulation of Spatially Developing, Compressible, Turbulent Mixing Layers, AIAA Paper 2001-0291.

12:29

413

P1: JZZ references

CUFX063/Wagner

414

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Lui, C. and Lele, S.K. (2002a): A Numerical Investigation of Broad-Band Shock Noise, AIAA Paper 2002-0074. Lui, C. and Lele, S.K. (2002b): A Numerical Study of Shock-Associated Noise, AIAA Paper 20022530. Lui, C. and Lele, S.K. (2003): Sound Generation Mechanism of Shock-Associated Noise, AIAA Paper 2003-3315. Lund, T.S., Wu, X. and Squires, K.D. (1998): Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations, Journal of Computational Physics, 140:233–258. Lush, P.A. (1971): Measurements of subsonic jet noise and comparison with theory, Journal of Fluid Mechanics, 46:477–500. Lyrintzis, A.S. (1994): Review: The use of Kirchhoff’s method in computational aeroacoustics, ASME Journal of Fluids Engineering, 116:665–676. Maestrello, L. (1976): Two-Point Correlations of Sound Pressure in the Far Field of a Jet: Experiment. NASA-TM-X-72835. Technical Report, NASA Langley Research Center, Hampton, VA. Magagnato, F. (1998): Kappa–Karlsruhe parallel program for aerodynamics, Tasc Quarterly, 1(4):215–270. Gdansk. Magagnato, F. and Gabi, M. (2000): A new adaptive turbulence model for unsteady flow fields in rotating machinery, Proceedings of the 8th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Volume I, Honolulu, Hawaii. Magagnato, F., Pantle, I. and Gabi, M. (2002): Prediction of aero-acoustic noise around a circular cylinder by large-eddy simulation, Proceedings of the International Workshop on “LES for Acoustics,” German Aerospace Center, DLR, G¨ottingen, Germany, DGLR-Report 2002-03. Magagnato, F., Pritz, B. and Gabi, M. (2005): LES on Internal and External Flow Fields Using a Compressible Solver. To be published at Proceedings of the ERCOFTAC International Symposium on Engineering Turbulence Modelling and Measurements -ETMM6-, Sardinia, Italy. Magagnato, F., Sorg¨uven, E. and Gabi, M. (2003): Far Field Noise Prediction by Large Eddy Simulation and Ffowcs Williams–Hawkings Analogy, AIAA Paper 2003-3206. Magnient, J.C., Sagaut, P. and Deville, M. (2001): A study of built-in filter for some, eddy viscosity models in large-eddy simulation, Physics of Fluids, 13(5):1440–1449. Mahesh, K., Lee, S., Lele, S. and Moin, P. (1995): The interaction of an isotropic field, of acoustic waves with a shock wave, Journal of Fluid Mechanics, 300:383–407. Mahesh, K., Lele, S. and Moin, P. (1997): The influence of entropy fluctuations on the interaction of turbulence with a shock wave, Journal of Fluid Mechanics, 334:353–379. Mandal, J.C. and Deshpande, S.M. (1994): Kinetic flux vector splitting for Euler equations, Computer, and Fluids, 23:447. Manhart, M. (2001): Analyzing near-wall behavior in a separating turbulent boundary layer by DNS, 4th Workshop on DNS & LES, Enschede, The Netherlands, July 18–20, 2001, ERCOFTAC Series, vol. 8, Direct and Large–Eddy Simulation IV, B.J. Geurts, R. Friedrich, and O. M´etais (eds.), Kluwer Academic Publishers. Dordrecht, pp. 81–88. Mani, M. and Paynter, G.C. (2002): Hybrid Turbulence Models for Unsteady Simulation of Jet Flows, AIAA Paper 2002-2959. Mankbadi, R.R., Hayder, M.E. and Povinelli, L.A. (1994): Structure of supersonic jet flow and its radiated sound, AIAA Journal, 32(5):897–906. Mankbadi, R.R., Hixon, R. and Povinelli, L.A. (2000): Very Large-Eddy Simulations of Jet Noise, AIAA Paper 2000-2008. Mankbadi, R.R., Hixon, R., Shih, S.-H. and Povinelli, L.A. (1995): On the Use of Linearized Euler Equations in the Prediction of Jet Noise, AIAA Paper 95-0505, 33rd Aerospace Sciences Meeting. Mankbadi, R.R., Hixon, R., Shih, S.-H. and Povinelli, L.A. (1998): Use of linearized Euler equations for supersonic jet noise prediction, AIAA Journal, 36(2):140–147. Mankbadi, R.R., Shih, S.-H., Hixon, R. and Povinelli, L.A. (1995): Direct Computation of Sound Radiation by Jet Flow Using Large-Scale Equations, AIAA Paper 95-0680, 33rd Aerospace Sciences Meeting. Mankbadi, R.R., Shih, S.H., Hixon, R. and Povinelli, L.A. (2000): Direct computation of jet noise produced by large-scale axisymmetric structures, Journal of Propulsion and Power, 16(2):207–215.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Manoha, E., Delahay, C., Redonnet, S., Ben Khelil, S., Guillen, P., Sagaut, P. and Mary, I. (2001): Numerical prediction of the unsteady flow and radiated noise from a 3D lifting airfoil, Proc. of NATO/RTO Applied Vehicle Technology Panel, Symposium on Aging Mechanisms and Control, Development in Computational Aero- and Hydro-Acoustics, Manchester, UK, 8-11 October. Manoha, E., Delahay, C., Sagaut, P., Mary, I., Khelil, S.B. and Guillen, P. (2001): Numerical Prediction of the Unsteady Flow and Radiated Noise from a 3D Lifting Airfoil, AIAA Paper 2001-2133. Manoha, E., Elias, G., Troff, B. and Sagaut, P. (1999): Towards the Use of Boundary Element Method in Computational Aeroacoustics, AIAA Paper 99-1980. Manoha, E., Herrero, C., Sagaut, P. and Redonnet, S. (2002): Numerical Prediction of Airfoil Aerodynamic Noise, AIAA Paper 2002-2573, 8th CEAS/AIAA Aeroacoustics Conference, Breckenridge (CO), USA, 17–19 June, 2002. Manoha, E., Troff, B. and Sagaut, P. (2000): Trailing-edge noise prediction using large-eddy simulation and acoustic analogy, AIAA Journal, 38:575–583. Marble, F.E. and Candel, S.M. (1977): Acoustic disturbance from gas non-uniformities convected through a nozzle, Journal of Sound and Vibration, 55(2):225–243. Mary, I. and Sagaut, P. (2001): Large-Eddy Simulation of Flow around an Airfoil near Stall, AIAA Paper 2001-2559. Mary, I. and Sagaut, P. (2002): Large-eddy simulation of the flow around an airfoil near stall, AIAA Journal, 40(2):1139–1145. Mason, P.J. and Brown, A.R. (1994): The sensitivity of large-eddy simulations of turbulent shear flow to subgrid models, Boundary Layer Meteorology, 70:133–150. Mason, P.J. and Callen, N.S. (1986): On the magnitude of the subgrid-scale eddy coefficient in largeeddy simulations of turbulent channel flow, Journal of Fluid Mechanics, 162:439–462. McCormack, R.W. (1969): The Effect of Viscosity in Hypervelocity Impact Cratering, AIAA Paper 69-354. McDonough, J.M. and Bywater, M. (1986): Large-scale effects on local small-scale chaotic solutions to Burgers’ equation, AIAA Journal, 24(12):1924–1930. Meier, G.E.A., Grabitz, G., Jungowski, W.M., Witczak, K.J. and Anderson, J.S. (1978): Oscillations of the supersonic flow downstream of an abrupt increase in duct cross-section, Mitteilungen aus dem Max Planck-Institut f¨ur Str¨omungsforschung und der aerodynamischen Versuchsanstalt, 65:1– 172. Mein, M., Dupuy, F. and Bohineust, X. (2002): Experimental Characterization of Aeroacoustic Sources: Subsonic Flow over a Forward-Backward Step, AIAA Paper 2002-2507. Meinke, M., El-Askary, W.A., Zhang, Q. and Schr¨oder, W. (2004): Large eddy simulation of airfoil flow, Fourth Aeroacoustics Workshop, W. Schr¨oder (ed.), TU Dresden Press, Aachen, Germany. Meinke, M., Schr¨oder, W., Krause, E. and Rister, T. (2001): A comparison of second- and sixth-order methods for large-eddy simulation, Computers and Fluids, 31:695–718. Meinke, M., Schr¨oder, W., Krause, E. and Rister, Th. (2002): A comparison of second- and sixth-order methods for large-eddy simulations, Computers and Fluids, 31(4):695–718. Mendon¸ca, F. (2002): Aeroacoustics DESTINY-AAC industrial project, In: STAR-CD Dynamics, p. 5, winter 2002/2003, URL: http://www.cd-adapco.com/news/19/destiny.htm. Mendon¸ca, F., Allen, R., de Charentenay, J. and Kirkenham, D. (2003): CFD Prediction of Narrowband and Broadband Cavity Acoustics at M = 0.85, AIAA Paper 2003-3303. Mendon¸ca, F., Read, A., Caro, S., Debatin, K. and Caruelle, B. (2005): Aeroacoustic Simulation of Double Diaphragm Orifices in an Aircraft Climate Cooling System, AIAA Paper 2005-2976. Meneveau, C. and Katz, J. (2000): Scale-invariance and turbulence models for large-eddy simulation, Annual Review of Fluid Mechanics, 32:1–32. Meneveau, C., Lund, T.S. and Cabot, W.H. (1996): A Lagrangian dynamic subgrid-scale model of turbulence, Journal of Fluid Mechanics, 319:353–385. Menter, F.R. (1997): Eddy viscosity transport models and their relation to the k- model, ASME Journal of Fluids Engineering, 119:876–884. Menter, F.R., Kuntz, M. and Bender, R. (2003): A Scale-Adaptive Simulation Model for Turbulent Flow Predictions, AIAA Paper 2003-0767. Michalke, A. (1984): Survey on jet instability theory, Progress in Aerospace Sciences, 21:159–199.

12:29

415

P1: JZZ references

CUFX063/Wagner

416

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Michalke, A. (1989): On the propagation of sound generated in a pipe of circular cross-section with uniform mean flow, Journal of Sound and Vibration, 134:203–234. Michard, M., Jacob, M.C. and Grosjean, N. (2002): An experimental characterization of the flow past an airfoil in the wake of a circular rod, Proceedings of ASME FEDSM’02, FEDSM2002-31344. Miles, J.W. (1947): The diffraction of sound due to right-angle joints in rectangular tubes, Journal of the Acoustical Society of America, 19:572–579. Mitchell, B.E., Lele, S.K. and Moin, P. (1995a): Direct computation of the sound from a compressible co-rotating vortex pair, Journal of Fluid Mechanics, 285:181–202. Mitchell, B.E., Lele, S.K. and Moin, P. (1995b): Direct Computation of the Sound Generated by Vortex Pairing in an Axisymmetric Jet, AIAA Paper 95-0504. Mitchell, B.E., Lele, S.K. and Moin, P. (1997): Direct computation of Mach wave radiation in an axisymmetric supersonic jet, AIAA Journal, 35(10):1574–1580. Mitchell, B.E., Lele, S.K. and Moin, P. (1999): Direct computation of the sound generated by vortex pairing in an axisymmetric jet, Journal of Fluid Mechanics, 383:113–142. Miyake, Y., Bando, K. and Hori, J. (1993): Analysis of sound source distribution by means of largeeddy simulation (Japanese), Transactions of the Japan Society of Mechanical Engineers, Part B, 59(567):3475–3481. M¨ohring, W. (1978): On vortex sound at low Mach number, Journal of Fluid Mechanics, 85:685–691. M¨ohring, W. (1979): Modelling low Mach number noise, Mechanics of Sound Generation in Flows, E.-A. M¨uller (ed.), Springer-Verlag. M¨ohring, W. (1999): A well posed acoustic analogy based on a moving acoustic medium, SWING, Aeroacoustic Workshop, P. K¨oltzsch and N. Kalitzin (eds.), Dresden, Germany. DFG-BMBF: University of Stuttgart, RWTH Aachen, Dresden University of Technology, DLR Braunschweig. M¨ohring, W. (2001): Energy conservation, time-reversal invariance and reciprocity in ducts with flow, Journal of Fluid Mechanics, 431:223–237. Mohsen, A.M. (1967): Experimental Investigation of the Wall Pressure Fluctuations in Subsonic Flows, Boeing Commercial Airplane Division Report, D6-17094. Moin, P. and Mahesh, K. (1998): Direct numerical simulation: A tool in turbulence research, Annual Review of Fluid Mechanics, 30:507–538. Moin, P., Squires, K., Cabot, W. and Lee, S. (1991): A dynamic subgrid-scale model for compressible turbulence and scalar transport, Physics of Fluids A, 3:2746–2757. Mollo-Christensen, E., Kolpin, M.A. and Martucelli, J.R. (1964): Experiments on jet flows and jet noise far-field spectra and directivity patterns, Journal of Fluid Mechanics, 18:285–301. Mongeau, L., Kook, H., Brown, D.V. and Zorea, S.I. (1997): Analysis of interior pressure oscillations induced by flow over vehicle openings, Noise Control Engineering Journal, 45:223–234. Montavon, C. (2002): A big bang in aero-acoustics, CFX Update, 22:10–11, Autumn 2002. Montavon, C., Jones, I., Szepessy, D., Henriksson, R., el-Hachemi, Z., Dequand, S., Piccirillo, M., Tournour, M. and Tremblay, F. (2002): Noise propagation from a cylinder in a cross flow: Comparison of SPL from measurements and from a CAA method based on a generalised acoustic analogy. IMA Conference on Computational Aeroacoustics, University of Greenwich, 9–11, April 2002. Moon, Y.J., Sung, R.K., Cho, Y. and Chung, J.M. (1999): Aeroacoustic computations of the unsteady flows over a rectangular cavity with a lip, 3rd Computational Aeroacoustics Workshop, pp. 347–353, Ohio Aerospace Institute, Cleveland, Nov. 8–10. Moore, C.J. (1977): The role of shear layer instabilities in jet exhaust noise, Journal of Fluid Mechanics, 80:321–367. Morfey, C.L. (1973): Amplification of aerodynamic noise by convected flow inhomogeneities, Journal of Sound and Vibration, 31:391–397. Morfey, C.L. (1976): Sound radiation due to unsteady dissipation in turbulent flows, Journal of Sound and Vibration, 48:95–111. Morfey, C.L. (2003): The role of viscosity in aerodynamic sound generation, International Journal of Aeroacoustics, 2:230–240. Morinishi, Y. and Vasilyev, O.V. (2002): Vector level identity for dynamic subgrid scale modeling in large eddy simulation, Physics of Fluids, 14(10):3616–3623.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Morley, A.W. (1939): Estimation of aeroplane noise level: Some empirical laws with an account of the present experiments on which they are based, Aircraft Engineering, 11(123):187– 189. Morris, P. J. and Farassat, F. (2002): Acoustic analogy and alternative theories for jet noise prediction, AIAA Journal, 40(4):671–680. Morris, P.J., Long, L.N., Bangalore, A. and Wang, Q. (1997): A parallel three-dimensional computational aeroacoustics method using non-linear disturbance equations, Journal of Computational Physics, 133:56–74. Morris, P.J., Long, L.N. and Scheidegger, T. (1999): Parallel Computations of High Speed Jet Noise, AIAA Paper 99-1873. Morris, P.J., Long, L.N., Scheidegger, T.E. and Boluriaan, S. (2002): Simulations of supersonic jet noise, International Journal of Aeroacoustics, 1(1):17–41. Morris, P.J., Scheidegger, T. and Long, L.N. (2000): Jet Noise Simulations for Circular Nozzles, AIAA Paper 2000-2080. Morris, P.J., Wang, Q., Long, L.N. and Lockhard, D.P. (1997): Numerical Predictions of High-Speed Jet Noise, AIAA Paper 97-1598. Morse, P.M. and Feshbach, H. (1953): Methods of Theoretical Physics Volume I, McGraw-Hill Book Company, New York. Morse, P.M. and Ingard, K.U. (1968): Theoretical Acoustics, McGraw-Hill Book Company, New York. Moss, W.D. and Baker, S. (1980): Re-circulating flows associated with two-dimensional steps, Aeronautical Quarterly, August: 151–172. Muchinsky, A. (1996): A similarity theory of locally homogeneous anisotropic turbulence generated by a Smagorinsky-type LES, Journal of Fluid Mechanics, 325:239–260. M¨uller, E.A. and Obermeier, F. (1988): Vortex sound, Fluid Dynamics Research, 3:43–51. Munt, R.M. (1977): The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe, Journal of Fluid Mechanics, 83(4):609–640. Munt, R.M. (1990): Acoustic radiation properties of a jet pipe with subsonic jet flow: I. The cold jet reflection coefficient, Journal of Sound and Vibration, 142(3):413–436. Murlis, J., Tsai, H.M. and Bradshaw, P. (1982): The structure of turbulent boundary layers at low Reynolds numbers, Journal of Fluid Mechanics, 122:13–56. Musafir, R.E. (1997): A discussion on the structure of aeroacoustic wave equations, Proceedings of the 4th Congress on Acoustics, Marseille, France, pp. 923–926, Toulouse, France, April 1997. Teknea edition. Myers, M.K. (1980): On the acoustic boundary condition in the presence of flow, Journal of Sound and Vibration, 71(3):429–434. Myers, M.K. (1986a): An exact energy corollary for homentropic flow, Journal of Sound and Vibration, 109:277–284. Myers, M.K. (1986b): Generalization and Extension of the Law of Acoustic Energy Conservation in a Nonuniform Flow, AIAA Paper 86-0471. Myers, M.K. (1991): Transport of energy by disturbances in arbitrary flows, Journal of Fluid Mechanics, 226:383–400. Najjar, F.M. and Tafti, D.K. (1996): Study of discrete test filters and finite difference approximations for the dynamic subgrid-scale stress model, Physics of Fluids, 8(4):1076–1088. Najm, H.N. and Ghoniem, A.F. (1991): Numerical simulation of the convective instability in a dump combustor, AIAA Journal, 29(6):911–919. Nance, D.V., Viswanathan, K. and Sankar, L.N. (1997): Low-dispersion finite volume scheme for aeroacoustic applications, AIAA Journal, 35:255–262. NASA Glenn Research Center. (2001): Proceedings of the Jet Noise Workshop, NASA/CP–2001211152, Cleveland, OH, 1–1070. Nederveen, C.J. (1998): Acoustical Aspects of Woodwind Instruments, Northern Illinois University Press, DeKalb, 2nd edition. Nelson, P.A., Halliwell, N.A. and Doak, P.E. (1983): Fluid dynamics of flow excited resonance, Part 2: Flow acoustic interaction, Journal of Sound and Vibration, 91:375–402.

12:29

417

P1: JZZ references

CUFX063/Wagner

418

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Nelson, P.M. (1987): Transportation Noise Reference Book, Butterworths & Co. (Publishers) Ltd., London, UK. Nicolopoulos, D., P´eri´e, F. and Jacques, A. (2004): Direct Numerical Simulation of Aero-Acoustic Phenomena, Whitepaper, M 3 , France. Nicoud, F. and Ducros, F. (1993): Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow, Turbulence and Combustion, 62(3):183–200. Norberg, C. (1992): Pressure forces on a circular cylinder in a cross flow, IUTAM Symposium on Bluff Body Wakes, Dynamics and Instabilities, G¨ottingen, pp. 275–278. Norton, M.P. (1989): Fundamentals of Noise and Vibration Analysis for Engineers, Cambridge University Press, Cambridge, UK. Oberai, A.A., Roknaldin, F. and Hughes, T.J.R. (2002): Trailing-Edge Noise Due to Turbulent Flows, Report No. 02-002, Boston University. Oberlack, M. (1997): Invariant modeling in large-eddy simulation of turbulence, Annual Research Briefs – Center for Turbulence Research, 3–22. Obermeier, F. (1985): Aerodynamic sound generation caused by viscous processes, Journal of Sound and Vibration, 99:111–120. Obrist, D., Nicolopoulos, D. and Jacques, A. (2002): Aero acoustic simulation of a side mirror compared with experimental results, Internoise, August 2002. Oh, K.J. and Colonius, T. (2002): Large Eddy Simulation of the Compressible Flow over an Open Cavity, Proceedings of ASME FEDSM’02, Montreal, Canada, July 14–18. ASME Paper FEDSM 2002-31352. Orkwis, P.D., Sekar, B., Chakravarthy, S. and Peroomian, O. (1998): Comparison of three Navier– Stokes solvers for supersonic open cavity simulations, AIAA Journal, 36(5):865–867. Orlanski, I. (1976): A simple boundary condition for unbounded flows, Journal of Computational Physics, 21:251–269. Ota, D.K., Chakravarthy, S.R., Becker, T. and Sturznegger, T. (1994): Computational study of resonance suppression of open sunroofs, ASME Journal of Fluids Engineering, 116:877–882. Ovenden, N.C. (2002): Near Cut-On/Cut-Off Transitions in Lined Ducts with Flow, AIAA Paper 2002-2445. Ovenden, N.C. (2005): A uniformly valid multiple scales solution for sound in flow ducts with cut-on cut-off transition, Journal of Sound and Vibration, 286:403–416. Ovenden, N.C., Eversman, W. and Rienstra, S.W. (2004): Cut-On Cut-Off Transition in Flow Ducts: Comparing Multiple-Scales and Finite-Element Solutions, AIAA Paper 2004-2945. ¨ or¨uk, Y. and Long, L.N. (1996a): A new efficient algorithm for computational aeroacoustics on Ozy¨ parallel processors, Journal of Computational Physics, 125:135–149. ¨ or¨uk, Y. and Long, L.N. (1996b): Computation of sound radiating from engine inlets, AIAA Ozy¨ Journal, 34:894–901. ¨ or¨uk, Y. and Long, L.N. (1997): Multigrid acceleration of a high-resolution computational aeroaOzy¨ coustics scheme, AIAA Journal, 35:428–433. Page, G.J., McGuirk, J.J., Behrouzi, P. and Hossain, M. (2001): A CFD coupled acoustics approach for the prediction of coaxial jet noise, Proceedings of the NATO RTO-AVT Symposium on Ageing Mechanisms and Control, Part A – Developments in Aero and Hydro-Acoustics, Manchester, UK, 8–11 Oct. 2001. Pagneux, V. and Froelich, B. (2001): Influence of low Mach number shearflow on acoustic propagation in ducts, Journal of Sound and Vibration, 246:137–155. Paliath, U. and Morris, P.J. (2003): Numerical Simulation of Two-Dimensional Laminar and Turbulent Cavity Flows, AIAA Paper 2003-3233. Panda, J. and Seasholtz, R.G. (2002): Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise, Journal of Fluid Mechanics, 450:97–130. Panda, J., Seasholtz, R.G., Elam, K.A., Mielke, A.F. and Eck, D.G. (2004): Effect of Heating on Turbulent Density Fluctuations and Noise Generation from High Speed Jets, AIAA Paper 20043016. Pantano, C. and Sarkar, S. (2001): A subgrid model for nonlinear functions of a scalar, Physics of Fluids, 13(12):3803–3819.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Pantle, I. (2002): Str¨omungsakustik auf der Basis akustischer Analogie mit LES und URANS, Ph.D. thesis, Karlsruhe University. Paterson, A.R. (1983): A First Course in Fluid Dynamics, Cambridge University Press, Cambridge, UK. Pereira, J.M.C., Kobayashi, M.H. and Pereira, J.C.F. (2001): A fourth-order-accurate finite volume compact method for the incompressible Navier–Stokes solutions, Journal Computational Physics, 167:217–243. Pereira, M.J., Kobayashi, M.H. and Pereira, C.J. (2000): High order compact schemes in finite volume context, ECCOMAS 2000, Barcelona, B. Suarez E. Onate, G. Bugeda (eds.), CIMNE. P´eri´e, F. (2002): A direct approach to CAA, Internoise 2002, Dearborn, MI, USA, August 19–21, 2002. Peroomian, O., Chakravarthy, S., Palaniswamy, S. and Goldberg, U.C. (1998): Convergence Acceleration for Unified-Grid Formulation Using Preconditioned Implicit Relaxation, AIAA Paper 98-0116. P´erot, F., Auger, J.M., Giardi, H., Bailly, C. and Juv´e, D. (2004): Computation of the noise generated by low Mach number flows around a cylinder and a wall-mounted half-cylinder, 10th AIAA/CEAS Aeroacoustic Conference, Manchester, UK, AIAA Paper 2004-2859. Perzon, S. and Davidson, L. (2000): On transient modelling of the flow around vehicles using the Reynolds equation, ACDF 2000, Beijing, China, pp. 720–727. Peters, M.C.A.M. and Hirschberg, A. (1993): Acoustically induced periodic vortex shedding at sharp edged open channel ends: Simple vortex models, Journal of Sound and Vibration, 161:281–299. Peters, M.C.A.M., Hirschberg, A., Reijnen, A.J. and Wijnands, A.P.J. (1993): Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers, Journal of Fluid Mechanics, 256:499–534. Petropoulos, P. (2000): Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell’s equations in rectangular, cylindrical and spherical coordinates, SIAM Journal of Applied Mathematics, 60:1037. Phillips, O.M. (1960): On the generation of sound by supersonic turbulent shear layers, Journal of Fluid Mechanics, 9(1):1–28. Pierce, A.D. (1981): Acoustics: An Introduction to Its Physical Principles and Applications, McGrawHill Book Company, New York. Also available from the Acoustical Society of America. Pierce, A.D. (1990): Wave equation for sound in fluids with unsteady inhomogeneous flow, Journal of the Acoustical Society of America, 87(6):2292–2299. Piller, M. and Stalio, E. (2004): Finite-volume compact schemes on staggered grids, Journal of Computational Physics, 197(1):299–340. Piomelli, U. (2001): Large-Eddy Simulation of the Sound Emission from Unsteady Flow Separation, Final Report on Grant NAG-1 1880, Department of Mechanical Engineering, College Park, MD. Piomelli, U. and Balaras, E. (2002): Wall-layer models for large-eddy simulation, Annual Review of Fluid Mechanics, 34:349–374. Piomelli, U., Ferziger, J.H., Moin, P. and Kim, J. (1989): New approximate boundary conditions for large-eddy simulations of wall-bounded flows, Physics of Fluids A, 1(6):1061–1068. Piomelli, U., Streett, G.L. and Sarkar, S. (1997): On the computation of sound by large-eddy simulation, Journal of Engineering Mathematics, 32(5):217–236. Poinsot, T.J. and Lele, S.K. (1992): Boundary conditions for direct simulations of compressible viscous flows, Journal of Computational Physics, 101:104–129. Povitsky, A. (2001): High-Order Compact Simulation of Wave Propagation in a Non-uniform Flow, AIAA Paper 2001-2628. Povitsky, A. and Morris, P.J. (2000): A higher-order compact method in space and time based on parallel implementation of the Thomas algorithm, Journal of Computational Physics, 161:182– 203. Powell, A. (1964): Theory of vortex sound, Journal of the Acoustical Society of America, 36(1):177– 195. Powell, A. (1990): Some aspects of aeroacoustics: From Rayleigh until today, Journal of Vibration and Acoustics, 112:145–159.

12:29

419

P1: JZZ references

CUFX063/Wagner

420

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Prandtl, L. and Tietjens, O.G. (1957): Fundamentals of Hydro- and Aeromechanics, Dover, New York. Prieur, J. and Rahier, G. (1998): Comparison of Ffowcs Williams–Hawkings and Kirchhoff Rotor Noise Calculations, AIAA Paper 98-2376. Pruett, C.D. (2000): Eulerian time-domain filtering for spatial large-eddy simulation, AIAA Journal, 38(9):1634–1642. Pruett, C.D., Zang, T., Chang, C.-L. and Carpenter, M.H. (1995): Spatial direct numerical simulation of high-speed boundary layer flows. Part I. Algorithmic considerations and validation, Theoretical Computational Fluid Dynamics, 7:49–76. Pulliam, T.H. and Chaussee, D.S. (1981): A diagonal form of an implicit approximate-factorization algorithm, Journal of Computational Physics, 39:347–363. Qu´em´er´e, P. and Sagaut, P. (2002): Zonal multi-domain RANS/LES simulation of turbulent flows, International Journal for Numerical Methods in Fluids, 40:903–925. Quinn, M.C. and Howe, M.S. (1984): On the production and absorption of sound by lossless liners in the presence of mean flow, Journal of Sound and Vibration, 97(1):1–9. Rahier, G. and Prieur, J. (1997): An efficient Kirchhoff integration method for rotor noise prediction starting indifferently from subsonically or supersonically rotating meshes, 53rd Annual Forum of the American Helicopter Society, Virginia Beach, VA, April–May 1997. Rai, M.M. and Moin, P. (1991): Direct Numerical Simulation of Transition and Turbulence in a Spatially Evolving Boundary Layer, AIAA Paper 91-1607. Ramboer, J. and Lacor, C. (2002): A finite volume formulation for compact upwind schemes , Proc. 2nd Int. Conf. on CFD, Sydney, July 2002. Ramboer, J., Smirnov, S. and Lacor, C. (2002): Use of compact schemes in CAA and LES applications, Proc. of Workshop LES for Acoustics, DLR Goettingen, 7–8 October 2002. Ramboer, J., Smirnov, S. and Lacor, C. (2003): Finite Volume Formulation of Upwind Compact Schemes for CAA and LES Applications, AIAA Paper 2003-3967. Ravichandran, K.S. (1997): Higher order KFVS algorithms using compact upwind difference operators, Journal of Computational Physics, 130:161–173. Read, A. and Mendon¸ca, F. (2004): Numerical simulation methods for aeroacoustics with an overview of industrial test cases, 5th MIRA International Vehicle Aerodynamics Conference, UK, October 2004. Read, A., Mendon¸ca, F., Barone, F., Durello, P., Carena, F., Gallez, X., Ploumhans, P. and Caro, S. (2004): Comparison between Measured and Predicted Tonal Noise from a Subsonic Fan Using a Coupled Computational Fluid Dynamics (CFD) and Computational Acoustics (CA) Approach, AIAA Paper 2004-2936. Redmore, T.L. and Mulholland, K.A. (1982): The application of mode coupling theory to the transmission of sound in the sidebranch of a rectangular duct system, Journal of Sound and Vibration, 85:323–331. Redonnet, S., Manoha, E. and Sagaut, P. (2001): Numerical Simulation of Propagation of Small Perturbations Interacting with Flows and Solid Bodies, AIAA Paper 2001-2223. Reichl, C., Grogger, H.A., Krenn, Ch., Kaltenhauser, A., H¨oll, M., Mann, M. and Lang, H. (2003): Aeroacoustics characterisation of an exterior mirror applying numerical and experimental techniques, Computational Aeroacoustics, Euromech Colloquium No. 449, Chamonix, France. Rembold, B. and Kleiser, L. (2003): Noise Prediction from Rectangular Jet Flow Using LES, AIAA Paper 2003-3281. Ricot, D., Maillard, V., Bailly, C. and Juv´e, D. (2001): Numerical Simulation of the Unsteady Flow past a Cavity and Application to Sun Roof-Buffeting, AIAA Paper 2001-2112. Rienstra, S.W. (1981a): Sound diffraction at a trailing edge, Journal of Fluid Mechanics, 108:443–460. Rienstra, S.W. (1981b): On the acoustical implications of vortex shedding from an exhaust pipe, Transactions of the ASME, Journal of Engineering for Industry, 103(4):378–384. Rienstra, S.W. (1983): A small Strouhal number analysis for acoustic wave–jet flow–pipe interaction, Journal of Sound and Vibration, 86(4):539–556. Rienstra, S.W. (1984): Acoustic radiation from a semi-infinite annular duct in a uniform subsonic mean flow, Journal of Sound and Vibration, 94(2):267–288.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Rienstra, S.W. (1985): Contributions to the theory of sound propagation in ducts with bulk-reacting lining, Journal of the Acoustical Society of America, 77(5):1681–1685. Rienstra, S.W. (1986): Hydrodynamic instabilities and surface waves in a flow over an impedance wall, G. Comte-Bellot and J.E. Ffowcs Williams (eds.), Proceedings IUTAM Symposium ‘Aeroand Hydro-Acoustics’ 1985 Lyon, Springer-Verlag, Heidelberg, pp. 483–490. Rienstra, S.W. (1988): 1-D reflection at an impedance wall, Journal of Sound and Vibration, 125:43– 51. Rienstra, S.W. (1999): Sound transmission in slowly varying circular and annular ducts with flow, Journal of Fluid Mechanics, 380:279–296. Rienstra, S.W. (2002): A classification of duct modes based on surface waves, Wave Motion, 37(2):119– 135. Rienstra, S.W. (2003): Sound propagation in slowly varying lined flow ducts of arbitrary cross section, Journal of Fluid Mechanics, 495:157–173. Rienstra, S.W. and Hirschberg, A. (2001): An Introduction to Acoustics, Report IWDE 01–03 May 2001, Eindhoven University of Technology, Eindhoven, the Netherlands. Rizzetta, D.P. (1988): Numerical simulation of supersonic flow over a three dimensional cavity, AIAA Journal, 26(7):799–807. Rizzetta, D.P. and Visbal, M.R. (2002): Large-Eddy Simulation of Supersonic Cavity Flowfields Including Flow Control, AIAA Paper 2002-2853. Rockwell, D. (1983): Oscillations of impinging shear layers, AIAA Journal, 21(5):645–664. Rockwell, D. and Naudascher, E. (1978): Review – Self-sustaining oscillations of flow past cavities, ASME Journal of Fluids Engineering, 100:152–165. Roe, P.L. (1981): Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43:357–372. Rogallo, R.S. (1981): Numerical Experiments in Homogeneous Turbulence, NASA Technical Memorandum, TM–81315. Rogallo, R.S. and Moin, P. (1984): Numerical simulation of turbulent flows, Annual Review of Fluid Mechanics, 16:99–137. Roger, M. (2004): Noise from moving surfaces. Advances in Aeroacoustics and Applications, J. Anthoine (ed.), Brussels, Von K´arm´an Institute, Lecture Notes Series 2004–5. Rona, A. and Dieudonn´e, W. (2000): A Flow-Resonant Model of Transonic Laminar Open Cavity Instability, AIAA Paper 2000-1967. Ronchi, C., Ypma, M. and Canuto, V.M. (1992): On the application of the Germano identity to subgrid-scale modeling, Physics of Fluids A, 4(12):2927–2929. Roozen, N.B., Bockholts, M., van Eck, P. and Hirschberg, A. (1998): Vortex sound in bass-reflex ports of loudspeakers: Parts i & ii, Journal of the Acoustical Society of America, 104:1914– 1924. Rossiter, J.E. (1964): Wind Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Transonic Speeds, Technical Report RAE-ARC-R&M 3438, Aeronautical Research Council Reports and Memoranda. Rowley, C.W., Colonius, T. and Basu, A.J. (2002): On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities, Journal of Fluid Mechanics, 455:315–346. Roy, S. and Cho, P. (1999): Designing Axial Flow Fan for Flow and Noise, SAE, September 1999. Saffman, P.G. (1992): Vortex Dynamics, Cambridge Monograph on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge, UK. Sagaut, P. (2001): Large-Eddy Simulations of Incompressible Flows. An Introduction, Springer-Verlag, Berlin. Sagaut, P. (2002): Large-Eddy Simulation for Incompressible Flows. An Introduction, Springer-Verlag, Berlin, 2nd edition. Sagaut, P. (2004): Private communication with M. Breuer. Sagaut, P., Garnier, E., Tromeur, E., Larchev`eque, L. and Labourasse, E. (2003): Turbulent Inflow Conditions for LES of Supersonic and Subsonic Wall-Bounded Flows, AIAA Paper 20030068.

12:29

421

P1: JZZ references

CUFX063/Wagner

422

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Sagaut, P. and Grohens, R. (1999): Discrete filters for large-eddy simulation, International Journal for Numerical Methods in Fluids, 31:1195–1220. Sagaut, P., Montreuil, E. and Labb´e, O. (1999): Assessment of some self-adaptive SGS models for wall-bounded flows, Aerospace Science and Technology, 6:335–344. Sakurai, M., Endo, M. and P´eri´e, F. (2002): Development of the exhaust systems radiation noise simulation technology, ASME 2003 Fluids Engineering Division Summer Meeting, Honolulu (USA), July 6–11, 2003. Salvetti, M.V. and Banerjee, S. (1994): A priori tests of a new dynamic subgrid-scale model for finite-difference large-eddy simulations, Physics of Fluids, 7(11):2831–2847. Sandham, N.D., Li, Q. and Yee, H.C. (2002): Entropy splitting for high-order numerical simulation of compressible turbulence, Journal of Computational Physics, 178:307–322. Scheichl, S. (2004): On the calculation of the transmission line parameters for long tubes using the method of multiple scales, Journal of the Acoustical Society of America, 115:534–555. Schlichting, H. (1968): Boundary-Layer Theory, McGraw-Hill Book Company, New York, 6th edition. Schram, C. and Hirschberg, A. (2003): Application of vortex sound theory to vortex-pairing noise: Sensitivity to errors in flow data, Journal of Sound and Vibration, 266:1079–1098. Schreiber, L. (1995): Straßenverkehrsl¨arm. Taschenbuch der Technischen Akustik, M. Heckl and H.A. M¨uller (eds.), Springer-Verlag, Berlin, 2nd edition, pp. 348–354. Schr¨oder, W., Meinke, M., Ewert, R. and El-Askary, W. (2001): LES of the turbulent flow around a sharp trailing edge, Direct and Large-Eddy Simulation IV, Kluwer Academic Publishers, Dordrecht, pp. 353–363. Schumann, U. (1975): Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli, Journal of Computational Physics, 18:376–404. Scotti, A. and Meneveau, C. (1997): Fractal model for coarse-grained nonlinear partial differential equations, Physical Review Letters, 78(5):867–870. Seror, C. and Sagaut, P. (2000): Subgrid-scale contribution to noise production in decaying isotropic turbulence, AIAA Journal, 38(10). Seror, C. and Sagaut, P. (2002): On the noise prediction computed by hybrid methods, Proc. Workshop LES for Acoustics, G¨ottingen. Seror, C., Sagaut, P., Bailly, C. and Juv´e, D. (1999): Subgrid Scale Contribution to Noise Production in Decaying Isotropic Turbulence, AIAA Paper 99-1979, 5th CEAS/AIAA Aeroacoustics Conference, Seattle, USA, 10–12 May 1999. Seror, C., Saguat, P., Bailley, C. and Juv´e, D. (2000): Subgrid-scale contribution to noise production in decaying isotropic turbulence, AIAA Journal, 38(10):1795–1803. Seror, C., Sagaut, P., Bailley, C. and Juv´e, D. (2001): On the radiated noise computed in large-eddy simulation, Physics of Fluids, 13(2):476–487. Shao, L., Sarkar, S. and Pantano, C. (1999): On the relationship between the mean flow and subgrid stresses in large-eddy simulation of turbulent shear flows, Physics of Fluids, 11(5):1229– 1248. Shapiro, A.H. (1953): The Dynamics and Thermodynamics of Compressible Flow, Volume I, The Ronald Press Co., New York. Shen, W.Z. and Sørensen, J. (2001): Acoustic modeling of turbulent airfoil flows, AIAA Journal, 39(6):1057–1064. Shieh, C.M. and Morris, P.J. (1999a): A parallel numerical simulation of automobile noise involving feedback, 3rd Computational Aeroacoustics Workshop, pp. 363–370, Ohio Aerospace Institute, Cleveland, Nov. 8–10. Shieh, C.M. and Morris, P.J. (1999b): Parallel Numerical Simulation of Subsonic Cavity Noise, AIAA Paper 99-1891. Shieh, C.M. and Morris, P.J. (2000): Parallel Computational Aeroacoustic Simulation of Turbulent Subsonic Cavity Flow, AIAA Paper 2000-1914. Shieh, C.M. and Morris, P.J. (2001): Comparison of Two- and Three-Dimensional Turbulent Cavity Flows, AIAA Paper 2001-0511. Shih, S.H., Hamed, A. and Yeuan, J.J. (1994): Unsteady supersonic cavity flow simulations using coupled k–ε and Navier–Stokes equations, AIAA Journal, 32(10):2015–2021.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Shih, T.-H., Povinelli, L.A. and Liu, N.-S. (2003): Application of generalized wall function for complex turbulent flows, Journal of Turbulence, 4:015. Shim, I.B., Kim, J.W. and Lee, D.J. (1999): Numerical Study on Radiation of Multiple Pure Tone Noise from an Aircraft Engine Inlet, AIAA Paper 99-1831. Shur, M.L., Spalart, P.R. and Strelets, M.K. (2003): Noise prediction for increasingly complex jets, Colloquium EUROMECH 449, Dec. 9–12, 2003, Chamonix, France. Siegert, R., Schwarz, V. and Reichenberger, J. (1999): Numerical Simulation of Aeroacoustic Sound Generated by Generic Bodies Placed on a Plate: Part II – Prediction of Radiated Sound Pressure, AIAA Paper 99-1895. Sijtsma, P., Oerlemans, S. and Holthusen, H. (2001): Location of Rotating Sources by Phased Array Measurements, AIAA Paper 2001-2169. Singer, B.A., Brentner, K.S., Lockard, D.P. and Lilley, G.M. (1999): Sound Prediction of an Airfoil in the Wake of a Cylinder, AIAA Paper 99-0231. Sinha, N., Arunajatesan, S. and Ukeiley, L.S. (2000): High-Fidelity Simulation of Weapons Bay Aeroacoustics and Active Flow Control, AIAA Paper 2000-1968. Slimon, S.A., Davis, D.W. and Wagner, C.A. (1998): Far-Field Aeroacoustic Computation of Unsteady Cavity Flow, AIAA Paper 98-0285. Smagorinsky, J. (1963): General circulation experiments with the primitive equations. I: The basic experiment, Monthly Weather Review, 91(3):99–165. Smirnov, S., Lacor, C. and Baelmans, M. (2001): A Finite Volume Formulation for Compact Schemes with Applications to LES, AIAA Paper 2001-2546. Smirnov, A., Shi, S. and Celik, I. (2001): Random flow generation technique for large-eddy simulations and particle-dynamics modeling, ASME Journal of Fluids Engineering, 123:359–371. Smith, B.R. (2001): Large-eddy simulation of supersonic cavity flow with an unstructured grid flow solver, 3rd AFOSR International Conference on Direct Numerical Simulation and Large-Eddy Simulation (TAICDL), Arlington, TX, Aug. 5–9. Smith, B.R., Jordan, J.K., Bender, E.E. and Rizk, S.N. (2000): Computational Simulation of Active Control of Cavity Acoustics, AIAA Paper 2000-1927. Smith, M.J.T. (1989): Aircraft Noise, Cambridge University Press, Cambridge, UK. Smol’yakov, A.V. and Tkachenko, V.M. (1991): Model of a field of pseudosonic turbulent wall pressures and experimental data, Soviet Physics – Acoustics, 37(6):289–292. Snyder, R.D. and Scott, J.N. (1999): Comparison of Numerical Schemes for the Analysis of Aeroacoustics, AIAA Paper 99-0354. Soemarwoto, B.I. and Kok, J.C. (2001): Computations of three-dimensional unsteady cavity flow to study the effect of different downstream geometries, AGARD RTO-AVT Symposium on Development in Computational Aero- and Hydroacoustics, Manchester, UK. Sorg¨uven, E., Magagnato, F. and Gabi, M. (2003): Simulation of acoustic scattering from a trailing edge, Computational Aeroacoustics: From Acoustic Sources Modeling to Far-Field Radiated Noise Prediction, Colloquium EUROMECH 449, December 9–12, 2003, Chamonix, France. Souliez, F.J., Long, L.N., Morris, P.J. and Sharma, A. (2002): Landing gear aerodynamics noise prediction using unstructured grids, International Journal of Aeroacoustics, 1(2):115–135. Spalart, P.R. (1988): Direct simulation of a turbulent boundary layer up to Reθ = 1410, Journal of Fluid Mechanics,187:61–98. Spalart, P.R. (2000): Strategies for turbulence modeling and simulations, International Journal of Heat and Fluid Flow, 21:252–263. Spalart, P.R., Jou, W.H., Strelets, M. and Allmaras, S.R. (1997): Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach, 1st AFOSR Int. Conf. on DNS/LES, August 4–8, 1997, Ruston, LA. In Advances in DNS/LES, C. Liu and Z. Liu (eds.), Greyden Press, Columbus, OH. Spalart, P.R. and Leonard, A. (1987): Direct numerical simulation of equilibrium turbulent boundary layers, Proc. 5th Sym. on Turbulent Shear Flows, Ithaca NY, Springer-Verlag, pp. 235–252. Spalart, P.R. and Squires, K.D. (2003): The status of detached-eddy simulation for bluff bodies, UEF Conf. Aerodynamics of Heavy Vehicles: Trucks, Buses and Trains, Asilomar, December 2–6, 2002, Springer-Verlag.

12:29

423

P1: JZZ references

CUFX063/Wagner

424

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Speziale, C.G. (1996): Computing non-equilibrium flows with time-dependent RANS and VLES, 15th International Conference on Numerical Methods in Fluid Dynamics, Monterey. Speziale, C.G. (1998): Turbulence modeling for time-dependent RANS and VLES: A review, AIAA Journal, 36(2):173–184. Speziale, C.G., Abid, R. and Anderson, E.C. (1990): A Critical Evaluation of Two-Equation Models for Near Wall Turbulence, ICASE Report, No. 90–46. Spille-Kohoff, A. and Kaltenbach, H.-J. (2001): Generation of turbulent inflow data using a prescribed shear-stress profile, DNS/LES Progress and Challenges, C. Liu, L. Sakell, and T. Beutner (eds.), Greyden, Columbus, OH, pp. 319–326. Sreedhar, M. and Ragab, S. (1994): Large-eddy simulation of longitudinal stationary vortices, Physics of Fluids, 6(7):2501–2514. Srinivasan, S. and Baysal, O. (1991): Navier–Stokes calculations of transonic flows past cavities, ASME Journal of Fluids Engineering, 113:368–376. Stanescu, D. and Habashi, W.G. (1998): 2N -storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 143:674–681. Stanescu, D., Hussaini, M.Y. and Farassat, F. (2002): Aircraft Engine Noise Scattering – A Discontinuous Galerkin Spectral Element Approach, AIAA Paper 2002-0800. Stanley, S. and Sarkar, S. (2000): Influence of nozzle conditions and discrete forcing on turbulent planar jets, AIAA Journal, 38:1615–1623. STAR-CD and pro-STAR User Guide (2004): STAR-CD and pro-STAR User Guide, CD-adapco, URL: http://www.cd-adapco.com. Stolz, S. and Adams, N.A. (1999): An approximate deconvolution procedure for large-eddy simulation, Physics of Fluids, 11(7):1699–1701. Stolz, S. and Adams, N.A. (2001): LES of supersonic boundary layers using the approximate deconvolution model, Direct and Large-Eddy Simulation IV, Kluwer Academic Publishers, Dordrecht, pp. 269–276. Stolz, S., Adams, N.A. and Kleiser, L. (2001a): An approximate deconvolution model for large-eddy simulations with application to incompressible wall-bounded flows, Physics of Fluids, 13(4):997– 1015. Stolz, S., Adams, N.A. and Kleiser, L. (2001b): The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction, Physics of Fluids, 13(10):2985–3001. Streett, C.L. and Macaraeg, M.G. (1989): Spectral multi-domain for large-scale fluid dynamic simulations, Applied Numerical Mathematics, 6:123–139. Strelets, M. (2001): Detached Eddy Simulation of Massively Separated Flows, AIAA Paper 20010879. Stromberg, J.L., McLaughlin, D.K. and Troutt, T.R. (1980): Flow field and acoustic properties of a Mach number 0.9 jet at a low Reynolds number, Journal of Sound Vibration, 72:159–176. Suhs, N.E. (1987): Computations of Three-Dimensional Cavity Flow at Subsonic and Supersonic Speeds, AIAA Paper 87-1208. Suhs, N.E. (1993): Unsteady Flow Computations for a Three Dimensional Cavity with or Without an Acoustic Suppression Device, AIAA Paper 93-3402. Ta’asan, S. and Nark, D.M. (1995): An Absorbing Buffer Zone Technique for Acoustic Wave Propagation, AIAA Paper 95-0164. Tam, C. and Dong, Z. (1993): A study of the short wave components in computational acoustics, Journal of Computational Acoustics, 1(1):1–30. Tam, C. and Dong, Z. (1994): Wall boundary condition for high-order finite-difference schemes in computational aeroacoustics, Theoretical Computational Fluid Dynamics, 6(6):303–322. Tam, C.-J., Orkwis, P.D. and Disimile, P.J. (1995): Comparison of Baldwin–Lomax turbulence models for two-dimensional open cavity computations, AIAA Journal, 34(3):629–631. Tam, C.K.W. (1995): Computational Aeroacoustics: Issues and Methods, AIAA Paper 95–0677. Tam, C.K.W. (1998a): Advances in numerical boundary conditions for computational aeroacoustics, Journal of Computational Acoustics, 6:377–402.

12:29

P1: JZZ references

CUFX063/Wagner

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Tam, C.K.W. (1998b): LES for Aeroacoustcs, AIAA Paper 98-2805. Tam, C.K.W. (2001): Noise from high-speed jets, J. Anthoine and C. Schram (eds.), Advances in Aeroacoustics, VKI Lecture Series 2001–2, Brussels, Von K´arm´an Institute for Fluid Dynamics, pp. 1–34. Tam, C.K.W. and Auriault, L. (1998): Mean flow refraction effects on sound radiated from localized sources in a jet, Journal of Fluid Mechanics, 370:149–174. Tam, C.K.W. and Auriault, L. (1999): Jet mixing noise from fine-scale turbulence, AIAA Journal, 37(2):145–153. Tam, C.K.W. and Block, P.J.W. (1978): On the tones and pressure oscillations induced by flow over rectangular cavities, Journal of Fluid Mechanics, 89(2):373–399. Tam, C.K.W., Golebiowski, M. and Seiner, J.M. (1996): On the Two Components of Turbulent Mixing Noise from Supersonic Jets, AIAA Paper 1996-1716. Tam, C.K.W. and Morris, P.J. (1980): The radiation of sound by the instability waves of a compressible plane turbulent shear layer, Journal of Fluid Mechanics, 98(2):349–381. Tam, C.K.W. and Shen, H. (1993): Direct Computation of Nonlinear Acoustic Pulses Using HighOrder Finite Difference Schemes, AIAA Paper 93-4325. Tam, C.K.W. and Webb, J.C. (1993): Dispersion-relation-preserving finite difference schemes for computational acoustics, Journal of Computational Physics, 107:262–281. Tam, C.K.W., Webb, J.C. and Dong, Z. (1993): A study of the short wave components in computational acoustics, Journal of Computational Acoustics, 1:1–30. Tang, L. and Baeder, J.D. (1997): Uniformly Accurate Compact Difference Schemes, AIAA Paper 97-2093. Temkin, S. (2001): Elements of Acoustics, Acoustical Society of America, New York. Reprint. Tennekes, H. and Lumley, J.L. (1972): A First Course in Turbulence, MIT Press, Cambridge, MA. Terracol, M., Labourasse, E., Manoha, E. and Sagaut, P. (2003): Simulation of the 3D Unsteady Flow in a Slat Cove for Noise Prediction, AIAA Paper 2003-3110. Terracol, M., Manoha, E., Herrero, C., Labourasse, E., Redonnet, S. and Sagaut, P. (2005): Hybrid methods for airframe noise numerical prediction, TCFD, 19(3):197–227. Terracol, M., Sagaut, P. and Basdevant, C. (2001): A multilevel algorithm for large-eddy simulation of turbulent compressible flows, Journal of Computational Physics, 167(2):439–474. Thompson, K.W. (1990): Time-dependent boundary conditions for hyperbolic systems II, Journal of Computational Physics, 89:439–461. Thompson, M.C., Hourigan, K. and Welsh, M.C. (1992): Acoustic sources in a tripped flow past a resonator tube, AIAA Journal, 30:1484–1491. Thomson, K.W. (1987): Time dependent boundary conditions for hyperbolic systems, Journal of Computational Physics, 68:1–24. Tolstykh, A.I. and Shirobokov, D.A. (1995): Fifth-order compact upwind method and its applications to three-dimensional compressible Navier–Stokes equations, Proc. 1st Asian CFD Conf. Toro, E.F. (1992): Viscous flux limiters, Notes on Numerical Fluid Mechanics, Vos, Rizzi, and Rhyming (eds.), Vieweg Verlag Wiesbaden, 35:592–600. Travin, A., Shur, M., Strelets, M. and Spalart, P.R. (2000): Detached-eddy simulations past a circular cylinder, International Journal of Flow Turbulence and Combustion, 63:293–313. Troff, B., Manoha, E. and Sagaut, P. (1997): LES of Trailing Edge Flow with Application to Radiated Noise, AIAA Paper 1997-120. Tyler, J.M. and Sofrin, T.G. (1962): Axial flow compressor noise studies, Transactions of the Society of Automotive Engineers, 70:309–332. Urbin, G. and Knight, D. (2001): Large-eddy simulation of a supersonic boundary layer using an unstructured grid, AIAA Journal, 39(7):1288–1295. Uzun, A. (2003): 3-D Large-Eddy Simulation for Jet Aeroacoustics. Ph.D. thesis, Purdue University, West Lafayette, IN. Uzun, A., Blaisdell, G.A. and Lyrintzis, A.S. (2002): Recent Progress Towards a Large-Eddy Simulation Code for Jet Aeroacoustics, AIAA Paper 2002-2598.

12:29

425

P1: JZZ references

CUFX063/Wagner

426

0 521 87144 1

printer: sheridan

November 17, 2006

REFERENCES

Uzun, A., Blaisdell, G.A. and Lyrintzis, A.S. (2003): 3-D Large-Eddy Simulation for Jet Aeroacoustics, AIAA Paper 2003-3322. van Kuiken, G.D.C. (1995): Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology, John Wiley & Sons, Chichester, UK. Van Leer, B. (1979): Towards the ultimate conservative difference scheme.V. A second-order sequel to Godunov’s method, Journal of Computational Physics, 32:101–136. van Lier, L., Dequand, S., Hirschberg, A. and Gorter, J. (2001): Aeroacoustics of diffusers: An experimental study of typical industrial diffusers at Reynolds numbers of O(105 ), Journal of the Acoustical Society of America, 109:108–117. van Wijngaarden, L. (1968): On the oscillations near and at resonance in open pipes, Journal of Engineering Mathematics, 11:225–240. van Wijngaarden, L. (1972): One-dimensional flow of liquids containing small gas bubbles, Annual Review of Fluid Mechanics, 4:369–396. Vasilyev, O., Lund, T.S. and Moin, P. (1998): A general class of commutative filters for LES in complex geometries, Journal of Computational Physics, 146:82–104. Verge, M.P., Fabre, B., Hirschberg, A. and Wijnands, A.P.J. (1997a): Sound production in recorderlike instruments, I: Dimensionless amplitude of the internal acoustic field, Journal of the Acoustical Society of America, 101:2914–2924. Verge, M.P., Hirschberg, A. and Causs´e, R.R. (1997b): Sound production in recorderlike instruments, II: A simulation model, Journal of the Acoustical Society of America, 101:2925–2939. Vergne, S., Auger, J.-M., G’Styr, N. and P´eri´e, F. (2002): Simulation of cavity aero-elastic noise induced by an external turbulent flow perturbed by a small ruler, Proceedings of the International Workshop on “LES for Acoustics,” 7–8 October, German Aerospace Center, DLR, G¨ottingen, Germany, DGLR-Report 2002-03. Verzicco, R., Iafrati, A., Riccardi, G. and Fatica, M. (1997): Analysis of the sound generated by the pairing of two axisymmetric co-rotating vortex rings, Journal of Sound and Vibration, 200:347–358. Vichnevetsky, R. (1981): Propagation through numerical mesh refinement for hyperbolic equations, Mathematics and Computers in Simulation, 23:344. Visbal, R.M. and Gaitonde, V.D. (1998): Higher-Order-Accurate Methods for Complex Unsteady Subsonic Flows, AIAA Paper 98-0131. Visbal, R.M. and Gaitonde, V.D. (1999): Higher-order-accurate methods for complex unsteady flows, AIAA Journal, 37:1231–1239. Visbal, M.R. and Gaitonde, D.V. (2001): Very high-order spatially implicit schemes for computational acoustics on curvilinear meshes, Journal of Computational Acoustics, 9(4): 1259–1286. Visbal, R.M. and Gaitonde, V.D. (2002): On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, Journal of Computational Physics, 181:155–185. Visbal, M.R. and Rizzetta, D.P. (2002): Large-eddy simulation on curvilinear grids using compact differencing and filtering schemes, ASME Journal of Fluids Engineering, 124:836–847. von Terzi, D.A. and Fasel, H.F. (2002): A New Flow Simulation Methodology Applied to the Turbulent Backward-Facing Step, AIAA Paper 2002-0429. Vreman, B. (1995): Direct and Large-Eddy Simulation of the Compressible Turbulent Mixing Layer. Ph.D. Thesis, Twente University. Vreman, B., Geurts, B. and Kuerten, H. (1997): Large-eddy simulation of the turbulent mixing layer, Journal of Fluid Mechanics, 339:357–390. Wagner, C. (2001): An eddy viscosity scaled similarity model, Z. Angew. Math. Mech., 81(Suppl. 3): 491–492. Wagner, C. (2003): A non-dissipative dynamic subgrid scale similarity model, Proc. of the Third International Symposium on Turbulence and Shear Flow Phenomena, Sendai, Japan, 25–27 June 2003. Wagner, C. and H¨uttl, T. (2002): LES for acoustics, Proceedings of the International Workshop on “LES for Acoustics, 7–8 October, German Aerospace Center, DLR, G¨ottingen, Germany, DGLRReport 2002-03. Wagner, S., Bareiss, R. and Guidati, G. (1996): Wind Turbine Noise, Springer-Verlag, Berlin.

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Walz, A. (1969): Boundary Layers of Flow and Temperature, MIT Press, Cambridge, MA. Wang, M., Lele, S.K. and Moin, P. (1996): Computation of Quadrupole Noise from Airfoil VortexShedding, Stanford University CTR Manuscript 158. Wang, M. and Moin, P. (2000): Computation of trailing-edge flow and noise using large-eddy simulation, AIAA Journal, 38(12):2201–2209. Weber, C. and Ducros, F. (2000): Large-eddy and Reynolds-averaged Navier–Stokes simulation of turbulent flow over an airfoil, International Journal of Computational Fluid Dynamics, 13:327– 355. Welsh, M.C. and Stokes, A.N. (1984): Flow-resonant sound interaction in a duct containing a plate, Part I: Semi-circular leading edge, Journal of Sound and Vibration, 95:305–323. Werner, H. and Wengle, H. (1993): Large-eddy simulation of turbulent flow over and around a cube in a plate channel, 8th Symp. on Turb. Shear Flows, F. Durst, R. Friedrich, B. Launder, F. Schmidt, U. Schumann, and J. Whitelaw (eds.), Springer Verlag, Berlin, pp. 155–168. Westley, R. and Lilley, G.M. (1952): An Investigation of the Noise Field from a Small Jet and Methods for Its Reduction, Technical Report 53, College of Aeronautics, Cranfield, UK. Wilson, R.V., Demuren, A.O. and Carpenter, M. (1998): Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows, ICASE Report 98-13. Wilson, T.A., Beavers, G.S., De Coster, M.A., Holger, D.K. and Regenfuss, D. (1971): Experiments on the fluid mechanics of whistling, Journal of the Acoustical Society of America, 50:366–372. Winant, C.D. and Browand, F.K. (1974): Vortex pairing: The mechanism of turbulent mixing layer growth at moderate Reynolds number, Journal of Fluid Mechanics, 63(2):237–255. Winckelmans, G.S., Wray, A.A., Vasilyev, O.V. and Jeanmart, H. (2001): Explicit-filtering largeeddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. Physics of Fluids, 13(5):1385–1403. Wu, X., Jacobs, R., Hunt, J. and Durbin, P. (1999): Simulation of boundary layer transition induced by periodically passing wake, Journal of Fluid Mechanics, 398:109–153. W¨ustenberg, H. and Hupertz, B. (1995): M¨oglichkeiten und Grenzen der Aerodynamik-Berechnung mit Zwei-Gleichungs-Turbulenzmodellen und Feinaufl¨osung der Grenzschicht. Seminar “Aerodynamik des Kraftfahrzeugs”, Haus der Technik, Essen. Xu, J., Stanescu, D., Hussaini, M.Y. and Farassat, F. (2003): Computation of Engine Noise Propagation and Scattering off an Aircraft, AIAA Paper 2003-0542. Yoon, S. and Jameson, A. (1987): An LU-SSOR Scheme for the Euler and Navier–Stokes Equation, AIAA-Paper 87-600. Yoshizawa, A. (1979): A statistical investigation upon the eddy viscosity in incompressible turbulence, Journal of the Physical Society of Japan, 47(5):1665–1669. Yoshizawa, A. (1984): Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation, Physics of Fluids, 27(6):1377–1387. Yu, S.-T., Hsieh, K.-C. and Tsai, Y.-L.P. (1995): Simulating waves in flows by Runge–Kutta and compact difference schemes, AIAA Journal, 33:1421–429. Zaman, K.B.M.Q. (1985): Effect of initial condition on subsonic jet noise, AIAA Journal, 23(9):1370– 1373. Zaman, K.B.M.Q. (1986): Flow field and near and far sound field of a subsonic jet, Journal of Sound and Vibration, 106(1):1–16. Zang, Y., Street, R.L. and Koseff, J.R. (1993): A dynamic mixed subgrid-scale model and its application to turbulent recirulating flows, Physics of Fluids A, 5(12):3186–3196. Zhang, H.L., Bachman, C.R. and Fasel, H.F. (2000a): Application of a New Methodology for Simulations of Complex Turbulent Flows, AIAA Paper 2000-2535. Zhang, H.L., Bachman, C.R. and Fasel, H.F. (2000b): Reynolds-Averaged Navier–Stokes Calculations of Unsteady Turbulent Flow, AIAA Paper 2000-0143. Zhang, X. (1995): Compressible cavity flow oscillation due to shear layer instabilities and pressure feedback, AIAA Journal, 33(8):1404–1411. Zhang, X., Chen, X.X., Rona, A. and Edwards, J.A. (1998): Attenuation of cavity flow oscillation through leading edge flow control, Journal of Sound and Vibration, 221(1):23–47.

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Zhang, X., Rona, A. and Lilley, G.M. (1995): Far-Field Noise Radiation from an Unsteady Supersonic Cavity, AIAA Paper 95-040. Zhao, H. and Voke, P.R. (1996): A dynamic subgrid-scale model for low-Reynolds-number channel flow, International Journal for Numerical Methods in Fluids, 23:19–27. Zhao, W., Frankel, S.H. and Mongeau, L. (2001b): Numerical simulations of sound from confined puslating axisymmetric jets, AIAA Journal, 39(10):1868–1874. Zhao, L. and Cangellaris, A.C. (1996): GT-PML: Generalized theory of perfectly match layers and its application to the reflectionless truncation of finite-difference time-domain grids, IEEE Transactions on Microwave Theory and Techniques, 44:2555–2563. Zhao, W., Frankel, S.H. and Mongeau, L. (2000): Large-Eddy Simulation of Sound Radiation from a Subsonic Turbulent Jet, AIAA Paper 00-2078. Zhao, W., Frankel, S.H. and Mongeau, L. (2001a): Large-eddy simulations of sound radiation from subsonic turbulent jets, AIAA Journal, 39(8):1469–1477. Zhong, X. (1998): High-order finite difference schemes for numerical simulation of hypersonic boundary-layer transition, Journal Computational Physics, 144:662–709. Zhou, Y. (1993): Interacting scales and energy transfer in isotropic turbulence, Physics of Fluids A, 5(10):2511–2524. Ziada, S. and B¨uhlmann, E.T. (1992): Self-excited resonances of two side-branches in close proximity, Journal of Fluids and Structures, 6:583–601.

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Index

90◦ bend, 207 a posteriori test, 18 a priori evaluation, 125 a priori test, 18 absorbing cell, 154 absorbing conditions, 353 absorbing zone, 217, 218, 219, 220, 222, 386 acoustic analogy, 16, 17, 21, 55, 128, 273, 296, 299, 300, 311, 334 acoustic feedback, 272, 356 acoustic flow feedback, 358 acoustic flow interaction, 264 acoustic perturbation equation (APE), 147, 224, 296, 297 acoustic resonances, 263, 349, 351 acoustic scattering, 174, 196, 294, 318, 329 acoustic wave propagation, 143, 295 acoustical admittance, 46 acoustical displacement, 34 acoustical flow separation, 34 acoustics receivers, 274 Adams-Bashforth scheme, 201, 329 advective upstream splitting method (AUSM), 173, 305 adverse pressure gradient, 228 aeroacoustic coupling, 354, 356 aeroacoustic instability, 354 aeroacoustic resonance, 357, 358 aerodynamic instabilities, 357 aerodynamic noise, 273 aeroelastic loading, 289 aeroelastic noise, 272, 273, 293 aeroelastic phenomena, 387 aeroengine, 58 aerospace cavity, 358 aerovibroacoustics, 273 Ahmed body, 345 air-conditioning, 345, 373 aircraft, 378

aircraft engine, 189, 197 aircraft industry, 222 aircraft manufacturer, 4, 378 aircraft noise, 2, 3, 39, 382 aircraft propeller, 52 aircraft turbine, 71 aircraft wing, 58, 381 airfoil, 209, 225, 294, 295, 318, 336, 371 airfoil leading edge, 325, 333, 387 airfoil noise, 318 airfoil trailing edge, 226 airframe noise, 222, 232, 294, 295, 296 airplane stability, 273 ALESSIA, 6 algebraic turbulence model, 172, 212 aliasing error, 109, 110 analogy of Howe, 67 analytic acoustic propagation model, 145 analytical transport technique, 11 anechoic pipe termination, 81 anechoic wind tunnel, 337 angle of attack, 318 annoyance, 2, 3 annular shear layer, 248, 254, 257, 261 annuli, 210 antidissipative effect, 124 A-pillar, 357, 360 approach, 294 approximation error, 9 arbitrary Lagrangian Eulerian formulation (ALE), 276 Archimedes, 63 artifical noise, 13 artificial boundary, 234 artificial damping, 219 artificial dispersion, 248 artificial dissipation, 110, 171, 175, 187, 189, 219, 336, 379 artificial inflow boundaries, 203, 204

429

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artificial inflow data, 207 artificial noise, 235 artificial reflection, 358 artificial selective damping (ASD), 173, 175, 301 artificial velocity fluctuations, 143 artificial viscosity, 110 ASMO model, 345 asymmetric DRP scheme, 175 asymptotic radiation boundary condition, 301 attached boundary layer, 209 attached eddy, 213 attached flow, 212 audio range, 33 autocorrelations, 205, 208 automobile, 360 automobile manufacturer, 357 automotive exhaust, 274, 358 automotive heating, 373 automotive intake, 358 autospectrum, 285, 286 auxiliary filter, 117 auxiliary simulation, 206, 207, 226 axisymmetric jet, 238 axisymmetric mixing layer, 239, 241 axisymmetric nozzle, 246 axisymmetric shear layer, 241, 243, 244 azimuthal correlation, 258 backscattering, 14, 113, 114 backward facing step, 213 backward scattering, 334 Baldwin-Lomax model, 263, 318 bell-shaped linear pulse, 189 benchmark problem, 181, 195, 197 benchmark validation case, 337 Bessel equation, 251 Bessel function, 69, 70 bidiagonal interpolation, 193 bidiagonal third-order accurate scheme, 193 bidimensional mixing layer, 239 bifurcation, 266, 276 bilinearly interpolation, 309 Biot-Savart law, 234, 235, 237 blade-passing frequency (BPF), 375 Blasius boundary layer, 268 blended wing-body combination, 196 blower fan, 373 blunt body, 333 blunted trailing edge, 320 bluntness, 318 boundary condition, 248 boundary element method (BEM), 294, 344, 348 boundary layer, 318, 356 Boussinesq closure, 135 Boussinesq eddy viscosity concept, 263 Boussinesq model, 134

Boussinesq-like model, 126 box filter, 94 breathing mode, 161 broadband, 350, 358, 361, 364, 371, 372, 373 broadband amplification phenomenon, 232 broadband flow excitations, 376 broadband noise, 6, 8, 15, 17, 128, 148, 156, 167, 252, 318, 364, 385 buffer zone, 218, 220 buffeting, 357 buoyancy, 63, 91, 214 bypass, 58 bypass ratio, 18 bypass stream, 18 bypass the wall layer, 267 cabin climate control device, 357 car, 154, 333, 334, 345, 387 car aerodynamics, 345 car manufacturer, 349 car roof, 346 car underbody, 346 Cartesian grid, 181 car-wing-mirror, 154 catastrophic failure, 273 cavity, 262, 263, 265, 266, 357, 364, 386, 387 cavity acoustics, 172 cavity ceiling, 364 cavity drag, 265 cavity flow, 15, 263, 266, 272 cavity noise, 262, 264, 265 cavity opening, 266 cavity oscillations, 263, 268, 271 cavity resonance, 263 cavity resonance frequency, 371, 372 cavity wall, 269 cell isotropy, 358 cell-centered finite volume, 317 cell-centered version, 192 centered scheme, 111 central difference scheme, 195, 305, 346 central scheme, 171, 194 centrifugal forces, 214 C-grid topology, 314 channel flow, 187, 206, 213, 226 chaotic acoustic signal, 342 chaotic structure, 340 characteristic decomposition, 217 characteristic nonreflecting boundary condition, 217 characteristic splitting, 217 chevron, 246, 254 Chimera technique, 14 Cholesky decomposition, 144 circular cylinder, 20, 21, 132, 186, 333, 334, 335, 336, 387

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INDEX

circular duct, 365 circular jet, 175, 239, 241, 250 closed pipe, 75 cockpit climate control device, 357 coexistence of different modes, 271, 272 coherence function, 285 coherency-squared function, 285 coherent structure, 87, 129, 209, 238, 242, 269, 272, 387 combustion, 36, 42, 63, 91 combustion noise, 3, 8 commercial software, 148, 164, 316, 344, 345, 348, 357, 374 compact central scheme, 176, 191 compact filtering, 266 compact finite difference, 176 compact finite element projection method, 195 compact finite volume, 261 compact region, 42 compact scheme, 175, 178, 182 compact source, 48 compact upwind scheme, 176, 190, 191 competitive mode, 271 compressible subgrid model, 125 compressor fan, 373 computational aeroacoustics (CAA), 6, 7, 223, 297 computational cost, 11 computational transport method, 11 condensation, 36 cone of silence, 52, 382 conservative finite volume method, 353 control device, 350 convecting point source, 236 convecting vortex problem, 237 convective boundary conditions, 204 convective ridge, 324, 325 convolution, 235 convolution filter, 90, 92, 93, 95, 96, 101, 379 convolution product, 316 cooling and climate system, 378 cooling fan, 357 coordinate transformation, 185, 206 Coriolis acceleration, 31 Coriolis force, 66 corotating vortex pair, 173, 241 correlation length and time, 151 counterclockwise rotating vortex, 236 counterrotating vortices, 236, 266 coupling, 149 coupling mechanisms, 10 coupling region, 10 coupling technique, 295, 330 Courant condition, 140, 277, 280 Courant number, 112 cover plate, 266

crackle, 251 Crank-Nicolson scheme, 201 cross correlation, 16, 205, 208, 285 cross spectrum, 285, 320 cross-spectral density (CSD), 286 cross-stress tensor, 97 cruise flight condition, 52, 209 C-type mesh, 306 Curle’s formulation, 20, 59, 60, 62, 142, 143, 264, 383 curved surface, 214 curvilinear coordinates, 185 curvilinear grid, 185, 301, 313, 353 cutoff, 266, 286, 379 cutoff frequency, 69, 72 cutoff length, 89, 90, 92, 93, 100, 101, 104, 111, 112, 115, 116, 117 cutoff mode, 69 cutoff time, 93 cutoff wave number, 17, 92 cuton mode, 69, 71 cyclical pattern, 271 cylinder wake, 371 cylindrical Rayleigh equation, 250 damping, 151, 219 damping coefficient, 154 damping function, 267 damping zone, 234, 235, 237 data injection process, 329 dealiasing, 171 decay of homogeneous isotropic flow, 202 decaying turbulence, 202 decomposition, 207 deconvolution model, 119, 120, 124, 380 deep cavity, 267 detached eddy simulation (DES), 131, 213, 264, 345, 364 deterministic vortex model, 380 diaphragm, 71, 75, 76, 81, 349, 350, 351 differentiation error, 109, 110 diffraction, 54, 325 diffusion, 113 diffusion error, 9, 10, 11, 13, 385 diffusor, 85, 354 digital filtering of random data, 208 dilatation field, 160, 163, 240, 241 dipole, 39, 42, 43, 44, 47, 48, 58, 59, 60, 294, 309, 344, 382 direct injection, 296 direct method, 9 direct noise computation (DNC), 239, 244, 264, 265, 386 direct numerical simulation (DNS), 7, 15, 18, 19, 58, 89, 129, 198, 239, 246, 249, 294, 334, 349, 379, 387

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directivity, 242 directivity function, 79 directivity pattern, 253 Dirichlet boundary conditions, 202, 204 discontinuity, 193 discontinuous boundary, 149 discontinuous Galerkin method, 176, 195 discrete tone, 148, 156 discretization error, 124 discretization grid, 111, 151 discretization scheme, 9, 11, 13, 167, 170, 179, 180 discretization stencil, 168, 172 dispersion, 113, 199, 248, 265 dispersion error, 9, 10, 11, 13, 111, 168, 169, 170, 173, 174, 177, 180, 189, 193, 196, 199, 200, 317, 380, 385 dispersion relation, 247 dispersion relation preserving (DRP), 173, 248, 301, 350 dispersive behavior, 178, 181, 192, 197, 201, 385 dissipation, 31, 113, 153, 195, 199, 248, 265, 267, 276 dissipation error, 111, 168, 169, 170, 175, 189, 193, 196, 199, 200, 317, 380 dissipation rate, 115, 132, 138, 144 dissipative behavior, 197, 201, 385 dissipative filter, 188 dissipative model, 266 dissipative weighted compact scheme, 193 divergence free, 249, 250 divergent solutions, 296 domain decomposition, 148, 163 door cavity, 266 Doppler amplification, 50, 303 Doppler effect, 11, 49, 52, 60, 382, 383, 384 Doppler factor, 50, 51, 52 double spiral structure, 241 downstream corner, 262, 272, 353 downstream edge, 269 downstream shedding, 372 downwind interpolation, 194 downwind-biased formulation, 194 driven cavity, 266 dual time stepping, 346 duct, 357 duct flow, 206, 350, 354 duct mode, 69 ducted cavity, 353 ducted turbofan, 171 dynamic model, 123 dynamic procedure, 121 dynamic Smagorinsky model, 243, 259, 266 dynamic subgrid scale model, 135, 253 eardrum, 34 eddy, 109

eddy viscosity, 263 eddy viscosity model, 18, 114, 123, 266, 267 edge-tone, 85 effective angular frequency, 201 effective filter, 92, 112 eigenfunction, 68, 69, 70 eigenfunction solutions, 250 eigenvalue, 69, 70, 71, 72 eighth-order filter, 188 Einstein’s convention, 26, 300 ejection boundary condition, 211 ejections, 211 elastic plate-cavity system, 274 eleven-point stencil, 269, 330 elliptic equations, 201 end correction, 78, 80, 83, 84, 85 energy aliasing, 189 energy cascade, 110, 111, 113 energy spectra, 208 engine cooling device, 345 engine noise, 294 engine nozzle, 248 entropy splitting, 111 environmental pollution, 1 equilibrium turbulent-boundary layer, 228, 229 error control reliability, 199 error reduction, 167 essentially nonoscillatory scheme (ENO), 173, 193 Etkin test case, 335 Euler flux, 317 Eulerian time derivative, 26 European Union, 5 exact amplification factor, 199, 200 exhaust plume, 246, 386 exit zone, 219, 220 explicit filter, 187 explicit scheme, 177 explicit time integration, 317 external mixer, 245 external vehicle noise, 378 extrapolation, 150, 203, 214 extremely loud jet, 251 factorization error, 183 fan blade, 374 fan noise, 62 far-field radiation, 149, 175 Favre filtering operator, 316 feedback, 6, 14, 24, 57, 60, 150, 337, 349, 351, 357, 383, 384 feedback loop, 262, 263, 266 Ffowcs Williams-Hawkings equation, 7, 11, 23, 216, 294, 334, 344, 384, 387 Ffowcs Williams-Hawkings method, 60, 62, 133, 175, 264, 374, 383

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INDEX

Ffowcs Williams-Hawkings surface, 133, 251, 253, 254, 371 Fifth Research Framework Program, 6 fifth-order accurate scheme, 190 filter kernel, 93, 100, 102, 118, 119 finite difference, 19, 105, 214, 269, 295 finite difference method, 92, 104, 248 finite difference scheme, 172, 173, 176, 181, 182, 186, 189, 192, 194 finite element, 173, 277, 278, 371 finite element method, 20, 92, 95, 105, 195 finite element scheme, 248 finite volume, 19, 95, 316, 317 finite volume method, 92, 104, 105, 195, 196, 253 finite volume scheme, 176, 181, 193, 194, 248 five stage algorithm, 201 five stage Runge-Kutta scheme, 200, 305 flap, 158, 294, 357 flat plate, 187, 231 flat plate boundary layer, 213 flow acoustic coupling, 349, 351 flow acoustic interaction, 349 flow simulation methodology (FSM), 213 flow with discontinuities, 191 flow-induced noise, 357 flow-induced vibration, 273 fluid acoustic computation, 353 fluid-structure interaction, 273, 274 flute, 85, 383 flux difference splitting, 172 flux splitting, 191, 192 flux vector splitting, 192, 193 Fourier analysis, 37, 41, 109, 277 Fourier decomposition, 359 Fourier difference scheme, 179, 180 Fourier footprint, 169 Fourier method, 95 Fourier mode, 167 Fourier mode analysis, 167 Fourier model, 144 Fourier reconstruction, 144 Fourier series, 188 Fourier space, 94, 235 Fourier spectral method, 109 Fourier synthesis, 49 Fourier transformation, 37, 47, 71, 73, 94, 156, 173, 202, 235, 285, 302, 303, 313, 323, 340, 386 Fourier wave, 168, 175, 187, 196 four-level time integration, 201 fourth-order accurate scheme, 171, 179, 182, 183, 186, 187, 199 free flight experiment, 205 free jet, 58, 59 free shear layer, 205, 262 free vortex, 67

free-stream condition, 212 frequency domain, 180, 197 frequency domain method, 371 frequency space, 269 frequency spectrum, 8 frozen spatial turbulence field, 207 functional modeling, 112, 113 fundamental equation of thermodynamics, 28 fuselage, 174 fuselage-nacelle configuration, 197 fuselage-wing-nacelle configuration, 197 Galerkin projection, 195 Galerkin-type procedures, 94 Galilean invariance, 93, 97 Gaussian filter, 94 Gaussian wave, 181 general curvilinear grids, 100 generalized acoustic analogy, 261 generalized coordinates, 102 generic car shape, 345, 346 generic mirror, 348 geometrical singularity, 333 Germano identity, 121, 122, 380 Germano’s consistent decomposition, 97 ghost cell, 150 global mass conservation, 204 Godunov theorem, 172 Godunov-type scheme, 172 gray area, 148 gray zone, 135 grazing flow, 265 Green’s function, 24, 36, 37, 38, 47, 48, 49, 59, 62, 70, 71, 72, 74, 75, 76, 309, 382, 383 Green’s theorem, 196, 382 grid fequency cutoff, 357 grid filter, 92 grid stretching, 203, 219, 220 gridpoints per wavelength (PPW), 178 grid-to-grid oscillation, 269 Gutin’s principle, 60 half-plane, 54 Hankel function, 252 Hardin-Pope two-step approach, 173 health, 1, 3 hearing loss, 1, 2 heated jet, 186, 250, 253 helicopter rotor, 8, 381 Helmholtz equation, 274 Helmholtz number, 83, 84, 223 Helmholtz resonator, 85 hemisperical half-cylinder, 154 Hermitian interpolation, 179, 181 hexahedral cell, 362 hexahedral elements, 154, 280, 292

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high resolution LES, 209 high-bypass turbofan aeroengine, 58 higher-order moments, 208 high-lift airfoil, 152, 158 high-lift device, 294, 356, 371 high-lift wing, 158 high-lift wing device, 154 high-order accuracy, 187 high-order explicit filtering, 266 high-pass filter, 277 high-pressure steam, 354 high-speed cavity, 368 high-speed train, 5, 15, 21, 22 homogeneous isotropic flow, 386 hot jet, 253, 255 hot wire anemometry, 20, 279 H-topology, 314 h-type method, 176 human hearing range, 361 hybrid approach, 355 hybrid CFD CAA method, 357 hybrid LES-Kirchhoff method, 173 hybrid LES-LEE approach, 349 hybrid LES-Lighthill method, 173 hybrid RANS-LES approach, 213, 216, 264, 267, 380, 381 hybrid RANS-LES method, 128, 130, 133, 138, 139, 140, 142, 143, 145, 147, 148, 156, 163, 164, 165 hybrid upwind-centered discretization scheme, 317 hybrid URANS-LES, 265 hyperbolic system, 196, 215 ideal gas law, 28 impedance, 40, 44, 46, 69, 71, 77, 78, 79, 81, 83 impermeability condition, 209, 210 impinging flow, 164 impinging shear layer, 262 implicit filter, 187 implicit hypothesis, 273 Implicit method, 112 implicit temporal integration, 316, 317 implicit tridiagonal interpolation, 193 impulsive noise, 8, 381 inclined wing, 203, 204 incoming boundary layer, 263 industrial RANS code, 263 inertial range, 107, 115 inflow boundary conditions, 203, 204 inflow section, 153 inherent damping, 195 inhomogeneous wave equation, 142 in-house code, 5 initial conditions, 201, 202 injection interface, 330 inlet conditions, 238, 248, 249

inlet disturbances, 249 instability, 141, 262, 298, 354 instability theories, 238 instability wave, 262, 350 insulator, 21, 22 integral error, 201 integral scale, 105 intensity level, 33 intermittency, 271, 272 internal flow, 349, 355 internal mixer, 245 internal vehicle noise, 272, 378 interpolation, 14, 144 interpolation coefficient, 182 interpolation function, 179 intrinsic dissipation, 317 invariants of the transformation, 184 inverse energy cascade, 113 inverse Fourier transform, 208 inverse Jacobian, 184 inverse problem, 38, 39, 382 isothermal jet, 253 isotropic decaying turbulence, 139 isotropic filter, 100, 101 isotropic mesh, 358, 362 isotropic turbulence, 16, 17, 105, 106, 113, 121, 266, 380 Jacobian derivatives, 183 Jacobian matrix, 215, 217 Jacobian transformation, 172, 181, 183, 185, 186 Jameson dissipation, 189 Jameson scheme, 336 Jameson-Schmidt-Turkel scheme, 175 jet aircraft, 197 jet centerline, 251 jet core, 241 jet engine, 4 jet engine manufacturer, 4, 378 jet flow, 198 jet noise, 58, 232 jet nozzle, 357 jet plume, 247, 253, 254, 258, 261 Kelvin-Helmholtz instability, 161 kinetic energy spectrum, 108, 115 Kirchhof integral, 11, 175, 264, 294, 315, 316, 331, 332, 384 Kirchhoff approach, 133 Kirchhoff integration, 387 Kirchhoff interface, 333 Kirchhoff surface, 133, 251, 252, 253 Kirchhoff’s method, 7, 19 Kolmogorov hypothesis, 108 Kolmogorov length scale, 131, 139, 222 Kolmogorov microscale, 334, 379

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435

INDEX

Kolmogorov scale, 89, 105, 107, 108, 266 Kolmogorov spectrum, 115, 117 Kolmogorov time scale, 106 Kutta condition, 32, 55, 82, 83, 87 Kutta-Zhukhovski theorem, 236 Lagrangian integral time scale, 256 Lagrangian interpolation, 179 Lagrangian time derivative, 26 laminarization, 138 laminar-turbulent transition, 306 landing, 225 landing gear, 294, 356, 358 Laplace eigenvalue problem, 68 Laplace’ equation, 41 Laplace transform, 201 large-scale instability wave, 258 latency factor, 134 latency parameter, 137 lateral interface, 152 Lattice Boltzmann method, 105 law of the wall, 210 Lax-Friedrichs splitting, 191 leading edge, 55, 159, 160, 333, 357, 387 length scale, 153, 167, 208 Leonard stress tensor, 96 Leonard’s decomposition, 96 Leray’s regularization, 119 LES-CAA coupling, 225 LES-Euler-coupling, 327, 328, 330 Li´enard-Wiechert potential, 50 Lighthill source term, 246 Lighthill stress tensor, 133 Lighthill tensor, 21, 59, 373 Lighthill-Curle’s equation, 20, 22 Lighthill’s analogy, 7, 16, 17, 18, 25, 55, 56, 57, 58, 59, 62, 64, 65, 223, 241, 246, 264, 304, 350, 383, 384 Lighthill’s equation, 20, 142 Lilley model, 344, 346 Lilley source, 360 Lilley’s equation, 246, 350 limited numerical scales (LNS), 133, 334 limiter, 172, 191, 196, 266 line integral, 186 linear combination model, 119 linear disturbance equations, 295 linear instability theory, 239 linear stochastic correlation tensor, 211 linearization error, 183 linearized acoustic perturbation equations, 296 linearized Euler approach, 7 linearized Euler equation (LEE), 11, 20, 21, 147, 170, 173, 174, 196, 198, 199, 216, 220, 221, 223, 241, 277, 295, 296, 298, 299, 334, 349, 387

linearized spatial instability problem, 250 lip line, 256, 257 listener, 37, 43, 47, 50, 51, 55, 56, 60, 63, 64, 65, 72, 73, 74, 383 local adaptive mesh refinement, 138 local isotropy hypothesis, 90 local polynomial expansion, 176 localized mesh clustering, 138 logarithmic buffer layer, 210 logarithmic law, 268 logarithmic outer layer, 210 Lorentz transformation, 52, 221 low dispersion, 295 low-dispersion scheme, 199, 200, 301 lower-upper symmetric Gauss-Seidel method, 317 low-frequency noise, 259, 260 low-frequency structures, 286 low-order scheme, 317 low-order trigonometric interpolation, 180 low-order upwind scheme, 266 low-pass filter, 187, 220, 319, 326 low-storage form, 198 low-storage scheme, 269 mass injection, 267 mass-conservation law, 27 material fatigue, 273 matrix inversion, 195 McCormack scheme, 201 mean flow-perturbation splitting, 330 medium resolution LES, 209 mesh smoothness, 197 mesh-frequency cutoff, 359, 360, 362, 376 message passing interface (MPI), 207 method of images, 73 military aircraft, 267 mixed model, 266 mixed scale model, 116 mixing layer, 16, 161, 162, 175, 186, 198, 238, 239, 241, 248, 266, 386 mixing zone, 381 mixing-length model, 213 modal analysis, 293, 387 modal decomposition, 250 mode, 68, 69, 70, 71, 72, 79, 82, 88, 144, 259 modified wave number, 109 molecular Prandtl number, 214 monoatomic gas, 99 monopole, 39, 42, 43, 44, 47, 48, 49, 51, 58, 59, 60, 63, 77, 277, 282, 371, 382, 383 monotone integrated LES (MILES), 110, 111, 126, 266, 267 monotone upstreamcentred scheme, 172 Morkovin’s hypothesis, 125, 229 moving body, 36, 51 moving surface, 8, 214

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436

CUFX063/Wagner

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printer: sheridan

October 26, 2006

INDEX

multiblock solver, 253 multidimensional stencil, 186 multilayer absorbing zone, 219 multilevel approach, 121 multilevel closure, 121 multilevel simulations, 112 multiple modes, 364 multiple streaky structures, 307 multiple tones, 263 multiple-rotating frames of reference, 374 multipole, 48, 49 multipole expansion, 47, 48, 49 Munt model, 83 Munt’s solution, 83, 84 NACA0012 airfoil, 19, 332, 387 nacelle, 197 narrow-band, 371 narrowband component, 321 narrowband excitations, 361, 371 narrow-band power, 372 narrowband spectra, 257 narrowband tone, 364 natural damping zone, 309 near-field noise, 149, 163 nearly axisymmetric disturbances, 249 near-wall damping, 213 near-wall region, 147, 212 near-wall structure, 210, 267 Neumann boundary conditions, 202, 203, 214, 215 Newton method, 317 Newtonian fluid, 29, 89, 91 NLDE decomposition, 151 noise carpet, 11 noise footprint, 5 noise reduction, 15, 18 noise regulations, 345 noise requirements, 378 non-Cartesian grid, 172, 181, 182, 186 noncompact upwind scheme, 190 nondissipative central difference stencil, 195 nondissipative numerical method, 209 nondissipative scheme, 379 non-Favre weighted variables, 253 nonisothermal flow, 214 nonlinear acoustics solvers, 130, 145, 154, 165 nonlinear discontinuities, 189 nonlinear disturbance equations, 146, 148, 149, 150, 165, 172, 175, 294, 299 nonlinear Euler equations, 295, 351, 353 nonlinear perturbation equations, 165 nonlinear saturation, 296 nonlinear stochastic estimation, 212 nonreflecting boundary conditions, 152, 204, 215, 217, 219, 248, 265, 302, 318, 338, 385 nonreflexive border, 329

nonresolveld wave number, 190 nonuniform grid, 172 nonuniqueness of source, 38 nonzonal approach, 212, 213 normal shock, 355 no-slip boundary condition, 208, 212, 308, 318 nozzle, 57, 77, 84, 204, 246, 248, 251, 252, 254, 257, 258, 261 nozzle shear layer, 261 nozzle-ejector mixer, 252 numerical amplification factor, 199 numerical bifurcation, 265 numerical dissipation, 108, 110, 111, 120, 126, 219, 266, 317, 334, 336 numerical error, 13, 92, 105, 110, 111, 124, 126, 151 numerical filter, 92, 112 numerical instability, 113, 114, 187 numerical noise, 13, 232 numerical reflection, 219 numerical stability, 110, 123 numerical stabilization, 111 Nyquist criterion, 363 Nyquist frequency, 113, 140, 153 Nyquist grid wavelength, 140 Nyquist length, 112 Nyquist requirement, 358 Nyquist theorem, 92 oblique shock, 355 obstacle, 207 open apertures, 357 open pipe end, 75 open pipe termination, 76, 77, 78, 79, 80, 81 open window, 358 optimization of dissipation and dispersion, 199 optimized Runge-Kutta scheme, 200 oscillating signal, 364 oscillation, 155, 161, 220, 263, 268, 286, 336, 353, 354, 355, 357, 372 oscillation frequency, 271 O-type mesh, 306 outer ear, 34 outer region, 212 outflow boundary conditions, 203, 204 outflow interface, 152 overall sound pressure level (OASPL), 253, 269 overdissipative model, 266 overlapping region, 150 overshoot, 174, 179 Pad´e derivative scheme, 248 Pad´e-type scheme, 176 pairing, 239 pairing frequency, 240 pantograph, 21

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P1: PJU 0521871441ind

CUFX063/Wagner

0 521 87144 1

printer: sheridan

October 26, 2006

437

INDEX

parabolic equations, 201 paradox of d’Alembert, 66 parallel computer, 207 particle image velocimetry (PIV), 279 passenger cabin, 358 passenger compartement, 349 perceived aircraft noise, 5 perfectly matched layer (PML), 217, 220, 222, 385 periodic boundary conditions, 202, 206, 226, 329, 350 periodic ejection, 265 periodic recycling, 153 perturbation injections, 329 phase shift, 78 Phillips’ analogy, 16 pipe flow, 206 piston engine, 5 plane channel, 150, 152, 209, 210, 212 plane jet, 238 plane mixing layer, 239 plane wave, 9, 33, 34, 35, 39, 41, 46, 53, 54, 68, 69, 72, 77, 83 plane wave mode, 72 POD-Galerkin method, 95 point source, 236 points per wavelength (PPW), 109 Poisson equation, 212 policeman’s whistle, 34 polyhedra, 362 poor-resolution LES, 209 postprocessing, 357, 358, 359, 360, 370 potential core, 254, 256, 258 power law, 211 power spectral density (PSD), 286, 311, 320 power train noise, 272, 273, 378 power-scaling law, 303 Prandtl number, 30, 35, 91, 126 Prandtl-Glauert factor, 332 Prandtl-Glauert transformation, 52, 221 predictor simulation, 150 predictor-corrector scheme, 195, 201 prefiltering technique, 111 prescribed energy spectrum, 208 pressure boundary conditions, 214 pressure pulse, 194 pressure reducing valve, 354 pressure spike, 381 prismatic, 134 projector operators, 94 propeller, 23, 36 propeller noise, 62 proper orthogonal disposition (POD), 95 pseudo noise, 273 pseudo sound, 357 pseudospectral methods, 95

Q-criterion, 268 quadrangular elements, 281 quadrilateral grid, 196 quadrupole, 39, 42, 44, 47, 48, 52, 63, 245, 255, 371, 382, 383 quasi-laminar annular shear layer, 248 quasi-laminar disturbances, 248, 249 quasi-laminar flow, 153 quasi-two-dimensional, 158 quiet boundary conditions, 272 q-wave, 189 radiation boundary condition, 218 radiator, 58 random error, 57, 67 random Fourier mode, 268 random velocity disturbance, 241 random-number generator, 208 Rayleigh scattering technique, 255 rear side-mirror vehicle, 274 reattaching flow, 283 reattachment, 13 reattachment length, 283, 285 reattachment point, 283, 284, 286 reattachment region, 140, 148 reattachment zone, 286 recirculating flow, 263, 266, 271 recirculation, 211 recirculation bubble, 161, 162 recirculation inside the cavity, 263 recirculation length, 286 recirculation region, 266, 283 recirculation zone, 266 reconstruction, 150 reconstruction step, 191 rectangular boxcar, 369 rectangular duct, 69, 70, 354 rectangular jet, 18, 173, 175, 250 rectangular nozzle, 246 recycled inlet plane, 143 recycling approach, 138 recycling techniques, 141 recycling zone, 139 reentry of fluid, 355 reflecting boundary conditions, 203 reflection, 204, 265, 269, 371, 385 reflections at the interfaces, 151 refraction, 82, 296, 316, 371 relaminarization, 13 rescaled boundary conditions, 206 rescaling, 138, 144, 150, 153, 206, 207, 226, 227, 228, 229, 231, 306 resolution error, 174 resolved scales, 266 resonance, 34, 148, 273, 356, 358 resonance frequency, 282, 372

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P1: PJU 0521871441ind

438

CUFX063/Wagner

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printer: sheridan

October 26, 2006

INDEX

resonant modes, 384 resonator, 88, 358, 372 resynthesizing noise, 357 reverse flow, 203 reverse-flow reciprocity principle, 74 Reynolds’ approach, 208 Reynolds averaging, 210 Reynolds stress, 30, 133 Reynolds stress tensor, 136 Riemann problem, 172 Riemann solver, 195, 196 Riemann’s invariants, 338 road traffic, 378 road-contact noise, 5 robustness, 259 robustness of the solver, 253 rocket engine, 85 Roe’s scheme, 172, 193, 194 Rossiter acoustic modes, 364 Rossiter formula, 262 rotating blade, 337 rotating channel flow, 212, 213 rotating dipole, 371 rotating mode, 70 rotating quadrupole, 241 rotating surface, 214 rotor blade, 383 rotor-stator interaction, 8, 60, 71, 381 ruler, 278, 286, 387 Runge-Kutta discontinuous Galerkin method, 196 Runge-Kutta method, 196 Runge-Kutta scheme, 170, 197, 198, 199, 200, 248, 268, 269, 301, 329 Runge-Kutta stage, 170 saturation, 88 scalar energy spectrum, 202 scalar transport equation, 214 scale reduction, 267 scale similarity model, 18, 118, 380 scaled boundary condition, 226 scaling law, 8, 58, 106, 107, 109, 207, 226, 227 scaling rule, 58, 59 scattering, 52, 54, 79, 316, 333, 382, 387 Schlieren visualization, 271 Schlieren-like view, 160, 268 second law of Newton, 26 second moments, 144, 153, 165 second-moment turbulence closure, 266 second-order accuracy, 167 second-order accurate scheme, 178, 179, 187, 317, 346, 357 second-order centered scheme, 267 second-order central scheme, 169, 171 second-order dissipation, 189 second-order filter, 188

second-order statistic, 208 selective mixed-scale model, 267, 316, 317 self-similar region, 253 self-sustained oscillations, 24, 84, 85, 88, 262, 263, 271, 349, 351, 365, 384 self-sustaining physical disturbances, 153 semianalytic model, 129 semidiscretized equation, 168, 169, 170 semiempirical model, 263 semiempirical sources, 10 semiperiodic boundary condition, 226 sensitivity, 151 separated flow, 263 separated flow regions (detached eddies), 213 separated shear layer, 264, 265, 362 separation, 66, 85, 87, 111, 132, 148, 164, 211, 212, 213 separation bubble, 284 separation zone, 159, 160 seven-point stencil, 174, 330 shallow cavity, 267 sharp cutoff filter, 94, 115, 117 sharp edge, 32, 58, 67, 71, 76, 85, 86, 87, 88, 381 sharp leading edge, 173 shear flow, 266 shear layer, 52, 80, 87, 88, 106, 238, 239, 241, 242, 244, 248, 249, 254, 255, 256, 261, 262, 266, 269, 381 shear layer instability, 272 shear layer oscillation, 372 shear region, 161 shedding, 337 shedding frequency, 339, 340 shedding of vorticity, 87 shifted boundary conditions, 211 shock, 3, 252 shock capturing, 108, 110, 111, 126 shock corrugation, 108, 109 shock deformation, 108 shock pattern, 355 shock sensor, 111 shock wave, 27, 34, 108, 126, 173, 265 shock-turbulence interaction, 108, 109 side mirror, 291 sideview mirror, 357, 360 sign convention, 37, 68 silencer, 81 silent artificial boundary, 234 similarity model of Bardina, 266 simplified mirror, 349 single-point statistics, 153 six stage algorithm, 201 six stage Runge-Kutta scheme, 200 Sixth Research framework Program, 6 sixth-order accurate scheme, 182, 183 sixth-order compact scheme, 186

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P1: PJU 0521871441ind

CUFX063/Wagner

0 521 87144 1

printer: sheridan

October 26, 2006

439

INDEX

sixth-order Fourier difference scheme, 180 sixth-order Taylor scheme, 180 skin friction, 153, 209 skin friction coefficient, 106 slat, 158, 196 slat cove, 152, 154, 158, 159, 371 slats, 294 slat’s trailing edge, 163 sleep, 1, 2 sliding interface, 374 Smagorinsky constant, 134, 140, 271 Smagorinsky model, 18, 19, 20, 22, 115, 116, 132, 134, 135, 239, 241, 259, 266, 269, 277, 334, 344, 346, 350, 388 Smagorinsky-Lilley model, 336 smooth flow, 191 solid surface, 329 solid wall boundary condition, 203, 208 solution of d’Alembert, 40, 41 Sommerfeld impedance, 277 Sommerfeld solution, 54 sound intensity, 33 sound pressure level, 1, 33, 156, 157, 302, 312, 336, 347, 363 sound pressure level correction, 302 Spalart-Allmaras model, 132, 158 spatial correlation, 153, 208, 338 spatial discretization scheme, 167, 168, 169, 170 spatial stability problem, 250 spatially developing mixing layer, 238 spational discretization scheme, 199 spational DRP scheme, 200 spectral broadening, 339 spectral element implementation, 197 spectral element method, 105, 197 spectral gap, 114, 115 spectral method, 105, 110, 176 spectral/hp element method, 176 spectral-like resolution, 317 spherical dipole field, 181 spherically symmetric wave, 39, 41, 42 spherical-shaped eddies, 144 spiraling mode, 69 splitter plate, 85 splitting, 116, 328, 382 sponge layer, 218, 219, 220, 309 spray nozzle, 365 spurious effects, 113 spurious noise, 249, 304, 313 spurious perturbation, 233, 234 spurious reflections, 14, 385 spurious signal, 314 spurious sound, 77, 232, 234, 235, 237, 304, 309, 312 spurious velocity field, 233 spurious vortical perturbations, 233

spurious wave, 9, 174, 301 square cylinder, 20, 334 stability, 199 stabilization, 110, 379 stagnation flow, 186 stagnation region, 140 standard central scheme, 177 statistical subgrid noise model, 261 stencil width, 171 stochastic modeling, 350 stochastic noise generation and radiation (SNGR), 7, 359 stochastik fluctuations, 208 Stokes assumption, 91 Stokes’ hypothesis, 29 Stoke’s no-slip boundary condition, 208 strain rate, 213 stratification, 91, 92 streak, 209, 358 streaky structure, 106, 307 streamline, 203, 339, 346 streamline curvature, 161 streamline upwind Petrov-Galerkin (SUPG) integration, 277, 289 streamwise vortices, 238 Strouhal number, 65, 79, 80, 81, 83, 84, 88, 223, 259, 263, 279, 299, 322, 339 structural modeling, 112, 113 structural response, 276 structure function model, 17, 116 structure-borne noise, 272, 378 subfilter scales, 112 subgrid dissipation, 110 subgrid kinetic energy, 117 subgrid mode, 115 subgrid model, 92, 96, 103, 104, 106, 110, 111, 113, 117, 239 subgrid model filter, 92 subgrid Reynolds stress tensor, 97 subgrid scale, 94, 112, 113, 115, 119 subgrid scale model, 16, 17, 19, 90, 110, 112, 129, 145, 166, 186, 253, 259, 262, 266, 316 subgrid scale noise model, 261, 262 subgrid tensor, 21, 97, 102, 112, 117, 118 subgrid terms, 111 successive grid refinement, 202 suction side, 161 sudden enlargement, 355 sudden expansion, 354 sunroof, 357, 358 super-bang phenomenon, 50 superposition, 153 superposition of perturbations, 202 supersonic jet, 15, 16, 18, 355 suppression device, 267 surface integral, 133

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P1: PJU 0521871441ind

440

CUFX063/Wagner

0 521 87144 1

printer: sheridan

October 26, 2006

INDEX

surface-pressure cross spectrum, 327 surrounding node, 168 Sutherland law, 91 sweeps, 211 switching, 271 symmetric DRP scheme, 173 symmetry breaking, 202 synthesize turbulent fluctuations, 150 synthesized disturbances, 154 synthesized fluctuations, 154 synthesizing turbulent data, 153 synthetic model, 144, 153 synthetic noise, 143 synthetic perturbations, 367 synthetic reconstruction, 138, 141, 143, 147, 153, 154, 156, 164, 165, 358, 359, 380 synthetic turbulence, 142, 145, 153, 154 synthetic velocity fluctuations, 143 synthetic vortical structure, 365 synthetical reconstruction, 165 T106 turbine blade, 152 tabs, 246 takeoff, 52, 225 Taylor difference scheme, 179, 180 Taylor expansion, 167, 171, 173, 177, 182, 185 Taylor microscale, 108 Taylor series, 188 Taylor series expansions, 47, 119 Taylor’s hypothesis, 207, 303 temperature boundary conditions, 214 temporal correlation, 153, 208 temporal discretization scheme, 167, 168, 170, 197, 248 temporally evolving mixing layer, 238, 239 tenth-order accurate scheme, 178 tenth-order filter, 330 test filter, 104, 116 tetrahedral mesh, 134, 197, 362 theoretical filter, 92, 112 thin boundary layer equations (TBLE), 212 third-order accurate scheme, 176, 190 third-order upwind scheme, 189 three-dimensional curvilinear grid, 295 three-dimsnsional mixing layer, 239 threshold of hearing, 34 threshold of pain, 34 thunder, 74 time advancement scheme, 248 time and length scales, 165 time and space correlation, 144 time discretization, 201 time integration scheme, 199 time scale, 153, 208 time series, 208 tire noise, 272, 273, 378

T-joint, 71 tonal flow excitations, 376 tonal noise, 3, 8, 15, 129, 189, 381 total variation diminishing (TVD), 353 trailing edge, 19, 20, 55, 58, 82, 152, 159, 161, 163, 225, 273, 294, 296, 302, 305, 306, 307, 309, 311, 315, 333, 381, 387 trailing edge noise, 201, 223, 231, 233, 293, 294, 295, 296, 297, 305 train, 3, 378 transient period, 153 transition, 132, 265, 266 transition process, 202 transition region, 320 transition to turbulence, 106, 205 transition triggering, 322 transitional flow, 238 transitional shear layer, 238, 239 transverse acoustic mode, 267 transverse electromagnetic wave, 181 transverse mode pattern, 353 transverse unstable mode, 158 triangular elements, 281 triangular mesh, 196 triple decomposition, 149 truncation error, 177 turbine blade, 206 turbine engine, 8 turbine engine fan, 381 turbocharger, 357 turbofan, 58 turbofan engine, 245 turbofan revolution, 58 turbomachine, 334, 357 TurboNoiseCFD, 6 turbulence intensity, 205 turbulence investigations, 202 turbulence level, 346 turbulence model, 9, 16, 19, 263 turbulence-generated noise, 349 turbulence-related noise, 148 turbulent boundary layer, 206, 264, 268, 273, 294 turbulent bubble, 163 turbulent eddy, 10, 167, 294, 327 turbulent energy cascade, 106 turbulent fluctuation, 358 turbulent structures, 153, 238, 327 two point correlation, 153 two-engine jet aircraft, 197 two-equation turbulence model, 334 two-layer model, 212, 213 two-point correlation, 105, 206, 211, 246 two-point statistic, 208 two-step method, 264 two-step Runge-Kutta scheme, 301 two-way coupling, 150

18:15

P1: PJU 0521871441ind

CUFX063/Wagner

0 521 87144 1

printer: sheridan

October 26, 2006

441

INDEX

unheated jet, 250, 252, 253, 255, 256, 262 uniform Euler flow, 194 uniform grid, 176 uniqueness of source, 38 United States, 5 unphysical streak, 209 unresolved scales, 154, 164, 243, 266 unstaggered formula, 192 unsteady Reynolds averaged Navier-Stokes (URANS), 150, 263, 294, 334 unwanted oscillations, 187 upstream edge, 262 upstream traveling perturbations, 203, 204 upwind biased, 172, 194 upwind compact scheme, 189 upwind discretization, 183 upwind scheme, 171, 190 valve, 349, 350 van Cittert iterative procedure, 118, 126 van Driest damping, 134, 269 van Driest function, 267, 271 van Driest transformation, 227 vehicle noise, 3 ventilation, 373, 381 ventilator, 8 very large eddy simulation (VLES), 130, 186, 200, 346, 349 vibration, 3, 273 vibroacoustics, 273, 274, 278, 387 vibroacoustics response, 279, 289 viscous boundary layer, 31 viscous damping, 296 viscous dissipation, 58 viscous limiter function, 166 viscous sublayer, 128, 210, 211 von K´arm´an constant, 213 von K´arm´an energy spectrum, 359 von K´arm´an length scale, 139, 141 von K´arm´an vortex street, 339 vortex, 194, 236, 266 vortex merging, 238 vortex method, 350 vortex pairing, 57, 161, 163, 239, 240, 242 vortex ring, 87, 88 vortex shedding, 35, 55, 67, 82, 83, 84, 132, 161, 162, 163, 265, 268, 318, 320, 321, 360, 361 vortex shedding airfoil, 304 vortex shedding frequency, 20, 322, 325, 327, 336 vortex shedding noise, 326 vortex sheet, 55, 82 vortex sound, 65 vortex sound theory, 25, 32, 80, 84, 86, 384 vortex streaks, 358 vortex structure, 339

vortex-blade interaction, 3 vortical structures, 245 wake, 20, 22, 60, 154, 203, 318, 336, 360, 367 wake mode, 265, 266 wake mode transition, 265 wake thickness, 322 wall boundary layer, 266 wall function, 141, 154, 210, 214 wall model, 107, 131, 210, 211, 212, 267, 268 wall resolving LES, 107 wall shear stress, 211, 212 wall stress model, 107 wall units, 209 wall-adapting local eddy viscosity model (WALE), 116 wall-damping function, 123 wall-modeling strategy, 214 wall-normal distance, 210 water tunnel, 265 water-air mixture, 64 wave extrapolation method, 264 wave-number-frequency spectrum, 323 wave-propagation zone, 247 wave-vector frequency spectrum, 285 weak-coupling assumption, 275, 279 weakly reflecting boundary conditions, 338 weapon bay, 267, 357, 358 weapons, 357 weighted essentially nonoscillatory scheme (WENO), 173 weighting function, 178 whistle, 84 whistler-nozzle, 84 whistling, 24, 77, 84, 88, 384 white noise, 153, 208, 323, 365 wideband continuum, 321, 322 Wiener-Hopf method, 79, 82 wiggle, 317 wiggle detection, 111 wind rotor, 383 wind tunnel, 203, 204, 205 wind turbine, 3, 8, 174, 378, 381 window, 291, 357 windshield wiper, 278 wing, 203 wing mirror, 154, 360, 362, 363, 366, 367 wing span, 302 wiperblade, 357, 360 zero-equation turbulence model, 263 zero-order truncation error, 180 zero-pressure gradient boundary layer, 229 zonal approach, 212 zonal hybrid method, 148

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