Large Eddy Simulation for Compressible Flows

279 Pages·2009·6.963 MB·English

by  Dr. E. Garnier, Prof. Dr. N. Adams, Prof. P. Sagaut (auth.)

Large Eddy Simulation for Compressible Flows
Scientific Computation
Editorial Board
J.-J. Chattot, Davis, CA, USA
P. Colella, Berkeley, CA, USA
W. Eist, Princeton, NJ, USA
R. Glowinski, Houston, TX, USA
Y. Hussaini, Tallahassee, FL, USA
P. Joly, Le Chesnay, France
H.B. Keller, Pasadena, CA, USA
J.E. Marsden, Pasadena, CA, USA
D.I. Meiron, Pasadena, CA, USA
O. Pironneau, Paris, France
A. Quarteroni, Lausanne, Switzerland
and Politecnico of Milan, Milan, Italy
J. Rappaz, Chicago, IL, USA
R. Rosner, Paris, France
P. Sagaut, Pasadena, CA, USA
J.H. Seinfeld, Pasadena, CA, USA
A. Szepessy, Stockholm, Sweden
M.F. Wheeler, Austin, TX, USA
For other titles published in this series, go to
www.springer.com/series/718
E. Garnier (cid:2) N. Adams (cid:2) P. Sagaut
Large Eddy Simulation
for Compressible Flows
Dr. E. Garnier Prof. Dr. N. Adams
Office National d’E´tudes Institute of Aerodynamics
et de Recherches A´erospatiales Technische Universit¨at Mu¨nchen
8 rue des Vertugadins 85747 Garching
92190 Meudon Germany
France e-mail: [email protected]
e-mail: [email protected]
Prof. P. Sagaut
Institut Jean Le Rond d’Alembert
Universit´e Pierre et Marie Curie
Boite 162
4, place Jussieu
75252 Paris Cedex 5
France
e-mail: [email protected]
ISBN 978-90-481-2818-1 e-ISBN 978-90-481-2819-8
DOI 10.1007/978-90-481-2819-8
LibraryofCongressControlNumber:2009933039
(cid:2)cSpringerScience+BusinessMediaB.V.2009
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Contents
1 Introduction............................................... 1
2 LES Governing Equations ................................. 5
2.1 Preliminary Discussion ................................... 5
2.2 Governing Equations..................................... 6
2.2.1 Fundamental Assumptions.......................... 6
2.2.2 Conservative Formulation .......................... 7
2.2.3 Alternative Formulations ........................... 9
2.3 Filtering Operator....................................... 9
2.3.1 Definition ........................................ 10
2.3.2 Discrete Representation of Filters ................... 14
2.3.3 Filtering of Discontinuities.......................... 16
2.3.4 Filter Associated to the Numerical Method ........... 18
2.3.5 Commutation Error ............................... 20
2.3.6 Favre Filtering .................................... 20
2.3.7 Summary of the Different Type of Filters............. 22
2.4 Formulation of the Filtered Governing Equations ............ 22
2.4.1 Enthalpy Formulation.............................. 23
2.4.2 Temperature Formulation .......................... 24
2.4.3 Pressure Formulation .............................. 24
2.4.4 Entropy Formulation .............................. 25
2.4.5 Filtered Total Energy Equations .................... 26
2.4.6 Momentum Equations ............................. 29
2.4.7 Simplifying Assumptions ........................... 30
2.5 Additional Relations for LES of Compressible Flows ......... 33
2.5.1 Preservation of Original Symmetries ................. 33
2.5.2 Discontinuity Jump Relations for LES ............... 35
2.5.3 Second Law of Thermodynamics .................... 37
2.6 Model Construction...................................... 38
2.6.1 Basic Hypothesis .................................. 38
2.6.2 Modeling Strategies................................ 39
E. Garnier et al., Large Eddy Simulation for Compressible Flows,
Scientific Computation,
(cid:2)c Springer Science + Business Media B.V. 2009
vi Contents
3 Compressible Turbulence Dynamics ....................... 41
3.1 Scope and Content of This Chapter........................ 41
3.2 Kovasznay Decomposition of Turbulent Fluctuations ......... 42
3.2.1 Kovasznay’s Linear Decomposition .................. 42
3.2.2 Weakly Nonlinear Kovasznay Decomposition.......... 45
3.3 Statistical Description of Compressible Turbulence........... 46
3.4 Shock-Turbulence Interaction ............................. 48
3.4.1 Introduction to the Linear Interaction Approximation
Theory........................................... 48
3.4.2 Vortical Turbulence-Shock Interaction................ 49
3.4.3 Mixed-Mode Turbulence-Shock Interaction ........... 57
3.4.4 Consequences for Subgrid Modeling.................. 62
3.5 Different Regimes of Isotropic Compressible Turbulence ...... 64
3.5.1 Quasi-Isentropic-Turbulence Regime ................. 65
3.5.2 Nonlinear Subsonic Regime ......................... 71
3.5.3 Supersonic Regime ................................ 74
3.5.4 Consequences for Subgrid Modeling.................. 75
4 Functional Modeling....................................... 77
4.1 Basis of Functional Modeling ............................. 77
4.1.1 Phenomenology of Scale Interactions................. 77
4.1.2 Basic Functional Modeling Hypothesis ............... 79
4.2 SGS Viscosity........................................... 79
4.2.1 The Boussinesq Hypothesis ......................... 79
4.2.2 Smagorinsky Model................................ 81
4.2.3 Structure Function Model .......................... 82
4.2.4 Mixed Scale Model ................................ 82
4.3 Isotropic Tensor Modeling ................................ 83
4.4 SGS Heat Flux.......................................... 84
4.5 Modeling of the Subgrid Turbulent Dissipation Rate ......... 85
4.6 Improvement of SGS models .............................. 85
4.6.1 Structural Sensors and Selective Models.............. 85
4.6.2 Accentuation Technique and Filtered Models.......... 87
4.6.3 High-Pass Filtered Eddy Viscosity................... 88
4.6.4 Wall-Adapting Local Eddy-Viscosity Model........... 88
4.6.5 Dynamic Procedure................................ 89
4.6.6 Implicit Diffusion and the Implicit LES Concept ...... 93
5 Explicit Structural Modeling .............................. 95
5.1 Motivation of Structural Modeling......................... 95
5.2 Models Based on Deconvolution ........................... 97
5.2.1 Scale-Similarity Model .............................100
5.2.2 Approximate Deconvolution Model ..................103
5.2.3 Tensor-Diffusivity Model ...........................105
5.3 Regularization Techniques ................................105
Contents vii
5.3.1 Eddy-Viscosity Regularization ......................106
5.3.2 Relaxation Regularization ..........................109
5.3.3 Regularization by Explicit Filtering..................111
5.4 Multi-Scale Modeling of Subgrid-Scales.....................113
5.4.1 Multi-Level Approaches ............................113
5.4.2 Stretched-Vortex Model ............................116
5.4.3 Variational Multi-Scale Model.......................117
6 Relation Between SGS Model and Numerical
Discretization..............................................119
6.1 Systematic Procedures for Nonlinear Error Analysis..........119
6.1.1 Error Sources .....................................119
6.1.2 Modified Differential Equation Analysis ..............121
6.1.3 Modified Differential Equation Analysis in Spectral
Space ............................................126
6.2 Implicit LES Approaches Based on Linear and Nonlinear
Discretization Schemes ...................................129
6.2.1 The Volume Balance Procedure of Schumamm ........129
6.2.2 The Kawamura-Kuwahara Scheme...................130
6.2.3 The Piecewise-Parabolic Method ....................131
6.2.4 The Flux-Corrected-Transport Method...............132
6.2.5 The MPDATA Method.............................136
6.2.6 The Optimum Finite-Volume Scheme ................138
6.3 Implicit LES by Adaptive Local Deconvolution..............140
6.3.1 Fundamental Concept of ALDM.....................140
6.3.2 ALDM for the Incompressible Navier-Stokes Equations.143
6.3.3 ALDM for the Compressible Navier-Stokes Equations ..148
7 Boundary Conditions for Large-Eddy Simulation of
Compressible Flows........................................155
7.1 Introduction ............................................155
7.2 Wall Modeling for Compressible LES.......................156
7.2.1 Statement of the Problem ..........................156
7.2.2 Wall Boundary Conditions in the Kovasznay
Decomposition Framework: an Insight................156
7.2.3 TurbulentBoundaryLayer:VorticityandTemperature
Fields............................................159
7.2.4 Turbulent Boundary Layer: Acoustic Field............163
7.2.5 Consequences for the Development of Compressible
Wall Models ......................................169
7.2.6 Extension of Existing Wall Models for Incompressible
Flows............................................170
7.3 Unsteady Turbulent Inflow Conditions for Compressible LES..172
7.3.1 Fundamentals.....................................172
7.3.2 Precursor Simulation: Advantages and Drawbacks .....174
viii Contents
7.3.3 Extraction-Rescaling Techniques ....................175
7.3.4 Synthetic-Turbulence-Based Models..................179
8 Subsonic Applications with Compressibility Effects ........185
8.1 Homogeneous Turbulence.................................185
8.1.1 Context ..........................................185
8.1.2 A Few Realizations ................................186
8.1.3 Influence of the Numerical Method ..................187
8.1.4 SGS Modeling ....................................190
8.2 Channel Flow...........................................191
8.2.1 Context ..........................................191
8.2.2 A Few Realizations ................................191
8.2.3 Influence of the Numerical Method ..................192
8.2.4 Influence of the SGS Model.........................194
8.3 Mixing Layer ...........................................195
8.3.1 Context ..........................................195
8.3.2 A Few Realizations ................................195
8.3.3 Influence of the Numerical Method ..................196
8.3.4 Influence of the SGS Model.........................197
8.4 Boundary-Layer Flow ....................................198
8.4.1 Context ..........................................198
8.4.2 A Few Realizations ................................198
8.5 Jets....................................................200
8.5.1 Context ..........................................200
8.5.2 A Few Realizations ................................201
8.5.3 Influence of the Numerical Method ..................202
8.5.4 Influence of the SGS Model.........................204
8.5.5 Physical Analysis..................................205
8.6 Flows over Cavities ......................................206
8.6.1 Context ..........................................206
8.6.2 A Few Realizations ................................206
8.6.3 Influence of the Numerical Method ..................207
8.6.4 Influence of the SGS Model.........................208
8.6.5 Physical Analysis..................................208
9 Supersonic Applications ...................................211
9.1 Homogeneous Turbulence.................................211
9.2 Channel Flow...........................................212
9.2.1 Context ..........................................212
9.2.2 A Few Realizations ................................212
9.2.3 Influence of the Numerical Method ..................213
9.2.4 Influence of the Grid Resolution.....................214
9.2.5 Influence of the SGS Model.........................215
9.3 Boundary Layers ........................................215
9.3.1 Context ..........................................215
Contents ix
9.3.2 A Few Realizations ................................216
9.3.3 Influence of the Numerical Method ..................216
9.3.4 Influence of the Grid Resolution.....................217
9.3.5 SGS Modeling ....................................219
9.4 Jets....................................................220
9.4.1 Context ..........................................220
9.4.2 A Few Realizations ................................220
9.4.3 Influence of the Numerical Method ..................221
9.4.4 Influence of the SGS Model.........................221
9.4.5 Physical Analysis..................................221
10 Supersonic Applications with Shock-Turbulence
Interaction ................................................223
10.1 Shock-Interaction with Homogeneous Turbulence ............224
10.1.1 Phenomenology of Shock-Interaction with
Homogeneous Turbulence...........................224
10.1.2 LES of Shock-Interaction with Homogeneous
Turbulence .......................................228
10.2 Shock-Turbulence Interaction in Jets.......................230
10.2.1 Phenomenology of Shock-Turbulence Interaction in
Jets .............................................230
10.2.2 LES of Shock-Turbulence Interaction in Jets ..........231
10.3 Shock-Turbulent-Boundary-Layer Interaction ...............233
10.3.1 Phenomenology of Shock-Turbulent-Boundary-Layer
Interaction .......................................233
10.3.2 LES of Compression-Ramp Configurations............237
References.....................................................255
Index..........................................................273
1
Introduction
Turbulent flows are ubiquitous in most application fields, ranging from en-
gineering to earth sciences and even life sciences. Therefore, simulation of
turbulent flows has become a key tool in both fundamental and applied re-
search. The complexity of Navier–Stokes turbulence, which is illustrated by
the fact that the number of degrees of freedom of turbulence grows faster
than O(Re11/4), where Re denotes the Reynolds number, renders the Direct
Numerical Simulation (DNS) of turbulence inapplicable to most flows of in-
terest. To alleviate this problem, truncated solutions in both frequency and
wavenumbermaybesought,whosecomputationalcostismuchlowerandmay
ideally be arbitrarily adjusted. The most suitable approach to obtain such a
low-cost three-dimensional unsteady simulation of a turbulent flow is Large-
EddySimulation(LES),whichwaspioneeredtocomputemeteorologicalflows
in the late 1950s and the early 1960s.
One of the main issues raised by LES is a closure problem: because of the
non-linearity of the Navier–Stokes equations, the effect of unresolved scales
must be taken into account to recover a reliable description of resolved scales
of motion (Chap. 2). This need to close the governing equations of LES has
certainly been the main area of investigation since the 1960s, and numerous
closures,alsoreferredtoassubgridmodels,havebeenproposed.Mostexisting
subgridmodelshavebeenbuiltusingsimplifiedviewsofturbulencedynamics,
themainphysicalphenomenontakenintoaccountbeingthedirectkineticen-
ergycascadefromlargetosmallscalesthatisobservedinisotropicturbulence
and high-Reynolds fully developed turbulent flows. The most popular para-
digm for interscale energy transfer modeling is subgrid viscosity (Chap. 4),
whichisaneasywaytoaccountforthenetpumpingofresolvedkineticenergy
by unresolved scales. Other models have been based on mathematical manip-
ulations of governing equations, such as approximate deconvolution, and are,
at least theoretically, more general since they are not based on a priori as-
sumptions on turbulence dynamics (Chap. 5). An important observation is
that the vast majority of existing works dealing with subgrid modeling is de-
voted to incompressible flows, the main extension being for variable-density
E. Garnier et al., Large Eddy Simulation for Compressible Flows,
Scientific Computation,
(cid:2)c Springer Science + Business Media B.V. 2009