Large Eddy Simulations for Incompressible Flows

The book is unique in that it is the only one of its kind devoted entirely to the subject of (large-eddy simulation). It presents a comprehensive account of the available knowledge in this field, and also the first unified view of the various existing approaches. Large-eddy simulation is the only efficient technique for approaching high Reynolds numbers when simulating industrial, natural or experimental configurations. The author concentrates on incompressible fluids. The book gives a complete account of this young but very rich discipline. The topics are well chosen and both the mathematical ideas and the applicatons are presented with care. The book addresses researchers as well as graduate students and engineers.

1. Introduction 1
1.1 Computational Fluid Dynamics 1
1.2 Levels of Approximation: General 2
1.3 Statement of the Scale Separation Problem 3
1.4 Usual Levels of Approximation 4
1.5 Large-Eddy Simulation 7
2. Formal Introduction to Scale Separation: Band-Pass Filtering 9
2.1 Definition and Properties of the Filter in the Homogeneous Case 9
2.1.1 Definition 9
2.1.2 Fundamental Properties 10
2.1.3 Characterization of Different Approximations 12
2.1.4 Differential Filters 13
2.1.5 Three Classical Filters for Large-Eddy Simulation 15
2.2 Extension to the Inhomogeneous Case 19
2.2.1 General 19
2.2.2 Non-uniform Filtering Over an Arbitrary Domain 20
3. Application to Navier­Stokes Equations 31
3.1 Navier­Stokes Equations 31
3.1.1 Formulation in Physical Space 31
3.1.2 Formulation in Spectral Space 32
3.2 Filtered Navier­Stokes Equations (Homogeneous Case) 33
3.2.1 Formulation in Physical Space 33
3.2.2 Formulation in Spectral Space 33
3.3 Decomposition of the Non-linear Term. Associated Equations 34
3.3.1 Leonard's Decomposition 34
3.3.2 Germano Consistent Decomposition 44
3.3.3 Germano Identity 47
3.3.4 Invariance Properties 49
3.3.5 Realizability Conditions 54
3.4 Extension to the Inhomogeneous Case 55
3.4.1 Second-Order Commuting Filter 56
3.4.2 High-Order Commuting Filters 58
3.5 Closure Problem 58
3.5.1 Statement of the Problem 58
3.5.2 Postulates 59
3.5.3 Functional and Structural Modeling 60
4. Functional Modeling (Isotropic Case) 63
4.1 Phenomenology of Inter-Scale Interactions 63
4.1.1 Local Isotropy Assumption: Consequences 64
4.1.2 Interactions Between Resolved and Subgrid Scales 65
4.1.3 A View in Physical Space 74
4.1.4 Summary 75
4.2 Basic Functional Modeling Hypothesis 76
4.3 Modeling of the Forward Energy Cascade Process 77
4.3.1 Spectral Models 77
4.3.2 Physical Space Models 81
4.3.3 Improvement of Models in the Physical Space 102
4.3.4 Implicit Diffusion 124
4.4 Modeling the Backward Energy Cascade Process 125
4.4.1 Preliminary Remarks 125
4.4.2 Deterministic Statistical Models 126
4.4.3 Stochastic Models 131
5. Functional Modeling: Extension to Anisotropic Cases 141
5.1 Statement of the Problem 141
5.2 Application of Anisotropic Filter to Isotropic Flow 141
5.2.1 Scalar Models 142
5.2.2 Tensorial Models 144
5.3 Application of an Isotropic Filter to an Anisotropic Flow 146
5.3.1 Phenomenology of Inter-Scale Interactions 146
5.3.2 Anisotropic Models 152
6. Structural Modeling 161
6.1 Formal Series Expansions 162
6.1.1 Models Based on Approximate Deconvolution 162
6.1.2 Non-linear Models 166
6.1.3 Homogenization Technique: Perrier and Pironneau Models 171
6.2 Differential Subgrid Stress Models 173
6.2.1 Deardorff Model 173
6.2.2 Link with the Subgrid Viscosity Models 174
6.3 Deterministic Models of the Subgrid Structures 175
6.3.1 General 175
6.3.2 S3/S2 Alignment Model 176
6.3.3 S3/omega Alignment Model 177
6.3.4 Kinematic Model 177
6.4 Scale Similarity Hypotheses and Models Using Them 177
6.4.1 Scale Similarity Hypotheses 177
6.4.2 Scale Similarity Models 178
6.4.3 A Bridge Between Scale Similarity and Approximate Deconvolution Models. Generalized Similarity Models 183
6.5 Mixed Modeling 183
6.5.1 Motivations 183
6.5.2 Examples of Mixed Models 185
6.6 Explicit Evaluation of Subgrid Scales 188
6.6.1 Fractal Interpolation Procedure 190
6.6.2 Chaotic Map Model 191
6.6.3 Subgrid Scale Estimation Procedure 194
6.6.4 Multilevel Simulations 196
6.7 Implicit Structural Models 198
6.7.1 L ocal Average Method 199
6.7.2 Approximate Deconvolution Procedure 201
6.7.3 Scale Residual Model 202
7. Numerical Solution: Interpretation and Problems 205
7.1 Dynamic Interpretation of the Large-Eddy Simulation 205
7.1.1 Static and Dynamic Interpretations: Effective Filter 205
7.1.2 Theoretical Analysis of the Turbulence Generated by Large-Eddy Simulation 207
7.2 Ties Between the Filter and Computational Grid. Pre-filtering 212
7.3 Numerical Errors and Subgrid Terms 214
7.3.1 Ghosal's General Analysis 214
7.3.2 Remarks on the Use of Artificial Dissipations 218
7.3.3 Remarks Concerning the Time Integration Method 220
8. Analysis and Validation of Large-Eddy Simulation Data 221
8.1 Statement of the Problem 221
8.1.1 Type of Information Contained in a Large-Eddy Simulation 221
8.1.2 Validation Methods 222
8.1.3 Statistical Equivalency Classes of Realizations 223
8.1.4 Ideal L ES and Optimal L ES 226
8.2 Correction Techniques 228
8.2.1 Filtering the Reference Data 228
8.2.2 Evaluation of Subgrid Scale Contribution 228
8.3 Practical Experience 229
9. Boundary Conditions 231
9.1 General Problem 231
9.1.1 Mathematical Aspects 231
9.1.2 Physical Aspects 231
9.2 Solid Walls 232
9.2.1 Statement of the Problem 232
9.2.2 A Few Wall Models 238
9.3 Case of the Inflow Conditions 243
9.3.1 Required Conditions 243
9.3.2 Inflow Condition Generation Techniques 244
10. Implementation 247
10.1 Filter Identification. Computing the Cutoff Length 247
10.2 Explicit Discrete Filters 249
10.2.1 Uniform One-Dimensional Grid Case 250
10.2.2 Extension to the Multidimensional Case 252
10.2.3 Extension to the General Case. Convolution Filters 253
10.2.4 High-Order Elliptic Filters 254
10.3 Implementation of the Structure Function Model 254
11. Examples of Applications 257
11.1 Homogeneous Turbulence 257
11.1.1 Isotropic Homogeneous Turbulence 257
11.1.2 Anisotropic Homogeneous Turbulence 258
11.2 Flows Possessing a Direction of Inhomogeneity 260
11.2.1 Time-Evolving Plane Channel 260
11.2.2 Other Flows 263
11.3 Flows Having at Most One Direction of Homogeneity 264
11.3.1 Round Jet 264
11.3.2 Backward Facing Step 272
11.3.3 Square-Section Cylinder 275
11.3.4 Other Examples 276
11.4 Lessons 277
11.4.1 General Lessons 277
11.4.2 Subgrid Model Efficiency 278
A. Statistical and Spectral Analysis
of Turbulence 281
A.1 Turbulence Properties 281
A.2 Foundations of the Statistical Analysis of Turbulence 281
A.2.1 Motivations 281
A.2.2 Statistical Average: Definition and Properties 282
A.2.3 Ergodicity Principle 282
A.2.4 Decomposition of a Turbulent Field 284
A.2.5 Isotropic Homogeneous Turbulence 285
A.3 Introduction to Spectral Analysis of the Isotropic Turbulent Fields 285
A.3.1 Definitions 285
A.3.2 Modal Interactions 287
A.3.3 Spectral Equations 288
A.4 Characteristic Scales of Turbulence 290
A.5 Spectral Dynamics of Isotropic Homogeneous Turbulence 291
A.5.1 Energy Cascade and Local Isotropy 291
A.5.2 Equilibrium Spectrum 291
B. EDQNM Modeling 293
B.1 Isotropic EDQNM Model 293
B.2 Cambon's Anisotropic EDQNM Model 295
Bibliography 299
Subject Index 317