Mathematics of Large Eddy Simulation of Turbulent Flows (eBook)

Preface 6
Index of Acronyms 9
Contents 10
Introduction 15
1 Introduction 16
1.1 Characteristics of Turbulence 19
1.2 What are Useful Averages? 21
1.3 Conventional Turbulence Models 27
1.4 Large Eddy Simulation 29
1.5 Problems with Boundaries 30
1.6 The Interior Closure Problem in LES 31
1.7 Eddy Viscosity Closure Models in LES 33
1.8 Closure Models Based on Systematic Approximation 35
1.9 Mixed Models 38
1.10 Numerical Validation and Testing in LES 39
2 The Navier–Stokes Equations 42
2.1 An Introduction to the NSE 42
2.2 Derivation of the NSE 45
2.3 Boundary Conditions 49
2.4 A Few Results on the Mathematics of the NSE 50
2.5 Some Remarks on the Euler Equations 75
2.6 The Stochastic Navier–Stokes Equations 78
2.7 Conclusions 81
Eddy Viscosity Models 82
3 Introduction to Eddy Viscosity Models 83
3.1 Introduction 83
3.2 Eddy Viscosity Models 84
3.3 Variations on the Smagorinsky Model 89
3.4 Mathematical Properties of the Smagorinsky Model 93
3.5 Backscatter and the Eddy Viscosity Models 114
3.6 Conclusions 115
4 Improved Eddy Viscosity Models 117
4.1 Introduction 117
4.2 The Gaussian–Laplacian Model 123
4.3 k - e Modeling 129
4.4 Conclusions 133
5 Uncertainties in Eddy Viscosity Models and Improved Estimates of Turbulent Flow Functionals 135
5.1 Introduction 135
5.2 The Sensitivity Equations of Eddy Viscosity Models 136
5.3 Improving Estimates of Functionals of Turbulent Quantities 139
5.4 Conclusions: Are u and p Enough? 142
Advanced Models 144
6 Basic Criteria for Subfilter-scale Modeling 145
6.1 Modeling the Subfilter-scale Stresses 145
6.2 Requirements for a Satisfactory Closure Model 146
7 Closure Based on Wavenumber Asymptotics 153
7.1 The Gradient (Taylor) LES Model 155
7.2 The Rational LES Model (RLES) 164
7.3 The Higher-order Subfilter-scale Model (HOSFS) 189
7.4 Conclusions 203
8 Scale Similarity Models 205
8.1 Introduction 205
8.2 Other Scale Similarity Models 210
8.3 Recent Ideas in Scale Similarity Models 211
8.4 The S4 = Skew-symmetric Scale Similarity Model 215
8.5 The First Energy-sponge Scale Similarity Model 223
8.6 The Higher Order, Stolz– Adams Deconvolution Models 229
8.7 Conclusions 233
Boundary Conditions 235
9 Filtering on Bounded Domains 236
9.1 Filters with Nonconstant Radius 238
9.2 Filters with Constant Radius 249
9.3 Conclusions 260
10 Near Wall Models in LES 262
10.1 Introduction 262
10.2 Wall Laws in Conventional Turbulence Modeling 263
10.3 Current Ideas in Near Wall Modeling for LES 265
10.4 New Perspectives in Near Wall Models 268
10.5 Conclusions 281
Numerical Tests 282
11 Variational Approximation of LES Models 283
11.1 Introduction 283
11.2 LES Models and their Variational Approximation 284
11.3 Examples of Variational Methods 289
11.4 Numerical Analysis of Variational Approximations 290
11.5 Introduction to Variational Multiscale Methods 293
11.6 Eddy Viscosity Acting on Fluctuations as a VMM 297
11.7 Conclusions 301
12 Test Problems for LES 303
12.1 General Comments 303
12.2 Turbulent Channel Flows 304
12.3 A Few Remarks on Isotropic Homogeneous Turbulence 328
12.4 Final Remarks 332
References 334
Index 352