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E-grāmata: Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis

William J. Layton, Leo G. Rebholz

Formāts: PDF+DRM
Sērija : Lecture Notes in Mathematics 2042
Izdošanas datums: 06-Jan-2012
Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Valoda: eng
ISBN-13: 9783642244094

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Formāts: PDF+DRM
Sērija : Lecture Notes in Mathematics 2042
Izdošanas datums: 06-Jan-2012
Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Valoda: eng
ISBN-13: 9783642244094

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This volume presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems. The resulting Approximate Deconvolution Models or ADMs have some advantages over more commonly used turbulence models – as well as some disadvantages. Our goal in this book is to provide a clear and complete mathematical development of ADMs, while pointing out the difficulties that remain. In order to do so, we present the analytical theory of ADMs, along with its connections, motivations and complements in the phenomenology of and algorithms for ADMs.

Recenzijas

From the reviews:

This monograph presents a mathematical approach to turbulence modeling and is aimed at graduate students and researchers in the field of computational fluid dynamics. The book presents the governing Navier-Stokes equations and the basics of large eddy simulation from a mathematical perspective without going into details of the flow physics. Difficulties of the models in the vicinity of walls with no-slip boundary conditions are mentioned, and numerical examples of different flows illustrate some properties of the approximate deconvolution models. (Kai Schneider, Zentralblatt MATH, Vol. 1241, 2012)

1 Introduction

(34)

1.1 The Navier--Stokes Equations

(10)

1.1.1 Integral Invariants

(2)

1.1.2 The K41 Theory of Homogeneous, Isotropic Turbulence

(6)

1.2 Large Eddy Simulation

(1)

1.3 Eddy Viscosity Closures

(3)

1.4 Closure by van Cittert Approximate Deconvolution

(8)

1.4.1 The Bardina Model

(1)

1.4.2 The Accuracy of van Cittert Deconvolution

(4)

1.5 Approximate Deconvolution Regularizations

(4)

1.5.1 Time Relaxation

(2)

1.5.2 The Leray-Deconvolution Regularization

(1)

1.5.3 The NS-Alpha Regularization

(1)

1.5.4 The NS-Omega Regularization

(1)

1.6 The Problem of Boundary Conditions

(3)

1.6.1 The Commutator Error

(1)

1.6.2 Near Wall Modeling

(1)

1.6.3 Changing the Averaging Operator to a Differential Filter

(1)

1.6.4 Ad Hoc Corrections and Regularization Models

(1)

1.6.5 Near Wall Resolution

(1)

1.7 Ten Open Problems in the Analysis of ADMs

(3)

2 Large Eddy Simulation

(26)

2.1 The Idea of Large Eddy Simulation

(2)

2.1.1 Differing Dynamics of the Large and Small Eddies

(1)

2.1.2 The Eddy-Viscosity Hypothesis/Boussinesq Assumption

(1)

2.2 Local Spacial Averages

(9)

2.2.1 Top Hat Filter

(1)

2.2.2 Discrete Filters

(1)

2.2.3 Weighted Discrete Filters

(1)

2.2.4 Other Filters

(1)

2.2.5 Weighted Compact Discrete Filter from [ SAK01a]

(1)

2.2.6 Differential Filters

(3)

2.2.7 Scale Space: What Is the Right Averaging?

(1)

2.3 The SFNSE

(2)

2.4 Eddy Viscosity Models

(3)

2.4.1 A First Choice of vT

(1)

2.5 The Smagorinsky Model

(3)

2.6 Some Smagorinsky Variants

(4)

2.6.1 Using the Q-Criterion

(1)

2.6.2 A Multiscale Turbulent Diffusion Coefficient

(1)

2.6.3 Localization of Eddy Viscosity in Scale Space

(1)

2.6.4 Vreman's Eddy Viscosity

(1)

2.7 A Glimpse into Near Wall Models

(1)

2.8 Remarks

(2)

3 Approximate Deconvolution Operators and Models

(28)

3.1 Useful Deconvolution Operators

(4)

3.1.1 Approximate Deconvolution

(2)

3.2 LES Approximate Deconvolution Models

(1)

3.3 Examples of Approximate Deconvolution Operators

(4)

3.3.1 Tikhonov Regularization

(1)

3.3.2 Tikhonov-Lavrentiev Regularization

(1)

3.3.3 A Variant on Tikhonov-Lavrentiev Regularization

(1)

3.3.4 The van Cittert Regularization

(1)

3.3.5 van Cittert with Relaxation Parameters

(1)

3.3.6 Other Approximate Deconvolution Methods

(1)

3.4 Analysis of van Cittert Deconvolution

(5)

3.4.1 Proof

(1)

3.5 Discrete Differential Filters

(3)

3.6 Reversibility of Approximate Deconvolution Models

(1)

3.7 The Zeroth Order Model

(9)

3.7.1 Proof

(6)

3.8 Remarks

(2)

4 Phenomenology of ADMs

(10)

4.1 Basic Properties of ADMs

(2)

4.2 The ADM Energy Cascade

(4)

4.2.1 Another Approach to the ADM Energy Spectrum

(1)

4.2.2 The ADM Helicity Cascade

(1)

4.3 The ADM Micro-Scale

(2)

4.3.1 Design of an Experimental Test of the Model's Energy Cascade

(1)

4.4 Remarks

(2)

5 Time Relaxation Truncates Scales

(22)

5.1 Time Relaxation

(3)

5.2 The Microscale of Linear Time Relaxation

102

(8)

5.2.1 Case 1: Fully Resolved

109

(1)

5.2.2 Case 2: Under Resolved

109

(1)

5.2.3 Case 3: Perfect Resolution

110

(1)

5.3 Time Relaxation Does Not Alter Shock Speeds

110

(2)

5.4 Nonlinear Time Relaxation

112

(2)

5.4.1 Open Question 1: Does Nonlinear Time Relaxation Dissipate Energy in All Cases?

113

(1)

5.4.2 Open Question 2: If not, What Is the Simplest Modification to Nonlinear Time Relaxation that Always Dissipates Energy?

113

(1)

5.4.3 Open Question 3: How Is Nonlinear Time Relaxation to be Discretized in Time so as to be Unconditionally Stable and Require Filtering Only of Known Functions?

113

(1)

5.5 Analysis of a Nonlinear Time Relaxation

114

(6)

5.5.1 Open Question 4: Is the Extra (I -- G) Necessary to Ensure Energy Dissipation or Just a Mathematical Convenience?

115

(1)

5.5.2 Open Question 5: If the Extra (I -- G) Is Necessary, How Is it to be Discretized in Time?

115

(2)

5.5.3 The N = 0 Case

117

(1)

5.5.4 The Analysis of Lilly

118

(2)

5.5.5 Open Question 8: Is It Possible to Extend the Above Calculation of the Optimal Relaxation Parameter to the Original Version of Nonlinear Time Relaxation?

120

(1)

5.6 Remarks

120

(1)

6 The Leray-Deconvolution Regularization

121

(24)

6.1 The Leray Regularization

121

(3)

6.2 Dunca's Leray-Deconvolution Regularization

124

(1)

6.3 Analysis of the Leray-Deconvolution Regularization

125

(5)

6.3.1 Existence of Solutions

125

(1)

6.3.2 Proof

126

(3)

6.3.3 Limits of the Leray-Deconvolution Regularization

129

(1)

6.4 Accuracy of the Leray-Deconvolution Family

130

(4)

6.4.1 The Case of Homogeneous, Isotropic Turbulence

131

(3)

6.5 Microscales

134

(2)

6.5.1 Case 1

135

(1)

6.5.2 Case 2

136

(1)

6.6 Discretization

136

(2)

6.7 Numerical Experiments with Leray-Deconvolution

138

(6)

6.7.1 Convergence Rate Verification

138

(1)

6.7.2 Two-Dimensional Channel Flow Over a Step

139

(3)

6.7.3 Three-Dimensional Channel Flow Over a Step

142

(2)

6.8 Remarks

144

(1)

7 NS-Alpha- and NS-Omega-Deconvolution Regularizations

145

(18)

7.1 Integral Invariants of the NSE

145

(2)

7.2 The NS-Alpha Regularization

147

(4)

7.2.1 The Periodic Case

147

(3)

7.2.2 Discretizations of the NS-alpha Regularization

150

(1)

7.3 The NS-Omega Regularization

151

(3)

7.3.1 Motivation for NS-ω: The Challenges of Time Discretization

153

(1)

7.4 Computational Problems with Rotation Form

154

(3)

7.5 Numerical Experiments with NS-α

157

(1)

7.5.1 Two-Dimensional Flow Over a Step

157

(1)

7.5.2 Three-Dimensional Flow Over a Step

157

(1)

7.6 Model Synthesis

158

(3)

7.6.1 Synthesis of NS-α and ω Models

159

(1)

7.6.2 Scale Truncation, Eddy Viscosity, VMMs and Time Relaxation

160

(1)

7.7 Remarks

161

(2)

A Deconvolution Under the No-Slip Condition and the Loss of Regularity

163

(12)

A.1 Regularity by Direct Estimation of Derivatives

164

(3)

A.2 The Bootstrap Argument

167

(3)

A.2.1 The Case k = 3

167

(1)

A.2.2 Observation

167

(3)

A.3 Examples

170

(2)

A.4 Application to Differential Filters

172

(1)

A.5 Remarks

173

(2)

References

175

(8)

Index

183

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E-grāmata: Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis

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