1.Introduction
1.1 Computational Fluid Dynamics
1.2 Levels of Approximation: General
1.3 Statement of the Scale Separation Problem
1.4 Usual Levels of Approximation
1.5 Large-Eddy Simulation: from Practice to Theory.Structure of the Book
2.Formal Introduction to Scale Separation:Band-Pass Filtering
2.1 Definition and Properties of the Filter in the Homogeneous Case
2.1.1 Definition
2.1.2 Fundamental Properties
2.1.3 Characterization of Different Approximations
2.1.4 Differential Filters
2.1.5 Three Classical Filters for Large-Eddy Simulation
2.1.6 Differential Interpretation of the Filters
2.2 Spatial Filtering: Extension to the Inhomogeneous Case
2.2.1 General
2.2.2 Non-uniform Filtering Over an Arbitrary Domain
2.2.3 Local Spectrum of Commutation Error
2.3 Time Filtering: a Few Properties
3.Application to Navier-Stokes Equations
3.1 Navier-Stokes Equations
3.1.1 Formulation in Physical Space
3.1.2 Formulation in General Coordinates
3.1.3 Formulation in Spectral Space
3.2 Filtered Navier-Stokes Equations in Cartesian Coordinate
(Homogeneous Case)
3.2.1 Formulation in Physical Space
3.2.2 Formulation in Spectral Space
3.3 Decomposition of the Non-linear Term. Associated Equations for the Conventional Approach
3.3.1 Leonard's Decomposition
3.3.2 Germano Consistent Decomposition
3.3.3 Germano Identity
3.3.4 Invariance Properties
3.3.5 Realizability Conditions
3.4 Extension to the Inhomogeneous Case for the Conventional Approach
3.4.1 Second-Order Commuting Filter
3.4.2 High-Order Commuting Filters
3.5 Filtered Navier-Stokes Equations in General Coordinates
3.5.1 Basic Form of the Filtered Equations
3.5.2 Simplified Form of the Equations
Non-linear Terms Decomposition
3.6 Closure Problem
3.6.1 Statement of the Problem
3.6.2 Postulates
3.6.3 Functional and Structural Modeling
4.Other Mathematical Models for the Large-Eddy Simulation Problem
4.1 Ensemble-Averaged Models
4.1.1 Yoshizawa's Partial Statistical Average Model
4.1.2 McComb's Conditional Mode Elimination Procedure
4.2 Regularized Navier-Stokes Models
4.2.1 Leray's Model
4.2.2 Holm's Navier-Stokes-α Model
4.2.3 Ladyzenskaja's Model
5.Functional Modeling (Isotropic Case)
5.1 Phenomenology of Inter-Scale Interactions
5.1.1 Local Isotropy Assumption: Consequences
5.1.2 Interactions Between Resolved and Subgrid Scales
5.1.3 A View in Physical Space
5.1.4 Summary
5.2 Basic Functional Modeling Hypothesis
5.3 Modeling of the Forward Energy Cascade Process
5.3.1 Spectral Models
5.3.2 Physical Space Models
5.3.3 Improvement of Models in the Physical Space
5.3.4 Implicit Diffusion: the ILES Concept
5.4 Modeling the Backward Energy Cascade Process
5.4.1 Preliminary Remarks
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