Large Eddy Simulation of Turbulent Incompressible Flows

1 Introduction.- 1.1 Short Remarks on the Nature and Importance of Turbulent Flows.- 1.2 Remarks on the Direct Numerical Simulation (DNS) and the k — ? Model.- 1.3 Large Eddy Simulation (LES).- 1.4 Contents of this Monograph.- 2 Mathematical Tools and Basic Notations.- 2.1 Function Spaces.- 2.2 Some Tools from Analysis and Functional Analysis.- 2.3 Convolution and Fourier Transform.- 2.4 Notations for Matrix-Vector Operations.- 3 The Space Averaged Navier-Stokes Equations and the Commutation Error.- 3.1 The Incompressible Navier-Stokes Equations.- 3.2 The Space Averaged Navier-Stokes Equations in the Case ? = ?d.- 3.3 The Space Averaged Navier-Stokes Equations in a Bounded Domain.- 3.4 The Gaussian Filter.- 3.5 Error Estimate of the Commutation Error Term in the Lp (?d) Norm.- 3.6 Error Estimate of the Commutation Error Term in the H-1 (?) Norm.- 3.7 Error Estimate for a Weak Form of the Commutation Error.- 4 LES Models Which are Based on Approximations in Wave Number Space.- 4.1 Eddy Viscosity Models.- 4.1.1 The Smagorinsky Model.- 4.1.2 The Dynamic Subgrid Scale Model.- 4.2 Modelling of the Large Scale and Cross Terms.- 4.2.1 The Taylor LES Model.- 4.2.2 The Second Order Rational LES Model.- 4.2.3 The Fourth Order Rational LES Model.- 4.3 Models for the Subgrid Scale Term.- 4.3.1 The Second Order Fourier Transform Approach.- 4.3.2 The Fourth Order Rational LES Model.- 4.3.3 The Smagorinsky Model.- 4.3.4 Models Proposed by Iliescu and Layton.- 5 The Variational Formulation of the LES Models.- 5.1 The Weak Formulation of the Equations.- 5.2 Boundary Conditions for the LES Models.- 5.2.1 Dirichlet Boundary Condition.- 5.2.2 Outflow or Do-Nothing Boundary Condition.- 5.2.3 Free Slip Boundary Condition.- 5.2.4 Slip With Linear Friction and No Penetration Boundary Condition.- 5.2.5 Slip With Linear Friction and Penetration With Resistance Boundary Condition.- 5.2.6 Periodic Boundary Condition.- 5.3 Function Spaces for the LES Models.- 6 Existence and Uniqueness of Solutions of the LES Models.- 6.1 The Smagorinsky Model.- 6.1.1 A priori error estimates.- 6.1.2 The Galerkin Method.- 6.2 The Taylor LES Model.- 6.3 The Rational LES Model.- 7 Discretisation of the LES Models.- 7.1 Discretisation in Time by the Crank-Nicolson or the Fractional-Step ?-Scheme.- 7.2 The Variational Formulation and the Linearisation of the Time-Discrete Problem.- 7.3 The Discretisation in Space.- 7.4 Inf-Sup Stable Pairs of Finite Element Spaces.- 7.5 The Upwind Stabilisation for Lowest Order Non-Conforming Finite Elements.- 7.6 The Implementation of the Slip With Friction and Penetration With Resistance Boundary Condition.- 7.7 The Discretisation of the Auxiliary Problem in the Rational LES Model.- 7.8 The Computation of the Convolution in the Rational LES Model.- 7.9 The Evaluation of Integrals, Numerical Quadrature.- 8 Error Analysis of Finite Element Discretisations of the LES Models.- 8.1 The Smagorinsky Model.- 8.1.1 The Variational Formulation and Stability Estimates.- 8.1.2 Goal of the Error Analysis and Outline of the Proof.- 8.1.3 The Error Equation.- 8.1.4 The Case ?w ? L3 (0, T; L3 (?)) and a0 (?).- 8.1.5 The Case ?w ? L3 (0, T; L3 (?)) and a0 (?).- 8.1.6 The Case ?w ? L2 (0, T; L? (?)) and a0 (?).- 8.1.7 Failures of the Present Analysis in Other Interesting Cases.- 8.1.8 A Numerical Example.- 8.2 The Taylor LES Model.- 9 The Solution of the Linear Systems.- 9.1 The Fixed Point Iteration for the Solution of Linear Systems.- 9.2 Flexible GMRES (FGMRES) With Restart.- 9.3 The Coupled Multigrid Method.- 9.3.1 The Transfer Between the Levels of the Multigrid Hierarchy.- 9.3.2 The Vanka Smoothers.- 9.3.3 The Standard Multigrid Method and the Multiple Discretisation Multilevel Method.- 9.3.4 Schematic Overview and Parameters.- 9.4 The Solution of the Auxiliary Problem in the Rational LES Model.- 10 A Numerical Study of a Necessary Condition for the Acceptability of LES Models.- 10.1 The Flow Through a Channel.- 10.2 The Failure of the Taylor LES Model.- 10.3 The Rational LES Model.- 10.3.1 Computations With the Smagorinsky Subgrid Scale Model.- 10.3.2 Computations With the Iliescu-Layton Subgrid Scale Model.- 10.3.3 Computations Without Model for the Subgrid Scale Term.- 10.4 Summary.- 11 A Numerical Study of the Approximation of Space Averaged Flow Fields by the Considered LES Models.- 11.1 A Mixing Layer Problem in Two Dimensions.- 11.1.1 The Definition of the Problem and the Setup of the Numerical Tests.- 11.1.2 The Smagorinsky Model and the Rational LES Model With Smagorinsky Subgrid Scale Term.- 11.1.3 The Rational LES Model With Iliescu-Layton Subgrid Scale Term.- 11.1.4 The Rational LES Model Without Model for the Subgrid Scale Term.- 11.1.5 A Comparison of the Smagorinsky Subgrid Scale Term and the Iliescu-Layton Subgrid Scale Term.- 11.2 A Mixing Layer Problem in Three Dimensions.- 11.2.1 The Definition of the Problem and the Setup of the Numerical Tests.- 11.2.2 Evaluation of the Numerical Results.- 12 Problems for Further Investigations.- 13 Notations.- References.