Dynamic Multilevel Methods and the Numerical Simulation of Turbulence

by Thierry Dubois , François Jauberteau , Roger Temam

• ISBN13: 9780521621656
• ISBN10: 0521621658
• Format: Hardcover
• Copyright: 1999-01-13
• Publisher: Cambridge University Press

Summary

This book describes the implementation of multilevel methods for the numerical simulation of turbulent flows. The general ideas for the algorithms presented stem from dynamical systems theory and are based on the decomposition of the unknown function into two or more arrays corresponding to different scales in the Fourier space. Before describing in detail the numerical algorithm, survey chapters on the mathematical theory of the Navier-Stokes equations and on the physics of the conventional theory of turbulence are included. The multilevel methods are applied here to the simulation of homogeneous isotropic turbulent flows as well as turbulent channel flows. The implementation issues are discussed in detail and numerical simulations of the flows cited above are presented and analyzed. The methods have been applied in the context of the direct numerical simulation and are therefore compared to such simulations.

Table of Contents

Preface	ix
Acknowledgments	xi
Introduction	xiii
The Navier-Stokes Equations and Their Mathematical Background	1	(16)
The Equations	1	(4)
Boundary Value Problems	5	(3)
The Functional Setting	8	(4)
The Main Results on Existence and Uniqueness of Solutions	12	(5)
The Physics of Turbulent Flows	17	(48)
Some Probabilistic Tools	17	(5)
An Idealized Model of Turbulent Flows: Homogenous (Isotropic) Turbulence	22	(35)
Two-Point Correlation Tensors and Their Spectral Representation	22	(11)
Homogeneous Isotropic Turbulence	33	(10)
Dynamical Equations for the Correlation Tensor and the Energy Spectrum Function	43	(4)
The Universal Theory of Equilibrium	47	(7)
The Statistical Properties of Homogeneous Turbulent Flows	54	(3)
Nonhomogeneous Turbulence: The Channel Flow Problem	57	(8)
Computational Methods for the Direct Simulation of Turbulence	65	(32)
The Fully Periodic Case: Homogeneous Turbulence	65	(7)
Flows in an Infinite Channel: Nonhomogenous Turbulence	72	(25)
Preliminary	72	(4)
Fourier-Galerkin Approximations and Time Discretizations	76	(5)
Methods Based on Pressure Solvers	81	(3)
Spectral Approximations of the Velocity Formulation	84	(7)
A Legendre--Galerkin Approximation of the NSE	91	(6)
Direct Numerical Simulation versus Turbulence Modeling	97	(12)
The Practical Limits of DNS	97	(5)
Homogeneous Turbulent Flows	97	(3)
Turbulent Channel Flows	100	(2)
A Different Approach: Turbulence Modeling	102	(7)
The Reynolds-Averaged Equations and the Closure Problem	102	(2)
The Large-Eddy Simulation	104	(3)
Spectral and Statistical Models	107	(2)
Long-Time Behavior. Attractors and Their Approximation	109	(6)
Attractors: Existence and Dimension	110	(2)
Intertial Manifolds and Approximate Intertial Manifolds	112	(3)
Separation of Scales in Turbulence	115	(19)
Scale Decomposition	115	(4)
Filter Functions	115	(2)
Projective Filters Based on Galerkin Approximations	117	(2)
The Separation of Scales	119	(15)
The Fully Periodic Case	119	(2)
Wall-Bounded Flows	121	(13)
Numerical Analysis of Multilevel Methods	134	(12)
A Simple Model	134	(1)
Two-Level Discretization Schemes	135	(5)
Multilevel Discretization Schemes	140	(6)
Dynamic Multilevel Methodologies	146	(41)
Behavior of the Small and Large Scales	147	(19)
The Homogeneous Case	147	(10)
The Nonhomogeneous Case	157	(9)
Multilevel Schemes	166	(21)
Periodic Flows	166	(14)
Wall-Bounded Flows	180	(7)
Computational Implementation of the Dynamic Multilevel Methods	187	(32)
General Description of the DML Methods	187	(4)
Survey of Related DML Methods	191	(2)
Dynamic Control of the Numerical Parameters	193	(5)
Synthesis of the Algorithm	198	(4)
The Channel Flows Problem	202	(17)
The DML Method Applied in the Homogeneous Directions	202	(12)
The DML Method Applied in the Nonhomogenous Direction	214	(5)
Numerical Results	219	(54)
Two-Dimensional Homogeneous Turbulent Flows	219	(8)
The Computational Parameters	219	(3)
Description and Analysis of the Numerical Results	222	(5)
Three-Dimensional Homogeneous Turbulent Flows	227	(28)
The Computational Parameters	229	(5)
Comparison and Analysis of the DNS and DML Simulations	234	(21)
Turbulent Channel Flows	255	(18)
Conclusion	273	(4)
References	277	(8)
Index	285