NAVIER-STOKES EQUATIONS AND TURBULENCE

By PATRICIA WILSON

CONTENTS CHAPTER I INTRODUCTION AND OVERVIEW OF TURBULENCE INTRODUCTION 1 VISCOUS FLUIDS THE NAVIER–STOKES EQUATIONS 2 TURBULENCE: WHERE THE INTERESTS OF ENGINEERS AND MATHEMATICIANS OVERLAP 3 ELEMENTS OF THE THEORIES OF TURBULENCE OF KOLMOGOROV AND KRAICHNAN 4 FUNCTION SPACES, FUNCTIONAL INEQUALITIES, AND DIMENSIONAL ANALYSIS CHAPTER II ELEMENTS OF THE MATHEMATICAL THEORY OF THE NAVIER–STOKES EQUATIONS 1 ENERGY AND ENSTROPHY 2 BOUNDARY VALUE PROBLEMS 3 HELMHOLTZ–LERAY DECOMPOSITION OF VECTOR FIELDS 4 WEAK FORMULATION OF THE NAVIER–STOKES EQUATIONS 5 FUNCTION SPACES 6 THE STOKES OPERATOR 7 EXISTENCE AND UNIQUENESS OF SOLUTIONS: THE MAIN RESULTS 8 ANALYTICITY IN TIME 9 GEVREY CLASS REGULARITY AND THE DECAY OF THE FOURIER COEF?CIENTS 10 FUNCTION SPACES FOR THE WHOLE-SPACE CASE 11 THE NO-SLIP CASE WITH MOVING BOUNDARIES 12 DISSIPATION RATE OF FLOWS 13 NONDIMENSIONAL ESTIMATES AND THE GRASHOF NUMBER APPENDIX A MATHEMATICAL COMPLEMENTS APPENDIX B PROOFS OF TECHNICAL RESULTS IN CHAPTER II CHAPTER III FINITE DIMENSIONALITY OF FLOWS 1 DETERMINING MODES 2 DETERMINING NODES 3 ATTRACTORS AND THEIR FRACTAL DIMENSION 4 APPROXIMATE INERTIAL MANIFOLDS APPENDIX A PROOFS OF TECHNICAL RESULTS IN CHAPTER III CHAPTER IV STATIONARY STATISTICAL SOLUTIONS OF THE NAVIER–STOKES EQUATIONS, TIME AVERAGES, AND ATTRACTORS 1 MATHEMATICAL FRAMEWORK, DE?NITION OF STATIONARY STATISTICAL SOLUTIONS, AND BANACH GENERALIZED LIMITS 2 INVARIANT MEASURES AND STATIONARY STATISTICAL SOLUTIONS IN DIMENSION 3 STATIONARY STATISTICAL SOLUTIONS IN DIMENSION 4 ATTRACTORS AND STATIONARY STATISTICAL SOLUTIONS 5 AVERAGE TRANSFER OF ENERGY AND THE CASCADES IN TURBULENT FLOWS APPENDIX A NEW CONCEPTS AND RESULTS USED IN CHAPTER IV APPENDIX B PROOFS OF TECHNICAL RESULTS IN CHAPTER IV APPENDIX C A MATHEMATICAL COMPLEMENT: THE ACCRETIVITY PROPERTY IN DIMENSION CHAPTER V TIME-DEPENDENT STATISTICAL SOLUTIONS OF THE NAVIER–STOKES EQUATIONS AND FULLY DEVELOPED TURBULENCE 1 TIME-DEPENDENT STATISTICAL SOLUTIONS ON BOUNDED DOMAINS 2 HOMOGENEOUS STATISTICAL SOLUTIONS 3 REYNOLDS EQUATION FOR THE AVERAGE FLOW 4 SELF-SIMILAR HOMOGENEOUS STATISTICAL SOLUTIONS 5 RELATION WITH AND APPLICATION TO THE CONVENTIONAL THEORY OF TURBULENCE 6 SOME CONCLUDING REMARKS APPENDIX A PROOFS OF TECHNICAL RESULTS IN CHAPTER V REFERENCES I INTRODUCTION AND OVERVIEW OF TURBULENCE INTRODUCTION IN THIS CHAPTER WE ?RST BRIE?Y RECALL, IN SECTION 1, THE DERIVATION OF THE NAVIER–STOKES EQUATIONS (NSE) STARTING FROM THE BASIC CONSERVATION PRINCIPLES IN MECHANICS: CON- SERVATION OF MASS AND MOMENTUM SECTION 2 CONTAINS SOME GENERAL REMARKS ON TURBU- LENCE, AND IT ALLUDES TO SOME DEVELOPMENTS NOT PRESENTED IN THE BOOK FOR THE BENE?T OF THE MATHEMATICALLY ORIENTED READER (AND PERHAPS OTHERS), SECTION 3 PROVIDES A FAIRLY DETAILED ACCOUNT OF THE KOLMOGOROV THEORY OF TURBULENCE, WHICH UNDERLIES MANY PARTS OF CHAPTERS III–V FOR THE PHYSICS-ORIENTED READER, SECTION 4 GIVES AN INTUITIVE INTRO- DUCTION TO THE MATHEMATICAL PERSPECTIVE AND THE NECESSARY TOOLS A MORE RIGOROUS PRESENTATION APPEARS IN THE ?RST HALF OF CHAPTER II AND THEREAFTER AS NEEDED FOR EACH OF THE ASPECTS THAT WE DEVELOP, THE PRESENT CHAPTER SHOULD PROVE MORE USEFUL FOR THE NONSPECIALIST THAN FOR THE SPECIALIST 1 VISCOUS FLUIDS THE NAVIER–STOKES EQUATIONS FLUIDS OBEY THE GENERAL LAWS OF CONTINUUM MECHANICS: CONSERVATION OF MASS, ENERGY, AND LINEAR MOMENTUM THEY CAN BE WRITTEN AS MATHEMATICAL EQUATIONS ONCE A REPRE- SENTATION FOR THE STATE OF A ?UID IS CHOSEN IN THE CONTEXT OF MATHEMATICS, THERE ARE TWO CLASSICAL REPRESENTATIONS ONE IS THE SO-CALLED LAGRANGIAN REPRESENTATION, WHERE THE STATE OF A ?UID “PARTICLE” AT A GIVEN TIME IS DESCRIBED WITH REFERENCE TO ITS INI- TIAL POSITION THE OTHER REPRESENTATION (ADOPTED THROUGHOUT THIS BOOK) IS THE SO-CALLED EULERIAN REPRESENTATION, WHERE AT EACH TIME T AND POSITION X IN SPACE THE STATE – IN PAR- TICULAR, THE VELOCITY U(X, T) – OF THE ?UID “PARTICLE” AT THAT POSITION AND TIME IS GIVEN IN THE EULERIAN REPRESENTATION OF THE ?OW, WE ALSO REPRESENT THE DENSITY ?(X, T) AS A FUNCTION OF THE POSITION X AND TIME T THE CONSERVATION OF MASS IS EXPRESSED BY THE CONTINUITY EQUATION ?? ?T + DIV(?U) = 0 (1 1) THE CONSERVATION OF MOMENTUM IS EXPRESSED IN TERMS OF THE ACCELERATION ? AND THE CAUCHY STRESS TENSOR ?: ??I = ??IJ ?XJ + FI, I = 1, 2, 3 (1 2) I INTRODUCTION AND OVERVIEW OF TURBULENCE HERE ? = (?1, ?2, ?3) AND ? = (?IJ)I,J=1,2,3, COMPONENTWISE IN THE 3-DIMENSIONAL CASE MOREOVER, F = (FI, F2, F3) REPRESENTS VOLUME FORCES APPLIED TO THE ?UID THE ACCELERATION VECTOR ? = ? (X, T) OF THE ?UID AT POSITION X AND TIME T CAN BE EXPRESSED, USING PURELY KINEMATIC ARGUMENTS, BY THE SO-CALLED MATERIAL DERIVATIVE ? = DU DT = ?U ?T + (U ?)U, (1 3) OR, COMPONENTWISE, ?I = ?UI ?T +