ADVANCED METHODS IN APPLIED MATHEMATICS
CONTENTS PART I OVERVIEW 1 MAIN OBJECTIVE 3 COMPUTATIONAL TURBULENT INCOMPRESSIBLE FLOW 3 2 MYSTERIES AND SECRETS MYSTERIES SECRETS 3 TURBULENT FLOW AND HISTORY OF AVIATION LEONARDO DA VINCI, NEWTON AND D’ALEMBERT CAYLEY AND LILIENTHAL KUTTA, ZHUKOVSKY AND THE WRIGHT BROTHERS 4 THE EULER EQUATIONS FOUNDATION OF FLUID DYNAMICS DERIVATION OF THE EULER EQUATIONS THE EULER EQUATIONS AS A CONTINUUM MODEL INCOMPRESSIBLE FLOW 5 THE INCOMPRESSIBLE EULER AND NAVIER–STOKES EQUATIONS THE INCOMPRESSIBLE EULER EQUATIONS THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS WHAT IS VISCOSITY? WHAT IS HEAT CONDUCTIVITY? FRICTION BOUNDARY CONDITIONS EINSTEIN’S IDEAL EULER AND NS AS DYNAMICAL SYSTEMS 6 TRIUMPH AND FAILURE OF MATHEMATICS TRIUMPH: CELESTIAL MECHANICS FAILURE: POTENTIAL FLOW 7 LAMINAR AND TURBULENT FLOW REYNOLDS APPLICATIONS AND REYNOLDS NUMBERS 8 COMPUTATIONAL TURBULENCE ARE TURBULENT FLOWS COMPUTABLE? TYPICAL OUTPUTS: DRAG AND LIFT WHAT ABOUT BOUNDARY LAYERS? APPROXIMATE WEAK SOLUTIONS: G2 G2 ERROR CONTROL AND STABILITY WHAT ABOUT MATHEMATICS OF EULER AND NS? WHEN IS A FLOW TURBULENT? G2 VS PHYSICS COMPUTABILITY AND PREDICTABILITY 9 A FIRST STUDY OF STABILITY THE LINEARIZED EULER EQUATIONS FLOW IN A CORNER OR AT SEPARATION COUETTE FLOW RESOLUTION OF SOMMERFELD’S MYSTERY RE?ECTIONS ON STABILITY AND PERSPECTIVES 10 D’ALEMBERT’S MYSTERY AND BERNOULLI’S LAW INTRODUCTION BERNOULLI, EULER, IDEAL FLUIDS AND POTENTIAL SOLUTIONS D’ALEMBERT’S MYSTERY A VECTOR CALCULUS IDENTITY BERNOULLI’S LAW POTENTIAL FLOW AROUND A CIRCULAR CYLINDER ZERO DRAG/LIFT OF POTENTIAL FLOW IDEAL FLUIDS AND VORTICITY D’ALEMBERT’S COMPUTATION OF ZERO DRAG/LIFT A REFORMULATION OF THE MOMENTUM EQUATION 11 PRANDTL’S RESOLUTION OF D’ALEMBERT’S MYSTERY QUOTATION FROM A STANDARD SOURCE QUOTATION FROM PRANDTL’S 1904 REPORT DISCUSSION OF PRANDTL’S RESOLUTION 12 NEW RESOLUTION OF D’ALEMBERT’S MYSTERY DRAG OF A CIRCULAR CYLINDER THE ROLE OF THE BOUNDARY LAYER ANALYSIS OF INSTABILITY OF THE POTENTIAL SOLUTION SUM UP OF THE NEW RESOLUTION PART II MATHEMATICS OF TURBULENCE 13 TURBULENCE AND CHAOS WEATHER AS DETERMINISTIC CHAOS PREDICTING THE TEMPERATURE IN M?ALILLA CHAOTIC DYNAMICAL SYSTEM THE HARMONIC OSCILLATOR AS A CHAOTIC SYSTEM RANDOMNESS AND FOUNDATIONS OF PROBABILITY NS CHAOTIC RATHER THAN RANDOM OBSERVABILITY VS COMPUTABILITY LORENZ SYSTEM LORENZ, NEWTON AND FREE WILL ALGORITHMIC INFORMATION THEORY STATISTICAL MECHANICS AND ROULETTE 14 A $1 MILLION PRIZE PROBLEM THE CLAY INSTITUTE IMPOSSIBLE $1 MILLION PRIZE TOWARDS A POSSIBLE FORMULATION WELL-POSEDNESS ACCORDING TO HADAMARD ?-WEAK SOLUTIONS EXISTENCE OF ?-WEAK SOLUTIONS BY REGULARIZATION OUTPUT SENSITIVITY AND THE DUAL PROBLEM REFORMULATION OF THE PRIZE PROBLEM THE STANDARD APPROACH TO UNIQUENESS 15 WEAK UNIQUENESS BY COMPUTATION UNIQUENESS OF CD AND CL NON-UNIQUENESS OF D(T) STABILITY OF THE DUAL SOLUTION WITH RESPECT TO TIME SAMPLING CONCLUSION 16 EXISTENCE OF ?-WEAK SOLUTIONS BY G2 THE BASIC ENERGY ESTIMATE FOR THE NAVIER–STOKES EQUATIONS EXISTENCE BY G2 A POSTERIORI OUTPUT ERROR ESTIMATE FOR G2 17 STABILITY ASPECTS OF TURBULENCE IN MODEL PROBLEMS THE LINEARIZED DUAL PROBLEM ROTATING FLOW A MODEL DUAL PROBLEM FOR ROTATING FLOW A MODEL DUAL PROBLEM FOR OSCILLATING REACTION MODEL DUAL PROBLEM SUMMARY THE DUAL SOLUTION FOR BLU?BODY DRAG DUALITY FOR A MODEL PROBLEM ENSEMBLE AVERAGES AND INPUT VARIANCE 18 A CONVECTION-DI?USION MODEL PROBLEM POINTWISE VS MEAN VALUE RESIDUALS ARTI?CIAL VISCOSITY FROM LEAST SQUARES STABILIZATION 19 G2 FOR EULER EG2 AS A MODEL OF THE WORLD SOLUTION OF THE EULER EQUATIONS BY G2 DRAG OF A SQUARE CYLINDER INSTABILITY OF THE POTENTIAL SOLUTION TEMPERATURE G2 AS DISSIPATIVE WEAK SOLUTIONS COMPARISON WITH VISCOUS REGULARIZATION FINITE LIMIT OF TURBULENT DISSIPATION THE 2ND LAW OF THERMODYNAMICS A GLOBAL FORM OF THE 2ND LAW UNDERSTANDING A BASIC FACT PROOF THAT EG2 IS A DISSIPATIVE WEAK SOLUTION 20 SUMMARY OF MATHEMATICAL ASPECTS OUTPUTS OF ?-WEAK SOLUTIONS CHAOS AND TURBULENCE COMPUTATIONAL TURBULENCE IRREVERSIBILITY PART III SECRETS 21 SECRETS OF BALL SPORTS DIMPLES OF A GOLF BALL: DRAG CRISIS TOPSPIN IN TENNIS: MAGNUS E?ECT ROBERTO CARLOS: MAGNUS E?ECT PITCHING: DRAG CRISIS AND MAGNUS E?ECT 22 SECRETS OF FLIGHT GENERATION OF LIFT SIMULATION OF TAKE-O? MORE ON GENERATION OF DRAG A CRITICAL VIEW ON KUTTA-ZHUKOVSKY THE CHALLENGE 23 SECRETS OF SAILING THE SAIL THE KEEL 24 SECRETS OF RACING DOWNFORCE THE WHEELS DRAG AND FUEL CONSUMPTION PART IV COMPUTATIONAL METHOD 25 REYNOLDS STRESSES IN AND OUT INTRODUCING REYNOLDS STRESSES REMOVING REYNOLDS STRESSES 26 SMAGORINSKY VISCOSITY IN AND OUT INTRODUCING SMAGORINSKY VISCOSITY REMOVING SMAGORINSKY VISCOSITY 27 FRICTION BOUNDARY CONDITION AS WALL MODEL A SKIN FRICTION WALL MODEL 28 G2 FOR NAVIER-STOKES EQUATIONS DEVELOPMENT OF G2 THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS G2 AS EULERIAN CG(P)DG(Q) NEUMANN BOUNDARY CONDITIONS NO SLIP AND SLIP BOUNDARY CONDITIONS OUT?OW BOUNDARY CONDITIONS SHOCK CAPTURING BASIC ENERGY ESTIMATE FOR CG(P)DG(Q) G2 AS EULERIAN CG(1)DG(0) EULERIAN CG(1)CG(1) BASIC ENERGY ESTIMATE FOR CG(1)CG(1) SLIP WITH FRICTION BOUNDARY CONDITIONS 29 A DISCRETE SOLVER FIXED POINT ITERATION USING MULTIGRID/GMRES 30 G2 AS ADAPTIVE DNS/LES AN A POSTERIORI ERROR ESTIMATE PROOF OF THE A POSTERIORI ERROR ESTIMATE INTERPOLATION ERROR ESTIMATES G2 AS ADAPTIVE DNS/LES COMPUTATION OF MULTIPLE OUTPUTS MESH RE?NEMENT 31 IMPLEMENTATION OF G2 WITH FENICS THE FENICS PROJECT 32 MOVING MESHES AND ALE METHODS G2 FORMULATION FREE BOUNDARY LAPLACIAN MESH SMOOTHING MESH SMOOTHING BY LOCAL OPTIMIZATION OBJECT IN A BOX SLIDING MESH PART V FLOW FUNDAMENTALS 33 BLU?BODY FLOW DRAG AND LIFT AN ALTERNATIVE FORMULA FOR DRAG AND LIFT A POSTERIORI ERROR ESTIMATION SURFACE MOUNTED CUBE FLOW PAST A CAR SQUARE CYLINDER CIRCULAR CYLINDER SPHERE 34 BOUNDARY LAYERS FLAT PLATE LAMINAR BOUNDARY LAYER SKIN FRICTION FOR LAMINAR BOUNDARY LAYERS SKIN FRICTION FOR TURBULENT BOUNDARY LAYERS COMPUTING SKIN FRICTION BY G2 SUMMARY 35 SEPARATION SIMULATION OF BLOOD FLOW DRAG REDUCTION FOR A SQUARE CYLINDER DRAG CRISIS DRAG CRISIS FOR A CIRCULAR CYLINDER EG2 AND TURBULENT EULER SOLUTIONS THE DUAL PROBLEM FOR EG2 EG2 FOR A CIRCULAR CYLINDER THE MAGNUS E?ECT FLOW PAST AN AIRFOIL FLOW DUE TO A CYLINDER ROLLING ALONG GROUND 36 TRANSITION TO TURBULENCE MODAL AND NON-MODAL SCHOOLS DI?CULTIES OF EXPERIMENTAL TRANSITION STUDIES POSSIBILITIES OF COMPUTATIONAL TRANSITION MODAL AND NON-MODAL PERTURBATION GROWTH DI?ERENT PERTURBATIONS AND THRESHOLD LEVELS ANALYTICAL STABILITY OF THE LINEARIZED NS COMPUTATIONAL TRANSITION IN SHEAR FLOWS POISEUILLE FLOW - REYNOLDS EXPERIMENT TAYLOR-G¨ORTLER PERTURBATIONS UNSTABLE JET FLOW TEST FOR OPTIMAL PERTURBATIONS A CRITICAL REVIEW OF CLASSICAL THEORY COMPARISON WITH BIFURCATION TOWARDS STABILITY AN ODE-MODEL FOR TRANSITION A BIFURCATING ODE-MODEL PART VI LOSCHMIDT’S MYSTERY 37 THERMODYNAMICS OBJECTIVE WHAT IS THERMODYNAMICS? EG2 AS A MODEL OF THERMODYNAMICS THE CLASSICAL LAWS OF THERMODYNAMICS WHAT IS THE ROLE OF THE 2ND LAW? 38 JOULE’S 1845 EXPERIMENT THE EXPERIMENT 39 COMPRESSIBLE EULER IN 1D THE COMPRESSIBLE EULER EQUATIONS IN 1D EULER IS FORMALLY REVERSIBLE ALL WRONG THE 2ND LAW IN LOCAL FORM THE 2ND LAW IN GLOBAL FORM IRREVERSIBILITY BY THE 2ND LAW COMPRESSION AND EXPANSION 40 BURGERS’ EQUATION A MODEL OF THE EULER EQUATIONS THE RANKINE-HUGONIOT CONDITION RAREFACTION WAVE SHOCK WEAK SOLUTIONS MAY BE NON-UNIQUE THE 2ND LAW FOR BURGERS’ EQUATION DESTRUCTION OF INFORMATION 41 COMPRESSIBLE EULER IN 3D TREND TOWARDS EQUILIBRIUM BY THE 2ND LAW COMPARISON WITH CLASSICAL ENTROPY HEAT CAPACITIES AND THE GAS CONSTANT 42 EG2 FOR COMPRESSIBLE FLOW G2 FOR THE COMPRESSIBLE EULER EQUATIONS EG2 SATIS?ES THE 2ND LAW EG2 AND THE CLASSICAL ENTROPY 43 PHILOSOPHY OF EG2 DIJKSTRA’S VISION THE ROLE OF LEAST SQUARES STABILIZATION IN G2 ASPECTS OF IRREVERSIBILITY IMPERFECT NATURE AND MATHEMATICS? A NEW PARADIGM OF COMPUTATION THE CLAY PRIZE PROBLEM AGAIN 44 DOES GOD REALLY PLAY DICE? EINSTEIN AND MODERN PHYSICS BOLTZMANN AND STATISTICAL MECHANICS