ASYMPTOTIC ANALYSIS AND BOUNDARY LAYERS
CONTENTS INTRODUCTION INTRODUCTION TO SINGULAR PERTURBATION PROBLEMS REGULAR AND SINGULAR PROBLEMS LINEAR OSCILLATOR SECULAR PROBLEM SINGULAR PROBLEM APPROXIMATION METHODS FOR SINGULAR PERTURBATION PROBLEMS METHOD OF MATCHED ASYMPTOTIC EXPANSIONS SUCCESSIVE COMPLEMENTARY EXPANSION METHOD MULTIPLE SCALE METHOD POINCARɖLIGHTHILL’S METHOD RENORMALIZATION GROUP METHOD CONCLUSION PROBLEMS BOUNDARY LAYER STRUCTURE STUDY OF A SECOND ORDER DI?ERENTIAL EQUATION ANALYSIS OF EACH CASE ASYMPTOTIC EXPANSIONS ORDER FUNCTIONS ORDER OF A FUNCTION DE?NITION OF AN ORDER FUNCTION COMPARISON OF ORDER FUNCTIONS TOTAL ORDERING ORDER OF A FUNCTION ASYMPTOTIC SEQUENCE DE?NITION OF AN ASYMPTOTIC SEQUENCE CLASS OF EQUIVALENCE GAUGE FUNCTIONS ASYMPTOTIC EXPANSION ASYMPTOTIC APPROXIMATION REGULAR FUNCTIONS REGULAR AND GENERALIZED ASYMPTOTIC EXPANSIONS CONVERGENCE AND ACCURACY OPERATIONS ON ASYMPTOTIC EXPANSIONS EXPANSION OPERATOR OUTER EXPANSION - INNER EXPANSION ASYMPTOTIC MATCHING BOUNDARY LAYER EXPANSION OPERATOR TO A GIVEN ORDER SIGNI?CANT APPROXIMATIONS INTERMEDIATE MATCHING KAPLUN’S EXTENSION THEOREM STUDY OF EXAMPLES RULE OF INTERMEDIATE MATCHING ASYMPTOTIC MATCHING PRINCIPLE VAN DYKE’S PRINCIPLE MODI?ED VAN DYKE’S PRINCIPLE EXAMPLES AND COUNTER-EXAMPLES EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 DISCUSSION OF THE MATCHING PRINCIPLE CORRECTIVE BOUNDARY LAYER THE MVDP FROM THE OVERLAP HYPOTHESIS PRINCIPLE EQUIVALENCE OF MVDP AND OF REGULAR SCEM APPLICATIONS OF SCEM ORDINARY DI?ERENTIAL EQUATIONS APPLICATION OF MMAE APPLICATION OF SCEM IDENTI?CATION WITH MMAE RESULTS NUMERICAL RESULTS STOKES-OSEEN’S FLOW MODEL TERRIBLE PROBLEM HIGH REYNOLDS NUMBER FLOWS BOUNDARY LAYER THEORIES PRANDTL’S BOUNDARY LAYER TRIPLE DECK ANALYSIS OF AN INTEGRAL METHOD INTEGRAL METHOD DIRECT MODE INVERSE MODE SIMULTANEOUS MODE VISCOUS-INVISCID INTERACTION INTERACTIVE BOUNDARY LAYER OUTER APPROXIMATION DETERMINATION OF A UNIFORMLY VALID APPROXIMATION GAUGE FOR THE PRESSURE FIRST ORDER INTERACTIVE BOUNDARY LAYER GENERALIZED BOUNDARY LAYER EQUATIONS BOUNDARY CONDITIONS ESTIMATE OF THE REMAINDERS OF EQUATIONS SECOND ORDER INTERACTIVE BOUNDARY LAYER DISPLACEMENT E?ECT REDUCED MODEL FOR AN IRROTATIONAL EXTERNAL FLOW APPLICATIONS OF INTERACTIVE BOUNDARY LAYER MODELS CALCULATION OF A FLOW WITH SEPARATION DE?NITION OF THE FLOW NUMERICAL METHOD RESULTS APPLICATION TO AERODYNAMIC FLOWS FLAT PLATE OF FINITE LENGTH AIRFOILS AT HIGH REYNOLDS NUMBERS IN?UENCE OF A ROTATIONAL EXTERNAL FLOW INVISCID FLOW METHOD OF RESOLUTION FLOWS STUDIED REGULAR FORMS OF INTERACTIVE BOUNDARY LAYER 10 1 SECOND ORDER BOUNDARY LAYER MODEL 10 1 1 SECOND ORDER INTERACTIVE BOUNDARY LAYER MODEL 10 1 2 VAN DYKE’S SECOND ORDER MODEL 10 2 TRIPLE DECK MODEL 10 2 1 FLOW ON A FLAT PLATE WITH A SMALL HUMP 10 2 2 REGULAR EXPANSIONS 10 3 SUMMARY OF APPROXIMATIONS OF NAVIER-STOKES EQUATIONS 10 4 CONCLUSION TURBULENT BOUNDARY LAYER 11 1 RESULTS OF THE STANDARD ASYMPTOTIC ANALYSIS 11 1 1 AVERAGED NAVIER-STOKES EQUATIONS 11 1 2 SCALES 11 1 3 STRUCTURE OF THE FLOW 11 2 APPLICATION OF SCEM 11 2 1 FIRST APPROXIMATION 11 2 2 CONTRIBUTION OF THE OUTER REGION OF THE BOUNDARY LAYER 11 2 3 CONTRIBUTION OF THE INNER REGION OF THE BOUNDARY LAYER 11 3 INTERACTIVE BOUNDARY LAYER 11 3 1 FIRST ORDER MODEL 11 3 2 SECOND ORDER MODEL 11 3 3 GLOBAL MODEL 11 3 4 REDUCED MODEL FOR AN IRROTATIONAL EXTERNAL FLOW 11 4 APPROXIMATION OF THE BOUNDARY LAYER: VELOCITY PRO?LE 11 4 1 FORMULATION OF THE PROBLEM 11 4 2 TURBULENCE MODEL 11 4 3 OUTER REGION 11 4 4 EQUATION TO SOLVE 11 4 5 EXAMPLES OF RESULTS 11 5 CONCLUSION CHANNEL FLOW 12 1 FORMULATION OF THE PROBLEM 12 2 UNIFORMLY VALID APPROXIMATION 12 3 IBL MODEL FOR THE LOWER WALL 12 4 GLOBAL IBL MODEL 12 5 NUMERICAL SOLUTION 12 5 1 GENERAL METHOD 12 5 2 SIMPLI?ED METHOD FOR THE PRESSURE 12 6 APPLICATION OF THE GLOBAL IBL MODEL 12 6 1 DISCUSSION OF THE NUMERICAL PROCEDURE 12 6 2 COMPARISONS WITH SMITH’S THEORY 12 6 3 COMPARISON WITH NAVIER-STOKES SOLUTIONS 12 7 CONCLUSION SOLUTIONS OF PROBLEMS REFERENCES