A TREATISE ON HYDRODYNAMICS

By W. BENAD

CONTENTS 1 INTRODUCTION 1 1 A BRIEF HISTORICAL OVERVIEW SUMMARY OF CONTENT 1 3 REFERENCES CHAPTER 1 2 HYDRODYNAMIC BACKGROUND REYNOLDS’ DECOMPOSITION OF PHYSICAL QUANTITIES DETERMINATION OF THE TURBULENT MEAN FLOW THE REYNOLDS EQUATIONS MODELLING OF TURBULENT STRESSES ENERGY FLUX IN A FLOW APPENDIX: TENSOR NOTATION REFERENCES - CHAPTER 2 3 LINEAR WAVES 3 1 ASSUMPTIONS AND THE SIMPLIFIED EQUATIONS BASIC SOLUTION FOR LINEAR WAVES SOLUTION FOR Q5 AND R] EVALUATION OF LINEAR WAVES PARTICLE MOTION THE PRESSURE VARIATION DEEP WATER AND SHALLOW WATER APPROXIMATIONS TIME AVERAGED PROPERTIES OF LINEAR WAVES IN ONE HORIZONTAL DIMENSION (1DH) INTRODUCTION MASS AND VOLUME FLUX MOMENTUM FLUX-RADIATION STRESS ENERGY DENSITY ENERGY FLUX DIMENSIONLESS FUNCTIONS FOR WAVE AVERAGED QUANTITIES SUPERPOSITION OF LINEAR WAVES STANDING WAVES WAVE GROUPS WAVE SPECTRA 3 5 LINEAR WAVE PROPAGATIONOVER UNEVENBOTTOM SHOALING AND REFRACTION SIMPLE SHOALING REFRACTION BY RAY TRACING THE GEOMETRICAL OPTICS APPROXIMATION KINEMATIC WAVE THEORY WAVE MODIFICATION BY CURRENTS DETERMINATION OF THE REFRACTION PATTERN THE ENERGY EQUATION THE ENERGY BALANCE FOR PERIODIC WAVES INTRODUCTION OF DIMENSIONLESS PARAMETERS FOR L-D WAVE MOTION A CLOSED FORM SOLUTION OF THE ENERGY EQUATION THE ENERGY EQUATION FOR STEADY IRREGULAR WAVES THE GENERAL ENERGY EQUATION: UNSTEADY WAVE-CURRENT MOTION THE WAVE ACTION EQUATION REFERENCES - CHAPTER 4 5 PROPERTIES OF BREAKING WAVES 5 1 INTRODUCTION THE HIGHEST POSSIBLE WAVE ON CONSTANT DEPTH 5 3 QUALITATIVE DESCRIPTION OF WAVE BREAKING TRANSITION (OR: WHY DO THE WAVES BREAK?) 5 4 WAVE CHARACTERISTICS AT THE BREAKPOINT 5 5 EXPERIMENTAL RESULTS FOR SURFZONE WAVES QUALITATIVE SURFZONE CHARACTERISTICS THE PHASE VELOCITY C SURFACE PROFILES V(T) THE SURFACE SHAPE PARAMETER BO THE CREST ELEVATION QC/H THE ROLLER AREA MEASUREMENTS OF PARTICLE VELOCITIES TURBULENCE INTENSITIES THE VALUES OF P,,, B, AND D 5 5 10 THE WAVE GENERATED SHEAR STRESS U,W, SURFZONE WAVE MODELLING SURFZONE ASSUMPTIONS ENERGY FLUX EF FOR SURFZONE WAVES RADIATION STRESS IN SURFZONE WAVES VOLUME FLUX IN SURFZONE WAVES THE PHASE VELOCITIES FOR QUASI-STEADY BREAKING WAVES THE ENERGY DISSIPATION IN QUASI STEADY SURFZONE WAVES 5 7 FURTHER ANALYSIS OF THE ENERGY DISSIPATION ENERGY DISSIPATION FOR RANDOM WAVES ENERGY DISSIPATION WITH A THRESHOLD A MODEL FOR ROLLER ENERGY DECAY ADVANCED COMPUTATIONAL METHODS FOR SURZONE WAVES SWASH 5 9 REFERENCES - CHAPTER 5 6 WAVE MODELS BASED ON LINEAR WAVE THEORY 6 1 INTRODUCTION 1DH SHOALING-BREAKING MODEL 2DH REFRACTION MODELS THE WAVE PROPAGATION PATTERN DETERMINATION OF THE WAVE AMPLITUDE VARIATION 6 4 WAVE ACTION MODELS 6 5 MODELS BASED ON THE MILD SLOPE EQUATION AND THE PARABOLIC APPROXIMATION REFERENCES - CHAPTER 6 7 NONLINEAR WAVES: ANALYSIS OF PARAMETERS 7 1 INTRODUCTION THE EQUATIONS FOR THE CLASSICAL NONLINEAR WAVE THEORIES 7 3 THE SYSTEM OF DIMENSIONLESS VARIABLES USED SOLUTION FOR THE LAPLACE EQUATION 9 10 FURTHER DEVELOPMENTS OF BOUSSINESQ MODELLING A TREATISE ON HYDRODYNAMICS 9 11 4 WAVE BREAKING MODELLED BY THE NONLINEAR SHALLOW WATER EQUATIONS 9 12 LARGE AMPLITUDE LONG WAVES U >> 1: THE NONLINEAR SHALLOW WATER EQUATIONS (NSW) 9 13 REFERENCES - CHAPTER 9 10 BOUNDARY LAYERS 10 1 INTRODUCTION 10 1 1 THE BOUNDARY LAYER EQUATIONS FORMULATION OF THE PROBLEM 10 1 2 PERTURBATION EXPANSION FOR U 10 1 3 THE 1ST ORDER SOLUTION 10 1 4 THE 2ND ORDER SOLUTION 10 1 5 THE STEADY STREAMING US IN WAVE BOUNDARY LAYERS 10 1 6 RESULTS FOR U, 10 2 ENERGY DISSIPATION IN A LINEAR WAVE BOUNDARY LAYER 10 3 TURBULENT WAVE BOUNDARY LAYERS 10 3 1 ROUGH TURBULENT FLOW 10 3 2 ENERGY DISSIPATION IN TURBULENT WAVE BOUNDARY LAYERS 10 4 BOTTOM SHEAR STRESS IN 3D WAVE-CURRENT BOUNDARY LAYERS 10 4 1 INTRODUCTION 10 4 2 FORMULATION OF THE PROBLEM 10 4 3 THE MEAN SHEAR STRESS 10 4 4 SPECIAL CASES 10 5 REFERENCES - CHAPTER 10 11 NEARSHORE CIRCULATION 11 1 INTRODUCTION 11 2 DEPTH INTEGRATED CONSERVATION OF MASS 11 2 1 SEPARATION OF WAVES AND CURRENTS 11 3 CONDITIONS AT FIXED AND MOVING BOUNDARIES, I1 11 3 1 KINEMATIC CONDITIONS 11 3 2 DYNAMIC CONDITIONS 11 4 DEPTH INTEGRATED MOMENTUM EQUATION 11 4 1 INTEGRATION OF HORIZONTAL EQUATIONS 11 4 2 INTEGRATION OF THE VERTICAL MOMENTUM EQUATION 12 11 5 THE NEARSHORE CIRCULATION EQUATIONS 11 5 1 THE TIME AVERAGED MOMENTUM EQUATION 11 5 2 THE EQUATIONS FOR DEPTH UNIFORM CURRENTS 11 6 ANALYSIS OF THE RADIATION STRESS IN TWO HORIZONTAL DIMENSIONS, 2DH 11 6 1 SAP EXPRESSED IN TERMS OF S, AND S, 11 6 2 RADIATION STRESS FOR LINEAR WAVES IN TWO HORIZONTAL DIMENSIONS (2DH) 11 7 EXAMPLES ON A LONG STRAIGHT BEACH 11 7 1 THE MOMENTUM BALANCE 11 7 2 THE CROSS-SHORE MOMENTUM BALANCE: SETDOWN AND SETUP 11 7 3 LONGSHORE CURRENTS 11 7 4 LONGSHORE CURRENT SOLUTION FOR A PLANE BEACH 11 7 5 DISCUSSION OF THE EXAMPLES 11 8 WAVE DRIVERS 11 9 CONDITIONS ALONG OPEN BOUNDARIES 11 9 1 INTRODUCTION ABOUT OPEN BOUNDARIES 11 9 2 ABSORBING-GENERATING BOUNDARY CONDITIONS 11 9 3 BOUNDARY CONDITIONS ALONG CROSS-SHORE BOUNDARIES 11 10 REFERENCES - CHAPTER CROSS-SHORE CIRCULATION AND UNDERTOW 12 1 THE VERTICAL VARIATION OF CURRENTS 12 1 1 INTRODUCTION 12 1 2 THE GOVERNING EQUATIONS FOR THE VARIATION OVER DEPTH OF THE 3D CURRENTS 12 2 THE CROSS-SHORE CIRCULATION, UNDERTOW 12 2 1 FORMULATION OF THE 2-D PROBLEM AND GENERAL SOLUTION 12 2 2 BOUNDARY CONDITIONS 12 2 3 SOLUTION FOR THE UNDERTOW PROFILES WITH DEPTH 12 2 4 DISCUSSION OF RESULTS AND COMPARISON WITH MEASUREMENTS 12 2 5 SOLUTIONS INCLUDING THE EFFECT OF THE BOUNDARY LAYER 12 2 6 UNDERTOW OUTSIDE THE SURFZONE 12 2 7 CONCLUSIONS 12 3 REFERENCES - CHAPTER 12 UNIFORM VT AND 011 13 6 2 RIP CURRENTS 13 6 3 CURVILINEAR VERSION OF THE SHORECIRC MODEL 13 7 REFERENCES - CHAPTER 13 MODELLING THE START-UP OF A LONGSHORE CURRENT 13 6 4 THE NEARSHORE COMMUNITY MODEL, NEARCOM 14 OTHER NEARSHORE FLOW PHENOMENA 14 1 INFRAGRAVITY WAVES 14 1 1 INTRODUCTION 14 1 2 BASIC EQUATIONS FOR INFRAGRAVITY WAVES 14 1 3 HOMOGENEOUS SOLUTIONS ~ FREE EDGE WAVES 14 1 4 IG WAVE GENERATION 14 2 SHEAR INSTABILITIES OF LONGSHORE CURRENTS 14 2 1 INTRODUCTION 14 2 2 THE DISCOVERY OF SHEAR WAVES 14 2 3 DERIVATION OF THE BASIC EQUATIONS 14 2 4 STABILITY ANALYSIS OF THE EQUATIONS 14 2 5 FURTHER ANALYSES OF THE INITIAL INSTABILITY 14 2 6 NUMERICAL ANALYSIS OF FULLY DEVELOPED SHEAR WAVES 14 3 REFERENCES - CHAPTER 14