AN ELEMENTARY TREATISE ON HYDRODYNAMICS AND SOUND

By E. PAIRY

CONTENTS EQUATIONS OF MOTION 1 1 THE ?UID STATE 1 2 THE MATERIAL DERIVATIVE 1 3 CONSERVATION OF MASS: EQUATION OF CONTINUITY 1 4 MOMENTUM EQUATION 1 4 1 RELATIVE MOTION OF NEIGHBOURING ?UID ELEMENTS 1 4 2 VISCOUS STRESS TENSOR 1 4 3 NAVIER–STOKES EQUATION 1 4 4 THE REYNOLDS EQUATION AND REYNOLDS STRESS 1 5 THE ENERGY EQUATION 1 5 1 ALTERNATIVE TREATMENT OF THE ENERGY EQUATION 1 5 2 ENERGY EQUATION FOR INCOMPRESSIBLE ?OW 1 6 SUMMARY OF GOVERNING EQUATIONS 1 7 BOUNDARY CONDITIONS PROBLEMS POTENTIAL FLOW OF AN INCOMPRESSIBLE FLUID 2 1 IDEAL ?UID 2 2 KELVIN’S CIRCULATION THEOREM 2 3 THE VELOCITY POTENTIAL 2 3 1 BERNOULLI’S EQUATION 2 3 2 IMPULSIVE PRESSURE 2 3 3 STREAMLINES AND INTRINSIC EQUATIONS OF MOTION 2 3 4 BERNOULLI’S EQUATION IN STEADY ?OW 2 4 MOTION PRODUCED BY A PULSATING SPHERE 2 5 THE POINT SOURCE 2 6 FREE-SPACE GREEN’S FUNCTION CONTENTS 2 7 MONOPOLES, DIPOLES, AND QUADRUPOLES 2 7 1 THE VIBRATING SPHERE 2 7 2 STREAMLINES 2 7 3 FAR ?ELD OF A MONOPOLE DISTRIBUTION OF ZERO STRENGTH 2 8 GREEN’S FORMULA 2 8 1 VOLUME AND SURFACE INTEGRALS 2 8 2 GREEN’S FORMULA 2 8 3 SOURCES ADJACENT TO A PLANE WALL 2 9 DETERMINANCY OF THE MOTION 2 9 1 FLUID MOTION EXPRESSED IN TERMS OF MONOPOLE OR DIPOLE DISTRIBUTIONS 2 9 2 DETERMINANCY OF CYCLIC IRROTATIONAL ?OW 2 9 3 KINETIC ENERGY OF CYCLIC IRROTATIONAL ?OW 2 10 THE KINETIC ENERGY 2 10 1 CONVERSE OF KELVIN’S MINIMUM-ENERGY THEOREM 2 10 2 ENERGY OF MOTION PRODUCED BY A TRANSLATING SPHERE 2 11 PROBLEMS WITH SPHERICAL BOUNDARIES 2 11 1 LEGENDRE POLYNOMIALS 2 11 2 VELOCITY POTENTIAL OF A POINT SOURCE IN TERMS OF LEGENDRE POLYNOMIALS 2 11 3 INTERPRETATION IN TERMS OF IMAGES 2 12 THE STOKES STREAM FUNCTION 2 12 1 STREAM FUNCTION EXAMPLES 2 12 2 RANKINE SOLIDS 2 12 3 RANKINE OVOID 2 12 4 DRAG IN IDEAL ?OW 2 12 5 AXISYMMETRIC ?OW FROM A NOZZLE 2 12 6 IRROTATIONAL ?OW FROM A CIRCULAR CYLINDER 2 12 7 BORDA’S MOUTHPIECE 2 13 THE INCOMPRESSIBLE FAR ?ELD 2 13 1 DEDUCTIONS FROM GREEN’S FORMULA 2 13 2 FAR ?ELD PRODUCED BY MOTION OF A RIGID BODY 2 13 3 INERTIA COEF?CIENTS 2 13 4 PRESSURE IN THE FAR ?ELD 2 14 FORCE ON A RIGID BODY 2 14 1 MOMENT EXERTED ON A RIGID BODY 2 15 SOURCES NEAR SOLID BOUNDARIES 2 15 1 THE RECIPROCAL THEOREM 2 16 FAR-?ELD GREEN’S FUNCTION 2 16 1 THE KIRCHHOFF VECTOR 2 16 2 FAR-?ELD GREEN’S FUNCTION FOR A SPHERE CONTENTS 2 17 FAR-?ELD GREEN’S FUNCTION FOR CYLINDRICAL BODIES 2 17 1 THE CIRCULAR CYLINDER 2 17 2 THE RIGID STRIP 2 18 SYMMETRIC FAR-?ELD GREEN’S FUNCTION 2 18 1 FAR ?ELD OF AN ARBITRARILY MOVING BODY 2 19 FAR-?ELD GREEN’S FUNCTION SUMMARY AND SPECIAL CASES 2 19 1 GENERAL FORM 2 19 2 AIRFOIL OF VARIABLE CHORD 2 19 3 PROJECTION OR CAVITY ON A PLANE WALL 2 19 4 RANKINE OVOID 2 19 5 CIRCULAR APERTURE 2 19 6 CIRCULAR DISC IDEAL FLOW IN TWO DIMENSIONS 3 1 COMPLEX REPRESENTATION OF ?UID MOTION 3 1 1 THE STREAM FUNCTION 3 1 2 THE COMPLEX POTENTIAL 3 1 3 UNIFORM ?OW 3 1 4 FLOW PAST A CYLINDRICAL SURFACE 3 2 THE CIRCULAR CYLINDER 3 2 1 CIRCLE THEOREM 3 2 2 UNIFORM ?OW PAST A CIRCULAR CYLINDER 3 2 3 THE LINE VORTEX 3 2 4 CIRCULAR CYLINDER WITH CIRCULATION 3 2 5 EQUATION OF MOTION OF A CYLINDER WITH CIRCULATION 3 3 THE BLASIUS FORCE AND MOMENT FORMULAE 3 3 1 BLASIUS’S FORCE FORMULA FOR A STATIONARY RIGID BODY 3 3 2 BLASIUS’S MOMENT FORMULA FOR A STATIONARY RIGID BODY 3 3 3 KUTTA–JOUKOWSKI LIFT FORCE 3 3 4 LEADING-EDGE SUCTION 3 4 SOURCES AND LINE VORTICES 3 4 1 LINE VORTICES 3 4 2 MOTION OF A LINE VORTEX 3 4 3 K´ARM´AN VORTEX STREET 3 4 4 KINETIC ENERGY OF A SYSTEM OF RECTILINEAR VORTICES 3 5 CONFORMAL TRANSFORMATIONS 3 5 1 TRANSFORMATION OF LAPLACE’S EQUATION 3 5 2 EQUATION OF MOTION OF A LINE VORTEX 3 5 3 NUMERICAL INTEGRATION OF THE VORTEX PATH EQUATION 3 6 THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 3 6 1 IRROTATIONAL ?OW FROM AN IN?NITE DUCT 3 6 2 IRROTATIONAL ?OW THROUGH A WALL APERTURE CONTENTS 3 7 FREE-STREAMLINE THEORY 3 7 1 COANDA EDGE ?OW 3 7 2 MAPPING FROM THE W PLANE TO THE T PLANE 3 7 3 SEPARATED ?OW THROUGH AN APERTURE 3 7 4 THE WAKE OF A ?AT PLATE 3 7 5 FLOW PAST A CURVED BOUNDARY 3 7 6 THE HODOGRAPH TRANSFORMATION FORMULA 3 7 7 CHAPLYGIN’S SINGULAR POINT METHOD 3 7 8 JET PRODUCED BY A POINT SOURCE 3 7 9 DE?ECTION OF TRAILING-EDGE ?OW BY A SOURCE 3 8 THE JOUKOWSKI TRANSFORMATION 3 8 1 THE ?AT-PLATE AIRFOIL 3 8 2 CALCULATION OF THE LIFT 3 8 3 LIFT CALCULATED FROM THE KIRCHHOFF VECTOR FORCE FORMULA 3 8 4 LIFT DEVELOPED BY A STARTING AIRFOIL 3 9 THE JOUKOWSKI AIRFOIL 3 9 1 STREAMLINE ?OW PAST AN AIRFOIL 3 10 SEPARATION AND STALL 3 10 1 LINEAR THEORY OF SEPARATION 3 11 SEDOV’S METHOD 3 11 1 BOUNDARY CONDITIONS 3 11 2 SEDOV’S FORMULA 3 11 3 TANDEM AIRFOILS 3 11 4 HIGH-LIFT DEVICES 3 11 5 PLAIN ?AP OR AILERON 3 11 6 POINT SOURCES AND VORTICES 3 11 7 FLOW THROUGH A CASCADE 3 12 UNSTEADY THIN-AIRFOIL THEORY 3 12 1 THE VORTEX SHEET WAKE 3 12 2 TRANSLATIONAL OSCILLATIONS 3 12 3 THE UNSTEADY LIFT 3 12 4 LEADING-EDGE SUCTION FORCE 3 12 5 ENERGY DISSIPATED BY VORTICITY PRODUCTION 3 12 6 HANKEL FUNCTION FORMULAE ROTATIONAL INCOMPRESSIBLE FLOW 4 1 THE VORTICITY EQUATION 4 1 1 VORTEX LINES 4 1 2 VORTEX TUBES 4 1 3 MOVEMENT OF VORTEX LINES: HELMHOLTZ’S VORTEX THEOREM CONTENTS 4 1 4 CROCCO’S EQUATION 4 1 5 CONVECTION AND DIFFUSION OF VORTICITY 4 1 6 VORTEX SHEETS 4 2 THE BIOT–SAVART LAW 4 2 1 THE FAR ?ELD 4 2 2 KINETIC ENERGY 4 2 3 THE BIOT–SAVART FORMULA IN THE PRESENCE OF AN INTERNAL BOUNDARY 4 2 4 THE BIOT–SAVART FORMULA FOR IRROTATIONAL ?OW 4 3 EXAMPLES OF AXISYMMETRIC VORTICAL ?OW 4 3 1 CIRCULAR VORTEX ?LAMENT 4 3 2 RATE OF PRODUCTION OF VORTICITY AT A NOZZLE 4 3 3 BLOWING OUT A CANDLE 4 3 4 AXISYMMETRIC STEADY ?OW OF AN IDEAL ?UID 4 3 5 HILL’S SPHERICAL VORTEX 4 4 SOME VISCOUS ?OWS 4 4 1 DIFFUSION OF VORTICITY FROM AN IMPULSIVELY STARTED PLANE WALL 4 4 2 DIFFUSION OF VORTICITY FROM A LINE VORTEX 4 4 3 CREEPING ?OW 4 4 4 MOTION OF A SPHERE AT VERY SMALL REYNOLDS NUMBER 4 4 5 THE OSEEN APPROXIMATION 4 4 6 LAMINAR ?OW IN A TUBE (HAGEN–POISEUILLE ?OW) 4 4 7 BOUNDARY LAYER ON A ?AT PLATE; K´ARM´AN MOMENTUM INTEGRAL METHOD 4 5 FORCE ON A RIGID BODY 4 5 1 SURFACE FORCE IN TERMS OF THE IMPULSE 4 5 2 THE KIRCHHOFF VECTOR FORCE FORMULA 4 5 3 THE KIRCHHOFF VECTOR FORCE FORMULA FOR IRROTATIONAL ?OW 4 5 4 ARBITRARY MOTION IN A VISCOUS ?UID 4 5 5 BODY MOVING WITHOUT ROTATION 4 5 6 SURFACE FORCE IN TWO DIMENSIONS 4 5 7 BLUFF BODY DRAG AT HIGH REYNOLDS NUMBER 4 5 8 MODELLING VORTEX SHEDDING FROM A SPHERE 4 5 9 FORCE AND IMPULSE IN ?UID OF NON-UNIFORM DENSITY 4 5 10 INTEGRAL IDENTITIES 4 6 SURFACE MOMENT 4 6 1 MOMENT FOR A NON-ROTATING BODY 4 6 2 AIRFOIL LIFT, DRAG, AND MOMENTS 4 7 VORTEX–SURFACE INTERACTIONS 4 7 1 PRESSURE EXPRESSED IN TERMS OF THE TOTAL ENTHALPY 4 7 2 EQUATION FOR B CONTENTS 4 7 3 SOLUTION OF THE B EQUATION 4 7 4 THE FAR ?ELD SURFACE GRAVITY WAVES 5 1 INTRODUCTION 5 1 1 CONDITIONS AT THE FREE SURFACE 5 1 2 WAVE MOTION WITHIN THE ?UID 5 1 3 LINEARISED APPROXIMATION 5 1 4 TIME HARMONIC, PLANE WAVES ON DEEP WATER 5 1 5 WATER OF ?NITE DEPTH 5 2 SURFACE WAVE ENERGY 5 2 1 WAVE-ENERGY DENSITY 5 2 2 WAVE-ENERGY ?UX 5 2 3 GROUP VELOCITY 5 3 VISCOUS DAMPING OF SURFACE WAVES 5 3 1 THE INTERIOR DAMPING 5 3 2 BOUNDARY-LAYER DAMPING 5 3 3 COMPARISON OF BOUNDARY-LAYER AND INTERNAL DAMPING FOR LONG WAVES 5 4 SHALLOW-WATER WAVES 5 4 1 WAVES ON WATER OF VARIABLE DEPTH 5 4 2 SHALLOW-WATER GREEN’S FUNCTION 5 4 3 WAVES GENERATED BY A LOCALISED PRESSURE RISE 5 4 4 WAVES APPROACHING A SLOPING BEACH 5 5 METHOD OF STATIONARY PHASE 5 5 1 FORMULATION OF INITIAL-VALUE DISPERSIVE-WAVE PROBLEMS 5 5 2 EVALUATION OF FOURIER INTEGRALS BY THE METHOD OF STATIONARY PHASE 5 5 3 NUMERICAL RESULTS FOR THE SURFACE DISPLACEMENT 5 5 4 CONSERVATION OF ENERGY 5 5 5 RAYLEIGH’S PROOF THAT ENERGY PROPAGATES AT THE GROUP VELOCITY 5 5 6 SURFACE WAVE-ENERGY EQUATION 5 5 7 WAVES GENERATED BY A SUBMARINE EXPLOSION 5 6 INITIAL-VALUE PROBLEMS IN TWO SURFACE DIMENSIONS 5 6 1 WAVES GENERATED BY A SURFACE ELEVATION SYMMETRIC ABOUT THE ORIGIN 5 6 2 THE ENERGY EQUATION IN TWO DIMENSIONS 5 7 SURFACE MOTION NEAR A WAVEFRONT 5 7 1 ONE-DIMENSIONAL WAVES 5 7 2 WAVES GENERATED BY MOTION OF THE SEABED CONTENTS 5 7 3 TSUNAMI PRODUCED BY AN UNDERSEA EARTHQUAKE 5 8 PERIODIC WAVE SOURCES 5 8 1 ONE-DIMENSIONAL WAVES 5 8 2 PERIODIC SOURCES IN TWO SURFACE DIMENSIONS 5 8 3 THE SURFACE WAVE POWER 5 8 4 SURFACE WAVE AMPLITUDE 5 9 SHIP WAVES 5 9 1 MOVING LINE PRESSURE SOURCE 5 9 2 WAVE-MAKING RESISTANCE 5 9 3 MOVING POINT-LIKE PRESSURE SOURCE 5 9 4 PLOTTING THE WAVE CRESTS 5 9 5 BEHAVIOUR AT THE CAUSTIC 5 9 6 WAVE-MAKING POWER 5 9 7 WAVE AMPLITUDE CALCULATED FROM THE POWER 5 10 RAY THEORY 5 10 1 KINEMATIC THEORY OF WAVE CRESTS 5 10 2 RAY TRACING IN AN INHOMOGENEOUS MEDIUM 5 10 3 REFRACTION OF WAVES AT A SLOPING BEACH 5 11 WAVE ACTION 5 11 1 VARIATIONAL DESCRIPTION OF A FULLY DISPERSED WAVE GROUP 5 11 2 FULLY DISPERSED WAVES IN A NON-UNIFORMLY MOVING MEDIUM 5 11 3 GENERAL WAVE-BEARING MEDIA 5 12 DIFFRACTION OF SURFACE WAVES BY A BREAKWATER 5 12 1 DIFFRACTION BY A LONG, STRAIGHT BREAKWATER 5 12 2 SOLUTION OF THE DIFFRACTION PROBLEM 5 12 3 THE SURFACE WAVE PATTERN 5 12 4 UNIFORM ASYMPTOTIC APPROXIMATION: METHOD OF STEEPEST DESCENTS INTRODUCTION TO ACOUSTICS 6 1 THE WAVE EQUATION 6 1 1 THE LINEAR WAVE EQUATION 6 1 2 PLANE WAVES 6 1 3 SPEED OF SOUND 6 2 ACOUSTIC GREEN’S FUNCTION 6 2 1 THE IMPULSIVE POINT SOURCE 6 2 2 GREEN’S FUNCTION 6 2 3 RETARDED POTENTIAL 6 2 4 SOUND FROM A VIBRATING SPHERE 6 2 5 ACOUSTIC ENERGY ?UX CONTENTS 6 2 6 GREEN’S FUNCTION IN ONE SPACE DIMENSION: METHOD OF DESCENT 6 2 7 WAVES GENERATED BY A ONE-DIMENSIONAL VOLUME SOURCE 6 3 KIRCHHOFF’S FORMULA 6 4 COMPACT GREEN’S FUNCTION 6 4 1 GENERALIZED KIRCHHOFF FORMULA 6 4 2 THE TIME HARMONIC WAVE EQUATION 6 4 3 THE COMPACT APPROXIMATION 6 4 4 RAYLEIGH SCATTERING: SCATTERING BY A COMPACT BODY 6 5 ONE-DIMENSIONAL PROPAGATION THROUGH JUNCTIONS 6 5 1 CONTINUITY OF VOLUME VELOCITY 6 5 2 CONTINUITY OF PRESSURE 6 5 3 RE?ECTION AND TRANSMISSION AT A JUNCTION 6 6 BRANCHING SYSTEMS 6 6 1 FUNDAMENTAL FORMULA 6 6 2 ENERGY TRANSMISSION 6 6 3 ACOUSTICALLY COMPACT CAVITY 6 6 4 THE HELMHOLTZ RESONATOR 6 6 5 ACOUSTIC ?LTER 6 6 6 ADMITTANCE OF A NARROW CONSTRICTION 6 7 RADIATION FROM AN OPEN END 6 7 1 RAYLEIGH’S METHOD FOR LOW-FREQUENCY SOUND 6 7 2 THE RE?ECTION COEF?CIENT 6 7 3 ADMITTANCE OF THE OPEN END 6 7 4 OPEN-END INPUT ADMITTANCE 6 7 5 FLANGED OPENING 6 7 6 PHYSICAL SIGNI?CANCE OF THE END CORRECTION 6 7 7 ADMITTANCE OF A CIRCULAR APERTURE 6 8 WEBSTER’S EQUATION 6 9 RADIATION INTO A SEMI-IN?NITE DUCT 6 9 1 THE COMPACT GREEN’S FUNCTION 6 9 2 WAVE GENERATION BY A TRAIN ENTERING A TUNNEL 6 10 DAMPING OF SOUND IN A SMOOTH-WALLED DUCT 6 10 1 TIME HARMONIC PROPAGATION IN A DUCT 6 10 2 THE VISCOUS CONTRIBUTION 6 10 3 THE THERMAL CONTRIBUTION 6 10 4 THE THERMO-VISCOUS DAMPING COEF?CIENT