Fluid mechanics third edition

Fluid Mechanics, Third Edition Founders of Modern Fluid Dynamics Ludwig Prandtl (1875–1953) G I Taylor (1886–1975) (Biographical sketches of Prandtl and Taylor are given in Appendix C.) Photograph of Ludwig Prandtl is reprinted with permission from the Annual Review of Fluid Mechanics, Vol 19, Copyright 1987 by Annual Reviews www.AnnualReviews.org Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted with permission from the AIP Emilio Segr`e Visual Archieves Copyright, American Institute of Physics, 2000 Fluid Mechanics Third Edition Pijush K Kundu Oceanographic Center Nova University Dania, Florida Ira M Cohen Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, Pennsylvania with a chapter on Computational Fluid Dynamics by Howard H Hu AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Elsevier Academic Press 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper ∞ Copyright © 2004, Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data A catalogue record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-12-178253-0 For all information on all Academic Press publications visit our Web site at www.academicpress.com Printed in the United States of America 04 05 06 07 08 The third edition is dedicated to the memory of Pijush K Kundu and also to my wife Linda and daughters Susan and Nancy who have greatly enriched my life “Everything should be made as simple as possible, but not simpler.” —Albert Einstein “If nature were not beautiful, it would not be worth studying it And life would not be worth living.” —Henry Poincar´e In memory of Pijush Kundu Pijush Kanti Kundu was born in Calcutta, India, on October 31, 1941 He received a B.S degree in Mechanical Engineering in 1963 from Shibpur Engineering College of Calcutta University, earned an M.S degree in Engineering from Roorkee University in 1965, and was a lecturer in Mechanical Engineering at the Indian Institute of Technology in Delhi from 1965 to 1968 Pijush came to the United States in 1968, as a doctoral student at Penn State University With Dr John L Lumley as his advisor, he studied instabilities of viscoelastic fluids, receiving his doctorate in 1972 He began his lifelong interest in oceanography soon after his graduation, working as Research Associate in Oceanography at Oregon State University from 1968 until 1972 After spending a year at the University de Oriente in Venezuela, he joined the faculty of the Oceanographic Center of Nova Southeastern University, where he remained until his death in 1994 During his career, Pijush contributed to a number of sub-disciplines in physical oceanography, most notably in the fields of coastal dynamics, mixed-layer physics, internal waves, and Indian-Ocean dynamics He was a skilled data analyst, and, in this regard, one of his accomplishments was to introduce the “empirical orthogonal eigenfunction” statistical technique to the oceanographic community I arrived at Nova Southeastern University shortly after Pijush, and he and I worked closely together thereafter I was immediately impressed with the clarity of his scientific thinking and his thoroughness His most impressive and obvious quality, though, was his love of science, which pervaded all his activities Some time after we met, Pijush opened a drawer in a desk in his home office, showing me drafts of several chapters to a book he had always wanted to write A decade later, this manuscript became the first edition of “Fluid Mechanics,” the culmination of his lifelong dream; which he dedicated to the memory of his mother, and to his wife Shikha, daughter Tonushree, and son Joydip Julian P McCreary, Jr., University of Hawaii Contents Preface xvii Preface to Second Edition xviii Preface to First Edition xx Author’s Notes xxiii Chapter Introduction 10 Fluid Mechanics Units of Measurement Solids, Liquids, and Gases Continuum Hypothesis Transport Phenomena Surface Tension Fluid Statics Classical Thermodynamics Perfect Gas Static Equilibrium of a Compressible Medium Exercises Literature Cited Supplemental Reading 12 16 17 22 23 23 Chapter Cartesian Tensors Scalars and Vectors Rotation of Axes: Formal Definition of a Vector 24 25 vii viii Contents 10 11 12 13 14 15 16 Multiplication of Matrices Second-Order Tensor Contraction and Multiplication Force on a Surface Kronecker Delta and Alternating Tensor Dot Product Cross Product Operator ∇: Gradient, Divergence, and Curl Symmetric and Antisymmetric Tensors Eigenvalues and Eigenvectors of a Symmetric Tensor Gauss’ Theorem Stokes’ Theorem Comma Notation Boldface vs Indicial Notation Exercises Literature Cited Supplemental Reading 28 29 31 32 35 36 36 37 38 40 42 45 46 47 47 49 49 Chapter Kinematics 10 11 12 13 14 Introduction Lagrangian and Eulerian Specifications Eulerian and Lagrangian Descriptions: The Particle Derivative Streamline, Path Line, and Streak Line Reference Frame and Streamline Pattern Linear Strain Rate Shear Strain Rate Vorticity and Circulation Relative Motion near a Point: Principal Axes Kinematic Considerations of Parallel Shear Flows Kinematic Considerations of Vortex Flows One-, Two-, and Three-Dimensional Flows The Streamfunction Polar Coordinates Exercises Supplemental Reading 50 51 53 54 56 57 58 59 61 64 65 68 69 72 73 75 ix Contents Chapter Conservation Laws 10 11 12 13 14 15 16 17 18 19 Introduction Time Derivatives of Volume Integrals Conservation of Mass Streamfunctions: Revisited and Generalized Origin of Forces in Fluid Stress at a Point Conservation of Momentum Momentum Principle for a Fixed Volume Angular Momentum Principle for a Fixed Volume Constitutive Equation for Newtonian Fluid Navier–Stokes Equation Rotating Frame Mechanical Energy Equation First Law of Thermodynamics: Thermal Energy Equation Second Law of Thermodynamics: Entropy Production Bernoulli Equation Applications of Bernoulli’s Equation Boussinesq Approximation Boundary Conditions Exercises Literature Cited Supplemental Reading 77 77 79 81 82 84 86 88 92 94 97 99 104 108 109 110 114 117 121 126 128 128 Chapter Vorticity Dynamics Introduction Vortex Lines and Vortex Tubes Role of Viscosity in Rotational and Irrotational Vortices Kelvin’s Circulation Theorem Vorticity Equation in a Nonrotating Frame Velocity Induced by a Vortex Filament: Law of Biot and Savart Vorticity Equation in a Rotating Frame Interaction of Vortices Vortex Sheet Exercises 129 130 130 134 138 140 141 146 149 150 Index Ackeret, Jacob, 687, 728 Acoustic waves, 689 Adiabatic density gradient, 565, 581 Adiabatic process, 23, 707, 717 Adiabatic temperature gradient, 19, 581 Advection, 53 Advective derivative, 53 Aerodynamics aircraft parts and controls, 654–657 airfoil forces, 657–659 airfoil geometry, 657 conformal transformation, 662–666 defined, 653 finite wing span, 669–670 gas, 653 generation of circulation, 660–662 incompressible, 653 Kutta condition, 659–660 lift and drag characteristics, 677–679 Prandtl and Lanchester lifting line theory, 670–675 propulsive mechanisms of fish and birds, 679–680 sailing, 680–682 Zhukhovsky airfoil lift, 666–669 Air, physical properties of, 735 Aircraft, parts and controls, 654–657 Airfoil(s) angle of attack/incidence, 657 camber line, 657 chord, 657 compression side, 659 conformal transformation, 662–666 drag, induced/vortex, 670, 673–674 finite span, 669–670 forces, 657–659 geometry, 657 lift and drag characteristics, 677–679 stall, 668, 677 suction side, 659 supersonic flow, 728–731 thin airfoil theory, 662 Zhukhovsky airfoil lift, 666–669 Alston, T M., 378, 381, 384 Alternating tensors, 35–36 Analytic function, 158 Anderson, John, D., Jr., 406, 450, 660, 662, 684 Angle of attack/incidence, 657, 672 Angular momentum principle/theorem, for fixed volume, 92–93 Antisymmetric tensors, 38–39 Aris, R., 49, 75, 95, 128 Ashley, H., 675, 684 Aspect ratio of wing, 655 Asymptotic expansion, 368–369 Atmosphere properties of standard, 736 scale height of, 21 Attractors aperiodic, 513 dissipative systems and, 509–511 fixed point, 509 limit cycle, 509 strange, 512–513 Autocorrelation function, 526 normalized, 526 of a stationary process, 526 Averages, 522–525 Axisymmetric irrotational flow, 187–189 Babuska–Brezzi stability condition, 414 Baroclinic flow, 136–137 Baroclinic instability, 639–647 Baroclinic/internal mode, 246, 608 Barotropic flow, 111, 135, 136 Barotropic instability, 637–638 Barotropic/surface mode, 245–246, 608 Baseball dynamics, 357 Batchelor, G K., 23, 99, 123, 124, 128, 152, 198, 280, 317, 337, 385, 577, 647, 684, 744 Bayly, B J., 454, 498, 504, 507, 518 Becker, R., 710, 711, 732 Berg´e, P., 509, 512, 515, 518 B´enard, H., 352 convection, 456 thermal instability, 455–466 Bender, C M., 384 Bernoulli equation, 110–114 applications of, 114–117 energy, 114 745 746 Index Bernoulli equation (continued) one-dimensional, 694–695 steady flow, 112–113 unsteady irrotational flow, 113–114 β-plane model, 588 Bifurcation, 510 Biot and Savart, law of, 140 Bird, R B., 276 Birds, flight of, 680 Blasius solution, boundary layer, 330–339 Blasius theorem, 171–172 Blocking, in stratified flow, 254 Body forces, 83 Body of revolution flow around arbitrary, 194–195 flow around streamlined, 193–194 Bohlen T., 342, 384 Bond number, 272 Boundary conditions, 121–125, 643 geophysical fluids, 606 at infinity, 156 kinematic, 206 on solid surface, 156 Boundary layer approximation, 319–324 Blasius solution, 330–339 breakdown of laminar solution, 337–339 closed form solution, 327–330 concept, 318–319 decay of laminar shear layer, 378–382 displacement thickness, 325–326 drag coefficient, 335–336 dynamics of sports balls, 354–357 effect of pressure gradient, 342–343, 500–501 Falkner–Skan solution, 336–337 flat plate and, 327–339 flow past a circular cylinder, 346–352 flow past a sphere, 353 instability, 503–505 Karman momentum integral, 339–342 momentum thickness, 326–327 perturbation techniques, 366–370 secondary flows, 365–366 separation, 343–346 simplification of equations, 319–324 skin friction coefficient, 335–336 technique, 2, 154 transition to turbulence, 344–345 two-dimensional jets, 357–364 u = 0.99U thickness, 324–325 Bound vortices, 671–672 Boussinesq approximation, 69, 81, 108–109 continuity equation and, 118–119 geophysical fluid and, 583–585 heat equation and, 119–121 momentum equation and, 119 Bradshaw, P., 566, 577 Brauer, H., 436, 450 Breach, D R., 317 Bridgeman, P W., 276 Brooks, A N., 403, 450 Brunt–V¨ais¨al¨a frequency, 249–250 Buckingham’s pi theorem, 268–270 Buffer layer, 556 Bulk strain rate, 57 Bulk viscosity, coefficient of, 96 Buoyancy frequency, 249, 583 Buoyant production, 539–540, 565 Bursting in turbulent flow, 563 Buschmann, M H., 557, 577 Camber line, airfoil, 657 Cantwell, B J., 562, 577 Capillarity, Capillary number, 272 Capillary waves, 219, 222 Carey G F., 414, 450 Cascade, enstrophy, 648 Casten R G., 385 Castillo, L., 557, 577–8 Cauchy–Riemann conditions, 155, 158 Cauchy’s equation of motion, 87 Centrifugal force, effect of, 102–103 Centrifugal instability (Taylor), 471–476 Chandrasekhar, S., 128, 454, 474, 484, 516, 518 Chang G Z., 436, 450 Chaos, deterministic, 508–516 Characteristics, method of, 232 Chester, W., 312, 317 Chord, airfoil, 657 Chorin, A J., 404, 410, 450 Chow C Y., 662, 684 Circular Couette flow, 285 Circular cylinder flow at various Re, 346–352 flow past, boundary layer, 346–352 flow past, with circulation, 168–171 flow past, without circulation, 165–168 Circular Poiseuille flow, 283–285 Circulation, 59–60 Kelvin’s theorem, 134–138 Clausius-Duhem inequality, 96 Cnoidal waves, 237 Coefficient of bulk viscosity, 96 Cohen I M., iii, xviii, 378, 381, 384, 711, 732 Coherent structures, wall layer, 562–564 Coles, D., 477, 518 Comma notation, 46–47, 141 Complex potential, 158 Complex variables, 157–159 Complex velocity, 159 747 Index Compressible flow classification of, 687–688 friction and heating effects, 717–720 internal versus external, 685 Mach cone, 720–722 Mach number, 686–687 one-dimensional, 692–696, 701–704 shock waves, normal, 705–711 shock waves, oblique, 722–726 speed of sound, 689–692 stagnation and sonic properties, 696–700 supersonic, 726–728 Compressible medium, static equilibrium of, 17–18 potential temperature and density, 19–21 scale height of atmosphere, 21 Compression waves, 200 Computational fluid dynamics (CFD) advantages of, 387–388 conclusions, 448 defined, 386 examples of, 416–448 finite difference method, 388–393 finite element method, 393–400 incompressible viscous fluid flow, 400–416 sources of error, 387 Concentric cylinders, laminar flow between, 285–288 Conformal mapping, 177–179 application to airfoil, 662–666 Conservation laws Bernoulli equation, 110–117 boundary conditions, 121–126 Boussinesq approximation, 117–121 differential form, 77 integral form, 77 of mass, 79–81 mechanical energy equation, 104–107 of momentum, 86–88 Navier-Stokes equation, 97–99 rotating frame, 99–104 thermal energy equation, 108–109 time derivatives of volume integrals, 77–79 Conservative body forces, 83, 136 Consistency, 390–393 Constitutive equation, for Newtonian fluid, 94–97 Continuity equation, 69–70, 79, 81 Boussinesq approximation and, 118–119 one-dimensional, 693 Continuum hypothesis, 4–5 Control surfaces, 77 Control volume, 77 Convection, 53 -dominated problems, 402–403 forced, 567 free, 567 sloping, 646 Convergence, 390–393 Conversion factors, 734 Corcos G M., 488, 518 Coriolis force, effect of, 103–104 Coriolis frequency, 587 Coriolis parameter, 587 Correlations and spectra, 525–529 Couette flow circular, 285 plane, 282, 500 Courant, R., 733 Cramer, M S., 720, 732 Creeping flow, around a sphere, 303–308 Creeping motions, 302 Cricket ball dynamics, 354–356 Critical layers, 497–498 Critical Re for transition over circular cylinder, 349–351 over flat plate, 337–339 over sphere, 353 Cross-correlation function, 529 Cross product, vector, 36–37 Curl, vector, 37 Curtiss, C F., 276 Curvilinear coordinates, 737–741 D’Alembert’s paradox, 167, 175 D’Alembert’s solution, 201 Davies, P., 509, 515, 518 Dead water phenomenon, 243 Decay of laminar shear layer, 378–382 Defect law, velocity, 554 Deflection angle, 723 Deformation of fluid elements, 105–106 Rossby radius of, 618 Degree of freedom, 509 Delta wings, 679 Dennis, S C R., 450 Density adiabatic density gradient, 565, 581 potential, 19–21 stagnation, 697 Derivatives advective, 53 material, 53–54 particle, 53 substantial, 53 time derivatives of volume integrals, 77–79 Deviatoric stress tensor, 94 Differential equations, nondimensional parameters determined from, 263–266 Diffuser flow, 701–703 748 Index Diffusion of vorticity from impulsively started plate, 288–294 from line vortex, 296–298 from vortex sheet, 295–296 Diffusivity eddy, 559–562 effective, 575–576 heat, 279 momentum, 279 thermal, 109, 120 vorticity, 136, 295–296 Dimensional homogeneity, 267 Dimensional matrix, 267–268 Dipole See Doublet Dirichlet problem, 182 Discretization error, 387 of transport equation, 389–390 Dispersion of particles, 569–573 relation, 209, 629–630, 634–637 Taylor’s theory, 568–576 Dispersive wave, 203, 221–225, 248–250 Displacement thickness, 319–320 Dissipation of mean kinetic energy, 513 of temperature fluctuation, 545 of turbulent kinetic energy, 517 viscous, 105–106 Divergence flux, 104–105 tensor, 37 theorem, 43, 80 vector, 37 Doormaal, J P., 411, 413, 450 Doppler shift of frequency, 199 Dot product, vector, 36 Double-diffusive instability, 467–471 Doublet in axisymmetric flow, 192 in plane flow, 162–164 Downwash, 672–673 Drag characteristics for airfoils, 677–679 on circular cylinder, 351 coefficient, 270, 335–336 on flat plate, 335–336 force, 657–659 form, 345, 678 induced/vortex, 670, 673–674 pressure, 658, 678 profile, 678 skin friction, 335–336, 658, 678 on sphere, 353 wave, 273–274, 673, 730 Drazin, P G., 454, 456, 466, 476, 482, 497, 498, 518 Dussan, V., E B., 128 Dutton J A., 560, 565–6, 577 Dyke, M., 366, 368, 369, 384 Dynamic pressure, 115, 279–280 Dynamic similarity nondimensional parameters and, 270–272 role of, 262–263 Dynamic viscosity, Eddy diffusivity, 559–562 Eddy viscosity, 559–562 Effective gravity force, 102 Eigenvalues and eigenvectors of symmetric tensors, 40–42 Einstein summation convention, 27 Ekman layer at free surface, 593–598 on rigid surface, 598–601 thickness, 595 Ekman number, 592 Ekman spiral, 595–596 Ekman transport at a free surface, 596 Elastic waves, 200, 689 Element point of view, 398–400 Elliptic circulation, 675–677 Elliptic cylinder, ideal flow, 179–180 Elliptic equation, 156 Energy baroclinic instability, 645–647 Bernoulli equation, 114 spectrum, 528 Energy equation integral form, 77 mechanical, 104–107 one-dimensional, 693–694 thermal, 108–109 Energy flux group velocity and, 224–227 in internal gravity wave, 256–259 in surface gravity wave, 215 Ensemble average, 523–524 Enstrophy, 647 Enstrophy cascade, 648 Enthalpy defined, 13 stagnation, 696 Entrainment in laminar jet, 358 turbulent, 547 Entropy defined, 14 production, 109–110 Epsilon delta relation, 36, 99 Equations of motion averaged, 529–535 Boussinesq, 119, 583–584 Cauchy’s, 87 749 Index for Newtonian fluid, 94–97 in rotating frame, 99–104 for stratified medium, 583–585 for thin layer on rotating sphere, 585–588 Equations of state, 13 for perfect gas, 16 Equilibrium range, 544 Equipartition of energy, 214 Equivalent depth, 603 Eriksen, C C., 488, 518 Euler equation, 98, 111, 317 one-dimensional, 694–695 Euler momentum integral, 91, 175 Eulerian description, 52 Eulerian specifications, 51–53 Exchange of stabilities, principle of, 455 Expansion coefficient, thermal, 15–16, 17 Falkner, V W., 336, 384 Falkner–Skan solution, 336–337 Feigenbaum, M J., 514, 515, 518 Fermi, E., 123, 128 Feynman, R P., 573, 577 Fick’s law of mass diffusion, Finite difference method, 388–392, 396–398 Finite element method element point of view, 398–400 Galerkin’s approximation, 394–396 matrix equations, 396–398 weak or variational form, 393–394 First law of thermodynamics, 12–13 thermal energy equation and, 108–109 Fish, locomotion of, 679–680 Fixed point, 509 Fixed region, mechanical energy equation and, 107 Fixed volume, 78 angular momentum principle for, 92–93 momentum principle for, 88–91 Fjortoft, R., 647, 651 Fjortoft’s theorem, 495–497 Flat plate, boundary layer and Blasius solution, 330–339 closed form solution, 327–330 drag coefficient, 335–336 Fletcher, C A J., 400, 403, 450 Fluid mechanics, applications, 1–2 Fluid statics, 9–12 Flux divergence, 105 Flux of vorticity, 60 Force field, 83 Force potential, 83 Forces conservative body, 83, 136 Coriolis, 103–104 on a surface, 32–35 Forces in fluid body, 83 line, 84 origin of, 82–84 surface, 83 Form drag, 345, 678 Fourier’s law of heat conduction, f-plane model, 588 Franca, L P., 403, 415, 450 Frequency, wave circular or radian, 203 Doppler shifted, 205 intrinsic, 204 observed, 204 Frey S L., 415, 450 Friction drag, 335–336, 658, 678 Friction, effects in constant-area ducts, 717–720 Friedrichs K O., 385, 733 Froude number, 233, 265, 274 internal, 274–275 Fry R N., 720, 732 Fully developed flow, 280 Fuselage, 654 Gad-el-Hak, M., 557, 577 Galerkin least squares (GLS), 415 Galerkin’s approximation, 394–396 Gallo, W F., 330, 384 Gas constant defined, 16–17 universal, 16 Gas dynamics, 653 See also Compressible flow Gases, 3–4 Gauge functions, 366–368 Gauge pressure, defined, Gauss’ theorem, 42–45, 77 Geophysical fluid dynamics approximate equations for thin layer on rotating sphere, 585–588 background information, 579–581 baroclinic instability, 639–647 barotropic instability, 637–638 Ekman layer at free surface, 593–598 Ekman layer on rigid surface, 598–601 equations of motion, 583–585 geostrophic flow, 588–593 gravity waves with rotation, 612–615 Kelvin waves, 615–619 normal modes in continuous stratified layer, 603–610 Rossby waves, 632–637 shallow-water equations, 601–603, 610–611 vertical variations of density, 581–583 vorticity conservation in shallow-water theory, 619–622 750 Index George W K., 557, 577–578 Geostrophic balance, 589 Geostrophic flow, 588–593 Geostrophic turbulence, 647–650 Ghia, U., 416, 450 Ghia, K N., 450 Gill, A E., 243, 254, 261, 612, 635, 636, 637, 651 Glauert, M B., 362, 384 Glowinski scheme, 413–414 Glowinski, R., 413, 414, 415, 450 Gnos, A V., 384 Goldstein, S., 384, 484 G¨ortler vortices, 476 Gower J F R., 350, 384 Grabowski, W J., 504, 518 Gradient operator, 37 Gravity force, effective, 102 Gravity waves deep water, 216–217 at density interface, 240–243 dispersion, 209, 227–231, 254–256 energy issues, 256–259 equation, 200–201 finite amplitude, 236–238 in finite layer, 244–246 group velocity and energy flux, 224–227 hydraulic jump, 233–235 internal, 251–254 motion equations, 248–251 nonlinear steepening, 231–233 parameters, 202–205 refraction, 218–219 with rotation, 612–615 shallow water, 217–219, 246–248 standing, 222–224 Stokes’ drift, 238–240 in stratified fluid, 251–254 surface, 205–209, 209–215 surface tension, 219–222 Gresho, P M., 401, 450 Group velocity concept, 215, 224–231 of deep water wave, 216–217 energy flux and, 224–227 Rossby waves, 635–636 wave dispersion and, 227–231 Half-body, flow past a, 164–165 Hardy, G H., Harlow, F H., 407, 450 Harmonic function, 156 Hatsopoulos, G N., 23 Hawking, S W., 675, 684 Hayes, W D., xvii, 711, 732 Heat diffusion, 279 Heat equation, 108–109 Boussinesq equation and, 119–121 Heat flux, turbulent, 535 Heating, effects in constant-area ducts, 717–720 Heisenberg, W., 500, 518, 521 Hele-Shaw, H S., 317 Hele–Shaw flow, 312–314 Helmholtz vortex theorems, 138 Herbert, T., 454, 498, 518 Herreshoff, H C., 682, 684 Hinze, J O., 578 Hodograph plot, 595 Holstein, H., 342, 384 Holton, J R., 99, 128, 632, 651 Homogeneous turbulent flow, 525 Hou, S., 450 Houghton, J T., 632, 638, 651 Howard, L N., 484, 488, 490, 491, 497, 518 Howard’s semicircle theorem, 488–490 Hughes T J R., 400, 403, 444, 450, 577 Hugoniot, Pierre Henry, 706 Huppert, H E., 467, 518 Hydraulic jump, 233–235 Hydrostatics, 11 Hydrostatic waves, 218 Hypersonic flow, 688 Images, method of, 148, 176–177 Incompressible aerodynamics See Aerodynamics Incompressible fluids, 81, 96 Incompressible viscous fluid flow, 400 convection-dominated problems, 402–403 Glowinski scheme, 413–414 incompressibility condition, 404 MAC scheme, 406–410 mixed finite element, 414–416 SIMPLE-type formulations, 410–413 Induced/vortex drag, 670, 673–674 coefficient, 676 Inertia forces, 302 Inertial circles, 615 Inertial motion, 614–615 Inertial period, 587, 614 Inertial sublayer, 555–557 Inertial subrange, 543–545 Inflection point criterion, Rayleigh, 495, 637 inf-sup condition, 414 Initial and boundary condition error, 387 Inner layer, law of the wall, 552–554 Input data error, 387 Instability background information, 453–454 baroclinic, 639–647 barotropic, 637–638 751 Index boundary layer, 500–501, 503–505 centrifugal (Taylor), 471–476 of continuously stratified parallel flows, 484–490 destabilizing effect of viscosity, 501–503 double-diffusive, 467–471 inviscid stability of parallel flows, 494–498 Kelvin–Helmholtz instability, 476–484 marginal versus neutral state, 455 method of normal modes, 454–455 mixing layer, 498–499 nonlinear effects, 505–506 Orr–Sommerfeld equation, 493–494 oscillatory mode, 455, 470–471 pipe flow, 500 plane Couette flow, 500 plane Poiseuille flow, 499–500 principle of exchange of stabilities, 455 results of parallel viscous flows, 498–503 salt finger, 467–470 sausage instability, 517 secondary, 506 sinuous mode, 517 Squire’s theorem, 484, 490, 492–493 thermal (B´enard), 455–466 Integral time scale, 527 Interface, conditions at, 122 Intermittency, 545–547 Internal energy, 12, 108–109 Internal Froude number, 274–275 Internal gravity waves, 200 See also Gravity waves energy flux, 256–259 at interface, 240–243 in stratified fluid, 251–254 in stratified fluid with rotation, 622–632 WKB solution, 624–627 Internal Rossby radius of deformation, 618 Intrinsic frequency, 204, 631 Inversion, atmospheric, 19 Inviscid stability of parallel flows, 494–498 Irrotational flow, 59 application of complex variables, 157–159 around body of revolution, 193–194 axisymmetric, 187–191 conformal mapping, 177–179 doublet/dipole, 162–164 forces on two-dimensional body, 171–176 images, method of, 148, 176–177 numerical solution of plane, 182–187 over elliptic cylinder, 179–180 past circular cylinder with circulation, 168–171 past circular cylinder without circulation, 165–168 past half-body, 164–165 relevance of, 153–155 sources and sinks, 161 uniqueness of, 181–182 unsteady, 113–114 velocity potential and Laplace equation, 155–157 at wall angle, 159–161 Irrotational vector, 38 Irrotational vortex, 66–67, 131–133, 162 Isentropic flow, one-dimensional, 701–704 Isentropic process, 17 Isotropic tensors, 35, 94 Isotropic turbulence, 532 Iteration method, 182–187 Jets, two-dimensional laminar, 357–356 Kaplun, S., 310, 317 Karamcheti, K., 684 Karman See under von Karman Karman number, 557 Keenan, J H., 23 Keller, H B., 436, 450 Kelvin–Helmholtz instability, 476–484 Kelvin’s circulation theorem, 134–138 Kelvin waves external, 615–618 internal, 618–619 Kim, John, 564, 577 Kinematics defined, 50 Lagrangian and Eulerian specifications, 51–53 linear strain rate, 57–58 material derivative, 53–54 one-, two-, and three-dimensional flows, 68–69 parallel shear flows and, 64–65 path lines, 54–55 polar coordinates, 72–73 reference frames and streamline pattern, 56–57 relative motion near a point, 61–64 shear strain rate, 58–59 streak lines, 56 stream function, 69–71 streamlines, 54–56 viscosity, vortex flows and, 65–68 vorticity and circulation, 59–60 Kinetic energy of mean flow, 535–537 of turbulent flow, 537–540 752 Index Kinsman, B., 220, 261 Klebanoff, P S., 507, 518 Kline, S J., 563, 564, 577 Kolmogorov, A N., 499 microscale, 543 spectral law, 272, 543–545 Korteweg–deVries equation, 237 Kronecker delta, 35–36 Krylov V S., 128 Kuethe, A M., 662, 684 Kundu, P K., 597, 651 Kuo, H L., 638, 651 Kutta condition, 659–660 Kutta, Wilhelm, 170 Kutta–Zhukhovsky lift theorem, 170, 173–175, 659 Lagerstrom, P A., 385 Lagrangian description, 52 Lagrangian specifications, 51–52 Lam, S H., 562, 577 Lamb, H., 113, 122, 128 Lamb surfaces, 113 Laminar boundary layer equations, Falkner–Skan solution, 336–337 Laminar flow creeping flow, around a sphere, 303–308 defined, 278 diffusion of vortex sheet, 295–296 Hele–Shaw, 312–314 high and low Reynolds number flows, 301–303 oscillating plate, 298–301 pressure change, 279 similarity solutions, 288–294 steady flow between concentric cylinders, 285–288 steady flow between parallel plates, 280–283 steady flow in a pipe, 283–285 Laminar jet, 357–364 Laminar shear layer, decay of, 378–382 Laminar solution, breakdown of, 337–339 Lanchester, Frederick, 660 lifting line theory, 670–675 Landahl, M., 543, 562, 577, 675, 684 Lanford, O E., 509, 518 Laplace equation, 155 numerical solution, 182–187 Laplace transform, 294 Law of the wall, 552–554 LeBlond, P H., 236, 261, 609, 651 Lee wave, 630–632 Leibniz theorem, 77, 78 Leighton, R B., 577 Lesieur, M., 520, 577 Levich, V G., 122, 128 Liepmann, H W., 232, 261, 686, 713, 732 Lift force, airfoil, 657–659 characteristics for airfoils, 677–679 Zhukhovsky, 666–669 Lifting line theory Prandtl and Lanchester, 670–675 results for elliptic circulation, 675–677 Lift theorem, Kutta–Zhukhovsky, 170, 173–175, 659 Lighthill, M J., 147, 151, 230–231, 232, 236, 261, 280, 317, 675, 679, 684 Limit cycle, 509 Lin C Y., 349, 384, 448, 450, 518 Lin, C C., 448, 500, 518 Linear strain rate, 57–58 Line forces, 84 Line vortex, 130, 296–298 Liquids, 3–4 Logarithmic law, 554–557 Long-wave approximation See Shallow-water approximation Lorenz, E., 452, 511, 512, 513, 515, 518 Lorenz, E model of thermal convection, 511–512 strange attractor, 512–513 Lumley J L., 541, 545, 549, 554, 561, 565, 577 MacCormack, R W., 386, 404, 405, 406, 416, 418, 420, 421, 423, 427, 449, 450 McCreary, J P., 636, 651 Mach, Ernst, 687 angle, 722 cone, 720–722 line, 722 number, 233, 276, 686–687 MAC (marker-and-cell) scheme, 406–410 Magnus effect, 171 Marchuk, G I., 407, 450 Marginal state, 455 Marvin, J G., 384 Mass, conservation of, 79–81 Mass transport velocity, 240 Material derivative, 53–54 Material volume, 78–79 Mathematical order, physical order of magnitude versus, 367 Matrices dimensional, 267–268 multiplication of, 28–29 rank of, 267–268 transpose of, 25 Matrix equations, 396–398 Mean continuity equation, 530 Mean heat equation, 534–535 Mean momentum equation, 530–531 753 Index Measurement, units of SI, 2–3 conversion factors, 734 Mechanical energy equation, 104–107 Mehta, R., 354, 355, 384 Miles, J W., 484, 518 Millikan, R A., 306, 317, 521 Milne-Thomson, L M., 198 Mixed finite element, 414–416 Mixing layer, 498–499 Mixing length, 559–562 Modeling error, 387 Model testing, 272–274 Moilliet, A., 577 Mollo-Christensen, E., 577 Momentum conservation of, 86–88 diffusivity, 279 thickness, 326–327 Momentum equation, Boussinesq equation and, 119 Momentum integral, von Karman, 339–342 Momentum principle, for control volume, 695 Momentum principle, for fixed volume, 88–91 angular, 92–93 Monin, A S., 521, 577 Monin–Obukhov length, 566–567 Moore, D W., 103, 128 Moraff, C A., 711, 732 Morton K W., 393, 450 Munk, W., 632, 651 Mysak L A., 236, 261, 609, 651 Narrow-gap approximation, 474 Navier–Stokes equation, 97–99, 264 convection-dominated problems, 402–403 incompressibility condition, 404 Nayfeh, A H., 366, 368, 383–384, 504, 518 Neumann problem, 182 Neutral state, 455 Newman J N., 682, 684 Newtonian fluid, 94–97 non-, 97 Newton’s law of friction, of motion, 86 Nondimensional parameters determined from differential equations, 263–266 dynamic similarity and, 270–272 significance of, 274–276 Non-Newtonian fluid, 97 Nonrotating frame, vorticity equation in, 136–140 Nonuniform expansion, 369–370 at low Reynolds number, 370 Nonuniformity See also Boundary layers high and low Reynolds number flows, 301–303 Oseen’s equation, 309–312 region of, 370 of Stokes’ solution, 308–312 Normal modes in continuous stratified layer, 603–610 instability, 454–455 for uniform N, 607–610 Normal shock waves, 705–711 Normal strain rate, 57–58 Normalized autocorrelation function, 526 No-slip condition, 278 Noye, J., 392, 450 Nozzle flow, compressible, 701–704 Numerical solution Laplace equation, 182–187 of plane flow, 182–187 Oblique shock waves, 722–726 Observed frequency, 631 Oden, J T., 414, 450 One-dimensional approximation, 68 One-dimensional flow area/velocity relations, 701–704 equations for, 692–695 Order, mathematical versus physical order of magnitude, 367 Ordinary differential equations (ODEs), 397 Orifice flow, 115–117 Orr–Sommerfeld equation, 493–494 Orszag S A., 366, 368, 384, 454, 498, 518, 562, 578 Oscillating plate, flow due to, 298–301 Oscillatory mode, 455, 470–471 Oseen, C W., 308, 309, 310, 312, 317, 346, 353 Oseen’s approximation, 309–312 Oseen’s equation, 309 Oswatitsch, K., 744 Outer layer, velocity defect law, 554 Overlap layer, logarithmic law, 554–557 Panofsky, H A., 560, 565, 577 Panton, R L., 385 Parallel flows instability of continuously stratified, 484–490 inviscid stability of, 494–498 results of viscous, 498–503 Parallel plates, steady flow between, 280–283 Parallel shear flows, 64–65 Particle derivative, 53 Particle orbit, 613–614, 627–629 Pascal’s law, 11 754 Index Patankar, S V., 410, 411, 412, 432, 450 Path functions, 13 Path lines, 54–56 Pearson J R A., 310, 317 Pedlosky, J., 99, 128, 152, 585, 598, 640, 647, 649, 651 Peletier, L A., 330, 384 Perfect differential, 181 Perfect gas, 16–17 Permutation symbol, 35 Perturbation pressure, 210 Perturbation techniques, 366 asymptotic expansion, 368–369 nonuniform expansion, 369–370 order symbols/gauge functions, 366–368 regular, 370–373 singular, 373–377 Perturbation vorticity equation, 640–642 Petrov–Galerkin methods, 395 Peyret, R., 409, 410, 450 Phase propagation, 636 Phase space, 509 Phenomenological laws, Phillips, O M., 226, 238, 261, 570, 577, 632, 652 Physical order of magnitude, mathematical versus, 367 Pipe flow, instability and, 500 Pipe, steady laminar flow in a, 283–285 Pitch axis of aircraft, 655 Pi theorem, Buckingham’s, 268–270 Pitot tube, 114–115 Plane Couette flow, 282, 500 Plane irrotational flow, 182–187 Plane jet self-preservation, 548–549 turbulent kinetic energy, 549–550 Plane Poiseuille flow, 282–283 instability of, 499–500 Planetary vorticity, 144, 145, 587 Planetary waves See Rossby waves Plastic state, Pohlhausen, K., 328, 339, 342, 384 Poincar´e, Henri, 515 Poincar´e waves, 612 Point of inflection criterion, 343 Poiseuille flow circular, 283–285 instability of, 499–500 plane laminar, 282–283 Polar coordinates, 72–73 cylindrical, 737–738 plane, 739 spherical, 739–741 Pomeau, Y., 509, 512, 515, 518 Potential, complex, 158 Potential density gradient, 21, 565 Potential energy baroclinic instability, 645–647 mechanical energy equation and, 106–107 of surface gravity wave, 214 Potential flow See Irrotational flow Potential temperature and density, 19–21 Potential vorticity, 621 Prager, W., 49 Prandtl, L., 2, 23, 75, 152, 171, 195, 198, 319, 331, 366, 381, 459, 484, 511, 514, 521, 522, 555, 561, 569, 580, 652, 671, 675, 684 biographical information, 742–743 mixing length, 559–562 Prandtl and Lanchester lifting line theory, 670–675 Prandtl–Meyer expansion fan, 726–728 Prandtl number, 276 turbulent, 566 Pressure absolute, coefficient, 165, 266 defined, 5, drag, 658, 678 dynamic, 115, 279–280 gauge, stagnation, 115 waves, 200, 689 Pressure gradient boundary layer and effect of, 342–343, 500–501 constant, 281 Principal axes, 40, 61–64 Principle of exchange of stabilities, 455 Probstein, R F., 122, 128 Profile drag, 678 Proudman theorem, Taylor-, 591–593 Proudman, I., 310, 317 Quasi-geostrophic motion, 633–634 Quasi-periodic regime, 515 Raithby, G D., 411, 413, 450 Random walk, 573–574 Rankine, W.J.M., 706 vortex, 67–68 Rankine–Hugoniot relations, 706 Rayleigh equation, 494 inflection point criterion, 495, 637 inviscid criterion, 471–472 number, 456 Rayleigh, Lord (J W Strutt), 124, 128 Reduced gravity, 247 Reducible circuit, 181 755 Index Refraction, shallow-water wave, 218–219 Regular perturbation, 370–373 Reid W H., 454, 456, 466, 476, 482, 497–8, 518, 651 Relative vorticity, 620 Relaxation time, molecular, 12 Renormalization group theories, 539 Reshotko, E., 505, 518 Reversible processes, 13 Reynolds analogy, 566 decomposition, 529–530 experiment on flows, 278 similarity, 549 stress, 531–534 transport theorem, 79 Reynolds W C., 278, 521, 562, 577 Reynolds, O., 498 Reynolds number, 154, 265, 274, 346 high and low flows, 301–303, 346, 349–352 Rhines, P B., 649, 652 Rhines length, 649–650 Richardson, L F., 522 Richardson number, 275, 565–566 criterion, 487–488 flux, 565 gradient, 275, 488 Richtmyer, R D., 450 Rigid lid approximation, 608–610 Ripples, 222 Roll axis of aircraft, 655 Root-mean-square (rms), 525 Rosenhead, L., 330, 342, 384 Roshko A., 232, 261, 686, 713, 732 Rossby number, 589 Rossby radius of deformation, 618 Rossby waves, 632–637 Rotating cylinder flow inside, 287–288 flow outside, 286–287 Rotating frame, 99–104 vorticity equation in, 141–145 Rotation, gravity waves with, 612–615 Rotation tensor, 62 Rough surface turbulence, 557–558 Ruelle, D., 515, 518 Runge–Kutta technique, 333, 397 Saad, Y., 416, 450 Sailing, 680–682 Salinity, 20 Salt finger instability, 467–470 Sands, M., 577 Sargent, L H., 507, 518 Saric W S., 504, 518 Scalars, defined, 24 Scale height, atmosphere, 21 Schlichting, H., 317, 321, 342, 384, 454, 500, 504 Schlieren method, 687 Schraub, F A., 577 Schwartz inequality, 526 Scotti, R S., 518 Secondary flows, 365–366, 476 Secondary instability, 506 Second law of thermodynamics, 14–15 entropy production and, 109–110 Second-order tensors, 29–31 Seiche, 223 Self-preservation, turbulence and, 547–549 Separation, 343–346 Serrin, J., 330, 384 Shallow-water approximation, 246–248 Shallow-water equations, 601–603 high and low frequencies, 610–611 Shallow-water theory, vorticity conservation in, 619–622 Shames, I H., 198 Shapiro, A H., 686, 733 Hele-Shaw, H S., 277, 312, 314, 317 Shear flow wall-bounded, 551–559 wall-free, 545–551 Shear production of turbulence, 537, 540–543 Shear strain rate, 55 Shen, S F., 500, 504, 505, 518 Sherman, F S., 342, 384 Shin, C T., 450 Shock angle, 723 Shock structure, 691, 705 Shock waves normal, 705–711 oblique, 722–726 structure of, 709–711 SI (syst`eme international d’unit´es), units of measurement, 2–3 conversion factors, 734 Similarity See also Dynamic similarity geometric, 264 kinematic, 264 Similarity solution, 263 for boundary layer, 330–337 decay of line vortex, 296–298 diffusion of vortex sheet, 295–296 for impulsively started plate, 288–295 for laminar jet, 357–364 SIMPLER formulation, 427–436 SIMPLE-type formulations, 410–413 Singly connected region, 181 Singularities, 158 Singular perturbation, 373–378, 500 Sink, boundary layer, 327–330 Skan, S W., 336, 384 756 Index Skin friction coefficient, 335–336 Sloping convection, 646 Smith, L M., 562, 577 Smits A J., 556, 578 Solenoidal vector, 38 Solid-body rotation, 65–66, 131 Solids, 3–4 Solitons, 237–238 Sommerfeld, A., 30, 49, 138, 149, 151, 521, 687 Sonic conditions, 697 Sonic properties, compressible flow, 696–700 Sound speed of, 15, 17, 689–692 waves, 689–692 Source-sink axisymmetric, 192 near a wall, 176–177 plane, 161 Spalding D B., 410, 450 Spatial distribution, 10 Specific heats, 13–14 Spectrum energy, 528 as function of frequency, 528 as function of wavenumber, 528 in inertial subrange, 543–545 temperature fluctuations, 568–569 Speziale, C G., 562, 577 Sphere creeping flow around, 303–305 flow around, 192–193 flow at various Re, 353 Oseen’s approximation, 309–312 Stokes’ creeping flow around, 303–305 Spiegel, E A., 117, 128 Sports balls, dynamics of, 354–357 Squire’s theorem, 484, 490, 492–493 Stability, 390–393 See also Instability Stagnation density, 697 Stagnation flow, 160 Stagnation points, 155 Stagnation pressure, 115, 696 Stagnation properties, compressible flow, 696–700 Stagnation temperature, 696 Standard deviation, 525 Standing waves, 222–224 Starting vortex, 661–662 State functions 13, 15 surface tension, 8–9 Stationary turbulent flow, 525 Statistics of a variable, 525 Steady flow Bernoulli equation and, 112–113 between concentric cylinders, 285–288 between parallel plates, 280–283 in a pipe, 283–285 Stern, M E., 467, 518 Stewart, R W., 577 Stokes’ assumption, 96 Stokes’ creeping flow around spheres, 297–302 Stokes’ drift, 238–240 Stokes’ first problem, 288 Stokes’ law of resistance, 271, 306 Stokes’ second problem, 299 Stokes’ stream function, 190 Stokes’ theorem, 45–46, 60 Stokes’ waves, 236–237 Stommel, H M., 103, 128, 467, 518, 601 Strain rate linear/normal, 57–58 shear, 58–59 tensor, 59 Strange attractors, 512–513 Stratified layer, normal modes in continuous, 603–610 Stratified turbulence, 522 Stratopause, 582 Stratosphere, 581–582 Streak lines, 56 Streamfunction generalized, 81–82 in axisymmetric flow, 190–191 in plane flow, 69–71 Stokes, 190 Streamlines, 54–56 Stress, at a point, 84–86 Stress tensor deviatoric, 94 normal or shear, 84 Reynolds, 532 symmetric, 84–86 Strouhal number, 348 Sturm–Liouville form, 605 Subcritical gravity flow, 233 Subharmonic cascade, 513–515 Sublayer inertial, 555–557 streaks, 563 viscous, 553–554 Subrange inertial, 543–545 viscous convective, 569 Subsonic flow, 276, 687 Substantial derivative, 53 Sucker, D., 436, 450 Supercritical gravity flow, 233 Supersonic flow, 276, 688 airfoil theory, 728–731 expansion and compression, 726–728 Surface forces, 83, 86 757 Index Surface gravity waves, 200, 205–209 See also Gravity waves in deep water, 216–217 features of, 209–215 in shallow water, 217–219 Surface tension, 8–9 Surface tension, generalized, 122 Sverdrup waves, 612 Sweepback angle, 655, 671 Symmetric tensors, 38–39 eigenvalues and eigenvectors of, 40–42 Takami, H., 436, 450 Takens F., 515, 518 Taneda, S., 348, 384 Tannehill, J C., 406, 450 Taylor T D., 2, 23, 409–410, 450, 471, 521, 561, 569, 574, 577, 600, 652, 743–744 Taylor, G I., 2, 23, 577, 652, 743–744 biographical information, 743–744 centrifugal instability, 471–476 column, 592 hypothesis, 529 number, 474, 476 theory of turbulent dispersion, 569–576 vortices, 476 Taylor–Goldstein equation, 484–486 Taylor–Proudman theorem, 591–593 TdS relations, 15 Temam, R., 410, 450 Temperature adiabatic temperature gradient, 19, 581 fluctuations, spectrum, 568–569 potential, 19–21 stagnation, 696 Tennekes, H., 541, 545, 549, 554, 577, 656 Tennis ball dynamics, 356–357 Tensors, Cartesian boldface versus indicial notation, 47 comma notation, 46–47 contraction and multiplication, 31–32 cross product, 36–37 dot product, 36 eigenvalues and eigenvectors of symmetric, 40–42 force on a surface, 32–35 Gauss’ theorem, 42–45 invariants of, 31 isotropic, 35, 94 Kronecker delta and alternating, 35–36 multiplication of matrices, 28–29 operator del, 37–38 rotation of axes, 25–28 scalars and vectors, 24–25 second-order, 29–31 Stokes’ theorem, 45–46 strain rate, 57–59 symmetric and antisymmetric, 38–39 vector or dyadic notation, 47–48 Tezduyar, T E., 415, 450 Theodorsen’s method, 662 Thermal conductivity, Thermal convection, Lorenz model of, 511–512 Thermal diffusivity, 109, 120 Thermal energy equation, 108–109 Boussinesq equation and, 119–121 Thermal energy, 12–13 Thermal expansion coefficient, 15–16, 17 Thermal instability (B´enard), 455–466 Thermal wind, 589–591 Thermocline, 583 Thermodynamic pressure, 94 Thermodynamics entropy relations, 15 equations of state, 13, 16 first law of, 12–13, 108–109 review of, 688–689 second law of, 14–15, 109–110 specific heats, 13–14 speed of sound, 15 thermal expansion coefficient, 15–16, 17 Thin airfoil theory, 662, 728–731 Milne-Thomson, L M., 198 Thomson, R E., 384 Thorpe, S A., 482, 518 Three-dimensional flows, 68–69 Thwaites, B., 342, 384 Tidstrom, K D., 507, 518 Tietjens, O G., 23, 75, 152, 684 Time derivatives of volume integrals general case, 77–78 fixed volume, 78 material volume, 78–79 Time lag, 526 Tip vortices, 670 Tollmien–Schlichting wave, 454, 500 Townsend, A A., 545, 547, 549, 550, 577 Trailing vortices, 670, 671–672 Transition to turbulence, 344–345, 506–508 Transonic flow, 687–688 Transport phenomena, 5–7 Transport terms, 105 Transpose, 25 Tropopause, 581 Troposphere, 581 Truesdell, C A., 96, 128 Turbulent flow/turbulence averaged equations of motion, 529–535 averages, 522–525 buoyant production, 539–540, 565 cascade of energy, 542 characteristics of, 520–521 coherent structures, 562–563 commutation rules, 524–525 758 Index Turbulent flow/turbulence (continued) correlations and spectra, 525–529 defined, 272 dispersion of particles, 569–573 dissipating scales, 542 dissipation of mean kinetic energy, 536 dissipation of turbulent kinetic energy, 540 eddy diffusivity, 560–562 eddy viscosity, 559–562 entrainment, 547 geostrophic, 647–650 heat flux, 535 homogeneous, 525 inertial sublayer, 555–557 inertial subrange, 543–545 integral time scale, 527 intensity variations, 558–559 intermittency, 545–547 isotropic, 532–533 in a jet, 548–551 kinetic energy of, 537–540 kinetic energy of mean flow, 535–537 law of the wall, 552–554 logarithmic law, 554–557 mean continuity equation, 530 mean heat equation, 534–535 mean momentum equation, 530–531 mixing length, 559–562 Monin–Obukhov length, 566–568 research on, 521–522 Reynolds analogy, 566 Reynolds stress, 531–534 rough surface, 557–558 self-preservation, 547–549 shear production, 537, 540–543 stationary, 525 stratified, 565–569 Taylor theory of, 569–576 temperature fluctuations, 568–569 transition to, 344–345, 506–508 velocity defect law, 554 viscous convective subrange, 569 viscous sublayer, 553–554 wall-bounded, 551–559 wall-free, 545–551 Turner J S., 235, 238, 254, 261, 467–8, 483, 518, 566–7, 578 Two-dimensional flows, 68–69, 171–176 Two-dimensional jets See Jets, two-dimensional, 357–364 Unbounded ocean, 615 Uniform flow, axisymmetric flow, 191 Uniformity, 109 Unsteady irrotational flow, 113–114 Upwelling, 619 Vallentine, H R., 198 Vapor trails, 670 Variables, random, 522–525 Variance, 525 Vector(s) cross product, 36–37 curl of, 37 defined, 24–28 divergence of, 37 dot product, 36 operator del, 37–38 Velocity defect law, 554 Velocity gradient tensor, 61 Velocity potential, 113, 155–157 Veronis G., 117, 128 Vertical shear, 589 Vidal, C., 509, 512, 515, 518 Viscoelastic, Viscosity coefficient of bulk, 96 destabilizing, 490 dynamic, eddy, 559–562 irrotational vortices and, 130–134 kinematic, net force, 132, 133 rotational vortices and, 129–130 Viscous convective subrange, 569 Viscous dissipation, 105–106 Viscous fluid flow, incompressible, 400–416 Viscous sublayer, 553–554 Vogel, W M., 577 Volumetric strain rate, 57 von Karman, 23, 384, 521–2, 660, 675, 684, 733, 744 constant, 555 momentum integral, 339–342 vortex streets, 254, 347–349 Vortex bound, 674–675 decay, 296–298 drag, 646, 670, 673–674 G¨ortler, 476 Helmholtz theorems, 138 interactions, 146–149 irrotational, 162 lines, 130, 296–298 sheet, 149–150, 295–296, 480, 670 starting, 661–662 stretching, 145, 621 Taylor, 476 tilting, 145, 598, 621 tip, 670 trailing, 670, 671–672 tubes, 130 von Karman vortex streets, 254, 347–349 759 Index Vortex flows irrotational, 66–67 Rankine, 67–68 solid-body rotation, 65–66 Vorticity, 59–60 absolute, 144, 620–621 baroclinic flow and, 136–138 diffusion, 136, 279, 295–296 equation in nonrotating frame, 138–140 equation in rotating frame, 141–146 flux of, 60 Helmholtz vortex theorems, 138 Kelvin’s circulation theorem, 134–138 perturbation vorticity equation, 640–642 planetary, 144, 145, 587 potential, 621 quasi-geostrophic, 633–634 relative, 620 shallow-water theory, 619–622 Wall angle, flow at, 159–161 Wall-bounded shear flow, 551–559 Wall-free shear flow, 545–551 Wall jet, 362–364 Wall, law of the, 552–554 Wall layer, coherent structures in, 562–564 Water, physical properties of, 735 Wavelength, 202 Wavenumber, 202, 203 Waves See also Internal gravity waves; Surface gravity waves acoustic, 689 amplitude of, 202 angle, 723 capillary, 219 cnoidal, 237 compression, 200 deep-water, 216–217 at density interface, 240–243 dispersive, 209, 227–231, 254–256 drag, 273, 673, 730–731 elastic, 200, 689 energy flux, 215, 227–231 equation, 200–202 group speed, 215, 227–231 hydrostatic, 218 Kelvin, 615–619 lee, 630–632 packet, 226–227 parameters, 202–205 particle path and streamline, 210–213 phase of, 200 phase speed of, 203 Poincar´e, 612 potential energy, 214 pressure, 200, 689 pressure change, 210 refraction, 218–219 Rossby, 632–637 shallow-water, 217–218 shock, 705–711 solitons, 237–238 solution, 642 sound, 689–692 standing, 222–224 Stokes’, 236–237 surface tension effects, 219–222 Sverdrup, 612 Wedge instability, 646–647 Welch J E., 450 Wen, C Y., 349, 384, 448, 450 Whitham, G B., 236, 261 Wieghardt K., 744 Williams, G P., 649, 652 Wing(s) aspect ratio, 655 bound vortices, 671–672 drag, induced/vortex, 670, 673–674 delta, 679 finite span, 669–670 lift and drag characteristics, 677–679 Prandtl and Lanchester lifting line theory, 670–675 span, 655 tip, 655 tip vortices, 670 trailing vortices, 670, 671–672 WKB approximation, 624–627 Woods J D., 481, 518, 566, 577 Wosnik, M., 556, 578 Yaglom A M., 521, 577 Yahya, S M., 733 Yakhot, V., 562, 578 Yanenko, N N., 407, 450 Yaw axis of aircraft, 655 Yih, C S., 342, 384, 500, 518 Zagarola, M V., 556, 578 Zhukhovsky, N., airfoil lift, 666–669 hypothesis, 660 lift theorem, 170, 173–175, 659 transformation, 663–666 Zone of action, 722 Zone of silence, 722 [...]... Deep Fluids Waves in a Finite Layer Overlying an Infinitely Deep Fluid Shallow Layer Overlying an Infinitely Deep Fluid Equations of Motion for a Continuously Stratified Fluid Internal Waves in a Continuously Stratified Fluid Dispersion of Internal Waves in a Stratified Fluid Energy Considerations of Internal Waves in a Stratified Fluid. .. Reading 14 15 15 15 16 17 19 21 22 23 23 1 Fluid Mechanics Fluid mechanics deals with the flow of fluids Its study is important to physicists, whose main interest is in understanding phenomena They may, for example, be interested in learning what causes the various types of wave phenomena in the atmosphere and in the ocean, why a layer of fluid heated from below breaks up into cellular patterns,... contrast, a fluid deforms 4 Introduction Figure 1.1 Deformation of solid and fluid elements: (a) solid; and (b) fluid continuously under the action of a shear force, however small Thus, the element of the fluid ABCD confined between parallel plates (Figure 1.1b) deforms to shapes such as ABC′ D′ and ABC′′ D′′ as long as the force F is maintained on the upper plate Therefore, we say that a fluid flows... own research interests The material selected is what I believe to be of the most interest in a book on general xx Preface to First Edition fluid mechanics It includes topics of special interest to geophysicists (for example, the chapters on Gravity Waves and Geophysical Fluid Dynamics) and to engineers (for example, the chapters on Aerodynamics and Compressible Flow) There are also chapters of common... the reader can follow up with specialized texts for a more comprehensive understanding An historical survey of fluid mechanics from the time of Archimedes (ca 250 B.C.E.) to approximately 1900 is provided in the Eleventh Edition of The Encyclopædia Britannica (1910) in Vol XIV (under “Hydromechanics,” pp 115–135) I am grateful to Professor Herman Gluck (Professor of Mathematics at the University of... high-speed motion of a viscous fluid was apparently too recent for its importance to have been realized IMC Chapter 1 Introduction Fluid Mechanics 1 Units of Measurement 2 Solids, Liquids, and Gases 3 Continuum Hypothesis 4 Transport Phenomena 5 Surface Tension 8 Fluid Statics ... computational fluid dynamics, graciously provided an entirely new chapter, Chapter 11, thereby providing the student with an entree into this exploding new field Both finite difference and finite element methods are introduced and a detailed worked-out example of each is provided I have been a student of fluid mechanics since 1954 when I entered college to study aeronautical engineering I have been teaching fluid. .. from below breaks up into cellular patterns, why a tennis ball hit with “top spin” dips rather sharply, how fish swim, and how birds fly The study of fluid mechanics is just as important to engineers, whose main interest is in the applications of fluid mechanics to solve industrial problems Aerospace engineers may be interested in designing airplanes that have low resistance and, at the same time, high... engineering I have been teaching fluid mechanics since 1963 when I joined the Brown University faculty, and I have been teaching a course corresponding to this book since moving to the University of Pennsylvania in 1966 I am most grateful to two of my own teachers, Professor Wallace D Hayes (1918–2001), who expressed xviii xix Preface to Second Edition fluid mechanics in the clearest way I have ever... Hypothesis A fluid, or any other substance for that matter, is composed of a large number of molecules in constant motion and undergoing collisions with each other Matter is therefore discontinuous or discrete at microscopic scales In principle, it is possible to study the mechanics of a fluid by studying the motion of the molecules themselves, as is done in kinetic theory or statistical mechanics However, .. .Fluid Mechanics, Third Edition Founders of Modern Fluid Dynamics Ludwig Prandtl (1875–1953) G I Taylor (1886–1975) (Biographical... of Physics, 2000 Fluid Mechanics Third Edition Pijush K Kundu Oceanographic Center Nova University Dania, Florida Ira M Cohen Department of Mechanical Engineering and Applied Mechanics University... Supplemental Reading 14 15 15 15 16 17 19 21 22 23 23 Fluid Mechanics Fluid mechanics deals with the flow of fluids Its study is important to physicists, whose main interest is