Water Wave Mechanics For Engineers and Scientists

 

WATER WAVE MECHANIC MECHANICSS FOR ENGI NEERS A N D SCIENTISTS

 

ADVANCED SERIES ON OCEAN ENGINEERING Series Editor-in-Chief Philip L- F Liu

Cornell University, USA

Vol.

1

The Applied Dynamics of Ocean Surface Waves

b y Chiang C Mei (MIT, USA) for Engineers and Scientists by Robert G Dean (Univ. Florida, USA) and Robert A (Univ. Delaware. U S A ) Vol. 3 Mechanics of Coastal Sediment Transport b y hrgen Fredsue and Rolf Deigaard (Tech. Univ. De Vol. 4 Coastal Bottom Boundary Layers and Sediment Transp by Peter Nielsen (Univ. Queensland, Australia)

Vol.

2 Water Wave Mechanics

Forlhcorning titles: Water Waves Propagation Over Uneven Bottoms by Maarten W Dingemans (Delft Hydraulics, The Netherlands) Ocean Outfall Design by / an R Wood (Univ. Canterbury, New Zealand)

Tsunami Run-up by Philip L - F Liu (Cornell Univ.), Cosras Synolakis (Univ. Sou Harry Yeh (Univ. Washington) a n d Nobu Shuto (Tohoku Univ.)

Physical Modules and Laboratory Techniques in Coastal Engin by Steven A . Hughes (Coastal Engineering Research Center,

 

Advan Adv ance ced dS Seri eries es on Ocean Engineering

- olume 2

Robert G . Dean University of Florida

Robert A . Dalrymple University of Delaware

World Scientific

ingapore .New .New Jersey. Jerse y. London. Hong Kong

 

Published by

W orld Scientific Publishing Co. R e . t t d . P 0 Box 128, Farrer Road, Singapore 912805 USA oftice: Suite IS, 1060 Main Main S treet, River Edge, N J 07561

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Library of Congress Cataloging-in-Publica~ianData Dean, Robert G . (Robert George), 1930W ater wave mechanics for engineers and scientists scientists Rob ea G. I and Robert A . Dalryrnple. p. cm. Includes bibliographical bibliographical referenc es and ind ex. ISBN 9810204205. - - I S B N 9810204213 (pbk.) 2. Fluid mechanics. I . Dalrym ple, Robert A , , 1 . Water waves. I I . Title 1945TC172.D4 1991 627'.042--dc20 90-27331 CIP

First published in 1984 by Prentice Ha l l , Inc

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Printed in Singapore.

 

xi

PREFACE

INTRODUCTION TO WA VE MECHANlCS MECHANlCS

1.1 1.2

Introduction, 2 Characteristics of Waves, 2

1.3

Historical and Present Literature, 4

A REVIEW OF HYDRODYNAMICS AND VECTOR ANALYSIS 2.1 introduction, 7 2.2 Review of Hydrodynamics, 7 2.3 Review of Vector Analysis, 79 2.4 Cylindrical Coordinates, 32 References, 36 Problems, 37

SMALL-AMPLITUDE WATER WAVE THEO THEORY RY FORMUL A TION A N D SOLUTION 3.1 3.1 intro introducti duction, on, 42 3.2 Boundary Value Problems, 42

1

6

41

vii

 

viii

Contents

3.3

Summary of the Two-Dimensional Two-Dimens ional Periodic Per iodic Water Water Wave Wave Boundary Value Problem, 52 Linearized ized Water Water Wave Boundary Bounda ry Value Value 3.4 Solution to Linear Problem for a Horizontal Bottom, 53 Appendix: ix: Approximate Solutions to the Dispersion Dis persion 3.5 Append Equation, 77 References, 73 Problems, 73

4 ENGINEERING WAVE PROPERTIES

4.1 4.2 4.3 4.4 4.5

Introduction, 79 Water Wat er Particle Kinematics for Progressive Waves Waves,, 79 Pressure Field Fi eld Under a Progressive Wave Wave,, 83 Waterr Particle Wate Parti cle Kinematic Kine matics s for Standing Standi ng Waves, Waves, 86 Pressure Field Under a Standing Wave, Wave, 89

4.6 4.7 4.8 4.9 4.10

Parti al Standing Partial Stan ding Waves, Waves, 90 Energy and Energy Propagation in Progressive Waves, Waves, 93 Transformation of Wave Waves s Entering Enter ing Shallow Water, Water, 700 Wave Diffraction, 176 Combined Refraction-Diffraction, Refraction-Diffracti on, 723 References, 124 Problems, 126

78

5 LONG LON G WA VES

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Introduction, 132 Asymptotic Long Waves, 732 Long Wave Theory, 133 One-Dimensional Tides in Idealized Ideal ized Channels, 738

Reflection and Transmission Past an Abrupt Transition, Transition, 147 Long Waves Waves with Bottom Friction, 146 Geostrophic Effects on Long Waves Waves,, 154 Long Wav Waves es in Irregular-Shaped Basins or Bays, 157 Storm Surge, 157 Long Wav Waves es Forced by a Moving Atmospheric Pressure Disturbance, Disturbance, 163 Waves s Forced Force d by a Translating Bottom Bot tom 5.11 Long Wave Displacement, 766 References, 167 Problems, 767

131

 

ix

Contents

WA VEMA K ER THEO THEORY RY 6.1 Introduction, 770

6.2

Simplified Theory for Plane Plane Wavemakers Wavemakers in Shallow Water, 777

6.3

Compl ete Wavema Complete Wavemaker ker Theory for Plane Waves Produced by a Paddle Paddle,, 772 Cylindric Cyli ndrical al Wavemakers, 780 Plunger Wavemakers, 784 References, 785 Problems, 785

6.4 6.5 6.5

170

7 WA VE STA TISTICS A ND SPECT SPECTRA RA 7.1 Introduction, 787 7.2 Wave Height Distri Dis tribut bution ions, s, 788 7.3 The Wave Spectr Spe ctrum, um, 793

7.4 7.5 7.6

The Directiona Direc tionall Wave Wave Spectrum, Spect rum, 202 Time-Series Simulation Simul ation,, 207 Examples of Use of Spectral Methods to Determine Momentum Flux, 208 References, 209 Problems, 270

WAVE FORCES

8.1

8.2 8.2 8.3 8.4

8.5

212 21 2

Introduction, 272 Potential Flow Approach, 273 Forces Due to Real Fluids, 227 Inertia Force Predominant Pr edominant Case, Case, 237 Spectral Approach to Wave Force Prediction, 254 References, 255 Problems, 257

WA VES OVER OVER REAL SEAB EDS 9.1 Introduction, 261

9.2 9.3

187

Waves Over Smooth, Rigid, Impermeable Bottoms, 262 Water Waves Over a Viscous Mud Bottom, 277

26 1

 

Contents

X

9.4

Waves Over Rigid, Porous Bottoms, 277 References, 282 Problems, 282

10 NONLINEAR NONLINE AR PROPERTIE PROPERTIES S DERIVA B L E FRO M SM AL L - AM PL I T UD UD E WA VES 10.1 introduction, 285 10.2 10 .2 Mass Transport and and Momentum Flux, 285 10.3 10 .3 Mean Water Level, Level , 287

284

10.4 Mean Pressure, 288 10.5 10 .5 Momentum Moment um Flux, 289 10.6 Summary, 293 References, 293 Problems, 294

11 NONLINE NON LINEAR AR WA VES 11.1 Introduction, 296 11.2 Perturbati Pertu rbation on Approach Approa ch of Stokes, 296 11.3 1.3 The Stream Str eam Function Funct ion Wave Theory, Theory , 305 11.4 1.4 Finite Fin ite-Am -Ampli plitud tude e Waves in Shallow Shall ow Water, Water, 309 11.5 1.5 The Validi Val idity ty of Nonlinear Wave Wave Theories, Theories , 322

295

References, 324 Problems, 325

A SERIES OF EXPERIMENTS FOR A L AB ORATORY ORATORY COURSE COU RSE COMPONE NT I N WATER WAVES 12.1 Introduction, 326 12.2 12 .2 Req Requir uired ed Equipment, Equip ment, 326 12.3 Experiments, 330

326

References, 345 SUBJECT INDEX

347

AUTHOR INDEX

35 1

 

C The initial substantive substan tive interest in an d contributions to water wave mechanics date from more th an a century ago, beginning with the analysis of of linear wave theory theo ry by Airy in 1845 an d continuing with higher order theories by Stokes in 1847, long wave wave theories by Boussinesq in 1872, an and d limiting wave heights by Michell i n 1893 an d McCowan McCowan i n 1894. Following that tha t half-century of pioneering pioneering developments, developm ents, research continued at a t a relatively relatively slow pace until the amphib am phibious ious landings in the Second World War emphasized the need for a much better understanding of wave initiation initiatio n and growth due to winds, the conservative conservative an d dissipative transfortransformation mechanisms source area to the attempt shoaling, shoaling, the breaking processes atoccurring the shore.from The the largely unsuccessful toand utilize portable and floating breakwaters in the surprise amphibious landing at Normandy, France, stimulated interest in wave interaction with fixed and floating objects. After the Second World War, the activity in water wave research probably would have subsided subsid ed without the rather ra ther explosive growth in oceanrelated engineering in scientific, industrial, and military activities. act ivities. From the 1950s to the 1980s, offshore drilling and production of petroleum resources progress prog ressed ed from water depths de pths of approximately app roximately 10 meters to over 300 meters, platforms for the latter being designed for wave heights on the order of 2 5 meters and costing in excess excess of $700,000,000 $700,000,000(U.S.). incentives (U.S.).The financial incentives of well-planned and comprehensive studies of water wave phenomena became much greater. Laboratory studies as well as much more expensive field fie ld programs were required req uired to validate design methodology methodology and to provide prov ide a better basis for describing the complex and nonlinear directional seas. A second and substantial impetus to nearshore research on water waves has been the interest i n coastal erosion, an area still only poorly poorly understood. For example, although the momentum flux concepts were systematized by LonguetLong uet-Higg Higgins ins an d Stewart and an d applied a pplied to a number of relevant problems xi

 

xi i

Preface

in the 1960s, the usual (spilling wave) assumption of the wave height inside the surf zone being proportional to the water depth avoids the important matter matt er of the distribu dist ribution tion of the applied longshore stress across the surf zone. This can only be reconciled through careful laboratory and field measurements of wave breaking. breaking. Wave energy provides another anot her example. In the last two decades remote sensing has indicated the potential of defining synoptic measures of wave intensity over very wide areas, are as, with the associated benefits to shipping effi efficien ciency. cy. Simple calculations of the magnitudes of the “standing “stand ing crop” of wave energy energy have stimulated sti mulated many scientists and engineer engineerss to devise ingenious mechanisms to harvest this energy. Still, these mechanisms must operate in a harsh environment known for its long-term corrosive and fouling effects an d the high-intensity high-intensity forces during duri ng severe storms. Th e problem problem of quantifying quantifying the wave climate, underst understanding anding the interaction of waves waves with with structures and/o a nd/o r sediment, an d predictin predicting g the associassociated responses of interest underlies almost every problem in coastal and ocean engineering. engineering. It is toward towa rd this goal tha t this book is directed. Although Although the book is intended for use primarily primarily as a text at the advanced undergraduate first-year ar graduate gra duate level, it is hoped that th at it will serve also as a reference reference and o r first-ye will assist one to t o learn l earn the field through throug h self-stu self-study. dy. Toward these objectives, each chapter concludes with a number numb er of problems developed to illu illustrate strate by by application the material presented. The references references included should aid the student stud ent and an d the practicing engineer to extend their knowledge further. The book is comprised of twelve twelve chapters. Chapter 1 presents a number of comm common on examples exa mples illustrating illustra ting the wide range of water water wave phenomena, many of which can be commonly observed. Chapter 2 offers a review of potential flow flow hydrodynamics an d vector anal analysis ysis.. This material is presented completene ss, even though it will be familiar famil iar to many readers. for the sake of completeness, Chapter 3 formulates the linear water wave theory and develops the simplest simplest two-dimensional solution for standing and progressive waves. Chapter 4 extends the the solutions developed in Chapter C hapter 3 to many m any features fe atures of engineering engineering relevance, including kinematics, kinematic s, pressure fields, energ energy, y, shoaling, refraction, and diffraction. Chapter 5 investigates long wave phenomena, such as kinematics, seiching, standing stan ding an d progressive waves with with friction, and an d long waves including geostrophic forces and storm surges. Chapter 6 explores various wavemaker problems, which ar e relevant to problems of wave tank and wave basin design and to problems of damping of floati floating ng bodies. bodies. The utility of spectral analysis to combine many elemental solutions is explored in Chapter 7. In this manner a complex sea comprising a spectrum of frequenciess and, frequencie and , at each frequency, a continu con tinuum um of directions can be represented. Chapter 8 examines the problem proble m of wave forces on struct structures. ures. A slight modification of the problem of two-dimensional idealized flow about a cylinder yields the well-known Morison equation. Both drag- and inertiadominant systems are discussed, including methods for data analysis, and some field dat a are ar e presented. This chapter concludes with a brief description of the Green’s function representation for calculating the forces on large

 

xiii

Preface

bodies. Chapter 9 considers the effects of waves propagating over seabeds which whic h may be porous, viscous, viscous, a nd /o r compressible compressible and at which which frictio frictional nal effects may occur in the bottom boundary layer. Chapter 10 develops a number of nonlinear (to second order in wave height) height) results results that, somewhat surprisingly, surprising ly, may be obtained from linear wave theory. These results, many of which whic h are of engine engineerin ering g concern, include mass transport, transp ort, momentum flux, flux, set-down an d set-up of the mean water wate r level, mean pressure under a progressive wave, wave, an and d the “microseisms,” in-pha in-phase se pressure fluctuations fluctuations that occur under two-dimensional standing waves. Chaper 11 introduces the perturbation method metho d to t o develop and solve various various nonlinear wave theori theories, es, including including the Stokes Stokes second order theory, theory, an d the solitary and an d cnoidal wave wave theories. theories. The procedure for developing numerical wave theories to high order is described, as are the analytical and physical validities of theories. Finally, Chapter 12 presents a number of water wave experiments (requiring only simple simp le instrumentation) that the th e authors have found useful for demonstrating the theory a nd introducing the student to t o wave experiment experimentation, ation, specifispecifically cal ly methodolo methodology, gy, instrumentation, an d frustrations. Each chapter is dedicated to a scientist who contributed importantly impor tantly to this field. Brief biographies biographies were gleaned from such su ch sources as T h eDictionary e Dictionary of National Biography (United Kindom scientists; Cambridge University Press), Dictionary of Scientific Biography (Charles Scribner’s Sons, New York), Neue D eutsche Biogruphie (Helmholtz; Duncker and an d Humblot, Berlin) and The London Times (Havelock). These productive a nd influential individuals are but a few few of those who have laid the foundations of our presentpresent-day day knowledge; however, the biographies illustrate the t he level of effort effort and a nd inteninte nsity of those people and an d the their ir eras, through which great scientif scientific ic stride stridess were made. The authors wish to acknowledge the stimulating discussions and inspiration provided by many of their colleagues and former professors. In particular, Professors R . 0 . Re Reid id,, B. W. Wilson, Wils on, A . T. Ippen, and C . L. Bretschneider were were central in introducing the authors aut hors to t o the field. Numerous focused discussions with M. I? O’Brien have crystallized understanding of water wave wave phenomena a nd their effects on sediment transport. Drs. Todd L. Walton Walt on and Ib A . Svendsen provided valuable reviews of the manuscript, as have a numbe n umberr of students stude nts who have taken the t he Water Water Wave Mechanics course at the University U niversity of Delaware. Mrs. Sue Thom pson deserves great great praise for her cheerful disposition and faultless typing of numerous drafts of the manuscript, as does Mrs. Con Connie nie Weber Weber,, who managed final revision. Finally the general support and encouragement provided by the University of Delaware is appreciated.

Robert G . Dean Robert A . Dalrymple

 

V

Dedication SIR HORACE L A M B Sir Horace Hor ace Lamb 1849-1934) is best known for his extremely extr emely thorough and well-written book, Hydrodynamics, which first appeared in 1879 and has has been reprinted numerous times. It still serves serv es as a compendium of useful usef ul information as well as the source for a great number of papers and books. If this present book has but a small fraction of the appeal of

Hydrodynamics, the authors would be well satisfied.

Sir Horace Lamb was born b orn in Stockport, England in 1849, educated at Owens College, Manchester, and then Trinity College, Cambridge University, where he studied with professors such as J . Clerk Maxwell and G. G. Stokes. After his graduation, he lectured at Trinity Adelaide, e, Australia, Australia, to become Profes1822-1825) and then moved to Adelaid sor of Mathematics. After ten years, he returned to Owens College (part of Victoria University of Manchester) as Professor of Pure Mathematics; he remained remaine d until unt il 1920. Professor Lamb was noted for his excellent teaching and writing abilities. In response to a student tribute on the occasion oc casion of his eightieth birthday, he replied: “I did try to make things clear clear,, first to myself. my self... .and then to my students, and somehow make these dry bones live.” His research areas encompassed tides, waves, and earthquake properties as well as mathematics. 1

 

2

Introduction to W av e Mechanics Mechanics

Chap. 1

1.I NTRODUCTION

Rarely can can one fin find d a body of water open to the atmosphere that does not have waves on its i ts surface. These waves are a manifestation of forces acting on the fluid tending to deform it against the action of gravity and surface tension, which which together act to m maint aintain ain a level fluid surface. surface.Thus Thus it requires a force of some kind, such as would be caused b by y a gust of wind or a falling stone impacting on the water, to create waves. Once these are created, gravitational and surface su rface tension forces are a activated ctivated tthat hat allow the waves to propagate, in the same manner as tension on a string causes the string to vibrate, much to our listening enjoymen enjoyment. t. Waves Wav es occur in all sizes and an d forms, depend depending ing on the magnitude of the forces acting on the wat water. er. A simple illustration is that a small stone and a large rock create different-size waves after impacting on water. Further, different speeds of impact create different-size waves, which indicates that the pressure forces forces ac acting ting on the fluid surface are important, as we well ll as the magnitude of the displaced fluid. The gravitational attraction of the moon, sun, and an d other astronomical bodies creates the longest known water w waves, aves, the tides. These wa waves ves circle half halfway way aro aroun und d the ear earth th from end to end a and nd travel with tremendous speeds. The shortest waves can be less than a centimeter in length. The length of the wave gives one an idea of the magnitude of the forces acting on the waves. For example, the longer the wave, wav e, the more impo importan rtantt gravity (compr (comprised ised o off the contributions from the earth, the moon, and the sun) is in relation to surface tension. The importance of waves cannot be overestimated. Anything that is near or in a body of water is subject to wave action. At the coast, this can result in the movement of sa sand nd along the shore, causing erosion or damage to structures during storms. In tthe he water, offshore oil platforms must be able to withstand severe storms without destruction. At present drilling depths exceeding 300 m, this th is requires enormous and expensive structur structures. es. On the water, all ships are subjected to wave attack, and countless ships have foundered due to t o waves which have been observed tto o be as large as 34 m in height. heig ht. Further, any ship moving through water creates a pressure field and, hence, waves. These waves create a significant portion of the resistance to motion enountered by the ships.

1.2 CHARACTERISTICS OF WAVES

The important parameters parameters to describe wav waves es are their leng length th and height, and the water depth over ove r which they ar are e propagating. Al Alll othe otherr parameters, such as wave-induced water velocities and accelerations, can be determined theoretically from these quantities. In Figure 1.1, a two-dimensional schematic of a wave propagating in the x direction is shown. The length of the wav wave, e,

 

Sec. 1.2

3

Characteristics of W a v e s

. I

h

Trough

Figure 1.1 Wave characteristics.

L , is the horizont horizontal al distance betwee between n two successive wave crest crests, s, or th the e high

points on a wave, or alternatively the distance between two wave troughs. The wave length will be show shown n later to be related to the water depth h and wave period T , which is the time required for two successi successive vecrests or troughs to pass a particular particular point. A s the wave, then, must move a distance L in time he speed of the wave, called the celerity, C, is defined as C L /T . While T , wave the form travels trav els with celerity C, the water that comprises the wave does not translate tran slate in th the e direction of the wave. The coordinate axis that will be used to describe wave motion will be located loca ted at the still w water ater line, z = 0. The bottom of the water body wi will ll b be e at z = -h. Waves in nature rarely appear to look exactly the same from wave to wave, nor do they always propagate in the same direction. If a device to measure the water surface elevation, 9 , as a function func tion of time was pla placed ced on a platform in the middle of the ocean, it might obtain a record such as that shown in Figure 1:2. This sea can be seen to be a superposition of a large number of sinusoids go going ing in dif different ferent directio directions. ns. For example, consider the two sine waves waves shown in Figure 1.3 and their sum. It is this superpo superposition sition of sinusoids that permits th the e use of Fourier analysis and spec spectral tral techniques to be used in describing the sea. Unfortunately, there is a great amount of randomness randomne ss in the sea, and st statistical atistical techniques need to be brought to bear. Fortunately, very large waves waves or, alterna alternatively, tively,waves waves in shallow water appear appea r

Figure 1.2

Example of a possib possible le recorded recorded wave form.

 

4

Introduction o Wave Mechanics

Chap. 1

Figure 1.3 Complex wave form resulting as the sum of two sinusoids.

to be more regular regu lar than smaller wa waves ves or those in dee deeper per water, water, and not so random. Therefore, in these cases, each wave is more readily described by one sinusoid, which repeats itself periodical periodically. ly. Re Realistica alistically, lly,due to shallow w water ater nonlinearities, more than t han one sinusoid, all of the same phase phase,, are necessary; however,, usin however using g one sinusoid h has as been shown to be reasonably accurate for some purposes. It is this thi s surprising accuracy and ease of applicat application ion that have maintained the popularity and the widespread usage of so-called linear, or small-amplitude, wave theory. The advantages are that it is easy to use, as opposed complicate complicated dmanipulations. nonlinear theories, and lends itsel itself f to superposition andtoother omore ther complicated Moreover, linear wave theory is an effective stepping-stone to some nonlinear theories. For this reason, this book is directed primarily to linear theory. 1.3 HISTORIC HISTORICAL AL A ND PR PRES ESEN ENT T LITERATURE

The field of water wave theory is ov over er 150 years old and, an d, of course, during this period of time numerous books and articles have been written about the subject. Perhaps the most outstanding is the seminal work of Sir Horace Lamb. His Hydrodynamics has served as a source book since its original publication in 1879. Other notable books with which which the reader shoul should d becom become e acquainted are R . L. Wiegel's Ocean Oceanographi ographical cal Engin Engineering eering and A . T. Ippen's Estuary

 

Sec. 1.3

Historic al and Present Literature

5

and Coastline Hydrodynamics. These two books, appearing in the 1960s,

provided the education educatio n of many of the practicing coastal and ocean engineers of today. The author au thorss also recommend for furthe furtherr studies on waves the book by G. B. Witham entitled Linear and Nonlinear Wa ves,from which a portion of Chapter 11 is deriv de rived, ed, a and nd the th e a arti rticle cle “§Surface Waves,” by J. V. Wehausen and E. V. Laitone, in the Handbuch der Physik. In terms of arti articles, cles, tthere here ar are e a number ofjo ofjourna urnals ls and proceed proceedings ings that will provide the reader with more up-to-date material on waves and wave theory and its applications. These include the American Society of Civil Engineers’ Journal of Waterway, Port, Coastal and Ocean Division, the Journal of Fluid Mechanics, the Proceedings of the International International Coastal Engineering Conferences, the Journal of Geophysical Research, Coastal Engin eering, Appli Applied ed Ocean Research, and the Proceedings of the Offshore

Technology C onference.

 

Dedication L EONH EONHA A RD EULE LER R Leonhar d Euler (170 Leonhard (1707-1 7-1783 783), ), born bo rn in Basel, Bas el, Switzerland, Switzer land, was one of the earliest practitioners o f applied mathematics, developing with others the theory of ordinary and partial differential equations and applying them to the physical world. The most frequent use o f his work here is the use of the Euler equations of motion, which describe the flow of an inviscid fluid. In 1722 1722 he graduated from the University of Basel with a degree in I. Arts. During this Bernoulli’s time, however, he and attended Bernoulli (Daniel father), turnedthe to lectures the studyofofJohan mathematics. In 1723 1723 he received a master’s level degree in philosophy and began to teach in i n the philosophy department. departm ent. In I n 1727 1727 he moved to St. Petersburg, Russia, and to the St. Peter Petersburg sburg Academy of Science, where he worked in physiology and mathematics m athematics and succeeded Daniel Daniel Bernoulli Bernou lli as Professo Prof essorr of Physics Physi cs in 17 1731. 31. In 174 1741 1 he he was invited i nvited to work in the Berlin Society of Sciences (founded by Leibniz). Some of his work there was applied as opposed to theoretical. He worked on the hydraulic hydr aulic works of Frederick the Great’s summer residence as well as in ballistics, ballisti cs, which was of national interest. In Berlin he published 380 works related to mathematical physics in such areas as geometry, optics, electricity, and magnetism. magnetism. In 1761 1761 he he published his monograph, “Principia motus fluidorum,” which put forth the now-familiar now-famil iar Euler and continuity equations. He returned to St. Petersburg in 1766 after a falling-out with

6

 

Sec. 2.2

7

Review of H y d r o d y n a m i c s

Frederick the Great and began to depend on coauthors for a numbe numberr of his works, as he was was going blind. He died there in 1783. In mathematics mathematics,, Euler Euler was responsible for introd introducing ucing numerou numerous s notations: for example, example, i = f i for base of of the th e natural natur al log, and and the finite difference b . 2.1 INTRODUCTION

In order to investigate water waves most effectively, a reasonably good background in fluid dynamics and mathematics is helpful. Although it is anticipated that the reader has this background, a review of the essential derivations and equations is offered here as a refresher and to acquaint the reader with the notation to be used through throughout out the book. A mathematical tool that will be used often is the Taylor series. Mathematically, it can be shown that if a continuous functionfix, y) of two independent variables x and y is known at at,, say say,, x equal to XO,hen XO,hen it can be approximated at another location on the x axis axis,, xo + A x , by theTaylor series.

+...

+ ~ " J T X O ~ Y ) ( ~.).". dx"

n

where the derivatives offix where offix,, y ) are all taken at x = xo, the location fo forr which the function is known. For very small values of A x , the terms involving (Ax) , where n > 1, are very much smaller than the first two terms on the right-hand side of the equation and often in practice can be neglected. If Ax, y ) varies linearly with x, for example,Ax, y ) = y 2 + mx + b , truncating the Taylor series to two terms involves no error, for all values of Ax.' Through the use of the Taylor to develop relationships is possible between betw een fluid properties a att two series, closelyitspac spaced ed locations. locations . 2.2 REVIEW OF HYDRODYNAMICS 2.2.1 Conservation o f Mass

In a real fluid, mass must be conserved; it cannot be created or destroyed. To develop a mathematical equation to express this concept, consider a very small cube located with its center at x, y , z in a Cartesian coordinate coordin ate system as shown in Figure 2.1. For the cube wi with th sides A x , Ay, and 'In fact, for for any nth-order function, the ex pression (2.1) is exact as long as (n + series are are ob tained.

1)

terms in the

 

8

A Review of Hydrodynam ics and Vector Analysis Analysis

Chap. 2

W

Velocity

components

Figure 2.1

Reference cube n a fluid.

A z , the rate at a t which fluid mass flo flows ws int into o the cube across the v various arious face facess

must equal the sum of the rate of mass accumula accumulation tion in the cube and the mass fluxes out of the t he face faces. s. Taking Taki ng first the x face at x - A x / 2 , the rate r ate a att which the fluid mass flows flows in is equal to the velocity velocity component in the x direction times the area through which whi ch it is crossing, all multiplied times tthe he density of the fluid, p. Therefore, the mass inflow inflow rate at x - A x / 2 , or side A C E G, is

where whe re the terms in parentheses denote the coordinate location. This mass flow flow rate c can an be related to tha thatt at a t the cen center ter of the cube b by y the truncated truncate d Ta Taylor ylor se series, ries, keeping in mind the smallnes smallnesss of the cube,

(x -

Ax 7

2

Y,Z

) W

-

Ax Y , z ) AY A.2 2 7

(2.3)

For conve convenienc nience, e, tthe he coord coordinate inatess o f p and u at the ce center nter of the cube w wil illl not be shown hereafter. The mass flow rate o out ut of the oth other er x face, at x + A x / 2 , face BDFH, can also be represen represented ted by the Tayl Taylor or series, [pu

+d@ x 2 u . . ) y Asz+

subtracting ing the mass flo flow w rate out from th the e mass flow rate in in,, the net flux By subtract

of mass into the cube in the x direction is obtained, that is, the rate of mass accumulation in the x direction:

where the term O ( A X ) ~enotes terms of higher order, or power, than (Ax)’

 

Sec. 2.2

9

Review o f H y d r o d y n a m i c s

and is stated as order of AX)^.'' This term is a result of neglected higherorder terms term s in the Tayl Taylor or series and implicitly implicitly assumes that A x , A y , and A z are the same order or der of magnitude. If the procedure is followed followed for the y and z directions, their contributions will also be obtained. The net rate of mass accumulation inside the control volume due to flux across all six faces is

Let us us now consider this accu accumulat mulation ion of mass to occur for a time increment At and evaluate the increase in mass within the volume. The mass of the volume at time t is p ( t ) A x A y Az and at time ( t + A l ) is f i t + A t ) A x A y A z . The increase in mass is therefore Lp(t

+ A t ) - A t ) ] Ax

Ay Az

=

9

L t

t

I

+ O(At)2

Ax A y A z

(2.7)

where O(At)* represents the higher-order terms in the Taylor series. Since mass must be conserved, this t his increase in mass must be due to the net inflow rate [Eq. (2.6)] occurring over a time increment increm ent A t , that is,

a@u) + a@v) ay ax

1

+ a@w' A x az

Ay Az At

+O

(2.8), A X ) ~t

Dividing both sides by A x Ay A z At and allowing allowing the time increment and size of the volume tto o approach zero, the following exact equation result results: s: p apu apv -+-+-+-= at ay ax

apw az

(2.9)

By expanding the product produ ct terms, terms , a different form of the continuity contin uity equation can be derived.

Recalling the definition for the total derivative from the calculus, the te Recalling term rm within brackets can be seen to be the total derivative* of p(x, y , z, t ) with respect respe ct to time, t ime, D p / D t or d p l d t , given u = d x / d t , v = d y / d t , and w = d z / d t . The first term is then ( l / p ) ( d p / d t ) nd is related to the change in pressure through the bulk modulus E of the fluid, where

E = p -P dP 'This is discussed later in the chapter.

(2.11)

 

A Review o f Hydrodynam ics and Vector Analysis

10

Chap. 2

compres sion of where d p is the incremental change in pressure, causing the compression

the fluid. Thus

-I -d=p- -

p dt

1 dp E dt

(2.12)

For water, E = 2.07 x 109Nm-2, a very large number. For example, a 1 x lo6 Nm-2 increase in pressure results in a 0.05% change in density of wate wa ter. r. Therefore, it will be assumed henceforth tthat hat water is incompress incompressible. ible. From Eq. (2.10 (2.10), ), the conservation co nservation of mass equati equation on for an incompresszble fluid can be stat stated ed simply as I

(2.13) I

I

which must be true at every location in the fluid. This equation is also referre refe rred d tto oa ass the continuity equation, and the flow field satisfling Eq. (2.13) is termed term ed a “nondivergent flow flow.” .” Referringback to the cube in Figure 2 2..1, this equation requires that if there is a change in the fl flow ow in a particular directio direction n across the cube, there must be a corresponding flow change in another direction,, to ensure no fluid accum direction accumulation ulation in the cube.

Example 2.1 An example of an incompressible flow is accelerating flow into a corner in two dimensions, as shown in Figure 2.2 The velocity components are u = -Axt and w = A z t . To determine if it is an incompiessible flow, substitute the velocity components into the continuity equation, - A t + A t = 0 . Therefore, it is incompressible. 2.2.2 Surface Stresses o n a Particle

The motion of a fluid particle is induced by the forces that act on the particle. These forces forces are of two types, as can be seen if w we e again refer to the fluid cube that was utilized in the precedin preceding g section. Surface forces include pressure and shear stresses which act on the surface of the volume. Body forces, for ces, on the other ha hand, nd, a act ct throughout the v volume olume of the cube. These for forces ces Z

0

Figure 2.2 Fluid flow in a corner. Flow is tangent to solid lines.

 

Sec. 2.2

11

Review of Hydrodynamics

include gravity, gravity, magnetic, and o other ther forces that act direct directly ly on eac each h individual particle in the volume under consideration. Alll of these forces which act on the c Al cube ube of fluid wi will ll cause it to move a ass predicte pred icted d by Newt Newton’s on’s sec second ond la law, w, F = ma, for a volume ofconsta ofconstant nt mass m . This law relates the resultan resultant t forces bodys and to iits tsaccelerations resu resultant ltant acceleration alaw, , is, awhich vector equation, being made upon ofaforces force in the x , y, and z coordinate directions, and therefore all for forces ces for convenience must be resol resolved ved into their components. By defi definition, nition, a fluid is a substan substance ce distinguished from solids by the fact that it deforms continuously under the action of shear stresses. This deformation occurs by the fluid‘s flowing. Therefore, for a still fluid, there ar are e no shear stresses and the normal stre stresses sses or forces must balance each other, F = 0. Normal (perpendicular) sstres tresses ses must be present p resent because w we e know tha thatt a fluid column has a weig weight ht and this weigh we ightt must be supported by a pressure times the area of the column. Using this static force balance, w we e wil willl show first tha thatt tthe he pressure is the same in al alll directions (i.e. (i.e.,, a scalar) and then derive the hydrostatic pressu pressure re relationship. For a container co ntainer of fluid, as illustr illustrated ated in Figure 2. 2.3a, 3a, the only forces that act are gravity and hydrostatic hy drostatic pressure. If we first isol isolate ate a stat stationary ionary pri prism sm Hydrostatic pressure.

of fluid fluid with dimensions A x , A z, A1 [= J ( A x ) * + (Az)’], we can examine the force balance on it. W We e wil willl only c cons onsider ider the x and z directio directions ns for now now;; the forces in the y direction do not contribute to the x direction. On the left side of the prism, there is a pressure force acting in the positive x direction, p x Az Ay. On the diagona diagonall face face,, there must be a balanc+z

+z

t

S

Figure 2.3 Hydrostatic pressures on (a) a prism and (b) a cube.

F

,

 

12

A Review o f Hydrodynam ics and Vector Analysis

Chap. Chap. 2

ing componen componentt ofp,, which yields the following form of New Newton's ton's second law:

p x A z A y = p n sin 8 A1 Ay

(2.14)

In the vert vertical ical direction, the fo force rce balance yields

p z Ax A y = p n cos 8 A1 Ay

+ &g

A z Ax Ay

(2.15)

where the se where second cond term on the ri right-han ght-hand d side corresp corresponds onds to the weight o off the prism, whi which ch also must be su supported pported by the vertical pressure force force.. From the geometry of the prism, sin 8 = Az/Al and cos 8 = Ax/Al, and after substitution we have Px = Pn ~z

= Pn

+ Pg

hrink k to z zero ero,, then If we let the prism sshrin Px =

Pz =

Pn

which indicates that the pressures in the x-z plane are the same at a point irrespective of the orientation of the prism's diagonal face, since the final equations do not involve the angle 8. This result would still be valid, of course, if the prism were oriented along th e y axis, and thus thu s w we e conclud conclude e at a point, Px =P

y =Pz

(2.16)

or, or, the pres pressure sure at a point is independent of direction. An importan importantt point to notice is that the pressure is not a vector; it is a scalar and thus has no direction associated with it. A Any ny su surface rface immersed in a fluid wi will ll have a force exerted on it by the hydro hydrostatic static pressure, and th the e force acts in the directio direction n of the normal, normal , or perpendicu perpendicular lar to the surface; tha thatt is, the direction directi on of the force depends on the orienta orientation tion of the face considered. conside red. Now to be cube consist consistent with conservation tion of mass derivation derivation, , let us examine a, small of ent siz size ewi 2.3b). However, this time A xth, Athe y , Aconserva z (see Figure we wi will ll not shrink the cube to a point. On th the e left-hand fac face e at x - A x / 2 there is a pressure pressur e acting on th the e face with a surface area of Ay Az. The total force tending to accelerate the cu cube be in the +x direction is aP A x Ay AZ = P ( X , y , Z ) Ay AZ - - - y AZ + . . (2.17) ax 2

where the truncated where truncatedTaylor Taylor series is used, assuming a small cube. On the oth other er face,, there must be a an n equal a and nd opposit opposite e force; otherwise, the cube would x face have to accelerate in this direction. The force in the minus x direction is exerted on the face located at x + A x / 2 .

P A x Ay A Z Ay AZ = p Ay AZ + -ax 2

(2.18)

 

Sec. 2.2

13

Review of Hydrodynamics

Equating the two forces yield yieldss

0

p -=

ax

(2.19)

For the y direction, a similar result is obtained,

In the vertical vertical,, z , direction the force acting upward is

which must be equal to the pres pressure sure force acting downward, and the weight o off the cube, pg A X A y A z , where g is the acce accelerat leration ion o off gra gravity. vity. Summing these forces yiel yields ds

or dividing by the volume of the small cube, we have

aP az

-pg

(2.22)

Integrating the three p partia artiall differenti differential al equations for the pressure results in the hydrostatic pressure equation

p

=

-pgz

4-

c

(2.23)

Evaluating the constant C at the free surface, z = 0, where p = 0 (gage pressure), P

= -P@

(2.24)

The pressure increases linearly with increasing increasin g dept depth h iinto nto the fluid.3 The buoyancy for force ce is jus t a result of the hydrostat hydrostatic ic pressure actin acting g over the surface of a bod body. y. In a co contai ntainer ner of fluid, imagine a sma small ll sphere o off fluid that could be denoted by some means such as dye. The spherical boundaries of this fluid would be acted upon by the hydrostatic pressure, which whic h would be greater at the bott bottom om of the sphere, as it is deeper there, than at the top of the sphere. The sphere does not move because the pressure differenc diffe rence e sup suppor ports ts the weight o off the ssphere phere.. N Now ow,, if we coul could d remov remove e the fluid sphere and replace it with a sphere of lesse lesserr d density, ensity, the same pressure forces would exist at its surface, yet the weig weight ht woul would d be less and therefore the hydrostatic force would push the object upward. Intuitively, we would say say 'Note that z is negative into the fluid and therefore Eq. (2.24) does yield positive pressure underwater.

 

14

is A Review of Hy dro dy nam ic s and V ec t or A naly s is

Chap. 2

that the buoyancy buoyancy force due to the flu fluid id pressure is equal to the t he weight of the fluid displac displaced ed b by y tthe he object. To examine this, let us look again a att the force balance in the z direction, Eq. (2.21):

-az A z A x A y

=

p g A x A y Az = pg AV

=

dF,

(2.25)

which states that the net force in the z direction for the incremental area Ax Ay equals the the weigh weightt of the incrementa incrementall volume volum e of fluid delimited by tthat hat area. There is no restriction on the size o off the cube due to the linear variation of hydrostatic pressure. If we we now integ integrate rate the pres pressur sure e force over th the e ssurfa urface ce of the ob object, ject, w we e obtain Fbuoyancy

= PgV

(2.26)

The buoyancy buoyancy force is equal to the weig weight ht of the fluid f luid displaced by the object, as discovered by Archimedes in about 250 B.C., and is in the positive z (vertica (vertical) acts cts throu through gh the center of gravity of the displaced fluid). l) direction (and it a Shear stresses also act on the surface; howe however, ver, they differ from the pressure in that they are not isotropic. Shear stresses are caused by by forces acting tangen tangentially tially to a surface; they a are re alwa always ys present in a real flowing fluid and, as pressures, have the units of force per unit area. If we we again exami examine ne our small vo volume lume (see Figure 2.4), we can see tha thatt there are th three ree possible stresses for each of the six faces faces of the cube; two shear stresses and a norma normall stress, perpendicu perpendicular lar to the face face.. An Any y other arb arbitrarily itrarily oriented stress can al always ways be expressed in te terms rms of these three. O n the x face at x + A x / 2 which will be designated the positive x face face,, the stresses are a, Shear stresses.

T~,,,

nd rXz.The The notation convention for stresses is that the first subscript

Figure 2.4 X

Shear and no rmal stresses

o n a fluid cube.

 

Sec. 2.2

16

Review of Hydrodynamics

refers to the axis to which the face is perpendicular and the second to the direction of the stre stress. ss. Far a positive fac face, e, the stresses point in the positive axes directio directions. ns. For the negative x fac face e at x - &/2, the stresses are again om, 7xy, and 7=, but they point in the direction of negative x , y, and z , re~pectively.~lthough these stresses have the same designation as those in the positive x face face,, in general tthey hey wil willl differ in magnitud magnitude. e. In fact, it iiss the difference in magnitude that leads to a net force on the cube and a corresponding correspo nding acceleratio acceleration. n. There are a re nine stresses that are exerted on th the e cube fac faces es.. Three of thes these e stresses include the p pressure, ressure, as the no normal rmal stresses are wriften as IY,=-p+7,

aw = -p ozz =

+ ,rw

(2.27)

-P + 7 2 2

where

for both still and flowing fluids. It is possible, howeve however, r, to show that some of

the shear stresses are identical. To do this we use Newton's second law as adapted to moments and angular momentum. If w we e examine the moments about the z axis, we have

M 2 = zzo2

(2.28)

where M , is the sum of the moments about the z axis, Z2 is the moment of inertia, and hz s the z component of the angular ac acceleratio celeration n of the bo body dy.. The moments about an axis through the center of the cube, parallel to the z axis, can be readily identified if a slice is taken through the fluid cube perpendicularly to the z axis. This is shown in Figure 2.5. Considering moments about the center of the element and positive in the clockwise direction, Eq. (2.28) is written, in terms of the stre stresses sses exist existing ing at the center of

Y

Figure 2.5 Shear stresses contributing to moments mome nts about about the z-axis. z-axi s. Note th that at rw, r,, are functions of x and y.

4Canyou identify the missing stresses o n the

-

X

Ayy/2) face and orient them correctly?

 

A Review o f H y d r o d y n a m i c sand s and Vector Vector A nalysis

16

Chap. 2

the cube,

(2.29)

Reducing the equat equation ion leaves

z A X Ay AZ - T,, A x Ay AZ

=

& p [ A X Ay A Z ( A x 2+ A y 2 ) ] O z (2.30)

For a nonzero difference, on the left-hand side, as the cube is taken to be smaller and smal smaller, ler, the accele acceleration ration hZmust become grea greater, ter, as the moment of inertia involves terms of length to the fifth power, whereas the stresses involve only the length to the third power. Therefore, in order that the angular acceleration of the fluid particle no nott unrealisticall unrealistically y be infin infinite ite as th the e cube reduces in size, we conclude that z = z (i.e., the two shear stresses must be equal). Further, similar logic will show that T, = zZx, T,, = T Therefor e, tthere Therefore, here are only six unknown stresses (axx,,,, z T,, a,,, and azz) n the element. These stresses depend on paramete parameters rs such as fluid viscosity and fluid turbulence turbulenc e and wi will ll b be e discus discussed sed later. 2.2.3 The Tra Translational nslational Equations o f M o t i o n

For the x dire directio ction, n, N Newto ewton’s n’s second law is, again, CF, = ma,, where a, is the particle part icle acceleration in the x directio direction. n. B By y definiti definition on a , = du/dt, where u is the velocity in the x direction direction.. This velo velocity city,, how however ever,, is a function of space and time, u = u ( x , y , z , t ) ; herefore, its total derivative is

d=u - + d- -u+ - -d+u-d- x

dudy

dudz

dt

ay dt

az at

at

ax dt

or, since dx/dt is u , and so forth,

au d u au au +u-+v-+w=ax ay dt at

au

az

(2.31)

(2.32)

This is the total accelerati a cceleration on and wil willl be denoted as Du/Dt. The derivative is composed of two types of terms, the local acceleration, du/ dt , which is the change of u observed at a point with time, and the convective accel accelerat eration ion terms

au

au ay

u-+v-+w-

ax

au az

which whic h ar are e the chang changes es of u that resul resultt due tto o the motion of the particle. For

I.

Sec. 2.2

Review of Hydr odynam ics

17

igure 2.6 Acceleration of flow through a convergent section.

 

example, if we follow a water particle in a steady flow (i.e., a flow which is independent of time so that &/at = 0) into a transiti transition on section as show shown n in Figure 2.6, it is clear that the fluid a accel ccelerat erates. es. The iimpor mportant tant terms applicaau au ble to the figure are the u - nd the w - erms. ax az The equation of motion in the x direction can ca n now be formulated:

Du CF..=mDt

From Figure 2.4, the surface fforces orces can be obtained on the six fa faces ces via th the e truncated trunc ated Tayl Taylor or series (0,

+

%$)

Ay Az

- (0,

A x Az

+

-

-ax 2

( + 2 $) zx

A x Ay

(2.33)

The capital X denotes any body force fo rce per un unit it mass acting in the x direction. Combining terms an and d dividing by the volume of the cube yield yieldss

DU p-

Dt

a0,

= -

ax

aTyx

-

ay

aTzx

-

az

px

(2.34)

or

(2.35) and, by by exactly simil similar ar developments, the equations of motion are obtained

 

18

A Review of Hydrodynam ics and Vector Analysis Analysis

Chap. 2

fo forr th ey and z directions:

D=V- - - I a p Dt pay

I ar,,

+ --+ -+ p

ax

az, ay

a Tz zy)

+Y

(2.36)

(2.37)

-=---

necessary ary to To apply the equations of motion for a fluid particle, it is necess

know something about stresses in a flui know fluid. d. The most convenient assumption, one that th at is reasonably valid fo forr most prob problems lems in water wave mechanics, is thatt the tha t he she shear ar stresses are zero, which results in the Euler equations. ExpressExpressing the body force per unit un it mass as -g in the z direction and zero in the x and y directio directions, ns, w we e hav have e

D _U Dt

- -l a- p

(2.38a)

pax

the Euler equation equationss

(2.38b)

(2.38~)

In many real flow case cases, s, the flow is turbulent an and d shear stresses are iinfluence nfluenced d by the turbulence and thus the previous stress terms must be retained. If the flow is laminar, that is there is no turbulence in the fluid, the stresses are governed by the Newtonian shear stress relationship and the accelerations are governed by (2.39a)

+Y

(2.39b) (2.39~)

and p is the dynamic dy namic (molec (molecular) ular) viscosity of the fluid. Often p / p is replaced by v , defined as a s the kin kinematic ematic visco viscosity sity.. For turbulent flows, where the velocities and pressure fluctuate about mean values due to the presence of eddies, these equations are modified to describe the mean and the fluctuating quantities separately, in order to

 

Sec. 2.3

19

Review of Vector Analysis

facilitate their use. We wil willl not not,, however, b be e using these turbulent forms oft o fthe he equationss dire equation directly ctly.. 2.3 R E V I E W OF V E C T O R ANALYSIS

Throughout the book, vector algebra will to and facilitate and minimize required requ ired algebra; therefore therefore, , the usebe ofused vectors vectorproofs analysis is reviewed briefly below. In a three-dime th ree-dimensional nsional Cartesian coor coordinate dinate system, a rreference eference system ( x , y , z ) as has been used before can be drawn (see Figure 2.7). For each coordinate direction, there is a unit vector, that is, a line segment of unit length oriented such that it is directed in the corresponding coordinate direction. These unit vectors are defined as (i, j, k ) in the (x, y , z ) directions. Thc boldface boldface type denotes vecto vectorr quantities. An Any y vec vector tor wi with th or orientation ientation and a length length can be expressed in tterms erms of unit vectors. For example example,, the ve vector ctor a can be represented represen ted as a = a,i

+ ayj+ a,k

(2.40)

where a,, up,and a, are the projections o off a on the x, , and z axes. 2.3.1 The Dot Product

The dot (or inner or o r scalar) product is defined as

a * b = a \bl cos8

(2.41)

where the absolute value sign ref where refers ers to th the e magni magnitude tude or leng length th of the vecto vectors rs and 8 refers to the angle between them. For the unit vectors, the following identiti iden tities es readi readily ly follow:

i.i = I

i.j = O

i*k=O

(2.42)

j.j = I

j.k=O k*k=l Z

k

 

A Review of Hydr odyn am ics and Ve Vector ctor Analysis

20

Chap. 2

P

A - igure 2.8

Projections of vector a.

These rules rules are commutative, also, so that reversing the or order der of the operation does not alter the results. For instance,

i . j.= j . i

(2.43)

or a b = b a. Consider taking a dot product produ ct of the vector with itself. itself.

-

a. a = (axi + ayj + a,k) (axi + ayj + a,k) = a;

(2.44)

+ a; + a f

-

A graphical interpretation of a a can be obtained obta ined fr from om Figure 2.8, where the magnitude of vector a is the length rom the Pythagorea Pythagorean n theor theorem, em, = OQ' + ut is just a, and a: + a; + a:. af + a;. Therefore, Therefore, the magnitude of vector a can be written as

m.

-

m.

m2

la1 =

m2

D =a.a

m2

(2.45)

The quantity a b as shown sh own before is a scalar quantity; that is, it has a magnitude, but no direction (therefore, it i s not a vector). Another way to express a is

.

a . b = la1 Ibl cos8=a.xbx+a$y+azbz

(2.46)

Note that if a b is zero, but neither a or b is the zero vector, defined as (Oi + O j + Ok), then cos 8 must be zero; the vectors are perpendicular to one another. An imp import ortant ant use of the dot product is in determining the projection of a vector onto another anot her vector. For example, the projection of vector a onto the general , the pr projection ojection of a onto the b vector direction would x axis is a . i . In general, bea-b/IbI. 2.3.2 The C r o s s Product

The cross product (or outer, or vector product) is a vector qualztity which is defined as a x b = 1 a I I b I sin 8, but with a direction perpendicular to the plane of a and b according to the right-hand rule rule.. For the unit vectors,

ixi=jxj=kxk=O;

i x j= k ,

jxk=i,

kxi=j

(2.47)

 

Sec. 2.3

Review of V ec t or A naly s is

21

but this rule is not commutative. So, for example, j x i = - k . A convenient method for evaluating the cross product of two vectors is to use a determinant form: ax

b=

i

j

a,

ay a,

(2.48) = (a$, - a,by)i + (a,b, - axbz)j+ (axby a$,)k

k

b, by b,

2.3.3 The Vector Differential Differential Operator Operator and th e Gradient

Consider a scalar field in space; for example, this might b be e the temperature T ( x ,y , y , z ) in a room. Because of uneven heating, it is logical to expect that the temperature will vary both with height and horizontal distance into the roo room. m. If th the e te te>x >xb b Ant n

.

H

?

truncated three-dimensional Tay Taylor lor seri series es can be used to estimate the tempertemp erature at a small distance dr (= dxi + dyj+ d z k ) away.

T ( x + Ax, y

+ Ay, z + Az)

(2.49)

The last three terms in this expression may be written as the dot product of two vectors:

($ + 5 +

k ) (Axi

+ Ayj + Azk)

(2.50)

The first term is defined as the gradient of the temperature and the second is the differential vector Ar. The gradient or gradient vector is often written as a s grad T or V T , nd can be further broken down to

(2.51) where the first ter term mo on n the rightright-hand hand side is defined as the vector differential operator V, and the second, of course, is is just the scalar tempe temperature. rature. The gradient always indicates the direction of maximum change of a scalar field' and can be used to indi indicate cate perpendicular, or normal, vectors to 'The total differential d T the direction of I VT I.

=

VT

.

r = I VT I I dr I cos & The maximum value occurs when dr is in

?

 

A Review of Hydrodyna mics and V e c t o r Analysis

22

Chap. 2

a surface. surface. For example, if the tem tempera perature ture in a room wa wass stably stratified, the temperature would be solely solely a function of elevation in the room, or T (x, , z ) = T(z). If we we move horiz horizontally ontally across the room tto o a new point, the change in temperature would be zero, as we have moved along a surface of constant temperature. Therefore,

(2.52) where

T

aT

ax

ay

-=-=

Ar

0,

=

dxi + dyj + Ok

(2.53)

or (2.54)

VT*Ar=O

which whic h means, usi using ng tthe he definition of ofth the e dot product, that V T s perpendicular to the surface of constant temperature. The unit normal vector will be defined here as the vector n , having a magnitude of 1 and directed per perpendicpendicular to the surface surface.. For this example,

(2.55) or n = Oi

+ Oj + l k

=

k

2.3.4 The Divergence

If the vector differential operator is applied to a vector using a dot product rather than to a scalar, as in the gradient, we have the divergence

(2.56) - da, _ ax

+ -day +ay

aa, a2

We have already see seen n this operator in the continuity eq equation, uation, Eq. (2.10), which whi ch can be rewritten as

(2.57) where u is the velocity ve vector, ctor, u

=

iu

+ v + kw,

v . u = -d+u - +avax

 

ay

aw az

(2.58)

Sec.

2.3

23

Review of Vector An a l ysi s

For an incompressible fluid, for which ( l / p ) ( D p / D t ) is equal to zero, the divergence of the veloci velocity ty is also zero, and therefore the fluid is divergen divergenceceless. Another useful result may be obtained by taking the divergence of a gradient,

V.VT=

=-d2T +- +- a2T d2T ax2

=

ay2

(2.59)

az2

V2T

Del square squared d (V2) is known as the Lapla Laplacian cian operator, named after the famous French mathematician Laplace (1749-1827).6 2.3.5 The Curl

If the vector differential operator is applied to a vector using the cross product, then the c u d of the vector vecto r results.

x (a$

+ ayj+ a,k)

(2 . 6 0 )

Carrying out the cross product, which can be done by evaluat evaluating ing the following determinant determ inant,, yiel yields ds

(2.61)

A s we wil willl see later, the curl of a velocity vecto vectorr is a m measure easure of the rot rotation ation in

the velocity field. A s an example of the curl operator, let us determine the divergence of the curl of a.

%3apter33 is dedicated to Laplace. %3apter

 

A Review of Hydrodynamics and V ec t or Analysis

24

Figure 2.9

Integration paths between

two points.

+

0

Chap. 2

This is an identity for any vector that has continuous first and second derivatives.

2.3.6 Li Line ne Integra Integrals ls In Figure 2.9, two points are ar e shown in the ( x - y )plane, P o and PI. ver this plane the vector a(x, y ) exists. Consider the integral int egral from P o to P I f the projection of the vector a on the contour lin line e C1.We w will ill denot de note e this int integra egrall as F

(2.62) It is anticipated that should w we e have chosen contour C2, differe different nt value of the integral would have resulted. The question is whether constraints constraint s can be prescribed presc ribed on the na nature ture of a such that it makes no diff difference erence whether w we e go from P O o P , n contour C , or C2. If Eq. (2.62) were rewritten as

F

=

$?dF

where dF is the exact differe differential ntial o f F , hen F would be equal to F(Pl) F(P0); that is, itrequire is on only lythat a function ofof thethe en end d points Therefore, erefore, if f he integration. dF,o independence we can a d l be form of Th path should ensue. ensu e. No Now, w, a.

dl

=

a, d x

+ a, d z

for two dimensions, as

dl

=

dxi

+ dzk

and the total differential o f F is

dF By equating a

dimensions,

.

=

aF X ax

-

aF +az

z

=

-

V F dl

(2.63)

l with d F , we see th that at ind independence ependence of path rrequires equires,, in two a,=-

aF ax

and

a,=-

aF az

or

a=VF

(2.64)

 

Sec. 2.3

Review of Vector Analysis

25

If this is true for ax and a,, it follows tha thatt

aa,_ aa, _ - -- 0 ax

az

(2.65)

as

2F ----

a2F - 0 axaz

azax

Therefore, in summary, independence of path of the line integral requires that Eq. (2.65) be satisfied. For three dimensions it can be shown that this condition requires that the curl of a must be zero. Example 2.2 What is the value of

if V x a

=

0 and where the

P

indicatess a complete circuit indicate circuit around the closed contour

composed of C , and C2? Do this by parts.

Solution. F

=

$" a - dl +

-

a dl = F ( P I )- F ( P o )+ F ( P o )- F ( P , ) = 0

PO

Alternatively, note that by Stokes's theorem, the integral can be cast into another form:

F

=

a

- dl

=

ss

V x a ) . n ds

where ds is a surface element contained within the perimeter of C , + CZ, nd n is an outward unit normal to ds. Therefore, if V x a is zero, F = 0.

2.3.7 Velocity Potential

Instead of discussing the vector a, let u s consider u, he vector ve ctor velocity, given b y

u(x, y , z , t ) = ui

+ vj + w k

(2.66)

Now, let u s define the value of the line integra integrall of u as -4:

-+=$;u.dl=$

(udx +vd y+ wdz)

(2.67)

The quantity u s l is a measure of the fluid velocity in the direction of the

 

26

A R e vi e w o f H y d r o d y n a m i c s a n d V e c t o r A n a llyy s i s

Chap. 2

contour a t each point. Therefore, Therefore, -4 is related to the product p roduct of the velocity and length along along the path between between tthe he two points poin ts P o and P I .The minus min us sign is a matter of definitional convenience; quite often in the literature it is not present. inde pendent of path, that th at is, for for the flow flow rate For the value of 4 to be independent between P o and P I o be the same no matte m atte r how the integration integration is carried out, the terms in the integral integral must be an exact differential differential d4, and therefore (2.68a) (2.68b)

(2.68~) To ensure that this scalar function mus t be zero:

4

exists, the curl of the velocity vector

The curl of the velocity velocity vector iiss refer referred red tto o a s the vorticity vorticity a. Th e velocity velocity vector u can therefore be conveniently represented as

u = -u$

(2.70)

That is, we can express the vec vector tor quan quantity tity by the gradient of a scalar function 4 for a flow with no vorticity. Further u flows “downhill,” that is, in the direction of decreasing decreasing 4.’ If 4 ( x , y , z , ) s known over ove r all space, then u , v , and w an be determined. determined . Note that 4 has the units oflength oflength squared divided by time. Let u s exam examine ine more mo re closely closely the line integral integral of the velocity component compon ent along the contour. conto ur. If we we consider the closed path from P o to P , nd then back again, we know, from before, that tha t the integral is zero.

I

u.dl=O

(2.71)

which means that if, for example, the path taken which ta ken from Po o P I nd back again were wer e circular, no fluid would travel this circular path. pa th. Therefore, we we expect no rotation of the fluid in circles if the curl of the velocity vector is zero. zero. To examine this irrota i rrotationa tionality lity concept more fully fully,, consider the average rate of rotation of a pair of orthogonal axes drawn on the small water mass ’This is the reason for the minus sign in the defintion of 4.

 

Sec. 2.3

Review o f Vector A n a l y s i s

27

f Az I

Figure 2.10

shown in Figure 2.1 2.10. De Denoting noting the po positive sitive rotati rotation on in tthe he counterclockwise directio dire ction, n, the aver average age rate of rotati rotation on of the axes w will ill be given by Eq. (2.72). (2.72) Now if u and w are known at ( X O , ZO), the coordinates coor dinates of the center of the fluid mass, then at the edges of the mass the velocities are approxim approximated ated as

and

Now the angular an gular velocity of the z axis can be expressed as

4

=

-

( x oo , + 62/21 - ~ ( x o O) , -

6212

au az

and similarly ffor or 8 b :

The average rate of rotation is therefore (2.73) Therefore, the j component of the curl of the velocity vector is equal to twice the rate of rotation of the fluid parti particles, cles, or V x u = 28 = o, here o s the fluid vorticity.

A mechanical analog to irrotational and rotational flows can be depicted by considering a carnival Ferns wheel. Under normal operating

 

A Review of Hydrodynamics and Vector Analysis

28

Chap. 2

Figure 2.11 (a) Irrotational motion of chairs on a Ferris wheel; (b) rotational rotati onal m otion of the chairs.

conditions the chairs do not rotate; they alwa always ys have the same orientat orientation ion with respect to the earth (see Figure 2.11a). A s far as the occupants are concerned, this is irro irrotational tational motion. If, on the other hand, tthe he cars wer were e fixed rigidly to the Ferris wheel, we would have, first, rotational motion (Figure (Fig ure 2..1 1binviscid )a and nd the then n pe perhaps rhaps a castast castastrophe For2 an and incompre incompressible ssiblerophe. ffluid, luid,.w where here the Euler equa equations tions a are re valid, there are a re only normal stresses (pressure (pressures) s) acting on the surface of a fluid flui d particle; since th the e shear stresses are zer zero, o, there a are re no stre stresses sses to impa impart rt a rotation rot ation on a fluid particle. Therefore Therefore,, in an inviscid flu fluid, id, a nonrotating particle remains nonrotating. However, if an initial vorticity exists in the fluid, the vorticity remains co constant. nstant. To see this, we writ write e the Euler equatio equations ns in vector form:

Du - 1 _ - - v p - gk

(2.74)

Dt

P Taking Taki ng the curl of this equation a and nd substitut substituting ing V x u (identically), we we have

DO -=o

Dt

= o

and V x V p

=

0

(2.75)

Therefore, there can be no c change hange in th the e vorticity o orr the rotation of the fluid with time. time. This theory is due tto o Lord Kelvin (1869).8 2.3.8 Stream Function

For the velocity potential, we defined 4 as (minus) (min us) the line integral o off the velocity vector vector projected onto the line element; let us now define the line integral composed of the velocity component perpendicular to the line *Chapter5 is dedicated to Lo Lord rd Kelv in.

 

Sec. 2.3

29

Review of Vector Analysis

element in two dimensions.

v=

ndl

“ti.o

(2.76)

where d l = Idl I. Consideration of the integrand integ rand above wi will ll demonstrate that ty represents the amount of fluid crossing the line C I between points P o and The unit vector n is perpendicular to the path of integration CI. P I .The To determine the unit normal vector n , it is necessary to find fi nd a nor normal mal vector N such that

N*dl=O

+ N , dz = 0

N , dx

r This is always tru true e if

N,

=

-dz

and

N,

=

dx

It would have been equally valid to take N , = d z and N , = - dx; however however,, this would have resulted in N directed to the right along the path of integration instead the the left.unit normal n , it remains only to normalize N . Tooffind

n=--

-dzi+dxk

N

IN1 -&Z-z?=

-dzi+dxk

dl

The integral integra l can thus be written as

(-u dz + w dx)

v/ =

(2.77)

For independence of path, so that the flow between P o and P I will be measured the same sa me w wa ay no m matter atter which w way ay w we e connect the p points, oints, the integrand must be an exact differential, dty. This requires that

u = - -av

av.

w=ax’

az

(2.78)

and thus the condition for independence of path [Eq. (2.65)] is

aw

au

-+-=O az ax

(2.79)

which is the two-dimensional form of the co which continuity ntinuity equation. Therefore, for two-dimensional incompressible flow, a stream function exists and if we know its functional funct ional form, w we e know tthe he velocit velocity y vector. In general, there can be no st stream ream functi function on for three-dimensionalflows, with the exception of axisymmetric flow flows. s. However, the velocity potential exists in any three-dimen three-dimensional sional flo flow w that is irrotational.

 

30

A Review of Hydrodynamics and Vector Analysis

Chap. 2

Note that the flow rate (per unit width) between points P o and P I s measured by the difference between and y/(Po). f an arbitrary constant is added to both values of the stream function, the flow rate is not affected. 2.3.9 Streamline

-

A streamline as a line everywhere the velocity vector, or, is ondefined a streamline, where normal to to the u nthat = 0, is n is the tangent streamline. From the earlier section,

dx -dz dz w or _ - _ (2.80) u w dx u along a streamline. Th These ese are the equations for a strea streamline mline in two dimensions. Streamlines are a physical concept and therefore must also exist in all three-dimensional flows and all compressible flow flows. s. From the definition of the stream function in two-dimensional flows, a y / / d l = 0 on a streamline, strea mline, and therefore the stream ffunction, unction, when it exists, is a constant along a streamline. This leads to the result Vy/ d l = 0 along a u - n = u dz+ wdx

=

or

0

streamline, and tion therefore thletto the gradient of v / isvecto perpendicular to the streaml streamlines ines and in the direction direc norma normal o the velocity vector. r. 2. 2.3. 3.10 10 Rela Relationship tionship bet w een Velocity Velocity Potential and Stream Function

For a three-dimensional flow, the velocity field may be determined from a veloc velocity ity potential if the fluid is irrotational. For some threedimensional flows and all two-dimensional flows for which the fluid is incompressible,a stream function v / exists. Each Each is a measure of the flow flow rate between two points: in either the normal or transverse direction. For twodimensional incompressible fluid fflo low, w, which is irrotational, irrota tional, both the stream s tream function and the velocity potential exist and must be related through the velocity components. The streamline, or line of constant stream function, and the lines of constant velocity potential are perpendicular, as can be seen from the fact that their gradients are perpendicular: n$.Vy/=O

as (a,i4

+ z 4k )

(-ui -

wk) (+wi

-uw

 

( E i

+ uw

=

+

$

-

0

) =

uk) =

(2.81)

Sec. 2.3

31

Review of Vector A naly s is

The primary advantage of either the stream function or the velocity potential is that they are scalar quantities fro m w hich the velocity velocity vect vector or field can be obtained. A s one can easi easily ly imagine, it is far easier to work with scalar rather than with vector funct functions. ions. Often, the stream function or the velocity potential is known an d the other is desir desired. ed. To obtain one fro m the o ther, it iiss necessary necessary to relate the two. Recalling Recall ing the defin ition of the velocit velocityy com pone nts u = - - =4 -dV az ax

a4w=--= az

a+ ax

we have (2.82a) (2.82b) These relati relationships onships are cal called led the Cauchy-Rieman n cond itions an d enabl enablee the hydrodynam icist to utili utilize ze the powerful techoiques of complex variable analysis. anal ysis. See See for example, Milne-Thom son (1949). Example 2.3 For the fol follow lowing ing v velo elocit city y potential, de determ termine ine the correspo corresponding nding stream function.

4(x, z , 2 )

=

(-3x

+ 5z) cos

2 nt T

Thi s veloc This velocity ity pot potential ential represents a to-and-fro mot ion of the fluid with the stre streamli amlines nes slanted with respect to the origin as shown in Figure 2.12. The vel veloci ocity ty components are

Solution.

From the Cauch Cauchy-Ri y-Riemann emann conditions

or, integrating, Y(X,

 

Z,

t ) = -3z cos 2nt T

C,(X, )

32

A Review o f H y d r o d y n a m i c s a n d Vector A n a l y s i s

Chap. 2

Figure 2.12

:

Note that because we integrated a partial differential, the unknown quantity that results is a function of both x and t. For the t he vertical v ertical velocity, velocity,

ary

-

x

-5 cos

2 nt T

or Y(X,

Z,

2nt t ) = - 5 ~ os T

G ~ ( z ,)

Comparing these two equations, which must be the same stream function, it is apparent that W(X,

Z,

t ) = - ( 5 ~ 3 Z) cos

2nt T

G(t)

The quantity G ( t ) s a constant with regard to the space variables x and z and can, in fact, vary with time.This time dependency, due to G ( t ) ,has n o bearing whatsoever on the flow field; hence G ( t ) can be set equal to zero without affecting the flow field.

2.4 CYLINDRI CYLINDRICAL CAL COORDINATES COORDINATES

The most appropriate coordinate system to describe a particular problem usually is that fo forr whi which ch constant values of a coordinate most nearly conform to the boundarie boundariess or response variables in the problem. Therefore Therefore,, for the case ofci ofcircul rcular ar waves, which might be generated whe when n a stone is dropped iint nto o a pond, it is not convenient to use Cartesian coordinates to describe the problem,, but cylindrica problem cylindricall coordina coordinates. tes. These coordinates are ( r , 8, z ) , which are shown in Fi Figure gure 2 2..13. Th The e tran transfor sformatio mation n between coordinates depends on these equations, x = r cos 0, y = r sin 8, and z = z . For a velocity potential defined in terms of ( r , 8,z), the velocity velocity components are (2.83a) (2.83b)

 

Sec. 2.4

33

Cylindri cal Coordin ates

i Figure 2.13 Relationsh ip between Cartesian and and cylindrical cylindrical coordinate systems r and 8 ie in the x - y plane.

(2.83~)

A s noted previously, the stream function exists only for those three-

dimensional flows which are axisymmetric. The stream function for an axisymmetric flow in cylindrical coordinates is called the “Stokes” stream function. The derivation of this stream function is presented in numerous references, referenc es, however this for form m is not used extensively in wave mechanics and therefore will will not be discussed furthe furtherr here.

2.5 THE BERNOULLI EQ The Bernoull Bernoullii equation is simply an integrate integrated d form of Euler equations of motion and provides a relationship between the pressure field and kinematics, and will be useful later. Retaining our assumptions of irrotational motion and a nd an incompres incompressible sible fluid, the governing equations of motion in the fluid for the x-z plane are ar e the Euler equations, Eqs Eqs.. (2.38).

(2.84a)

(2.84b) Substituting in the two-dimension two-dimensional al irrotationality conditio cond ition n [Eq. [Eq. (2.69)],

au

- aw az ax

(2.85)

the equations can be rewritten as

au at

w at

a(u2/2) -a(w2/2) ax

ax

a(u2/2) -a(w2/2) az

az

I ap P ax

(2.86)

1 ap P az

(2.87)

 

34

A Review o f Hydrodyn amics and Vector Analysis

Chap. 2

Now, since since a velocity potential exists for the fluid, we have =--

a4.

w

=--

ax'

a4

(2.88)

az

Therefore, ifw e substitute these definitions i nto Eqs. (2. (2.86) 86) an d (2.87) (2.87),, we get (2.89a) (2.89b) where it has been assumed that the density is uniform throughout the flui fluid. d. Integrating Integrat ing the x equation yiel yields ds _ 4+ A

u2

2

at

+ w2) + P- = C ( Z , t )

(2.90)

P

where, as indicated, indicated, the constan t of integrati integration on C' ( z , t ) varies only w ith ith z a n d t.

Integrating Integrati ng th e z equation yiel yields ds

-4 + - (u2 + w 2 ) + P at

2

=

-gz

+ C( X , t )

P

(2.91)

Examining these two equations, which have the same quantity on the lefthand side sides, s, shows clea clearly rly that C ( z , t ) = -gz

+ C(X, t )

Thus C cannot be a function of x , as neither C' nor (gz) depend on x . Therefore, C' ( z , t ) = -gz + C ( t ) . Th e resu resulti lting ng equa tion is

1-Tt

+ L(u2 + w 2 ) + P + g z = C ( t )

(2.92)

-

2

P

The steady-stat steady-statee form of tthis his equa tion, the integrated form of the equation s of motion, is called the Bernoulli equation, which is valid throughout the fluid. In this book we will refer to Eq. (2.92) as the unsteady form of the Bernoulli equation or, for brevity, as simply the Bernoulli equation. The function C ( t ) s refe referred rred to as the Bernou lli term an d is a constant for st steady eady flows. The Bernoull Bernoullii equation c an also be written as

-[(>'

-4 + Pp + 2 aa4x at

 

+

31

gz

=

C(t)

(2.93)

See. 2.4

35

Cy lindric lindric al Coord inat es

which interrelates the fluid pressure, particle elevation, and velocity potential. Between any two points in the fluid of known elevation and velocity potential, pressure differences can be obtained by th this is equation; for example, for points A and B at elevatio elevations ns zAand z ~ he , pressure at A is

(2.94)

Notice that the Bernoulli constant is the same at both locations and thus dropped out of the last equation. [Another me method thod to elimina eliminate te the constant is to absorb it into the velocity potential. Starting with Eq. (2.93) for the Bernoulli equation, w we e can defi define ne a function functionJt) Jt) such that

Therefore, the Bernoulli equation can be written as as

-

at

+-

Now, if we define &(x, z , t) =

(2.95)

P

(x, z , t ) + At),' (2.96)

Often we will use the & form of the velocity potential, or, equivalently, we will take the Bernoulli constant as zero.] For three-dimensional flows, Eq. (2.96) would be modified only by the addition of (1/2>(d$/~3y)~ n the lefthand side. In the following paragraphs a form of the Bernoulli equation will be derived for two-dimensional steady flow flow in which the density is uniform and the shear stresses are zero; however, in contrast to the previous case, the results apply to rotational flow fields (i.e., the velocity potential does not exist). In Figure 2.14 the velocity vector at a po point int on a streamline is shown, as is a coordinate system, s and n, in the streamline tangential and normal directions. By definition of a streaml streamline, ine, at A a tangential velocity exists, us,but there is no normal velocity to the streamline u n . Referring to Eq. (2.84), the steady-state form of the equation of motion for a particle at A would be 9The kinematicss associated with @ (x, z , 9Thekinematic easily by the reader.

)

are exactly the sa me as $ ( x , z ,

),

as can be shown

 

36

A Review o f Hydrodyn amics and Vector Vector Analysis

Chap. 2

2

I

-g sin OL = forcelunit mass in s direction

Figure 2.14 Definition sketch for derivation of steady-state two-dimensional Bernou lli equation for rotational flows. flows.

written as

au, us-

.

I ap

g sin a

= ---

(2.97)

as p as where sin a accounts for the fact that the streamline coordinate system is inclined with respect respect to the horizont horizontal al plane. From the figure, sin a = dz/ds, and therefore the the equation of moti motion on is

+-+gz .-( .: 1= o 2

s

p

where again we have assumed the density p to be a constant along the streamline. Integrating Integratin g along the stre streamline, amline, w we e have

uf

-

P

-

+gz

=

C(y)

(2.98)

2 P This is nearly the familiar form of the Bernoulli equation, except that the time-dependent term te rm resulti resulting ng from the local acceleration is not present due to the assumption assump tion of steady flow an and d also, the Bernoulli constant is a func function tion of the streamline on which we integrated the equation. In contrast to the Bernoulli equation for an ideal flow, in this case we cannot apply the Bernoulli equation everywhere, only at points along the same streamline.

REFERENCES MILNE-THOMSON, ed.,, The Macmillan C o . , Theoret etic ical al H ydrodyn am ics, 4th ed. . M., Theor

N.Y., 1960.

Chap. 2

37

Problems

PROBLEMS

 

2.1

Consider the following following transition section: +lorn&

L m

- i - t - - -

-

+--+

L

’

’m

(a) The flow from left to right is constant at Q = 12n m3/s. What is the total

acceleration of a water particle in the x direction at x = 5 m? Assu Assume me that tha t the water is incompressible and that the x component of velocity is uniform across each cross section. (b) The flow of water from right to left is given by

Q ( t > = nt2 Calculate the total acceleration at x assumptions as i n part (a).

2.2

=

5 m for t

=

2.0 s. Make the same

Consider the followingtransition section:

-----

y-sj /

,

(a) If the flow of water from left to right is constant at at

Q

=

.1 m’/ m’/s, s, what is the

total acceleration of a water particle at x = 0.5 m? Assume Assume that the water is incompressible incompressi ble and that the x component of velocity is uniform across each cross section. (b) The flow of water from fro m right to left is expressed by

Q < t >= t 2 / 1 0 0

Calculate the total acceleration a cceleration at x assumptions as in part (a).

 

=

0.5 m fort = 4.48 s. Make the same

38

2.3

A Review of H y d r o d y n a m i c s a n d V ec t or Analysis

Chap. 2

The velocity potential for f or a particular two-dimensional flo flow w fie field ld in which the density is uniform is is

2n

(b = (-3x + 5z) cos T where the z axis is oriented orient ed vertically upward. (a) Is the flow flow irrotational? (b) Is the flow nondivergent? If so, derive the stream function and sketch sketch any two stream streamlines lines fo rt = T/8.

2.4

If the water (assumed inviscid) inviscid ) in the U-tube is displaced from its equilibrium position, it will oscillate about this position with its natural period. Assume thatt the displacement of the surface is tha

where the amplitude amplit ude A is 10 cm and an d the natural period T is 8 s. What will be the pressure at a distance 20 cm below the instantaneous instanta neous water surface for tj = +lo, 0, and -10 cm?Assume that g = 980 cm cm/s /s2 2 an dp = 1 g/cm’.

2.5

Suppose that we measure the mass density p at function of time and a nd observe the following: following:

fixed point (x, y , z )

a

From this information alone, is it possible to determine whether the flow is nondivergent?

 

Chap. 2

2.6

39

Problems

Deriv e the fol following lowing equa tion for an inviscid fluid fluid a nd a nondiv ergent steady steady flow: 1

ap

a (uw ) a (vw ) a (w 2 ) ax ay az

g---=paz

2.7

~

Expan d the fol follo lowing wing expression so tha t gradients of products of scalar functions d o not appear in the re resul sult: t:

v +wf) where 4, ty, n d f are scalar functions. 2.8 The velocity components in a two-dimensional flow of an inviscid fluid are u=- Kx

x 2+ z2

w = - Kz x 2+ z2 flow nond ivergent? a) I s the flow (b) I s the flow irrotational?

(c) Sketch the two streamlines passing through points A a n d R , where the coordinates of these these poin ts are:

2.9

Point A : x = 1 , z = 1 Point B : x = 1,.z = 2 For a particular flui fluidd flow, the vel velocit ocityy co mp one nts u , v , a n d w i n t h e .x, y , a n d z directio ns, rrespective espectively, ly, are

+ 8y + 6 f z + t 4 8~ - y + 6~

u = X

v

=

w = 1 2 ~ 6y

2at

+ 1 2 ~os --

T

+ 1’

a) Are there an y tim es for which the flo flow w is nondivergen t? If so, when?

(b) Are there any tim es for which the flo flow w is irrotational? If so, when? (c) (c) Dev elop the expression for the pre pressur ssuree gradient in the vert vertica icall ( z ) irection as a functi function on of spa space ce an d time.

inviscid fl flui uidd fl flow ow is 2.10 Th e stream function for a n inviscid

w = AX2Zt where x ,

z, t

0. (a) Sket Sketch ch the stream lines w = 0 a n d II/ = 6 A f o r t = 3 s. (b) F o r t = 5 s, what are the coordinates oft he point wher wheree the streamline sslo lope pe d z / d x is -5 for the particu lar streamline w = IOOA? ( c ) W h a t is t h e pressure g r a d i e n t a t x = 2 , z = 5 a n d a t t i m e t = 3 s ? A = 1.0, p = 1.0. for sinh x a n d c o s h x for small values of x . using the 2.11 Develop expressions for Taylor series series expansio n.

 

40

A Review of Hydrodynamics and Vector Analysis

Chap. Chap. 2

2.12 The pressures p d ( f ) and p B ( t ) act on the massless pistons containing the inviscid, incompressible incom pressible fluid in the horizontal hori zontal tube shown bel below ow.. Develop an expression for the velocity of the fluid as a function of time p = I gm/cm3.

p

100 cm

Note:

p&)

= CA in

P&)

=

at

CB sin (at + a)

a = 0.5 rad/s

where

c d

=

C , = 10dyn/cm3

2.13 An early experimenter of waves and other two-dimensional fluid motions closely approximating irrotational flows noted that at an impermeable horizontal boundary, the gradient of horizonta horizontall velocity in the vertical direction is alwayszero. Is this finding in accordance with with hydrodynamic fun fundamentals? damentals?If so, prove your answer.

t X

 

Small-Amplitude Water Small-Amplitude Wave Dedication

PIERRE SIMON LAPLACE Pierre Simon Laplace L aplace ((17 1749 49-1 -182 827) 7) is well known for the equation that bears his name. The Laplace equation is one of the most ubiquitous equations of mathematical mathematical physics (the (the Helmholtz, the diffusion, and the wave equation being others); others); it appears in i n electrostatics, hydrodynamics, groundwater flow flow,, thermostatics, and other fields. As had Euler, Euler, Laplace worked worke d in a great variety of areas, applying his knowledge of mathematics to physical problems. He has been called the the Newton of France. He was born in Beaumont-en-Auge, Normandy, France, and educated at Capn Capn (1765(1765-176 1767). 7). In 1768 1768 he he became Professo Pro fessorr of MatheMathe matics at the Ecole Militaire in Paris. Later he moved to the Ecole Normale, also in Paris. Napoleon appointed him Minister of the Interior Interio r in 1799 1799,, and he became a Count Cou nt in 1806 1806 and a Marqu Marquis is in i n 1807 1807,, the same year year that tha t he assumed the presidency p residency of the French Frenc hAcademy Academy of Sciences. A large portion of Laplace’s research was devoted to astronomy. He wrote on the orbital motion of the planets and celestial mechanics mechanics and on the stability of the solar sola r system. He also developed developed the hypothesis that the solar system coalesced out of a gaseous nebula. In other areas of physics, p hysics, he developed the theory of tides which bears his name, name, worked with Lavoisier on specific heat o off solids, studied capillary action, surface tension, ten sion, and electric theory, and with Legendre, introduced partial differential equations into the study of probability. probabilit y. He also developed and applied numerous solutions (potential functions) fun ctions) of of the Laplace equation. equation. 41

 

42

Small-AmplitudeWat Small-Amplitude Water er Wave Theory Formulation an and d Solution Solut ion

Chap Chap.. 3

3.1 INTRODUCTION

Real water waves propagate in a viscous fluid over an irregular bottom of varying permeab permeability. ility. A remarkable fact, however, is that in most cases the main body of the fluid motion is nearly irrotational. This is because the viscous eff effects ects are usually c concentrate oncentrated d in thin “boundary” layers near the surface and the bottom. Since water can also be considered reasonably incompressible, a velocity potential and a stream function should exist for waves. wav es. To simplify the mathematical analysis, numerous other ot her ass assumptions umptions must and an d wil willl be made as the development of the theory proceeds. 3.2 BOUNDARY VALUE PRO PROBL BL EMS

In formulating the small-amplitude water wave problem, it is useful to revie review, w, in very general terms terms,, the stru structur cture eo off boundary value problems, of which the present problem of interest is an example. Numerous classical

t

Boundary conditions (B.C.) specified

I

t5B’c‘

Region of interest (in general, an be any shape)

X

\

B.C. specified (a) Kinematic free surface boundary condition Dynamic free surface boundary condition

Lateral (LBO

I

Velocit Vel ocity y components

I

Bottom boundary conditi condition on (kinematic requirement) (b)

Figure 3.1 (a) Gener al structure of two-dimensional boundary value problems. (Note:The (Note: The num ber of boundary conditions conditions required required depends on the order of the differential equation.) (b) Tw o-dimen sional water water wa ves spe cified as a bound ary value problem.

 

Sec. 3.2

Boundary Value Problems

43

problems of physics and most analytical problems in engineering may be posed as boundary value problems; however, in some developments, this may not be apparent. The formulation of a boundary boun dary value problem is simply simply the expression expression in mathematical terms of the physical physical situation situatio n such that a unique solution exists. This generally consists of first establishing a region of interest and specifyinga differential equatio equ ation n that th at must be satisfied within the th e region (see Figure Figu re 3.la). Often, Often, there a re a n infinite number of solutions to the differen differen-tial equation and the remaining task is selecting the one or more solutions that ar e relevant to the physical physical problem under investigation. investigation. This selection selection is effecte effected d through the boundary boun dary conditions, that is, rejecting those solutions solu tions that a re not compatible with these conditions. conditions. In addition to the spatial (or geometric) boundary conditions, there are temporal boundary conditions which specify the state of the variable of interest at some point in time. time . This temporal condition is termed a n “initial “initial condition.” If we are interested in water waves, which are periodic in space, then we might specify, for example, that the waves are propagating in the positive x dire directio ction n a nd that at t = 0, the wave crest is located at x = 0. In the following development of linear water wave theory, it will be helpful help ful to relate each each major step to the general general structure struc ture of boundary value problems discussed previously. Figure Figure 3.lb presents the t he region of interest, the governing differential equations, and indicates in a general manner the important boundary conditions. 3.2.1 The Governing Dif feren tial Equation Equation

With the assumption of irrotational motion and an incompressible co ntinuity uity equation fluid, a velocity potential exists which should satisfy the contin

o.u=o

(3.la)

O*Vi$=O

(3.lb)

or

A s was shown in Chapter Chapt er 2, the divergence of a gradient leads to the Laplace equation, which must hold throughout the fluid.

The Laplace equation occurs frequently in many fields of physics and engineering engineerin g and numerous nume rous solutions to this equati eq uation on exist (see, e.g., e.g., the book by Bland, 1961), and therefore it is necessary to select only those which are applicab appl icable le to the particular water wave motion of interest. interest. In addition, for flows that are nondivergent and irrotational, the Laplace Lapl ace equation also applies to the stream function. Th e incompress incompressibil ibility ity

 

44

Small-Amplitude Water Wave Theory Formulation and Solution

Chap. Chap. 3

or, equivalently, the nondivergent condition for two dimensions guarantees the existence of a stre stream am function, ffrom rom which the velocities under the wave can be determined. Substituting these velocities into the irrotationality condition again yields the Laplace equation, except for the stream function this time, (3.3a) or (3.3b) This equation must hold throughout the fluid. If the motion had been rotational, ye yett fiiction fiictionless, less, the governing equatio equation n would b be e V2y/ =

(3.4)

where o s the vortici vorticity. ty. A few few comme comments nts on the velocity potential a and nd the stream stre am function may help in obtaining a better understanding for later applications. First, as mentioned earlier, the velocity potential can be defined for both two and three dimensions, dimensions, whereas the defi definition nition of the stream function is such that it can only be defined for three dimensions if the flow is symmetric about an axis (in this case although the flow occurs in three dimensions, it is mathematically two-dimensional). It ttherefore herefore follow followss that the stream function is of greatest use in cases where the wave motion occurs in one plane. Second, Seco nd, the Laplac Laplace e equation is linear; that is, it involves no products a and nd thus has the interesting and valuable property of superposition; that is, if 4 , and 42 each satisfy the Laplace equation, then 43 = A 4 , + B42 also will solve the equati equation, on, where A and B are ar e arbitr arbitrary ary constants. Therefore, we can add and subtract solutions to build up solutions applicable for different problems of interest. 3.2.2 Bounda Boundary ry Condition s

A t w e t h e r t is fixed, such as the bo bottom, ttom, or free, such as the water surface, which is free to deform under the influence of forces, certa certain in physical conditions must be satisfied by the fluid velocities. These conditions on the water particle kinematics are called kinema tic boun dary conditi conditions. ons. At any surface or fluid interface, it is clear that there t here must be no flow across the interfa interface; ce; otherwise, there would be no interface. This is most obvious in the case of an impermeable fixed surface such as a sheet pile seaw seawall all.. The mathematical expression for the kinematic boundary condition Kinematic trorrndat-y c a n d i t h

may be derived from the equation which describes the surface that constitutes the th e boundary. A Any ny fixed or moving ssurface urface can be expressed in tterms erms of

 

Sec. 3.2

Boundary Value Problems

45

a mathematical expression of the form F ( x , y , z , t ) = 0. For example, for a stationary sphere of fixed radius a , F ( x , y , z , t ) = x 2+ y 2 + z 2 - a 2 = 0. If the surface surf ace vanes with time, as would the water surface, then the total derivative of the surface with respect to time would be zero on the surface. In other words, if w we e move with the surface, it does not change.

aF dF x , , z , 0 = o = -aF +u-+v-+wax at Dt av

or

- -a F- - u . V F = u . n l V F I

(3.5a) Z

al

on F ( x . y , r , f ) = ~

at

(3.5b)

where the unit vector normal to the surface has been introduced as

n

=

VF/IVFI.

Rearranging the kinematic boun boundary dary con condition dition results:

where

This condition requires req uires that the co component mponent of the fluid v velocity elocity norm normal al to the surface be related tto o the local velo velocity city of the surface. If the ssurface urface does not change with time, then u n = 0; that is, i s, the veloci velocity ty component normal to the surface is zero.

-

Example 3.1 Fluid in a U-tube has been forced to oscillate sinusoidally due to an oscillating pressure on one leg of the tube (see Figure 3.2). Develop the kinematic boundary condition for the free surface in leg leg A .

Solution. The still water level level in the U-tube U -tube is located locat ed at z = O.Th O.The e motion m otion of the free surface can be described by z = q ( t ) = a cos t , where a is the the amplitude ofth e vanation of q. If we we examine closely the motion of a fluid particle partic le at the surface (Figure 3. 3.2b), 2b), as the surface drops, with velocity w , t follows that the t he particle has to t o move with the speed of the surface or else the particle leaves the surf surface. ace.The The same sam e is true for a rising surface.Therefore, we would postulate on physical grounds that

 

46

Cha Chap. p. 3

Small-Am SmallAm plitude Wat er Wa ve Theory Formulat Formulation ion and Soluti Solution on Oscillating pressure

z= o

(a)

Figure 3.2 (a) Oscillating flow in a U-tube; (b) details of free surface.

where dqfdt = the rate of rise or fall of the surface. To ensure that this is formally correct, we we follow the equation for the kinematic boundary bounda ry condition, Eq. (3.6), where F ( z , t ) = z - qt)= 0. Therefore,

where n = O i + O j + 1k, directed vertically upward and u = u i out the scalar product, we we find that tha t

+ v j + wk,

nd carrying

w = - rl

at which is the same as obtained o btained previously, when we realize that dqfdt = a q f a t , as q is only a function of time.

The Bottom Boundary Boundary Condition (BBC ).

In general, general, the lower lower boundary of our region of interest is described as z = - h ( x ) for a two-dimensional case where the origin is located at the still water level and h represents the depth. If the bottom botto m is impermeable, we we expect that u n = 0, as the bottom does not move with time. (For some cases, such as earthquake motions, obviously obvious ly the time dependency of of the bottom must be included.) in cluded.) The surface equation for the bottom is F ( x , z) = z + h ( x ) = 0. Therefore,

u.n=O

(3.7)

where

VF

 

dh -i+lk dx

(3.8)

Sec. 3.2

47

Boundary Value Value Problems

Carrying out the dot do t product and a nd multiplying mult iplying through by by the square sq uare root, we we have u

dh dx

w

-

=

0

on

z

=

-h(x)

(3.9a)

or

w

=

-u

dh dx

-

on z = - h ( x )

(3.9b)

For a horizontal bottom, then, w = 0 on z = -h. For a sloping bottom, we have

w u

-=--

dh dx

(3.10)

Referring to Figure 3.3, it is clear that the kinematic condition states that the Referring fl flow ow at the bottom bot tom is tangent to the bottom. bottom . In fact, we we could treat the bottom bo ttom as a streamline, as the flow is everywhere tangential to it. The bottom boundary condition, Eq. (3.7), also applies directly to flows in three dimendi mensions in which h is h ( x , y ) .

Kinem atic Fre Free Surface Bou ndary Condition (K FS BC ). The free surface of a wave can be described as F ( x , y , z , t ) = z - q ( x , y , t ) = 0, where q ( x , y , t ) s the displacement of the free surface about the horizontal plane, z = 0. The kinematic boundary condition at a t the free surface is u.n=

lllat

J ( W W 2 (WW2

1

on z = q(x,y , t )

(3.112

i

Figure 3.3 Illustration o f bottom boundary condition for the two-dimensional case.

 

S m a l l -A -A m p l i t u d e W a t e r W a v e T h e o r y F o r m u l a t i o n a n d S o l u t i o n

48

Chap. 3

where

(3.1 b) Carrying out the dot product yields (3.1 c) This condition, the KFSBC, is a more complicated expression than that obtained for (l), the U-tube, where the flow was normal to the surface and (2) the bottom, bo ttom, where the flow was tangential. In fact, inspection ofEq. (3.11~) will verify that the KFSBC is a combination of the other two conditions, which whi ch a are re just special cases of this more general type of condition.’

Dynamic Free Surface Boundary Condition.

The boundary conditions for fixed surfaces arexelatively easy to prescribe, as shown in the preceding section, and they apply on the known surface. A distinguishing feature of fixed, (in space)that surfaces is that they support pressure variations. However, However surfaces are ar e “free,” ssuch uch can as the air-w air-water ater interface, cannot support variations iin n pressure2across tthe he interfac interface e and henc hence e must respond in order to maintain the pressure as uniform. A second boundary condition, termed a dynamic boundary condition, is thus required on any free surface or interface, to prescribe the pressure distribution pressures on this boundary. An An interesting effect o off the displacement of the free surface is that the position of the upper boundary is not known a priori in the water wave problem. This aspect causes considerable difficulty in the attempt to obtain accurate accu rate solutions that apply for large wave heigh heights ts (Chapter 11). A s the dynamic free surface surface boundary condition is a requirem requirement ent that the pressure on the free surface be uniform along the wave form, the Bernoulli equation [Eq. (2.92)] with p q surface, z = q(x, t ) ,

-_ at

+1 2

u2 + w’)

=

constant is applied on the free

+ P3

gz

=

C(t)

P

(3.12)

where p q s a constant and a nd usually taken as ga gage ge pressure, p t l = 0.

Con diti ditions ons at “Responsive”Boundaries. “Responsive”Boundaries.

As noted previously, an ad addiditional condition must be imposed on those boundaries that can respond to spatial or temporal variatio vari ations ns in pr pressure. essure. In the case of w wind ind blowing across ’Th e reader reader is urged urged to de velop the general general kinema tic free free surface boundary cond ition for a wave propagating in the x direction alone. ‘Neglecting surface tension.

 

Sec. 3.2

49

Boun dary Value Problems

a water surface and generating waves, if the pressure relationship were known, know n, the Bernoulli equation woul would d serve to couple that wind field with the kinematics of the wave. The wave and wind field would be interdependent and the wave motio motion n would be term termed ed “coupled. “ coupled.” ” If the wave were driven b by y, but did not affect the applied surface pre pressure ssure distribution, this would be be a case cas e of “f o rc ed wav wave e motion and agai again n the Bernoulli equation would would serve to express the boundary condition. For the simpler cas case e that is explo explored red in some detail in this thi s chapter, the pressure wi will ll be considered to be uniform a and nd hence a case of “free” wave wave motion exists. Figure 3. 3.4 4 depict depictss various degrees of coupling betw between een the wind an and d wave fields.

__Jt

Wind

Surface pressure distribution affected by interaction of wind and waves

X

Translating pressure field

p = atmospheric everywhere

Figure 3.4 Various degrees of air-water boundary interaction and coupling to atmosp heric pressure pressure field: field: (a) coupled wind a nd wa ves; (b) forced waves d ue to mo ving pressure field; field; (c) free free wav es-not affected by pressure variations at airairwater interface.

 

50

S m all-A mp lit ude Wat er Wa v e Theory Formulat Formulation ion and S olut ion ion

Chap. 3

The boundary condition for free waves is termed the “dynamic free surface boundary condition” (DFSBC), which the Bernoulli equation expresses as Eq. (3.13) with a uniform surface pressurep,: -

at

+5 +I p

2

[(3ax7 + (

I2]

gz

=

C(t),

z = ~ ( x ,)

(3.13)

=

p,, 0centimeters), whereIfp,, pthe ,, is wave a consta constant nt hs an and dare usually taken as ga gage geor pressure, , lengt lengths very shor short t (on the order der of several centimeters) , the surface is no lon longer ger “free.” Although the pressure is uniform above the water surface, as a result of the surface curvature, a nonuniform nonu niform pressure will will occur within the water immediately below the surface film. Denoting the coefficient of surface tension as o’, he tension per unit length T is simply

T

(3.14)

= 0‘

Consider now a surface for whi which ch a curvat curvature ure exists as shown in Figure 3.5. Denoting p as the pressure under the free surface, a free-body force analysis in the vertical direction direct ion yields T [-sin a J , sin C Y ~ ~

~ X ] (p

-pa) A x

+ terms of order A x 2 = 0

in which the approximation d q / d x = sin a will be made. Expanding by Taylor’s series and allowing the size of the element to shrink to zero yields (3.15)

Thus for cases in which surface tension forces are important, the dynamic free surface surface boundary conditio condition n is modified to - dq5

at

p 2 p

+’d2q + _’d2q +- 1 p ax2 2

[(*I2 + (?I2] + g z ax

=

C(t),

z

=

~ ( x ,)

(3.16)

which will be of use in our later examination of capillary water waves. Lateral Boundary Conditions. At this stage boundary conditions have been been discussed for the bottom and upper surfaces. I n order to complete complete specification of the bound boundary ary value problem, conditions condi tions must also a lso be be speci-

x

 

+ Ax

Defin ition sketch sketch for surface surf ace elemen t. Figure 3.5

Sec. 3.2

Boundary Value Value Problem s

51

fied on the remaining lateral boundaries. boundaries. There a are re several several situations that must be considered. If the waves are propagating in one direction (say the x direction), conditions ar e two-dimensional two-dimensional and then “no-fl “no-flow” ow” conditions are appropriate for the velocities in the y direction. The boundary conditions to be applied in the th e x direct direction ion depend on the th e problem under consideration. consideration. If the wave motion results from a prescribed distur d isturbance bance of, sa say, an object at x = 0, which is the classical wavemaker problem, then at the object, the usual kinematic boundary bou ndary condition condi tion is expressed by Figu Figure re 3.6a. Consider a vertical paddle acting as a wavemaker in a wave tank. If the displacement of the paddle may be described as x = S ( z , ) , the kinematic boundary condition is

where

as

li--k

-

z

t

Outgoing waves only

(b)

Figure 3.6 (a) Schematic of wavemaker i n a wave tank; b) radiation condition for wavemaker problem problem for region unbounded in x direction.

 

52

Small-Amplitude Water Wave Theory Formulation and Solution

Cha Chap. p. 3

or, carrying carrying out the dot product, product, (3.17) which, of course, requires that tha t tthe he fluid particles at the moving wall follow follow the wall. wall. Two beach different conditions occur the of other possible at a fixed as shown at the rightatside Figure 3.6a, lateral whe where re boundaries: a kinematic condition would be applied, or as in Figure 3.6b, where a “radiation” boundary conditi condition on is applied which which requires that th at only outgoing waves occur at infinity. This precludes incoming waves which would not be physically meaningful in a wavemaker problem. For wave wavess that are ar e periodic in space space and time, the boundary b oundary co condition ndition is expressed as a periodicity condition,

+(x, 0 = +(x + L , t )

( 3.18a)

+