Gravity Capillary Free Surface Flows

8/3/2019 Gravity Capillary Free Surface Flows

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G R A V I T YC A P I L L A R Y F R E E - S U R F A C E F L O W S

Free-surface problems occur in many aspects of science and of everyday life, for

example in the waves on a beach, bubbles rising in a glass of champagne, melting

ice, pouring fl ws from a container and sails billowing in the wind. Consequently,

the theory of gravitycapillary free-surface fl ws continues to be a fertile fiel of

research in applied mathematics and engineering.

Concentrating on applications arising from flui dynamics, Vanden-Broeck

draws upon his years of experience in the fiel to address the many challenges

involved in attempting to describe such fl ws mathematically. Whilst careful

numerical techniques are implemented to solve the basic equations, an empha-sis is placed upon the reader developing a deep understanding of the structure of

the resulting solutions. The author also reviews relevant concepts in flui mechan-

ics to enable readers from other scientifi field to develop a working knowledge

of free-boundary problems.


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O T H E R T I T L E S I N T H I S S E R I E S

All the titles below can be obtained from good booksellers or from

Cambridge University Press. For a complete series listing visit

http://www.cambridge.org/uk/series/sSeries.asp?code=CMMA

Waves and Mean Flows

OLIVER BUHLER

Lagrangian Fluid Dynamics

ANDREW BENNETT

Plasticity

SIA NEMAT-NASSER

Reciprocity in Elastodynamics

JAN D. ACHENBACH

Theory and Computation in Hydrodynamic Stability

W. O. CRIMINALE, T. L. JACKSON AND R. D. JOSLIN

The Physics and Mathematics of Adiabatic Shear Bands

T. W. WRIGHT

Theory of Solidificatio

STEPHEN H. DAVIS

The Dynamics of Fluidized Particles

ROY JACKSON

Turbulent Combustion

NORBERT PETERS

Acoustics of FluidStructure Interactions

M. S. HOWE

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

PHILIP HOLMES, JOHN L. LUMLEY AND GAL BERKOOZ

Topographic Effects in Stratifie Flows

PETER G. BAINES

Ocean Acoustic Tomography

WALTER MUNK, PETER WORCESTER AND CARL WUNSCH

Benard Cells and Taylor Vortices

E. L. KOSCHMIEDER


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GRAVITYCAPILLARY

FREE-SURFACE

FLOWS

JEAN-MARC VANDEN-BROECKUniversity College London


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CAMBRIDGE UNIVERSITY PRESS

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So Paulo, Delhi, Dubai, Tokyo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-81190-3

ISBN-13 978-0-511-72961-4

Cambridge University Press 2010

2010

Information on this title: www.cambridge.org/9780521811903

This publication is in copyright. Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any part

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Cambridge University Press has no responsibility for the persistence or accuracyof urls for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (NetLibrary)

Hardback
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To Mirna and Ada


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Contents

Preface page xi

1 Introduction 1

2 Basic concepts 7

2.1 The equations of fluid mechanics 7

2.2 Free-surface flows 8

2.3 Two-dimensional flows 11

2.4 Linear waves 15

2.4.1 The water-wave equations 15

2.4.2 Linear solutions for water waves 17

2.4.3 Superposition of linear waves 24

3 Free-surface flows that intersect walls 31

3.1 Free streamline solutions 33

3.1.1 Forced separation 33

3.1.2 Free separation 43

3.2 The effects of surface tension 58

3.2.1 Forced separation 58

3.2.2 Free separation 66

3.3 The effects of gravity 733.3.1 Solutions with 1 = 0 (funnels) 83

3.3.2 Solutions with 1 = 0 (nozzles and bubbles) 88

3.3.3 Solutions with 1 = /2 (flow under a gate with

gravity) 94

3.4 The combined effects of gravity and surface tension 98

3.4.1 Rising bubbles in a tube 99

3.4.2 Fingering in a Hele Shaw cell 103

3.4.3 Further examples involving rising bubbles 1083.4.4 Exponential asymptotics 112

vii


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viii Contents

4 Linear free-surface flows generated by moving

disturbances 114

4.1 The exact nonlinear equations 115

4.2 Linear theory 116

4.2.1 Solutions in water of finite depth 116

4.2.2 Solutions in water of infinite depth 122

4.2.3 Discussion of the solutions 124

5 Nonlinear waves asymptotic solutions 129

5.1 Periodic waves 129

5.1.1 Solutions when condition (5.55) is satisfied 135

5.1.2 Solutions when condition (5.55) is not satisfied 138

5.2 The Kortewegde Vries equation 142

6 Numerical computations of nonlinear water waves 148

6.1 Formulation 148

6.2 Series truncation method 151

6.3 Boundary integral equation method 152

6.4 Numerical methods for solitary waves 156

6.4.1 Boundary integral equation methods 157

6.5 Numerical results for periodic waves 160

6.5.1 Pure capillary waves (g = 0, T = 0) 1606.5.2 Pure gravity waves (g

= 0, T = 0) 164

6.5.3 Gravitycapillary waves (g = 0, T = 0) 1756.6 Numerical results for solitary waves 181

6.6.1 Pure gravity solitary waves 181

6.6.2 Gravitycapillary solitary waves 186

7 Nonlinear free-surface flows generated by moving

disturbances 191

7.1 Pure gravity free-surface flows in water of finite depth 192

7.1.1 Supercritical flows 192

7.1.2 Subcritical flows 1957.2 Gravitycapillary free-surface flows 201

7.2.1 Results in finite depth 201

7.2.2 Results in infinite depth (removal of the

nonuniformity) 203

7.3 Gravitycapillary free-surface flows with Wilton ripples 206

8 Free-surface flows with waves and intersections with

rigid walls 210

8.1 Free-surface flow past a flat plate 2118.1.1 Numerical results 212


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Contents ix

8.1.2 Analytical results 215

8.2 Free-surface flow past a surface-piercing object 218

8.2.1 Numerical results 219


8.3 Flow under a sluice gate 226

8.3.1 Formulation 228

8.3.2 Numerical procedure 231

8.3.3 Discussion of the results 233

8.4 Pure capillary free-surface flows 236



9 Waves with constant vorticity 244

9.1 Solitary waves with constant vorticity 245

9.1.1 Mathematical formulation 2459.1.2 Numerical procedure 248

9.1.3 Discussion of the results 249

9.2 Periodic waves with constant vorticity 267

9.2.1 Mathematical formulation 268

9.2.2 Numerical procedure 270


9.2.4 Discussion 276

10 Three-dimensional free-surface flows 27810.1 Greens function formulation for two-dimensional

problems 278

10.1.1 Pressure distribution 278

10.1.2 Two-dimensional surface-piercing object 283

10.2 Extension to three-dimensional free-surface flows 286

10.2.1 Gravity flows generated by moving disturbances

in water of infinite depth 286

10.2.2 Three-dimensional gravitycapillary free-surfaceflows in water of infinite depth 293

10.3 Further extensions 298

11 Time-dependent free-surface flows 301

11.1 Introduction 301

11.2 Nonlinear gravitycapillary standing waves 301

References 308

Index 318


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Preface

This book is concerned with the theory of gravitycapillary free-surface

flows. Free-surface flows are flows bounded by surfaces that have to be foundas part of the solution. A canonical example is that of waves propagating

on a water surface, the latter in this case being the free surface.

Many other examples of free-surface flows are considered in the book (cav-

itating flows, free-surface flows generated by moving disturbances, rising

bubbles etc.). I hope to convince the reader of the beauty of such problems

and to elucidate some mathematical challenges faced when solving them.

Both analytical and numerical methods are presented. Owing to space lim-

itations, some topics could not be covered. These include interfacial flowsand the effects of viscosity, compressibility and surfactants. Some further

developments of the theories described in the book can be found in the list

of references.

Many results presented in the book have grown out of my research over the

last 35 years and, of course, out of the research of the whole fluid mechanics

community. References to the original papers are given. For this book, I

have repeated the older numerical calculations with larger numbers of grid

points than was possible at the time. I am pleased to report that the new

results are in agreement with the earlier ones!I am deeply indebted to my mentors, coworkers, students and friends who

participated in the research. I feel very fortunate to have known them and I

look forward to continuing these collaborations in the future. Special thanks

are due to Scott Grandison for help with the figures, to David Tranah for his

patience over the years in waiting for the manuscript, to Susan Parkinson for

very careful copy-editing, to Caroline Brown for her help during production

and to Cambridge University Press for publishing the book.

xi


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1

Introduction

Free-surface problems occur in many aspects of science and everyday life.

They can be defined as problems whose mathematical formulation involvessurfaces that have to be found as part of the solution. Such surfaces are

called free surfaces. Examples of free-surface problems are waves on a beach,

bubbles rising in a glass of champagne, melting ice, flows pouring from a

container and sails blowing in the wind. In these examples the free surface

is the surface of the sea, the interface between the gas and the champagne,

the surface of the ice, the boundary of the pouring flow and the surface of

the sail.

In this book we concentrate on applications arising in fluid mechanics. Wehope to convince the reader of the beauty of such problems and to present

the challenges faced when one attempts to describe these flows mathemati-

cally. Many of these challenges are resolved in the book but others are still

open questions. We will always attempt to present fully nonlinear solutions

without restricting assumptions on the smallness of some parameters. Our

techniques are often numerical. However, it is the belief of the author that

a deep understanding of the structure of the solutions cannot be gained

by brute-force numerical approaches. It is crucial to combine numerical

methods with analytical techniques, especially when singularities are present.Therefore analytical treatments will be presented whenever appropriate. We

hope that the techniques discussed will be useful not only to researchers in

the field but also to those working in areas other than fluid mechanics.

For completeness the relevant concepts of fluid mechanics are reviewed in

Chapter 2.

Free-surface flows fall into two main classes. The first is the class of such

flows for which there are intersections between the free surface and a rigid

surface. The classic example in this class is the flow due to a ship moving atthe surface of a lake, which involves an intersection between the free surface

1


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2 Introduction

(i.e. the surface of the lake) and a rigid surface (i.e. the hull of the ship).

Other examples are jets leaving a nozzle, cavitating flows past an obstacle,

bubbles attached to a wall and flows under a sluice gate. In each case there is

a rigid surface (the nozzle, the obstacle, the wall or the gate) that intersects

a free surface.

The second class contains free-surface flows for which there are no inter-

sections between the free surface and a rigid wall. Here the classic example

is the flow due to a submerged object moving below the surface of a lake. If

the object is small compared with the size of the lake, it is then reasonable

to regard the lake as being of infinite horizontal extent; then there is no

intersection between the free surface and a rigid surface. Other examples

include free bubbles rising in a fluid and solitary waves.

Chapter 3 is concerned with the theory of free-surface flows of the first

class. We use the classical assumptions of potential flow theory (irrota-tional flows of inviscid and incompressible fluids) and proceed in stages of

increasing complexity. In addition we restrict our attention to steady and

two-dimensional flows (time-dependent and three-dimensional flows are con-

sidered in the last two chapters). In the first stage the effects of gravity

and surface tension are neglected. Such free-surface flows are called free

streamline flows. They are characterized by a constant velocity along the

free surfaces. Conformal mapping techniques can then be used to find ex-

act nonlinear solutions. This situation is fortunate since there are very fewsuch solutions for free-surface flows. The main results of the free streamline

theory are summarized in Section 3.1. The most important result for the

remaining part of Chapter 3 is that the velocity and slope of the free surface

must be continuous at a separation point (i.e. the intersection between a

free surface and a rigid surface in two dimensions) but the curvature of the

free surface is in general infinite. Since this curvature only enters the equa-

tions when surface tension is included in the dynamic boundary condition,

we expect a gravity flow with small gravity to be a regular perturbation of

a free streamline flows, and a capillary flow with small surface tension tobe a singular perturbation. This is confirmed by the numerical results pre-

sented in Sections 3.2 and 3.3. It is shown in Section 3.2 that the presence

of surface tension does not remove the infinite curvature at the separation

points. On the contrary it makes the problem more singular by introducing

a discontinuity in slope at the separation point. Depending on the angle

between the free surface and the rigid boundary, the velocity is infinite or

equal to zero at the separation point. The appearance of an infinite ve-

locity is a limitation of the model. We show that a basic way to removethis singularity is to take into account the finite thickness of the rigid walls,


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Introduction 3

i.e. to consider the walls as thin objects with a continuous slope. When

surface tension is neglected, the free surfaces leave the wall tangentially but

the position of the separation point along the walls is free. We then have

a one-parameter family of solutions (the parameter defines the position of

the separation point) and the question is to determine which value of the

parameter is physically relevant. This is an example of a selection problem.

Selection problems are usually resolved by imposing an extra constraint on

the problem or by including a previously neglected effect and taking the

limit as this effect approaches zero. Here both approaches work. A unique

position for the separation point is obtained by neglecting surface tension

and imposing a constraint known as the BrillouinVillat condition. Equiva-

lently, the same position for the separation point is obtained by solving the

problem with surface tension and then taking the limit as the surface tension

approaches zero. We will show that the mechanism by which solutions witha small amount of surface tension are selected is related to the fact that, for

each value of the surface tension, the separation point has only one position

for which the free surface leaves the wall tangentially.

In Section 3.3 we turn our attention to the effects of gravity on free stream-

line solutions. We assume that gravity is acting vertically downwards and

neglect surface tension (the combined effects of gravity and surface tension

are covered in Section 3.4). We show that it is again possible for the free

surface not to leave the walls tangentially, but the angle between the freesurface and the wall must be such that the velocity is finite at the separa-

tion point (infinite velocities cannot occur on a free surface in the absence

of surface tension). Local analysis shows that there are only three possible

behaviours at the separation point. In the first there is a horizontal free

surface at the separation point, in the second there is an angle of 120 be-

tween the free surface and the wall and in the third the free surface leaves

the wall tangentially. We show by examples (e.g. flows emerging or pour-

ing from containers) that these three possibilities occur in free-surface flows

with gravity. The restriction to three local behaviours is to be contrastedwith the cases including surface tension discussed in Section 3.2, where all

angles between the walls and the free surfaces are in principle possible. This

contrast suggests that some interesting behaviours might emerge if we com-

bine the effects of gravity and surface tension; this is confirmed in Section

3.4. We show in this section that some free-surface flow problems possess

a continuum of solutions when surface tension is neglected and an infinite

discrete set of solutions when surface tension is taken into account. This

discrete set reduces to a unique solution as the surface tension approacheszero. Therefore a small amount of surface tension can again be used to select


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4 Introduction

solutions. One difference between this selection mechanism and that de-

scribed in Section 3.2 is that there is a infinite discrete set of solutions when

surface tension is included instead of one solution. Another difference is

that the selection is associated with exponentially small terms in the surface

tension. This implies that exponential asymptotics is required to predict

the selected solutions analytically. As we shall see, exponential asymptotics

plays an important role in many other free-surface flow problems such as

the study of gravitycapillary waves (see Chapter 6) and free-surface flows

generated by moving disturbances for small values of the Froude number or

small values of the surface tension (see Chapter 8).

The results presented in Chapter 3 were obtained by a combination of var-

ious numerical schemes that the author has used successfully over the years

to obtain highly accurate solutions for free-surface flow problems. They in-

clude series truncation techniques and boundary integral equation methods.The idea of the series truncation methods is to identify a rapidly convergent

series representation for the solution that satisfies all the appropriate partial

differential equations (for example the Laplace equation for potential flows)

and all the linear boundary conditions. This often requires a local analysis

to identify and remove the singularities associated with corners, stagnation

points etc. The series is then truncated after a finite number of terms and

the unknown coefficients are determined by satisfying the remaining non-

linear boundary condition (the pressure condition for free-surface flows) atappropriately chosen collocation points. This leads to a system of nonlinear

algebraic equations which can be solved by iteration (for example by using

Newtons method). Boundary integral equation methods are based on a

reformulation of the problem as a system of nonlinear integro-differential

equations for the unknown quantities on the free surface. These equations

are then discretised and the resultant nonlinear algebraic equations solved

by iteration. Such boundary integral equation methods have been used ex-

tensively by many researchers.

Insight into free-surface flows of the second class can be gained by study-ing the limitations of the classical linear theories. In particular we study in

Chapter 4 the waves generated by a disturbance moving at a constant veloc-

ity (for example a submerged object or a pressure distribution). The results

are qualitatively independent of the type of disturbance, and so most results

in Chapter 4 are presented just for a pressure distribution with bounded sup-

port. A frame of reference moving with the pressure distribution is chosen

and the flow is assumed to be steady. In the linear theory it is assumed

that the disturbance is small enough for the flow to be a small perturbationof a uniform stream. The equations are then linearised (around a uniform


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Introduction 5

stream) and the resulting linear equations solved by separation of variables

and using Fourier transforms. These linear solutions can be expected to

be a good approximation when the disturbance is small. In other words, if

denotes the size of the disturbance, we expect the nonlinear solutions to

approach the linear solutions as

0. This is usually the case, but the

problem is complicated by the fact that the solutions depend not only on

but also on other parameters such as the Froude number

F =U

(gH)1/2

and the capillary number

=T g

U4.

Here U is the velocity of the disturbance, H the depth of the fluid, T the

surface tension, g the acceleration of gravity and the density of the fluid.

This leads to nonuniformities when these parameters approach critical

values. These nonuniformities appear in an obvious way in the linear solu-

tions. For example the linear theory for pure gravity flows (i.e. flows with

T = 0) predicts infinite displacements of the free surface as F 1. Thisis unacceptable since the linear theory assumes small perturbations around

a uniform stream and in particular small displacements of the free surface.More precisely, for any F = 1 the linear theory provides a good approxima-tion of the nonlinear problem as 0. However, for any , no matter howsmall, the linear solutions become invalid as F 1. A similar situationoccurs for gravitycapillary flows. As approaches a critical value H, the

linear theory again predicts infinite displacements of the free surface. The

critical value H depends on the depth H (for example H = 0.25 in water

of infinite depth).

The resolution of these nonuniformities requires a nonlinear theory. We

develop in Chapter 7 such a theory by solving the fully nonlinear equa-tions numerically. This approach has the advantage of not pre-assuming a

particular type of expansion. Furthermore it gives solutions without any

assumption on the size of . It also provides a valuable guide in deriving

appropriate perturbation expansions for small or moderate values of . We

show in Chapter 7 that the resolution of the nonuniformities is associated

with solitary waves. Near the critical values of and F, there are not only

solutions that are perturbations of a uniform stream (the nonlinear equiv-

alent of the linear solutions mentioned earlier) but also solutions that areperturbations of solitary waves. Some of these solitary waves are of the


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6 Introduction

well-known Kortewegde Vries type but others are solitary waves with de-

caying oscillatory tails.

As a preparation for the nonlinear results of Chapter 7 we present in

Chapters 5 and 6 analytical and numerical solutions for nonlinear periodic

and solitary waves. Such solutions describe the far-field behaviour of the

nonlinear free-surface flows past disturbances described in Chapter 7. They

are also interesting canonical free-surface flow problems. In particular we

show that waves of the Kortewegde Vries type have oscillatory tails of con-

stant amplitude when surface tension is included. These waves are referred

to as generalised solitary waves to distinguish them from true solitary waves,

which are flat in the far field.

In Chapter 8 we consider further free-surface flows of the first class (i.e.

flows for which the free surface intersects rigid walls). The solutions of

Chapter 3 approach either an infinitely thin jet in the far field or are wave-less. We study in Chapter 8 various extensions for which the free surface is

characterised by a train of nonlinear waves in the far field. An attractive

feature of some of these flows is that exact formulae can be derived for the

amplitude of the waves in the far field. These relations provide analytical

insight and can be used to check the accuracy of the numerical codes.

All the flows in Chapters 38 are assumed to be steady, two-dimensional

and irrotational. The final three chapters of the book describe some ex-

tensions in which these assumptions are removed. In Chapter 9 we studysolitary and periodic waves with constant vorticity. We show that there are

new solution branches that do not have an equivalent for irrotational waves.

In Chapter 10 we study some three-dimensional free-surface flows. In partic-

ular we calculate three-dimensional gravitycapillary solitary waves. These

waves are characterised by decaying oscillations in the direction of propaga-

tion and monotonic decay in the direction perpendicular to the direction of

propagation.

Chapter 11 is concerned with time-dependent free-surface flows. This is a

very large subject involving problems such as breaking waves, stability, thebreaking of jets etc. Here we limit our attention to the subject of gravity

capillary standing waves. This choice is motivated by the fact that these

standing waves have properties similar to those of the travelling gravity

capillary waves presented in Chapter 6.


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2

Basic concepts

2.1 The equations of fluid mechanics

We start with a brief introduction to the equations of fluid mechanics. For

further details see for example Batchelor [8] or Acheson [1].

All the fluids considered in this book are assumed to be inviscid and to

have constant density (i.e. to be incompressible).

Conservation of momentum yields the Euler equations

Du

Dt= 1

p + X, (2.1)

where u is the vector velocity, p is the pressure and X is the body force.Here

D

Dt=

t+ u (2.2)

is the material derivative. We assume that the body force X derives from a

potential , i.e. that

X = . (2.3)

In most applications considered in this book, the flow is assumed to beirrotational. Therefore

u = 0. (2.4)

Relation (2.4) implies that we can introduce a potential function such that

u = . (2.5)

Conservation of mass gives

u = 0. (2.6)

7


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8 Basic concepts

Then (2.5) and (2.6) imply that satisfies Laplaces equation

2 = 0. (2.7)Flows that satisfy (2.4)(2.7) are referred to as potential flows. Using the

identity

u u = 12(u u) + ( u) u, (2.8)

(2.4) and (2.2) yield

Du

Dt=

u

t+

1

2(u u). (2.9)

Substituting (2.9) into (2.1) and using (2.3) and (2.5) we obtain

t+

u u2

+p

+ = 0. (2.10)

After integration, (2.10) gives the well-known Bernoulli equation

t+

u u2

+p

+ = F(t). (2.11)

Here F(t) is an arbitrary function of t. It can be absorbed in the definition

of , and then (2.11) can be rewritten as

t+

u u2

+p

+ = B, (2.12)

where B is a constant. For steady flows (2.12) reduces to

u u2

+p

+ = B. (2.13)

2.2 Free-surface flows

We introduce the concept of a free surface by contrasting the flow past a

rigid sphere (see Figure 2.1) with that of the flow past a bubble (see Figure2.2). Both flows are assumed to be steady and to approach a uniform stream

with a constant velocity U as x2 + y2 + z2 ; the effects of gravity areneglected. They can interpreted as the flows due to a rigid sphere or a

bubble rising at a constant velocity U, when viewed in a frame of reference

moving with the sphere or the bubble. The pressure pb in the bubble is

constant. We denote by S the surface of the sphere or bubble and by n the

outward unit normal.

The flow past a sphere can be formulated as follows:

xx + yy + zz = 0 outside S, (2.14)


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2.2 Free-surface flows 9

z

y

xS

U

Fig. 2.1. The flow past a rigid sphere. The surface S of the sphere is described byx2 + y2 + z2 = R2 , where R is the radius of the sphere.

z

y

x

S

U

Fig. 2.2. The flow past a bubble. The surface S of the bubble is not known a prioriand has to be found as part of the solution.

n= 0 on S (2.15)

(x, y , z ) (0, 0, U) as x2 + y2 + z2 . (2.16)

Equation (2.14) is Laplaces equation (2.7) expressed in cartesian coordi-

nates. The boundary condition (2.15) is known as the kinematic boundary

condition. It states that the normal component of the velocity vanisheson S.


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10 Basic concepts

Equations (2.14)(2.16) form a linear boundary value problem whose

solution is

= U

z +R3z

2(x2 + y2 + z2)3/2

. (2.17)

Here R is the radius of the sphere.We note that we have derived the solution (2.17) without using the Bernoulli

equation (2.13), which for the present problem can be written as

1

2(2x +

2y +

2z ) +

p

=

1

2U2 +

p

. (2.18)

Here p denotes the pressure as x2 + y2 + z2 . Equation (2.18) holds

everywhere outside the sphere. In deriving (2.18) we have set = 0 in

(2.13) and evaluated B by taking the limit x2

+ y2

+ z2

in (2.13).Then, using (2.16) gives B = U2/2 +p/.

Equation (2.18) is nonlinear but it is only used if we want to calculate the

pressure p inside the fluid. In other words the main problem is to find by

solving the linear set of relations (2.14)(2.16). We may then substitute the

values (2.17) of into the nonlinear equation (2.18) if we wish to compute

the pressure.

We now show that we need to use the nonlinear boundary condition (2.18)

to solve for the potential for a flow past the bubble of Figure 2.2. This

implies that, because of its nonlinearity, the flow past a bubble is a much

harder problem to solve than the flow past a sphere. The potential function

still satisfies (2.14)(2.16). However, the main difference is that the shape

of the surface S of the bubble is not known and has to be found as part of the

solution. In other words the equation of the surface S is no longer given as it

was for the flow past a sphere. Therefore we need an extra equation to find

S. This equation uses (2.18) and can be derived as follows. First we relate

the pressure p on the fluid side of S to the pressure pb inside the bubble

by using the concept of surface tension. If we draw a line on a fluid surface(such as S), the fluid on the right of the line is found to exert a tension T,

per unit length of the line, on the fluid to the left. We call T the surface

tension coefficient. It depends on the fluid and also on the temperature. It

can be shown (see for example Batchelor [8]) that

p pb = T K = T

1

R1+

1

R2

. (2.19)

Here R1 and R2 are the principal radii of curvature of the fluid surface: theyare counted positive when the centres of curvature lie inside the fluid. The


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quantity

K =

1

R1+

1

R2

(2.20)

is referred to as the mean curvature of the fluid surface. In most applications

presented in this book the surface tension T is assumed to be constant.We now apply the Bernoulli equation (2.18) to the fluid side of the surface

S and use (2.19). This gives

1

2(2x +

2y +

2z ) +

T

K =

1

2U2 +

p pb

on S. (2.21)

Equation (2.21) is known as the dynamic boundary condition. This is the

extra equation needed to find S. To solve the bubble problem we seek the

function and the equation of the surface S such that (2.14)(2.16) and(2.21) are satisfied. It is a nonlinear problem that requires the solution of a

partial differential equation (here the Laplace equation (2.14)) in a domain

whose boundary (here S) has to be found as part of the solution. This

is a typical free-surface flow problem. In this book we will describe various

analytical and numerical methods for investigating such nonlinear problems.

We note that the problem of Figure 2.2 is an idealised one, in which

the viscosity and gravity and the wake behind the bubble are neglected.

Bubbles with wakes and the effect of including gravity will be considered in

Section 3.4.3. Readers interested in the effects of viscosity are referred to,for example, [117].

The dynamic boundary condition (2.21) is valid for steady flows with

= 0. Combining (2.12) and (2.19) we find that the general form of the

dynamic boundary condition (for unsteady flows) with = 0 is

t+

u u2

+ +T

K = B. (2.22)

Here B is the Bernoulli constant. For steady flows, (2.22) reduces tou u

2+ +

T

K = B. (2.23)

2.3 Two-dimensional flows

As we shall see, many interesting free-surface flows can be modelled as two-

dimensional flows. We then introduce cartesian coordinates x and y with

the y-axis directed vertically upwards (at present we reserve the letter z todenote the complex quantity x + iy). In most applications considered in


26/331

12 Basic concepts

this book, the potential (see (2.3)) is due to gravity. Assuming that the

acceleration of gravity g is acting in the negative y-direction, we write as

= gy. (2.24)

An example is the two-dimensional free-surface flow past a semicircularobstacle at the bottom of a channel (see Figure 2.3). This two-dimensional

configuration provides a good approximation to the three-dimensional free-

surface flow past a long half-cylinder perpendicular to the plane of the figure

(except near the ends of the cylinder). The cross section of the cylinder is

the semicircle shown in Figure 2.3.

x

y

Fig. 2.3. Two-dimensional free-surface flow past a submerged semicircle.

For two-dimensional potential flows, (2.4) and (2.6) become

u

y v

x= 0, (2.25)

u

x+

v

y= 0. (2.26)

Here u and v are the x- and y- components of the velocity vector u.

We can introduce a streamfunction by noting that (2.26) is satisfied by

u =

y, (2.27)

v = x

. (2.28)

It then follows from (2.25) that

2 = 2x2

+ 2y2

= 0. (2.29)


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For two-dimensional flows, equations (2.5) and (2.7) give

u =

x, (2.30)

v =

y (2.31)

and

2 = 2

x2+

2

y2= 0. (2.32)

Combining (2.27), (2.28), (2.30) and (2.31) we obtain

x=

y, (2.33)

y=

x. (2.34)

Equations (2.33) and (2.34) can be recognised as the classical Cauchy

Riemann equations. They imply that the complex potential

f = + i (2.35)

is an analytic function of z = x + iy in the flow domain. This result is

particularly important since it implies that two-dimensional potential flowscan be investigated by using the theory of analytic functions. This applies in

particular to all two-dimensional potential free-surface flows with or without

gravity and/or surface tension included in the dynamic boundary condition.

It does not apply, however, to axisymmetric and three-dimensional free-

surface flows. Since the derivative of an analytic function is also an analytic

function, it follows that the complex velocity

u

iv =

x i

y=

y+ i

x=

df

dz(2.36)

is also an analytic function of z = x + iy.

The theory of analytic functions will be used intensively in the follow-

ing chapters to study two-dimensional free-surface flows. In particular the

following important tools will be useful.

The first tool is conformal mappings. These are changes of variable defined

by analytic functions. For example, if h(t) is an analytic function of t, the

change of variables z = h(t) enables us to seek the complex velocity u iv asan analytic function of t (since an analytic function of an analytic functionis also an analytic function). Such conformal mappings are used to redefine


28/331

14 Basic concepts

a problem in a new complex t-plane in which the geometry is simpler than

in the original z-plane.

The second tool is Cauchys theorem: If h(z) is analytic throughout a

simply connected domain D then, for every closed contour C within D,

C

h(z)dz = 0. (2.37)

The third tool is the Cauchy integral formula: Let h(z) be analytic every-

where within and on a closed contour C, taken in the positive sense (coun-

terclockwise). Then the integral

1

2i

C

h(z)

z z0 dz (2.38)

takes the following values:

0 if z0 is outside C, (2.39)

h(z0) if z0 is inside C, (2.40)

1

2h(z0) if z0 is on C. (2.41)

When z0 is on C the integral (2.38) is a Cauchy principal value.

We now show that for steady flows the streamfunction is constant along

streamlines. A streamline is a line to which the velocity vectors are tangent.Let us describe a streamline in parametric form by x = X(s), y = Y(s),

where s is the arc length. Then we have

vX(s) + uY(s) = 0, (2.42)where the primes denote derivatives with respect to s. Using (2.27) and

(2.28) we have

x X

(s) +

y Y

(s) =

d

ds = 0, (2.43)

which implies that is constant along a streamline. For steady flows the

kinematic boundary condition implies that a free surface is a streamline.

The streamfunction is therefore constant along a free surface.

For two-dimensional flows the dynamic boundary condition (2.22) be-

comes

t+

1

2(2x +

2y ) + gy +

T

K = B. (2.44)

If we denote by the angle between the tangent to the free surface and the


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2.4 Linear waves 15

horizontal then the curvature K can be defined by

K = dds

(2.45)

where again s denotes the arc length. In particular if the (unknown) equation

of the free surface is y = (x, t) then

tan = x anddx

ds=

1

(1 + 2x)1/2

. (2.46)

Using (2.45), (2.46) and the chain rule gives the formula

K = xx(1 + 2x)

3/2. (2.47)

2.4 Linear waves

2.4.1 The water-wave equations

Many fre-surface flows involve waves on their free surfaces. When dissi-

pation is neglected and the flow is assumed to be two-dimensional, these

waves often approach uniform wave trains in the far field (see for example

Figure 2.3). Therefore a fundamental problem in the theory of free-surface

flows is the study of a uniform train of two-dimensional waves of wavelength

extending from x = to x = and travelling at a constant velocity c.The flow configuration is illustrated in Figure 2.4.

x

y

y = y1

y = y2

Fig. 2.4. A two-dimensional train of waves viewed in a frame of reference movingwith the wave. The free-surface profile has wavelength . The fluid is bounded

below by a horizontal bottom with equation y = h. Also shown is the rectangularcontour used in (2.56).


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16 Basic concepts

For convenience we have chosen a frame of reference moving with the wave,

so that the flow is steady. Using the notation of Section 2.3, we formulate

the problem as

xx + yy = 0, h < y < (x), (2.48)

y = xx on y = (x), (2.49)

y = 0 on y = h, (2.50)

1

2(2x +

2y ) + gy

T

xx

(1 + 2x)3/2

= B on y = (x), (2.51)

(x + , y) = (x, y), (2.52)(x + ) = (x), (2.53)

0

(x)dx = 0, (2.54)

1

0

xdx = c on y = constant. (2.55)

Here g is the acceleration of gravity (assumed to act in the negative y-

direction), T is the surface tension, is the density, y = h is the equationof the bottom and y = (x) is the equation of the (unknown) free surface.

Equations (2.49) and (2.50) are the kinematic boundary conditions on the

free surface and on the bottom respectively. Equation (2.51) is the dynamic

boundary condition on the free surface. We have used (2.44) and the formula

(2.47) for the curvature of a curve y = (x). Relations (2.52) and (2.53)

are periodicity conditions, which require the solution to be periodic withwavelength . Equation (2.54) fixes the origin of the y-coordinates as the

mean water level. Finally, (2.55) defines the velocity c as the average value

of u = x at a level y = constant in the fluid. The value of c is independent

of the constant chosen; this can be seen by applying Stokes theorem to

the vector velocity (u, v) using a contour C consisting of two horizontal

lines y = y1, y = y2 and two vertical lines separated by a wavelength (see

Figure 2.4). Since the flow is irrotational,

C

udx + vdy = 0. (2.56)


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2.4 Linear waves 17

The contributions from the two vertical lines cancel by periodicity and (2.56)

gives 0

[u]y=y1 dx =

0

[u]y=y2 dx. (2.57)

Since y1 and y2 are arbitrary, the integral on the left-hand side of (2.55) isindependent of the level y = constant chosen in the fluid.

The relations (2.48)(2.55) are referred to as the water-wave equations

because they model waves travelling at the interface between water and air

(although they apply also to other fluids).

2.4.2 Linear solutions for water waves

A trivial solution of the system (2.48)(2.55) is

= cx, (x) = 0 and B =c2

2. (2.58)

This solution describes a uniform stream with constant velocity c, bounded

below by a horizontal bottom and above by a flat free surface.

Linear waves are obtained by seeking a solution as a small perturbation

of the exact solution (2.58). Therefore we write

(x, y) = cx + (x, y) (2.59)and assume that both |(x, y)| and |(x)| are small. Substituting (2.59) into(2.48)(2.55) and dropping nonlinear terms in and , we obtain the linear

system

xx + yy = 0, h < y < 0, (2.60)

y = cx, y = 0, (2.61)

y = 0, y = h, (2.62)

T

xx + cx + g = 0, y = 0, (2.63)

(x + , y) = (x, y), (2.64)

(x + ) = (x), (2.65)

0(x)dx = 0, (2.66)


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18 Basic concepts

1

0

xdx = 0 on y = constant. (2.67)

We choose the origin ofx at a crest and assume that the wave is symmetric

about x = 0. Thus we impose the conditions

(x, y) = (x, y), (2.68)(x) = (x). (2.69)

Using the method of separation of variables, we seek a solution of (2.60)

in the form

(x, y) = X(x)Y(y). (2.70)

Substituting (2.70) into (2.60), (2.68) and (2.62) yields

X(x) = X(x), (2.71)the ordinary differential equations

X(x)

X(x)= Y

(y)

Y(y)= constant = 2, (2.72)

and the boundary condition

Y(h) = 0. (2.73)Here we have chosen a negative separation constant in (2.72), so that the

solution is periodic in x. Solutions of the two ordinary differential equations

(2.72) satisfying (2.71) and (2.73) are written as

X(x) = sin x, (2.74)

Y(y) = cosh (y + h). (2.75)

The periodicity condition (2.64) implies that

= nk, (2.76)

where n is a positive integer and

k =2

(2.77)

is the wavenumber. Multiplying (2.74) and (2.75) and taking a linear com-

bination of the solutions corresponding to the values (2.76) of , we obtain

(x, y) =

n=1

Bn cosh nk(y + h)sin nkx. (2.78)


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2.4 Linear waves 19

Here the Bn are constants.

Using the periodicity and the symmetry conditions (2.69) and (2.65), we

express (x) as the Fourier series

(x) = A0 +

n=1

An cos nkx (2.79)

where the An are constants. The condition (2.66) implies that A0 = 0.

Substituting (2.78) and (2.79) into (2.61) and equating the coefficients of

sin nkx yields

cAn = Bn sinh nkh, n = 1, 2, . . . (2.80)Similarly substituting (2.78) and (2.79) into (2.63) gives

T

Ann2

k2

+ gAn + cBnnk cosh nkh = 0, n = 1, 2, . . . (2.81)

Eliminating Bn between (2.80) and (2.81) yieldsg +

T

n2k2 c

2nk

sinh nkhcosh nkh

An = 0, n = 1, 2, . . . (2.82)

Since we seek a nontrivial periodic solution (x) = 0, we can assume withoutloss of generality that A1 = 0; then (2.82) with n = 1 implies that

c2

=g

k +

T

k

tanh kh. (2.83)

Relation (2.82) for n > 1 gives

An = 0, n = 2, 3, . . . , (2.84)

provided that

g +T

n2k2 c

2nk

sinh nkhcosh nkh = 0, n = 2, 3, . . . (2.85)

If (2.85) is satisfied, the solution of the linear problem is

= cA1sinh kh

cosh k(y + h)sin kx, (2.86)

= A1 cos kx. (2.87)

If the condition (2.85) is not satisfied for some integer value m of n, the

solution of the linear problem is

=

cA1

sinh khcosh k(y + h)sin kx

cAm

sinh mkh,cosh mk(y + h)sin mkx,

(2.88)


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20 Basic concepts

1 = A1 cos kx + Am cos mkx, (2.89)

where Am is an arbitrary constant. In the theory of linear waves, it is

usually assumed that Am = 0. However, when we are developing nonlinear

theories for water waves in Chapters 5 and 6, i.e. improving the linear

approximations (2.88) and (2.89) by adding nonlinear corrections or solvingthe fully nonlinear problem (2.48)(2.55) numerically, we shall see that Am =0. Two consequences are the existence of many different families of nonlinear

periodic gravitycapillary waves and the existence of solitary waves with

oscillatory tails.

The velocity c is called the phase velocity and equation (2.83) is the

(linear) dispersion relation. Relation (2.83) implies that waves of differ-

ent wavenumbers and therefore of different wavelengths travel at different

phase velocities c.

It is convenient to rewrite (2.83) in the dimensionless form

F2 =

1

kh+ kh

tanh kh, (2.90)

where

F =c

(gh)1/2(2.91)

is the Froude number and

= Tgh2

(2.92)

is the Bond number. Relation (2.90) is shown graphically in Figure 2.5,

where we present values of F2 versus 1/(kh) = /(2h) for four values of .

The curves of Figure 2.5 illustrate that F2 is a monotonically decreasing

function of /h when > 1/3 and that it has a minimum for < 1/3.

As /h , F 1. The different behaviours for < 1/3 (minimum)and > 1/3 (monotone decay) in Figure 2.5 have many implications, in

particular for the study of nonlinear periodic and solitary gravitycapillarywaves (see Chapters 5 and 6).

We now examine two particular cases.

The first is the case of water of infinite depth. This is obtained by taking

the limit kh in (2.83), (2.86) and (2.87) and leads to = cA1eky sin kx, (2.93)

= A1 cos kx, (2.94)

c2 = gk

+ T

k. (2.95)


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2.4 Linear waves 21

0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14

Fig. 2.5. Values of F2 versus 1/(kh). The curves correspond from top to bottomto = 1.3, = 1/3, = 0.1 and = 0.05. For < 1/3 the curves have a minimum

whereas for > 1/3 the curves are monotonically decreasing.

Since kh = 2h/, the infinite-depth results (2.93)(2.95) provide an ap-

proximation to the finite-depth results (2.83), (2.86) and (2.87) when the

wavelength is small compared with the depth h.

Waves with g = 0, T = 0 are referred to as pure gravity waves. They arecharacterised in the case of infinite depth by the dispersion relation

c2 =g

k. (2.96)

Similarly, waves with g = 0, T = 0 are called pure capillary waves and arecharacterised in the infinite-depth case by the dispersion relation

c2 =T

k. (2.97)

A simple calculation based on (2.95) shows that c2 reaches a minimum value

given by

cmin =

4T g

1/4

(2.98)

when

k = kmin =g

T

1/2. (2.99)

Graphs ofc versus in units ofcmin and min = 2/kmin are shown in Figure

2.6. The solid curve corresponds to (2.95), the dotted curve to (2.97) and

the broken curve to (2.96). These curves show that waves with > minare dominated by gravity and can be approximated by pure gravity waves


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22 Basic concepts

0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6

Fig. 2.6. Values of c versus = 2/k in units of cmin and min . The solid curvecorresponds to (2.95), the dotted curve to (2.97) and the broken curve to (2.96).

for large. Waves with < min are dominated by surface tension and can

be approximated by pure capillary waves for small.

The second particular case is that of pure gravity waves (i.e. T = 0) in

water of finite depth. Then (2.90) reduces to

khF2 = tanh kh. (2.100)

Since

dd (kh)

tanh kh 1, (2.101)

equation (2.100) has a solution kh > 0 when F < 1. For F > 1 the only

real solution of (2.100) is kh = 0. This implies that linear gravity waves

only exist when F < 1; for F > 1, linear gravity waves are not possible.

Flows characterised by F < 1 are called subcritical and those characterised

by F > 1 are called supercritical. The distinction between subcritical and

supercritical flows will appear often in this book.

So far we have discussed linear waves in a frame of reference movingwith the wave. This is a convenient choice because the flow is then steady.

However, it is also useful to look at waves from the point of view of a fixed

frame of reference in which the wave moves to the left at a constant velocity

c. The nonlinear governing equations are then

xx + yy = 0, h < y < (x, t), (2.102)

t = y

xx on y = (x, t), (2.103)

y = 0 on y = h, (2.104)


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2.4 Linear waves 23

t +1

2(2x +

2y ) + gy

T

xx

(1 + 2x)3/2

= B on y = (x, t), (2.105)

(x + ,y,t) = (x,y,t), (2.106)

(x + , t) = (x, t), (2.107)0

(x, t)dx = 0. (2.108)

A trivial solution of the system (2.102)(2.108) is

= 0, = 0 and B = 0. (2.109)

We can construct linear waves by assuming a small perturbation of the exact

solution (2.109) in the form of a wave travelling to the left at a constantvelocity c. Therefore we rewrite and in terms of two new functions

and :

(x,y,t) = (x + ct,y) and (x, t) = (x + ct) (2.110)

Substituting (2.110) into the system (2.102)(2.108) and dropping nonlinear

terms in and , we obtain the linear system

xx + yy = 0, h < y < 0, (2.111)

cx = y on y = 0, (2.112)

y = 0 on y = h, (2.113)

cx + g T

xx = 0 on y = 0, (2.114)

(x + + ct,y) = (x + ct,y), (2.115)

(x + + ct) = (x + ct), (2.116)0

(x + ct)dx = 0. (2.117)

Following the derivation of (2.86)(2.89), we find that the solution of

(2.111)(2.117) is

= cA1sinh kh

cosh k(y + h)sin k(x + ct), (2.118)

= A1 cos k(x + ct) (2.119)


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24 Basic concepts

if (2.85) is satisfied and

= cA1sinh kh

cosh k(y + h)sin k(x + ct),

cAm

sinh mkh cosh mk(y + h)sin mk(x + ct), (2.120)

= A1 cos k(x + ct) + Am cos mk(x + ct) (2.121)

if for n = m (2.85) is not satisfied. The dispersion relation is given as before

by (2.83).

2.4.3 Superposition of linear waves

Since the system (2.111)(2.117) is linear, new solutions can be obtained bysuperposing solutions corresponding to different values of k and/or of A1.

We consider two particular superpositions for the solution (2.118), (2.119).

The first corresponds to the superposition of two waves of the same am-

plitude travelling at the same velocity but in opposite directions. This gives

= A1 cos k(x + ct) + A1 cos k(x ct), (2.122)

= cA1

sinh kh cosh k(y + h)sin k(x + ct)

+cA1

sinh khcosh k(y + h)sin k(x ct). (2.123)

Using the trigonometric identities

cosp + cos q = 2 cosp + q

2cos

p q2

(2.124)

sinp + sin q = 2sinp + q

2 cosp

q

2 (2.125)

we can rewrite (2.122), (2.123) as

= 2A1 cos kx cos kct, (2.126)

= 2 cA1sinh kh

cosh k(y + h)cos kx sin kct. (2.127)

The solution defined by (2.126), (2.127) is known as a linear standing wave

because the position of its nodal points and of the maximum displacementof the free surface are fixed as t varies. The wave does not propagate and


39/331

2.4 Linear waves 25

its free surface moves periodically up and down as t varies. The period of

this motion is

Ts =2

kc. (2.128)

Since u = x = 0 along the lines x = 0 and x = /k = /2, we can replace

these two lines by walls (the kinematic boundary condition on them is then

automatically satisfied). The resulting flow models the periodic sloshing of

a liquid in a container (see Figure 2.7).

0

0.1

0.2

0 1 2 3

Fig. 2.7. Standing wave for A1 = 0.1 and k = 1. The broken line is the free-surfaceprofile at t = 0 and the dotted line is the free-surface profile at t = Ts /2. This flow

models liquid sloshing in a container bounded by vertical walls at x = 0 and x = .

An interesting question is whether there are similar nonlinear solutions.

This question is addressed in Chapter 11, where we construct analytical

approximations to such solutions.

The second example of superposition that we consider is that of two wave

trains of the same amplitude travelling in the same direction but with slightly

different wavenumbers k and k. We first introduce the angular frequency

= kc (2.129)

and write = W(k). Using (2.83) we have

W(k) = k

g

k+

T

k

tanh kh

1/2. (2.130)

Next we rewrite (2.118) and (2.119) as

= A1 cos[kx + W(k)t], (2.131)

= cA1sinh kh

cosh k(y + h)sin[kx + W(k)t]. (2.132)


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26 Basic concepts

The superposition described above then yields

= A1 cos[kx + W(k)t] + A1 cos[kx + W(k)t]. (2.133)

Using the identity (2.124), we can rewrite (2.133) as

= 2A1 cos

12

[(k + k)]x +12

[W(k) + W(k))t

cos

1

2[(k k)]x + 1

2[(W(k) W(k)t]

. (2.134)

For k close to k, we may approximate (2.134) by

= (x, t) cos[kx + W(k)t], (2.135)

where

(x, t) = 2A1 cos

1

2(k k)x + 1

2[W(k) W(k)]t

. (2.136)

The expression (2.135) is the same as (2.131) except that the constant am-

plitude A1 has been replaced by the variable amplitude (x, t).

Differentiating (2.136) with respect to x and t yields

x

= A1(k k)sin

12

(k k)x + 12

[W(k) W(k)]t

(2.137)

and

t= A1[W(k) W(k)]sin

1

2(k k)x + 1

2[W(k) W(k)]t

. (2.138)

The derivatives (2.137) and (2.138) are of order k k and W(k) W(k)respectively. They are therefore small for k close to k. This implies that the

amplitude (x, t) is a slowly varying function ofx and t. In other words, thesolution is a wave of wavenumber k, travelling at velocity c, whose amplitude

(x, t) is slowly modulated. The amplitude (x, t) is itself a wave travelling

at velocity

W(k) W(k)k k . (2.139)

For k close to k, the velocity (2.139) becomes

cg =dW(k)

dk. (2.140)


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2.4 Linear waves 27

The velocity cg is called the group velocity. In general it differs from the

phase velocity

c =W(k)

k. (2.141)

For water waves, (2.130) and (2.140) give

cg =1

2

gk +

T

k3

tanh kh

1/2

g +3T k2

tanh kh +

g +

T

k2

kh

cosh2 kh

. (2.142)

We now examine in more detail the case of infinite depth. The phase

velocity c is then given by (2.95). Taking the limit kh in (2.142), weobtain

cg =1

2

gk +

T k3

1/2 g +

3T k2

. (2.143)

In particular, we have for pure gravity waves (g = 0, T = 0)

cg =1

2 g

k1/2

=c

2(2.144)

and for pure capillary waves (T = 0, g = 0)

cg =3

2

T k

1/2=

3c

2. (2.145)

The phase velocity c is the velocity at which the wave travels. The group

velocity cg is the velocity at which the slowly varying amplitude travels.

This phenomenon is illustrated in Figure 2.8, where we present the solution

(2.135) for pure gravity waves of infinite depth.

Here we have assumed g = 1, k = 1, k = 1.1 and A1 = 0.2 and havechosen t = 0. The outside curves are the envelope of the wave train. Both

the wave train and the envelope travel to the left. Using (2.96) we find

that the wave train travels at the speed c = 1 whereas (2.144) shows that

the envelope travels at the speed cg = 1/2. Since cg < c the waves will

advance in their envelope, and as they approach the nodal points of their

envelope they will progressively die out. However, waves are born just ahead

of the nodal points of the envelope. These graphical results illustrate that

the wave travels at the velocity c whereas the envelope of the wave (i.e. theamplitude) travels at the velocity cg.


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28 Basic concepts

0

0 20 40

0.4

60

0.2

Fig. 2.8. The solution (2.135) for A1 = 0.2, k = 1 and k = 1.1.

A simple relation between c and cg can be derived by combining (2.129)

and (2.140) to give

cg = c + kdc

dk. (2.146)

Relation (2.146) shows that if c has a minimum for some value of k, then

c = cg at this minimum (since dc/dk = 0 at a minimum). For example, in

water of infinite depth c has a minimum for k = kmin, where kmin is given by

(2.99) (see also Figure 2.6), and cg = c when k = kmin. On the right of theminimum in Figure 2.6 we have dc/dk < 0, and (2.146) implies that cg < c.

Similarly, dc/dk > 0 on the left of the minimum in Figure 2.6 and cg > c.

One important property of the group velocity is that it is the speed at

which the energy of a linear wave travels. We will demonstrate this property

in the particular case of pure gravity waves in water of finite depth. The

analysis is similar to that presented in Billingham and King [13]. At a fixed

value of x, the rate at which the fluid on the left does work on the fluid on

the right is given by 0h

p

xdy. (2.147)

The average of (2.147) over one period is

Ef =

2

t+2/t

0h

p

xdydt, (2.148)

where t

is an arbitrary value of t and is the angular frequency. The valueof p is obtained by linearising (2.12) (with = gy) around u = 0. This


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2.4 Linear waves 29

gives

p = t

gy + constant. (2.149)

Using (2.118) we findt

+2/

t

x

dt = cA1ksinh kh

cosh k(y + h)

t

+2/

tcos k(x + ct)dt = 0.

(2.150)

Therefore (2.148) simplifies to

Ef = 2

t+2/t

0h

t

xdydt. (2.151)

Substituting (2.118) into (2.151) and evaluating the integral yields

Ef =A21k2c3

4 sinh2kh

h + sinh 2kh

2k

. (2.152)

We now define the kinetic and potential energy per unit horizontal length

by 0h

1

2

x

2+

y

2dy (2.153)

and

0gydy = 1

2g2. (2.154)

Averaging the quantities (2.153) and (2.154) over a wavelength gives the

mean kinetic energy

K =

2

0

0h

x

2+

y

2dydx (2.155)

and the mean potential energy

V =g

2

0

2dx. (2.156)

Substituting (2.118) and (2.119) into (2.155) and (2.156) gives, after inte-

gration,

K = V =1

4gA21. (2.157)

Thus the total energy is

E = K+ V = 12

gA21. (2.158)


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30 Basic concepts

Combining (2.152) and (2.158) we obtain

Ef = E1

2c

1 +

2kh

sinh 2kh

. (2.159)

Using (2.142) with T = 0 gives

cg =c

2

1 +

2kh

sinh 2kh

. (2.160)

Therefore comparing (2.159) and (2.160) yields

Ef = Ecg. (2.161)

This shows that the energy in the wave travels at the group velocity cg.

This property will be used in Chapter 4, where we discuss the radiation

condition.


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3

Free-surface flows that intersect walls

We continue our study of free-surface flows by considering the two-

dimensional flow shown in Figure 3.1. The flow domain is bounded be-low by the horizontal wall AB and above by the inclined walls CD and DE

and the free surface EF. The fluid is assumed to be incompressible and

inviscid and the flow is assumed to be irrotational and steady. We introduce

cartesian coodinates with the x-axis along the horizontal wall AB and the

y-axis through the separation point E (here a separation point refers to an

intersection of a free surface and a rigid wall). The angles between the walls

CD and DE and the horizontal are denoted by 1 and 2 respectively.

A B

D

E

F

x

y

1

2

C

Fig. 3.1. A two-dimensional free-surface flow bounded by the walls CD, DE andAB and the free surface EF. The separation point E is defined as the point atwhich the free surface EF meets the wall DE. Points C, A, F, B are at an infinitedistance from E. The flow is from left to right.

The configuration of Figure 3.1 was chosen because it can be used todescribe many properties of free-surface flows that intersect, i.e. adjoin,

31


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32 Free-surface flows that intersect walls

rigid walls. These properties when understood for the flow of Figure 3.1 can

then be used to describe locally flows with more complex geometries.

There are various illustrations of the flow of Figure 3.1. The first is the

flow emerging from a container bounded by the walls CD, DE and AB.

When 1 = 2 = /2, the configuration of Figure 3.1 models the flow under

an infinitely high gate (see Figure 3.2). Here the point D is irrelevant and

has been omitted from the figure.

A B

C

E

F

Fig. 3.2. The free-surface flow under a gate. The flow is from left to right.

When 1 = 0 and 2 < 0, Figure 3.1 describes locally the flow near thebow or the stern of a ship (see Figure 3.3). A clear distinction between

the stern and bow flows will be introduced in Chapter 8, when we discuss

gravity flows with a train of waves in the far field. Further particular cases

of Figure 3.1, which model bubbles rising in a fluid and jets falling from a

nozzle, are described in Section 3.3.2.

A

DC

E

F

B

Fig. 3.3. A model for the free-surface flow near the bow or stern of a ship.

As mentioned in Chapter 1 we will proceed with problems of increasingcomplexity. Section 3.1 is devoted to free-surface flows with g = 0 and


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T = 0. Such flows are called free streamline flows and the corresponding

free surfaces are called free streamlines. In Section 3.2 we will study the

effect of surface tension (T = 0, g = 0). In Section 3.3 we will examine theeffect of gravity (T = 0, g = 0). The combined effects of gravity and surfacetension (T

= 0, g

= 0) are considered in Section 3.4.

3.1 Free streamline solutions

3.1.1 Forced separation

We consider the flow configuration of Figure 3.1. Here the effects of gravity

and surface tension will be neglected (T = 0, g = 0). We refer to this

problem as one of forced separation because the free surface is forced to

separate at the point E where the wall DE terminates. We denote by u

and v the horizontal and vertical components of the velocity. Using theincompressibility of the fluid and the irrotationality of the flow, we define

a potential function (x, y) and a streamfunction (x, y). As shown in

Section 2.3, the complex potential f = + i and the complex velocity

w = u iv = df/dz are both analytic functions of z = x + iy.The wall AB is a streamline along which we choose = 0. The walls CD

and DE and the free surface EF define another streamline, along which

the constant value of is denoted by Q. We also choose = 0 at the

separation point E. These two choices ( = 0 on AB and = 0 at E)can be made without loss of generality because and are defined up to

arbitrary additive constants. Bernoullis equation (2.13) with = 0 yields

1

2(u2 + v2) +

p

= constant (3.1)

everywhere in the fluid. The free surface EF separates the fluid from the

atmosphere which is assumed to be characterised by a constant pressure pa.

In the absence of surface tension, which we are assuming, the pressure is

continuous across the free surface (see (2.19)). Therefore p = pa on the freesurface. It follows from (3.1) that

u2 + v2 = U2 on EF, (3.2)

where U is a constant.

A significant simplification in the formulation of the problem is obtained

by using and as independent variables. This choice was used by Stokes

[144], to study gravity waves, and by Helmholtz [71] and Kirchhoff [90]

(see also [19] and [69]) to investigate free streamline flows. We shall useit extensively in our studies of gravitycapillary free-surface flows. The


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simplification comes from the fact that the flow domain is mapped into the

strip 0 < < Q shown in Figure 3.4. The free surface EF (whose position

was unknown in the physical plane z = x + iy of Figure 3.1) is now part

of the known boundary = Q in the f-plane. Since u iv is an analyticfunction of z and z is an analytic function of f (the inverse of an analytic

function is also an analytic function), u iv is an analytic function of f.

BA

C D E F

= 0

= Q

Fig. 3.4. The flow configuration of Figure 3.1 in the complex potential plane f = + i.

A remarkable result is that many free streamline problems can be solved

in closed form (see Birkhoff and Zarantonello [19] and Gurevich [69]). These

exact solutions are obtained by using conformal mappings, and several meth-

ods have been derived to calculate them. The method we now choose to de-

scribe uses a mapping of the flow domain into the unit circle. It was chosen

because it yields naturally to the series truncation methods used in Sec-

tions 3.23.4 to solve numerically problems with gravity and surface tension

included.

In the absence of gravity and surface tension, the flow approaches a uni-form stream of constant depth H as x . It follows from the dynamicboundary condition (3.2) that this uniform stream is characterised by a

constant velocity U. Since = 0 on AB and = Q on EF, H = Q/U.

We define the logarithmic hodograph variable i by the relationw = u iv = ei . (3.3)

The function i has some interesting properties. First, the quantity =

12 ln(u

2

+v2

) is constant along free streamlines (see (3.2)). Second, canbe interpreted as the angle between the vector velocity and the horizontal.


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Third, (3.3) leads, for steady flows, to a very simple formula for the curvature

of a streamline. This formula can be derived as follows. Since the vector

velocity is tangent to streamlines, is the angle between the tangent to a

streamline and the horizontal. The curvature K of a streamline is given by

(2.45). Using the chain rule, we can rewrite (2.45) as

K =

s

s. (3.4)

Along a streamline is constant and therefore

s= 0 and

s= e. (3.5)

Subsituting (3.5) into (3.4) yields the simple formula

K = e

. (3.6)

We now introduce dimensionless variables by using U as the reference

velocity and H as the reference length. Therefore = 1 on the walls CD

and DE and on the free surface EF. The dynamic boundary condition (3.2)

becomes

u2 + v2 = 1 on EF. (3.7)

We map the strip ABFC shown in Figure 3.4 into the unit circle in the

t-plane by the conformal mapping

ef =(1 t)2

4t. (3.8)

The flow configuration in the t-plane is shown in Figure 3.5. It can easily

be checked that the points A and C are mapped into t = 0 and the points

B and F are mapped into t = 1. The value of t at the point D is denoted

by d. The free surface EF is mapped onto the portion

t = ei , 0 < < , (3.9)

of the unit circle. This can easily by shown by noting that the substitution

of (3.9) into (3.8) gives, after some algebra,

= 1

ln sin2

2on = 1. (3.10)

As varies from 0 to , varies from to 0, so that (3.9) is the image ofthe free surface in the t-plane.


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ACDE

F

B

Fig. 3.5. The flow configuration of Figure 3.1 in the complex t-plane.

One might attempt to represent the complex velocity w = u iv by theseries

w =

n=0

antn. (3.11)

However, the series will not converge inside the unit circle

|t

| 1, because

singularities can be expected at the corner D and as x (i.e. at t = 0).Nevertheless we can generalise the representation (3.11) by writing

w = G(t)

n=0

antn, (3.12)

where the function G(t) contains all the singularities of w. As we shall see

in Sections 3.23.4, this type of series representation enables the accurate

calculation of many free-surface flows with gravity and surface tension in-

cluded. For the present problem we require G(t) to behave like w as t 0and as t d. We can then expect the series in (3.12) to converge for |t| 1.

To construct G(t), we find the asymptotic behaviour of w near the singu-

larities by performing local asymptotic analysis near D and as x .The flow near D is a flow inside a corner. We will find the nature of the

singularity at D by considering the general problem of a flow inside a corner

of angle (see Figure 3.6).

We introduce cartesian coordinates with the origin at the apex G of the

corner. We choose = 0 on the streamline HGL and = 0 at x = y = 0.Assuming without loss of generality that the flow is in the direction of the


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H

Gx

y

L

Fig. 3.6. Flow in a corner bounded by the walls GH and GL.

arrow, we have < 0 along the wall HG, > 0 along the wall GL and

> 0 in the flow domain. The flow configuration in the complex potential

plane is shown in Figure 3.7.

H G

L

Fig. 3.7. The flow configuration of Figure 3.6 in the complex potential plane. Theflow domain is the upper half-plane > 0.

We seek a solution of the form

z = Aei f, (3.13)

where A > 0, and are real constants. On the wall GL (where > 0),

the kinematic boundary condition can be written as arg z = 0. Therefore

(3.13) implies that

= 0. (3.14)

On the wall GH (where < 0), the kinematic boundary condition can be


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written as arg z = . Writing = ei || and using (3.13), we find that + = . (3.15)

Relations (3.14) and (3.15) imply that

=

; (3.16)

therefore (3.13) gives

z = Af/ . (3.17)

Since

w = dz

df1

(3.18)

we obtain the formula

w =

Af1/ (3.19)

or, eliminating f between (3.17) and (3.19),

w =

A/z/1. (3.20)

Flows inside corners will occur in many flow configurations described in this

book and we will refer often to the above local analysis. We note that theformulae (3.17), (3.19) and (3.20) still hold if the boundary GL in Figure

3.6 is an arbitrary straight line through G (i.e. if the angle HGL is rotated).

The only difference is that is then different from zero.

The velocity at the point G is equal to zero when < and is unbounded

when > see (3.20). We will refer to the flow of Figure 3.6 as a flow

inside a corner when < and as a flow around a corner when > .

For the flow of Figure 3.1, = 2 + 1 and (3.19) impliesw = O[(f D i)(21 )/ )] as f D + i, (3.21)

where D is the value of at the point D. Here we have used the classical

O notation to indicate an estimate of the behaviour of a function. We recall

that writing

f(x) = O[g(x)] as x x0 (3.22)means that

f(x)g(x)

A as x x0, (3.23)


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where A is a constant. Similarly,

f(x) = o[g(x)] as x x0 (3.24)means that

f(x)g(x)

0 as x x0. (3.25)

Using (3.8) yields

f D i = O[t d] as f D + i. (3.26)Combining (3.21) and (3.26) gives

w = O[(t d)(21 )/ )] as t d. (3.27)This concludes our local analysis near the point D.

As x , the flow behaves like that due to a sink at x = y = 0.Therefore

f B ln z as x (3.28)where B is a positive constant. Differentiating (3.28) with respect to z gives

w =df

dz

=

B

z

. (3.29)

Since the flux of the fluid coming from is 1 and the angle between thewalls CD and AB is 1, we have

B =1

1. (3.30)

Eliminating z between (3.28) and (3.29) gives

w = O[e1 f] as f

, (3.31)

and relation (3.8) then implies that

ef = O(t) as f . (3.32)Therefore (3.31) and (3.32) give

w = O(t1 / ) as t 0. (3.33)Combining (3.27) and (3.33), we can choose

G(t) = (t d)(21 )/ t1 / (3.34)


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and write (3.12) as

w = (t d)(21 )/ t1 /

n=0

antn. (3.35)

There are, of course, many other possible choices for G(t). For example G(t)in (3.34) can be multiplied by any function analytic in |t| 1.We now need to determine coefficients an in (3.35) such that the dynamic

boundary condition (3.7) is satisfied. This can be done numerically by trun-

cating the infinite series in (3.35) after N terms and finding the coefficients

an , n = 0, . . . , N 1 by collocation. This is the approach we will use whensolving problems where the effects of gravity or surface tension are included

in the dynamic boundary condition. However, it can checked that

n=0

antn =

11 td

(21 )/(3.36)

and therefore the present problem has the exact solution

w =

t d

1 td(21 )/

t1 / . (3.37)

The existence of an exact solution for the flow of Figure 3.1 follows from

the general theory of free streamline flows. This theory was developed by

Kirchhoff [90] and Helmholtz [71]; see Birkhoff and Zarantonello [19] orGurewich [69] for details.

The free-surface profile is obtained by setting = 1 in (3.8) and (3.37),

calculating the partial derivatives x and y from the identity

x + iy =1

w(3.38)

and integrating with respect to .

As a first example let us assume that 1 = 2 = /2 (see Figure 3.2).

Then (3.37) reduces to

w = t1/2 (3.39)

and (3.9), (3.38) and (3.39) yield

x + iy = ei/2, 0 < < , (3.40)

along the free surface EF. Differentiating (3.10) with respect to and

applying the chain rule to (3.40) gives

x + iy = 1

cotan 2

ei/2. (3.41)


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Integrating (3.41) with respect to and taking the real and imaginary parts

gives

x =2

cotan

2+

1 (3.42)

y = 2

sin 2

+ 1. (3.43)

Relations (3.42) and (3.43) define the free-surface profile in parametric form.

It is shown in Figure 3.8.

0

0.5

1.0

1.5

2.0

0 2 4 6 8

Fig. 3.8. Free-surface profile for the flow configuration of Figure 3.2. The positionof the separation point E is indicated by a small horizontal line. The vertical scalehas been exaggerated to show clearly the free-surface profile.

A classical parameter associated with this flow is the contraction ratio Cc,

defined as the ratio yF/yE of the ordinates of the points F and E. Using

(3.43) with = and = 0, we obtain

Cc

=

+ 2 0.611. (3.44)

As a second example, let us assume 1 = 0 and 2 = /2 (see Figure 3.9).

Then (3.37) becomes

w =

t d

1 td1/2

. (3.45)

Proceeding as in the previous example, we obtain

x + iy =

1

cotan

21 e

i d

ei

d 1/2

(3.46)

on the free surface EF.


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B

CD

E

F

Fig. 3.9. A free-surface flow emerging from a container bounded by the horizontalwalls CD and AB and by the verical wall DE.

Integrating (3.46) gives x and y on the free surface as functions of .There is a solution for each value of1 < d < 0; the parameter d measuresthe length of the vertical wall DE in the complex t-plane. This is an inverse

formulation in the sense that for each value of d the length of the wall DE

in the physical plane is found at the end of the calculations, in the following

way. We first calculate y for 1 < t < d by using (3.38) and (3.45). Wethen evaluate yt for 1 < t < d by using (3.8) and the chain rule. Thelength of the wall DE is then obtained by integrating with respect to t from

1 to d. A typical solution for d = 0.5 is shown in Figure 3.10.

0

0.4

0.8

1.2

1.6

0 1 2 3 4

Fig. 3.10. Computed free-surface profile for the flow configuration of Figure 3.9with d = 0.5. The position of the separation point E is indicated by a smallhorizontal line. The vertical scale has been exaggerated to show clearly the free-

surface profile.


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As d 0, the length of the vertical wall DE tends to infinity and theflow reduces to that of Figure 3.2. As d 1, the length of the verticalwall DE tends to zero and the flow reduces to a uniform stream.

As a third example, we assume 2 < 0 and 1 = 0 (see Figure 3.3). As

mentioned at the begining of this chapter, this configuration models the

flow due to a surface-piercing obstacle moving at a constant velocity when

viewed in a frame of reference moving with the obstacle. In particular it is

a simple model for the flow near the stern or the bow of a ship. Again using

(3.37), we obtain

w =

t d

1 td2 /

. (3.47)

As in the previous two examples we use (3.47) to calculate x + iy on the

free surface. After integration we obtain the shape of the free surface inparametric form. A typical free-surface profile for d = 0.2 and 2 = /3is shown in Figure 3.11.

0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Fig. 3.11. Computed free-surface profile for the flow configuration of Figure 3.3with d = 0.2 and 2 = /3. The position of the separation point E is indicatedby a small horizontal line. The vertical scale has been exaggerated to show clearlythe free-surface profile.

3.1.2 Free separation

In Figures 3.1 and 3.9, on the one hand, the free surface is forced to separate

from the rigid wall DE at E because the wall DE terminates at E. We referto this situation as forced separation. On the other hand, if the infinitely


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thin wall DE is replaced by a wall of finite thickness bounded by a smooth

curve then in principle the point of separation E can be any point on the

smooth curve (see Figure 3.12). We refer to this situation as free separation.

A B

F

C

D

E

Fig. 3.12. The flow configuration of Figure 3.9 but with the vertical wall DE re-

placed by a wall bounded by a smooth curve.

We note that any solution corresponding to free separation represents

also a solution with forced separation if the smooth curve is cut along a line

through the separation point (see Figure 3.13). As we shall see in Section

3.2, the distinction between forced and free separation is important when

studying the effects of surface tension.

A B

F

C

E

D

Fig. 3.13. The flow configuration of Figure 3.12 when the smooth curve is cut by avertical line.

3.1.2.1 Open cavities

We now consider some solutions with free separation which will be useful

in Section 3.2 when we consider the effects of surface tension. Figure 3.14

shows a particular case of Figure 3.12 for which the vertical rigid wall DE

of Figure 3.9 has been replaced by a smooth elliptical wall with equation

xa

1/2+

yb

1/2= 1. (3.48)


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Here x and y refer to coordinates with the origin at the centre of the ellipse

and a and b are the semi-axes of the ellipse.

A B

E

F

C D

Fig. 3.14. The flow of Figure 3.9 when the vertical wall DE is replaced by a smooth

semi-elliptical wall.

If a b, the semi-ellipse is thin and the configuration of Figure 3.14 canbe viewed as that of Figure 3.9 but with the infinitely thin wall DE replaced

by a smooth wall of finite thickness. In other words, Figure 3.14 takes into

account the finite thickness of any real wall but approaches the configuration

of Figure 3.9 as a/b 0. However, to study flows with free separation weshall assume that b = a (i.e. that the semi-ellipse is a semicircle), so that

flows corresponding to different positions of the separation point E can beclearly distinguished on the profiles.

The flow of Figure 3.14 can be reflected in the wall CD. This yields the

flow of Figure 3.15. It models a flow past a circular cylinder with a cavity

behind it (see for example Batchelor [8] for a discussion of cavitating flows).

Fig. 3.15. Cavitating flow past a circular object in a domain bounded by two hor-izontal walls.

We shall study the flow of Figure 3.15 when the radius of the circle is very

small compared with the distance between the horizontal walls, so that the

circle can be assumed to be in a fluid unbounded in the vertical directio

Gravity Capillary Free Surface Flows

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