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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
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
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
may take place without the written permission of Cambridge University Press.
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
http://www.cambridge.org/9780521811903http://www.cambridge.org/http://www.cambridge.org/9780521811903http://www.cambridge.org/8/3/2019 Gravity Capillary Free Surface Flows
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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
8.2.2 Analytical results 221
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
8.4.1 Numerical results 236
8.4.2 Analytical results 239
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
9.2.3 Numerical results 271
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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2.3 Two-dimensional flows 11
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
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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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2.3 Two-dimensional flows 13
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
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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
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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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3.1 Free streamline solutions 33
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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34 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 35
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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36 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 37
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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38 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 39
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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40 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 41
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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42 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 43
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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44 Free-surface flows that intersect walls
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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3.1 Free streamline solutions 45
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
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