Gravity-Capillary Free-Surface Flows

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CUUK1062-Vanden

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12:43

G R A V I T Y–C 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 gravity–capillary 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 emphasis is placed upon the reader developing a deep understanding of the structure of the resulting solutions. The author also reviews relevant concepts in flui mechanics to enable readers from other scientifi field to develop a working knowledge of free-boundary problems.

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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 ¨ HLER OLIVER BU 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 Fluid–Structure 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 B´enard Cells and Taylor Vortices E. L. KOSCHMIEDER

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GRAVITY – CAPILLARY FREE-SURFACE FLOWS JEAN-MARC VANDEN-BROECK University College London

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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521811903 © Cambridge University Press 2010 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. First published in print format 2010 ISBN-13

978-0-511-72961-4

eBook (NetLibrary)

ISBN-13

978-0-521-81190-3

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of 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.

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To Mirna and Ada

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vi

Contents

Preface

page xi

1

Introduction

1

2

Basic concepts 2.1 The equations of ﬂuid mechanics 2.2 Free-surface ﬂows 2.3 Two-dimensional ﬂows 2.4 Linear waves 2.4.1 The water-wave equations 2.4.2 Linear solutions for water waves 2.4.3 Superposition of linear waves

7 7 8 11 15 15 17 24

3

Free-surface ﬂows that intersect walls 3.1 Free streamline solutions 3.1.1 Forced separation 3.1.2 Free separation 3.2 The eﬀects of surface tension 3.2.1 Forced separation 3.2.2 Free separation 3.3 The eﬀects of gravity 3.3.1 Solutions with β1 = 0 (funnels) 3.3.2 Solutions with β1 = 0 (nozzles and bubbles) 3.3.3 Solutions with β1 = π/2 (ﬂow under a gate with gravity) 3.4 The combined eﬀects of gravity and surface tension 3.4.1 Rising bubbles in a tube 3.4.2 Fingering in a Hele Shaw cell 3.4.3 Further examples involving rising bubbles 3.4.4 Exponential asymptotics

31 33 33 43 58 58 66 73 83 88

vii

94 98 99 103 108 112

viii

4

Contents

Linear free-surface ﬂows generated by moving disturbances 4.1 The exact nonlinear equations 4.2 Linear theory 4.2.1 Solutions in water of ﬁnite depth 4.2.2 Solutions in water of inﬁnite depth 4.2.3 Discussion of the solutions

114 115 116 116 122 124

5

Nonlinear waves – asymptotic solutions 5.1 Periodic waves 5.1.1 Solutions when condition (5.55) is satisﬁed 5.1.2 Solutions when condition (5.55) is not satisﬁed 5.2 The Korteweg–de Vries equation

129 129 135 138 142

6

Numerical computations of nonlinear water waves 6.1 Formulation 6.2 Series truncation method 6.3 Boundary integral equation method 6.4 Numerical methods for solitary waves 6.4.1 Boundary integral equation methods 6.5 Numerical results for periodic waves 6.5.1 Pure capillary waves (g = 0, T = 0) 6.5.2 Pure gravity waves (g = 0, T = 0) 6.5.3 Gravity–capillary waves (g = 0, T = 0) 6.6 Numerical results for solitary waves 6.6.1 Pure gravity solitary waves 6.6.2 Gravity–capillary solitary waves

148 148 151 152 156 157 160 160 164 175 181 181 186

7

Nonlinear free-surface ﬂows generated by moving disturbances 7.1 Pure gravity free-surface ﬂows in water of ﬁnite depth 7.1.1 Supercritical ﬂows 7.1.2 Subcritical ﬂows 7.2 Gravity–capillary free-surface ﬂows 7.2.1 Results in ﬁnite depth 7.2.2 Results in inﬁnite depth (removal of the nonuniformity) 7.3 Gravity–capillary free-surface ﬂows with Wilton ripples

8

Free-surface ﬂows with waves and intersections with rigid walls 8.1 Free-surface ﬂow past a ﬂat plate 8.1.1 Numerical results

191 192 192 195 201 201 203 206 210 211 212

Contents

8.2

8.3

8.4

8.1.2 Analytical results Free-surface ﬂow past a surface-piercing object 8.2.1 Numerical results 8.2.2 Analytical results Flow under a sluice gate 8.3.1 Formulation 8.3.2 Numerical procedure 8.3.3 Discussion of the results Pure capillary free-surface ﬂows 8.4.1 Numerical results 8.4.2 Analytical results

ix

215 218 219 221 226 228 231 233 236 236 239

9

Waves with constant vorticity 9.1 Solitary waves with constant vorticity 9.1.1 Mathematical formulation 9.1.2 Numerical procedure 9.1.3 Discussion of the results 9.2 Periodic waves with constant vorticity 9.2.1 Mathematical formulation 9.2.2 Numerical procedure 9.2.3 Numerical results 9.2.4 Discussion

244 245 245 248 249 267 268 270 271 276

10

Three-dimensional free-surface ﬂows 10.1 Green’s function formulation for two-dimensional problems 10.1.1 Pressure distribution 10.1.2 Two-dimensional surface-piercing object 10.2 Extension to three-dimensional free-surface ﬂows 10.2.1 Gravity ﬂows generated by moving disturbances in water of inﬁnite depth 10.2.2 Three-dimensional gravity–capillary free-surface ﬂows in water of inﬁnite depth 10.3 Further extensions

278

11

Time-dependent free-surface ﬂows 11.1 Introduction 11.2 Nonlinear gravity–capillary standing waves References Index

278 278 283 286 286 293 298 301 301 301 308 318

Preface

This book is concerned with the theory of gravity–capillary free-surface ﬂows. Free-surface ﬂows are ﬂows bounded by surfaces that have to be found as 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 (cavitating 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 limitations, some topics could not be covered. These include interfacial ﬂows and 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

1 Introduction

Free-surface problems occur in many aspects of science and everyday life. They can be deﬁned as problems whose mathematical formulation involves surfaces 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. We hope to convince the reader of the beauty of such problems and to present the challenges faced when one attempts to describe these ﬂows mathematically. 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 at the surface of a lake, which involves an intersection between the free surface 1

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 intersections 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 (irrotational 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 considered 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 exact nonlinear solutions. This situation is fortunate since there are very few such 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 equations 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’ to be a singular perturbation. This is conﬁrmed by the numerical results presented 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 velocity is a limitation of the model. We show that a basic way to remove this singularity is to take into account the ﬁnite thickness of the rigid walls,

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 Brillouin–Villat condition. Equivalently, 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 with a 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 streamline 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 free surface and the wall must be such that the velocity is ﬁnite at the separation 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◦ between 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 pouring from containers) that these three possibilities occur in free-surface ﬂows with gravity. The restriction to three local behaviours is to be contrasted with 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 combine 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 approaches zero. Therefore a small amount of surface tension can again be used to select

4

Introduction

solutions. One diﬀerence between this selection mechanism and that described 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 gravity–capillary 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 various numerical schemes that the author has used successfully over the years to obtain highly accurate solutions for free-surface ﬂow problems. They include 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 nonlinear boundary condition (the pressure condition for free-surface ﬂows) at appropriately chosen collocation points. This leads to a system of nonlinear algebraic equations which can be solved by iteration (for example by using Newton’s 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 extensively by many researchers. Insight into free-surface ﬂows of the second class can be gained by studying the limitations of the classical linear theories. In particular we study in Chapter 4 the waves generated by a disturbance moving at a constant velocity (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 support. 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 perturbation of a uniform stream. The equations are then linearised (around a uniform

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 α=

Tg . ρU 4

Here U is the velocity of the disturbance, H the depth of the ﬂuid, T the surface tension, g the acceleration of gravity and ρ 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 solutions. 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. This is 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 approximation of the nonlinear problem as → 0. However, for any , no matter how small, the linear solutions become invalid as F → 1. A similar situation occurs for gravity–capillary ﬂows. As α approaches a critical value αH , the linear theory again predicts inﬁnite displacements of the free surface. The critical value αH depends on the depth H (for example αH = 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 equations 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 α and F , there are not only solutions that are perturbations of a uniform stream (the nonlinear equivalent of the linear solutions mentioned earlier) but also solutions that are perturbations of solitary waves. Some of these solitary waves are of the

6

Introduction

well-known Korteweg–de Vries type but others are solitary waves with decaying 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 Korteweg–de Vries type have oscillatory tails of constant 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 waveless. 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 3–8 are assumed to be steady, two-dimensional and irrotational. The ﬁnal three chapters of the book describe some extensions in which these assumptions are removed. In Chapter 9 we study solitary 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 particular we calculate three-dimensional gravity–capillary solitary waves. These waves are characterised by decaying oscillations in the direction of propagation 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, the breaking 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.

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 ρ (i.e. to be incompressible). Conservation of momentum yields the Euler equations 1 Du = − ∇p + X, Dt ρ

(2.1)

where u is the vector velocity, p is the pressure and X is the body force. Here D ∂ = +u·∇ (2.2) Dt ∂t is the material derivative. We assume that the body force X derives from a potential Ω, i.e. that X = −∇Ω.

(2.3)

In most applications considered in this book, the ﬂow is assumed to be irrotational. Therefore ∇ × u = 0.

(2.4)

Relation (2.4) implies that we can introduce a potential function φ such that u = ∇φ.

(2.5)

∇ · u = 0.

(2.6)

Conservation of mass gives

7

8

Basic concepts

Then (2.5) and (2.6) imply that φ satisﬁes Laplace’s equation ∇2 φ = 0.

(2.7)

Flows that satisfy (2.4)–(2.7) are referred to as potential ﬂows. Using the identity 1 (2.8) u · ∇u = ∇(u · u) + (∇ × u) × u, 2 (2.4) and (2.2) yield Du ∂u 1 = + ∇(u · u). (2.9) Dt ∂t 2 Substituting (2.9) into (2.1) and using (2.3) and (2.5) we obtain ∂φ u · u p + + + Ω = 0. (2.10) ∇ ∂t 2 ρ After integration, (2.10) gives the well-known Bernoulli equation ∂φ u · u p + + + Ω = F (t). ∂t 2 ρ

(2.11)

Here F (t) is an arbitrary function of t. It can be absorbed in the deﬁnition of φ, and then (2.11) can be rewritten as ∂φ u · u p + + + Ω = B, ∂t 2 ρ

(2.12)

where B is a constant. For steady ﬂows (2.12) reduces to u·u p + + Ω = B. 2 ρ

(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 Figure 2.2). Both ﬂows are assumed to be steady and to approach a uniform stream with a constant velocity U as x2 + y 2 + z 2 → ∞; the eﬀects of gravity are neglected. 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: φxx + φyy + φzz = 0 outside

S,

(2.14)

2.2 Free-surface ﬂows

9

U z

y

S

x

Fig. 2.1. The ﬂow past a rigid sphere. The surface S of the sphere is described by x2 + y 2 + z 2 = R2 , where R is the radius of the sphere.

U z

y S x

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

∂φ =0 ∂n

on S

(φx , φy , φz ) → (0, 0, −U ) as x2 + y 2 + z 2 → ∞.

(2.15) (2.16)

Equation (2.14) is Laplace’s equation (2.7) expressed in cartesian coordinates. The boundary condition (2.15) is known as the kinematic boundary condition. It states that the normal component of the velocity vanishes on S.

10

Basic concepts

Equations (2.14)–(2.16) form a linear boundary value problem whose solution is R3 z φ = −U z + . (2.17) 2(x2 + y 2 + z 2 )3/2 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 1 2 p p∞ (φx + φ2y + φ2z ) + = U 2 + . 2 ρ 2 ρ

(2.18)

Here p∞ denotes the pressure as x2 + y 2 + z 2 → ∞. Equation (2.18) holds everywhere outside the sphere. In deriving (2.18) we have set Ω = 0 in (2.13) and evaluated B by taking the limit x2 + y 2 + z 2 → ∞ in (2.13). Then, using (2.16) gives B = U 2 /2 + p∞ /ρ. 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 φ by solving the linear set of relations (2.14)–(2.16). We may then substitute the values (2.17) of φ 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 φ 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 φ 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 1 1 . (2.19) + p − pb = T K = T R1 R2 Here R1 and R2 are the principal radii of curvature of the ﬂuid surface: they are counted positive when the centres of curvature lie inside the ﬂuid. The

2.3 Two-dimensional ﬂows

quantity

K=

1 1 + R1 R2

11

(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 T p ∞ − pb 1 2 1 (φx + φ2y + φ2z ) + K = U 2 + 2 ρ 2 ρ

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 φ 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 Ω = 0. Combining (2.12) and (2.19) we ﬁnd that the general form of the dynamic boundary condition (for unsteady ﬂows) with Ω = 0 is T ∂φ u · u + + Ω + K = B. ∂t 2 ρ

(2.22)

Here B is the Bernoulli constant. For steady ﬂows, (2.22) reduces to u·u T + Ω + K = B. 2 ρ

(2.23)

2.3 Two-dimensional ﬂows As we shall see, many interesting free-surface ﬂows can be modelled as twodimensional ﬂows. We then introduce cartesian coordinates x and y with the y-axis directed vertically upwards (at present we reserve the letter z to denote the complex quantity x + iy). In most applications considered in

12

Basic concepts

this book, the potential Ω (see (2.3)) is due to gravity. Assuming that the acceleration of gravity g is acting in the negative y-direction, we write Ω as Ω = gy.

(2.24)

An example is the two-dimensional free-surface ﬂow past a semicircular obstacle at the bottom of a channel (see Figure 2.3). This two-dimensional conﬁguration provides a good approximation to the three-dimensional freesurface 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.

y x

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

For two-dimensional potential ﬂows, (2.4) and (2.6) become ∂u ∂v − = 0, ∂y ∂x

(2.25)

∂u ∂v + = 0. ∂x ∂y

(2.26)

Here u and v are the x- and y- components of the velocity vector u. We can introduce a streamfunction ψ by noting that (2.26) is satisﬁed by u=

∂ψ , ∂y

v=−

∂ψ . ∂x

(2.27)

(2.28)

It then follows from (2.25) that ∇2 ψ =

∂2ψ ∂2ψ + = 0. ∂x2 ∂y 2

(2.29)

2.3 Two-dimensional ﬂows

13

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

∂φ , ∂x

(2.30)

v=

∂φ ∂y

(2.31)

and ∇2 φ =

∂2φ ∂2φ + 2 = 0. ∂x2 ∂y

(2.32)

Combining (2.27), (2.28), (2.30) and (2.31) we obtain ∂φ ∂ψ = , ∂x ∂y

(2.33)

∂φ ∂ψ =− . ∂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 = φ + iψ

(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 ﬂows can 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 freesurface ﬂows. Since the derivative of an analytic function is also an analytic function, it follows that the complex velocity u − iv =

∂φ ∂ψ ∂ψ df ∂φ −i = +i = ∂x ∂y ∂y ∂x dz

(2.36)

is also an analytic function of z = x + iy. The theory of analytic functions will be used intensively in the following 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 as an analytic function of t (since an analytic function of an analytic function is also an analytic function). Such conformal mappings are used to redeﬁne

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 Cauchy’s theorem: If h(z) is analytic throughout a simply connected domain D then, for every closed contour C within D, h(z)dz = 0. (2.37) C

The third tool is the Cauchy integral formula: Let h(z) be analytic everywhere within and on a closed contour C, taken in the positive sense (counterclockwise). Then the integral h(z) 1 dz (2.38) 2iπ C z − z0 takes the following values: 0 if z0 h(z0 ) if z0 1 h(z0 ) if z0 2

is outside C,

(2.39)

is inside C,

(2.40)

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 ψ 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 ∂ψ ∂ψ dψ X (s) + Y (s) = = 0, ∂x ∂y ds

(2.43)

which implies that ψ 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) becomes ∂φ 1 2 T + (φx + φ2y ) + gy + K = B. (2.44) ∂t 2 ρ If we denote by θ the angle between the tangent to the free surface and the

2.4 Linear waves

15

horizontal then the curvature K can be deﬁned by K=−

dθ ds

(2.45)

where again s denotes the arc length. In particular if the (unknown) equation of the free surface is y = η(x, t) then tan θ = ηx

and

1 dx = . ds (1 + ηx2 )1/2

(2.46)

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

ηxx . (1 + ηx2 )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 dissipation 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 λ extending from x = −∞ to x = ∞ and travelling at a constant velocity c. The ﬂow conﬁguration is illustrated in Figure 2.4. y

x

y = y1 y = y2

Fig. 2.4. A two-dimensional train of waves viewed in a frame of reference moving with the wave. The free-surface proﬁle has wavelength λ. The ﬂuid is bounded below by a horizontal bottom with equation y = −h. Also shown is the rectangular contour used in (2.56).

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 φxx + φyy = 0, φy = φx ηx

−h < y < η(x), on y = η(x),

on y = −h,

φy = 0

1 2 ηxx T (φx + φ2y ) + gy − =B 2 ρ (1 + ηx2 )3/2

on y = η(x),

(2.48) (2.49) (2.50) (2.51)

∇φ(x + λ, y) = ∇φ(x, y),

(2.52)

η(x + λ) = η(x),

(2.53)

λ

η(x)dx = 0,

(2.54)

0

1 λ

λ

φx dx = c

on y = constant.

(2.55)

0

Here g is the acceleration of gravity (assumed to act in the negative ydirection), T is the surface tension, ρ is the density, y = −h is the equation of the bottom and y = η(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 = η(x). Relations (2.52) and (2.53) are periodicity conditions, which require the solution to be periodic with wavelength λ. 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 = φx 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, udx + vdy = 0. (2.56) C

2.4 Linear waves

17

The contributions from the two vertical lines cancel by periodicity and (2.56) gives λ λ [u]y=y1 dx = [u]y=y2 dx. (2.57) 0

0

Since y1 and y2 are arbitrary, the integral on the left-hand side of (2.55) is independent 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 φ = cx,

η(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 φ(x, y) = cx + ϕ(x, y)

(2.59)

and assume that both |ϕ(x, y)| and |η(x)| are small. Substituting (2.59) into (2.48)–(2.55) and dropping nonlinear terms in ϕ and η, we obtain the linear system ϕxx + ϕyy = 0, ϕy = cηx , ϕy = 0, −

−h < y < 0,

(2.60)

y = 0,

(2.61)

y = −h,

(2.62)

T ηxx + cϕx + gη = 0, ρ

y = 0,

(2.63)

∇ϕ(x + λ, y) = ∇ϕ(x, y),

(2.64)

η(x + λ) = η(x),

(2.65)

λ

η(x)dx = 0, 0

(2.66)

18

1 λ

Basic concepts λ

ϕx dx = 0

on y = constant.

(2.67)

0

We choose the origin of x at a crest and assume that the wave is symmetric about x = 0. Thus we impose the conditions ϕ(−x, y) = −ϕ(x, y),

(2.68)

η(−x) = η(x).

(2.69)

Using the method of separation of variables, we seek a solution of (2.60) in the form ϕ(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) Y (y) =− = constant = −α2 , X(x) Y (y)

(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 αx,

(2.74)

Y (y) = cosh α(y + h).

(2.75)

The periodicity condition (2.64) implies that α = nk,

(2.76)

2π λ

(2.77)

where n is a positive integer and k=

is the wavenumber. Multiplying (2.74) and (2.75) and taking a linear combination of the solutions corresponding to the values (2.76) of α, we obtain ϕ(x, y) =

∞ n=1

Bn cosh nk(y + h) sin nkx.

(2.78)

2.4 Linear waves

19

Here the Bn are constants. Using the periodicity and the symmetry conditions (2.69) and (2.65), we express η(x) as the Fourier series η(x) = A0 +

∞

An cos nkx

(2.79)

n=1

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 An n2 k 2 + gAn + cBn nk cosh nkh = 0, ρ

n = 1, 2, . . .

Eliminating Bn between (2.80) and (2.81) yields c2 nk T 2 2 g+ n k − cosh nkh An = 0, ρ sinh nkh

n = 1, 2, . . .

(2.81)

(2.82)

Since we seek a nontrivial periodic solution η(x) = 0, we can assume without loss of generality that A1 = 0; then (2.82) with n = 1 implies that g T + k tanh kh. (2.83) c2 = k ρ Relation (2.82) for n > 1 gives An = 0,

n = 2, 3, . . . ,

(2.84)

provided that g+

c2 nk T 2 2 n k − cosh nkh = 0, ρ sinh nkh

n = 2, 3, . . .

(2.85)

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

cA1 cosh k(y + h) sin kx, sinh kh η = A1 cos kx.

(2.86) (2.87)

If the condition (2.85) is not satisﬁed for some integer value m of n, the solution of the linear problem is ϕ=−

cAm cA1 cosh k(y + h) sin kx − cosh mk(y + h) sin mkx, sinh kh sinh mkh, (2.88)

20

Basic concepts

η1 = 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 solving the 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 gravity–capillary 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 diﬀerent wavenumbers and therefore of diﬀerent wavelengths travel at diﬀerent phase velocities c. It is convenient to rewrite (2.83) in the dimensionless form 1 2 + τ kh tanh kh, (2.90) F = kh where F =

c (gh)1/2

(2.91)

T ρgh2

(2.92)

is the Froude number and τ=

is the Bond number. Relation (2.90) is shown graphically in Figure 2.5, where we present values of F 2 versus 1/(kh) = λ/(2πh) for four values of τ . The curves of Figure 2.5 illustrate that F 2 is a monotonically decreasing function of λ/h when τ > 1/3 and that it has a minimum for τ < 1/3. As λ/h → ∞, F → 1. The diﬀerent behaviours for τ < 1/3 (minimum) and τ > 1/3 (monotone decay) in Figure 2.5 have many implications, in particular for the study of nonlinear periodic and solitary gravity–capillary waves (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 ϕ = −cA1 eky sin kx,

(2.93)

η = A1 cos kx,

(2.94)

g T + k. k ρ

(2.95)

c2 =

2.4 Linear waves

21

2.0

1.5

1.0

0.5

0

0

2

4

6

8

10

12

14

Fig. 2.5. Values of F 2 versus 1/(kh). The curves correspond from top to bottom to τ = 1.3, τ = 1/3, τ = 0.1 and τ = 0.05. For τ < 1/3 the curves have a minimum whereas for τ > 1/3 the curves are monotonically decreasing.

Since kh = 2πh/λ, the inﬁnite-depth results (2.93)–(2.95) provide an approximation to the ﬁnite-depth results (2.83), (2.86) and (2.87) when the wavelength λ is small compared with the depth h. Waves with g = 0, T = 0 are referred to as pure gravity waves. They are characterised 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 are characterised in the inﬁnite-depth case by the dispersion relation c2 =

T k. ρ

(2.97)

A simple calculation based on (2.95) shows that c2 reaches a minimum value given by 4T g 1/4 (2.98) cmin = ρ when k = kmin =

ρg 1/2 T

.

(2.99)

Graphs of c versus λ in units of cmin and λmin = 2π/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 λ > λmin are dominated by gravity and can be approximated by pure gravity waves

22

Basic concepts 2.0

1.5

1.0

0.5

0

0

1

2

3

4

5

6

Fig. 2.6. Values of c versus λ = 2π/k in units of cm in and λm in . The solid curve corresponds to (2.95), the dotted curve to (2.97) and the broken curve to (2.96).

for λ large. Waves with λ < λmin are dominated by surface tension and can be approximated by pure capillary waves for λ 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 khF 2 = tanh kh.

(2.100)

d tanh kh ≤ 1, d (kh)

(2.101)

Since

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 moving with 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 φxx + φyy = 0, ηt = φy − φx ηx φy = 0

−h < y < η(x, t),

(2.102)

on y = η(x, t),

(2.103)

on y = −h,

(2.104)

2.4 Linear waves

ηxx 1 T φt + (φ2x + φ2y ) + gy − =B 2 ρ (1 + ηx2 )3/2

23

on y = η(x, t),

(2.105)

φ(x + λ, y, t) = φ(x, y, t),

(2.106)

η(x + λ, t) = η(x, t),

(2.107)

λ

η(x, t)dx = 0.

(2.108)

0

A trivial solution of the system (2.102)–(2.108) is φ = 0,

η = 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 constant velocity c. Therefore we rewrite φ and η in terms of two new functions φ¯ and η¯: ¯ + ct, y) and η(x, t) = η¯(x + ct) φ(x, y, t) = φ(x

(2.110)

Substituting (2.110) into the system (2.102)–(2.108) and dropping nonlinear terms in φ¯ and η¯, we obtain the linear system φ¯xx + φ¯yy = 0, c¯ ηx = φ¯y φ¯y = 0 cφ¯x + g η¯ −

−h < y < 0,

(2.111)

on y = 0,

(2.112)

on y = −h,

(2.113)

T η¯xx = 0 on y = 0, ρ

(2.114)

¯ + λ + ct, y) = φ(x ¯ + ct, y), φ(x

(2.115)

η¯(x + λ + ct) = η¯(x + ct),

(2.116)

λ

η¯(x + ct)dx = 0.

(2.117)

0

Following the derivation of (2.86)–(2.89), we ﬁnd that the solution of (2.111)–(2.117) is cA1 φ¯ = − cosh k(y + h) sin k(x + ct), sinh kh

(2.118)

η¯ = A1 cos k(x + ct)

(2.119)

24

Basic concepts

if (2.85) is satisﬁed and cA1 cosh k(y + h) sin k(x + ct), φ¯ = − sinh kh −

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

η¯ = A1 cos k(x + ct) + Am cos mk(x + ct)

(2.120) (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 by superposing 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 amplitude travelling at the same velocity but in opposite directions. This gives η = A1 cos k(x + ct) + A1 cos k(x − ct), φ=− +

(2.122)

cA1 cosh k(y + h) sin k(x + ct) sinh kh cA1 cosh k(y + h) sin k(x − ct). sinh kh

(2.123)

Using the trigonometric identities cos p + cos q = 2 cos

p+q p−q cos 2 2

(2.124)

sin p + sin q = 2 sin

p+q p−q cos 2 2

(2.125)

we can rewrite (2.122), (2.123) as η = 2A1 cos kx cos kct, φ = −2

cA1 cosh k(y + h) cos kx sin kct. sinh kh

(2.126) (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 displacement of the free surface are ﬁxed as t varies. The wave does not propagate and

2.4 Linear waves

25

its free surface moves periodically up and down as t varies. The period of this motion is 2π Ts = . (2.128) kc Since u = φx = 0 along the lines x = 0 and x = π/k = λ/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). 0.2

0.1

0

0

1

2

3

Fig. 2.7. Standing wave for A1 = 0.1 and k = 1. The broken line is the free-surface proﬁle 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 = π.

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 ¯ We ﬁrst introduce the angular frequency diﬀerent wavenumbers k and k. ω = kc and write ω = W (k). Using (2.83) we have 1/2 g T + k tanh kh . W (k) = k k ρ

(2.129)

(2.130)

Next we rewrite (2.118) and (2.119) as η¯ = A1 cos[kx + W (k)t],

(2.131)

cA1 φ¯ = − cosh k(y + h) sin[kx + W (k)t]. sinh kh

(2.132)

26

Basic concepts

The superposition described above then yields ¯ + W (k)t]. ¯ η¯ = A1 cos[kx + W (k)t] + A1 cos[kx

(2.133)

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

1 1 ¯ + [W (k) + W (k))t ¯ [(k + k)]x η¯ = 2A1 cos 2 2 × cos

1 1 ¯ ¯ [(k − k)]x + [(W (k) − W (k)t] . 2 2

(2.134)

For k¯ close to k, we may approximate (2.134) by η¯ = α(x, t) cos[kx + W (k)t], where

α(x, t) = 2A1 cos

1 1 ¯ + [W (k) − W (k)]t ¯ (k − k)x . 2 2

(2.135)

(2.136)

The expression (2.135) is the same as (2.131) except that the constant amplitude A1 has been replaced by the variable amplitude α(x, t). Diﬀerentiating (2.136) with respect to x and t yields

1 ∂α 1 ¯ ¯ ¯ = −A1 (k − k) sin (k − k)x + [W (k) − W (k)]t (2.137) ∂x 2 2 and ∂α ¯ sin = −A1 [W (k) − W (k)] ∂t

1 1 ¯ + [W (k) − W (k)]t ¯ (k − k)x . (2.138) 2 2

¯ 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 α(x, t) is a slowly varying function of x and t. In other words, the solution is a wave of wavenumber k, travelling at velocity c, whose amplitude α(x, t) is slowly modulated. The amplitude α(x, t) is itself a wave travelling at velocity ¯ W (k) − W (k) . (2.139) k − k¯ For k¯ close to k, the velocity (2.139) becomes cg =

dW (k) . dk

(2.140)

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

For water waves, (2.130) and (2.140) give −1/2 1 T 3 cg = gk + k tanh kh 2 ρ kh T 2 3T k 2 tanh kh + g + k × g+ . ρ ρ cosh2 kh

(2.141)

(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), we obtain −1/2 1 3T k 2 T k3 g+ gk + . (2.143) cg = 2 ρ ρ In particular, we have for pure gravity waves (g = 0, T = 0) 1 g 1/2 c = cg = 2 k 2 and for pure capillary waves (T = 0, g = 0) 3 T k 1/2 3c cg = = . 2 ρ 2

(2.144)

(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 have chosen 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. the amplitude) travels at the velocity cg .

28

Basic concepts 0.4

0.2

0

0

20

40

60

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 + k

dc . 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 the minimum 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 0 ∂φ (2.147) p dy. −h ∂x The average of (2.147) over one period is ω Ef = 2π

t∗ +2π/ω

t∗

0

p −h

∂φ dydt, ∂x

(2.148)

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

2.4 Linear waves

29

gives p = −ρ

∂φ − ρgy + constant. ∂t

(2.149)

Using (2.118) we ﬁnd t∗ +2π/ω t∗ +2π/ω ∂φ cA1 k cos k(x + ct)dt = 0. dt = − cosh k(y + h) ∂x sinh kh t∗ t∗ (2.150) Therefore (2.148) simpliﬁes to t∗ +2π/ω 0 ∂φ ∂φ ω dydt. (2.151) Ef = −ρ 2π t∗ −h ∂t ∂x Substituting (2.118) into (2.151) and evaluating the integral yields ρA21 k 2 c3 sinh 2kh Ef = h+ . (2.152) 2k 4 sinh2 kh We now deﬁne the kinetic and potential energy per unit horizontal length by 2 0 ∂φ 2 ∂φ 1 ρ + dy (2.153) 2 ∂x ∂y −h and

η

0

1 ρgydy = ρgη 2 . 2

(2.154)

Averaging the quantities (2.153) and (2.154) over a wavelength gives the mean kinetic energy λ 0 2 2 ∂φ ∂φ ρ ¯ = + dydx (2.155) K 2λ 0 −h ∂x ∂y and the mean potential energy ρg V¯ = 2λ

λ

η 2 dx.

(2.156)

0

Substituting (2.118) and (2.119) into (2.155) and (2.156) gives, after integration, ¯ = V¯ = 1 ρgA21 . (2.157) K 4 Thus the total energy is ¯ + V¯ = 1 ρgA21 . E=K 2

(2.158)

30

Basic concepts

Combining (2.152) and (2.158) we obtain 2kh 1 . Ef = E c 1 + 2 sinh 2kh Using (2.142) with T = 0 gives cg =

c 2

1+

2kh sinh 2kh

(2.159)

.

(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.

3 Free-surface ﬂows that intersect walls

We continue our study of free-surface ﬂows by considering the twodimensional ﬂow shown in Figure 3.1. The ﬂow domain is bounded below 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 γ1 and γ2 respectively. C γ1 D γ2

E y x

A

F B

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

The conﬁguration of Figure 3.1 was chosen because it can be used to describe many properties of free-surface ﬂows that intersect, i.e. adjoin, 31

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 γ1 = γ2 = π/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. C

E F

A

B

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

When γ1 = 0 and γ2 < 0, Figure 3.1 describes locally the ﬂow near the bow 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. F E C A

D 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 increasing complexity. Section 3.1 is devoted to free-surface ﬂows with g = 0 and

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 the eﬀect of gravity (T = 0, g = 0). The combined eﬀects of gravity and surface tension (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 the incompressibility of the ﬂuid and the irrotationality of the ﬂow, we deﬁne a potential function φ(x, y) and a streamfunction ψ(x, y). As shown in Section 2.3, the complex potential f = φ + iψ 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 ψ = 0. The walls CD and DE and the free surface EF deﬁne another streamline, along which the constant value of ψ is denoted by Q. We also choose φ = 0 at the separation point E. These two choices (ψ = 0 on AB and φ = 0 at E) can be made without loss of generality because φ and ψ are deﬁned up to arbitrary additive constants. Bernoulli’s equation (2.13) with Ω = 0 yields p 1 2 (u + v 2 ) + = constant 2 ρ

(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 free surface. It follows from (3.1) that u2 + v 2 = U 2

on

EF ,

(3.2)

where U is a constant. A signiﬁcant simpliﬁcation in the formulation of the problem is obtained by using φ and ψ 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 use it extensively in our studies of gravity–capillary free-surface ﬂows. The

34

Free-surface ﬂows that intersect walls

simpliﬁcation comes from the fact that the ﬂow domain is mapped into the strip 0 < ψ < 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 ψ = Q in the f -plane. Since u − iv is an analytic function 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 . ψ

C

D

E

ψ=Q

F

ψ=0

A

φ

B

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

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 methods have been derived to calculate them. The method we now choose to describe 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 Sections 3.2–3.4 to solve numerically problems with gravity and surface tension included. In the absence of gravity and surface tension, the ﬂow approaches a uniform stream of constant depth H as x → ∞. It follows from the dynamic boundary condition (3.2) that this uniform stream is characterised by a constant velocity U . Since ψ = 0 on AB and ψ = Q on EF , H = Q/U . We deﬁne the logarithmic hodograph variable τ − iθ by the relation w = u − iv = eτ −iθ .

(3.3)

The function τ − iθ has some interesting properties. First, the quantity τ = 12 ln(u2 +v 2 ) is constant along free streamlines (see (3.2)). Second, θ can be interpreted as the angle between the vector velocity and the horizontal.

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, θ 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=−

∂θ ∂φ ∂θ ∂ψ − . ∂φ ∂s ∂ψ ∂s

(3.4)

Along a streamline ψ is constant and therefore ∂ψ = 0 and ∂s

∂φ = eτ . ∂s

(3.5)

Subsituting (3.5) into (3.4) yields the simple formula K = −eτ

∂θ . ∂φ

(3.6)

We now introduce dimensionless variables by using U as the reference velocity and H as the reference length. Therefore ψ = 1 on the walls CD and DE and on the free surface EF . The dynamic boundary condition (3.2) becomes u2 + v 2 = 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 e−πf =

(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 = eiσ ,

0 < σ < π,

(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 σ φ = − ln sin2 π 2

on ψ = 1.

(3.10)

As σ varies from 0 to π, φ varies from ∞ to 0, so that (3.9) is the image of the free surface in the t-plane.

36

Free-surface ﬂows that intersect walls

E

D

F B

C A

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 the series ∞ an tn . (3.11) w= n=0

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)

∞

an tn ,

(3.12)

n=0

where the function G(t) contains all the singularities of w. As we shall see in Sections 3.2–3.4, this type of series representation enables the accurate calculation of many free-surface ﬂows with gravity and surface tension included. For the present problem we require G(t) to behave like w as t → 0 and 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 singularities 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 γ (see Figure 3.6). We introduce cartesian coordinates with the origin at the apex G of the corner. We choose ψ = 0 on the streamline HGL and φ = 0 at x = y = 0. Assuming without loss of generality that the ﬂow is in the direction of the

3.1 Free streamline solutions

37

y

H

γ G

L

x

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

arrow, we have φ < 0 along the wall HG, φ > 0 along the wall GL and ψ > 0 in the ﬂow domain. The ﬂow conﬁguration in the complex potential plane is shown in Figure 3.7. ψ

G

H

L

φ

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

We seek a solution of the form z = Aeiα f µ ,

(3.13)

where A > 0, µ and α are real constants. On the wall GL (where φ > 0), the kinematic boundary condition can be written as arg z = 0. Therefore (3.13) implies that α = 0.

(3.14)

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

38

Free-surface ﬂows that intersect walls

written as arg z = γ. Writing φ = eiπ |φ| and using (3.13), we ﬁnd that α + πµ = γ.

(3.15)

Relations (3.14) and (3.15) imply that µ=

γ ; π

(3.16)

therefore (3.13) gives z = Af γ/π . Since

w=

dz df

(3.17)

−1 (3.18)

we obtain the formula w=

π 1−γ/π f Aγ

(3.19)

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

π −π/γ π/γ−1 A z . γ

(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 the formulae (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 α is then diﬀerent from zero. The velocity at the point G is equal to zero when γ < π and is unbounded when γ > π see (3.20). We will refer to the ﬂow of Figure 3.6 as a ﬂow inside a corner when γ < π and as a ﬂow around a corner when γ > π. For the ﬂow of Figure 3.1, γ = π − γ2 + γ1 and (3.19) implies w = O[(f − φD − i)(γ2 −γ1 )/π) ]

as

f → φD + i,

(3.21)

where φD is the value of φ 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) →A g(x)

as

x → x0 ,

(3.23)

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) → 0 as g(x)

x → x0 .

(3.25)

Using (3.8) yields f − φD − i = O[t − d] as

f → φD + i.

(3.26)

Combining (3.21) and (3.26) gives w = O[(t − d)(γ2 −γ1 )/π) ] 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

x → −∞

as

(3.28)

where B is a positive constant. Diﬀerentiating (3.28) with respect to z gives w=

df B =− . dz z

(3.29)

Since the ﬂux of the ﬂuid coming from −∞ is 1 and the angle between the walls CD and AB is γ1 , we have B=

1 . γ1

(3.30)

Eliminating z between (3.28) and (3.29) gives as

f → −∞,

(3.31)

eπf = O(t) as

f → −∞.

(3.32)

t → 0.

(3.33)

G(t) = (t − d)(γ2 −γ1 )/π tγ1 /π

(3.34)

w = O[eγ1 f ] and relation (3.8) then implies that

Therefore (3.31) and (3.32) give w = O(tγ1 /π ) as Combining (3.27) and (3.33), we can choose

40

Free-surface ﬂows that intersect walls

and write (3.12) as (γ 2 −γ1 )/π γ1 /π

w = (t − d)

t

∞

an tn .

(3.35)

n=0

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 truncating 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 when solving problems where the eﬀects of gravity or surface tension are included in the dynamic boundary condition. However, it can checked that (γ2 −γ1 )/π ∞ 1 n an t = (3.36) 1 − td n=0

and therefore the present problem has the exact solution t − d (γ2 −γ1 )/π γ1 /π w= t . 1 − td

(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] or Gurewich [69] for details. The free-surface proﬁle is obtained by setting ψ = 1 in (3.8) and (3.37), calculating the partial derivatives xφ and yφ from the identity xφ + iyφ =

1 w

(3.38)

and integrating with respect to φ. As a ﬁrst example let us assume that γ1 = γ2 = π/2 (see Figure 3.2). Then (3.37) reduces to w = t1/2

(3.39)

and (3.9), (3.38) and (3.39) yield xφ + iyφ = e−iσ/2 ,

0 < σ < π,

(3.40)

along the free surface EF . Diﬀerentiating (3.10) with respect to σ and applying the chain rule to (3.40) gives 1 σ xσ + iyσ = − cotan e−iσ/2 . π 2

(3.41)

3.1 Free streamline solutions

41

Integrating (3.41) with respect to σ and taking the real and imaginary parts gives 2 σ σ x = cotan + − 1 (3.42) π 2 π σ 2 sin + 1. (3.43) π 2 Relations (3.42) and (3.43) deﬁne the free-surface proﬁle in parametric form. It is shown in Figure 3.8. y=

2.0

1.5

1.0

0.5

0

0

2

4

6

8

Fig. 3.8. Free-surface proﬁle for the ﬂow conﬁguration of Figure 3.2. The position of the separation point E is indicated by a small horizontal line. The vertical scale has 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 σ = π and σ = 0, we obtain π ≈ 0.611. (3.44) Cc = π+2 As a second example, let us assume γ1 = 0 and γ2 = π/2 (see Figure 3.9). Then (3.37) becomes t − d 1/2 . (3.45) w= 1 − td Proceeding as in the previous example, we obtain 1/2 1 σ 1 − eiσ d xσ + iyσ = − cotan (3.46) π 2 eiσ − d on the free surface EF .

42

Free-surface ﬂows that intersect walls C

D

E F

B

A

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

Integrating (3.46) gives x and y on the free surface as functions of σ. There is a solution for each value of −1 < d < 0; the parameter d measures the 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 yφ for −1 < t < d by using (3.38) and (3.45). We then evaluate yt for −1 < t < d by using (3.8) and the chain rule. The length 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. 1.6

1.2

0.8

0.4

0

0

1

2

3

4

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

3.1 Free streamline solutions

43

As d → 0, the length of the vertical wall DE tends to inﬁnity and the ﬂow reduces to that of Figure 3.2. As d → −1, the length of the vertical wall DE tends to zero and the ﬂow reduces to a uniform stream. As a third example, we assume γ2 < 0 and γ1 = 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 t − d γ2 /π . (3.47) w= 1 − td As in the previous two examples we use (3.47) to calculate xσ + iyσ on the free surface. After integration we obtain the shape of the free surface in parametric form. A typical free-surface proﬁle for d = −0.2 and γ2 = −π/3 is shown in Figure 3.11. 1.0

0.8

0.6

0.4

0.2

0

0

1

2

3

4

Fig. 3.11. Computed free-surface proﬁle for the ﬂow conﬁguration of Figure 3.3 with d = −0.2 and γ2 = −π/3. The position of the separation point E is indicated by a small horizontal line. The vertical scale has been exaggerated to show clearly the 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 refer to this situation as forced separation. On the other hand, if the inﬁnitely

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. D C

E F A

B

Fig. 3.12. The ﬂow conﬁguration of Figure 3.9 but with the vertical wall DE replaced 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. D C

E F A

B

Fig. 3.13. The ﬂow conﬁguration of Figure 3.12 when the smooth curve is cut by a vertical 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 1/2 1/2 y˜ x ˜ + = 1. (3.48) ˜b a ˜

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. C

D

E F A

B

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 can be 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 we shall 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 be clearly 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 horizontal 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 direction (see Figure 3.16). The angle between the free surface and the circle at the

46

Free-surface ﬂows that intersect walls

separation points is denoted by β. For free streamline solutions β = 0. This follows from (3.20) with γ = β, which shows that a value β = 0 would generate a zero or an inﬁnite velocity at the separation points. This would contradict (3.2). However, we shall see in Sections 3.2 and 3.4 that values β = 0 can occur when surface tension is taken into account. G y B

E

C

x

A

D

Fig. 3.16. The cavitating ﬂow past a circle in an unbounded ﬂuid domain. When the surface tension T is zero, the free surfaces leave the circle tangentially and β = 0. When T = 0, the angle β can be diﬀerent from zero.

We deﬁne dimensionless variables by using the radius R of the circle as the reference length and the constant velocity U far upstream as the reference velocity. We introduce the potential function bφ, the streamfunction bψ and the complex potential f = bφ + ibψ. Without loss of generality we may choose φ = 0 at the point C and ψ = 0 on the streamlines ECAD and ECBG. Here and in the remaining part of this section, the letters E, C, B, G, A and D refer to Figure 3.16. The constant b is deﬁned so that φ = 1 at the separation points A and B. The ﬂow conﬁguration in the complex potential plane is illustrated in Figure 3.17. We introduce the complex velocity u − iv and deﬁne the function τ − iθ by the relation (3.3). Using (3.6), we have K=−

eτ ∂θ . b ∂φ

(3.49)

We shall seek τ − iθ as an analytic function of φ + iψ in the half-plane ψ < 0 (see Figure 3.17). The solution in ψ > 0 can then be obtained by symmetry. The boundary conditions on ψ = 0 are given by θ=0

on ψ = 0,

−∞ < φ < 0,

(3.50)

3.1 Free streamline solutions

C

E

47

B

G

+1 A

D

Fig. 3.17. The ﬂow of Figure 3.16 in the complex potential plane.

eτ ∂θ = 1 on ψ = 0, b ∂φ τ =0

on ψ = 0,

0 < φ < 1,

1 < φ < ∞.

(3.51) (3.52)

The condition (3.50) follows from symmetry. Equation (3.51) follows from (3.49) and the fact that the curvature of the rigid boundary ACB is 1. Relation (3.52) is the dynamic boundary condition rewritten in terms of τ . This completes the formulation of the problem. We seek τ − iθ as an analytic function of φ + iψ in ψ < 0 satisfying (3.50)–(3.52). We will solve the problem by following the series truncation method introduced in Section 3.1.1 (see (3.12)). First we map the ﬂow domain into the unit circle in the complex t-plane by the transformation 1 1 1/2 . (3.53) = t− f t 2i The ﬂow conﬁguration in the t-plane in shown in Figure 3.18. The rigid surface ACB is mapped onto the circle |t| = 1 and the free surfaces AD and BG are mapped onto the imaginary axis. The conditions (3.50)–(3.52) become θ=0 eτ ∂θ =1 b ∂φ τ =0

on

0 < t < 1,

on t = eiσ , on t = ir,

−π/2 < σ < 0, −1 < r < 0.

(3.54) (3.55) (3.56)

Here we have described the unit circle |t| = 1 by t = eiσ , where σ is a real parameter. Following Brodetsky [23] and Vanden-Broeck [160] we represent τ − iθ by

48

Free-surface ﬂows that intersect walls B

C

G E D

A

Fig. 3.18. The ﬂow of Figure 3.16 in the complex t-plane.

an expansion, as follows: ∞

τ − iθ = − ln

1+t − Bn tn . 1−t

(3.57)

n=0

The derivation of (3.57) follows that leading to (3.12). There is a singularity at the point C where locally we have a ﬂow inside a right angle corner (see Figure 3.16). Therefore, (3.19) yields u − iv ∼ f 1/2

as

f → 0.

(3.58)

Using (3.3) and (3.53) yields τ − iθ ∼ ln(1 − t) as t → 1. Thus τ − iθ + ln

1+t 1−t

(3.59)

is not singular and can be represented in the unit circle of the t-plane by a Taylor expansion. This leads to (3.57). One might argue that other singularities occur at the separation points A and B. However, these singularities are automatically taken into account by (3.53). We note that (3.57) implies

∞ Bn tn , (3.60) u − iv = G(t) exp − n=0

where G(t) =

1−t . 1+t

(3.61)

3.1 Free streamline solutions

49

Therefore (3.60) is similar to (3.12). The only diﬀerence is that the series has been rewritten as the exponential of a series. It can easily be checked that (3.54) and (3.56) are satisﬁed by assuming that the coeﬃcients Bn are real and that Bn = 0 when n is even. Therefore we can rewrite (3.57) as ∞

τ − iθ = −ln

1+t An t2n−1 . + 1−t

(3.62)

n=1

We now determine coeﬃcients An such that (3.55) is satisﬁed. This is done numerically by series truncation and collocation. Thus we truncate the inﬁnite series in (3.62) after N terms, i.e. we write 1+t An t2n−1 . + τ − iθ ≈ −ln 1−t N

(3.63)

n=1

Next, we satisfy (3.55) at the mesh points σ = σI , where σI = −

π I, 2N

I = 1, 2, . . . , N.

(3.64)

This is achieved by using (3.62) to evaluate the values of τ , θ and ∂θ/∂φ at the mesh points (3.64) and substituting these values into (3.55). This leads to a system of N equations for the N + 1 unknowns An , n = 1, 2, . . . , N , and b. The last equation is obtained by ﬁxing the position of the separation point A. This is done by imposing π (3.65) θ(σN ) = γ¯ − , 2 where the angle γ¯ is deﬁned in Figure 3.16. The system of N + 1 nonlinear algebraic equations with N + 1 unknowns needs to be solved numerically by iteration. In most problems considered in this book, this is done by Newton’s method. This method can be described as follows. Assume that we want to solve a system of M nonlinear algebraic equations fi (x1 , x2 , . . . , xM ) = 0,

i = 1, 2, . . . , M, (n)

(n)

xj −

(n) xj

(3.66)

(n)

with M unknowns x1 , x2 , . . . , xM . Let (x1 , x2 , . . . , xM ) be the approximation of the solution at iteration n. Then we linearise the left-hand side of (3.66) around this iteration as (n) (n) (n) fi (x1 , x2 , . . . , xM )

+

M j=1

∂fi ∂xj

(n) .

(3.67)

50

Free-surface ﬂows that intersect walls (n+1)

(n+1)

(n+1)

The next approximation, (x1 , x2 , . . . , xM ), is obtained by equating (3.67) to zero and solving the resulting linear system for x1 , x2 , . . . , xM . Each iteration is expensive since it requires solving a linear system of equations. However, the iterations usually converge quadratically so that only a few iterations are needed to obtain an accurate solution. The method (0) (0) (0) also requires an initial guess (x1 , x2 , . . . , xM ) to start the iterative process. When facing a problem with several solutions, the solution obtained after convergence will depend on the initial guess chosen. The matrix with elements ∂fi (3.68) ∂xj is called the Jacobian matrix; an attractive feature of Newton’s method is that bifurcations from branches of solutions can be found by monitoring the sign of its determinant. This is a consequence of the fact that the determinant vanishes at a bifurcation point (Keller [84]). The free-surface proﬁles are then obtained by integrating numerically the identity ∂y 1 ∂x +i (3.69) = e−τ +iθ . b ∂φ ∂φ The numerical results can be described in terms of the angle γ¯ (see Figure 3.16). Solutions can be obtained for all values 0 < γ¯ < π. However, only the solutions for γ ∗ < γ¯ < γ ∗∗ , where γ ∗ ≈ 55◦ and γ ∗∗ ≈ 124◦ , have a physical meaning for cavitating ﬂow past a circle. For γ¯ < γ ∗ the solutions are not acceptable because then the free surfaces would enter the body (see the solution for γ¯ = 25◦ in Figure 3.19). They are nevertheless useful in describing the cavitating ﬂow past the body obtained by cutting the circle along the straight line AB in Figure 3.16 and retaining only the portion on the left of AB. This cutting of the circle is similar to the cutting seen in Figure 3.13. For γ¯ > γ ∗∗ the solutions are not acceptable because the free surfaces cross each other (see the solution for γ¯ = 150◦ in Figure 3.19). The last acceptable solution, at γ¯ = γ ∗∗ , has free surfaces that approach the x-axis asymptotically as x → ∞ (see Figure 3.20). Physically acceptable solutions for γ¯ > γ ∗∗ can be obtained by considering cusped cavities. Cusped cavities were introduced numerically by Southwell and Vaisey [142] and analytically by Lighthill [98] and [99]. They will be calculated numerically by a boundary integral equation method in the next section.

3.1 Free streamline solutions

51

1.5

1.0

0.5

0

0

1

2

3

4

5

Fig. 3.19. Computed free-surface proﬁles for γ¯ = 25◦ , γ¯ = γ ∗ and γ¯ = 150◦ .

3

2

1

0

0

2

4

6

8

10

Fig. 3.20. The cavitating ﬂow corresponding to γ¯ = γ ∗∗ ≈ 124◦ .

3.1.2.2 Cusped cavities Unwanted intersections of free surfaces, such as those described above for γ¯ > γ ∗∗ , occur in many applications. A classical example is the exact solution of Crapper [37] for nonlinear capillary waves travelling at a constant

52

Free-surface ﬂows that intersect walls

velocity at the surface of a ﬂuid of inﬁnite depth (see Section 6.5.1 and Figures 6.8–6.11). Crapper’s solutions form a one-parameter family of solutions. The parameter can be chosen as the steepness s of the waves (i.e. the diﬀerence in height of the crests and the troughs divided by the wavelength). For small values of s, the waves are close to linear sine waves (see Figure 6.8). As s increases the waves develop rounded crests and sharp troughs (see Figure 6.9). When s reaches the critical value s∗ ≈ 0.73, the free surface develops a point of contact with itself and a small trapped bubble forms at the trough of the wave (see Figure 6.10). For s > s∗ , the free surface is self-intersecting and the solutions lose their physical meaning (see Figure 6.11). Vanden-Broeck and Keller [185] showed that physically acceptable solutions for s > s∗ can be obtained by preventing the free surface from self-intersecting. The resulting free-surface proﬁles for s > s∗ have trapped bubbles at the troughs, as in Crapper’s solution for s = s∗ . Since preventing self-intersection imposes an extra constraint on the solutions, an extra unknown is needed. This is provided by the pressure in the trapped bubble, which is found as part of the solution. The calculations of Vanden-Broeck and Keller [185] will be described in Section 6.5.1. Here we use a similar approach to ﬁnd physically acceptable cavitating ﬂows for γ¯ > γ ∗∗ , by preventing the crossing of the streamlines and seeking a family of cusped cavities (see Figure 3.21). y

B

E

C

γ

M

G

L

D

x

A

Fig. 3.21. Flow past a circle giving rise to a cusped cavity.

As we shall see there is a cusped cavity for each value of γ¯ > γ ∗∗ . These solutions approach the solution in Figure 3.20 as γ¯ → γ ∗∗ . In other words the x-coordinate of the cusp in Figure 3.21 tends to ∞ as γ¯ → γ ∗∗ and the corresponding solution approaches that of Figure 3.20. As γ¯ → 180◦ , the x-coordinate of the cusp tends to 2 and the cavity collapses to a point.

3.1 Free streamline solutions

53

Following the work of Vanden-Broeck and Keller [185], as mentioned above we need to identify a new unknown to prevent the intersection of the free streamlines. A natural choice is the pressure pc in the cavity. This is motivated by the fact that cusped cavities are closed (they do not extend to inﬁnity as do the open cavities of Figure 3.19) and so we do not have to require that pc = pb . Therefore our dynamic boundary condition on the free surfaces AL and BM of Figure 3.21 is τ=

1 ln(1 + C), 2

(3.70)

where the cavitation number C is found as part of the solution. We deﬁne the potential function bφ and the streamfunction bψ and choose b so that φ = 1 at the separation points B and A. The ﬂow conﬁguration in the complex (φ, ψ)-plane is illustrated in Figure 3.22.

B E

C

+1 A

G D

Fig. 3.22. The ﬂow of Figure 3.21 in the complex potential plane.

We solve the problem by a boundary integral equation method. This technique will be used extensively in the remaining part of the book. The basic idea is to reformulate the problem as a system of integro-diﬀerential equations that involves only unknowns on the boundary of the ﬂow domain. This system is then discretised and the resultant algebraic equations are solved by iteration (usually Newton iteration). The obvious advantage is that mesh points are only needed on the boundary rather than in the whole ﬂow domain. In other words the two-dimensional ﬂow problem of Figure 3.21 is reduced to a one-dimensional problem on the boundary ECALD. As we shall see in Chapter 10, boundary integral equation methods can also be used for solving fully three-dimensional problems. There the three-dimensional problem is reduced to a two-dimensional problem on the boundary. A convenient way of deriving the system of integro-diﬀerential equations for two-dimensional ﬂows is to use the Cauchy’s integral equation formula (see (2.38)–(2.41)). An alternative way which does not rely on complex variables is to use Green’s theorem and Green’s functions. For

54

Free-surface ﬂows that intersect walls

three-dimensional problems, complex variables are not available and Green’s theorem and Green’s functions are the only way to derive the system of integro-diﬀerential equations. We can derive such a system for the problem of Figure 3.21 by applying the Cauchy integral equation formula in the (φ, ψ)-plane of Figure 3.22 to the function τ (φ, ψ) − iθ(φ, ψ) with a contour consisting of the axis ψ = 0 and a semicircle in ψ < 0 centred on φ = ψ = 0 and of arbitrary large radius. Since τ (φ, ψ) − iθ(φ, ψ) → 0 as ψ → −∞, there is no contribution from the semicircle and we obtain ∞ τ (ϕ, 0) − iθ(ϕ, 0) 1 dϕ when ψ < 0. (3.71) τ (φ, ψ) − iθ(φ, ψ) = − 2iπ −∞ ϕ − φ − iψ (see (2.40)). On the free surface, (2.41) gives 1 ∞ τ (ϕ, 0) − iθ(ϕ, 0) dϕ. τ (φ, 0) − iθ(φ, 0) = − iπ −∞ ϕ−φ

(3.72)

The integral in (3.72) is a Cauchy principal value. Taking the real and imaginary parts of (3.72) gives 1 ∞ θ(ϕ, 0) dϕ, (3.73) τ (φ, 0)) = π −∞ ϕ − φ 1 ∞ τ (ϕ, 0) θ(φ, 0) = − dϕ. (3.74) π −∞ ϕ − φ Relations (3.73) and (3.74) are known as Hilbert transforms. It can be shown that one implies the other. Therefore we are free to choose either (3.73) or (3.74). It turns out that (3.73) is the better choice because it leads to a relation between τ and θ on the portion CAL of the streamline ψ = 0. This follows from the fact that θ = 0 on EC and on LD. Therefore (3.73) simpliﬁes to 1 l θ(ϕ, 0) dϕ. (3.75) τ (φ, 0) = π 0 ϕ−φ Here l is the value of φ at the cusp L. If we restrict the values of φ in (3.75) to 0 < φ < l, then (3.75) is a relation between τ and θ on CAL. The kinematic boundary condition on CA and the dynamic boundary condition (3.70) imply that eτ ∂θ = 1, b ∂φ τ=

0 < φ < 1,

1 ln(1 + C), 2

1 < φ < l.

(3.76) (3.77)

3.1 Free streamline solutions

Finally, we impose y = 0 at the cusp by writing l e−τ (ϕ,0) sin θ(ϕ, 0)dϕ = 0.

55

(3.78)

0

This completes the reformulation of the problem as a system of nonlinear integro-diﬀerential equations. We seek τ (φ, 0) and θ(φ, 0) such that (3.75)– (3.78) are satisﬁed. Once τ (φ, 0) and θ(φ, 0) are known for 0 < φ < l, then the shape of the cusped cavity and the velocity ﬁeld in the ﬂow domain can be calculated by integration in the following way. First the shape of the cavity is obtained in the parametric form x(φ, 0), y(φ, 0) by integrating the identity 1 (3.79) = e−τ (φ,0)+iθ(φ,0) . xφ + iyφ = u − iv Next, τ (φ, 0) for φ < 0 and φ > l can be calculated from (3.75). The values of τ (φ, 0) and θ(φ, 0) are then known for all −∞ < φ < ∞. Substituting these values in (3.71), we can evaluate by integration τ (φ, ψ) and θ(φ, ψ) everywhere in the ﬂow domain. The velocity ﬁeld is then given by (3.3). We will solve the problem numerically. First we deﬁne the mesh points φI =

I −1 , N −1

I = 1, . . . , M,

(3.80)

I = 1, . . . , M,

(3.81)

and the corresponding unknowns θI = θ(φI , 0),

where M and N are positive integers and l = (M − 1)/(N − 1). Since l > 1, we require M > N . We also use the midpoints φm I =

φI + φI+1 , 2

I = 1, . . . , M − 1.

(3.82)

We calculate τ (φm I ) in terms of the unknowns (3.81) by applying the trapezoidal rule to the integral in (3.75) and summing over the points (3.80). We justify this discretisation by showing that the symmetry of the quadrature and of the distribution of mesh points enable us to calculate the Cauchy principal value as if it were an ordinary integral. First we rewrite the integral on the right-hand side of (3.75) (evaluated at φm I ) as φI φI + 1 l θ(ϕ, 0) θ(ϕ, 0) θ(ϕ, 0) dϕ + dϕ + (3.83) m m m dϕ. ϕ − φI ϕ − φI 0 φI φ I + 1 ϕ − φI The ﬁrst and third integrals in (3.83) are ordinary integrals and can therefore

56

Free-surface ﬂows that intersect walls

be evaluated by the trapezoidal rule. The second integral in (3.83) is a Cauchy principal value, which we rewrite as

φI + 1

φI

θ(ϕ, 0) − θ(φm I , 0) dϕ + θ(φm I , 0) m ϕ − φI

φI + 1

φI

dϕ . ϕ − φm I

(3.84)

The second integral in (3.84) is also a Cauchy principal value. Simple integration shows that its value is zero. The ﬁrst integral in (3.84) is an ordinary integral and can be evaluated by the trapezoidal rule as θI − θ(φm θI+1 − θ(φm h h θI+1 θI I , 0) h I , 0) h + = + . (3.85) m m m m φI − φI 2 φI+1 − φI 2 φI+1 − φI 2 φI − φI 2 The right-hand side of (3.85) is just the integral

φI + 1

φI

θ(ϕ, 0) dϕ ϕ − φm I

evaluated by the trapezoidal rule. Therefore the Cauchy principal value on the right-hand side of (3.75) can be evaluated by the trapezoidal rule as if it were an ordinary integral. This approach to evaluating Cauchy principal values will be used often in this book. We note that the derivation (3.83)– (3.85) can easily be extended to other integration formulae such as Simpson’s rule or for mesh points φM I that are not midpoints. The only diﬀerences are that the second integral in (3.84) might not be zero and that the left-hand side of (3.85) should be used instead of the right-hand side. We now return to our problem and satisfy (3.76) at the mesh points φm I , , I = N, . . . , M − 2. The I = 2, . . . , N − 1, and (3.77) at the mesh points φm I last three equations are given by (3.78) and by the geometric conditions π θ1 = − , 2

θM = 0.

(3.86)

This system of algebraic equations is solved by Newton’s method. Typical free-surface proﬁles are shown in Figure 3.23. It can be seen that as γ¯ → γ ∗∗ , C → 0. We note that the numerical procedure presented here is not restricted to a circular obstacle and can be generalised to ones of arbitrary shapes in the following way. First we denote by F (x, y) = 0 the equation of the rigid boundary CA (see Figure 3.21) and calculate x and y on CA by the formulae x(φ, 0) = 0

φ

e−τ (ϕ,0) cos θ(ϕ, 0)dϕ

(3.87)

3.1 Free streamline solutions

57

3

2

1

0

0

2

4

6

8

10

Fig. 3.23. Three computed cusped cavities. The cavitation numbers C from the smallest cavity to the largest are −0.55, −0.29 and −0.1 respectively.

and y(φ, 0) =

φ

e−τ (ϕ,0) sin θ(ϕ, 0)dϕ.

(3.88)

0

We then apply the numerical procedure described above, the equations obtained by satisfying (3.76) at the mesh points φm I , I = 2, . . . , N − 1, being replaced by the new equations m F [x(φm I , 0), y(φI , 0)] = 0,

I = 2, . . . , N − 1,

(3.89)

where x(φ, 0) and y(φ, 0) are deﬁned by (3.87) and (3.88). The solutions derived in this section are examples of cavitating ﬂows with C < 0. Such cavities were considered analytically by Lighthill [98], [99]. Batchelor [8] notes that such cavities have not been observed, perhaps because the boundary layer at the rigid surface would separate before reaching the low-velocity region where the free streamlines begin. Before concluding this section, let us mention that there are many cavity models with C > 0 (the Riabouchinsky model, the re-entrant jet model, the Roskho model etc). The reader interested in these models is referred to the books of Birkhoﬀ and Zarantonello [19] and Gurevich [69].

58

Free-surface ﬂows that intersect walls

3.2 The eﬀects of surface tension In this section we will investigate the eﬀects of the surface tension T on the free streamline solutions of Section 3.1. We show that the limit T → 0 is singular. When T = 0, discontinuities in slope can appear at the separation points. In particular, values of β = 0 can occur in Figure 3.16. We shall also show that the limit T → 0 can be used to select solutions. 3.2.1 Forced separation We start our study by investigating the local behaviour of the ﬂow of Figure 3.1 near the separation point E, in the absence of surface tension. For simplicity we assume γ1 = γ2 = π/2, i.e. we consider the ﬂow shown in Figure 3.2. The point E corresponds to t = −1, ψ = 1 and φ = 0. Using (3.8) we ﬁnd 1 (3.90) φ ≈ − (t + 1)2 as t → −1. 4π Relation (3.39) gives i w ≈ i − (t + 1) as 2

t → −1.

(3.91)

Furthermore (3.3) gives w ≈i+θ+

π 2

as

t → −1.

Combining (3.90)–(3.92), we obtain π θ ≈ − + (πφ)1/2 . 2

(3.92)

(3.93)

Since eτ = 1 at E, (3.6) implies that K ≈ −Sφ−1/2

as

φ → 0,

(3.94)

where 1 S = π 1/2 . 2

(3.95)

Therefore the ﬂow leaves the wall DE tangentially (see (3.93)) but the curvature of the free surface at E is unbounded (see (3.94)). It can be shown that (3.94) holds for γ1 = π/2 and γ2 = π/2. Of course, the value of S depends on γ1 and γ2 . These free streamline results show that an inﬁnite curvature can occur at the separation points. This singularity does not invalidate the free streamline theory because the curvature does not appear explicitly in the equations

3.2 The eﬀects of surface tension

59

and does not have a direct physical meaning. However, when surface tension is taken into account, the condition p = pa on the free surface is replaced by p = pa + T K,

(3.96)

where T is the surface tension and K is the curvature of the free surface (see (2.19)). It follows from (3.1) and (3.96) that the dynamic boundary condition on the free surface becomes 1 2 T (u + v 2 ) + K = constant. (3.97) 2 ρ Equation (3.97) shows that an inﬁnite curvature at the separation point E implies an inﬁnite velocity at E. This implies that solutions with T = 0 are qualitatively diﬀerent from the solutions with T = 0 of Section 3.1.1. It also suggests that the limit T → 0 is a singular limit. These two properties are conﬁrmed by the calculations below. Ackerberg [2], Cumberbatch and Norbury [39], Vanden-Broeck ([159], [160], [163], [164], [176]) and others studied the ﬂow conﬁguration of Figure 3.2 (and related free-surface ﬂows) in the limit as T → 0. The results of Vanden-Broeck showed that the inclusion of surface tension in the free streamline ﬂows of Section 3.1.1 does not remove the inﬁnite curvature at the separation points. On the contrary, it makes the ﬂow more singular by introducing a discontinuity in slope at the separation points. In other words there is an angle β = 0 between the tangent to the free surface at the separation point and the wall (see Figure 3.24). C

E β

A

F

B

Fig. 3.24. The ﬂow under a gate with surface tension included in the dynamic boundary condition. The free surface does not leave the gate tangentially: there is an angle β = 0 between the free surface and the gate at the separation point E.

This angle β is a function of the surface tension. We now demonstrate

60

Free-surface ﬂows that intersect walls

these ﬁndings by presenting asymptotic results, for T small. We will present later fully nonlinear computations for arbitrary values of T . We assume that the ﬂow in Figure 3.24 is characterised by a uniform stream with constant velocity U as x → ∞. As in Section 3.1.1, we deﬁne dimensionless variables by taking U as the unit velocity and H = Q/U (for a deﬁnition of Q see Figure 3.4) as the unit length. The dynamic boundary condition (3.97) in dimensionless form is then 1 2 1 (u + v 2 ) + K = constant, 2 α

(3.98)

where α=

ρU 2 H . T

(3.99)

Since u2 + v 2 → 1 and K → 0 as φ → ∞, the constant on the right-hand side of (3.98) is equal to 1/2. Using (3.3) and (3.6), we rewrite (3.98) as 1 1 2τ 1 ∂θ e − eτ = . 2 α ∂φ 2

(3.100)

If we assume φ = 0 at the separation point then the free streamline solution (i.e. the solution for α = ∞) for the conﬁguration of Figure 3.1 can be described near the separation point by θ ≈ θ0 − Cφ1/2

as

φ → 0,

(3.101)

where C is a constant. Here θ0 is the value of θ at the separation point when α = ∞. For example, for the ﬂow of Figure 3.24, (3.93) shows that C = −π 1/2

and θ0 = −π/2.

(3.102)

Relation (3.101) implies that the curvature of the free surface near E behaves like ∂θ 1 ≈ Cφ−1/2 as φ → 0. K = −eτ (3.103) ∂φ 2 Therefore the curvature of the free surface is unbounded at the separation point E unless C = 0. Following Ackerberg [2] we introduce the following scaling of the variables: f ∗ = αf,

(3.104)

τ ∗ − iθ∗ = α1/2 (τ − iθ + iθ0 ).

(3.105)

The function τ ∗ satisﬁes Laplace’s equation in ψ ∗ < 0. Thus ∂2τ ∗ ∂2τ ∗ + = 0. ∂φ∗ 2 ∂ψ ∗ 2

(3.106)

3.2 The eﬀects of surface tension

61

The kinematic and dynamic boundary conditions linearise in the limit α → ∞, so that the boundary conditions on ψ ∗ = 0 are ∂τ ∗ =0 ∂ψ ∗

on ψ ∗ = 0,

φ∗ < 0,

(3.107)

∂τ ∗ =0 ∂ψ ∗

on ψ ∗ = 0,

φ∗ > 0.

(3.108)

|f ∗ | → ∞,

(3.109)

Relation (3.101) gives the outer behaviour τ ∗ ≈ C(f ∗ )1/2

as

where indicates the imaginary part of a function. Cumberbatch and Norbury [39] showed that the solution of (3.106)–(3.108) not containing waves and having the weakest singularity at the separation point φ∗ = 0 is given on the free surface by 1 C θ∗ (φ∗ ) = Cπ 1/2 + 1/2 φ∗ ln φ∗ 2 2π τ ∗ (φ∗ ) =

1 C ln φ∗ 2π 1/2

as

as

φ∗ → 0,

φ∗ → 0.

(3.110)

(3.111)

The solution (3.110), (3.111) is not valid near φ∗ = 0 because τ ∗ is unbounded at φ∗ = 0 (an unbounded value of τ ∗ invalidates the linearisation). The asymptotic scheme can now be described as follows. For φ large we have an outer solution whose ﬁrst term is the free streamline solution (i.e. the solution without surface tension). This solution merges with the solution (3.110), (3.111) obtained for φ∗ ≈ 1, i.e. for φ ≈ α−1 . Since the solution (3.110), (3.111) becomes invalid as φ∗ → 0, we follow Vanden-Broeck [159] and seek a local solution that corresponds to a ﬂow past a corner of angle γ0 . Thus on the one hand, using (3.19) and (3.3), we can write π π/γ0 −1 φ , Aγ0

(3.112)

γ0 ln φ. τ ≈ 1− π

(3.113)

eτ = which implies that, near φ = 0,

On the other hand (3.105) and (3.111) give τ=

τ∗ 1 1 ≈ C ln φ. 1/2 2 (πα)1/2 α

(3.114)

62

Free-surface ﬂows that intersect walls

A comparison of (3.113) and (3.114) yields C π 1/2 . γ0 = π − 2 α

(3.115)

Thus we have matched the solution (3.110), (3.111) with a local solution corresponding to a ﬂow in a corner of angle γ0 (cf. Figure 3.6). The value of θ at the separation point is C π 1/2 . (3.116) θ = θ0 − 2 α If we denote by β the angle between the wall and the free surface at E (see Figure 3.24) then (3.116) implies that C π 1/2 . (3.117) β=− 2 α For the particular ﬂow of Figure 3.24, (3.102) and (3.116) yield π β≈ as α → ∞. (3.118) 2α1/2 We now present fully nonlinear solutions for arbitrary values of α. The presence of surface tension changes drastically the dynamic boundary condition and invalidates the techniques used for streamline ﬂows. Exact solutions can no longer be expected and fully nonlinear solutions have to be calculated numerically. There are, however, a few examples of exact solutions. Those will be considered in Section 6.5.1. We can calculate nonlinear solutions for the ﬂow conﬁguration of Figure 3.1 by modifying appropriately the series representation (3.35) to accommodate the singularity at t = −1. Using insight given by the asymptotic result (3.117), we assume that the ﬂow near t = −1 is a ﬂow in an angle π + β. Using (3.19) and (3.90) we obtain w ∼ f −β/π

φ→0

as

(3.119)

and w ∼ (t + 1)−2β/π

as

t → −1.

Therefore (γ 2 −γ1 )/π γ1 /π

w = (t − d)

t

−2β/π

(t + 1)

∞

(3.120)

an tn

(3.121)

n=0

is the appropriate generalisation of (3.35) when surface tension is included. The asymptotic solution (3.117) for α large suggests that β should be found as part of the solution.

3.2 The eﬀects of surface tension

63

We now present explicit calculations in the particular case γ1 = γ2 = π/2. In other words we consider the ﬂow conﬁguration of Figure 3.24. Then the expression (3.121) becomes w = t1/2 (t + 1)−2β/π

∞

an tn .

(3.122)

n=0

The dynamic boundary condition is given in dimensionless variables by (3.100), where α is deﬁned in (3.99). We truncate the inﬁnite series in (3.122) after N terms and calculate the coeﬃcients an , n = 0, . . . , N − 1, and β by satisfying (3.100) at the N + 1 equally spaced mesh points 1 π I− , I = 1, . . . , N + 1. (3.123) σI = N +1 2 This leads to a system of N + 1 equations with N + 1 unknowns, which can be solved by Newton’s method. We present numerical results in terms of the parameter αv = 2α.

(3.124)

The factor 2 in (3.124) has been introduced so that αv coincides with the parameter α used by Vanden-Broeck [163]. Typical free-surface proﬁles are shown in Figure 3.25. For αv = ∞, the free-surface proﬁle reduces to the free streamline solution of Figure 3.8. As αv → 0, the free-surface proﬁle approaches the horizontal line y = 1 (i.e. the horizontal line through the stagnation point E). This is consistent with the fact that the dynamic boundary condition (3.100) predicts that the curvature of the free surface tends to zero as αv → 0 (the straight line y = 1 has zero curvature). Numerical values of β versus αv are shown in Figure 3.26. As αv varies from 0 to ∞, β varies continuously from π/2 to 0. For αv = ∞, the dynamic boundary condition (3.100) reduces to the free streamline condition u2 +v 2 = 1. The solution is then given by β = 0;

a0 = 1,

an = 0,

n = 1, 2, . . .

(3.125)

Subsituting (3.125) into (3.122) we obtain w∞ = t1/2 .

(3.126)

This is the free streamline solution (3.39). Here the subscript ∞ refers to the case αv = ∞.

64

Free-surface ﬂows that intersect walls 1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Fig. 3.25. Computed free-surface proﬁles for the ﬂow conﬁguration of Figure 3.24. The proﬁles from top to bottom correspond to αv = ∞, αv = 50, αv = 25, αv = 10 and αv = 5; αv is deﬁned by (3.99) and (3.124). 1.6 1.4 1.2 1.0 0.8 0.6 0.4

0

5

10

15

20

Fig. 3.26. Values of the angle β between the free surface and the wall at the separation point E (see Figure 3.24) versus αv .

As αv → 0, the free surface approaches a horizontal straight line. The solution is then π β = , a0 = 2; an = 0, n = 1, 2, . . . (3.127) 2 Substituting (3.127) into (3.122) we obtain the following exact solution: w0 =

2t1/2 . t+1

(3.128)

3.2 The eﬀects of surface tension

65

The subscript 0 refers to the case αv = 0. As in Section 3.1.1, we can deﬁne the contraction ratio Cc as the ratio of the ordinate of the free surface as x → ∞ and the ordinate of the separation point E. For αv = ∞, we have π . (3.129) Cc∞ = π+2 (see (3.42)–(3.44) for an explicit derivation). For αv = 0, the free surface is the horizontal line y = 1 and the ordinates of E and of the level of the free surface as x → ∞ are equal to 1. Therefore Cc0 = 1.

(3.130)

In Figure 3.27 we present numerical values of Cc versus αv . As αv decreases from inﬁnity, the contraction ratio increases monotonically from Cc∞ to Cc0 . 1.0

0.9

0.8

0

5

10

15

20

Fig. 3.27. Values of the contraction ratio Cc versus αv for the ﬂow conﬁguration of Figure 3.24.

As αv → ∞, the values of β are described by the asymptotic formula (3.117). Combining (3.117) and (3.124) we obtain β≈

π (2αv )1/2

as

αv → ∞.

(3.131)

The numerical values in Figure 3.26 are in good agreement with (3.131) for αv large. For αv = 20, the value of β predicted by (3.131) agrees with the numerical results within two per cent.

66

Free-surface ﬂows that intersect walls

We can also construct a perturbation solution for αv small by writing τ = τ0 + αv τ1 + O(α2 ),

(3.132)

θ = θ0 + αv θ1 + O(α2 ).

(3.133)

Here τ0 and θ0 are deﬁned by τ0 − iθ0 = ln w0 ,

(3.134)

where w0 is given by (3.128). Representing the free surface EF by (3.9), we ﬁnd from (3.134) that σ (3.135) τ0 = − ln cos , θ0 = 0, 0 < σ < π, 2 on the free surface EF . Substituting (3.132) and (3.133) into (3.100) and equating coeﬃcients of αv , we obtain ∂θ1 1 sin2 (σ/2) = . ∂φ 2 cos(σ/2)

(3.136)

Using (3.10) and the chain rule we can rewrite (3.136) as dθ1 1 σ = − sin . dσ 2π 2

(3.137)

Integrating (3.137) and using the condition θ1 = 0 at σ = 0, we obtain σ 1 θ1 = cos − 1 . (3.138) π 2 In particular, (3.138) implies that θ1 = −

1 π

at

σ = π.

Combining (3.117), (3.133) and (3.139) we obtain π αv β= − as αv → 0. 2 2π

(3.139)

(3.140)

For αv = 1, the value of β predicted by (3.140) agrees with the numerical results within two per cent.

3.2.2 Free separation The behaviour (3.103) occurs for all free streamline problems near the intersection of a free surface with a rigid wall. The constant C depends on the geometry of the problem. When C < 0, as in the ﬂow of Figure 3.24, the asymptotic analysis of Section 3.2.1 for α large shows that there is locally

3.2 The eﬀects of surface tension

67

a ﬂow around a corner near the separation point E. The velocity at the separation point E is then inﬁnite. An interesting question is whether there are free streamline ﬂows for which C ≥ 0. Of particular interest are ﬂows for which C = 0 and therefore for which the singular behaviour (3.103) disappears. For C > 0, the asymptotic analysis of the previous section suggests that, when surface tension is included, the ﬂow near the separation point is a ﬂow inside an angle with a stagnation point at the separation. In this section we will show that there are free streamline ﬂows with C ≥ 0 when the rigid boundaries are curved. The consideration of such ﬂows will enable us to introduce the concept of selection, which will be very useful when we are studying gravity–capillary ﬂows. We consider the open-cavity model of Figure 3.16 but now with the eﬀect of the surface tension T included in the dynamic boundary condition. Proceeding as in Section 3.1.2 we seek τ − iθ as an analytic function of φ + iψ in the lower half-plane, ψ < 0, of the domain shown in Figure 3.17. This function must satisfy (3.50), (3.51) and α eτ ∂θ = (e2τ − 1) b ∂φ 2

on ψ = 0,

1 < φ < ∞.

(3.141)

Here α is deﬁned by α=

ρU 2 R . T

(3.142)

We start our investigation by reconsidering the solutions of Section 3.1.2 and by calculating the curvature KA of the free surface at the separation point A. Using (3.49) and (3.62), we ﬁnd that KA = −

1 1 ∂θ ≈ C(bφ − b)−1/2 b ∂φ 2

as

φ → 1,

(3.143)

where C = −b−1/2 − b−1/2

∞

(−1)n+1 (2n − 1)An .

(3.144)

n=1

Since the An are functions of γ, (3.144) deﬁnes C as a function of γ¯ . A graph of C versus the angular position γ¯ of the separation points is shown in Figure 3.28. If the angle β is counted positive when shown as in Figure 3.16, a comparison of Figures 3.24 and 3.16 shows that (3.117) implies that C π 1/2 . (3.145) β= 2 α In other words β has opposite signs in Figures 3.16 and 3.24.

This

68

Free-surface ﬂows that intersect walls

diﬀerence in sign has been maintained to be consistent with previously published results. 1.5 1.0 0.5 0

20

40

60

80

100

120

140

Fig. 3.28. Values of the constant C versus γ¯ .

The constant C vanishes when γ¯ = γ ∗ ≈ 55◦ (see Figure 3.28). Thus (3.143) shows that the curvature of the free surface at the separation points is inﬁnite unless γ¯ = γ ∗ . For γ¯ > γ ∗ , Figure 3.28 shows that C > 0 and (3.145) predicts β > 0. The ﬂow near B in Figure 3.16 is a ﬂow inside an angle with zero velocity at B. For γ¯ < γ ∗ , Figure 3.28 shows that C < 0 and the values of β predicted by (3.145) are then negative. The ﬂow near B is then a ﬂow around a corner with inﬁnite velocity at B. These results are only valid for α large. As α → 0, (3.141) shows that the curvature of the free surfaces tends to zero. Since the ﬂows are characterised by a constant velocity at inﬁnity, the free surfaces must approach two horizontal straight lines. Therefore β → γ¯ −

π 2

as

α → 0.

(3.146)

Relation (3.146) shows that β < 0 in the limit α → 0 when γ¯ < π/2. Relation (3.145) shows that β > 0 in the limit α → ∞ when γ¯ > γ ∗ . If we assume that, for a given value of γ¯ , β is a continuous function of α then there must exist for each value of α a particular value of γ ∗ < γ¯ < π/2 for which β = 0 (i.e. for which the ﬂow leaves the circle tangentially). We describe these particular values of γ¯ by the function γ¯ = g(α).

(3.147)

This conjecture is conﬁrmed by the nonlinear computations below.

In

3.2 The eﬀects of surface tension

69

particular these results show that g(α) → γ ∗

as

α → ∞.

(3.148)

This implies that the limit T → 0 can be used to select a particular solution with T = 0. In Section 3.1.2 we calculated solutions for T = 0. Then the dynamic boundary condition implies β = 0. We obtained solutions for all values of γ¯ . When T = 0, solutions with β = 0 exist only for values of γ¯ satisfying (3.147). Taking the limit α → ∞, (3.148) shows that we should select the solution corresponding to γ¯ = γ ∗ , which is known as the one satisfying the Brillouin–Villat condition (see [22], [194], [19] and [69]). This condition was introduced to select the position of the separation points in the case of free separation without surface tension. It requires the pressure to be minimal in the cavity. By Bernoulli’s equation (3.1), this is equivalent to the condition that the velocity is a maximum on the free streamlines. For the conﬁguration of Figure 3.16, the Brillouin–Villat condition yields γ¯ = γ ∗ . The above analysis shows that the selection mechanism based on the limit T → 0 provides a new physical interpretation of the Brillouin–Villat condition. We will now solve the problem numerically, calculate g(α) and demonstrate (3.148). We ﬁrst map the ﬂow of Figure 3.16 into the unit circle in the t-plane by the transformation t=

1 + if 1/2 . 1 − if 1/2

(3.149)

The ﬂow conﬁguration in the t-plane is shown in Figure 3.29. A

D E

C

Fig. 3.29. The ﬂow conﬁguration of Figure 3.16 in the t-plane deﬁned by (3.149).

Next we note that the ﬂow near A is locally a ﬂow inside a corner with angle π − β and that the ﬂow near C is a ﬂow inside a right-angle corner (see Figure 3.16). Therefore, using (3.149), we obtain u − iv ∼ (t − i)β/π

as

t → i,

(3.150)

70

Free-surface ﬂows that intersect walls

u − iv ∼ t − 1

as

t → 1.

(3.151)

Following the series truncation method of Section 3.1.2, we represent the complex velocity by u − iv = eτ −iθ = (1 − t)(1 + t2 )β/π

∞

an tn .

(3.152)

n=0

The multiplicative factors in front of the series in (3.152) remove the singularity (3.150). Therefore we can expect the series in (3.152) to converge in the unit circle of the t-plane. If we describe points on the unit circle by t = eiσ , 0 < σ < π, we can rewrite (3.141) and (3.51) as −

eτ cos3 (σ/2) dθ α = (e2τ − 1), b sin(σ/2) dσ 2 eτ cos3 (σ/2) dθ = 1, b sin(σ/2) dσ

π < σ < π, 2

0<σ<

π . 2

(3.153)

(3.154)

Coeﬃcients an in (3.152) are found such that (3.153) and (3.154) are satisﬁed. This is achieved by series truncation as in Section 3.1.2. We truncate the inﬁnite series in (3.152) after N terms and ﬁnd the N coeﬃcients a0 , a1 , . . . , aN −1 , the constant b and the angle β by collocation. Thus we introduce the N mesh points 1 π , I = 1, . . . , N. (3.155) σI = I − 2 N In order to avoid the value σ = π/2 at which the expression (3.152) is singular, we choose N to be even. Using (3.152) and (3.155) we obtain ˜ τ˜(σ) and θ(σ) at the points σI in terms of the coeﬃcients an . Substituting these expressions into (3.153) and (3.154) we obtain N equations. An extra equation is obtained by requiring the velocity to be unity at inﬁnity. This leads to τ˜(π) = 0.

(3.156)

The last equation relates γ and the angle θ at the separation point: π ˜ + β. (3.157) γ¯ − [θ(σ)] σ=π/2 = 2 This system of N + 1 equations with N + 1 unknowns can be solved by Newton’s method. We used this numerical scheme to compute solutions for various values of α and γ¯ . The coeﬃcients an were found to decrease rapidly as n increases. For example, for γ¯ = 30◦ and α = 10, a1 ≈ 0.6,

3.2 The eﬀects of surface tension

71

a10 ≈ −0.3×10−2 and a40 ≈ 0.3×10−3 . In Figure 3.30 we present numerical values of β/π versus γ¯ for α = 1. 0.10

0.05

0

60

70

80

90

100

Fig. 3.30. Values of β/π versus γ¯ (in degrees) for α = 1.

The curve in Figure 3.30 shows that for α = 1 there is exactly one value of γ¯ at which β = 0. Similar results were found for other values of α (see [176] and [183]). As α → 0, the free surfaces in ﬁgure 3.16 approach two horizontal lines. Therefore the curve corresponding to α = 0 in Figure 3.30 is the straight line (not shown in the ﬁgure) of equation β = γ¯ −

π . 2

(3.158)

For α = ∞, the angle β is equal to zero for all values of γ¯ and the curve corresponding to α = ∞ in Figure 3.30 is the horizontal line β = 0 (not shown). These results imply that, for each value of α = ∞, there is a particular value γ˜ for which β = 0 (i.e. for which the free surface leaves the obstacle tangentially). We denote these particular values of γ¯ by the function (3.147). In Figure 3.31 we present computed values of γ¯ = g(α) versus α−1 . As α → 0, γ˜ → 90◦ . As α → ∞, γ˜ → γ ∗ . Therefore the particular solution that satisﬁes the Brillouin–Villat condition in the absence of surface tension can be viewed as the limit of the family of solutions in Figure 3.31 as the surface tension approaches zero. So far we have mainly considered solutions with β = 0. It is of interest to look at solutions with β = 0. The angle β can then be interpreted as a contact angle whose value depends on the properties of the ﬂuid and of the rigid boundary. In Figure 3.32 we present values of γ¯ versus α−1 for β = 0.04π.

72

Free-surface ﬂows that intersect walls 80

70

60

50

0

0.2

0.4

0.6

0.8

1.0

Fig. 3.31. Values of γ˜ versus α−1 . The corresponding free-surface proﬁles leave the circular object tangentially, i.e. β = 0. 98 96 94 92 90 88 86 84 82 80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 3.32. Values of γ¯ versus α−1 for β = 0.04π.

This curve can be viewed as the equivalent of the curve (3.147) but with β = 0.04π instead of β = 0. One interesting property to note in Figure 3.32 is that now γ¯ is not a monotonic function of α−1 . However, only one value of γ¯ corresponds to each value of the surface tension (i.e. α−1 ). For α−1 small, γ¯ increases rapidly. This behaviour can be described by substituting β = 0.04π into (3.145) and noting that α 1/2 C = 2(0.04π) . (3.159) π

3.3 The eﬀects of gravity

73

Together with Figure 3.28, this predicts that γ¯ increases as α−1 decreases, for α large.

3.3 The eﬀects of gravity In this section we study solutions for the ﬂow conﬁguration of Figure 3.1 with surface tension neglected but with gravity included in the dynamic boundary condition. We assume that gravity acts in the direction deﬁned by the angle β1 (see Figure 3.33) and write the dynamic boundary condition as 1 2 (u + v 2 ) + gy sin β1 − gx cos β1 = B, (3.160) 2 where B is the Bernoulli constant. In this section we will look for solutions without waves on EF . Solutions with waves on EF will be studied in Chapter 8. C β1

D

g

E y A

F x

B

Fig. 3.33. The ﬂow conﬁguration of Figure 3.1 with the eﬀects of gravity included.

Two situations of particular interest are β1 = π/2 and β1 = 0. When β1 = π/2, gravity is acting in the negative y-direction. One interpretation of the ﬂow of Figure 3.33 is then the ﬂow emerging from a container with gravity included. Another is the ﬂow under a sluice gate, a classical topic in hydraulics (see [11], [54], Larock [93], Chung [30], Vanden-Broeck [181] and Binder and Vanden-Broeck [15], [16]). There are now two free surfaces CD and EF (see Figure 3.34). The model consists of replacing the upper free surface CD by a rigid lid (see Figure 3.35). The conﬁguration of Figure 3.35 is that of Figure 3.33 with γ1 = 0. An accurate numerical study of the complete free-surface ﬂow of Figure 3.34 is presented in Section 8.3.

74

Free-surface ﬂows that intersect walls D C

g

γ2

E F A

B

Fig. 3.34. A free-surface ﬂow under a sluice gate. D

C

g

γ2

E F A

B

Fig. 3.35. A model for the ﬂow under a sluice gate, in which the free surface CD has been replaced by a rigid lid.

When β1 = 0, gravity is acting in the positive x-direction. A realistic view of the ﬂow is obtained by rotating Figure 3.33 by 90◦ clockwise. If the ﬂow is then reﬂected in the wall AB, the result corresponds to a jet falling from a nozzle (see Figure 3.36). In Figures 3.33 and 3.36 we have assumed that the free surface EF leaves the wall DE tangentially. As we shall see, there are in addition solutions for which EF does not leave the wall DE tangentially. There is then an angle µ between the wall DE and the free surface EF at the separation point E (see Figure 3.37). There are only three possible values for µ. One of them is µ = π; it corresponds to the case already mentioned where the free surface leaves the wall tangentially. The existence of these three values of µ is to be contrasted with the problems including surface tension discussed in Section 3.2, where all values of µ were in principle possible, and with the free streamline problems in Section 3.1, where only the value µ = π was acceptable. The existence of the two values of µ = π can be established by deriving a local solution valid in the neighbourhood of the separation point E. The

3.3 The eﬀects of gravity A

75

C

g

γ2

D

E

B

F

Fig. 3.36. The free-surface ﬂow emerging from a nozzle. This is the ﬂow conﬁguration of Figure 3.33 rotated by 90◦ and reﬂected in the wall AB. Here γ1 = 0 and β1 = 0. C γ1

β1

D

g

γ2

µ

E F

y A

x

B

Fig. 3.37. The ﬂow of Figure 3.33 with an angle µ between the free surface and the wall at the separation point E.

analysis follows the work of Dagan and Tulin [40]. We deﬁne new local coordinates x and y with the origin at the separation point and such that gravity is acting in the negative y-direction. The local ﬂow is illustrated in Figure 3.38. Here the separation point is denoted by G, the wall by HG and the free surface by GL. It can easily be seen that the ﬂow of Figure 3.38 describes the ﬂow near E in Figure 3.37 if µ2 =

π − β 1 + γ2 . 2

(3.161)

76

Free-surface ﬂows that intersect walls

If µ = π, the free surface LG leaves the wall HG tangentially and the velocity at G is ﬁnite and diﬀerent from zero. If µ < π, the ﬂow is locally a ﬂow inside a corner and the velocity at G is zero (i.e. G is a stagnation point). If µ > π, the ﬂow is locally a ﬂow around a corner and the velocity at G is inﬁnite (see Figure 3.6 and (3.19)). Therefore values of µ > π are not possible, since (3.160) requires the velocity at G to be ﬁnite. H y

µ2

g

G

µ

x

L

Fig. 3.38. Local-gravity free-surface ﬂow near the intersection of a wall HG with a free surface GL.

We shall now determine the allowed values of µ when µ < π. We deﬁne a potential function φ and a streamfunction ψ and choose φ = ψ = 0 at the point G. The complex potential plane is shown in Figure 3.39. ψ

H

G

L

φ

Fig. 3.39. The ﬂow of Figure 3.38 in the complex potential plane. The ﬂow domain is ψ < 0.

Using (3.17) we express the local solution in the form z = Af µ/π .

(3.162)

We write the complex constant A as A = aeiα ,

(3.163)

3.3 The eﬀects of gravity

77

where a and α are real and α is such that the kinematic boundary condition on the wall HG is satisﬁed. On the wall HG, arg z = π − µ2 . However, the argument of (3.162) evaluated on ψ = 0, φ < 0, gives arg z = −µ + α. Equating the two expressions for arg z gives α = π + µ − µ2 .

(3.164)

Substituting (3.163) and (3.164) into (3.162) yields z = af µ/π ei(π+µ−µ2 ) .

(3.165)

Before satisfying the dynamic boundary condition we need to improve the local solution (3.165) by writing explicitly the next-order correction: z = af µ/π ei(π+µ−µ2 ) + bf ν eiδ + · · ·

(3.166)

Here ν, b and δ are real constants. The second term in (3.166) takes into account the deviation of the free surface from the straight line GL as φ increases. Since we require the second term in (3.166) to be a small correction to (3.165) in the limit f → 0, we impose ν>

µ . π

(3.167)

Taking the real and imaginary parts of (3.166) on ψ = 0, φ > 0, we obtain x = aφµ/π cos(π + µ − µ2 ) + bφν cos δ + · · · ,

(3.168)

y = aφµ/π sin(π + µ − µ2 ) + bφν sin δ + · · · .

(3.169)

Diﬀerentiating (3.168) and (3.169) with respect to φ yields aµ 2 aµ x2φ + yφ2 = φ2µ/π−2 + 2 bνφµ/π+ν−2 π π × cos(π + µ − µ2 − δ) + b2 ν 2 φ2ν−2 + · · · .

(3.170)

Since 2µ µ − 2 < + ν − 2 < 2ν − 2, π π

(3.171)

the last term in (3.170) is of lower order and can be neglected. We can rewrite (3.170) as aµ 2 2π 2 2 2µ/π−2 ν−µ/π φ cos(π + µ − µ2 − δ) + · · · 1+ bνφ xφ + yφ = π aµ (3.172)

78

Free-surface ﬂows that intersect walls

and express u2 + v 2 in the limit f → 0 as u2 + v 2 = =

π aµ

2

x2φ

1 + yφ2

2π bνφν−µ/π cos(π + µ − µ2 + δ) + · · · . φ2−2µ/π 1 − aµ

(3.173)

We now substitute (3.169) and (3.173) into the dynamic boundary condition (3.160). First we note that β1 = π/2 since g is acting in the negative y-direction in Figure 3.38 and that B = 0 since u = v = 0 at the point G. This gives 1 π 2 2−2µ/π φ + gaφµ/π sin(π + µ − µ2 ) + gbφν sin δ + · · · = 0. (3.174) 2 aµ We then equate the coeﬃcients of the leading-order terms in (3.174). If π + µ − µ2 = π then the ﬁrst and second terms in (3.174) give 2µ µ =2− π π and

or

µ=

2π 3

1 π2 1 a= 2 µ2 g sin(µ2 − µ − π)

(3.175)

1/3 .

(3.176)

Since µ = 2π/3, we require −π/3 < µ2 < 2π/3, for otherwise the free surface in Figure 3.38 would descend towards the stagnation point G and this would be in contradiction with the dynamic boundary condition (3.160) with β1 = π/2, which implies that a stagnation point on a free surface is the highest point on it. In the remaining part of this chapter we will assume µ2 > 0; however, solutions with µ2 < 0 will be considered in Chapter 8 (see Figure 8.5, where γ3 = −µ2 ). Furthermore, the solution with µ2 = −π/6 will be used in Section 6.5.2 to describe the singularity near the crests of the highest gravity waves. If π + µ − µ2 = π

or

µ = µ2 ,

(3.177)

the second term in (3.174) vanishes and the balance of the remaining terms gives 2µ . (3.178) ν =2− π The conditions (3.167), (3.177) and (3.178) yield µ = µ2 <

2π . 3

(3.179)

3.3 The eﬀects of gravity

79

The condition (3.177) implies that the free surface LG is horizontal. In summary we have the following possibilities. If µ2 ≤ 2π/3, there are three possible values for µ: π, 2π/3 and µ2 . However, if µ2 ≥ 2π/3 then the only possible value for µ is π. We note that on the one hand the solution (3.162) with µ = 2π/3 and A deﬁned by (3.163), (3.164) and (3.176) is an exact solution for the ﬂow conﬁguration of Figure 3.38 for all values of z inside the angle HGL. On the other hand the solution (3.162) with µ = µ2 is only a local solution in the limit z → 0. We shall solve the ﬂow problem of Figure 3.37 numerically by using a series truncation method similar to that used in Section 3.2. All the solutions constructed in this section are waveless as x → ∞. Solutions with waves as x → ∞ will be computed in Chapter 8. The ﬂow conﬁguration in the complex potential plane is shown in Figure 3.4. As in Section 3.1.1, we map the complex potential plane onto the inside of the unit circle in the t-plane by using the transformation (3.8). The ﬂow conﬁguration in the t-plane is illustrated in Figure 3.5. Next we represent the complex velocity w = u − iv by the expansion (3.12) where G(t) contains all the singularities of w in |t| ≤ 1. In this case there are two singularities: one is at the separation point E and the other at t = 1 (i.e. as x → ∞). The former corresponds to a ﬂow inside an angle µ (see Figure 3.37) and is described by w ≈ f 1−µ/π

f →0

as

(3.180)

(see (3.19)). Using (3.8) we have w ≈ (1 + t)2−2µ/π

as

t → −1.

(3.181)

The singularity at t = 1 depends on the value of β1 . It can be seen from (3.160) that u2 + v 2 ≈ 2gx cos β1

as

x→∞

when

β1 = π/2

(3.182)

as

x→∞

when

β1 = π/2.

(3.183)

and that u2 + v 2 ≈ constant

In (3.183) we have used the fact that there are no waves as x → ∞. We ﬁrst examine the case β = π/2 (see Figure 3.37). Relation (3.182) shows that u2 + v 2 → ∞ as x → ∞. Since the ﬂux between the free surface EF and the wall AB is ﬁnite (and equal to Q), it follows by conservation of mass that the free surface approaches the wall asymptotically as x → ∞.

80

Free-surface ﬂows that intersect walls

In other words the ﬂow reduces to an arbitrarily thin jet in the x-direction as x → ∞. Therefore u v as x → ∞, and (3.182) implies that u ≈ (2gx cos β1 )1/2 .

(3.184)

If we denote by y = η(x) the equation of the free surface then from the conservation of mass we have as x → ∞ uη(x) = Q.

(3.185)

Combining (3.184) and (3.185) we have η(x) ≈ Q(2gx cos β)−1/2 .

(3.186)

Writing successively φx and ψy for u in (3.184) gives ∂φ = (2gx cos β)1/2 , ∂x

(3.187)

∂ψ = (2gx cos β)1/2 . ∂y

(3.188)

Integrating (3.187) with respect to x and (3.188) with respect to y gives expressions for φ and ψ. Combining them gives 1/2 2 3/2 1/2 (3.189) x + ix y . f = φ + iψ = (2g cos β1 ) 3 We note that, for x large, iy 3/2 3y 3x1/2 y 3/2 3/2 3/2 1+ 1+i = x3/2 + i . =x ≈x z x 2x 2

(3.190)

Combining (3.189) and (3.190) gives f ∼ z 3/2

or

z ∼ f 2/3 .

(3.191)

Diﬀerentiating (3.191) with respect to z and eliminating z by using the second of the relations (3.191) yields df = u − iv ∼ z 1/2 ∼ f 1/3 . dz

(3.192)

Next we examine the case β1 = π/2 (see Figure 3.37). Then the ﬂow approaches a uniform stream with constant velocity U and constant depth H as x → ∞ (see (3.183)). The exact equations that describe the ﬂow as

3.3 The eﬀects of gravity

81

x → ∞ are, in dimensional variables, φxx + φyy = 0, 2 1 2 (φx

+

φ2y )

+ gy =

0 < y < H + η(x),

(3.193)

φy = φx ηx

on

y = H + η(x),

(3.194)

1 2 2U

on

y = H + η(x),

(3.195)

+ gH

φy = 0

on y = 0.

(3.196)

Here y = H + η(x) is the equation of the free surface. Equations (3.194) and (3.196) are the kinematic boundary conditions on the free surface and on the bottom, and equation (3.195) is the dynamic boundary condition on the free surface. We write w =U +w ˜ + ··· ,

(3.197)

φ = U x + φ˜ + · · · ,

(3.198)

η = η˜ + · · · ,

(3.199)

where w, ˜ φ˜ and η˜ are assumed to be small perturbations. We have assumed as before that the ﬂow approaches a uniform stream as x → ∞. Substituting (3.197)–(3.199) into (3.193)–(3.196) and linearising yields φ˜xx + φ˜yy = 0, U η˜x = φ˜y

on

y = H,

U φ˜x + g η˜ = 0 on y = H, φ˜y = 0

on y = 0.

(3.200) (3.201) (3.202) (3.203)

Eliminating η between (3.201) and (3.202) gives U φ˜xx +

g ˜ φy = 0 U

on y = H.

(3.204)

We use separation of variables, to ﬁnd a solution of (3.200) in the form ˜ y) = X(x)Y (y). φ(x,

(3.205)

Substituting (3.205) in (3.200) and in (3.203) yields the ordinary diﬀerential equations X (x) Y (y) =− =µ ˜2 X(x) Y (y)

(3.206)

82

Free-surface ﬂows that intersect walls

and the boundary condition Y (0) = 0.

(3.207)

Here µ ˜2 is the separation constant. The solutions of the two diﬀerential equations (3.206) satisfying (3.207) are X(x) = Be−˜µx + Ceµ˜ x ,

(3.208)

Y (y) = D cos µ ˜y,

(3.209)

where B, C and D are constants. We set C = 0, so that φ˜ remains bounded as x → ∞. Multiplying (3.208) and (3.209) yields the solution φ˜ = Ae−¯µx cos µ ¯y,

(3.210)

where A = DB is a constant. Substituting (3.210) into (3.204) yields µ ¯H =

1 tan µ ¯H, F2

(3.211)

U (gH)1/2

(3.212)

where F =

is the Froude number. We note that the derivation leading to (3.210) is similar to that leading to (2.74) and (2.75) in the theory of linear waves. The main diﬀerence is that we chose a negative separation constant in (2.72) whereas we have chosen a positive one in (3.206). Substituting (3.210) into (3.198) and diﬀerentiating with respect to x gives w=

∂φ ∂φ −i = U − A¯ µe−¯µz . ∂x ∂y

(3.213)

Next we rewrite (3.213) in terms of t. Using (3.8) we obtain, as x → ∞ or equivalently as t → 1, e−f ≈ (1 − t)2/π

as

t → 1.

(3.214)

Combining (3.213) and (3.214) and using dimensionless variables, i.e. setting U = 1 and H = 1, we have w ≈ 1 − A¯ µ(1 − t)2µ¯ /π

as

t → 1.

(3.215)

Similarly (3.211) gives in dimensionless variables µ ¯=

1 tan µ ¯. F2

(3.216)

3.3 The eﬀects of gravity

83

Relations (3.215) and (3.216) deﬁne the singular behaviour of w as t → 1 for β1 = π/2. Finally we use (3.214) to rewrite (3.192) as w ≈ [−ln(1 − t)]1/3

as

t → 1.

(3.217)

This demonstrates the singular behaviour of w as t → 1 when β = π/2. 3.3.1 Solutions with β1 = 0 (funnels) We ﬁrst consider the conﬁguration of Figure 3.37 with β1 = 0. Following the approach of Sections 3.1.1 and 3.2, we ﬁnd that the complex velocity w can be represented by the expression t − d (γ2 −γ1 )/π γ1 /π [− ln Cp (1 − t)]1/3 ¯ (3.218) t (1 + t)2−2µ/π G(t), w= 1 − td (− ln Cp )1/3 where t is deﬁned by (3.8). Here Cp is an arbitrary constant. The vari¯ ous factors appearing in the numerator of the expression multiplying G(t) remove the singularities in w at t = 0 (see (3.33)), t = 1 (see (3.217)), t = d (see (3.27)) and t = −1 (see (3.181)). The factors appearing in ¯ the denominator of the expression multiplying G(t) are not essential and the computations could have been performed without them. The function ¯ G(t) in (3.218) is then free of singularities and can be written as the Taylor expansion ∞ ¯ an tn . (3.219) G(t) = n=0

¯ There are of course alternative representations for G(t). For example we ¯ shall see in Section 3.3.2 that another convenient representation for G(t) is

∞ ¯ = exp (3.220) an tn G(t) n=0

(see also (3.60)). We choose 0 < Cp < 0.5 in (3.218), so that [− ln Cp (1 − t)]1/3 is real for −1 < t < 1. Then it can be easily checked that the kinematic boundary conditions on the walls CD, DE and AB are automatically satisﬁed by assuming that the coeﬃcients an in (3.219) are real. For the computation presented we chose Cp = 0.2. We note that diﬀerent values of Cp and ¯ diﬀerent choices for G(t) (see (3.219) or (3.220)) will yield diﬀerent values

84

Free-surface ﬂows that intersect walls

for the coeﬃcients an . However these various series representations yield the same values of w provided that, all the singularities in the unit circle of the complex t-plane have been properly removed. We present explicit solutions for γ1 = γ2 = 0 (the analysis follows Lee and Vanden-Broeck [94]). After reﬂection in the wall AB, this models a jet of ﬂuid emerging from a funnel (see Figure 3.40). A

γ2

µ

y

E

x

B

C

F

Fig. 3.40. A free-surface ﬂow emerging from a funnel. Gravity is acting vertically downwards. This ﬂow can be obtained by rotating the ﬂow of Figure 3.33 by 90◦ and reﬂecting it in the wall AB.

The expression (3.218) reduces to w = tγ2 /π

[− ln Cp (1 − t)]1/3 ¯ (1 + t)2−2µ/π G(t), (− ln Cp )1/3

(3.221)

¯ where G(t) is deﬁned by (3.219). The dynamic boundary condition (3.160) with β1 = 0 gives 1 2 2 (u

+ v 2 ) − gx = B.

(3.222)

We deﬁne dimensionless variables by using (Q2 /g)1/3 as the unit length and (Qg)1/3 as the unit velocity; here Q is the value of ψ on the streamline CEF. This scaling is diﬀerent from that used in Sections 3.1.1 and 3.2 since the velocity tends to inﬁnity as x → ∞ instead of approaching a constant U . In dimensionless variables (3.222) becomes 1 2 2 (u

+ v 2 ) − x = B.

(3.223)

The ﬂow domain in the f = (φ + iψ)-plane is, as before, the strip 0 < ψ < 1 (see Figure 3.4). Here the point D is irrelevant since γ1 = γ2 .

3.3 The eﬀects of gravity

85

We will ﬁnd the coeﬃcients an by truncating the series in (3.221) and satisfying (3.223) at suitably chosen collocation points. Therefore we need to express x in (3.223) in terms of w = u−iv. This is achieved by diﬀerentiating (3.223) with respect to σ and using (3.10) and (3.38); σ is deﬁned in (3.9). Then the chain rule gives uuσ + vvσ +

u 1 σ = 0. cotan 2 π 2 u + v2

(3.224)

The local analysis near E described at the beginning of this chapter shows that there are three possible values of µ: π, 2π/3 and π/2 + γ2 . Numerical experimentation shows that, for a given value of γ1 = γ2 , the solutions corresponding to µ = π and µ = π/2 + γ2 form a one-parameter family of solutions whereas there is only one solution corresponding to µ = 2π/3. A convenient choice for the parameter is ¯ = 1 , H W

(3.225)

where W is the dimensionless distance between the separation point E and the wall AB. Therefore ¯ = 1 , (3.226) H y(π) where y(π) denotes the value of y at σ = π. For µ = π and µ = π/2 + γ2 , we truncate the inﬁnite series within (3.221) after N terms and satisfy (3.224) at the N − 1 collocation points π 1 I− , I = 1, 2, . . . , N − 1. (3.227) σI = 2(N − 1) 2 ¯ is given. The An extra equation is obtained by satisfying (3.226), where H value of y(π) in (3.226) is obtained by integrating numerically the identity (3.38), where w is deﬁned by (3.221). This gives a system of N algebraic equations for the N unknowns a0 , a1 , . . . , aN −1 . This system is solved by Newton’s method. For β = 2π/3, there is one fewer equation since now we do not have to satisfy (3.226). Therefore again we truncate the inﬁnite series within (3.221) after N terms but satisfy (3.224) at the N collocation points 1 1 I− , I = 1, 2, . . . , N. (3.228) σI = 2N 2 As before, this gives a system of N nonlinear algebraic equations that can be

86

Free-surface ﬂows that intersect walls

solved by Newton’s method. We ﬁnd that there are solutions for all values ¯ > 0. Diﬀerent behaviours are found for γ2 > π/6 and γ2 < π/6. of H • For γ2 < π/6, there is a solution with µ = 2π/3 for a particular value, say ¯ Solutions with µ = π and µ = π/2 + γ2 occur for H ¯ >H ¯ c and ¯ c , of H. H ¯ ¯ H < Hc respectively. Therefore µ=π

µ=

µ=

¯ >H ¯c, when H

2π 3

π + γ2 2

¯ =H ¯c, when H

when

¯
(3.229)

(3.230)

(3.231)

The solutions corresponding to (3.229) and (3.231) approach the solution ¯ approaches H ¯ c from above and from below corresponding to (3.230) as H ¯ respectively. The value of Hc depends on the value of γ2 . ¯ > 0 are characterised by µ = π. This • For γ2 > π/6, all solutions with H is consistent with the fact that the local analysis (see Figure 3.38) shows that there are no solutions with µ = 2π/3 or µ = π/2 + γ2 when γ2 > π/6. These results are illustrated in Figure 3.41, where we plot values of ¯ for various values the velocity qE at the separation point E versus H of γ2 . The solutions with µ = 2π/3 and µ = π/2 + γ2 have stagnation point at E and are therefore characterised by qE = 0. For γ2 ≥ π/6, ¯ > 0 and qE → 0 as H ¯ → 0. For γ < π/6, qE → 0 as qE = 0 for all H ¯ ¯ ¯ ¯ H → Hc and qE = 0 for H < Hc . Therefore the curves corresponding to ¯ → 0, whereas the curve correγ2 = π/4 and γ2 = π/6 approach 0 as H ¯ =H ¯c ≈ sponding to γ2 = π/12 intersects the horizontal axis at the value H 0.528. Typical free-surface proﬁles are shown in Figures 3.42–3.45. The three proﬁles of Figures 3.42–3.44 are for γ2 = π/12. Figure 3.42 ¯ = 0.81, i.e. the case (3.229). The free surface leaves the corresponds to H wall tangentially. Both the free surface and the wall position can be seen in the ﬁgure (the separation point E corresponds to the point on the curve with ordinate y = 0). ¯ =H ¯ c ≈ 0.528 (case (3.230)). There is Figure 3.43 shows the proﬁle for H a 2π/3 angle between the free surface and the wall at the separation point.

3.3 The eﬀects of gravity

87

1.0

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 3.41. Values of the dimensionless velocity qE at the separation point E versus ¯ (see (3.225)). The values of γ2 corresponding to the curves from left to right are H π/4, π/6 and π/12.

¯ = 0.25. Since H ¯
1

0

1

2

3

4

5

6

Fig. 3.42. A free-surface proﬁle EF of Figure 3.40 for γ2 = π/12. The ordinate of the separation point is zero. The dimensionless velocity qE at the separation point is 0.5. The free surface leaves the wall tangentially and µ = π.

88

Free-surface ﬂows that intersect walls 1

0

1

2

3

4

5

6

Fig. 3.43. A free-surface proﬁle EF of Figure 3.40 for γ2 = π/12. The ordinate of ¯ =H ¯c ≈ the separation point is zero. This proﬁle corresponds to µ = 2π/3 and H 0.528.

1

0

0

1

2

3

4

5

6

¯ = 0.25. Fig. 3.44. A free-surface proﬁle EF of Figure 3.40 for γ2 = π/12 and H The ordinate of the separation point is zero. The free surface is horizontal at the separation point.

3.3.2 Solutions with β1 = 0 (nozzles and bubbles) Next we present computations for γ1 = γ2 = 0 (see Figure 3.46). This conﬁguration diﬀers from that of Figure 3.40 because the ﬂow approaches a uniform stream with constant velocity U as x → −∞ instead of approaching a zero velocity. This problem was considered by many investigators (see [18], [62], [63], [165], [166], [169] and others). The study presented below follows Vanden-Broeck [165], [166] and [169]. We will describe the problem by reverting to dimensionless variables in which the unit length is H = Q/U and the unit velocity is U . Then the

3.3 The eﬀects of gravity

89

1

0

1

2

3

4

5

Fig. 3.45. A free-surface proﬁle EF of Figure 3.40 for γ2 = π/6. The ordinate of the separation point is zero. The dimensionless velocity qE at the separation point is 0.5. The free surface leaves the wall tangentially. C

A

µ y

x

B

E

F

Fig. 3.46. The free-surface ﬂow from a nozzle.

dynamic boundary condition (3.160) becomes, in dimensionless variables, u2 + v 2 −

1 x = B, F2

(3.232)

where F is the Froude number deﬁned by F =

U . (2gH)1/2

(3.233)

90

Free-surface ﬂows that intersect walls

¯ in (3.225) by We note that F is related to H ¯ 3 1/2 H . F = 2

(3.234)

The factor 21/2 in (3.233) was introduced for consistency with previous calculations. Similarly (3.224) becomes 2uuσ + 2vvσ +

u 1 σ cotan = 0. 2 2 πF 2 u + v2

(3.235)

The ﬂow of Figure 3.46 has interesting applications. First, it clearly models a jet emerging from a nozzle. Second, it models a bubble rising in a tube when viewed in a frame of reference moving with the bubble (see Figure 3.47). This follows from the symmetry of the ﬂow: the portion EF of the bubble surface in Figure 3.47 is identical, for the same value of the Froude number F , to the portion EF of the jet surface in Figure 3.46. A

C

y

E x B

F

Fig. 3.47. A ‘bubble’ rising in a tube, viewed in a frame of reference moving with the bubble. Physical bubbles are characterised by a continuous slope at the apex.

Returning to our calculation, we represent the complex velocity w by the expansion w=

[− ln Cp (1 − t)]1/3 ¯ (1 + t)2−2µ/π G(t). 1/3 (− ln Cp )

(3.236)

¯ by (3.220) instead of (3.219). Therefore we In this section we represent G(t) write

∞ n ¯ = exp an t , (3.237) G(t) n=1

where we have set a0 = 0 so that w = 1 at t = 0.

3.3 The eﬀects of gravity

91

As for the solutions with γ1 = γ2 = 0 of Section 3.3.1, there are solutions with µ = π, µ = 2π/3 and µ = π/2 + γ2 = π/2. The solution for µ = 2π/3 corresponds to a critical value Fc ≈ 0.3578 of the Froude number F . Solutions with µ = π occur for F > Fc , and those with µ = π/2 + γ2 = π/2 occur for F < Fc . Therefore we have µ=π

when

F > Fc ,

(3.238)

2π when F = Fc , (3.239) 3 π µ= when F < Fc . (3.240) 2 As expected these solutions are the limit of those of Figure 3.40 as γ2 → 0. The three relations (3.229)–(3.231) reduce to (3.238)–(3.240) as γ2 → 0, with µ=

Hc = (2Fc2 )1/3 .

(3.241)

The computations were performed as follows. For the solutions with µ = π and µ = π/2, we truncated the series (3.237) after N terms and satisﬁed (3.235) at the N collocation points (3.228). For a given value of F , this gives a system of N algebraic equations with N unknowns. This system is solved by Newton’s method. For µ = 2π/3, F is one unknown. Therefore we truncate the series (3.237) after N − 1 terms and satisfy (3.235) at the N collocation points (3.228). This leads again to a system of N algebraic equations with N unknowns. Typical free-surface proﬁles are shown in Figures 3.48–3.52. 0

0

0.2

0.4

Fig. 3.48. Rising bubble in a tube for F = 0.1.

The solutions that we have computed model the ﬂow emerging from a nozzle (see Figure 3.46) or a bubble rising in a tube (see Figure 3.47). On

92

Free-surface ﬂows that intersect walls 0

0

0.2

0.4

Fig. 3.49. Rising bubble in a tube for F = 0.3. 0

0

0.2

0.4

Fig. 3.50. Rising bubble in a tube for F = Fc . There is a 120◦ angle at the apex of the bubble.

physical grounds we expect a bubble to be characterised by a continuous slope at its apex. This implies that µ = π/2. Therefore all the solutions for 0 < F < Fc should model a rising bubble. However, experiments (see [32] and [107]) showed that bubbles are only observed for a unique value, Fe ≈ 0.25,

(3.242)

of the Froude number. Clearly the value Fe is in the interval 0 < F < Fc for which we have computed bubbles but the question is to ﬁnd what is special about the value Fe . This is an example of a ‘selection problem’. We have already encountered such a problem in Section 3.2.2. There we found that cavitating ﬂow past a circular cylinder could be calculated for all values γ ∗ < γ¯ < γ ∗∗ when surface tension was neglected. We then showed that a unique solution, for γ¯ ≈ 55◦ , could be selected by solving the problem with

3.3 The eﬀects of gravity

93

0

0

0.2

0.4

Fig. 3.51. Rising bubble in a tube for F = 0.4. A cusp is appearing at the apex. 0

0

0.2

0.4

Fig. 3.52. Rising bubble in a tube for F = 1. There is a pronounced cusp at the apex of the free-surface proﬁle.

T = 0 and then taking the limit as T → 0. We shall show in Section 3.4 that a unique bubble can again be selected by introducing surface tension and then taking the limit as T → 0. We conclude this section by mentioning that our ﬁndings with T = 0 are consistent with analytical results derived by Garabedian [62]. Garabedian [62] proved that there are mathematical solutions describing ‘bubbles’ with a continous slope at the apex for all values of F smaller than a critical value Fc . He then used an energy argument to suggest that the only physically signiﬁcant solution is the one for which F = Fc . In addition he showed that Fc > 0.2363. However, our computations using series truncation showed that Fc ≈ 0.36. This value is about 40 percent higher than the experimental value (3.242). Furthermore the solution corresponding to F = Fc does not have

94

Free-surface ﬂows that intersect walls

a continuous slope at the apex since µ = 2π/3 (see Figure 3.50). Therefore Garabedian’s energy argument does not select the relevant solution. 3.3.3 Solutions with β1 = π/2 (ﬂow under a gate with gravity) In this section we consider solutions for the ﬂow conﬁguration of Figure 3.37 with β1 = π/2. The analysis follows [171] and [94]. We introduce dimensionless variables by using U as the unit velocity and H as the unit length. The Bernoulli equation (3.160) can be written as 1 1 2 (u + v 2 ) + 2 y = B, 2 F

(3.243)

where U . (gH)1/2

F =

(3.244)

C

µ

E

F

y A

B

x

Fig. 3.53. The ﬂow under a gate showing, the angle µ between the free surface EF and the wall CE.

We represent the complex velocity w by t − d (γ2 −γ1 )/π γ1 /π (t + 1) 2−2µ/π A(1−t)2 µ˜ / π t e G(t), w= 1 − td 2

(3.245)

where A is a real constant to be found as part of the solution. The function G(t) is free of singularities. The condition w = 1 at t = 1 implies that G(1) = 0. Therefore we can write ∞

G(t) = e

n=1

a n (tn −1)

.

(3.246)

Following the derivation leading to (3.224) we rewrite (3.243) as uuσ + vvσ −

v 1 σ cotan = 0. πF 2 2 u2 + v 2

(3.247)

3.3 The eﬀects of gravity

95

We now present explicit computations for γ1 = γ2 = π/2 (i.e. for the ﬂow conﬁguration of Figure 3.53). As we shall see, these numerical computations show that there are solutions corresponding to the three possible values π, π/2 and 2π/3 of µ. We ﬁrst calculate the solutions corresponding to µ = π (i.e. solutions for which the free surface leaves the wall tangentially). We truncate the inﬁnite series in (3.245) after N −2 terms and introduce the N −1 collocation points 1 π I− , I = 1, 2, . . . , N − 1. (3.248) σI = N −1 2 We then use (3.245) to evaluate u − iv at the mesh points (3.248) and obtain N − 1 algebraic equations by satisfying (3.247) at these points. One more equation is given by (3.216). The ﬁnal equation is obtained by ﬁxing a parameter characterising the ﬂow. An obvious choice for this parameter is the Froude number F . However, the computations reveal that there can be several solutions corresponding to the same value of the Froude number; a better choice for the parameter is the dimensionless velocity at the separation point E. Therefore we set qE = |(w)t=−1 |.

(3.249)

This leads to a system of N + 1 equations for the N + 1 unknowns a1 , a2 , . . . , ¯. aN −2 , A, F and µ Typical proﬁles are presented in Figures 3.54 and 3.55. 2

1

0

0

1

2

3

Fig. 3.54. Computed free-surface proﬁle for the ﬂow under a gate with F = 2. The angle µ is equal to π.

The values of the dimensionless velocity qE at the separation point E versus the Froude number F are shown in Figure 3.56.

96

Free-surface ﬂows that intersect walls

2

1

0

0

1

2

3

Fig. 3.55. Computed free-surface proﬁle for the ﬂow under a gate with F = 1.8. The angle µ is equal to π. 1.0

0.8

0.6

0.4

0.2

0 1.6

2.0

2.4

2.8

3.2

Fig. 3.56. Values of the dimensionless velocity qE at the separation E versus the Froude number F .

As F → ∞, so qE → 1 and the solution reduces to the free streamline solution (3.39). As qE decreases from 1, the Froude number F ﬁrst decreases to a minimum value F2 ≈ 1.8 and then increases up to the value F1 ≈ 1.87. The value F = F1 corresponds to qE = 0. These results show that a unique solution with µ = π exists for all values of F > F1 . For F2 < F < F1 , two diﬀerent solutions with µ = π are possible. For F < F2 , there are no solutions with µ = π. The coeﬃcients an were found to decrease rapidly as n increases. However, as qE approches zero, the rate of convergence of the series deteriorates and larger and larger values of N are needed to obtain accurate solutions. This is due to the fact that solutions with qE = 0 must correspond to µ = π/2 or µ = 2π/3. Therefore they cannot be computed by the expansion (3.245), (3.246) with µ = π.

3.3 The eﬀects of gravity

97

We now consider solutions with µ = 2π/3. Numerical experimentation shows that there is only one solution. This is the limit of the family of solutions with µ = π as qE → 0. To calculate it, we set µ = 2π/3 in (3.245), truncate the inﬁnite series in (3.246) after N − 3 terms and satisfy (3.247) at the mesh points (3.248). This leads to N − 1 equations for the N unknowns ¯. The last equation is (3.216). The resulting a1 , a2 , . . . , aN −3 , A, F and µ numerical solution is shown in Figure 3.57.

3

2

1

0

0

1

2

3

Fig. 3.57. Computed free-surface proﬁle for the ﬂow under a gate with µ = 2π/3. The solution is unique and the corresponding value of the Froude number is F = F1 ≈ 1.86.

Finally, we look for solutions with µ = π/2. Numerical experimentation shows that there is a one-parameter family of solutions (the parameter can be chosen as the Froude number F ). This family exists for F > F1 . As F → F1 , the solutions approach the solution corresponding to F = F2 and qE = 0. A typical free-surface proﬁle is shown in Figure 3.58. To compute these solutions we set µ = π/2 in (3.245), truncate the inﬁnite series in (3.246) after N − 2 terms and again satisfy (3.247) at the mesh points (3.248). For a given value of F , this leads to N − 1 equations for the ¯. The last equation is given as before N unknowns a1 , a2 , . . . , aN −2 , A and µ by (3.216). Although these solutions are mathematically interesting they are unstable since in them the heavy ﬂuid is lying on top of the light one. Values of the contraction ratio Cc versus F for the solution branch with µ = π are shown in Figure 3.59. Budden and Norbury [24] derived the following asymptotic solution for Cc : (4j + 2)π 2 + π 3 π − α − 0.0007α2 + · · · (3.250) Cc = π+2 (π + 2)5

98

Free-surface ﬂows that intersect walls

3

2

1

0

0

1.0

2.0

Fig. 3.58. Computed free-surface proﬁle for the ﬂow under a gate with µ = π/2. The value of the Froude number is F ≈ 1.9. 0.6

0.5

0.4

1.6

2.0

2.4

2.8

3.2

Fig. 3.59. Values of the contraction ratio Cc versus the Froude number F .

Here j = 0.915 0965 · · · and α is deﬁned by 2 3 F4 2 α= 1−δ + 2 4 F

(3.251)

where δ = qE . For F = 2, the value of the contraction ratio predicted by (3.250) agrees with the numerical results within one per cent.

3.4 The combined eﬀects of gravity and surface tension When gravity is included in the dynamic boundary condition and surface tension is neglected, only three values of the angle µ between the free surface EF and the rigid wall DE are allowed for the ﬂow conﬁguration of Figure 3.37. When surface tension is included and gravity neglected, all values of

3.4 The combined eﬀects of gravity and surface tension

99

µ are in principle possible. Therefore we can expect interesting behaviours to occur when both gravity and surface tension are taken into account, especially in the limit T → 0.

3.4.1 Rising bubbles in a tube We present our main ﬁndings for the ﬂow of Figure 3.46. The analysis follows Vanden-Broeck [166]. We recall that this conﬁguration also models the ﬂow past a bubble in a tube (see Figure 3.47). As in Section 3.3.2 we assume without loss of generality that there is an angle µ between the wall CE and the free surface EF at the point E (see Figure 3.46). If the free surface leaves the wall tangentially then µ = π. If µ < π then the ﬂow near E is locally a ﬂow inside an angle and the speed at E is zero. If µ > π then the ﬂow near E is locally a ﬂow around a corner and the speed at E is inﬁnite. Following the analysis in Section 3.3.2, we introduce dimensionless variables by taking the constant velocity U at x = −∞ as the reference velocity and the distance H between AB and CE as the reference length. The dynamic boundary condition on the free surface EF can be written in dimensionless variables as 1 2 1 2 q − (3.252) x + K = B. 2 2F 2 α Here q is the magnitude of the velocity variable, K the curvature of the free surface EF , B the Bernoulli constant, F the Froude number deﬁned in (3.233) and α the Weber number, deﬁned by α=

2ρU 2 H . T

(3.253)

As noted in Section 3.3.2 only values of µ ≤ π are allowed when T = 0 because the dynamic boundary condition (3.160) requires the velocity qE at the point E in Figure 3.46 to be ﬁnite. However, when T = 0, values of µ > π are in principle possible because an inﬁnite value of q in (3.252) can be balanced by an inﬁnite value of the curvature. Examples of such ﬂows with surface tension included and gravity neglected were covered in Section 3.2. In this section we show examples of gravity–capillary ﬂows for which qE is inﬁnite. The physical relevance of such ﬂows can of course be questioned. However, we shall see that their consideration is crucial in constructing systematically other physical solutions. The ﬂow conﬁguration in the complex potential plane is shown in Figure 3.4. We map it into the complex t-plane by using (3.8). The complex t-plane is illustrated in Figure 3.5. Proceeding as in Section 3.3.1, we write

100

Free-surface ﬂows that intersect walls

the complex velocity w as w=

[− ln Cp (1 − t)]1/3 (1 + t)2−2µ/π G(t), (− ln Cp )1/3

where

∞

G(t) = e

n=1

a n tn

.

(3.254)

(3.255)

1.0

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1.0

1.2

Fig. 3.60. Values of ν versus F for the ﬂow conﬁguration of Figure 3.47 when α = ∞, i.e. T = 0.

1.0

0.8

0.6

0.4

0.2 0

0.2

0.4

0.6

0.8

1.0

1.2

Fig. 3.61. Values of ν versus F for the ﬂow conﬁguration of Figure 3.47 when α = 10.

Next we diﬀerentiate (3.252) with respect to φ and use (3.10) and the chain rule to obtain u σ uvσ − vuσ 1 σ 4π ∂ 2uuσ + 2vvσ + tan cotan + = 0. πF 2 2 u2 + v 2 α ∂σ 2 (u2 + v 2 )1/2 (3.256)

3.4 The combined eﬀects of gravity and surface tension

101

1.1

1.0

0.9

0.8

0.10

0.15

0.20

0.25

0.30

Fig. 3.62. Enlargement of part of Figure 3.61 showing clearly oscillations around 1.0. 0.0

0

0.2

0.4

Fig. 3.63. The selected bubble for T = 0. The value of the Froude number is F = F ∗ ≈ 0.23.

We truncate the inﬁnite series in (3.255) after N −1 terms and satisfy (3.256) at the mesh points 1 σI = 2N

1 I− 2

,

I = 1, . . . , N.

(3.257)

This is achieved by substituting (3.254) and its derivative with respect to σ into (3.256). It leads to N nonlinear equations for the N unknowns a1 , . . . , aN −1 and µ. This system is solved by Newton’s method for given values of F and α. We start the presentation of the numerical results by recalling the ﬁndings of Section 3.3.2 when T = 0 (i.e. α = ∞). They are shown graphically in

102

Free-surface ﬂows that intersect walls

Figure 3.60 where we plot the parameter ν=2

π−µ π

(3.258)

versus F . There is a unique solution with µ = 2π/3 (i.e. ν = 2/3) for F = Fc ≈ 0.36. Solutions with µ = π/2 (i.e. ν = 1) and µ = π (i.e. ν = 0) occur for F < Fc and F > Fc respectively. The curve of Figure 3.60 is discontinuous with a jump at F = Fc . When surface tension is included in the dynamic boundary condition, the discontinuity disappears and the curve of Figure 3.60 is replaced by a continuous one. This is illustrated in Figure 3.61, where we present values of ν versus F for α = 10. An interesting feature is that the curve of Figure 3.61 oscillates inﬁnitely often around ν = 1. An enlargement of Figure 3.61 is shown in Figure 3.62. As F decreases, the amplitude and wavelength of the oscillations decrease. These results suggest that there is a countably inﬁnite set of values of F for which ν = 1. We denote this set by Fi∗ ,

i = 1, 2, 3, . . . ,

(3.259)

where F1∗ > F2∗ > F3∗ > · · · . We recall that physically relevant bubbles are identiﬁed as those for which ν = 1. Therefore all bubbles with F < Fc are physically relevant when T = 0 whereas only those corresponding to the set (3.259) are physically relevant when T = 0. The numerical computations show that, for each given value of i, Fi∗ → F ∗

as

α → ∞,

(3.260)

where F ∗ ≈ 0.23.

(3.261)

This is illustrated in Figure 3.64, where we plot values of F1 versus α−1 . As T → 0 (i.e. α−1 → 0), F1 → F ∗ in agreement with (3.260). Our ﬁndings can be summarised as follows. When T = 0, there is a bubble (with µ = π/2) for each value of 0 < F < Fc . When T = 0, there is a bubble (with µ = π/2) for a discrete set of values of F (see (3.259)). As T → 0, the discrete set reduces to a unique value F = F ∗ of F . Therefore we have succeeded in selecting a unique solution by including surface tension and taking the limit as T → 0. Moreover the selected value F ∗ ≈ 0.23 is close to the experimental value Fe ≈ 0.25 (see [32] and [107]).

3.4 The combined eﬀects of gravity and surface tension

103

2.0

1.5

1.0

0.5

0

0

0.1

0.2

0.3

Fig. 3.64. Values of α−1 versus F1∗ . As α−1 → 0, F1 → F ∗ ≈ 0.23.

3.4.2 Fingering in a Hele Shaw cell Another classical example of the selection of solutions via surface tension occurs in the study of ﬁngering in a Hele Shaw cell, This problem can be motivated as follows. It is well known that an instability may occur in a porous medium when a less viscous ﬂuid drives a more viscous ﬂuid (Saﬀman and Taylor [132]). To study this instability, experiments have been performed in a Hele Shaw cell, a channel formed by two closely spaced parallel glass plates; this provides a model of a two-dimensional ﬂow through a porous medium. It was found that the unstable interface develops a number of ‘ﬁngers’. After some time, one ﬁnger dominates and suppresses the growth of the others and the ﬂow reaches a steady state in which a single ﬁnger propagates without change of shape. McLean and Saﬀman [109] modelled this ﬁnger by a two-dimensional potential ﬂow with surface tension included at the interface (see Figure 3.65). They denoted by U the velocity of the ﬁnger, 2a the lateral width of the channel, b the transverse thickness, T the surface tension and µ ˜ the viscosity (here we use a tilde to avoid confusion with the angle µ used earlier in this section). In addition they denoted the ratio of the width of the ﬁnger and the width of the channel by λ. Taking a as the unit length and (1 − λ)U as the unit velocity, McLean and Saﬀman derived a nonlinear integro-diﬀerential equation for the unknown shape S of the free surface: S ln q(S) = − π

0

1

θ(S ) dS , S (S − S)

(3.262)

104

Free-surface ﬂows that intersect walls y q

A

x B

Fig. 3.65. Model for a ﬁnger in a Hele Shaw cell. Only half the ﬁnger is shown.

κqS

d dS

qS

dθ dS

θ(0) = 0, π θ(1) = − , 2

− q = − cos θ,

(3.263)

q(0) = 1,

(3.264)

q(1) = 0.

(3.265)

Here

κ=

θ = θˆ − π,

(3.266)

q = (1 − λ)ˆ q,

(3.267)

T b2 π 2 . 12µU a2 (1 − λ)2

(3.268)

The integral in (3.262) is of Cauchy principal-value form. The variables θˆ and qˆ in (3.266) and (3.267) are deﬁned in terms of the dimensionless complex velocity u ˆ − iˆ v by the relation ˆ

u ˆ − iˆ v = qˆe−iθ .

(3.269)

The ﬂow conﬁguration is illustrated in Figure 3.65. The x ˆ-axis is parallel to the walls of the channel and is the axis of symmetry of the ﬁnger. The points A and B correspond to S = 0 and S = 1 respectively. After a solution for θ and q is obtained, the shape of the ﬁnger is given by 1 − λ 1 eiθ x ˆ(S) + iˆ y (S) = − dS. (3.270) π S Sq

3.4 The combined eﬀects of gravity and surface tension

105

For κ = 0 (i.e. in the absence of surface tension), Saﬀman and Taylor [132] obtained the following exact solution:

(1 − S)/(1 − λ)2 q= (1 − λ)2 + S(2λ − 1) θ = cos−1 q.

1/2 ,

(3.271)

(3.272)

The solution (3.271), (3.272) leaves the parameter λ undetermined. In other words a solution can be found for each value of 0 < λ < 1. This ﬁnding is not consistent with experiments, which show that for small values of the surface tension there is only one ﬁnger, corresponding to λ ≈ 0.5. This is again a selection problem that can be resolved by solving the problem with surface tension and then taking the limit as the surface tension approaches zero. Early numerical calculations with κ = 0 were performed by McLean and Saﬀman [109], who identiﬁed one family of solutions. Romero [129] then found two other families. As we shall see there is in fact a countably inﬁnite set of families of solutions. There is a strong analogy between the ﬁngering problem and the bubble problem of Figure 3.47. The procedure to ﬁnd the discrete set (3.259) allowed the angle µ to be found as part of the solution. Therefore we shall use a similar approach for the ﬁngering problem. The analysis follows VandenBroeck [161]. We deﬁne a modiﬁed problem that has solutions for all values of λ and κ. This modiﬁed problem is obtained by replacing (3.265) simply by q(1) = 0.

(3.273)

Therefore θ(1) becomes an unknown to be found as part of the solution. We solve the modiﬁed problem deﬁned by (3.262)–(3.264) and (3.273) and obtain solutions for all values of λ and κ. We will then obtain solutions of the original problem by selecting among the solutions of the modiﬁed problem those for which θ(1) = −π/2. Following McLean and Saﬀman [109] we introduce the change of variables Sτ = 1 − ζ γ .

(3.274)

1 cotan πτ = κ. τ2

(3.275)

Here τ is the smallest root of

106

Free-surface ﬂows that intersect walls

With (3.274), θ is twice diﬀerentiable with respect to ζ at both end points. McLean and Saﬀman [109] chose γ = 2 in (3.274). In order to solve the modiﬁed problem we will choose γ = 4. We introduce the N mesh points I −1 , N

ζI =

I = 1, . . . , N.

(3.276)

We also deﬁne the unknowns θI = θ(1 − ζIν ),

I = 1, . . . , N.

(3.277)

We discretise the system (3.262)–(3.264) and (3.273) by following the procedure outlined by McLean and Saﬀman [109]. Thus we obtain N − 1 nonlinear algebraic equations for the N − 1 unknowns θI , I = 2, . . . , N . For given values of λ and κ this system is solved by Newton’s method. In Figure 3.66 we present numerical values of θ(1) versus λ for κ = 0.273. As λ approaches zero, θ(1) → −π and the ﬁnger collapses on the negative x ˜-axis. As λ approaches unity, θ(1) oscillates inﬁnitely often around −π/2. An enlargement of part of Figure 3.66 is shown in Figure 3.67 to illustrate the oscillations clearly. Figures 3.66 and 3.67 show that there is a countably inﬁnite set of values of λ for which θ(1) = −π/2. We denote this set by λi ,

i = 1, 2, 3, . . .

(3.278)

where λ1 > λ2 > λ3 · · · . 1.72

1.68

1.64

1.60

1.56 0.5

0.6

0.7

0.8

0.9

1.0

Fig. 3.66. Values of θ(1) versus λ for κ = 0.273.

3.4 The combined eﬀects of gravity and surface tension

107

The solutions corresponding to the values (3.278) of λ are the solutions of the original problem.

1.572

1.571

1.570

1.569 0.70

0.75

0.80

0.85

0.90

Fig. 3.67. Enlargement of part of Figure 3.66 showing clearly the oscillations around π/2.

Vanden-Broeck [161] showed numerically that, for a given value of i, λi →

1 2

as

κ → 0.

(3.279)

It can be seen from relation (3.279) that a unique solution corresponding to λ = 1/2 is selected in the limit as the surface tension tends to zero. We note that (3.279) is comparable with (3.260). This ﬁnding is illustrated in Figure 3.68, where values of λ1 , λ2 and λ3 versus κ are plotted. As κ → 0, the three curves approach λ = 1/2.

0.82

0.74

0.66

0.58

0.50

0

0.2

0.4

0.6

0.8

1.0

Fig. 3.68. Values of λ versus κ.

1.2

108

Free-surface ﬂows that intersect walls

3.4.3 Further examples involving rising bubbles We now return to potential ﬂows and present three other examples of freesurface ﬂows for which a unique solution can be selected by taking the limit T → 0. These ﬂows are similar to the ﬂow past a bubble shown in Figure 3.47. The ﬁrst two are given in Figures 3.69 and 3.70. They model a two-dimensional bubble rising at a constant velocity U in an unbounded ﬂuid when viewed in a frame of reference moving with the bubble. Both conﬁgurations include a model for the wake behind the bubble. In that sense they are improvements on the ﬂow of Figure 2.2, in which the wake was neglected. The free-surface ﬂows of Figures 3.69 and 3.70 can be solved by series truncation methods similar to those used earlier in this section. Details can be found in [170] and [174] respectively. We summarise here the main results.

E A

U y

x D

S

S'

E'

J'

J

Fig. 3.69. A free streamline model for a rising bubble.

In Figure 3.69, a free streamline model (similar to the cavitating models of Section 3.1.2) is used. This implies that the velocity is equal to U on the boundaries SJ and S J of the wake. When surface tension is neglected there is one solution for each value of the Froude number F =

U , (gD)1/2

(3.280)

where D is deﬁned in Figure 3.69. If we denote by µ the angle between the

3.4 The combined eﬀects of gravity and surface tension

109

E U

A

y x

S

S'

L

J

J'

Fig. 3.70. Joukovskii’s model for a rising bubble.

symmetry line EA and the free surface AS then it is found that µ=

π 2

when

F < Fc ,

(3.281)

µ=

2π 3

when

F = Fc ,

(3.282)

µ=π

when

F > Fc ,

(3.283)

where Fc ≈ 0.9.

(3.284)

These ﬁndings are very similar to those obtained in (3.238)–(3.240) for the ﬂow of Figure 3.47. Introducing the surface tension T on the surface SAS of the bubble yields a discrete set of values of F for which µ = π/2. A unique solution for which F ≈ 0.51 is then obtained by taking the limit T → 0 (see [170] for details). The ﬂow of Figure 3.70 is similar to that of Figure 3.69 except that the boundary of the wake is now approximated by two vertical lines, SJ and S J . This ﬂow can be characterised by the Froude number F =

U , (gL)1/2

(3.285)

110

Free-surface ﬂows that intersect walls

where L is deﬁned in Figure 3.70. This is a crude model for the wake. It is, however, of mathematical interest because Joukovskii (see [69]) found an exact solution in the absence of surface tension. This solution is characterised by F = (2π)−1/2 . The results obtained for the ﬂow of Figure 3.69 suggest by analogy that there is a solution for each value of F and that Joukosvkii’s solution is just a member of this family of solutions. This was conﬁrmed by the numerical computations in [174], where it was shown that there is a solution for the ﬂow conﬁguration of Figure 3.70 for 0 < F < ∞ satisfying (3.281)–(3.283), where µ is the angle between EA and AS in Figure 3.70 and where Fc ≈ 0.66. As before a unique solution with µ = π/2 can be selected by introducing surface tension and taking the limit T → 0. Interestingly, the numerical computations suggest that the selected solution is Joukovskii’s exact solution (see [174]). The third example is a generalisation of the problem of a bubble rising in a tube considered in Sections 3.3.2 and 3.4.1. The ﬂow conﬁguration is shown in Figure 3.71(a). Gravity is acting vertically downwards. The angle between the left-hand wall and the horizontal is denoted by β, and the angle between the negative x-axis and the tangent line to the free surface JS at J is denoted by γ. When β < π/2, Figure 3.71(a) models a physical bubble rising in an inclined tube. 1.5

1.5 I'

0.5 y

0

I'

I

U

1.0

0.5

x

y

0

S

I

U

1.0

x S

J

J H

J'

J' 0

(a)

0.5

0

0.5

(b)

Fig. 3.71. The ﬂow domain and the coordinates. This is a computed proﬁle for β = 7π/12, F = 0.11 and ω = 10.

This problem was studied experimentally by Maneri [107] and has been studied theoretically in [36] and [95]. The ﬂow conﬁguration of Figure 3.71(a)

3.4 The combined eﬀects of gravity and surface tension

111

also describes a jet emerging from a nozzle and falling down along a wall. In this case the ﬂow is viewed as bounded on the left by an inﬁnite wall and on the right by a semi-inﬁnite wall and a free surface (Figure 3.71(b)). The ﬂow can be characterised by the Froude number F =

U (gH)1/2

(3.286)

and the Weber number ρU 2 H . (3.287) T Here ρ is the density of the ﬂuid, U the velocity as x → ∞ and H the width of the tube. When T = 0 (i.e. ω = ∞), the admissible values of γ depend on β and can be predicted by the local analysis of Section 3.3 (see Figure 3.38). When 0 < β < 2π/3, there is a critical value Fc of the Froude number F such that solutions with γ = 0, γ = π/3 and γ = π − β occur for F > Fc , F = Fc and F < Fc respectively. Values of Fc versus β are shown in Figure 3.72. However, for 2π/3 ≤ β ≤ π there is no such critical value of F , and γ = 0 for all 0 < F < ∞ (see [95]). ω=

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

30

60

90

120

Fig. 3.72. Values of Fc versus β.

We note that these results include those of Section 3.3.2 as a particular case, since the ﬂow of Figure 3.71(a) is simply half that √of Figure 3.47. Figure 3.72 predicts Fc ≈ 0.506. Dividing this value by 2 to take into account the diﬀerent deﬁnitions of H in Figures 3.3.2 and 3.71(a) gives 0.36, in agreement with the value obtained in Section 3.3.2.

112

Free-surface ﬂows that intersect walls

We now consider in more detail the problem of a bubble rising in an inclined tube (i.e. the ﬂow of Figure 3.71(a)). The experimental data of Manieri [107] showed that for each value of 0 < β < π/2, there is only one value of F for which a bubble exists. This does not agree with the numerical results, which predict a solution for each value of F and 0 < β < π/2. This discrepancy can be removed by generalising the procedure described in Section 3.4.1. Thus again we introduce surface tension and take the limit as the surface tension tends to zero. Cou¨et and Strumolo [36] chose for each value of β and ω the particular solution corresponding to the largest value of F for which γ = π/2. The selected results were found to be in good agreement with experiment. The results in [95] summarised above show that the only possible values of γ when T = 0 are π − β, π/3 and 0. Therefore there are no solutions with γ = π/2 when T = 0 unless β = π/2. This implies that the criterion, in [36] has to be used with T small but diﬀerent from zero. Here we follow [95] and use a diﬀerent selection criterion, in which we take the limit T → 0 instead of keeping T = 0 as in [36]. The numerical results in [95] show that for each value of 0 < β ≤ π/2 and ω there is a discrete set of values Fi∗ ,

i = 1, 2, 3, . . . ,

(3.288)

where F1∗ > F2∗ > F3∗ > · · · ,

(3.289)

for which γ = π − β. This ﬁnding reduces to (3.259) when β = π/2. The numerical computations show that for a given value of i Fi∗ → F ∗

as

ω → ∞.

(3.290)

The selected values of F ∗ and the corresponding proﬁles are found to be in close agreement with the experimental data of Manieri [107] (see [95] for details). A typical selected proﬁle for β = π/3 is shown in Figure 3.73. 3.4.4 Exponential asymptotics Since the ﬁngering problem of Section 3.4.2 has an exact solution when T = 0 (see (3.271), (3.272)), it is tempting to try to construct asymptotic solutions for T small in the form of a power series in T . This was achieved by McLean and Saﬀman [109], who showed that an arbitrary number of terms can be calculated. However, they found that this expansion leads to solutions for each value of 0 < λ < 1. This is to be contrasted with the discrete set of solutions (3.278) found by direct numerical computation.

3.4 The combined eﬀects of gravity and surface tension

113

Fig. 3.73. Computed solution for β = π/3 and F = F ∗ ≈ 0.527. The dots are the experimental values of Manieri [107].

This paradox was resolved by noting that the selection mechanism leading to (3.278) is associated with exponentially small terms in T . Such terms are smaller than any positive integer power of T as T → 0 and therefore cannot be calculated by a power series in T . Exponential asymptotics has been used by many investigators to study this problem analytically (see Saﬀman [131] for a review and references). Vanden-Broeck showed that the selection mechanism for a rising bubble that leads to (3.259) (see Section 3.4.1) is also associated with exponentially small terms. Over the last 30 years, it has been found that exponentially small terms play a surprisingly signiﬁcant part in free-surface ﬂow problems. We will encounter further examples later in this book. One example concerns the eﬀect of surface tension on solitary waves (see Chapter 6). Other examples are the free-surface ﬂows generated by moving disturbances for small values of the Froude number or small values of the surface tension (see Chapter 8).

4 Linear free-surface ﬂows generated by moving disturbances

In this chapter we consider free-surface ﬂows generated by a disturbance moving at a constant velocity U . We choose a frame of reference moving with the disturbance and assume that the ﬂow is steady in that frame. We will restrict our attention to two-dimensional linear ﬂows. Nonlinear, three-dimensional and unsteady free-surface ﬂows will be discussed in Chapters 7, 10 and 11 respectively. The disturbance could be a submerged object, say a submarine (see Figure 4.1), an obstacle at the bottom of a channel (see Figure 4.2), a surface-piercing object, say a ship (see Figure 4.3), or a distribution of pressure (see Figure 4.4).

Fig. 4.1. The two-dimensional free-surface ﬂow generated by a submerged object moving at a constant velocity U , when viewed in a frame of reference moving with the object.

Fig. 4.2. The two-dimensional free-surface ﬂow generated by an object moving at a constant velocity U along the bottom of a channel, when viewed in a frame of reference moving with the object.

114

4.1 The exact nonlinear equations

115

Fig. 4.3. The two-dimensional free-surface ﬂow generated by a surface-piercing object moving at a constant velocity U , when viewed in a frame of reference moving with the object. y x

Fig. 4.4. The two-dimensional free-surface ﬂow generated by a pressure distribution moving at constant velocity U , when viewed in a frame of reference moving with the pressure distribution.

4.1 The exact nonlinear equations We introduce cartesian coordinates x, y with the x-axis parallel to the bottom (see Figure 4.4). The fully nonlinear problem can be formulated in terms of a potential function φ(x, y) as φxx + φyy = 0, φy = φx ηx

−h < y < η(x), on y = η(x),

ηxx 1 2 T P (x) (φx + φ2y ) + gy − =B + 3/2 2 2 ρ (1 + ηx ) ρ

(4.2) on

y = η(x),

(4.3)

on y = −h,

(4.4)

on the wetted surfaces of the objects (if any).

(4.5)

φy = 0 φn = 0

(4.1)

Here y = η(x) is the equation of the free surface and φn denotes the normal derivative of φ. The bottom of the channel is at y = −h. Equations (4.2), (4.4) and (4.5) are the kinematic boundary conditions on the free surface, on the bottom and on the surface of any object present, respectively. Equation (4.3) is the dynamic boundary condition on the free surface. The quantity P (x) is the prescribed pressure distribution; the parameter measures its size. We ﬁx the Bernoulli constant B in (4.3) by choosing y = 0 as the

116

Linear free-surface ﬂows generated by moving disturbances

constant level of the free surface corresponding to P = 0 and to a uniform stream with constant velocity U . This leads to 1 B = U 2. (4.6) 2 If the object is fully submerged (i.e. if its surface does not intersect the free surface y = η(x)) and does not touch the bottom y = −h, we need to specify, in addition to equations (4.1)–(4.5), the circulation of the vector velocity around the object. 4.2 Linear theory 4.2.1 Solutions in water of ﬁnite depth A ﬁrst insight into the problem will be gained by developing a linear theory. The analysis presented in this chapter follows Rayleigh [128]. Nonlinear investigations are presented in Chapter 7. The results are qualitatively independent of the nature of the disturbance; therefore we will restrict our attention to the case where the disturbance is a pressure distribution (see Figure 4.4). The nonlinear problem is described by (4.1)–(4.4). To derive a linear theory we assume that the parameter in (4.3) is small and seek a solution in the form of a small perturbation around a uniform stream with constant velocity U and constant depth h. This is motivated by the fact that φ(x, y) = U x, η(x) = 0 is an exact solution of (4.1)–(4.4) when = 0. The linear equations are obtained by expanding the solutions in powers of and retaining only the terms of order . Therefore we write φ(x, y) = U x + φ1 (x, y) + O(2 ),

(4.7)

η(x) = η1 (x) + O(2 ).

(4.8)

Substituting (4.7) and (4.8) into (4.1)–(4.4), using (4.6) and equating terms of order yields φ1xx + φ1yy = 0, φ1y = U η1x U φ1x + gη1 −

−h < y < 0, on

y = 0,

T P η1xx + = 0 on y = 0, ρ ρ

φ1y = 0

on y = −h.

(4.9) (4.10) (4.11) (4.12)

When P = 0 the system (4.9)–(4.12) reduces to the system (2.60)–(2.63) with c = U . Therefore if φ1 = φ1p , η1 = η1p is a particular solution of

4.2 Linear theory

117

(4.9)–(4.12) with P = 0, the general solution of (4.9)–(4.12) with P = 0 is obtained by adding to the particular solution φ1p , η1p an arbitrary multiple of a solution of (2.60)–(2.63). When the condition (2.85) is satisﬁed, (2.86) and (2.87) give φ1 = φ1p −

U A1 cosh k(y + h) sin k(x + δ), sinh kh

η1 = η1p + A1 cos k(x + δ)

(4.13) (4.14)

where A1 is an arbitrary amplitude and δ an arbitrary phase. The constant k in (4.13) and (4.14) satisﬁes the equation (2.83) with c = U . Relations (4.13) and (4.14) mean that we can add a train of periodic waves of wavenumber k and of arbitrary amplitude A1 to any particular solution of (4.9)–(4.12) and obtain a new solution. Cases for which the condition (2.85) is not satisﬁed will be considered in Section 7.3. Rayleigh [128] showed that a unique solution of the linear problem (4.9)– (4.12) can be obtained by using the radiation condition. This condition requires that there is no energy coming from inﬁnity. In other words, if there is a train of waves at x = ∞ in Figure 4.4 then the energy in this wave train must be travelling to the right. Similarly, if there is a wave train at x = −∞ then the energy must be travelling to the left. The discussion at the end of Section 2.4.3 shows that the energy of a wave train travels at the group velocity cg . In a frame of reference in which the disturbance moves to the left at a constant velocity U , the waves and the energy in the waves move also to the left at velocities c = U and cg respectively. In the frame of reference moving with the disturbance, the waves are not travelling and the energy travels to the left at velocity cg − U . Therefore we require cg > U for a wave train at x = −∞ and cg < U for a wave train at x = ∞. We note that for τ < 1/3 and U > cmin there are exactly two possible wave trains (see Figure 2.5). The wave train with k > kmin is characterised by c < cg and can therefore appear at x = −∞. The wave train with k < kmin is characterised by c > cg and can appear at x = ∞. Following Rayleigh [128] we shall see that the radiation condition is automatically satisﬁed by introducing some dissipation in the equations (4.9)– (4.12), solving the problem and then taking the limit as the coeﬃcient of dissipation tends to zero. One way to do this is to introduce the so-called Rayleigh viscosity µ > 0 and to rewrite (4.11) as U φ1x + gη1 −

T P η1xx + + µφ1 = 0 on y = 0. ρ ρ

(4.15)

We now solve the system (4.9), (4.10), (4.15) and (4.12) and then take

118

Linear free-surface ﬂows generated by moving disturbances

the limit µ → 0. Eliminating η1 between (4.10) and (4.15) gives U φ1xx +

g T Px φ1y − φ1yxx + + µφ1x = 0 U ρU ρ

on y = 0.

We represent φ1 (x, y) by the Fourier integral ∞ φ1 (x, y) = F (a, y)eiax da.

(4.16)

(4.17)

−∞

Substituting (4.17) into (4.9) and (4.12) gives ∂2F − a2 F = 0, ∂y 2 ∂F =0 ∂y

−h < y < 0,

on y = −h.

(4.18) (4.19)

The solution of (4.18) satisfying (4.19) is F (a, y) = A(a) cosh a(y + h), and (4.17) can be rewritten as ∞ φ1 (x, y) = A(a) cosh a(y + h)eiax da.

(4.20)

(4.21)

−∞

Here A(a) is an arbitrary function of a. Next we write the pressure distribution as the Fourier integral ∞ B(a)eiax da. (4.22) P (x) = −∞

The inverse Fourier transform gives ∞ 1 P (x)e−iax dx. B(a) = 2π −∞

(4.23)

Substituting (4.21) and (4.22) into (4.16) gives A(a) =

iB(a) , ρU D(a) cosh ah

(4.24)

where D(a) = a −

T a2 g tanh ah − tanh ah − iµ1 . U2 ρU 2

Here µ1 = µ/U . Combining (4.10), (4.21) and (4.25) yields after integration ∞ iax e B(a) tanh ah 1 η1 (x) = da. 2 ρU −∞ D(a)

(4.25)

(4.26)

4.2 Linear theory

119

To simplify the presentation we assume that P (x) is an even function (i.e. P (−x) = P (x)). It follows from (4.23) that 1 ∞ P (x) cos ax dx. (4.27) B(a) = π 0 Therefore B(a) is a real even function of a, and the integrand in (4.26) can be written as the ratio of a complex numerator and a complex denominator: RN + iIN RN RD + IN ID RD IN − RN ID = +i ; 2 + I2 2 + I2 RD + iID RD RD D D

(4.28)

here RN and RD are real odd functions of a and IN and ID are real even functions of a. The real part of (4.28) is an even function of a and the imaginary part of (4.28) is an odd function of a. Therefore (4.26) can be written as ∞ iax e B(a) tanh ah 2 η1 (x) = da, (4.29) 2 ρU D(a) 0 where denotes the real part. We now take the limit of (4.29) as µ1 tends to zero. If D(a) = 0 for 0 < a < ∞, we obtain ∞ eiax B(a) tanh ah 2 da. η1 (x) = ρU 2 a − (g/U 2 ) tanh ah − (T a2 /ρU 2 ) tanh ah 0

(4.30)

If D(a) vanishes for some values of a, some care has to be taken in taking the limit. This occurs when there are values a∗ such that a∗ −

g T a∗ 2 ∗ tanh a h − tanh a∗ h = 0. U2 ρU 2

(4.31)

Comparing (4.31) with (2.83) we see that a∗ = k, where k is a wavenumber satisfying the dispersion relation (2.83) with c = U . If τ ≤ 1/3, there is a minimum value cmin of c (see Figure 2.5). On the one hand there are then two values a∗1 and a∗2 of a∗ for U > cmin ; without loss of generality we can assume that 0 < a∗1 < a∗2 . On the other hand there are no values a∗ when U < cmin , and the solution is given by (4.30). Let us consider the case τ < 1/3. We rewrite D(a) in the form D(a) =

a [U 2 − C 2 (a)] − iµ1 , U2

where

2

C (a) =

g T + a tanh ah. a ρ

(4.32)

(4.33)

120

Linear free-surface ﬂows generated by moving disturbances

If a∗ is a root of D(a), we seek a root of D(a) for µ1 → 0 by writing a = a∗ + ˜(a∗ ),

(4.34)

where |˜ (a∗ )| is small. Subsituting (4.34) into (4.32) and expanding in a Taylor expansion for ˜(a∗ ) small gives (a∗ ) D(a∗ + ˜(a∗ )) ≈ −˜

2a∗ C(a∗ )C (a∗ ) − iµ1 , U2

(4.35)

where the prime denotes the derivative with respect to a. Therefore ˜(a∗ ) =

−iµ1 U 2 2a∗ C(a∗ )C (a∗ )

(4.36)

to leading order. Since C(a∗ ) = U , we can rewrite (4.36) as ˜(a∗ ) = i0 (a∗ ),

(4.37)

where 0 (a∗ ) = −

µ1 U . ∗ 2a C (a∗ )

(4.38)

We now evaluate the integral in (4.29) by contour integration in the complex plane ξ = a + ib. The quantity C(a) is the phase velocity of linear gravity– capillary waves of wavenumber a (see (4.33)). Relation (2.146) then shows that the group velocity Cg (a) is greater than C(a) when C (a) > 0 and smaller than C(a) when C (a) < 0. The ﬁrst possibility occurs for a = a∗2 and the second for a = a∗1 (since we have assumed that 0 < a∗1 < a∗2 ). Relation (4.38) shows that 0 (a∗ ) > 0 when C (a∗ ) < 0 and 0 (a∗ ) < 0 when C (a∗ ) > 0. Therefore the pole a∗1 +i0 (a∗1 ) is in the upper half of the complex ξ-plane whereas the pole a∗2 + i0 (a∗2 ) is in the lower half. Similarly, it can be shown that there is also a pole at −a∗1 + i0 (a∗1 ) in the upper half-plane and a pole at −a∗2 + i0 (a∗2 ) in the lower half-plane. We now consider the integral 2 eiξx B(ξ) tanh ξh dξ (4.39) ρU 2 C c D(ξ) where ξ = a + ib. The deﬁnition of the contour C depends on the sign of x. For x > 0, the contour C is anticlockwise and consists of the a-axis and a semicircle of radius R in the upper half-plane. For x < 0, the contour C is clockwise and consists of the a-axis and a semicircle of radius R in the lower half-plane. We will refer to the contours C for x > 0 and x < 0 as C + and C − respectively. These contours were

4.2 Linear theory

121

chosen so that the contributions from the semicircles will tend to zero as R → ∞, which follows from the fact that the exponential eiξx = eiax e−bx

(4.40)

appearing in the integrand of (4.39) tends to zero as b → ∞ (b → −∞) when x > 0 (x < 0). Therefore (4.39) reduces to 2 ρU 2

C±

eiξx B(ξ) tanh ξh 2 dξ = D(ξ) ρU 2

∞

−∞

eiax B(a) tanh ah da. D(a)

(4.41)

We now evaluate the integral in (4.39) by using the residue theorem. This gives eiξx B(ξ) tanh ξh 2 dξ = 2iπRI+ + 2iπ[residue at a∗1 + i0 (a∗1 )] ρU 2 C + D(ξ) + 2iπ[residue at −a∗1 + i0 (a∗1 )]

(4.42)

for x > 0 and eiξx B(ξ) tanh ξh 2 dξ = −2iπRI− − 2iπ[residue at a∗2 + i0 (a∗2 )] ρU 2 C − D(ξ) − 2iπ[residue at −a∗2 + i0 (a∗2 )] (4.43) for x < 0. Here RI+ and RI− denote the sums of the residues on the positive and negative parts of the imaginary ξ-axis. Noting that the real part of the ﬁrst integral on the right-hand side of (4.41) is 2η1 (x) (see (4.29)), combining (4.41) with (4.42) and (4.43) and taking the limit µ1 → 0, we obtain η1 (x) = (iπRI+ ) +

2π B(a∗1 ) tanh a∗1 h sin a∗1 x ρU a∗1 C (a∗1 )

for x > 0

(4.44)

η1 (x) = −(iπRI− ) −

2π B(a∗2 ) tanh a∗2 h sin a∗2 x ρU a∗2 C (a∗2 )

for x < 0.

(4.45)

x→∞

(4.46)

and

Taking the limit |x| → ∞ we ﬁnd η1 (x) ≈

2π B(a∗1 ) tanh a∗1 h sin a∗1 x ρU a∗1 C (a∗1 )

as

122

Linear free-surface ﬂows generated by moving disturbances

and η1 (x) ≈ −

2π B(a∗2 ) tanh a∗2 h sin a∗2 x ρU a∗2 C (a∗2 )

as

x → −∞.

(4.47)

Relations (4.46) and (4.47) show that the free-surface proﬁle is characterised by a periodic wave train in the far ﬁeld. These waves have wavelengths 2π/a∗1 and 2π/a∗2 respectively. Since a∗1 < a∗2 , the train of longer wavelength occurs as x → ∞ and the train of shorter wavelength as x → −∞. In conclusion, the linear free-surface proﬁles are given by (4.30) for U < cmin and by (4.44) and (4.45) for U > cmin .

4.2.2 Solutions in water of inﬁnite depth We now examine the particular case of water of inﬁnite depth. The exact relations are given by (4.1)–(4.3), (4.6) and φx → U,

φy → 0 as y → −∞.

(4.48)

Proceeding as in Section 4.2.1, we assume (4.7) and (4.8) with small and derive the linearised system φ1xx + φ1yy = 0,

y < 0,

(4.49)

φ1y = U η1x

y = 0,

(4.50)

U φ1x + gη1 −

on

T P η1xx + + µφ1 = 0 on y = 0, ρ ρ

φ1x → 0

φ1y → 0 as

y → −∞.

(4.51) (4.52)

Eliminating η1 (x) between (4.50) and (4.51) yields (4.16). The remaining analysis follows closely that of Section 4.2.1. We ﬁnd ∞ A(a)e|a|y eiax da (4.53) φ1 (x, y) = −∞

with A(a) =

iB(a) , ˜ ρU D(a)

(4.54)

where B(a) is given by (4.27) and g |a| T a2 |a| ˜ − − iµ1 . D(a) =a− 2 U a ρU 2 a

(4.55)

4.2 Linear theory

The expression for the proﬁle η1 (x) is ∞ iax e B(a) 2 da. η1 (x) = 2 ˜ ρU 0 D(a)

123

(4.56)

Since only positive values of a are involved in (4.56) we can now rewrite ˜ D(a) as g T a2 ˜ D(a) =a− 2 − − iµ1 . U ρU 2

(4.57)

˜ We now take the limit µ1 → 0. If D(a) = 0 for 0 < a < ∞, we obtain ∞ eiax B(a) 2 da. (4.58) η1 (x) = ρU 2 a − (g/U 2 ) − (T a2 /ρU 2 ) 0 The expression (4.58) is the equivalent of (4.30) in water of inﬁnite depth. As expected, it can be formally derived from (4.30) by replacing the tanh factors by 1. ˜ If D(a) vanishes for some values of a then some care is needed in taking the limit µ1 → 0, and we follow the approach of Section 4.2.1. This situation occurs when there are values a∗ such that a∗ −

g T a∗2 − = 0. U2 ρU 2

(4.59)

Comparing (4.59) and (2.95), we see that a∗ = k, where k is a wavenumber satisfying the dispersion relation (2.95) with c = U . Referring to Figure 2.6, we see that there are two values a∗1 < a∗2 of a∗ for U > cmin , where cmin is given by (2.98). These values are given explicitly by solving the quadratic equation (4.59); they are 1/2 2 4T g ρU , (4.60) 1− 1− a∗1 = 2T ρU 4 1/2 2 4T g ρU a∗2 = 1+ 1− . 2T ρU 4

(4.61)

However, there are no values a∗ when U < cmin ; then the solution is given by (4.58). ˜ in the form When U > cmin , we rewrite D(a) a ˜ D(a) = 2 [U 2 − C 2 (a)] − iµ1 , (4.62) U where g T C 2 (a) = + a. (4.63) a ρ

124

Linear free-surface ﬂows generated by moving disturbances

The remaining part of the calculations follows step by step that of Section 4.2.1, with the tanh factors replaced by 1. This leads to η1 (x) = (iπRI+ ) +

2π B(a∗1 ) sin a∗1 x, ρU a∗1 C (a∗1 )

x > 0,

(4.64)

η1 (x) = (iπRI− ) −

2π B(a∗2 ) sin a∗2 x, ρU a∗2 C (a∗2 )

x < 0.

(4.65)

and

Taking the limit |x| → ∞ we ﬁnd η1 (x) ≈

2π B(a∗1 ) sin a∗1 x ρU a∗1 C (a∗1 )

as

x→∞

(4.66)

x → −∞.

(4.67)

and η1 (x) ≈ −

2π B(a∗2 ) sin a∗2 x as ρU a∗2 C (a∗2 )

As in water of ﬁnite depth, (4.66) and (4.67) and the inequality a∗1 < a∗2 imply that the solutions are characterised by a wave train of ‘long’ wavelength as x → ∞ and a wave train of ‘short’ wavelength as x → −∞. For U values not close to cmin , the long waves are dominated by gravity and the short waves are dominated by surface tension. The solution in the absence of surface tension is obtained by taking the limit T → 0. Relations (4.60) and (4.61) show that a∗1 → g/U 2

and a∗2 → ∞

as

T → 0.

(4.68)

It follows that the right-hand side of (4.67) tends to zero. Therefore the free surface is ﬂat as x → −∞ and is characterised by wave trains of wavenumber g/U 2 as x → ∞.

4.2.3 Discussion of the solutions We illustrate these results ﬁrst in the particular case of water of inﬁnite depth; then cmin is given by (2.98). Following Vanden-Broeck [184] and others, we deﬁne the parameter α=

Tg . ρU 4

(4.69)

Then U < cmin and U > cmin correspond respectively to α > 0.25 and α < 0.25.

4.2 Linear theory

125

We present in Figures 4.5 and 4.6 typical free-surface proﬁles predicted by (4.58) for (see (4.7), (4.8)) equal to ±0.001 and α equal to 0.27. In these calculations we chose P (x) =

ρU 2 −5g 2 x2 /U 4 . e 2

(4.70)

Since α > 0.25, the free surface is ﬂat in the far ﬁeld (i.e. there are no wave trains in the far ﬁeld). Since the pressure distribution is symmetric about x = 0, the free-surface proﬁles are also symmetric about x = 0. They are characterised by oscillations of decaying amplitude as |x| → ∞. This is consistent with the fact that (4.59) does not have real roots a∗ for α > 1/4. In fact solving (4.59) for a∗ yields a∗ =

ρU 2 [1 ± (1 − 4α)1/2 ], 2T

(4.71)

and a∗ is complex when α > 0.25. 0.002

0.001

0

0

10

20

Fig. 4.5. Linear free-surface proﬁle for the pressure distribution (4.70) with = 0.001. The value of α is 0.27.

In Figure 4.7 we present a typical proﬁle calculated from (4.64) and (4.65) with = 0.001 and α = 0.222. The pressure distribution is deﬁned by (4.70). Since α < 1/4, there are wave trains in the far ﬁeld, in accordance with (4.66) and (4.67). We now examine some limitations and nonuniformities of the linear theory. Those limitations will be resolved in Chapter 7 where we develop a nonlinear theory. First, the linear result (4.58) predicts unbounded displacements of the free surface as α → 1/4+ . This is shown analytically by noting that when

126

Linear free-surface ﬂows generated by moving disturbances 0.001

0

0

10

20

Fig. 4.6. Linear free-surface proﬁle for the pressure distribution (4.70) with = −0.001. The value of α is 0.27. 0.002

0.001

0

0

5

10

15

Fig. 4.7. Linear free-surface proﬁle for the pressure distribution (4.70) with = 0.001. The value of α is 0.222.

α = 0.25, the denominator of the integrand in (4.58) can be written as T − ρU 2

ρU 2 a− 2T

2 (4.72)

and so the integral in (4.58) diverges as α → 0.25+ , because of the double zero in (4.72). Similarly a∗1 →

ρU 2 2T

and a∗2 →

Furthermore, C

ρU 2 2T

ρU 2 2T

as

α → 0.25− .

(4.73)

= 0.

(4.74)

4.2 Linear theory

127

Therefore (4.66) and (4.67) predict waves of unbounded amplitude as α → 0.25− . This behaviour is shown graphically in Figure 4.8, where we present values of the elevation of the free surface at x = 0, i.e. ∞ B(a) 2 da, (4.75) η1 (0) = 2 ) − (T a2 /ρU 2 ) ρU 2 a − (g/U 0 versus α for α > 1/4. Here P (x) is deﬁned by (4.70) and = ±0.001. As α → 1/4+ , η1 (0) tends to inﬁnity in accordance with (4.72). 0.008

0.004

0

0.25

0.26

0.27

0.28

0.29

0.30

Fig. 4.8. Values of the amplitude of the free-surface proﬁle versus α for the pressure distribution (4.70) with = −0.001 (upper curve) and = 0.001 (lower curve). The unit length in the vertical scale is U 2 /g.

These results show that, for a given value of (however small), η1 (x) and φ1 can be made arbitrarily large by taking α suﬃciently close to 1/4. This is in contradiction with the assumptions leading to (4.7) and (4.8). In other words, the expansions (4.7) and (4.8) are nonuniform as α → 1/4. As we shall see in Chapter 7, appropriate solutions for α close to 1/4 can be obtained by using a nonlinear theory. We note that the nonuniformity of the linear theory in water of inﬁnite depth is associated with the existence of a minimum phase velocity (see (2.98)). Similar nonuniformities occur in water of ﬁnite depth and are associated with the minimum occurring in the phase velocity when τ < 1/3 (see Figure 2.5). Second, a similar nonuniformity occurs for pure gravity ﬂows (g = 0, T = 0) in ﬁnite depth as F → 1. This can be shown by setting T = 0 in (4.30) and noting that for a small the denominator of the integrand in (4.30) then behaves like 1 a3 h2 1 + ··· (4.76) 1− 2 a+ 2 F F 3

128

Linear free-surface ﬂows generated by moving disturbances

Therefore the integral in (4.30) diverges as F → 1. This phenomenon is illustrated graphically in Figure 4.9, where we plot values of η1 (0) versus F for F > 1. The curve has a vertical asymptote at F = 1, and the linear theory is not valid for F close to 1. Again, this nonuniformity will be removed in Chapter 7 where we develop nonlinear theories. 0.04

0.03

0.02

0.01

0 0.4

0.6

0.8

1.0

1.2

1.4

Fig. 4.9. Values of the amplitude of the free-surface proﬁle versus F for the pressure distribution (4.70) with = 0.001. The unit of length in the vertical scale is U 2 /g.

As we shall see in Chapter 7 the nonuniformities in Figures 4.8 and 4.9 are associated with the existence of solution branches corresponding to solitary waves.

5 Nonlinear waves – asymptotic solutions

The linear results of Chapter 4 show examples of free-surface ﬂows that approach either periodic wave trains or ﬂat free surfaces in the far ﬁeld. As we shall see this is also the case for nonlinear free-surface ﬂows. However, periodic waves if present are then nonlinear. This motivates a study of nonlinear travelling gravity–capillary waves since they describe the far-ﬁeld behaviour of nonlinear ﬂows past disturbances. These solutions are useful in nonlinear computations because they can be employed to derive eﬃcient truncation procedures (see for example Section 7.2.1). They are also interesting canonical free-surface ﬂow problems. This chapter is concerned with asymptotic solutions for nonlinear waves. Fully nonlinear numerical solutions will be presented in Chapter 6. 5.1 Periodic waves As shown in Section 2.4.1, nonlinear travelling gravity–capillary periodic waves are described by the equations φxx + φyy = 0, φy = φx ηx

−h < y < η(x), on y = η(x),

ηxx 1 2 T =B (φx + φ2y ) + gy − 2 ρ (1 + ηx2 )3/2

on y = η(x),

(5.1) (5.2) (5.3)

∇φ(x + λ, y) = ∇φ(x, y),

(5.4)

η(x + λ) = η(x),

(5.5)

1 λ

λ

φx dx = c, 0

129

(5.6)

130

Nonlinear waves – asymptotic solutions

λ

η(x)dx = 0.

(5.7)

0

The ﬂow conﬁguration is illustrated in Figure 2.4. We will derive analytical approximations by improving on the linear approximations of Section 2.4. We ﬁrst introduce an amplitude parameter to measure the size of the waves (a precise deﬁnition will be given later) and seek solutions in the form of expansions in powers of . Thus we write φ(x, y) = cx + φ1 (x, y) + 2 φ2 (x, y) + 3 φ3 (x, y) + O(4 ),

(5.8)

η(x) = η1 (x) + 2 η2 (x) + 3 η3 (x) + O(4 ),

(5.9)

c = c0 + c1 + 2 c2 + 3 c3 + O(4 ),

(5.10)

B = B0 + B1 + 2 B2 + 3 B3 + O(4 ).

(5.11)

Such expansions were ﬁrst used by Stokes (see [143]) and (5.8)–(5.11) are often referred to as the Stokes expansion. As we shall see, φ1 (x, y), η1 (x) are the linear approximations (2.88), (2.89) and the terms of order 2 , 3 , . . . are nonlinear corrections. One diﬃculty is that the boundary conditions (5.2) and (5.3) are expressed on y = η(x), where η(x) is itself expressed as an expansion in powers of (see (5.9)). Therefore it is not possible to equate powers of directly. We resolve this diﬃculty by expanding expanding φy (x, η(x)) and φx (x, η(x)) in Taylor expansions about y = 0. This leads to φy (x, η(x)) = φy (x, 0) + φyy (x, 0)η(x) + φyyy (x, 0) + φyyyy (x, 0)

η 3 (x) + ··· , 6

φx (x, η(x)) = φx (x, 0) + φxy (x, 0)η(x) + φxyy (x, 0) + φxyyy (x, 0)

η 2 (x) 2 (5.12) η 2 (x) 2

η 3 (x) + ··· . 6

(5.13)

Subsituting the expansions (5.8)–(5.10) into (5.12) and (5.13) gives φy (x, η(x)) = φ1y + 2 (φ2y + φ1yy η1 ) 3 2 φ1yyy + φ3y + φ2yy η1 + η2 φ1yy + η1 + O(4 ). 2

(5.14)

5.1 Periodic waves

131

Similarly, we obtain φx (x, η(x)) = c0 + (c1 + φ1x ) + 2 (c2 + φ2x + φ1xy η1 ) 3 2 φ1xyy + c3 + φ3x + φ2xy η1 + η2 φ1xy + η1 + O(4 ). 2 (5.15) Substituting (5.8)–(5.11) into (5.1)–(5.7), using (5.14) and (5.15) and equating equal powers of yields B0 =

c20 2

(5.16)

and the three linear problems φ1xx + φ1yy = 0,

−h < y < 0,

φ1y − c0 η1x = 0, φ1y = 0, −

y = 0,

(5.19) y = 0;

−h < y < 0,

φ2y − c0 η2x = c1 η1x − η1 φ1yy + φ1x η1x , φ2y = 0,

(5.18)

y = −h,

T η1xx + c0 φ1x + gη1 = B1 − c0 c1 , ρ φ2xx + φ2yy = 0,

(5.17)

(5.20) (5.21)

y = 0,

y = −h,

(5.22) (5.23)

T − η2xx + c0 φ2x + gη2 = −c0 η1 φ1xy − c1 φ1x − 12 (φ21x + φ21y ) ρ + B2 − 12 c21 − c0 c2 , y = 0; (5.24) φ3xx + φ3yy = 0,

−h < y < 0,

(5.25)

φ3y − c0 η3x = −η1 φ2yy − η2 φ1yy − 12 η12 φ1yyy + c1 η2x + c2 η1x + η1x φ2x + η1x η1 φ1xy + φ1x η2x , φ3y = 0,

y = −h,

y = 0,

(5.26) (5.27)

132

Nonlinear waves – asymptotic solutions

T 3T 2 − η3xx + c0 φ3x + gη3 = − η1xx η1x − c1 c2 − c1 φ2x − c1 φ1xy η1 ρ 2ρ − φ1x c2 − φ1x φ2x − φ1x φ1xy η1 − c0 c3 − c0 φ2xy η1 − c0 η2 φ1xy − 12 c0 η12 φ1xyy − φ1y φ2y − φ1y φ1yy η1 + B3 ,

y = 0. (5.28)

We shall restrict our attention to solutions that are symmetric. Assuming that the origin of x is chosen so that the wave is symmetric about the axis y = 0, we impose φ(−x, y) = −φ(x, y),

(5.29)

η(−x) = η(x).

(5.30)

Using the expansions (5.8) and (5.9) we have φi (−x, y) = −φi (x, y), ηi (−x) = ηi (x),

i = 1, 2, . . . ,

i = 1, 2, . . . ,

(5.31) (5.32)

Applying the method of separation of variables, we seek a solution of (5.17) in the form φ1 (x, y) = X(x)Y (y).

(5.33)

Substituting (5.33) in (5.17) yields X (x) Y (y) =− = constant = −α2 . X(x) Y (y)

(5.34)

Here we choose the separation constant to be negative, so that the solution is periodic in x. Relations (5.33) and (5.19) yield Y (−h) = 0.

(5.35)

Equation (5.34) gives two ordinary diﬀerential equations. Using the conditions (5.31) and (5.35), we write their solutions as X(x) = sin αx,

(5.36)

Y (y) = cosh α(y + h).

(5.37)

The periodicity condition (5.4) implies that α = nk,

(5.38)

5.1 Periodic waves

133

where n is a positive integer and k=

2π λ

(5.39)

is the wavenumber. Multiplying (5.36) by (5.37) and taking a linear combination of the solutions corresponding to the values (5.38) of α, we obtain φ1 (x, y) =

∞

Bn cosh nk(y + h) sin nkx.

(5.40)

n=1

Similarly, we can write the solutions of (5.21) and (5.25) as φ2 (x, y) =

∞

Fn cosh nk(y + h) sin nkx,

(5.41)

Gn cosh nk(y + h) sin nkx.

(5.42)

n=1

φ3 (x, y) =

∞ n=1

Here Bn , Fn and Gn are constants. Using (5.7), the periodicity condition (5.5) and the symmetry conditions (5.32), we may express η1 (x), η2 (x) and η3 (x) as Fourier series: η1 (x) =

∞

An cos nkx,

(5.43)

En cos nkx,

(5.44)

Dn cos nkx,

(5.45)

n=1

η2 (x) =

∞ n=1

η3 (x) =

∞ n=1

where An , En and Dn are constants. We now deﬁne the parameter as =

a , λ

where a is the ﬁrst Fourier coeﬃcient of η(x), i.e. 2 λ η(x) cos kx dx. a= λ 0

(5.46)

(5.47)

It follows from (5.43)–(5.45) that A1 = λ

(5.48)

134

Nonlinear waves – asymptotic solutions

and E1 = D1 = 0.

(5.49)

Substituting (5.40) and (5.43) into (5.18) and equating coeﬃcients of sin nkx yields c0 An = −Bn sinh nkh.

(5.50)

Similarly, substituting (5.40) and (5.43) into (5.20) gives B1 = c0 c1 and T An n2 k 2 + gAn + c0 Bn nk cosh nkh = 0. ρ Eliminating Bn between (5.50) and (5.51) yields T 2 2 c20 nk g+ n k − An = 0, ρ tanh nkh

n = 1, 2, . . .

Since A1 = 0 (see (5.48)), (5.52) with n = 1 implies that g T 2 c0 = + k tanh kh. k ρ

(5.51)

(5.52)

(5.53)

Relation (5.52) with n > 1 gives An = 0,

n = 2, 3, . . . ,

(5.54)

provided that g+

c20 nk T 2 2 n k − = 0. ρ tanh nkh

(5.55)

The solution of the ﬁrst order problem is then φ1 = −

c0 A1 cosh k(y + h) sin kx, sinh kh η1 = A1 cos kx.

(5.56) (5.57)

However, if for some value m of the integer n, the condition (5.55) is not satisﬁed i.e. if T c20 mk = 0, (5.58) g + m2 k 2 − ρ tanh mkh then the solution is c0 Am c0 A1 cosh k(y + h) sin kx − cosh mk(y + h) sin mkx, φ1 = − sinh kh sinh mkh (5.59) η1 = A1 cos kx + Am cos mkx,

(5.60)

5.1 Periodic waves

135

where Am is arbitrary at this stage of the calculations. As we shall see the values of Am are determined by solvability conditions at higher order in . We conclude this subsection by noting that in the particular case of water of inﬁnite depth, so that h → ∞, the condition (5.55) and the solutions (5.56), (5.57) and (5.59), (5.60) reduce to T 2 2 n k − c20 nk = 0, ρ

(5.61)

φ1 = −c0 A1 eky sin kx,

(5.62)

η1 = A1 cos kx,

(5.63)

φ1 = −c0 A1 eky sin kx − c0 Am emky sin mkx,

(5.64)

η1 = A1 cos kx + Am cos mkx.

(5.65)

g+

Using (5.53), equation (5.61) reduces further to 1 , n

(5.66)

4π 2 T . ρgλ2

(5.67)

κ = where κ is deﬁned by κ=

5.1.1 Solutions when condition (5.55) is satisﬁed We consider ﬁrst the case when the condition (5.55) is satisﬁed. Substituting (5.56) and (5.57) into the right-hand sides of (5.22) and (5.24), we obtain, after some algebra, φ2y − c0 η2x = −c1 A1 k sin kx + c0 A21 k 2 coth kh sin 2kx cosh2 kh T 1 c2 A2 k 2 − η2xx + c0 φ2x + gη2 = 0 1 − c20 k 2 A21 ρ 4 4 sinh2 kh (5.68) cosh kh cos kx + c0 c1 A1 k sinh kh 3 2 2 2 1 2 2 cosh2 kh cos 2kx + B2 . c0 A1 k − c0 k + 4 4 sinh2 kh Using the expansions (5.41) and (5.44) in the left-hand sides of the previous two relations and equating coeﬃcients of sin nkx and cos nkx yields B2 =

A21 1 c21 + c20 k 2 , 2 4 sinh2 kh

(5.69)

136

Nonlinear waves – asymptotic solutions

F1 k sinh kh + c0 E1 k = −c1 A1 k,

(5.70)

F1 k cosh kh tanh kh = c1 A1 k,

(5.71)

2kF2 sinh 2kh + 2kc0 E2 = 2kF2 cosh 2kh + 4k

2T

ρ

+g

A21 k 2 c0 , tanh kh

E2 3 1 c0 A21 k 2 = c0 A21 k 2 − , c0 4 4 tanh2 kh

nkFn sinh nkh + nkc0 En = 0,

n = 3, 4, . . .

T En n2 k 2 + c0 nkFn cosh nkh + gEn = 0, ρ

n = 3, 4, . . .

(5.72)

(5.73) (5.74) (5.75)

Equations (5.49), (5.70) and (5.71) give F1 = 0,

c1 = 0.

(5.76)

Solving (5.72) and (5.73) yields E2 =

A21 k 2 c20 4 tanh kh − 3 tanh2 kh tanh 2kh + tanh 2kh , 4 tanh2 kh[2kc20 − (4k 2 T /ρ + g) tanh 2kh] 1 F2 = 2k sinh 2kh

A21 k 2 c0 − 2kc0 E2 . tanh kh

(5.77)

(5.78)

Solving (5.74) and (5.75) gives Fn = 0,

En = 0,

n = 3, 4, . . .

(5.79)

Therefore η2 = E2 cos 2kx,

(5.80)

φ2 = F2 cosh 2nk(y + h) sin 2nkx,

(5.81)

where E2 and F2 are deﬁned by (5.77) and (5.78). Combining (5.9), (5.47) and (5.80) gives η = A1 cos kx + 2 E2 cos 2kx.

(5.82)

To conclude the calculation up to order 2 we need to calculate c2 . This is done by substituting the expressions (5.43), (5.46), (5.47), (5.76), (5.80) and (5.81) for η1 , η2 , φ1 , φ2 , c0 and c1 into the right-hand sides of (5.26) and (5.28) and the series (5.42) and (5.45) into the left-hand sides. Equating the

5.1 Periodic waves

137

coeﬃcients of cos kx and those of sin kx gives two equations. Eliminating G1 between these two equations yields, after some algebra, c2 =

c0 A21 k 2 (coth kh + 4 coth 2kh − 3 tanh kh)S2 4 3T k 2 + 3 − 2 coth kh coth 2kh − tanh kh , 4ρgS1

where

S1 =

T k2 1+ ρg

(5.83)

tanh kh

(5.84)

and (T k 2 + ρg) tanh kh[tanh 2kh − 2 coth kh − tanh 2kh (2 sinh2 kh)−1 ] . (16T k 2 + 4ρg) tanh 2kh − 8(T k2 + ρg) tanh kh (5.85) Therefore the phase velocity c is given up to order 2 by (5.10), where c0 , c1 and c2 are deﬁned by (5.53), (5.76) and (5.83) respectively. We conclude this section by examining the behaviour of the solution (5.10) in the shallow-water limit (i.e. as kh → 0). Using the properties S2 = 2

sinh kh ≈ kh

as

kh → 0

and cosh kh ≈ 1 as kh → 0, we ﬁnd that (5.83) yields c2 1 ≈ c0 (kh)2

as

kh → 0.

(5.86)

Since c1 = 0, (5.10) then implies that, on the one hand, the ratio of the ﬁrst two terms of the expansion (5.10) for c satisﬁes 2 c2 2 ≈ c0 (kh)2

as

kh → 0;

(5.87)

on the other hand, the expansion (5.10) is only valid asymptotically if the ratio 2 c2 /c0 is small in the limit → 0. Relation (5.87) shows that for any value of > 0 the ratio 2 c2 /c0 can be made arbitrarily large by taking kh suﬃciently small. In other words the asymptotic expansions derived in this section become invalid as kh → 0. Appropriate asymptotic expansions that are valid when both and kh are small will be derived in Section 5.2.

138

Nonlinear waves – asymptotic solutions

5.1.2 Solutions when condition (5.55) is not satisﬁed We now assume that the condition (5.55) is not satisﬁed for some value m of n. We will present explicit calculations for m = 2. Therefore we have 2kc20 T cos 2kh = 0. g + 4 k2 − ρ sinh 2kh

(5.88)

The solutions for η1 and φ1 are obtained by setting m = 2 in (5.59) and (5.60). This gives φ1 = −

c0 A2 c0 A1 cosh k(y + h) sin kx − cosh 2k(y + h) sin 2kx, (5.89) sinh kh sinh 2kh η1 = A1 cos kx + A2 cos 2kx.

(5.90)

We determine the constants A2 and c1 by substituting (5.89) and (5.90) into the right-hand sides of (5.22) and (5.24). To simplify the algebra and to illustrate the main ideas we shall present these calculations in the particular case of inﬁnite depth. Taking the limit h → ∞ in (5.53), (5.88) and (5.89) gives g T (5.91) c20 = + k, k ρ T g + 4 k 2 − 2kc20 = 0, ρ

(5.92)

φ1 = −c0 A1 eky sin kx − c0 A2 e2ky sin 2kx.

(5.93)

As noted earlier, the condition (5.92) can be rewritten in the simple form 1 κ= , 2

(5.94)

where κ is deﬁned by (5.67). In water of inﬁnite depth, φ ≈ cx as y → −∞. Therefore (5.8) implies that φi → 0,

i = 1, 2, . . .

as

y → −∞.

(5.95)

Using (5.95) with i = 2 and following the derivation of (5.40) we obtain φ2 (x, y) in the form φ2 (x, y) =

∞ n=1

F˜n enky sin nkx.

(5.96)

5.1 Periodic waves

139

Substituting (5.44), (5.96), (5.90) and (5.93) into (5.22) and (5.24) and equating coeﬃcients of sin kx and cos kx yields 3 k F˜1 = −c1 A1 k + c0 A1 A2 k 2 , 2 c2 A1 A2 k 2 + c0 c1 A1 k. c0 F˜1 k = 0 2 Similarly, equating the coeﬃcients of sin 2kx and cos 2kx yields 2k F˜2 + 2kc0 E2 = −2c1 A2 k + c0 A21 k 2 , c2 A2 k 2 T 4 k 2 E2 + gE2 + 2F˜2 kc0 = 0 1 + 2c0 c1 A2 k. ρ 2

(5.97) (5.98)

(5.99) (5.100)

Solving (5.97) and (5.98) gives c1 =

c0 A2 k . 2

(5.101)

Similarly (5.99) and (5.100) give T c2 k 2 − 4 k 2 E2 − gE2 + 2kc20 E2 = 0 A21 − 4c0 c1 kA2 ρ 2 and thus

E2

2kc20

4T k 2 −g − ρ

= −4c0 c1 A2 k +

c20 A21 k 2 . 2

(5.102)

(5.103)

Using (5.91) and (5.92), it can easily be checked that the contents of the parentheses on the left-hand side of (5.103) equal zero. Therefore c1 A2 =

A21 kc0 . 8

(5.104)

A1 2

(5.105)

Combining (5.101) and (5.104) yields A2 = ±

1 η1 = A1 cos kx ± A1 cos 2kx, 2 φ1 = −c0 A1 eky sin kx ∓

c0 A1 e2ky sin 2kx, 2

1 c1 = ± c0 kA1 . 4

(5.106) (5.107) (5.108)

140

Nonlinear waves – asymptotic solutions

We deﬁne the steepness s of the wave by the equation s=

η(0) − η(λ/2) λ

(5.109)

Noting that = s/2 to leading order and using (5.9), (5.10), (5.48), (5.106) and (5.108) we obtain π (5.110) c = c0 ± s c 0 + · · · , 4 s s η = cos kx ± cos 2kx + · · · . λ 2 4

(5.111)

Relations (5.106) and (5.111) show that there are two solutions when (5.94) is satisﬁed (one corresponding to the + sign and one to the − sign). The corresponding free-surface proﬁles in water of inﬁnite depth are shown in Figures 5.1 and 5.2. 1.5

1.0

0.5

0

0

5

10

15

20

Fig. 5.1. The Wilton ripple (5.106) with A1 = 1 and k = 1. This solution corresponds to the + sign in (5.106).

They are usually referred to as the Wilton ripples (because Wilton [198] was one of the ﬁrst to calculate them). One solution is characterised by a crest dimple and the other by a trough dimple. We note that (5.82) and (5.111) both involve cos kx and cos 2kx. However, a crucial diﬀerence is that the cos 2kx term is of order 2 in (5.82) whereas it is of order s, and therefore of order , in (5.111). This implies that (5.82) approaches A1 cos kx as → 0 and that (5.111) approaches (s/2)[cos kx ± (1/2) cos 2kx] as → 0. Therefore the solution (5.111) exhibits the trough and crest dimples shown in Figures 5.1 and 5.2 however small and s are. The above calculations illustrate the nonuniqueness of gravity–capillary periodic waves: we obtain two diﬀerent solutions if the condition (5.58) is

5.1 Periodic waves

141

1.0

0.5

0

0

5

10

15

20

Fig. 5.2. The Wilton ripple (5.106) with A1 = 1 and k = 1. This solution corresponds to the − sign in (5.106).

satisﬁed for m = 2. Similar nonuniqueness occurs when (5.58) is satisﬁed for values of m > 2. Calculations were performed by Nayfeh [116] for m = 3. The analysis can also be extended to ﬁnite depth, and solutions valid when (5.58) is satisﬁed can be derived. However, the calculations for larger values of m become quickly intractable because it is necessary to calculate the solution to high order in in order to ﬁnd the values of Am in (5.60). Such solutions will be calculated numerically in Sections 6.5.3.1 and 6.5.3.2. We also note that the denominator in (5.77) vanishes when (5.58) is satisﬁed for m = 2. Similarly, the denominators in ηn and φn for n ≥ 3 are also found to vanish when (5.58) is satisﬁed for some integer value m. This is consistent with the solutions of Section 5.1.1, which were derived under the assumption (5.55). However, the expansions of Section 5.1.1 are nonuniform in the sense that the ratios of successive terms (for example the ratio of 2 η2 and η1 ) can be made arbitrarily large by making the left-hand side of (5.55) suﬃciently small by appropriate choices of the parameters. For future comparison of the present results with the numerical computations of Chapter 6 we now rewrite (5.91) and (5.110) in terms of the parameter κ (see (5.67)) and a parameter µ deﬁned by 2πc2 . gλ

(5.112)

µ=1+κ

(5.113)

µ= This gives

142

Nonlinear waves – asymptotic solutions

and 3 3 πs (1 + κ) = ± πs + · · · (5.114) 2 2 4 Finally let us mention that Wilton [198] calculated further terms in the expansion and found 3 3 s2 µ = ± πs − π 2 ··· (5.115) 2 4 4 µ=1+κ±

5.2 The Korteweg–de Vries equation As we saw in Section 5.1.1, the Stokes expansion becomes nonuniform as h → 0. This suggests that a diﬀerent expansion should be used when both the depth h and the amplitude of the waves are small. We study in this section asymptotic expansions valid when h is small with various assumptions on the size of the amplitude of the waves. For the sake of generality we will derive these equations in the time-dependent case. Travelling waves will then be found as particular cases. The exact time-dependent-potential free-surface ﬂow equations are given by (2.102)–(2.105). We rewrite these equations by choosing the origin of y on the bottom. This gives φxx + φyy = 0,

0 < y < h + η,

(5.116)

φy = 0 on y = 0, ηt + φx ηx − φy = 0

(5.117)

on y = h + η,

ηxx 1 T gη + φt + (φ2x + φ2y ) − =C 2 ρ (1 + ηx2 )3/2

on

y = h + η,

(5.118) (5.119)

where h is the depth of the ﬂuid. Here we have written the elevation of the free surface as y = h + η, where h is the undisturbed depth. We introduce the velocity scale c0 = (gh)1/2 and denote the typical amplitude of η by a. We also deﬁne the amplitude parameter a (5.120) α= h and the depth parameter h2 β= 2. (5.121) l Next we rescale the variables as x y c0 x ˜ = , y˜ = , t˜ = t, (5.122) l h l

5.2 The Korteweg–de Vries equation

η˜ =

η , a

c0 φ˜ = φ. gla

143

(5.123)

That the scalings in the x- and y- directions in (5.122) are diﬀerent is crucial in deriving shallow-water approximations. We now rewrite the nonlinear equations (5.116)–(5.119) in terms of the new variables (5.122) and (5.123). After dropping the tilde this leads to βφxx + φyy = 0, φy = 0 ηt + αφx ηx −

0 < y < 1 + αη, on y = 0,

1 φy = 0 on y = 1 + αη, β

(5.124) (5.125) (5.126)

1 1α 2 ηxx φy − τ β = 0 on y = 1 + αη, (5.127) η + φt + αφ2x + 2 2β (1 + α2 βηx2 )3/2 where τ=

T . ρgh2

We seek φ in the form of an expansion in powers of y. Thus we write φ(x, y, t) =

∞

y n fn (x, t).

(5.128)

n=0

Substituting (5.128) into (5.124) and (5.125) yields fn = 0 fn = −

for n = 1, 2, 3, . . . ,

∂ 2 fn−2 β , n(n − 1) ∂x2

n = 2, 4, 6, . . .

(5.129) (5.130)

Solving (5.130) recursively gives φ=

∞ n=0

(−1)n

y 2n ∂ 2n f n β , (2n)! ∂x2n

(5.131)

where f = f0 .

(5.132)

Substituting (5.131) into the boundary conditions (5.126) and (5.127) gives ηt + [(1 + αη)fx ]x − [ 16 (1 + αη)3 fxxxx + 12 α(1 + αη)2 ηx fxxx ]β + O(β 2 ) = 0, (5.133)

144

Nonlinear waves – asymptotic solutions 2 η + ft + 12 αfx2 − 12 (1 + αη)2 (fxxt + αfx fxxx − αfxx )β ηxx + O(β 2 ) = 0. −τ β (1 + α2 βηx2 )3/2

(5.134)

If all terms of order β are dropped and (5.134) is diﬀerentiated with respect to x, we obtain ηt + [(1 + αη)w]x = 0,

(5.135)

wt + αwwx + ηx − τ βηxxx = 0,

(5.136)

w = fx .

(5.137)

where

Equations (5.135) and (5.136) are the shallow-water equations. They deﬁne a hyperbolic system of partial diﬀerential equations that does not admit travelling waves (see Whitham [195]). The Korteweg–de Vries equation with the eﬀects of surface tension included can be derived from (5.133) and (5.134) by specialising to waves moving to the right. To lowest order in α and β (i.e. to order α0 and β 0 ), (5.135) and (5.136) reduce to ηt + wx = 0,

(5.138)

wt + ηx = 0

(5.139)

and a solution moving to the right is w = η,

ηt + ηx = 0.

(5.140)

Following Whitham [195], we seek solutions correct to order α and order β in the form w = η + αA + βB + O(α2 + β 2 ).

(5.141)

Here A and B are functions of η and its derivatives. Equations (5.133) and (5.134) become ηt + ηx + α(Ax + 2ηηx ) + β Bx − 16 ηxxx + O(α2 + β 2 ) = 0, (5.142) ηt + ηx + α(At + ηηx ) + β Bt − 12 ηxxt − βτ ηxxx + O(α2 + β 2 ) = 0. (5.143) Since ηt = −ηx + O(α, β), we can replace the t-derivatives by minus the x-derivatives in the ﬁrst-order terms and rewrite (5.143) as ηt +ηx +α(−Ax +ηηx )+β −Bx + 12 ηxxx −βτ ηxxx +O(α2 +β 2 ) = 0. (5.144)

5.2 The Korteweg–de Vries equation

145

Equations (5.142) and (5.144) are consistent if the coeﬃcients of α and β in both equations are the same. This yields Ax + 2ηηx = −Ax + ηηx ,

(5.145)

Bx − 16 ηxxx = −Bx + 12 ηxxx − τ βηxxx .

(5.146)

Integrating (5.145) and (5.146) gives A = − 14 η 2 , B = 13 − 12 τ ηxx .

(5.147) (5.148)

Substituting (5.147) and (5.148) into (5.144) gives the Korteweg–de Vries equation, ηt + ηx + 32 αηηx + 16 (1 − 3τ )βηxxx + O(α2 + β 2 ) = 0.

(5.149)

We can now rewrite (5.149) in terms of the unscaled dimensional variables (see (5.120)–(5.123)) as 3 c0 1 (5.150) ηηx + c0 h2 (1 − 3τ )ηxxx = 0. 2h 6 We construct travelling-wave solutions of (5.150) by assuming that ηt + c0 ηx +

η = hζ(X),

where

X = x − U t.

(5.151)

The function η(X) represents a wave travelling to the right with constant velocity U . Substituting (5.151) into (5.150) gives the ordinary diﬀerential equation 3 1 − hU ζ + c0 hζ + c0 hζζ + c0 h2 (1 − 3τ )hζ = 0, (5.152) 2 6 where the primes denote derivatives with respect to X. Integrating (5.152) and dividing by c0 h yields U 3 2 1 2 h (1 − 3τ )ζ + ζ − − 1 ζ + G = 0, (5.153) 6 4 c0 where G is a constant of integration. Multiplying (5.153) by ζ and integrating again gives U 1 2 2 3 (1 − 3τ )h ζ + ζ − 2 − 1 ζ 2 + 4Gζ + J = 0, (5.154) 3 c0 where J is another constant of integration. Equation (5.154) admits periodic travelling solutions. These were calculated analytically by Korteweg and de Vries [91] (see also [195] and [191]).

146

Nonlinear waves – asymptotic solutions

These solutions are called cnoidal waves because they are expressed in terms of the Jacobian elliptic function cn. As the ratio of their wavelength and the depth h tends to inﬁnity, the cnoidal waves approach solitary waves. These solitary waves consist of a single hump or a single depression, depending on the value of the parameter τ . As x → ±∞, η and its derivatives tend to zero. It then follows that G = J = 0, and (5.154) reduces to U 1 2 2 3 (1 − 3τ )h ζ + ζ − 2 − 1 ζ 2 = 0. (5.155) 3 c0 It can easily be checked that the solution of (5.155) is then ζ=a ˜ sech

2

where

3˜ a 2 4h (1 − 3τ )

a ˜=2

1/2 X,

U −1 . c0

(5.156)

(5.157)

When τ < 1/3, a ˜ > 0 and (5.156) is an elevation wave with U > c0 . The value of ζ increases from ζ = 0 at X = ∞, rises to a maximum ζ = a ˜ and then returns symmetrically to ζ = 0 at X = −∞. When τ > 1/3, a ˜ < 0 and (5.156) is a depression wave with U < c0 . The value of ζ decreases from ζ = 0 at X = ∞, reaches a minimum ζ = a ˜ and then returns symmetrically to ζ = 0 at X = −∞. The solution (5.156) predicts that dζ/dX becomes unbounded as τ → 1/3. This is due to the fact that the dispersive term ηxxx in (5.150) disappears when τ = 1/3. We note that in the expansion procedure leading to (5.150) it was assumed that α and β are both small and of the same order of magnitude, i.e. α = , β = where 0 < 1. To obtain a model equation valid near τ = 1/3, we need to include higher-order dispersive terms. This can be achieved by assuming that α = 2 ,

β=

(5.158)

and expanding τ near 1/3 by writing τ=

1 + τ1 + · · · 3

(5.159)

The resulting equation rewritten in terms of the dimensional variables is ηt + c0 ηx +

3 c0 1 c0 h4 ηηx + c0 h2 (1 − 3τ )ηxxx + ηxxxxx = 0. 2h 6 90

(5.160)

5.2 The Korteweg–de Vries equation

147

This is known as the ﬁfth-order Korteweg–de Vries equation. It was derived by Hunter and Vanden-Broeck [76] in the steady case and by Hunter and Scherule [75] in the unsteady case (see also [83]). An important question is whether there are fully nonlinear solutions consistent with (5.156) when their amplitude is small. We shall see in Chapter 6 that the answer is positive when τ = 0 or τ > 1/3. However, there are no fully nonlinear solutions of small amplitude consistent with (5.156) when 0 < τ < 1/3. The fully nonlinear solutions are then characterised by a train of ripples in the far ﬁeld. Such waves are often referred to as generalised solitary waves, to distinguish them from true solitary waves, which are ﬂat in the far ﬁeld.

6 Numerical computations of nonlinear water waves

6.1 Formulation We consider again a train of periodic waves like that shown in Figure 2.4. Suppose that the waves have wavelength λ and propagate to the left in a channel bounded below by a horizontal bottom. We take a frame of reference moving with the wave and seek steady solutions. We will present the numerical methods of interest in terms of cartesian coordinates with x = y = 0 at a crest of the wave (see Figure 6.1). Some alternative choices for the origin of coordinates will be used in Sections 6.4 and 6.5. Gravity is taken as acting in the negative y-direction. ψ=0

ψ = −Q

y A

x

C

B

λ

B'

ψ = −2Q

A'

C'

Fig. 6.1. A wave of wavelength λ travelling in a channel bounded below by a horizontal bottom. A frame of reference moving with the wave is chosen. The free surface ψ = 0, the bottom ψ = −Q and the image of the free surface ψ = −2Q are shown.

The potential function φ and the streamfunction ψ were introduced in Section 2.3. We choose φ = 0 at x = y = 0 and ψ = 0 on the free surface 148

6.1 Formulation

149

and denote by −Q the constant value of ψ on the horizontal bottom and by u and v the horizontal and vertical components of the velocity. As in Section 2.4 and Chapter 5, we deﬁne the phase velocity as the average horizontal velocity at a constant level of y in the ﬂuid. Using (5.6), we obtain 1 λ 1 1 λ 1 c= u(x, y)dx = φx dx = [φ(λ, y) − φ(0, y)] = φ(λ, y). (6.1) λ o λ 0 λ λ Therefore φ(λ, y) = cλ and it follows from the assumed periodicity and symmetry of the wave that nλ nλ , y = c , n = 0, ±1, ±2, . . . (6.2) φ 2 2 Following Stokes [144] and the analysis in Chapter 3, we will use the potential function and the streamfunction as independent variables. We deﬁne the complex potential f = φ + iψ. The kinematic boundary condition on the horizontal bottom can then be satisﬁed by using the method of images. Thus we reﬂect the ﬂow in the bottom (see Figure 6.1). The ﬂow conﬁguration in the complex potential plane is shown in Figure 6.2: it is the strip −∞ < φ < ∞, −2Q < ψ < 0. ψ

ψ=0 A

B

C

A'

B'

C'

φ

ψ = −2Q

Fig. 6.2. The ﬂow conﬁguration of Figure 6.1 in the complex potential plane.

As noted in Chapter 3, the obvious advantage of working in the complex f -plane is that the unknown free surface and its image in Figure 6.1 have been mapped onto the known boundaries ψ = 0 and ψ = −2Q.

150

Numerical computations of nonlinear water waves

We deﬁne the function δ + iβ by the relations δ + iβ =

1 = xφ + iyφ . u − iv

(6.3)

Since u − iv is an analytic function of f , δ + iβ is also an analytic function of f . Next we map the f -plane into the unit disk in the complex t-plane by the transformation t = e−2iπf /cλ = e−2iπφ/cλ e2πψ/cλ .

(6.4)

The ﬂow conﬁguration in the t-plane is shown in Figure 6.3.

B

B'

C' A'

C A

Fig. 6.3. The ﬂow conﬁguration of Figures 6.1 and 6.2 in the complex t-plane. The solid curve |t| = 1 corresponds to the free surface and the broken curve |t| = r02 to the image of the free surface.

The free surface is mapped into the circle |t| = 1, the bottom into the circle |t| = e−2πQ/cλ and the image of the free surface into the circle |t| = e−4πQ/cλ . To compute solutions, we need to express the fact that δ +iβ is an analytic function of t in the annulus r02 < |t| < 1, where r0 = e−2πQ/cλ .

(6.5)

There are two methods for doing this. The ﬁrst is to represent δ + iβ as a Laurent expansion in powers of t. This leads to the numerical procedure we call series truncation. The second is to apply the Cauchy integral formula to the function δ + iβ in the annulus of Figure 6.3. This leads to the boundary integral equation method. Both methods were introduced in Chapter 3. They are described within the framework of water waves in Sections 6.2 and 6.3 respectively.

6.2 Series truncation method

151

6.2 Series truncation method The function δ + iβ is an analytic function of t in the annulus r02 < |t| < 1. Therefore it can be represented by the Laurent expansion δ + iβ = a0 + = a0 +

∞ n=1 ∞

an tn +

∞

bn t−n

n=1

an e−2iπnf /cλ +

n=1

∞

bn e2iπnf /cλ .

(6.6)

n=1

Since ψ = −2Q is the image of the free surface ψ = 0 in the bottom we have δ(φ, −2Q) + iβ(φ, −2Q) = δ(φ, 0) − iβ(φ, 0),

(6.7)

and it follows from (6.6) that bn = an r02n ,

n = 1, 2, . . .

(6.8)

Furthermore (6.1) implies that 1 a0 = . c

(6.9)

Substituting (6.8) and (6.9) into (6.6) gives ∞

∞

n=1

n=1

1 an e−2iπnf /cλ + an r02n e2iπnf /cλ . δ + iβ = + c

(6.10)

We shall determine coeﬃcients an in (6.10) such that the dynamic boundary condition (5.3) is satisﬁed. To do so, we need to rewrite (5.3) in terms of δ and β. Using (3.3), (3.6) and (6.3) we obtain φ 1 1 T δ˜β˜φ − δ˜φ β˜ ˜ β(ϕ)dϕ − +g = B, (6.11) 2 δ˜2 + β˜2 ρ (δ˜2 + β˜2 )3/2 0 ˜ ˜ where B is the Bernoulli constant and δ(φ) = δ(φ, 0) and β(φ) = β(φ, 0) denote the values of δ and β on the free surface ψ = 0. In the remaining part of this section, we introduce dimensionless variables by using λ as the unit length and c as the unit velocity. In terms of the dimensionless variables, (6.11) becomes 1 κ δ˜β˜φ − δ˜φ β˜ 2π φ ˜ 1 β(ϕ)dϕ − + = B, (6.12) 2 δ˜2 + β˜2 µ 0 2πµ (δ˜2 + β˜2 )3/2 where κ and µ are deﬁned by (5.67) and (5.112) respectively, and (6.10)

152

Numerical computations of nonlinear water waves

becomes δ + iβ = 1 +

∞

an e−2iπnf +

n=1

∞

an r02n e2iπnf .

(6.13)

n=1

We truncate the inﬁnite series in (6.13) after N − 1 terms and determine the N + 1 unknowns B, µ and an , n = 1, . . . , N − 1, by collocation. Thus we introduce the N collocation points φI =

1 I −1 , 2N −1

I = 1, . . . , N,

(6.14)

and satisfy the dynamic boundary condition (6.12) at these points. This gives N nonlinear algebraic equations. The ﬁnal equation is obtained by ﬁxing the amplitude of the wave, for example by imposing 1/2 ˜ β(ϕ)dϕ = −s, (6.15) 0

where s is given and is called the steepness of the wave. It is deﬁned as the diﬀerence in height between a crest and a trough of the wave proﬁle divided by the wavelength. In the particular case of water of inﬁnite depth, r0 = 0 and (6.8) implies bn = 0, n = 1, 2, . . . The Laurent series (6.13) reduces then to the Taylor series ∞ ∞ an tn = an e−2iπnf , (6.16) δ + iβ = n=0

n=0

where a0 = 1. The representation (6.16) can be derived directly by noting that when r0 = 0, (6.4) maps the ﬂow domain into the unit circle 0 < |t| < 1 of the t-plane. Therefore δ + iβ can be represented by the Taylor expansion (6.16). Various variations of the series truncation method have been proposed. For example, Vanden-Broeck [172] calculated waves in water of inﬁnite depth by expanding the complex velocity w = u − iv instead of δ + iβ in powers of t, i.e. by writing ∞ cn e−2iπnf . (6.17) w =1+ n=1

6.3 Boundary integral equation method Instead of using the series representation (6.10), we now derive a relation between δ and β by using Cauchy integral formula (see (2.38)). Such an

6.3 Boundary integral equation method

153

approach was used in Chapter 3 to compute cusped cavities (see Section 3.1.2.2). The presentation in this section follows [135], [162] and [76]. We apply the Cauchy integral formula to the function δ + iβ in the complex t-plane with the contour C chosen as the boundaries of the annulus in Figure 6.3. This yields δ(t) + iβ(t) 1 dt. (6.18) δ(t0 ) + iβ(t0 ) = − iπ C t − t0 The contour C consists of the circle |t| = 1 oriented clockwise and of the circle |t| = r02 oriented anticlockwise. Here t0 is assumed to be on the unit circle |t| = 1 of Figure 6.3. Therefore the integral in (6.18) is a Cauchy principal value. Applying (6.4) with ψ = 0 as a change of variables and using (6.7), we can rewrite (6.18) as

ξ 1 e−iπϕ/ξ ˜ ˜ ˜ ˜ δ(φ) + iβ(φ) = [δ(ϕ) + iβ(ϕ)] dϕ ξ e−iπϕ/ξ − e−iπφ/ξ −ξ ξ 2 e−iπϕ/ξ r 0 ˜ ˜ − r02 [δ(ϕ) − iβ(ϕ)] dϕ , (6.19) 2 e−iπϕ/ξ − e−iπφ/ξ r −ξ 0 where ξ = cλ/2. Taking the real part of (6.19), we obtain the following ˜ ˜ relation between δ(φ) and β(φ):

ξ −iπϕ/ξ 1 e ˜ ˜ ˜ dϕ δ(φ) = [δ(ϕ) + iβ(ϕ)] ξ e−iπϕ/ξ − e−iπφ/ξ −ξ ξ 2 e−iπϕ/ξ r 2 0 ˜ ˜ −r0 dϕ , (6.20) [δ(ϕ) − iβ(ϕ)] r02 e−iπϕ/ξ − e−iπφ/ξ −ξ where denotes the real part. Relations (6.11) and (6.20) deﬁne a system of nonlinear integro-diﬀerential ˜ ˜ equations for the unknown functions δ(φ) and β(φ). We will solve this system numerically. First we introduce the mesh points cλ cλ (I − 1), I = 1, . . . , N, (6.21) φI = − + 2 N −1 the midpoints φI + φI−1 φm , I = 1, . . . , N − 1, (6.22) I = 2 and the unknowns ˜ I ), I = 1, . . . , N, (6.23) δ˜I = δ(φ ˜ I ), β˜I = β(φ

I = 1, . . . , N.

(6.24)

154

Numerical computations of nonlinear water waves

Since β˜1 = 0 = β˜N = 0 by symmetry, there are only 2N − 2 unknowns δ˜I and β˜I . The integrals in (6.20) can be approximated by the trapezoidal rule (which is spectrally accurate for periodic functions) with ϕ = φJ , J = 1, . . . , N , and φ = φm I , I = 1, . . . , N − 1. The symmetry of the quadrature and of the discretisation enable us to evaluate the Cauchy principal value as if it were an ordinary integral (see Section 3.1.2). This gives N − 1 equations. Next we evaluate the derivatives δ˜φ and β˜φ at the midpoints by centred diﬀerence formulae. For example, using two points we have δ˜I+1 − δ˜I , ) = δ˜φ (φm I h

I = 1, . . . , N − 1,

(6.25)

and using four points we have δ˜I−1 − 27δ˜I + 27δ˜I+1 − δ˜I+2 δ˜φ (φm ) = , I 24h

I = 2, . . . , N − 2.

(6.26)

Here h=

cλ 2ξ = N −1 N −1

is the mesh size. We also evaluate δ˜ and β˜ at the midpoints by centred interpolation formulae. For example, using two points we have ˜ ˜ ˜ m ) = δI+1 + δI , δ(φ I 2

I = 1, . . . , N − 1,

(6.27)

and using four points we have ˜ ˜ ˜ ˜ ˜ m ) = −δI−1 + 9δI + 9δI+1 − δI+2 , δ(φ I 16

I = 2, . . . , N − 2.

(6.28)

We note that formulae like (6.26) and (6.28) can be extended for I = 1 and I = N − 1 since δ˜I can be deﬁned for I < 1 and I > N by using the assumed periodicity and symmetry of the wave. Now we need to approximate the integral in (6.11). A simple way to do this is ﬁrst to deﬁne φ ˜ β(ϕ)dϕ. (6.29) Y (φ) = 0

If N is odd we can calculate Y (φI ) by using Y (φI+1 ) = Y (φI ) +

β˜I+1 + β˜I h, 2

I=

N +1 , . . . , N − 1, 2

(6.30)

6.3 Boundary integral equation method

155

and Y (φI−1 ) = Y (φI ) −

β˜I−1 + β˜I h, 2

I=

N +1 N −1 , , . . . , 2, 2 2

(6.31)

where Y (φ(N +1)/2 ) = 0.

(6.32)

A similar formula can be derived when N is even. The values of Y (φm I ) can then be evaluated by interpolation, for example by using the interpolation formula Y (φI ) + Y (φI+1 ) , I = 1, . . . , N − 1. (6.33) Y (φm I )= 2 To describe the numerical procedure further we need to introduce dimensionless variables. There are several possible choices. For example, we could choose Q/c as the unit length and c as the unit velocity. Here, however, we will follow the choice of Section 6.2 and choose λ as the unit length and c as the unit velocity. We then set c = 1 and λ = 1 in (6.20) and use (6.12) instead of (6.11). By satisfying (6.12) and (6.20) at the midpoints (6.22) we obtain 2N −2 algebraic equations for the 2N unknowns δ˜I , I = 1, . . . , N , β˜I , I = 2, . . . , N −1, B and µ. A further equation expresses that the wavelength is λ (here λ = 1): 1 ˜ δ(ϕ)dϕ = 1. (6.34) 0

The ﬁnal equation ﬁxes the amplitude of the wave, for example by writing 1/2 ˜ β(ϕ)dϕ = −s, (6.35) 0

where s is the steepness. The integrals in (6.34) and (6.35) are evaluated numerically (for example by the trapezoidal rule). The resulting system of 2N nonlinear algebraic equations with 2N unknowns is solved by Newton’s method for given values of κ and s. When r0 = 0, the water is of inﬁnite depth and, using ξ = cλ/2, (6.20) reduces to the simple integral relation cλ/2 1 1 π ˜ ˜ δ(φ) = − (6.36) β(ϕ) cot (ϕ − φ) dϕ. c cλ −cλ/2 cλ We conclude this section by mentioning that the eﬃciency of the numerical method can be improved by using the assumed periodicity and symmetry of the waves. The symmetry implies that ˜ ˜ ˜ ˜ β(−ϕ) = −β(ϕ) and δ(−ϕ) = δ(ϕ).

(6.37)

156

Numerical computations of nonlinear water waves

Therefore the integrals in (6.20) and (6.36) can be reduced to integrals from 0 to cλ/2. For example, we can rewrite (6.36) as cλ/2 1 π 1 π ˜ ˜ δ(φ) = − β(ϕ) cot (ϕ − φ) + cot (ϕ + φ) dϕ. (6.38) c cλ 0 cλ cλ We can then solve problems by following the numerical scheme described above but with the mesh points restricted to the interval 0 < φ < cλ/2, i.e. by replacing (6.21) by φI =

cλ (I − 1), 2(N − 1)

I = 1, . . . , N.

(6.39)

6.4 Numerical methods for solitary waves Some solitary waves have already been introduced as solutions of the Korteweg–de Vries equation (see Section 5.2). As we shall see, there are three diﬀerent types of solitary wave. The ﬁrst type is a solitary wave with a free-surface proﬁle that approaches monotonically a constant level in the far ﬁeld (see Figure 6.4).

Fig. 6.4. A solitary wave with monotonic decay to a constant level.

The solution (5.156) of the Kortweg–de Vries equation is an example of a solitary wave of this type. Solitary waves of the second type also have a free-surface proﬁle that approaches a constant level in the far ﬁeld but with decaying oscillations. Examples of such waves occur when both gravity and surface tension are included. They are described in Section 7.2.2 and an extension to three-dimensional waves is presented in Chapter 10. There are both elevation and depression waves (see Figures 6.5 and 6.6). The third type of solitary wave does not approach a uniform stream in the far ﬁeld but is characterised by oscillations of constant amplitude (see Figure 6.7). We will refer to these waves as generalised solitary waves to contrast them with true solitary waves, whose proﬁles approach constant levels in the far ﬁeld. As we shall see these generalised solitary waves occur when both gravity and surface tension are taken into account.

6.4 Numerical methods for solitary waves

157

Fig. 6.5. An elevation solitary wave with a decaying oscillatory tail.

Fig. 6.6. A depression solitary wave with a decaying oscillatory tail.

There are two approaches to the numerical investigation of solitary waves. The ﬁrst is to derive schemes to compute them directly. The second is to compute periodic waves by using the schemes outlined in Sections 6.2 and 6.3 and then to take larger and larger values of the wavelength. The ﬁrst approach works particularly well for solitary waves of the ﬁrst two types (see Figures 6.4–6.6). The inﬁnite domain −∞ < x < ∞ is then truncated to −A < x < A, where A is a large real number, and the contributions from A < x < ∞ and from −∞ < x < −A are either neglected or approximated by asymptotic solutions. The second approach is more appropriate for computing generalised solitary waves, i.e. solitary waves of the third type (see Figure 6.7). This reason is that no approximations are made about the behaviour of the ﬂow in the far ﬁeld and we can then be conﬁdent that the oscillations in the far ﬁeld are not generated by the truncation of the domain.

6.4.1 Boundary integral equation methods In this section we follow the ﬁrst approach and derive a boundary integral equation method to compute directly the solitary waves of Figures 6.4– 6.6. Therefore we assume that the ﬂow approaches a uniform stream with constant velocity U and constant depth H as |x| → ∞. The value −Q of

158

Numerical computations of nonlinear water waves

Fig. 6.7. A generalised solitary wave.

the streamfunction ψ on the bottom is then given by Q = U H.

(6.40)

˜ ˜ To obtain an integral relation between δ(φ) and β(φ) on the free surface, we take the limit λ → ∞ in (6.20). This leads to 1 ˜ δ(φ) − U ˜ ˜ ˜ 1 ∞ β(ϕ) − φ) + 2Q[δ(ϕ) − 1/U ] 1 ∞ −β(ϕ)(ϕ =− dϕ + dϕ. (6.41) 2 2 π −∞ ϕ − φ π −∞ (ϕ − φ) + 4Q To proceed further we need to deﬁne dimensionless variables. A natural choice is to take H as the unit length and U as the unit velocity. In terms of these dimensionless variables (6.41) becomes ˜ ˜ ˜ − φ) + 2[δ(ϕ) − 1] 1 ∞ −β(ϕ)(ϕ 1 ∞ β(ϕ) ˜ dϕ + dϕ. δ(φ) − 1 = − 2 π −∞ ϕ − φ π −∞ (ϕ − φ) + 4 (6.42) We also write the dynamic boundary condition (6.11) in dimensionless form as φ δ˜β˜φ − δ˜φ β˜ 1 1 1 ˜ β(ϕ)dϕ −τ + 2 = B, (6.43) 2 δ˜2 + β˜2 F 0 (δ˜2 + β˜2 )3/2 where F =

U (gH)1/2

(6.44)

T ρHU 2

(6.45)

is the Froude number and τ=

is the Bond number, already introduced in Section 2.4.2 (see (2.92)). ˜ Relations (6.42) and (6.43) deﬁne an integro-diﬀerential system for δ(φ) ˜ and β(φ). It was solved numerically by Hunter and Vanden-Broeck [76] for

6.4 Numerical methods for solitary waves

159

pure gravity solitary waves. Another, similar, scheme was derived by ByattSmith and Longuet-Higgins [25]. Here we will follow the approach in [76] and present a numerical procedure to solve the general problem with both gravity and surface tension included in the dynamic boundary condition. We introduce the N mesh points φI = (I − 1)h,

I = 1, . . . , N,

(6.46)

where h is the interval of discretisation. We also deﬁne the midpoints φI + φI+1 , I = 1, . . . , N − 1. (6.47) 2 We satisfy (6.42) and (6.43) at the midpoints (6.47). This leads to 2N − 2 ˜ I ), ˜ I ) and β(φ nonlinear algebraic equations for the 2N +2 unknowns F , B, δ(φ I = 1, . . . , N . The details of the discretisation follow those of Section 6.3. The integrals in (6.42) are approximated by the trapezoidal rule with ˜ m ϕ = φJ , J = 1, . . . , N , and φ = φm I , I = 1, . . . , N − 1. The values of δ(φI ), m m m ˜ ˜ ˜ β(φI ), δφ (φI ) and δφ (φI ) needed to satisfy (6.43) at the mesh points (6.47) are approximated by formulae similar to (6.25)–(6.28). The integral in (6.43) is calculated using (6.29)–(6.33). Four more equations are needed. One is obtained by imposing the symmetry condition φm I =

˜ 1 ) = 0. β(φ

(6.48)

Another equation ﬁxes the amplitude of the wave. Several choices are possible for this; for example the distance between the crest (or trough) and the level of the free surface at inﬁnity can be ﬁxed by writing ∞ ˜ β(ϕ)dϕ = −A, (6.49) 0

where A is given. Here A > 0 for an elevation wave (see Figures 6.4 and 6.5) and A < 0 for a depression wave (see Figure 6.6). An extra equation is obtained by satisfying the dynamic boundary condition (6.43) as |x| → ∞. This leads to A 1 (6.50) B = − 2. 2 F There are also various possible choices for the ﬁnal equation. One possibility is to relate an unknown on the free surface to unknowns at neighbouring ˜ M) points by an extrapolation formula. For example, we could express β(φ ˜ ˜ in terms of β(φM −1 ) and β(φM −2 ) by a linear extrapolation formula. A variation is to use an extrapolation formula that reﬂects the fact that θ˜ decays exponentially as |φ| → ∞ (see [76]). The particular choice is not crucial provided that the formulae used become exact in the limit h → 0.

160

Numerical computations of nonlinear water waves

The resulting system of 2N + 2 equations with 2N + 2 unknowns is solved by Newton’s method. Series truncation methods have also been used to compute solitary waves. We will describe one of them in Section 6.6.1.

6.5 Numerical results for periodic waves We describe in this section the numerical computation of nonlinear periodic waves. We examine successively pure capillary waves, pure gravity waves and gravity–capillary waves.

6.5.1 Pure capillary waves (g = 0, T = 0) Numerical solutions for pure capillary waves can be computed using the numerical procedures of Sections 6.2 and 6.3. However, Crapper [37] derived an exact solution for pure capillary waves in water of inﬁnite depth. His work was generalised to water of ﬁnite depth by Kinnersley [89]. A simpler derivation of Kinnersley’s solutions was provided by Crowdy [38]. Crapper’s exact solution can be written as z=

4i 1 4i f + − , −2iπf /λc c k 1 + Ae k

(6.51)

2πT 1 − A2 . ρλ 1 + A2

(6.52)

c2 =

The steepness s of the wave is related to the constant A by s=

4A . π(1 − A2 )

(6.53)

Relation (6.52) is a dispersion relation. It reduces to the linear dispersion (2.97) when A → 0. The nonlinear dispersion (6.52) is also consistent with the asymptotic solution (5.82). To check this, we expand (6.52) and (6.53) in powers of A. This yields on the one hand s2 π 2 2πT 2 1− + o(A2 ). (6.54) c = ρλ 8 On the other hand, taking the limits h → ∞, g → 0 in (5.83) gives c2 = −

c0 A21 k 2 . 16

(6.55)

6.5 Numerical results for periodic waves

161

Using (5.43) with g = 0 and h → ∞, (5.48), (5.10) and (6.55) yields 2 π 2 2πT 2 1− + o(2 ). (6.56) c = ρλ 2 The deﬁnition (5.46) of implies that = s/2 to leading order. Therefore relation (6.56) agrees with (6.54). Fully nonlinear free-surface proﬁles are obtained by taking the real and imaginary parts of (6.51). This gives a proﬁle in the parametric form x = x(φ) and y = y(φ). Typical free-surface proﬁles are shown in Figures 6.8– 6.11. 0.6

0.4

0.2

0

0

0.2

0.4

0.6

Fig. 6.8. Crapper’s free-surface proﬁle for k = 2π. The steepness s = 0.1.

0.6

0.4

0.2

0

0

0.2

0.4

0.6

Fig. 6.9. Crapper’s free-surface proﬁle for k = 2π. The steepness s = 0.55.

For small values of the steepness s, the waves are close to linear sine waves (see Figure 6.8). As s increases the waves develop more rounded crests and

162

Numerical computations of nonlinear water waves 0.6

0.4

0.2

0

0

0.2

0.4

0.6

Fig. 6.10. Crapper’s free-surface proﬁle for k = 2π. The steepness s = 0.73. There is a small trapped bubble at the trough of the wave. 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

Fig. 6.11. Crapper’s free-surface proﬁle for k = 2π. The steepness s = 0.9. This proﬁle is unphysical because it is self-intersecting.

sharp troughs and ultimately become overhanging near the troughs (see Figure 6.9). A limiting conﬁguration is reached for s = s∗ ≈ 0.73, when the free-surface proﬁle develops a point of contact with itself and a small trapped bubble at the trough (see Figure 6.10). For s > s∗ , (6.51) predicts unphysical self-intersecting free-surface proﬁles (see Figure 6.11). Vanden-Broeck and Keller [186] computed new physical solutions for s > s∗ . They all have proﬁles with a point of contact with themselves and a small trapped bubble at their troughs, just like Crapper’s highest wave with s = s∗ . As described at the beginning of Section 3.1.2.2, preventing the self-intersection of the proﬁle (see Figure 6.11) imposes an extra constraint on the solutions. Therefore an extra unknown is needed. This is achieved by ﬁnding the pressure P (s) in the trapped bubble as part of the solution. Such

6.5 Numerical results for periodic waves

163

solutions can be computed by an appropriate modiﬁcation of the boundary integral equation technique of Section 6.3. Details can be found in [186]. The pressure P (s) is found to be an increasing function of s, with P = 0 at s = s∗ (see Figure 6.12). P 1.0

0.5

0.5

1.0

1.5

s

Fig. 6.12. The bubble pressure P (in units of ρc2 ) versus the steepness s for the family of waves computed by Vanden-Broeck and Keller [186]. Taken from J. Fluid Mech. 98, 161–169 (1980). The axis P = 0 corresponds to Crapper’s solution, which is physically meaningful for 0 ≤ s ≤ 0.73 and unphysical for s > 0.73.

The waves exist for s∗∗ < s < s∗ . Here s∗∗ ≈ 0.663. Some typical proﬁles of the waves and bubbles are shown in Figure 6.13

s = 0.6635

s = 0.7

s = 0.75 s = 1.05

The bubble for s = 0.6635, expanded by a factor of 10

s = 1.4

Fig. 6.13. Free-surface proﬁles for the family of waves computed by Vanden-Broeck and Keller [186]. Taken from J. Fluid Mech. 98, 161–169 (1980).

The exact solutions of Kinnersley [89] are more complicated than Crapper’s and involve elliptic integrals. Kinnersley solved the more general

164

Numerical computations of nonlinear water waves

problem of waves propagating on a sheet of ﬁnite thickness. The ﬂow conﬁguration is shown in Figure 6.14. y ψ=0 Fluid sheet

x ψ=Q

n

Fig. 6.14. A travelling wave on a ﬂuid sheet.

The ﬂuid sheet is surrounded by regions of constant pressure (e.g. air) and gravity is neglected. Dimensionless variables are introduced by choosing the wavelength λ as the unit length and c as the unit velocity. The solutions are then characterised by two parameters. The ﬁrst is the value Q of the streamfunction on the lower boundary of the sheet (see Figure 6.14). The second is the parameter T . (6.57) W = ρλc2 Kinnersley obtained exact solutions for both symmetric waves (see Figure 6.15) and antisymmetric waves (see Figure 6.16). The symmetric waves satisfy by symmetry the condition φy = 0 on y = 0. Therefore they describe also capillary waves in water of ﬁnite depth when the ﬂows of Figure 6.15 are restricted to the domain y > 0. Figures 6.15 and 6.16 show that both symmetric and antisymmetric waves have limiting conﬁgurations with trapped bubbles. Solutions for higher values of the amplitude can be obtained by following the approach of Vanden-Broeck and Keller [186] described earlier. An interesting question is whether the solutions of Crapper [37] and Kinnersley [89] are unique. In other words, are there additional branches of solutions? Blyth and Vanden-Broeck [20] showed that the answer is in fact positive. They computed new branches of solutions that bifurcate from the symmetric solution branches. Their numerical procedure follows that of Section 6.2. Typical free-surface proﬁles for these new solutions are shown in Figures 6.17–6.19. 6.5.2 Pure gravity waves (g = 0, T = 0) Using the asymptotic results of Section 5.1, Stokes [143], [144] noticed that pure gravity waves tend to develop sharp crests and broad troughs as their

6.5 Numerical results for periodic waves 0.8

(a)

(b)

0.8

0.4

165

0.4 0

0

0 1.2

0.5

1.0

1.5

2.0 1.2

(c)

0.8

0

0.4

0

0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

(d)

0.8

0.4

0

0.5

1.0

x

1.5

2.0

0

x

Fig. 6.15. Symmetric waves for Q = −0.5: (a) W = 0.1013, (b) W = 0.1154, (c) W = 0.129 49, (d) W = 0.141 03. Part (d) shows the limiting conﬁguration with trapped bubbles. Taken from J. Fluid Mech. 507, 255–264 (2004).

amplitude is increased. Furthermore the heights of the crests increase with the amplitude. It follows from (5.3) (with T = 0) that the velocity at the crests decreases as the amplitude increases. This prompted Stokes to conjecture that, as the amplitude increases, the waves ultimately reach a limiting conﬁguration with a stagnation point at their crests. Stokes then showed that the ﬂow near the crest of this limiting conﬁguration is locally the same as the ﬂow inside a corner with an enclosed angle of 120◦ (see Figure 6.20). This conclusion can easily be derived by noticing that the ﬂow of Figure 6.20 is a particular case of the ﬂow of Figure 3.38 for µ2 = −π/6. Satisfying the dynamic boundary condition on GL yields the exact solution (3.162) with µ = 2π/3 and A deﬁned by (3.163), (3.164) and (3.176). In deriving this solution in Chapter 3, we satisﬁed the dynamic boundary condition on GL but not on HG. However, the symmetry of the ﬂow of Figure 6.20 implies that the dynamic boundary condition is then automatically satisﬁed on HG by symmetry. The existence of a limiting conﬁguration with a stagnation point at the crest suggests that we should characterise gravity waves by the

166

Numerical computations of nonlinear water waves (a)

0.6

(b)

0.8

0.4 0.4

0.2 0

0

0

0.5

1.0

1.5

2.0

(c)

0 1.2

0.8

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

(d)

0.8

0.4

0.4 0

0

0

0.5

1.0

x

1.5

2.0

0

x

Fig. 6.16. Antisymmetric waves for Q = −0.5: (a) W = 0.1853, (b) W = 2.038, (c) W = 0.2223, (d) W = 0.2426. Part (d) shows the limiting conﬁguration with trapped bubbles. Taken from J. Fluid Mech. 507, 255–264 (2004).

parameter 0 = 1 −

1 ˜ δ(1/2) ˜ c2 δ(0)

(6.58)

˜ instead of the steepness s (see [28] and [172]). Since 1/δ(0) is the speed at the crest of the wave, 0 = 1 for the limiting conﬁguration. Furthermore, 0 → 0 as the steepness s tends to zero since 1/δ˜ → c as s → 0. Therefore the advantage of the parameter 0 is that the range 0 < 0 < 1 of possible values of 0 is known a priori. Similar parameters were used by Chen and Saﬀman [28] and Vanden-Broeck [162]. For example, Vanden-Broeck [162] used the parameter 1 (6.59) 1 = 1 − 2 ˜ c δ 2 (0) instead of the parameter 0 . It is easily checked that 0 < 1 < 1. Numerical computations of the limiting conﬁgurations with a stagnation point at the crest were performed as early as 1883 (see [112]). However, it was not until 1982 that a proof of the existence of waves with a 120◦ angle at their crest was provided (see [5]).

6.5 Numerical results for periodic waves 0.2

(a)

167

(b)

0.1

0.1 0 0

0

0.5

1.0

0

1.5

(c)

0.1

0.5

1.0

1.5

1.0

1.5

(d) 0

0

0

0.5

1.0

x

1.5

0

0.5

x

Fig. 6.17. New free-surface proﬁles for Q = −0.1: (a) W = 0.2783, (b) W = 0.2794, (c) W = 0.2840, (d) W = 0.2943. Part (d) shows the limiting conﬁguration with trapped bubbles. The vertical scale has been exaggerated to show the trapped bubbles clearly. Taken from J. Fluid Mech. 507, 255–264 (2004).

We ﬁrst present numerical results for pure gravity waves in water of inﬁnite depth. Results for pure gravity waves in water of ﬁnite depth will be described after that. 6.5.2.1 Waves in water of inﬁnite depth The asymptotic results (5.10), (5.76) and (5.83) predict that the speed of a nonlinear gravity wave (i.e. µ) is an increasing function of . A tempting conjecture is then that µ is a monotonically increasing function of that reaches its maximum value for the limiting conﬁguration with a 120◦ angle. However, this is not the case and it was shown in the 1970s (see [133], [101] and [106]) that the values of µ oscillates inﬁnitely often as the limiting conﬁguration is approached. This is illustrated in Figure 6.21, where values of µ versus 0 are shown. The ﬁgure shows clearly the ﬁrst maximum. This maximum is followed by an inﬁnite succession of minima and maxima as 0 → 1. This is a very ﬁne structure and only the ﬁrst maximum can be seen on the scale of Figure 6.21.

168

Numerical computations of nonlinear water waves 0.2

0.2

(a)

0.1

0.1

0

0

0

0.5

1.0

1.5

(c)

0.2

0 0.2

0.1

0.1

0

0

0

0.5

x

1.0

1.5

(b)

0.5

1.0

1.5

1.0

1.5

(d)

0

0.5

x

Fig. 6.18. New free-surface proﬁles for Q = −0.1: (a) W = 0.2900, (b) W = 0.2913, (c) W = 0.2977, (d) W = 0.3025. Part (d) shows the limiting conﬁguration with trapped bubbles. The vertical scale has been exaggerated to show the trapped bubbles clearly. Taken from J. Fluid Mech. 507, 255–264 (2004).

Similarly it can be shown that the parameter used in the asymptotic analysis of Chapter 5 oscillates inﬁnitely often as the wave of maximum steepness is approached. In Figure 6.22 we present values of the steepness s versus 0 . These results show that the steepness s is a monotonically increasing function of 0 . Therefore s is a suitable parameter to describe steep waves since there is only one solution for each value of s. However, it is not as convenient as 0 because the maximum steepness (and therefore the range of s) is not known a priori. Typical free-surface proﬁles for 0 = 0.4, 0.6 and 0.99 are shown in Figure 6.23. The proﬁle corresponding to 0 = 0.99 is close to the limiting conﬁguration with a 120◦ angle at the crest. The results presented in Figures 6.21–6.23 can be computed by using either of the two numerical procedures outlined in Sections 6.2 and 6.3. The boundary integral equation method of Section 6.3 works well for waves of small or moderate amplitude. However, the accuracy deteriorates as the limiting conﬁguration is approached and larger and larger numbers

6.5 Numerical results for periodic waves 0.2

0.2

(a)

0.1

169

(b)

0.1

0

0

0

0.25

0.50

0.75

0

1.00

0.50

0.75

1.00

0.25

0.50

0.75

1.00

(d)

(c)

0.2

0.25

0.2

0.1

0.1

0

0

0

0.25

0.50

x

0.75

1.00

0

x

Fig. 6.19. New free-surface proﬁles for Q = −0.1: (a) W = 0.2960, (b) W = 0.2990, (c) W = 3083, (d) W = 0.3176. Part (d) shows the limiting conﬁguration with trapped bubbles. The vertical scale has been exaggerated to show the trapped bubbles clearly. Taken from J. Fluid Mech. 507, 255–264 (2004). y π 6

H

G 2π 3

x π 6

L

Fig. 6.20. The local ﬂow near the crest of the highest pure gravity wave.

N of mesh points are required. This is due to the fact that the velocity at the crest decreases and tends to zero as 0 → 1. Since the magnitude of the velocity is |∇φ|, the equally distributed mesh points (6.21) or (6.39) lead to fewer and fewer points near the crest of the wave in the physical (x, y)-plane as 0 → 1. This is particularly bad for accuracy because the curvature at the crest gets larger and larger as 0 → 1. These problems can be overcome by concentrating mesh points near the crest. There are several ways to do

170

Numerical computations of nonlinear water waves 1.20

1.16

1.12

1.08

1.04

1.00

0

0.2

0.4

0.6

0.8

1.0

Fig. 6.21. Values of the parameter µ versus 0 . 0.16

0.12

0.08

0.04

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 6.22. Values of the steepness s versus 0 .

this. Schwartz and Vanden-Broeck [135] used the change of variable φ = νn

(6.60)

where n is an integer greater than 1. In their computations, they chose n = 3 and reformulated the problem in terms of ν. The problem was then solved numerically by using points equally spaced in ν. The change of variables (6.60) concentrates points near the crest in the physical plane and enables computations for values of 0 very close to 1. Chen and Saﬀman [28] used another change of variable, ξ¯ sin 2πν , φ = cλ ν − 2π

(6.61)

6.5 Numerical results for periodic waves

171

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

Fig. 6.23. Typical free-surface proﬁles for 0 = 0.4, 0.6 and 0.99 (respectively the bottom, middle and top lines).

where the variable ν and the parameter ξ¯ are between 0 and 1. They then used equally spaced points in ν and concentrated mesh points near the crest by choosing ξ¯ very close to 1 (for example ξ¯ = 0.99 or even ξ¯ = 0.999). The advantage of (6.61) over (6.60) is that variables that are periodic in φ, with period cλ, are still periodic in ν, with period 1. The series truncation method of Section 6.2 also gives accurate results for waves of small and moderate amplitude. For the limiting conﬁguration, the ﬂow near the crest at f = 0 is a ﬂow inside a 120◦ angle, and (3.19) implies that w ≈ f 1/3

as

f → 0.

(6.62)

Since the crests are mapped to t = 1 under the transformation (6.4), (6.62) implies that w ≈ (1 − t)1/3

as

t → 1.

(6.63)

Because of the singularity of (6.63) at t = 1, we cannot expect the expansions (6.16) or (6.17) to converge when 0 = 1 (i.e. for the limiting conﬁguration). Also, we can expect these expansions to converge more and more slowly as 0 → 1. Michell [112], Olfe and Rottman [118], Vanden-Broeck [172] and VandenBroeck and Miloh [188] calculated the wave of maximum steepness by writing

∞ (6.64) dn e−2inπf , w = (1 − t)1/3 1 + n=1

172

Numerical computations of nonlinear water waves

where t is deﬁned by t = e−2iπf . The expansion (6.64) satisﬁes (6.63) and is therefore convergent. The philosophy here is similar to that used in Chapter 3: the singularities of w in the unit circle |t| ≤ 1 are removed by writing w in the form (3.12), where G(t) contains all the singularities of w. In this case G(t) = (1 − t)1/3 , since w is not analytic at t = 1. The expansion (6.64) leads to very accurate results. In particular it predicts the value µ = 1.1931 for the dimensionless wave speed. Havelock [70] and Vanden-Broeck [172] calculated waves of arbitrary amplitude up to the limiting conﬁguration by expressing w as

∞ (6.65) w = (1 − βt)1/3 1 + gn e−2inπf n=1

and ﬁnding the coeﬃcients gn and β by series truncation and collocation. By taking the cube of (6.17), (6.64) and (6.65), we see that these three expressions are particular cases of w3 = 1 +

∞

fn e−2iπnf

(6.66)

n=1

for appropriate choices of the coeﬃcients fn . Therefore gravity waves of arbitrary amplitude (including the highest) can also be calculated by using (6.66) and series truncation. This approach was used by Vanden-Broeck and Miloh [188]. We conclude this section by asking whether there are other solution branches. As in Section 6.5.1, the answer is positive. Chen and Saﬀman [28] discovered new branches that bifurcate from the solutions described above. We will refer to the solutions described above as the basic branches. The new solutions can be calculated as follows. We ﬁrst recompute the basic branch by deﬁning the wavelength as a multiple of the fundamental wavelength (here the fundamental wavelength refers to the smallest value of λ for which (5.4) and (5.5) are satisﬁed). We then choose λ as the unit length. As an example, Figure 6.24 shows a solution with the wavelength chosen as twice the fundamental wavelength. Since the wavelength is taken as the unit length, the distance between every second crest is 1. We then follow the solution branch associated, for example, with the solution of Figure 6.24 by the boundary integral equation method of

6.5 Numerical results for periodic waves

173

0.04

0.02

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 6.24. Free-surface proﬁle on the basic solution branch. The wavelength was chosen as twice the fundamental wavelength. This solution corresponds to the bifurcation point.

Section 6.3, using the solution in Figure 6.24 as the initial guess in the Newton iterations, together with continuation. This means that a previously computed solution is used as the initial guess to compute a new solution for a slightly smaller or larger value of the steepness. Following the general discussion of Newton’s method (see (3.66)–(3.68)), we look for possible bifurcation points by monitoring the sign of the determinant of the Jacobian matrix. Chen and Saﬀman [28] identiﬁed such a bifurcation point, and the solution shown in Figure 6.24 in fact corresponds to this bifurcation point. The next step is to compute the bifurcating branches. Chen and Saﬀman [28] used a general numerical method developed by Keller [84] to move from the solution of Figure 6.24 to the new branches. Here we use a simpler approach, which consists of using the solution of Figure 6.24 as the initial guess and replacing the amplitude equation (6.35) or (6.58) by Y (0) − Y (0.5) = γ,

(6.67)

where Y (φ) is deﬁned by (6.29) and γ is given. Figures 6.25 and 6.26 show two computed solutions, with γ > 0 and γ < 0 respectively. As |γ| increases the waves ultimately approach the limiting conﬁguration with a 120◦ angle, at x = 0 and x = 1 for γ > 0 and at x = 0.5 for γ < 0. Chen and Saﬀman [28] and Saﬀman [130] computed further solutions by starting with basic solutions in which the wavelength is n times the fundamental wavelength, for n > 2. The existence of these new branches of waves is related to the presence of the ﬁrst maximum, shown in Figure 6.21. This maximum implies that two

174

Numerical computations of nonlinear water waves 0.04

0.02

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 6.25. Free-surface proﬁle for a solution on the bifurating branch with γ > 0.

0.04

0.02

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 6.26. Free-surface proﬁle for a solution on the bifurating branch with γ < 0.

waves of diﬀerent amplitudes can travel at the same speed. As mentioned in Section 5.1.2, this phenomenon occurs for linear gravity–capillary waves. In that case linear superposition leads to new solutions (see (5.89) and (5.90)). Although linear superposition is not applicable for nonlinear solutions, the new solutions of Figures 6.25 and 6.26 can be roughly interpreted in that way. This suggests that further bifurcation points might be associated with the inﬁnite sequence of maxima and minima of µ as 0 → 1. This was conﬁrmed by the calculations of Vanden-Broeck [162], who identiﬁed a bifurcation point, between the ﬁrst maximum and the ﬁrst minimum, leading to solutions similar to those of Figures 6.25 and 6.26.

6.5 Numerical results for periodic waves

175

6.5.2.2 Waves in water of ﬁnite depth All the computations and results presented in the previous subsection extend to the case of ﬁnite depth. In particular, waves in water of ﬁnite depth also have a limiting conﬁguration with a 120◦ angle at their crests and values of µ oscillating inﬁnitely often as 0 → 1. For a given value of r0 (i.e. of the depth), the waves approach the linear waves of Section 2.4 as 0 → 0. However, a new feature of ﬁnite depth is that for any given value of 0 (no matter how small) the waves deviate from the linear waves of Section 2.4 as r0 approaches 1 (i.e. as the depth tends to zero). This behaviour is consistent with the nonuniformity of the expansions of Section 5.1 as h → 0 (see the end of Section 5.1). It is also consistent with the asymptotic analysis of Section 5.2: the numerical calculations show that the waves (with 0 small) approach the cnoidal waves mentioned in Section 5.2 as r0 → 1. For r0 = 1, these waves become solitary waves. Vanden-Broeck [162] showed that the results of Chen and Saﬀman [28] can be extended to ﬁnite depth and that there are again solution branches that bifurcate from the basic branches. These results can be described as follows. Using the notation in [162] we characterise the amplitude of the waves by the parameter 1 (see (6.59)) instead of 0 . In Figures 6.27 and 6.28 we present values of τ∗ =

c2 c20

(6.68)

versus 1 for r02 = 0.4 and r02 = 0.8. Here c is the nonlinear wave speed and c0 =

g k

1/2 tanh kh

is the phase speed corresponding to linear waves (see (2.83)). In these ﬁgures, the solid curves correspond to the basic solution branch (see Figure 6.24 and the associated discussion for a deﬁnition of the basic branch). The broken curves are the bifurcating branches. A typical freesurface proﬁle on the bifurcating branch for r02 = 0.4 is shown in Figure 6.29.

6.5.3 Gravity–capillary waves (g = 0, T = 0) The analysis of Section 5.1 illustrates the nonuniqueness of gravity–capillary waves when the condition (5.58) is satisﬁed for some integer value of m.

176

Numerical computations of nonlinear water waves ␶ 0.24

0.23

0.22

0.85

0.90

0.95

1.0

1

Fig. 6.27. Values of the parameter τ ∗ versus 1 for r02 = 0.4. Taken from Phys. Fluids 26, 2385–2387 (1983). ␶ 0.48 0.47 0.46 0.45

0.85

0.90

0.95

1.0

1

Fig. 6.28. Values of the parameter τ ∗ versus 1 for r02 = 0.8. Taken from Phys. Fluids 26, 2385–2387 (1983).

For example, (5.106) and (5.107) predict two diﬀerent solutions in water of inﬁnite depth when m = 2. The numerical computations of Schwartz and Vanden-Broeck [135], Chen and Saﬀman [29] and Hogan [74] show that there are indeed many diﬀerent families of solutions. We illustrate these results by considering ﬁrst gravity–capillary waves in water of inﬁnite depth. Results for water of ﬁnite depth will be presented later. 6.5.3.1 Waves in water of inﬁnite depth We use λ as the unit length and c as the unit velocity. The gravity–capillary waves are characterised by the three parameters κ, µ (see (5.67) and (5.112))

6.5 Numerical results for periodic waves

177

0.04 0.03 0.02

y

0.01 0

0.2

0.4

0.6

0.8

1.0

x

Fig. 6.29. Computed free-surface proﬁle for a wave on the bifurcating branch for r02 = 0.4. Taken from Phys. Fluids 26, 2385–2387 (1983).

and the steepness s. Using these parameters and assuming inﬁnite depth we can rewrite the linear solutions (5.57) and (5.53) as s (6.69) η(x) = cos 2πx, 2 µ = 1 + κ.

(6.70)

These solutions hold for s suﬃciently small provided that the condition (5.55) is satisﬁed. We recall that in water of inﬁnite depth (5.55) takes the simple form 1 (6.71) κ = , n where n > 1 is an integer. When (6.71) is not satisﬁed for some value n = m, i.e. when 1 , (6.72) m the results of Section 5.1 shows that several solutions are possible. For example when m = 2 (i.e. κ = 0.5), there are two solutions, described by (5.106) and (5.108). In terms of the dimensionless variables, these solutions can be written as s s (6.73) η(x) = cos 2πx ± cos 4πx + · · · , 2 4 κ=

3 3 ± πs − (πs)2 + · · · (6.74) 2 4 The above results are valid only for s suﬃciently small. We now extend them by fully nonlinear calculations obtained by using the boundary integral equation method of Section 6.3. µ=

178

Numerical computations of nonlinear water waves

Figure 6.30 shows numerical values of µ versus κ for s = 0.03.

2.0

2 1

1.5

4 3

µ

1.0

0.5

0.5

1.0

κ

Fig. 6.30. Values of µ versus κ for s = 0.03. Taken from J. Fluid Mech. 95, 119–139 (1979). The long-broken line corresponds to µ = 1 + κ.

These results show that there are many diﬀerent families of solutions. Four families, numbered 1 to 4, are shown in Figure 6.30. The long-broken straight line corresponds to the linear solution (6.70). Relations (6.74) predict µ = 1.42 and µ = 1.56 for s = 0.03. The point with coordinates (0.5, 1.42) in Figure 6.30 is close to family 1 whereas the point with coordinates (0.5, 1.56) is close to family 2. This illustrates that the Wilton ripples are members of two diﬀerent families of solutions (families 1 and 2). The corresponding free-surface proﬁles are shown in Figure 6.31. Using the notation of Schwartz and Vanden-Broeck [135] we have plotted 2πy versus 2πx. Only half a wavelength is shown in Figure 6.31. Solutions on the other branches of solutions are characterised by a larger number of dimples on their proﬁles. This is clearly shown in Figure 6.32, where we present free-surface proﬁles for κ = 0.33 and s = 0.03. Further proﬁles for other values of κ and s can be found in [135].

6.5 Numerical results for periodic waves

0.10

179

2

0.05

1 0

Fig. 6.31. Free-surface proﬁles for κ = 0.5 and s = 0.03. Taken from J. Fluid Mech. 95, 119–139 (1979).

0.2

2 0.1

0

1 3

4

Fig. 6.32. Four free-surface proﬁles for κ = 0.33 and s = 0.03. Taken from J. Fluid Mech. 95, 119–139 (1979).

In Figure 6.33 we plot values of µ versus for κ = 0.5. For small the values of µ agree with (6.74) as expected. 6.5.3.2 Waves in water of ﬁnite depth It is convenient to describe the properties of gravity–capillary waves in water of ﬁnite depth in terms of the parameters τ , r0 and F deﬁned in (2.92), (6.5) and (2.91) respectively. We deﬁne dimensionless variables by taking the undisturbed depth Q/c as the reference length and c as the reference

180

Numerical computations of nonlinear water waves 2

1.5

1.0

1

m 0.5

0.05

0.10

0.15

s

Fig. 6.33. Values of µ versus s for κ = 0.5. The solid curves correspond to the numerical computation and the broken curves to the asymptotic solution (6.74). Taken from J. Fluid Mech. 95, 119–139 (1979).

velocity. In this section the amplitude of the waves is measured by the dimensionless velocity u0 at x = y = 0 (see Figure 6.1). Values of u0 close to 1 correspond to waves of small amplitude. The condition (5.55) is satisﬁed when τ > 1/3. Periodic waves are then unique and described for small values of s by the expansions of Section 5.1.1. When τ < 1/3, the condition (5.55) is not satisﬁed for some values of the parameters, and numerical calculations show the existence of many diﬀerent families of solutions. The situation is then similar to that described in the previous section for waves in water of inﬁnite depth. We examine in more detail the properties of waves with τ < 1/3 for r0 close to 1. Figure 6.34 shows values of F versus l=−

2π ln r0

(6.75)

for τ = 0.24 and u0 = 0.97. Here l is the dimensionless wavelength. There are many diﬀerent families of waves and three of them are presented, as (a),(b) and (c). Typical free-surface proﬁles for each family are shown in Figure 6.35. As expected there are dimples on the free-surface proﬁles, as was found for water of inﬁnite depth. However, for values of r0 close to 1 these dimples tend to concentrate in the troughs of the waves. The results of Figure 6.35 show that as one moves to the right in Figure 6.34, from one family to the next, two crests or two troughs are added to the proﬁles. For example there are 11 crests and 12 troughs per wavelength on the proﬁle on Figure 6.35(a), 13 crests and 12 troughs per wavelength in the proﬁle of Figure 6.35(b) and

6.6 Numerical results for solitary waves

181

F 1.010

(a)

(b)

(c)

1.008

1.006

30

32

34

36

l

Fig. 6.34. Computed values of the Froude number F versus the wavelength l for periodic waves with τ = 0.24 and u0 = 0.97. Taken from J. Fluid Mech. 134, 205–219 (1983).

13 crests and 14 troughs per wavelength on the proﬁle of Figure 6.35(c). A most interesting question is what happens as one moves further to the right of Figure 6.34 and considers the limit l → ∞ (i.e. r0 → 1). We shall see in Section 6.6.2 that generalised solitary waves are obtained in this limit.

6.6 Numerical results for solitary waves Many of the periodic solutions described in Section 6.5 have nontrivial limits as λ → ∞. These limiting solutions are solitary waves, which belong to one of the three types illlustrated in Figures 6.4–6.7 (i.e. solitary waves with monotonic decay, solitary waves with decaying oscillatory tails and generalised solitary waves.). No solitary waves were found for pure capillary waves (i.e. T = 0, g = 0). In the next two sections, we explore the properties of pure gravity solitary waves and of gravity–capillary solitary waves.

6.6.1 Pure gravity solitary waves In Section 6.5.2.2, we mentioned that periodic gravity waves develop ﬂat troughs for λ/h large. As λ/h → ∞ these waves approach solitary waves. For small values of the amplitude, they agree with the solution (5.156) of the Korteweg–de Vries equation. As the amplitude increases, the waves deviate from (5.156) and ultimately approach the limiting conﬁguration with a 120◦ angle at its crest. An existence proof for solitary waves of ﬁnite amplitude was given by Friedrichs [61]. Numerical computations of solitary waves of ﬁnite amplitude were performed by many investigators (see Schwartz and

182

Numerical computations of nonlinear water waves 0

(a)

y˜

0

y˜

5

10

0

15

(b)

0

5

10

0

15

(c)

y˜

0

5

10

15

x˜

Fig. 6.35. (a) Computed free-surface proﬁle of a periodic wave with τ = 0.24 and u0 = 0.97. The wavelength is l = 32. (b) Same as (a) but with l = 34. (c) Same as (a) but with l = 36.6. Taken from J. Fluid Mech. 134, 205–219 (1983).

Fenton [134] for a review and Tanaka et al. [149] for more recent references). Here we follow mainly the numerical approach of Hunter and Vanden-Broeck [77] (see also Section 6.4.1). It is found that there is a one-parameter family of solutions. This parameter can be chosen as the quantity A deﬁned in (6.49). However, a better

6.6 Numerical results for solitary waves

183

choice is ω =1−

F2 , δ˜2 (0)

(6.76)

where F is deﬁned by (6.44). This choice is similar to (6.58) and (6.59). ˜ Since 1/δ(0) is the dimensionless speed at the crest, ω = 1 for the limiting ˜ → 1 and F → 1 as A → 0. Therefore (6.76) implies conﬁguration. Also, δ(0) that ω → 0 as A → 0. The advantage of the parameter ω over A is that the range of possible values 0 < ω < 1 is known a priori. We present in Figure 6.36 numerical values of F versus ω for waves close to the limiting conﬁguration (i.e. for ω close to 1). 1.295

b

a c

1.290

x

F 1.285

0.80

0.85

0.90

0.95

1.00

Fig. 6.36. Values of F versus ω. Taken from J. Fluid Mech. 136, 63–71 (1983).

The curve a corresponds to numerical values obtained by the method described in Section 6.4.1. It shows that F is not a monotonic function of ω for ω close to 1. In fact there is an inﬁnite number of maxima and minima as ω → 1 (only the ﬁrst maximum is shown in Figure 6.36). This behaviour is similar to that already encoutered for periodic waves (see Figure 6.21). The curves c and b in Figure 6.36 correspond to earlier results obtained by Longuet-Higgins and Fenton [105] and by Byatt-Smith and LonguetHiggins [25]. These authors also found the ﬁrst maximum. A comparison of the curves b and c with the curve a show that the solutions of Byatt-Smith and Longuet-Higgins are more accurate than those of Longuet-Higgins and Fenton. The cross in Figure 6.36 corresponds to the limiting conﬁguration, which will be calculated by series truncation in what follows.

184

Numerical computations of nonlinear water waves

6.6.1.1 A series truncation method for gravity solitary waves We now describe how the series truncation method is used to compute solitary waves up to the limiting conﬁguration. The ﬂow is shown in Figure 6.37. We introduce cartesian coordinates such that the level of the free y

x 0

1.0

2.0

Fig. 6.37. The ﬂow and of the coordinates. The proﬁle is the computed free surface of the highest solitary wave. Taken from J. Fluid Mech. 136, 63–71 (1983).

surface in the far ﬁeld corresponds to y = 0. As previously, we introduce the complex potential f = φ + iψ and the complex velocity w = u − iv. We choose φ = 0 at the crest and ψ = 0 on the free surface. As |x| → ∞, the ﬂow approaches a uniform stream characterised by a constant velocity U and a constant depth H. We introduce dimensionless variables by taking U as the unit velocity and H as the unit length. On the free surface the Bernoulli equation yields u2 + v 2 +

2 y = 1, F2

(6.77)

where F is deﬁned by (6.44). We map the ﬂow domain in the region |t| < 1 of the complex t-plane by the transformation f=

2 1+t ln − i. π 1−t

(6.78)

The transformation (6.78) maps the bottom of the channel onto the real diameter −1 < t < 1 and the free surface onto the half-circumference |t| = 1 in the upper half of the t-plane. We will use the notation t = reiσ , so that the free surface is described by r = 1 and 0 < σ < π. Hunter and Vanden-Broeck [77] calculated the highest solitary wave by representing the complex velocity w by the expansion 1/3 ∞ 1 + t2 exp A(1 − t2 )2λ + an (t2n − 1) , (6.79) w= 2 n=1

6.6 Numerical results for solitary waves

185

where λ is the smallest positive root of πλ −

tan πλ = 0. F2

(6.80)

The term A(1−t2 )2λ in (6.79) comes from the fact that the complex velocity on the free surface satisﬁes u − iv ≈ 1 + Ae−πλ|φ|

as

|φ| → ∞.

(6.81)

The constants an and A must satisfy the free-surface condition (6.77). The analysis leading to (6.65) and (6.66) shows that solitary waves of arbitrary amplitude (including the highest) can be calculated by representing w3 as follows: ∞ an t2n−2 . (6.82) w3 = 1 + (1 − t2 )2λ n=1

Alternatively, we can set 1/3 ∞ 1 + βt2 2 2λ 2n w= exp A(1 − t ) + an (t − 1) . 1+β

(6.83)

n=1

Here β is another constant to be found as part of the solution. The expansion (6.79) is similar to the Michell expansion for the periodic wave of largest steepness (6.66) and was used before by Lenau [96]. The factor (1 + t2 )1/3 removes the singularity associated with the 120◦ angle at the crest of the highest wave. The expansions (6.82) and (6.83) are equivalent to the Davies approximation [41] and the Havelock expansion [70] (see Vanden-Broeck and Miloh [188]). As shown by Vanden-Broeck and Miloh [188], equivalent numerical results are obtained by using (6.82) or (6.83). Here we describe results based on (6.83). First we diﬀerentiate (6.77) with respect to σ. Then using (6.78) we obtain F 2 [u(σ)uσ (σ) + v(σ)vσ (σ)] −

v(σ) 1 2 = 0. π u2 (σ) + v 2 (σ) sin σ

(6.84)

We will characterise the amplitude of the solitary wave by the parameter ω deﬁned in (6.76). We now truncate the inﬁnite series in (6.82) after N terms and determine the N + 4 unknowns a1 , a2 , . . . , aN , A, β, λ and F 2 by collocation. Thus we introduce the N + 2 collocation points σI =

E + (I − 1)E, 2

I = 1, . . . , N + 2,

(6.85)

186

Numerical computations of nonlinear water waves

where E = π/(2N + 4), and satisfy (6.84) at the collocation points (6.85). This yields N + 2 nonlinear algebraic equations. The ﬁnal two equations are given by (6.80) and (6.76). Thus for a given value of ω we have a system of N + 4 equations with N + 4 unknowns. This system is solved by Newton iteration. In some calculations, it is more convenient to use a variant of the scheme in which β is ﬁxed and ω is found as part of the solution. As an example we present numerical results for the highest wave. Then we ﬁx ω = 1 (or β = 1 if we use the above-mentioned variant). The scheme is then equivalent to that used by Hunter and Vanden-Broeck [77]. We note that for the highest wave (for which u = v = 0 at the crest), the elevation α of the crest is related to F by the simple relation F2 =

2 α

(6.86)

(see 6.77). Hunter and Vanden-Broeck [77] computed solutions for values of N ≤ 100 and concluded that α = 0.833 22. This estimate can be improved by increasing N . They found the values 0.833 217 02, 0.833 208 77, 0.833 205 44, 0.833 203 15 and 0.833 202 00 for N = 200, 400, 600, 900 and 1200 respectively. An extrapolation for N → ∞ gives the value 0.833 198 6. This value is in close agreement with the value 0.833 197 obtained by Williams [197] and the value 0.833 199 79 obtained by Evans and Ford [53]. The computed proﬁle of the highest solitary wave is shown in Figure 6.37.

6.6.2 Gravity–capillary solitary waves The numerical results of Section 6.5.3.2 suggest that periodic gravity– capillary waves with τ < 1/3 approach generalised solitary waves as l → ∞ (see Figure 6.35). To conﬁrm this idea we repeated the calculations of Figure 6.35 for larger values of l (i.e. for r0 closer to 1) and for various values of u0 and τ . The calculations follow closely Vanden-Broeck [177]. In Figure 6.38, we present values of F versus l for τ = 0.24 and u0 = 0.99. Curves (a) and (b) correspond to two computed families of solutions; corresponding proﬁles of the waves for these two families are shown in Figures 6.39 and 6.40 respectively. Comparing the proﬁles in Figures 6.39 and 6.40, we see that the number of inﬂexion points (and therefore the number of ripples) increases as one moves in Figure 6.38 from one family to another family further to the right. Furthermore it follows from the symmetry of the ﬂow that the last point on the graphs in Figures 6.39 and 6.40 is either a crest or a trough of the ripples. These facts indicate that, for l suﬃciently large, there is an inﬁnite

6.6 Numerical results for solitary waves

187

F

1.010 1.008 1.006

(b)

(a)

1.004

×

×

(c)

1.002

96

104

112

120

128

l

Fig. 6.38. Values of the Froude number F versus the dimensionless wavelength l for periodic waves with τ = 0.24 and u0 = 0.99. The solid curves (a) and (b) are the numerical solutions. The broken curves show families obtained by translation. Taken from Phys. Fluids A 3, 263–266 (1991).

number of families of solutions. The corresponding curves in Figure 6.38 can be obtained from any particular curve by translating it horizontally by a multiple of the wavelength of the ripples. Generalised solitary waves are then obtained by jumping from one curve to the next as we take the limit l → ∞. After each jump, two more crests or troughs appear on each wavelength of the proﬁles, one on the right and one on the left. In the limit l → ∞, we obtain a generalised solitary wave with inﬁnite trains of ripples in the far ﬁeld. For each value of 0 < τ < 1/3, these generalised solitary waves form a two-parameter family of solutions. The numerical results in Figures 6.38–6.40 are for periodic waves with l > 90. We will use these long waves to approximate generalised solitary waves. The proﬁles in Figures 6.39 and 6.40 show that the steepness of the ripples (i.e. the diﬀerence in height between a crest and a trough of the ripples divided by their wavelength) is small for all the proﬁles presented. Therefore the dimensionless wavelength L of the ripples satisﬁes the linear dispersion relation L 2π 4π 2 2 1 + τ 2 tanh (6.87) F = 2π L L (see (2.90)). Starting with curve (a) in Figure 6.38, we used (6.87) to calculate the wavelength L corresponding to each value of F . Then we performed the translations mentioned earlier. The resulting curves are the broken lines in

188

Numerical computations of nonlinear water waves 0.010

0.006

(a)

(b) 0.008

0.004

0.006

0.002

0.004 0.000

0.002 0.000

0

10

20

30

40

50

60

0.010

0

10

20

30

40

50

0.010

(c)

(d)

0.008

0.008

0.006

0.006

0.004

0.004

0.002

0.002

0.000

0.000

0

10

20

30

40

50

0

10

20

30

40

50

0.025

(e)

0.020 0.015 0.010 0.005 0.000

0

10

20

30

40

50

Fig. 6.39. (a) Computed free-surface proﬁles for a periodic wave with τ = 0.24, u0 = 0.99 and F = 1.0005. Only half a wavelength is shown. (b) Same as (a) but with F = 1.0035. (c) Same as (a) but with F = 1.003 58. (d) Same as (a) but with F = 1.0036. (e) Same as (a) but with F = 1.008. Taken from Phys. Fluids A 3, 263–266 (1991).

Figure 6.38. We note that the 12th broken line coincides within graphical accuracy with curve (b). This constitutes a check on our calculations. The solution corresponding to the proﬁle in Figure 6.39(a) is very close to a train of periodic waves extending from x = −∞ to x = ∞. This shows that the generalised solitary waves bifurcate from a train of periodic waves.

6.6 Numerical results for solitary waves

189

As one moves away from the bifurcation point, one crest of the train of periodic waves is progressively lifted. 0.010

0.010

(b)

(a) 0.008

0.008

0.006

0.006

0.004

0.004

0.002

0.002

0.000

0.000

0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

0.010

(c) 0.008 0.006 0.004 0.002 0.000

0

10

20

30

40

50

60

70

Fig. 6.40. (a) Computed free-surface proﬁles for a periodic wave with τ = 0.24, u0 = 0.99 and F = 1.0038. Only half a wavelength is shown. (b) Same as (a) but with F = 1.003 87. This proﬁle is close to that corresponding to the right-hand cross in Figure 6.38. (c) Same as (a) but with F = 1.003 95. Taken from Phys. Fluids A 3, 263–266 (1991).

Figure 6.39 shows that the amplitude of the ripples ﬁrst decreases and then increases as one moves along the of solution branch. There is one point on each curve in Figure 6.38 for which the amplitude of the ripples is a minimum. These points are indicated by the crosses in Figure 6.38, and the proﬁle corresponding to the cross on the left is shown in Figure 6.39(c). Within graphical accuracy, the amplitude of the ripples in Figure 6.39(c) is zero and the corresponding solution looks like a true solitary wave (i.e. a wave without ripples in the far ﬁeld). However, the calculations of Champneys et al. [26] showed that the amplitude of the ripples is always diﬀerent from zero (although it can be very small) for 9/50 < τ < 1/3. This was also shown theoretically by Sun [146] for τ suﬃciently close to 1/3. One feature of the ripples is that their amplitude is an exponentially small function of F − 1; this was shown by exponential asymptotics [147] and

190

Numerical computations of nonlinear water waves

by application of centre-manifold and normal-form theory [97]. For further references see the review [43]. We conclude this chapter by comparing our numerical results with the predictions of the Korteweg–de Vries equation (5.154). The curve (c) in Figure 6.38 corresponds to the periodic solutions of (5.154) (i.e. the cnoidal waves). As l → ∞, curve (c) approches the solitary-wave solution. The solutions of the Korteweg–de Vries equation (see curve (c)) do not have ripples in the far ﬁeld. However, the above discussion shows that the numerical solutions of the complete nonlinear problem (curves (a) and (b) and the broken curves) always have ripples of nonzero amplitude in the far ﬁeld. Figure 6.38 shows that the numerical solutions of the complete nonlinear problem (curves (a) and (b) and the broken curves) intersect curve (c) at a discrete set of points. The intersections with curves (a) and (b) are very close to the crosses. Therefore, for u0 close to 1 the solutions for which the amplitude of the ripples is a minimum are closely approximated by the cnoidal wave solutions, although ripples are not predicted for the cnoidal waves.

7 Nonlinear free-surface ﬂows generated by moving disturbances

We now return to the free-surface ﬂow generated by a moving disturbance and extend the results of Chapter 4 to the nonlinear regime. We shall see that the wave trains in the far ﬁeld (if they exist) are then described by the nonlinear theories of Chapters 5 and 6. Furthermore we will show that the nonuniformities of Figures 4.9 and 4.8 are removed when a nonlinear theory is used. Some nonlinear solutions described in this chapter approach the linear solutions of Chapter 4 as the size of the disturbance approaches zero, while others approach solitary waves. We have organised the results in the following way. In Section 7.1 we present pure gravity free-surface ﬂows (i.e. g = 0, T = 0) in water of ﬁnite depth and show in Section 7.1.1 that the nonuniformity of the linear supercritical solutions near F = 1 (see Figure 4.9) is removed when a nonlinear theory is used. Subcritical ﬂows are considered in Section 7.1.2. In Section 7.2 we consider gravity–capillary free-surface ﬂows. Solutions in water of ﬁnite depth are described in Section 7.2.1. We show in Section 7.2.2 that the nonuniformity of the linear theory near α = 0.25 (see Figure 4.8) is removed when a nonlinear theory is used. We examine in Section 7.3 the implications of the existence of multiple branches of periodic gravity– capillary waves (see Sections 5.1.2 and 6.5.3) for free-surface ﬂows generated by moving disturbances in water of inﬁnite depth. Qualitatively similar numerical results were obtained for the three types of disturbance shown in Figures 4.1–4.4. Therefore, in each section or subsection we will restrict our attention to one type of disturbance (due to a pressure distribution in Sections 7.1.1, 7.2.2 and 7.3 and due to a submerged object in Sections 7.1.2 and 7.2). 191

192

Nonlinear free-surface ﬂows generated by moving disturbances

7.1 Pure gravity free-surface ﬂows in water of ﬁnite depth 7.1.1 Supercritical ﬂows We will assume that the disturbance is a pressure distribution, and we will neglect the eﬀect of surface tension. The ﬂow conﬁguration is shown in Figure 4.4. The fully nonlinear problem is described by equations (4.1)– (4.4) with T = 0. In this section we restrict our attention to supercritical free-surface ﬂows. Therefore the ﬂow approaches a uniform stream with constant velocity U and constant depth H in the far ﬁeld. We deﬁne the Froude number U . (7.1) F = (gH)1/2 Since the ﬂow is assumed to be supercritical, F > 1. We introduce dimensionless variables by taking U as the reference velocity and U 2 /g as the reference length. In terms of the dimensionless variables, the dynamic boundary condition (4.3) (with T = 0) becomes 1 2 P (x) ˜ (φ + φ2y ) + y + =B 2 x ρU 2

on y = η(x).

(7.2)

˜ = 1. ˜ = B/U 2 . It follows from the choice of the origin of y that B Here B In order to compare the nonlinear computations with the linear results of Chapter 4, we assume that the pressure distribution is given by (4.70). Therefore, using the dimensionless variables we write P (x) e−5x . = ρU 2 2 2

(7.3)

Since P (−x) = P (x), the solutions are symmetric with respect to x = 0. We solve the problem numerically by using a boundary integral equation method similar to those used in Sections 3.1.2.2 and 6.4.1. We ﬁrst deﬁne the complex potential f = φ + iψ. We choose ψ = 0 on the free surface and φ = 0 at the point x = 0 on the free surface. The value −Q of ψ on the bottom satisﬁes 1 (7.4) Q = 2, F where F is deﬁned by (7.1). Next we deﬁne the complex function δ + iβ by (6.3). We seek δ + iβ as an analytic function of the complex potential f = φ + iψ. We satisfy the kinematic boundary condition β = 0 on ψ = −Q by reﬂecting the ﬂow in the bottom surface. Therefore we seek δ + iβ as an analytic function in the strip −2Q < ψ < 0. The values of δ(φ, ψ) and of

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

193

β(φ, ψ) on the free surface ψ = 0 and on its image ψ = −2Q are related by δ(φ, −2Q) = δ(φ, 0) and β(φ, −2Q) = −β(φ, 0).

(7.5)

We derive an integral equation on the free surface by applying the Cauchy integral formula to the function δ − 1 + iβ in the complex f -plane with a contour consisting of the free surface, its image and two vertical lines at φ = ±∞. Since δ − 1 + iβ → 0 as φ → ±∞, there are no contributions from the two vertical lines at φ = ±∞. The contributions from the free surface and its image give, after taking the real part and using (7.5), ˜ ˜ ˜ − φ) + 2Q[δ(ϕ) − 1] 1 ∞ −β(ϕ)(ϕ 1 ∞ β(ϕ) ˜ dϕ + dϕ. δ(φ) − 1 = − 2 2 π −∞ ϕ − φ π −∞ (ϕ − φ) + 4Q (7.6) ˜ ˜ Here δ(φ) = δ(φ, 0) and β(φ) = β(φ, 0). We can rewrite (7.2) in terms of δ˜ and β˜ as 1 + 2y + 2p(φ) = 1, 2 ˜ δ + β˜2

(7.7)

where p(φ) = and

∞

y=−

e−5φ 2

2

(7.8)

˜ β(ϕ)dϕ.

(7.9)

φ

Comparing (7.3) and (7.8), we see that P (x) has been replaced by p(φ). This change in the nonlinear dynamic boundary condition is compatible with the linear theory of Chapter 4, since φ = x + O() (see (4.7) and (4.8)) implies that p(φ) = p(x) + O(2 ) as → 0. Therefore the nonlinear problem deﬁned by (7.6) and (7.7) reduces formally to that of Chapter 4 in the limit → 0. We introduce the N mesh points φI = −A + 2A

I −1 , N −1

I = 1, . . . , N,

(7.10)

I = 1, . . . , N − 1,

(7.11)

the midpoints φm I =

φI + φI+1 , 2

and the unknowns ˜ I ), δI = δ(φ

˜ I ), βI = β(φ

I = 1, . . . , N.

(7.12)

194

Nonlinear free-surface ﬂows generated by moving disturbances

We now obtain 2N − 2 equations by satisfying (7.6) and (7.7) at the midpoints (7.11). The details of the discretisation follow that used in Sections 6.3 and 6.4.1. We force the free surface to be ﬂat in the far ﬁeld by imposing the conditions β1 = βN = 0.

(7.13)

This gives two more equations. The ﬁnal equation ﬁxes the elevation of the free surface at φ = 0: ∞ ˜ β(ϕ)dϕ = a, (7.14) − 0

where a is given. This system of 2N + 1 nonlinear equations for the 2N + 1 unknowns δI , βI and Q is solved by Newton’s method. We note that after the value of Q has been calculated, the corresponding value of F can be deduced from (7.4). Numerical values of a versus F are shown in Figure 7.1 for = 0.001. The corresponding linear results are shown in Figure 4.9. Figure 7.1 shows that the nonuniformity of the linear theory (i.e. the vertical asymptote in Figure 4.9) has been removed in the nonlinear calculations: all the values of a are now ﬁnite. 0.04

0.03

0.02

0.01

0

1

1.01

1.02

1.03

Fig. 7.1. Values of a versus F for = 0.001.

The curve of Figure 7.1 has a turning point at F = Ft ≈ 1.004. For F > Ft , the lower portion of the curve is close to that predicted by the linear theory of Chapter 4. The corresponding solutions are perturbations of a uniform stream. This can be checked in the following way. Select a solution corresponding to a point on the lower part of the curve of Figure 7.1 (let us say for F = 1.02). Then use this solution as an initial guess to compute a new solution for F = 1.02 and a value 1 of slightly smaller than

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

195

0.001. Then use this solution as an initial guess to compute a new solution for a value of slightly smaller than 1 , and so on. Ultimately we obtain a solution corresponding to = 0. This solution is a uniform stream. We shall use this procedure, of computing solutions for a decreasing sequence of values of , several times in this chapter and in those that follow. We shall refer to it as a ‘continuation in ’ (or in whatever parameter we are using). An interesting question is what happens if, using continuation in , we start with a solution on the upper portion of the curve of Figure 7.1. The answer is that the resulting solution for = 0 is a solitary wave. Solitary waves form a solution branch bifurcating from F = 1 in Figure 7.1 (see Chapter 5 for analytical solutions for solitary waves of small amplitude and Chapter 6 for numerical computations of solitary waves of arbitrary amplitude). These solitary waves are solutions of the present problem corresponding to = 0. The upper portion of the curve of Figure 7.1 is a perturbation of that branch of solitary waves. Our ﬁndings can be summarised as follows. For = 0, the solution in Figure 7.1 consists of the F -axis (a uniform stream) and of the branch of solitary waves bifurcating from the F -axis at F = 1 and extending for F > 1. For = 0, the curve in Figure 7.1 has a turning point at F = Ft > 1 (the value of Ft depends on ). The solutions corresponding to the lower portion of the curve are perturbations of a uniform stream, while the solutions corresponding to the upper portion of the curve are perturbations of solitary waves. In Figure 7.2 we present two free-surface proﬁles corresponding to F = 1.02 and = 0.001. The proﬁle of lower amplitude is a perturbation of a uniform stream. The proﬁle of larger amplitude is a perturbation of a solitary wave. Results similar to those presented in this section were obtained in [173], [45] and [48] for other disturbances.

7.1.2 Subcritical ﬂows We now consider subcritical ﬂows (i.e. we assume that F < 1). Although the results could be described by using the pressure distribution of Section 7.1.1, we will choose to present them for another type of disturbance, namely a submerged semicircular obstacle on the bottom of a channel (see Figure 7.3). We introduce cartesian coordinates with the x-axis along the bottom and the origin at the centre of the semicircular obstacle. The acceleration of gravity g is acting in the negative y-direction. The linear theory of Chapter 4 indicates that waves should be expected on the free surface as

196

Nonlinear free-surface ﬂows generated by moving disturbances 0.04

0.03

0.02

0.01

0 0

10

20

Fig. 7.2. Two free-surface proﬁles generated by a moving pressure distribution for = 0.001 and F = 1.02. The lower proﬁle is a perturbation of a uniform stream while the upper proﬁle is a perturbation of a solitary wave.

x → ∞. As x → −∞, the ﬂow approaches a uniform stream with a constant velocity U and constant depth H. We introduce dimensionless variables by taking H as the unit length and U as the unit velocity.

H

U y

x

Fig. 7.3. The ﬂow conﬁguration. When surface tension is neglected, as in the ﬁgure, the free surface is ﬂat as x → −∞ and wavy as x → ∞. If surface tension were taken into account, there would be in addition a train of waves as x → −∞.

We then deﬁne the following complex variables, z = x + iy, the complex potential f = φ + iψ and the complex velocity u − iv. Without loss of generality we choose ψ = 0 on the bottom, so that ψ = 1 on the free surface. The condition of constant pressure on the free surface can be written as 1 2 2 1 F (u + v 2 ) + y = F 2 + 1. (7.15) 2 2 Here F is the Froude number deﬁned in (7.1). Following Forbes and Schwartz [59] we deﬁne the new variable ζ = ξ + iγ

(7.16)

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

197

by the relation 1 ξ + iγ = 2

α2 z+ z

,

(7.17)

where α = R/H is the dimensionless radius of the semicircle. Relation (7.17) is the classical Joukowskii transformation; it maps the z-plane into the ζ-plane, in which the bottom streamline is a straight line. We shall seek ξ + iγ as an analytic function of f = φ + iψ in the strip 0 < ψ < 1. In terms of the variable ζ(φ, ψ), (7.15) becomes F2

(z 2 − α2 )(z 2 − α2 ) 1 + z − F 2 − 1 = 0, 2 2 8(zz) ζφ ζφ

(7.18)

where an overbar signiﬁes the complex conjugate and denotes the imaginary part. By using the Cauchy integral equation formula, Forbes and Schwartz [59] derived the following integral relation between ξ(φ, 1) and γ(φ, 1): dθ 2 +∞ 1 1 − ξφ (θ, 1) − ξφ (φ, 1) − 2 π −∞ 2 (θ − φ)2 + 4

+∞ +∞ γθ (θ, 1)(θ − φ) γθ (θ, 1) 1 =− dθ + dθ . π (θ − φ)2 + 4 −∞ −∞ (θ − φ)

(7.19)

Their derivation is similar to that leading to (7.6), therefore the details will not be repeated here. The second integral on the right-hand side of (7.19) is a Cauchy principal value. This concludes the formulation of the problem. For given values of α and F , we seek two functions ξ and γ satisfying (7.18) and (7.19). The linear theory of Chapter 4 indicates that waves should be expected on the free surface. Furthermore the radiation condition requires the waves to be behind the obstacle and the free surface to be ﬂat as x → −∞. This suggests that an easy way to impose the radiation condition in the nonlinear computations is to truncate the integrals from −∞ to ∞ in (7.19) at some ﬁnite values −t1 and t2 and force the free surface to be ﬂat at φ = −t1 . This approach was used sucessfully by Forbes and Schwartz [59], VandenBroeck [158], Mekias and Vanden-Broeck [110] and others. A more accurate approach is to approximate the integrals from −∞ to −t1 and from t2 to ∞ by asymptotic solutions for |φ| large. These two approaches are described for the ﬂow conﬁguration of Figure 7.3 in Sections 7.1.2.1 and 7.1.2.2.

198

Nonlinear free-surface ﬂows generated by moving disturbances

7.1.2.1 Results obtained using truncation of the integrals The numerical method follows closely the work of Forbes and Schwartz [59], Forbes [55] and Grandison and Vanden-Broeck [65]. The reader is referred to those papers for details. We ﬁrst truncate the three integrals from −∞ to ∞ in (7.19) to integrals from −t1 to t2 , where t1 and t2 are large positive numbers. Then we deﬁne equally spaced points φI = −t1 +

t2 + t1 I, N

I = 0, . . . , N,

and corresponding unknowns ξI = ξφ (φI , 1),

I = 0, . . . , N,

ηI = γφ (φI , 1),

I = 0, . . . , N.

The problem is then discretised by following the approach used in Section 7.1.1. Relations (7.18) and (7.19) yield a system of nonlinear algebraic equations for the unknowns ξI and ηI . The radiation condition in the absence of surface tension is equivalent to the requirement that the free-surface proﬁle is ﬂat as x → −∞. This is imposed by forcing γ = 1/2 and γφ = 0 at the ﬁrst mesh point φ0 . The system of nonlinear algebraic equations is solved by Newton iteration. The above numerical scheme gives very good results (see Forbes and Schwartz [59] for examples). However, as noted by Forbes and Schwartz, spurious small waves appear in front of the obstacle (i.e. in the region x < 0). These waves do not satisfy the radiation condition and are an artifact of the numerical procedure. There is also a distortion in approximately the last quarter wavelength of the downstream waves, which, although insigniﬁcant, can prevent the numerical method from converging for values of the Froude number close to 1. We present below a numerical method which improves the accuracy, removes the spurious waves and reduces the distortion. 7.1.2.2 Results obtained without using truncation of the integrals The inaccuracies in the scheme of Section 7.1.2.1 are caused by the replacement of the integrals from −∞ to ∞ in (7.19) by integrals from φ = −t1 to φ = t2 . We shall show that these inaccuracies are greatly reduced by including suitable approximations of the integrals from −∞ to −t1 and from t2 to ∞.

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

199

We consider ﬁrst the integrals from −∞ to −t1 . As φ → −∞, the ﬂow approaches a uniform stream with velocity 1. An asymptotic solution describing a waveless perturbation of a uniform stream was derived in Section 3.3 (see formulae (3.210) and (3.211)). Using (3.210), (3.211) and the properties φ ≈ x and y ≈ 2γ as φ → −∞ we obtain γφ ≈ Aeλφ

as

φ → −∞,

where A is a constant and λ is the smallest positive root of tan λ = 0. λ− F2 We also have ξφ ≈ C as φ → −∞,

(7.20)

(7.21) (7.22)

where C is a constant. We now use (7.22) and (7.20) to approximate the three integrals from −∞ to −t1 as follows: −t1 dθ 1 1 −(t1 + φ) π 1 ≈ C− arctan + , ξφ (θ, 1) − 2 (θ − φ)2 + 4 2 2 2 4 −∞ (7.23) −t1 −t1 λθ γθ (θ, 1)(θ − φ) e (θ − φ) dθ ≈ A dθ, (7.24) 2 2 (θ − φ) + 4 −∞ −∞ (θ − φ) + 4 −t1 λθ −t1 γθ (θ, 1) e dθ ≈ A dθ. (7.25) −∞ θ − φ −∞ θ − φ The constants A and C are found by requiring that (7.20) and (7.22) are satisﬁed at the last mesh point φN . We now consider the integrals from t2 to ∞. This is more complicated than the previous case, because there are nonlinear waves on the free surface for φ > 0. However, the waves ultimately approach a train of periodic waves as φ → ∞. Therefore we can assume that the waves repeat themselves without change of shape or amplitude for φ > t2 . This is a good approximation when t2 is large. We deﬁne at each iteration two values p1 < p2 of φ that correspond to the two crests of the wave trains closest to φ = t2 (i.e. the two crests further to the right). We deﬁne γφ for φ > t2 by γφ (t2 + s) = γφ (t2 + p1 − p2 + s)

for s < p2 − p1

(7.26)

and, more generally, γφ (t2 + s) = γφ (t2 + p1 − p2 + s − n(p2 − p1 )) for

n(p2 − p1 ) < s < (n + 1)(p2 − p1 ).

(7.27)

200

Nonlinear free-surface ﬂows generated by moving disturbances 1.01

1.00

0.99

0.98

0.97 −12

0

4

8

Fig. 7.4. Free-surface ﬂow for α = 0.2, F = 0.4.

Here s > 0 and n is a positive integer. Similarly, we write ξφ (t2 + s) = ξφ (t2 + p1 − p2 + s − n(p2 − p1 )) for n(p2 − p1 ) < s < (n + 1)(p2 − p1 ).

(7.28)

Relations (7.26)–(7.28) deﬁne ηφ and ξφ for φ > t2 in terms of their values for φ < t2 . This enables us to extend the trapezoidal-rule (or Simpson-rule) approximation of the integrals to values of φ much larger than t2 without introducing extra unknowns ξI and ηI or increasing signiﬁcantly the computing time. Figure 7.4 shows a typical free-surface proﬁle. We note the elimination of waves upstream of the disturbance. All the results presented were obtained with N = 300 mesh points. We checked that these results are correct within graphical accuracy by varying N . Making the above downstream approximation removes the distortion in the last quarter wavelength experienced by Forbes and Schwartz [59] and allows us to compute solutions for ﬂow speeds greater than those used by these authors. Convergence was obtained for values of the Froude number up to F = 0.555, for α = 0.2 (see Figure 7.5). The proﬁle of Figure 7.5 contains nonlinear waves with sharp crests and broad troughs. This proﬁle demonstrates that the radiation condition is eﬀectively satisﬁed and that there is no downstream distortion.

7.2 Gravity–capillary free-surface ﬂows

201

1.10

1.05

1.00

0.95 0

2

4

6

Fig. 7.5. Free-surface ﬂow for α = 0.2, F = 0.555.

7.2 Gravity–capillary free-surface ﬂows In this section, we include the eﬀects of both the gravity g and the surface tension T . We will consider solutions for water of ﬁnite depth in Section 7.2.1. Results for inﬁnite depth are presented in Section 7.2.2.

7.2.1 Results in ﬁnite depth We generalised the calculations of Section 7.1.2 by including the eﬀect of surface tension. The ﬂow conﬁguration is sketched in Figure 7.3. The dynamic boundary condition (7.15) now becomes 1 1 2 2 F (u + v 2 ) + y + W K = F 2 + 1, 2 2

(7.29)

where K is the curvature at the free surface and W is the Weber number, deﬁned by T . W = ρgH 2 Following Forbes [56] we ﬁnd that (7.29) can be replaced by 1 2 (z 2 − α2 )(z 2 − α2 ) (z 2 − α2 )1/2 (z 2 − α2 )1/2 1 2 1 F F iW + [z] − − 1 − 8 2 4 (zz)2 ζφ ζ φ (zz)(ζφ ζ φ )3/2 zζ zζ φ φ = 0. (7.30) × (ζφ ζ φφ − ζ φ ζφφ ) + 4α2 ζφ ζ φ − 2 (z 2 − α2 )2 (z − α2 )2 This expression reduces to (7.18) when W = 0.

202

Nonlinear free-surface ﬂows generated by moving disturbances

The linear theory of Chapter 4 shows that in general there are wave trains both upstream and downstream of the disturbance. We use the method described in Section 7.1.2 to approximate the integrals from t2 to +∞; the method used in Section 7.1.2 to approximate the integral from −∞ to −t1 is no longer applicable because of the upstream train of waves. Here we use the approach developed by Vanden-Broeck [182] and Grandison and Vanden-Broeck [65]. It consists in approximating γφ for −∞ < φ < −t1 by a linear wave. We will assume in this section that the condition (5.55) is satisﬁed (solutions when (5.55) is not satisﬁed will be considered in Section 7.3). Using (5.57) and noting that to leading order φ ≈ x we can write, for −∞ < φ < −t1 , γφ = D sin(kφ + ω),

(7.31)

where D, k and ω are to be found as part of the solution. Relation (7.31) approximates the waves in −∞ < φ < −t1 by linear waves. It is therefore a good approximation if these waves are of small amplitude. An extension of (7.31) consists of writing γφ =

˜ N

Dn sin(nkφ + θ).

(7.32)

n=1

Relation (7.32) is now an approximation for nonlinear waves and is exact ˜ → ∞. The constants D1 , D2 , . . . , D ˜ , k and θ are found by in the limit N N ﬁtting values of γφ at points on the discretised mesh close to −t1 . ˜ and checked that We repeated the calculations for diﬀerent values of N ˜ the results presented are independent of N . This truncation procedure enabled us to conﬁrm and improve the results of Forbes [56]. The results presented by Forbes do not have a constant mean upstream height. Furthermore the distortion that occurs in the T = 0 case is even more pronounced in the T = 0 case and extends over two wavelengths. In Figure 7.6 we present results in the absence of surface tension to contrast them with the gravity–capillary solution of Figure 7.7. As can be seen in Figure 7.7, the modiﬁed model not only demonstrates the existence of a solution with wave trains both upstream and downstream but also that the previous problems associated with this computation, namely the decreasing mean height of the upstream waves and the violation of the radiation condition, has been eliminated. The linear dispersion relation (2.83) predicts an upstream wavelength λ ≈ 0.8796 and a downstream wavelength λ ≈ 3.4247. These predictions agree to within 2% with the values predicted by the nonlinear computations. We note that for λ ≈ 3.4247 and W = 0.07 the condition (5.55) is satisﬁed. Further solutions, which involve Wilton ripples

7.2 Gravity–capillary free-surface ﬂows

203

1.02

1.01

1.00

0.99

0.98 −15

−10

−5

0

5

10

15

Fig. 7.6. Free-surface ﬂow for α = 0.05, F = 0.8, W = 0. 1.02

1.01

1.00

0.99

0.98 −15

−10

−5

0

5

10

15

Fig. 7.7. Free-surface ﬂow for α = 0.05, F = 0.8, W = 0.07.

and which are valid when condition (5.55) is not satisﬁed, will be presented in Section 7.3. The approximations of the far ﬁeld described in this section are general and can be used for any steady two-dimensional gravity–capillary free-surface ﬂow past a disturbance. 7.2.2 Results in inﬁnite depth (removal of the nonuniformity) In this section, we show that the nonuniformity of the linear gravity–capillary free-surface ﬂows shown in Figure 4.8 as α → 1/4 is removed when nonlinear solutions are calculated. Here the parameter α is deﬁned by (4.69). The linear theory of Chapter 4 predicts solutions with waves when α < 1/4 and solutions with a ﬂat free surface in the far ﬁeld when α > 1/4. In this section we restrict our attention to values α > 1/4 and show that the

204

Nonlinear free-surface ﬂows generated by moving disturbances

nonuniformity of Figure 4.8 as α → 1/4+ is removed when a nonlinear theory is developed. In the far ﬁeld we have φx → U

as |x| → ∞.

(7.33)

We introduce dimensionless variables by using U 2 /g as the reference length and U as the reference velocity and assume that the disturbance is the pressure distribution (7.8). The formulation and numerical procedure follow that of Section 7.1.1. The only diﬀerences are that surface tension needs to be included in the dynamic boundary condition and that the ﬂuid is of inﬁnite depth. Therefore we rewrite the dynamic boundary condition (7.7) as 2 δ˜β˜φ − δ˜φ β˜ 1 + 2y + 2p(φ) − = 1. α (δ˜2 + β˜2 )3/2 δ˜2 + β˜2

(7.34)

When α = ∞ (i.e. T = 0), (7.34) reduces to (7.7). The appropriate integral relation between δ˜ and β˜ can be derived by taking the limit Q → ∞ in (7.6). This yields ˜ 1 ∞ β(ϕ) ˜ δ(φ) − 1 = − dϕ. (7.35) π −∞ ϕ − φ Equations (7.34) and (7.35) deﬁne a system of equations for the unknown ˜ To solve it numerically we introduce the mesh points functions δ˜ and β. (7.10), the midpoints (7.11) and the unknowns (7.12). We obtain 2N − 2 equations by satisfying (7.34) and (7.35) at the midpoints (7.11); three more equations are provided by (7.13) and (7.14). The resulting system of 2N + 1 equations for the 2N + 1 unknowns δI , βI and α is solved by Newton’s method for given values of and a. Details about the numerical procedure can be found in Vanden-Broeck and Dias [184] and Dias et al. [44]. In Figure 7.8 we present numerical values of a (see (7.14) for a deﬁnition) versus α. The upper curve corresponds to = −0.001 and the lower curve to = 0.001. The corresponding linear results are shown in Figure 4.8. Figure 7.8 shows that the nonuniformity of the linear theory (i.e. the vertical asymptote at α = 0.25) has been removed in the nonlinear calculations: all the values of a are now ﬁnite. The curves in Figure 7.8 have turning points at α = α± . Here α+ and α− correspond to the upper and lower curves respectively. The values α± are greater than 0.25 and depend on the value of . The solutions corresponding to the lower portion of the upper curve for α > α+ and to the upper portion of the lower curve for α > α− are perturbations of a uniform stream, in the sense that continuation in (see Section

7.2 Gravity–capillary free-surface ﬂows

205

0.04

0

0.25

0.26

0.27

0.28

0.29

0.30

Fig. 7.8. Values of a versus α for = −0.001 (upper curve) and = 0.001 (lower curve). 0.1

0

0

20

40

Fig. 7.9. Depression solitary wave for α = 0.264.

7.1.1 for a deﬁnition) ultimately leads to a uniform stream. These solutions are described by the linear theory of Chapter 4 when || is small. If we use continuation in by starting with a solution on the upper portion of the upper curve in Figure 7.8 for α > α+ , we obtain elevation solitary waves with a decaying tail; similarly, if we use continuation in by starting with a solution on the lower portion of the lower curve in Figure 7.8 for α > α− , we obtain depression solitary waves with a decaying tail. The shapes of such solitary waves are indicated in Figures 6.5 and 6.6. The solitary waves with decaying tails form solution branches that bifurcate from α = 0.25 in Figure 7.8. Typical free-surface proﬁles are shown in Figures 7.9 and 7.10. The fact that the solitary waves bifurcate from α = 0.25 can be explained

206

Nonlinear free-surface ﬂows generated by moving disturbances

intuitively as follows. The proﬁles of Figures 7.9 and 7.10 look like a wave train whose amplitude is slowly varying. According to linear theory we should expect these waves of varying amplitude to travel at the phase velocity while their envelopes travel at the group velocity (see Section 2.4.3). Since in general phase and group velocities have diﬀerent values, the waves of Figures 7.9 and 7.10 cannot be expected to be steady unless the phase and group velocity are equal. This is exactly what happens when α = 0.25.

0.1

0

0

20

40

Fig. 7.10. Elevation solitary wave for α = 0.261.

We recall that nonuniformities similar to that of Figure 4.8 occur in water of ﬁnite depth. Here we have restricted our attention to inﬁnite depth in order to keep the presentation simple. However, similar calculations can be performed for water of ﬁnite depth (see [44]). In particular it can be shown that branches of solitary waves bifurcate from the minima of the curves in Figure 2.5 when τ < 1/3. Existence proofs for solitary waves with decaying tails were given by Iooss and Kirchgassner [79] and others (see Dias and Khariﬀ [43] for a review). Furthermore, analytical approximations valid near the bifurcation points were obtained by Dias and Iooss [42], Akylas [3] and Longuet-Higgins [103].

7.3 Gravity–capillary free-surface ﬂows with Wilton ripples When α < 1/4, linear solutions in water of inﬁnite depth for the ﬂow conﬁguration of Figure 4.4 are characterised in the far ﬁeld by trains of linear periodic waves. Here α is deﬁned by (4.69). There are two wave trains. The one of shorter wavelength occurs as x → −∞ and the one of longer wavelength as x → ∞. These linear waves are described by the formulae (4.66) and (4.67) of Chapter 4. They are consistent with the formulae (5.62)

7.3 Gravity–capillary free-surface ﬂows with Wilton ripples

207

and (5.63) derived in Chapter 5. However, we showed in Section 5.1 that the formulae (5.62) and (5.63) are only valid when the condition (5.66) is satisﬁed. Therefore the linear solutions of Section 4.2.2 are not valid when κ=

1 4π 2 T = , 2 ρgλ m

m = 2, 3, . . .

(7.36)

because there is no nonlinear train of periodic waves that can approach a linear wave train at x = ∞ as the size of the disturbance approaches zero. A similar diﬃculty arises in problems in water of ﬁnite depth when the condition (5.55) is not satisﬁed (nonlinear solutions when (5.55) is satisﬁed were described in Section 7.2.1). We describe in this section the appropriate solutions in water of inﬁnite depth when κ = 1/m in the particular case m = 2. Then the linear waves as x → ∞ should be described by (5.107) and (5.106) instead of (5.62) and (5.63). In this section we use the numerical procedure of Section 7.2.2 together with the truncation techniques of Section 7.2.1 to compute solutions when κ = 1/2 (see also Vanden-Broeck [182]). We note that (5.112), (5.113) and (5.67) imply that α=

(1 + κ)2 9 µ2 = = . κ κ 8

(7.37)

The problem is formulated in equations (7.34) and (7.35). To solve them numerically we introduce the mesh points φI = −

(N − 1)E + (I − 1)E, 2

I = 1, . . . , N,

(7.38)

and the unknowns ˜ I ). βI = β(φ

(7.39)

Here E is the interval of discretisation. The integro-diﬀerential equations are discretised and satisﬁed at the intermediate mesh points φI+1/2 =

φI + φI+1 , 2

I = 1, . . . , N − 1.

(7.40)

This leads to N − 1 nonlinear algebraic equations. The Cauchy principal value in (7.35) is approximated by the trapezoidal rule with a summation over the mesh points (7.38) (see Section 3.1.2.2). This yields N βJ 1 ˜ wJ , δ(φI+1/2 ) = 1 − π φJ − φI+1/2 J =1

(7.41)

208

Nonlinear free-surface ﬂows generated by moving disturbances

where w1 = wN = E/2 and wJ = E otherwise. The symmetry of the quadrature has enabled us to evaluate the Cauchy principal value integral in (7.35) as if it were an ordinary integral. The approximation (7.41) replaces the integral from −∞ to ∞ in (7.35) by an integral from −A to A, where A = (N − 1)E/2. To obtain the results to be presented here, we improve this truncation by modifying appropriately the approach of Section 7.2.1. We ﬁrst rewrite (7.35) as ˜ 1 −A β˜ (ϕ) 1 B β˜r (ϕ) 1 A β(ϕ) ˜ dϕ − dϕ − dϕ, (7.42) δ(φ) =1− π −A ϕ − φ π −B ϕ − φ π A ϕ−φ before applying the trapezoidal rule. Here B A. For A suﬃciently large, β˜ and β˜r are the periodic wave trains of Chapter 5. We present results for the value α = 9/8 derived in (7.37). For this value of α, we showed that the linear solution of Chapter 5 fails in the nonlinear regime because there are no nonlinear periodic waves approaching the linear wave on the far right as → 0. The correct nonlinear solutions for α = 9/8 should have a train of Wilton ripples on the far right. As we shall see this is conﬁrmed by our numerical calculations. We will restrict our calculations to 1, so that a formula for β˜r can be derived from (5.106) and (5.107). This yields β˜r = −Ak− sin k− (φ + δ) ∓ Ak− sin 2k− (φ + δ).

(7.43)

Similarly, (5.62) and (5.63) give β˜ (φ1 ) = −A∗ k+ sin k+ (φ + δ ∗ ).

(7.44)

The constants A, A∗ , δ, δ ∗ are found by imposing the continuity conditions β˜r (φN ) = βN ,

β˜r (φN −1 ) = βN −1 , β˜ (φ1 ) = β1 ,

β˜r (φN −2 ) = βN −2 ,

β˜ (φ2 ) = β2 .

(7.45) (7.46)

Relation (7.44) imposes a train of linear sine waves of short wavelength at the far left. Relation (7.43) imposes a train of Wilton ripples at the far right. Relations (7.45) and (7.46) together with the N − 1 equations obtained by discretising the integro-diﬀerential equations deﬁne a system of N + 4 nonlinear algebraic equations for the N + 4 unknowns A, A∗ , δ, δ ∗ and βI , I = 1, . . . , N . This system is solved by Newton’s method. There are two nonlinear solutions, corresponding to the plus and minus signs in (7.43). These are shown in Figures 7.11 and 7.12. As x → ∞, these solutions approach Wilton ripples. A further discussion of this problem can be found in [182].

7.3 Gravity–capillary free-surface ﬂows with Wilton ripples

209

10

5

0

0

5

10

Fig. 7.11. Free-surface proﬁle of a nonlinear solution for α = 9/8 and = 10−5 . The vertical scale is in units of 10−5 . 10

5

0

0

5

10

Fig. 7.12. Free-surface proﬁle of a nonlinear solution for α = 9/8 and = 10−5 . The vertical scale is in units of 10−5 .

8 Free-surface ﬂows with waves and intersections with rigid walls

In Chapter 3 we calculated pure gravity ﬂows (g = 0, T = 0) which approach either an inﬁnitely thin jet in the far ﬁeld (see Figure 3.36) or a uniform stream characterised by a Froude number F greater than unity (see Figure 3.53). In the latter case the ﬂow is waveless in the far ﬁeld, in accordance with the linear theory of Section 2.4. We now consider pure gravity ﬂows for the conﬁguration of Figure 3.37, for which the ﬂow is subcritical in the far ﬁeld (i.e. for which F < 1). The linear theory of Section 2.4 and the nonlinear computations of Chapter 7 suggest that we should expect a wave train in the far ﬁeld. We shall present computations for three particular cases of Figure 3.37. In the ﬁrst two cases (see Figures 8.1 and 8.2) it will be assumed that the ﬂow is of inﬁnite depth (i.e. that the distance between the bottom AB and the streamline CEF in Figure 3.37 is so large that the layer of ﬂuid can be assumed to be of inﬁnite depth). The value of the Froude number F is then 0. The analysis follows Vanden-Broeck [158], Vanden-Broeck et al. [190], Vanden-Broeck and Tuck [192] and Vanden-Broeck [168]. y SF A

B SP SL

x

C

β SR

U SH

Fig. 8.1. The free surface past a ﬂat plate. The contour used in Sections 8.1.2 and 8.4 is also shown. The ﬂow leaves the plate tangentially (i.e. β = 0) when surface tension is neglected.

210

8.1 Free-surface ﬂow past a ﬂat plate B

γ3

211

C

D

A U

Fig. 8.2. The free-surface ﬂow past a surface-piercing obstacle. The free surface separates tangentially from the inclined wall at point B.

The third case is the ﬂow under a sluice gate, shown in Figure 3.34 (see also Vanden-Broeck [181] and Binder and Vanden-Broeck [15], [16]). This is a ﬂow in water of ﬁnite depth which is subcritical as x → −∞ and supercritical as x → ∞. The ﬂow of Figure 8.1 is probably the simplest ﬂow for which a free surface intersects a rigid surface. As we shall see, an attractive feature of this ﬂow is that some exact formulae for the amplitude of the waves in the far ﬁeld can be derived. These formulae provide analytical insights and can be used to check the accuracy of numerical codes. It is tempting to try to compute solutions for the ﬂow conﬁgurations of Figures 8.1, 8.2 and 3.34 by using a series representation similar to (3.245), (3.246). However, it is not easy to remove the singularity of the ﬂow as x → ∞ in Figures 8.1 and 8.2 because the complex velocity w = u − iv oscillates inﬁnitely often as x → ∞. The singularity at t = 1 corresponds to a nonlinear wave train. This is to be contrasted with the corresponding supercritical ﬂows of Section 3.3, where the ﬂow approaches a uniform stream as x → ∞ and can therefore be described by linear theory (which assumes a small perturbation around a uniform stream). In view of these remarks it is preferable to use a boundary integral equation method to compute freesurface ﬂows having a wave train in the far ﬁeld. Solutions for the ﬂow conﬁgurations of Figures 8.1 and 8.2 are presented in Sections 8.1 and 8.2. The ﬂow under a sluice gate is treated in Section 8.3. Finally, an extension of the ﬁndings of Section 8.1 to pure capillary ﬂows is presented in Section 8.4.

8.1 Free-surface ﬂow past a ﬂat plate We start with the ﬂow conﬁguration of Figure 8.1. Numerical results are described in Section 8.1.1. The exact analytical results are derived in Section 8.1.2. The presentation follows [158].

212

Free-surface ﬂows with waves and intersections with rigid walls

8.1.1 Numerical results Let us consider the ﬂow conﬁguration of Figure 8.1. As y → −∞, the ﬂow is uniform and characterised by a constant velocity U . We introduce dimensionless variables by using U 2 /g as the reference length and U as the reference velocity. We deﬁne cartesian coordinates with the y-axis directed vertically upwards. Gravity is acting in the negative y-direction. We introduce a potential function φ and a streamfunction ψ. We choose φ = 0 at the separation point B and ψ = 0 on the streamline ABC. The ﬂow is then mapped into the lower half-plane ψ < 0 of the complex potential plane f = φ + iψ. We introduce the complex velocity u − iv and deﬁne the function τ − iθ by the formula (3.3). Since u − iv → 1 as ψ → −∞, τ − iθ → 0 as ψ → −∞.

(8.1)

The dynamic boundary condition in dimensionless variables yields ˜ e2τ + 2y = B.

(8.2)

˜ = 1. We choose the origin of y such that B The solutions are characterised by the Froude number F =

U , (gH)1/2

(8.3)

where y = −H is the dimensional ordinate of the separation point B in Figure 8.1. Next we apply Cauchy’s integral equation formula (2.41) to τ − iθ in the complex f -plane with a contour consisting of the ψ-axis and a circle CR of radius R in the lower half-plane, ψ < 0. This yields R ˜ τ˜(ϕ) − iθ(ϕ) τ (f ) − iθ(f ) 1 1 ˜ dϕ − df, (8.4) τ˜(φ) − iθ(φ) = − iπ −R ϕ−φ iπ C R f −φ ˜ where the ﬁrst integral is a Cauchy principal value. Here τ˜(φ) and θ(φ) denote the values of τ and θ on the streamline ψ = 0. We now take the limit R → ∞ in (8.4). It can be shown by using (8.1) that the integral over CR in (8.4) tends to 0 as R → ∞. Therefore (8.4) yields ∞ ˜ 1 τ˜(ϕ) − iθ(ϕ) ˜ τ˜(φ) − iθ(φ) = − dϕ. (8.5) iπ −∞ ϕ−φ Taking the real and imaginary parts of (8.5) gives ˜ 1 ∞ θ(ϕ) τ˜(φ) = dϕ π −∞ ϕ − φ

(8.6)

8.1 Free-surface ﬂow past a ﬂat plate

and 1 ˜ θ(φ) =− π

∞

−∞

τ˜(ϕ) dϕ. ϕ−φ

213

(8.7)

Relations (8.6) and (8.7) are similar to the Hilbert transforms (3.73) and (3.74). As mentioned in Section 3.1.2.2, each implies the other. Therefore we are free to choose either (8.6) or (8.7) for our boundary integral equation formulation. However, (8.6) is more convenient than (8.7) because it only involves unknowns on the free surface ψ = 0, φ > 0. This follows from the fact that the kinematic boundary condition on the wall AB can be written as θ=0

on ψ = 0,

φ < 0.

(8.8)

Substituting (8.8) into (8.6) gives 1 τ˜(φ) = π

0

∞

˜ θ(ϕ) dϕ. ϕ−φ

(8.9)

If we assume that φ > 0 then (8.9) involves only values of θ and τ on the free surface. Next we eliminate y from (8.2) by diﬀerentiating the latter with respect to φ. Using the identity xφ + yφ =

1 = e−τ +iθ u − iv

(8.10)

gives e2˜τ

∂ τ˜ + e−˜τ sin θ˜ = 0. ∂φ

(8.11)

Relations (8.9) and (8.11) deﬁne a system of integro-diﬀerential equations ˜ for the values of τ˜(φ) and θ(φ) on the free surface ψ = 0, φ > 0. An equation ˜ involving only θ(φ) can be obtained by substituting (8.9) into (8.11). We solve this equation numerically by ﬁnite diﬀerences. We ﬁrst deﬁne the mesh points φI = (I − 1)E,

I = 1, . . . , N,

(8.12)

and the corresponding unknowns θI = θ(φI ).

(8.13)

As before, E is the interval of discretisation. We shall require (8.11) to be satisﬁed at the midpoints φI + φI+1 , I = 1, . . . , N − 1. (8.14) 2 Following the analysis in Section 3.1.2.2, we ﬁrst evaluate τ˜(φm I ) by applying φm I =

214

Free-surface ﬂows with waves and intersections with rigid walls

the trapezoidal rule to the integral (8.9) with a summation over the points (8.12). Next we evaluate the values of θ˜ and ∂ τ˜/∂φ at the midpoints (8.14) by linear interpolation and centred ﬁnite diﬀerence formulae. There are N unknowns θI , I = 1, . . . , N . One equation forces the free surface to leave the plate tangentially: θ1 = 0.

(8.15)

Another deﬁnes the value of F by the requirement e2˜τ (0) = 1 +

2 , F2

(8.16)

where F is prescribed. Equation (8.16) follows because (8.2) holds at φ = ψ = 0 and the dimensionless ordinate of the separation point B is −1/F 2 . In (8.16) the value of τ˜(0) is approximated by the two-point extrapolation formula τ˜(0) ≈ 3˜ τ (φm τ (φm 1 ) − 2˜ 2 ).

(8.17)

The ﬁnal N − 2 equations are obtained by satisfying (8.11) at φm I ,I = 1, . . . , N − 2. The resulting system of N equations with N unknowns is solved by Newton iteration. Typical free-surface proﬁles for F = 3.34 and F = 6.3 are shown in Figure 8.3. For convenience the origin of cartesian coordinates has been chosen as the separation point B. The ﬁgure shows that there is a wave train on the free surface. As x → ∞, the amplitude of the waves tends to a constant. When F → ∞, the ﬂow reduces to a uniform stream. For F large, the waves are of small amplitude and are close to linear sine waves. As F decreases, the amplitude of the waves increases and the wave proﬁles develop sharp crests and broad troughs in accordance with the nonlinear properties of steep gravity waves (see Chapter 6). There are two sources of error in the numerical scheme. The ﬁrst is that the inﬁnite domain in the integral (8.9) has been truncated at φ = N E. The second is the size of E. Accurate solutions are obtained in the limit E → 0 and N E → ∞. One way to take this double limit is to ﬁx a value of N E and compute solutions for smaller and smaller values of E until they become independent of E (for example, within graphical accuracy) and then to repeat the procedure for larger and larger values of N E until the solutions become independent of N E as well. The above numerical method truncates the integral from 0 to ∞ to an integral from 0 to N E. In other words θ˜ is assumed to be zero from N E to ∞. As was noted in Section 7.1.2, this truncation has a pronounced eﬀect only on the last wavelength of the wave train, and so very accurate solutions

8.1 Free-surface ﬂow past a ﬂat plate

215

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

35

40

Fig. 8.3. Free-surface proﬁles for the ﬂow shown in Figure 8.1. The solid curve corresponds to F = 3.34 and the broken curve to F = 6.3. The plate is along the negative x-axis.

for gravity free-surface ﬂows with waves can be obtained. Furthermore, the distortion in the last wavelength can be removed by using the approach of Section 7.1.2 to approximate θ for N E < φ < ∞. 8.1.2 Analytical results In this section we supplement the numerical computations of Section 8.1.1 with exact analytical results. We ﬁrst note that for two-dimensional steady ﬂows the Euler equations (2.1) and the conservation of mass equation (2.6) imply p V(V · n) + gyn + n dσ = 0. (8.18) ρ S Here S is a closed contour inside the ﬂuid region, V the vector velocity, p the pressure, ρ the density, σ the arc length along S and n the unit normal exterior to the contour S. Relation (8.18) expresses the conservation of momentum; it follows from (2.1) and (2.6) by application of the divergence theorem. We now choose S to consist of the plate Sp , the free surface SF , a vertical line SR at x = ∞, a horizontal line SH at y = −∞ and a vertical line SL at x = −∞ (see Figure 8.1). We take the component of (8.18) along the x-axis. This gives p u(V · n) + gynx + nx dσ = 0. (8.19) ρ S

216

Free-surface ﬂows with waves and intersections with rigid walls

Here u and nx are the components of V and n along the x-axis. It is convenient to replace the line SH by a horizontal line at y = −d, where d is arbitrarily large. Without loss of generality we may assume that SR intersects the free surface at the level y = 0. The integrals over Sp and SH in (8.19) do not contribute since V · n = 0 and nx = 0 along them. The integration over SF , SR and SL in (8.19) gives −H gH 2 p gd2 SW ∗2 dy − u + + gy − + = 0. (8.20) 2 ρ 2 ρ −d x=−∞ Here u∗ is the uniform velocity at x = −∞. The quantity SW is deﬁned by 0 SW = (p + ρu2 )x=∞ dy (8.21) −d

and represents the momentum ﬂux per unit span of the waves far from the plate. Using Bernoulli’s equation we can rewrite the integral in (8.20) as −H 1 ∗2 1 2 p ∗2 u + + gy dy = (8.22) u + u (−H + d). ρ 2 2 −d We note that SW involves only quantities deﬁned as x → ∞ (see (8.21)). Since the ﬂow as x → ∞ is characterised by a train of periodic waves, SW can be calculated as a function of the amplitude of the waves by using the computations of periodic waves described in Chapter 6. Longuet-Higgins [101] showed that SW can be written as 1 SW = −3V + ρdU 2 + ρd2 g, 2 where gρ V = 2λ

s˜+λ

η 2 (x)dx

(8.23)

(8.24)

s˜

is the mean potential energy per unit horizontal area of the waves. Here s˜ is a large number and y = η(x) is the equation of the free surface in the far ﬁeld. We note that V = V¯ , (8.25) where V¯ is given by (2.156). Substituting (8.22) and (8.23) into (8.20) we have 3V 1 1 1 gH 2 = − U 2 H − u∗2 H + (u∗2 d − U 2 d). (8.26) 2 ρ 2 2 2 Longuet-Higgins [101] also showed that the conservation of mass can be

8.1 Free-surface ﬂow past a ﬂat plate

217

written as u∗ (d − H) = U d − where ρ K= 2λ

s˜

s˜+λ

η

−d

∂φ ∂x

2K , Uρ

2 +

∂φ ∂y

(8.27) 2 dydx

(8.28)

is the mean kinetic energy per unit horizontal area of the waves. We note that when the waves are of small amplitude the upper limit of integration η can be replaced by zero, and so ¯ K = K,

(8.29)

¯ is deﬁned by (2.155). Multiplying (8.27) ﬁrst by u∗ and then by U where K and adding the results yields u∗2 d − U 2 d = u∗2 H + u∗ U H −

2K 2K u∗ − . ρ ρ U

(8.30)

Substituting (8.30) into (8.26), we obtain gH 2 3V K K u ∗ u∗ U H U2 = − − + − . 2 ρ ρ ρ U 2 2

(8.31)

As d → −∞ we have u∗ = U . Thus gH 2 3V 2K = − . 2 ρ ρ

(8.32)

Using the deﬁnition (8.3) of the Froude number we can rewrite (8.32) as −1/4 6g 4g V − K . (8.33) F =U ρ ρ As shown in Section 6.5.2, periodic gravity waves can be characterised by their steepness s, deﬁned as the diﬀerence in ordinates between a crest and a trough divided by the wavelength. The quantities U , V and K are functions of the steepness s of the waves. Values of U versus s were calculated in Section 6.5.2 (see Figure 6.21). Similarly values of V and K versus s were calculated by Longuet-Higgins [101], Cokelet [31] and Schwartz and VandenBroeck [135]. Thus (8.33) amounts to a relation between the Froude number F and the steepness s of the waves far from the plate. This relation is shown graphically in Figure 8.4 (see the solid curve). It was obtained by substituting Cokelet’s results into (8.33). The numerical results of Section 8.1.1 show that the steepness of the waves decreases away from the plate and reaches a constant value after a few periods. The crosses in Figure 8.4

218

Free-surface ﬂows with waves and intersections with rigid walls

show these constant values for a few values of the Froude number. These numerical results are in close agreement with those predicted by the exact relation (8.33). s

0.14

0.10

0.05

0

2

4

6

F

Fig. 8.4. Values of the steepness s versus F . The solid curve shows the exact relation (8.33) in which Cokelet’s [31] results are used. The crosses give the numerical results obtained in Section 8.1.1. The broken line corresponds to the numerical results derived in Section 8.2.1.

We showed in Section 6.5.2 that the values of U oscillate inﬁnitely often as the wave of maximum steepness is approached. Longuet-Higgins [101] and Cokelet [31] showed that the values of V and K have a similar property. Therefore we can expect the values of F predicted by (8.33) also to oscillate inﬁnitely often as the waves in the far ﬁeld approach their maximum steepness. This was conﬁrmed by the calculations of Vanden-Broeck [158]. We note that these oscillations are too small to be seen on the scale of Figure 8.4.

8.2 Free-surface ﬂow past a surface-piercing object We now turn our attention to the ﬂow shown in Figure 8.2. For the ﬂow conﬁguration of Figure 8.1, the free surface leaves the plate AB tangentially at the point B. The numerical and analytical calculations of Sections 8.1.1 and 8.1.2 show that there is a one-parameter family of solutions, for the ﬂow of Figure 8.1. By analogy we can expect that there is a two-parameter family of solutions for the ﬂow of Figure 8.2. A possible choice for the ﬁrst

8.2 Free-surface ﬂow past a surface-piercing object

219

parameter is the Froude number F deﬁned by (8.3), where again −H is the dimensional ordinate of corner D in Figure 8.2. The second parameter is then the length lw of the wall DB. As lw → 0, the ﬂow of Figure 8.2 reduces to the ﬂow of Figure 8.1. As lw increases, the velocity at the separation B decreases, and a limiting conﬁguration is reached when a stagnation point (i.e. a point with u = v = 0) occurs at the separation point B. B

C

γ3

A

D U

Fig. 8.5. Free-surface ﬂow past a surface-piercing object with a stagnation point at B. When γ3 > π/3, the free surface is horizontal at the separation point B. When γ3 < π/3, there is an angle of 2π/3 between the wall DB and the free surface.

The local analysis of Section 3.3 can be used to determine the angle between the free surface and the wall DB when B is a stagnation point. Comparing Figures 3.38 and 8.5, we ﬁnd that γ3 = −µ2 . Therefore the discussion following (3.175) and (3.176) implies that an angle of 2π/3 between the free surface and the wall at the stagnation point B will occur if γ3 < π/3. If γ3 > π/3, the free surface has to be horizontal at the stagnation point.

8.2.1 Numerical results We now present numerical results for the limiting conﬁguration of Figure 8.5 with γ3 = π/2. Since π/2 > π/3, as discussed above the free surface is then horizontal at B. We choose φ = 0 at the stagnation point B and denote by −b the value of φ at the corner D. The formulation and the numerical method follow that of Section 8.1.1. Our analysis also follows that in [190], [192] and [168]. For consistency we shall use the same notation as that used in these publications. In particular we introduce dimensionless variables by taking b/U as the unit length and U as the unit velocity. The dynamic boundary condition on the free surface then becomes e2˜τ + y = ,

φ > 0,

(8.34)

220

Free-surface ﬂows with waves and intersections with rigid walls

where =

U3 . 2gb

(8.35)

Diﬀerentiating (8.34) with respect to φ and using the identity (8.10) yields 2e2˜τ

∂ τ˜ + e−˜τ sin θ˜ = 0, ∂φ

φ > 0.

(8.36)

The ﬂow is mapped into the lower half-plane ψ < 0 of the complex potential f -plane. We then introduce the complex velocity u − iv and deﬁne ˜ the function τ − iθ by (3.3). The values of τ˜(φ) and θ(φ) on the streamline ψ = 0 are related by (8.6). The kinematic boundary conditions on the walls AD and DB yield θ˜ = 0, −∞ < φ < −1, (8.37) θ˜ = π/2,

−1 < φ < 0,

(8.38)

since the dimensionless value of φ at corner D is −1. Substituting (8.37) and (8.38) into (8.6) yields ˜ 1 ∞ θ(ϕ) 1 0 1 dϕ + dϕ. (8.39) τ˜(φ) = 2 −1 ϕ − φ π 0 ϕ−φ Evaluating the ﬁrst integral on the right-hand side of (8.39) gives ˜ 1 φ 1 ∞ θ(ϕ) dϕ. τ˜(φ) = ln + 2 φ+1 π 0 ϕ−φ

(8.40)

˜ Relations (8.36) and (8.40) deﬁne an integro-diﬀerential equation for θ(φ). We discretise the problem by following the procedure used in Section 8.1.1 for the ﬂow past a ﬂat plate. We introduce the mesh points (8.12), the unknowns (8.13) and the midpoints (8.14) and evaluate τ˜(φm I ) by applying the trapezoidal rule to the integral on the right-hand side of (8.40) with a ˜ m ) by linear summation over the mesh points (8.12). We then evaluate θ(φ i interpolation and ∂τ /∂φ at the midpoints (8.14) by centred diﬀerences. We then satisfy (8.36) at the midpoints (8.14). This leads to N − 1 equations. The ﬁnal equation is obtained by expressing θ˜3 in terms of θ˜2 : 1/2 φ3 ˜ ˜ . (8.41) θ 3 = θ2 φ2 Equation (8.41) becomes exact in the limit E → 0, since θ˜ ≈ φ1/2

as

φ → 0.

8.2 Free-surface ﬂow past a surface-piercing object

221

The motivation for the choice (8.41) is given below in Section 8.2.2 (see also [190], [192] and [168]). A typical free-surface proﬁle is shown in Figure 8.6. Further free surface proﬁles can be found in [190], [192] and [168]. 0

0

5

10

15

20

25

30

Fig. 8.6. Values of y − versus x for = 0.45.

There is again a wave train in the far ﬁeld. As → 0 (i.e. as F → 0), the amplitude of the waves approaches zero and the free surface approaches a horizontal straight line. This solution is often referred to as the rigid-lid solution. An exact solution is obtained on setting θ˜ = 0 in (8.40). This gives 1 φ (8.42) , θ˜0 = 0 for φ > 0. τ˜0 (φ) = ln 2 φ + 1 Here the subscript 0 signiﬁes that = 0. Relation (8.42) and the identity (8.10) imply φ + 1 1/2 dz0 = for φ > 0. df φ

(8.43)

8.2.2 Analytical results It is tempting to try to construct analytically a solution for F small (or equivalently for small) by perturbing (8.42) and then seeking dz/df as an expansion in powers of . Thus we write ∞

dz n zn (f ), = e−τ +iθ = df n=0

where z0 is given on the free surface by the right-hand side of (8.43).

(8.44)

222

Free-surface ﬂows with waves and intersections with rigid walls

Substituting (8.44) into the system (8.36), (8.40) and equating coeﬃcients of n leads to a recurrence relation for the coeﬃcients zn (f ), and an arbitrary large number of coeﬃcients can in principle be calculated. However, the series is divergent and furthermore the partial sums predict a ﬂat free surface in the far ﬁeld, without wave trains. This discrepancy between the expansion (8.44) and the numerical results is due to the fact that the amplitude of the waves is an exponentially small function of . This means that as → 0 the amplitude aw of the waves tends to zero faster than any powers of , i.e. aw = o(m )

(8.45)

for any positive integer m, where the o symbol was deﬁned in (3.24) and (3.25). The work of [190], [192] and [168] showed that the expansion (8.44) diverges as n!. More precisely it was found that zn ≈ C(φ)[ζ1 (φ)]n (n + a)! for n large.

(8.46)

Here a ≈ −0.6. The functions ζ1 (φ) and C(φ) are complex valued and were calculated in [190], [192] and [168]. Combining (8.46) and (8.44) we obtain N ∞ dz n zn + C(φ) n [ζ1 (φ)]n (n + a)!. ≈ df n=0

(8.47)

n=N +1

Using the deﬁnition of the gamma function, ∞ (n + a)! = e−t tn+a dt, 0

we can rewrite the inﬁnite sum in (8.47): ∞ ∞ ∞ n n −t a N +1 ζ1 (n + a)! = C e t (ζ1 t) (ζ1 t)m dt. C n=N +1

0

(8.48)

m=0

The inﬁnite series ∞

(ζ1 t)m

(8.49)

m=0

appearing in the integrand of (8.48) is immediately recognizable as the Taylor expansion of the function 1 . 1 − ζ1 t

(8.50)

8.2 Free-surface ﬂow past a surface-piercing object

Therefore we have ∞ n C (ζ1 ) (n + a)! ≈ C 0

n=N +1

∞

e−t ta (ζ1 t)N +1 dt. 1 − ζ1 t

223

(8.51)

We note that the Taylor expansion (8.49) converges only if |t| < (|ζ1 |)−1 . The above procedure, known as Borel summation, is based on the assumption that the series (8.49) can be identiﬁed with the function (8.50) for all t. Using the identity (ζ1 t)N +1 (ζ1 t)N = − (ζ1 t)N , 1 − ζ1 t 1 − ζ1 t we rewrite the right-hand side of (8.51) in the form ∞ −t a e t (ζ1 t)N1 C dt − C(ζ1 )N (N + a)! − · · · − C(ζ1 )N 1 (N1 + a)!, 1 − ζ1 t 0

(8.52)

where N1 is the smallest integer such that N1 + a > −1. Thus, substituting (8.52) into (8.51) we obtain ∞ −t a N e t (ζ1 t)N1 z (φ, ) ≈ C [n zn −C(ζ1 )n (n+a)!]+FN1 , (8.53) dt+ 1 − ζ1 t 0 n=N 2

where N2 = N1 if N1 > 0 and N2 = 0 otherwise, and FN1 = −C

−1

(ζ1 )n (n + a)!

if N1 < 0,

n=N 1

FN1 = 0 FN 1 = C

N 1 −1

if N1 = 0, n zn

if N1 > 0.

n=0

Using terminology introduced by Dingle [51] we describe the function ∞ −t a e t (ζ1 t)N1 1 =C dt (8.54) TN 1 − ζ1 1 − ζ1 t 0 as the ‘terminant’ of the divergent series expansion. The main diﬃculty about the terminant (8.54) is its lack of uniqueness. We can describe this nonuniqueness in terms of a branch-cut location. For this purpose we rewrite (8.54) in the form 1 1 N 1 −1 −1/ζ1 e WN 1 − = −C(ζ1 ) , (8.55) TN1 − ζ1 ζ1

224

Free-surface ﬂows with waves and intersections with rigid walls

where the function WN1 (t) can be represented by the following convergent expansions (see [51]): t t 1 −t 1− WN1 (t) = (N1 + a)!e t N1 + a N1 + a − 1

t2 ··· + (N1 + a − 1)(N1 + a − 2) N +a πt 1 − (8.56) sin π(N1 + a) when a is not an integer and WN1 (t) = e−t

∞ i t [ψ(i) − ln t] i=0

i!

,

(8.57)

when a is an integer. Here ψ(i) =

d (ln i!). di

The branch cut of WN1 (t) is deﬁned as the branch cut of ln t appearing explicitly in the expansion for integer a and implicitly in the expansion for noninteger a, since tN 1 +a = e(N1 +a) ln t . We note that when a is an integer the function WN1 (t) reduces to the exponential integral function E1 (t). Diﬀerent functions WN1 (t) can be obtained by specifying diﬀerent cuts in the complex t-plane. In the present problem, we are interested in the following three choices for WN1 (t), with t < 0: ∞ (µ − t)N 1 +a e−µ 0 dµ; (8.58) WN 1 (t) = WN1 (t) = µ t WN1 (t) = WN+1 (t) = WN0 1 (t), WN1 (t) = WN1 (t) = WN1 (t) =

WN+1 (t) WN−1 (t) WN−1 (t)

= = =

WN0 1 (t) − WN0 1 (t) + WN0 1 (t),

t > 0, 2iπ(−t)N1 +a ,

t < 0,

(8.59)

2iπ(−t)N1 +a ,

t > 0,

(8.60)

t < 0.

Here (−t)N1 +a is deﬁned as the principal value of the power. The function WN 1 (t) is deﬁned in the complex t-plane cut along the negative real t-axis, whereas the functions WN+1 (t) and WN−1 (t) are deﬁned in the complex t-plane cut along the positive real t-axis.

8.2 Free-surface ﬂow past a surface-piercing object

225

The usefulness of the original asymptotic expansion can now be increased by substituting (8.54) and (8.55) into (8.53). This yields z (φ, ) ≈ FN1 +

N

[n zn − C(ζ1 )n (n + a)!]

n=N 2

− C(ζ1 )N 1 −1 e−1/ζ1 WN 1 (−1/ζ1 ) .

(8.61)

We note that the three solutions (8.61) corresponding to the three choices (8.58)–(8.61) diﬀer by a term that is exponentially small in the limit → 0. Let us ﬁrst consider the choice (8.58). Since WN0 1 (t) is deﬁned in the complex t-plane cut along the negative real t-axis, the corresponding function in (8.61) is deﬁned in the complex ζ1 -plane cut along the positive real ζ1 -axis. The numerical results show that this branch cut is crossed at some point on the free surface. Hence the corresponding free surface (8.61) has a discontinuity at that point and is therefore not physical. Let us now consider the choices (8.59) and (8.61). Since the functions WN+1 and WN−1 are deﬁned in a t-plane cut along the positive real t-axis, the corresponding function in (8.61) is deﬁned in the complex ζ1 -plane cut along the negative real ζ1 -axis. This branch cut is not crossed at any point along the free surface. Hence the corresponding free surfaces are continuous and can be expected to have a physical meaning. Vanden-Broeck [168] showed that the solution (8.61) corresponding to WN+1 has a wave train in the far ﬁeld and agrees with the numerical solutions of Section 8.2.1 when (or equivalently F ) is small. By using (8.59) and (8.61), Vanden-Broeck [168] estimated the steepness of the waves in the far ﬁeld as (0.5/0.4 )e−1/D ,

(8.62)

where D ≈ 0.4. The quantity (8.62) is exponentially small in the limit → 0, in accordance with the discussion at the beginning of this section. An improved estimate (which also shows that the steepness of the waves is exponentially small) has recently been derived by Trinh, Chapman and Vanden-Broeck. Before discussing the solution corresponding to the choice WN−1 , let us mention that the ﬂows of Figures 8.1 and 8.2 can be interpreted as freesurface ﬂows caused by the plate AB or the object ADB moving at a constant velocity U to the left. This models the ﬂow near the stern of a ship. For these reasons the ﬂows of Figures 8.1 and 8.2 are referred to as stern ﬂows. The presence of waves on the free surface is consistent with the radiation condition (see Chapter 4), which requires gravity waves (if present) to

226

Free-surface ﬂows with waves and intersections with rigid walls

be at the back of a moving disturbance. Since potential ﬂows are reversible, we can reverse the direction of U in Figures 8.1 and 8.2. Therefore the ﬂows of Figures 8.1 and 8.2 can also be interpreted as free-surface ﬂows caused by the plate AB or the object ADB moving at a constant velocity U to the right; the ﬂows are then referred to as bow ﬂows. However, the presence of waves on the free surface is now incompatible with the radiation condition: appropriate bow ﬂows should be characterised by a uniform stream as x → ∞. In other words, none of the free-surface ﬂows computed so far in this chapter can model bow ﬂows. The solution (8.61) corresponding to WN−1 is in fact characterised by a uniform stream as x → ∞. However, the summation procedure based on Borel summation fails to yield reliable solutions near the stagnation point. Vanden-Broeck and Tuck [192] conjectured that bow ﬂows are characterised by a jet or spray near the rigid wall DB. This conjecture was conﬁrmed by Dias and Vanden-Broeck [47], who computed such solutions by series truncation methods similar to those introduced in Chapter 3. A typical free-surface proﬁle is shown in Figure 8.7. 8

4

y

0

0

x

4

8

12

Fig. 8.7. Computed bow ﬂow with a spray.

We conclude this section by mentioning that bow ﬂows without spray exist in water of ﬁnite depth (see Vanden-Broeck [175]). 8.3 Flow under a sluice gate The free-surface ﬂow under a sluice gate is a classical problem of ﬂuid mechanics. The ﬂow conﬁguration is shown in Figure 3.34. We shall ﬁrst discuss results for γ2 = π/2 (see Figure 8.8). Extensions for γ2 = π/2 will be considered at the end of this section. The ﬂow of Figure 8.8 is bounded

8.3 Flow under a sluice gate A

227

B

C D

y

H

U

x

Fig. 8.8. The free-surface ﬂow past a vertical sluice gate.

below by a horizontal bottom and above by two free surfaces AB and CD and a vertical wall BC. Far downstream there is uniform ﬂow with a constant velocity U and a constant depth H. As we shall see, the problem can be characterized by the downstream Froude number F =

U , (gH)1/2

(8.63)

where g is the acceleration of gravity. Over the years, many analytical and numerical approximations have been obtained (see [11], [54], [93], [30], [181], [15] and [16]). It is usually assumed that B is a stagnation point and that there is a uniform stream far upstream. This last assumption is the radiation condition, which requires that there are no waves far upstream. It is then possible to deﬁne an upstream Froude number, V (8.64) FU = (gD)1/2 where V and D are the constant velocity and depth far upstream. The Froude numbers F and FU are related by the identity 3 1/2 8 F2 2 +1 −1 (8.65) FU = 8 F2 (see [17] for a derivation). Many previous results suggest that there is a one-parameter family of solutions and that the ﬂow is subcritical upstream (i.e. FU < 1) and supercritical downstream (i.e. F > 1). In this section we will compute accurate numerical solutions for the fully nonlinear problem. The analysis follows Vanden-Broeck [181] closely. The problem is ﬁrst formulated as an integral equation for the unknown shapes of the free surfaces. This equation is then discretised and the resulting

228

Free-surface ﬂows with waves and intersections with rigid walls

algebraic equations are solved by Newton’s method, following the approaches of Sections 3.1.2.2, 6.3 and Chapter 7. We again ﬁnd a one-parameter family of solutions. However, our results show that the solutions do not satisfy the radiation condition: there is a wave train on the upstream free surface. The amplitude of the waves comes as part of the solution. Our numerical ﬁndings indicate the nonexistence of solutions that satisfy the radiation condition. For F > 2.4, the waves are of very small amplitude and the upstream free surface is essentially ﬂat. However, for F < 2.4 the waves are quite noticeable and ultimately become large nonlinear waves as F is decreased. As mentioned in Sections 3.1.1 and 3.3.3, an interesting quantity is the contraction ratio H . (8.66) Cc = yC Here yC is the distance from the separation point C to the bottom. We will show that our computed values of Cc are in good agreement with those of Fangmeier and Strelkoﬀ [54] for F large. This is consistent since the scheme of Fangmeier and Strelkoﬀ neglects the waves upstream and our calculations show that the waves are indeed very small for F large. The problem is formulated in Section 8.3.1. The numerical procedure is described in Section 8.3.2 and the results are discussed in Section 8.3.3.

8.3.1 Formulation The ﬂuid is assumed to be inviscid and incompressible and the ﬂow to be irrotational. The ﬂow domain is bounded below by a horizontal bottom and above by free surfaces AB and CD and a vertical wall BC (see Figure 8.8). We introduce cartesian coordinates with the origin on the bottom and the y-axis along the vertical wall. Gravity g is acting in the negative y-direction. Far downstream the ﬂow approaches a uniform stream with constant velocity U and constant depth H. We deﬁne dimensionless variables by choosing H as the unit length and U as the unit velocity. We introduce the complex potential function f = φ + iψ

(8.67)

w = u − iv.

(8.68)

and the complex velocity

Here as before u and v are the horizontal and vertical components of the velocity. Following Benjamin [11] and others, we assume that B is a stagnation point (i.e. u = v = 0 at B). Without loss of generality, we choose

8.3 Flow under a sluice gate

229

φ = 0 at B and ψ = 0 on the streamline ABCD. It follows from our choice of dimensionless variables that ψ = −1 on the bottom. We denote by φC the value of φ at the separation point C. The ﬂow conﬁguration in the f -plane is shown in Figure 8.9.

A

B

C

A

D φ

D

Fig. 8.9. The ﬂow conﬁguration of Figure 8.8 in the complex potential plane.

In terms of the dimensionless variables, the dynamic boundary condition on the free surfaces AB and CD can be written as 2 2 u2 + v 2 + 2 y = 1 + 2 . (8.69) F F Here F is the Froude number deﬁned by (8.63). The kinematic conditions on the bottom and on the gate BC imply that v=0

on ψ = −1

(8.70)

and u=0

on ψ = 0,

0 < φ < φC .

(8.71)

This concludes the formulation of the problem. We seek w as an analytic function of f in the strip −1 < ψ < 0. This function must approach 1 as φ → ∞ and must satisfy (8.69)–(8.71). As we shall see, there is a oneparameter family of solutions. It is convenient to choose this parameter as φC . We now reformulate the problem as an integral equation. First we deﬁne the function τ − iθ by w = eτ −iθ

(8.72)

and we map the ﬂow domain onto the lower half of the ζ-plane by the transformation ζ = α + iβ = eπf . The ﬂow in the ζ-plane is shown in Figure 8.10.

(8.73)

230

Free-surface ﬂows with waves and intersections with rigid walls

D

A

B

C

D

Fig. 8.10. The ﬂow conﬁguration of Figure 8.8 in the ζ-plane.

Next we apply the Cauchy integral formula (2.41) to the function τ − iθ in the complex ζ-plane. We choose a contour consisting of the real axis and a semicircle of arbitrary large radius in the lower half-plane. After taking the real part we obtain 1 ∞ θ(α ) (8.74) dα . τ (α) = π −∞ α − α Here τ (α) and θ(α) denote the values of τ and θ on the axis β = 0. The integral in (8.74) is a Cauchy principal value. The kinematic conditions (8.70) and (8.71) imply θ(α) = 0 for α < 0

(8.75)

and θ(α) = −π/2

for

1 < α < αC ,

where αC = eπφC . Substituting (8.75) and (8.76) into (8.74), we obtain |αC − α| 1 1 θ(α ) 1 ∞ θ(α ) 1 + dα + dα . τ (α) = − ln 2 |1 − α| π 0 α − α π α C α − α

(8.76)

(8.77)

Equation (8.77) provides a relation between τ and θ on the free surfaces. We can obtain another relation between τ and θ on the free surfaces in the following way. First we substitute (8.72) into (8.69). This yields 2 2 y = 1+ 2. (8.78) 2 F F Next we evaluate the values of y on the free surfaces by using (8.73) and integrating the identity d(x + iy) = w−1 . (8.79) df e2τ +

8.3 Flow under a sluice gate

231

This gives y(α) = 1 +

1 F2 + 2 π

and 1 y(α) = 1 + π

α

1

α

∞

e−τ (α 0 ) sin θ(α0 ) dα0 α0

e−τ (α 0 ) sin θ(α0 ) dα0 α0

for

0<α<1

for α > αC .

(8.80)

(8.81)

Equations (8.77), (8.78), (8.80) and (8.81) deﬁne a nonlinear integral equation for the unknown function θ(α) on the free surfaces 0 < α < 1 and α > αC . 8.3.2 Numerical procedure We will solve the integral equation deﬁned by (8.77), (8.78), (8.80) and (8.81) numerically, using equally spaced points in the potential function φ. Thus we introduce the change of variables α = eπφ

(8.82)

and rewrite (8.77) as |αC − eπφ | 1 + τ (φ) = − ln 2 |1 − eπφ |

0

−∞

θ (φ0 )eπφ0 dφ0 + eπφ0 − eπφ

∞

φC

θ (φ0 )eπφ0 dφ0 . eπφ0 − eπφ (8.83)

Similarly we rewrite (8.80) and (8.81) as 1 φ −τ (φ0 ) F2 y (φ) = 1 + + e sin θ (φ0 ) dφ0 , 2 π 0

φ

y (φ) = 1 +

e−τ

(φ

0)

∞

sin θ (φ0 ) dφ0 .

(8.84)

(8.85)

Here τ (φ) = τ (eπφ ), θ (φ) = θ(eπφ ) etc. Next we introduce the equally spaced mesh points φU I = −(I − 1)∆1 ,

I = 1, . . . , N1 ,

(8.86)

and φD I = φC + (I − 1)∆2 ,

I = 1, . . . , N2

(8.87)

on the upstream and downstream free surfaces. Here ∆1 > 0 and ∆2 > 0 are the mesh sizes. The corresponding unknowns are θIU = θ (φU I ),

I = 1, . . . , N1 ,

(8.88)

232

Free-surface ﬂows with waves and intersections with rigid walls

and θID = θ (φD I ),

I = 1, . . . , N2 .

(8.89)

Since θ1U = 0 and θ1D = −π/2, there are only N1 + N2 − 2 unknowns θIU and θID . U D We evaluate the values τI+1/2 and τI+1/2 of τ (φ) at the midpoints; thus U φU I + φI+1 , 2

I = 1, . . . , N1 − 1,

(8.90)

D φD I + φI+1 , = 2

I = 1, . . . , N2 − 1.

(8.91)

φU I+1/2 = and φD I+1/2

Now the trapezoidal rule is applied to the integrals in (8.83) with summaD tions over the points φU I and φI . The symmetry of the quadrature and of the distribution of mesh points enables us to evaluate the Cauchy principal values as if they were ordinary integrals (see Section 3.1.2.2). In the calculations presented here, we follow Hocking and Vanden-Broeck [72] and ﬁrst rewrite the last integral in (8.83) as φD πφ D πφ 0 N 2 [θ (φ0 ) − θI+1/2 ]e θI+1/2 |e N 2 − eπφ | dφ0 + ln (8.92) eπφ0 − eπφ π |αC − eπφ | φC before applying the trapezoidal rule. The values θI+1/2 of θ at the mesh points (8.90), (8.91) are evaluated in terms of the unknowns (8.88), (8.89) by four-point interpolation formulae. D D Next we evaluate yIU = y (φU I ) and yI = y (φI ) by applying the trapezoidal rule to (8.84) and (8.85). This yields y1U = 1 + −τ IU+ 1 / 2

U −e yIU = yI−1

F2 , 2

U (sin θI+1/2 )∆1 ,

I = 2, 3, . . . , N1 − 1,

D yN = 1, 2 −τ ID+ 1 / 2

D −e yID = yI+1

D (sin θI+1/2 )∆2 ,

I = N2 − 1, N2 − 2, . . . , 1.

(8.93) (8.94) (8.95) (8.96)

We use these values to evaluate y (φ) at the midpoints (8.90) and (8.91) by interpolation formulae. We now satisfy (8.78) at the midpoints (8.90) and (8.91). This yields N1 + N2 − 2 nonlinear algebraic equations for the N1 + N2 − 1 unknowns F , θIU , I = 2, . . . , N1 , and θID , I = 2, . . . , N2 .

8.3 Flow under a sluice gate

233

The ﬁnal equation is obtained by ﬁxing the length of the plate BC. Using (8.72), (8.76) and (8.79), we obtain ∂y = −e−τ ∂φ

on

0 < φ < φC .

(8.97)

We use (8.83) to evaluate τ (φ) for 0 < φ < φC and integrate (8.97) numerically. This yields the length L of the plate BC in terms of the unknowns. The last equation is then y1U − y1D − L = 0.

(8.98)

For a given value of φC this system of N1 + N2 − 1 equations with N1 + N2 − 1 unknowns is solved by Newton’s method. 8.3.3 Discussion of the results We used the numerical scheme described in Section 8.3.2 to compute solutions for various values of φC . Most calculations were performed with N1 = 320, N2 = 360, ∆1 = 0.02 and ∆2 = 0.01. We also calculated solutions with smaller values of ∆1 and ∆2 and larger values of N1 and N2 and checked that the results to be presented here were independent of these parameters within graphical accuracy. An example of such a check is presented at the end of this section. Typical free-surface proﬁles are shown in Figure 8.11. There is a wave train on the upstream free surface. For large values of F (let us say F > 2.4), the waves are so small that they cannot be seen on the ﬁgures and the proﬁles are essentially ﬂat far upstream (see Figures 8.11(a), (b)). However, for F < 2.4, the waves are clearly noticeable on the proﬁles (see Figures 8.11(c), (e)). As F decreases, the waves become large-amplitude nonlinear waves with broad troughs and sharp crests (this tendency is beginning to shown in Figure 8.11(e)). We expect that they will ultimately reach the limiting conﬁguration with a 120◦ angle at their crests (see Figure 6.20) as F is further decreased. The proﬁles of Figures 8.11(a)–(e) show that the mean elevation of the upstream free surface increases as F increases. Since the ﬂux U H is normalized to 1, it follows that the mean velocity far upstream decreases as F increases (this is also consistent with (8.65), which is a valid approximation when the waves are of small amplitude). Therefore the phase velocity of the waves decreases as F increases. This explains why the wavelength of the waves increases as we move from Figure 8.11(a) to Figure 8.11(e).

234

Free-surface ﬂows with waves and intersections with rigid walls 4

7 6 5

3

4 3 2

2

(a)

(b)

1

0

5

1

0

5

2.6 3 2.2 1.8

2

1.4

(c)

(d)

1

0

5

0

2.2 2.0 1.8 1.6 1.4 1.2

(e) 1.0

0

5

Fig. 8.11. Computed proﬁles of the free surfaces in the vicinity of the gate. The arrows indicate the position of the point at which the downstream free surfaces separate from the gate. (a) φC = 0.71 and F = 3.25; (b) φC = 0.41 and F = 2.41; (c) φC = 0.26 and F = 2.03; (d) φC = 0.19 and F = 1.83; (e) φC = 0.075 and F = 1.51.

In Figure 8.12, we show values of the contraction ratio Cc (deﬁned in (8.66) versus yC /yB . Here yC and yB are the ordinates of the points C and B (see Figure 8.8). As yC /yB approaches zero, yB → ∞ and F → ∞. The problem reduces then to a classical free streamline ﬂow (3.39) and π Cc = π+2

8.3 Flow under a sluice gate

235

0.615

0.610

0.605

Cc 0.600

0.595

0.590

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yC /yB

Fig. 8.12. Values of the contraction ratio Cc versus yC /yB . The symbols correspond to the calculations of Fangmeier and Strelkoﬀ [54].

(see (3.44)). The dots in Figure 8.12 are numerical values taken from Figure 13 in Fangmeier and Strelkoﬀ [54]. These numerical values are in good agreement with ours for small values of yC /yB (i.e. for large values of F ). This is consistent, since Fangmeier and Strelkoﬀ assumed that there are no waves upstream and our results show that the waves are of very small amplitude for F large. In Figure 8.13, we present an example of the checks that we used to test the accuracy of the numerical results. The two curves, which lie close together, are computed upstream free surfaces for φC = 0.19. Figure 8.13 shows that the results are independent of φU N1 . The results presented in this section show that there is a wave train on the upstream free surface. The amplitude of these waves is diﬀerent from zero, except in the limit F → ∞. This indicates the nonexistence of solutions satisfying the radiation condition, which requires there to be no waves far upstream. In particular the numerical procedure of Section 8.3.2 diverges if we try to impose the radiation condition by, for example, forcing the free surface to be ﬂat far upstream. In the calculations presented here we have assumed that B is a stagnation point. We could have assumed instead that the upstream free surface is tangential to the gate at the separation point (see Figure 8.2). Such solutions were calculated by Binder and Vanden-Broeck [15]. Their results showed

236

Free-surface ﬂows with waves and intersections with rigid walls 2.7

2.6

2.5

2.4

2.3

0

Fig. 8.13. Computed proﬁles of the upstream free surface with N1 = 320, N2 = 360, ∆1 = 0.04 and ∆2 = 0.01 (solid curve) and N1 = 320, N2 = 360, ∆1 = 0.02 and ∆2 = 0.01 (broken curve).

that there is always a train of waves upstream. Recently Binder and VandenBroeck [16] showed analytically and numerically that these upstream waves can be eliminated if a disturbance is introduced in front of the gate.

8.4 Pure capillary free-surface ﬂows In this section we supplement the ﬁndings of Section 8.1 by considering the ﬂow of Figure 8.1 with gravity neglected but surface tension included in the dynamic boundary condition (i.e. g = 0 and T = 0). As we shall see, there is then a train of pure capillary waves as x → ∞. Pure capillary free-surface ﬂows were considered in Chapter 3 (e.g. the ﬂow emerging from the container of Figure 3.1 and the cavitating ﬂow of Figure 3.16). The absence of waves in these ﬂows is justiﬁed because a wave train as x → ∞ would violate the radiation condition. However, waves are compatible with the radiation condition if the direction of the ﬂow (i.e. the direction of U ) is reversed in Figure 8.1. Following the discussion at the end of Section 8.2.2, we refer to such ﬂow as a bow ﬂow. It models the ﬂow in front of the semi-inﬁnite plate AB when it moves to the right at a constant velocity U , as viewed in a frame of reference moving with the obstacle.

8.4.1 Numerical results We compute solutions for the ﬂow of Figure 8.1 by a boundary integral equation method similar to that used in Section 8.1.1. The numerical procedure

8.4 Pure capillary free-surface ﬂows

237

follows that described in Anderson and Vanden-Broeck [6]. As y → −∞, the ﬂow approaches a uniform stream with constant velocity U . We introduce dimensionless variables by taking T /(ρU 2 ) as the reference length and the velocity U as reference velocity. We introduce the complex velocity u − iv and deﬁne τ and θ by (3.3). The dynamic boundary condition gives in dimensional variables 1 2 (8.99) (u + v 2 ) + K = B, 2 where K is the curvature of the free surface and B is the Bernoulli constant. Using (3.3) and (3.6) we rewrite (8.99) in terms of τ and θ as ∂ θ˜ 1 2˜τ e − eτ˜ = B, 2 ∂φ

(8.100)

˜ where τ˜(φ) and θ(φ) are the values of τ and θ on the free surface ψ = 0. For the present problem B = 1/2 since, for periodic waves of wavelength λ in inﬁnite depth, λ λ 2 2 (u + v )dx = λ and Kdx = 0. (8.101) 0

0

The latter is a general property of periodic waves; the former is only valid for inﬁnite depth and may be found in Wehausen and Laitone [196]. Equation (8.100) provides a relation between τ and θ on the free surface ψ = 0, φ > 0. A second relation is provided by (8.9). Our experience with capillary ﬂows in Section 3.2 suggests that the free surface does not leave the plate tangentially: there is an angle β = 0 between the tangent to the free surface and the plate at the separation point B (see Figure 8.1). This expection is conﬁrmed by the analytical results presented below in Section 8.4.2. There we show analytically that there are no solutions with β = 0 except for the trivial solution corresponding to a uniform stream with constant velocity 1 and a ﬂat free surface. However, we will not use this analytical insight when solving the problem numerically; rather, we will assume that there is an angle β (measured counterclockwise from the x-axis) at the separation point and ﬁnd its value numerically. This allows solutions that leave the plate tangentially (β = 0) and also solutions with a discontinuity in slope (β = 0). The computed values of β are then compared with the exact analytical values to check the accuracy of the numerical procedure. We introduce the mesh points φI = (I − 1)E,

I = 1, . . . , N + 1,

(8.102)

238

Free-surface ﬂows with waves and intersections with rigid walls

the midpoints 1 φM I = 2 (φI + φI+1 ),

I = 1, . . . , N,

(8.103)

and the unknowns θI = θ(φI ),

I = 1, . . . , N + 1.

(8.104)

We deﬁne the angle β by imposing θ1 = β.

(8.105)

Now we evaluate τ (φM I ) by applying the trapezoidal rule to (8.9) with summation over the points (8.102) (see Section 3.1.2 for a justiﬁcation of the discretisation); a truncation procedure similar to that presented in Section 8.1.1 is used. We then require (8.100) to hold at the midpoints (8.103). This leads to N equations for the N + 1 unknowns θI , I = 1, . . . , N + 1. The last equation ﬁxes an extra parameter (for example the value of θ at some point on the free surface). Once again the system is solved by Newton’s method. Typical free-surface proﬁles are shown in Figures 8.14 and 8.15. 0.15

(a)

0.10 0.05

y 0

0 0.10

5

10

5

10

x

15

20

25

15

20

25

(b)

0.05 0

y

0

x

Fig. 8.14. Computed free-surface proﬁles for (a) δ = π/48 and (b) δ = −π/48. Taken from Proc. Roy. Soc. A 452, 1985–1997 (1996).

8.4 Pure capillary free-surface ﬂows 2

239

(a)

1

y 0

0 2

5

10

5

10

x

15

20

25

15

20

25

(b)

1

y 0

0

x

Fig. 8.15. Computed free-surface proﬁles for (a) δ = π/4 and (b) δ = −π/4. Taken from Proc. Roy. Soc. A 452, 1985–1997 (1996).

For |β| small, the waves are close to linear sine waves (see Figure 8.14). As |β| increases, the waves become nonlinear with broad crests and narrow troughs (see Figure 8.15), in accordance with Crapper’s exact solution (see Section 6.5.1).

8.4.2 Analytical results We use the conservation of momentum equation (8.19) with g = 0, i.e. p u(V · n) + nx dσ = 0, (8.106) ρ S to derive an exact relationship between the angle β, the steepness s and the wavelength λ of the waves in the far ﬁeld. The derivation below is similar to that used in Section 8.1.2 (see also [6]). However, it is entirely analytical because the train of waves in the far ﬁeld is described by Crapper’s exact solution (see Section 6.5.1). We will derive the exact relation between s, λ

240

Free-surface ﬂows with waves and intersections with rigid walls

and β for inﬁnite depth by ﬁrst considering the ﬂow past a plate in ﬁnite depth d and then taking the limit as d → ∞. We let y = 0 represent the mean free-surface level of the waves in the far ﬁeld and y = −H the level of the plate. We deﬁne a reference velocity U such that u = U + u ˜, where x0 +λ u ˜dx = 0 (8.107) x0

as x0 → ∞, the integral being taken at an arbitrary level y inside the ﬂow region. Bernoulli’s equation gives in the ﬂow domain p 1 2 1 (u + v 2 ) + = U ∗∗ 2 , 2 ρ 2

(8.108)

where the right-hand side is Bernoulli’s constant. As d → ∞, U ∗∗ → U . As in Section 8.1.2, we choose S in (8.106) to consist of the plate Sp , a vertical line SR at x = M , a horizontal line SH at y = −d and a vertical line SL at x = −M (see Figure 8.1). Here M is a large positive real number. There is no contribution to (8.106) from Sp and SH since on these curves nx = 0 and u · n = 0. On SL , nx = −1 and the contribution to (8.106) is −H p ∗2 dy, (8.109) − U + ρ −d where U ∗ is the constant velocity of the ﬂuid as x → −∞. On SR , nx = 1 and we have likewise the contribution η p 2 u + dy. ρ −d

(8.110)

Using the fact that p = T K along the free surface, where K denotes its curvature, we ﬁnd that pnx dσ = T (cos θ − cos β). (8.111) − SF

Here θ denotes the angle between the free surface and the x-axis at the point C where SR intersects the free surface SF (see Figure 8.1). Since u · n = 0 on SF , we have p T u(u · n) + nx dσ = − (cos θ − cos β). (8.112) ρ ρ SF Altogether, then, from (8.106) we obtain p u(u · n) + nx dσ 0= ρ S L +S R +S F

=−

8.4 Pure capillary free-surface ﬂows

−H

U ∗2 +

−d

p ρ

η

dy +

u2 +

−d

p ρ

dy −

241

T (cos θ − cos β) . (8.113) ρ

Using (8.108) we have

−H

U

−d

−H

= −d

∗2

p + ρ

−H

dy = −d

1 ∗2 U + 2

1 ∗2 p U + 2 ρ

dy

1 ∗ 2 1 ∗∗ 2 1 U + U dy = (U ∗ 2 + U ∗∗ 2 )(d − H). 2 2 2

(8.114)

Following Hogan [74], we deﬁne the mean wave momentum

η

I=

u ˜dy

(8.115)

−d

and the excess momentum ﬂux due to the waves

η

T p u ˜2 + dy + (1 − cos θ), ρ ρ −d

Sxx =

(8.116)

where the overbar denotes an average over one wavelength. These are related to the average kinetic energy K and the average potential energy V by the relations I=−

2K , U

(8.117)

Sxx = 4K − V + (U ∗∗ 2 − U 2 )d

(8.118)

(see Hogan [74]). Hence

η

p u2 + dy = ρ −d

η

p (U + u ˜)2 + dy ρ −d η η 2 =U dy + 2U u ˜ dy + −d

−d

= U 2 d + 2U I + Sxx − = −V + U ∗∗ 2 d −

η

u ˜2

−d

p + ρ

dy

T (1 − cos θ) ρ

T (1 − cos θ). ρ

(8.119)

242

Free-surface ﬂows with waves and intersections with rigid walls

Taking the average in equation (8.113) over one wavelength, we have 1 T ∗2 ∗∗ 2 ∗∗ 2 0 = − (d − H)(U + U ) + −V + dU − (1 − cos θ) 2 ρ T T + (1 − cos θ) − (1 − cos β) ρ ρ 1 1 ∗2 = −V − (U − U 2 )d + (U ∗∗ 2 − U 2 )d 2 2 1 ∗2 T (8.120) − (1 − cos β) + (U + U ∗∗ 2 )H. ρ 2 Equating the mass ﬂux on the left- and right-hand sides of the surface S, we have η 2K ∗ . (8.121) u ˜ dy = U d − U (d − h) = U d + U −d Multiplying by U and U ∗ and adding the two resulting equations gives 1 ∗2 1 U∗ 2 − (U − U )d = K 1 + − (U ∗ 2 + U U ∗ )H (8.122) 2 U 2 Substituting this into equation (8.120) gives T 1 1 U∗ − (1 − cos β) + (U ∗∗ 2 − U U ∗ )H + (U ∗∗ 2 − U 2 )d. 0 = −V + K 1 + U ρ 2 2 (8.123) This is an exact relation for ﬂow in ﬁnite depth. As d → ∞, we have U ∗ → U , U ∗∗ → U and (U ∗∗ 2 − U 2 )d → 0. Hence for inﬁnite depth 0 = −V + 2K −

T (1 − cos β). ρ

(8.124)

The energies V and K were calculated by Hogan [74] using Crapper’s exact solution. They are given by 1 2 2 1/2 2T 1+ π s −1 (8.125) V = ρ 4 and π 2 s2 T K= 4ρ

1 1 + π 2 s2 4

−1/2 .

Substituting these into equation (8.124) gives 1 2 2 −1/2 cos β = 2 1 + π s − 1. 4

(8.126)

(8.127)

8.4 Pure capillary free-surface ﬂows

243

Equation (8.127) shows that if there are waves on the free surface (i.e. s = 0) then the ﬂow will not leave the plate tangentially. The numerical results of Section 8.4.1 conﬁrm that are indeed waves on the free surface. Equation (8.127) provides a check on the numerical results. For example, on the one hand the numerical proﬁles of Figures 8.14 show waves of steepness 0.0292 in the far ﬁeld. On the other hand, (8.127) with β = ±π/48 predicts s = 0.0295. The theoretical and numerical values of the steepness agree within one per cent.

9 Waves with constant vorticity

Two fundamental approaches have been used in the previous chapters to calculate free-surface ﬂows. The ﬁrst involves perturbing known exact solutions. Often these exact solutions are trivial, e.g. a uniform stream. To leading order this approach gives a linear theory (see for example the calculations of Chapter 4) and at higher order a weakly nonlinear theory (see for example the small-amplitude expansions and the Korteweg–de Vries equation of Chapter 5). In the second approach fully nonlinear solutions are computed. This approach involves a discretisation leading to a system of nonlinear algebraic equations, which is then solved by iteration (e.g. using Newton’s method). Iteration requires the choice of an initial guess. These initial guesses are often trivial solutions or asymptotic solutions derived in the ﬁrst approach. After convergence of the iterations, the solution obtained is then used as an initial guess to compute a new solution for slightly diﬀerent values of the parameters. For example, the linear solutions of Section 2.4 were used as an initial guess in Chapter 6 to compute a nonlinear solution of small amplitude. This solution was then used as an initial guess to compute a solution of larger amplitude and so on. This method of ‘continuation’ leads to families of solutions; an application is the ‘continuation in ’ used in Section 7.1.1. We can then investigate whether other solution branches bifurcate from these branches (see Section 6.5.2.1 for an example). A limitation of this second approach is that we can only compute branches of solutions that extend the simple initial guesses we have tried. It is always possible that we are missing solution branches that could be calculated by starting with other, unknown, initial guesses. In this chapter we will solve a problem for which additional solution branches (not connected by continuation to simple initial guesses) are found: the eﬀect of vorticity on water waves. The assumption of irrotationality used 244

9.1 Solitary waves with constant vorticity

245

in Chapters 5 and 6 is often a very good approximation. However, vorticity can be generated by wind stress or by friction at the bottom of a channel. Here we assume for simplicity that the vorticity is constant in the ﬂuid. This enables us to reduce the problem to one described by Laplace’s equation. Solitary waves and periodic waves are considered in Sections 9.1 and 9.2 respectively.

9.1 Solitary waves with constant vorticity 9.1.1 Mathematical formulation We consider a two-dimensional solitary wave in an inviscid incompressible ﬂuid (see Figure 9.1). 5

4

3

2

y

1

0

0

x

5

10

15

Fig. 9.1. The ﬂow and the coordinates. The free surface shown is a computed solution for ω = −0.474, A = 3.8 and G = 0. Taken from J. Fluid Mech. 274, 340 (1994).

The ﬂuid is bounded below by a horizontal bottom. The ﬂow is assumed to be rotational and characterized by a constant vorticity Ω. We take a frame of reference with the x-axis along the horizontal bottom and in which the ﬂow is steady. We assume that the ﬂow is symmetric with respect to the y-axis. We denote by H the depth at inﬁnity and by c the velocity on the free surface at inﬁnity. The variables are made dimensionless by choosing H as the unit length and c as the unit velocity.

246

Waves with constant vorticity

The ﬂow is characterized by the following parameters: ω=

ΩH , c

(9.1)

G=

gH , c2

(9.2)

α=

A . H

(9.3)

Here g is the acceleration of gravity, counted as positive when acting vertically downwards, and A is the elevation of the wave crests above the level of the free surface at inﬁnity. The parameter ω is the dimensionless vorticity, G is the dimensionless gravity and α is an amplitude parameter. When ω = 0, the problem reduces to that considered in Sections 5.2 and 6.6.1. Benjamin [12] derived asymptotic solutions for solitary waves of small amplitude. His results showed that for each value of ω there is a oneparameter family of solutions that bifurcates from the uniform shear ﬂow at the critical value G = 1 + ω.

(9.4)

Teles da Silva and Peregrine [150], Pullin and Grimshaw [127] and VandenBroeck [178], [179] calculated numerically solutions for solitary waves with constant vorticity. Their results extended Benjamin’s results for waves of ﬁnite amplitude. We shall show that there are in addition solution branches that are not connected by continuation to Benjamin’s solutions. The analysis below follows [178] and [179]. It is convenient to describe the problem in terms of a streamfunction ψ(x, y) satisfying ∇2 ψ = −ω

(9.5)

in the ﬂow domain. Following [141], [150], [127], [178] and [179], we can reduce the problem to one described by Laplace’s equation by subtraction of a particular solution of (9.5). Thus if we write ψ =Ψ−

ω 2 y + (1 + ω)y 2

(9.6)

then ∇2 Ψ(x, y) = 0. Therefore the quantity w(z) = u − iv = Ψy + iΨx is an analytic function of z = x + iy, where the ﬂuid velocity vector is (u − ω(y − 1) + 1, v). We satisfy the kinematic condition that v must equal zero on the bottom by reﬂecting the ﬂow in the bottom. The function w(z)

9.1 Solitary waves with constant vorticity

247

vanishes at inﬁnity. Hence the Cauchy integral formula (2.41), when z is on the free surface, gives w(ζ)dζ 1 w(z) = − (9.7) πi C ζ − z with a Cauchy principal-value interpretation. Here C denotes the free surface and its image in the bottom. We parametrize the free surface by x = X(t), y = Y (t), where t is the arc length with t = 0 at the crest of the wave. Then X (t)2 + Y (t)2 = 1, X(0) = 0,

Y (0) = 1 + α,

(9.8) (9.9)

where α is the amplitude parameter deﬁned in (9.3). We now consider u and v to be functions of t. Taking the real part of (9.7) and using the symmetry of the wave with respect to the y-axis, we obtain, after some algebra, ∞ πu(t) = − 0

∞ − 0

∞ + 0

∞ + 0

Xs−t [u(s)Y (s) − v(s)X (s)] − Ys−t [u(s)X (s) + v(s)Y (s)] ds (Xs−t )2 + (Ys−t )2 Xs+t [u(s)Y (s) − v(s)X (s)] − Ys−t [u(s)X (s) + v(s)Y (s)] ds (Xs+t )2 + (Ys−t )2 Xs−t [v(s)X (s) − u(s)Y (s)] + Ys+t [u(s)X (s) + v(s)Y (s)] ds (Xs−t )2 + (Ys+t )2 Xs+t [v(s)X (s) − u(s)Y (s)] + Ys+t [u(s)X (s) + v(s)Y (s)] ds, (Xs+t )2 + (Ys+t )2 (9.10)

where Xs±t = X(s) ± X(t) and Ys±t = Y (s) ± Y (t); equation (9.10) holds for all 0 < t < ∞. On the free surface, the kinematic condition and Bernoulli equation yield (u + ω + 1 − ωY )Y (s) = vX (s),

(9.11)

[u + 1 − ω(Y − 1)]2 + v 2 + 2GY = 1 + 2G.

(9.12)

Here G is the gravity parameter deﬁned in (9.2). Equation (9.12) expresses the fact that the pressure is constant on the free surface.

248

Waves with constant vorticity

We ﬁx two of the three parameters ω, G and α and seek four functions u, v, X and Y and the value of the third parameter such that the system (9.8), (9.10)–(9.12) is satisﬁed.

9.1.2 Numerical procedure We are seeking a numerical solution of the nonlinear integro-diﬀerential system (9.8), (9.10)–(9.12). First we deﬁne N distinct mesh points on the free surface by specifying values of the arc length parameter t = SI , where SI = E(I − 1),

I = 1, . . . , N ;

(9.13)

here as before E is the interval of discretization. We shall also make use of the intermediate mesh points SI−1/2 = (SI+1 + SI )/2, I = 1, . . . , N − 1. We now deﬁne the 4N corresponding fundamental unknown quantities uI = u(SI ),

I = 1, . . . , N,

(9.14)

vI = v(SI ),

I = 1, . . . , N,

(9.15)

XI = X (SI ),

I = 1, . . . , N

(9.16)

YI = Y (SI ),

I = 1, . . . , N.

(9.17)

and

We will estimate the values of the x- and y- coordinates XI = X(SI ), YI = Y (SI ) in terms of the fundamental unknowns by the trapezoidal rule, i.e. X1 = 0, Y1 = 1 + α and XI = XI−1 + X (SI−3/2 )E, YI = YI−1 + Y (SI−3/2 )E,

I = 2, . . . , N, I = 2, . . . , N,

(9.18) (9.19)

where X (SI−3/2 ) and Y (SI−3/2 ) are evaluated from XI and YI by a fourpoint interpolation formula. We satisfy (9.11) and (9.12) at the mesh points SI , I = 1, . . . , N − 1 and (9.8) at SI , I = 1, . . . , N . This yields 3N − 2 nonlinear equations. Next we evaluate X(SI−1/2 ) and Y (SI−1/2 ) by fourpoint interpolation. We then satisfy (9.10) at the point t = SI−1/2 , I = 1, . . . , N − 1, by applying the trapezoidal rule to (9.10), with a sum over the points s = SJ , J = 1, . . . , N . The symmetry of the discretisation and of the trapezoidal rule with respect to the singularity of the integrand at s = t enables us to evaluate this Cauchy principal-value integral by ignoring the singularity,

9.1 Solitary waves with constant vorticity

249

with an accuracy no less than for a nonsingular integral (see Section 3.1.2.2 for a justiﬁcation). This yields N − 1 extra nonlinear equations. Four more equations are obtained by imposing the symmetry condition Y1 = 1

(9.20)

YN = 0,

(9.21)

uN = 0,

(9.22)

vN = 0.

(9.23)

and the far-ﬁeld conditions

We now have 4N + 1 equations for the 4N + 1 unknowns (9.14)–(9.17) and G. This system is solved by Newton’s method for given values of α and ω. To obtain the results presented in the next section, we also used a variation of the scheme in which we ﬁxed G and ω and found α as part of the solution.

9.1.3 Discussion of the results We used the numerical scheme of Section 9.1.2 to compute solutions for various values of the parameters. In all cases presented we checked that the numerical results were independent of E and N within graphical accuracy. 9.1.3.1 Solitary wave branches Numerical values of α versus G are shown in Figure 9.2 for various values of ω. For all values of ω and G, the uniform shear ﬂow α = 0, u = 0, v = 0, X = 1 and Y = 0 is a trivial solution. This branch of trivial solutions corresponds to the G-axis in Figure 9.2. The solitary waves are solution branches that bifurcate from this trivial solution at the critical values (9.4). Benjamin [12] derived an asymptotic solution for small values of α. In particular he obtained the following relation between G, α and ω (see his equation (46)): ω2 . (9.24) G=1+ω−α 1+ω+ 3 Our numerical values agree with (9.24) as α approaches zero. As we progress along the solution branches, we ﬁnd that there is a critical value ωc ≈ −0.32 of ω such that diﬀerent limiting conﬁgurations occur for ω < ωc and for ω > ωc . For ω > ωc , the limiting conﬁguration is

250

Waves with constant vorticity

8

4

0 0.4

0.8

G

Fig. 9.2. Values of α versus G for various values of ω. The curves from right to left correspond to ω = 0.11, −0.11, −0.32, −0.35 and −0.8. The broken curve corresponds to (9.25). Taken from J. Fluid Mech. 274, 344 (1994).

characterized by a stagnation point at the crest with a 120◦ angle. Using (9.12), we ﬁnd that these limiting conﬁgurations satisfy α=

1 . 2G

(9.25)

Relation (9.25) corresponds to the broken line in Figure 9.2. These solution branches were studied by Pullin and Grimshaw [127] and Teles Da Silva and Peregrine [150]. In the particular case ω = 0, the solutions agree with previous numerical calculations for solitary waves in the absence of vorticity (see Section 6.6.1). Typical proﬁles are shown in Figures 9.3(a), (b). For ω < ωc a new phenomena occurs: the solution branches in Figure 9.2 extend for arbitrarily large values of α without intersecting the broken curve. As α → ∞, the wave approaches a closed region of constant vorticity in contact with the bottom of the channel. Below we consider these limiting conﬁgurations further. A typical proﬁle for ω < ωc is shown in Figure 9.3(c). Figure 9.2 shows that there are solitary waves with constant vorticity in the absence of gravity (i.e. for G = 0). These solutions form a one-parameter family of solutions. Numerical values of α versus ω for G = 0 are shown in Figure 9.4. There is again a trivial solution corresponding to a uniform

9.1 Solitary waves with constant vorticity

251

1.20

(a)

2.0

1.10

(b)

1.6

1.2 1.00

0

5

3.0

10

0.8

15

0

5

10

15

(c)

2.0

1.0

0

5

10

15

Fig. 9.3. Computed free-surface proﬁle for: (a) ω = −0.11, A = 0.2 and G = 0.754; (b) ω = −0.11, A = 1.0 and G = 0.49 (this solution is close to the limiting conﬁguration with a 120◦ angle at the crest); (c) ω = −0.8, A = 2.5 and G = −0.19.

12

8

4

0

0.0

Fig. 9.4. Values of α versus ω for solitary waves without gravity, i.e. for G = 0.

shear ﬂow. This trivial solution corresponds to the ω-axis in Figure 9.4. The solitary waves bifurcate from this trivial solution at ω = −1. A typical free-surface proﬁle for G = 0 is shown in Figure 9.1. As α → ∞, the wave approaches a circular region made up of ﬂuid in rigid body rotation and in contact with the bottom of the channel at a single point. Such

252

Waves with constant vorticity

a conﬁguration was suggested by Teles da Silva and Peregrine [150] as a possible limiting conﬁguration for solitary waves without gravity. The solution branches for −1 < ω < ωc in Figure 9.2 intersect the α-axis. As discussed above, the corresponding solutions are the solutions without gravity of Figure 9.4. For ωc < ω < 0, the solution branches of Figure 9.2 do not intersect the α-axis and the solutions without gravity are members of a new solution branch. These branches were computed by continuation: we used the solution without gravity as the initial guess to compute a solution for a small value of G. This solution was then used as the initial guess to compute a solution for a larger value of G and so on. Values of α versus G for the new branch corresponding to ω = −0.11 are shown in Figure 9.5. 60

50

40

×

30

0

G

Fig. 9.5. Values of α versus G for the new branch with ω = −0.11. The cross corresponds to the solution (9.26).

Typical free-surface proﬁles are shown in Figures 9.6(a)–(d). The new branches do not bifurcate from a uniform shear ﬂow and evolve between two diﬀerent limiting conﬁgurations. The ﬁrst limiting conﬁguration is a wave whose proﬁle has a point of contact with itself (see Figure 9.6(a)). It is a wave without gravity (i.e. G = 0) and the closed region is a circular domain made up of ﬂuid in rigid body rotation. The radius of the circular domain is A/2. Since the velocity on the free surface is 1 when G = 0, we have A=

−4 . ω

(9.26)

9.1 Solitary waves with constant vorticity 60

60

(a)

(b)

40

40

20

20

0

0

20

120

40

0

0

20

40

120

(c)

(d)

80

80

40

40

0

253

0

40

0

0

40

Fig. 9.6. Computed free-surface proﬁles for (a) ω = −0.11, A = 38 and G = 0.0016; (b) ω = −0.11, A = 35.5 and G = 0.0064; (c) ω = −0.11, A = 41.5 and G = −0.02; (d) ω = −0.11, A = 50.5 and G = −0.034.

For ω = −0.11, (9.26) gives A = 36.364. This value corresponds to the cross in Figure 9.5. It is consistent with our numerical calculations. The second limiting conﬁguration is the same as for the solution branches with ω < ωc in Figure 9.2. It is a closed region of constant vorticity in contact with the bottom of the channel along a segment. The approach to this limiting conﬁguration is apparent in Figures 9.6(c), (d). As ω increases, the curve in Figure 9.5 moves upwards. We expect that this curve will coalesce into the point α = ∞, G = 0 as ω → 0 and that the new solutions do not exist for ω > 0. This is suggested by the fact that the solution corresponding to G = 0, α = ∞ and ω = 0 has the properties of both the ﬁrst and the second limiting conﬁguration (i.e. the proﬁle of the wave has a point of contact with itself and is also in contact with the bottom). As ω approaches ωc from above, we expect the new family of solutions to merge with the solution branches that bifurcate from a uniform shear ﬂow. The results ﬁrst discussed show that there are solitary waves of arbitrary large amplitude. Figures 9.6(c), (d) suggest that these waves approach

254

Waves with constant vorticity

closed regions of constant vorticity touching the bottom along a segment as α → ∞. These closed regions are diﬃcult to compute accurately using the previous numerical scheme because more and more mesh points are needed on the free surface as α increases. Therefore we shall calculate these limiting conﬁgurations directly (see Figure 9.7(a)). 1.2

(a)

1.2

0.8

0.8

0.4

0.4

0

0

0.4

1.2

(b)

0

0.8

0

0.5

1.0

(c)

0.8

0.4

0

0

1.0

Fig. 9.7. Computed solutions for a closed region of constant vorticity in contact with a wall with (a) µ = −0.045 and ζ = 3.04; (b) µ = −0.6 and ζ = 3.33; (c) µ = −0.21 and ζ = 4.34. Taken from J. Fluid Mech. 258, 105–113 (1994).

We ﬁrst rescale the variables by choosing the velocity Q at the separation points (i.e. the extremities of the segment of contact) as the unit velocity and the unit length such that the total length of the free surface is equal to 4. We choose cartesian coordinates with the origin in the middle of the segment of contact. Then the problem is characterized by the dimensionless gravity 2gL (9.27) µ= 2 Q and the dimensionless vorticity ζ=

ΩL . Q

(9.28)

9.1 Solitary waves with constant vorticity

255

Following the formulation of Section 9.1.1 and Vanden-Broeck and Tuck [193], we introduce the streamfunction ψ and write ζ ψ = Ψ − y2 ; 2

(9.29)

then ∇2 Ψ(x, y) = 0. As in Section 9.1.1, w(z) = u−iv = Ψy +iΨx is an analytic function of z = x + iy, where the ﬂuid velocity is now (u − ζy, v). We parametrise the free surface by x = X(t), y = Y (t), where t is the arc length, with t = 0 at the left-hand separation point. Then X (t) and Y (t) satisfy (9.8) and X(2) = 0. Proceeding as in Section 9.1.1, we ﬁnd that the integrodiﬀerential equation (9.10) still holds if the upper limit ∞ is replaced by 2. On the free surface, the kinematic condition and Bernoulli equation yield (u − ζY )Y (s) = vX (s),

(9.30)

(u − ζY )2 + v 2 + µY = 1.

(9.31)

This completes the formulation of the problem. For a given value of µ we seek four functions, u, v, X and Y , satisfying (9.8)–(9.10), (9.30) and (9.31). The value of ζ comes as part of the solution. We solved the problem using a numerical procedure similar to that described in Section 9.1.2. Most results were obtained with 60 equally spaced mesh points between s = 0 and s = 2. Typical free-surface proﬁles are shown in Figures 9.7(a)–(c). For µ = 0, the solution is a circular region of ﬂuid in rigid body rotation with one point of contact with the bottom. The corresponding value of ζ is π. A graph of ζ versus −µ is shown in Figure 9.8.

7

ζ 5

3

1

3

5

7

Fig. 9.8. Values of ζ versus −µ.

We found that there is a solution for each value of µ < 0. As µ becomes more negative, ζ increases and the length of the segment of contact increases.

256

Waves with constant vorticity

Vanden-Broeck and Tuck [193] considered closed regions of constant vorticity in contact with curved boundaries. The results that we have discussed generalize some of their ﬁndings by including the eﬀect of gravity.

9.1.3.2 More branches of solitary waves In Section 9.1.3.1 it was shown that there are branches of solitary waves that do not bifurcate from a uniform shear ﬂow. These branches start from a limiting conﬁguration consisting of a circular region of ﬂuid in rigid body rotation with no gravity, i.e. with G = 0 (see Figure 9.10). As one progresses along the solution branch in Figure 9.5 G increases and then decreases back to 0. The corresponding solution is a solitary wave with the familiar shape of a single hump (see Figure 9.9). As one continues along the branch G becomes negative and the wave ultimately approaches the form of a closed region of constant vorticity in contact with the bottom of the channel. Solutions with G negative were calculated in the Section 9.1.3.1 and will not be considered further. 30

25

20

15

10

y 5

0

x 0

5

10

15

20

Fig. 9.9. Computed solution for ω = −0.245 and G = 0. Taken from J. Mech. B (Fluids) 14, 761–771 (1995).

The two solutions without gravity shown in Figures 9.9 and 9.10 can be combined to generate further solutions without gravity. For example, we can have two circular regions of ﬂuid in solid body rotation on top of each other (see Figure 9.11(b)) or the solitary wave of Figure 9.9 with one or two circular regions of ﬂuid in rigid body rotation on top of its crest (see Figures 9.11(a), (c).

9.1 Solitary waves with constant vorticity

257

30

25

20

15

10

5

0

0

5

10

15

20

Fig. 9.10. Solution for G = 0 consisting of a circular region of ﬂuid in rigid body rotation. The value of ω is = −0.245.

In fact, an arbitrary number of circular regions of ﬂuid in rigid body rotation can be added in a similar way. Since the velocity is constant on the free surface when G = 0, the radius R of the circular regions in rigid body rotation are related to ω by the simple formula R 2 = . H |ω|

(9.32)

The solutions of Figures 9.11(a)–(c) are trivial extensions of the solutions of Figures 9.9 and 9.10. However, it is not obvious that these solutions can be perturbed to generate solutions with G = 0. Below, however, we will show that there are branches of solitary waves with gravity that approach the solutions in Figures 9.11(a)–(c) as G → 0. These new branches do not bifurcate from a uniform shear ﬂow. We found them by ﬁrst calculating a solution close to the limiting conﬁguration with a 120◦ angle at the crest of the wave. We then computed the corresponding family of solutions by a continuation method. All the solutions presented were obtained by using the numerical scheme described in Section 9.1.2. To construct the new families of solutions, we look ﬁrst at solutions which are close to the limiting conﬁguration with a 120◦ degree angle at the crest of the wave. These limiting conﬁgurations are characterized by a stagnation point at their crest. Using this property and (9.12), we ﬁnd that they satisfy 1 . (9.33) α= 2G

258

Waves with constant vorticity 60

(a)

40

20

0 60

(b)

40

20

0 60

(c)

40

20

0 –40

–20

0

20

40

Fig. 9.11. (a) Solution for G = 0 consisting of the solitary wave of Figure 9.9 with one circular region of ﬂuid in rigid body rotation on top of its crest. The value of ω is −0.245. (b) Solution for G = 0 consisting of two circular regions of ﬂuid in rigid body rotation on top of each other. The value of ω is −0.245. (c) Solution for G = 0 consisting of the solitary wave of Figure 9.9 with two circular regions of ﬂuid in rigid body rotation on top of its crest. The value of ω is −0.245.

9.1 Solitary waves with constant vorticity

259

We deﬁne waves close to the limiting conﬁguration with a 120◦ angle as solutions for which 2αG is close to 1. In Figure 9.12, we show values of α versus ω for such waves with 2αG = 0.96. 40 35 30

c

25

b

20

a

15 10 5

d

0

Fig. 9.12. Values of α versus ω for solitary waves with 2αG = 0.96. These waves are close to the limiting conﬁguration with a 120◦ angle at the crest. The branches associated with points a, b, c and d are shown in Figure 9.13.

As α increases, the values of ω oscillate between a succession of minima and maxima. The lowest minimum in ω coincides with the critical value ωc introduced in the Section 9.1.3.1. For ω > ωc , the solution branches that bifurcate from a uniform shear ﬂow at the critical values (9.4) have a limiting conﬁguration with a 120◦ angle at their crests; see Teles da Silva and Peregrine [150], Pullin and Grimshaw [127] and the discussion in Section 9.1.3.1. Each limiting conﬁguration corresponds to a point on the lower part of the curve in Figure 9.12 with ω > ωc . For ω < ωc , the solution branches that bifurcate from a uniform shear ﬂow at the critical values (9.4) do not have limiting conﬁgurations with a 120◦ angle at their crests; see the references mentioned above. As we showed, these branches approach a conﬁguration similar to that of Figure 9.9 as G → 0 and ultimately tend to closed regions of constant vorticity in contact with the bottom of the channel if G is allowed to become negative. This discussion shows that the lower portion of the curve in Figure 9.12 for ω > ωc consists of limiting conﬁgurations of branches that bifurcate from a uniform shear ﬂow at the critical values (9.4). Each point on the

260

Waves with constant vorticity

remaining part of the curve of Figure 9.12 corresponds to a solution close to the limiting conﬁguration for a new family of waves, which does not bifurcate from a uniform shear ﬂow. We calculate these new families by continuation, i.e. we ﬁx a value of ω associated with a point on the curve of Figure 9.12 and use the corresponding solution as the initial guess to compute a solution for a slightly perturbed value of α or G; this solution is then used as the initial guess for a further calculation. Explicit calculations were performed for the points a, b, c and d of Figure 9.12. The solution branches are shown in Figure 9.13, where we present values of α versus G.

× 42

c

36

×

b

30 ×

24

a

18

e 12

6

d 0.03

0.06

0.09

G

Fig. 9.13. Values of α versus G for ω = −0.245 (curves a, b, c), ω = −0.3 (curve d) and ω = −0.267 (curve e). The curves a, b, c and d intersect the curve of Figure 9.12 at points a, b, c and d respectively. The broken curve corresponds to the relation 2αG = 0.96. The three crosses correspond, from bottom to top, to the solutions without gravity of Figures 9.11(a)–(c).

9.1 Solitary waves with constant vorticity

261

The broken curve corresponds to the relation 2αG = 0.96. For convenience we denote the branches associated with points a, b, c and d in Figure 9.12 by the same letters in Figure 9.13. Figure 9.13 shows that the new solution branches do not bifurcate from a uniform shear ﬂow (i.e. from the G-axis in Figure 9.13). Furthermore the new branches can be extended up to G = 0. However, our numerical scheme becomes more sensitive as G approaches zero; the numerical calculations for curves a, b and c had to be stopped at some small positive value of G (see Figure 9.13). Typical free-surface proﬁles are shown in Figures 9.14–9.16. The proﬁles of Figure 9.14 are for ω = −0.245 and correspond to the family of solutions starting at the point a in Figure 9.12 (see curve a in Figure 9.13). As G → 0, the proﬁles of Figure 9.14 suggest that the wave approaches a conﬁguration consisting of a ‘single hump’ wave with a circular region of ﬂuid in rigid body rotation at its crest. This conﬁguration is shown in Figure 9.11(a) and the corresponding value of α is indicated by the lowest cross on the α-axis in Figure 9.13. The position of this cross is consistent with our numerical calculations. The proﬁles of Figure 9.15 are for ω = −0.245 and correspond to the family of solutions starting at the point b in Figure 9.12 (see curve b in Figure 9.13). As G → 0, the wave approaches two circular regions of ﬂuid in rigid body on top of each other. This conﬁguration is shown in Figure 9.11(b) and the corresponding value of α is indicated by the middle cross on the α-axis in Figure 9.13. The proﬁles of Figure 9.16 are also for ω = −0.245 but correspond to the family of solutions starting at the point c in Figure 9.12 (see curve c in Figure 9.13). As G → 0, the wave approaches a ‘single hump’ wave with two circular regions of ﬂuid in rigid body rotation at its crest. This limiting proﬁle is shown in Figure 9.11(c) and the corresponding value of α is indicated by the top cross on the α-axis in Figure 9.13. The proﬁles of Figure 9.17 correspond to ω = −0.3. They are on the solution branch starting at point d in Figure 9.12 (see curve d in Figure 9.13). As G → 0, the wave proﬁles approach a solitary wave with the shape of a single hump. The corresponding proﬁle is shown in Figure 9.17(c). Next we show the connection between these new solutions and the solutions in Section 9.1.3.1. There it was shown that there are solution branches for ωc < ω < 0 that do not bifurcate from a uniform shear ﬂow. These branches evolve between two conﬁgurations with G = 0. The ﬁrst is a circular region of ﬂuid in rigid body rotation (see Figure 9.10) and the second is a ‘single hump’ conﬁguration (see Figure 9.9). These branches were obtained by ﬁrst calculating ‘single hump’ solutions for G = 0 and ωc < ω < 0 and

262

Waves with constant vorticity 60

(a) 50

40

30

20

10

0 60

(b) 50

40

30

20

10

0 60

(c) 50

40

30

20

10

0

0

20

40

Fig. 9.14. Computed free-surface proﬁles for solutions on curve a of Figure 9.13. The value of ω is −0.245. The values of G are 0.024 (Figure 9.14(a)); 0.018 (Figure 9.14(b)) and 0.005 (Figure 9.14(c)).

9.1 Solitary waves with constant vorticity

263

60

(a) 50

40

30

20

10

0 60

(b) 50

40

30

20

10

0 60

(c) 50

40

30

20

10

0

0

20

40

Fig. 9.15. Computed free-surface proﬁles for solutions on curve b of Figure 9.13. The value of ω is −0.245. The values of G are 0.02 (Figure 9.15(a)); 0.011 (Figure 9.15(b)) and 0.0045 (Figure 9.15(c)).

264

Waves with constant vorticity 60

(a)

40

20

0 60

(b)

40

20

0 60

(c)

40

20

0

0

20

40

Fig. 9.16. Computed free-surface proﬁles for solutions on curve c of Figure 9.13. The value of ω is −0.245. The values of G are (a) 0.015; (b) 0.01 and (c) 0.0085.

9.1 Solitary waves with constant vorticity 16

265

(a)

12

8

4

0 16

(b)

12

8

4

0

16

(c)

12

8

4

0

0

5

10

Fig. 9.17. Computed free-surface proﬁles for solutions on curve d of Figure 9.13. The value of ω is −0.3. Part (a) corresponds to the intersection of curve d in Figure 9.13 with the broken line in Figure 9.13. Part (b) is for G = 0.05 and Part (c) is for G = 0.

then constructing the corresponding family of solutions by continuation; see Section 9.1.3.1 for details. Such families are shown in Figure 9.5 for ω = −0.11 and in Figure 9.13 for ω = −0.267 (see curve e). Corresponding proﬁles for ω = −0.267 are presented in Figure 9.18. These results indicate that, as ω decreases, the branches in Section 9.1.3.1 approach the broken curve in Figure 9.13 and ultimately intersect it at ω = ωd . Here ωd is between −0.3 and −0.267. For ωc < ω < ωd , the branch

266

Waves with constant vorticity 30

(a)

20

10

0 30

(b)

20

10

0 30

(c)

20

10

0

0

10

20

Fig. 9.18. Computed free-surface proﬁles for solutions on curve e of Figure 9.13. The value of ω is −0.267. The proﬁles correspond to (a) α = 8.5, G = 0.054, (b) α = 15.9, G = 0.015 and (c) α = 16.14, G = 0.0065.

in Section 9.1.3.1 becomes a branch that evolves between a ‘single hump’ solution and a point on the broken curve of Figure 9.13 (i.e. a solution close to the limiting conﬁguration with a 120◦ angle at its crest); an example of such a family is curve d in Figure 9.13. These families exist up to ω = ωc . For ω < ωc there are still families of solutions that start from the ‘single hump’ conﬁguration with G = 0. However, they do not intersect the broken

9.2 Periodic waves with constant vorticity

267

curve of Figure 9.13 and can be extended up to the G-axis. These are the solutions for ω < ωc that bifurcate from a uniform shear ﬂow. Such families of solutions are shown in Figure 9.2 for ω = −0.35 and ω = −0.8. We have shown that there are families of solutions with gravity that approach the conﬁgurations of Figure 9.11 as G → 0. Our solutions are not exhaustive. Indeed, we expect that there are an inﬁnite number of families of solutions that, as G → 0, approach conﬁgurations similar to those of Figures 9.11(a)–(c) but with an arbitrary number of regions of ﬂuid in rigid body rotation. These could be obtained by extending the calculations in Figure 9.12 for higher values of α. Similarly we expect that there are an inﬁnite number of branches similar to that computed in Section 9.1.3.1 (i.e. similar to curve e in Figure 9.13). These families should connect the conﬁgurations of Figure 9.11, and their generalization with an arbitrary number of circular regions, without intersecting the broken curve of Figure 9.13. 9.2 Periodic waves with constant vorticity Simmen and Saﬀman [141] obtained numerical solutions for periodic waves with constant vorticity in water of inﬁnite depth. Their results showed that the waves have either a limiting conﬁguration with a 120◦ angle at the crest or a trapped bubble at their troughs. The results of Section 9.1.3 showed that there are solitary waves with constant vorticity that approach limiting conﬁgurations with trapped bubbles at their crests as the acceleration of gravity approaches zero (see Figure 9.11). In this section we extend the calculations of Simmen and Saﬀman [141] by showing that there are additional families of periodic solutions that, like the solitary waves of Section 9.1.3.2, approach limiting conﬁgurations with trapped bubbles at their crests as the acceleration of gravity tends to zero. Each bubble is circular and contains ﬂuid in rigid body rotation. The main diﬀerence between solitary and periodic waves is that there are no limiting conﬁgurations with trapped bubbles at the troughs for solitary waves. The problem is solved numerically by a boundary integral equation method. The numerical approach is similar to that used in Section 9.1.2 for solitary waves. It is also similar to the approach used by Simmen and Saﬀman [141]. The essential diﬀerence between our scheme and that of Simmen and Saﬀman is that we do not scale the variables so that the acceleration of gravity is 1. This enables us to consider waves in the limit as the acceleration of gravity approaches zero. The new families of solutions exist only for suﬃciently large values of the amplitude. As we shall see, they are constructed by a continuation method similar to that used in Section 9.1.3.

268

Waves with constant vorticity

The problem is formulated in Section 9.2.1. The numerical procedure is discussed in Section 9.2.2 and the results are presented in Section 9.2.3. The presentation follows Vanden-Broeck [180] closely.

9.2.1 Mathematical formulation We consider a two-dimensional periodic wave in an inviscid incompressible ﬂuid. The ﬂuid is of inﬁnite depth. The ﬂow is assumed to be rotational and characterized by a constant vorticity −Ω. We take a frame of reference with the x-axis along the mean water level and in which the ﬂow is steady. Gravity is acting in the negative y-direction. We assume that the ﬂow is symmetric with respect to the y-axis. The ﬂow is described in terms of a streamfunction ψ(x, y) satisfying ∇2 ψ = Ω

(9.34)

in the ﬂow domain. As before, we reduce the problem to one described by Laplace’s equation by subtracting a particular solution of (9.34). Thus if we write ψ =Ψ+

Ω 2 y − cy 2

(9.35)

then ∇2 Ψ(x, y) = 0. We require that Ψ→0

as

y → −∞.

(9.36)

This deﬁnes uniquely the quantity c in (9.35). Following Simmen and Saﬀman [141], we refer to c as the wave speed. The variables are made dimensionless by choosing the wavelength λ as the unit length and c as the unit velocity. In terms of the dimensionless variables, (9.35) becomes ψ =Ψ+

ω 2 y − y, 2

(9.37)

Ωλ . c

(9.38)

where ω=

The quantity w(z) = u − iv = Ψy + iΨx is an analytic function of z = x + iy, and the ﬂuid velocity vector is (u + ωy − 1, v). The function w(z) vanishes at inﬁnity.

9.2 Periodic waves with constant vorticity

269

We map the ﬂow domain within the wavelength −1/2 < x < 1/2 from the z-plane into the interior of a domain of the ζ-plane by the transformation ζ = e−2iπz .

(9.39)

Since w(z) is periodic in x with period 1, it gives rise to a single-valued analytic function of ζ. Applying the Cauchy integral formula (2.38) to the function u − iv in the ζ-plane we obtain w(µ)dζ 1 , (9.40) w(ζ) = − πi C µ − ζ where C denotes the closed curve (9.40), ζ is on C and the integral is Suppose that the free surface is where t is the arc length with t = 0

bounding the ﬂow in the ζ-plane. In a Cauchy principal value. parametrised by x = X(t), y = Y (t), at the crest of the wave. Then

X (t)2 + Y (t)2 = 1, X(0) = 0,

Y (0) = α,

(9.41) (9.42)

where α is the dimensionless elevation of the crests. We now consider u and v to be functions of t. Taking the real part of (9.40) and using the symmetry of the wave with respect to the y-axis, we obtain, after some algebra, 1/2 [u(p) − iv(p)][−2iπx (p) + 2πy (p)] dp πu(t) = − 1 − exp{−2iπ[x(t) − x(p)] + 2π[y(t) − y(p)]} 0 1/2 [u(p) + iv(p)][−2iπx (p) − 2πy (p)] dp, (9.43) − 1 − exp{−2iπ[x(t) + x(p)] + 2π[y(t) − y(p)]} 0 where as before denotes the imaginary part. On the free surface, the kinematic condition and Bernoulli equation yield (u − 1 + ωY )Y (s) = vX (s),

(9.44)

[u + ωY − 1]2 + v 2 + 2GY = B.

(9.45)

Here B the Bernoulli constant and G is the gravity parameter deﬁned by G=

gλ . c2

(9.46)

Equation (9.45) states that the pressure is constant on the free surface. For given values of ω and α we seek four functions, u, v, X and Y , satisfying (9.41)–(9.45). The parameters G and B are found as part of the solution.

270

Waves with constant vorticity

9.2.2 Numerical procedure We seek a numerical solution of the nonlinear integro-diﬀerential system (9.41)–(9.45). First we deﬁne N distinct mesh points on the free surface by specifying values of the arc length parameter t = SI , where SI = b

I −1 , 2N − 2

I = 1, . . . , N.

(9.47)

Here b is the length of the free surface between two successive crests. We shall also make use of the intermediate mesh points SI−1/2 = (SI+1 + SI )/2, I = 1, . . . , N − 1. We now deﬁne the 4N corresponding fundamental unknown quantities uI = u(SI ),

I = 1, . . . , N,

(9.48)

vI = v(SI ),

I = 1, . . . , N,

(9.49)

XI = X (SI ),

I = 1, . . . , N,

(9.50)

YI = Y (SI ),

I = 1, . . . , N.

(9.51)

and

We estimate the values of the x- and y- coordinates XI = X(SI ) and YI = Y (SI ) in terms of the fundamental unknowns by the trapezoidal rule, i.e. X1 = 0, Y1 = α and XI = XI−1 + X (SI−3/2 )

YI = YI−1 + Y (SI−3/2 )

b , 2N − 2

b , 2N − 2

I = 2, . . . , N,

I = 2, . . . , N.

(9.52)

(9.53)

and YI−3/2 are evaluated from XI and YI by a four-point Here XI−3/2 interpolation formula. We satisfy (9.44) and (9.45) at the mesh points SI , I = 1, . . . , N , and (9.41) at the mesh points SI , I = 1, 2, . . . , N . This yields 3N nonlinear equations. Next we satisfy (9.43) at the point t = SI , I = 2, . . . , N − 1, by applying the trapezoidal rule to (9.43), with a sum over the points s = SJ −1/2 , J = 1, . . . , N − 1. The symmetry of the discretisation and of the trapezoidal rule with respect to the singularity of the integrand at s = t enables us to evaluate this Cauchy principal-value integral by ignoring the singularity, with an accuracy no less than that of a nonsingular integral (see Section 3.1.2.2). This yields N − 2 extra nonlinear equations.

9.2 Periodic waves with constant vorticity

271

Two equations are obtained by imposing the symmetry conditions v1 = 0

(9.54)

vN = 0.

(9.55)

and

We obtain two more equations by ﬁxing the origin of Y at the mean water level and by requiring that w vanishes at inﬁnite depth. This leads to b/2 Y X ds = 0 (9.56) 0

and

b/2

(uX + vY ) ds = 0.

(9.57)

0

The ﬁnal equation ﬁxes the wavelength by imposing XN = 1/2.

(9.58)

We now have 4N + 3 equations for the 4N + 3 unknowns (9.48)–(9.51), B, b and G. This system is solved by Newton’s method for given values of α and ω. To obtain the results presented in the next section, we also used a variation of the scheme in which we ﬁxed G and ω and found α as part of the solution.

9.2.3 Numerical results We used the numerical scheme of Section 9.2.2 to compute solutions for various values of ω and α. Most calculations were performed with N = 121. In all cases we checked that the numerical results were independent of N within graphical accuracy. We ﬁrst conﬁrmed the ﬁndings of Simmen and Saﬀman [141]. There are solution branches that bifurcate from the uniform shear ﬂow u = v = 0. As one progresses along the solution branches, the waves ultimately reach limiting conﬁgurations with a trapped bubble at the trough or a 120◦ angle at the crest. We now construct the new families of solutions. Following the approach of Section 9.1.3, we look ﬁrst at solutions that are close to the limiting conﬁguration with a 120◦ angle at the crest. Such limiting conﬁgurations are characterized by a stagnation point at the crest. We deﬁne waves close to the limiting conﬁguration with a 120◦ angle at the crest as solutions for

272

Waves with constant vorticity

which the velocity at the crest is equal to a small quantity . Therefore they satisfy u1 + ωα − 1 = .

(9.59)

Here we choose = 0.1. In Figure 9.19, we show values of the steepness s (i.e. the diﬀerence in heights between a crest and a trough divided by the wavelength) versus ω for waves close to the limiting conﬁguration. We see that there is a minimum value ωc such that there are no solutions for ω < ωc and denote the corresponding value of s by sc . We now calculate families of solutions by continuation. To do this we ﬁx a value of ω associated with a point on the curve of Figure 9.19 and use the corresponding solution as the initial guess to compute a solution for a slightly perturbed value of α or G. This solution is then used as an initial guess for a further calculation. s 3.0 ×

c

2.0 ×b

1.0 ×

a

0.0

0

Fig. 9.19. Values of the steepness s versus ω for waves close to the limiting conﬁguration with a 120◦ angle at the crest. The waves are characterised by = 0.1. The arrow indicates the point (ωc , sc ). Taken from IMA J. Appl. Math. 56, 207–217 (1996).

We ﬁnd that the points on the lower part of the curve in Figure 9.19, for s < sc , are limiting conﬁgurations, or more precisely solutions close to the limiting conﬁguration, of branches that bifurcate from the uniform shear ﬂow u = v = 0. These branches were described by Simmen and Saﬀman [141] and will not be considered further here. The points on the upper part of the curve in Figure 9.19, with s > sc , are limiting conﬁgurations of new families of waves. Explicit calculations were performed for points a and b in Figure 9.19. The corresponding solution

9.2 Periodic waves with constant vorticity

273

branches are shown in Figures 9.20 and 9.21, where we present values of s versus G for the values ω = −2.202 and ω = −1.045 corresponding to the points a and b of Figure 9.19. s 0.74

0.72

0.70

0.68

0.66

0

0.10

0.20

0.30

0.40

G

Fig. 9.20. Values of s versus G for ω = −2.202. The highest point on the curve corresponds to point a in Figure 9.19. Taken from IMA J. Appl. Math. 56, 207–217 (1996).

s 1.70

1.60

1.50

0

0.010

0.020

0.030

0.040

G

Fig. 9.21. Values of s versus G for ω = −1.045. The highest point on the curve corresponds to point b in Figure 9.19. Taken from IMA J. Appl. Math. 56, 207–217, (1996).

Figures 9.20 and 9.21 show that the new branches do not bifurcate from the uniform shear ﬂow (i.e. the G-axis in Figures 9.20 and 9.21). Furthermore the new branches can be extended up to G = 0. However, the

274

Waves with constant vorticity

numerical calculations in Figure 9.21 become more sensitive for G small and had to be stopped at a small positive value of G. Typical free-surface proﬁles corresponding to points on the curves of Figures 9.20 and 9.21 are shown in Figures 9.22 and 9.23 respectively. y

0.8

(a)

0.4

0.0

y

0.8

0

0.5

1.0

1.5

0

0.5

1.0

1.5

x

(b)

0.4

0.0

x

Fig. 9.22. Typical free-surface proﬁles for points on the curve of Figure 9.20. The value of ω is −2.202. The proﬁles correspond to (a) G = 0.38 and (b) G = 0. Two wavelengths are shown in both parts (a) and (b). Taken from IMA J. Appl. Math. 56, 207–217 (1996).

The proﬁles of Figure 9.22 show that as G → 0 the waves approach, a wave similar to those computed by Simmen and Saﬀman [141]. The proﬁle of Figure 9.22(b) is close to a conﬁguration with a trapped bubble at the trough. Our calculations show that there is a wave with a trapped bubble when G = 0 and that the corresponding value s1 of the steepness is approximately 0.7. This value is consistent with the results in Figure 12 of Simmen

9.2 Periodic waves with constant vorticity y

1.0

275

(a)

0.5

0.0

y

1.0

0

0.5

1.0

1.5

2.0

0

0.5

1.0

1.5

2.0

0

0.5

1.0

1.5

2.0

x

(b)

0.5

0.0

y

1.0

x

(c)

0.5

0.0

x

Fig. 9.23. Typical free-surface proﬁles for points on the curve of Figure 9.21. The value of ω is −1.045. The proﬁles correspond to (a) G = 0.03, (b) G = 0.011 and (c) G = 0.007. Two wavelengths are shown in each of parts (a) and (b). Taken from IMA J. Appl. Math. 56, 207–217 (1996).

and Saﬀman [141]. Their broken curve has a vertical asymptote at about s = s1 (the value of the parameter Ω∗ used by Simmen and Saﬀman is equal to −∞ when G = 0 because their solutions are scaled such that g = 1). The proﬁles of Figure 9.23 are more exotic. A comparison of the proﬁles in Figures 9.23(a)–(c) show that as G → 0 the waves ultimately approach,

276

Waves with constant vorticity

a conﬁguration consisting of a wave with a circular region at the crest. The ﬂuid in the circular region is in rigid body rotation. Our new families of waves can also be limited by trapped bubbles at their troughs. This is illustrated in Figure 9.24, where we present the free-surface proﬁle corresponding to the point c in Figure 9.19. The wave has almost developed a closed region at the crests and a trapped bubble at the troughs.

9.2.4 Discussion We have used a boundary integral equation method to compute periodic waves with constant vorticity are have shown that there are solution branches that approach conﬁgurations with a closed region of ﬂuid in rigid body rotation as G → 0. Our results supplement those of Simmen and Saﬀman [141], which show that there are limiting conﬁgurations with a 120◦ angle at the crest or a trapped bubble at the trough. In Section 9.1, we showed that there are solitary waves with circular closed regions at their crests. Since solitary waves can be viewed as the limit of periodic waves as the ratio of the wavelength and the depth goes to inﬁnity, our results suggest that conﬁgurations with circular regions at the crest are a general property of waves in water of arbitrary depth. y

1.0

0.5

0.0

0

0.4

0.8

1.2

1.6

x

Fig. 9.24. Free surface corresponding to the point c in Figure 9.19. The value of ω is −0.52. Two wavelengths are shown. Taken from IMA J. Appl. Math. 56, 207–217 (1996).

The results presented are only representative. We expect that there are a large number of families similar to the family shown in Figure 9.23. In particular we should expect waves with several closed regions of ﬂuid in rigid

9.2 Periodic waves with constant vorticity

277

body rotation at their crests (such conﬁgurations were computed for solitary waves in Section 9.1.3.2). However, for periodic waves the admissible solutions are limited by the appearance of trapped bubbles at the troughs. This is to be contrasted with solitary waves, for which there are no conﬁgurations with trapped bubbles at the troughs. Finally we should point out that some of our ‘exotic’ waves, such as those in Figure 9.23(c), are likely to be unstable. However, others such as those in Figure 9.23(a) have more classical shapes.

10 Three-dimensional free-surface ﬂows

As shown in the previous chapters, eﬃcient methods for two-dimensional free-surface ﬂows can be derived by using the theory of analytic functions. In particular, free streamline problems, series truncation methods and boundary integral equation methods based on the Cauchy integral formula can be used to obtain highly accurate solutions. Unfortunately such techniques are not available for three-dimensional free-surface ﬂows. However, as we shall see in this chapter, boundary integral equation methods can still be derived using Green’s theorem (see also [122]–[125]) . Boundary integral equation methods based on Green’s theorem can also be used for two-dimensional free-surface ﬂows as an alternative to methods based on the Cauchy integral formula. We ﬁrst show this for twodimensional free-surface ﬂows generated by moving disturbances in water of inﬁnite depth. Gravity is included in the dynamic boundary condition but surface tension is neglected.

10.1 Green’s function formulation for two-dimensional problems We describe the numerical method based on Green’s functions by considering the free-surface ﬂows generated by a moving pressure distribution (see Figure 4.4) or by a moving surface-piercing object (see Figure 4.3). We will assume that the water is of inﬁnite depth. The corresponding method based on the Cauchy theorem was described in Chapter 7 for a moving pressure distribution.

10.1.1 Pressure distribution We consider the two-dimensional free-surface ﬂow generated by a pressure distribution moving at a constant velocity U at the surface of a ﬂuid of 278

10.1 Green’s function formulation for two-dimensional problems

279

inﬁnite depth. The ﬂuid is assumed to be inviscid and incompressible and the ﬂow is assumed to be irrotational. We choose a cartesian frame of reference moving with the pressure distribution and assume that the ﬂow is steady. Gravity is acting in the negative y-direction and the eﬀect of surface tension is neglected. Following the formulation of Section 4.1, we introduce the potential function φ(x, y), so that the velocity is given by (φx , φy ). In the ﬂow domain, φ satisﬁes ∇2 φ = 0,

−∞ < x < ∞,

−∞ < y < η(x),

(10.1)

with the condition (φx , φy ) → (U, 0)

as

y → −∞.

(10.2)

Here we denote by y = η(x) the equation of the free surface. The kinematic and dynamic boundary conditions give φx ηx = φy

on

y = η(x)

(10.3)

and 1 2 U2 p (φx + φ2y ) + gη + = 2 ρ 2

on

y = η(x),

(10.4)

where g is the acceleration of gravity, ρ is the ﬂuid density and p is the prescribed pressure distribution. The choice of the Bernoulli constant on the right-hand side of (10.4) ﬁxes the origin of y. The upstream radiation condition gives (φx , φy ) → (U, 0)

and η → 0

as x → −∞.

(10.5)

The physical quantities are made dimensionless by taking U as the unit velocity and the length L of the support of the pressure distribution as the unit length. The Froude number is deﬁned by U F =√ . gL The formulation involves Green’s second identity, ∂α ∂β −β ds. (α∇2 β − β∇2 α)dV = α ∂n ∂n V C

(10.6)

(10.7)

Here C is a closed curve bounding a region V of the plane. The curve C is characterised by its arc length s and its outward normal n. Assuming that α satisﬁes Laplace’s equation and that β is the two-dimensional free space

280

Three-dimensional free-surface ﬂows

Green’s function g = 1/4π ln[(x − x∗ )2 + (y − y ∗ )2 ], (10.7) gives ∂α ∂g ∗ ∗ −g ds. α(x , y ) = r α ∂n ∂n C

(10.8)

Here r = 1 when (x∗ , y ∗ ) is inside C and r = 1/2 when (x∗ , y ∗ ) is on C. We now choose α = Φ − x and assume that C consists of the free surface and a semicircle of arbitrarily large radius in the region y < η(x). Using the arc length s and describing the free surface parametrically by x = X(s) and y = Y (s), we obtain 1 F(s∗ ) = 2

∞ −∞

∂G ∗ ∗ ∂F(s) F(s) (s, s ) − G(s, s ) ds. ∂n ∂n

(10.9)

Here φ(s) = Φ(X(s), Y (s)), F(s) = φ(s)−X(s), G(s, s∗ ) = (1/4π) ln{[X(s)− X(s∗ )]2 + [Y (s) − Y (s∗ )]2 } and n = (−Y (s), X (s)). The deﬁnition of the arc length requires that X 2 + Y 2 = 1.

(10.10)

The kinematic and dynamic boundary conditions on the free surface are rewritten in terms of the dimensionless variables as ∂φ =0 (10.11) ∂n and 1 2 Y 1 φ + (10.12) + P = , 2 s F2 2 where P is the dimensionless pressure. We choose 1/(s2 −1) e for |s| < 1, P (s) = (10.13) 0 otherwise. The unknown functions φ(s), X(s) and Y (s) are obtained by solving numerically the nonlinear equations (10.9)–(10.12) subject to the radiation condition, which requires that there is no energy coming from inﬁnity (see Chapter 4 for a discussion of the radiation condition). We deﬁne N equally spaced points s1 = −e(N − 1)/2, si = s1 + e(i − 1), i = 2, . . . , N , where e is the interval of discretisation. We choose N to be odd. Here s1 approximates −∞ and sN = −s1 approximates +∞. We use the notation xi = X(si ), yi = Y (si ) etc. The domain of integration for (10.9) is (s1 , sN ). In order to satisfy the Bernoulli equation at the ﬁrst point, we impose y1 = 0,

x1 = φ1 = 1,

x1 = φ1 = s1 .

(10.14)

10.1 Green’s function formulation for two-dimensional problems

Equations (10.10)–(10.12) and the trapezoidal rule yield xk = 1 − yk2 , 1 xk = xk−1 + e xk + xk−1 , 2 1 , yk = yk−1 + e yk + yk−1 2 yk φk = 1 − 2 2 − 2pk , F 1 φk = φk−1 + e(φk + φk−1 ), 2

281

(10.15)

for k = 2, . . . , N . The values of the functions at the midpoints are calculated by interpolation with two or four points (xk−1/2 = 12 (xk−1 + xk ) etc.). Equation (10.9) is evaluated at the midpoints si−1/2 , i = 2, . . . , N − 1. The integral is approximated by the trapezoidal rule with a summation over the mesh points si , i = 2, . . . , N . Substituting (10.15) yields N −2 nonlinear algebraic equations. The ﬁnal two equations are obtained by imposing the radiation condition, using the relations y1 = 0

and

− 3y1 + 4y2 − y3 = 0.

(10.16)

The second of these relations imposes y1 = 0, approximately. This system of is solved by Newton’s N nonlinear equations for the N unknowns y1 , . . . , yN method. The initial guess for the unknowns yi is zero when ε 1 or previous computed solutions obtained for slightly diﬀerent values of F and ε when ε is large. The numerical accuracy of the scheme was checked by varying N and e (see Figure 10.1). We found that the solutions presented here are independent of N and e within graphical accuracy for N ≥ 200 and e ≤ 0.1. In the numerical calculations, the integral from −∞ to ∞ in (10.9) was replaced by an integral from s1 to sN . We found that these upstream and downstream truncations only aﬀect the ﬁrst and last half-wavelength of the free-surface proﬁles. A similar numerical behavior was found in Asavanant and Vanden-Broeck [7]. Improved truncation methods similar to those used in Section 7.1.2 can be developed. We compared our numerical solutions with those obtained using the method of Chapter 7. A typical pair of proﬁles is shown in Figure 10.2. Similar results were found for other values of ε and F . The conclusion of

282

Three-dimensional free-surface ﬂows 8

x 10

4

0

−4

−8

−8

−4

0

4

8

Fig. 10.1. Computed free-surface proﬁles for F = 0.7, ε = 0.001 with grids N = 721, e = 0.025 (broken line) and N = 361, e = 0.05 (solid line).

the comparison is that numerical results as accurate as those of Chapter 7 can be obtained without using complex variables. This suggests that accurate results for three-dimensional free-surface ﬂows can be obtained by generalising the Green’s function formulation of Section 10.1.1 to three dimensions. This is done in Section 10.2. 8

x 10

4

0

−4

−8

−8

−4

0

4

8

Fig. 10.2. Computed free-surface proﬁles obtained with an algorithm based on Green’s functions (solid line) and with the algorithm of Chapter 7 (dotted line). The parameters are F = 0.7, = 0.001. The grid used is N = 721 and e = 0.025.

10.1 Green’s function formulation for two-dimensional problems

283

10.1.2 Two-dimensional surface-piercing object As noted in Chapter 7, once a solution of (10.9)–(10.12) has been computed for a given pressure distribution (10.13), we can replace the free surface under the support −1 < s < 1 of the pressure distribution by a rigid surface. Here, the support refers to the interval over which the pressure is nonzero. Therefore the schemes described in Section 10.1.1 and in Chapter 7 provide an inverse method for calculating free-surface ﬂows past surface-piercing objects, e.g. two-dimensional ‘ships’ (see Figure 4.3). The shape of the ship is given at the end of the calculations by the shape of the streamline under the support of the pressure distribution. One drawback of this approach is that the shape of the ship depends on the Froude number F . It is therefore desirable to have an approach that enables a direct calculation of the free-surface ﬂow past a given surface-piercing object. This was achieved by Asavanant and Vanden-Broeck [7] using complex variables. In the present section we will explore a corresponding approach using the Green’s function formulation. As before we will assume that the water is of inﬁnite depth. We shall present results for a parabolic object deﬁned by the equation ε y = (x2 − 1). 2

(10.17)

In general we might expect a spray or splash at the front of the object (see for example Dias and Vanden-Broeck [47]). Here, we restrict our attention to ﬂows that separate smoothly from the object. Let us denote by sa and sb the values of s at the left- and right-hand separation points. Since we need to ﬁnd sa and sb as part of the solution, we introduce a new variable t by s = sa + (sb − sa )t.

(10.18)

The relation (10.18) maps the unknown interval (sa , sb ) into the ﬁxed ˜ = φ(s), X(t) ˜ interval (0, 1). The new unknown functions are φ(t) = X(s), ˜ Y (t) = Y (s), where s is deﬁned by (10.18). The system of nonlinear equations to be solved is now obtained by substituting (10.18) into (10.9), (10.10) and (10.12). This yields the

284

Three-dimensional free-surface ﬂows

integro-diﬀerential equation ˜ ∗ ) − X(t ˜ ∗ )] 2π[φ(t ∞ ˜ − X(t ˜ ∗ )][−Y˜ (t)] + [Y˜ (t) − Y˜ (t∗ )]X ˜ (t) [X(t) ˜ − X(t)] ˜ 2[φ(t) = ˜ − X(t ˜ ∗ )]2 + [Y˜ (t) − Y˜ (t∗ )]2 [X(t) −∞ ˜ − X(t ˜ ∗ )]2 + [Y˜ (t) − Y˜ (t∗ )]2 Y˜ (t) dt, − ln [X(t) −∞ < t∗ < ∞, (10.19) the Bernoulli equation 2

Y˜ φ˜t 1 1 + 2 = , 2 sb − sa F 2

for

t < 0 or t > 1

(10.20)

and the arc length equation ˜ 2 + Y˜ 2 = (sb − sa )2 . X

(10.21)

In addition the kinematic condition boundary condition on the object gives ε ˜2 Y˜ = (X − 1) for 0 < t < 1. (10.22) 2 At the separation points t = 0 and t = 1 we must satisfy both (10.20) and (10.22), so we have 2

˜ 2 − 1) φ˜t ε(X 1 1 + = at t = 0 or t = 1. (10.23) 2 2 sb − sa 2F 2 For the numerical computation, again we introduce N equally spaced points t1 = −e(N − 1)/2, ti = t1 + e(i − 1), i = 2, . . . N , and use the notation ˜ i ), x = X ˜ i ), yi = Y˜ (ti ), φi = φ(t ˜ (ti ), y = Y˜ (ti ) and φ = φ˜t (ti ). xi = X(t i i i The values of φ˜t at the surface of the object cannot be determined, as in (10.15), by using the Bernoulli equation. At the surface of the object, between t = 0 and t = 1 there are M = 1/e − 1 mesh points (we will choose e such that 1/e is integer, but this is not a necessary condition). At each mesh point there are two unknowns, φi and yi . So we have N + M + 2 , φ unknowns: y1 , . . . , yN (N +1)/2+1 , . . . , φ(N +1)/2+M and sa , sb . The integral equation is evaluated at the midpoints ti−1/2 , i = 2, . . . , N −1, as before, so we obtain N − 2 equations. Another M equations are given by ε 2 x(N +1)/2+j − 1 , j = 1, . . . , M. (10.24) y(N +1)/2+j = 2 The equations at the separation points (10.23) give another two equations and the radiation condition (10.16) provides the ﬁnal two equations.

10.1 Green’s function formulation for two-dimensional problems

285

The values of φ˜ at the separation points, φ(N +1)/2 and φ(N +1)/2+M +1 , are obtained using an extrapolation formula with four points (taken from the object). The usual initial guess is yi = 0, i = 1, . . . , N , sa = −1, sb = 1, φ(N +1)/2+j = sb − sa , j = 1, . . . , M . At the ﬁrst point we impose y1 = 0,

x1 = φ1 = sb − sa ,

x1 = φ1 = sa + (sb − sa )t1 ,

(10.25)

and the remaining functions are calculated as before, using equations (10.20), (10.21) and the trapezoidal rule. Again, the values of the functions at the midpoints are calculated by interpolation with two points. The numerical scheme described above was used to calculate solutions for various values of F and ε. The accuracy of the results was checked by varying N and e. We present typical free surfaces for ε > 0 and ε < 0 in Figures 10.3 and 10.4.

0.008

0.004

0

−0.004

−0.008 −40

−20

0

20

40

Fig. 10.3. Computed free-surface proﬁles with N = 421, e = 0.1 for F = 1.5, ε = 0.004. The parabolic object and the separation points (the two crosses) are also shown.

We observe that if F is held constant and the value of ε is varied then the wavelength of the waves downstream does not change much, but their amplitude is aﬀected. Our calculations cannot be directly compared with those of Asavanant and Vanden-Broeck [7] because their study was for ﬁnite depth. Also, they chose the position of the separation points and obtained the position of the vertex of the obstacle as part of the solution. In our case the position of

286

Three-dimensional free-surface ﬂows 0.015

0.010

0.005

0

−0.005

−0.010

−0.015

−40

−20

0

20

40

Fig. 10.4. Computed free-surface proﬁles for F = 1.5, ε = −0.006 (dotted line), ε = −0.004 (broken line) and ε = −0.001 (solid line).

the vertex is known and we calculate the position of the separation points as part of the solution.

10.2 Extension to three-dimensional free-surface ﬂows The results of Sections 10.1.1 and 10.1.2 show that two-dimensional freesurface ﬂows can be computed accurately by using the Green’s function formulation. In this section we extend this approach to three-dimensional ﬂows. Our method follows Forbes [58] and Parau and Vanden-Broeck [122]. We ﬁrst present explicit results for pure gravity ﬂows generated by moving pressure distributions or submerged disturbances in water of inﬁnite depth. Extensions to ﬂuids with surface tension are discussed in Section 10.2.2.

10.2.1 Gravity ﬂows generated by moving disturbances in water of inﬁnite depth 10.2.1.1 Formulation We consider a three-dimensional pressure distribution moving at constant velocity U at the surface of a ﬂuid of inﬁnite depth. The ﬂow is shown in Figure. 10.5. As in Section 10.1.1, we choose a frame of reference moving with the pressure distribution and assume that the ﬂow is steady. We introduce cartesian coordinates x, y, z with the z-axis directed vertically upwards and the x-axis in the opposite direction to the velocity U . We denote by

10.2 Extension to three-dimensional free-surface ﬂows

287

z

P

x

y U

Fig. 10.5. The ﬂow in the three-dimensional case. The pressure distribution is moving to the left at a constant velocity U .

z = ζ(x, y) the equation of the free surface. The potential function Φ(x, y, z) satisﬁes the Laplace equation, −∞ < z < ζ(x, y), (10.26) in the ﬂow domain. The kinematic boundary condition (10.3), the dynamic boundary condition (10.4) and the radiation condition (10.5) are now ∇2 Φ = 0,

−∞ < x < ∞,

−∞ < y < ∞,

Φx ζx + Φy ζy = Φz

on

U2 1 2 p (Φx + Φ2y + Φ2z ) + gζ + = 2 ρ 2 no waves

as

z = ζ(x, y), on

z = ζ(x, y),

x → −∞.

(10.27) (10.28) (10.29)

Equation (10.7) holds in three dimensions, where V now represents a volume bounded by the surface C. Proceeding as in Section 10.1.1 and using the three-dimensional free surface Green’s function G=

1 1 , ∗ 2 4π [(x − x ) + (y − y ∗ )2 + (z − z ∗ )2 ]1/2

(10.30)

we obtain 1 ∗ ∗ 2 [φ(x , y )

− U x∗ ]

[φ(x, y) − U x]

= R2

ζ(x, y) − ζ(x∗ , y ∗ ) − (x − x∗ )ζx − (y − y ∗ )ζy 1 dxdy 4π {(x − x∗ )2 + (y − y ∗ )2 + [ζ(x, y) − ζ(x∗ , y ∗ )]2 }3/2 1 U ζx + dxdy, ∗ 2 ∗ 2 4π {(x − x ) + (y − y ) + [ζ(x, y) − ζ(x∗ , y ∗ )]2 }1/2

×

R2

(10.31)

288

Three-dimensional free-surface ﬂows

where φ(x, y) = Φ(x, y, ζ(x, y)). The pressure is chosen as  L2 L2  + , |x| < L and |y| < L, P0 exp p(x, y) = (x2 − L2 ) (y 2 − L2 )  0 otherwise. We introduce dimensionless variables by using U as the unit velocity and L as the unit length. Combining equations (10.27) and (10.28) and using the chain rule we obtain ζ 1 1 (1 + ζx2 )φ2y + (1 + ζy2 )φ2x − 2ζx ζy φx φy + 2 + εP = , 2 1 + ζx2 + ζy2 F 2

(10.32) 1

where F = U/(gL)1/2 and ε = P0 /(ρU 2 ). Now P (x, y) is e x 2 −1 |x| < 1 and |y| < 1, and 0 otherwise. Equation (10.31) can be rewritten as 2π[φ(x∗ , y ∗ ) − x∗ ] = I1 + I2 ,

+

1 y 2 −1

for

(10.33)

where ∞ ∞ I1 =

[φ(x, y) − φ(x∗ , y ∗ ) − x + x∗ ]K1 dxdy,

(10.34)

0 −∞

∞ ∞ I2 =

ζx (x, y)K2 dxdy,

(10.35)

0 −∞

K1 =

K2 =

ζ(x, y) − ζ(x∗ , y ∗ ) − (x − x∗ )ζx − (y − y ∗ )ζy {(x − x∗ )2 + (y − y ∗ )2 + [ζ(x, y) − ζ(x∗ , y ∗ )]2 }3/2 ζ(x, y) − ζ(x∗ , y ∗ ) − (x − x∗ )ζx − (y + y ∗ )ζy + {(x − x∗ )2 + (y + y ∗ )2 + [ζ(x, y) − ζ(x∗ , y ∗ )]2 }3/2 1 ∗ 2 ∗ 2 (x − x ) + (y − y ) + [ζ(x, y) − ζ(x∗ , y ∗ )]2 +

1 (x − x∗ )2 + (y + y ∗ )2 + [ζ(x, y) − ζ(x∗ , y ∗ )]2

(10.36)

.

(10.37)

In deriving (10.33) we used the fact that the solutions are symmetric in the y-direction. We note that the integral I2 is singular, whereas I1 is not.

10.2 Extension to three-dimensional free-surface ﬂows

289

10.2.1.2 The numerical scheme We truncate the intervals −∞ < x < ∞ and 0 < y < ∞ to x1 < x < xN and y1 < y < yM and introduce the mesh points xi = x1 +(i−1)∆x, i = 1, . . . , N and yj = (j − 1)∆y, j = 1, . . . , M . Following Forbes [58] and Parau and Vanden-Broeck [122], the integral I2 is written in the form I2 = I2 + I2 , where yM xN [ζx (x, y)K2 − ζx (x∗ , y ∗ )S2 ]dxdy, I2 = y 1 x1

∗

∗

yM xN

I2 = ζx (x , y )

S2 dxdy, y 1 x1

with S2 =

1 A(x −

+

x∗ )2

+ B(x − x∗ )(y − y ∗ ) + C(y − y ∗ )2 1

A(x − x∗ )2 − B(x − x∗ )(y + y ∗ ) + C(y + y ∗ )2

and A = 1 + ζx2 (x∗ , y ∗ ),

B = 2ζx (x∗ , y ∗ )ζy (x∗ , y ∗ ),

C = 1 + ζy2 (x∗ , y ∗ ).

The integral I2 , which contains the singularity, can be calculated using dsdt t √ = √ ln[2As + Bt + 2 A(As2 + Bst + Ct2 )] As2 + Bst + Ct2 A s + √ ln[2Ct + Bs + 2 C(As2 + Bst + Ct2 )]. C The 2N M unknowns are u = (ζx1 1

ζx1 2

···

ζxN , M −1

ζxN M

φx1 1

···

φxN M )T .

The integrals and the Bernoulli equation are evaluated at the points (xi+1/2 , yj ), i = 1, . . . , N − 1, j = 1, . . . , M , so we have 2(N − 1)M equations. Another 2M equations are obtained from the radiation condition ζx1 j = 0, φx1 j = 1, j = 1, . . . , M . The values of ζ and φ are obtained by integrating ζx and φx with respect to x by the trapezoidal rule. The integration is started by using the values derived from the radiation condition (10.29) and the free-surface condition (10.32) satisﬁed at the ﬁrst row: ζ1j = 0,

ζy1 j = 0,

φ1j = x1 ,

φy1 j = 0,

j = 1, . . . , M.

290

Three-dimensional free-surface ﬂows

The values of ζy and φy are then calculated by central diﬀerences. The values of the variables ζ and φ at (xi+1/2 , yj ) are obtained by interpolation. The 2N M nonlinear equations are solved by Newton’s method. In most calculations we chose ζxi j = 0, φxi j = 1 for i = 1, . . . , N , j = 1, . . . , M , as the initial guess. 10.2.1.3 Numerical results We used the scheme described in Section 10.2.1.2 to calculate solutions for diﬀerent values of the Froude number F and of the parameter ε; the results are qualitatively similar. We present a typical free-surface proﬁle for F = 0.7 and ε = 1 (see Figure 10.6).

0 2 4

z

0.2 0 −0.2 −5

6 8 −4

−3

−2

−1

0

10 1

2

y

3

4

5

x

12

Fig. 10.6. The solution for the wave ﬁeld due to a moving pressure distribution for F = 0.7 and ε = 1. The grid parameters are N = 75, M = 25, ∆x = 0.2, ∆y = 0.2. The transverse waves are perpendicular to the direction of the velocity U , which is parallel to the x-axis. In this graph and the following three-dimensional ﬁgures the darker parts correspond to the troughs and the brighter parts to the peaks of the waves.

The wake and the two diﬀerent families of waves (transverse waves and short-length divergent waves) can easily be observed. When F increases, the amplitude of the divergent waves becomes greater than that of the transverse waves (see Figure 10.7). The wavelength of the transverse waves increases with the Froude number (see Figure 10.8). The inﬂuence of the truncation upstream and downstream is seen to be negligible (see Figure 10.9). Here we show the centreline (i.e. the intersection

10.2 Extension to three-dimensional free-surface ﬂows

291

0 5 10 15 20

0.5

z

25

0 −0.5 −15

30 −10

−5

0

5

10

y

x

35 15

Fig. 10.7. These waves were generated for a higher Froude number (F = 1.2) than in Figure 10.6. The grid used was N = 61, M = 19, ∆x = ∆y = 0.6. The divergent waves can be observed more easily than in Figure 10.6; their amplitudes are larger than those of the transverse waves. 5 4 3 2 1

y

0 −1 −2 −3 −4 −5 −2

0

2

4

x

6

8

10

12

Fig. 10.8. The wake in the cases F = 0.7 (lower half) and F = 0.5 (upper half). In both cases ε = 1.

of the free surface with the plane y = 0) and two curves corresponding to diﬀerent truncations, x = (−3, 12) and x = (−6, 6). The accuracy of the solutions was tested by varying the number of grid points and the intervals ∆x and ∆y between grid points (see an example in

292

Three-dimensional free-surface ﬂows 0.20 0.15 0.10 0.05

z 0

0

2

x

4

6

8

10

12

Fig. 10.9. The free-surface elevation at the plane y = 0 for two diﬀerent truncations, x = (−3, 12) (the broken curve) and x = (−6, 6) (the solid curve), is shown. In both cases F = 1 and = 1.

2

z

0 −2 5 4 3 2 1 0 −1 −2

y

−3 −4 −5

−2

0

2

6

4

8

10

12

14

16

x

Fig. 10.10. The accuracy check: F = 0.7, = 0.0001; N = 89, M = 13; ∆x = ∆y = 0.2 (lower half); N = 61, M = 17, ∆x = ∆y = 0.3 (upper half).

Figure 10.10). The upper part of Figure 10.10, y > 0, was calculated for N = 61, M = 17, ∆x = ∆y = 0.3 and the lower part, y < 0, for M = 89, M = 13, ∆x = ∆y = 0.2. The values of the parameters are the same in both cases: F = 0.7, ε = 1. The algorithm can be easily modiﬁed to include two or more pressure distributions and to study the interaction of the wakes produced by each of

10.2 Extension to three-dimensional free-surface ﬂows

293

them. We present an example in Figure 10.11 for two pressure disturbances moving in parallel. 0.05

z

0 −0.05 5

4

3

2

y

1

0

−2

−1

0

1

2

3

4

5

6

x

Fig. 10.11. The case of two pressure distributions moving in parallel (F = 0.4).

The V-shapes of the waves become a W-shape downstream. This case can be viewed as representing the wave interactions between ships moving in parallel in deep water. There are various possible generalisations of our code. For example, it can be used to consider moving submerged objects. An inverse method to compute the resulting waves is by superposing singularities. An example of the waves generated by a source and a sink is given in Figure 10.12.

10.2.2 Three-dimensional gravity–capillary free-surface ﬂows in water of inﬁnite depth In this section we consider steady three-dimensional fully nonlinear gravity– capillary free-surface ﬂows. We show how to extend the results of Section 7.2.2 to three dimensions and start by considering free-surface ﬂows generated by a moving pressure distribution. We ﬁnd that some solutions are perturbations of a uniform stream while others are perturbations of solitary waves. The solitary waves have decaying oscillations in the direction of propagation and monotone decay in the direction perpendicular to the direction of propagation. They travel at a velocity U smaller than the minimum velocity cmin of linear gravity–capillary waves. It is shown that the structure

294

Three-dimensional free-surface ﬂows 0.6 0.4

z

0.2 0

−0.2 4

2

0

−2

y

−4 −4

−2

0

2

x

4

6

8

10

12

14

16

Fig. 10.12. The waves generated by a source–sink pair (F = 0.7). The source is at (0, 0, −1) and the sink at (1, 0, −1).

of the solutions in three dimensions is similar to that found in Section 7.2.2 for two-dimensional waves (see Figure 7.8). 10.2.2.1 Formulation We consider a three-dimensional solitary wave travelling at a constant velocity U at the upper surface of a ﬂuid of inﬁnite depth. The ﬂuid is incompressible and the ﬂow is irrotational. We choose a frame of reference moving with the wave and assume that the ﬂow is steady. We introduce cartesian coordinates x, y, z with the z-axis directed vertically upwards (opposite to the direction of gravity) and the x-axis in the direction of wave propagation. As in Section 10.2.1.1 we denote by z = ζ(x, y) the equation of the free surface and deﬁne the potential function Φ(x, y, z). We introduce dimensionless variables by using U as the unit of velocity and T /(ρU 2 ) as the unit of length. Here T is the surface tension. The velocity potential function Φ(x, y, z) satisﬁes Laplace’s equation ∇2 Φ = 0,

−∞ < x < ∞,

−∞ < y < ∞,

−∞ < z < ζ(x, y), (10.38)

in the ﬂow domain. The kinematic and dynamic boundary conditions can be written as Φx ζx + Φy ζy = Φz

on

z = ζ(x, y),

(10.39)

10.2 Extension to three-dimensional free-surface ﬂows







295



ζx 1 2  −  ζy  =1 (Φx + Φ2y + Φ2z ) + αζ −  2 2 1 + ζ2 + ζ2 1 + ζ2 + ζ2 x

on

y

x

x

z = ζ(x, y);

y

y

(10.40)

we have used the conditions (Φx , Φy , Φz ) → (1, 0, 0) and ζ → 0 as (x2 + y 2 )1/2 → ∞

(10.41)

to ﬁx the value of Bernoulli’s constant in (10.40). The parameter α in (10.40) is deﬁned by gT . α= ρU 4 We solve the problem numerically by a boundary integral equation method similar to that used in Section 10.2.1.2. Details can be found in [123] and [124]. One diﬀerent feature for the numerical scheme is that the surface tension term introduces higher derivatives in (10.40). They are approximated by centred-diﬀerence formulae. Another new feature is that no radiation condition is needed. Instead the solutions are assumed to be symmetric about the x- and y-axes. The discretisation involves a regular grid with N points in the x-direction and M points in the y-direction. The uniform mesh sizes on the x- and y-axes are denoted by ∆x and ∆y. The algebraic equations obtained after discretization are solved by Newton’s method. A suitable initial guess to compute solitary waves is obtained by adapting the method of Section 7.2.2 from two dimensions to three dimensions. This consists of ﬁrst obtaining solutions for the problem with an extra pressure term in equation (10.40) and then taking the limit as the magnitude of the pressure tends to zero. 10.2.2.2 Results We computed solutions for various values of α by using the numerical scheme described in the previous section. In all cases α is assumed to be greater than 1/4, which corresponds to waves moving steadily with a constant velocity U smaller than the minimum phase speed cmin deﬁned by (2.98). In this case only a highly localised disturbance of the water surface is predicted. In two dimensions, capillary–gravity waves were calculated in the same regime of parameters (α > 1/4). Most computations were performed with ∆x = ∆y = 0.8 and N = 40, M = 50. The accuracy of the solutions was tested by varying the number of grid points and the intervals ∆x and ∆y between grid points. To indicate the numerical consistency, we found that, for α = 0.35, if we changed ∆x and ∆y from 0.8 to 0.6 the corresponding change in the

296

Three-dimensional free-surface ﬂows 0.5

z

0

−0.5

−1.0

−1.5 40 20

y

0 −20 −40

−30

−20

−10

0

10

20

30

x

Fig. 10.13. Solitary gravity–capillary wave with central depression for α = 0.35. The surface elevation is vertically exaggerated by a factor 20.

wave surface elevation was less than 5% everywhere. The smallness of ∆x and ∆y is limited by memory capacity in order to obtain a truncated domain large enough to encompass nearly all the wave disturbance. We present typical free-surface proﬁles in Figures 10.13–10.15 and 10.17. As expected, there is a localised disturbance of the water surface. For each α there is a central-depression wave (for such a wave ζ(0, 0) < 0; see Figure 10.13 for the full solution and Figure 10.14 for half the solution) and a central-elevation wave (for this wave ζ(0, 0) > 0; see Figure 10.15). The waves have decaying oscillations in the direction of propagation and monotonic decay in the direction perpendicular to the direction of propagation. Figure 10.16 shows the zx and zy cross-sections and a close-up with the same horizontal and vertical scale for α = 0.35. As α decreases and approaches 1/4, more and more oscillations appear in front and behind the main disturbance. The amplitude of the free capillary–gravity waves decreases to zero as α decreases to 1/4. Their form (see Figure 10.17) suggests that they approach a train of two-dimensional (constant in the y-direction) periodic waves in the limit as α decreases to 1/4. When α is increased, the solitary capillary– gravity wave elevation decreases quickly in every direction and the surface has the form of a central-depression or central-elevation three-dimensional fully localised solitary wave. The solitary-wave branches are shown in Figure 10.18. These branches bifurcate from the uniform stream at α = 1/4.

10.2 Extension to three-dimensional free-surface ﬂows

297

0.5

z

0

−0.5

−1.0

40 30

−1.5 −30

20 −20

10

−10

0

10

20

x

y

0

30

Fig. 10.14. Central-depression solitary gravity–capillary wave for α = 0.35, showing half the solution. The surface elevation is vertically exaggerated by a factor 20.

0.5

z

0

−0.5

40

−1.0 30 −1.5 −30

20 −20

−10

10 0

x

10

20

30

0

y

Fig. 10.15. Central-elevation solitary gravity–capillary wave for α = 0.35. Only half the solution is shown. The vertical exaggeration is by a factor 20.

For the two-dimensional problem, the solutions are described in Section 7.2.2. The numerical results presented in this section are consistent with the asymptotic ﬁndings of Kim and Akylas [87], [88] and of Milewski [115] and with the rigorous results of Groves and Sun [68]. The graphs presented in this section are based on [123]. We have recently repeated the calculations with a ﬁner grid and found results which are qualitatively similar: there are still two branches bifurcating from α = 0.25 in Figure 10.18. However, the shape of the curves is slightly diﬀerent. These reﬁned computations will be reported elsewhere.

298

Three-dimensional free-surface ﬂows (a) 0.5

z

0

−0.5

−1.0

−1.5 −30

−20

−10

0

10

20

30

(b)

8

4

z

0

−4

−8

−8

−4

0

4

8

Fig. 10.16. (a) The centreline in the Ox direction (solid line) and in the Oy direction (broken line) for a central depression wave and α = 0.35. The vertical exaggeration is by a factor 20. (b) A close-up with the same scaling in the horizontal and vertical directions.

10.3 Further extensions An extension of the numerical method of Section 10.2.1.2 to ﬁnite depth was developed in [124]. The basic idea is to use the formulation of Section 10.2.1.2 but with the free space Green’s function replaced by a Green’s function that satisﬁes the kinematic boundary condition on the bottom. Three-dimensional gravity–capillary solitary waves similar to those in water of inﬁnite depth were obtained.

10.3 Further extensions

299

0.5

0

−0.5

−1.0

40 20

−1.5 −30

−20

−10

0

10

20

30

0

Fig. 10.17. Solitary capillary–gravity waves for α = 0.266. Only half the solution (y ≥ 0) is shown. The vertical exaggeration is by a factor 20. 0.4 0.2 0 −0.2

z(0,0) −0.4 −0.6 −0.8 −1.0 −1.2 0.24

0.26

0.28

0.30

0.32

0.34

a

Fig. 10.18. Values of the amplitude ζ(0, 0) versus α.

The three-dimensional solitary waves of Section 10.2.2 were calculated by considering free-surface ﬂows generated by a moving pressure distribution when α > 0.25. In [125], the properties of three-dimensional gravity– capillary free-surface ﬂows generated by a moving pressure distribution when α < 0.25 were studied. There are then two wave trains on the free surface. The radiation condition forces waves of longer wavelength to occur at the back of the disturbance and waves of shorter wavelength to occur at the front of the disturbance. This radiation condition cannot be as easily imposed as in the pure gravity case, where we simply required the free surface to be ﬂat at some distance in front of the disturbance (see Section 10.2.1).

300

Three-dimensional free-surface ﬂows

It was shown in [125] that an eﬃcient way to impose the radiation condition when α < 0.25 is to introduce some dissipation in the dynamic boundary condition in the form of a Rayleigh viscosity. The Rayleigh viscosity was introduced in Chapter 4 as a way of satisfying the radiation condition for linearised problems. The results in [125] showed that it can also be used to compute nonlinear solutions. Another extension is to consider three-dimensional waves propagating at the interface of two ﬂuids of constant density (see [126]). Finally, let us mention that the Green’s function formulation employed in this chapter can be used to compute axisymmetric free-surface ﬂows. The idea is to use cylindrical coordinates r, ϕ, z with the z-axis along the axis of symmetry and to integrate the Green’s function over ϕ (see [113], [114], [60], [187] and [73] for examples of such computations).

11 Time-dependent free-surface ﬂows

11.1 Introduction The ﬁrst ten chapters of this book were devoted to steady free-surface ﬂows. An equally important topic is that of time-dependent free-surface ﬂows. Boundary integral equation methods can still be used to investigate these problems. The idea is to ‘march in time’ and to solve at each time step a linear integral equation similar to those derived in the previous chapters, by using Cauchy integral equation formula or Green’s theorem. Such methods have been developed and used by many authors (see for example [104], [9], [35], [136] and the references cited in these papers). In particular, results have been obtained for breaking waves. An obvious use of time-dependent codes is to study the stability of steady solutions. In this chapter we will conﬁne our attention to one type of time-dependent free-surface ﬂow, namely gravity–capillary standing waves. We will solve the problem by a series expansion similar to that used in Section 5.1 to study periodic travelling waves. The analysis follows Vanden-Broeck [167] closely. The choice of this problem is motivated by the fact that gravity–capillary standing waves have properties similar to those of Wilton ripples (see Section 6.5.3.1). We note that a proof of the existence of nonlinear gravity standing waves was provided only recently [81].

11.2 Nonlinear gravity–capillary standing waves The concept of linear standing waves was introduced in Section 2.4.3. Here we extend the theory of standing waves to the nonlinear regime. More precisely we consider the time-periodic two-dimensional ﬂow of a ﬂuid bounded below by a horizontal bottom and above by a free surface. We assume the 301

302

Time-dependent free-surface ﬂows

motion to be periodic in the horizontal direction with wavelength λ and measure lengths in units of k −1 = λ/(2π). Following Concus [33], we deﬁne the parameters γ and δ by the relations γ=

T k2 , ρg

(11.1)

δ=

γ . 1+γ

(11.2)

Here T is the surface tension, ρ is the density and g is the acceleration of gravity. For δ 1, the capillary eﬀects are small, whereas for |1 − δ| 1 they predominate. We deﬁne cartesian coordinates such that the motion is symmetric about the vertical axis, x = 0, and such that y = 0 corresponds to the mean level. Let k−1 h denote the mean depth, [kg(1 + γ)]1/2 ω the angular frequency, [kg(1 + γ)]−1/2 ω −1 t the time and a the amplitude of the linearised surfacewave motion. Then we deﬁne = ak and let k −1 η(x, t) be the elevation of the free surface above the mean level and [g(1 + γ)]1/2 k −3/2 φ the velocity potential. In dimensionless variables, the nonlinear problem is described by the equations φxx + φyy = 0 (1 − δ)η − δ

in 0 < x < π,

−h < y < η(x, t),

ηxx + ωφt + (φ2x + φ2y ) = 0 3/2 2 2 2 (1 + ηx ) φy = ωηt + φx ηx

ηx = 0

on y = η(x, t), (11.4)

on y = η(x, t),

∂φ = 0 on x = 0 and x = π, ∂n

(11.3)

y = −h,

on x = 0 and x = π, π η(x, t)dx = 0,

(11.5) (11.6) (11.7) (11.8)

0

∇φ(x, y, t + 2π) = ∇φ(x, y, t),

0

π

2π

φ(x, y, t) sin t cos x dtdxdy = 0, −h

0

−h

0

π

0

0

(11.9) (11.10)

0

2π

φ(x, y, t) cos t cos x dtdxdy = 12 π 2 (tanh h)1/2 .

(11.11)

11.2 Nonlinear gravity–capillary standing waves

303

As noted by Tadjbakhsh and Keller [148] and Concus [33], a unique solution does not exist for values of h for which the frequency of the nth spatial harmonic, {n[1 + δ(n2 − 1)] tanh nh}1/2 , is an integral multiple of the fundamental frequency (tanh h)1/2 . This yields the uniqueness condition n[1 + δ(n2 − 1)] tanh nh = j 2 , n = 2, 3, . . . , j = 1, 2, . . . (11.12) tanh h Concus [34] showed that the values of h for which the uniqueness condition (11.12) is not satisﬁed form a denumerably inﬁnite set that is densely distributed over the entire positive real line. Following Tadjbakhsh and Keller [148], Concus [33] sought a solution as an expansion in powers of . Thus η = η 0 (x, t) + 2 η 1 (x, t) + 12 3 η 2 (x, t) + O(4 ),

(11.13)

φ = φ0 (x, y, t) + 2 φ1 (x, y, t) + 12 3 φ2 (x, y, t) + O(4 ),

(11.14)

ω = ω0 + ω1 + 12 2 ω2 + O(3 ).

(11.15)

Substituting (11.13)–(11.15) into (11.3)–(11.11) and equating powers of gives a succession of linear systems of equations, whose solutions are: η 0 = sin t cos x, φ0 =

ω0 cos t cos x cosh(y + h), sinh h ω02 = tanh h;

1 η = 8 1

ω0−2 − 3ω0−6 ω02 + ω0−2 + cos 2t cos 2x, 1 + 3δ 1 − 3δω0−4

(11.16) (11.17) (11.18) (11.19)

1 1 φ1 = β0 + (ω0 − ω0−3 )t − (3ω0 + ω0−3 ) sin 2t 8 16 3[ω0 − 2δω0−3 − (1 + 2δ)ω0−7 ] − sin 2t cos 2x cosh(2y + h), 16(1 − 3δω0−4 ) cosh 2h (11.20) ω1 = 0.

(11.21)

Higher-order terms can be found in Concus [33] for δ = 0 and in [148] for δ = 0. We note that η1 is unbounded when 1 − 3δω0−4 = 0.

(11.22)

304

Time-dependent free-surface ﬂows

This critical value corresponds to n = j = 2 in (11.12). We now derive a perturbation solution valid when (11.22) holds. We substitute the expansions (11.13), (11.14) into the system (11.3)–(11.11) and collect all terms of like power in . The terms of order in (11.4) and (11.5) give 0 + ω0 φ0t = 0, (1 − δ)η 0 − δηxx

φ0y − ω0 ηt0 = 0

on y = 0,

on y = 0.

(11.23) (11.24)

Equations (11.3) and (11.6)–(11.11) remain unchanged in form as equations for η 0 and φ0 . The terms of order 2 in (11.4), (11.5) and (11.11) give 1 (1 − δ)η 1 − δηxx + ω0 φ1t = F0

φ1y − ω0 ηt1 = G0

0

−h

0

π

2π

on

on y = 0, y = 0,

φ1 cos t cos x dtdxdy = 0.

(11.25) (11.26) (11.27)

0

Here F 0 and G0 are deﬁned by F 0 = − 12 [(φ0x )2 + (φ0y )2 ] − ω0 η 0 φ01y − ω1 φ0t ,

(11.28)

G0 = ηx0 φ0x − η 0 φ0yy + ω1 ηt0 .

(11.29)

Equations (11.3) and (11.6)–(11.10) remain of the same form as the equations in η 1 , φ1 and ω1 . The solution of the zeroth-order problem deﬁned by (11.3), (11.23), (11.24), (11.6)–(11.11) and (11.22) is η 0 = sin t cos x + A cos 2t cos 2x, φ0 =

(11.30)

Aω0 ω0 cos t cos x cosh(y + h) − sin 2t cos 2x cosh 2(y + h), sinh h sinh 2h (11.31) ω02 = tanh h.

(11.32)

Here A is an arbitrary constant. Thus the solution of the zeroth-order problem is not unique when (11.22) is satisﬁed. Diﬀerentiating (11.25) with respect to t, substituting ηt1 from (11.26) and 1 ηxxt from (11.26) after diﬀerentiating twice with respect to x, we obtain − δφ1yxx + (1 − δ)φ1y + ω02 φ1t = H0

on y = 0.

(11.33)

11.2 Nonlinear gravity–capillary standing waves

305

Here H0 is deﬁned by H0 = ω0 Ft0 + (1 − δ)G0 − δG0xx .

(11.34)

Separation of variables yields, for the solution of (11.3) subject to (11.6), 1

φ (x, y, t) =

∞

Am (t) cos mx cosh m(y + h).

(11.35)

m=0

Substituting (11.35) into (11.33) we obtain ω02 cosh mhAm (t) + [(1 − δ)m + δm3 ] sinh mh Am (t) =

1 µπ

2π

H0 cos mx dx.

(11.36)

0

Here µ = 1 for m > 0 and µ = 2 for m = 0. Using (11.28)–(11.32), we can rewrite (11.36) in the form ω02 Am (t) = 14 (3ω02 + ω0−1 ) sin 2t − 2A2 (ω02 coth2 2h + 3ω02 ) sin 4t,

(11.37)

ω02 cosh h A1 (t) + sinh h A1 = [2ω1 + 14 A(ω0−1 − 3ω03 + 4ω0 coth 2h)] cos t + 14 A(4ω0 coth 2h + ω0−1 + 21ω03 ) cos 3t,

(11.38) ω02 cosh 2h A2 (t) + 2(1 + 3δ) sinh 2h A2 (t) = { 34 [ω02 − (1 + 2δ)ω0−1 ] − 2Aω1 − 6Aω1 δ − 4Aω1 ω02 coth 2h} sin 2t, (11.39) ω02 cosh 3h A3 (t) + (3 + 5δ) sinh 3h A3 (t) = 14 Aω0 cos t[(4 + 48δ) coth 2h − (3 + 24δ)ω0−2 − 3ω02 ],

− 14 Aω0 cos 3t[(12 + 48δ) coth 2h + (3 + 24δ)ω0−2 − 21ω02 ],

(11.40)

ω02 cosh 4h A4 (t) + (4 + 60δ) sinh 4h A4 (t) = A2 ω0 sin 4t[(2 + 30δ) coth 2h + 2ω0 coth2 2h − 6ω02 ], ω02 cosh mh Am (t) + [(1 − δ)m + δm3 ] sinh mh Am (t) = 0,

(11.41) m = 5, 6, . . . (11.42)

From (11.9) and (11.35) it follows that Am (t) must be periodic in t with period 2π for m ≥ 1 and from (11.22) and (11.42) that Am = 0 for m ≥ 5.

306

Time-dependent free-surface ﬂows

The periodicity of A1 requires the coeﬃcient of cos t in (11.38) to be equal to zero. Thus ω1 = 16 A(3ω03 − ω0−1 − 4ω0 coth 2h).

(11.43)

If we set A = 0 in (11.37)–(11.43), the solution of (11.39) is A2 = −

3[ω0 − 2δω0−3 − (1 + 2δ)ω0−7 ] sin 2t, 16(1 − 3δω0−4 ) cosh 2h

(11.44)

in agreement with (11.20). As mentioned earlier this solution is unbounded when (11.22) is satisﬁed. We shall determine the constant A in such a way that the solution of (11.39) is bounded. The appropriate compatibility condition is obtained by multiplying (11.39) by sin 2t, integrating with respect to t from 0 to 2π, applying integration by parts twice to the term containing A2 (t) and using (11.22). Thus we ﬁnd that the coeﬃcient of sin 2t on the right-hand side of (11.39) must be equal to zero. This yields the relation Aω1 =

3[ω03 − (1 + 2δ)ω0−1 ] . 8 + 24δ + 16ω02 coth 2h

(11.45)

Substituting (11.43) into (11.45) we obtain A=±

3[ω03 − (1 + 2δ)ω0−1 ] (1 − 3δ + 2ω02 coth 2h)(3ω03 − ω0−1 − 4ω0 coth 2h)

1/2 .

(11.46)

The remaining part of the calculation follows closely the work of Tadjbaksh and Keller [148] and Concus [33]. Integrating (11.37)–(11.41), we obtain A0 = −

1 1 (3ω0 + ω0−3 ) sin 2t + A2 (ω0 coth2 2h + 3ω0 ) sin 4t + α0 t + β0 , 16 8 (11.47) A1 =

−A(4ω0 coth 2h + ω0−1 + 21ω03 ) cos 3t, 32 sinh h A2 = α2 sin 2t,

A3 =

(11.48) (11.49)

Aω0 cos t[(4 + 48δ) coth 2h − (3 + 24δ)ω0−2 − 3ω02 ] (12 + 20δ) sinh 3h − ω02 cosh 3h −

Aω0 cos 3t[(12 + 48δ) coth 2h + (3 + 24δ)ω0−2 − 21ω02 ] , (12 + 20δ) sinh 3h − 36ω02 cosh 3h

(11.50)

11.2 Nonlinear gravity–capillary standing waves

A4 =

A2 ω0 sin 4t[(2 + 30δ) coth 2h + 2ω0 coth2 2h − 6ω02 ] . (4 + 60δ) sinh 4h − 16ω02 cosh 4h

307

(11.51)

Here α0 , β0 and α2 are constants to be determined. Substituting (11.35) into (11.25) we obtain (1 − δ)η − 1

1 δηxx

= F0 − ω0

4

Am (t) cos mx cosh mh,

(11.52)

m=0

where F0 and Am (t) are deﬁned by (11.28) and (11.47)–(11.51). The function η1 is therefore deﬁned as the solution of (11.52) subject to (11.7). The constant α0 in (11.47) is evaluated by integrating (11.52) with respect to x between 0 and π and using (11.7) and (11.8). Thus we ﬁnd α0 = 18 ω0 − 18 ω0−3 + 12 A2 ω0 (1 − coth2 2h).

(11.53)

This completes the determination of the ﬁrst-order solution. It contains an arbitrary constant α2 . This constant may be determined at second order in the way in which A was determined at ﬁrst order. The way in which the calculations proceed should be clear to the reader. Equation (11.46) shows that there that there are two solutions when (11.22) is satisﬁed. These solutions are very similar to the Wilton ripples of Section 6.5.3.1. Vanden-Broeck [167] showed that the two solutions corresponding to (11.46) are members of two diﬀerent families. In conclusion, we have illustrated in this chapter that gravity–capillary standing waves have properties qualitatively similar to those of the gravity– capillary travelling waves considered in Section 6.5.3.1. In particular, there are many diﬀerent families of solutions with dimples on their proﬁles.

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Index

Bernoulli constant, 11, 73 Bernoulii equation, 8, 94 Bond number, 20, 158 boundary integral, 53 boundary integral equation, 150, 157 bow ﬂow, 226, 236 Brillouin–Villat condition, 3, 69, 71

Jacobian matrix, 50 Joukowskii models, 110 Joukowskii transformation, 197 kinematic boundary condition, 9 Korteweg–de Vries equation, 156 ﬁfth-order, 147

capillary wave, pure, 21 Cauchy’s integral formula, 14 Cauchy’s theorem, 14 cavitation number, 53 cnoidal wave, 146, 190 conformal mapping, 13 contraction ratio, 41, 91, 227 Crapper’s solution, 52, 160, 242

linear standing wave, 24 logarithmic hodograph, 34 mean curvature of ﬂuid surface, 11 Newton’s method, 49 phase velocity, 19, 26, 27 potential ﬂows, 8 pure capillary wave, 21 pure gravity wave, 21

dispersion relation, linear, 20 dynamic boundary condition, 11 Euler’s equations, 7 ﬂow bow, 32 stern, 32 subcritical, 22 supercritical, 22 free surface, 1, 8, 14 Froude number, 5, 20, 82, 91, 92, 95, 158, 229, 279 Garabedian energy argument, 93 gravity wave, pure, 21 group velocity, 27

radiation condition, 30, 117, 225 Rayleigh viscosity, 300 series truncation, 40, 47, 70, 150 shallow water equations, 144 solitary wave, generalised, 147 stern ﬂow, 225 subcritical ﬂow, 22 supercritical ﬂow, 22 terminant, 223

Hele Shaw cell, 103 Hilbert transform, 54 intersection of ﬂows and walls, 31

wave speed, 268 Weber number, 99, 201 Wilton ripple, 140, 206

318