Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems

Abstract

Numerical solutions for three-dimensional gravity capillary waves in water of finite depth are presented. The full Euler equations are used and the waves are calculated by a boundary integral equation method. The findings generalize previous results of Parau, Vanden-Broeck, and Cooker [J. Fluid Mech. 536, 99 (2005)] in water of infinite depth. It is found that there are both lumps that bifurcate from linear sinusoidal waves and other fully localized solitary waves which exist for large values of the Bond number. These findings are consistent with rigorous analytical results and asymptotic calculations. The relation between the solitary waves and free surface flows generated by moving disturbances is also explored.

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Three-dimensional gravity-capillary solitary waves in water of
finite depth and related problems
E.I. P˘ ar˘ au, J.-M. Vanden-Broeck,∗and M.J. Cooker
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
(Dated: 11 July 2005)
Abstract
Numerical solutions for three-dimensional gravity capillary waves in water of finite depth are
presented. The full Euler equations are used and the waves are calculated by a boundary integral
equation method. The findings generalise previous results of E. I. Parau, J.-M. Vanden-Broeck
and M.J. Cooker [J. Fluid Mech. 536, 99 (2005)] in water of infinite depth. It is found that
there are both lumps which bifurcate from linear sinusoidal waves and other fully localised solitary
waves which exist for large values of the Bond number. These findings are consistent with rigorous
analytical results and asymptotic calculations. The relation between the solitary waves and free
surface flows generated by moving disturbances is also explored.
PACS numbers: 47.20.Ky; 47.35.+i
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I. INTRODUCTION
Two-dimensional gravity solitary waves have been investigated analytically and numer-
ically since the pioneering work of Korteweg and de Vries.1The study of two-dimensional
gravity capillary solitary waves is more recent although Korteweg and de Vries already
included surface tension in the derivation of their classical equation.
It was found both numerically and analytically that two-dimensional gravity solitary
waves develop in the far field an oscillatory tail of constant amplitude as soon as a small
amount of surface tension is introduced (see Hunter and Vanden-Broeck2, Champneys et al3
for numerical calculations and Dias and Iooss4for a review of analytical work). These waves
are often referred to as generalised solitary waves to contrast them from true solitary waves
which are flat in the far field.
Other studies have shown that there are also two-dimensional gravity capillary solitary
waves with decaying oscillatory tails in the far field. These waves exist both in water of
finite depth and in water of infinite depth.5–7
In a previous paper8, we computed three-dimensional gravity capillary solitary waves in
water of infinite depth for the full Euler equations. These waves are left-right and fore-aft
symmetric, have decaying oscillatory tails in the direction of propagation but have monotonic
decay in the direction perpendicular to that of propagation. They can be viewed as the
three-dimensional equivalent of the the two-dimensional gravity capillary solitary waves
with decaying oscillatory tails described in the previous paragraph.
Here we consider three-dimensional gravity-capillary solitary waves in water of finite
depth. These waves have been considered recently by Kim and Akylas9and Milewski10
who derived weakly nonlinear models, and by Groves and Sun (“Fully localised solitary-
wave solutions of the three-dimensional gravity-capillary water-wave problem”, preprint)
who proved rigorously their existence. We show that there are two types of solutions. One
type is similar to those obtained in water of infinite depth in the sense that the waves
approach those in infinite depth in the limit as the depth tends to infinity. The other type
has profiles with monotonic decay in all directions and exists only for sufficiently large values
of the Bond number.
Introductory analytical results based on the dispersion relation are discussed in Section II.
The boundary integral equation method is described in Section III and results are presented
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in Section IV. Finally we show in Section V that some free surface flows generated by moving
disturbances can be viewed as perturbations of three-dimensional solitary waves.
II. DISPERSION RELATION
The classical problem of three-dimensional water waves in finite depth when both gravity
and surface tension are present is considered. Some insight into the problem can be gained
from the dispersion relation for linearized capillary-gravity waves travelling at a constant
velocity c in water of finite depth h. It can be written as (see for instance Lamb,11Wehausen
and Laitone12)
D(κ;λ,β) ≡ (λ + βκ2)tanhκ − κ = 0(1)
where κ = k∗h is the dimensionless wavenumber. This equation involves two dimensionless
numbers
λ =gh
c2
and β =
T
ρhc2,
where T is the constant coefficient of surface tension, g is the acceleration due to gravity, ρ
is the fluid density. The parameter λ is the inverse of the square of the Froude number and
both parameters are related to the Bond number B = T/ρgh2by the relation B = β/λ.
Kim and Akylas9have shown that three-dimensional solitary waves (which they call
lumps) can bifurcate from linear sinusoidal waves with wavenumber corresponding to the
minimum of the phase speed which is also a double root of the dispersion relation (1). This
minimum corresponds, in the plane (β,λ), to a a curve Γ, given in parametric form (see
Dias and Iooss6) by
β =
1
2κsinh2κ(sinhκcoshκ − κ),
κ
2sinh2κ(sinhκcoshκ + κ).
(2)
λ =(3)
In the limit as κ approaches zero, β ∼ 1/3 − 2κ2/45, λ ∼ 1 + κ4/45 and as κ approaches
infinity, β ∼ 1/(2κ), λ ∼ κ/2. The curve is shown in Fig. 1. Lumps are predicted to exist
on the region above the curve Γ. It is worth noting that two-dimensional gravity-capillary
waves are shown to exist in the same region of parameters.13Results similar to those of Kim
and Akylas9were obtained by Milewski10for a weakly nonlinear model equation.
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Groves and Sun have shown that fully localised solitary waves also exist for another set
of parameters
β > 1/3,0 < λ − 1 ? 1.
In this region of strong surface tension Kadomtsev and Petviashvili14derived the well-known
KP-I equation as a long-wave approximation for solutions of the steady gravity-capillary
water wave problem which has fully localized solitary-wave solutions.
The results of Kim and Akylas9and Milewski10are restricted to the weakly nonlinear
regime. Here we present fully nonlinear computations of the waves described in the two
previous paragraphs.
III.FORMULATION AND NUMERICAL SCHEME
The fluid is assumed to be incompressible and inviscid, and the flow to be irrotational.
We are interested in steady waves which travel at a constant velocity c in water of finite
depth h, and we choose a frame of reference moving with the wave, so 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 direction of wave propagation. We denote by z = ζ(x,y) the equation
of the free surface. Dimensionless variables are introduced by taking the unit length to be
T/ρc2and the unit velocity to be c. In terms of the velocity potential function Φ(x,y,z),
the problem is formulated as follows:
∇2Φ = 0,x,y ∈ R,−1
β< z < ζ(x,y), (4)
with the boundary conditions
Φxζx+ Φyζy= Φz,on z = ζ(x,y),(5)
1
2(Φ2
x+ Φ2
y+ Φ2
z) + λβζ−


−


ζx
?
1 + ζ2
x+ ζ2
y


x
−


ζy
?
1 + ζ2
x+ ζ2
y
y
=1
2,on z = ζ(x,y), (6)
and
Φz= 0, on z = −1
β.
(7)
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Equations (5) and (7) are the kinematic boundary conditions on the free surface and
on the bottom. Equation (6) is the dynamic condition with the effect of surface tension
included.
Since we are looking for fully localised three-dimensional solitary waves, we impose the
conditions
(Φx,Φy,Φz) → (1,0,0), ζ → 0, as(x2+ y2)1/2→ ∞.(8)
to fix the value of Bernoulli’s constant in (6).
The numerical scheme is an extension to finite depth of the scheme used by P˘ ar˘ au,
Vanden-Broeck and Cooker8for the computation of the fully localised gravity-capillary waves
in deep water and it is based on a boundary integral equation method introduced by Forbes15
for three dimensional gravity free surface flows due to a source. Only the main points of
the formulation and of the numerical procedure are presented here. The reader is referred
to P˘ ar˘ au and Vanden-Broeck16for details.
The formulation involves applying Green’s second identity for the functions Φ−x and G
where G is the three dimensional free space Green function
G =
1
4π
1
((x − x∗)2+ (y − y∗)2+ (z − z∗)2)1/2,(9)
for a volume V which consists of a cylinder bounded by the free surface (except a small
hemisphere around the point P(x∗,y∗,z∗)), and its image SF? on the other side of the
bottom z = −1/β (see Fig. 2). In that way, by symmetry, the condition of no flow normal
to the bottom (7) is satisfied. This technique was applied for the problem of a withdrawal
through a point sink by Hocking, Vanden-Broeck and Forbes17and others.
After some manipulation of the surface integrals and after projecting them onto the Oxy
plane, we obtain
1
2(φ(x∗,y∗) − x∗) =
ζ(x,y) − ζ(x∗,y∗) − (x − x∗)ζx(x,y) − (y − y∗)ζy(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2)3/2
ζx(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2)1/2dxdy+
ζ(x,y) + ζ(x∗,y∗) + 2/β − (x − x∗)ζx(x,y) − (y − y∗)ζy(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2)3/2
ζx(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2)1/2dxdy
=
? ?
R2(φ(x,y) − x)1
4π
dxdy+
+
? ?
R2
1
4π
+
? ?
R2(φ(x,y)−x)1
4π
dxdy+
+
? ?
R2
1
4π
(10)
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where φ(x,y) = Φ(x,y,ζ(x,y)).
We look for solutions which are symmetric in x and y. Therefore we assume
ζ(x,y) = ζ(−x,y) = ζ(x,−y) = ζ(−x,−y),
φ(x,y) = −φ(−x,y) = φ(x,−y) = −φ(−x,−y).
Following Forbes15, P˘ ar˘ au and Vanden-Broeck16and P˘ ar˘ au et al.8we rewrite equation (10)
as
2π(φ(x∗,y∗) − x∗) = I1+ I2+ I3+ I4
(11)
where
I1=
∞
?
0
∞
?
0
[(φ(x,y) − φ(x∗,y∗) − x + x∗)K1a(x,y,x∗,y∗)+
+(−φ(x,y) − φ(x∗,y∗) + x + x∗)K1b(x,y,x∗,y∗)]dxdy,
∞
?
0
0
(12)
I2=
∞
?
(ζx(x,y)K2a(x,y,x∗,y∗) − ζx(x,y)K2b(x,y,x∗,y∗))dxdy,(13)
I3=
∞
?
0
∞
?
0
[(φ(x,y) − x)K3a(x,y,x∗,y∗)+
+(−φ(x,y) + x)K3b(x,y,x∗,y∗)]dxdy,(14)
I4=
∞
?
0
∞
?
0
(ζx(x,y)K4a(x,y,x∗,y∗) − ζx(x,y)K4b(x,y,x∗,y∗))dxdy, (15)
K1a(x,y,x∗,y∗) =ζ(x,y) − ζ(x∗,y∗) − (x − x∗)ζx(x,y) − (y − y∗)ζy(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2)3/2
+ζ(x,y) − ζ(x∗,y∗) − (x − x∗)ζx(x,y) − (y + y∗)ζy(x,y)
((x − x∗)2+ (y + y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2)3/2
K2a(x,y,x∗,y∗) =
?
+
?
K3a(x,y,x∗,y∗) =ζ(x,y) + ζ(x∗,y∗) + 2/β − (x − x∗)ζx(x,y) − (y − y∗)ζy(x,y)
((x − x∗)2+ (y − y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2)3/2
+ζ(x,y) + ζ(x∗,y∗) + 2/β − (x − x∗)ζx(x,y) − (y + y∗)ζy(x,y)
((x − x∗)2+ (y + y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2)3/2
K4a(x,y,x∗,y∗) =
?
6
+
,
1
(x − x∗)2+ (y − y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2+
1
(x − x∗)2+ (y + y∗)2+ (ζ(x,y) − ζ(x∗,y∗))2,(16)
+
,
1
(x − x∗)2+ (y − y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2+

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+
1
?
(x − x∗)2+ (y + y∗)2+ (ζ(x,y) + ζ(x∗,y∗) + 2/β)2,
and Kib(x,y,x∗,y∗) = Kia(−x,y,x∗,y∗), i = 1,..,4.
We note that the integrand in I2is singular whereas those in I1, I3and I4are not.
We truncate the intervals 0 < x < ∞ and 0 < y < ∞ to x1< x < xN, and y1< y < yM
and introduce the mesh points xi= (i−1)∆x, i = 1,...,N and yj= (j−1)∆y, j = 1,...,M.
The integral I2 (which contains the singularity) can be calculated using some indefinite
integral (see Forbes15). The 2NM unknowns are
u = (ζx11,ζx12,...,ζxN,M−1,ζxNM,φ11,...,φNM)T,
where we use the notation ζxij= ζx(xi,yj), φij= φ(xi,yj), ζij= ζ(xi,yj) etc.
The integrals and the Bernoulli equation are evaluated at the points (xi+1/2,yj), i =
1,...,N − 2, j = 1,...,M so we have 2(N − 2)M equations. Another 2M equations are
obtained from the truncation conditions ζxNj= 0,φxNj= 1, j = 1,...,M and another 2M
equations are given by the symmetry conditions ζx1j= 0 and by φ1j= 0. The values of ζ
are obtained by integrating ζxwith respect to x by the trapezoidal rule. The values of φx,
ζy and φy are then calculated by central differences. The values of the variables ζ and φ
at (xi+1/2,yj) were obtained by interpolation and the values of the other derivatives which
appears in the problem were computed by finite differences.
The 2NM nonlinear equations are solved by Newton’s method. Most of the computations
were performed with ∆x = ∆y = 0.8 and N = 40,M = 50. The accuracy of the solutions
have been tested by varying the number of grid points and the intervals ∆x and ∆y between
grid points.
The formulation in infinite depth was given in P˘ ar˘ au, Vanden-Broeck and Cooker8and
was obtained by applying Green’s second identity on a volume bounded by the free surface
and a half sphere of arbitrarily large radius in the fluid. The main difference from the
present case is that equation (11) does not contain the integrals I3and I4. We also use the
parameter
α =gT
ρc4,
instead of the term λβ in the equation (6), since we cannot use the parameters λ and β,
which are based on h.
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Generally, in order to obtain a solitary-wave solution in finite depth we first compute
forced gravity-capillary waves for fixed λ and β (see Section V), then we remove gradually
the pressure, by keeping the amplitude of the solution constant. The solution obtaned is
then used as an initial guess or further computations.
IV.GRAVITY-CAPILLARY SOLITARY WAVES
For small surface tension (β < 1/3) we found that the three dimensional problem is qual-
itatively similar to the two dimensional problem. In particular there are two branches of
fully localised three-dimensional gravity-capillary solitary waves. One branch has a central
depression, the other branch has a central elevation. These waves have decaying oscillations
in the direction of propagation and are monotonically decaying perpendicular to the direc-
tion of propagation. In Fig.3 we show a typical central depression gravity-capillary wave
solution and in Fig.4 a typical central elevation gravity-capillary wave solution. The curves
obtained by cutting the free surface with planes parallel to the direction of propagation are
qualitatively similar to the two dimensional profiles obtained by Vanden-Broeck and Dias5
and by Dias et al.7. The solutions are quite similar to the fully localised solitary waves found
on deep water (see Parau et al.8).
For strong surface tension (β > 1/3) we found only fully localised depression gravity
capillary solitary waves. They are similar to fully-localized solitary-wave solutions of the
KP-I equation as shown in Fig. 5 for λ = 1.14 and β = 1. The lump solution for the KP-I
equation is, as given by Milewski10and rewritten in term of our parameters, is
ζ(x,y) =−16µ
β
3 − 2µ
3 + 2µ
νβ2x2+ 4µ2
νβ2x2+ 4µ2
νβ2y2
νβ2y2?2,
?
where µ = 1−
fully-localized solitary-wave solution (solid line) and the lump solution for the KP-I equation
1
√λand ν =β
λ−1
3. Figure 6 shows the x and y cross-section of the computed
(dotted line). It can be observed that there is a very good agreement between the two
solutions for the centreline on the Ox direction, except at the last points, where the effect
of the truncation is visible. There is also a good agreement on the Oy direction between the
numerical and analytical solution, but the numerical solution seems to decay faster than the
analytical solution. This is likely to be caused by truncation, as the algebraic decay of the
solution in this direction is slower than on the Ox direction.
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The agreement between the amplitude of the KP-I solution and our solution of the full
equations can be observed better if we keep λ constant and vary β (see Fig. 7). We can
follow continuously a branch of central depression solitary-wave solutions from the region of
strong surface tension to the region of weak surface tension, by passing through β = 1/3, as
predicted by Milewski10. There is a maximum amplitude for the branch of central depression
waves (with λ constant) for β near 1/3.
V.FORCED CAPILLARY-GRAVITY WAVES
In this section we consider three-dimensional free surface flows due to a moving distur-
bance. As we shall see some of the solutions can be viewed as perturbations of the solitary
waves of Section III. We choose a frame of reference moving with the disturbance and we
seek steady solutions. The disturbance can be submerged (e.g. a submarine) or surface
piercing (e.g. a ship or a probe). The results presented are qualitatively independent of the
disturbance chosen. Therefore we assume for simplicity that the disturbance is a distribution
of pressure εP(x,y) where ε is a given parameter and
P(x,y) =





e
1
(x2−1)+
1
(y2−1), |x| < 1 and |y| < 1
otherwise0,
(17)
The problem is described mathematically by the equations (4)-(7) with the term ?P(x,y)
added on the left hand side of (6). For finite depth numerical solutions are calculated by
using the numerical procedure of Section III. For infinite depth the numerical scheme is
similar to that in P˘ ar˘ au, Vanden-Broeck and Cooker8.
We calculated solutions both in finite and infinite depth and obtained qualitatively similar
results. In Fig. 8 and 9 we present two such profiles for λ = 1.132 and β = 1. The first one
is obtained by imposing a positive pressure (ε > 0) on the surface and the second one by
imposing a negative pressure distribution (ε < 0). To simplify the problem, we will present
from now on results obtained in infinite depth only, as the profiles are quite similar to the
ones in finite depth and the number of parameters is decreased by one (α and ε).
In all cases α is assumed to be greater than 1/4, which corresponds to flows where the
distribution of pressure moves steadily with a constant velocity c smaller than the minimum
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phase speed cmin, which in infinite depth is
cmin=
?4gT
ρ
?1/4
.(18)
In this case no wave pattern was observed and only a highly localised disturbance of the water
surface is predicted. As α decreases and approaches 1/4 more and more oscillations appear
in front and behind the main disturbance, the area disturbed by the pressure increasing
considerably. In two dimensions, Vanden-Broeck & Dias5computed capillary-gravity waves
in the same regime of parameters (α > 1/4).
We show in Figure 10 values of ζ(0,0) versus α for various values of ε. The dashed lines
corresponds to ε = ±1. For each ε ?= 0, there is a critical number αεsuch that there is no
solution for 1/4 < α < αε. In order to compute solutions near the turning point we used a
variation of the scheme of Section III in which the amplitude is fixed and α is found as part
of the solution.
Linear solutions for the flow due to a moving pressure distribution were calculated by
Rapha¨ el and de Gennes18, Sun and Keller19and others. We have recalculated these solutions
numerically by running the code with the boundary conditions linearized. The corresponding
values of ζ(0,0) versus α are shown in Figure 10 by dotted lines. It shows that |ζ(0,0)| → ∞
as α → 1/4. Therefore the linear theory is not valid near α = 1/4. Our nonlinear solutions
do not blow up as α → 1/4 but have a turning point at α = αe > 1/4. The solutions
corresponding to the portions of the broken curves closest to the α-axis and extending from
αeto α = ∞ are close to the linear solutions corresponding to the dotted lines in Figure
10. These solutions are perturbations of a uniform stream in the sense that they approach
a uniform stream with constant velocity U as |?| → 0.
The remaining portions of the broken curves (i.e. the portions of the curves further away
from the α-axis and extending to the right of the turning point αe) are perturbations of
three-dimensional gravity-capillary solitary waves. In other words the remaining portions of
the broken curves approach solitary waves as |?| → 0. The remaining portion of the broken
curve for ? = 1 is already almost on the branch of depression solitary waves. The remaining
portion for ? = −1 is still not very close to the branch of elevation solitary waves, but it
approaches it when |?| decreases.
The branches of solitary waves correspond to the solid curves in Figure 10 and they were
discussed in detail in Parau et al.8.
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Figure 10 shows forced solutions that are elevation solitary waves perturbed by a negative
distribution of pressure (? < 0) and forced solutions that are depression solitary waves
perturbed by a positive distribution of pressure (? > 0). We found that there are also forced
capillary gravity waves that are elevation solitary waves perturbed by positive pressure
distributions and depression solitary waves perturbed by negative pressure distributions. A
typical example is shown in Figure 11.
VI. CONCLUSION
We have calculated three dimensional gravity capillary solitary waves of the full Euler
equations in finite depth. Two branches of solutions were obtained. One is bifurcating at
the minimum value of the phase speed of the linear periodic waves. There are both elevation
and depression waves. The other one is an extension of the solution of the KP-I equation
to the fully nonlinear regime. We have shown that some of the solutions corresponding to
flows due to moving disturbances can be viewed as perturbations of these solitary waves.
Acknowledgments
This work was supported by EPSRC, under Grant Number GR/S47786/01.
∗Author to whom the correspondence should be addressed. Electronic mail: J.Vanden-
broeck@uea.ac.uk
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FIG. 1. Curves in the β −λ plane where solitary waves can bifurcate (the grey area): above
the curve Γ for β < 1/3 and above the curve λ = 1 for β < 1/3. The curve Γ is shown as a
dark solid line.
FIG. 2. The surfaces used for the application of the Green’s second identity.
FIG. 3. Central depression solitary gravity-capillary wave for β = 0.235,λ = 1.13. Only
half of the solutions (y ≥ 0) are shown.
FIG. 4. Central elevation solitary gravity-capillary wave for β = 0.28,λ = 1.13. Only half
of the solutions (y ≥ 0) are shown.
FIG. 5. Central depression solitary gravity-capillary wave for β = 1,λ = 1.14 (x ≤ 0)
compared with the fully localised solitary wave solution for KP-I (x ≥ 0), as given by
Milewski10. Only half of the solutions (y ≥ 0) are shown.
FIG. 6. (a) The centreline in the Ox direction of the computed solitary gravity-capillary
wave (solid line) and the fully localised solitary wave solution for KP-I (dotted line). The
parameters are β = 1,λ = 1.14. (b) The centreline in the Oy direction of the computed
solitary gravity-capillary wave (solid line) and the fully localised solitary wave solution for
KP-I (dotted line). The parameters are β = 1,λ = 1.14.
FIG. 7. The maximum amplitude of the computed solution (solid line) and KP-I solutions
(dashed line) for a fixed λ = 0.132
FIG. 8. Forced capillary-gravity waves for λ = 1.132, β = 1 and ε = 0.3.
FIG. 9. Forced capillary-gravity waves for λ = 1.132, β = 1 and ε = −0.3.
FIG. 10. Values of the amplitude (ζ(0,0)) versus α. The dashed lines corresponds to ε = 1
in the negative half of the plane and to ε = −1 on the positive half of the plane. The solid
line corresponds to free capillary-gravity waves (ε = 0). The dotted line corresponds to
linear solutions for ε = ±1.
FIG. 11. Free surface profile for α = 0.35,ε = 3 which is a perturbation of an elevation
solitary wave. Only half of the solution is shown.
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00.1 0.20.3 0.40.5
β
0.6 0.70.8 0.91
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
λ
Γ
FIG. 1: Parau, Physics of Fluids
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P*
SF
SF’
z=−1/β
x
z
y
FIG. 2: Parau, Physics of Fluids
15