Mechanics of Ocean Waves 4.7 Computational Method for Fully Nonlinear Waves 93
Part A | 4.7
et al. [4.54] showed that the infragravity wave energy
tends to be a function of h
1:1 , an almost linear dependence. However, coupling (4.97) with Green’s Law
shoaling yielded a growth rate closer to h
5 . The discrepancy is due to how the wave environment in the
surf zone is treated. Van Dongeren and Svendsen [4.55],
using a quasi-3-D (three-dimensional) nearshore circulation model, showed that the growth rate of infragravity waves can be dictated by manipulating the phase
difference between the bound wave (locked with the
wave group) and a free wave in the surf zone. Later,
Janssen et al. [4.56] developed an analytical solution
for the phase shift, leading to growth rates more consistent with measurements [4.57].
Oltman-Shay et al. [4.58], examining wave-like
structures in nearshore current data taken at Duck, NC
(USA), found that these structures did not correspond
to any known wave theory. Bowen and Holman [4.59]
applied stability theory to the equations governing
nearshore circulation (with several simplifying assumptions) and determined that the most unstable modes had
frequencies that were in the range of those observed
in [4.58]. These instabilities were termed shear waves
since they seemed to be caused by the shear instabilities
of the longshore current. These motions only appeared
when the longshore current was present and sufficiently
energetic. The speed of these waves was independent of
their frequency (a nondispersive phenomenon) and their
signature in frequency longshore wave number space is
both distinct and distinctly different from gravity wave
phenomena.
Since the original papers there have been numerous
studies on these shear waves. Reniers et al. [4.60] recreated these waves in the laboratory. Analytical means of
studying the growth of these phenomena were advanced
by Dodd and Falques [4.61], Shrira et al. [4.62], and
Feddersen [4.63], among others. Additionally, the modeling of shear waves became a motivating factor for
the development and application of numerical models
of nearshore circulation ( [4.64, 65], among others).
4.7 Computational Method for Fully Nonlinear Waves
With the development of powerful computers, numerical methods have been developed to tackle nonlinear
wave problems and problems involving arbitrary body
boundaries. Boundary integral methods are found to
be more efficient for solving wave problems formulated using the potential flow theory, in particular to
handle steep and overturning waves and complex body
boundary shapes. For linear problems, boundary integral methods based on the free surface Green’s function
have been developed for analysis in both the frequency
and time domains. Frequency domain analysis using
a simple Rankine source has also been used to studying
linear wave–body interaction problems in the frequency
domain. An excellent review on numerical methods for
free surface flows is given in [4.66].
Longuet-Higgins and Cokelet [4.4] developed
a boundary integral method to solve the fully nonlinear inviscid wave motion problem. The method involves
solution of Green’s theorem, which is based on the
Eulerian description of flow and the nonlinear free
surface boundary conditions in the Lagrangian form;
the method is, therefore, considered to be based on
the mixed Eulerian–Lagrangian (MEL) formulation. To
illustrate the method, let us consider a wave–body interaction problem such as that depicted in Fig. 4.1. Let
the lateral extent of the domain be truncated by an open
boundary ˙. Let us say that the flow has been started
from rest with the initial condition being D 0 and
Á D 0 at time t D 0. Since ˙ is not a physical boundary, it has to be modeled so that waves incident on it
may pass through without any reflection. There are several ways to achieve that approximately, as through use
of nonlinear wave equations, free surface damping etc.
Here let us consider a simple model by which it is assumed that D 0 on ˙ during the duration of flow
simulation; in other words, simulation will be carried
out only until the radiating waves reach the vicinity of
the open boundary ˙.
Per Green’s theorem,
2.P/ C
Z
@@
@
@n
1
r PQ
d@@
Z
@@
1
r PQ
@
@n
d@@ D 0 ;
(4.98)
where @˝ is the union of all boundaries; i. e., @˝ D BC
S B C F C ˙. Here Green’s function 1=r PQ corresponds
to the potential at P due to the point source at Q. On
B and S B the normal velocity @=@n is known based
on the no-flux condition, while is not known. On the
open boundary, is known, here set to be zero, while
@=@n is not known. On the free surface, one can time
integrate the fully nonlinear free surface conditions at
each time step to determine the free surface deformation
and the velocity potential on the free surface; in other
words, time integrate the dynamic condition
D
Dt
D
1
2
jrj
2
gY ;
to advance from discrete time n to n C 1 and time
integrate the free surface (material surface) kinematic
Part A | 4.7
et al. [4.54] showed that the infragravity wave energy
tends to be a function of h
1:1 , an almost linear dependence. However, coupling (4.97) with Green’s Law
shoaling yielded a growth rate closer to h
5 . The discrepancy is due to how the wave environment in the
surf zone is treated. Van Dongeren and Svendsen [4.55],
using a quasi-3-D (three-dimensional) nearshore circulation model, showed that the growth rate of infragravity waves can be dictated by manipulating the phase
difference between the bound wave (locked with the
wave group) and a free wave in the surf zone. Later,
Janssen et al. [4.56] developed an analytical solution
for the phase shift, leading to growth rates more consistent with measurements [4.57].
Oltman-Shay et al. [4.58], examining wave-like
structures in nearshore current data taken at Duck, NC
(USA), found that these structures did not correspond
to any known wave theory. Bowen and Holman [4.59]
applied stability theory to the equations governing
nearshore circulation (with several simplifying assumptions) and determined that the most unstable modes had
frequencies that were in the range of those observed
in [4.58]. These instabilities were termed shear waves
since they seemed to be caused by the shear instabilities
of the longshore current. These motions only appeared
when the longshore current was present and sufficiently
energetic. The speed of these waves was independent of
their frequency (a nondispersive phenomenon) and their
signature in frequency longshore wave number space is
both distinct and distinctly different from gravity wave
phenomena.
Since the original papers there have been numerous
studies on these shear waves. Reniers et al. [4.60] recreated these waves in the laboratory. Analytical means of
studying the growth of these phenomena were advanced
by Dodd and Falques [4.61], Shrira et al. [4.62], and
Feddersen [4.63], among others. Additionally, the modeling of shear waves became a motivating factor for
the development and application of numerical models
of nearshore circulation ( [4.64, 65], among others).
4.7 Computational Method for Fully Nonlinear Waves
With the development of powerful computers, numerical methods have been developed to tackle nonlinear
wave problems and problems involving arbitrary body
boundaries. Boundary integral methods are found to
be more efficient for solving wave problems formulated using the potential flow theory, in particular to
handle steep and overturning waves and complex body
boundary shapes. For linear problems, boundary integral methods based on the free surface Green’s function
have been developed for analysis in both the frequency
and time domains. Frequency domain analysis using
a simple Rankine source has also been used to studying
linear wave–body interaction problems in the frequency
domain. An excellent review on numerical methods for
free surface flows is given in [4.66].
Longuet-Higgins and Cokelet [4.4] developed
a boundary integral method to solve the fully nonlinear inviscid wave motion problem. The method involves
solution of Green’s theorem, which is based on the
Eulerian description of flow and the nonlinear free
surface boundary conditions in the Lagrangian form;
the method is, therefore, considered to be based on
the mixed Eulerian–Lagrangian (MEL) formulation. To
illustrate the method, let us consider a wave–body interaction problem such as that depicted in Fig. 4.1. Let
the lateral extent of the domain be truncated by an open
boundary ˙. Let us say that the flow has been started
from rest with the initial condition being D 0 and
Á D 0 at time t D 0. Since ˙ is not a physical boundary, it has to be modeled so that waves incident on it
may pass through without any reflection. There are several ways to achieve that approximately, as through use
of nonlinear wave equations, free surface damping etc.
Here let us consider a simple model by which it is assumed that D 0 on ˙ during the duration of flow
simulation; in other words, simulation will be carried
out only until the radiating waves reach the vicinity of
the open boundary ˙.
Per Green’s theorem,
2.P/ C
Z
@@
@
@n
1
r PQ
d@@
Z
@@
1
r PQ
@
@n
d@@ D 0 ;
(4.98)
where @˝ is the union of all boundaries; i. e., @˝ D BC
S B C F C ˙. Here Green’s function 1=r PQ corresponds
to the potential at P due to the point source at Q. On
B and S B the normal velocity @=@n is known based
on the no-flux condition, while is not known. On the
open boundary, is known, here set to be zero, while
@=@n is not known. On the free surface, one can time
integrate the fully nonlinear free surface conditions at
each time step to determine the free surface deformation
and the velocity potential on the free surface; in other
words, time integrate the dynamic condition
D
Dt
D
1
2
jrj
2
gY ;
to advance from discrete time n to n C 1 and time
integrate the free surface (material surface) kinematic