Part A | 4.8
94 Part A Fundamentals
condition
DX
Dt
D r ;
to advance position of free surface nodes X D .X; Y; Z/
from discrete time n to n C 1. Algorithms such as
the fourth-order Runge–Kutta and Adams–Bashforth
schemes [4.67, 68] may be used for the time integration.
Thus, knowing on the free surface, one may rewrite
the above Green’s theorem with known terms on the
right-hand side and unknown terms on the left-hand side
2.P 2 B C S B /
C
Z
BCSB
@
@n
1
r PQ
d@˝
Z
†CF
1
r PQ
@
@n
d@˝
D D2.P 2 F C †/
Z
F C†
@
@n
1
r PQ
d@˝ C
Z
BCSB
1
r PQ
@
@n
d@˝ : (4.99)
The above integral equation is discretized and the
resulting algebraic (matrix) equation solved either directly or iteratively for on solid boundaries B and S B ,
and for @=@n on the free surface F and open boundary ˙. Upon determining on the body, one can use the
Euler integral (unsteady Bernoulli’s equation) to determine pressure and through integration of pressure the
hydrodynamic force on the body. The solution is thus
advanced in time.
The mixed Eulerian–Lagrangian formulation has
become a standard approach for solving fully nonlinear inviscid wave and wave–body interaction problems
and it has been adopted in field discretization methods
such as the finite difference method. Works on nonlinear wave and wave–body interaction problems based on
the MEL formulation include those by Vinje and Brevig [4.69], Grosenbaugh and Yeung [4.67], Dommermuth et al. [4.68], Saout and Ananthakrishnan [4.70],
Ananthakrishnan [4.71] and Xue et al. [4.72].
4.8 Wave Forces on Fixed and Floating Structures
In this section, the methods to determine the wave
exciting force (which consists of incident and bodydiffracted wave forces) on a body, and in the case
of a freely floating body the additional wave radiation force due to the body motion generated waves are
presented. Both theoretical and numerical methods to
determine the wave forces will be discussed. We shall
take the mean forward of the bodies to be zero here.
One can find the nonzero forward speed cases in the
literature on ship hydrodynamics and naval architecture. Empirical and exact methods to determine the
viscous drag force is also discussed. Particular emphasis is given to parameters that govern ratios of various
wave component and viscous drag forces.
4.8.1 Incident Wave Force:
Froude–Krylov Force
Let us consider a body (submerged or floating) in a wave
field as illustrated in Fig. 4.1. Let the incident wave be
of small amplitude and be propagating in the positive x
direction with elevation and potential given by
Á
i
D
H
i
2
cos .kx !t/ ;
i
D
H
i
2
cosh k.z C h/
cosh kh
sin .kx !t/ ;
where the superscript i denotes incident wave. Using the
Euler integral, one can find the dynamic pressure of the
incident wave as
p
i
D D
@
i
@t
D g
H
i
2
cosh k.z C h/
cosh kh
cos .kx !t/ :
By integrating the incident wave pressure about the
body surface (mean surface if the body is undergoing
oscillation) one can determine the incident wave force,
which is also known as the Froude–Krylov force, F
i
F
i
D
Z
So
p
i dS o
D
Z
So
g
H
i
2
cosh k.z C h/
cosh kh
O
n cos .kx !t/ dS o ;
where S o denotes the body surface and O
n the unit normal
vector into the body. Using the Gauss theorem one may
write the above as a volume integral
F
i
D
Z
So
p
i
O
ndS o D D
Z
8o
rp
i d8 o ;
where 8 o denotes the volume occupied by the body. In
the component form, the incident wave forces are then
F
i
x D gk
H
i
2
1
cosh kh
Z
8o
cosh k.z C h/ sin .kx !t/d8 o ;
94 Part A Fundamentals
condition
DX
Dt
D r ;
to advance position of free surface nodes X D .X; Y; Z/
from discrete time n to n C 1. Algorithms such as
the fourth-order Runge–Kutta and Adams–Bashforth
schemes [4.67, 68] may be used for the time integration.
Thus, knowing on the free surface, one may rewrite
the above Green’s theorem with known terms on the
right-hand side and unknown terms on the left-hand side
2.P 2 B C S B /
C
Z
BCSB
@
@n
1
r PQ
d@˝
Z
†CF
1
r PQ
@
@n
d@˝
D D2.P 2 F C †/
Z
F C†
@
@n
1
r PQ
d@˝ C
Z
BCSB
1
r PQ
@
@n
d@˝ : (4.99)
The above integral equation is discretized and the
resulting algebraic (matrix) equation solved either directly or iteratively for on solid boundaries B and S B ,
and for @=@n on the free surface F and open boundary ˙. Upon determining on the body, one can use the
Euler integral (unsteady Bernoulli’s equation) to determine pressure and through integration of pressure the
hydrodynamic force on the body. The solution is thus
advanced in time.
The mixed Eulerian–Lagrangian formulation has
become a standard approach for solving fully nonlinear inviscid wave and wave–body interaction problems
and it has been adopted in field discretization methods
such as the finite difference method. Works on nonlinear wave and wave–body interaction problems based on
the MEL formulation include those by Vinje and Brevig [4.69], Grosenbaugh and Yeung [4.67], Dommermuth et al. [4.68], Saout and Ananthakrishnan [4.70],
Ananthakrishnan [4.71] and Xue et al. [4.72].
4.8 Wave Forces on Fixed and Floating Structures
In this section, the methods to determine the wave
exciting force (which consists of incident and bodydiffracted wave forces) on a body, and in the case
of a freely floating body the additional wave radiation force due to the body motion generated waves are
presented. Both theoretical and numerical methods to
determine the wave forces will be discussed. We shall
take the mean forward of the bodies to be zero here.
One can find the nonzero forward speed cases in the
literature on ship hydrodynamics and naval architecture. Empirical and exact methods to determine the
viscous drag force is also discussed. Particular emphasis is given to parameters that govern ratios of various
wave component and viscous drag forces.
4.8.1 Incident Wave Force:
Froude–Krylov Force
Let us consider a body (submerged or floating) in a wave
field as illustrated in Fig. 4.1. Let the incident wave be
of small amplitude and be propagating in the positive x
direction with elevation and potential given by
Á
i
D
H
i
2
cos .kx !t/ ;
i
D
H
i
2
cosh k.z C h/
cosh kh
sin .kx !t/ ;
where the superscript i denotes incident wave. Using the
Euler integral, one can find the dynamic pressure of the
incident wave as
p
i
D D
@
i
@t
D g
H
i
2
cosh k.z C h/
cosh kh
cos .kx !t/ :
By integrating the incident wave pressure about the
body surface (mean surface if the body is undergoing
oscillation) one can determine the incident wave force,
which is also known as the Froude–Krylov force, F
i
F
i
D
Z
So
p
i dS o
D
Z
So
g
H
i
2
cosh k.z C h/
cosh kh
O
n cos .kx !t/ dS o ;
where S o denotes the body surface and O
n the unit normal
vector into the body. Using the Gauss theorem one may
write the above as a volume integral
F
i
D
Z
So
p
i
O
ndS o D D
Z
8o
rp
i d8 o ;
where 8 o denotes the volume occupied by the body. In
the component form, the incident wave forces are then
F
i
x D gk
H
i
2
1
cosh kh
Z
8o
cosh k.z C h/ sin .kx !t/d8 o ;