Part A | 4.8
96 Part A Fundamentals
From the above discussion, the following points are
worth recapitulating:
The ratio of drag force to inertia force depends on
(i) the wave height to body diameter ratio and (ii)
wave length to water depth ratio.
In the case of small wave height to body diameter ratio H
i
=D in deep water waves (i. e., tanh kh
= 1), the Morison force will be equal to the Froude–
Krylov force for appropriate value of C i .
When using the Morison equation method for wave
force on a moving body, the velocity and accelerations are taken to be relative to the body motion.
If the body dimension is large compared to wave
height and not small compared to the incident wave
length, then the drag force will not be important, but the
Morison equation cannot be used to determine the inertial force because the scattering (diffraction) of waves
by the body will become significant. One can use the
potential flow theory to solve the diffraction problem
(as done in [4.75] for a vertical cylinder) and also determine the wave diffraction force. The wave exciting
force will then be the sum of the Froude–Krylov and the
diffraction wave forces. The diffraction wave problem
pertaining to linear wave–body interaction is discussed
in the next section.
4.8.3 Wave Diffraction over a Body
Let the body shown in Fig. 4.1 be stationary. The mere
presence of the body will cause the incident waves
to scatter. For small amplitude waves, governed by
linearized free surface conditions, one may solve the
diffraction problem separately and determine the total
potential as
D
i
C
d
;
where the superscripts i and d denote incidence and
diffraction, respectively. The solution of the incident
wave potential is simply that of a free periodic wave as
presented in Sect. 3.2. The diffraction potential is governed by the following set of equations
r
2
d
D 0 ;
@
d
@z
D 0; on the sea bottom z D Dh ;
@
d
@n
D D
@
i
@n
on the body surface S B ;
@
2
d
@t 2 C g
@
d
@z
D 0; on the mean free surface z D 0 :
Moreover, in the far field, the diffraction potential must
satisfy the Sommerfeld radiation condition [4.9]
p
R i!
d
C C
@
d
@R
!
D 0; as R ! 1 ;
where R D
p
x 2 C y 2 , which denotes the radial distance
from the body and C the wave phase speed. The above
diffraction problem was solved by McCamy and Fuchs
for the case of a vertical circular cylinder [4.75]. The x
component wave exciting force with diffraction is given
by
F x D
g
H
2
D
2 tanh.kh/
k 2 R 2 H
.2/ 0
1 .kR/
;
where H
.2/
1
denotes the Hankel function of the second
kind and order 1 and R = D=2 the radius of the cylinder.
In the case of small H=D and not so small L=D, as
in the case of waves incident on a supertanker or a large
gravity platform, the diffraction force will be more significant than the drag force. On the other hand, drag
force will be the predominant part of the wave exciting
force for a mooring cable (small diameter) in a similar
sea.
4.8.4 Wave Radiation Force
on an Oscillating Body
In the case of a compliant or freely floating body, the
body will undergo oscillatory motion when subject to
the wave exciting force. The force due to waves caused
by body motion is referred to as the wave radiation
force [4.9]. Now let the body considered in Fig. 4.1 undergo rigid body motion such that the normal velocity
of the body may be written as
V n D U O
n C .˝ r/ O
n
D U O
n C ˝ .r O
n/
D
iD6
X
iD1
U i n i ;
where O
n denotes the unit normal vector on the body
surface and r the position vector from the axis of rotation through the center of gravity. Moreover, U i D
U 1 ; U 2 ; U 3 ; ˝ 1 ; ˝ 2 ; ˝ 3 and n i D n 1 ; n 2 ; n 3 ; .r O
n/ 1 ; .r
O
n/ 2 ; .r O
n/ 3 for i D 1; 2; 3; 4; 5; 6, respectively, corresponding to the sixth degree of freedom rigid body
motion. For the linear small amplitude body and wave
motion problem, the radiation wave potential can be decomposed as per Kirchoff modal decomposition [4.9]
96 Part A Fundamentals
From the above discussion, the following points are
worth recapitulating:
The ratio of drag force to inertia force depends on
(i) the wave height to body diameter ratio and (ii)
wave length to water depth ratio.
In the case of small wave height to body diameter ratio H
i
=D in deep water waves (i. e., tanh kh
= 1), the Morison force will be equal to the Froude–
Krylov force for appropriate value of C i .
When using the Morison equation method for wave
force on a moving body, the velocity and accelerations are taken to be relative to the body motion.
If the body dimension is large compared to wave
height and not small compared to the incident wave
length, then the drag force will not be important, but the
Morison equation cannot be used to determine the inertial force because the scattering (diffraction) of waves
by the body will become significant. One can use the
potential flow theory to solve the diffraction problem
(as done in [4.75] for a vertical cylinder) and also determine the wave diffraction force. The wave exciting
force will then be the sum of the Froude–Krylov and the
diffraction wave forces. The diffraction wave problem
pertaining to linear wave–body interaction is discussed
in the next section.
4.8.3 Wave Diffraction over a Body
Let the body shown in Fig. 4.1 be stationary. The mere
presence of the body will cause the incident waves
to scatter. For small amplitude waves, governed by
linearized free surface conditions, one may solve the
diffraction problem separately and determine the total
potential as
D
i
C
d
;
where the superscripts i and d denote incidence and
diffraction, respectively. The solution of the incident
wave potential is simply that of a free periodic wave as
presented in Sect. 3.2. The diffraction potential is governed by the following set of equations
r
2
d
D 0 ;
@
d
@z
D 0; on the sea bottom z D Dh ;
@
d
@n
D D
@
i
@n
on the body surface S B ;
@
2
d
@t 2 C g
@
d
@z
D 0; on the mean free surface z D 0 :
Moreover, in the far field, the diffraction potential must
satisfy the Sommerfeld radiation condition [4.9]
p
R i!
d
C C
@
d
@R
!
D 0; as R ! 1 ;
where R D
p
x 2 C y 2 , which denotes the radial distance
from the body and C the wave phase speed. The above
diffraction problem was solved by McCamy and Fuchs
for the case of a vertical circular cylinder [4.75]. The x
component wave exciting force with diffraction is given
by
F x D
g
H
2
D
2 tanh.kh/
k 2 R 2 H
.2/ 0
1 .kR/
;
where H
.2/
1
denotes the Hankel function of the second
kind and order 1 and R = D=2 the radius of the cylinder.
In the case of small H=D and not so small L=D, as
in the case of waves incident on a supertanker or a large
gravity platform, the diffraction force will be more significant than the drag force. On the other hand, drag
force will be the predominant part of the wave exciting
force for a mooring cable (small diameter) in a similar
sea.
4.8.4 Wave Radiation Force
on an Oscillating Body
In the case of a compliant or freely floating body, the
body will undergo oscillatory motion when subject to
the wave exciting force. The force due to waves caused
by body motion is referred to as the wave radiation
force [4.9]. Now let the body considered in Fig. 4.1 undergo rigid body motion such that the normal velocity
of the body may be written as
V n D U O
n C .˝ r/ O
n
D U O
n C ˝ .r O
n/
D
iD6
X
iD1
U i n i ;
where O
n denotes the unit normal vector on the body
surface and r the position vector from the axis of rotation through the center of gravity. Moreover, U i D
U 1 ; U 2 ; U 3 ; ˝ 1 ; ˝ 2 ; ˝ 3 and n i D n 1 ; n 2 ; n 3 ; .r O
n/ 1 ; .r
O
n/ 2 ; .r O
n/ 3 for i D 1; 2; 3; 4; 5; 6, respectively, corresponding to the sixth degree of freedom rigid body
motion. For the linear small amplitude body and wave
motion problem, the radiation wave potential can be decomposed as per Kirchoff modal decomposition [4.9]