Environmental Engineering Reference
In-Depth Information
Note:
The Laplace differential equation and the boundary conditions given above are satisfied
in accordance with the definition by the potential of the undisturbed primary wave (
F
0
,
see Section 2.5.3).
In addition, the perturbation potential must satisfy the Sommerfeld general radiation
condition:
p
x
2
p
2
g
w
diff
@w
diff
@
r
v
lim
r
!1
¼
0 where
r
¼
þ
y
2
þ
z
2
The kinematic boundary condition must be satisfied for the body held in the wave, that
is the velocity component normal to the surface of the body must disappear with respect
to the total potential (
F
0
þF
diff
):
S
0
¼
@ F
0
þ F
diff
S
0
¼
0 r
@w
S
0
¼
@ w
0
þ w
diff
S
0
¼
0
@F
@
ð
Þ
ð
Þ
n
@
n
@
n
@
n
It follows from this that
S
0
¼
@F
0
S
0
S
0
¼
@w
0
S
0
@F
diff
@
@w
diff
@
or
n
@
n
n
@
n
Here, S
0
is the wetted surface at rest, and n is the exterior normal of S
0
acting towards
the fluid.
2.6.4 Integral equation method (singularity method)
The integral equation method is based on the principle that the flow around fixed
bodies is described by choosing a suitable distribution of pulsating singularities.
The perturbation potential
w
diff
required for the diffraction problem can be
expressed as
ZZ
1
4
p
w
diff
x
ð
;
y
;
z
Þ¼
Q
ð
j; h; z
Þ
Gx
ð
;
y
;
z
; j; h; z
Þ
dS
S
0
for the three-dimensional case. In this integral equation, Q describes the singularity
density at point {
} on the surface of the body (S
0
) and the Green (or influence)
function G describes how the potential of a pulsating unit source at that point affects the
point being considered {x, y, z}. The reader is referred to [17] for details of the
analytical description.
According to potential theory, the kinematic boundary condition at the surface of the
body is
j
,
h
,
z
ZZ
@w
0
@
n
¼
@w
diff
Þ
@
ð
;
;
; j; h; z
Þ
1
2
Qx
1
4
p
Gx
y
z
n
¼
ð
;
y
;
z
Þ
Q
ð
j; h; z
dS
@
@
n
S
0
