Environmental Engineering Reference
In-Depth Information
b) The kinematic boundary condition at the surface of the structure
@F
@
n
¼
0
for S
ð
x
;
y
;
z
;
t
Þ¼
0
c) The dynamic boundary condition at the surface of the water (Bernoulli)
"
#
2
þ
2
þ
2
@F
@
@F
@
@F
@
r
@F
@
t
þ
2
þ r
g
z ¼
0
x
y
z
for
z ¼ z
ð
x
;
y
;
t
Þ
d) The kinematic boundary condition at the surface of the water
x
þ
@z
y
@F
t
þ
@z
x
@F
@F
@
z
¼
@z
@
@
y
@
@
@
for
z ¼ z
ð
x
;
y
;
t
Þ
e) The radiation condition
p
@F
@
r
i
v
2
g
F
lim
r
!1
¼
0
In linear theory the underlined higher-order terms may be neglected provided the
dynamic field may be considered as an (infinitesimally) small disturbance of the steady
state. In this case it is permissible to satisfy the kinematic and dynamic boundary
conditions at still water level (z
¼
0) and not at the wave profile (z
¼z
). From this we get
the combined linearised boundary condition at the surface of thewater (see Section 2.6.3):
2
@
t
2
þ
g
@F
F
z
¼
0
@
@
In non-linear theory we have to describe the pressures on the wetted surface (S) at
time t. The surface is bounded by the unknown wave surface
(x, y, t). We solve this
problem by developing the partial derivations of the potential function at the still
water level (z
¼
0) in Taylor series. That results in the approximate solutions of the
first, second, . . . , nth order.
For example, the second-order boundary value problem gives us the pressure field from
the Bernoulli equation - as was already used in 2.6.4:
z
"
#
2
2
2
p
i
¼
2
@F
i
@
@F
i
@
@F
i
@
r
@F
i
@
þ
þ
t
r
g
z
x
y
z
2.6.7 Wave loads on large-volume offshore structures
It is essential to consider the diffraction effects that occur if we are to calculate the
hydrodynamic loads on compact,
large-volume offshore structures reliably [32].
