Practical Ship Hydrodynamics

z
y
Figure 7.5 Coordinate system used; sources i are located inside contour
z
Numerical example for BEM 251
2. There is atmospheric pressure everywhere on the free surface z D
(dynamic condition). Then Bernoulli’s equation yields
t C 1
2⊲r ⊳2 g D 0
3. There is no flow through the free surface (kinematic condition), i.e. the
local vertical velocity of a particle coincides with the rate of change of the
surface elevation in time:
z D t
4. Differentiation of the dynamic condition with respect to time and combination
with the kinematic condition yields
tt C y yt C z zt g z D 0
This expression can be developed in a Taylor expansion around z D 0.
Omitting all non-linear terms yields then
tt g z D 0
5. There is no flow through the body contour, i.e. the normal velocity of the
water on the body contour coincides with the normal velocity of the hull
(or, respectively, the relative normal velocity between body and water is
zero):
En Ðr DEn ÐEv
Here Ev is the velocity of the body, En is the outward unit normal vector.
6. Waves created by the body must radiate away from the body:
lim
jyj!1 D Re⊲ Oϕe kz e i⊲ωet kjyj⊳
⊳
Oϕ is here a yet undetermined, but constant, amplitude.
Using the harmonic time dependency of the potential, we can reformulate
the Laplace equation and all relevant boundary conditions such that only the
time-independent complex amplitude of the potential O appears:
y
i
X k
X k+1
k
a ik
k+1
n