Practical Ship Hydrodynamics

Numerical example for BEM 239
In each iterative step, wave elevation and potential are updated yielding
successively better approximations for the solution of the non-linear problem.
The equations are formulated here in a right-handed Cartesian coordinate
system with x pointing forward towards the ‘bow’ and z pointing upward. For
the assumed ideal fluid, there exists a velocity potential such that Ev Dr .
The velocity potential fulfils Laplace’s equation in the whole fluid domain:
1 D xx C zz D 0
The hull condition requires vanishing normal velocity on the body:
En Ðr D 0
En is the inward unit normal vector on the body hull.
The kinematic condition (no penetration of water surface) gives at z D :
r Ðr D z
For simplification, we write ⊲x, y, z⊳ with z D ∂ /∂z D 0.
The dynamic condition (atmospheric pressure at water surface) gives at z D
:
1
2⊲r ⊳2 C gz D 1
2V2 with g D 9.81 m/s 2 . Combining the dynamic and kinematic boundary conditions
eliminates the unknown wave elevation z D :
1
2r Ðr⊲r ⊳2 C g z D 0
This equation must still be fulfilled at z D . If we approximate the potential
and the wave elevation by arbitrary approximations 8 and , linearization
about the approximated potential gives at z D :
r8 Ðr⊲ 1
2⊲r8⊳2 Cr8Ðr⊲ 8⊳⊳ Cr⊲ 8⊳ Ðr⊲ 1
2⊲r8⊳2⊳ C g z D 0
8 and 8 are developed in a Taylor expansion about . The Taylor expansion
is truncated after the linear term. Products of
8 are neglected. This yields at z D :
with derivatives of
r8 Ðr⊲ 1
2 ⊲r8⊳2 Cr8 Ðr⊲ 8⊳⊳ Cr⊲ 8⊳ Ðr⊲ 1
2 ⊲r8⊳2 ⊳ C g z
C [ 1
2r8 Ðr⊲r8⊳2 C g8z]z⊲ ⊳ D 0
A consistent linearization about 8 and substitutes by an expression
depending solely on , 8⊲ ⊳ and ⊲ ⊳. For this purpose, the original expression
for is also developed in a truncated Taylor expansion and written at z D :
D 1
2g ⊲ ⊲r8⊳2 C 2r8 Ðr C 2r8 Ðr8z⊲ ⊳ V 2 ⊳
D
1
2 ⊲2r8 Ðr ⊲r8⊳2 V 2 ⊳ g
g Cr8 Ðr8z