Practical Ship Hydrodynamics

240 Practical Ship Hydrodynamics
Substituting this expression in our equation for the free-surface condition gives
the consistently linearized boundary condition at z D :
r8r[ ⊲r8⊳ 2 Cr8Ðr ] C 1
2r r⊲r8⊳2 1
[ 2 C g z C r8r⊲r8⊳2 C g8z]z
g Cr8Ðr8z 1
ð ⊲ 2 [ ⊲r8⊳2 C 2r8 Ðr V 2 ] g ⊳ D 0
The denominator in the last term becomes zero when the vertical particle
acceleration is equal to gravity. In fact, the flow becomes unstable already at
0.6 to 0.7g both in reality and in numerical computations.
It is convenient to introduce the following abbreviations:
Ea D 1
2 r⊲⊲r8⊳2 � �
8x8xx C 8z8xz
⊳ D
8x8xz CC8z8zz
B D
[ 1
2 r8r⊲r8⊳2 C g8z]z
g Cr8 Ðr8z
D
D 1
⊲8
g C a2
2 x8xxz C 8 2 z 8zzz C g8zz
[r8Ea C g8z]z
g C a2
C 2[8x8z8xzz C 8xz Ð a1 C 8zz Ð a2]⊳
Then the boundary condition at z D becomes:
2⊲Ear C 8x8z xz⊳ C 8 2 x xx C 8 2 z zz C g z Br8r
D 2Ear8 B⊲ 1
2 ⊲⊲r8⊳2 C V 2 ⊳ g ⊳
The non-dimensional error in the boundary condition at each iteration step is
defined by:
ε D max⊲jEar8 C g8zj⊳/⊲gV⊳
Where ‘max’ means the maximum value of all points at the free surface.
For given velocity, Bernoulli’s equation determines the wave elevation:
z D 1
2g ⊲V2 ⊲r ⊳ 2 ⊳
The first step of the iterative solution is the classical linearization around
uniform flow. To obtain the classical solutions for this case, the above equation
should also be linearized as:
z D 1
2g ⊲V2 C ⊲r8⊳ 2
2r8r ⊳
However, it is computationally simpler to use the non-linear equation.
The bottom, radiation, and open-boundary conditions are fulfilled by the
proper arrangement of elements as described below. The decay condition – like
the Laplace equation – is automatically fulfilled by all elements.