Practical Ship Hydrodynamics

Numerical example for BEM 247
is changed by 1z and 1 giving the final sinkage and trim when converged.
The coefficients should actually change as the ship trims and sinks and thus
its actual waterline changes from the still waterline. However, this error just
slows down the convergence, but (for convergence) does not change the final
result for trim and sinkage.
7.3.2 Numerical implementation
The velocity potential is approximated by parallel flow superimposed by
a finite number of elements. These elements are for STEADY higher-order
panels lying on the ship surface, linear panels (constant strength) in a layer
above part of the free surface, and vortex elements lying on the local centre
plane of any side floater. However, the choice of elements is rather arbitrary.
If just wave resistance computations are performed, first-order elements are
sufficient and actually preferable due their greater robustness.
The free-surface elements are again usually ‘desingularized’. We place them
approximately one panel length above the still-water plane ⊲z D 0⊳.
We formulate the potential as the sum of parallel uniform flow (of speed
V) and a residual potential which is represented by the elements:
D Vx C �
iϕ
i is the strength of the ith element, ϕ the potential of an element of unit
strength. The index i for ϕ is omitted for convenience but it should be understood
in the equations below that ϕ refers to the potential of only the ith
element. The expression ‘element’ refers to one panel or vortex and all its
mirror images.
Then the no-penetration boundary condition on the hull becomes:
�
i⊲En Ðrϕ⊳ D Vn1
The Kutta condition becomes:
�
iϕy D 0
The transom stern condition becomes:
�
iϕx D V
�
max⊲0,V 2 2gzT⊳
The linearized free surface condition becomes then:
�
i⊲2⊲Earϕ C 8x8yϕxy C 8x8zϕxz C 8y8zϕyz⊳ C 8 2 x ϕxx
C 8 2 y ϕyy C 8 2 z ϕzz C gϕz Br8rϕ⊳
D 2⊲Ear8 C a1V⊳ B⊲ 1
2 ⊲⊲r8⊳2 C V 2 ⊳ g C V8x⊳
These four equations form a linear system of equations in the unknown element
strengths i. Once the system is solved, the velocities (and higher derivatives of
the potential) are determined on the water surface and the error ε is determined.
A special refinement accelerates and stabilizes to some extent the iteration