Practical Ship Hydrodynamics

Ship seakeeping 129
to zero. Then the boundary conditions form a system of linear equations for
the unknown element strengths which is solved, e.g., by Gauss elimination.
Once the element strengths are known, all potentials and derivatives can be
computed.
For the computation of the total potential t , the motion amplitudes
ui remain to be determined. The necessary equations are supplied by the
momentum equations:
� �p ⊲1⊳
m⊲R Eu C R E˛ ðExg⊳ D E˛ ð EG C
[EuEa g CE˛⊲Ex ðEa g ⊳] � En dS
m⊲Exg ð R Eu⊳ C IR �
E˛ D Exg ð ⊲E˛ ð EG⊳ C ⊲p ⊲1⊳
[EuEa g CE˛⊲Ex ðEa g ⊳]⊳
ð ⊲Ex ðEn⊳ dS
G D gm is the ship’s weight, Exg its centre of gravity and I the matrix of the
moments of inertia of the ship (without added masses) with respect to the
coordinate system. I is the lower-right 3 ð 3 sub-matrix of the 6 ð 6matrix
M given in the section for the strip method.
The integrals extend over the average wetted surface of the ship. The
harmonic pressure p ⊲1⊳ can be decomposed into parts due to the incident wave,
due to diffraction, and due to radiation:
p ⊲1⊳ D p w C p d C
6�
iD1
p i ui
The pressures p w ,p d and p i , collectively denoted by p j , are determined from
the linearized Bernoulli equation as:
p j D ⊲ j t Cr ⊲0⊳ r j ⊳
The two momentum vector equations above form a linear system of equations
for the six motions ui which is easily solved.
The explicit consideration of the steady potential s changes the results for
computed heave and pitch motions for wave lengths of similar magnitude as
the ship length – these are the wave lengths of predominant interest – by as
much as 20–30% compared to total neglect. The results for standard test cases
such as the Series-60 and the S-175 agree much better with experimental data
for the ‘fully three-dimensional’ method. For the standard ITTC test case of the
S-175 container ship, in most cases good agreement with experiments could
be obtained (Fig. 4.17). Only for low encounter frequencies, the antisymmetric
motions are overpredicted, probably because viscous effects and autopilot were
not modelled at all in the computations.
If s is approximated by double-body flow, similar results are obtained as
long as the dynamic trim and sinkage are small. However, the computational
effort is nearly the same.
Japanese experiments at a tanker model indicate that for full hulls the diffraction
pressures in the forebody for short head waves ( /L D 0.3 and 0.5) are
predicted with errors of up to 50% if s is neglected (as typically in GFM or
strip methods). Computations with and without consideration of s yield large