Practical Ship Hydrodynamics
Practical Ship Hydrodynamics
Practical Ship Hydrodynamics
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214 <strong>Practical</strong> <strong>Ship</strong> <strong>Hydrodynamics</strong><br />
�<br />
1<br />
�<br />
x2 x<br />
dx D<br />
C a23 a 2�<br />
x2 C C<br />
C a2 �<br />
1 1<br />
dx D p arctan 2<br />
a C bx ab bx<br />
p<br />
ab<br />
1<br />
for b>0<br />
The numerical evaluation of the induced velocities has to consider some<br />
special cases. As an example: the finite accuracy of computers can lead to<br />
problems for the above given expression of the x component of the velocity,<br />
when for small values of x and z the argument of the logarithm is rounded<br />
off to zero. Therefore, for ( p x 2 C z 2 − y) thetermr1 y must be substituted<br />
by the approximation ⊲x 2 C z 2 ⊳/2x. The other velocity components<br />
require similar special treatment.<br />
Hess and Smith (1964) pioneered the development of boundary element<br />
methods in aeronautics, thus also laying the foundation for most subsequent<br />
work for BEM applications to ship flows. Their original panel used constant<br />
source strength over a plane polygon, usually a quadrilateral. This panel is<br />
still the most popular choice in practice.<br />
The velocity is again given in a local coordinate system (Fig. 6.3). For<br />
quadrilaterals of unit source strength, the induced velocities are:<br />
∂<br />
∂x D y2 y1<br />
ln<br />
d12<br />
C y4 y3<br />
d34<br />
∂<br />
∂y D x2 x1<br />
ln<br />
d12<br />
s<br />
2<br />
C x4 x2<br />
d34<br />
y<br />
3<br />
� r1 C r2 d12<br />
ln<br />
r1 C r2 C d12<br />
� r3 C r4 d34<br />
r3 C r4 C d34<br />
� r1 C r2 d12<br />
ln<br />
4<br />
r1 C r2 C d12<br />
� r3 C r4 d34<br />
x<br />
r3 C r4 C d34<br />
1<br />
−s /2<br />
2<br />
s /2 s /2<br />
�<br />
C y3 y2<br />
d23<br />
ln<br />
�<br />
C y1 y4<br />
d41<br />
�<br />
C x3 x2<br />
d23<br />
ln<br />
� r2 C r3 d23<br />
ln<br />
�<br />
C x1 x4<br />
ln<br />
d41<br />
−s /2 −s /2<br />
3<br />
s /2<br />
r2 C r3 C d23<br />
�<br />
� r4 C r1 d41<br />
r4 C r1 C d41<br />
�<br />
� r2 C r3 d23<br />
1<br />
r2 C r3 C d23<br />
� r4 C r1 d41<br />
r4 C r1 C d41<br />
s /2<br />
�<br />
�<br />
−s /2<br />
4 4<br />
Figure 6.3 A quadrilateral flat panel of constant strength is represented by Hess and Smith as<br />
superposition of four semi-infinite strips
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