Practical Ship Hydrodynamics

232 Practical Ship Hydrodynamics
1. Two-dimensional case
The potential of a two-dimensional point source is:
ϕ D 1
ln r2
4
The integral zero-flow condition for a patch is:
V Ð nx Ð l C �
i
iMi D 0
nx is the x component of the unit normal (from the body into the fluid),
l the patch area (length). The flow through a patch is invariant of the
coordinate system. Consider a local coordinate system x, z, (Fig. 6.11). The
patch extends in this coordinate system from s to s. The flow through the
patch is:
M D
� s
s
z dx
A
x
B
Figure 6.11 Patchin2d
z
x q,z q
A Rankine point source of unit strength induces at x, z the vertical velocity:
z D 1
2
z zq
⊲x xq⊳ 2 C ⊲z zq⊳ 2
Since z D 0 on the patch, this yields:
M D
� s
s
1
2
zq
⊲x xq⊳ 2 C z 2 q
dx D 1
2 arctan
lzq
x 2 q C z2 q
The local zq transforms from the global coordinates:
zq D nx Ð ⊲xq xc⊳ nz Ð ⊲zq zc⊳
xc, zc are the global coordinates of the patch centre, xq, zq of the source.
s 2