Practical Ship Hydrodynamics

234 Practical Ship Hydrodynamics
2. Three-dimensional case
The potential of a three-dimensional source is:
S
1
ϕ D
⊲1⊳
4 jEx Exqj
Figure 6.12 shows a triangular patch ABC and a source S. Quadrilateral
patches may be created by combining two triangles. The zero-flow condition
for this patch is
V ⊲Ea ð Eb⊳1
2
A
b
c
C �
C
i
a
B
iMi D 0
Figure 6.12 Source point S and patch ABC
The first term is the volume flow through ABC due to the uniform flow; the
index 1 indicates the x component (of the vector product of two sides of
the triangle). The flow M through a patch ABC induced by a point source
of unit strength is ˛/⊲4 ⊳. ˛ is the solid angle in which ABC is seen
from S. The rules of spherical geometry give ˛ as the sum of the angles
between each pair of planes SAB, SBC, and SCA minus :
˛ D ˇSAB,SBC C ˇSBC,SCA C ˇSCA,SAB
where, e.g.,
ˇSAB,SBC D arctan
[⊲EA ð EB⊳ ð ⊲EB ð EC⊳] Ð EB
⊲EA ð EB⊳ Ð ⊲EB ð EC⊳jEBj
Here EA, EB, EC are the vectors pointing from the source point S to the panel
corners A, B, C. The solid angle may be approximated by AŁ /d2 if the
distance d between patch centre and source point exceeds a given limit.
AŁ is the patch area projected on a plane normal to the direction from the
source to the patch centre:
Ed D 1
3⊲EA C EB C EC⊳
A Ł D 1
2 ⊲Ea ð Eb⊳ Ed
d
With known source strengths i, one can determine the potential and its
derivatives r at all patch corners. From the values at the corners A, B,
C, the average velocity within the triangle is found as:
Ev D r D
A C
En 2 EnAB C
AB
B A
En 2 EnAC
AC