Practical Ship Hydrodynamics

with:
Boundary element methods 235
EnAB D Eb
Ec Ð Eb
Ec 2 Ec and EnAC DEc
Eb ÐEc
Eb
Eb
2
With known Ev and corner velocities EvA, EvB, EvC, the pressure force on the
triangle can be determined:
�
�
1 Ef DEn p dA DEn
2
V 2 �
Ð A Ev 2 �
dA
where Ev is not constant! A D 1
2 jEa ð Ebj is the patch area. To evaluate this
equation, the velocity within the patch is approximated by:
Ev D Ev C ⊲EvA Ev⊳⊲2r 2
r⊳ C ⊲EvB Ev⊳⊲2s 2
s⊳ C ⊲EvC Ev⊳⊲2t 2
r is the ‘triangle coordinate’ directed to patch corner A: r D 1atA,and
r D 0 at the line BC. s and t are the corresponding ‘triangle coordinates’
directed to B resp. C. Using this quadratic Ev formula, the integral of Ev 2
over the triangle area is found after some algebraic manipulations as:
�
Ev 2 dA D A Ð
�
Ev 2 C 1
30 ⊲EvA Ev⊳ 2 C 1
30 ⊲EvB Ev⊳ 2 C 1
30 ⊲EvC Ev⊳ 2
1
90 ⊲EvA Ev⊳⊲EvB Ev⊳
1
90 ⊲EvC
�
Ev⊳⊲EvA Ev⊳
1
90 ⊲EvB Ev⊳⊲EvC Ev⊳
t⊳