Practical Ship Hydrodynamics

216 Practical Ship Hydrodynamics
computed by simple numerical integration if the integrands are transformed
analytically to remove singularities. In the formulae for this element, En is the
unit normal pointing outward from the body into the fluid, c � the integral over
S excluding the immediate neighbourhood of Exq, andr the Nabla operator
with respect to Ex.
1. Two-dimensional case
A Rankine source distribution on a closed body induces the following potential
at a field point Ex:
�
⊲Ex⊳ D ⊲Exq⊳G⊲Ex, Exq⊳ dS
S
S is the surface contour of the body, the source strength, G⊲Ex, Exq⊳ D
⊲1/2 ⊳ ln jEx Exqj is the Green function (potential) of a unit point source.
Then the induced normal velocity component is:
�
vn⊲Ex⊳ DEn⊲Ex⊳r ⊲Ex⊳ D C ⊲Exq⊳En⊲Ex⊳rG⊲Ex, Exq⊳ dS C
S
1
2 ⊲Exq⊳
Usually the normal velocity is given as boundary condition. Then the important
part of the solution is the tangential velocity on the body:
�
vt⊲Ex⊳ D Et⊲Ex⊳r ⊲Ex⊳ D C ⊲Exq⊳Et⊲Ex⊳rG⊲Ex, Exq⊳ dS
S
Without further proof, the tangential velocity (circulation) induced by a
distribution of point sources of the same strength at point Exq vanishes:
�
C rG⊲Ex, Exq⊳Et⊲Ex⊳ dS D 0
S⊲Ex⊳
Exchanging the designations Ex and Exq and using rG⊲Ex, Exq⊳ D rG⊲Exq, Ex⊳,
we obtain:
�
C rG⊲Ex, Exq⊳Et⊲Exq⊳ dS D 0
S
We can multiply the integrand by ⊲Ex⊳ – which is a constant as the integration
variable is Exq – and subtract this zero expression from our initial
integral expression for the tangential velocity:
�
�
vt⊲Ex⊳ D C ⊲Exq⊳Et⊲Ex⊳rG⊲Ex, Exq⊳ dS C ⊲Ex⊳rG⊲Ex, Exq⊳Et⊲Exq⊳ dS
S
S
� �� �
�
D0
D C [ ⊲Exq⊳Et⊲Ex⊳
S
⊲Ex⊳Et⊲Exq⊳]rG⊲Ex, Exq⊳ dS
For panels of constant source strength, the integrand in this formula tends to
zero as Ex !Exq, i.e. at the previously singular point of the integral. Therefore
this expression for vt can be evaluated numerically. Only the length 1S of
the contour panels and the first derivatives of the source potential for each
Ex, Exq combination are required.