Practical Ship Hydrodynamics

7
Numerical example for BEM
7.1 Two-dimensional flow around a body in infinite fluid
One of the most simple applications of boundary element methods is the
computation of the potential flow around a body in an infinite fluid. The
inclusion of a rigid surface is straightforward in this case and leads to the
double-body flow problem which will be discussed at the end of this chapter.
7.1.1 Theory
We consider a submerged body of arbitrary (but smooth) shape moving with
constant speed V in an infinite fluid domain. For inviscid and irrotational flow,
this problem is equivalent to a body being fixed in an inflow of constant speed.
For testing purposes, we may select a simple geometry like a circle (cylinder
of infinite length) as a body.
For the assumed ideal fluid, there exists a velocity potential such that
Ev Dr . For the considered ideal fluid, continuity gives Laplace’s equation
which holds in the whole fluid domain:
1 D xx C zz D 0
In addition, we require the boundary condition that water does not penetrate
the body’s surface (hull condition). For an inviscid fluid, this condition can be
reformulated requiring just vanishing normal velocity on the body:
En Ðr D 0
En is the inward unit normal vector on the body hull. This condition is mathematically
a Neumann condition as it involves only derivatives of the unknown
potential.
Once a potential and its derivatives have been determined, the forces on the
body can be determined by direct pressure integration:
�
f1 D pn1 dS
S
�
f2 D pn2 dS
S
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