Practical Ship Hydrodynamics

Numerical example for BEM 237
S is the wetted surface. p is the pressure determined from Bernoulli’s equation:
p D 2 ⊲V 2
⊲r ⊳ 2 ⊳
The force coefficients are then:
Cx D f1
2 V2 S
Cz D f2
2 V2 S
7.1.2 Numerical implementation
The velocity potential is approximated by uniform flow superimposed by
a finite number N of elements. These elements are in the sample program
DOUBL2D desingularized point sources inside the body (Fig. 6.10). The
choice of elements is rather arbitrary, but the most simple elements are selected
here for teaching purposes.
We formulate the potential as the sum of parallel uniform flow (of speed
V) and a residual potential which is represented by the elements:
D Vx C �
iϕ
i is the strength of the ith element, ϕ the potential of an element of unit
strength. The index i for ϕ is omitted for convenience but it should be
understood in the equations below that ϕ refers to the potential of only the
ith element.
Then the Neumann condition on the hull becomes:
N�
iD1
i⊲En Ðrϕ⊳ D Vn1
This equation is fulfilled on N collocation points on the body forming thus
a linear system of equations in the unknown element strengths i. Once the
system is solved, the velocities and pressures are determined on the body.
The pressure integral for the x force is evaluated approximately by:
�
pn1 dS ³
S
N�
iD1
pin1,isi
The pressure, pi, and the inward normal on the hull, ni, are taken constant
over each panel. si is the area of one segment.
For double-body flow, an ‘element’ consists of a source at z D zq and its
mirror image at z D zq. Otherwise, there is no change in the program.