Practical Ship Hydrodynamics

Boundary element methods 223
the source density ( x and y terms) are expressed in terms of the source
strength density at the panel collocation point and at the collocation points
of panels bordering the panel in question. First the collocation points of the
adjacent panels are transformed into the local coordinate system of the panel
in question. Then the above equation for the source strength is fitted in a
least squares sense to the values of source density at the collocation points of
the adjacent panels to determine 0, x,and y. For a four-sided panel which
does not lie on a boundary of the body, four adjacent panel collocation
points will be available for performing the least squares fit, (Fig. 6.6). In
(x o3 , h o3 )
3
(x o2 , h o2 )
2
(x o0 , h o0 )
0
(x o1 , h o1 )
1
(x o4 , h o4 )
Figure 6.6 Adjacent panels used in the least-squares fit for the source density derivatives
4
other cases only three or possibly two adjacent panels will be available. The
procedure expresses the unknown source strength derivatives in terms of the
source density at the collocation point of the adjacent panels. If the higherorder
terms are set to zero, the element reduces to the regular first-order
panel. A corresponding option is programmed in our version of the panel.
6.3 Vortex elements
Vortex elements are useful to model lifting flows, e.g. in the lifting-line method
for propellers and foils, see section 2.3, Chapter 2.
1. Two-dimensional case
Consider a vortex of strength 0 at xw, zw and a field point x, z. Denote
1x D x xw and 1z D z zw. The distance between the two points is
r D p 1x 2 C 1z 2 . The potential and velocities induced by this vortex are:
D 0
2
arctan z zw
x xw