Panel Methods -- Boundary Conditions
After the AIC matrix is computed, we specify the boundary conditions.
The total normal velocity at panel i is then given by the expression below.
This must be zero if the flow is tangent to the surface of the body and
constitutes the boundary conditions of the problem.
Here, {n} is a vector of surface unit normals {sigma} represents the unknown
singularity strengths. Each element of these vectors is associated with
one panel of the geometry.
The boundary conditions for panel methods must express the requirement that
streamlines follow the surface contour. But they do not have to explicitly
set V·n = 0. In fact, the method currently more in vogue is to specify
the B.C.'s in terms of the potential. This is called a Dirichlet (as opposed
to the von Neumann) type of boundary condition. It works as follows on the
doublet panel method.
The total potential in the interior of the section is set to 0. If the
total potential is 0 everywhere inside the body (in practice it is set to
0 just inside at each panel control point) then the velocity there is 0
also. In particular the velocity normal to the panel, on the inside of
the panel is 0. Since doublets produce no jump in normal velocity (see next
section) then V·n = 0 in the external flow as well. This form of the
B.C.'s is often better behaved (numerically) than the direct (Neumann) type
of B.C..
