Panel Methods -- End Notes
After the AIC matrix and boundary conditions are computed, we can solve
for the unknown singularity strengths, and finally the complete flow field
and pressures.
In practical cases we usually do not invert the AIC matrix, but rather solve
the linear system for the singularity strengths.
It seems then that the solution is unique: we have a linear system of equations
and as long as the number of boundary conditions and unknowns is equal (AIC
is a square matrix) and the matrix is not singular, then the answer is unique.
This is not quite true in that we have simply decided where to put the panels
and this decision is not unique.
We chose to put panels on the body surface because that is where we might
have rotational flows and the panels can be used to model the vorticity
in shear layers such as boundary layers. But in 3D, lifting surfaces shed
vortex wakes so panels must be put in this region too. The location of this
wake can affect the answer to some extent. It is especially important to
position the wake properly in the case of interfering lifting surfaces.
We do not necessarily have to restrict the placement of singularities to
the surface panels or thin wake. In cases for which some vorticity exists
in the flow it is possible to model the flow with singularities in the flow.
This is the idea behind vortex methods. These are especially useful in 2-D
where point vortices are shed into the flow and are allowed to move with
the local flow velocity. The effect of each singularity on the others (and
on the body) is computed and the process is integrated forward in time (See
Spalart, 1988).
Flow with separation over a four element airfoil. Vortex method with 1300
vortices from Spalart 1988.
Memory requirements
Panel methods have the advantage that we need to solve only for quantities
on the surfaces in which we are interested and do not need to keep track
of velocities throughout the flow field as we do with finite difference
methods. But this does not always mean that the memory requirements are
smaller. The problem is that we need to compute the effect of each panel,
not on just the neighboring panels, but on all other panels. The AIC matrix
is Npanels x Npanels, so a wing with 20 chordwise panels and 40 spanwise
has 1600 panels (upper and lower surfaces) and 2.56 million influence coefficients.
This requires 10.2MB to store, forcing many older codes to use disk space for
storage and slowing down the process.
The moral is that for complex geometries, panel methods may not be faster
than finite difference methods and require "serious" computing
power.