Panel Methods
AME 60639
Spring 2008
Prepared by Lee Neuharth
Panel methods are often referred to as vortex panel methods; this refers to the cases where each
panel (or line segment from node to node) has an associated vortex, γ, at its centerpoint. A panel
method may be performed with only sources and sinks, q, but this is generally for a non lifting case
or a very thick airfoil1. When concerned with airfoils, though, it is necessary to employ the Kutta
condition.
According to the methods outlined by Moran2, the Kutta condition may be defined by assigning
a universal value of γ: the same value of γis used on each panel. This provides another variable to
accompany the extra equation given by the Kutta condition.
Using the definitions of sources and sinks (q) and vortices (γ) the velocity potential,
φ=V∞(xcos α+ysin α) +
N
X
j=1 Zjth q(s)
2πln r−
γ
2πds (1)
can then be used to find the velocity everywhere. Each source, sink, and vortex contributes a certain
amount to the velocity at each point. Through a geometric investigation, the velocity components
can be used to find the tangential velocity on each panel, and the normal velocity. This reduces to
N
X
j=1
Ai,jqj+Ai,N +1γ=bi.(2)
The right hand side is the tangent velocity, and the Kutta condition is augmented into the matrix
at the (N+ 1)th row and column, where Nis the number of nodes. These two conditions are shown
as
bi=V∞sin(θi−α),(3)
and N
X
j=1
AN+1,jqj+AN+1,N +1γ=bN+1.(4)
This discussion closely follows the equations and methods presented by Moran; a code named
‘PANEL,’ employing the inviscid impermeability and the Kutta conditions is presented in chapter
four of this text.
1J. Katz and A. Plotkin. Low-Speed Aerodynamics. New York: Wiley, 1984. Chapter 4.
2J. Moran. An Introduction to Theoretical and Computational Aerodynamics. McGraw-Hill, 1991. Page 317.