Panel methods are numerical models for solving potential flow over an object, usually of relatively complex geometry. These methods are based on a boundary-integral solution to Laplace's equation. Source, vortex, or doublet singularities are placed on the boundary of the object and suitable boundary conditions are applied. Refer to [2] for details.

The panel method implemented in this Demonstration uses vortex panels and external Neumann boundary conditions. The discretized airfoil is specified by

straightline panels, where the circulation density for each panel varies linearly with distance from the beginning of the panel.

for

.

Strength parameters are denoted by

,

, and the panel-local coordinate is denoted by

. Strength parameters for the discretized airfoil are grouped as lists,

,

.

There are

unknown strength parameters in the formulation of this model. Tangent-flow boundary conditions at collocation points provide

equations. Collocation points are placed at mid-panel, just outside the airfoil. Continuity of circulation density between panels (except at trailing edge) provides

equations, and a Kutta condition (zero circulation density at trailing edge) provides the final equation for a system of

linear algebraic equations involving the unknown strength parameters. The system of equations is solved using the

*Mathematica* function

LinearSolve.

This Demonstration uses dimensionless parameters, with the characteristic length of the problem being the chord of the airfoil and the characteristic speed being that of the uniform onset flow.

[1] I. H. Abbott and A. E. von Doenhoff,

*Theory of Wing Sections*, New York: Dover Publications, 1949.

[2] J. Katz and A. Plotkin,

*Low-Speed Aerodynamics*, 2nd ed., Cambridge: Cambridge University Press, 2001.