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
straight-line panels, where the circulation density for each panel varies linearly with the 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.
The family of NACA four-digit airfoils is used in this Demonstration. These airfoils were defined in the 1930s based on algebraic equations for camber and thickness distributions. As an example, consider the NACA 3412 airfoil whose chord is denoted by
. The first digit in the identification number specifies that the maximum camber is
, the second digit specifies that the maximum camber is located
from the leading edge, and the last two digits specify that the maximum thickness of the airfoil is
.
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. The second digit of the airfoil identification number is fixed at 4, and sliders control the maximum camber and thickness digits of the airfoil identification number. The special case of a flat plate is obtained by setting the maximum camber and thickness equal to zero.
[1] I. H. Abbott and A. E. von Doenhoff,
Theory of Wing Sections, New York: Dover, 1949.
[2] J. Katz and A. Plotkin,
Low-Speed Aerodynamics, 2nd ed., Cambridge: Cambridge University Press, 2001.
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