Potential Flow over a NACA Four-Digit Airfoil

Flow visualization and surface pressure coefficient are plotted for potential flow over a member of the NACA four-digit family of airfoils. The geometry of the airfoil is controlled by varying the maximum camber and thickness. Controls are also available for discretizing the airfoil for the numerical computation, and for the angle of attack (α) the airfoil makes with the uniform onset flow. A panel method using straight-line vortex panels of linearly varying strength and external Neumann boundary conditions provides the potential-flow solution.
The default graphic is a plot of the surface pressure coefficient. The plot includes values for the lift coefficient , pitching moment coefficient about the leading edge , and location of the center of pressure . The yellow dot on the graph shows the location of the center of pressure. Alternatively, you can chose to display a flow visualization graphic showing a stream plot of the velocity field. This is an important check on the plausibility of the potential-flow solution.



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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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