Potential Flow over a NACA Four-Digit Airfoil


	Streamlines and the surface pressure coefficient are plotted for potential flow over some members of the NACA four-digit family of airfoils. Each plot includes values for the lift coefficient () and pitching moment coefficient about the leading edge (). Camber and thickness of the airfoil are controlled by varying digits of the identification number. Controls are also available to digitize the airfoil for the numerical computation, and for the angle of attack 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 numerical solution to the model.


	Contributed by: Richard L. Fearn

The streamline plot is a compromise between computational time and smoothness of streamlines. Implausible streamlines while slider controls are being moved are numerical artifacts. Snapshot 2 shows a flat plate at an angle of attack of 7°, and the agreement of the results shown for lift and moment coefficients with those of thin-airfoil theory can serve as a plausibility check for the numerical model.

The family of NACA four-digit 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 c. The first digit in the identification number specifies that the maximum camber is 0.03 c, the second digit specifies that the maximum camber is located 0.4 c from the leading edge, and the last two digits specify that the maximum thickness of the airfoil is 0.12 c.

Panel methods are numerical models based on simplifying assumptions about the physics and properties of the flow. The viscosity of air in the flow field is neglected, and the net effect of viscosity is summarized by requiring that the flow leave the sharp trailing edge of the airfoil smoothly (Kutta condition). The compressibility of air is neglected, and the curl of the velocity field is assumed to be zero (no vorticity in the flow field). Under these assumptions, the vector velocity describing the flow field can be represented as the gradient of a scalar velocity potential,

, and the resulting flow is referred to as potential flow. A statement of conservation of mass in the flow field leads to Laplace's equation as the governing equation for the velocity potential,

To solve the problem of potential flow over an airfoil, Laplace's equation is solved subject to the boundary condition that there be no flow across the contour of the airfoil. Additionally, the flow far from the airfoil is required to be uniform. The basic solution procedure consists of discretizing the contour of the airfoil with straight-line panels, and selecting a piecewise continuous linear distribution of circulation density over the panels with unknown strength parameters. The velocity potential associated with each panel satisfies Laplace's equation. The tangent-flow boundary condition is satisfied at a discrete number of collocation points, one for each panel of the discretized airfoil. This process leads to a system of linear algebraic equations to be solved for the unknown singularity-strength parameters.

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 the convention is used that symmetric airfoils are designated as 00xx, where xx denotes the thickness of the airfoil.