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NACA 2412 at 4° by a 160-panel linear-vortex method: lift, moment, pressure distribution, critical Mach number and a profile-drag estimate

Tool	Airfoil Panel Method
Source	Jacobs, E. N., Ward, K. E., and Pinkerton, R. M. (1933). The Characteristics of 78 Related Airfoil Sections from Tests in the Variable-Density Wind Tunnel. NACA Report 460 (thickness and mean-line equations, pp. 4-5; ordinates of the 2412, Figure 18); Abbott, I. H., von Doenhoff, A. E., and Stivers, L. S. (1945). Summary of Airfoil Data. NACA Report 824 (velocity tables of the basic thickness forms, p. 71; NACA 2412 section data, p. 136).
Level	Undergraduate

Problem

Find the inviscid lift and moment coefficients, the zero-lift angle, the lift-curve slope, the surface pressure distribution, the stagnation point and suction peak, the ideal angle of attack and the critical Mach number of the NACA 2412 section at an angle of attack of 4°, and estimate its profile drag at a chord Reynolds number of 6 million.

Inputs to Enter

NACA 2412 generated from its number (Report 460 equations), 160 panels with cosine spacing, the 0.252 % trailing-edge gap taken out as a wedge
Angle of attack 4°; linear-strength vortex panels with the Kutta condition; sweep from -4° to 12° in steps of 2°
Boundary-layer estimate on: Reynolds number 6,000,000, free transition

Open the Airfoil Panel Method tool (loaded with this example)

Answers

c_l = 0.7407 from the circulation (0.7406 from the pressures, pressure drag 0.0004); Hess-Smith panels on the same nodes give 0.7425
c_m about the quarter chord -0.0609, about the leading edge -0.2456; centre of pressure at 0.3325 chord; aerodynamic centre at 0.262 chord
Zero-lift angle -2.146° and lift-curve slope 6.918 per radian, against -2.077° and 2π from thin-airfoil theory of the mean line
Stagnation point on the lower surface at x/c = 0.0045; minimum C_p = -1.444 on the upper surface at x/c = 0.014 (v/V = 1.563)
Ideal angle of attack 0.44° (c_l 0.312; thin-airfoil theory 0.26° and 0.256); least peak suction at -0.86° (minimum C_p -0.492, critical Mach number 0.703)
Critical Mach number at 4°: 0.520 by the Karman-Tsien rule (sonic C_p -1.928), 0.543 by the Prandtl-Glauert rule
Sweep: c_l = -0.224, 0.259, 0.741, 1.219, 1.691 at -4°, 0°, 4°, 8°, 12°
Boundary-layer estimate: Michel transition at 0.110 chord (upper) and 0.644 chord (lower), no turbulent separation, Squire-Young c_d = 0.00717 (0.0061 at 0°; Report 824 measured about 0.006 at the minimum)

Notes

The tool's default inputs are this case; press Submit. The numbers are the tool's own potential-flow solution, not a printed example: no public report prints a panel-method solution of the 2412 to these digits. The method is checked by test_airfoil_panel_method.php (1,074 checks) against the exact Karman-Trefftz and Joukowski sections (lift within 0.03 % at 200 panels), the circle, the ordinate tables of NACA Reports 460 and 824, the velocity tables of Report 824 (within 0.012 of the free-stream velocity for the NACA 0012), the mean-line table of Report 537 and an independent numpy solution. Potential flow has no stall: the lift at 12° is not attainable. The drag is an estimate without viscous-inviscid interaction.