Tools › Aerodynamics › Airfoils and Wings › Airfoil and Wing Lift › Example
NACA 2412 wing of aspect ratio 8, taper 0.5 and 3° washout at 5° angle of attack: thin-airfoil section and Prandtl's lifting line at Multhopp stations
| Tool | Airfoil and Wing Lift |
| Source | Glauert, H. (1926). The Elements of Aerofoil and Airscrew Theory. Cambridge University Press, Cambridge. Section 11.4 (the monoplane equation, Tables 11–13); Abbott, I. H., von Doenhoff, A. E., and Stivers, L. S. (1945). Summary of Airfoil Data. NACA Report 824 (mean-line data pp. 89–90, eq. (16) and Example 2 p. 14, NACA 2412 section data p. 136); Anderson, R. F. (1936). Determination of the Characteristics of Tapered Wings. NACA Report 572 (additional and basic distributions, Tables I–IV). |
| Level | Graduate |
Problem
A straight-tapered wing of 36 ft span, 6 ft root chord and 3 ft tip chord (S = 162 ft², aspect ratio 8, taper 0.5) uses the NACA 2412 section throughout and has 3° of linear washout. Find the section's zero-lift angle, design lift coefficient and quarter-chord moment from thin-airfoil theory, then the wing's lift coefficient, induced drag, span efficiency, lift-curve slope and zero-lift angle at 5° angle of attack of the root chord, the span loading, the station where a section c_l,max of 1.6 is reached first and the wing C_L and angle at that stall onset, and compare Schrenk's approximation with the lifting-line loading.
Inputs to Enter
- Section: NACA four-digit camber line, m = 2 % c at p = 4/10 c (the NACA 2412); c_l,max 1.6
- Planform: straight taper, b 36 ft, c_r 6 ft, c_t 3 ft, tip twist −3° (linear), quarter-chord sweep 0
- Angle of attack 5°; 20 Fourier terms (Multhopp stations θ_k = k π/40); 11 table stations; wing-body factor off
Answers
- Section (thin-airfoil closed form): a_0 = 2π per radian, α_L0 = −2.077° (Report 824 Example 2 prints −2.0° for the same camber line; measured about 0.93 × theory = −1.93°), α_i = 0.257°, c_li = 0.256, c_m,c/4 = −0.0531
- Planform: S 162 ft², A 8, λ 0.5, mean chord 4.5 ft, MAC 4.667 ft at y = 8 ft, chord-weighted mean twist −1.333°
- Lifting line (20 terms): A_1 0.019936, A_3 −0.001212, A_5 0.001146; C_L = π A A_1 = 0.5011; C_Di = 0.01028; δ 0.0290, e 0.972 (additional loading alone δ 0.0172, e 0.983); a = 4.965 per radian (0.0867 per degree), Glauert's τ 0.0619; wing α_L0 = −0.782° from the root chord
- Span loading: c c_l 3.288 ft at the root, 2.343 ft at mid semi-span; section c_l peaks at 0.561 at y/s = 0.16; basic (twist) c_l +0.065 at the root, −0.080 at 90 % semi-span
- Stall onset for c_l,max 1.6: C_L = 1.507 at α = 16.6°, first at y/s = 0.35 (6.23 ft): a root stall
- Schrenk's additional loading is within 2.1 % of the lifting line over the inner 90 % of the semi-span (rms 1.2 %); Schrenk's slope with κ = 0.020 is 5.007 per radian against 4.965
- Converged values (200 terms): C_L 0.5007, C_Di 0.01026, e 0.9723, a 4.964 per radian; SI: b 10.97 m, S 15.05 m², stall station 1.899 m
Notes
The tool's default inputs are this wing; press Submit. The library behind it reproduces Glauert's Tables 11–13 (rectangular aerofoils, four terms at 22.5°, 45°, 67.5°, 90°: τ 0.163 and δ 0.046 at aspect ratio 6), NACA Report 572's L_a and L_b tables and example factors for aspect ratio 6, taper 0.5 (u 0.995, J −0.408, w 0.0039, C_Di 0.0786 at C_L 1.2) and the Report 824 mean-line data in test_lifting_line.php (279 checks); this case is checked against an independent numpy solution (References/Lifting-Line/checks/liftline_check.py) at 20 and 200 terms.