ReferenceAirfoil and Wing Geometry

Formula Sheet

Every key result from the module in one place. Statistics are from Raymer, Chapters 4 and 6; check the values against your edition.

Fits on one page of Letter or A4. For a digital copy, choose “Save as PDF” as the printer.

Airfoils

NACA \(mptt\) (fractions of the chord)

\[ y_c = \frac{m}{p^2}(2px - x^2)\ \ (x \le p) \] \[ y_c = \frac{m}{(1 - p)^2}\left[1 - 2p + 2px - x^2\right]\ \ (x > p) \] \[ \begin{aligned} y_t = 5t\,\big(&0.2969\sqrt x - 0.1260x - 0.3516x^2 \\ &+ 0.2843x^3 - 0.1015x^4\big) \end{aligned} \]
  • Thickest at \(x = 0.30\); leading-edge radius \(r = 1.1019\,t^2c\).
  • Five-digit: first digit \(\times 0.15\) = design \(c_l\). Six-series \(64_2\)-215: design \(c_l\) 0.2, 15%. SC(2)-0410: design \(c_l\) 0.4, 10%.

More in Lesson 1

Lift, moment, Reynolds number

Thin-airfoil theory and \(Re\)

\[ c_l = 2\pi(\alpha - \alpha_{L0}), \qquad M' = q\,c^2c_{m,c/4} \] \[ Re = \frac{\rho Vc}{\mu} \]
  • NACA four-digit with \(p = 0.4\): \(\alpha_{L0} \approx -1.04^\circ\) and \(c_{m,c/4} \approx -0.027\) per 1% camber. Aerodynamic center at \(c/4\); center of pressure \(x_{cp}/c = 0.25 - c_{m,c/4}/c_l\).
  • Stall: thick (\(\gtrsim 14\%\)) trailing-edge, gentle; moderate (9 to 13%) leading-edge, abrupt; thin, long bubble, low \(c_{l,\max}\).

More in Lesson 2

Design lift coefficient and thickness

Cruise \(C_L\) and the Korn equation

\[ C_L = \frac{W/S}{q}, \qquad q = \tfrac12\gamma pM^2, \qquad C_L^{*} = \sqrt{\CDz/K} \] \[ \MDD = \frac{\kappa_A}{\cos\Lambda} - \frac{t/c}{\cos^2\Lambda} - \frac{c_l}{10\cos^3\Lambda} \] \[ \frac tc = \kappa_A\cos\Lambda - \frac{c_l}{10\cos\Lambda} - \MDD\cos^2\Lambda \]
  • Jet best range at \(C_L^{*}/\sqrt3\); propeller at \(C_L^{*}\). Airfoil design \(c_l\) near the cruise \(C_L\).
  • \(\kappa_A = 0.87\) conventional, 0.95 supercritical.

More in Lesson 3

Aspect ratio

Span, induced drag, lift-curve slope

\[ A = \frac{b^2}{S}, \qquad \bar c_g = \frac Sb, \qquad K = \frac{1}{\pi Ae} \] \[ \CLa = \frac{2\pi A}{2 + \sqrt{4 + \dfrac{A^2\beta^2}{\eta^2}\left(1 + \dfrac{\tan^2\Lambda}{\beta^2}\right)}} \]
  • \(\beta^2 = 1 - M^2\), \(\eta \approx 0.95\). Prandtl (elliptic, low speed): \(\CLa = a_0/[1 + a_0/(\pi A)]\), too high below \(A \approx 6\). Statistics: propeller 6 to 9.2; jets \(A = aM_{\max}^C\) (transport 7.5, trainer \(4.737M^{-0.979}\), fighter \(5.416M^{-0.622}\)); sailplanes \(0.19\LDmax^{1.3}\).

More in Lesson 4

Sweep

Normal Mach number, Mach cone, chord lines

\[ M_n = M\cos\Lambda, \qquad \Lambda_{LE} > 90^\circ - \arcsin\frac1M \] \[ \tan\Lambda_n = \tan\Lambda_m - \frac4A\left[(n - m)\frac{1 - \lambda}{1 + \lambda}\right] \]
  • Costs: lower \(\CLa\) and \(C_{L,\max}\), tip stall and pitch-up, weight, aileron reversal. About \(1^\circ\) of effective dihedral per \(10^\circ\) of sweep.

More in Lesson 5

Planform

Trapezoidal wing and Schrenk loading

\[ b = \sqrt{AS}, \qquad c_r = \frac{2S}{b(1 + \lambda)}, \qquad c_t = \lambda c_r \] \[ \bar c = \frac23c_r\frac{1 + \lambda + \lambda^2}{1 + \lambda}, \qquad \bar y = \frac b6\,\frac{1 + 2\lambda}{1 + \lambda} \] \[ \bar x_{LE} = \bar y\tan\Lambda_{LE}, \qquad \frac{c_l}{C_L} = \frac{c + c_e}{2c},\ \ c_e = \frac{4S}{\pi b}\sqrt{1 - \eta^2} \]
  • \(\lambda \approx 0.45\) unswept, 0.2 to 0.35 swept. Aerodynamic center at \(\bar c/4\).

More in Lesson 6

Twist, incidence, dihedral

Incidence for a level fuselage

\[ i_w = \alpha_{L0} + C_L/\CLa \]
  • Washout 0 to \(5^\circ\). Incidence about \(2^\circ\) light aircraft, \(1^\circ\) transports.
  • Dihedral (low/mid/high): unswept 5 to 7 / 2 to 4 / 0 to 2; subsonic swept 3 to 7 / \(-2\) to 2 / \(-5\) to \(-2\); supersonic 0 to 5 / \(-5\) to 0 / \(-5\) to 0.

More in Lesson 7

Tails

Tail volume coefficients

\[ S_{HT} = \frac{c_{HT}\bar c_WS_W}{L_{HT}}, \qquad S_{VT} = \frac{c_{VT}b_WS_W}{L_{VT}} \]
  • \(c_{HT},\ c_{VT}\): GA single 0.70, 0.04; GA twin 0.80, 0.07; twin turboprop 0.90, 0.08; jet trainer 0.70, 0.06; fighter 0.40, 0.07; jet transport 1.00, 0.09.
  • Arm: 60% of the fuselage (nose engine), 50 to 55% (wing engines), 45 to 50% (rear engines).

More in Lesson 8

Common mistakes

  • Degrees in \(2\pi(\alpha - \alpha_{L0})\). Convert to radians.
  • The sign of \(\alpha_{L0}\). It is negative for positive camber: \(\alpha - \alpha_{L0}\) adds.
  • kg/m² for \(W/S\). Multiply by \(g\) for \(C_L\).
  • Forgetting \(\gamma\). \(q = 0.7pM^2\), not \(0.5pM^2\).
  • MAC for the fin. The vertical tail uses the span \(b_W\).
  • Mean geometric chord for the MAC. \(S/b \ne \bar c\) unless \(\lambda = 1\).