Finite Wings, Induced Drag & the Drag Polar

Real wings have tips. Learn wing geometry (span, aspect ratio, taper, sweep, dihedral), tip vortices, downwash and induced drag, lifting-line theory and the Oswald efficiency factor, elliptic lift distribution, winglets, the components of drag (skin friction, form, interference, wave), the drag polar C_D = C_D0 + kC_L², and maximum L/D — with Python computing the polar and minimum-drag speed.

Air leaks around a wing tip from the high-pressure bottom to the low-pressure top, forming tip vortices. They push the air behind the wing down (downwash), tilting the lift backwards — a drag that exists only because lift is being produced: induced drag.

Total drag is the sum of parasite drag (which grows with V²) and induced drag (which falls with V²). The balance gives a best speed and a maximum lift-to-drag ratio that determines range and glide performance.

An analogy

Induced drag is the 'price of lift': like a rower pushing water backwards to move forwards, a wing must push air down to stay up. Longer, slimmer wings push more air down more gently — gliders and albatrosses use exactly this trick.

Wing planform

Span b, area S, aspect ratio AR = b²/S (airliners 8–10, gliders 20–40, fighters 2–4), taper ratio λ = tip chord/root chord, sweep Λ (delays compressibility effects), dihedral (roll stability) and twist/washout (tip stalls last).

Tip vortices, downwash and induced drag

The pressure difference drives flow around the tips into two trailing vortices. They induce downwash at the wing, reducing the effective angle of attack and tilting the lift vector backwards by the induced angle α_i = C_L/(πeAR). The rearward component is induced drag. Prandtl's lifting-line theory shows an elliptic lift distribution minimises it (e = 1).

Drag breakdown

Total drag components:

Reducing drag

Winglets and raked tips increase effective span without the full structural penalty (~3–5% fuel savings). Laminar flow control, riblets and smooth surfaces cut friction. Area ruling reduces transonic wave drag. Swept wings reduce the effective Mach number normal to the leading edge (M cos Λ). Truss-braced and very high aspect-ratio wings are being developed for next-generation airliners.

Formulas

Worked example: Aspect ratio

Given: Span 36 m, area 124 m²

Find: AR.

Answer: AR = 10.5

Worked example: Induced drag

Given: C_L = 0.5, e = 0.8, AR = 9

Find: C_Di.

Answer: 0.0111

Worked example: Best L/D

Given: C_D0 = 0.025, k = 0.0442

Find: (L/D)max.

Answer: 15.0

Drag polar and best L/D

import math
# Drag polar CD = CD0 + CL^2/(pi e AR) and best L/D
CD0, e, AR = 0.025, 0.8, 9.0
k = 1 / (math.pi * e * AR)
CL_md = math.sqrt(CD0 / k)
LD_max = 1 / (2 * math.sqrt(CD0 * k))
print(f"k = {k:.4f}, CL for (L/D)max = {CL_md:.3f}, (L/D)max = {LD_max:.2f}")
# Aircraft: W = 12 kN, S = 16 m^2, sea level
W, S, rho = 12000.0, 16.0, 1.225
V_md = math.sqrt(2 * W / (rho * S * CL_md))
print(f"minimum-drag speed {V_md:.1f} m/s, drag there {W/LD_max:.0f} N")
for V in (30, 45, 70):
    CL = 2 * W / (rho * V**2 * S)
    D = 0.5 * rho * V**2 * S * (CD0 + k * CL**2)
    print(f"V={V} m/s: CL={CL:.3f}  D={D:.0f} N")
# k = 0.0442, CL for (L/D)max = 0.752, (L/D)max = 15.04
# minimum-drag speed 40.4 m/s, drag there 798 N
# V=30 m/s: CL=1.361  D=942 N
# V=45 m/s: CL=0.605  D=817 N
# V=70 m/s: CL=0.250  D=1333 N

Below V_md, drag rises because induced drag dominates ('back side of the drag curve' — slowing down needs more thrust). At V_md parasite and induced drag are equal. The best glide ratio equals (L/D)max ≈ 15 here; a sailplane with AR = 25 reaches 40–60.

Common mistakes

Applications

Key takeaways

Going further