Oswald-Efficiency Induced Drag
Induced drag coefficient and force from lift, aspect ratio, and Oswald efficiency.
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The engineering
Induced drag is the price of making lift: the trailing vortices tilt the local lift vector aft, and the leftover component points downstream as drag. It scales with the square of lift coefficient and inversely with aspect ratio, so it dominates at low speed and high angle of attack — climb, loiter, and the back side of the drag curve.
The Oswald efficiency e (typically 0.7–0.85 for a real wing, 1.0 for an ideal elliptical loading) lumps together non-elliptical span loading and the lift-dependent part of parasite drag. If your induced drag looks too small, check that you used the finite-wing e and not 1.0 — Raymer's fitted values run closer to 0.8 for straight-wing subsonic aircraft.
Bump aspect ratio and the induced penalty falls linearly, which is why sailplanes and long-endurance UAVs carry such slender wings — at the cost of structural weight and roll response.
Where this math comes from
Ludwig Prandtl's lifting-line theory (1918–1919) gave the first rigorous account of why a finite wing drags even in inviscid flow: the shed vortex sheet induces downwash, and the induced drag is minimized by an elliptical spanwise lift distribution. His C_Di = C_L²/(πAR) is the ideal-wing floor.
Real wings never hit that floor, so W. Bailey Oswald introduced the span efficiency factor e in a 1933 NACA report to fold non-ideal loading and lift-dependent parasite drag into one multiplier. Daniel Raymer's Aircraft Design textbook popularized the practical e estimates that conceptual designers still plug in today.
- 1918Ludwig PrandtlLifting-line theory derives induced drag from trailing vortices.
- 1933W. Bailey OswaldNACA Report 408 introduces the airplane efficiency factor e.
- 1989Daniel P. RaymerAircraft Design: A Conceptual Approach codifies practical e estimates.
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