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Wingtip Vortices and Induced Drag

Source lecture(s): PC316 Ch. 8

Intuition

A finite wing generating lift carries bound circulation \(\Gamma\). At the tips that circulation cannot simply stop — Helmholtz theorem 3 forbids a vortex tube ending in the fluid. So it turns and trails downstream as a pair of counter-rotating wingtip vortices, closing a giant horseshoe with the starting vortex left behind at takeoff.

Everything below is a consequence of that one topological constraint.

Induced drag: the unavoidable price of lift

The trailing pair is a counter-rotating vortex pair, so between the tips it drives air downward — the downwash. The wing therefore flies through air it has itself deflected, and the local relative wind is tilted downward. Since lift is by definition perpendicular to the local relative wind, the lift vector tilts backwards, and its rearward component is a drag.

This is induced drag, and it exists in a perfectly inviscid fluid. It is not friction; it is the momentum you gave the air in order to be held up.

It decreases with aspect ratio — long thin wings put the tips further apart, weakening the downwash over the span. Hence the wings of gliders and albatrosses, and the winglets bolted onto airliners: a winglet is an attempt to make the tip behave as if it were further away.

Upwash, and birds

Outboard of each tip the induced flow is upward. A bird flying just outside and behind the wingtip of the one ahead sits in rising air and gets lift for free — which is why pelicans, geese and ibises fly in V-formation. Measured heart rates of formation-flying pelicans confirm the saving, and the optimal station is predicted by exactly this vortex model.

Wake turbulence

A heavy airliner's trailing pair has \(\Gamma \sim 500\) m²/s. As a counter-rotating pair they descend by mutual induction at a few m/s and persist for minutes, drifting with the wind. A light aircraft that flies into one can be rolled uncontrollably.

This is why air-traffic control enforces wake-turbulence separation — several nautical miles behind a heavy, more for a light aircraft following. The rules are a direct application of point-vortex dynamics: the descent rate \(\Gamma/2\pi d\) and the decay time determine the spacing.

Common mistakes

  • "Induced drag is friction at the tips." It is inviscid, and it would exist in an ideal fluid. Skin friction is a separate contribution.
  • "Winglets eliminate induced drag." They reduce it by modifying the tip loading. Lift always costs induced drag; the only way to zero is zero lift.
  • Assuming the vortices dissipate quickly. They persist for minutes and kilometres, which is exactly why separation rules exist.
  • Confusing the horseshoe with a closed loop of fluid. The vortex lines close; the fluid does not travel around them.

Knowledge graph position

Prerequisites: circulation, Helmholtz theorems, point vortices. Leads to: lifting-line theory, aircraft design, wake-turbulence separation standards.

Quiz

Q1 (conceptual). Why does induced drag exist even in an inviscid fluid?

Answer

Because it is not a friction effect. Generating lift requires deflecting air downward; the downwash tilts the local relative wind, and hence tilts the lift vector backward. The rearward component is drag. Momentum given to the air is paid for in drag whether or not the fluid is viscous.

Q2 (computational). An airliner has \(\Gamma = 500\) m²/s and a tip separation of 50 m. How fast does the vortex pair descend?

Answer

A counter-rotating pair translates at \(V = \Gamma/2\pi d = 500/(2\pi \times 50) = 1.6\) m/s — a slow, persistent sink, which is why the wake settles below the flight path and lingers.

Q3 (MCQ). Birds fly in a V-formation to:

  • (a) reduce air resistance by drafting directly behind
  • (b) sit in the upwash outboard of the wingtip vortex of the bird ahead
  • (c) improve visibility
  • (d) reduce the starting vortex
Answer

(b). Directly behind is downwash — the worst place to be. The gain is outboard and slightly behind, where the trailing vortex induces rising air.