Finite wings, downwash and induced drag

Why the lift vector tilts aft

A finite lifting wing creates a trailing vortex system and downwash across the span.

Effective angle of attack

Downwash tilts the local relative airflow downward. The effective section angle of attack is smaller than the geometric angle referred to undisturbed flow. The aerodynamic resultant tilts aft relative to that undisturbed flow.

The aft component is induced drag. Vortices are evidence of the three-dimensional lifting flow; treating them as an unrelated friction effect misses why induced drag changes with lift.

Lift distribution matters

For the same area and CL, greater aspect ratio usually reduces induced drag. Span efficiency e describes departure from an ideal elliptical loading in a simple lifting-line model; it is not propeller efficiency.

Winglets can reduce induced losses through their effect on the wake and spanwise loading. Their benefit must be weighed against wetted area, interference, structure and operating conditions.

Compare span at equal supported lift

Take CL = 1.0 and e = 0.8. A wing of aspect ratio 8 gives CDi = 1/(π × 8 × 0.8) = 0.0497. At aspect ratio 12, the value becomes 0.0332. The reduction is one third when reference area, supported lift and dynamic pressure are unchanged. The force comparison follows because both coefficients are multiplied by the same qS.

If the reference area were changed at the same time, that simple force ratio would no longer follow directly. Increasing span, area and mass together requires a new complete comparison. The aspect-ratio equation isolates a geometrical effect; it does not by itself establish which complete aircraft is more economical.

In the laboratory, keep lift and speed fixed while raising aspect ratio. Observe the induced coefficient and force together. Explain that the downwash effect changes the effective angle and resultant direction rather than adding an unrelated backward friction force. Then vary lift to see why the induced term can dominate slow, heavily loaded flight.

Separate coefficient from drag force

The coefficient relation and the fixed-lift speed relation answer different questions.

Coefficient scaling

CDi = CL²/(πARe). At CL = 1, AR = 8 and e = 0.8, CDi ≈ 0.0497. Doubling CL multiplies CDi by four, provided the model remains applicable.

Induced drag force is Di = qSCDi. At fixed load L, substitute CL = L/(qS): Di = L²/(qSπARe). At fixed density it therefore falls as 1/V².

Apply the correct constraint

At fixed CL and geometry, raising speed increases lift and induced drag force with q. At fixed weight in steady level flight, raising speed reduces CL and induced drag. Always state which variable is held fixed.

A level turn at the same speed requires L = nW. Induced drag then scales as n², which explains a significant thrust demand during sustained high-load manoeuvring.

A turn consumes thrust margin

Suppose the straight-level induced drag is 2 kN at a particular speed. A sustained level turn at load factor 1.5 needs 1.5 times the lift and approximately 2.25 times the induced drag: 4.5 kN. If parasite drag stays approximately 3 kN, total drag rises from 5 to 7.5 kN. Doubling only the induced term would underestimate the thrust demand.

The same load-factor increase raises the required coefficient and brings the wing closer to CLmax. Available thrust and stall margin can therefore deteriorate together. If the aircraft cannot supply the required thrust, it cannot sustain that speed, bank and level path indefinitely; height or speed must change unless the demand is reduced.

First predict the induced-drag ratio before moving the controls. Hold speed fixed and increase required lift by 50%. Then restore the original lift and raise speed by 50%. The opposite directions of change reveal why a statement about “more speed gives more drag” must name the drag component and the supported-load constraint.

Can you explain and apply the mechanism?

Four original questions. Work from the stated facts, then compare your reasoning with the explanation.