Wing Design: Span, Sweep, and the Shape of Flight

simulator intermediate ~10 min
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AR = 10.7 — induced drag factor K = 0.037

A 30m span wing with 4m root chord and 0.4 taper ratio gives aspect ratio 10.7 and wing area 84 m². The Oswald efficiency factor is about 0.80, yielding an induced drag factor of 0.037.

Formula

Aspect ratio: AR = b² / S
Wing area: S = b × c_root × (1 + λ) / 2
Induced drag coefficient: CDi = CL² / (π × e × AR)

The Wing Planform

A wing's planform — its shape viewed from above — determines most of its aerodynamic character. Aspect ratio controls induced drag, sweep governs transonic behavior, and taper ratio affects the spanwise lift distribution. Every aircraft represents a specific compromise among these parameters, optimized for its mission. A long-range airliner needs high AR and moderate sweep; a dogfighter needs low AR and high sweep for agility.

Aspect Ratio and Induced Drag

Induced drag is the penalty for generating lift in a finite wing. Air leaks around the wingtip from high pressure below to low pressure above, creating trailing vortices. Higher aspect ratio wings have less tip leakage relative to their span, producing weaker vortices and less induced drag. The relationship is direct: doubling AR approximately halves the induced drag coefficient for the same lift. This is why albatrosses and sailplanes share the same slender wing shape.

Sweep: Cheating the Sound Barrier

When an aircraft approaches Mach 1, shock waves form on the wing and drag increases dramatically. Wing sweep reduces the effective Mach number felt by the wing cross-section — if the wing is swept 30°, the perpendicular flow component is only cos(30°) ≈ 0.87 times the flight speed. This allows higher cruise speeds before drag divergence. The Boeing 707 (1958) was the first successful swept-wing airliner, establishing the template for all modern jets.

Taper and Lift Distribution

An elliptical lift distribution minimizes induced drag — Ludwig Prandtl proved this in 1920. While an elliptical wing planform achieves this naturally (like the Spitfire), it is expensive to manufacture. A trapezoidal wing with a taper ratio near 0.4 closely approximates the elliptical distribution at far lower cost. Modern wings fine-tune the distribution with twist (washout), varying airfoil sections along the span, and carefully positioned winglets.

FAQ

What is aspect ratio and why does it matter?

Aspect ratio (AR) is the wingspan squared divided by wing area — essentially how long and slender a wing is. Higher AR reduces induced drag because longer wings produce weaker wingtip vortices. Sailplanes (AR > 20) minimize induced drag for efficient soaring, while fighter jets (AR 3-5) prioritize maneuverability and structural strength.

Why do jet airliners have swept wings?

Wing sweep delays the formation of shock waves as the aircraft approaches the speed of sound. By sweeping the wing back, the component of airflow perpendicular to the leading edge is slower than the aircraft's speed, allowing higher cruise Mach numbers before drag-divergence occurs. Most airliners use 25-35° of sweep.

What is taper ratio?

Taper ratio (λ) is the ratio of tip chord to root chord. A taper ratio of 1.0 means a rectangular wing; lower values mean the wing narrows toward the tip. A taper ratio of ~0.4 approximates an elliptical lift distribution, which minimizes induced drag. Too much taper can cause tip stall.

Why do some aircraft have winglets?

Winglets reduce induced drag by weakening wingtip vortices. They act like a small increase in aspect ratio without extending the physical span (which would increase bending loads and may not fit airport gates). Modern blended winglets and split-tip devices can reduce fuel burn by 3-5% on long flights.

Sources

Embed

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