Advanced10 min read7 / 9

Drag & finite wings

Lift is never free. The air also pushes backward, and that drag sets how much thrust you need and how far a glider travels. Drag has several distinct sources — and one of them only exists because real wings have ends.

The two-dimensional drag of a section

For a 2-D section (our solver’s world) the profile drag has two viscous pieces, since the inviscid part is zero:

c_d = c_{d,\text{friction}} + c_{d,\text{form}}

(1)

Skin-friction drag — the tangential shear of the boundary layer. For a flat plate it scales with Reynolds number as $C_f \approx 1.328/\sqrt{Re}$ (laminar) or $C_f \approx 0.074/Re^{1/5}$ (turbulent). Turbulent layers rub harder.
Form (pressure) drag — when the flow separates, the pressure no longer fully recovers at the rear, leaving a net rearward push. It grows rapidly once separation sets in near stall.

Finite wings: the price of having tips

A real wing is not infinitely long. The high pressure below and suction above leak around each tip, rolling up into two trailing tip vortices. These induce a downward velocity — a downwash $w$ — over the wing, tilting the effective oncoming flow downward.

Fig. 1. Pressure leaks around the tips into trailing vortices. Their downwash tilts the local flow down, so the lift vector tilts back — that backward component is induced drag.

Because lift is perpendicular to the local flow, tilting the flow down tilts the lift backward, and that rearward component is a brand-new drag that exists even with zero friction: induced drag, the unavoidable cost of making lift with a finite wing.

C_{D,i} = \frac{C_L^2}{\pi\, e\, AR}, \qquad AR = \frac{b^2}{S}

(2)

$AR$: aspect ratio (span² / area) — long thin wings have high AR [—]
$b$: wingspan [m]
$e$: span efficiency factor (≈0.7–0.95; 1 for an ideal elliptical wing) [—]

Finiteness also reduces the lift slope: the downwash means a finite wing needs a larger geometric angle for the same lift,

a = \frac{a_0}{1 + a_0/(\pi\, e\, AR)}

(3)

where $a_0 = 2\pi$ is the 2-D (infinite-wing) slope from thin-aerofoil theory. As $AR \to \infty$ , $a \to a_0$ , recovering the 2-D result our section model computes.

Lift-to-drag ratio and gliding

The single best measure of aerodynamic efficiency is the lift-to-drag ratio $L/D = C_L/C_D$ . It has a vivid meaning for an unpowered glider in steady descent, where the glide angle $\gamma$ satisfies

\tan\gamma = \frac{D}{L} = \frac{1}{L/D}

(4)

So a wing with $L/D = 50$ glides 50 m forward for every 1 m it descends. Modern sailplanes reach 50–70; an airliner cruises near 17–20; a brick is about 0.