Lift-to-Drag Ratio⚠ unverified
Aerospace / Aerodynamics · Lift-to-drag ratio
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| CL | CL | — | 0.5 | Lift coefficient |
| CD | CD | — | 0.03 | Drag coefficient |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| ld | L/D | — | Lift-to-drag ratio |
The science & history
Understanding the Parameters
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The ratio, not the magnitudes — because $L = qSC_L$ and $D = qSC_D$ share the same $qS$, efficiency depends only on $C_L/C_D$. You can fly the same $L/D$ at many speeds and altitudes by holding the same $C_L$ (same angle of attack).
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The maximum $L/D$ — occurs at a specific angle of attack where induced and parasite drag are balanced: $(L/D)_{\max} = \tfrac12\sqrt{\pi\,AR\,e/C_{D0}}$. It rises with aspect ratio and falls with parasite drag — the two levers of clean, long-winged efficient aircraft.
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Glide angle — in an unpowered glide the glide ratio (horizontal distance per unit height lost) equals $L/D$; a glider at $L/D = 40$ travels 40 m forward for every 1 m of descent (Glide Ratio).
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Range link — for a jet, range is proportional to $L/D$ (Breguet); flying at best $L/D$ (or near it) maximises how far the fuel goes.
Derivation (Approaching a Proof)
Divide the lift equation by the drag equation (Lift Force, Drag Force):
$$\frac{L}{D} = \frac{\tfrac12 C_L\,\rho V^2 S}{\tfrac12 C_D\,\rho V^2 S} = \frac{C_L}{C_D}.$$
The dynamic pressure $\tfrac12\rho V^2$ and area $S$ cancel exactly, leaving a pure coefficient ratio. To find the best $L/D$, substitute the drag polar $C_D = C_{D0} + C_L^2/(\pi AR e)$ and maximise $C_L/C_D$ over $C_L$; setting the derivative to zero gives the optimum at $C_L^\ast = \sqrt{\pi AR e\,C_{D0}}$ (parasite drag equal to induced drag), and
$$\left(\frac{L}{D}\right)_{\max} = \frac{1}{2}\sqrt{\frac{\pi\,AR\,e}{C_{D0}}}.$$
Dimensional check. $L/D = C_L/C_D = (\text{–})/(\text{–})$ = dimensionless — a pure ratio, as required.
History and Development
The lift-to-drag ratio has been the yardstick of aerodynamic quality since the earliest aeronautics: Otto Lilienthal and the Wright brothers measured it to compare wings, and it remains the headline figure of merit. Its maximisation drives high-aspect-ratio wings (gliders, the U-2, modern airliners with winglets) and clean airframes. The result that best $L/D$ occurs when induced drag equals parasite drag is a cornerstone of performance design.
Related Concepts: Lift Force, Drag Force, Induced Drag, Parasite Drag, Glide Ratio, Lift Curve Slope
Notes: $qS$ cancels → efficiency $= C_L/C_D$. Best $L/D$ where induced = parasite drag, $(L/D)_{\max}=\tfrac12\sqrt{\pi AR e/C_{D0}}$. Equals the glide ratio (Glide Ratio); range $\propto L/D$ for jets. Sailplanes $>50$, airliners $\sim17$–$20$.