Induced Drag⚠ unverified
Aerospace / Aerodynamics · Compute the induced drag coefficient
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| CL | CL | — | 1.0 | Lift coefficient (dimensionless) |
| AR | AR | — | 1.0 | Wing aspect ratio (dimensionless) |
| e | e | — | 0.8 | Oswald span efficiency factor (dimensionless). Default is 0.8 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | CDi | — | Induced drag coefficient (dimensionless) |
The science & history
Understanding the Parameters
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Lift coefficient squared $C_L^2$ — induced drag rises with the square of lift, so it explodes at high $C_L$ (slow flight, hard turns) and is negligible at cruise. In a level turn at load factor $n$, $C_L$ (and hence induced drag) scales with $n$.
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Aspect ratio $AR = b^2/S$ — the key design lever. High aspect ratio (long, thin wings) spreads the lift over a longer span, weakening the tip vortices and slashing induced drag ($C_{Di} \propto 1/AR$). Gliders and the U-2 have very high $AR$; fighters trade it away for maneuverability and structure.
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Oswald efficiency $e$ — how close the spanwise lift distribution is to the ideal elliptical shape that minimises induced drag ($e = 1$). Real wings have $e \approx 0.7$–$0.9$; twist, taper, and winglets push it up.
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Why it exists — a finite wing sheds vortices at the tips; these induce a downwash (Downwash Angle) that tilts the local lift vector backward, and that backward component is induced drag. An infinite wing has none.
Derivation (Approaching a Proof)
Prandtl's lifting-line theory models the wing as a bound vortex shedding a sheet of trailing vortices. The trailing vorticity induces a downward velocity (downwash) at the wing, tilting the effective flow by the induced angle $\alpha_i$. The lift, being perpendicular to the local (tilted) flow, gains a small streamwise component — that is the induced drag: $D_i = L\,\alpha_i$.
For the minimum-drag elliptical lift distribution, the downwash is uniform and $\alpha_i = C_L/(\pi AR)$ (Downwash Angle), giving
$$C_{Di} = C_L\,\alpha_i = \frac{C_L^2}{\pi\,AR}.$$
Real (non-elliptical) wings do slightly worse, captured by dividing by the Oswald efficiency $e \le 1$:
$$C_{Di} = \frac{C_L^2}{\pi\,AR\,e}.$$
Dimensional check. $C_{Di} = \dfrac{C_L^2}{\pi\,AR\,e}$ — all of $C_L$, $AR$, $e$ are dimensionless, so $C_{Di}$ is dimensionless, as a drag coefficient must be.
History and Development
Induced drag and its $C_L^2/(\pi AR)$ law are among the crowning results of Ludwig Prandtl's lifting-line theory (1918–19), which explained why finite wings drag more than 2-D airfoils and why span matters. It made aspect ratio a primary design variable and, a century later, still drives high-$AR$ wings and winglets (which raise the effective $AR$/$e$) across gliders, airliners, and UAVs.
Related Concepts: Downwash Angle, Drag Force, Parasite Drag, Lift-to-Drag Ratio, Lift Force, Lift Curve Slope
Notes: Drag-due-to-lift: $\propto C_L^2$, $\propto 1/AR$ — dominant at low speed/high $C_L$. $e \le 1$ (Oswald), $=1$ for elliptical loading. From Prandtl lifting-line via downwash (Downwash Angle). Total drag $= C_{D0} + C_{Di}$ (drag polar).