Downwash Angle⚠ unverified
Aerospace / Aerodynamics · Compute the induced downwash angle
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| CL | CL | — | 1.0 | Lift coefficient (dimensionless) |
| AR | AR | — | 1.0 | Wing aspect ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | ε | rad | Downwash angle, in radians (rad) |
The science & history
Understanding the Parameters
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Lift coefficient $C_L$ — more lift means stronger trailing vortices and more downwash, so $\varepsilon$ grows with $C_L$. At high angle of attack the downwash is largest.
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Aspect ratio $AR$ — the downwash falls as $1/AR$: a long, slender wing sheds weaker tip vortices and deflects the flow less. An infinite wing ($AR\to\infty$) has zero downwash, which is why 2-D airfoil data represent the ideal.
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Induced angle of attack — $\varepsilon$ is the angle by which the local flow is tilted below the freestream, so the wing's effective angle of attack is reduced by $\varepsilon$. To make the same lift, a finite wing must be pitched to a higher geometric angle than an airfoil — the origin of the reduced lift-curve slope.
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Elliptical assumption — this $C_L/(\pi AR)$ result is exact only for the minimum-drag elliptical lift distribution (uniform downwash); other distributions vary the downwash across the span, captured on average by the Oswald factor.
Derivation (Approaching a Proof)
In Prandtl's lifting-line theory, the wing sheds a sheet of trailing vortices whose induced velocity at the wing is downward — the downwash $w$. For the elliptical lift distribution this downwash is uniform across the span, and the induced angle it produces is
$$\varepsilon = \frac{w}{V} = \frac{C_L}{\pi\,AR}.$$
The result follows from the elliptical circulation distribution $\Gamma(y) = \Gamma_0\sqrt{1-(2y/b)^2}$: the constant downwash it induces, when integrated for total lift, ties the induced angle directly to the overall $C_L$ and the aspect ratio. This uniform downwash is exactly why the elliptical wing has the minimum possible induced drag — the induced drag is $C_{Di} = C_L\,\varepsilon = C_L^2/(\pi AR)$ (Induced Drag).
Dimensional check. $\varepsilon = \dfrac{C_L}{\pi\,AR}$ — $C_L$ and $AR$ are dimensionless, and the small angle is in radians, as required (the induced angle is a pure number in the small-angle sense).
History and Development
The downwash concept is central to Prandtl's finite-wing (lifting-line) theory, which unified the seemingly separate effects of induced drag and reduced lift-curve slope under one cause: the induced flow of the trailing vortex system. It explains the horizontal-tail sizing problem too — the wing's downwash changes the tail's effective angle of attack, a key term in longitudinal stability. It remains a staple of every aerodynamics curriculum.
Related Concepts: Induced Drag, Lift Curve Slope, Lift Force, Lift-to-Drag Ratio, Static Margin, Moment Coefficient
Notes: Elliptical-loading result ($\varepsilon = C_L/\pi AR$); $\to 0$ as $AR\to\infty$ (2-D airfoil). Reduces effective angle of attack → lower finite-wing Lift Curve Slope; tilts lift → Induced Drag ($C_{Di}=C_L\varepsilon$). Also drives tail downwash in stability.