 Induced angle of attack

The induced angle of attack ${\alpha }_i$ is the change between the geometric flow direction and the local effective flow direction caused by the wing's induced velocity. In the usual small-angle lifting-line theory convention, downward downwash is positive andHere $U$ is the reference streamwise speed and $w$ the local downward speed. The originating lecture notes explicitly distinguish geometric, induced and effective incidence.^[ref]

If the local velocity has only these two components, its geometric inclination is $\arctan (w/U)$. The ratio $w/U$ is the small-angle approximation, with angles in radians. An original example uses $U=40$ m/s and $w=1.0$ m/s: ${\alpha }_i=\arctan (0.025)\simeq 1.432{}^{\circ }$. A geometric incidence of $5{}^{\circ }$ then gives effective incidence about $3.568{}^{\circ }$. This does not subtract the zero-lift angle of attack, which is a separate term in the section lift law.

For ideal elliptic span loading, the angle is constant across the span and approximatelyThe latter relation is the lift-induced drag contribution, with unit span efficiency factor.^[ref] For stipulated $C_L=0.60$ and $AR=8$, the approximation gives ${\alpha }_i\simeq 0.02387$ radians and $C_{D_i}\simeq 0.01432$.

A nonuniform wing generally has different local induced angles. Its total induced drag involves the loading-weighted integral $D_i\simeq \int L^{\prime }(y){\alpha }_i(y)dy$; an unweighted arithmetic mean of wake angles need not reproduce that integral. A wake velocity traverse and sectional surface pressure integration for airfoil lift test different pieces of this prediction.

The angle concerns induced flow at the modeled lifting line. A velocity measurement far downstream requires the corresponding wake model before being substituted into this formula. The small-angle, attached-flow approximation also does not account for all sources of drag or for strong crossflow.