Inlet Flow and Crosswind: Method

Oblique Inflow and Lip Separation

Momentum theory with an oblique free stream

When the air arrives at an angle α to the duct axis, the disk still adds an induced velocity v_i along the axis, but the mass that carries the momentum change is set by the resultant speed through the disk, U = √((V_∞ sin α)² + (V_∞ cos α + v_i)²). Glauert's relation T = 2 ρ A_e U v_i, written with the duct's exit area A_e = σ_d A, is solved for v_i. The induced power is T v_i, and the axial component of the free stream adds the climb work T V_∞ cos α. As the free stream grows, more mass passes the disk and v_i falls: a crosswind lowers the induced power of a hovering fan, the same effect that gives a helicopter its translational lift.

Momentum drag and the ram moment

The captured air arrives with a cross-stream momentum ṁ V_∞ sin α and leaves along the duct axis, so the duct feels a drag of that size in the free-stream direction. It acts where the flow is turned, at the inlet lip, ahead of the duct's center: the tool takes half the duct chord as the arm and reports the pitching moment. Martin and Tung measured this moment on a 10-inch duct in a crosswind and found it grows with speed in the destabilizing sense (upwind lip up), the "speed instability" of ducted-fan vehicles; a sharper lip that separates early reduced it at the price of hover performance.

Lip separation

At a large angle the upstream lip has to turn the flow through nearly the full angle, and it separates when the turning is too severe for the lip radius and the suction the fan provides. NASA's test of a full-scale 7-foot ducted propeller (TN D-4142) mapped the duct angle of attack at which local separation begins and at which the whole upstream lip is separated, as functions of 1/T_c, the inverse of the thrust coefficient on free-stream dynamic pressure and disk area. High thrust at low speed (small 1/T_c) lets the lip turn the flow through 80 to 90°; at 1/T_c = 1 the onset falls to about 38° and full separation to about 61°. The tool reads both curves at the operating point and reports the margin. The 7-foot model's lip radius was about 2.4 percent of the diameter; Figure 15(b) of the report shows lips of 0.4 to 0.9 inch on a 4-foot duct separating fully at 40 to 45° instead of 60°, so for a thin lip the margins here are optimistic.

Limits

Momentum theory contains no lip losses, no duct lift and no wake skew; beyond separation it does not apply at all. The hub, the airframe and the fan's own blade stall in the asymmetric inflow are not included.