Effect of flow shear on induced drag.

Abstract

V Karman and Tsien formulated the lifting line theory for a wing in nonuniform flow in 1945. In their formulation, the approaching flow may have arbitrary variations in the plane perpendicular to the flight direction. By working with the perturbation pressure rather than the perturbation velocities, they have obtained the condition for the minimum induced drag and the method of calculation for the induced drag of a lifting line with arbitrary spanwise lift distribution. No specific examples, however, were given. In this note, we present the calculation of the induced drag of an elliptically loaded lifting line in an exponentially sheared flow in the vertical direction. The result applies equally well to a linearly sheared flow provided the shear rate is small. Consider a lifting line with a unit semispan in an incompressible, inviscid stream which is sheared in the vertical direction. Let the rectangular coordinate system be such that the y axis is in the direction of the span with the lifting line extending from y = — 1 to y = +1, the x axis in the direction of the approaching flow U(z), and the z axis pointing upwards. By considering the flow in the Trefftz plane (x -> oo ), von Karman and Tsien give the following governing equation: ] = 0 (1)