A Lifting-Line Theory and a Critical Shear Parameter of a Wing in Uniform Shear Flow

A lifting-line theory of a wing in uniform shear flow is presented and also a critical shear parameter where aerodynamic characteristics abruptly change is found by obtaining analytical expressions of the characteristics. The governing equation with respect to potential function defined by Karman is derived and reduced to two ordinary differential equations by separation of variables. A general solution is obtained by linearly combining the fundamental solutions of the ordinary differential equations. An induced attack angle is derived from the general soultion and a condition of far downstream. A relation of a lift force to an effective attack angle is described by the Taylor series expansion in a vicinity of geometrical attack angle. From the induced attack angle and the lift force, a lifting-line equation is obtained, and solved by Gaussian elimination method. Effects of shear parameter and aspect ratio on the local lift, total lift and induced drag coefficients are clarified through some numerical calculations.