Some exact and numerical results for plane steady sheared flow of an incompressible inviscid fluid

Abstract

Analytical and numerical solutions are presented for the steady flow of an inviscid fluid about symmetric lifting profiles at an angle of attack in a plane sheared onset flow for which conformal mapping plays a critical role. For uniform shear (i.e. the onset flow speed varies linearly with position) in two dimensions, the disturbance field is potential and hence a solution based on the conformal transformation technique may be constructed. The Moriya transformation, which employs a leading-term transformation coefficient that stretches and rotates the field at great distances from the foil (as distinct from other classical transformations which leave the far field unchanged) is used and, with a limited number of terms selected for the transformation, a simple elegant solution is obtained that may be easily evaluated at arbitrary points on the foil contour. An additional investigation is reported for the field solution — involving a locally similar but globally non-uniform sheared onset flow — about one of the foils for which a simple O-type grid is analytically generated from the mapping function. These data indicate that the uniform-shear solution overpredicts the lift and surface speed on the suction side of the foil relative to the more realistic onset flow: the numerical solution predicts surface speeds that generally lie between those for the uniform flow and the uniformly sheared flow solutions.