Direct Approach to Aerodynamic Design Problems

Abstract

“Direct” small disturbance approaches to inviscid aerodynamic “inverse” or “design” problems in two and three-dimensional, subsonic, supersonic, and transonic flow, are given which extend the stream function method of Chin and Rizzetta. Our shape solutions generally involve nonlinear differential equations of mixed type for scalar “streamlike” functions Ψ which are solved with Neumann conditions related to prescribed surface pressure and “Kutta-type” constraints on edge closure or included angle specified using jumps in ψ or its streamwise derivative. Because edge constraints are enforced automatically the methods are direct. They resemble “analysis” potential function (φ) methods for flows past fixed shapes where the jumps [φ] are chosen “directly” to insure smooth trailing edge flow; thus computational methods for analysis apply with minor change to design. Dualities relating the analysis problem for camber to the design problem for thickness, and the design problem for camber to the analysis problem for thickness, are given for planar flow; both closed trailing edges, and cusped ones, generally opened, which model the displacement effects of viscous wakes, are considered. These are extended to three dimensions and consequences of an inverse analogy to Prandtl’s lifting line for analysis problems are explored. Transonic inverse formulations for airfoils, wings, fans, cascades, inlets, and nacelles are discussed and preliminary numerical results are presented.