Boundary Integral Equation Methods for Aerodynamics

T HE objective of this chapter is to review the boundary-integral-equation methods in potential aerodynamics of airplanes and helicopter rotors, with emphasis on the "direct velocity-potential formulation," which was introduced by Morino' and further developed by him and his collaborators, in particular by Gennaretti. For the sake of clarity, the formulation is presented at levels of increasing complexity, starting with incompressible nonlifting problems and ending with the most recent developments: a boundary-integral-equation formulation for the velocity potential equation for compressible flows, in a frame of reference in arbitrary motion, and with applications to aerodynamics of airplanes and helicopter rotors. The formulation is given in terms of the velocity potential, for which an explicit treatment of the wake is required; special emphasis is given to the formulation for the wake transport. Recently obtained numerical results are included. Other methods, in particular those by Hess' and by Maskew, are also presented. In the remainder of this section we present an outline of this chapter. In Sec. n we present a review of the development of boundary-integral methods (in airplane and helicopter-rotor aerodynamics) with emphasis on the "direct velocity-potential formulation." Next, we present the mathematical aspects of the methodology. In order to introduce some fundamental concepts of the boundary-integral-equation methodology, we begin with incompressible potential flows (Sec. IE). Lifting flows require the introduction of potential wakes; this concept, which may not be familiar to the reader, is presented in Sec. IV, which includes a detailed discussion of the treatment of the wake, of the condition at the trailing edge, and of the numerical discretization of the problem. Then we extend the potential-flow formulation to compressible flows (Sec. V); the integral formulation for airplanes is treated in Sec. VI and its extension to