S-plane aerodynamics of nonplanar lifting surfaces

Abstract

A compressible unsteady panel for predicting generalized force transfer functions for nonplanar lifting surfaces is described. The scheme is suitable for both subsonic and supersonic flow, and is valid everywhere in the complex Laplace s-plane. The effort of constructing the influence coefficient matrix is minimized by using the symmetry and scale properties of the kernel function. Results of steady and unsteady analyses for numerous configurations are compared with a lattice method, a point method, and a hybrid lattice-doublet point method, showing excellent agreement. With a view toward application to rotating ma- chinery, an analysis is given of a circular duct with guide vanes in both steady and unsteady flow. NSTEADY nonplanar lifting surface methods1'3 are routinely used in the aeroelastic analysis of aircraft to obtain the generalized aerodynamic forces resulting from vi- bration and gust loading. Although such methods have a long history, dating to the 1950s4-8 papers on the subject still ap- pear occasionally in literature.9-12 This article describes a scheme with several novel features that has been developed as part of an effort to integrate rotating blades with the rest of the aircraft. Only the nonrotating methodology will be described here; its integration with a rotor analysis will be discussed separately. The notable features are 1) both subsonic and supersonic flows are included; 2) the formulation is valid everywhere in the complex Laplace plane; and 3) the nu- merical overhead of constructing the influence coefficient ma- trix is minimized by taking advantage of the symmetries of the unsteady kernel function. The uniform validity in the s- plane has applications in aeroservoelas tic modeling, and to some extent removes the need for Fade (or other) curve-fitting techniques of analytic continuation from the imaginary axis (or at least allows the use of off-axis data in the curve fit). While both subsonic and supersonic flows are dealt with, any strong nonlinearities, whether of transonic, hypersonic, or high incidence origin, invalidate the model. In the interest of brevity, only subsonic results are presented here. For steady flow, the discretization used reduces to a piece- wise constant load, collocated panel method. This gives rea- sonable smoothness at supersonic speeds and considerable insensitivity to control point placement. However, nonsteady effects are included by a point approximation, thereby re- ducing the computational cost. In this latter aspect, the scheme resembles the doublet point method of Ueda and Dowell.2 However, since the point approximation is applied only to nonsingular quantities, no special care need be exercised near singularities. The basic structure is such that practically any steady panel code could easily be modified for unsteady ef- fects, simply by multiplying the influence coefficients by ap- propriate phase factors. Formulation