Doubled-point method for supersonic unsteady aerodynamics of nonplanar lifting surfaces

Abstract

A new method is devised for the calculation of pressures and aerodynamic influence-coefficients on nonplanar lifting-surface configurations oscillating in a supersonic freestream. The method is an extension of the meth- odology introduced in the planar supersonic doublet-point scheme, which is based upon the concept of concen- trated lift forces and uses the acceleration potential doublet as an elementary solution of the wave equation. These features make the method capable of being incorporated in a unified code for both subsonic and supersonic speeds, as well as amenable to rapid aeroelastic calculations. Results on various lifting-surface configurations are in agreement with other supersonic oscillatory methods, validating the doublet-point approximation in nonplanar supersonic applications. OR studying the aeroelastic characteristics and aeroser- voelastic interactions of modern supersonic aircraft, an efficient method of predicting the unsteady supersonic aero- dynamics of practical lifting configurations is required. The unsteady aerodynamic loads due to a general motion of the structure can be derived from those arising out of simple harmonic motion, by using analytic continuation in the La- place domain. Hence, an oscillatory supersonic lifting-surface theory becomes necessary. The supersonic counterpart of the subsonic doublet-lattice method,1 which is well established for its amenability to aeroelastic calculations, has been sought in the past2 without much success. Nevertheless, motivation for a supersonic scheme with enough commonality with the subsonic doublet-lattice method to have a unified code for both the speed regimes, has persisted. Garrick and Rubinow3 presented an integral equation for the supersonic velocity potential source-strength. The most common velocity potential solution procedure is the Mach box method introduced by Pines et al., 4 and further refined by several authors.59 Difficulties with the Mach box method include the necessity to evaluate velocity potential in dia- phragm regions off the lifting surface and the dependence of the grid on Mach number. The refinements suggested to al- leviate these difficulties add complexity to the method. An- other velocity potential approach is the extension of Eward's10 steady-state theory to the oscillatory case by Burkhart.11 This scheme eliminates the need for diaphragm regions, but is limited to planar applications. Jones and Appa12 proposed the potential-gradient method. They used a series expansion of the kernel function, which made the scheme less accurate at high frequencies. Hounjet13 avoided the series expansion by using an integration scheme similar to that of the subsonic doublet-lattice method1 for directly downstream receiving points. Chen and Liu14 applied another approach to avoid the series expansion by using a parabolic curve-fit for the exponential part of the integrand, in order to integrate the dipole spanwise singularity of the planar kernel. The schemes of Refs. 12-14 needed to consider the wake region between lifting surfaces. Also, the compu-