Approximate factorization schemes for three-dimensional nonlinear supersonic potential flow

Abstract

The nonconservative finite difference analog of the three-dimensional full-potential equation is solved implicitly by marching in a radial coordinate system with the marching initiated by a conical flow at the apex. The crossflow is a mixed type and transonic type-dependent differencing is used to determine each spherical plane solution. Two types of approximate factorizations are studied, the AFl or ADI and the AF2 factorization. In the AF2 factorization, one of the second derivatives is split between the two factors. The type of AFl factorization that was found to work for the conical problem was also found to be more sensitive to the coordinate transformations and strong shocks than the AFl scheme. Both schemes were found to require some form of temporal damping for stability for the multishock conical flow problem. The AFl factorization was then extended to include three-dimensional flows with the bow shock fit by explicitly including the hyperbolic third-dimension terms. The AFl factorization was shown to improve the convergence rate markedly for most cases.