Approximate analysis of regularized compressible flow using the fictitious gas concept

Abstract

THE MOST direct extension of linear incompressible potential flow theory is to consider compressibility effects in the potential flow formulation. The resulting governing potential flow equation is nonlinear and may be of mixed type. If the speed of the flow is everywhere subsonic the governing equation is elliptic. However, when the flow accelerates near a profile such as an airfoil, the velocity may be locally supersonic in which case we have both a supersonic (hyperbolic) region and a subsonic (elliptic) region. Further, the nonlinearity of the potential equation permits the formation of shock discontinuities terminating the supersonic region. If the flow is entirely subsonic (elliptic) it is said io be subcritical and is termed supercritical if part of the flow is supersonic (hyperbolic). The earliest studies of compressible flow problems employed classical methods to determine asymptotic approximations for problems with small compressibility effects. For example, Rayleigh [l] used an iterative approach to replace the nonlinear potential flow equation by a sequence of linear Poisson equations. The incompressible solution and first-order compressibility correction were determined for simple cyclindrical and spherical shapes using classical complex variable techniques. This method is actually a fore-runner of the RayleighJantzen perturbation technique in which the potential solution is written as a perturbation expansion in the incident Mach number (the speed of the incident flow relative to the speed of sound). Other, similarly motivated, perturbation analyses were based on the thickness parameter for thin airfoils to yield a simpler linearized problem amenable to classical analysis as in the Prandtl-Glauert method [2]. An interesting observation resulting from the Rayleigh-Jantzen perturbation expansion procedure is that the first order compressibility correction is independent of the ratio of specific heats y for the gas. This implies that this first-order effect can be determined for any convenient choice of y and it was noted that the choice of a fictitious gas with y = -1 simplifies the mathematical problem statement. This choice of a fictitious gas, devised by Chaplygin [3] andsometimes termed the Chaplygin flow was later used by von Kdrman [4]. Hence, the idea of a fictitious gas first arose in the context of analytical studies of compressible subcritical flow to simplify the governing equation and make it more amenable to classical analytic solution methods. The concept has resurfaced in recent years in connection with finite difference and finite element methods for computing compressible flows. Carey and Pan [5] used the fictitious gas to improve the convergence of gradient solution techniques for finite element approximation