Entropy condition satisfying approximations for the full potential equation of transonic flow

Abstract

Abstract. We shall present a new class of conservative difference approximations for the steady full potential equation. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. We prove rigorously that these new schemes satisfy a new discrete "entropy inequality", which rules out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in our analysis is the construction of an "entropy inequality" for the full potential equation itself. We conclude by presenting results of some numerical experiments using our new schemes.I. Introduction. The full potential equation is a common model for describing supersonic and subsonic flow close to the speed of sound. The flow is assumed to be that of a perfect gas, and the assumptions of irrotationality and constant entropy are made. The resulting equation is a single nonlinear partial differential equation of second order, which changes type from hyperbolic to elliptic, as the flow goes from supersonic to subsonic. Flows with a supersonic component generally have solutions with shocks, so the conservation form of the equation is important.This formulation, (FP), is one of three conservative formulations used for inviscid transonic flows. The other two are transonic small-disturbance equation, (TSD), and Euler equation, (EU), which is the exact inviscid formulation. The FP formulation is the most efficient of the three in terms of accuracy-to-cost ratio for a wide range of inviscid transonic flow applications for real geometries. TSD is valid for thin wings at free stream Mach numbers near unity, and EU, while the least restrictive, involves the most complicated system of equations.During the last few years, many numerical calculations using FP have been presented, e.g., [19], [14], [17], and [6]. The object of our present investigation is twofold. First, we wish to put the theory of nonlinear difference approximations to FP on a sound theoretical basis, via an "entropy condition", as described below. Second, we introduce a new class of entropy condition satisfying approximations, which are, in general, no more complicated to program than the usual density

A convergent finite element formulation for transonic flow

“…Similar results could not yet be obtained for the finite difference method. Engquist and Osher [8], Osher [24][25][26] and Osher et al [27] propose in their papers a consistent finite difference method for the full potential equation and the transonic small disturbance equation. Under the a priori assumption that the difference solutions converge they show that they must converge to a physical solution.…”

“…For the finite difference method a similar result is not yet known. Engquist and Osher [8], Osher [24][25][26] and Osher et al [27] prove in their papers the existence of finite difference approximations satisfying an entropy condition. But there the convergence of the difference solutions is still an open problem.…”

Least squares finite element simulation of transonic flows

“…Following [11] (see also [12] - [14]) the modified density takes the form P = P -UP 6.s, s (2.13) where is the derivative of the density P along the streamwise direction. Since the density has the form P P ( I grad <P I ), the derivative ps formally involves second derivatives of <p.…”

On Some Numerical Schemes for Transonic Flow Problems

“…In [2], [19], [18], [14] and [3], a number of shock-capturing finite difference approximations for solving the TSD and the FP equations have been proposed. These schemes satisfy properties (iii) and (v), and with the inclusion of flux limiters, property (iv) as well.…”

“…For the FP equation, the upwinding can be performed separately for the ^-dependent term and the ^-dependent term. This approach was labeled directional flux biasing in [14]. Recently, this approach was refined by introducing the method of streamwise flux biasing (see [17]) in which the upwinding is performed in a direction close to that of the actual flow.…”