Global relaxation procedure for compressible solutions of the steady-state euler equations

Abstract

Transonic flow solutions of the steady-state Euler equations are obtained based on the global relaxation procedure developed by Rubin and Lin for the Reduced Navier-Stokes (RNS). The same code written for the viscous RNS equations is used to predict inviscid flows by neglecting the viscous term in these equations and deleting the no-slip boundary condition. Results for compressible flows have been computed without any stability limitation or need for explicit artificial viscosity or compressibility. A variety of solutions is presented in this paper. This method provides an alternative approach to solving the Euler equations for certain cases and for capturing of simple imbedded shock waves; although, the primary purpose of the present study is to demonstrate the applicability of this procedure for accurately representing inviscid behavior when the full RNS system is solved. The global relaxation procedure considered here has been developed primarily for solutions of the RNS equations describing flows in which strong viscous-inviscid interaction occurs. Although for the Euler equations viscous-inviscid interaction is not present, the same methodology is still applicable since the boundary value nature in these two sets of equations is predominantly associated with the unknown axial pressure gradient term (Px). Note that the RNS equations contain an exact (Euler) representation of the inviscid flowfield and the presence of the viscous term does not alter this fact. The boundary value character associated with the Px term is easily determined for the steady-state Euler equations (characteristic analysis), and is a result of wave (elliptic) propagation when this term is treated as unknown. Rubin and Lin [1, 21] presented a global relaxation technique for this elliptic problem by using a simple difference form for the numerical approximation of Px. Using boundary-layer type differencing for the convective terms, the method requires a downstream pressure boundary condition which is a characteristic of the boundary value problem. The resulting procedure consists of several forward marching sweeps for pressure relaxation. This procedure has been applied to many complex subsonic and transonic flows. With this simple treatment of the px term, the method is capable of capturing smooth imbedded shock waves, or smooth detached shock waves with imbedded subsonic regions. These results have been presented in Refs [2-5]. Application of the global relaxation procedure to inviscid flows has been demonstrated for incompressible flow over a boattail afterbody in Ref. [3]. This paper will focus on inviscid subsonic, transonic and supersonic solutions of the steady-state Euler equations, with a modified treatment of the Px term to include the compressibility effect at high Math number. The numerical method has been previously developed for viscous compressible flows. The viscous code developed in Ref. [5] is used to compute the inviscid solutions by ignoring the viscous term in the axial momentum equation and the associated no-slip boundary condition. The compressibility correction employed for the px term is based on the results obtained from the eigenvalue analysis by Vigneron et al. [6] and the stability