Computation of hypersonic laminar separated flows using an iterated PNS algorithm

Abstract

Hypersonic laminar separated flows over hollow cylinder-flare configurations have been computed using the time-iterated parabolized Navier-Stokes (TIPNS) algorithm. The hollow cylinder-flare test cases correspond to experiments recently performed by Holden in the Calspan-University of Buffalo Research Center Large Energy National Shock facility. The TIPNS algorithm solves the parabolized NavierStokes (PNS) equations using multiple streamwise sweeps in order to compute flows with upstream influences. A typical flow-field is computed using a combination of the traditional single-sweep PNS method and the multiple-sweep TIPNS approach. Two hollow cylinder-flare test cases have been computed. The effects on the solutions due to grid refinement, leadingedge slip, streamwise viscous derivative terms, and second-order accuracy in the streamwise direction are studied. Introduction The study of hypersonic laminar separated flows is important because of the high pressures and large heat transfer rates that are associated with these flows. These peak conditions are the result of very strong shock/shock and shock/boundary-layer interactions that occur in most of these flowfields. Hypersonic separated flowfields have been studied both experimentally and numerically. Recently, experiments have been performed in the ONERA R5Ch hypersonic wind tunnel [1,2] and in the Calspan-University of Buffalo Research Center Large Energy National Shock (LENS) facility [3,4] in order to provide a database for code validation studies. The laminar experiments performed by Holden in the LENS facility include shock/shock and shock/boundary-layer interactions over cylindrical leading edges, cone/cone, and hollow cylinder-flare configurations. Hypersonic, rarefied, separated flowfields are normally computed using either a Navier-Stokes (NS) or 'Graduate Student, AIAA Student Member tManager, Computational Fluid Dynamics Center, and Professor, Dept. of AEEM. Fellow AIAA Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. a direct simulation Monte Carlo (DSMC) numerical scheme. Recently, a new numerical approach has been developed '[5-9] that iteratively solves the parabolized Navier-Stokes (PNS) equations. In this approach, the standard single-sweep PNS method is used to march the solution in the streamwise direction in regions where there are negligible upstream influences. In regions where upstream influences are present (such as near flow separations), the governing equations are solved using multiple streamwise sweeps in order to duplicate the results that would be obtained with the NS equations. As a result of this approach, a complete flow-field can be computed much more efficiently (in terms of computer time and storage) than with a standard NS solver which marches the solution in time using the time-dependent approach. Two iterative PNS algorithms (called IPNS and TIPNS) have been developed. The IPNS (iterative PNS) algorithm [5-7] splits the streamwise flux vector using the Vigneron splitting [10] and can be applied to flows with moderate upstream influences and small streamwise separated regions. The TIPNS (time iterative PNS) algorithm [8,9] splits the streamwise flux vector using the Steger-Warming splitting [11] and may retain the time derivative terms. The TIPNS algorithm can be used to compute flows with strong upstream influences including large streamwise separated regions. The TIPNS approach has been used to successfully compute hypersonic separated flows over 2-D compression ramps. [12] In the present study, the TIPNS algorithm has been used to compute two of Holden's hollow cylinder-flare test cases. [4] The effects on the numerical solutions due to grid refinement and leading edge slip are studied. In addition, the changes in the results due to the inclusion of the streamwise viscous terms and secondorder accuracy in the streamwise direction are examined. Governing Equations The thin-layer Navier-Stokes (TLNS) equations are obtained from the compressible Navier-Stokes equations by dropping all viscous terms except the normal viscous terms. If the crossflow viscous terms are also retained, the resulting equations can be written in a AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. general nonorthogonal coordinate system (£,77,0 as 1 dU <9E OF dG where J is the Jacobian of the transformation and U = [p,pu,pv,pw,Et} where