Iterative PNS algorithms for solving 3-D supersonic flows with upstream influences

Abstract

Two iterative parabolized Navier-Stokes (PNS) algorithms (called IPNS and TIPNS) have been developed to efficiently solve three-dimensional (3-D) supersonic viscous flowfields with upstream influences. These algorithms compute the flow in regions where upstream influences are present by solving the PNS equations using multiple streamwise sweeps. In regions where there are negligible upstream influences, the standard single-sweep PNS method is used. As a result, a complete 3-D flowfield can be computed more efficiently than with a full Navier-Stokes (NS) solver while maintaining the same accuracy. The IPNS (iterated PNS) algorithm splits the streamwise flux vector using the Vigneron splitting while the TIPNS (time iterated PNS) algorithm splits the streamwise flux vector using the Steger-Warming splitting and can retain the time derivative terms. Both algorithms have been successfully incorporated into NASA’s upwind PNS (UPS) code. The new 3-D algorithms have been validated by applying them to test cases consisting of the flow over a cone-cylinder and a cone-cylinder-flare geometry both at 5” angle of attack and a 3-D cone-cylinder-flare-fin geometry at 0’ angle of attack. The present numerical results are in excellent agreement with results obtained using the NS code OVERFLOW. *Manager, Computational Fluid Dynamics Center, and Professor, Dept. of AEEM, also President, Engineering Analysis Inc., Ames, IA. Fellow AIAA. t Research Scientist; currently Visiting Scientist, Air Force Research Lab., Wright-Patterson AFB, OH 45433. Senior Member AIAA. fResearch Scientist, Integrated Systems Technologies Branch. Member AIAA. Copyright @ZOO0 by the American Institute of Aeronaul tics and Astronautics, Inc. All rights reserved. Introduction The present atithors have recently developed very efficient algorithms for solving 2-D/axisymmetric, supersonic, viscous flowfields with upstream influences [l-4]. Th ese algorithms achieve their efficiency by solving the parabolized Navier-Stokes (PNS) equations throughout the entire flowfield. In regions where there are negligible upstream influences, the standard single-sweep PNS method is used to march the solution in the streamwise direction. In regions where upstream influences are present (such as near flow separations), the governing equations are solved using multiple streamwise sweeps to duplicate the results that would be obtained with the complete Navier-Stokes (NS) equations. As a result of this approach, a complete flowfield 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. These algorithms are based in part on the prior work of several investigators [5-161 who d eveloped iterative PNS methods. The IPNS (iterative PNS) algorithm [l-4] splits the streamwise flux vector using the Vigneron splitting [17] and can be applied to flows with moderate upstream influences and small streamwise separated regions. The TIPNS (time iterative PNS) algorithm [4] splits the streamwise flux vector using the Steger-Warming splitting [18] 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. In order for a PNS-type code to be able to solve the entire flowfield surrounding a super1 American Institute of Aeronautics and Astronautics (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. sonic/hypersonic vehicle, it must be able to automatically detect and measure the extent of embedded regions that produce significant upstream effects. Innovative techniques have been developed [l-3] to automatically detect and measure the extent of these embedded regions by examining the known body geometry before the flowfield is computed. Using correlation functions which give the extent of the upstream/downstream regions of influence, the parts of the flowfield where the IPNS/TIPNS algorithm must be applied are known apriori. In the present study, the IPNS and TIPNS algorithms have been extended to 3-D flowfields. These algorithms have been successfully incorporated into NASA’s upwind PNS (UPS) code [19, 201. The new 3-D code has been validated by applying it to several supersonic laminar flow test cases. These test cases include flow over a cone-cylinder geometry at 0” and 5’ angle of attack and flow over a cone-cylinder-flare geometry at the same angles of attack. In addition, flow over a 3-D cone-cylinder-flare-fin geometry at 0” angle of attack has been computed. The present numerical results are compared with results obtained using the OVERFLOW NS code [21] Governing Equations The thin-layer Navier Stokes (TLNS) equations are obtained from the compressible Navier-Stokes equations by retaining only the normal viscous terms. If all the crossflow viscous terms are retained, the resulting equations can be written in a general nonorthogonal coordinate system ( ( ‘> G = (;);::;)+($)(&+ ($) (Gi-G:) dropped. These same viscous terms are also dropped in the PNS equations and the boundary-layer equations. The inviscid (subscript i) and viscous (subscript v) flux vectors are given by Ei = [PU, pu2 + p, ~~21, PUW, (Et + P) u]’ Fi = [pv, PUV, pv2 + P, P’7JW, (Et •IP) ~1’ Gi = [pW,PUW,PvW,PW2 +P,(Et +P)w]~ E” = [O, TX,, Tzy, TLC,, UT,, + “Tzy + wrxz q,lT Fv = 1% Liz, ~yy, ryyzr u~~z + “~yy + w~yz qylT G, = [O, rz,, czar rzz, urzz + “~zy ‘t w~zz qzlT where Et = p[e + t(u” + u2 + w”)] . The PNS equations are obtained from Eq. (1) by dropping the unsteady terms. The PNS equations expressed in a general nonorthogonal coordinate system (E, 7, <) are given by Ec+F,+G<=O (2) The E vector is frequently split for PNS applications using the Vigneron [17] parameter w. The parameter w is given by: W = min 1, 1 PYq 1+(7-1)M~ 1 where MC is the local Mach number ,in the [ direction and ,B is a safety factor that accounts for nonlinearities in the analysis. The E vector can then be written as: E=E*+W=A*U+APU where E’ = Ez 7 PU PU2 + wp PU v PU w (Et + P)U P w PU w PV w PW2 + wp (Et +P)W PV PUV pv2 + wp PV w (Et + P~J