A Review of Marching Procedures for Parabolized Navier-Stokes Equations
It has now been generally accepted that boundary-layer methodology can be extended to the so-called parabolized Navier-Stokes (PNS) equations for a significant variety of flow problems. In a recent paper, Davis and Rubin [2.82] have reviewed several visco
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A Review of Marching Procedures for Parabolized Navier-Stokes Equations
s. G. Rubin*
1 Introduction It has now been generally accepted that boundary-Iayer methodology can be extended to the so-called parabolized Navier-Stokes (PNS) equations for a significant variety of flow problems. In arecent paper, Davis and Rubin [2.82] have reviewed several viscous flow computations in which parabolized or thin-Iayer techniques have been applied in order to accurately determine the flow characteristics. This publication also reviews some of the early history of the PNS development. The purpose of the present paper is to discuss some recent investigations using the PNS equations. In particular, we are concerned here with efficient three-dimensional algorithms, a clearer understanding of the limits of applicability ofPNS marching techniques, and pressure interaction relaxation for separated flows and other problems where upstream influence is of importance. In this regard, three solution procedures are considered: (a) single-sweep "boundary-Iayer-like" marching for two- and threedimensional flows; (b) multiple-sweep iteration or global pressure relaxation, where upstream influence and possibly axial flow separation are important, but regions of subsonic flow are smalI; and (c) global relaxation where subsonic flow domains are large. For the last two classes of problems, the analysis draws heavily on that of interacting boundary layers and inviscid subsonic relaxation methods where applicable. • Department of Aerospace Engineering and Applied Mechanics, University of Cincinnati, Cincinnati, OH 45221.
171 T. Cebeci (ed.), Numerical and Physical Aspects of Aerodynamic Flows © Springer Science+Business Media New York 1982
2 Interactive Steady Boundary Layers
172
In the course of proceeding to specific examples, a brief review of the limitations associated with PNS marching or relaxation is necessary. The PNS equations, which for simplicity are given here only for "two-dimensional incompressible flow", are as follows: (la)
VI
+ UV x + vV y Ux
= -Py
+ vy =
O.
+
(*
Vyy ).
(lb) (1c)
R is the Reynolds number; x is defined as the axial flow direction, and y as
the normal direction. The system (1) differs from the complete NavierStokes equations only by the omission of the axial diffusion terms. Strictly speaking the inclusion ofthe V yy term in (lb) is inconsistent with the omission of U xx in (la). Either the former should be neglected (this is probably a more appropriate definition of the PNS equations) or the latter retained. In fact, these terms have little effect on any of the results presented here. The mathematical character of (1) is controlled by the Px term in (la). When Px is prescribed (assumed known), the system is parabolic. This was the case in the original merged-Iayer analysis of Rudman and Rubin [2.83J, where for hypersonic cold-wall flow Px can, in fact, be neglected in (1a). It should be emphasized, however, that axial pressure gradients are still present and are evaluated through the moment um equat
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