Quasi-Parabolic Technique for Spatial Marching Navier-Stokes Computations

Abstract

ACOMPUTATIONAL technique is presented for obtaining flowfield solutions to a parabolic form of the Navier-Stokes equations. The point of departure is the general interpolants method (GIM), which provides a discretization for partial differential equations on arbitrary threedimensional geometries. The new scheme, termed quasiparabolic (QP), treats the parabolized equations but with time-like terms appended. Addition of these extra terms, which are relaxed by iteration, avoids many of the singularities inherent in classical parabolic Navier-Stokes methods. Solutions are presented for viscous flows in boundary layers and free shear layers as computed with the GIM/QP scheme. Contents The full three-dimensional Navier-Stokes equations are elliptic in character and require either time-dependent or relaxation/iteration schemes to integrate the complete spatial flowfield simultaneously. This can require large amounts of computer storage and relatively long run times. For situations in which a region of the flow is inviscid and entirely supersonic, a spatial hyperbolic marching algorithm would be efficient. There are also many viscous problems of interest in which parabolic marching solutions are acceptable. Certain assumptions must be made in using a spatial marching technique. There must exist a dominant flow direction in which to march, and there can be no flow back upstream, i.e., no recirculation in the streamwise direction. Stress terms are not allowed to act on the cross planes, i.e., there can be no second-order terms (diffusion, viscosity) in the marching coordinate. The downstream pressure field also must not be allowed to propagate upstream. Reference 1 contains a review of the approaches to the solution of the parabolic Navier-Stokes (PNS) equations. The paper of Lin and Rubin2 also discusses many of the approaches and difficulties involved in PNS solutions. The intent of this research is to provide a parabolic spatial marching technique which can be readily incorporated into the general interpolants method (GIM). The GIM code provides a discretization of partial differential equations on arbitrary three-dimensional geometries. The GIM treatment of arbitrary geometric shapes in three dimensions and the method of solution of the full Navier-Stokes equations is described fully in Refs. 1,3, and 4 and will not be repeated