Coupled space-marching method for the Navier-Stokes equations for subsonic flows

Abstract

An implicit space-marching finite-difference procedure is described for solving the compressible form of the steady, two-dimensional Navier-Stokes equations in body-fitted curvilinear coordinates. The coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space-marching procedure. A new pressure correction method has been developed that significantly accelerates convergence of the iterative process. The scheme is used to compute incompressible flows by taking the low Mach number limit of the compressible formulation. Computed results are compared with other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and external flow over a cylinder. NUMBER of different methods have been developed to solve the Navier-Stokes equations in primitive variables for steady, subsonic flows. Many of these algorithms are described in Anderson et al. 1 The present methods are not entirely adequate for all problems, and advanced techniques continue to be developed. New procedures and recent im- provements to existing methods are briefly reviewed in the following paragraphs. The most frequently used primitive variable algorithms for incompressible flows solve the momentum equations for the velocity components in an uncoupled (segregated) manner, holding pressure fixed. Although different in detail, the segre- gated solution schemes (e.g., Refs. 2-4) use the continuity equation indirectly in the formulation of a separate Poisson equation that is solved for the pressure. Since the velocity components and pressure are computed from different algo- rithms, rather than in a coupled manner, convergence is slowed. Furthermore, the solution of the elliptic Poisson equa- tion consumes a large part of the total computaion time for each global iteration. Van Doormaal et al.5 recently have eval- uated improved algorithms for solving the pressure Poisson equation, and Rhie4 has applied a multigrid method to acceler- ate the solution of the momentum equations and the pressure equation. Several strategies have been advanced for solving the cou- pled momentum and continuity equations for the primitive variables, including the pressure. The schemes all share the advantage that a separate procedure for imposing the continu- ity constraint and determining the pressure is not required. The choice of variables and the algebraic approach used to solve the system of nonlinear equations distinguish the direct, time- marching, and space-marching algorithms. Vanka and Leaf6 and recently Patankar et al. 7 have com- pared solutions for two-dimensional flows obtained by direct and segregated methods. A large sparse matrix solver is used to invert a system of equations spanning the entire flow domain.