The Segregated Approach to Predicting Viscous Compressible Fluid Flows

Abstract

The SIMPLE method of Patankar and Spalding and its variants such as SIMPLER, SIMPLEC and SIMPLEX are segregated methods for solving the discrete algebraic equations representing the equations of motion for an incompressible fluid flow. The present paper presents the extension of these methods to the solution of compressible fluid flows within the context of a generalized segregated approach. To provide a framework for better understanding the segregated approach to solving viscous compressible fluid flows an interpretation of the role of pressure in the numerical method is presented. With this interpretation it becomes evident that the linearization of the equation for mass conservation and the approach used to solve the linearized algebraic equations representing the equations of motion are important in determining the performance of the numerical method. The relative performance of the various segregated methods are compared for several subsonic and supersonic compressible fluid flows.

An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids

“…Van Doormaal et al (1987) and Bai et al (1987) have extended the SIMPLE algorithms to handle compressible flows. The SIMPLE algorithms are formulated in terms of a pressure-correction variable, the difference between "predicted" pressure and "corrected" pressure.…”

“…The semi-implicit method for pressure-linked equations (SIMPLE) scheme (Patankar and Spalding, 1972) and its variants SIMPLER (Patankar, 1980), SIMPLEC (Van Doormaal and Raithby, 1984), SIMPLEX (Van Doormaal and Raithby, 1985), and SIMPLEST (Sha, 1985) have proven to be popular pressure-based algorithms for solving incompressible flows. Van Doormaal et al (1987) and Bai et al (1987) have extended the SIMPLE algorithms to handle compressible flows.…”

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

“…Later on, pressure correction equations appeared in numerical schemes proposed by several researchers, essentially in the finite-volume framework, using either a collocated [10,23,26,30,33,34] or a staggered arrangement [2,4,7,22,24,25,37,38,[40][41][42] of unknowns; in the first case, some corrective actions are to be foreseen to avoid the usual odd-even decoupling of the pressure in the low Mach number regime. Some of these algorithms are essentially implicit, since the final stage of a time step involves the unknown at the end-of-step time level; the end-of-step solution is then obtained by SIMPLE-like iterative processes [10,23,25,26,30,34,39]. The other schemes [2,7,22,24,33,37,38,40,42,43] are predictor-corrector methods, where basically two steps are performed sequentially: first a semi-explicit decoupled prediction of the momentum or velocity (and possibly energy, for non-barotropic flows) and, second, a correction step where the end-of step pressure is evaluated and the momentum and velocity are corrected, as in projection methods for incompressible flows (see [5,36] for the original papers, [29] for a comprehensive introduction and [19] for a review of most variants).…”

Simulations of Two-dimensional High Speed Turbulent Compressible Flow in a Diffuser and a Nozzle Blade Cascade

“…Therefore, this is the common scheme employed by the modern commercial CFD codes due to the robustness of the numerical procedure. Known as Pressure Based Method (PBM), this algorithm has been applied extensively in the incompressible flow field originally and has been extended to compressible flow by Issa and Lockwood [1] , Van Doormaal et al [2] , McGuirk and Page [3] , Watterson [4] and Jasak [5] . Notwithstanding this, due to the fact that the momentum, continuity and pressure equations are solved in an uncoupled approach, this may result in convergence problems, especially in situations where the gradients of flow variables are relatively large such as the stagnation point at the leading edge.…”