The Jameson’s numerical method for solving the incompressible viscous and inviscid flows by means of artificial compressibility and preconditioning method Vahid Esfahanian, Pooria Akbarzadeh * Mechanical Engineering Department, University of Tehran, Tehran, Iran article info Keywords: Incompressible flow Artificial viscosity Artificial compressibility Artificial dissipation Preconditioning Finite volume Backward facing-step abstract A computational code is developed using cell-centered finite volume method with a non- uniform grid for solving the incompressible viscous and inviscid flows. The method has been used to determine the steady incompressible inviscid flows past a cylinder in free stream, the steady incompressible inviscid flows past a circular bump through a channel, and also the steady incompressible viscous flows past a backward facing-step. In this method, the 2D Navier–Stokes equations (or 2D incompressible Euler equations for inviscid flow), which are modified by artificial compressibility and preconditioning concepts, are solved with the Jameson’s artificial dissipation and viscosity terms under the form of a fourth- and second-order x-derivative, respectively. An explicit fourth-order Runge–Kutta integration algorithm is applied to find the steady state condition. The effects of CFL num- ber, artificial viscosity coefficient, and pseudo-compressibility parameter in convergence of solution are investigated. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction The solution of incompressible flow problems is important in a range of applications including heat exchanger design, low speed aerodynamics, hydrodynamic systems, cooling of electronic systems, bio-medical systems, etc [12]. It is increasingly important to be able to deal with complicated geometries in these applications. Time marching schemes provide good sta- bility and convergence characteristics when solving compressible flows at transonic and supersonic Mach numbers. How- ever, time marching schemes are less effective for solving incompressible flows because the incompressible system is not fully hyperbolic and pressure cannot be updated from an equation of state. To overcome this difficulty, Chorin [1] modified the continuity equation by adding an artificial compressibility term under the form of the time derivative of the pressure. With this modification, the system of equations changes to a symmetric hyperbolic system for the inviscid terms, so time marching schemes can be useful. Since we were not interested in transient case, one can use acceleration techniques which might destroy the time accuracy but enables one to reach the steady state faster. Such methods can be considered as pre- conditioning to accelerate the convergence to a steady state. This concept was developed by Turkel [2]. He added a pressure time derivative to the all momentum equations. Preconditioning methods have been developed with the aim of solving nearly incompressible flow problems with numerical algorithms designed for the compressible flows. In this study, first, the modified Navier–Stokes equations (or incompressible Euler equations for inviscid flows) are presented and the finite vol- ume discretization is described. Next, finite volume approaches to evaluating viscous terms are examined, and issues related 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.05.087 * Corresponding author. E-mail addresses: evahid@ut.ac.ir (V. Esfahanian), akbarzad@ut.ac.ir (P. Akbarzadeh). Applied Mathematics and Computation 206 (2008) 651–661 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc