A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

Abstract

AbstractA bridge is built between projection methods and SIMPLE type methods (Semi‐Implicit Method for Pressure‐Linked Equation). A general second‐order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non‐linear update of pressure term as Θn∇pn+1+(I−Θn)∇pn, where Θn is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three‐ and four‐step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second‐order temporal accuracy for unsteady flows. The classical second‐order projection method and SIMPLE type methods are united within the framework of the general second‐order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge–Kutta technique to update the convective term and Crank–Nicholson scheme for the diffusion term. Copyright © 2007 John Wiley & Sons, Ltd.