Abstract

A numerical algorithm for enforcing the conservation of mass in incompressible flow simulation is discussed and the details of the implementations in terms of standard finite volumes and finite elements are given. It is also demonstrated that both segregated and coupled iterative techniques are applicable and numerical results of test cases for two and three dimensional cavity flows are presented.

Introduction

In many applications, numerical simulations of three dimensional incompressible flows are needed. The problem of satisfying exactly the continuity equation, for these flows, is well known [1] [2].

It is conceivable to update the velocity field using the momentum equations but it is not clear how to update the pressure to conserve mass, since no pressure term appears in the continuity equation. Various methods have been introduced to tackle this problem including the penalty method, the artificial compressibility method, the artificial viscosity methods, the projection and the pressure correction methods. (The discussion, here, is limited to methods based on primitive variables. The vector potential and/or velocity vorticity formulations are not covered, see for example the work of Osswald, Ghia & Ghia [19] which still requires staggered orthogonal grids to enforce mass conservation.)

In the primitive variable methods, either staggered grids are used or the continuity equation is modified. For example in the penalty method of Temam [3], a small term proportional to the pressure is added to the continuity equation, while in Chorin's artificial compressibility method [4], the continuity equation is modified by an artificial time dependent term proportional to the time derivative of the pressure.