Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, 2nd Edition

11 INCOMPRESSIBLE FLOW SOLVERS

Among the flows that are of importance and interest to mankind, the category of low Mach-number or incompressible flows is by far the largest. Most of the manufactured products we use on a daily basis will start their life as an incompressible flow (polymer extrusion, melts, a large number of food products, etc.). The air which surrounds us can be considered, in almost all instances, as an incompressible fluid (airplanes flying at low Mach numbers, flows in and around cars, vans, buses, trains and buildings). The same applies to water (ships, submarines, torpedoes, pipes, etc.) and most biomedical liquids (e.g. blood). Given this large number of possible applications, it is not surprising that numerical methods to simulate incompressible flows have been developed for many years, as evidenced by an abundance of literature (various conferences¹, Thomasset (1981), Gunzburger and Nicolaides (1993), Hafez (2003)).

The equations describing incompressible, Newtonian flows may be written as

Here p denotes the pressure, v the velocity vector and both the pressure p and the viscosity μ have been normalized by the (constant) density ρ. By taking the divergence of (11.1) and using (11.2) we can immediately derive the so-called pressure-Poisson equation