GENERAL

The basic concepts in FDM and applications to simple partial differential equations have been presented in the previous chapters. This chapter will focus on incompressible viscous flows in which the physical property of the fluid, incompressibility, requires substantial modifications of computational schemes discussed in Chapter 4.

In general, a flow becomes incompressible for low speeds, that is, M <0.3 for air, and compressible for higher speeds, that is, M ≥0.3, although the effect of compressibility may appear at the Mach number as low as 0.1, depending on pressure and density changes relative to the local speed of sound. Computational schemes are then dictated by various physical conditions: viscosity, incompressibility, and compressibility of the flow. The so-called pressure-based formulation is used for incompressible flows to keep the pressure field from oscillating, which may arise due to difficulties in preserving the conservation of mass or incompressibility condition as the sound speed becomes so much higher than convection velocity components. The pressure-based formulation for incompressible flows uses the primitive variables (p, vi, T), whereas the densitybased formulation applicable for compressible flows utilizes the conservation variables (ρ, ρvi, ρE).

Incompressible viscous flows are usually computed by means of the continuity and momentum equations. If temperature changes in natural and/or forced convection heat transfer are considered, then the energy equation is also added. For simplicity in demonstrating the computational strategies for incompressible flows in general, we shall consider only the isothermal case in this chapter. In Chapter 6, it will be shown that computational schemes for incompressible flows can also be developed from preconditioning processes of the density-based formulation which is originally intended for compressible flows.