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

8 SIMPLE EULER/NAVIER–STOKES SOLVERS

This chapter describes some simple numerical solution schemes for the compressible Euler/Navier–Stokes equations. The discussion is restricted to central difference and Lax–Wendroff schemes, with suitable artificial viscosities. The more advanced schemes that employ limiters, such as flux-corrected transport (FCT) or total variation diminishing (TVD) techniques, are discussed in subsequent chapters.

Let us recall the compressible Navier–Stokes equations:

where

Here ρ, p, e, T, k and v_i denote the density, pressure, specific total energy, temperature, conductivity and fluid velocity in direction x_i, respectively. This set of equations is closed by providing an equation of state, e.g. for a polytropic gas

where γ and c_v are the ratio of specific heats and the specific heat at constant volume, respectively. Furthermore, the relationship between the stress tensor σ_ij and the deformation rate must be supplied. For water and almost all gases, Newton's hypothesis

complemented with Stokes' hypothesis

is an excellent approximation. The compressible ...