Compressible Navier-Stokes Simulations

The compressible Navier-Stokes equations in conservative form are:

$$\diff{{\bf S}}{t} + \diff{{\bf F}^{Euler}_x}{x} + \diff{{\b... ...x}{x} + \diff{{\bf F}^{Visc}_y}{y} + \diff{{\bf F}^{Visc}_z}{z}$$

with following definitions:

Variable Description

$$\rho({\bf x}, t)$$ Density

$${\bf v}({\bf x}, t)=(u,v,w)({\bf x}, t)$$ Velocity

$$E = \frac{p}{(\gamma-1)} +\frac{1}{2}(\rho {\bf v}\cdot{\bf v})$$ Total energy

p Pressure

$$T = \frac{p}{\rho}$$ Temperature

$$Pr = \frac{c_p \mu}{\kappa}$$ Prandtl number

$$\mu$$ Dynamic viscosity

$$\lambda$$ Bulk viscosity

$$\kappa$$ Thermal conductivity

The formulation of the Riemann solver for the Euler fluxes is identical to that proposed by Lomtev [80]. We will omit them here for brevity. We will go into more detail for the compressible MHD equations that we develop in the next chapter. The implementation of the upwinding routine is independent of the element type.