The compressible Navier-Stokes equations

In two dimensions the compressible Navier-Stokes equations can be written as

\[\begin{equation} \vec{Q}_t + \frac{\partial}{\partial x}(\vec{f}(\vec{Q})+\vec{F}(\vec{Q})) + \frac{\partial}{\partial y} (\vec{g}(\vec{Q})+\vec{G}(\vec{Q})) = 0. \end{equation}\]

Where the conservative state variables are \(\vec{Q} = [\rho, \rho u, \rho v, E]^T\) and the hyperbolic fluxes are

\[\begin{equation} \vec{f}(\vec{Q}) = \left( \begin{array}{c} \rho u\\ \rho u^2 + p \\ \rho u v \\ u (E+p) \end{array} \right), \ \ \ \vec{g}(\vec{Q}) = \left( \begin{array}{c} \rho v\\ \rho u v \\ \rho v^2 + p \\ v (E+p) \end{array} \right) \end{equation}\]

Here we assume an equation of state \(p = (\gamma-1) \rho e\) and note that we can relate \(E\) and \(p\) through \(\rho e = E - \rho (u^2+v^2)/2\).

The viscous fluxes are:

\[\begin{equation} \vec{F}(\vec{Q}) = - \left( \begin{array}{c} 0\\ \frac{4 \mu}{3} u_x+(\frac{\mu}{3}+\mu_b) v_y \\ \mu (u_y+v_x) \\ u F_2 + v F_3 + \frac{\mu \gamma}{Pr} T_x \end{array} \right), \ \ \ \vec{G}(\vec{Q}) = - \left( \begin{array}{c} 0\\ \mu (u_y+v_x) \\ \frac{4 \mu}{3} v_y + (\frac{mu}{3} + \mu_b) u_x\\ u G_2 + v G_3 + \frac{\mu \gamma}{Pr} T_y \end{array} \right). \end{equation}\]

Here \(\rho\) is the density, \((u,v)\) is the velocity vector, \(E\) is the total energy, \(e\) is the specific internal energy, \(\mu\) is the dynamic shear viscosity, \(\mu_b\) is the bulk viscosity, \(\gamma\) is the ratio of specific heats, \(Pr\) is the Prandtl number, \(T = (\gamma-1)e/R\) is the temperature and \(R\) is the gas constant.

Isentropic vortex

Couette flow

Lid driven cavity

A pair of vortices