2-D vortex in isentropic flow

The test case involves convection of an isentropic vortex in inviscid flow. The free-stream conditions are

$$\begin{eqnarray*} \rho &=& 1 \\ u &=& 0.5\\ v &=& 0\\ p &=& 1/\gamma \end{eqnarray*}$$

Perturbations are added to the free-stream in such a way that there is no entropy gradient in the flow-field. The perturbations are given by

$$\begin{eqnarray*} (\delta u, \delta v) &=& \frac{\beta}{2\pi} \exp\left( \frac{1... ...ght]^{\frac{1}{\gamma-1}} \\ p &=& \frac{ \rho^\gamma }{\gamma} \end{eqnarray*}$$

where

$$r = [ (x-x_o)^2 + (y-y_o)^2 ]^{1/2}$$

is distance from the vortex origin $(x_o, y_o)$. One choice for the domain and parameters are

$$\begin{eqnarray*} \Omega &=& [0,10] \times [-5,5] \\ (x_o, y_o) &=& (5,0) \\ \beta &=& 5 \end{eqnarray*}$$

As a result of isentropy, the exact solution corresponds to a pure advection of the vortex at the free-stream velocity. Further details can be found in
Yee, H-C., Sandham, N. and Djomehri, M., ``Low dissipative high order shock-capturing methods using characteristic-based filters", JCP, 150, 1999.