# 2-D vortex in isentropic flow

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

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

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

$\begin{matrix} (\delta u, \delta v) &=& \frac{\beta}{2\pi} \exp\left( \frac{1-r^2}{2} \right) [ -(y-y_o), (x-x_o) ] \\ \rho &=& \left[ 1 - \frac{ (\gamma-1)\beta^2}{8\gamma\pi} \exp\left( 1-r^2\right) \right]^{\frac{1}{\gamma-1}} \\ p &=& \frac{ \rho^\gamma }{\gamma} \end{matrix}$

where

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

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

$\Omega = [0,10] \times [-5,5], \quad (x_o, y_o) = (5,0), \quad \beta = 5$

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 et al. (1999).

## References

• Yee, H-C., Sandham, N. and Djomehri, M., (1999), "Low dissipative high order shock-capturing methods using characteristic-based filters", JCP, Vol. 150.