2-D vortex in isentropic flow

This is one of the few exact solutions for the compressible Euler equations. The test case involves [[convection]] of an [[isentropic]] [[vortex]] in [[inviscid flow]]. It is used to show the ability of numerical schemes to accurately capture vortical flows. The [[free-stream conditions]] are 

:<math>
\begin{matrix}
\rho &=& 1 \\
u &=& u_\infty \\
v &=& v_\infty \\
p &=& 1
\end{matrix}
</math>

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

<math>
(\delta u, \delta v)  = \frac{\beta}{2\pi} \exp\left( \frac{1-r^2}{2}
\right) [ -(y-y_o), (x-x_o) ]
</math>

while the temperature, density and pressure are given by

<math>
\begin{matrix}
T &=& 1 - \frac{ (\gamma-1)\beta^2}{8\gamma\pi^2} \exp\left(
1-r^2\right)\\
\rho &=& T^{\frac{1}{\gamma-1}} \\
p &=& \rho T = \rho^\gamma
\end{matrix}
</math>

where 

:<math>
r = [ (x-x_o)^2 + (y-y_o)^2 ]^{1/2}
</math>

is distance from the [[vortex]] center <math>(x_o, y_o)</math>. 

One choice for the domain and parameters is: 

:<math>
\Omega = [0,10] \times [-5,5], \quad
(x_o, y_o) = (5,0), \quad
\beta = 5
</math>

As a result of [[isentropy]], the exact solution corresponds to a pure [[advection]]
of the [[vortex]] at the [[free-stream velocity]]. This problem is usually solved with periodic boundary conditions. Further details can be found in Yee et al. (1999).

==References==

*{{reference-paper | author=Yee, H-C., Sandham, N. and Djomehri, M., | year=1999 | title=Low dissipative high order shock-capturing methods using characteristic-based filters| rest=JCP, Vol. 150}}


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