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2-D vortex in isentropic flow

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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^2} \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 is:


\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.


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