# 2-D vortex in isentropic flow

(Difference between revisions)
 Revision as of 07:52, 16 September 2005 (view source)Praveen (Talk | contribs)← Older edit Revision as of 14:35, 6 January 2012 (view source)Gazaix (Talk | contribs) m (Correct bug)Newer edit → (2 intermediate revisions not shown) Line 1: Line 1: - The test case involves convection of an isentropic vortex in inviscid flow. + The test case involves [[convection]] of an [[isentropic]] [[vortex]] in [[inviscid flow]]. - The free-stream conditions are + The [[free-stream conditions]] are :$:[itex] Line 11: Line 11:$ [/itex] - Perturbations are added to the free-stream in such a way that there is no + 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 + [[entropy]] gradient in the [[flow-field]]. The perturbations are given by :$:[itex] Line 18: Line 18: (\delta u, \delta v) &=& \frac{\beta}{2\pi} \exp\left( \frac{1-r^2}{2} (\delta u, \delta v) &=& \frac{\beta}{2\pi} \exp\left( \frac{1-r^2}{2} \right) [ -(y-y_o), (x-x_o) ] \\ \right) [ -(y-y_o), (x-x_o) ] \\ - \rho &=& \left[ 1 - \frac{ (\gamma-1)\beta^2}{8\gamma\pi} \exp\left( + \rho &=& \left[ 1 - \frac{ (\gamma-1)\beta^2}{8\gamma\pi^2} \exp\left( 1-r^2\right) \right]^{\frac{1}{\gamma-1}} \\ 1-r^2\right) \right]^{\frac{1}{\gamma-1}} \\ p &=& \frac{ \rho^\gamma }{\gamma} p &=& \frac{ \rho^\gamma }{\gamma} Line 30: Line 30:$ [/itex] - is distance from the vortex center $(x_o, y_o)$. One choice for the domain + is distance from the [[vortex]] center $(x_o, y_o)$. - and parameters are + + One choice for the domain and parameters is: :$:[itex] Line 39: Line 40:$ [/itex] - As a result of isentropy, the exact solution corresponds to a pure advection + 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). + of the [[vortex]] at the [[free-stream velocity]]. Further details can be found in Yee et al. (1999). ==References== ==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}} *{{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}} + + + {{stub}}

## Revision as of 14:35, 6 January 2012

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.