# Favre averaging

(Difference between revisions)
 Revision as of 10:36, 7 September 2005 (view source)Jola (Talk | contribs)m← Older edit Revision as of 18:37, 4 October 2006 (view source)Tsaad (Talk | contribs) (fixed and added some information)Newer edit → Line 1: Line 1: - Let $\Phi$ be any dependent variable. This variable can be decomposed into a fluctuating part $\Phi''$ and mean part $\overline{\Phi}$ using a density weighted average in the following way: + Let $\Phi$ be any dependent variable. This variable can be decomposed into a mean part $\widetilde{\Phi}$ and a fluctuating part $\Phi''$ using a density weighted average in the following way: +
+ :$\Phi \equiv \widetilde{\Phi} + \Phi''$
:$\widetilde{\Phi} \equiv \frac{ \int_T \rho(t) \Phi(t) dt} :[itex]\widetilde{\Phi} \equiv \frac{ \int_T \rho(t) \Phi(t) dt} Line 7: Line 9: (1) (1) - :[itex]\Phi'' \equiv \Phi - \widetilde{\Phi}$ + where the overbars (e.g. $\overline{\rho \Phi}$) denote averages using the Reynolds decomposition. -
+ auxiliary relations include +
+ $\overline{\rho \Phi''}=0$ +
+ [itex]\overline{\rho \widetilde {\Phi}}=\overline{\rho}\widetilde {\Phi}=\overline{\rho \Phi}
+

## Revision as of 18:37, 4 October 2006

Let $\Phi$ be any dependent variable. This variable can be decomposed into a mean part $\widetilde{\Phi}$ and a fluctuating part $\Phi''$ using a density weighted average in the following way:

 $\Phi \equiv \widetilde{\Phi} + \Phi''$ $\widetilde{\Phi} \equiv \frac{ \int_T \rho(t) \Phi(t) dt} { \int_T \rho(t) dt } \equiv \frac{\overline{\rho \Phi}}{\overline{\rho}}$ (1) where the overbars (e.g. $\overline{\rho \Phi}$) denote averages using the Reynolds decomposition. auxiliary relations include $\overline{\rho \Phi''}=0$ $\overline{\rho \widetilde {\Phi}}=\overline{\rho}\widetilde {\Phi}=\overline{\rho \Phi}$

Favre averaging is sometimes used in compressible flow to separate turbulent fluctuations from the mean-flow. In most cases it is not necessary to use Favre averaging though, since turbulent fluctuations most often do not lead to any signigicant fluctuations in density. In that case the more simple Reynolds averaging can be used. Only in highly compressible flows and hypersonic flows is it necassary to perform the more complex Favre averaging.

Favre averaging can be used to derive the Favre averaged Navier-Stokes equations.