# Favre averaging

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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 necessary to perform the more complex Favre averaging.

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