# Favre averaging

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 $\overline{\rho \Phi''}=0$ $\overline{\rho \widetilde {\Phi}}=\overline{\rho}\widetilde {\Phi}=\overline{\rho \Phi}$