# Favre averaging

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:
 $\widetilde{\Phi} \equiv \frac{ \int_T \rho(t) \Phi(t) dt} { \int_T \rho(t) dt } \equiv \frac{\overline{\rho \Phi}}{\overline{\rho}}$ (1) $\Phi'' \equiv \Phi - \widetilde{\Phi}$