# Conditional filtering

A conditional filtering operation of a variable $\Phi$ is defined as

$\overline{\Phi|\eta} \equiv \frac{\int_V \Phi \psi_\eta \left( \xi(\mathbf{x'},t) - \eta \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'}{\bar{P}(\eta)}$

where $G$is a positive defined space filter with filter width $\Delta$ (see LES filters), $\psi_\eta$ is a fine-grained probability density function, which is taken as a Dirac delta $\psi_\eta \equiv \delta ( \xi - \eta )$. The probability density function $\bar{P}(\eta)$ is a subgrid PDF and $\eta$ is the sample space of the passive scalar $\xi$. In variable density flows, conditional density-weighted (Favre) filtering is used, $\bar{\rho}_\eta \tilde{\Phi}_\eta=\overline{\rho \Phi|\eta}$,