# Conditional filtering

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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, the conditional density-weighted (Favre) filtering is used. Using the density-weighted PDF , $\tilde {P}(\eta)$, the conditionally Favre filtered operation is

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

The relation between Favre and conventional PDF's is

$\bar{\rho} \tilde{P}(\eta) = \overline{\rho|\eta}\bar{P}(\eta)$