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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 ,  \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} ,

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