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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 (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)
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