Conditional filtering
From CFD-Wiki
(Difference between revisions)
m |
m |
||
(One intermediate revision not shown) | |||
Line 6: | Line 6: | ||
</math> | </math> | ||
- | where <math>G </math>is a positive defined space filter with filter width <math> \Delta </math>, | + | where <math>G </math>is a positive defined space filter with filter width <math> \Delta </math> (see [[LES filters]]), |
<math> \psi_\eta </math> is a fine-grained [[probability density function]], | <math> \psi_\eta </math> is a fine-grained [[probability density function]], | ||
which is taken as a Dirac delta <math> \psi_\eta \equiv \delta ( \xi - \eta ) </math>. | which is taken as a Dirac delta <math> \psi_\eta \equiv \delta ( \xi - \eta ) </math>. | ||
The probability density function | The probability density function | ||
<math>\bar{P}(\eta) </math> is a [[subgrid PDF]] and <math> \eta </math> is the sample space of the passive scalar | <math>\bar{P}(\eta) </math> is a [[subgrid PDF]] and <math> \eta </math> is the sample space of the passive scalar | ||
- | <math> \xi </math>. In variable density flows, conditional density-weighted | + | <math> \xi </math>. |
- | (Favre) filtering is used, | + | In variable density flows, the conditional density-weighted (Favre) filtering is used. |
- | <math> \bar{\rho} | + | Using the density-weighted PDF , <math> \tilde {P}(\eta) </math>, the conditionally Favre filtered operation is |
+ | |||
+ | :<math> | ||
+ | \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)} | ||
+ | </math> | ||
+ | |||
+ | The relation between Favre and conventional PDF's is | ||
+ | :<math> | ||
+ | \bar{\rho} \tilde{P}(\eta) = \overline{\rho|\eta}\bar{P}(\eta) | ||
+ | </math> |
Latest revision as of 12:38, 8 May 2006
A conditional filtering operation of a variable is defined as
where is a positive defined space filter with filter width (see LES filters), is a fine-grained probability density function, which is taken as a Dirac delta . The probability density function is a subgrid PDF and is the sample space of the passive scalar . In variable density flows, the conditional density-weighted (Favre) filtering is used. Using the density-weighted PDF , , the conditionally Favre filtered operation is
The relation between Favre and conventional PDF's is