CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Subgrid PDF

Subgrid PDF

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
m
m
Line 1: Line 1:
-
A subgrid [[probability density function]] <math> \bar{P}(\eta) </math>
+
A subgrid [[probability density function]] <math> \bar{P}(\eta) </math> ,
 +
also known as filtered density function (FDF),
is the distribution function of scalar <math> Z </math> at subgrid scales.
is the distribution function of scalar <math> Z </math> at subgrid scales.

Revision as of 12:25, 8 May 2006

A subgrid probability density function  \bar{P}(\eta) , also known as filtered density function (FDF), is the distribution function of scalar  Z at subgrid scales.

The probability of observing values between  \eta < Z < \eta + d\eta within the filter volume is   \bar{P}(\eta) d\eta


\bar{P}(\eta) 
\equiv \int_V  \delta \left(
Z(\mathbf{x'},t) - \eta
\right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'

where  \delta is the Dirac delta function,  G is a positive defined filter function with filter width  \Delta .

The joint subgrid PDF of  N scalars is


\bar{P}(\underline{\psi}) 
\equiv \int_V
\prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right)
G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'

where  \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N) is the phase space for the scalar variables  \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)

My wiki