Subgrid PDF
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where <math> \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)</math> | where <math> \underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)</math> | ||
is the phase space for the scalar variables | is the phase space for the scalar variables | ||
| - | <math> \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)</math> | + | <math> \underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)</math>. |
| + | |||
| + | A density weighted FDF, <math> \tilde{P}(\eta) </math>, can be obtained as | ||
| + | |||
| + | :<math> | ||
| + | \bar{\rho} \tilde{P}(\eta) | ||
| + | \equiv \int_V \rho \delta \left( | ||
| + | Z(\mathbf{x'},t) - \eta | ||
| + | \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' | ||
| + | </math> | ||
| + | |||
| + | and in the same manner for the joint FDF | ||
| + | :<math> | ||
| + | \bar{\rho} \tilde{P}(\underline{\psi}) | ||
| + | \equiv \int_V | ||
| + | \prod_i^N \rho \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) | ||
| + | G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' | ||
| + | </math> | ||
Revision as of 12:29, 8 May 2006
A subgrid probability density function 
,
also known as filtered density function (FDF),
is the distribution function of scalar 
at subgrid scales.
The probability of observing values between 
within the filter volume is 
where 
is the Dirac delta function,
is a positive defined filter function
with filter width 
.
The joint subgrid PDF of 
scalars is
where 
is the phase space for the scalar variables
.
A density weighted FDF, 
, can be obtained as
and in the same manner for the joint FDF
