# Subgrid PDF

(Difference between revisions)
 Revision as of 12:08, 10 November 2005 (view source)Salva (Talk | contribs)← Older edit Revision as of 12:17, 10 November 2005 (view source)Salva (Talk | contribs) mNewer edit → Line 21: Line 21: \bar{P}(\underline{\psi}) \bar{P}(\underline{\psi}) \equiv \int_V \equiv \int_V - \Pi \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) + \prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' [/itex] [/itex]

## Revision as of 12:17, 10 November 2005

A subgrid probability density function $\bar{P}(\eta)$ is the distribution function of scalar $Z$ at subrid 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 $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$