# Subgrid PDF

(Difference between revisions)
 Revision as of 12:17, 10 November 2005 (view source)Salva (Talk | contribs)m← Older edit Revision as of 11:09, 15 November 2005 (view source)Salva (Talk | contribs) mNewer edit → Line 1: Line 1: A subgrid [[probability density function]] $\bar{P}(\eta)$ A subgrid [[probability density function]] $\bar{P}(\eta)$ - is the distribution function of scalar $Z$ at subrid scales. + is the distribution function of scalar $Z$ at subgrid scales. The probability of observing values between $\eta < Z < \eta + d\eta$ The probability of observing values between $\eta < Z < \eta + d\eta$

## Revision as of 11:09, 15 November 2005

A subgrid probability density function $\bar{P}(\eta)$ 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 $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$