# Subgrid PDF

(Difference between revisions)
 Revision as of 12:08, 10 November 2005 (view source)Salva (Talk | contribs)← Older edit Latest revision as of 07:39, 12 April 2007 (view source)Jola (Talk | contribs) m (Reverted edits by YvrHqo (Talk); changed back to last version by Salva) (6 intermediate revisions not shown) 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. + also known as filtered density function (FDF), + is the distribution function of scalar $Z$ at subgrid scales. The probability of observing values between $\eta < Z < \eta + d\eta$ The probability of observing values between $\eta < Z < \eta + d\eta$ Line 21: Line 22: \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] Line 27: Line 28: where $\underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)$ where $\underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)$ is the phase space for the scalar variables is the phase space for the scalar variables - $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$ + $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$. + + A density weighted FDF, $\tilde{P}(\eta)$, can be obtained as + + :$+ \bar{\rho} \tilde{P}(\eta) + \equiv \int_V \rho \delta \left( + Z(\mathbf{x'},t) - \eta + \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' +$ + + and in the same manner for the joint FDF + :$+ \bar{\rho} \tilde{P}(\underline{\psi}) + \equiv \int_V \rho + \prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) + G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV' +$

## Latest revision as of 07:39, 12 April 2007

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 $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$.

A density weighted FDF, $\tilde{P}(\eta)$, can be obtained as

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

and in the same manner for the joint FDF

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