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Subgrid PDF

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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'
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