Spatial transport of the velocity-scalar NDF

As a final case, we will again consider the joint velocity-scalar NDF governed by Eq. (8.122), but without specifying a functional form for the scalar-conditioned velocity. The moment-transport equation is again given by Eq. (8.122). However, we will now use a scalar-conditioned multivariate EQMOM to reconstruct the joint NDE  

Eor the scalar, the kernel density function will be defined by 

As for the bivariate velocity moments in Section 8.4, 13 bivariate moments must be transported to And the parameters in Eq. (8.143) mo,o, mop, mop, mop, mop, m p, mip,m2p, mpp, mop,mop,mAfi,mpp, where the moments are defined by mjp = Jv n(v, )dvd. The weights Wap), abscissas a, Vap), and width parameters (cri, cria.) are determined from the 13 bivariate moments using the ECQMOM as described in Section 3.3.4. Eor clarity, the steps required for bivariate ECQMOM for the case under consideration are as follows. 

As in Eq. (8.102), the weights and abscissas in this expression come from (i) the ECQMOM, (ii) Gauss-Laguerre quadrature, and (iii) Gauss-Hermite quadrature. Likewise, Mi and M2 can be chosen as large as needed in order to minimize the quadrature error in the spatial fluxes and drag terms. 

The numerical solution of the moment-transport equation is again done by using operator splitting to update first for the spatial fluxes and then for the drag. The realizable scheme 

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