Realizable FVM

Qn a structured Cartesian grid, the grid cell centered at the point (i Ax, j Ay, k Az) can be denoted by ilijt and its volume by = AxAyAz. The volume-average NDF in Qijk at time t = m At, denoted by n (v, ), is defined by (Leveque, 2002) 

Appendix B. Kinetics-based finite-volume methods 

In general, the volume-average transported moment set is related to a volume-average NDF. Thus, in order to show that is a realizable moment set, it suffices to show that is nonnegative. This idea of using the NDF to show the realizability of the moment set is the basis of kinetics-based FVM. 

as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form  

In QBMM, the volume-average NDF is reconstructed from the volume-average transported moment set using a moment-inversion algorithm (see Chapter 3 for details). The resulting NDF has the form  

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In summary, the realizable FVM solves for the volume-average transported moment set at each time step and grid cell, M , and uses these moment sets at each time step to... 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

The reader should note that in Eqs. (B.2)-(B.5) the spatial derivative appears on the right-hand side, and therefore it will be necessary to define a realizable high-order FVM for each case. In contrast, the source term S in Eq. (B.l) contains no spatial derivatives and hence is local in each finite-volume grid cell. In other words, with operator splitting the source term leads to a (stiff) ordinary differential equation (ODE) for each grid cell. 

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