Moment-transport equation conservation form

The moment-transport equations discussed above become more and more complicated as the order increases. Moreover, these equations are not closed. In quadrature-based moment methods, the velocity-distribution function is reconstructed from a finite set of moments, thereby providing a closure. In this section, we illustrate how the closure hypothesis is applied to solve the moment-transport equations with hard-sphere collisions. For clarity, we will consider the monodisperse case governed by Eq. (6.131). Formally, we can re-express this equation in conservative form ... 

Before looking at specific examples for which the DQMOM is likely to fail, we should note that the direct solution of the moment-transport equations using the QMOM is very robust. In the context of conservation equations, the moment-transport equations (and, for that matter, the NDF) are written in a conservative form ... 

Strictly speaking, the conservative form refers to conservation equations without a source term. When one is solving moment-transport equations, operator splitting wherein the transport step is solved with S = 0 is often employed. 

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