Micromixing models multi-environment

The CFD model developed above is an example of a moment closure. Unfortunately, when applied to reacting scalars such as those considered in Section III, moment closures for the chemical source term are not usually accurate (Fox, 2003). An alternative approach that yields the same moments can be formulated in terms of a presumed PDF method (Fox, 1998). Here we will consider only the simplest version of a multi-environment micromixing model. Readers interested in further details on other versions of the model can consult Wang and Fox (2004). 

In Section 5.6, Lagrangian micromixing models based on mixing environments were introduced. In terms of the joint composition PDF, nearly all such models can be expressed mathematically as a multi-peak delta function. The principal advantage of this type of model is the fact that the chemical source term is closed, and thus it is not necessary to integrate with respect to the joint composition PDF in order to evaluate the... 

In a multi-environment micromixing model, the presumed composition PDF has the following form ... 

Multi-environment presumed PDF models can also be easily extended to treat cases with more than two feed streams. For example, a four-environment model for a flow with three feed streams is shown in Fig. 5.24. For this flow, the mixture-fraction vector will have two components, 2 and 22- The micromixing functions should thus be selected to agree with the variance transport equations for both components. However, in comparison with multi-variable presumed PDF methods for the mixture-fraction vector (see Section 5.3), the implementation of multi-environment presumed PDF models in CFD calculations of chemical reactors with multiple feed streams is much simpler. 

The results from (B.43) can be combined with the multi-environment presumed PDF models to define a new class of micromixing models that agree with the DQMOM result for inhomogeneous flows. Formally, (B.43) can be written as... 

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