Conditional moment closures limitations

Rank(Cg) = 1. In this case, all components of Cg are either 1 or —1, and it has only one independent row (column). If the allowable region at t = 0 is one-dimensional, then it will remain one-dimensional for all time (assuming that the rank does not change). This limiting case will occur when all scalars can be written as a function of the mixture fraction (e.g., the conditional-moment closure). 

Note that the vector functions go and gi will normally be time-dependent, but can be found from the conditional moments (01 %). In the transported PDF context, the latter can be computed directly from the joint composition PDF so that g0 and gi will be well defined functions.110 The FP model in this limit is thus equivalent to a transported PDF extension of the conditional-moment closure (CMC) discussed in Section 5.8.111 The FP model (including the chemical source term S(0, f)) becomes... 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

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