Multi-environment conditional PDF models

In a multi-environment conditional PDF model, it is assumed that the composition vector can be partitioned (as described in Section 5.3) into a reaction-progress vector y rp and a mixture-fraction vector . The presumed conditional PDF for the reaction-progress vector then has the form 155 

In Section 5.8, transport equations for the conditional moments ( 10 were presented. These equations represent the limit Ne = 1, and must be generalized for cases where Ne 1. Defining the weighted conditional moments by156 

154 Ad hoc extensions may be possible for this case by fixing the compositions in one environment at the stoichiometric point, and modeling the probability. On the other hand, by making Ne large, the results will approach those found using transported PDF methods. 

155 In a more general formulation, pn could depend on the mixture-fraction vector. 

156 Note that the conditional means can be found by summing over all environments Q = (v rpIC) = 1 Q - 

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The multi-environment conditional PDF model thus offers a simple description of the effect of fluctuations about the conditional expected values on the chemical source term. 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

As compared with the other closures discussed in this chapter, computation studies based on the presumed conditional PDF are relatively rare in the literature. This is most likely because of the difficulties of deriving and solving conditional moment equations such as (5.399). Nevertheless, for chemical systems that can exhibit multiple reaction branches for the same value of the mixture fraction,162 these methods may offer an attractive alternative to more complex models (such as transported PDF methods). Further research to extend multi-environment conditional PDF models to inhomogeneous flows should thus be pursued. 

Note finally that, for any given value of the mixture fraction (i.e., f f), the multienvironment presumed PDF model discussed in this section will predict a unique value of 4>. In this sense, the multi-environment presumed PDF model provides a simple description of the conditional means (0 f) at Ve discrete values of f. An obvious extension of the method would thus be to develop a multi-environment conditional PDF to model the conditional joint composition PDF / (-i/d x, / ). We look at models based on this idea below. 

Multi-environment presumed PDF models can be developed for the LES composition PDF using either the unconditional, (5.341), or conditional, (5.396), form. However, in order to simplify the discussion, here we will use the unconditional form to illustrate the steps needed to develop the model. The LES composition PDF can be modeled by163... 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

The connection between the multi-environment conditional and unconditional PDF models can be made by noting that... 

As discussed in Section 4.2, the conditional mean compositions will, in general, depend on the filter so that 4> V, 4>) = 4> need not be true. However, if the equality does not hold, it is then necessary to model the difference. Given the simplicity of the multi-environment presumed PDF, such a complication does not seem warranted. 

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