Multi-environment presumed PDF models

In summary, given the covariance matrix C, an eigenvalue/eigenvector decomposition can be carried out to find U and A. These matrices define a linear transformation 

The Reynolds-averaged chemical source term in (5.328) is then given by 

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 

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Multi-environment presumed PDF models are generally not recommended for homogeneous flows with uniform mean gradients. Indeed, their proper formulation will require the existence of a mixture-fraction vector that, by definition, cannot generate a uniform mean scalar gradient in a homogeneous flow. 

Five examples of multi-environment presumed PDF models are summarized in Tables 5.1-5.5. For each case, the flow is assumed to be non-premixed with a mixture... 

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 ability of multi-environment presumed PDF models to predict mean compositions in a turbulent reacting flow will depend on a number of factors. For example, their use with chemical kinetic schemes that are highly sensitive to the shape (and not just the low-order moments) of the joint composition PDF will be problematic for small Ne. Such... 

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 DQMOM results for the IEM model can be compared with the multi-environment presumed PDF models in Section 5.10. In particular, (5.374) on p. 226 can be compared with (B.44), and (5.375) can be compared with (B.45). First, we can note that for the IEM model G = 0 and Gs = a. Likewise, y AT"1 + pnSa 4>)n) = 7 .,z and = Ban. Of the four models introduced in Tables 5.1-5.5, only the symmetric two-environment model in Table 5.2 has G = 0 and yM(an) + pnSa = 7Zan. However, because the spurious dissipation terms only ensure that the mixture-fraction variance is correctly predicted, the symmetric two-environment model does not have Gs = a and Mg 1 = Ban. Thus, the covariance matrix is not predicted correctly, as it would be if (B.43) were used. We can thus conclude that the multi-environment presumed PDF models are incomplete in the sense that they do not control as many of the moments as possible for a given choice of -/Ve. 

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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