Molecular mixing models

The composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

Before discussing in detail specific molecular mixing models, it is useful to first state a few important constraints that can be derived by computing expected values. The first constraint follows from45... 

I) The molecular mixing model must leave the scalar mean unchanged. 

Note, however, that in the presence of a mean scalar gradient the local isotropy condition is known to be incorrect (see Warhaft (2000) for a review of this topic). Although most molecular mixing models do not account for it, the third constraint can be modified to... 

We shall see that constraints (I) and (II) are always taken into consideration when developing molecular mixing models, but that constraint (III) has been largely ignored. This is most likely because almost all of the existing molecular mixing models have been developed in the context of the joint composition PDF, i.e., for... 

In general, we would like a molecular mixing model to have the following properties. 

The reasoning behind each of these properties will be illustrated in the next section. We will then look at three simple molecular mixing models (namely, the CD, the IEM, and the FP models) and discuss why each is not completely satisfactory. For convenience, the list of constraints and desirable properties is summarized in Table 6.1. 

Although much effort has gone into searching for molecular mixing models that improve upon the existing models,54 no model completely satisfies all of the desirable properties... 

Table 6.1. Constraints and desirable properties of molecular mixing models. 

In another DNS study of two-scalar mixing (Juneja and Pope 1996), similar conclusions were drawn for the joint PDF of two inert scalars. These observations suggest that the development of molecular mixing models can proceed in two separate steps. 

Most molecular mixing models concentrate on step (1). However, for chemical-reactor applications, step (2) can be very important since the integral length scales of the scalar and velocity fields are often unequal (L / Lu) due to the feed-stream configuration. In the FP model (discussed below), step (1) is handled by the shape matrix H, while step (2) requires an appropriate model for e. 

Like the IEM model, the FP model weakly satisfies property (iv). Likewise, property (v) can be built into the model for the joint scalar dissipation rates (Fox 1999), and the Sc dependence in property (vi) is included explicitly in the FP model. Thus, of the three molecular mixing models discussed so far, the FP model exhibits the greatest number of desirable properties provided suitable functional forms can be found for (e 0). 

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. 

When used in a molecular mixing model, an important property of estimation algorithms is the ability to leave the mean composition unchanged. For example, with the (constant >) LIEM model, global mean conservation requires... 

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