Presumed PDF methods

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). 

As discussed in the present chapter, the closure of the chemical source term lies at the heart of models for turbulent reacting flows. Thus, the material on chemical source term closures presented in Chapter 5 will be of interest to all readers. In Chapter 5, attention is given to closures that can be used in conjunction with standard CFD-based turbulence models (e.g., presumed PDF methods). For many readers, these types of closures will be sufficient to model many of the turbulent-reacting-flow problems that they confront in real applications. Moreover, these closures have the advantage of being particularly simple to incorporate into existing CFD codes. 

The most widely used approach for approximating /g(C x, t) is the presumed PDF method, in which a known distribution function is chosen to represent the mixture-fraction PDF. We will look at the various possible forms in Section 5.9, where presumed PDF... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

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. 

Unlike presumed PDF methods, transported PDF methods do not require a priori knowledge of the joint PDF. The effect of chemical reactions on the joint PDF is treated exactly. The key modeled term in transported PDF methods is the molecular mixing term (i.e., the micromixing term), which describes how molecular diffusion modifies the shape of the joint PDF. 

This formulation can be compared with the multi-environment (ME) presumed PDF method discussed in Section 5.10. The principal difference between the two approaches is the treatment of turbulent convection. In the transported PDF simulation, turbulent convection is simulated by a random process. In the ME—PDF approach, it is handled using standard FV/FD discretization. 

The quadrature method of moments (QMOM) is a presumed PDF approach that determines the unknown parameters by forcing the lower-order moments of the presumed PDF to agree with the moment transport equations (McGraw 1997 Barrett and Webb 1998 Marchisio et al. 2003a Marchisio et al. 2003b). As with the multi-environment presumed PDF method discussed in Section 5.10, the form of the presumed PDF is... 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

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