Transported PDF methods

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

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We develop the CFD equations using the DQMOM model for micromixing. Nevertheless, care must also be taken when using other micromixing models, including transported PDF methods. 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

Starting from these initial conditions, the composition PDF will evolve in a non-trivial manner due to turbulent mixing and molecular diffusion.13 This process is illustrated in Fig. 3.10, where it can be seen that the shape of the composition PDF at early and intermediate times is far from Gaussian.14 As discussed in Chapter 6, one of the principal challenges in transported PDF methods is to develop mixing models that can successfully describe the change in shape of the composition PDF due to molecular diffusion. 

In general, given /u, (V, Ve x, f), transport equations for one-point statistics can be easily derived. This is the approach used in transported PDF methods as discussed in Chapter 6. In this section, as in Section 2.2, we will employ Reynolds averaging to derive the one-point transport equations for turbulent reacting flows. 

As we shall see in Chapter 4, models for izf have much in common with those used for IZij in the Reynolds stress transport equation. Indeed, as shown using transported PDF methods in Chapter 6, the model for uniquely determines the model for 7zf. 

This is the approach taken in transported PDF methods, as discussed in detail in Chapter 6. Most of the other closure methods discussed in Chapter 5 require knowledge of the scalar variance, which can be found from a transport equation as shown next. 

In Chapter 5, we will review models referred to as moment methods, which attempt to close the chemical source term by expressing the unclosed higher-order moments in terms of lower-order moments. However, in general, such models are of limited applicability. On the other hand, transported PDF methods (discussed in Chapter 6) treat the chemical source term exactly. 

Thus, the ASM scalar flux in a first-order reacting flow will decrease with increasing reaction rate. For higher-order reactions, the chemical source term in (3.102) will be unclosed, and its net effect on the scalar flux will be complex. For this reason, transported PDF methods offer a distinct advantage terms involving the chemical source term are closed so that its effect on the scalar flux is treated exactly. We look at these methods in Chapter 6. 

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. 

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. 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

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. 

The Monte-Carlo solvers developed for transported PDF methods are highly paral-lelizable and scale nearly linearly on multiprocessor computers. 

Transported PDF methods are continuing to develop in various directions (i.e, compressible flows, FES turbulence models, etc.). A detailed overview of transported PDF methods is presented in the following two chapters. 

The transport equation for /u, (V, ip x,t) will be derived in the next section. Before doing so, it is important to point out one of the key properties of transported PDF methods ... 

Note that the expected value on the left-hand side is an implicit function of x and t. This is generally true of all expected values generated by transported PDF methods. However, for notational simplicity, an explicit dependence is indicated only when needed to avoid confusion. In that case, we would write (Q)(x, t). 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

Because the chemical source term is treated exactly in transported PDF methods, the conditional reaction/diffusion term,... 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

In a particle implementation of transported PDF methods (see Chapter 7), it will be necessary to estimate go(f) using, for example, smoothing splines, gi (f) will then be found by differentiating the splines. Note that this implies that estimates for the conditional moments (i.e., go) are found only in regions of composition space where the mixture fraction occurs with non-negligible probability. 

The PDF codes presented in this chapter can be (and have been) extended to include additional random variables. The most obvious extensions are to include the turbulence frequency, the scalar dissipation rate, or velocity acceleration. However, transported PDF methods can also be applied to treat multi-phase flows such as gas-solid turbulent transport. Regardless of the flow under consideration, the numerical issues involved in the accurate treatment of particle convection and coupling with the FV code are essentially identical to those outlined in this chapter. For non-orthogonal grids, the accurate implementation of the particle-convection algorithm is even more critical in determining the success of the PDF simulation. 

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