Joint PDF

The CFD model for the reaction given in Eq. (60) in the limit where the first reaction is very fast must account for fluctuations in and Y2 due to turbulent mixing. In general, this is done by solving for their joint PDF (Fox, 2003), denoted here by / ((, y). There are several ways this can be accomplished ... 

The multi-environment model for the joint PDF generalizes Eq. (33) by writing... 

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. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

In this book, an alternative description based on the joint probability density function (PDF) of the species concentrations will be developed. (Exact definitions of the joint PDF and related quantities are given in Chapter 3.) The RTD function is in fact the PDF of the fluid-element ages as they leave the reactor. The relationship between the PDF description and the RTD function can be made transparent by defining a fictitious chemical species... 

At steady state, die left-hand side of this expression has independent variables ip. For fixed ip = ip, the integral on the right-hand side sweeps over all fluid elements in search of those whose concentrations ),atcll are equal to ip Af these fluid elements have the same age (say, a = o ), then the joint PDF reduces to Xoutiet) = E(a ),... 

For non-interacting fluid elements, the RTD function is thus equivalent to the joint PDF of the concentrations. In composition space, the joint PDF would he on a one-dimensional sub-manifold (i.e., have a one-dimensional support) parameterized by the age a. The addition of micromixing (i.e., interactions between fluid elements) will cause the joint PDF to spread in composition space, thereby losing its one-dimensional support. 

Likewise, in the limit of large numbers of samples and bins, the bi-variate histogram can be used to compute an estimate for the joint PDF of [Pg.40]

The procedure presented above can be easily extended to estimate the joint PDF of a vector of K composition variables fchemical source term (S) inside a CFD grid cell can be estimated by sampling the chemical source term at every... 

For example, all information is lost concerning the relative spatial locations of two random samples. As discussed in Chapter 2, this fact implies that all information concerning the spatial derivatives of the scalar fields is lost when the scalar field is described by its one-point joint PDF. 

In fully developed homogeneous turbulence,7 the one-point joint velocity PDF is nearly Gaussian (Pope 2000). A Gaussian joint PDF is uniquely defined by a vector of expected values (j, and a covariance matrix C ... 

The unconditional mean can be found from the conditional mean by averaging with respect to the joint PDF of U2 and U3 ... 

As discussed in Chapter 1, given a sample mixture-fraction field16 f (x, t) x e L3, this joint PDF could be estimated and plotted in terms of z and as a two-dimensional contour plot. 

More precisely, the Fourier coefficients in (4.27) can be replaced by random variables with the following properties k. U = 0 and (U ) = 0 for all k such that k > kc. An energy-conserving scheme would also require that the expected value of the residual kinetic energy be the same for all choices of the random variable. The LES velocity PDF is a conditional PDF that can be defined in die usual manner by starting from die joint PDF for the discrete Fourier coefficients U. ... 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

The relationship between age-based models and models based on the composition PDF can be understood in terms of the joint PDF of composition ([Pg.214]

As shown in Section 5.10, it is possible to reformulate multi-environment models in terms of a multi-peak presumed joint PDF. In this case, it is possible to specify the... 

Application of the SLF model thus reduces to predicting the joint PDF of the mixture fraction and the scalar dissipation rate. As noted above, in combusting flows flame extinction will depend on the value of x Thus, unlike the equilibrium-chemistry method (Section 5.4), the SLF model can account for flame extinction due to local fluctuations in the scalar dissipation rate. 

In almost all reported applications of the flamelet model, f and x have been assumed to be independent so that the joint PDF can be written in terms of the marginal PDFs ... 

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction . In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132... 

This assumption leads to relationships for a and b of the form of (5.144) and (5.145), respectively, but with the mixture-fraction mean and variance replaced by the mean and variance of Y. The joint PDF can thus be closed by solving four transport equations for... 

The method just described for treating multiple reacting-progress variables has the distinct disadvantage that the upper bounds must be found a priori. For a complex reaction scheme, this may be unduly difficult, if not impossible. This fact, combined with the desire to include the correlations between the reacting scalars, has led to the development of even simpler methods based on a presumed joint PDF for the composition vector... 

Given the strong assumptions needed to model the joint PDF, the use of more refined models would not be justified. 

A simple functional form that can be used to approximate the joint PDF of the Ns composition variables is the joint Gaussian PDF ... 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

Note that, by definition, = 0. Moreover, starting from (5.336), it is easily shown that (0in0 > = I and (0deP0dep) = 0- It then follows from the assumed joint Gaussian PDF for 0 that 0dep = 0, and that the joint PDF of 0m is given by... 

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 numerical methods employed to solve the transported PDF transport equation are very different from standard CFD codes. In essence, the joint PDF is represented by a large collection of notional particles. The idea is similar to the presumed multi-scalar PDF method discussed in the previous section. The principal difference is that the notional particles move in real and composition space by well defined stochastic models. Some of the salient features of transported PDF codes are listed below. 

The joint PDF contains no information concerning local velocity and/or scalar gradients. A two-point description would be required to describe the gradients.1... 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

From this expression, it can be seen that the joint PDF evolves by transport in... 

However, in order to use (6.19) to solve for the joint PDF, closures must be supplied for the conditional acceleration and reaction/diffusion terms. For simplicity, we will refer to these terms as the conditional fluxes. 

The scalars must remain bounded (e.g., the mixture fraction must lie between 0 and 1), but as the variances decrease to zero the shape of the joint PDF about ) should be asymptotically Gaussian. 

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. 

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