Moment closure

The simplest closure for the chemical source term is to assume that the joint composition PDF can be represented by its moments. In general, this assumption is of limited validity. Nevertheless, in this section we review methods based on moment closures in order to illustrate their limitations. 

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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). 

The next step would be to implement the CFD transport equation for the state variables in a CFD code. This is a little more difficult for the two-environment model (due to the gradient terms on the right-hand sides of Eqs. 37 and 38) than for the moment closure. Nevertheless, if done correctly both models will... 

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

The limiting case where the chemical time scales are all large compared with the mixing time scale r, i.e., the slow-chemistry limit, can be treated by a simple first-order moment closure. In this limit, micromixing is fast enough that the composition variables can be approximated by their mean values (i.e., the first-order moments (0)). We can then write, for example,... 

Or, to put it another way, the simplest hrst-order moment closure is to assume that all scalar covariances are zero ... 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

This observation suggests that a moment-closure approach based on the conditional scalar moments may be more successful than one based on unconditional moments. Because adequate models are available for the mixture-fraction PDF, conditional-moment closures focus on the development of methods for finding a general expression for Q( x, t). 

Obviously, conditional moments of higher order could also be modeled. However, as with moment closures, the unclosed terms in the higher-order transport equation are more and more difficult to close. 

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. 

Thus, the final product mixture will depend on the relative importance of mixing and reaction in determining (T )i(f). Finally, note that since the second environment was necessary to describe the ignition source, this simple description of ignition and extinction would not be possible with a one-environment model (e.g., the conditional moment closure). 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

In general, the scalar fields will not be Gaussian. Thus, analogous to what is done in second-moment closure methods, a gradient-diffusion model is usually employed to close this term ... 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

Rank(Cg) = 1. In this case, all components of Cg are either 1 or —1, and it has only one independent row (column). If the allowable region at t = 0 is one-dimensional, then it will remain one-dimensional for all time (assuming that the rank does not change). This limiting case will occur when all scalars can be written as a function of the mixture fraction (e.g., the conditional-moment closure). 

Note that the vector functions go and gi will normally be time-dependent, but can be found from the conditional moments (01 %). In the transported PDF context, the latter can be computed directly from the joint composition PDF so that g0 and gi will be well defined functions.110 The FP model in this limit is thus equivalent to a transported PDF extension of the conditional-moment closure (CMC) discussed in Section 5.8.111 The FP model (including the chemical source term S(0, f)) becomes... 

Conditional moment closure for turbulent reacting flow. Physics of Fluids A Fluid Dynamics 5,436 -44. 

Bushe, W. K. and H. Steiner (1999). Conditional moment closure for large eddy simulation of nonpremixed turbulent reacting flows. Physics of Fluids 11, 1896-1906. 

Cha, C. M., G. Kosaly, and H. Pitsch (2001). Modeling extinction and reignition in turbulent nonpremixed combustion using a doubly-conditional moment closure approach. Physics of Fluids 13, 3824-3834. 

Note on the conditional moment closure in turbulent shear flows. Physics of Fluids 7, 446 148. 

Klimenko, A. Y. and R. W. Bilger (1999). Conditional moment closure for turbulent combustion. Progress in Energy and Combustion Science 25, 595-687. 

Wouters, H. A., T. W. J. Peters, and D. Roekaerts (1996). On the existence of a generalized Langevin model representation for second-moment closures. Physics of Fluids 8, 1702-1704. 

Pope, S.B. 1994. On the relation between stochastic Lagrangian models of turbulence and second-moment closures. J. Physics Fluids 6(2) 973-85. 

The model given above is called the k- -kp model, which can be used for dilute, non-swirling, nonbuoyant gas-solid flows. For strongly anisotropic gas-solid flows, the unified second-order moment closure model, which is an extension of the second-order moment closure model for single-phase flows [Zhou, 1993], may be used. 

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