Conditional-moment closures

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.,... 

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). 

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. 

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. 

There are a few other non-PDF approaches to simulating reactive flow processes (for example, the linear eddy model of Kerstein, 1991 and the conditional moment closure model of Bilger, 1993). These approaches are not discussed here as most of the engineering simulations of reactive flow processes can be achieved by the approaches discussed earlier. The discussion so far has been restricted to single-phase turbulent reactive flow processes. We now briefly consider modeling multiphase reactive flow processes. 

Bilger, R.W. (1993), Conditional moment closure for turbulent reacting flow, Phys. Fluids A,5(2), 436-444. Brodkey, R.S. (1975), Mixing in turbulent field, in Turbulence in Mixing Operation (Ed. R.S. Brodkey), Academic Press, London. 

A. Vikhansky and S. M. Cox. Conditional moment closure for chemical reactions in laminar chaotic flows. AIChE J., 53 19-27, 2007. 

Klimenko, A. BUger R. (1999). Conditional Moment Closure for Turbulent Combustion, Prog. Ener. Combustion, Vol. 25, pp 595-687... 

Navarro-Martinez, S., Kronenburg A. and di Mare, F. (2005). Conditional moment closure for large eddy simulations. Flow, Turbulence and Combustion Vol. 75, pp 245-274... 

---