Linear-eddy model

The modeling ideas used in the LEM have recently been extended to include a onedimensional description of the turbulence (Kerstein 1999a). This one-dimensional turbulence (ODT) model has been applied to shear-driven (Kerstein 1999a Kerstein and Dreeben 2000) and buoyancy-driven flows (Kerstein 1999b), as well as to simple reacting shear flows (Echekki et al. 2001 Hewson and Kerstein 2001 Kerstein 2002). Since the 

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At present, there exists no completely general RANS model for differential diffusion. Note, however, that because it solves (4.37) directly, the linear-eddy model discussed in Section 4.3 can describe differential diffusion (Kerstein 1990 Kerstein et al. 1995). Likewise, the laminar flamelet model discussed in Section 5.7 can be applied to describe differential diffusion in flames (Pitsch and Peters 1998). Here, in order to understand the underlying physics, we will restrict our attention to a multi-variate version of the SR model for inert scalars (Fox 1999). 

The PDF of an inert scalar is unchanged by the first two steps, but approaches the well mixed condition during step (3).108 The overall rate of mixing will be determined by the slowest step in the process. In general, this will be step (1). Note also that, except in the linear-eddy model (Kerstein 1988), interactions between Lagrangian fluid particles are not accounted for in step (1). This limits the applicability of most mechanistic models to cases where a small volume of fluid is mixed into a much larger volume (i.e., where interactions between fluid particles will be minimal). 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

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. 

Desjardin, P. E. and S. H. Frankel (1996). Assessment of turbulent combustion submodels using the linear eddy model. Combustion and Flame 104, 343-357. 

Kerstein, A. R. (1988). A linear-eddy model of turbulent scalar transport and mixing. 

Linear-eddy modeling of turbulent transport. II Application to shear layer mixing. 

Linear-eddy modelling of turbulent transport. Part 3. Mixing and differential diffusion in round jets. Journal of Fluid Mechanics 216, 411 —4-35. 

McMurtry, P. A., S. Menon, and A. R. Kerstein (1993). Linear eddy modeling of turbulent combustion. Energy and Fuels 7, 817-826. 

Frankel, S.H., C.K. Madnia, P. A. McMurtry, and P. Givi. 1993. Binary scalar mixing and reaction in homogeneous turbulence Some linear eddy model results. 

Zimberg, M. J., S. H. Prankel, J. P. Gore, and Y. R. Sivathanu. 1998. A study of coupled turbulence, soot chemistry and radiation effects using the linear eddy model. Combustion Flame 113 454-69. 

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. 

Kerstein, A.R. (1991), Linear eddy modeling of turbulent transport, J. Fluid Mech., 231, 361-394. 

Among the other models proposed are 3- and 4-environment models (Ritchie and Togby, 1979 Mehta and Tarbell, 1983), stretch or laminar mixing model (Ou and Ranz, 1983a,b Chella and Otino, 1984 Ranz, 1985), linear eddy model (Kerstein, 1991), and a generalized IFM model for nonpremixed turbulent reacting flows (Tsai and Fox, 1998). 

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