Application to turbulent reacting flows

The reduction of the turbulent-reacting-flow problem to a turbulent-scalar-mixing problem represents a significant computational simplification. However, at high Reynolds numbers, the direct numerical simulation (DNS) of (5.100) is still intractable.86 Instead, for most practical applications, the Reynolds-averaged transport equation developed in 

85 Typically, solving (5.151) to find tpfc(oo ) is not the best approach. For example, in combusting systems Srp(0 4)1 1 so that convergence to the equilibrium state will be very slow. Thus, equilibrium thermodynamic methods based on Gibbs free-energy minimization are preferable for most applications. 

86 For a laminar flow problem, (5.100) is tractable however, the assumption that all scalars have the same molecular diffusivity needed to derive it will usually not be adequate for laminar flows. 

Chapter 3 will be employed. Thus, in lieu of (x, ( ), only the mixture-fraction means ) and covariances ffj) (i, j el,A7mfJ will be available. Given this information, we would then like to compute the reacting-scalar means ( rp) and covariances ) 

From (5.152), it follows that the Reynolds-averaged reacting scalars can be found from the mixture-fraction PDF x, t) by integration 87 i 

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In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

The choice of models to include in this book was dictated mainly by their ability to treat the wide range of turbulent reacting flows that occur in technological applications of interest to chemical engineers. In particular, models that cannot treat general chemical... 

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. 

The FM approach to modeling turbulent reacting flows had as its initial focus the description of turbulent combustion processes (e.g., Chung 1969 Chung 1970 Flagan and Appleton 1974 Bilger 1989). In many of the early applications, the details of the chemical reactions were effectively ignored because the reactions could be assumed to be in local chemical equilibrium.26 Thus, unlike the early emphasis on slow and finite-rate reactions... 

As discussed in the present chapter, the closure of the chemical source term lies at the heart of models for turbulent reacting flows. Thus, the material on chemical source term closures presented in Chapter 5 will be of interest to all readers. In Chapter 5, attention is given to closures that can be used in conjunction with standard CFD-based turbulence models (e.g., presumed PDF methods). For many readers, these types of closures will be sufficient to model many of the turbulent-reacting-flow problems that they confront in real applications. Moreover, these closures have the advantage of being particularly simple to incorporate into existing CFD codes. 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

If AW AW the process of finding a linear-mixture basis can be tedious. Fortunately, however, in practical applications Nm is usually not greater than 2 or 3, and thus it is rarely necessary to search for more than one or two combinations of linearly independent columns for each reference vector. In the rare cases where A m > 3, the linear mixtures are often easy to identify. For example, in a tubular reactor with multiple side-injection streams, the side streams might all have the same inlet concentrations so that c(2) = = c(iVin). The stationary flow calculation would then require only AW = 1 mixture-fraction components to describe mixing between inlet 1 and the Nm — I side streams. In summary, as illustrated in Fig. 5.7, a turbulent reacting flow for which a linear-mixture basis exists can be completely described in terms of a transformed composition vector ipm( defined by... 

McMurtry, P. A., S. Menon, and A. R. Kerstein. 1992. A linear eddy sub-grid model for turbulent reacting flows Application to hydrogen-air combustion. 24th Symposium (International) on Combustion Proceedings. Pittsburgh, PA The Combustion Institute. 271-78. 

Given their complexity and practical importance, it should be no surprise that different approaches for dealing with turbulent reacting flows have developed over the last 50 years. On the one hand, the chemical-reaction-engineering (CRE) approach came from the application of chemical kinetics to the study of chemical reactor design. In this approach, the details of the fluid flow are of interest only in as much as they affect the product yield and selectivity of the reactor. In many cases, this effect is of secondary importance, and thus in the CRE approach greater attention has been paid to other factors that directly affect the chemistry. On the other hand, the fluid-mechanical (FM) approach developed as a natural extension of the statistical description of turbulent flows. In this approach, the emphasis has been primarily on how the fluid flow affects the rate of chemical reactions. In particular, this approach has been widely employed in the study of combustion (Rosner 1986 Peters 2000 Poinsot and Veynante 2001 Veynante and Vervisch 2002). 

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