Transport equation scalar, reacting

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

In Section 3.3, we will use (3.16) with the Navier-Stokes equation and the scalar transport equation to derive one-point transport equations for selected scalar statistics. As seen in Chapter 1, for turbulent reacting flows one of the most important statistics is the mean chemical source term, which is defined in terms of the one-point joint composition PDF +(+x, t) by... 

The final form for the scalar mean transport equation in a turbulent reacting flow is given by... 

The transport equation for the scalar flux of a reacting scalar [Pg.103]

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

Table 3.3. The one-point scalar statistics and unclosed quantities appearing in the transport equations for inhomogeneous turbulent mixing of multiple reacting scalars at high Reynolds numbers. 

The one-point scalar statistics used in engineering calculations of high-Reynolds-number turbulent mixing of reacting scalars is summarized in Table 3.3 along with the unclosed terms that appear in their transport equations. In Chapter 4, we will discuss methods for modeling the unclosed terms in the RANS transport equations. 

This chapter is devoted to methods for describing the turbulent transport of passive scalars. The basic transport equations resulting from Reynolds averaging have been derived in earlier chapters and contain unclosed terms that must be modeled. Thus the available models for these terms are the primary focus of this chapter. However, to begin the discussion, we first review transport models based on the direct numerical simulation of the Navier-Stokes equation, and other models that do not require one-point closures. The presentation of turbulent transport models in this chapter is not intended to be comprehensive. Instead, the emphasis is on the differences between particular classes of models, and how they relate to models for turbulent reacting flow. A more detailed discussion of turbulent-flow models can be found in Pope (2000). For practical advice on choosing appropriate models for particular flows, the reader may wish to consult Wilcox (1993). 

Note that the right-hand side of this expression is an E x I null matrix, and thus element conservation must hold for any choice of e e l,E and i e 1Moreover, since the element matrix is constant, (5.10) can be applied to the scalar transport equation ((1.28), p. 16) in order to eliminate the chemical source term in at least E of the K equations.9 The chemically reacting flow problem can thus be described by only K - E transport equations for the chemically reacting scalars, and E transport equations for non-reacting (conserved) scalars.10... 

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. 

Note that Nr = 2. Thus, by applying an appropriate linear transformation, it should be possible to rewrite the scalar transport equation in terms of two reacting and two conserved scalars. 

The interest in reformulating the conserved-variable scalars in terms of the mixture-fraction vector lies in the fact that relatively simple forms for the mixture-fraction PDF can be employed to describe the reacting scalars. However, if < /Vmf, then the incentive is greatly diminished since more mixture-fraction-component transport equations (Nmf) would have to be solved than conserved-variable-scalar transport equations (/V, << ). We will thus assume that N m = Nmf and seek to define the mixture-fraction vector only for this case. Nonetheless, in order for the mixture-fraction PDF method to be applicable to the reacting scalars, they must form a linear mixture defined in terms of the components of the mixture-fraction vector. In some cases, the existence of linear mixtures is evident from the initial/inlet conditions however, this need not always be the case. Thus, in this section, a general method for defining the mixture-fraction vector in terms of a linear-mixture basis for arbitrary initial/inlet conditions is developed. 

Having demonstrated the existence of a mixture-fraction vector for certain turbulent reacting flows, we can now turn to the question of how to treat the reacting scalars in the equilibrium-chemistry limit for such flows. Applying the linear transformation given in (5.107), the reaction-progress-vector transport equation becomes... 

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... 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

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. 

For each test case, a non-reacting scalar (e.g., mixture fraction) should be used to determine the spatial distribution of its mean and variance (i.e., (f) and (f/2 . These results can then be compared with those found by solving the RANS transport equations (i.e., (4.70), p. 120 and (4.90), p. 125) with identical values for (U) and Tt. Fike-wise, the particle-weight distribution should be compared with the theoretical value (i.e., (7.74)). While small fluctuations about the theoretical value are to be expected, a systematic deviation almost always is the result of inconsistencies in the particle-convection algorithm. 

For complex chemistry, in many cases, a conserved scalar or a mixture fraction approach can be used, in which a single conserved scalar (mixture fraction) is solved instead of transport equations for individual species. The reacting system is treated using either chemical equilibrium calculations or by assuming infinitely fast reactions (mixed-is-reacted approach). The mixture fraction approach is applicable to non-premixed situations and is specifically developed to simulate turbulent diffusion flames containing one fuel and one oxidant. Such situations are illustrated in Fig. 5.6. The basis for the mixture fraction approach is that individual conservation equations for fuel and oxidant can be combined to eliminate reaction rate terms (see Toor, 1975 for more details). Such a combined equation can be simplified by defining a mixture... 

In scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

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