Mixture-fraction vector

33 The case where the covariance is always positive can be handled simply by changing the sign on the right-hand side. 

34 This set of reactions is auto-catalytic, i.e., production of B by the second step enhances the reaction rate of the first step. 

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The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

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. 

From the definition of ol in (5.65), it is clear that when = N-m we can define the mixture-fraction vector by (x, t) = a(x, f ).51 For this case, (5.73) defines an invertible, constant-coefficient linear transformation S(0) ... 

By definition of a and /3, 54 the sum of the Nm components of 7 is unity. Thus, by extending the definition of linear mixture to include the condition that (3 must be nonnegative,55 the last Nmf = Nm - I components of 7 can be used to define the mixture-fraction vector.56 The transformation matrix that links to ccv can be found by rewriting (5.84) using the fact that the components of 7 sum to unity ... 

In most practical applications, all but one component of ft will be non-negative. One can then choose the vector with the negative component to be the left-hand side of (5.82). However, the fact that fi must be non-negative greatly restricts the types of linear dependencies that can be expressed as a mixture-fraction vector of length... 

In order for 7 to be a mixture-fraction vector, all its components must be non-negative and their sum must be less than or equal to one.60 From the initial/inlet conditions on a(x, r), it is easily shown using (5.95) that in order for 7 to be a mixture-fraction vector, the components of B (for some k (),. Nw) must satisfy61... 

Once a mixture-fraction basis has been found, the linear transformation that yields the mixture-fraction vector is... 

Note that (5.101) holds under the assumption that the initial/inlet conditions have been renumbered so that the first AW correspond to the mixture-fraction basis. By definition, the mixture-fraction vector is always null in the reference stream. 

In words, this condition states that plJ) for j e Nmf + 1,..., Ain (except j = k) must be a linear mixture of p for i e 1,..., AW with the same coefficient matrix B needed for the mixture-fraction vector. Hereinafter, a mixture-fraction basis that satisfies (5.104) will be referred to as a linear-mixture basis. 

Note that thus far the reacting-scalar vector tpt has not been altered by the mixture-fraction transformation. However, if a linear-mixture basis exists, it is possible to transform the reacting-scalar vector into a new vector whose initial and inlet conditions are null ip = 0 for all i e 0,..., A7m. In terms of the mixture-fraction vector, the linear transformation can be expressed as... 

Figure 5.7. When the initial and inlet conditions admit a linear-mixture basis, the molar concentration vector c of length K can be partitioned by a linear transformation into three parts a reaction-progress vector of length NT , a mixture-fraction vector of length Nmf and 0, a null vector of length K — Nr — Nmf. The linear transformation matrix depends on the reference...

Note that, due to the choice of c(1) as the reference vector, the mixture-fraction vector l (third and fourth components of y> ) is null. The first component of the mixture-fraction vector thus describes mixing between the initial contents of the reactor and the two inlet streams, and the second component describes mixing with the second inlet stream. For a stationary flow (0) -> 0, and only one mixture-fraction component ( 2) will be required to describe the flow. Note, however, that if c(0) had been chosen as the reference vector, a similar reduction would not have occurred. As expected, the inlet and initial values of the two reaction-progress variables are null. 

Note that the reaction-progress vector in the first column is non-zero. Thus, as we suspected, the mixture-fraction basis is not a linear-mixture basis. The same conclusion will be drawn for all other mixture-fraction bases found starting from (5.118). For these initial and inlet conditions, a two-component mixture-fraction vector can be found however, it is of no practical interest since the number of conserved-variable scalars is equal to Nq,m = 1 (k e 0, 1, 2). In conclusion, although the mixture fraction can be defined for the... 

No other linear dependency is apparent, and thus we should expect to find a mixture-fraction vector with two components. 

The fact that no two-component mixture-fraction vector exists does not, however, change the fact that the flow can be described by two conserved scalars. 

None of these matrices satisfies both (5.96) and (5.97). We can thus conclude that no mixture-fraction basis exists for this set of initial and inlet conditions. Since Win = 3, a three-component mixture-fraction vector exists,75 but is of no practical interest. 

In a turbulent flow for which it is possible to define the mixture-fraction vector, turbulent mixing can be described by the joint one-point mixture-fraction PDF MC x, t). The mean mixture-fraction vector and covariance matrix are defined, respectively, by76 i... 

When the mixture-fraction vector has more than one component, the presumed form for the mixture-fraction PDF must be defined such that it will be non-zero only when AW... 

For elementary chemical reactions, it is sometimes possible to assume that all chemical species reach their chemical-equilibrium values much faster than the characteristic time scales of the flow. Thus, in this section, we discuss how the description of a turbulent reacting flow can be greatly simplified in the equilibrium-chemistry limit by reformulating the problem in terms of the mixture-fraction vector. 

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... 

In order to simplify the notation, we drop the superscript (k). Nevertheless, the reader should keep in mind that Sjp will depend on the linear-mixture basis chosen to define the mixture-fraction vector. 

Thus, in the equilibrium-chemistry limit, the reacting scalars depend on space and time only through the mixture-fraction vector ... 

Note that the numerical simulation of the turbulent reacting flow is now greatly simplified. Indeed, the only partial-differential equation (PDE) that must be solved is (5.100) for the mixture-fraction vector, which involves no chemical source term Moreover, (5.151) is an initial-value problem that depends only on the inlet and initial conditions and is parameterized by the mixture-fraction vector it can thus be solved independently of (5.100), e.g., in a pre(post)-processing stage of the flow calculation. For a given value of , the reacting scalars can then be stored in a chemical lookup table, as illustrated in Fig. 5.10. 

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. 

Chemical reactions for which the rank of the reaction coefficient matrix T is equal to the number of reaction rate functions R, (i. e 1,..., I) (i.e., Nr = I), can be expressed in terms of / reaction-progress variables Y, (i. e 1,...,/), in addition to the mixture-fraction vector . For these reactions, the chemical source terms for the reaction-progress variables can be found without resorting to SVD of T. Thus, in this sense, such chemical reactions are simple compared with the general case presented in Section 5.1. 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

The scalar mean conditioned on the mixture-fraction vector can be denoted by... 

By definition, the unconditional scalar means can be found from Q(C x, t) and the mixture-fraction-vector PDF ... 

Given Reynolds-averaged chemical source term then depends only on (p ), ( ), and (% a% p). 

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