Mixture-fraction vector initial/inlet conditions

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. 

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... 

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. 

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 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... 

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. 

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. 

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