Definition of mixture fraction

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 iVmf, 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. 

49 iVmf = 1 is the case that is usually associated with non-premixed turbulent reacting flows. Most commercial CFD codes are designed to treat only this case. 

50 The names binary and ternary come from the number of feed streams that are required for a stationary flow, i.e., two and three, respectively. Some authors also use ternary for the rank-deficient case where Nm = 3 but 

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)  

Furthermore, the transport equation for (x, t) is then exactly the same as (5.65). 

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