Mixture-fraction vector definition

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 ... 

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. 

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

Multi-environment presumed PDF models are generally not recommended for homogeneous flows with uniform mean gradients. Indeed, their proper formulation will require the existence of a mixture-fraction vector that, by definition, cannot generate a uniform mean scalar gradient in a homogeneous flow. 

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