Treatment of reacting scalars

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 

Note that, given prp and , the inverse transformation, (5.109), can be employed to find the original composition vector c. In order to simplify the notation, we will develop the theory in terms of (prp. However, it could just as easily be rewritten in terms of c using the inverse transformation. 

The Ny eigenvalues of the Jacobian of Srp will be equal to the Nr non-zero eigenvalues of the Jacobian of Sc. Thus, in the equilibrium-chemistry limit, the chemical time scales will obey 

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

84 Chemical equilibrium implies that all reactions are reversible. The method discussed in this section is thus usually applied to elementary reactions of the form given in (5.1). 

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