Reaction-progress vector equilibrium

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... 

The solution to (5.151) as t oo is the local equilibrium reaction-progress vector 85... 

An ad hoc extension of the method presented above can be formulated for complex chemistry written in terms of yip and . In the absence of chemical reactions, y>rp = 0. Thus, if a second limiting case can be identified, interpolation parameters can be defined to be consistent with the unconditional means. In combusting flows, the obvious second limiting case is the equilibrium-chemistry limit where yip = y>eq( ) (see Section 5.4). The components of the conditional reacting-progress vector can then be approximated by (no summation is implied on a)... 

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