TABLE OF CONTENTS For non-premixed turbulent reacting flows, it is often possible to define a mixture-fraction vector ( ?) that can be employed to develop chemical-source-term closures that are much more successful than moment closures. In this section, we discuss a general method for finding the mixture-fraction vector (when it exists) for a given set of initial/inlet conditions. We also show that when a mixture-fraction vector exists, it is possible to transform the reacting-scalar vector into a reaction-progress vector ( ? rp) that is null for all initial and inlet conditions. Thus, the turbulent reacting flow can be most simply described in terms of the reaction-progress vector and the mixture-fraction vector. These vectors are found using a constant-coefficient linear transformation matrix, (5.107), that depends only on the inlet/initial conditions and the reaction coefficient matrix. We conclude the section by introducing the joint PDF of the mixture-fraction vector, which is needed to compute the Reynolds-averaged chemical source term in closures based on the conditional composition PDF.
In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix ? into a vector c* that has N ? reacting components c r and N conserved components c c. [35] In the limit of equilibrium chemistry, the behavior of the N ? reacting scalars will be dominated by the transformed chemical source...
