TABLE OF CONTENTS Various authors have attempted to extend mixture-fraction PDF methods to handle finite-rate reactions. The principal difficulty lies in the fact that the joint PDF of the reaction-progress variables must be presumed to close the model. The transport equations for the moments of the reaction-progress variables contain unclosed chemical-reaction source terms. Thus, strong assumptions must be made concerning the statistical dependence between the mixture fraction and the reaction-progress variables in order to close the problem. In most cases, it is extremely difficult to justify both the assumed form of the joint reaction-progress-variable PDF and the assumed statistical dependence of the reaction-progress variables on the mixture-fraction vector. For example, in many of the proposed closures, the reaction-progress variables are assumed to be independent of the mixture fraction, which is unlikely to be the case. Nevertheless, in this section we will first review presumed PDF methods for kinetic schemes that can be described by a single reaction-progress variable (e.g., see Section 5.5), and then look at possible extensions to more complex schemes requiring multiple reaction-progress variables.
For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction. [131] For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction ?. In general, it is easiest to...
