5.8 Conditional-Moment Closures

For the equilibrium-chemistry limit, we have seen in Section 5.4 that the reaction-progress vector can be re-parameterized in terms of the mixture fraction, i.e., ? _rp( x, t) = ? _eq( ?( x, t)). It has been observed experimentally and from DNS that even for many finite-rate reactions the scatter plot of ? _?( x, t) versus ?( x, t) = ? for all ? often exhibits relatively small fluctuations around the mean conditioned on a given value of the mixture fraction, i.e., around ( ?( x, t) ?( x, t) = ?). An example of such a scatter plot is shown in Fig. 5.21. In the limit where fluctuations about the conditional mean are negligible, the Reynolds-averaged chemical source term can be written as ( S( ?) ?) = S(( ? ?)). Thus, a model for the conditional means, combined with a presumed mixture-fraction PDF, would suffice to close the Reynolds-averaged turbulent-reacting-flow equations. In this section, we will thus look at two methods for modeling the conditional means. First, however, we will review the general formulation of conditional random variables.

Figure 5.21: Scatter plot of concentration in a turbulent reacting flow conditioned on the value of the mixture fraction. Although large fluctuations in the unconditional concentration are present, the conditional fluctuations are considerably smaller. In the...