In the turbulent flow field, the Favre averaged (tilde superscript) mixture fraction equation is solved:

(7–32)

The Favre averaging is extensively explained in the theory documentation of CFX. Statistical information on the mixture fraction is obtained from the variance of Z.

(7–33)

The structure of this equation is similar to the mixture fraction equation itself except for last two terms on the right hand side. The first source term is the production and the second source term models the dissipation of the variance. Here, stands for the scalar dissipation rate and is modeled in turbulent flow using the empirical relation:

(7–34)

It includes the effects of strain as well as mixture fraction fluctuations. The standard set of model coefficients in CFX is , and

The mean composition of the fluid is computed as a function of mean mixture fraction, mixture fraction variance and scalar dissipation rate by look-up in a flamelet library:

(7–35)

The integration over the probability density function (PDF) is not carried out during the CFD calculation, but is reflected in the flamelet library. For details, see Flamelet Libraries. The CFD solver looks up the preintegrated values from the library.

In principle, many types of PDF could be applied, but the most commonly agreed choice is the Beta-PDF. The shape of is presumed to be that of a normalized beta function (-function):

(7–36)

(7–37)

The Beta-PDF is used for the Flamelet libraries shipped with CFX.

Note that for the table look-up, the solver is actually using instead of ,that is, the local value of the scalar dissipation rate is applied instead of the value at stoichiometric mixture fraction. This is exact only for stoichiometric mixture or for vanishing variance of mixture fraction (perfectly premixed case). However, many radicals of interest, for example, OH radicals, have significant concentrations only at stoichiometry and in its surrounding, which makes this approximation acceptable. In principle, could be derived from the solution fields, but this would introduce errors, too, because it would require either additional modeling or averaging over the computational domain.