Moment closures scalar fields

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

In general, the scalar fields will not be Gaussian. Thus, analogous to what is done in second-moment closure methods, a gradient-diffusion model is usually employed to close this term ... 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

---