Scalar covariance chemical source term

For higher-order reactions, a model must be provided to close the covariance source terms. One possible approach to develop such a model is to extend the FP model to account for scalar fluctuations in each wavenumber band (instead of only accounting for fluctuations in In any case, correctly accounting for the spectral distribution of the scalar covariance chemical source term is a key requirement for extending the LSR model to reacting scalars. 

In principle, the same equation could be used for a reacting scalar. However, one would need to know the spectral distribution of the covariance chemical source term Sai, (3.141), in order to add the corresponding covariance-dissipation-range chemical source term to (3.165). 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

The Reynolds-averaged chemical source terms can be written in terms of the scalar means and covariances. For example,... 

The terms on the right-hand side of this expression are the unknown covariance source terms, in particular wavenumber bands due to chemical reactions. For example, for a fast non-premixed, one-step reaction one would expect the product to be formed in the scalar dissipation range, so that (Sa Q is dominant. 

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