10.2.1 Calculation of Scalar Quantities

Density weighted mean scalars (such as species fractions and temperature), denoted by $\overline{\phi}$, are calculated from the probability density function (PDF) of $f$ and $c$ as

Under the assumption of thin flames, so that only unburnt reactants and burnt products exist, the mean scalars are determined from

where the subscripts $b$ and $u$ denote burnt and unburnt, respectively.

The burnt scalars, $\phi_b$, are functions of the mixture fraction and are calculated by mixing a mass $f$ of fuel with a mass $(1-f)$ of oxidizer and allowing the mixture to equilibrate. When non-adiabatic mixtures and/or laminar flamelets are considered, $\phi_b$ is also a function of enthalpy and/or strain, but this does not alter the basic formulation. The unburnt scalars, $\phi_u$, are calculated similarly by mixing a mass $f$ of fuel with a mass $(1-f)$ of oxidizer, but the mixture is not reacted.

Just as in the non-premixed model, the chemistry calculations and PDF integrations for the burnt mixture are performed in ANSYS FLUENT, and look-up tables are constructed.

Turbulent fluctuations are neglected for the unburnt mixture, so the mean unburnt scalars, $\overline{\phi}_u$, are functions of $\overline{f}$ only. The unburnt density, temperature, specific heat, and thermal diffusivity are fitted in ANSYS FLUENT to third-order polynomials of $\overline{f}$ using linear least squares:

Since the unburnt scalars are smooth and slowly-varying functions of $\overline{f}$, these polynomial fits are generally accurate. Access to polynomials is provided in case you want to modify them.