An age extended progress variable for conditioning reaction rates

An aging progress variable (APV) is proposed as a convenient tool for conditioning quantities used to calculate reaction rates in premixed turbulent combustion. The APV is defined to obey an advection-diffusion-reaction equation where the source term is linearly related to the fuel consumption rate when the APV is less than a threshold representative of the trailing edge of the fuel consumption layer. Above this threshold, the APV has a constant source term. To test the proposal, three-dimensional fully compressible direct numerical simulation (DNS) is performed in an inflow-outflow configuration with inlet turbulence forcing using a modified version of the DNS code SENGA. A model chemical mechanism involving two steps and four species is used which has an order of magnitude difference between the time scales associated with the two steps. The inlet Taylor scale Reynolds number is 51 and the Damköhler numbers are 2.0 and 0.29 for the two steps. When conditionally averaged on the proposed APV the scalar fluctuations about the conditional average are negligible. Further, it is shown that the probability density function of the APV can be reasonably approximated based on the first two moments of the APV and the fuel mass fraction. The APV probability density function (PDF) is approximated on rectangular slabs of cells normal to the flow direction using only these moments as a test case. Convolution of the PDF so approximated with conditional mean reaction rates—calculated from conditionally averaged scalar fields where the averaging was carried out over the entire domain—leads to an approximation for the unconditional mean reaction rates on each of these slabs typically within 10% of the true value for both steps. That the correlation between the reaction rates and the APV is strong, and that the PDF can be approximated for a situation where approximating the PDF for a product based progress variable is nontrivial, makes the proposed APV a strong alternative to traditional progress variables for both flamelet models and premixed conditional moment closure (CMC) approaches.