Direct numerical simulations (DNS) of low and high Karlovitz number (Ka) flames are analysed to investigate the behaviour of the reactive scalar sub-grid scale (SGS) variance in premixed combustion under a wide range of combustion conditions (regimes). An order of magnitude analysis is performed to assess the importance of various terms in the variance evolution equation and the analysis is validated using the DNS results. This analysis sheds light on the relative behaviour among turbulent transport and production, scalar dissipation and chemical processes involved in the evolution of the SGS variance at different Ka. The common expectation is that the variance equation shifts from a reaction-dissipation balance at low Ka to a production–dissipation balance at high Ka with diminishing reaction contribution. However, in large eddy simulation (LES), a high Ka alone does not make the reaction term negligible, as the relative importance of the reaction term has a concurrent increase with filter size. The filter size can be relatively large compared with the Kolmogorov length scale in practical LES of high Ka flames, and as a consequence a reaction–production–dissipation balance may prevail in the variance equation even in a high Ka configuration, and this possibility is quantified using the DNS analysis in this work. This has implications from modelling perspectives, and therefore two commonly used closures in LES for the SGS scalar dissipation rate are investigated a priori to estimate the importance of the above balance in LES modelling. The results are explained to highlight the interplay among turbulence, chemistry and dissipation processes as a function of Ka.

Direct numerical simulations (DNS) of statistically planar flames at moderate and high Karlovitz number (Ka) have been used to perform an a priori evaluation of a presumed-PDF model approach for filtered reaction rate in the framework of large eddy simulation (LES) for different LES filter sizes. The model is statistical and uses a presumed shape, based here on a beta-distribution, for the sub-grid probability density function (PDF) of a reaction progress variable. Flamelet tabulation is used for the unfiltered reaction rate. It is known that presumed PDF with flamelet tabulation may lead to over-prediction of the modelled reaction rate. This is assessed in a methodical way using DNS of varying complexity, including single-step chemistry and complex methane/air chemistry at equivalence ratio 0.6. It is shown that the error is strongly related to the filter size. A correction function is proposed in this work which can reduce the error on the reaction rate modelling at low turbulence intensities by up to 50%, and which is obtained by imposing that the consumption speed based on the modelled reaction rate matches the exact one in the flamelet limit. A second analysis is also conducted to assess the accuracy of the flamelet assumption itself. This analysis is conducted for a wide range of Ka, from 6 to 4100. It is found that at high Ka this assumption is weaker as expected, however results improve with larger filter sizes due to the reduction of the scatter produced by the fluctuations of the exact reaction rate.