Abstract
Flame surface density (FSD) based reaction rate closure is an important methodology of turbulent premixed flame modelling in the context of Large Eddy Simulations (LES). The transport equation for the Favre-filtered reaction progress variable needs closure of the filtered reaction diffusion imbalance (FRDI) term (i.e. filtered value of combined reaction rate and molecular diffusion rate) and the sub-grid scalar flux (SGSF). A-priori analysis of the FRDI and SGSF terms has in the past revealed advantages and disadvantages of the specific modelling attempts. However, it is important to understand the interaction of the FRDI and SGSF closures for a successful implementation of the FSD based closure. Furthermore, it is not known a-priori if the combination of the best SGSF model with the best FRDI model results in the most suitable overall modelling strategy. In order to address this question, a variety of SGSF models is analysed in this work together with one well established and one recent FRDI closure based on a-priori analysis. It is found that the success of the combined FRDI and SGSF closures depends on subtle details like the co-variances of the FRDI and SGSF terms. It is demonstrated that the gradient hypothesis model is not very successful in representing the SGSF term. However the gradient hypothesis provides satisfactory performance in combination of a recently proposed FRDI closure, whereas unsatisfactory results are obtained when used in combination with another existing closure, which was shown to predict the FRDI term satisfactorily in several previous analyses.
Highlights
The complexity of the system of partial differential equations describing turbulent reactive flows can be simplified assuming single-step chemistry and a unity Lewis number (=thermal diffusivity/mass diffusivity)
Arrhenius type irreversible chemistry has been considered for the current analysis, consisting of five flames with global Lewis number Le = 0.34, 0.6, 0.8, 1.0 and 1.2
It turns out that the sum of the three covariances in Eq 18 is on average of the order of 5 % of Cov TDFNRSDI, TLFERSDI and one can expect that the correlation coefficients of the combined model expressions (SGSF+filtered reaction diffusion imbalance (FRDI)) are within a few percentages of those for FRDI listed in Table 2 where the small differences observed are to a large extent due to the interaction of the 3 co-variances shown in Eq 18
Summary
The complexity of the system of partial differential equations describing turbulent reactive flows can be simplified assuming single-step chemistry and a unity Lewis number (=thermal diffusivity/mass diffusivity). In Eq 1, is the filter width and Q = ρQ/ρrefers to the well-known Favre filtering operation In this framework, the transport equation for the filtered reaction progress variable becomes:. The numerical solution requires closure for both terms on the right hand side of Eq 2, i.e. for the sub-grid scalar flux (SGSF) denoted as: TiSGSF := ρui c − ρuic (where ui is the ith component of velocity) as well as the filtered reaction diffusion imbalance (FRDI) term T F RDI = ∇ · (ρD∇c) + ωc. The closures of T F RDI and TiSGSF used in this work will be briefly summarized This will be followed by the analysis of the models on term by term basis in order to understand how they interact.
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