A simple algebraic model for the Favre averaged scalar dissipation rate, c, in high Damkohler number premixed flames is obtained from its transport equation by balancing the leading order terms. Recently proposed models for the dominant terms in the transport equation are revisited and revised. The algebraic model incorporates essential physics of turbulent premixed flames, namely, dilatation rate, its influence on turbulence-scalar interaction, chemical reactions, and dissipation processes. A realizability analysis is carried out to show that the algebraic model is always unconditionally realizable. The model predictions of dissipation rate are compared with the DNS results, and the agreement is good over a range of flame conditions. Application of the Kolmogorov-Petrovski-Piskunov (KPP) theorem along with the above algebraic model gives an expression for the turbulent flame speed. Its prediction compares well with a range of experimental data with no modifications to the model constants.

The influence of differential diffusion on the statistical behavior of the local displacement speed $\left(S_d\right)$ in relation to flame curvature is studied based on three-dimensional compressible direct numerical simulations (DNS) of statistically planar flames with single-step Arrhenius-type chemistry. Three different Lewis number cases ($\mathrm{Le}=0.8$ , 1.0, and 1.2) are considered. In order to study the influence of differential diffusion on curvature effects in flame propagation, temperature statistics are presented in terms of standard probability density functions (pdfs) and also joint pdfs with curvature for the nonunity Lewis number cases. Temperature statistics are found to be consistent with previous incompressible combustion DNS studies. It is found that both dilatation and tangential strain rate are negatively correlated with curvature. The relative strength of these two correlations determines the nature of the correlation between surface density function (SDF) $\left(\mid \nabla c\mid \right)$ and curvature. It is also found that the variations of temperature and SDF on an isosurface of reaction progress variable $\left(c\right)$ have significant influence on local displacement speed behavior. Displacement speed statistics are presented in terms of standard pdfs as well as joint pdfs with curvature for the three Lewis number cases. The curvature response of the displacement speed and its different components is found to be nonlinear which is consistent with previous two-dimensional DNS with detailed chemistry. The observed nonlinear behavior in the present study in the absence of a detailed chemical mechanism is explained through the influence of differential diffusion.

Scalar dissipation rate is a central quantity in turbulent flame modeling as it is closely related to the reaction rate. It is well known that turbulence-scalar interaction plays a vital role in turbulent flows with scalar mixing and thus on the scalar dissipation rate. This interaction process is characterized by the tensor inner product between the scalar gradient vector and the turbulence strain rate tensor and it is found to depend strongly on the Damköhler number, Da. Two direct numerical simulation data sets are analyzed in detail in order to understand the physics of Da dependence. The well known alignment of scalar gradient with the most compressive principal strain rate resulting in production of the scalar gradient by turbulence is observed for low $(\mathrm{Da}<1)$ Damköhler number flame, whereas the turbulence dissipates the scalar gradient in high Da flame. This dissipation of the scalar gradient in the high Da flame is because of its preferential alignment with the most extensive principal strain rate. Even for $\mathrm{Da}<1$ flame, in the regions of intense heat release the scalar gradient has a tendency to align with the most extensive strain rate. This distinct change is because of strong competition between chemical strain rate, due to dilatation from the flame front and the local fluid dynamic strain rate. The alignment characteristics affect flame normal and tangential strain rate statistics. The flame normal strain rate is found to be positive throughout the flame brush when $\mathrm{Da}>1$ , whereas it is predominantly negative when $\mathrm{Da}<1$ . Despite this difference the mean tangential strain rate remains positive yielding flame surface area production. However, the production results via different physical mechanisms. Possible implications of these differences on the modeling of turbulent premixed flames are identified and explained.