Flamelet Connection to Turbulence Kinetic Energy Dissipation Rate

The turbulence kinetic energy dissipation rate $\epsilon$, from a turbulent combustion computation using either Reynolds-averaged Navier-Stokes (RANS) or large-eddy simulation (LES), is proposed for closure with a sub-grid non-premixed flamelet model. The intentions are to avoid the creation of artificial tracking or progress variables and to relate accurately the physics of turbulent non-premixed combustion at the resolved length scales to the small-scale physics where the mixing and chemical reactions occur. The analysis addresses the relations between $\epsilon$ and the strain rate, vorticity, viscous dissipation rate, scalar gradients, scalar dissipation rate, and burning rate at the smallest turbulence length scales where diffusion-controlled burning is faster than at larger length scales and thereby dominant. The imposed strain rate and vorticity on these smallest eddies are determined from the kinetic energy dissipation rate. Thus, an $\epsilon$ value at a specific time and location determines the two mechanical constraints (vorticity and strain rate) on the inflow to the counterflow flamelet. $\epsilon$ affects the sign of the Laplacian of pressure, which must be negative to allow the existence of the counterflow. Using different flamelet models, with and without vorticity, different results for maximum flamelet temperature, integrated flamelet burning rate, and maximum flamelet scalar dissipation rate are obtained. Flamelet models that consider the centrifugal effect of vorticity produce substantial enhancements in the accuracy and completeness of information for a turbulent combustion computation. $\epsilon$ may be used as a tracking variable that connects the sub-grid flamelet model to resolved-scale RANS or LES computations.