A DNS Study of Closure Relations for Convection Flux Term in Transport Equation for Mean Reaction Rate in Turbulent Flow

A N Lipatnikov,V A Sabelnikov,S Nishiki,N Chakraborty,T Hasegawa

doi:10.1007/s10494-017-9833-y

A N Lipatnikov, V A Sabelnikov + Show 3 more

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https://doi.org/10.1007/s10494-017-9833-y

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Abstract

The present work aims at modeling the entire convection flux overline {rho mathbf {u}W} in the transport equation for a mean reaction rate overline {rho W} in a turbulent flow, which (equation) was recently put forward by the present authors. In order to model the flux, several simple closure relations are developed by introducing flow velocity conditioned to reaction zone and interpolating this velocity between two limit expressions suggested for the leading and trailing edges of the mean flame brush. Subsequently, the proposed simple closure relations for overline {rho mathbf {u}W} are assessed by processing two sets of data obtained in earlier 3D Direct Numerical Simulation (DNS) studies of adiabatic, statistically planar, turbulent, premixed, single-step-chemistry flames characterized by unity Lewis number. One dataset consists of three cases characterized by different density ratios and is associated with the flamelet regime of premixed turbulent combustion. Another dataset consists of four cases characterized by different low Damköhler and large Karlovitz numbers. Accordingly, this dataset is associated with the thin reaction zone regime of premixed turbulent combustion. Under conditions of the former DNS, difference in the entire, overline {rho {u}W}, and mean, tilde {u}overline {rho W}, convection fluxes is well pronounced, with the turbulent flux, overline {rho u^{prime prime }W^{prime prime }}, showing countergradient behavior in a large part of the mean flame brush. Accordingly, the gradient diffusion closure of the turbulent flux is not valid under such conditions, but some proposed simple closure relations allow us to predict the entire flux overline {rho mathbf {u}W} reasonably well. Under conditions of the latter DNS, the difference in the entire and mean convection fluxes is less pronounced, with the aforementioned simple closure relations still resulting in sufficiently good agreement with the DNS data.

Highlights

The critical point of the turbulent combustion theory stems from averaging reaction rates subject to fluctuations in the local temperature T and concentrations
The turbulent flux ρu W shows the countergradient behavior in the leading parts (c < 0.5) of turbulent flame brushes characterized by sufficiently low ratios of u /SL
In the trailing parts (0.8 < c) of all seven investigated flame brushes, the magnitude of the turbulent flux ρu W is much smaller than the magnitude of the mean convection flux ρuW

Summary

Introduction

The critical point of the turbulent combustion theory stems from averaging reaction rates subject to fluctuations in the local temperature T and concentrations. N ) characterized by unity Lewis number (i.e. Le = κ/D = 1) and a low Mach number, κ is the heat diffusivity of the mixture, W is the rate of production of the combustion progress variable in the flame, SL is the laminar flame speed, cm = ρcW /ρW is commonly assumed to be constant [6, 12], q and q = ρq/ρare the Reynolds and Favreaveraged values of a quantity q, respectively, and subscripts u and b designate unburned and burned mixture, respectively It is worth noting, that even the precise knowledge of χordoes not allow us to precisely evaluate W , because the linear relations between these three quantities are just assumptions, which are best justified if unburned and fully burned gases are separated by a thin zone (flamelet) that retains the structure of the laminar flame. It would be of interest to straightforwardly evaluate the mean rate W by solving an appropriate transport equation without invoking extra assumptions regarding a relation between W and χor

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