A Priori Direct Numerical Simulation Analysis of the Closure of Cross-Scalar Dissipation Rate of Reaction Progress Variable and Mixture Fraction in Turbulent Stratified Flames

Abstract

The cross-scalar dissipation rate of reaction progress variable and mixture fraction $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochiometric mixture. A parametric analysis has been conducted by considering a number of DNS cases with a varying initial root-mean-square velocity fluctuation $$u^{\prime }$$ u ′ and initial scalar integral length scale $$\ell_{\phi }$$ ℓ ϕ . The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ is inadequate for most cases, but its performance appears to improve with increasing initial $$\ell_{\phi }$$ ℓ ϕ and $$u^{\prime }$$ u ′ values. An exact transport equation for $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ has been derived from the first principle, and the budget of the unclosed terms of the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation has been analysed in detail. It has been found that the terms arising from the density variation, scalar-turbulence interaction, chemical reaction rate and molecular dissipation rate play leading order roles in the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport. These observations have been justified by a scaling analysis, which has been utilised to identify the dominant components of the leading order terms to aid model development for the unclosed terms of the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation. The performances of newly proposed models for the unclosed terms have been assessed with respect to the corresponding terms extracted from DNS data, and the newly proposed closures yield satisfactory predictions of the unclosed terms in the $$\widetilde{{\varepsilon_{c\xi } }}$$ ε c ξ ~ transport equation.

Implications of Using Scalar Forcing to Sustain Reactant Mixture Stratification in Direct Numerical Simulations of Turbulent Combustion

“…Explicit time advancement uses low storage 3rd-order Runge-Kutta scheme [28]. The numerical technique used in this study is standard practice for combustion DNS and has been used in different DNS analyses in the past [1][2][3][4][5][6][7][8][9][10][11][12][13]. The four boundaries transverse to the mean direction of the flame propagation are considered periodic, while the two boundaries in the direction of the mean flame propagation are considered to be a partially non-reflecting turbulent inflow and outflow in line with the Navier-Stokes characteristic boundary conditions (NSCBC) [29].…”

“…Without any external scalar fluctuation generating mechanism, a statistically quasi-stationary state can only be achieved once the reactants have become perfectly mixed (i.e., premixed combustion). As such, many DNS studies of turbulent stratified-mixture combustion have relied on unsteady decaying turbulence conditions (e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). However, mixture inhomogeneity in the unburned gas can be sustained with scalar forcing by including an additional term in the species conservation equations, which can be utilised to reach a quasi-steady state for stratified-mixture combustion.…”

“…Lipatnikov [14] and Domingo et al [15] provided an extensive review of experimental and numerical findings of stratified-mixture combustion. High-performance computing has enabled DNS of stratified-mixture combustion [1][2][3][4][5][6][7][8][9][10][11][12][13][16][17][18][19], and the vast majority of these studies have been conducted for statistically planar flames in canonical 'flame-in-a-box' configuration [1][2][3][4][5][6][7][8][9][10][11][12][13]. Richardson and Chen [16] and Brearley et al [19] analysed turbulent stratified-mixture combustion using three-dimensional DNS to analyse the effects of the relative alignments of equivalence ratio and reaction progress variable gradients on the global behaviours of burning, flame propagation rate and flame surface area and their modelling implications.…”

“…Proch et al [17] and Inanc et al [18] carried out flame-resolved three-dimensional high-fidelity simulations of turbulent stratified-mixture combustion of a laboratory-scale configuration used in experiments by Barlow et al [20] and Sweeney et al [21,22]. The aforementioned DNS analyses on turbulent stratified-mixture combustion offered fundamental physical information on the stratified flame structure [1][2][3]5,8,13,16,18,19] and flame propagation rate [4,12,16,18]. This information was subsequently utilised to develop closure methodologies for scalar variances and co-variances [3,5,11], turbulent scalar flux [9], flame surface density [10,19], and scalar dissipation and cross-scalar dissipation rates [7,8,13].…”

“…The aforementioned DNS analyses on turbulent stratified-mixture combustion offered fundamental physical information on the stratified flame structure [1][2][3]5,8,13,16,18,19] and flame propagation rate [4,12,16,18]. This information was subsequently utilised to develop closure methodologies for scalar variances and co-variances [3,5,11], turbulent scalar flux [9], flame surface density [10,19], and scalar dissipation and cross-scalar dissipation rates [7,8,13].…”