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![Variations of T1,T2,T3,T4,D1,-D2×D02/ρ0Sbϕ=14ξst2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {T_{1} ,T_{2} ,T_{3} ,T_{4} ,D_{1} , - D_{2} } \right\} \times D_{0}^{2} /\rho_{0} S_{{b\left( {\phi = 1} \right)}}^{4} \xi_{st}^{2}$$\end{document} with c~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{c}$$\end{document} across the flame brush for the εcξ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{{\varepsilon_{c\xi } }}$$\end{document} transport equation in cases (a–f) A-F](publication/361440880/figure/fig4/AS:1183001383837713@1659060917508/Variations-of-T1-T2-T3-T4-D1-D2D02-r0Sbph14xst2documentclass12ptminimal.png)
Variations of T1,T2,T3,T4,D1,-D2×D02/ρ0Sbϕ=14ξst2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {T_{1} ,T_{2} ,T_{3} ,T_{4} ,D_{1} , - D_{2} } \right\} \times D_{0}^{2} /\rho_{0} S_{{b\left( {\phi = 1} \right)}}^{4} \xi_{st}^{2}$$\end{document} with c~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{c}$$\end{document} across the flame brush for the εcξ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{{\varepsilon_{c\xi } }}$$\end{document} transport equation in cases (a–f) A-F
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![Variations of c″2~\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/361440880/figure/fig2/AS:1183001383841792@1659060917254/Variations-of-c2documentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Variations of {εcξ~\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/361440880/figure/fig3/AS:1183001383829506@1659060917382/Variations-of-ecxdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
The cross-scalar dissipation rate of reaction progress variable and mixture fraction εcξ~ plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of εcξ~ have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochio...
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Citations
... 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. ...
... 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. ...
A recently proposed scalar forcing scheme that maintains the mixture fraction mean, root-mean-square and probability density function in the unburned gas can lead to a statistically quasi-stationary state in direct numerical simulations of turbulent stratified combustion when combined with velocity forcing. Scalar forcing alongside turbulence forcing leads to greater values of turbulent burning velocity and flame surface area in comparison to unforced simulations for globally fuel-lean mixtures. The sustained unburned gas mixture inhomogeneity changes the percentage shares of back- and front-supported flame elements in comparison to unforced simulations, and this effect is particularly apparent for high turbulence intensities. Scalar forcing does not significantly affect the heat release rates due to different modes of combustion and the micro-mixing rate within the flame characterised by scalar dissipation rate of the reaction progress variable. Thus, scalar forcing has a significant potential for enabling detailed parametric studies as well as providing well-converged time-averaged statistics for stratified-mixture combustion using Direct Numerical Simulations in canonical configurations.
... Interested readers are referred to the review paper by Lipatnikov (2017) for an extensive review of the experimental and numerical investigations into stratified mixture combustion. The advances in high-performance computing have enabled Direct Numerical Simulations (DNS) of stratified mixture combustion (Hélie and Trouvé, 1998;Haworth et al. 2000;Malkeson and Chakraborty 2010a, b, c;2011a, b, c, 2012, 2013aRichardson and Chen 2017;Proch et al. 2017a, b;Inanc et al. 2022), but the vast majority of these studies have been conducted for statistically planar flames (Hélie and Trouvé, 1998;Haworth et al. 2000;Malkeson and Chakraborty 2010a, b, c;2011a, b, c, 2012, 2013aBrearley et al. 2020Brearley et al. , 2022. Moreover, one-dimensional laminar flame analyses (Egolfopoulos and Campbell 1996;Da Cruz et al. 2000;Lauvergne et al. 2000;Sankaran and Chen 2002;Kang and Kyritsis 2007;Richardson et al. 2010;Vena et al. 2011;Inanc et al. 2020) also provided important physical insights into stratified mixture combustion. ...
... The laboratory-scale stratified flame burners are mostly analysed numerically based on Reynolds-averaged Navier-Stokes/Large Eddy Simulations (Ribert et al. 2005;Robin et al. 2006;Darbyshire et al. 2010;Marincola et al. 2013;Proch and Kempf 2014;Mercier et al. 2015;Shi et al. 2016;Proch et al. 2017b;Turkeri et al. 2019;Zhang et al. 2021) but recently, Proch et al. (2017a, b) and Inanc et al. (2022) carried out flame-resolved three-dimensional flame-resolved high-fidelity simulations for turbulent stratified mixture combustion of a laboratory-scale configuration, which was analysed extensively using experiments (Barlow et al. 2009;Sweeney et al. 2012a, b). The aforementioned DNS and flame-resolved high-fidelity simulation studies provided important physical insights into the stratified flame structure (Hélie and Trouvé, 1998;Haworth et al. 2000;Chakraborty 2010a, c, 2011b;Richardson and Chen 2017;Brearley et al. 2020Brearley et al. , 2022Inanc et al. 2022) and flame propagation rate (Malkeson and Chakraborty 2010a, c;Richardson and Chen 2017;Inanc et al. 2022), which gave rise to closure methodologies for scalar variances and co-variances (Malkeson and Chakraborty 2010a, b;2013b), turbulent scalar flux (Malkeson and Chakraborty 2012), Flame Surface Density (Malkeson and Chakraborty 2013a), and scalar dissipation and cross-scalar dissipation rates (Malkeson and Chakraborty 2011a, c). The effects of different types of mixture stratification in the presence of shear in non-canonical flow configurations were analysed recently by Richardson and Chen (2017) based on three-dimensional DNS, where the effects of the relative alignments of mixture fraction and reaction progress variable gradients (i.e. ...
... Thus, a modelled transport equation of c′′ ′′ may need to be solved in RANS simulations of turbulent stratified mixture combustion. This approach also necessitates a closure for the cross-scalar dissipation rate Ñ c = D∇c ⋅ ∇ ∕ , which is addressed elsewhere (Brearley et al. 2022;Inanc et al. 2022) and thus is not discussed here. ...
The influence of mixture stratification on the development of turbulent flames in a slot-jet configuration has been analysed using Direct Numerical Simulation data. Mixture stratification was imposed at the inlet by varying the equivalence ratio between 0.6 and 1.0 with different alignments to the reaction progress variable gradient: aligned gradients (back-supported), opposed gradients (front-supported) and misaligned gradients. An additional premixed case with a global equivalence ratio of 0.8 was simulated for comparison. The flame is shortest for the front-supported case, followed by the premixed flame, with the back-supported and misaligned gradient flames being the tallest and of comparable size. This behaviour has been explained in terms of the variations of the mean equivalence ratio within the flame and the volume-integrated reaction rate in the streamwise direction. The difference in mixture composition for these cases results in significant variations in the burning rate, flame area, flame wrinkling and flame brush thickness in the streamwise direction. The globally front-supported case has the highest volume-integrated burning rate and flame area, while the back-supported case has the lowest. The misaligned scalar gradient case exhibits qualitatively similar behaviour to that of the globally back-supported case. The burning intensity is unity for a major part of the flame length but assumes values greater than unity towards the flame tip where the effects of flame curvature become strong. All cases predominantly exhibit the premixed mode of combustion within the flamelet regime, so flamelet assumption-based reaction rate closures, originally proposed for premixed combustion, were evaluated using a priori analysis. The terms which require improved closures have been identified and existing closures have been improved where necessary. It was found that the global nature of mixture stratification does not influence the performance of the mean reaction rate closures or the parameterisation of marginal probability density functions of scalars in turbulent stratified mixture combustion.
