Abstract
Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios < ϕ > = 0.7 and < ϕ > = 1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y F and mixture fraction ξ (i.e. \(\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })\) for Reynolds Averaged Navier Stokes simulations where \(\overline q \), \(\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }\) and \(q^{\prime\prime}= q-\tilde{q}\) are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) may need to be solved. The statistical behaviours of \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T 1 , reaction rate T 4 and molecular dissipation \(\left( {-D_2 } \right))\) have been analysed in detail. The contribution of T 1 remains important for all cases considered here. The term T 4 acts as a major contributor in < ϕ > = 1.0 cases, but plays a relatively less important role in < ϕ > = 0.7 cases, whereas the term \(\left( {-D_2 } \right)\) acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T 1 , T 4 and \(\left( {-D_2 } \right)\) have been addressed in detail. A model has been identified for the turbulent transport term T 1 which satisfactorily predicts the corresponding term obtained from DNS data. The models for T 4 , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T 4 obtained from DNS data. It has been found that existing algebraic models for \(D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} \) (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate \(\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }\) for the closure of the \(\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}\) transport equation.
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References
Ribert, G., Champion, M., Gicquel, O., Darabiha, N., Veynante, D.: Modeling nonadiabatic turbulent premixed reactive flows including tabulated chemistry. Combust. Flame 141, 271–280 (2005)
Robin, V., Mura, A., Champion, M., Plion, P.: A multi-Dirac presumed PDF model for turbulent reacting flows with variable equivalence ratio. Combust. Sci. Technol. 178, 1843–1870 (2006)
Mura, A., Robin, V., Champion, M.: Modeling of scalar dissipation in partially premixed turbulent flames. Combust. Flame 149, 217–224 (2007)
Malkeson, S.P., Chakraborty, N.: A-priori Direct Numerical Simulation analysis of algebraic models of variances and scalar dissipation rates for Reynolds Averaged Navier Stokes Simulations for low Damköhler number turbulent partially-premixed combustion. Combust. Sci. Technol. 182, 960–999 (2010)
Tarrazo, E., Sanchez, A., Linan, A., Williams, F.: A simple one-step chemistry model for partially premixed hydrocarbon combustion. Combust. Flame 147, 32–38 (2006)
Hélie, J., Trouvé, A.: Turbulent flame propagation in partially premixed combustion. Proc. Combust. Inst. 27, 891–898 (1998)
Grout, R., Swaminathan, N., Cant, R.S.: Effects of compositional fluctuations on premixed flames. Combust. Theory Model. 13, 823–832 (2009)
Eswaran, V., Pope, S.B.: Direct numerical simulations of the turbulent mixing of a passive scalar. Phys. Fluids 31, 506–520 (1988)
Chen, J.H., Choudhary, A., de Supinski, B., Devries, M., Hawkes, E.R., Klasky, S., Liao, W.K., Ma, K.L., Mellor-Crummey, J., Podhorski, N., Sankaran, R., Shende, S., Yoo, C.S.: Terascale direct numerical simulations of turbulent combustion using S3D. Comput. Sci. Discov. 2, 015001 (2009)
Bray, K.N.C., Domingo, P., Vervisch, L.: Role of the progress variable in models for partially premixed turbulent combustion. Combust. Flame 141, 431–437 (2005)
Veynante, D., Trouvé, A., Bray, K.N.C., Mantel, T.: Gradient and counter-gradient turbulent scalar transport in turbulent premixed flames. J. Fluid Mech. 332, 263–293 (1997)
Chakraborty, N., Cant, R.S.: Effects of Lewis number on turbulent scalar transport and its modelling in turbulent premixed flames. Combust. Flame 156, 1427–1444 (2009)
Malkeson, S.P., Chakraborty, N.: A-Priori DNS Modelling of the Turbulent Scalar Fluxes for Low Damköhler Number Stratified Flames. Combust. Sci. Technol. 184, 1680–1707 (2012). doi:10.1080/00102202.2012.690668
Jenkins, K.W., Cant, R.S.: DNS of turbulent flame kernels. In: Knight, D., Sakell, L., Beautner, T. (eds.) Proc. Second AFOSR Conf. on DNS and LES, pp. 192–202. Rutgers University, Kluwer Academic Publishers (1999)
Rogallo, R.S.: Numerical Experiments in Homogeneous Turbulence. NASA Technical Memorandum 81315. NASA Ames Research Center (1981)
Batchelor, G.K., Townsend, A.A.: Decay of turbulence in the final period. Proc. R. Soc. A 194, 527–543 (1948)
Peters, N.: Turbulent Combustion. Cambridge University Press, Cambridge (2000)
Poinsot, T., Lele, S.K.: Boundary conditions for direct simulation of compressible viscous flows. J. Comput. Phys. 101, 104–129 (1992)
Wray, A.A.: Minimal Storage Time Advancement Schemes for Spectral Methods. NASA Ames Research Center, California (1990). Report No. MS 202 A-1
Haworth, D., Blint, R., Cuenot, B., Poinsot, T.: Numerical simulation of turbulent propane-air combustion with nonhomogeneous reactants. Combust. Flame 121, 395–417 (2000)
Jimenez, C., Cuenot, B., Poinsot, T., Haworth, D.: Numerical simulation and modelling for lean stratified propane-air flames. Combust. Flame 128, 1–21 (2002)
Hawkes, E.R., Chen, J.H.: Direct numerical simulation of hydrogen-enriched lean premixed methane–air flames. Combust. Flame 138, 242–258 (2004)
Hawkes, E.R., Chen, J.H.: Evaluation of models for flame stretch due to curvature in the thin reaction zones regime. Proc. Combust. Inst. 30, 647–653 (2005)
Hawkes, E.R., Chen, J.H.: Comparison of direct numerical simulation of lean premixed methane–air flames with strained laminar flame calculations. Combust. Flame 144, 112–125 (2006)
Swaminathan, N., Bray, K.N.C.: Effect of dilatation on scalar dissipation in turbulent premixed flames. Combust. Flame 143, 549–565 (2005)
Swaminathan, N., Grout, R.: Interaction of turbulence and scalar fields in premixed flames. Phys. Fluids 18, 045102 (2006)
Echekki, T., Chen, J.H.: Unsteady strain rate and curvature effects in turbulent premixed methane-air flames. Combust. Flame 106, 184–202 (1996)
Echekki, T., Chen, J.H.: Analysis of the contribution of curvature to premixed flame propagation. Combust. Flame 118, 303–311 (1999)
Peters, N., Terhoeven, P., Chen, J.H., Echekki, T.: Statistics of flame displacement speeds from computations of 2-D unsteady methane-air flames. Proc. Combust. Inst. 27, 833–839 (1998)
Malkeson, S.P., Chakraborty, N.: The modeling of fuel mass fraction variance transport in turbulent stratified flames: a direct numerical simulation study. Numer. Heat Transf. A 58, 187–206 (2010)
Malkeson, S.P., Chakraborty, N.: Statistical analysis of scalar dissipation rate transport in turbulent partially premixed flames: a direct numerical simulation study. Flow Turbulence Combust. 86, 1–44 (2011)
Malkeson, S.P., Chakraborty, N.: Statistical analysis of cross scalar dissipation rate transport in turbulent partially premixed flames: a direct numerical simulation study. Flow Turbulence Combust. 87, 313–349 (2011)
Chakraborty, N., Swaminathan, N.: Influence of the Damköhler number on turbulence-scalar interaction in premixed flames. II. Model development. Phys. Fluids 19, 045104 (2007)
Chakraborty, N., Rogerson, J.W., Swaminathan, N.: A priori assessment of closures for scalar dissipation rate transport in turbulent premixed flames using direct numerical simulation. Phys. Fluids 20, 045106 (2008)
Kolla, H., Rogerson, J., Chakraborty, N., Swaminathan, N.: Scalar dissipation rate modelling and its validation. Combust. Sci. Technol. 181, 518–535 (2009)
Yamashita, H., Shimada, M., Takeno, T.: Numerical study on flame stability at the transition point of jet diffusion flames. Proc. Combust. Inst. 26, 27–34 (1996)
Libby, P.A., Williams, F.A.: A presumed PDF analysis of partially premixed turbulent combustion. Combust. Sci. Technol. 161, 351–390 (2000)
Robin, V., Mura, A., Champion, M., Plion, P.: Modélisation de la combustion turbulente des mélanges hétérogènes en richesse: Des flammes de prémélange aux flammes de diffusion. C. R. Mecanique 337, 596 (2009)
Nishiki, S., Hasegawa, T., Borghi, R., Himeno, R.: Modeling of flame generated turbulence using Direct Numerical Simulation databases. Proc. Combust. Inst. 29, 2017–2022 (2002)
Nguyen, P.D., Vervisch, L., Subramanian, V., Domingo, P.: Multidimensional flamelet-generated manifolds for partially premixed combustion. Combust. Flame 157, 43–61 (2010)
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Malkeson, S.P., Chakraborty, N. A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames. Flow Turbulence Combust 90, 243–267 (2013). https://doi.org/10.1007/s10494-012-9430-z
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DOI: https://doi.org/10.1007/s10494-012-9430-z