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Self-similarly developing, premixed, turbulent flames: A theoretical study
1.B. V. Raushenbakh, S. A. Belyi, I. V. Bespalov, V. Y. Borodachev, M. S. Volynskii, and A. G. Prudnikov, Physical Basis of Processes in Combustion Chambers of Air-Breathing Engines (Mashinostroenie, Moscow, 1964), Chap. 5, pp. 255–347 (in Russian) (Chap. 5 was written by A. G. Prudnikov).
2.K. N. C. Bray, “Turbulent transport in flames,” Proc. R. Soc. London, Ser. A 451, 231 (1995).
4.P. Bailly, M. Champion, and D. Garreton, “Counter-gradient diffusion in a confined turbulent premixed flame,” Phys. Fluids 9, 766 (1997).
5.R. P. Lindstedt and E. M. Váos, “Second moment modeling of premixed turbulent flames stabilizing in impinging jet geometries,” Combust. Flame 116, 461 (1999).
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19.A. N. Lipatnikov and J. Chomiak, “Turbulent flame speed and thickness: phenomenology, evaluation, and application in multi-dimensional simulations,” Prog. Energy Combust. Sci. 28, 1 (2002).
20.When the present work was completed, V. A. Sabel’nikov kindly drew our attention to an early technical report of A. G. Prudnikov, Report No. 1950, NII-1, Moscow, 1958 (in Russian). Prudnikov sought self-similar solutions to mass and temperature balance equations in the one-dimensional case. From a mathematical standpoint, the problem investigated by him is similar to the problem described by one-dimensional equation (2) with and by Eqs. (8), (9), and (14). Prudnikov has derived an analytical expression for the growth of flame thickness. This expression is similar to Eq. (56). Equation (7) investigated in the present paper is more general than the temperature balance equation investigated by Prudnikov. Accordingly, Eq. (56) is a particular case of a more general solution given by Eq. (55).
21.A. N. Lipatnikov and J. Chomiak, “Dependence of heat release on the progress variable in premixed turbulent combustion,” Proc. Combust. Inst. 28, 227 (2000).
22.A. N. Lipatnikov and J. Chomiak, “Developing premixed turbulent flames part I: a self-similar regime of flame propagation,” Combust. Sci. Technol. 162, 85 (2001).
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33.In English, the model is discussed in V. L. Zimont, “Gas premixed combustion at high turbulence. Turbulent flame closure combustion model,” Exp. Therm. Fluid Sci. 21, 179 (2000).
34.H. G. Weller, Imperial College of Science, Technology and Medicine, London, Report No. TF 9307, 1993.
35.N. Peters, Turbulent Combustion (Cambridge University Press, Cambridge, UK, 2000), p. 161.
36.It is worth noting that the division of turbulent scalar flux into turbulent diffusion and “flame induced” parts was first suggested by Prudnikov (Ref. 1) in 1964, i.e., about 15 years before the beginning of the systematic studies of the phenomenon of countergradient transport in flames. Equation (5.33a) written by Prudnikov in the quoted book reads , where and are the probabilities of finding unburned and burned mixture, respectively.
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53.A. N. Lipatnikov and J. Chomiak, “A study of the effects of pressure-driven transport on developing turbulent flame structure and propagation,” Combust. Theory Modell. 8, 211 (2004).
54.Strictly speaking, our study substantiates Eq. (6) solely in the statistically planar, one-dimensional case but provides no support to a gradient diffusion approximation for scalar fluxes tangential to a mean flame surface. In a multidimensional case, one can use such an approximation for the flux normal to the flame surface side by side with a countergradient closure for the tangential fluxes, e.g., see F. Biagioli, “Position, thickness and transport properties of turbulent premixed flames in stagnating flows,” Combust. Theory Modell. 8, 533 (2004).
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60.Note that the diffusion term does not vanish even at the limit of large . Since the flame thickness decreases as , the magnitude of term VII in Eq. (7) increases with , whereas term IX becomes negligible as compared with terms VII and VIII. This reasoning further justifies the keeping of the diffusion term in Eq. (7) even in the case of .
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