Abstract
We study the evolution of perturbations on the surface of a stationary plane flame front in a divergent flow of a combustible mixture incident on a plane wall perpendicular to the flow. The flow and its perturbations are assumed to be two-dimensional; i.e., the velocity has two Cartesian components. It is also assumed that the front velocity relative to the gas is small; therefore, the fluid can be considered incompressible on both sides of the front; in addition, it is assumed that in the presence of perturbations the front velocity relative to the gas ahead of it is a linear function of the front curvature. It is shown that due to the dependence (in the unperturbed flow) of the tangential component of the gas velocity on the combustion front on the coordinate along the front, the amplitude of the flame front perturbation does not increase infinitely with time, but the initial growth of perturbations stops and then begins to decline. We evaluate the coefficient of the maximum growth of perturbations, which may be large, depending on the problem parameters. It is taken into account that the characteristic spatial scale of the initial perturbations may be much greater than the wavelengths of the most rapidly growing perturbations, whose length is comparable with the flame front thickness. The maximum growth of perturbations is estimated as a function of the characteristic spatial scale of the initial perturbations.
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References
L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon, Oxford, 1987).
G. H. Markstein, “Experimental and theoretical studies of flame front stability,” J. Astronaut. Sci. 18(3), 199–209 (1951).
A. G. Kulikovskii and I. S. Shikina, “Conditions of instability of a flame front in a weakly inhomogeneous flow,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 12–19 (2000) [Fluid Dyn. 35, 635–641 (2000)].
S. G. Kotenkov and I. S. Shikina, “Absolute and convective instability of an oblique flame front in a combustible gas mixture flow,” Fiz. Goreniya Vzryva 37(4), 9–14 (2001) [Combust. Explos. ShockWaves 37, 372–377 (2001)].
G. I. Sivashinsky, “Nonlinear analysis of hydrodynamic instability in laminar flames. I: Derivation of basic equations,” Acta Astronaut. 4, 1177–1206 (1977).
E. A. Kuznetsov and S. S. Minaev, “Velocity of coherent structure propagation on the flame surface,” in Advanced Computation and Analysis of Combustion (ENAS Publ., Moscow, 1997), pp. 397–403.
Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions (Nauka, Moscow, 1980; Consultants Bureau, New York, 1985).
M. A. Liberman, V. V. Bychkov, and S. M. Gol’berg, “Stability of a flame in a gravitational field,” Zh. Eksp. Teor. Fiz. 104(8), 2685–2703 (1993) [J. Exp. Theor. Phys. 77 (2), 227–236 (1993)].
A. G. Kulikovskii, A. V. Lozovskii, and N. T. Pashchenko, “Evolution of perturbations on a weakly inhomogeneous background,” Prikl. Mat. Mekh. 71(5), 761–774 (2007) [J. Appl. Math. Mech. 71, 690–700 (2007)].
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Original Russian Text © A.G. Kulikovskii, N.T. Pashchenko, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 281, pp. 55–67.
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Kulikovskii, A.G., Pashchenko, N.T. Stability of a flame front in a divergent flow. Proc. Steklov Inst. Math. 281, 49–61 (2013). https://doi.org/10.1134/S0081543813040068
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DOI: https://doi.org/10.1134/S0081543813040068