Darrieus–Landau instability

The Darrieus–Landau instability or hydrodynamic instability is an instrinsic flame instability that occurs in premixed flames, caused by the density variation due to the thermal expansion of the gas produced by the combustion process. In simple terms, the stability inquires whether a steadily propagating plane sheet with a discontinuous jump in density is stable or not. It was predicted independently by Georges Jean Marie Darrieus and Lev Landau.[1][2] Yakov Zeldovich notes that Lev Landau generously suggested this problem to him to investigate and Zeldovich however made error in calculations which led Landau himself to complete the work.[3][4]

The instability analysis behind the Darrieus–Landau instability considers a planar, premixed flame front subjected to very small perturbations.[5] It is useful to think of this arrangement as one in which the unperturbed flame is stationary, with the reactants (fuel and oxidizer) directed towards the flame and perpendicular to it with a velocity u1, and the burnt gases leaving the flame also in a perpendicular way but with velocity u2. The analysis assumes that the flow is an incompressible flow, and that the perturbations are governed by the linearized Euler equations and, thus, are inviscid. With these considerations, the main result of this analysis is that, if the density of the burnt gases is less than that of the reactants, which is the case in practice due to the thermal expansion of the gas produced by the combustion process, the flame front is unstable to perturbations of any wavelength. Another result is that the rate of growth of the perturbations is inversely proportional to their wavelength; thus small flame wrinkles (but larger than the characteristic flame thickness) grow faster than larger ones. In practice, however, diffusive and buoyancy effects that are not taken into account by the analysis of Darrieus and Landau may have a stabilizing effect.[6][7][8][9]

Dispersion relation

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If the disturbances to the steady planar flame sheet are of the form  , where   is the transverse coordinate system that lies on the undisturbed stationary flame sheet,   is the time,   is the wavevector of the disturbance and   is the temporal growth rate of the disturbance, then the dispersion relation is given by[10]

 

where   is the laminar burning velocity (or, the flow velocity far upstream of the flame in a frame that is fixed to the flame),   and   is the ratio of burnt to unburnt gas density. In combustion   always and therefore the growth rate   for all wavenumbers. This implies that a plane sheet of flame with a burning velocity   is unstable for all wavenumbers. In fact, Amable Liñán and Forman A. Williams quote in their book[11][12] that in view of laboratory observations of stable, planar, laminar flames, publication of their theoretical predictions required courage on the part of Darrieus and Landau.

If the buoyancy forces are taken into account (in others words, accounts of Rayleigh–Taylor instability are considered) for planar flames that are perpendicular to the gravity vector, then some level of stability can be anticipated for flames propagating vertically downwards (or flames that held stationary by a vertically upward flow) since in these cases, the denser unburnt gas lies beneath the lighter burnt gas mixture. Of course, flames that are propagating vertically upwards or those that are held stationary by a vertically downward flow, both the Darrieus–Landau mechanism and the Rayleigh–Taylor mechanism contributes to the destabilizing effect. The dispersion relation when buoyance forces are included becomes

 

where   corresponds to gravitational acceleration for flames propagating downwards and   corresponds to gravitational acceleration for flames propagating upwards. The above dispersion implies that gravity introduces stability for downward propagating flames when  , where   is a characteristic buoyancy length scale.

Darrieus and Landau's analysis treats the flame as a plane sheet to investigate its stability with the neglect of diffusion effects, whereas in reality, the flame has a definite thickness, say the laminar flame thickness  , where   is the thermal diffusivity, wherein diffusion effects cannot be neglected. Accounting for the flame structure, as first envisioned by George H. Markstein, are found to stabilize the flames for small wavelengths  , except when fuel diffusion coefficient and thermal diffusivity differ from each other significantly leading to the so-called (Turing) diffusive-thermal instability.

Darrieus–Landau instability manifests in the range   for downward propagating flames and   for upward propagating flames.

See also

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References

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  1. ^ Darrieus, G. (1938). "Propagation d'un front de flamme". La Technique Moderne and Congrés de Mécanique Appliquée Paris.
  2. ^ Landau, L. D. (1944). "On the theory of slow combustion". Acta Physicochim.
  3. ^ Zeldovich, Ya. B. (1987) Remembering a teacher. For the Eightieth birthday of L. D. Landau: In: Selected Works of Yakov Borisovich Zeldovich, Volume II.
  4. ^ Zeldovich, Ya. B. (1989) Recollections of the teacher: In: Landau: the physicist & the man.
  5. ^ Clavin, Paul; Searby, Geoff (2016). Combustion Waves and Fronts in Flows. Cambridge: Cambridge University Press. doi:10.1017/cbo9781316162453. ISBN 9781316162453.
  6. ^ Markstein, G. H. Non-steady flame Propagation,(1964). P22, Pergarmon, New York.
  7. ^ Frankel, M. L.; Sivashinsky, G. I. (December 1982). "The Effect of Viscosity on Hydrodynamic Stability of a Plane Flame Front". Combustion Science and Technology. 29 (3–6): 207–224. doi:10.1080/00102208208923598. ISSN 0010-2202.
  8. ^ Matalon, M.; Matkowsky, B. J. (November 1982). "Flames as gasdynamic discontinuities". Journal of Fluid Mechanics. 124: 239–259. Bibcode:1982JFM...124..239M. doi:10.1017/S0022112082002481. ISSN 1469-7645. S2CID 121744586.
  9. ^ Pelce, P.; Clavin, P. (November 1982). "Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames". Journal of Fluid Mechanics. 124: 219–237. Bibcode:1982JFM...124..219P. doi:10.1017/S002211208200247X. ISSN 1469-7645. S2CID 102965398.
  10. ^ Williams, F. A. (2018). Combustion theory. CRC Press. page 353
  11. ^ Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.
  12. ^ Crighton, D. G. (1997). Fundamental Aspects of Combustion. By A. Liñan & FA Williams. Oxford University Press, 1993, 167 pp. ISBN 019507626 5.£ 25. Journal of Fluid Mechanics, 331, 439-443.