Darrieus–Landau instability

The Darrieus–Landau instability is an intrinsic flame instability that occurs in premixed flames due to the thermal expansion of the gas produced by the combustion process. It was predicted independently by Georges Jean Marie Darrieus and Lev Landau.[1][2]

The instability analysis behind the Darrieus–Landau instability considers a planar, premixed flame front subjected to very small perturbations.[3] 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.[4][5][6][7]

Amable Liñán and Forman A. Williams quote in their book[8][9] that it was indeed courageous of Darrius (1938) and Landau (1944) to publish - in the face of laboratory evidence of the existence of stable planar laminar flames - analyses showing the instability of a plane flame sheet to disturbances of all wavenumbers.

Dispersion relation

If the disturbances to the steady planar flame sheet are of the form ${\displaystyle e^{i\mathbf {k} \cdot \mathbf {x} _{\bot }+\omega t}}$, where ${\displaystyle \mathbf {x} _{\bot }}$ is the transverse coordinate system perpendicular to the steady flame sheet, ${\displaystyle t}$ is the time, ${\displaystyle \mathbf {k} }$ is the wavevector of the disturbance and ${\displaystyle \omega }$ is the temporal growth rate of the disturbance. Then the dispersion relation is given by[10]

${\displaystyle {\frac {\omega }{S_{L}k}}={\frac {R}{1+R}}\left({\sqrt {1+{\frac {R^{2}-1}{R}}}}-1\right)}$

where ${\displaystyle S_{L}}$ is the laminar burning velocity (or, the flow velocity far upstream of the flame in a frame that is fixed to the flame), ${\displaystyle k=|\mathbf {k} |}$ and ${\displaystyle R=\rho _{u}/\rho _{b}}$ is the ratio of unburnt to burnt gas density.

Since ${\displaystyle R>1}$ always in combustion due to the thermal expansion by heat release, the growth rate ${\displaystyle \omega >0}$ is also always positive for all wavenumbers. The whole analysis becomes invalid when the wavelength becomes comparable to the flame thickness, i.e., ${\displaystyle k^{-1}\sim l_{F}=\alpha /S_{L}}$, where ${\displaystyle \alpha }$ is the thermal diffusivity.

If the buoyancy forces are taken into account for vertically propagating planar flames so that the gravity vector with magnitude ${\displaystyle g}$ points from the unburnt side to burnt gas side, the dispersion relation becomes

${\displaystyle {\frac {\omega }{S_{L}k}}={\frac {R}{1+R}}\left[{\sqrt {1+\left({\frac {R^{2}-1}{R}}\right)\left(1-{\frac {g}{S_{L}^{2}kR}}\right)}}-1\right]}$

This implies that gravity introduces stability when ${\displaystyle k^{-1}>l_{b}=S_{L}^{2}R/g}$.

The range of instability region due to the density jump across the flame becomes ${\displaystyle l_{F}<k^{-1}<l_{b}}$. If ${\displaystyle l_{b}<l_{c}}$, then there is no hydrodynamic instability.

References

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. Clavin, Paul; Searby, Geoff (2016). Combustion Waves and Fronts in Flows. Cambridge: Cambridge University Press. doi:10.1017/cbo9781316162453. ISBN 9781316162453.
4. Markstein, G. H. Non-steady flame Propagation,(1964). P22, Pergarmon, New York.
5. 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.
6. 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.
7. 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.
8. Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.
9. 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.
10. Williams, F. A. (2018). Combustion theory. CRC Press. page 353