Darrieus-Landau instability

All premixed flames are xmconditionally unstable to a hydrodynamic instability that has its origin in the expansion of the gas through the flame (but the flame may remain planar for other reasons). This phenomenon was first recognized by George Darrieus [1] and independently by Lev Landau [2], and is usually referred to as the Darrieus-Landau instability. The full derivation of the instability is arduous here we will give a simple heuristic explanation. 

G. Glanet and G. Searby. First experimental study of the Darrieus-Landau instability. Physical Review Letters, 80(17) 3867-3870, 1998. 

J.M. Truffaut and G. Searby. Experimental study of the Darrieus-Landau instability on an inverted- V flame and measurement of the Markstein number. Combustion Science and Technology, 149 35-52, 1999. 

Nevertheless, despite all these remarkable achievements, some open questions still remain. Among them is the influence of the molecular transport properties, in particular Lewis number effects, on the structure of turbulent premixed flames. Additional work is also needed to quantify the flame-generated turbulence phenomena and its relationship with the Darrieus-Landau instability. Another question is what are exactly the conditions for turbulent scalar transport to occur in a coimter-gradient mode Finally, is it realistic to expect that a turbulent premixed flame reaches an asymptotic steady-state of propagation, and if so, is it possible, in the future, to devise an experiment demonstrating it ... 

Figure 7.1 Four stages of propagation found in the TC burner Stage A — wrinkled flame, due to the Darrieus-Landau instability Stage B — flat flame, due to the primary pyroacoustic instability Stage C — pulsating cellular flame, due to the secondary pyroacoustic instability and Stage D — turbulent flame.

Lewis number based on the limiting reactant is sufficiently large [3]. Therefore, the Darrieus-Landau instability is expected to result in enhanced flame-surface wrinkling only when it is able to overcome stabilizing influences of buoyancy and thermal diffusion. Additionally, a flame under confinement may be influenced by acoustic waves, and coupling between the flame and acoustic-wave dynamics may result in unstable flame propagation as well. 

Experimental studies of downward flame propagation in tubes of circular and annular cross-section ([4] and [5], respectively) away from the open end and toward the closed end have revealed four distinct stages of propagation as the flame traverses the length of the tube. Upon ignition, the flame surface is initially wrinkled due to the Darrieus-Landau instability, and, if the amplitude of the wrinkles is sufficiently large as the flame enters... 

The model predicts that methane-air flames with an equivalence ratio of 0.8 propagating downward are always unstable as a result of either the Darrieus-Landau instability or the secondary pyroacoustic instability, also known as the parametric instability, yet flames with an equivalence ratio of 1.3 will be stabilized by a small band of normalized acoustic-velocity amplitudes between 3.7 and 4. Although it was found experimentally that a methane-air flame with an equivalence ratio of 0.8 can be stabilized, the results of the model agree qualitatively with the experimental findings, specifically that a methane-air flame with an equivalence ratio of 1.3 is stable over a larger range of acoustic amplitudes than one with an equivalence ratio of 0.8. 

Premixed flames may be influenced by the Darrieus-Landau hydrodynamic instability [1, 2] when the chemical heat release is sufficiently large. Hydrodynam-ically unstable premixed flames are not always observed, however, because of stabilizing influences of buoyancy and thermal diffusion. The long-wavelength flame-surface wrinkles are attenuated by buoyancy for downward propagating flames, and thermal-diffusive effects stabilize small-wavelength wrinkles when the... 

Landau and Darrieus pioneering works on hydrodynamic instability of a flat laminar flame are well known [1]. According to Darrieus-Landau theory, small perturbations, independent of the wave length, make a flat flame unstable. 

In Chapter 5.3, D. Dunn-Rankin discusses the shape of deflagrations in closed tubes and the conditions under which it assumes the form of a tulip. The propagation of a premixed flame in closed vessels has been studied from the nineteenth century. The tulip flame is an interesting example of flame-flow interaction originating from the Landau-Darrieus instability. 

Consider initially the hydrodynamic instability—that is, the one due to the flow—first described by Darrieus [52], Landau [53], and Markstein [54], If no wrinkle occurs in a laminar flame, the flame speed SL is equal to the upstream unbumed gas velocity U0. But if a minor wrinkle occurs in a laminar flame, the approach flow streamlines will either diverge or converge as shown in Fig. 4.45. Considering the two middle streamlines, one notes that, because of the curvature due to the wrinkle, the normal component of the velocity, with respect to the flame, is less than U(). Thus, the streamlines diverge as they enter the wrinkled flame front. Since there must be continuity of mass between... 

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