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# Flame cellular structure and wrinkling

During experimental measurements of the H2-air burning velocities three kind of instabilities, leading to appearance of flame wrinkling, were usually observed: preferential-diffusion instability (when Markstein length L and Markstein number Ma are below zero), hydrodynamic instability and buoyant instability (AungKT:1997), (KwonOC:2001).

The preferential-diffusion instability creates irregular (chaotic) distortion, leading to formation of irregular flame surface, larger flame front area and faster propagation of the stretched flames.

Under the stable preferential-diffusion conditions propagation of the initially smooth flame front is followed by onset of the regular cellular structure, formed by hydrodynamic instabilities. Detailed review and analysis of the hydrodynamic instabilities in the spherically propagating premixed flames are given in the work by Bradley and co-workers {[DB, BradleyD:1994b]}, (BradleyD:1999), (BradleyD:2000). They considered a spherical premixed flame as a fractal surface, formed by the hydrodynamics instabilities. Hydrodynamic instability appears first as a cracking of the flame surface, then followed by development of a cellular flame structure and flame wrinkling. As the flame propagates the wrinkling increases and creates a larger surface area and consequent flame acceleration. As a result, the flame speed increases as a square root of elapsed time. If a flame radius is large enough, the turbulent flame appears.

According to Bradley (BradleyD:1999) hydrodynamic instabilities arise at the larger wavelengths and cascade down through a range of ever decreasing wavelengths to be stabilised at the smallest wave-length by thermo-diffusive effects. The largest instability is bounded only by the physical scale of the problem and the outer cut-off is expected to be of the order of the flame front radius. The inner cut-off of is expected to be of the order of the real flame front thickness. Semi-empirical correlation is given in (BradleyD:1999) for the critical Peclet number Pe_{cl} , characterising the onset of cellular structure for freely propagating flames:

Pe_{cl} = 177Ma+2177 for -5 <Ma<8 ,

where Pe=r s_u / \nu_u - Peclet number, r - flame radius, s_u - burning velocity, \nu_u - kinematic viscosity of unburnt (fresh) mixture. It is possible to show that for a wide range of mixtures the onset of instabilities occurs at a flame radius of order 10 cm and the minimum cell size is below 1 cm. Conclusions of this analysis can be applied only to the stable preferential-diffusion conditions. The flames with unstable preferential- diffusion conditions do not have the coherent cellular structure and the mechanism of their surface distortion cannot be explained by hydrodynamic instability alone (KwonOC:1992).

Fig. 1. Shadowgraph of the H2/O2/N2 flame surface for stable
preferential-diffusion conditions ( \Phi=3.27 ).
Left image for Pe > Pe_{cl} ; right image for Pe > Pe_{cl} .

Fig. 2. Shadowgraph of the H2/O2/N2 flame surface for unstable
preferential-diffusion conditions ( \Phi=1.00 ).
Left image for Pe > Pe_{cl} ; right image for Pe > Pe_{cl} .

The characteristic flame shape, corresponding to the thermodiffusively stable and unstable flames, is shown in Fig.1, 2, adopted from (KwonOC:1992) (with permission).

Freely propagating flames are subject to natural convection. Analysis of convection instabilities was conducted by Babkin et. al. (BabkinVS:1984). It was demonstrated that the natural convection deforms the flame front in the case when the Froude number is above the critical value:

Fr = {{s_u^2 } / {\left( {g\,D} \right)}} > 0.11,

where D is the critical diameter of the flame front. Under effect of natural convection the flame front initially becomes flatter at the lower part, the flame front rises as a whole in upward direction. Then the lower flat part of the flame bends inside, forming vortical flame structure. The fluid flow around naturally convected vortical flame creates the flame stretch, which is known to suppress formation of the cellular structure and flame wrinkling due to hydrodynamic instabilities.

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Kwon O.C. and Faeth G.M. (2001) Flame/stretch interactions of premixed hydrogen-fueled flames: measurements and predictions. Combustion and Flame, 124:590-610.(BibTeX)
Bradley D. and Harper C.M. (1994) The development of instabilities in laminar explosion flames. Combustion and Flame, 99:562-572.(BibTeX)
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Bradley D., Sheppard, C.G.W., Woolley R., Greenhalgh D.A. and Lockett R.D. (2000) The development and structure of flame instabilities and cellularity at low Markstein numbers in explosions. Combustion and Flame, 122:195-209.(BibTeX)
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