Cellular flames diffusive-thermal instabilities

There are three basic distinct types of phenomena that may be responsible for intrinsic instabilities of premixed flames with one-step chemistry body-force effects, hydrodynamic effects and diffusive-thermal effects. Cellular flames—flames that spontaneously take on a nonplanar shape—often have structures affected most strongly by diffusive-thermal... 

Rq oo. Since the density ratio Rq typically has a value of about 5, with the estimate 10 < < 20, the corresponding value of Le certainly does not exceed 0.5, a value that may be compared with the limit 0.9 (which is obtained at = 20 for Rq = 1) to demonstrate the large influence of gas expansion. In fact, the influence probably is somewhat greater than would be implied by equation (102) because the decrease in diffusivity with decreasing temperature helps to make the reaction sheet less accessible to reactants, as has been shown in a further investigation of variable-property effects [223]. Thus it seems safe to conclude that Lei > Le for most real combustible mixtures and, therefore, that the observed instabilities, which lead to cellular-flame structures affected mainly by diffusive-thermal phenomena, are in fact first initiated by the hydrodynamic instability. 

Such diffusion-driven instabilities have been observed earlier in combustion systems. As early as 1892, Smithells reported the observation of cellular flames in fuel-rich mixtures [40]. An example is shown in figure A3.14.14. These were explained theoretically by Sivashinsky in terms of a thermodifflisive mechanism [41]. The key feature here involves the role played by the Lewis number, Le, the ratio of the thermal to mass diffusivity. If Le <1, which may arise with fuel-rich flame, for which H-atoms are the relevant species, of relatively low thermal conductivity (due to the high hydrocarbon content), a planar flame is unstable to spatial perturbations along the front. This mechanism has also been shown to operate for simple one-off chemical wave fronts, such as the iodate-arsenite system [42] and for various pH-driven fronts [43], if... 

For a given conduction-diffusion system, therefore, the balance between these two effects determines the overall local stability to spatial perturbation. If the conductive influences are dominant, the planar front will be stable. This arises if the thermal diffusivity is greater than the molecular diffusion coefficient, i.e., if the Lewis number (Le) is less than unity. Instability and the growth of flame curvature occurs under the opposite conditions, when molecular diffusion is dominant and (Le) > 1. This latter situation can arise with light, mobile fuels such as H2 or if light, mobile chain carriers such as H-atoms are produced (lean hydrocarbon flames have (Le) < 1 rich flames have (Le) >1). The effect of this instability is to produce cellular flames. (It should also be mentioned that a different instability, leading to oscillatory flame speeds, can arise for (Le) < 1.)... 

In his study of the cellular structure of flames, Zeldovich [6] is among the first who has stressed the importance of this mechanism. He has indeed shown that a plane flame front can undergo an instability if the diffusivity of the species limiting the reaction (inhibitor) exceeds the thermal diffusivity (activator) of the gas mixture. 

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