Flame Propagation in Tubes: Hydrodynamics and Stability

Abstract

>The steady propagation of a convex laminar flame front in long channels is under consideration. In comparison to the previous theoretical efforts by Zel'dovich (1944), Tsien (1951), Chernyi (1954) and Borisov (1978 the existence of a stagnation zone fixed with respect to the flame front is taken into account. The flame front is supposed to be a hydrodynamic discontinuity with the known normal rate of propagation through the cold gas; the boundary surface of the stagnation zone is considered as a discontinuity of the tangential component of velocity, i.e, as a vortex sheet.The upstream flow field of cold gases can be treated as a potential one; however, the downstream flow of combustion products must be rotational. Vortices generated at the curved flame front drift along the streamlines and fill the whole flow area behind the flame. The gas viscosity, thermal conductivity, and diffusivity of the reacting components and heat losses at walls are neglected in this hydrodynamic model. The integral equation deduced under the simplified flow field description in front of the flame yields the dependence of the flame propagation velocity along the channel on its normal speed and on the ratio of initial and final temperatures, The results of an analytical approximation and numerical solution of this equation are compared.Special attention is given to the hydrodynamic instability of the curved flame front with respect to disturbances of small amplitude. It appears to be much more stable than the plane flame front because of the supplementary stabilizing effects: stretching of the wavelength of the disturbance drifting by the tengential velocity component along the flame front and quenching by the channel walls. An analysis performed shows that a critical Reynolds number corresponding to the appearance of the flame instability can attain values over a few hundred and is in accordance with the experimental observations.

On-shell description of unsteady flames

“…This cutoff ensures smoothness of the functions under consideration. In particular, it prevents the development of singularities of the front shape such as the edge points which would occur otherwise [15], leading to discontinuities in the values of the flow variables or their derivatives. That λ c often exceeds the actual thickness of the flame preheat zone significantly [16] has yet another virtue: the Reynolds number based on λ c and the fuel properties is typically over ∼ 10 2 , and hence is fairly large when based upon the width (> λ c or ≫ λ c ) of the channel where the flame studied below is meant to propagate.…”

“…Note that having imposed the boundary condition f ′ (0) = f ′ (1) = 0, we thereby exclude the possibility of stagnation zone formation near the end points of the flame front (see [15] for detail). We also assume that the flame is stable with respect to the short wavelength perturbations i.e., that there is a short wavelength cutoff, λ c .…”

The cellular burning regime in type Ia supernova explosions

“…The basic physical effects that determine flame propagation here are the hydrodynamical Landau-Darrieus instability (Darrieus 1938;Landau 1944) and the counteracting nonlinear stabilization of the flame (Zel'dovich 1966). The balance of the two effects gives rise to a stable cellular flame structure characterizing the cellular burning regime.…”

Magnetic Field Effects on the Thermonuclear Combustion Front of Chandrasekhar Mass White Dwarfs

“…LD instabilities lead to an acceleration no higher than 2% near the center of the star because they are nonlinearly stabilized (Khokhlov 1995). On the other hand, at the macroscopic scales, there is a critical wavelength above which the nonlinear stabilization fails 2 (Zeldovich et al 1980(Zeldovich et al , 1985.…”