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.