Propagation of curved stationary flames in tubes

Dynamics of a curved flame propagating in a tube is investigated by means of two-dimensional numerical simulations. The complete system of hydrodynamical equations including thermal conduction, viscosity, equation of chemical kinetics, and fuel diffusion is solved with the ideally adiabatic and slippery boundary conditions at the tube walls. It is found that only a planar flame can propagate in a narrow tube of width smaller than a half of the cutoff wavelength determined from the linear theory of the hydrodynamic instability of a flame front. In a wider tube, stationary curved flames are obtained, which propagate with the velocities larger than the corresponding velocity of a planar flame. The velocity of a curved flame front is studied as a function of the tube width and the expansion coefficient of the fuel. The influence of viscosity on the velocity of a curved flame front is found to be negligible. The configuration of a curved flame propagating upwards in a gravitational field is also investigated. It is shown that gravity leads to an additional increase of the flame velocity due to the effect of rising bubbles of light burning products. The analytical formulas for the velocity of a dame front are proposed for the cases of both zero and nonzero gravity.