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The analytical theory of premixed laminar flames accelerating in tubes is developed, which is an important part of the fundamental problem of flame transition to detonation. According to the theory, flames with realistically large density drop at the front accelerate exponentially from a closed end of a tube with nonslip at the walls. The acceleration is unlimited in time; it may go on until flame triggers detonation. The analytical formulas for the acceleration rate, for the flame shape and the velocity profile in the flow pushed by the flame are obtained. The theory is validated by extensive numerical simulations. The numerical simulations are performed for the complete set of hydrodynamic combustion equations including thermal conduction, viscosity, diffusion, and chemical kinetics. The theoretical predictions are in a good agreement with the numerical results. It is also shown how the developed theory can be used to understand acceleration of turbulent flames.

We explain the physical mechanism of ultrafast flame acceleration in obstructed channels used in modern experiments on detonation triggering. It is demonstrated that delayed burning between the obstacles creates a powerful jetflow, driving the acceleration. This mechanism is much stronger than the classical Shelkin scenario of flame acceleration due to nonslip at the channel walls. The mechanism under study is independent of the Reynolds number, with turbulence playing only a supplementary role. The flame front accelerates exponentially; the analytical formula for the growth rate is obtained. The theory is validated by extensive direct numerical simulations and comparison to previous experiments.

a b s t r a c tIt was demonstrated recently in Bychkov et al. [Bychkov et al., Phys. Rev. Lett. 101 (2008) 164501], that the physical mechanism of flame acceleration in channels with obstacles is qualitatively different from the classical Shelkin mechanism. The new mechanism is much stronger, and is independent of the Reynolds number. The present study provides details of the theory and numerical modeling of the flame acceleration. It is shown theoretically and computationally that flame acceleration progresses noticeably faster in the axisymmetric cylindrical geometry as compared to the planar one, and that the acceleration rate reduces with increasing Mach number and thereby the gas compressibility. Furthermore, the velocity of the accelerating flame saturates to a constant value that is supersonic with respect to the wall. The saturation state can be correlated to the Chapman-Jouguet deflagration as well as the fast flames observed in experiments. The possibility of transition from deflagration-to-detonation in the obstructed channels is demonstrated.

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 flame front are proposed for the cases of both zero and nonzero gravity.