Theory and modeling of accelerating flames in tubes

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.