Hamilton–Jacobi equation for a wrinkled flame front

A model equation of the Hamilton–Jacobi type is proposed to describe the propagation of a wrinkled premixed flame front. For periodic front disturbances, the proposed equation has simple analytical solutions. Using the model equation, the dynamic properties and shape of the wrinkled flame front are investigated. It is shown that three characteristic periods can be distinguished in the process of development of a periodic wrinkled front: the induction period, the period of accelerated front motion, and the period of steady-state motion of the cellular flame. It is shown that during the period of accelerated flame motion, the process of its propagation can be approximately considered self-similar, in which the average velocity of the wrinkled front changes with time according to the law t1/2. The shape of stationary flame front cells is obtained. Two-dimensional stationary flame front cells are investigated. The results of calculations by the proposed model equation are compared with the results of calculations by the well-known Michelson–Sivashinsky (MS) equation and with calculations within the context of a hydrodynamic theory. It is shown that the normalized shapes of stationary cells calculated by all three models are very close, but the cell amplitudes predicted by them differ significantly. Thus, the cell amplitudes calculated by the model equation agree well with the calculation within the context of a hydrodynamic theory for realistic expansion ratios of combustion products but are approximately 1.5–1.7 times greater for unrealistically small expansion ratios, while the amplitudes calculated by the MS equation agree well with the calculation within the context of a hydrodynamic theory for unrealistically small expansion ratios of combustion products (for which the MS equation was obtained), but give significantly overestimated results for realistic values of expansion ratios. A comparison of the calculation results using the model equation with known experimental data on the stationary and non-stationary propagation of cellular flames was performed.