Theoretical analysis on the transient ignition of premixed expanding flame in a quiescent mixture
The present theory considers the transient evolution of the temperature and fuel mass fraction distributions, and it can determine the critical heating power and minimum ignition energy for successful ignition. The flame initiation process subject to central heating can be approximately decomposed into four stages, i.e., the fast establishment of the ignition kernel, the ignition-energy-supported flame kernel propagation, unsteady transition of the flame kernel, and quasi-steady spherical flame propagation. Comparison between present transient theory and previous quasi-steady theory indicates that the unsteady effects directly lead to the appearance of flame kernel establishing stage and considerably affect the subsequent flame kernel development by lowering the flame propagation speed. Time scale analysis is conducted, and it is found that the transient formulation completely degenerates to the quasi-steady theory either in the neighborhood of stationary flame ball or as approaching the planar flame. Previous quasi-steady theory shows that the critical heating power for successful ignition is proportional to the cube of the critical flame radius. However, the critical heating power predicted by the present transient formulation deviates this scaling law, which can be attributed to the unsteady thermal conduction from heating center to the flame front. Furthermore, in the transient formulation the central heating with finite duration is considered and the minimum ignition energy is obtained. For the first time, the memory effect that persistently drives flame propagation subsequent to central heating switching-off is examined. It is found that as the heating power grows, the memory effect becomes increasingly important and it can greatly reduce the minimum ignition energy.