Modelling studies of spiral waves and target patterns in premixed flames

Spiral waves and target patterns have recently been observed experimentally in freely propagating and burner-stabilised flames, typically in systems close to the lean flammability limit and for which the Lewis number (Le, the ratio of thermal to mass diffusivity) exceeds unity. An interpretation of these observations based on a simple two-reaction model (the Sal'nikov model) is developed here in terms of the excitability of the system rather than by ascribing any special role to the Lewis number. The conditions for excitability are derived in terms of the various experimental parameters (reaction rate coefficients, exothermicity, initial concentrations, heat transfer coefficient, etc.) and this information is then used to direct a numerical search for wave behaviour in the reaction-diffusion equations. The conditions for wave propagation are then established in terms of the experimental parameters and examples of target and spiral waves computed. Waves are shown to exist for a wide range of Lewis number, including Le < 1, but it is also observed that high values of Le appear to increase the range of the conditions over which waves can be supported.