The stability of expanding and converging premixed flames is investigated theoretically. These configurations are inherently unsteady and represent simple examples of positively (expanding) and negatively (converging) stretched flames. A new model is employed in this study that incorporates both hydrodynamic and diffusional-thermal effects. The model expands earlier works by incorporating variable transport properties, variations in the mixture strength and a more realistic dependence on viscosity.
The expanding flame is shown to remain stable at small radii provided the thermal diffusivity exceeds the mass diffusivity of the deficient reactant. However, once the flame achieves a critical size, a cellular instability appears, consistent with experimental observations. The effects of viscosity and equivalence ratio on the onset of instability as well as the subsequent development of a cellular structure is studied. Theoretical predictions are given for critical flame size, cell size and growth rate of a disturbance. It is shown that, as the flame continues to expand indefinitely, there is an ever increasing range of unstable wavelengths. This cascade is fractal in nature and a fractal analysis is pursued that results in an expression for the turbulent flame speed. Results are shown to be in good agreement with experiments.
The converging flame is shown to be unconditionally unstable to disturbances of all wavelength. This instability is attributed to thermal expansion. The contracting surface area of this negatively stretched flame serves to enhance the growth rate of a disturbance. For this configuration, it is shown that diffusive- thermal effects are secondary to hydrodynamic effects, even at small radii.