An analytical model based on the G-equation for the response of technically premixed flames to perturbations of equivalence ratio

A model for the response of technically premixed flames to equivalence ratio perturbations is proposed. The formulation, which is an extension of an analytical flame tracking model based on the linearized G-equation, considers the flame impulse response to a local, impulsive, infinitesimal perturbation that is transported by convection from the flame base towards the flame surface. It is shown that the contributions of laminar flame speed and heat of reaction to the impulse response exhibit a local behavior, i.e. the flame responds at the moment when and at the location where the equivalence ratio perturbation reaches the flame surface. The time lag of this process is related to a convective time scale, which corresponds to the convective transport of fuel from the base of the flame to the flame surface. On the contrary, the flame surface area contribution exhibits a non-local behavior: albeit fluctuations of the flame shape are generated locally due to a distortion of the kinematic balance between flame speed and the flow velocity, the resulting wrinkles in flame shape are then transported by convection towards the flame tip with the restorative time scale. The impact of radial non-uniformity in equivalence ratio perturbations on the flame impulse response is demonstrated by comparing the impulse responses for uniform and parabolic radial profiles. Considerable deviation in the phase of the flame transfer function, which is important for thermo-acoustic stability, is observed.