Linear response of stretch-affected premixed flames to flow oscillations

The linear response of 2D wedge-shaped premixed flames to harmonic velocity disturbances was studied, allowing for the influence of flame stretch manifested as variations in the local flame speed along the wrinkled flame front. Results obtained from analyzing the G-equation show that the flame response is mainly characterized by a Markstein number over(σ, ̂)_C, which measures the curvature effect of the wrinkles, and a Strouhal number, St_f, defined as the angular frequency of the disturbance normalized by the time taken for the disturbance to propagate the flame length. Flame stretch is found to become important when the disturbance frequency satisfies over(σ, ̂)_C St_f² ∼ O (1), i.e. St_f ∼ O (over(σ, ̂)_C^{- 1 / 2}). Specifically, for disturbance frequencies below this order, stretch effects are small and the flame responds as an unstretched one. When the disturbance frequencies are of this order, the transfer function, defined as the ratio of the normalized fluctuation of the heat release rate to that of the velocity, is contributed mostly from fluctuations of the flame surface area, which is now affected by stretch. Finally, as the disturbance frequency increases to St_f ∼ O (over(σ, ̂)_C^{- 1}), i.e. over(σ, ̂)_C St_f ∼ O (1), the direct contribution from the stretch-affected flame speed fluctuation to the transfer function becomes comparable to that of the flame surface area. The present study phenomenologically explains the experimentally observed filtering effect in which the flame wrinkles developed at the flame base decay along the flame surface for large frequency disturbances as well as for thermal-diffusively stable and weakly unstable mixtures.