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Nonlinear Flame-Flow Transfer Function Calculations: Flow Disturbance Celerity Effects Part 2

1which determined the key parameters influencing the nonlinear flame characteristics. It was shown that the linear and nonlinear characteristics of the flame dynamics were controlled by the superposition of two sources of flame disturbances: those originating at the flame anchoring point due to boundary conditions and from flow non-uniformities. These disturbances do not generally propagate along the flame at the same speed. Consequently, they may either constructively or destructively superpose with each other, depending upon frequency, flame length, and the ratio of their propagation velocities. The response in the linear regime was shown to be characterized by two Strouhal numbers, namely, St2 and Stc. These two Strouhal numbers are related to the amount of time taken for a flow (Stc) and flame front (St2) disturbance to propagate the flame length, normalized by the acoustic period. This study considers the same problem analytically in order to obtain more insight into the underlying nonlinear flame dynamics. We have performed a third order perturbation analysis of the G-equation, which allows an explicit evaluation of the coupling between various parameters influencing flame nonlinearities, such as the flame length and flow disturbance phase speed. Explicit analytical expressions for the mean flame length, as well as the flame response at the forcing frequency and its harmonics have been obtained. It is shown that the nonlinear flame response is controlled by two additional characteristic time scales, apart from the flow and disturbance convective times represented by the nondimensional parameters St2 and Stc. The non-linearity in the flame response is found to be higher when the phase speed of the disturbances is less than the mean flow speed. One interesting prediction of this analysis concerns the impact of flow forcing on average flame length. We show that in most cases, the flame length decreases with increasing perturbation amplitude, as has been experimentally observed; e.g., see Durox 32 . However, the analysis also predicts that the flame can also lengthen under certain conditions, a result that has apparently not been experimentally observed.