A unified model for the prediction of laminar flame transfer functions : comparisons between conical and V-flame dynamics

International audienceTransfer functions of premixed laminar flames submitted to incident flow perturbations are envisaged and a unified model is derived analytically. This model, based on a linearization of the G-equation for an inclined flame, includes convective effects of the flow modulations propagating upstream of the flame. It is shown that the flame dynamics is governed by two relevant parameters, a reduced frequency, omega*, and the ratio of the flame burning velocity to the mean flow velocity, SL/V, or equivalently the flame angle alpha with respect to the flow direction. In the limit of low driving frequencies, the flame motion is only controlled by omega* and the unified model reduces to previous kinematic formulations derived for rim stabilized conical flames and V-flames anchored on a central rod. Flame transfer functions for these flame geometries with the different velocity models proposed are derived and limiting cases are examined. In the conical flame case, the low-frequency model gives a good approximation of the gain, but only a fair approximation of the phase. Convective effects are shown to induce an increasing phase lag, while low-frequency models predict a saturation phenomenon. The convective model derived in this article improves results for the gain and the phase which agree with numerical simulations and experiments. It is shown in particular that 1) the correct transfer function phase trend is retrieved and depends on the flame angle a; 2) the reduced cut-off frequency corresponds to a situation where the convective wavelength along the flame front lambda = Vcos(alpha)/f equals the flame length L; and 3) the flame response is weakly affected by the amplitude of these perturbations. In the V-flame case, the low-frequency model yields a good approximation of the phase but does not feature gain values in excess of one found in the simulations. This behavior is correctly predicted by the convective model and is shown to depend on the flame angle alpha. A V-flame behaves as an amplifier in a certain range of frequencies. It is shown that these types of flames are more susceptible to combustion instabilities than conical flames. The V-flame response is also shown to strongly depend on the amplitude of the fluctuations even for moderate perturbation levels