Modeling of premixed swirling flames transfer functions

An analytical model is derived for the linear response of swirling flames submitted to velocity disturbances. The flame dynamics is represented by a linearized version of the G-equation. Turbulent fluctuations are first averaged in time to obtain a kinematic equation in which the flame is represented by a wrinkled sheet. The variables are then phase averaged to describe acoustic perturbations and obtain a perturbed G  -equation. It is first concluded that the flame motion results from the combined effects of axial and azimuthal velocity perturbations. The latter disturbances formed at the swirler outlet when this element is submitted to axial velocity fluctuations are convected by the flow and impinge on the flame. In this disturbance field the swirl number is perturbed and this is effectively modeled by assuming that the turbulent burning velocity is modulated by the axial and azimuthal velocity perturbations. It is then shown that the response of swirling flames can be deduced from the transfer function of inverted conical flames submitted to axial velocity perturbations. It is however important to account for the phase shift resulting from the propagation of axial and azimuthal disturbances on the downstream side of the swirler. This phase shift, due to the difference in propagation velocity of acoustic and convective perturbations, is determined experimentally. Theoretical transfer functions are compared with measurements corresponding to two bulk velocities at a constant swirl number S=0.55S=0.55. A good agreement is obtained. It is shown in particular that minima and maxima of the flame response are suitably retrieved and the Strouhal number can be used to collapse the data.