Acoustically perturbed Bunsen flames : modelling, analytical investigations and numerical simulations

The design of noise-free, low-pollutant combustion devices requires the understanding and the prediction of the combustion noise. The noise originates from the coupling between perturbations in the heat release rate with the oscillating gas (acoustic velocity). Prediction of combustion noise requires the transfer function (TF) that correlates the variation in the heat release rate of the ame with the perturbation (acoustic velocity). Computational costs prohibit an e? cient calculation of the TF by simulating all combustion phenomena associated with the ame dynamics. Simpli? ed theoretical models, such as the hydrodynamic models, allow a signi? cant reduction of the computation time and capture the main features of the ame front dynamics. The fundamental assumption of the hydrodynamic models is that the Bunsen ame, which is described as a thin layer ( ame front) separating the burnt from the unburnt gas, is attached at the burner rim. Within the thin layer approximation, the computation of the TF requires the ame area and, consequenctly, the time-dependent ame front position. By de? ning the ame front as the zero-level set of a distance function, the evolution of the ame front is described by a level set equation. The existent experiments and theoretic kinematic models based on the thin layer approximation (e.g., the G-equation model) provide contradictory information on the phase of the TF. Experiments suggest a low-pass ? lter behaviour of the gain of the TF and an increase of the TF's phase up to several ? as the frequency increases. In contrast, most of the theoretical models indicate a saturation of the phase of the TF at the level of ? /2. Although the use of a convective wave model for the ow [80] leads to a better agreement between theory and experiments, the origin of the convective wave remains little understood. The discrepancies between theory and experiments most likely reside in the simpli? cations employed in the theoretical models. To understand the behaviour of the experimental TF, here we extend the theoretical models based on the G-equation approach. In the ? rst step, we extend the models by addressing ames that have an arbitrary cone angle and a burning velocity with variable direction. To carry out this ? rst extension we analyse the evolution of a Bunsen ame in a Poiseuille ow from an initially at pro? le to a stationary conical shape. The analysis leads to the correct derivation of the boundary conditions in the extended model employed for the ame evolution in a perturbed Poiseuille ow. We demonstrate that the new model proposed here improves the description of the front close to the boundary and, hence, the behaviour of the ame response to velocity perturbations. Because recent measurements [52] suggest that a better understanding of the TF's phase behaviour requires a hydrodynamic model that accounts for the moving edge of the ame, in the second step we extend the hydrodynamic model to account for non-attached ames. We model the ame front as an open curve by using two level set functions whose zero-level sets are locally normal at the edge point. The movement of the edge along the ame front is accounted for via a novel model for the edge speed. During computer simulations of the ame kinematics with the new extended model the local orthogonality of the level sets is preserved by employing a novel algorithm. The simulations demonstrate that our extended model qualitatively accounts for the ame stabilisation, the ame front and edge kinematics, and the stand-o? distance of the ame above the burner rim.