| Combustion instabilities observed in rockets and afterburners of jet engines involve the acoustic modes of the cavity in which the combustion takes place. The mechanisms that produce combustion instabilities are now relatively well understood, but when the problem occurs, a solution is still difficult to obtain. It is the purpose of this study to describe the basic principles required to develop a finite element model (FEM) of thermoacoustic oscillations in ducted flow, including three-dimensional geometries and nonuniform mean flows with steady heat addition. The thermoacoustic problem is not modeled by a form of the convected wave equation, but instead by the conservation equations in primitive form. As part of this investigation the three-dimensional model is reduced to one dimension, applicable to a uniform geometry, but allowing an axially nonuniform flow with sharp gradients in temperature, density, and Mach number. The steady mean flow is computed as a Rayleigh flow. Calculations are shown for standing waves in several flows with different heating rates and acoustically-rigid upstream and downstream conditions to emphasize the effect of the heat input on the modal structure in the cavity. Limited comparisons are made to relevant experimental work available in the literature. It is found that some of the features of modal structure in unstable oscillations reported in the literature can be reproduced using the FEM. A general method to implement the duct boundary conditions upstream or downstream is proposed, and it is illustrated for the case of a circular duct and a nozzle. Modeling of acoustic treatment as a way to attenuate the acoustic resonance amplitudes excited by the heat sources is shown for axisymmetric and rectangular cross section ducts. FEM using modal expansions on the convected wave equation and the primitive variable formulation are presented, and comparisons with the analytical solution and the standard finite element functions are carried out when possible. |
