The Two-Mode Approximation to Nonlinear Acoustics in Combustion Chambers. I. Exact Solution for Second Order Acoustics

Abstract

The behavior of unsteady pressure fluctuations in combustion chambers is examined. An approximate method used in the derivation of the amplitude equations is based on the spatial and time domain averaging of the conservation equations, and follows the analytical framework introduced by Culick (1976a,b). The first order perturbation terms retained in the analysis correspond to linear contributions from the combustion processes, gas/particle interactions, mean flow and boundary conditions, as well as second order nonlinear gas dynamics terms. Further simplification of these equations is obtained by an appropriate change of variables. Following this step, the analysis based on two longitudinal modes is reduced to the solution of a three-dimensional system of nonlinear equations. This enables derivation of exact results for the existence, stability and the amplitude of the limit cycle, in the general case of frequency shifted periodic oscillations. Consideration is also given to the transfer of energy within the spectrum of the acoustic modes. In agreement with experimental observations, it is shown analytically that the preferred direction of energy transfer is from the lower to the higher acoustic modes. Validation of the results is accomplished by comparison with numerical results obtained when higher numbers of modes are treated. Finally, it is shown that combustion instabilities can be treated analytically using the center manifold theory.