Linear and Nonlinear Acoustics with Nonuniform Entropy in Combustion Chambers

Abstract

A one-dimensional analytical model is presented for calculating the longitudinal acoustic modes of idealized dump-type ramjet engines. The geometry considered is the coaxial flow type with the inlet flow opening to the combustor at a simple dump plane. Since the frequencies are very low, the dominant modes are the one-dimensional longitudinal modes and allow the predictions to be extended to more complicated geometries (such as side dump combustors) with good success. A plane flame has been studied and incorporated into the combustor model where the flame is allowed to move or oscillate in the combustor. This provides three mechanisms of interaction at the flame sheet: change in mean temperature in the combustor, energy conversion at the sheet due to upstream fluctuations, and fluctuating heat release. A supersonic inlet upstream contains a shock wave in its diffuser section while the downstream exit is terminated by a choked nozzle. The linear coupling of the acoustic and entropy waves at the inlet shock, flame sheet, and exit nozzle along with acoustic admittances at the inlet and exit are combined to determine the stability of the system as well as the acoustic modes. Since the acoustic and entropy waves travel at different velocities, the geometry is a critical factor in determining stability. Typical values of the admittances will produce damped solutions when the entropy is neglected, but, as the ratio of the entropy to acoustic fluctuations is increased, the coupling can either feed acoustic energy into or out of different modes independently. This transfer of energy has a destabilizing or stabilizing effect on the acoustic modes of the system depending on the relative phases between the acoustic and entropy waves. In the linear case, the entropy and acoustics are decoupled in the flow field. All linear coupling occurs at the boundary conditions. For cases where the entropy fluctuations are of the same order of magnitude as the pressure oscillations and the coupling is of comparable order, the linear stability of the acoustic field is strongly dependent upon the entropy fluctuations. The linear acoustics are predominantly governed by the boundary conditions; thus it is imperative that the entire system of inlet, combustor, and exit be considered together to determine the characteristic eigenvalues (resonant frequencies) and eigenfunctions (mode shapes). In addition, there are two modes of acoustic pressure oscillations: the classical acoustic mode and the entropy-induced mode of pressure oscillation. The nonlinear case treats the quadratic nonlinear fluid mechanic interactions in the coupling of two acoustic modes. The result is that the nonlinear acoustic-entropy interactions are much smaller than the acoustic-acoustic interactions for this case. Hence, the nonlinear acoustic field is influenced by the nonuniform entropy only by its dependence upon the linear solution which can be strongly dependent upon the entropy. The energy in the acoustics of this model is controlled by the energy loss (gain) at the boundaries balanced with the energy gain (loss) at the flame front. Acoustic energy is typically lost at both the inlet and exit, but fluctuating entropy waves convecting with the mean flow velocity that impinge upon a choked nozzle generate acoustic waves that can, under the proper conditions, feed acoustic energy into the system. In addition, the Rayleigh condition for driving the system with a fluctuating heat release can also contribute to the stability of the system. The plane flame mechanism also contributes to the acoustic energy from the interaction of entropy and acoustic waves at a flame sheet. This allows a systematic study of the influence of entropy-acoustic wave interactions on the linear stability and modes of this combustor system.