Nonlinear Acoustics Instabilities in Combustion Chambers

Citation

Awad, Elias A. (1983) Nonlinear Acoustics Instabilities in Combustion Chambers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/h9sr-tq76. https://resolver.caltech.edu/CaltechETD:etd-12182006-131617

Abstract

In this report, we show, following a second order expansion in the pressure amplitude, analytical expressions for the amplitude, and the conditions for existence and stability of limit cycles for pressure oscillations in combustion chambers. Two techniques are used. The first technique is an asymptotic-perturbation technique where the asymptotic oscillatory behavior is sought by expanding the asymptotic solution in a measure of the amplitude of the wave, mainly the amplitude of the fundamental. The second technique is a perturbation-averaging technique where an approximate solution is sought by applying a perturbation method followed by an expansion of the solution in the normal modes of the acoustic field in the chamber. It is shown, to third order in the amplitude of the wave, that both techniques yield the same results regarding the amplitude and the conditions for existence and stability of the limit cycle. However, while the first technique can be extended to higher orders in the pressure amplitude, the second technique suffers serious difficulties. The advantage of the second technique is in its ability to handle easily a large number of modes.

A stable limit cycle seems to be unique. The conditions for existence and stability are found to be dependent only on the linear parameters. The nonlinear parameter affects only the wave amplitude. In very special cases, the initial conditions can change the stability of the limit cycle. The imaginary parts of the linear responses, to pressure oscillations, of the different processes in the chamber play an important role in the stability of the limit cycle. They also affect the direction of flow of energy among modes. In the absence of the imaginary parts, in order for an infinitesimal perturbation in the flow to reach a finite amplitude, the lowest mode must be unstable while the highest must be stable; thus energy flows from the lowest mode to the highest one. The same case exists when the imaginary parts are non-zero, but in addition, the contrary situation is possible. There are conditions under which an infinitesimal perturbation may reach a finite amplitude if the lowest mode is stable while the highest is unstable. Thus energy can flow "backward" from the highest mode to the lowest one. It is also shown that the imaginary parts increase the final wave amplitude.

Second, the triggering of pressure oscillations in solid propellant rockets is discussed. In order to explain the triggering of the oscillations to a non-trivial stable limit cycle, the treatment of two modes and the inclusion in the combustion response of either a second order nonlinear velocity coupling or a third order nonlinear pressure coupling seem to be sufficient. Moreover, some mechanisms which are likely to be responsible for triggering are identified.

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