Non-Linear Growth and Limiting Amplitude of Acoustic Oscillations in Combustion Chambers

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Abstract
Due to non-linear loss or gain of energy, unstable oscillations in combustion chambers cannot grow indefinitely. Very often the limiting amplitudes are sufficiently low that the wave motions appear to be sinusoidal without discontinuities. The analysis presented here is based on the idea that the gasdynamics throughout most of the volume can be handled in a linear fashion. Non-linear behavior is associated with localized energy losses, such as wall losses and particle attenuation, or with the interaction between the oscillations and the combustion processes which sustain the motions. The formal procedure describes the non-linear growth and decay of an acoustic mode whose spatial structure does not change with time. Integration of the conservation equations over the volume of the chamber produces a single non-linear ordinary differential equation for the time-dependent amplitude of the mode. The equation can be solved easily by standard techniques, producing very simple results for the non-linear growth rate, decay rate, and limiting amplitude. Most of the treatment is developed for unstable motions in solid propellant rocket chambers. Other combustion chambers can be represented as special cases of the general description.

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