Analytical model of nonlinear thermo-acoustic effects in a matrix burner

This paper considers a fundamental thermo-acoustic test rig developed by Noiray (Linear and Nonlinear Analysis of Combustion Instabilities, Application to Multipoint Injection Systems and Control Strategies”, Ph.D. Thesis, École Centrale Paris, 2007) and models it with an entirely analytical approach. The measured flame describing function is represented by a surprisingly simple time-lag law, which is then used to derive the governing equation for a single acoustic mode in the test rig. This equation turns out to be that of a harmonic oscillator with a damping/amplification coefficient that depends on the velocity amplitude. On this basis we find analytically the pattern of the oscillation regimes in parameter space, in particular the frequency and amplitude of limit cycles at various tube lengths. There is good qualitative agreement with some, but not all, features of Noiray’s observations.