A Green’s function approach to model the pseudo-active control of thermoacoustic instabilities

• A Rijke tube is the prototype apparatus that exhibits thermoacoustic instabilities • A pseudo-active instability control method is studied with a Green's function approach • Amplitude death, i.e. complete instability control, occurs for large parameter ranges • Mutual coupling can lead to beating and is an effective control strategy • Self-coupling is a very effective control strategy and easy to implement This paper investigates a new method of pseudo-active control of thermoacoustic instabilities. We apply this control method to a Rijke tube, which is very suitable for analytical modelling. The heat source in the Rijke tube is assumed to be nonlinear. The control is achieved by two alternative types of coupling, which both add a secondary sound source to the acoustic field in the Rijke tube. The first type is "mutual coupling"; this involves a secondary Rijke tube, which is connected with the primary Rijke tube by a narrow duct. The second type is "self-coupling"; this involves only a single Rijke tube, which has the two ends of a narrow duct attached to different points on its wall. Two coupling parameters are of particular interest: the time delay and the coupling strength. We use a tailored Green's function approach to model the complete configuration and predict how the coupling changes the amplitude. We visualise the results with two-bifurcation diagrams in terms of the two coupling parameters. Three different regions are observed in these diagrams: amplitude increase, amplitude decrease and zero amplitude (called "amplitude death"). Amplitude death is achieved for a large range of values of the coupling parameters. Self-coupling is more effective in leading to amplitude death than mutual coupling.