An adjoint Green’s function approach for thermoacoustic instabilities in a duct with mean flow

Thermoacoustic instabilities arise from the feedback between an acoustic field and the unsteady heat released in a burner, yielding self-sustained oscillations. A fundamental framework for modelling thermoacoustic instabilities in systems where a mean flow is present is introduced, based on the definition of the adjoint Green's function which permits to convert the acoustic analogy equation into an integral equation. The adjoint Green's problem produces sensitivity functions which quantify the response of the system to initial, boundary or other forcing terms. A simple one-dimensional system is examined; it includes a steady uniform mean flow and a nonlinear heat source with an amplitude-dependent time-delay heat release model. The versatility of the approach is demonstrated by applying it to two resonators characterized by different acoustic boundary conditions: a Rijke tube and a quarter-wave resonator. The control parameters are: heat source position, heater power and tube length. The results reveal that the proposed analytical framework successfully captures the limit cycles, triggering phenomena, hystereses, and Hopf bifurcations observed in experiments. We show that the mean flow velocity cannot be discarded in the study of such systems; by increasing it, a stabilization generally ensues, with a modification of the bistability characteristics of the system.