Experimental and theoretical bifurcation study of a nonlinear standing-wave thermoacoustic system

In this work, both theoretical and experimental bifurcation studies of nonlinear standing-wave thermoacoustic systems are performed. To characterize the unsteady heat release, two different models are proposed to approximate experimental measurements of the response of a premixed flame to acoustic waves. Model A is formulated with 3rd order polynomial equation, while model B is 5th order one. The expansion method of Galerkin series is applied to decouple the unsteady acoustic field in the spatial and time domains. Multiple scales method is proposed to extract the essential dynamic behaviors from the governing ordinary differential equations. The process of the thermoacoustic system's transition to instability is evaluated by continuation methods. Model A is shown to be associated with a supercritical Hopf bifurcation. However, subcritical Hopf bifurcation occurs in the higher-order modelled nonlinear thermoacoustic system, i.e. model B. Furthermore, hysteresis and bistable zones are observed. In addition, the effects of 1) the heating power K, 2) the axial location of the heat source x_f/L and 3) the overall acoustic damping/loss ξ on supercritical and subcritical bifurcations are studied one at a time. To validate our theoretical findings on supercritical and subcritical bifurcation, experimental investigation on a standing-wave Rijke tube with a premixed flame confined is conducted. It is observed that varying the flame axial location x_f/L can destabilize the thermoacoustic system via either subcritical or supercritical bifurcation on two different operation conditions. In addition, the measured bifurcation behaviors show a good qualitative agreement with the theoretical findings.