Abstract
The prediction of thermoacoustic instabilities is fundamental for combustion systems such as domestic burners and industrial gas turbine engines. High-amplitude pressure oscillations cause thermal and mechanical stress to the equipment, leading to premature wear or even critical damage. In this paper we present a new approach to produce nonlinear (i.e. amplitude-dependent) stability maps of a combustion system as a function of various parameters. Our approach is based on the tailored Green’s function of the combustion system, which we calculate analytically. To this end, we assume that the combustor is one-dimensional, and we describe its boundary conditions through reflection coefficients. The heat release is modelled by a generalised $n{\it\tau}$ law. This includes a direct-feedback term in addition to the usual time-lag term; moreover, its parameters (time lag, coupling coefficients) depend on the oscillation amplitude. The model provides new insight into the physical mechanism of the feedback between heat release rate and acoustic perturbations. It predicts the key nonlinear features of the thermoacoustic feedback, such as limit cycles, bistability and hysteresis. It also explains the frequency shift in the acoustic modes.
Highlights
Instabilities in fluid dynamics are caused by some feedback mechanism and are typically described by governing equations in the form of coupled differential equations
They performed time-domain simulations with a Galerkin approach and found that higher Galerkin modes can have a significant effect on predictions of limit cycle amplitude
Equation (5.7) is an equation for the frequencies Ωm. Once this has been solved for Ωm, the solution can be put into (5.8) and its complex conjugate to obtain the solution for the velocity amplitudes um anEdquu∗ma.tions (5.7) and (5.8) show that the eigenmodes of the thermoacoustic system depend on the parameters in the heat release model, in particular the heater power K, the time lag τ and the coupling constants n0 and n1
Summary
Instabilities in fluid dynamics are caused by some feedback mechanism and are typically described by governing equations in the form of coupled differential equations. Kashinath, Hemchandra & Juniper (2013) studied a 2D ducted flame whose heat release rate was described by a nonlinear kinematic model (the G-equation) They performed time-domain simulations with a Galerkin approach and found that higher Galerkin modes can have a significant effect on predictions of limit cycle amplitude. Heckl (2015) modelled the nonlinear stability behaviour of a matrix burner (a laboratory burner consisting of a quarter-wave oscillator with a 2D array of small flames near the open end) and correctly predicted the behaviour that was observed experimentally Despite their advantages, Green’s function methods have not been exploited to their full potential. We will analyse the thermoacoustic instability of this burner for several parameters, in particular the heat source position, tube length and heater power We treat these as control parameters and predict the stability boundaries for them as a function of perturbation amplitude. We note that (3.4) is valid for both linear and nonlinear thermoacoustic systems
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