Matrix-free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems

In order to define the nonlinear behaviour of a thermoacoustic system, it is important to find the regions of parameter space where limit cycles exist. Continuation methods find limit cycles numerically in the time domain, with no additional assumptions other than those used to form the governing equations. Once the limit cycles are found, these continuation methods track them as the operating condition of the system changes. Most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory. In the literature, there are therefore only a few applications of continuation methods to thermoacoustics, all with low-order models. Matrix-free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics, but are as yet unused in thermoacoustics. These matrix-free methods are shown to converge quickly to limit cycles by implicitly using a ‘reduced order model’ property. This is because the methods preferentially use the influential bulk motions of the system, whilst ignoring the features that are quickly dissipated in time. The matrix-free methods are demonstrated on a model of a ducted 2D diffusion flame, and the stability limits are calculated as a function of the Peclet number and the heat release parameter. Both subcritical and supercritical Hopf bifurcations are found. Physical information about the flame-acoustic interaction is found from the limit cycles and Floquet modes. Invariant subspace preconditioning, higher order prediction techniques, and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces. Two types of shooting are compared, and two types of matrix-free evaluation are compared.