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Subcritical bifurcation and bistability in thermoacoustic systems
Published in Cambridge University Press
2013
Volume: 715

Pages: 210 - 238
Abstract
Abstract This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart-Landau equation. From the Stuart-Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation. © Cambridge University Press 2013.
Journal Journal of Fluid Mechanics Cambridge University Press 00221120 No
Concepts (39)
• Analytical estimates
• Analytical method
• BIFURCATION PARAMETER
• Critical value
• FOLD POINTS
• HARMONIC BALANCE
• Heat release rate (hrr)
• Hopf point
• Linear instabilities
• Linear stability
• METHOD OF MULTIPLE SCALE
• Nonlinear instability
• NUMERICAL CONTINUATION
• PARAMETER REGIONS
• Parameter values
• Rijke tube
• STUART-LANDAU EQUATION
• SUB-CRITICAL BIFURCATIONS
• Subcritical hopf bifurcation
• SUBCRITICAL TRANSITION
• THERMOACOUSTIC SYSTEMS
• Unstable limit cycles
• Bifurcation (mathematics)
• Estimation
• Nonlinear analysis
• Nonlinear dynamical systems
• Numerical methods
• Parameter estimation
• Stability
• Thermoacoustics
• Tubes (components)
• Hopf bifurcation
• Acoustics
• Bifurcation
• EQUATION
• Flow stability
• Instability
• Nonlinearity
• Numerical model