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

Waugh, Iain C.; Illingworth, Simon J.; Juniper, Matthew P.

doi:10.1016/j.jcp.2012.12.034

Cited by 27 publications

(18 citation statements)

References 48 publications

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“…Using iterative techniques, 54 continuation methods have now been applied to a Burke-Schumann flame in an acoustic duct with a few hundred degrees of freedom 53 and to a G-equation flame in an acoustic duct with a few thousand degrees of freedom. 104 By examining the Floquet multipliers of the periodic solutions that are found, this reveals stable periodic solutions, unstable periodic solutions, stable and unstable period-2 periodic solutions, Hopf bifurcations, Neimark-Sacker bifurcations, fold bifurcations, and period-doubling bifurcations.…”

Section: Methods Of Multiple Scalesmentioning

confidence: 99%

“…Until recently, they have been limited to a few tens of degrees of freedom but systems with several thousand degrees of freedom can be considered now. 53,54 Periodic solutions are not the only possible solutions to nonlinear differential equations. There can also be multi-periodic, quasi-periodic, and chaotic solutions.…”

Section: Nonlinearity and Bifurcation Diagramsmentioning

confidence: 99%

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Non-normality and nonlinearity in thermoacoustic instabilities

Sujith

Juniper

Schmid

2016

International Journal of Spray and Combustion Dynamics

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Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances. The short-term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for nonmodal stability analysis is provided. The applicability of the tools from nonmodal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The article closes with a brief description of how to characterize bifurcations in thermoacoustic systems.

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Section: Methods Of Multiple Scalesmentioning

confidence: 99%

Section: Nonlinearity and Bifurcation Diagramsmentioning

confidence: 99%

Non-normality and nonlinearity in thermoacoustic instabilities

Sujith

Juniper

Schmid

2016

International Journal of Spray and Combustion Dynamics

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“…In the fluid dynamics context numerical bifurcation analysis has been successfully applied in recent years to a great variety of problems [18,[23][24][25][26]. Computations based on continuation of periodic orbits of nontrivial time dependence [27] and even tori [28,29] or other invariant objects [30] have provided useful information to clarify the dynamics.…”

Section: Introductionmentioning

confidence: 99%

Continuation and stability of convective modulated rotating waves in spherical shells

2016

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Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E = 10 −4 , and a low Prandtl number fluid Pr = 0.1 in a moderately thick shell of radius ratio η = 0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined.

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“…The problems treated included the plane Couette flow [62], the pipe flow [63], thermal convection in a cubical cavity [64], the thermoacoustics of flames [65,66], and the continuation of modulated waves for the thermal convection in rotating spherical shells in [67], among others. This multiplicity of works is due to the fact that the continuation and eigenvalue computation algorithms are functionally separated from that of the time evolution, and can therefore be developed pretty much independently allowing the use of adapted legacy simulation codes.…”

Section: Applications and Conclusionmentioning

confidence: 99%

Numerical continuation methods for large-scale dissipative dynamical systems

Umbría

Net

2016

Eur. Phys. J. Spec. Top.

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Abstract. A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.

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

Cited by 27 publications

References 48 publications

Non-normality and nonlinearity in thermoacoustic instabilities

Non-normality and nonlinearity in thermoacoustic instabilities

Continuation and stability of convective modulated rotating waves in spherical shells

Numerical continuation methods for large-scale dissipative dynamical systems

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