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Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube

Published online by Cambridge University Press:  22 September 2016

Alessandro Orchini ,
Georgios Rigas  and
Matthew P. Juniper
Alessandro Orchini*
Affiliation:
Department of Engineering, University of Cambridge, CambridgeCB2 1PZ, UK
Georgios Rigas
Affiliation:
Department of Engineering, University of Cambridge, CambridgeCB2 1PZ, UK
Matthew P. Juniper
Affiliation:
Department of Engineering, University of Cambridge, CambridgeCB2 1PZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations’ amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Low-dimensional models Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , pp. 523 - 550
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Copyright
© 2016 Cambridge University Press 

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