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Symmetry breaking of azimuthal thermo-acoustic modes in annular cavities: a theoretical study

Published online by Cambridge University Press:  11 November 2014

M. Bauerheim ,
P. Salas ,
F. Nicoud  and
T. Poinsot
M. Bauerheim*
Affiliation:
CERFACS, CFD team, 42 Av Coriolis, 31057 Toulouse, France Société Nationale d’Etude et de Construction de Moteurs d’Aviation, 77550 Reau, France
P. Salas
Affiliation:
INRIA Bordeaux – Sud Ouest, HiePACS Project, Joint INRIA-CERFACS lab. on High Performance Computing, 33405 Talence, France
F. Nicoud
Affiliation:
Université Montpellier 2. I3M UMR CNRS 5149, France
T. Poinsot
Affiliation:
IMF Toulouse, INP de Toulouse and CNRS, 31400 Toulouse, France
*
†Email address for correspondence: bauerheim@cerfacs.fr
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Abstract

Many physical problems containing rotating symmetry exhibit azimuthal waves, from electromagnetic waves in nanophotonic crystals to seismic waves in giant stars. When this symmetry is broken, clockwise (CW) and counter-clockwise (CCW) waves are split into two distinct modes which can become unstable. This paper focuses on a theoretical study of symmetry breaking in annular cavities containing a number $N$ of flames prone to azimuthal thermo-acoustic instabilities. A general dispersion relation for non-perfectly-axisymmetric cavities is obtained and analytically solved to provide an explicit expression for the frequencies and growth rates of all azimuthal modes of the configuration. This analytical study unveils two parameters affecting the stability of the mode: (i) a coupling strength corresponding to the cumulative effects of the $N$ flames and (ii) a splitting strength due to the symmetry breaking when the flames are different. This theory has been validated using a 3D Helmholtz solver and good agreement is found. When only two types of flames are introduced into the annular cavity, the splitting strength is found to depend on two parameters: the difference between the two burner types and the pattern used to distribute the flames along the azimuthal direction. To first order, this theory suggests that the most stable configuration is obtained for a perfectly axisymmetric configuration. Therefore, breaking the symmetry by mixing different flames cannot improve the stability of an annular combustor independently of the flame distribution pattern.

JFM classification

Acoustics: Acoustics Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 431 - 465
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© 2014 Cambridge University Press 

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