Résumé
Many physical problems containing rotating symmetry exhibit azimuthal waves, fromelectromagnetic waves in nanophotonic crystals to seismic waves in giant stars. Whenthis symmetry is broken, clockwise (CW) and counter-clockwise (CCW) waves aresplit into two distinct modes which can become unstable. This paper focuses ona theoretical study of symmetry breaking in annular cavities containing a numberN of flames prone to azimuthal thermo-acoustic instabilities. A general dispersionrelation for non-perfectly-axisymmetric cavities is obtained and analytically solved toprovide an explicit expression for the frequencies and growth rates of all azimuthalmodes of the configuration. This analytical study unveils two parameters affectingthe stability of the mode: (i) a coupling strength corresponding to the cumulativeeffects of the N flames and (ii) a splitting strength due to the symmetry breakingwhen the flames are different. This theory has been validated using a 3D Helmholtzsolver and good agreement is found. When only two types of flames are introducedinto 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 theflames along the azimuthal direction. To first order, this theory suggests that the moststable configuration is obtained for a perfectly axisymmetric configuration. Therefore,breaking the symmetry by mixing different flames cannot improve the stability of anannular combustor independently of the flame distribution pattern.