Abstract
Since its introduction in the late 19th century, symmetry breaking has been found toplay a crucial role in physics. In particular, it appears as one key phenomenoncontrolling hydrodynamic and acoustic instabilities in problems with rotationalsymmetries. A previous paper investigated its desired potential application to thecontrol of circumferential thermoacoustic modes in one annular cavity coupled withmultiple flames (Bauerheim et al., J. Fluid Mech., vol. 760, 2014, pp. 431–465).The present paper focuses on a similar problem when symmetry breaking appearsunintentionally, for example when uncertainties due to tolerances are taken intoaccount. It yields a large uncertainty quantification (UQ) problem containing numerousuncertain parameters. To tackle this well-known ‘curse of dimensionality’, a novel UQmethodology is used. It relies on the active subspace approach to construct a reducedset of input variables. This strategy is applied on two annular cavities coupled by 19flames to determine its modal risk factor, i.e. the probability of an azimuthal acousticmode being unstable. Since each flame is modelled by two uncertain parameters, itleads to a large UQ problem involving 38 parameters. An acoustic network model isthen derived, which yields a nonlinear dispersion relation for azimuthal modes. Thisnonlinear problem, subject to bifurcations, is solved quasi-analytically. Results showthat the dimension of the probabilistic problem can be drastically reduced, from 38uncertain parameters to only 3. Moreover, it is found that the three active variablesare related to physical quantities, which unveils underlying phenomena controllingthe stability of the two coupled cavities. The first active variable is associated witha coupling strength controlling the bifurcation of the system, while the two otherscorrespond to a symmetry-breaking effect induced by the uncertainties. Thus, anadditional destabilization effect appear caused by the non-uniform pattern of theuncertainty distribution, which breaks the initial rotating symmetry of the annular cavities. Finally, the active subspace is exploited by fitting the response surface withpolynomials (linear, quadratic and cubic). By comparing accuracy and cost, resultsprove that 5% error can be achieved with only 30 simulations on the reduced space, whereas 2000 are required on the complete initial space. It exemplifies that this novelUQ technique can accurately predict the risk factor of an annular configuration atlow cost as well as unveil key parameters controlling the stability.