Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by up to $\sim\mathcal{O}(M)$, as many as the Monte Carlo samples.European Research Council (Project ALORS 2590620), Royal Academy of Engineering (Research Fellowships Scheme

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non-self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.The authors are grateful to the 2014 Center for Turbulence Research Summer Program (Stanford University) where the ideas of this work were born. L.M. and M.P.J acknowledge the European Research Council – Project ALORS 2590620 for financial support. L.M gratefully acknowledges the financial support received from the Royal Academy of Engineering Research Fellowships scheme. The authors thank Prof. Franck Nicoud for fruitful discussions. Fig. 1 was adapted from the article of S.R. Stow and A.P. Dowling, A time-domain network model for nonlinear thermoacoustic oscillations, ASME Turbo Expo, GT2008-50770 [9] with permission of the original publisher ASME

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by as much as the Monte Carlo samples