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

Applied Mechanics Reviews

2019

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In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.

Section: Thermoacousticsmentioning

confidence: 99%

“…The substitution of the modal decomposition (54) into the governing equations, in general, results in a nonlinear eigenproblem (NEP) [40,163,165], in contrast to most cases in hydrodynamic stability where linear eigenproblems 19 govern the stability. The nonlinear eigenproblem reads…”

Section: Nonlinear Eigenproblemmentioning

confidence: 99%

Adjoint Methods as Design Tools in Thermoacoustics

Applied Mechanics Reviews

2019

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“…The thermoacoustic stability problem is generally non-Hermitian because of the flame response term and dissipative boundary conditions. On numerical discretization or travelling-wave decomposition [5], thermoacoustic stability is governed by a nonlinear eigenvalue problem [6,7] L(ω; ε)p = 0,…”

Section: Thermoacoustic Instabilitiesmentioning

confidence: 99%

“…On the one hand, eigenvalues of single-flame longitudinal thermoacoustic systems are typically simple [5,8]. On the other hand, systems with discrete rotational symmetry, such as annular and can-annular combustors, feature semi-simple degenerate eigenvalues [6,9], with fewer simple eigenvalues.…”

Section: Eigenvalue Classificationmentioning

confidence: 99%

Exceptional points in the thermoacoustic spectrum

Mensah

Journal of Sound and Vibration

Silva

et al. 2018

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Exceptional points are found in the spectrum of a prototypical thermoacoustic system as the parameters of the flame transfer function are varied. At these points, two eigenvalues and the associated eigenfunctions coalesce. The system's sensitivity to changes in the parameters becomes infinite. Two eigenvalue branches collide at the exceptional point as the interaction index is increased. One branch originates from a purely acoustic mode, whereas the other branch originates from an intrinsic thermoacoustic mode. The existence of exceptional points in thermoacoustic systems has implications for physical understanding, computing, modeling and control.

“…The aim of this paper is to reduce such a computational effort by combining first-and second-order adjoint-based eigenvalue sensitivities (Section 2), detailed in Part I of this paper [16], with a standard Monte Carlo method (Section 3) and a Monte Carlo method integrated with Active Subspace Identification (Section 4) to predict the probabilities that two annular-combustor configurations are unstable.…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Computational Physics

Bauerheim

Nicoud

et al. 2016

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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 nonlineareigenproblem calculations by up to ∼ O(M ), as many as the Monte Carlo samples.