Exceptional points in the thermoacoustic spectrum

Mensah, Georg A.; Magri, Luca; Silva, Camilo F.; Buschmann, Philip E.; Moeck, Jonas P.

doi:10.1016/j.jsv.2018.06.069

Cited by 32 publications

(38 citation statements)

References 11 publications

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“…Although not correlated with exceptional points, these regions of large sensitivities in the complex plane were identified previously by Juniper & Sujith (2018). At the exceptional points the eigenvalue sensitivities can no longer be evaluated by a first-order expansion (Mensah et al 2018). Further implications for the modelling of the system at such parameter configurations are still open and unresolved.…”

Section: Introductionmentioning

confidence: 75%

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

2019

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This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

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Section: Introductionmentioning

confidence: 75%

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

2019

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“…In linear eigenproblems, the algebraic multiplicities add up to the dimension of the problem. However, in NEPs there may exist an infinite number of eigenvalues, and an eigenvalue may have any algebraic multiplicity greater than the dimension of the problem 23 ; • An eigenvalue is semi-simple if a = g. If a = 1, the eigenvalue is simple; • An eigenvalue is defective if a > g. An important class of defective eigenvalues are branch-point solutions of the characteristic function 24 , which are known as exceptional points [246,247]. Exceptional points have infinite sensitivity to infinitesimal perturbations to the system (Fig.…”

Section: Features Of Nonlinear Eigenproblemsmentioning

confidence: 99%

“…In thermoacoustics, simple gradient-based iteration algorithms are typically utilized (e.g., [40]). Contour integration was applied to solve dispersion relations in [250] and to find all the defective and non-defective eigenvalues in a given circle of the complex plane in [247,251]. The contour integration method proved numerically more robust and stable than gradient-based iteration methods, with the advantage of finding all the eigenvalues in a defined domain in the complex plane;…”

Section: Features Of Nonlinear Eigenproblemsmentioning

confidence: 99%

“…Although beyond the scope of this review paper, it is interesting to characterize the sensitivity of defective branch-point eigenvalues because their rate of change is infinitely larger than the rate of change of the parameters. In other words, the thermoacoustic stability at an exceptional point is infinitely more sensitive to perturbations than the thermoacoustic stability of a non-defective system [163,247]. 35 Private communication with G. Mensah and A.…”

Section: Sensitivity To One Parametermentioning

confidence: 99%

“…Exceptional points. As thoroughly explained in [247], the set of parameters and the relevant eigenvalue at which intrinsic and acoustic modes cross each other are an exceptional point: The eigenvalue is a branch-point solution of the characteristic function. The eigenvalue at this exceptional point is two-fold defective, has infinite sensitivity to firstorder perturbations and the behaviour in its neighbourhood can be described by a Puiseux series (see footnote 34).…”

Section: The Thermoacoustic Spectrummentioning

confidence: 99%

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Adjoint Methods as Design Tools in Thermoacoustics

Magri

2019

Applied Mechanics Reviews

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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.

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Passive control of combustion instabilities

Zhao¹

2023

Thermoacoustic Combustion Instability Control

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Exceptional points in the thermoacoustic spectrum

Cited by 32 publications

References 11 publications

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

Adjoint Methods as Design Tools in Thermoacoustics

Passive control of combustion instabilities

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