Degenerate perturbation theory in thermoacoustics: high-order sensitivities and exceptional points

Orchini, Alessandro; Magri, Luca; Silva, Camilo F.; Mensah, Georg A.; Moeck, Jonas P.

doi:10.1017/jfm.2020.586

Cited by 9 publications

(9 citation statements)

References 71 publications

(185 reference statements)

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“…By using adjoint-based perturbation theory, it is possible to obtain a power series representation of the dependency of the eigenvalue trajectories on the parameter that, within the region of convergence, accurately approximates the actual eigenvalues. This was demonstrated recently by and Orchini et al (2020) in a compressible-flow configuration relevant for thermoacoustic applications in which the base flow was frozen. It is, however, known that a variation in the base flow affects the stability of the flow (see e.g.…”

Section: Eigenvalue Sensitivity Including Base Flow Variationsmentioning

confidence: 65%

“…The EPs are spectral singularities at which not only two (or more) eigenvalues coalesce, becoming degenerate, but also two (or more) eigenvectors coalesce, causing the linear operator to be defective. The EPs are expected to be found for complex-valued parameters, making also the linear operator complex-valued, implying that they are often not physically realizable (Seyranian, Kirillov & Mailybaev 2005). Still, their existence in the complex plane poses limits to the convergence of perturbation expansions.…”

Section: Arbitrary-order Sensitivities Of the Eigenvaluesmentioning

confidence: 99%

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Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

Knechtel,

Kaiser,

Orchini

et al. 2024

J. Fluid Mech.

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First-order sensitivities and adjoint analysis are used widely to control the linear stability of unstable flows. Second-order sensitivities have recently helped to increase accuracy. In this paper, a method is presented to calculate arbitrary high-order sensitivities based on Taylor expansions of the incompressible base flow and its eigenproblem around a scalar parameter. For the incompressible Navier–Stokes equations, general expressions for the sensitivities are derived, into which parameter-specific information can be inserted. The computational costs are low since, for all orders, a linear equation system has to be solved, of which the left-hand-side matrix stays constant and thus its preconditioning can be exploited. Two flow scenarios are examined. First, the cylinder flow equations are expanded around the inverse of the Reynolds number, enabling the prediction of the two-dimensional cylinder base flow and its leading eigenvalue as a function of the Reynolds number. This approach computes accurately the base flow and eigenvalue even in the unstable regime, providing, when executed subsequently, a mean to calculate unstable base flows. This case gives a clear introduction into the method and allows us to discuss its constraints regarding convergence behaviour. Second, a small control cylinder is introduced into the domain of the cylinder flow for stabilization. Higher-order sensitivity maps are calculated by modelling the small cylinder with a steady forcing. These maps help to identify stabilizing areas of the flow field for Reynolds numbers within the laminar vortex shedding regime, with the required number of orders increasing as the Reynolds number rises. The results obtained through the proposed method align well with numerically calculated eigenvalues that incorporate the cylinder directly into the grid.

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Section: Eigenvalue Sensitivity Including Base Flow Variationsmentioning

confidence: 65%

Section: Arbitrary-order Sensitivities Of the Eigenvaluesmentioning

confidence: 99%

Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

Knechtel,

Kaiser,

Orchini

et al. 2024

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

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“…2018; Orchini et al. 2020). In the nonlinear regime, small changes to the system's parameters can cause a variety of nonlinear bifurcations of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…Third, thermoacoustics is sensitive to perturbations to the system. In the linear regime, small changes to the system's parameters, such as the flame time delay, can cause arbitrarily large changes of the eigenvalue growth rates at exceptional points (Mensah et al 2018;Orchini et al 2020). In the nonlinear regime, small changes to the system's parameters can cause a variety of nonlinear bifurcations of the solution.…”

Section: Introductionmentioning

confidence: 99%

Real-time thermoacoustic data assimilation

Andrea

Magri

2022

J. Fluid Mech.

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Low-order thermoacoustic models are qualitatively correct, but typically, they are quantitatively inaccurate. We propose a time-domain bias-aware method to make qualitatively low-order models quantitatively (more) accurate. First, we develop a Bayesian ensemble data assimilation method for a low-order model to self-adapt and self-correct any time that reference data become available. Second, we apply the methodology to infer the thermoacoustic states and heat-release parameters on the fly without storing data (real time). We perform twin experiments using synthetic acoustic pressure measurements to analyse the performance of data assimilation in all nonlinear thermoacoustic regimes, from limit cycles to chaos, and interpret the results physically. Third, we propose practical rules for thermoacoustic data assimilation. An increase, reject, inflate strategy is proposed to deal with the rich nonlinear behaviour; and physical time scales for assimilation are proposed in non-chaotic regimes (with the Nyquist–Shannon criterion) and in chaotic regimes (with the Lyapunov time). Fourth, we perform data assimilation using data from a higher-fidelity model. We introduce an echo state network to estimate in real time the forecast bias, which is the model error of the low-fidelity model. We show that: (i) the correct acoustic pressure, parameters, and model bias can be inferred accurately; (ii) the learning is robust as it can tackle large uncertainties in the observations (up to 50 % of the mean values); (iii) the uncertainty of the prediction and parameters is naturally part of the output; and (iv) both the time-accurate solution and statistics can be inferred successfully. Data assimilation opens up new possibility for real-time prediction of thermoacoustics by combining physical knowledge and experimental data synergistically.

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“…Cremer's concept received little further attention until quite recently when a number of papers exploring the idea in the context of noise attenuation within circular and/or annular ducts with flow [8][9][10][11][12][13] appeared in the literature. It is worth noting that EPs also occur in phenomena such as zeros in group velocity [14] or instability in, for example, thermoacoustics [15]. However, in all contexts, exact mathematical treatment requires some care since, close to the EP, the usual set of eigenmodes do not form a complete basis, and this is manifest as degeneracy in the mathematical system under-pinning the scattering process.…”

Section: Introductionmentioning

confidence: 99%

Analytic mode-matching for accurate handling of exceptional points in a lined acoustic waveguide

2022

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Exceptional points (which occur when two or more modes coalesce) have long been associated with optimal attenuation in lined acoustic waveguides. In recent years, with a view to optimizing sound absorption, some effort has gone into designing liners that generate exceptional points (EPs) at specified frequencies. However, analytic modelling of acoustic scattering in the presence of an EP is not well developed, with most authors relying on standard methods applied close to (but not at) EP conditions. Indeed, exact treatment requires care since the mathematical system under-pinning the scattering process is degenerate. This article presents an analytic mode-matching approach to modelling the scattering of a plane wave travelling towards the junction of a rigid duct with a lined duct at EP conditions. Both EP2 and EP3 (coalescence of two and three modes respectively) are considered. The enhanced mode-matching scheme is shown to be valid and numerically robust, and it is anticipated that it will be straightforward to adapt to a wide range of applications involving complex symmetric operators.

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Degenerate perturbation theory in thermoacoustics: high-order sensitivities and exceptional points

Abstract: Abstract

Cited by 9 publications

References 71 publications

Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

Real-time thermoacoustic data assimilation

Analytic mode-matching for accurate handling of exceptional points in a lined acoustic waveguide

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