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Optimisation of chaotically perturbed acoustic limit cycles

Abstract

In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time-averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. By embedding the sensitivities into a gradient-update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification.

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Notes

  1. For example, in large-eddy simulations, the nonlinear operator \({\varvec{F}}\) in the dynamical system formulation in Eq. (4) is provided by the spatial discretisation of the momentum, mass and energy conservation laws and chemistry equations [63].

  2. It has been hypothesised by Gallavotti and Cohen [68], Gallavotti [69] that most high-dimensional physical systems develop asymptotically on an attracting set, the dynamics of which can be regarded as hyperbolic. This is called the chaotic hypothesis, which stems from measure theory of turbulence of [70].

  3. This paper provides a method to optimise deterministic thermoacoustic systems. Including stochastic processes in the optimisation of chaotic thermoacoustic systems is beyond the scope of this paper and is left for future work.

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Acknowledgements

F. Huhn is supported by Fundação para a Ciência e Tecnologia in Portugal under the Research Studentship No. SFRH/BD/134617/2017. L. Magri gratefully acknowledges financial support from the Royal Academy of Engineering Research Fellowships. Fruitful discussions with Prof. Q. Wang, Dr. P. Blonigan, N. Chandramoorthy and A. Ni are gratefully acknowledged. We are grateful to Dr. Patrick Blonigan for his insightful suggestions on using the filtering parameter.

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Uncertainty on the estimation of infinitely time-averaged cost functionals

Uncertainty on the estimation of infinitely time-averaged cost functionals

We wish to estimate the value of an infinitely time-averaged cost functional, \(\langle \mathcal {J}\rangle \), from a collection of N independent samples of the time-averaged cost functional, \(\{\langle \mathcal {J}\rangle _T^{(i)}\}\), where the subscript T and superscript (i) represent the finite time used in the averaging and the index of the sample, respectively. From the Central Limit Theorem, we assume that the error in the estimation of \(\langle \mathcal {J}\rangle \) decays with the number of samples used to the power of \(-1/2\). Thus, we set a nonlinear constrained minimisation problem

$$\begin{aligned} \begin{aligned}&\underset{a, b}{\text {minimise}}&b \\&\text {subject to}&a - \frac{b}{\sqrt{n}}< \frac{1}{n}\sum _{i=1}^n \langle \mathcal {J}\rangle _T^{(i)} < a + \frac{b}{\sqrt{n}} \\&&\quad \quad \quad \quad \quad \quad \quad \; \forall \, n \in \{1, \cdots , N\} \end{aligned}. \end{aligned}$$
(36)

Thus, \(\langle \mathcal {J}\rangle \) should be in the range \(\big (a-b/\sqrt{N}, a+b/\sqrt{N}\big )\).

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Huhn, F., Magri, L. Optimisation of chaotically perturbed acoustic limit cycles. Nonlinear Dyn 100, 1641–1657 (2020). https://doi.org/10.1007/s11071-020-05582-x

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Keywords

  • Acoustics
  • Thermoacoustics
  • Optimization of chaotic systems
  • Time delayed models
  • Shadowing methods