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.

Gradient-free optimization of chaotic acoustics with reservoir computing

“…Figure 4 also shows the time series of the two cost functionals, acoustic energy and Rayleigh index. The Rayleigh index oscillates substantially more than the acoustic energy because the time derivative of the acoustic energy is equal to the sum of the Rayleigh index and the dissipation from damping [17]. Time-wise, the hESN is able to time-accurately predict these modes for the whole time span shown, whereas the ESN deviates from the truth signal at t ≈ 10.…”

“…The cost functionals that we wish to obtain and minimize are the time averages of the acoustic energy and the Rayleigh index [17] E ac (t) =…”

“…Hence, W in can be generated by sampling uniformly from the range [−1, 1] and scaled by σ in directly in eq. (17). Similarly, W is generated by sampling from the uniform distribution in the range [−1, 1], with each node being on average connected to (1 − sp)N r other nodes, where sp is the desired sparseness.…”

Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity

“…In recent times, simultaneous advances in simulation capabilities and computing power have led to a proliferation of scale-resolving simulations of chaotic systems [29,30,31,32]. For many of the above target applications, the relevant observables, or outputs, in chaotic systems are statistically stationary or infinitely long-time averaged functions [30,31,32,33,34]. Useful gradients of ensemble averages, which are equal to infinite time averages in ergodic systems, cannot be obtained by time-averaging the instantaneous gradients in chaotic systems [35].…”

“…Unstable thermoacoustic systems have intricate nonlinear behaviours when design parameters are varied, from periodic, through quasi periodic to chaotic oscillations [55]. Although methods to investigate the sensitivity of fixed points (with eigenvalue analysis) and periodic solutions (with Floquet analysis) are well-established [13], a stability and sensitivity framework to tackle chaotic acoustic oscillations is only at its infancy [33,34]. In themoacoustics, sensitivity analysis quantitatively informs the practitioner on how to optimally change design parameters, such as geometric quantities; which passive device is most stabilizing; and how large is the uncertainty of the stability calculations [56,57,58], as reviewed by [13].…”