Steady-state statistics, emergent patterns and intermittent energy transfer in a ring of oscillators

Tiemo Pedergnana,Nicolas Noiray

doi:10.1007/s11071-022-07275-z

Abstract

Networks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker–Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for the intermittent energy transfer between Bloch waves observed in acoustic pressure spectrograms observed of real-world gas turbines.

Highlights

1.1 Thermoacoustic instabilities in can-annular combustorsIn a rigid enclosed volume, the heat release rate response of an compact unsteady flame to acoustic perturbations forms a feedback loop with the acoustics of the enclosure
We propose a symmetric model where all reactive effects of the flame response and the aeroacoustic coupling are neglected, and only resistive effects are taken into account. We show that this simple model is able to describe a wide variety of possible emergent patterns, including synchronization, rotating waves as well as quasi-steady superpositions of clockwise (CW) and counterclockwise (CCW) waves, and that it reproduces the intermittent energy transfer between different Bloch mode components of the acoustic pressure observed in real-world gas turbines
Due to the symmetry of the system considered in this work, we find that different emergent patterns occur at similar amplitudes and are mainly distinguished by different phase dynamics

Summary

Introduction

1.1 Thermoacoustic instabilities in can-annular combustorsIn a rigid enclosed volume, the heat release rate response of an compact unsteady flame to acoustic perturbations forms a feedback loop with the acoustics of the enclosure. At a given condition, this interaction between the sound field and the flame exceeds the dissipation due to radiation losses and viscous effects in the fluid, occurring mainly at the boundary of the enclosure, it can give rise to so-called thermoacoustic instabilities. The study of this physical phenomenon goes back to the work of Rayleigh [1]. When an instability is insufficiently damped, it can lead to high-amplitude acoustic pressure oscillations in the enclosure. Applied studies on Helmholtz dampers to suppress thermoacoustic instabilities in gas turbine combustors are found in Refs. Synchronization phenomena and nonlinear energy pumping in two coupled thermoacoustic oscillators are studied experimentally in [8]

Objectives

Findings

Discussion

Conclusion