Weakly nonlinear oscillations of gas column driven by self-sustained sources

Viktor Hruška,Michal Bednarřík,Milan Červenka,V Goussev,J Yin

doi:10.1051/matecconf/201928306001

Viktor Hruška, Michal Bednarřík + Show 3 more

Open Access

https://doi.org/10.1051/matecconf/201928306001

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Abstract

Self-sustained sources coupled to some sort of resonator have drawn attention recently as a subject of nonlinear dynamics with many practical applications as well as interesting mathematical problems from the chaos theory and the theory of synchronizations. In order to mimic the self-sustainability arising from physical background the van der Pol equation is commonly used as a model (e.g. vortex induced noise, flowstructure interactions, vocal folds motion etc.). In many cases the sound field inside the resonator is strong enough for weakly nonlinear formulation based on the Kuznetsov model equation to be employed. An array of sources governed by the inhomogeneous van der Pol equation coupled to the nonlinear acoustic wave equation is studied. The one dimensional constant cross-section open resonator with zero radiation impedance is assumed. The focus is on the main features such as mode-locking, harmonics generation and build-up from infinitesimal fluctuations.

Highlights

Many complex phenomena from the fields of aeroacoustics and fluid-structure interactions involve self-sustained sound sources [1,2,3,4,5]
We expect that the self-suistained source should be able to start oscillations only from omnipresent infinitesimal fluctuations and further that there should be some sort of saturation mechanism so the oscillations do not grow infintely
From the perspective of nonlinear acoustics we shall remain within the weakly nonlinear formulation

Summary

Introduction

Many complex phenomena from the fields of aeroacoustics and fluid-structure interactions involve self-sustained sound sources [1,2,3,4,5]. We expect that the self-suistained source should be able to start oscillations only from omnipresent infinitesimal fluctuations and further that there should be some sort of saturation mechanism so the oscillations do not grow infintely (note that this saturation should be a feature of the oscillator, not the acoustic medium). The both conditions are satisfied when the van der Pol oscillator is used. We choose the local acoustic velocity to be the quantity mediating the feedback The reason for this choice is e.g. the Howe’s formula (Howe’s energy corollary) describing the feedback effect of the existing sound field on the generated vortex sound power The reason for this choice is e.g. the Howe’s formula (Howe’s energy corollary) describing the feedback effect of the existing sound field on the generated vortex sound power (see e.g. [8])

Theory

Numerical experiment

Discussion

Summary