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Transition to chaos in a reduced-order model of a shear layer

Published online by Cambridge University Press:  14 December 2021

André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Erico L. Rempel Open the ORCID record for Erico L. Rempel [Opens in a new window]  and
Petrônio A.S. Nogueira Open the ORCID record for Petrônio A.S. Nogueira [Opens in a new window]
André V.G. Cavalieri*
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil
Erico L. Rempel
Affiliation:
Divisão de Ciências Fundamentais, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil
Petrônio A.S. Nogueira
Affiliation:
Department of Mechanical and Aerospace Engineering, Laboratory for Turbulence Research in Aerospace and Combustion, Monash University, Clayton, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$, leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Chaos Acoustics: Jet noise
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A43
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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