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Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Published online by Cambridge University Press:  20 March 2024

Matthew McCormack Open the ORCID record for Matthew McCormack [Opens in a new window] ,
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window]  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Matthew McCormack*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
André V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
Yongyun Hwang
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: matthew.mccormack@ed.ac.uk
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Abstract

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

JFM classification

Turbulent Flows: Turbulent boundary layers Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 983 , 25 March 2024 , A33
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© The Author(s), 2024. Published by Cambridge University Press

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