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The state space and travelling-wave solutions in two-scale wall-bounded turbulence

Published online by Cambridge University Press:  30 August 2022

Patrick Doohan Open the ORCID record for Patrick Doohan [Opens in a new window] ,
Yacine Bengana Open the ORCID record for Yacine Bengana [Opens in a new window] ,
Qiang Yang Open the ORCID record for Qiang Yang [Opens in a new window] ,
Ashley P. Willis Open the ORCID record for Ashley P. Willis [Opens in a new window]  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Patrick Doohan
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Yacine Bengana
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Qiang Yang
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Ashley P. Willis
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
Yongyun Hwang*
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: y.hwang@imperial.ac.uk
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Abstract

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.

JFM classification

Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
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Journal of Fluid Mechanics , Volume 947 , 25 September 2022 , A41
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© The Author(s), 2022. Published by Cambridge University Press

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