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Streamwise-localised, symmetric invariant solutions in square-duct flow

Published online by Cambridge University Press:  07 August 2025

Stanisław Wojciech Gepner Open the ORCID record for Stanisław Wojciech Gepner [Opens in a new window] ,
Shinya Okino Open the ORCID record for Shinya Okino [Opens in a new window]  and
Genta Kawahara Open the ORCID record for Genta Kawahara [Opens in a new window]
Stanisław Wojciech Gepner*
Affiliation:
Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics, Nowowiejska 24, Warsaw 00-665, Poland
Shinya Okino
Affiliation:
Graduate School of Engineering, Kyoto University, 4 Kyoto daigaku-katsura, Nishikyo, Kyoto 615-8540, Japan Department of Mechanical Engineering, Tokyo Denki University, 5 Senju Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan
Genta Kawahara
Affiliation:
Graduate School of Engineering Science, University of Osaka, 1–3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
*
Corresponding author: Stanisław Wojciech Gepner, stanislaw.gepner@pw.edu.pl
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Abstract

Flow through a square-duct at a moderate Reynolds number is investigated. We first employ an edge-tracking procedure in the $\pi$-rotationally symmetric sub-space of state space and identify a streamwise-localised invariant solution for square-duct flow, which is a steady travelling wave with mirror symmetries across bisectors of the duct walls. The identified invariant solution features four vortices placed in pairs at opposite duct walls and exhibits significant streamwise localisation making it the first reported localised solution in the square-duct flow. Additionally, this solution remains very close to the laminar attractor in the sense of the velocity perturbation energy and the corresponding hydraulic losses. Stability analysis of this solution demonstrates that the identified state is an edge state in the $\pi$-rotationally symmetric sub-space but not in the full space. Next, a long-time turbulence behaviour and its relevance to the symmetric streamwise-localised invariant solution are discussed. We focus on the characteristics of the averaged flow and the recurring patterns of eight- or four-vortex states, typical for the square-duct flow and related to Prandtl’s secondary flows of the second type. Through heuristic arguments, we illustrate that turbulent flow exhibits relatively quiescent interludes of increased symmetry of the velocity field across wall bisectors. We show that those periods correlate to episodes where, statistically, a four-vortex flow configuration emerges from the otherwise eight-vortex state, which is also associated with decreased symmetry of the flow field. Our results suggest that the four-vortex state appearing in the relatively quiescent periods in the flow time history, accompanied by flow field symmetrisation and the onset of streamwise localisation of turbulent flow, bears a striking similarity to the found symmetric streamwise-localised invariant solution.

JFM classification

Turbulent Flows: Turbulent transition

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1016 , 10 August 2025 , A54
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© The Author(s), 2025. Published by Cambridge University Press

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