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Exact coherent states in plane Couette flow under spanwise wall oscillation

Published online by Cambridge University Press:  15 August 2022

Yacine Bengana*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
Qiang Yang
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Guohua Tu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Yongyun Hwang
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
*
Email addresses for correspondence: y.bengana14@imperial.ac.uk, ghtu@skla.cardc.cn
Email addresses for correspondence: y.bengana14@imperial.ac.uk, ghtu@skla.cardc.cn

Abstract

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$. It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$. The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

Type
JFM Papers
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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Bengana et al. supplementary movie 1

\item \textbf{Caption for Movie 1} Dynamical variation of the streamwise-averaged low-speed (blue contour) and high-speed (red contour) streaks and in-plan vortex structure (pink arrows) during one spanwise wall oscillation period for case A ($Re=400$, $T^+=166.4$, $A_w^+=2.6$). The streaks are visualised by the contour of $\langle u' \rangle _x$, and the vortex structure is visualised by the vector of $\left(\langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z}\right)$.

Download Bengana et al. supplementary movie 1(Video)
Video 4 MB

Bengana et al. supplementary movie 2

\item \textbf{Caption for Movie 2} Dynamical variation of the streamwise-averaged low-speed (blue contour) and high-speed (red contour) streaks and in-plan vortex structure (pink arrows) during one spanwise wall oscillation period for case B ($Re=400$, $T^+=166.4$, $A_w^+=4.4$). The streaks are visualised by the contour of $\langle u' \rangle _x$, and the vortex structure is visualised by the vector of $\left(\langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z}\right)$.

Download Bengana et al. supplementary movie 2(Video)
Video 3 MB

Bengana et al. supplementary movie 3

\item \textbf{Caption for Movie 3} Exact coherent state under spanwise wall oscillation for case A ($Re=400$, $T^+=166.4$, $A_w^+=2.6$). The time variation during one spanwise wall oscillation period is shown. Yellow iso-surface ($u'^+=2.0$) indicates the high-speed streak, and green iso-surface ($v'^+=-0.2$) represents the vortical structure. Viewed from the wall into the flow.

Download Bengana et al. supplementary movie 3(Video)
Video 1 MB

Bengana et al. supplementary movie 4

\item \textbf{Caption for Movie 4} Exact coherent state under spanwise wall oscillation for case B ($Re=400$, $T^+=166.4$, $A_w^+=4.4$). The time variation during one spanwise wall oscillation period is shown. Yellow iso-surface ($u'^+=2.0$) indicates the high-speed streak, and green iso-surface ($v'^+=-0.1$) represents the vortical structure. Viewed from the wall into the flow.

Download Bengana et al. supplementary movie 4(Video)
Video 871 KB
Supplementary material: PDF

Bengana et al. supplementary material

Captions for movies 1-4

Download Bengana et al. supplementary material(PDF)
PDF 75 KB

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Exact coherent states in plane Couette flow under spanwise wall oscillation
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