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Shear stress-driven flow: the state space of near-wall turbulence as $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$

Published online by Cambridge University Press:  11 July 2019

Patrick Doohan Open the ORCID record for Patrick Doohan [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
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: [email protected]
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Abstract

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height $y^{+}<40$. A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$, the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high-$Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.

JFM classification

Turbulent Flows: Turbulent Flows Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 606 - 638
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Copyright
© 2019 Cambridge University Press 

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