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Stochastic Lagrangian dynamics of vorticity. Part 2. Application to near-wall channel-flow turbulence

Published online by Cambridge University Press:  19 August 2020

Gregory L. Eyink Open the ORCID record for Gregory L. Eyink [Opens in a new window] ,
Akshat Gupta  and
Tamer A. Zaki Open the ORCID record for Tamer A. Zaki [Opens in a new window]
Gregory L. Eyink*
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD21218, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD21218, USA Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD21218, USA
Akshat Gupta
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD21218, USA
Tamer A. Zaki
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

We use an online database of a turbulent channel-flow simulation at $Re_\tau =1000$ (Graham et al. J. Turbul., vol. 17, issue 2, 2016, pp. 181–215) to determine the origin of vorticity in the near-wall buffer layer. Following an experimental study of Sheng et al. (J. Fluid Mech., vol. 633, 2009, pp.17–60), we identify typical ‘ejection’ and ‘sweep’ events in the buffer layer by local minima/maxima of the wall stress. In contrast to their conjecture, however, we find that vortex lifting from the wall is not a discrete event requiring $\sim$1 viscous time and $\sim$10 wall units, but is instead a distributed process over a space–time region at least $1\sim 2$ orders of magnitude larger in extent. To reach this conclusion, we exploit a rigorous mathematical theory of vorticity dynamics for Navier–Stokes solutions, in terms of stochastic Lagrangian flows and stochastic Cauchy invariants, conserved on average backward in time. This theory yields exact expressions for vorticity inside the flow domain in terms of vorticity at the wall, as transported by viscous diffusion and by nonlinear advection, stretching and rotation. We show that Lagrangian chaos observed in the buffer layer can be reconciled with saturated vorticity magnitude by ‘virtual reconnection’: although the Eulerian vorticity field in the viscous sublayer has a single sign of spanwise component, opposite signs of Lagrangian vorticity evolve by rotation and cancel by viscous destruction. Our analysis reveals many unifying features of classical fluids and quantum superfluids. We argue that ‘bundles’ of quantized vortices in superfluid turbulence will also exhibit stochastic Lagrangian dynamics and satisfy stochastic conservation laws resulting from particle relabelling symmetry.

JFM classification

Vortex Flows: Vortex dynamics Turbulent Flows: Turbulent boundary layers Complex Fluids: Quantum fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A3
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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