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Exact coherent states with hairpin-like vortex structure in channel flow

Published online by Cambridge University Press:  15 June 2018

Ashwin Shekar  and
Michael D. Graham Open the ORCID record for Michael D. Graham [Opens in a new window]
Ashwin Shekar
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA
Michael D. Graham*
Affiliation:
Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: mdgraham@wisc.edu
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Abstract

Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier–Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number $Re=666$. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At $Re=1800$, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

JFM classification

Boundary Layers: Boundary layer structure Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Turbulent Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 76 - 89
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Copyright
© 2018 Cambridge University Press 

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