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Spanwise-localized solutions of planar shear flows

Published online by Cambridge University Press:  17 March 2014

J. F. Gibson  and
E. Brand
J. F. Gibson*
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
E. Brand
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present several new spanwise-localized equilibrium and travelling-wave solutions of plane Couette and channel flows. The solutions exhibit concentrated regions of vorticity that are centred over low-speed streaks and flanked on either side by high-speed streaks. For several travelling-wave solutions of channel flow, the vortex structures are concentrated near the walls and form particularly isolated and elemental versions of coherent structures in the near-wall region of shear flows. One travelling wave appears to be the invariant solution corresponding to a near-wall coherent structure educed from simulation data by Jeong et al. (J. Fluid Mech., vol. 332, 1997, pp. 185–214) and analysed in terms of transient growth modes of streaky flow by Schoppa & Hussain (J. Fluid Mech., vol. 453, 2002, pp. 57–108). The solutions are constructed by a variety of methods: application of windowing functions to previously known spatially periodic solutions, continuation from plane Couette to channel flow conditions, and from initial guesses obtained from turbulent simulation data. We show how the symmetries of localized solutions derive from the symmetries of their periodic counterparts, analyse the exponential decay of their tails, examine the scale separation and scaling of their streamwise Fourier modes, and show that they develop critical layers for large Reynolds numbers.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014 , pp. 25 - 61
Copyright
© 2014 Cambridge University Press 

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