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Generalized stability of a shear flow with a free surface with respect to three-dimensional perturbations

Christos Mallios and Nikolaos A. Bakas*

  • Laboratory of Meteorology, Department of Physics, University of Ioannina, Ioannina 45110, Greece

  • *nbakas@uoi.gr
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Phys. Rev. Fluids 2, 023901 – Published 6 February, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.023901

Abstract

Modal and nonmodal growth of three-dimensional perturbations in a shear flow with a free surface are examined for a wide range of Froude numbers. By approximating the mean flow with a piecewise linear profile, the modal instabilities are shown to arise from the interaction of three-dimensional edge waves supported at the interfaces of density discontinuity at the surface and mean vorticity discontinuity at the edges of the shear layer. The mechanism and properties of the instability are explained in terms of the dynamics of the edge-wave interactions. Previously reported modal stability analysis restricted to two-dimensional perturbations in the plane of the flow accurately predicts the fastest growing perturbations but underestimates the range of length scales for the unstable structures. Robust nonmodal transient growth of perturbations within a few advective time units is found. For low Froude numbers or low values of the shear, three-dimensional perturbations with small horizontal scales exhibit the largest growth through a synergy between the Orr and the lift-up mechanisms and produce large streamwise streaks in the shear flow without an effect on the free surface. For large Froude numbers or large values of the shear, planar perturbations with larger horizontal scales exhibit the largest energy growth by effectively instigating the modal instability and excite surface waves at large amplitude.

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Phys. Rev. Fluids 2, 023901– Published 6 February, 2017
  • Received 10 July 2016

©2017 American Physical Society

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