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Kelvin–Helmholtz instability with a free surface

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Abstract

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.

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Authors and Affiliations

  1. LAMA, UMR5127 CNRS, 73376, Le Bourget du Lac cedex, France

    Didier Bresch

  2. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061-0123, USA

    Michael Renardy

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  1. Didier Bresch
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  2. Michael Renardy
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Correspondence to Michael Renardy.

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This research was supported by Agence Nationale de La Recherche under Grant ANR-08-BLAN-0301-01 and by the National Science Foundation under Grant DMS-1008426.

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Bresch, D., Renardy, M. Kelvin–Helmholtz instability with a free surface. Z. Angew. Math. Phys. 64, 905–915 (2013). https://doi.org/10.1007/s00033-012-0270-4

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  • DOI: https://doi.org/10.1007/s00033-012-0270-4

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