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Nonlinear interaction of shear flow with a free surface

Published online by Cambridge University Press:  26 April 2006

Athanassios A. Dimas  and
George S. Triantafyllou
Athanassios A. Dimas
Affiliation:
The Benjamin Levich Institute and Department of Mechanical Engineering, The City College of CUNY, New York, NY 10031, USA
George S. Triantafyllou
Affiliation:
The Benjamin Levich Institute and Department of Mechanical Engineering, The City College of CUNY, New York, NY 10031, USA
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Abstract

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution.

Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break.

Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit.

It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 260 , 10 February 1994 , pp. 211 - 246
Copyright
© 1994 Cambridge University Press

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References

Bernal, L. P. & Kwon, J. T. 1989 Vortex ring dynamics at a free surface. Phys. Fluids A 1, 449–451.Google Scholar
Dimas, A. A. 1991 Nonlinear interaction of shear flows with the free surface PhD Thesis, Massachusetts Institute of Technology.
Dommermuth, D. G. & Yue, D. K.-P. 1990 A numerical study of three-dimensional viscous interactions of vortices with a free surface. In Proc. 18th Symp. on Naval Hydrodyn. pp. 727–788. National Academy of Sciences, Washington, DC.
Dommermuth, D. G., Yue, D. K.-P., Lin, W. M., Rapp, R. J., Chan, E. S. & Melville, W. K. 1988 Deep-water plunging breakers: a comparison between potential theory and experiments. J. Fluid Mech. 189, 423–442.Google Scholar
Gotlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Haidvogel, D. B., Robinson, A. R. & Schulman, E. E. 1980 The accuracy, efficiency, and stability of three numerical models with application to open ocean problems. J. Comput. Phys. 34, 1–53.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51, 233–272.Google Scholar
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J. 1987 Secondary instability of a temporally growing mixing layer. J. Fluid Mech. 184, 207–243.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech 19, 543–556.Google Scholar
Milinazzo, F. A. & Saffman, P. G. 1990 Effect of a surface shear layer on gravity and gravity–capillary waves of permanent form. J. Fluid Mech. 216–93–101.Google Scholar
Myers, R. B., Taylor, T. D. & Murdock, J. W. 1981 Pseudo-spectral simulation of a two-dimensional vortex flow in a stratified, incompressible fluid. J. Comput. Phys. 43, 180–188.Google Scholar
Ohring, S. & Lugt, H. J. 1991 Interaction of a viscous vortex pair with a free surface. J. Fluid Mech. 227, 47–70.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. hys. 54, 468–488.Google Scholar
Patnaik, P. C., Sherman, F. S. & Corcos, G. M. 1976 A numerical simulation of Kelvin–Helmholtz waves of finite amplitude. J. Fluid Mech. 73, 215–240.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Sarpkaya, T. 1986 Trailing-vortex wakes on the free surface. In Proc. 16th Symp. on Naval Hydrodyn., pp. 38–50. National Academy Press.
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Triantafyllou, G. S. & Dimas, A. A. 1989 Interaction of two-dimensional separated flows with a free surface at low Froude numbers. Phys. Fluids A 1, 1813–1821.Google Scholar
Tryggvason, G. 1988 Deformation of a free surface as a result of vortical flows. Phys. Fluids 31, 955–957.Google Scholar
Willmarth, W. W., Tryggvason, G., Hirsa, A. & Yu, D. 1989 Vortex pair generation and interaction with a free surface. Phys. Fluids A 1, 170–172.Google Scholar
Yu, D. & Tryggvason, G. 1990 The free-surface signature of unsteady, two-dimensional vortex flows. J. Fluid Mech. 218, 547–572.Google Scholar