3 results
Nonlinear interaction of shear flow with a free surface
- Athanassios A. Dimas, George S. Triantafyllou
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- Journal:
- Journal of Fluid Mechanics / Volume 260 / 10 February 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 211-246
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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.
Surface ripples due to steady breaking waves
- James H. Duncan, Athanassios A. Dimas
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- Journal:
- Journal of Fluid Mechanics / Volume 329 / 25 December 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 309-339
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Breaking waves generated by a two-dimensional hydrofoil moving near a free surface at constant speed (U∞), angle of attack and depth of submergence were studied experimentally. The measurements included the mean and fluctuating shape of the breaking wave, the surface ripples downstream of the breaker and the vertical distribution of vertical and horizontal velocity fluctuations at a single station behind the breaking waves. The spectrum of the ripples is highly peaked and shows little variation in both its peak frequency and its shape over the first three wavelengths of the wavetrain following the breaker. For a given speed, as the breaker strength is increased, the high-frequency ends of the spectra are nearly identical but the spectral peaks move to lower frequencies. A numerical instability model, in conjunction with the experimental data, shows that the ripples are generated by the shear flow developed at the breaking region. The spectrum of the vertical velocity fluctuations was also found to be highly peaked with the same peak frequency as the ripples, while the corresponding spectrum of the horizontal velocity fluctuations was found not to be highly peaked. The root-mean-square (r.m.s.) amplitude of the ripples (νrms) increases with increasing speed and with decreasing depth of submergence of the hydrofoil, and decreases as x-1/2 with increasing distance x behind the breaker. The quantity (gνrms)/(U∞Vrms) (where Vrms is the maximum r.m.s. vertical velocity fluctuation and g is the gravitational acceleration) was found to be nearly constant for all of the measurements.
Incipient breaking of steady waves in the presence of surface wakes
- MATTHEW MILLER, TOBIAS NENNSTIEL, JAMES H. DUNCAN, ATHANASSIOS A. DIMAS, STEPHAN PRÖSTLER
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- Journal of Fluid Mechanics / Volume 383 / 25 March 1999
- Published online by Cambridge University Press:
- 25 March 1999, pp. 285-305
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The effect of free-surface drift layers on the maximum height that a steady wave can attain without breaking is explored through experiments and numerical simulations. In the experiments, the waves are generated by towing a two-dimensional fully submerged hydrofoil at constant depth, speed and angle of attack. The drift layer is generated by towing a plastic sheet on the water surface ahead of the hydrofoil. It is found that the presence of this drift layer (free-surface wake) dramatically reduces the maximum non-breaking wave height and that this wave height correlates well with the surface drift velocity. In the simulations, the inviscid two-dimensional fully nonlinear Euler equations are solved numerically. Initially symmetric wave profiles are superimposed on a parallel drift layer whose mean flow characteristics match those in the experiments. It is found that for large enough initial wave amplitudes a bulge forms at the crest on the forward face of the wave and the vorticity fluctuations just under the surface in this region grow dramatically in time. This behaviour is taken as a criterion to indicate impending wave breaking. The maximum non-breaking wave elevations obtained in this way are in good agreement with the experimental findings.