Nonlinear interaction of shear flow with a free surface

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.

Short surface waves on surface shear

“…Surface shear flows with sech type profiles A well-studied case is the linear instability of a shear flow with a continuous velocity profile of the form U = U o sech 2 (bz). The shear flow with this profile was numerically studied by Triantyfallou & Dimas (1989) and Dimas & Triantyfallou (1994), and was found to fit experimental data for the shear flow in the wake of a hydrofoil. For the linear instability analysis, the neutral modes form the stability boundaries.…”

“…The instability of a horizontal shear flow with a free surface has important applications in the study of the surface wakes of ships or near-surface bodies (Dimas & Triantyfallou 1994); of the stability of the crest of a spilling wave breaker (Coakley & Duncan 1997); and of wind-drift currents (Stern & Adam 1973;Morland, Saffman & Yuen 1991). There are only a few cases where an analytic solution can be found; thus, the solutions of realistic surface shear profiles largely depend on elaborate numerical calculations.…”

Large-Wave Simulation (LWS) of Free-Surface Flows Developing Weak Spilling Breaking Waves

“…The nonlinear growth of the instability of this shear flow and its free-surface manifestation develop spilling breakers due to an increasingly steep freesurface slope with a small amplitude. This flow has been studied by Dimas and Triantafyllou [6], who show that DENS is unable to continue past the breaking point. No experimental measurements are available for this flow.…”

“…In terms of numerical simulations of free-surface flows, though, most present methodologies are unable to numerically predict the evolution of spilling breakers past the breaking point. The development of breaking waves is associated with a continuous slope increase of the free-surface elevation toward an infinite slope and overturning, which causes floating-point problems to numerical methods [5,6] that require the resolution of all the free-surface flow scales during and after breaking. The most promising method, so far, has been the surface marker and micro-cell (SMMC) method by Chen et al [3] that overcomes problems with free-surface overturnings but is also based on the requirement that all flow scales are resolved during and after a wave breaking event.…”

Statistical properties of nonlinear one-dimensional wave fields

“…This system of coordinates is unsteady and non-orthogonal, so equations of motion become complicated. Still, this method has been applied for the simulation of wave interaction with a shear flow (Dimas and Triantafyllou, 1994). Evidently, this approach may be combined with the MAC method, applied locally in the intervals with large steepness.…”