[1] The unidirectional, subcritical flow over fixed dunes is studied numerically using large-eddy simulation, while the immersed boundary method is implemented to incorporate the bed geometry. Results are presented for a typical dune shape and two Reynolds numbers, Re = 17,500 and Re = 93,500, on the basis of bulk velocity and water depth. The numerical predictions of velocity statistics at the low Reynolds number are in very good agreement with available experimental data. A primary recirculation region develops downstream of the dune crest at both Reynolds numbers, while a secondary region develops at the toe of the dune crest only for the low Reynolds number. Downstream of the reattachment point, on the dune stoss, the turbulence intensity in the developing boundary layer is weaker than in comparable equilibrium boundary layers. Coherent vortical structures are identified using the fluctuating pressure field and the second invariant of the velocity gradient tensor. Vorticity is primarily generated at the dune crest in the form of spanwise ''roller'' structures. Roller structures dominate the flow dynamics near the crest, and are responsible for perturbing the boundary layer downstream of the reattachment point, which leads to the formation of ''horseshoe'' structures. Horseshoe structures dominate the near-wall dynamics after the reattachment point, do not rise to the free surface, and are distorted by the shear layer of the next crest. The occasional interaction between roller and horseshoe structures generates tube-like ''kolk'' structures, which rise to the free surface and persist for a long time before attenuating.

It is shown that the low Froude number wake of floating two-dimensional objects is convectively unstable. This is shown to be true for bluff objects, like a circular cylinder, and for streamlined objects, like a thin airfoil. As a result, the wake behind a floating object remains steady, even at high Reynolds numbers, characterized by a long region of recirculating flow. It is concluded that the presence of the free surface has a stabilizing effect at low Froude numbers, suppressing the unsteadiness of the wake. The structure of the dispersion relation, shows that, at low Froude numbers, a short wavelength interaction between the wake instability and ambient waves is possible in the form of a spatially growing response.

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.

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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.

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.

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The linear, inviscid, spatial instability of a mixing layer uniformly laden with a dilute concentration of heavy particles is studied numerically. The effect of the particles is modeled using an ensemble averaged Eulerian description of the velocity field and Stokes’ drag formula to compute an averaged force, and the carrier fluid and the particle motions are assumed to be fully coupled. The behavior of the linear instability (for a given mean shear) depends on two dimensionless parameters: Cf, representing the product of the inverse Stokes number and mass loading, and Cp, representing the inverse Stokes number. For finite values of Cf and large values of Cp, the particles respond as fluid elements and the growth rate is equal to the one of the single-phase flow, while decreasing Cp results in a growth rate decrease. The growth rate also decreases with increasing Cf. Beyond certain critical values of increasing Cf and decreasing Cp, a second unstable low-frequency mode appears which is distinct from the fundamental mode. The fully coupled character of the instability reveals three important aspects of the particle effect on the flow structure: (1) the particle concentration field is organized into alternating bands of increased and decreased concentration corresponding to the braid and core regions of the vortices, respectively, with peak perturbations occurring at intermediate Cp values (0.01⩽Cp⩽0.1), (2) the streamwise particle velocity is higher than the streamwise fluid velocity for a substantial range of Cp values and every finite Cf, and (3) the modification of the fluid vorticity field structure with respect to the corresponding field in single-phase flow is driven by the divergence of the particle velocity field.