We use direct numerical simulation of stress-driven turbulent Couette flows over waving surfaces to study turbulence in the vicinity of water waves. Mechanistic study is performed through systematic investigation of different wavy surface conditions including plane progressive Airy and Stokes waves with and without wind-induced surface drift, as well as stationary wavy walls and vertically waving walls for comparison. Two different wave steepness values ak = 0.1 and 0.25 are considered, where a is the wave amplitude and k is the wavenumber. For effects of wave age, defined as the ratio between the wave phase speed c and the turbulence friction velocity u*, we consider three values, namely c/u* = 2, 14 and 25, corresponding to slow, intermediate and fast waves, respectively. Detailed analysis of turbulence structure and statistics shows their dependence on the above-mentioned parameters. Our result agrees with previous measurement and simulation results and reveals many new features unreported in the literature. Over progressive waves, although no apparent flow separation is found in mean flow, considerable intermittent separations in instantaneous flow are detected in slow waves with large steepness. The near-surface coherent vortical structures are examined. We propose two conceptual vortex structure models: quasi-streamwise and reversed horseshoe vortices for slow waves and bent quasi-streamwise vortices for intermediate and fast waves. Detailed examination of Reynolds stress with quadrant analysis, turbulent kinetic energy (TKE) and TKE budget with a focus on production shows large variation with wave phase; analysis shows that the variation is highly dependent on wave age and wave nonlinearity. Comparison between Airy waves and Stokes waves indicates that although the nonlinearity of surface water waves is a high-order effect compared with the wave age and wave steepness, it still makes an appreciable difference to the turbulence structure. The effect of wave nonlinearity on surface pressure distribution causes substantial difference in the wave growth rate. Wind-induced surface drift can cause a phase shift in the downstream direction and a reduction in turbulence intensity; this effect is appreciable for slow waves but negligible for intermediate and fast waves. In addition to providing detailed information on the turbulence field in the vicinity of wave surfaces, the results obtained in this study suggest the importance of including wave dynamics in the study of wind–wave interaction.

Direct numerical simulation (DNS) is used to examine low Froude number free-surface turbulence (FST) over a two-dimensional mean shear flow. The Navier–Stokes equations are solved using a finite-difference scheme with a grid resolution of 1283. Twenty separate simulations are conducted to calculate the statistics of the flow. Based on the velocity deficit and the vertical extent of the shear of the mean flow, the Reynolds number is 1000 and the Froude number is 0.7. We identify conceptually and numerically the surface layer, which is a thin region adjacent to the free surface characterized by fast variations of the horizontal vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or ‘source’) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the turbulence intensity, i.e. in the increase of the horizontal velocity fluctuations at the expense of the vertical velocity fluctuation. A prominent feature of FST is vortex connections to the free surface which occur inside the surface layer. It is found that as hairpin-shaped vortex structures approach the free surface, their ‘head’ part is dissipated quickly in the surface layer, while the two ‘legs’ connect almost perpendicularly to the free surface. Analysis of the evolution of surface-normal vorticity based on vortex surface-inclination angle shows that both dissipation and stretching decrease dramatically after connection. As a result, vortex structures connected to the free surface are persistent and decay slowly relative to non-connected vorticities. The effects of surface and blockage layers on the turbulence statistics of length scales, Reynolds-stress balance, and enstrophy dynamics are examined, which elucidate clearly the different turbulence mechanisms operating in the respective near-surface scales. Finally we investigate the effect of non-zero Froude number on the turbulence statistics. We show that the most significant effect of the presence of the free surface is a considerable reduction of the pressure–strain correlation at this surface, compared to that at a free-slip at plate. This reduction is finite even for very low values of the Froude number.

Observations from a suite of platforms deployed in the coastal ocean are being combined with numerical models and simulations to investigate the processes that couple the atmosphere and ocean.

Direct numerical simulation is used to study the turbulent flow over a smooth wavy wall undergoing transverse motion in the form of a streamwise travelling wave. The Reynolds number based on the mean velocity U of the external flow and wall motion wavelength λ is 10 170; the wave steepness is 2πa/λ = 0.25 where a is the travelling wave amplitude. A key parameter for this problem is the ratio of the wall motion phase speed c to U , and results are obtained for c/U in the range of −1.0 to 2.0 at 0.2 intervals. For negative c/U , we find that flow separation is enhanced and a large drag force is produced. For positive c/U , the results show that as c/U increases from zero, the separation bubble moves further upstream and away from the wall, and is reduced in strength. Above a threshold value of c/U ≈ 1, separation is eliminated; and, relative to smallc/U cases, turbulence intensity and turbulent shear stress are reduced significantly. The drag force decreases monotonically as c/U increases while the power required for the transverse motion generally increases for large c/U ; the net power input is found to reach a minimum at c/U ≈ 1.2 (for fixed U). The results obtained in this study provide physical insight into the study of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive efficiency.

Vortex connections at the surface are fundamental and prominent features in free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon that surface. We perform numerical simulations of the Navier–Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.