An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Part 1. Regular waves

Abstract

This paper describes an experimental study of two-dimensional surface water waves propagating on a depth-varying current with a non-uniform vorticity distribution. The investigation is divided into two parts. The first concerns the ‘equilibrium’ conditions in which the oscillatory wave motion and the current co-exist. Measurements of the water-surface elevation, the water-particle kinematics, and the near-bed pressure fluctuations are compared to a number of wave and wave–current solutions including a nonlinear model capable of incorporating the vertical structure of the current profile. These comparisons confirm that the near-surface vorticity leads to an important modification of the dispersion equation, and thus affects the nature of the wave-induced orbital motion over the entire water depth. However, the inclusion of vorticity-dependent terms within the dispersion equation is not sufficient to define the combined wave–current flow. The present results suggest that vorticity may lead to a significant change in the water-surface profile. If a current is positively sheared, dU/dz > 0, with negative vorticity at the water surface, as would be the case in a wind-driven current, a wave propagating in the same direction as the current will experience increased crest–trough asymmetry due to the vorticity distribution. With higher and sharper wave crests there is a corresponding increase in both the maximum water-particle accelerations and the maximum horizontal water-particle velocities. These results are consistent with previous theoretical calculations involving uniform vorticity distributions (Simmen & Saffman 1985 and Teles da Silva & Peregrine 1988).The second part of the study addresses the ‘gradually varying’ problem in which there are changes in the current, the wavelength and the wave height due to the initial interaction between the wave and the current. These data show that there is a large and non-uniform change in the current profile that is dependent upon both the steepness of the waves and the vorticity distribution. Furthermore, comparisons between the measured wave height change and a number of solutions based on the conservation of wave action, confirm that the vorticity distribution plays a dominant role. In the absence of a conservation equation for wave action appropriate for nonlinear waves on a depth-varying current, an alternative approach based on the conservation of total energy flux, first proposed by Longuet-Higgins & Stewart (1960), is shown to be in good agreement with the measured data.

Observed Variability of Ocean Wave Stokes Drift, and the Eulerian Response to Passing Groups

“…As noted in the introduction, Eulerian counterflows have also been observed in laboratory experiments (e.g., Groeneweg and Klopman 1998; also Kemp andSimons 1982, 1983;Swan 1990;Jiang and Street 1991;S. G. Monismith et al 1996, unpublished manuscript).…”

“…G. Monismith et al 1996, unpublished manuscript;Groeneweg and Klopman 1998;Swan et al 2001). The changes in the Eulerian current profile from case to case (no waves, down-current waves, up-current waves) are comparable to (and oppose) the Stokes drift profile calculated from the wave parameters as given (in both magnitude and shape).…”

On the Recovery of the Free Surface from the Pressure within Periodic Traveling Water Waves

“…Advances of the state-of-the-art depend on a fruitful interaction between theoretical studies (building upon recent insights obtained in [4,6,11]), numerical simulations (as in [14]) and experimental evidence gathered in the laboratory (see e.g. [20]) or by collection of field data. This is work in progress.…”

Regularity of Steady Periodic Capillary Water Waves with Constant Vorticity

“…The papers [13,20,21] make clear that nonuniform currents give rise to water flows with vorticity. There is an extensive research literature in the area of water waves with vorticity, see [2,3,10] for existence results, [14,22] for matters of uniqueness, [6][7][8]19] for symmetry results and [9,15,18] for regularity results.…”