Computational Model of Unsteady Hydromechanics of Large Amplitude Gerstner Waves

The computational experiments in the ship fluid mechanics involve the nonstationary interaction of a ship hull with wave surfaces that include the formation of vortices, surfaces of jet discontinuities, and discontinuities in the fluid under the influence of negative pressure. These physical phenomena occur not only near the ship hull, but also at a distance where the waves break as a result of the interference of the sea waves with waves reflected from the hull. In the study reported here we simulate the wave breaking and reflection near the ship hull. The problem reduces to determining the wave kinematics on the moving boundary of a ship hull and the free boundary of the computational domain. We build a grid of large particles having the form of a parallelepiped and, in the wave equation instead of the velocity field we integrate streams of fluid represented by functions as smooth as the wave surface elevation field. We assume that within the boundaries of the computational domain the waves do not disperse, i.e. their length and period stay the same. Under this assumption, we simulate trochoidal Gerstner waves of a particular period. This approach allows to simulate the wave breaking and reflection near the ship hull. The goal of the research is to develop a new method of taking the wave reflection into account in the ship motion simulations as an alternative to the classic method which uses added masses.

Gerstner waves in the presence of mean currents and rotation

We present a Lagrangian analysis of nonlinear surface waves propagating zonally on a zonal current in the presence of the Earth’s rotation that shows the existence of two modes of wave motion. The first, ‘fast’ mode, one with wavelengths commonly found for wind waves and swell in the ocean, represents the wave–current interaction counterpart of the rotationally modified Gerstner waves found first by Pollard (J. Geophys. Res., vol. 75, 1970, pp. 5895–5898) that quite closely resemble Stokes waves. The second, slower, mode has a period nearly equal to the inertial period and has a small vertical scale such that very long, e.g. $O(10^{4}~\text{km})$ wavelength, waves have velocities etc. that decay exponentially from the free surface over a scale of $O(10~\text{m})$ that is proportional to the strength of the mean current. In both cases, the particle trajectories are closed in a frame of reference moving with the mean current, with particle motions in the second mode describing inertial circles. Given that the linear analysis of the governing Eulerian equations only captures the fast mode, the slow mode is a fundamentally nonlinear phenomenon in which very small free surface deflections are manifestations of an energetic current.

Laboratory observations of mean flows under surface gravity waves

In this paper we present mean velocity distributions measured in several different wave flumes. The flows shown involve different types of mechanical wavemakers, channels of differing sizes, and two different end conditions. In all cases, when surface waves, nominally deep-water Stokes waves, are generated, counterflowing Eulerian flows appear that act to cancel locally, i.e. not in an integral sense, the mass transport associated with the Stokes drift. No existing theory of wave–current interactions explains this behaviour, although it is symptomatic of Gerstner waves, rotational waves that are exact solutions to the Euler equations. In shallow water (kH ≈ 1), this cancellation of the Stokes drift does not hold, suggesting that interactions between wave motions and the bottom boundary layer may also come into play.