Material Transport under a Wave Train in Interaction with Constant Wind: A Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model

The interaction of a developed train of gravity deep water waves with suddenly applied winds is investigated in this manuscript. The direction of the wind is the same as that of the wave train (i.e., following) and its imposed surface shear stress is constant and steady. The focus of this study is on a micro-scale water wave field where the time scale is on the order of ten wave periods and the length scale is on the order of ten wave lengths. Accurate 2D Reynolds-averaged Navier–Stokes (RANS) multi-phase simulations of Navier–Stokes equations are performed in a Eulerian framework to capture the flow features, induced by the wave field and the surface wind. The sufficiently large spatial domain in the horizontal direction, combined with the sufficiently long simulation time, permits the development of surface currents and the consequent formation of a near-surface shear layer. The interaction of surface currents with the wave orbital velocity field results in the generation of spilling breaking waves. Downstream of the domain, vertical turbulent structures are observed as the result of such breaking waves. Lagrangian particle tracking is performed, using the RANS simulation velocity and eddy diffusivity data. A second order random acceleration particle tracking method is applied with the vitally important spatial gradients of the eddy diffusivity (Journal of Marine Science and Engineering, 2018, 6, 10.3390/jmse6010007.) also included in calculations. The spatial gradients of the eddy diffusivity were proved to be a key factor in material transport simulations. Our particle tracking results exhibit strong vertical mixing downstream of the domain and by means of visualizing the spiral trajectory of neutrally buoyant particles. Such enhanced vertical mixing (caused by horizontal winds) is the result of the strong near-surface advection (induced by currents) and the turbulence (induced by breaking waves). The objectives of this paper are twofold. Firstly, a numerical approach to simulate wind breaking waves is proposed based on: using K − ϵ RANS model to capture turbulence features, employing the Volume of Fluid Method (VOF) to model the free surface flow, and applying a constant shear stress body force at the interfacial cells to simulate the wind force. Such treatment of the winds eliminates the need for fully resolving the air phase. The computed eddy viscosity profiles are in good agreement with the experimental profiles reported in the literature ( ν t = − κ u * z , Journal of Physical Oceanography, 1977, 7, pp. 248–255; Journal of Physical Oceanography, 1984, 14, pp. 855–863). Secondly, effects of the horizontally applied wind on the vertical mixing and eddy viscosity profiles on the water column are studied. It is observed that, away from the surface and outside the shear layer, the negative horizontal gradient of eddy diffusivity (induced by the dampening effect of breaking surface waves), combined with the downwards advection velocities (induced by breaking waves), results in an enhanced vertical mixing and reduced horizontal drift of transported material