School of Mechanical Engineering J. Touboul & K. Belibassakis
School of Mechanical Engineering Seminar
Wednesday, February 13, 2019 at 14:00
Wolfson Building of Mechanical Engineering, Room 206
Recent advances in phase-resolved modelling of water waves propagating in the presence of sheared current, over variable bathymetry
J. Touboul & K. Belibassakis
University of Toulon
Propagation of water waves in coastal zones is mainly affected by the influence of currents and bathymetry variations. Phase-resolved models describing wave propagation in coastal zones are often based on the numerical solution of the Mild Slope equation (Kirby, 1984). This equation (and its modern formulations) considers currents, uniform with depth, varying horizontally, on a length scale similar to the one relative to the bathymetry variations. It is, though, important to take into account the vorticity included within the mean flow field.
In this work, we will first emphasize the motivation for this work. Then, the various attempts made to bypass this new difficulty will be presented. First, an extension of the mild slope equation is derived, taking into account the linear variation of the current with depth, which results in a constant horizontal vorticity, slowly varying horizontally, within the background current field. This approach is based on the asymptotic expansion of the depth-integrated Lagrangian, assuming the linear variation of the background current with depth. With the aid of selected examples, the role of this horizontal vorticity, associated with the assumed background current velocity profile, is then illustrated and emphasized, demonstrating its effect on the propagation of water waves in coastal areas.
A coupled-mode model is then developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, here again with sheared current presenting a linear vertical velocity profile, over general bottom topography. The wave potential is represented by a series of local vertical modes containing the propagating and evanescent modes, plus additional terms accounting for the satisfaction of the boundary conditions. Using the latter representation, in conjunction with a variational principle, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown modal amplitudes. The above models are examined for predicting the characteristics of normally incident waves propagating over a bar and sinusoidal bottom topography in the presence of opposing shearing currents. It is shown that MMSs are able to provide good predictions, however, in the case of Bragg scattering of waves over rippled bathymetry without a current fail to provide good predictions concerning the resonant frequency in the presence of the current. In order to resolve the above mismatch, a two-equation mild-slope system (CMS2) is derived from a variational principle based on the representation of the wave potential expressed as a superposition of the forward and backward components.
Another limitation of the above models deals with the structure of the mean flow considered. Indeed, as long as potential theory is considered, the mean flow structure will remain limited to linear variations with respect to depth. Recent attempts to bypass this difficulty will finally be presented here.