A Semi-Lagrangian solver for the free surface Euler system with application to rotational wave flows

Even though the majority of the classical water wave theory is restricted to potential flows, vortical flows are abundant in nature. This necessitates the need for the development of accurate and efficient methods for the simulation of rotational phenomena, such as the propagation of waves over bathymetry in the presence of a sheared current [1, 2].

In the present work, a numerical method for the free surface Euler system with constant density and general bathymetry is developed within the framework
of classical Computational Fluid Dynamics (CFD). Specifically, using the well known σ coordinate system, a layer-wise integration followed by an operator
splitting is performed. The resulting horizontal advection component is, treated as a multilayered Shallow Water Equations (mSWE) system (see, e.g. [3]) 
and it is solved with a conventional Finite Volume solver. The vertical counterpart (that works similar to remeshing operator) regulates if the system is treated with a Lagrangian or an Eulerian approach. Finally, the dynamic pressure component coupled with the incompressibility constraint is treated using the well-known projection of Chorin [4].

The method’s main advantages stem from its highly modular character that makes it both robust and easy to implement. The method’s performance is tested in the case of waves propagating on top of a sheared current. Results concerning the dispersion and propagation characteristics for general current profiles are presented and compared with other models [1,2,5].

References
[1] Julien Touboul and Kostas Belibassakis. A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry. Journal of Ocean Engineering and Marine Energy, 5(4):333–350, 2019
[2] Kostas Belibassakis and Julien Touboul. A nonlinear coupled-mode model for waves propagating in vertically sheared currents in variable
bathymetry—collinear waves and currents. Fluids, 4(2):61, 2019.
[3] Fracois Bouchut and Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete and Continuous Dynamical
Systems-Series B, 13(4):739–758, 2010.
[4] Zhe Liu, Lei Lin, Lian Xie, and Huiwang Gao. Partially implicit finite difference scheme for calculating dynamic pressure in a terrain-following coordinate
non-hydrostatic ocean model. Ocean Modelling, 106:44–57, 2016. [5] Ellingsen SA, Li Y (2017) Approximate dispersion relations for waves on arbitrary shear flows. J Geophys Res Oceans 122(12):9889–9905
[5] Ellingsen SA, Li Y (2017) Approximate dispersion relations for waves on arbitrary shear flows. J Geophys Res Oceans 122(12):9889–9905