## A Numerical Wave Tank for Complex Current and Wave

### Organization:

*Funded by: *STW*Postdoc:* Vijaya R. Ambati*Supervisor: *Onno Bokhove*Collaboration*: Maritime Research Institute Netherlands (MARIN).

### Description:

A new variational finite element method is developed for nonlinear free surface waves. This method stems from Miles’ variational formulation using Galerkin polynomial expansion of flow field for nonlinear surface waves. Symplectic time-stepping techniques were further employed to devise a stable and energy conserving finite element scheme to simulate nonlinear waves for long time. A semi-analytic nonlinear wave solution following Fenton’s theory is simulated using the present nonlinear scheme for 100 time periods. The initial (Fig. 1) and final wave (Fig. 2) profiles obtained from the numerical scheme demonstrates that there is no decay in amplitude and discrete wave energy in the domain is conserved (see animation). Our future work now includes accommodating the wave maker movement in our nonlinear numerical schemes and validating the scheme with the experimental data provided by MARIN. Some preliminary numerical results after accommodating a piston type wave maker in a wave basin with solid boundaries are shown in the animation here.

Fig. 1: Initial wave profile of Fig. 2: Final wave profile of nonlinear

nonlinear Fenton wave. Fenton that is after simulating for 100 time

periods.

### Publications:

### Papers:

O. Bokhove and V.R. Ambati,* "Hybrid Rossby-shelf modes in a laboratory ocean,"* J. Phys. Ocean.,39(10), 2523--2542, 2009. [DOI], [[EPRINTS].

V.R. Ambati, J.J.W. van der Vegt and O. Bokhove, *"Variational space-time (dis)continuous Galerkin method for linear free surface waves,"* Under revision. [EPRINTS]

V.R. Ambati "*Forecasting water waves and currents: A space-time approach*,"Ph.D. Thesis, University of Twente, The Netherlands, 2008. ISBN: 978-90-365-2632-6, [EPRINTS].

V.R. Ambati and O. Bokhove, *"Space-time discontinuous Galerkin discretization of rotating shallow water equations,"* J. Comput. Phys., 225(2), 1233-1261, 2007. [DOI], [EPRINTS]

L. Pesch, A. Bell, W.E.H. Solie, V.R. Ambati, O. Bokhove and J.J.W. van der Vegt, *"hpGEM --- A sofware framework for discontinuous Galerkin finite element Methods,"* ACM Transactions on Mathematical Software, 33(4), 2007. [DOI], [EPRINTS]

V.R. Ambati and O. Bokhove, *"Space-time discontinuous Galerkin finite element method for shallow water flows,"* J. Comput. Appl. Math., 204(2), 452-462, 2007. [DOI], [EPRINTS]