HAWASSI – VBM Variational Boussinesq Model

Finite Element implementation with Optimized Dispersion

Application Areas

• HAWASSI – VBM1 for (1HD) simulations of irregular waves in wave tanks above flat and varying bottom.
• HAWASSI – VBM2 for (2HD) simulations with short-crested waves in coastal areas with harbours and strongly varying bathymetry, and for simulation of oceanic waves.

The interior fluid motion is modelled by a combination of a few (Airy-type) depth profiles; this makes it possible to optimize the dispersion properties depending on the specific case to be simulated. Nonlinear effects are accounted for in a weakly nonlinear way that is sufficient for most applications.

Features of the 1&2HD code include

• The quality of dispersion is optimized for the specific wave problem to be simulated, which makes it possible to simulate deep ocean waves or very short waves (kh=15 or more) and infragravity waves
• Use of an unstructured grid with mesh-size depending on bathymetry
• Various methods for wave influx from multiple interior lines using sources in the dynamic equations
• Use of efficient damping zones, (partially reflecting) walls for harbour lay-outs, etc
• Interior Flow Module for the calculation of interior fluid velocities, fluid accelerations and the (dynamic and total) pressure at user defined positions between surface and bottom

Facilities of the software include

• GUI for input of wave characteristics and model parameters, GUI for post processing
• For VBM2 easy generation by using a GUI of an unstructured mesh based on local depth for given bathymetry and geometric structures
• Time partitioned simulation is possible to reduce hardware requirements
• Project examples with harmonic, focusing and irregular waves above bathymetry in various geometries
• Manual for easy operation and scientific description of the equations with literature references

Underlying Modelling Methods

Just like HAWASSI-AB, VBM is based on the following principles

1. The free surface dynamics for inviscid, incompressible fluid is governed by a set of Hamilton equations for the surface elevation $\eta$ and the potential $\phi$ at the surface.
2. By approximating the kinetic energy functional $K(\phi,\eta)$ explicitly as an expression in $\eta$, and $\phi$, the simulation of the interior flow can be avoided, the Boussinesq character of the codes.
3. The way of approximating $K(\phi,\eta)$ is based on Dirichlet’s principle for the boundary-value problem in the fluid domain. By restricting the set of competing functions in the minimization, an approximation of $K(\phi,\eta)$ is obtained. The variational derivative $\delta_\phi K(\phi,\eta)=\partial_N\Phi$ is the corresponding consistent approximation of the Dirichlet-to-Neumann operator.
4. The (approximate) Hamilton system conserves the (approximate) positive definite total energy exactly, avoiding sources of instability.
The time dynamics is explicit, no CFL-conditions are required. Time stepping is done with matlab ode45 code, with automatic variable time step.

In VBM the interior flow is approximated by using a linear combination of vertical Airy profiles, characterized by the values of wave numbers $\kappa_m$. The choice of these values determines the dispersion relation. Given the spectrum of the influx signal (or initial profile) of the case under investigation, the values of $\kappa_m$ are optimized for best performance for the relevant frequency interval. Therefore, VBM can have excellent, tailor-made, dispersive properties; deep water waves can be simulated just as well as infragravity waves.

Implementation

A Finite Element method using piece-wise linear splines can deal with the first order differentiations that appear in the approximate KE. In addition a system of elliptic equations in the horizontal variables has to be solved for the amplitudes of the Airy functions. VBM uses variable grid size depending on the local depth, making 2HD simulations more efficient.