Dispersion in Shallow Moment Equations

Shallow flow models are of high practical relevance, however, in certain situations they reach their limits. Due to an averaging process, information on the vertical profile of the flow variables is generally lost. In this talk we will see a framework for the systematic derivation of dimensionally reduced dispersive equation systems that hold information on the vertical profiles of the flow variables. The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in vertical direction. Dimensional reduction follows via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface.

After a quick look at the dispersive properties of the resulting equation systems and a comparison to the well-known Green-Naghdi system I will present a numerical technique. The special structure of the equation systems makes a splitting technique necessary where we solve a Poisson equation in every step. We will see simulation results for the stationary case on a finite domain with a set of  mixed boundary conditions and results for the instationary case on a periodic domain.