Hydrodynamic module
The depth-integrated shallow water equations are the basis of the flow module. For an overview on shallow water equations see Vreugdenhil (1994).
The model equations are the continuity equation:
In addition to the effect of advection and pressure gradients, external forces like the Coriolis force, bottom shear stress, wind shear stress and radiation stress due to surface waves can be taken into account.
It is noted that turbulent shear stresses are not taken into account: the application is therefore restricted to advection dominated flows only. As a solution method, the discontinuous Galerkin method is adopted (Hughes, 1987) in which the flow variables are taken constant in each moment. This method has advantages in dealing with drying elements.
As the momentum equations contain first order derivatives in space, they can be written as:
The problem is now reduced to the determination of the fluxes F along the boundaries. As the variables are determined at the elements and
not at the sides, the flux F is not known beforehand, but involves the solution of a local Riemann problem.
An approximate Riemann solver according to Roe (Glaister, 1993) is applied. This method guarantees strict mass and momentum conservation, but suffers from some numerical diffusion in stream-wise direction. An explicit time integration scheme is used.
As this method restricts the time step, the time step is controlled automatically for optimum performance.
A special problem in shallow waters like for example estuaries is the drying and flooding of large areas during a tidal cycle. A discontinuous discretisation is used in combination with an explicit time-stepping. In this way this flooding and drying of the elements can be treated relatively easily. If an element tends to dry, the corresponding characteristic wave is partially reflected from this element which guarantees mass conservation.
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