Shallow Flow Hydrodynamics

A shallow flow is one in which the horizontal dimensions are much larger than the vertical extent and the vertical component of water particle acceleration is negligible compared with the horizontal acceleration components so that the pressure variation can be assumed hydrostatic.  Typical examples include wide rivers, lakes, coastal lagoons, estuaries, and so on.  The majority of the whole world’s population reside near to these areas.  Hence, better understanding of the shallow flow hydrodynamics and related processes, such as flooding, sediment transport, spreading and mixing of pollutant and its effects on water quality, is of great importance.   Mathematically, shallow flow hydrodynamics may be approximated by the shallow water equations, which include the continuity equation and the x- and y-direction momentum equations, and can be derived by depth integrating the three-dimensional Navier-Stokes equations.

The shallow water equations are solved using a Godunov-type finite volume method on dynamically adaptive quadtree grids with Cartesian cut-cell technique implemented for boundary fitness.  Interface fluxes are evaluated by the HLLC approximate Riemann solver, which is selected due to its ability to deal with the wet-dry front and the fact that it automatically satisfies the entropy condition.  The overall second-order accuracy is then achieved by using an unsplit MUSCL-Hancock method, which updates the conservative variables using a predictor and corrector step.  The resulting numerical model captures both subcritical and supercritical flow and automatically simulates wetting and drying.  Based on dynamically adaptive quadtree grids, the model is more than six times faster than its counterpart on a structured Cartesian grid for similar resolution.  Therefore the present shallow flow model is a robust and efficient tool for simulating a wide range of shallow flow hydrodynamics.

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