Numerical modelling of shallow flows with horizontal density variation
A numerical model is presented of vertically homogeneous shallow flows with variable horizontal density. The governing equations represent mass and momentum conservation of a liquid-species mixture, and mass conservation of the species within a control volume. Here, the term species refers to material transported with the liquid flow. For example, when the species is taken to be suspended sediment, the model provides an idealised simulation of hyper-concentrated sediment-laden flows. The volumetric species concentration acts as an active scalar, allowing the species dynamics to influence the flow structure. The model can simulate flows driven by depth and density differences in the horizontal. The governing equations are written in a deviatoric, hyperbolic form to facilitate their solution by means of a Godunov-type finite volume scheme appropriate for flows containing sharp fronts. The deviatoric governing equations ensure that flux gradient and source terms are balanced (and there is no need for further numerical balancing). The numerical model is first verified for constant density cases, for which the governing equations reduce to the conventional coupled shallow water and species transport equations. Close agreement between numerical predictions and benchmark test solutions illustrate the model's ability to capture rapidly-varying flow features over uniform and non-uniform bathymetries. For variable-density cases, analytical steady-state solutions are derived for two simple cases, one with uniform bathymetry and the other with sinusoidal bathymetry. Detailed parameter studies are then undertaken to examine the effects of varying the initial density and depth in different regions. The shock-capturing scheme resolves all sharp features in the flow such as bore, shear waves, shock diamond like features, contact discontinuities and locally intense vortices. These interesting and novel nonlinear features are unique to variable density flows. The validated numerical model is applied to an idealised case of a hyperconcentrated sediment-laden debris-type flow along a tributary entering a river. The predicted evolution of the free surface flow field is qualitatively similar to observations of an actual debris flows into a river connected to the Upper Yangtze.