Two-dimensional finite volume numerical models for unsteady free surface flows, solute transport and erosion/deposition processes

Zusammenfassung

In the last years the numerical modelling of shallow water flow in two dimensions in complex geometries involving transient flow and movable boundaries has been a challenge for modellers. Upwind finite volume methods based on Roe's approximate solver, initially developed for solving problems in gas dynamics, have been accepted as reliable and accurate for the numerical solution of the shallow water equations. The aim of this thesis consists on generating robust and accurate methods to solve realistic hydraulic problems. For that reason, in this work a transient 2D coupled vertically averaged flow/transport model with variable bed elevation surface generated by erosion/deposition processes is presented. The model presented deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. It is based on Roe's approximate Riemann solver for finite volume schemes. The convenience of considering the fully coupled system of equations is demonstrated along the thesis as it allows a correct upwinding of the source terms ensuring the exact balance of the numerical fluxes, reproducing exactly steady state cases with flow in movement and situations of still water for first and second order approximation. To test the efficiency of higher order resolution methods, the structure of several slope limited second order explicit finite volume schemes is presented as an extension of the first order explicit upwind scheme. One contribution of this work is the importance of including in the slope limiters new conditions to provide the correct balance of the source terms in steady state cases. The sign of the redefined advection velocities must be kept equal to the one obtained for first order approximation. Also the interpolation function for the water depth must be obtained interpolating the water level surface. Another remarkable contribution of this work is the extension of the exp