Development of robust, physically-based numerical models for transport processes and geomorphodynamics changes

Bed changes in rivers may occur under several morphodynamics and hydrodynamics conditions. The modeling of this type of phenomena can be performed coupling the Shallow Water Equations (SWE) for the hydrodynamic part and the Exner equation for the morphodynamic part. The Exner equation states that the time variation of the sediment layer is due to the sediment transport discharge through the boundaries of the volume. Considering that sediment transport discharge are computed by means of sediment capacity formulae based on 1D experimental steady flows, the assessment of these empirical relations under unsteady 1D and 2D situations must be studied. In order to ensure the reliability of the numerical experimentation, the numerical scheme must handle correctly the coupling between the 2D SWE and the Exner equation under any condition. If possible, it is convenient to express the formulation of different empirical laws under a general framework. In consequence, a finite-volume numerical scheme that includes these two main features has been chosen as a benchmark for comparing the 1D and 2D results obtained when using several well known sediment transport formulae: Meyer-Peter and M\"uller, Ashida and Michiue, Engelund and Fredsoe, Fernandez Luque and Van Beek, Parker, Smart, Nielsen, Wong and Camenen and Larson. In addition, a new interpretation of the Smart empirical law is presented in order to cope with bed load transport over irregular beds of changing slope. Detailed results for this new modified empirical law together with the ones obtained with Meyer-Peter and M\"uller (which is the sediment capacity formula more used in hydraulic engineering) are provided for every test case analyzed. Furthermore, the Root Mean Square Error (RMSE) associated to every formula at each experimental condition is calculated with the purpose of evaluating quantitatively the overall behavior of each one. The results point out that the new interpretation of the Smart formula reaches the most accurate results in all cases, but in a genuinely 2D flow, that is, a situation involving more than one flow direction, the differences among sediment transport formulae are not as noticeable as in the 1D studied situations. Once the forecasting capacity of each sediment transport formula has been studied, another concern is the computational cost. The coupling between the SWE and the Exner equation by means of an augmented Jacobian matrix involves a high number of algebraic operations for computing the eigenvalues and the eigenvectors. Therefore, the computational cost is increased significantly, limiting the applicability of the numerical scheme to realistic situations where large domains are involved. In order to improve the computational efficiency, the coupling technique is modified, not decreasing the number of waves involved in the Riemann Problem but simplifying their definitions. The approach proposed in this thesis is a new strategy to combine concepts from hyperbolic conservation laws and conservative finite volume schemes. With the aim to control numerical stability in the most efficient form possible, a numerical eigenvalue is defined to control the discrete Exner equation in the explicit scheme. This bed wave celerity helps mainly to ensure conservation and to control automatically the numerical stability of the explicit scheme. The effects of the numerical coupling strategy proposed in this thesis are tested against exact solutions and 1D and 2D experimental data. The results emerging from this analysis show that efficiency and accuracy can be obtained when choosing an adequate sediment transport law and the stability condition is augmented by including a new celerity associated to the bed changes. On the other hand, in environmental and civil engineering applications, geomorphological changes are not only present in rivers but also in steep areas where massive mobilizations of poorly sorted material can occur. This sliding material is usually composed by a mixture of sand and water. For simplifying the phenomenon, dry granular flows have been considered as a starting point for the understanding of the physics involved within the landslides. The hypothesis of Saint-Venant equations are considered valid for modeling these land movements. Taking advantage of this approach, in this thesis approximate augmented Riemann solvers are formulated providing appropriate numerical schemes for mathematical models of granular flow on irregular steep slopes. Fluxes and source terms are discretized to ensure steady state configurations including correct modeling of start/stop flow conditions, both in a global and a local system of coordinates. The weak solutions presented involve the effect of bed slope in pressure distribution and frictional effects by means of the adequate gravity acceleration components. The numerical solvers proposed are first tested against 1D cases with exact solution and then are compared with 2D experimental data in order to check the suitability of the mathematical models described in this thesis. Comparisons between results provided when using global and local system of coordinates are presented. Both the global and the local system of coordinates can be used to predict faithfully the overall behavior of the landslides. The performance of the numerical scheme has been studied using novel experimental situations. These laboratory works include bidimensional configurations, the inclusion of obstacles in the flow path and a variable slope in the domain. Hence, a further step in mimicking realistic situations is obtained, since the behavior of the granular flow is affected by the presence of natural elements such as boulders or trees. Three situations have been considered. The first experiment is based on a single obstacle, the second one is performed against multiple obstacles and the third one study the influence of a dike when an overtopping situation takes place. Due to the impact of the flow against the obstacles, fast moving shocks appear, and a variety of secondary waves emerge. Comparisons between computed and experimental data are presented for the three cases. The computed results show that the numerical tool previously developed is able to predict faithfully the overall behavior of this type of complex dense granular flow.