Abstract

In the mountain territory the majority of the population and of the productive activities are concentrated in the proximity of torrents or over alluvial fans. Here, when intense rainfall occurs, debris flow or hyper-concentrated flow events can produce serious problems to the population with possible casualties. On the other hand, the majority of these problems could be overcome with accurate hazard mapping, disaster prevention planning and mitigation structures (e.g. silt check dams, paved channels, weirs ...). Good and reliable mathematical and numerical models, able to accurately describe these phenomena are therefore necessary. Debris flows and hyper-concentrated flows can be adequately represented by means of a mixture of a fluid (usually water) and a solid phase (granular sediment, e.g. sand, gravel ...), flowing over complex and composite topography. Complex topography is related to complicated bed elevation variety inasmuch as there are slopes, channels, human artifacts and so on. On the other hand, topography is composite because every type of flow can encounter two different bed behaviors: the mobile bed and the fixed bed. In the first case, mass can be exchanged between the bed and the flow, so the bottom elevation can change in time. In the second case (fixed bed case), this mass transfer is inhibited, due to the presence of a rigid bottom, such as bedrock or concrete, and the bottom cannot change in time. The first objective of the work presented in this thesis concerns the development of a new type of hyperbolic mathematical model for free-surface two-phase hyper-concentrated flows able to describe in a single way the fixed bed, the mobile bed and also the transition between them. The second objective, strictly connected with the first, is the development of a numerical scheme that implements this mathematical model in an accurate and efficient way. In the framework of finite-volume methods with Godunov approach, the fluxes are evaluated solving a Riemann Problem (RP). A RP is an initial value problem related to a set of PDEs equations wherein, in a certain point, there is a discontinuity separating different left and right initial constant states. However, if the topography is composite, a new type of Riemann problem, called Composite Riemann Problem (CRP), occurs. In a CRP, not only the initial constant states, but also the relevant PDEs systems change across the discontinuity. This additional complexity makes the general solution of the CRP quite challenging to obtain. The first part of the work is devoted to the derivation of the PDEs systems describing the fixed- and mobile-bed behaviors. Starting from the 3D discrete equations valid for each phase (continuous fluid and solid granular) and using suitable average processes the 3D continuous equations (continuous fluid and solid) are obtained. Introducing the shallow water approximation and performing the depth average process, the 2D fully two-phase models for free-surface flow over fixed- and mobile-bed are derived. The isokinetic approximation, which states the equality between the velocity of the solid phase and the liquid phase, is then used, ending up with the so-called two-phase isokinetic models. Finally, an exhaustive comparison between the fixed- and the mobile-bed fully two-phase models, the two-phase isokinetic models and others models proposed in the literature is presented. The second part of the work concerns the definition and, mainly, the solution of the CRP from a mathematical point of view. Firstly, a general strategy for the CRP solution is developed. It allows to couple different hyperbolic systems that are physically compatible (e.g. fixed-bed with mobile-bed systems, free-surface flow with pressurized flow), also if they have a different number of equations. The resulting CRP solution is composed of a single PDEs system, called Composite PDEs system, whose properties, under some assumptions, degenerate to the properties of the original PDEs systems. The general strategy is developed using the simplest 1D isokinetic models for the fixed bed and the mobile bed (i.e. PDEs systems valid only for low concentration). Coherently with the generality of the CRP solution method, the low concentration constraint is then relaxed, ending up with a Composite PDEs system describing also high concentrated flows. From the numerical point of view, all the developed Composite systems are integrated using the finite-volume method with Godunov fluxes. These fluxes are evaluated using three different approximated Riemann solvers: the Generalized Roe solver, the LHLL solver and the Universal Osher solver. All the solvers are analyzed and an exhaustive comparison between them is performed, highlighting pros and cons. The schemes are second order accurate in space and time, and this has been achieved by means of the MUSCL approach. Finally numerical schemes have been parallelized using OpenMP standard. All the models are then tested comparing analytical and numerical solutions. The results are satisfactory, with an accurate agreement between the two solutions in the majority of the physically-based test cases. There is only some small issue when the simulations are performed in a few resonant cases. However, these problems arise in not realistic situations, so it is impossible to encounter them in real situations. Also a realistic application is presented (i.e. the evolution of a trench over partially paved channel), proving the capabilities of both the mathematical approach and the numerical scheme.