Lorenzo Begnudelli

The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations

In this paper, the solution of the Riemann Problem for the one-dimensional, free-surface Shallow Water Equations over a bed step is analyzed both from a theoretical and a numerical point of view. Particular attention has been paid to the wave that is generated at the location of the bed discontinuity. Starting from the classical Shallow Water Equations, considering the bed level as an additional variable, and adding to the system an equation imposing its time invariance, we show that this wave is a contact wave, across which one of the Riemann invariants, namely the energy, is not constant. This is due to the fact that the relevant problem is nonconservative. We demonstrate that, in this type of system, Riemann Invariants do not generally hold in contact waves. Furthermore, we show that in this case the equations that link the flow variables across the contact wave are the Generalized Rankine-Hugoniot relations and we obtain these for the specific problem. From the numerical point of view, we present an accurate and efficient solver for the step Riemann Problem to be used in a finite-volume Godunov-type framework. Through a two-step predictor-corrector procedure, the solver is able to provide solutions with any desired accuracy. The predictor step uses a well-balanced Generalized Roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution. In order to show the effectiveness and the accuracy of the proposed approach, we consider several step Riemann Problems and compare the exact solutions with the numerical results obtained by using a standard Roe approach far from the step and the novel two-step algorithm for the fluxes over the step, achieving good results.

A closure-independent Generalized Roe solver for free-surface, two-phase flows over mobile bed

Several different natural phenomena can be studied in the framework of free-surface, two-phase flows over mobile bed. Mathematically, they can be described by the same set of highly nonlinear, hyperbolic nonconservative PDEs but they differ in the possible algebraic closure relations. These affect significantly the relevant eigenvalues and consequently, all finite-volume numerical methods based on upwind Godunov-type fluxes. In this work the Generalized Roe solver, introduced in 29] for the case of a specific closure, is reformulated in a complete closure-independent way. This gives the solver a quite general applicability to the class of problems previously mentioned. Moreover, the new method maintains all the desirable features shown by the original one: full two-dimensionality and exact well-balanceness. This result is made possible thanks to the development of a novel Multiple Averages (MAs) approach that allows a straightforward determination of the matrices required by the solver. Several tests show the capabilities of the proposed numerical strategy.

Hyperconcentrated 1D Shallow Flows on Fixed Bed with Geometrical Source Term Due to a Bottom Step

Free-surface, two-phase flows constituted by water and a significant concentration of sediments (i.e. hyperconcentrated flows), are often studied under the classical 1D shallow-water assumptions. This paper deals with the problem of the numerical simulation of this type of flows in presence of a geometrical source term caused by a discontinuity in the bed. Since we work in the framework of the finite-volume Godunov-type numerical schemes, we focus on the exact and approximated solution to Riemann Problems at cell interfaces with a step-like bed discontinuity. The paper has a twofold aim. First, it provides insights into the properties of the Step Riemann Problems (SRPs), highlighting in particular that a contact wave with special features may develop. Second, it presents a two-step predictor-corrector algorithm for the solution of the SRPs. In the predictor step, a Generalized Roe solver is used in order to provide an estimation of the wave structure arising from the cell-interface, while in the corrector step the exact SRP is solved by means of an efficient iterative procedure. The capabilities of the proposed solver are tested by comparing numerical and exact solutions of some SRPs.

Single or Two-Phase Modelling of Debris-Flow? A Systematic Comparison of the Two Approaches Applied to a Real Debris Flow in Giampilieri Village (Italy)

A comparison between the performances of two different debris flow models has been carried out. In particular, a mono-phase model (FLO-2D) and a two phase model (TRENT2D) have been considered. In order to highlight the differences between the two codes, the alluvial event of October 1, 2009 in Sicily in the Giampilieri village has been simulated. The predicted time variation of several quantities (as the flow depth and the velocity) has been then analyzed in order to investigate the advantages and disadvantages of the two models in reproducing the global dynamic of the event. Both models seem capable of reproducing the depositional pattern on the alluvial fan in a fairly way. Nevertheless, for the FLO-2D model the tuning of the parameters must be done empirically, with no evidence of the physics of the phenomena. On the other hand, for the TRENT2D, which is based on more sophisticated theories, the parameters are physically based and can be estimated from laboratory experiments.

Discussion of “Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms” by Soumendra Nath Kuiry, Kiran Pramanik, and Dhrubajyoti Sen

The paper addresses a very important topic in the field of the numerical modeling of free surface shallow flows: the treatment of the bed slope source term in numerical methods for the solution of two-dimensional 2D shallow water equations. Using an accurate and robust method for the treatment of this term is in fact essential to achieving good results when dealing with complex topography, that is, in the large majority of practical applications. However a few points in the paper need to be discussed. In 2D shallow water equations, the bottom slope source term represents the horizontal component of the force acted on the fluid by an uneven bottom. Writing the equations in differential form, the bottom slope source term appears as S0 = 0� ghzb /x �ghzb /y T . In a finite volume model, the source term is integrated over the computational cell, giving