P. Brufau

1D Mathematical modelling of debris flow

Debris flow is modelled using the equations governing the dynamics of a liquid-solid mixture. An upwind finite volume scheme is applied to solve the resulting differential equations in one dimension. These equations have a structure similar to those of the monophasic water flow, differing from them by the presence of some terms characteristic of the bifasic nature of the mixture, such as granular bed erosion velocity, sediment concentration, bed shear stress, etc. The model and the system of equations to be solved are presented with the description of the implementation of the upwind scheme for the resulting hyperbolic conservation system. The numerical method is first order in both space and time. The treatment of the source terms is specified in detail and some comparison with laboratory experiments are presented.

Two‐dimensional dam break flow simulation

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests. Free surface flow in channels can be described mathematically by the shallow-water system of equations. These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two-dimensional dam break flows. A cell centred finite volume method based on Roes approximate Riemann solver across the edges of the cells is presented and the results are compared for first- and second-order accuracy. Special treatment of the friction term has been adopted and will be described. The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, i.e. that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC-IST). Comparisons of experimental and numerical results are shown.

Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique

In the context of numerical techniques for solving unsteady free surface problems, finite element and finite volume approximations are widely used. A class of upwind methods which attempts to model the equations in a genuinely multidimensional manner has been recently introduced as an alternative. Multidimensional upwind schemes (MUS) were developed initially for the approximation of steady-state solutions of the two-dimensional Euler equations on unstructured grids, although they can be applicable to any system of hyperbolic conservation laws, such as the shallow water equations. The formal analogy between the two systems of equations is useful for simple cases. However, in practical applications of interest in hydraulics, complex geometries and bottom slope variation can lead to important numerical errors produced by an inadequate source term discretization. This problem has been analyzed and, in this work, the necessity of a multidimensional upwind discretization of the source terms is justified. The basis of the numerical method is stated and the particular adaptation to unsteady shallow water flows over irregular geometry is described. As test cases, laboratory experimental data are used together with academic tests for validation.

2D modelling of erosion/deposition processes with suspended load using

This work describes a finite volume model applied to solve the coupled shallow water/load transport equations, where low concentrations of sediment load are assumed so that the concentration of the solid fraction is assumed not to influence the equations governing the dynamics. The unsteady simulation of erodible bed requires special attention in the discretization of the bed and friction source terms as important instabilities can arise in some situations. The upwind discretization of the time variable bed slope source terms is presented to provide an exact balance of the numerical fluxes and to guarantee steady-state solutions contrary to the pointwise formulation that leads to instabilities destroying the computation. Robust numerical schemes are presented for both moving and fixed boundaries. In the last case a numerical technique is provided to keep the concentration load bounded for values of CFL (Courant–Friedrichs–Lewy) greater than one and therefore decreasing the necessary computational cost.

A Riemann coupled edge (RCE) 1D–2D finite volume inundation and solute transport model

A novel 1D–2D shallow water model based on the resolution of the Riemann problem at the coupled grid edges is presented in this work. Both the 1D and the 2D shallow water models are implemented in a finite volume framework using approximate Roe’s solvers that are able to deal correctly with wet/dry fronts. After an appropriate geometric link between the models, it is possible to define local Riemann problems at each coupled interface and estimate the contributions that update the cell solutions from the interfaces. The solute transport equation is also incorporated into the proposed procedure. The numerical results achieved by the 1D–2D coupled model are compared against a complete 2D model, which is considered the reference solution. The computational time is also examined.

Diffusive-Wave Based Hydrologic-Hydraulic Model with Sediment Transport. II: Validation and Practical Application

AbstractThe development of a distributed two-dimensional (2D) hydrologic-hydraulic simulation model was presented in Paper I. The simulation model combined overland flow (kinematic/diffusive wave models), hillslope sediment transport, and groundwater flow apart from the water exchange mechanisms between zones. Particular attention was paid to the upwind discretization of the surface flow equations. In this paper, the proposed model is validated by using four test cases with exact solutions, one academic test case, and two laboratory test cases. The model adequately reproduced front advance over dry beds of any slope and water table evolution in simple cases. As practical application of the model, the simulation of real events in two experimental basins is also presented. The work is focused on the influence of the choice of the empirical parameters on the model results concerning solid and liquid discharges. Also, because of the lack of information referring the boundary and initial conditions of the grou...

Multidimensional Upwind Schemes: Application to Hydraulics

In the field of the numerical simulation of conservation laws, upwind and TVD techniques have progressively gained acceptance. Originally, they were derived for homogeneous scalar equations or systems of equations in one spatial dimension. Their extension to more than one spatial dimension is not straightforward.

Application of an adjoint-based optimization procedure for the optimal control of internal boundary conditions in the shallow water equations

ABSTRACT The shallow water equations have been extensively studied and used to model unsteady open channel flows in different applications. They belong to the category of hyperbolic partial differential equations and their treatment has recently been benefited from many numerical contributions leading to robust, accurate and well-balanced solutions. The correct formulation of external and internal boundary conditions is required to achieve a useful model in practical application. Control of internal boundary conditions may be useful in water distribution facilities. Therefore, development of a control strategy based on the fully dynamical mathematical model becomes attractive and justified. This work is devoted to the implementation of an adjoint based sensitivity analysis for the control of sluice gates in open-channel flow, formulated as internal boundary conditions. This is one of the most complex tasks in multiple regulated water delivery situations. Based on gradient method optimizers, the control of the whole channel to satisfy different requirements at several points of the channel is discussed. One of the achievements in this work is that the whole optimization process is performed two orders of magnitude faster than in real time. Moreover, the numerical results show this promising technique is a feasible way to obtain a robust and efficient control method.