P. García-Navarro

On numerical treatment of the source terms in the shallow water equations

Abstract Upwind schemes are very well adapted to advection dominated flows and have become popular for applications involving the Euler system of equations. Recently, Riemann solver-based techniques such as Roe’s scheme have become a successful tool for numerical simulation of other conservation laws like the shallow water equations. One of the disadvantages of this technique is related to the treatment of the source terms of the equations. The conservativity of the scheme can be seriously damaged if a careless treatment is applied. Previous papers studied the way to treat the terms arising from bed level changes. This paper deals with the analysis of the main reasons leading to a correct treatment of the geometrical source terms, that is, those representing the changes in cross-section which may be linked to the specific dependence of the flux function on the geometry.

Weak solutions for partial differential equations with source terms: Application to the shallow water equations

Weak solutions of problems with m equations with source terms are proposed using an augmented Riemann solver defined by m+1 states instead of increasing the number of involved equations. These weak solutions use propagating jump discontinuities connecting the m+1 states to approximate the Riemann solution. The average of the propagated waves in the computational cell leads to a reinterpretation of the Roes approach and in the upwind treatment of the source term of Vazquez-Cendon. It is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities, as usually done in the homogeneous case. Positivity requirements over the values of the intermediate states are the only way to control the global stability of the method. Also it is shown that the definition of well-balanced equilibrium in trivial cases is not sufficient to provide correct results: it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded. The one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study. The stability region is shown to differ from the one defined for the case without source terms, and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet/dry front is a particular case, of the wet/wet fronts. The consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case. Starting from this result, 1D and 2D numerical schemes are developed for both quadrilateral and triangular grids, enforcing conservation and positivity over the solution, allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size.

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.

McCormack's method for the numerical simulation of one-dimensional discontinuous unsteady open channel flow

This paper describes the use of the McCormack explicit finite difference scheme and the treatment of the boundary problem in the development of a one-dimensional simulation model that solves the St. Venant equations of the unsteady open channel flow. External and internal boundaries are considered. Various illustrative cases are presented to show the efficiency of this technique.

Godunov-type methods for free-surface shallow flows: A review

This review paper concerns the application of numerical methods of the Godunov type to the computation of approximate solutions to free-surface gravity flows modelled under a shallow-water type assumption. In the absence of dissipative processes the resulting governing equations are, with rare exceptions, of hyperbolic type. This mathematical property has, in the main, been responsible for the transfer of the Godunov-type numerical methodology, initially developed for the compressible Euler equations of gas dynamics in the aerospace community, to hydraulics and related areas of application. Godunov methods offer distinctive advantages over other methods. For example, they give correct representation of discontinuous waves (bores); this means the correct propagation speed (the methods are conservative), sharp definition of transitions and absence of unphysical oscillations in the vicinity of the wave. Future trends include (i) the use of these methods to deal with physically more complete models without the shallow water assumption and (ii) implementation of very-high order versions of these methods

An implicit method for water flow modelling in channels and pipes

An implicit time integration method for the simulation of steady and unsteady flow in pipes and channels is presented. It is based on the theory of Total Variation Diminishing (TVD) methods. A conservative linearization leads to a block tridiagonal system of equations which can be cheaply solved by means of a non-iterative matrix decomposition method. It keeps the advantages of classical implicit schemes, and properly deals with all kinds of flow. It is specially suited for steady flows including phenomena such as hydraulic jumps and also gives satisfactory results for time dependent problems. Several test cases are shown to illustrate the performance of this implicit technique in single channels and networks.

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.

An Exner-based coupled model for two-dimensional transient flow over erodible bed

Transient flow over erodible bed is solved in this work assuming that the dynamics of the bed load problem is described by two mathematical models: the hydrodynamic model, assumed to be well formulated by means of the depth averaged shallow water equations, and the Exner equation. The Exner equation is written assuming that bed load transport is governed by a power law of the flow velocity and by a flow/sediment interaction parameter variable in time and space. The complete system is formed by four coupled partial differential equations and a genuinely Roe-type first order scheme has been used to solve it on triangular unstructured meshes. Exact solutions have been derived for the particular case of initial value Riemann problems with variable bed level and depending on particular forms of the solid discharge formula. The model, supplied with the corresponding solid transport formulae, is tested by comparing with the exact solutions. The model is validated against laboratory experimental data of different unsteady problems over erodible bed.

Dam-break flow simulation : some results for one-dimensional models of real cases

Abstract In many countries, the determination of the parameters of the wave, likely to be produced after the failure of a dam, is required by law, and systematic studies are mandatory. There is a necessity to develop adequate numerical solvers which are able to reproduce situations originated from the irregularities of a non-prismatic bed and to model the complete equations that progress despite the irregular character of the data. Many hydraulic situations can be described by means of a one-dimensional (1D) model, either because a more detailed resolution is unnecessary or because the flow is markedly 1D. Many techniques have been developed recently for systems of conservation laws in 1D (in the context of gas dynamics). Some years after their adoption for solving problems in gas dynamics, upwind and total variation diminishing (TVD) numerical schemes have been successfully used for the solution of the shallow water equations, with similar advantages. Their use is nevertheless only gradually gaining acceptance in this sector. The performance of some of these techniques for practical applications in river flow is reported in this work.