A finite volume solver for 1D shallow‐water equations applied to an actual river

This paper describes the numerical solution of the 1D shallow‐water equations by a finite volume scheme based on the Roe solver. In the first part, the 1D shallow‐water equations are presented. These equations model the free‐surface flows in a river. This set of equations is widely used for applications: dam‐break waves, reservoir emptying, flooding, etc. The main feature of these equations is the presence of a non‐conservative term in the momentum equation in the case of an actual river. In order to apply schemes well adapted to conservative equations, this term is split in two terms: a conservative one which is kept on the left‐hand side of the equation of momentum and the non‐conservative part is introduced as a source term on the right‐hand side. In the second section, we describe the scheme based on a Roe Solver for the homogeneous problem. Next, the numerical treatment of the source term which is the essential point of the numerical modelisation is described. The source term is split in two components: one is upwinded and the other is treated according to a centred discretization. By using this method for the discretization of the source term, one gets the right behaviour for steady flow. Finally, in the last part, the problem of validation is tackled. Most of the numerical tests have been defined for a working group about dam‐break wave simulation. A real dam‐break wave simulation will be shown. Copyright © 2002 John Wiley & Sons, Ltd.     

Lattice Boltzmann modeling of shallow water flows over discontinuous beds

Numerical modeling of shallow water flows over discontinuous beds is presented. The flows are described with the shallow water equations and the equations are solved using the lattice Boltzmann method (LBM) with single relaxation time (Bhatnagar–Gross–Krook‐LBM (BGK‐LBM)) and the multiple relaxation time (MRT‐LBM). The weighted centered scheme for force term together with the bed height for a bed slope is described to improve simulation of flows over discontinuous bed. Furthermore, the resistance stress is added to include the local head loss caused by flow over a step. Four test cases, one‐dimensional tidal over regular bed and steps, dam‐break flows, and two‐dimensional shallow water flow over a square block, are considered to verify the present method. Agreements between predictions and analytical solutions are satisfactory. Furthermore, the performance and CPU cost time of BGK‐LBM and MRT‐LBM are compared and studied. The results have shown that the lattice Boltzmann method is simple and accurate for simulating shallow water flows over discontinuous beds. This demonstrates the capability and applicability of the lattice Boltzmann method in modeling shallow water flows on bed topography with a discontinuity in practical hydraulic engineering. Copyright © 2014 John Wiley & Sons, Ltd.     

Weighted essential non‐oscillatory schemes for tidal bore on unstructured meshes

In this paper, the third‐order weighted essential non‐oscillatory (WENO) schemes are used to simulate the two‐dimensional shallow water equations with the source terms on unstructured meshes. The balance of the flux and the source terms makes the shallow water equations fit to non‐flat bottom questions. The simulation of a tidal bore on an estuary with trumpet shape and Qiantang river is performed; the results show that the schemes can be used to simulate the current flow accurately and catch the stronger discontinuous in water wave, such as dam break and tidal bore effectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

A wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

This paper describes a numerical solver of well‐balanced, 2D depth‐averaged shallow water‐sediment equations. The equations permit variable horizontal fluid density and are designed to model water‐sediment flow over a mobile bed. A Godunov‐type, Harten–Lax–van Leer contact (HLLC) finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws that describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi‐analytical solutions for bedload transport and suspended sediment transport, respectively. The well‐balanced property of the equations is verified for a variable‐density dam break flow over discontinuous bathymetry. Simulations of an idealised dam‐break flow over an erodible bed are in excellent agreement with previously published results, validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable‐density governing equations. Flow hydrodynamics and final bed topography of a laboratory‐based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water‐sediment models to the choice of closure relationships. Copyright © 2016 John Wiley & Sons, Ltd.     

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite volume solution to integrated shallow surface–saturated groundwater flow

This paper describes development of an integrated shallow surface and saturated groundwater model (GSHAW5). The surface flow motion is described by the 2‐D shallow water equations and groundwater movement is described by the 2‐D groundwater equations. The numerical solution of these equations is based on the finite volume method where the surface water fluxes are estimated using the Roe shock‐capturing scheme, and the groundwater fluxes are computed by application of Darcy's law. Use of a shock‐capturing scheme ensures ability to simulate steady and unsteady, continuous and discontinuous, subcritical and supercritical surface water flow conditions. Ground and surface water interaction is achieved by the introduction of source‐sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The numerical solutions and coupling algorithms are explained. The model has been applied to 1‐D and 2‐D test scenarios. The results have shown that the model can produce very accurate results and can be used for simulation of situations involving interaction between shallow surface and saturated groundwater flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Shoreline tracking and implicit source terms for a well balanced inundation model

The HyFlux2 model has been developed to simulate severe inundation scenario due to dam break, flash flood and tsunami‐wave run‐up. The model solves the conservative form of the two‐dimensional shallow water equations using the finite volume method. The interface flux is computed by a Flux Vector Splitting method for shallow water equations based on a Godunov‐type approach. A second‐order scheme is applied to the water surface level and velocity, providing results with high accuracy and assuring the balance between fluxes and sources also for complex bathymetry and topography. Physical models are included to deal with bottom steps and shorelines. The second‐order scheme together with the shoreline‐tracking method and the implicit source term treatment makes the model well balanced in respect to mass and momentum conservation laws, providing reliable and robust results. The developed model is validated in this paper with a 2D numerical test case and with the Okushiri tsunami run up problem. It is shown that the HyFlux2 model is able to model inundation problems, with a satisfactory prediction of the major flow characteristics such as water depth, water velocity, flood extent, and flood‐wave arrival time. The results provided by the model are of great importance for the risk assessment and management. Copyright © 2009 John Wiley & Sons, Ltd.     

Two‐dimensional shallow water equations by composite schemes

Composite schemes are formed by global composition of several Lax–Wendroff steps followed by a diffusive Lax–Friedrichs or WENO step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. These schemes are applied to the shallow water equations in two dimensions. The Lax–Friedrichs composite is also formulated for a trapezoidal mesh, which is necessary in several example problems. The suitability of the composite schemes for the shallow water equations is demonstrated on several examples, including the circular dam break problem, the shock focusing problem and supercritical channel flow problems. Copyright © 1999 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

Fourth order finite volume solution to shallow water equations and applications

This study presents the fourth order accurate finite volume solution to shallow water equations. Fourth order accuracy in space was provided by using the Monotone Upstream‐centered Schemes for Conservation Laws–Total Variation Diminishing scheme, whereas fourth order accurate solution in time was achieved by using the third order predictor scheme of Adams–Basforth followed by the fourth order corrector scheme of Adams–Moulton. The applicability and accuracy of the solution algorithm were explored on complex flow conditions. These flow conditions cover a theoretical well‐known partial two‐dimensional dam break problems and an experimental flow in a compound channel with or without a bridge. The applicability limits of the solution algorithm were discussed. The overall performance of the solution was found to be reasonably good. Copyright © 2013 John Wiley & Sons, Ltd.     

Development of a generalized multi‐layer model for 3‐D simulation of free surface flows

A mathematical model was developed for three‐dimensional (3‐D) simulation of free surface flows. In this model, the flow depth is divided into a number of layers and shallow water equations are integrated in each layer to derive the hydrodynamic equations. To give a general form to this model, each layer is assumed to be non‐horizontal with varying thickness in the flow domain. A non‐orthogonal curvilinear coordinate system is employed in the model, to allow for flexibility in dealing with the irregular geometry of natural watercourses. Due to the similarity in governing equations, two‐dimensional (2‐D) depth averaged programs can be developed into a multi‐layer model. The development for a depth averaged program and its numerical scheme is described in this paper. Experimental data and semi‐analytical solutions are used to evaluate the performance of the model. Three different cases of open channel flow are tested: 1‐flow in a straight open channel, 2‐the flow development region in a channel, and 3‐flow in a meandering channel. It is shown that the model has the capability to predict velocity distribution and secondary flows in complex 3‐D flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of dam‐ and dyke‐break hydrodynamics on dynamically adaptive quadtree grids

Flooding due to the failure of a dam or dyke has potentially disastrous consequences. This paper presents a Godunov‐type finite volume solver of the shallow water equations based on dynamically adaptive quadtree grids. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) scheme is used to evaluate interface fluxes in both wet‐ and dry‐bed applications. The numerical model is validated against results from alternative numerical models for idealized circular and rectangular dam breaks. Close agreement is achieved with experimental measurements from the CADAM dam break test and data from a laboratory dyke break undertaken at Delft University of Technology. Copyright © 2004 John Wiley Sons, Ltd.     

Simulation of rapidly varying flow using an efficient TVD–MacCormack scheme

An efficient numerical scheme is outlined for solving the SWEs (shallow water equations) in environmental flow; this scheme includes the addition of a five‐point symmetric total variation diminishing (TVD) term to the corrector step of the standard MacCormack scheme. The paper shows that the discretization of the conservative and non‐conservative forms of the SWEs leads to the same finite difference scheme when the source term is discretized in a certain way. The non‐conservative form is used in the solution outlined herein, since this formulation is simpler and more efficient. The time step is determined adaptively, based on the maximum instantaneous Courant number across the domain. The bed friction is included either explicitly or implicitly in the computational algorithm according to the local water depth. The wetting and drying process is simulated in a manner which complements the use of operator‐splitting and two‐stage numerical schemes. The numerical model was then applied to a hypothetical dam‐break scenario, an experimental dam‐break case and an extreme flooding event over the Toce River valley physical model. The predicted results are free of spurious oscillations for both sub‐ and super‐critical flows, and the predictions compare favourably with the experimental measurements. Copyright © 2006 John Wiley & Sons, Ltd.     

HLLC scheme with novel wave‐speed estimators appropriate for two‐dimensional shallow‐water flow on erodible bed

This paper presents a first‐order HLLC (Harten‐Lax‐Van Leer with contact discontinuities) scheme to solve the Saint‐Venant shallow‐water equations, including morphological evolution of the bed by erosion and deposition of sediments. The Exner equation is used to model the morphological evolution of the bed, while a closure equation is needed to evaluate the rate of sediment transport. The system of Saint‐Venant–Exner equations is solved in a fully coupled way using a finite‐volume technique and a HLLC solver for the fluxes, with a novel wave‐speed estimator adapted to the Exner equation. Wave speeds are usually derived by computing the eigenvalues of the full system, which is highly time‐consuming when no analytical expression is available. In this paper, an eigenvalue analysis of the full system is conducted, leading to simple but still accurate wave‐speed estimators. The new numerical scheme is then tested in three different situations: (1) a circular dam‐break flow over movable bed, (2) an one‐dimensional bed aggradation problem simulated on a 2D unstructured mesh and (3) the case of a dam‐break flow in an erodible channel with a sudden enlargement, for which experimental measurements are available. Copyright © 2010 John Wiley & Sons, Ltd.     

A Godunov method for the computation of erosional shallow water transients

A Godunov method is proposed for the computation of open‐channel flows in conditions of rapid bed erosion and intense sediment transport. Generalized shallow water equations govern the evolution of three distinct interfaces: the water free‐surface, the boundary between pure water and a sediment transport layer, and the morphodynamic bottom profile. Based on the HLL scheme of Harten, Lax and Van Leer (1983), a finite volume numerical solver is constructed, then extended to second‐order accuracy using Strang splitting and MUSCL extrapolation. Lateralisation of the momentum flux is adopted to handle the non‐conservative product associated with bottom slope. Computational results for erosional dam‐break waves are compared with experimental measurements and semi‐analytical Riemann solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A mesh patching method for finite volume modelling of shallow water flow

A new mesh‐patching model is presented for shallow water flow described by the 2D non‐linear shallow water (NLSW) equations. The mesh‐patching model is based on AMAZON, a high‐resolution NLSW engine with an improved HLLC approximate Riemann solver. A new patching algorithm has been developed, which not only provides improved spatial resolution of flow features in particular parts of the mesh, but also simplifies and speeds up the (structured) grid generation process for an area with complicated geometry. The new patching technique is also compatible with increasingly popular parallel computing and adaptive grid techniques. The patching algorithm has been tested with moving bores, and results of test problems are presented and compared to previous work. Copyright © 2005 John Wiley & Sons, Ltd.     

An unstructured finite volume model for dam‐break floods with wet/dry fronts over complex topography

A robust, well‐balanced, unstructured, Godunov‐type finite volume model has been developed in order to simulate two‐dimensional dam‐break floods over complex topography with wetting and drying. The model is based on the nonlinear shallow water equations in hyperbolic conservation form. The inviscid fluxes are calculated using the HLLC approximate Riemann solver and a second‐order spatial accuracy is achieved by implementing the MUSCL reconstruction technique. To prevent numerical oscillations near shocks, slope‐limiting techniques are used for controlling the total variation of the reconstructed field. The model utilizes an explicit two‐stage Runge–Kutta method for time stepping, whereas implicit treatments for friction source terms. The novelties of the model include the flux correction terms and the water depth reconstruction method both for partially and fully submerged cells, and the wet/dry front treatments. The proposed flux correction terms combined with the water depth reconstruction method are necessary to balance the bed slope terms and flux gradient in the hydrostatical steady flow condition. Especially, this well‐balanced property is also preserved in partially submerged cells. It is found that the developed wet/dry front treatments and implicit scheme for friction source terms are stable. The model is tested against benchmark problems, laboratory experimental data, and realistic application related to dam‐break flood wave propagation over arbitrary topography. Numerical results show that the model performs satisfactorily with respect to its effectiveness and robustness and thus has bright application prospects. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparison between the theory and the numerical experiment in the problem of dam break on a jump of the cross-sectional area of a rectangular channel

Within the framework of the first approximation of the spatially one-dimensional shallow water theory the problem of flow generated by the dam break on a jump of the cross-sectional area of a rectangular channel is solved in the case in which the upper pool is wider than the lower one. The self-similar solutions constructed contain an euristic parameter related with the amount of the total flow energy lost on the cross-sectional area jump. The parameter is determined by means of comparing the one-dimensional solutions with the results of the numerical modeling of the problem on the basis of spatially two-dimensional equations of the shallow water theory.