Explicit Schemes for Dam-Break Simulations

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.     

Flux-Difference Splitting Schemes for 2D Flood Flows

Two numerical models for 2D flood flows are presented. One model is first-order accurate and another is second-order accurate. Roe's numerical flux is used to develop the first-order accurate model, while second-order accuracy, in space and time, is obtained by using the Lax-Wendroff numerical flux. A simple operator splitting is found to yield the same results as that obtained by using more complicated, and thus, time consuming, operator splitting. Roe's approximate Jacobian is used for conservative properties and Harten and Hyman's procedure is followed for the entropy inequality condition. Flux limiter is used in the second-order accurate model that removes oscillations while maintaining the order of accuracy. The models are verified against available experimental data of a 2D flood wave due to partial dam-break. Numerical experiments are conducted to verify the models' ability to correctly predict behavior of the free surface, in addition to prediction of depth and velocity.     

Numerical Solution of the Dam-Break Problem with a Discontinuous Galerkin Method

A discontinuous Galerkin method for the solution of the dam-break problem is presented. The scheme solves the shallow water equations with spectral elements, utilizing an efficient Roe approximate Riemann solver in order to capture bore waves. The solution is enhanced by a projection limiter that eliminates spurious oscillations near discontinuities. The main advantage of the model is the flexibility in approximating smooth solutions with high-order polynomials and resolving at the same time discontinuous shock waves. Furthermore, the finite element discretization is capable of handling complex geometries and producing correct results near the boundaries. Both the h- and p-type extensions are investigated for the one-dimensional dam break, and the results are verified by comparison with analytical solutions. The application to a two-dimensional dam-break problem shows the efficiency and stability of the method.     

Computational Dam-Break Hydraulics over Erodible Sediment Bed

This paper presents one of the first dedicated studies on mobile bed hydraulics of dam-break flow and the induced sediment transport and morphological evolution. A theoretical model is built upon the conservative laws of shallow water hydrodynamics, and a high-resolution numerical solution of the hyperbolic system is achieved using the total-variation-diminishing version of the second-order weighted average flux method in conjunction with the HLLC approximate Riemann solver and SUPERBEE limiter. It is found that a heavily concentrated and eroding wavefront first develops and then depresses gradually as it propagates downstream. In the early stage of the dam-break, a hydraulic jump is formed around the dam site due to rapid bed erosion, which attenuates progressively as it propagates upstream and eventually disappears. While the backward wave appears to migrate at the same speed as over a fixed bed, the propagation of the forward wavefront shows a complex picture compared to its fixed-bed counterpart as a result of the domination of rapid bed erosion initially, the density difference between the wavefront and the downstream ambient water in the intermediate period, and the pattern of the deformed bed profile in the long term. It is also found that the free surface profiles and hydrographs are greatly modified by bed mobility, which has considerable implications for flood prediction. The computed wave structure in the intermediate period exhibits great resemblance to available experiments qualitatively, and yet the existence of a shear wave is found in lieu of a secondary rarefaction postulated in an existing analysis. Finally, the use of the complete, rather than simplified, conservation equations is shown to be essential for correct resolution of the wave and bed structures, which suggests that previous models need reformulating.     

Dam-Break Waves in Power-Law Channel Section

The aim of this work is to highlight the effects of cross-sectional shape on dam-break wave propagation along channels by the solution of 1D conservative equations assuming a power-law variation of the channel width. An exact Riemann solution that allows a second-order accuracy of the solution for the power-law section shape is provided and is applied to the dam-break problem in valleys with different shapes but the same dam area. The streamflow state variables upstream of the bore and the bore speed for some typical sectional shapes (rectangular, triangular, concave, and convex banks) are determined as functions of variable flow depth differences and of the power law index.     

Weighted Averaged Flux-Type Scheme for Shallow-Water Equations with Fractional Step Method

A numerical model describing two-dimensional fluid motions has been developed on an unstructured grid system. By using a fractional step method, a two-dimensional problem governed by the two-dimensional shallow-water equations is treated as two one-dimensional problems. Thus it is possible to simulate two-dimensional numerical problems with a higher computational efficiency. One-dimensional problems are solved by using an upwind total variation diminishing version of the second-order weighted averaged flux method with an approximate Riemann solver. Numerical oscillations commonly observed in second-order numerical schemes are controlled by exploiting a flux limiter. For the general purpose, the model can simulate on an arbitrary topography, treat a moving boundary, and resolve a shock. Five ideal and practical problems are tested. Very accurate results are observed.     

Applicability of Sediment Transport Capacity Formulas to Dam-Break Flows over Movable Beds

In this paper, we investigate the extent to which well-known sediment transport capacity formulas can be used in one-dimensional (1D) numerical modeling of dam-break waves over movable beds. The 1D model considered here is a one-layer model based on the shallow-water equations, a bed update (Exner) equation, a space-lag equation for the nonequilibrium sediment transport and an empirical formula calculating the sediment transport capacity of the flow. The model incorporates a variety of sediment transport capacity formulas proposed by Meyer-Peter and Müller, Bagnold, Engelund and Hansen, Ackers and White, Smart and Jaeggi, van Rijn, Rickenmann, Cheng, Abrahams and Camenen, and Larson. We examine the performance of each formula by simulating four idealized laboratory cases on dam-break waves over sandy beds. Comparisons between numerical results and measurements show that for each case better predictions are obtained using a particular formula, but overall, formulas proposed by Meyer-Peter and Müller (with the factor 8 being replaced by 12), Smart and J?ggi, Cheng, Abrahams and Camenen, and Larson rank as the best predictors for the entire range of conditions studied here. Moreover, results show that in the cases where a bed step exists, implementing a mass failure mechanism in the numerical modeling plays an important role in reproducing the bed and water profiles.     

Dam Break in Channels with 90° Bend

In practice, dam-break modeling is generally performed using a one-dimensional (1D) approach for its limited requirements in data and computation. However, for valleys with multiple sharp bends, such a 1D model may fail for predicting as well the maximum water level as the wave arrival time. This paper presents an experimental study of a dam-break flow in an initially dry channel with a 90° bend, with refined measurements of water level and velocity field. The measured data are compared to some numerical results computed with finite-volume schemes associated with Roe-type flux calculation. The 1D approach reveals the expected limits, while a full two-dimensional (2D) approach provides fine level prediction and rather satisfactory information about the arrival time. A hybrid approach is now proposed, mixing the 1D model for the straight reaches and local 2D models for the bends. The compatibility of the Roe fluxes at the interfaces requires a careful formulation, but the resulting scheme seems able to capture reflection and diffraction processes in such a way that the results are really good in what concerns the water level.     

Investigation of the Importance of Spatial Resolution for Two-Dimensional Shallow-Water Model Accuracy

The importance of spatial resolution for two-dimensional shallow-water model accuracy has been investigated by testing the effect of mesh refinement on two test cases based on laboratory dam-break experiments. A balanced first-order accurate upwind Q-Scheme and a second-order accurate upwind Hancock Monotone Upstream-centered Scheme for Conservation Laws scheme were both first validated on an analytical test, and then applied to the experimental dam-break test cases on four meshes of different density. Simulation results were evaluated through comparison of experimental and computed water level values at several available gauge points. Model sensitivity analysis showed that (1) mesh density was not critical for results accuracy; (2) excessive mesh refinement somewhat deteriorated the results; and (3) optimal spatial resolution was relatively low. Response is shown to be highly complex and no simple relation between spatial resolution and model accuracy has been found.     

One-Dimensional Modeling of Dam-Break Flow over Movable Beds

A one-dimensional model has been established to simulate the fluvial processes under dam-break flow over movable beds. The hydrodynamic model adopts the generalized shallow water equations, which consider the effects of sediment transport and bed change on the flow. The sediment model computes the nonequilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. In particular, a correction factor is proposed to modify the Van Rijn formulas of equilibrium bed-load transport rate and near-bed suspended-load concentration for the simulation of sediment transport under high-shear flow conditions. The governing equations are solved by an explicit finite-volume method with the first-order upwind scheme for intercell fluxes. The model has been tested in two experimental cases, with fairly good agreement between simulations and measurements. The sensitivities of the model results to parameters such as the sediment nonequilibrium adaptation length, Manning’s roughness coefficient and the proposed correction factor have been verified. The proposed model has also been compared to an existing model and the results indicate the new model is more reliable.     

Cartesian Cut Cell Two-Fluid Solver for Hydraulic Flow Problems

A two-fluid solver which can be applied to a variety of hydraulic flow problems has been developed. The scheme is based on the solution of the incompressible Euler equations for a variable density fluid system using the artificial compressibility method. The computational domain encompasses both water and air regions and the interface between the two fluids is treated as a contact discontinuity in the density field which is captured automatically as part of the solution using a high resolution Godunov-type scheme. A time-accurate solution has been achieved by using an implicit dual-time iteration technique. The complex geometry of the solid boundary arising in the real flow problems is represented using a novel Cartesian cut cell technique, which provides a boundary fitted mesh without the need for traditional mesh generation techniques. A number of test cases including the classical low amplitude sloshing tank and dam-break problems, as well as a collapsing water column hitting a downstream obstacle have been calculated using the present approach and the results compare very well with other theoretical and experimental results. Finally, a test case involving regular waves interacting with a sloping beach is also calculated to demonstrate the applicability of the method to real hydraulic problems.     

Numerical Solution of Boussinesq Equations to Simulate Dam-Break Flows

To investigate the effect of nonhydrostatic pressure distribution, dam-break flows are simulated by numerically solving the one-dimensional Boussinesq equations by using a fourth-order explicit finite-difference scheme. The computed water surface profiles for different depth ratios have undulations near the bore front for depth ratios greater than 0.4. The results obtained by using the Saint Venant equations and the Boussinesq equations are compared to determine the contribution of individual Boussinesq terms in the simulation of dam-break flow. It is found that, for typical engineering applications, the Saint Venant equations give sufficiently accurate results for the maximum flow depth and the time to reach this value at a location downstream of the dam.     

Practical Aspects in Comparing Shock-Capturing Schemes for Dam Break Problems

The results of a survey aimed at comparing the performances of first-order and total variation diminishing (TVD) second-order upwind flux difference splitting schemes, first-order space-centered schemes, and second-order space-centered schemes with the TVD artificial viscosity term are reported here. The schemes were applied to the following dam-break wave cases: in a dry frictionless horizontal channel; in a dry, rough and sloping channel; and in a nonprismatic channel. Among first-order schemes, the diffusive scheme provides only slightly less accurate results than those obtained by the Roe scheme. For TVD second-order schemes, no significant difference between the upwind scheme and central schemes are reported. In the case of a dam break in a dry frictionless horizontal channel, the second-order schemes were two- to five-fold more accurate than the diffusive scheme and Roe’s scheme. These differences in scheme performances drastically reduce when the results obtained for the rough sloping channel test and for the nonprismatic channel test are analyzed. In particular, the accuracy of the diffusive and Roe’s schemes is similar to second-order schemes when such features of dam break wave, relevant from an engineering viewpoint, like wave peak arrival time and maximum water depths, are considered.     

Simulation of the St. Francis Dam-Break Flood

A two-dimensional (2D) simulation of flooding from the 1928 failure of St. Francis Dam in southern California is presented. The simulation algorithm solves shallow-water equations using a robust unstructured grid Godunov-type scheme designed for wetting and drying and achieves good results. Flood extent and flood travel time are predicted within 4 and 10% of observations, respectively. Representation of terrain by the mesh is identified as the dominant factor affecting accuracy, and an iterative process of mesh refinement and convergence checks is implemented to minimize errors. The most accurate predictions are achieved with a uniformly distributed Manning n = 0.02. A 50% increase in n increases travel time errors to 25% but has little effect on flood extent predictions. This highlights the challenge of a priori travel time prediction but robustness in flood extent prediction when topography is well resolved. Predictions show a combination of subcritical and supercritical flow regimes. The leading edge of the flood was supercritical in San Francisquito Canyon, but due to channel tortuosity, the wetting front reflected off canyon walls causing a transition to subcritical flow, considerably larger depths, and a standing wave in one particular reach that accounts for a 30% fluctuation in discharge. Elsewhere, oblique shocks locally increased flood depths. The 2D dam-break model is validated by its stability and accuracy, conservation properties, ability to calibrate with a physically realistic and simple resistance parametrization, and modest computational cost. Further, this study highlights the importance of a dynamic momentum balance for dam-break flood simulation.     

Two-Dimensional Simulation of Subcritical Flow at a Combining Junction: Luxury or Necessity?

Classically, in open-channel networks, the flow is numerically approximated by the one-dimensional Saint Venant equations coupled with a junction model. In this study, a comparison between the one-dimensional (1D) and two-dimensional (2D) numerical simulations of subcritical flow in open-channel networks is presented and completely described allowing for a full comprehension of the modeling of water flow. For the 1D, the mathematical model used is the 1D Saint Venant equations to find the solution in branches. For junction, various models based on momentum or energy conservation have been developed to relate the flow variables at the junction. These models are of empirical nature due to certain parameters given by experimental results and moreover they often present a reduced field of validity. In contrast, for the 2D simulation, the junction is discretized into triangular cells and we simply apply the 2D Saint Venant equations, which are solved by a second-order finite-volume method. In order to give an answer to the question of luxury or necessity of the 2D approach, the 1D and 2D numerical results for steady flow are compared to existing experimental data.     

Discretization of Integral Equations Describing Flow in Nonprismatic Channels with Uneven Beds

Application of the finite-volume method in one dimension for open channel flow predictions mandates the direct discretization of integral equations for mass conservation and momentum balance. The integral equations include source terms that account for the forces due to changes in bed elevation and channel width, and an exact expression for these source term integrals is presented for the case of a trapezoidal channel cross section whereby the bed elevation, bottom width, and inverse side slope are defined at cell faces and assumed to vary linearly and uniformly within each cell, consistent with a second-order accurate solution. The expressions may be used in the context of any second-order accurate finite-volume scheme with channel properties defined at cell faces, and it is used here in the context of the Monotone Upwind Scheme for Conservation Laws (MUSCL)-Hancock scheme which has been adopted by many researchers. Using these source term expressions, the MUSCL-Hancock scheme is shown to preserve stationarity, accurately converge to the steady state in a frictionless flow test problem, and perform well in field applications without the need for upwinding procedures previously reported in the literature. For most applications, an approximate, point-wise treatment of the bed slope and nonprismatic source terms can be used instead of the exact expression and, in contrast to reports on other finite-volume-based schemes, will not cause unphysical oscillations in the solution.     

Flood Simulation Using a Well-Balanced Shallow Flow Model

This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against an analytical case of flow sloshing in a domain with a parabolic bed profile, the model is applied to simulate an inundation event in a 36?km2 floodplain in Thamesmead near London. The numerical predictions are compared with analytical solutions and alternative numerical results.     

Dispersive and Nonhydrostatic Pressure Effects at the Front of Surge

Undular bores and shocks generated by dam-break flows or tsunamis are examined considering nonhydrostatic pressure and dispersive effects in one- and two-horizontal-dimensional space. The fully nonlinear Boussinesq-type equations based on a weakly nonhydrostatic pressure assumption are chosen as the governing equations. The equation set is solved by a fourth-order accurate finite-volume method with an approximate Riemann solver. Several typical benchmark problems such as dam-break flows and tsunami wave fission are tested in one- and two-horizontal-dimensional space. The computed results by the Boussinesq-type model are at least as accurate as the results by the hydrostatic shallow water equations. This is particularly evident near the steep front of the wave, where frequency dispersion can play an important role. The magnitude of this nonhydrostatic pressure and dispersive effect near the front is quantified, and the engineering implications of neglecting these physics, as would be done through the use of a hydrostatic model, are discussed.     

Discontinuous Galerkin Finite-Element Method for Simulation of Flood in Crossroads

A numerical solution of the two-dimensional Saint Venant equations is presented for the study of the propagation of the floods through the crossroads of the city. The numerical scheme is a Runge-Kutta discontinuous Galerkin method (RKDG) with a slope limiter. The work studies the robustness and the stability of the method. The study is organized around three aspects: the prediction of the water depths, the location of the right and oblique hydraulic jumps in the crossing, and especially the distribution of the flow discharges in the downstream branches. The objective of this paper was to use the RKDG method in order to simulate supercritical flow in crossroads and to compare these simulations with experimental results and to show the advantage of this RKDG method compared to a second-order finite-volume method. A good agreement between the proposed method and the experimental data was found. The method is then able to simulate the flow patterns observed experimentally and to predict accurately the water depths, the location of the hydraulic jumps, and the discharge distribution in the downstream branches.     

Upwind Conservative Scheme for the Saint Venant Equations

An upwind conservative scheme with a weighted average water-surface-gradient approach is proposed to compute one-dimensional open channel flows. The numerical scheme is based on the control volume method. The intercell flux is computed by the one-sided upwind method. The water surface gradient is evaluated by the weighted average of both upwind and downwind gradients. The scheme is tested with various examples, including dam-break problems in channels with rectangular and triangular cross-sections, hydraulic jump, partial dam-break problem, overtopping flow, a steady flow over bump with hydraulic jump, and a dam-break flood case in a natural river valley. Comparisons between numerical and exact solutions or experimental data demonstrated that the proposed scheme is capable of accurately reproducing various open channel flows, including subcritical, supercritical, and transcritical flows. The scheme is inherently robust, stable, and monotone. The scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients or discontinuities in the solution.