Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flows

A total variation diminishing Runge Kutta discontinuous Galerkin finite-element method for two-dimensional depth-averaged shallow water equations has been developed. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally as efficient as comparable finite-volume schemes. For smooth parts of the solution, the scheme is second order for linear elements and third order for quadratic shape functions both in time and space. Shocks are usually captured within only two elements. Several steady transcritical and transient flows are investigated to confirm the accuracy and convergence of the scheme. The results show excellent agreement with analytical solutions. For investigating a flume experiment of supercritical open-channel flow, the method allows very good decoupling of the numerical and mathematical model, resulting in a nearly grid-independent solution. The simulation of an actual dam break shows the applicability of the scheme to nontrivial bathymetry and wave propagation on a dry bed.     

Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-property, and the good resolution properties of the resulting scheme, in the cases of both continuous and discontinuous bottoms. Finally, a comparison between the fourth-order scheme proposed here and a well-established second-order scheme emphasizes the improvement achieved using the higher-order approach.     

Lagrangian Modeling of Weakly Nonlinear Nonhydrostatic Shallow Water Waves in Open Channels

A Lagrangian, nonhydrostatic, Boussinesq model for weakly nonlinear and weakly dispersive flow is presented. The model is an extension of the hydrostatic model—dynamic river model. The model uses a second-order, staggered grid, predictor-corrector scheme with a fractional step method for the computation of the nonhydrostatic pressure. Numerical results for solitary waves and undular bores are compared with Korteweg-de Vries analytical solutions and published numerical, laboratory, and theoretical results. The model reproduced well known features of solitary waves, such as wave speed, wave height, balance between nonlinear steepening and wave dispersion, nonlinear interactions, and phase shifting when waves interact. It is shown that the Lagrangian moving grid is dynamically adaptive in that it ensures a compression of the grid size under the wave to provide higher resolution in this region. Also the model successfully reproduced a train of undular waves (short waves) from a long wave such that the predicted amplitude of the leading wave in the train agreed well with published numerical and experimental results. For prismatic channels, the method has no numerical diffusion and it is demonstrated that a simple second-order scheme suffices to provide an efficient and economical solution for predicting nonhydrostatic shallow water flows.     

Finite-Difference TVD Scheme for Computation of Dam-Break Problems

A second-order hybrid type of total variation diminishing (TVD) finite-difference scheme is investigated for solving dam-break problems. The scheme is based upon the first-order upwind scheme and the second-order Lax-Wendroff scheme, together with the one-parameter limiter or two-parameter limiter. A comparative study of the scheme with different limiters applied to the Saint Venant equations for 1D dam-break waves in wet bed and dry bed cases shows some differences in numerical performance. An optimum-selected limiter is obtained. The present scheme is extended to the 2D shallow water equations by using an operator-splitting technique, which is validated by comparing the present results with the published results, and good agreement is achieved in the case of a partial dam-break simulation. Predictions of complex dam-break bores, including the reflection and interactions for 1D problems and the diffraction with a rectangular cylinder barrier for a 2D problem, are further implemented. The effects of bed slope, bottom friction, and depth ratio of tailwater∕reservoir are discussed simultaneously.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.     

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.     

Hydraulic Jet Control for River Junction Design of Yuen Long Bypass Floodway, Hong Kong

The Yuen Long Bypass Floodway (YLBF) was designed to collect flows from the Sham Chung River (SCR) and the San Hui Nullah (SHN) and to serve as a diversion channel of the Yuen Long Main Nullah (YLMN). Under a 200-year return period design condition, the floodway was designed (1) to divert a flow of approximately 38?m3/s from the supercritical YLMN flow and (2) to convey a total combined flow of 278?m3/s to downstream within acceptable flood levels. The success of the design depends critically on complicated junction flow interactions that cannot be resolved by 1D unsteady flow models. These features include the supercritical-subcritical flow transition at the San Hui-Floodway (SHN-YLBF) junction and the diversion of part of the supercritical flow from the Main Nullah (YLMN). A laboratory Froude scale physical model was constructed to study water stages and flow characteristics in the floodway and to investigate optimal design arrangements at channel junctions and transitions. This paper summarizes the main features of the unique river junction network, in particular the use of the hydraulic jet principle at the SHN-YLBF junction to lower flood levels. In addition, a numerical flow model is employed to study flow details at the river junctions. The model is based on the general 2D shallow water equations in strong conservation form. The equations are discretized using the total variation diminishing finite-volume method which captures the discontinuity in hydraulic jumps. The numerical model predictions are well supported by the laboratory data, and the theoretical and experimental results offer useful insights for the design of urban flood control schemes under tight space constraints.     

2D Numerical Model of Flooding in East Bangkok

We propose a new numerical flooding model oriented to water systems comprising large canal networks. Our approach is based on the diffusion wave equation related to a modified version of the so-called “calculations with apparent cross-sectional flow areas.” Our mathematical formulation is based on the analogy of a porous medium characterized by a permeability depending on parameters and directions of the canal network. Next, we combine the “continuous medium” approach with the nonnegative stable numerical algorithm (initially developed for the 1D diffusion wave equation). A numerical solution of our resulting finite-difference equations requires a tangible generalization of the nonnegative algorithm to the case of coupled 2D river-surface flows. Finally, we simulate the flood evolution in the lower Chao-Praya river basin (located in the eastern areas of Bangkok), and we demonstrate the computational efficiency of our proposed method.     

2D Shallow-Water Model Using Unstructured Finite-Volumes Methods

This paper presents a two-dimensional (2D) shallow-water numerical model, which is based on the resolution of the Saint–Venant equation using the unstructured finite-volumes method, combined with Green’s theorem technique. The model has been validated by several benchmarks. The numerical results obtained from the model are in good agreement with the analytical or experimental ones. The paper also presents an application of this model to flood diversion from the Red River into a water-retention zone for the purpose of reducing flood threat at Hanoi, capital of Vietnam.     

Dam-Breach Flood Wave Propagation Using Dimensionless Parameters

An analytical model of flood wave propagation is used to study the sensitivity of dam-breach flood waves to breach-outflow hydrograph volume, peak discharge, and downstream-channel bed slope. Dimensionless parameters are identified for discharge and distance along the channel. A dam-breach Froude number is defined to enable analysis through a wide range of site and flow conditions. It is found that, at a certain distance downstream, the attenuated peak discharge is independent of the magnitude of the discharge at the breach site. This attenuated peak discharge is termed the “ultimate discharge,” and the distance associated with it, the “ultimate distance.” These ultimate values are a function, primarily of the downstream-channel bed slope and, secondarily, of the breach-outflow hydrograph volume.     

Effect of Bed Armoring on Bed Topography of Channel Bends

The two-dimensional numerical model previously developed by the writers for modeling the bed variations in a channel bend with uniform sediment is upgraded to incorporate the nonuniformity of sediment particles as well as bed armoring. In this model, the two-dimensional, depth-averaged, unsteady flow equations along with the bed-load mass conservation equation are solved in a body-fitted coordinate system by using the Beam and Warming alternating-direction implicit (ADI) scheme. A one-dimensional bed surface armoring approach is extended herein for application to a two-dimensional domain. The model is applied to a 180° bend with a constant radius under unsteady flow conditions. Numerical simulations are carried out to study the effect of bed armoring on the bed deformations in channel bends. Results show that bed armoring reduces scour in channel bends.     

Numerical and Experimental Study on Two-Dimensional Flood Flows with and without Structures

The behavior of two-dimensional (2D) flood flows and the hydrodynamic force acting on structures are investigated numerically and experimentally. Numerical simulations are performed using a model based on the finite-volume method with an unstructured grid system and the flux-difference splitting technique. Experiments on flood propagation in a flood plain, with and without structures were conducted so as to obtain a comprehensive verification of the model. Front positions, depths, and surface velocities of flood flows as well as hydrodynamic forces on structures were observed. Comparisons of numerical results against these experimental data show that the model can predict 2D flood flows and the force on structures with reasonable accuracy.     

3D Numerical Modeling of Flow and Sediment Transport in Open Channels

A 3D numerical model for calculating flow and sediment transport in open channels is presented. The flow is calculated by solving the full Reynolds-averaged Navier-Stokes equations with the k ? ε turbulence model. Special free-surface and roughness treatments are introduced for open-channel flow; in particular the water level is determined from a 2D Poisson equation derived from 2D depth-averaged momentum equations. Suspended-load transport is simulated through the general convection-diffusion equation with an empirical settling-velocity term. This equation and the flow equations are solved numerically with a finite-volume method on an adaptive, nonstaggered grid. Bed-load transport is simulated with a nonequilibrium method and the bed deformation is obtained from an overall mass-balance equation. The suspended-load model is tested for channel flow situations with net entrainment from a loose bed and with net deposition, and the full 3D total-load model is validated by calculating the flow and sediment transport in a 180° channel bend with movable bed. In all cases, the agreement with measurements is generally good.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.     

Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography

A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe’s approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.     

Integrated Two-Dimensional Surface and Three-Dimensional Subsurface Contaminant Transport Model Considering Soil Erosion and Sorption

To investigate the complex hydrological, morphodynamic, and environmental processes in watersheds, a physically-based integrated two-dimensional (2D) surface and three-dimensional (3D) subsurface model for flow, soil erosion and transport, and contaminant transport in the surface-subsurface system is presented in this paper. The model simulates the rainfall-induced surface flow by solving the depth-averaged 2D diffusion wave equation and the variably-saturated subsurface flow by solving the 3D mixed-form Richards equation. The surface and subsurface flow equations are coupled using the continuity conditions of pressure and exchange flux at the ground surface. The model uses the concept of nonequilibrium in the depth-averaged 2D simulation of nonuniform total-load sediment transport in upland fields, considering detachments by rainsplash and hydraulic erosion driven by surface flow. The integrated 2D surface and 3D subsurface contaminant transport model takes into account the contaminant changes due to sediment sorption and desorption, as well as exchanges between surface and subsurface domains due to infiltration, diffusion, and bed change. The model applies the same set of surface equations of flow, sediment, and contaminant transport for describing both upland areas and streams, so that no special treatments are required at their interface. The established model has been evaluated by comparisons with published experimental, numerical, and analytical data and then applied in an agricultural watershed. The model is suitable for wetland areas and agricultural watersheds in which streams are not very narrow and deep, and meanwhile a relatively fine mesh that can distinguish the streams is preferred.     

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.     

Well-Balanced Scheme between Flux and Source Terms for Computation of Shallow-Water Equations over Irregular Bathymetry

Although many numerical techniques such as approximate Riemann solvers can be used to analyze subcritical and supercritical flows modeled by hyperbolic-type shallow-water equations, there are some difficulties in practical applications due to the numerical unbalance between source and flux terms. In this study, a revised surface gradient method is proposed that balances source and flux terms. The new numerical model employs the MUSCL–Hancock scheme and the HLLC approximate Riemann solver. Several verifications are conducted, including analyses of transcritical steady-state flows, unsteady dam break flows on a wet and dry bed, and flows over an irregular bathymetry. The model consistently returns accurate and reasonable results comparable to those obtained through analytical methods and laboratory experiments. The revised surface gradient method may be a simple but robust numerical scheme appropriate for solving hyperbolic-type shallow-water equations over an irregular bathymetry.