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.     

Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids

A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge–Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.     

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.     

Field Verification of Two-Dimensional Surface Irrigation Model

A two-dimensional (2D) model of unsteady shallow-water flow in surface irrigation was developed to evaluate the influence of field-grading precision on surface irrigation performance. This paper presents field data for verification of this 2D model. No attempt was made here to evaluate irrigation performance. Verification of such models relies on independent estimates of parameters for infiltration and roughness. To accomplish this, water surface elevations were measured at 26 points within a 3 ha level basin. A double-bubbler system was used to obtain relative water depths. Field surveys were used to convert these to water surface elevations and field water depths, from which surface water volumes over time were computed. The infiltration function was determined by matching inflow minus surface volume over time with computed subsurface volume. A value of Manning n (0.05) was found for which advance and water depth hydrographs were both well predicted with the 2D model. Differences in advance for a plane versus undulating field surface were minor, except near the end of advance.     

Finite Volume Model for Two-Dimensional Shallow Environmental Flow

This paper presents the development of a two-dimensional, depth integrated, unsteady, free-surface model based on the shallow water equations. The development was motivated by the desire of balancing computational efficiency and accuracy by selective and conjunctive use of different numerical techniques. The base framework of the discrete model uses Godunov methods on unstructured triangular grids, but the solution technique emphasizes the use of a high-resolution Riemann solver where needed, switching to a simpler and computationally more efficient upwind finite volume technique in the smooth regions of the flow. Explicit time marching is accomplished with strong stability preserving Runge-Kutta methods, with additional acceleration techniques for steady-state computations. A simplified mass-preserving algorithm is used to deal with wet/dry fronts. Application of the model is made to several benchmark cases that show the interplay of the diverse solution techniques.     

Modeling of Conjunctive Two-Dimensional Surface-Three-Dimensional Subsurface Flows

In the rainfall-runoff process, interaction between surface and subsurface flow components plays an important role, especially in rainwater abstraction and overland flow initiation at the early stage of rainfall events. Coupling of surface and subsurface flow submodels, therefore, is necessary for advanced comprehensive and sophisticated rainfall-runoff simulation. This article presents a conjunctive two-dimensional (2D) surface and three-dimensional (3D) subsurface flow model, which uses the noninertia approximation of the Saint-Venant equations for 2D unsteady surface flow and a modified version of the Richards equation for 3D unsteady unsaturated and saturated subsurface flows. The equations are written in the form of 2D and 3D heat diffusion equations, respectively, and solved numerically. The surface and subsurface flow components are coupled interactively using the common boundary condition of infiltration through the ground surface. The conjunctive model is verified with Smith and Woolhiser’s experimental data (reported in 1971) of initially dry and initially wet soil. Subsequently the model is applied to a hypothetical soil plot of clay or sand to simulate the overland flow, infiltration, and subsurface flow for four different rainfalls. The conjunctive model contributes as a tool for improved detailed simulation of 2D surface and 3D subsurface flows and their interaction.     

Two-Dimensional Basin Flow with Irregular Bottom Configuration

Two-dimensional flow from a point or line source is simulated in an irrigated basin with a nonlevel soil surface; the goal is to predict the distribution uniformity of infiltrated depths. The zero-inertia approximation to the equations of motion allows computation in both wet and dry areas. A fully implicit, nonlinear finite-difference scheme is developed for the solution, but practical numerical considerations suggest local linearization instead. Both isotropic and anisotropic resistance to flow are considered. Results in basins with irregular bottom configurations and small inflows show stream advance confined to the lowest elevations in the basin.     

Riemann Solvers with Runge–Kutta Discontinuous Galerkin Schemes for the 1D Shallow Water Equations

The spectrum of this survey turns on the evaluation of some eminent Riemann solvers (or the so-called solver), for the shallow water equations, when employed with high-order Runge–Kutta discontinuous Galerkin (RKDG) methods. Based on the assumption that: The higher is the accuracy order of a numerical method, the less crucial is the choice of Riemann solver; actual literature rather use the Lax-Friedrich solver as it is easy and less costly, whereas many others could be also applied such as the Godunov, Roe, Osher, HLL, HLLC, and HLLE. In practical applications, the flow can be dominated by geometry, and friction effects have to be taken into consideration. With the intention of obtaining a suitable choice of the Riemann solver function for high-order RKDG methods, a one-dimensional numerical investigation was performed. Three traditional hydraulic problems were computed by this collection of solvers cooperated with high-order RKDG methods. A comparison of the performance of the solvers was carried out discussing the issue of L1-errors magnitude, CPU time cost, discontinuity resolution and source term effects.     

Optimal Cross-Sectional Spacing in Preissmann Scheme 1D Hydrodynamic Models

Choosing a suitable set of cross sections for the representation of the natural geometry of a river is important for the efficiency of one-dimensional (1D) hydraulic models, but only few guidelines are available for the selection of the most suitable distance between cross sections, depending on the hydraulic problem at hand. This issue is investigated by examining models of a ～ 55?km reach of the River Po, Italy, and a ～ 16?km reach of the River Severn, United Kingdom, for both of which high quality laser scanning altimetry are available. The high-resolution digital terrain models of the two river reaches enabled the construction of a series of hypothetical topographical ground surveys with different spacing between cross sections, which could be used as input to a standard 1D model (UNET). Both historical and synthetic flood events for the two river reaches were simulated, and the results were then analyzed to quantify the accuracy associated with each resolution and to assess how survey resolution impacts the performance of standard 1D models. The study results agree with the available suggestions in the literature and provide useful guidelines for 1D hydrodynamic modeling.     

Three-Dimensional and Depth-Averaged Large-Eddy Simulations of Some Shallow Water Flows

Three-dimensional (3D) and two-dimensional (2D) depth-averaged (DA) large-eddy simulations (LES) are presented for three different shallow-water flows involving large-scale horizontal structures: a mixing layer, the flow around a circular cylinder, and the flow in a groyne field. The results are compared with each other and also with experiments. In the 3D-LES, most of the energy-containing turbulent motions, including the larger subdepth-scale motions, are resolved, while in the 2D-DA-LES the effect of the 3D subdepth-scale turbulence is represented by a quadratic bottom-friction model and a simple eddy-viscosity model. In the case of the mixing layer, an additional stochastic backscatter model is necessary to account for the energy transfer from the subdepth-scale turbulence to the 2D structures in order to generate the latter. The 3D-LES results are generally in good agreement with the experiments, including the evolution of the horizontal structures. The much more economic 2D-DA-LES are somewhat less realistic in detail but also produce results that are generally of sufficient accuracy for practical purposes.     

Two-Dimensional Fast-Response Flood Modeling: Desktop Parallel Computing and Domain Tracking

Emergency flood management is enhanced by using models that can estimate the timing and location of flooding. Typically, flood routing and inundation prediction is accomplished by using one-dimensional (1D) models. These have been the models of choice because they are computationally simple and quick. However, these models do not adequately represent the complex physical processes present for shallow flows located in the floodplain or in urban areas. Two-dimensional (2D) models developed on the basis of the full hydrodynamic equations can be used to represent the complex flow phenomena that exist in the floodplain and are, therefore, recommended by the National Research Council for increased use in flood analysis studies. The major limitation of these models is the increased computational cost. Two-dimensional flood models are prime candidates for parallel computing, but traditional methods/equipment (e.g., message passing paradigm) are more complex in terms of code refactoring and hardware setup. In addition, these hardware systems may not be available or accessible to modelers conducting flood analyses. This paper presents a 2D flood model that implements multithreading for use on now-prevalent multicore computers. This desktop parallel computing architecture has been shown to decrease computation time by 14 times on a 16-processor computer and, when coupled with a wet cell tracking algorithm, has been shown to decrease computation by as much as 310 times. These accomplishments make high-fidelity flood modeling more feasible for flood inundation studies using readily available desktop computers.     

Dam-Break Flows: Acquisition of Experimental Data through an Imaging Technique and 2D Numerical Modeling

This paper presents experimental and two-dimensional (2D) numerical results of four tests concerning rapidly varying flows induced by the sudden removal of a sluice gate. For the acquisition of the experimental data, an imaging technique capable of providing spatially distributed information was adopted: a coloring agent was added to the water, the opalescent bottom of the facility was backlighted, and photographs of the area of interest were taken. The gray tones of the acquired images were converted into water depths by means of transfer functions derived from a static calibration. The potential sources of error of the proposed procedure are discussed. A local comparison with an ultrasonic device showed a 20% maximum deviation in 95% of the observations. The tests were simulated through a 2D MUSCL-Hancock finite volume numerical model, based on the classical shallow water approximations, in which the intercell water depths are estimated according to the surface gradient method. A global analysis of the relative frequency distributions of the deviation between numerical and experimental results is performed. Despite some evident differences at a local scale, the adopted 2D numerical model is capable of reproducing the main features of the flow fields under investigation.     

Two-Dimensional Nonlinear Analysis of Long Cables

A simple finite-difference iterative model is presented here for the nonlinear analysis of a long inclined cable due to its self-weight and other concentrated loads in between. The input requires the end coordinates, area of cross section, modulus of elasticity, and initial tension of the cable. If the ends of the cable are subjected to displacements, the tension in the cable varies nonlinearly and the new deformed shape is computed using an iterative procedure. Unlike finite-element methods, initial cable geometry is not required here, and the method automatically computes the initial geometry even with concentrated loads.     

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.     

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.     

Correction Factors in Series Solutions for One- and Two-Dimensional Boundary Value Problems

A Fourier cum polynomial series solution with correction factors is presented herein for differential equations with variable coefficients. The differential equations correspond to a wide range of boundary value problems. The correction factors included herein are: (1) modified Lanczos correction; (2) Bessel J; and (3) loading correction factor. These correction factors are introduced in terms of Fourier and polynomial series. The main purpose of using correction factors through a set of series is to improve convergence of the proposed solution, using the first two terms of the series. For the loading correction factor, a Fourier series expansion coupled with orthogonality conditions leads to evaluating undetermined Fourier coefficients of arbitrarily applied loads using concepts of summation equations. Representative boundary value problems are provided to demonstrate the efficiency and accuracy of the first two terms of the proposed solution with correction factors.     

Approximate Solutions for Forchheimer Flow to a Well

An exact solution for transient Forchheimer flow to a well does not currently exist. However, this paper presents a set of approximate solutions, which can be used as a framework for verifying future numerical models that incorporate Forchheimer flow to wells. These include: a large time approximation derived using the method of matched asymptotic expansion; a Laplace transform approximation of the well-bore response, designed to work well when there is significant well-bore storage and flow is very turbulent; and a simple heuristic function for when flow is very turbulent and the well radius can be assumed infinitesimally small. All the approximations are compared to equivalent finite-difference solutions.     

Two-Dimensional Analyses of Thermoplastic Culvert Deformations and Strains

Tests have been performed in a biaxial pipe test cell to develop baseline information on profiled pipe behavior under biaxial loading. These include a lined corrugated high-density polyethylene pipe and a helically wound ribbed polyvinyl chloride pipe. Results of the tests are utilized to examine the effectiveness of the two-dimensional methods of buried pipe analysis. Calculations of pipe responses by the two-dimensional finite element method and a set of simplified design equations are compared with the measurements of pipe strains and deflections. The study reveals that the two-dimensional finite element analysis can effectively be used to calculate pipe deflections and circumferential strains. The simplified equations appear suitable as design tools for standard buried thermoplastic pipe installations. Janbu’s nonlinear soil model with Mohr–Coulomb plasticity provided an effective simulation of the nonlinear soil behavior A study of pipes with low stiffness soil support under the haunches shows that this leads to strain concentrations in the pipe walls near that zone. Higher values of empirical strain factor, Df, are estimated to include this strain concentration during design.     

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.     

Brief Analysis of Shallow Water Equations Suitability to Numerically Simulate Supercritical Flow in Sharp Bends

This work deals with the suitability of two-dimensional shallow water equations for the numerical simulation of supercritical free surface flows in bends, when the usual hypothesis of small width/curvature radius ratio does not hold. Here, a very reliable and accurate finite-volume, Godunov-type scheme is adopted for the numerical integration of the governing equations. Comparison with a selected set of experimental laboratory data and asymptotic analytical solutions shows that several aspects concerning the physics of the phenomenon are well reproduced, such as the blocking of the stream when the Froude number of the undisturbed flow is not large enough and the bend is sufficiently sharp, while maximum water depth in the bend is systematically underestimated.