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.     

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.     

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.     

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.     

Flood Inundation Modeling with an Adaptive Quadtree Grid Shallow Water Equation Solver

Flood risk studies require hydraulic modeling in order to estimate flow depths and other hydraulic variables in the floodplain for a wide range of input conditions. Currently there is a need to improve the computational efficiency of fully two-dimensional numerical models for large-scale flood simulation. This paper describes an adaptive quadtree grid-based shallow water equation solver and demonstrates its capability for flood inundation modeling. Due to the grid dynamically adapting to dominant flow features such as steep water surface gradients and wet-dry fronts, the approach is both efficient and accurate. The quadtree model is applied to a realistic scenario of flood inundation over an urban area of 36?km2, resulting from the flood defenses breaching at Thamesmead on the River Thames, United Kingdom. The results of the simulation are in close agreement with alternative predictions obtained using the commercially available software TUFLOW.     

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.     

Numerical Simulation of Shallow-Water Flow Using a Modified Cartesian Cut-Cell Approach

The Cartesian cut-cell method can be used to represent irregular and complex computational domains with less computational efforts by cutting the grid cells on the boundary surfaces in a background uniform Cartesian mesh. In this study, a modified Cartesian cut-cell grid technique is proposed to better represent complex physical geometries. A point shifting treatment was employed to determine the start and end points of a line segment in cut-cell grids. This led to an improved representation of sharply-shaped corners in surface polygons. Numerical simulation to solve a set of shallow-water equations was performed by incorporating a finite volume approach into the Cartesian cut-cell mesh. The advective fluxes at intercells were first estimated by a Harten, Lax and van Leer for contact wave approximate Riemann solver. In order to improve the model accuracy to the second order, a total variation diminishing-weighted average flux method was applied to work adaptively with the cut-cell mesh. The numerical model was then employed to simulate dam-break flow propagation in a small channel with a rectangular obstacle or a 45° bend. The numerical results show good agreement with available laboratory measurements.     

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.     

Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.     

Case Study: Hydraulic Modeling of Runoff Processes in Ghanaian Inland Valleys

The inland valleys of West Africa are strategic in terms of food security and poverty alleviation, but scientific studies on hydrologic processes happening in these environments have not been well documented. Modeling approaches presented in this paper are an attempt to better comprehend hydraulic phenomena occurring in inland valleys. An inland valley situated in the Northern Region of Ghana is set as the study site. The inland valley comprises well-drained uplands and hydromorphic valley bottoms. There are several earthen dams across the valley bottoms, which are at the same time seasonal wetlands cultivated to rice during the rainy season. A finite volume model for the shallow water equations is developed to numerically simulate surface runoff flows in the valley bottoms during flood events. Innovation is necessitated to handle a series of different hydraulic phenomena. Flux-splitting and data reconstruction techniques are used to achieve stable computation in the complex topography of the valley bottoms. Standard problems of oblique hydraulic jump and dam break flows are used to test the accuracy of the numerical model. The Manning’s roughness coefficient is determined from calibration in another Ghanaian watershed located in the Eastern Region. Using actually observed time series data of rainfall intensity, surface flows during the rainfall events are simulated in the computational domain representing the valley bottoms of the study area. Observed data of water levels in the dams are compared to predictions, and discrepancies between them are examined from the hydrological point of view. In the case of a hypothetical flood event, cascading collapses of the dams and flooding of cultivated fields are reproduced.     

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.     

Shallow Turbulent Flow Simulation Using Two-Length-Scale Model

Numerical simulations of the planar starting jets were conducted using a two-length-scale turbulence model and a hydraulic code to study the effect of friction on 2D turbulence in shallow open-channel flow. The simulation results were compared with the data of the starting jets obtained in a recent series of laboratory experiments conducted in a large tank of small thickness. Dividing the turbulence energies into large and small scales, and calculating the energies with separate models, the observed friction effects on the 2D large-scale turbulent motion were correctly simulated by a two-length-scale turbulence model. To maintain the large-scale turbulence in the shallow shear flow, the production of turbulence energy by the transverse shear must be greater than the dissipation of the energy by friction. The critical gradient bed-friction number obtained from the simulations of the starting jets was Sc ? 0.08, which is consistent with the experimental observations in other shallow turbulent flows.     

Divergence Form for Bed Slope Source Term in Shallow Water Equations

A novel technique is presented for the treatment of the bed slope source terms within the numerical solution of the shallow water equations. The proposed method consists of writing the bed slope source term as the divergence of a proper matrix, related to the static force due to bottom slope. Such a technique is founded on analytical reasoning and is physically based, so that it can be easily applied to a wide range of numerical models, as it is completely independent of any adopted discretization technique, and requires a minimum computational effort. Herein, we show an application to a Godunov-type model, second order accurate both in space and time, based on the finite-volume method. The presented technique leads to a strong improvement in the source terms numerical treatment, especially for steady states, in which flux gradients are exactly balanced by source terms. A surprising degree of simplicity is maintained, with respect to different existing methods. The numerical model has been applied to a set of classical test cases and to a selected laboratory experiment, in order to verify its stability, accuracy, and applicability to practical real-world cases.     

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.     

Methodology for Estimating Specific Yield in Shallow Water Environment Using Continuous Soil Moisture Data

A methodology for estimating specific yield in shallow water environments using continuous soil moisture data is introduced. An accurate estimate of specific yield is critical for management of water resources affecting well yields, water levels, and rates of water level decline. The study area is located in an intensive small-scale hydrologic monitoring field site in Hillsborough County, Florida. Data from four soil moisture monitoring stations were studied. The objectives of this paper are to describe a methodology to obtain precise estimates of the specific yield that can be obtained by a fitting procedure and to determine functional relationships between measured soil parameters and soil moisture storage variability. Estimated specific yield values varied from zero, when the water table was near land surface, to a maximum that was close to drainable porosity consistent with a humid shallow water environment, where sensitive ecosystems and wetlands are dependent on a seasonal hydroperiod of water table fluctuations. The results will be incorporated into regional integrated surface and ground water models that are being applied successfully in West-Central Florida.     

Computing Nonhydrostatic Shallow-Water Flow over Steep Terrain

Flood and dambreak hazards are not limited to moderate terrain, yet most shallow-water models assume that flow occurs over gentle slopes. Shallow-water flow over rugged or steep terrain often generates significant nonhydrostatic pressures, violating the assumption of hydrostatic pressure made in most shallow-water codes. In this paper, we adapt a previously published nonhydrostatic granular flow model to simulate shallow-water flow, and we solve conservation equations using a finite volume approach and an Harten, Lax, Van Leer, and Einfeldt approximate Riemann solver that is modified for a sloping bed and transient wetting and drying conditions. To simulate bed friction, we use the law of the wall. We test the model by comparison with an analytical solution and with results of experiments in flumes that have steep (31°) or shallow (0.3°) slopes. The law of the wall provides an accurate prediction of the effect of bed roughness on mean flow velocity over two orders of magnitude of bed roughness. Our nonhydrostatic, law-of-the-wall flow simulation accurately reproduces flume measurements of front propagation speed, flow depth, and bed-shear stress for conditions of large bed roughness.     

Case Study: Flood Mitigation of the Muda River, Malaysia

The 2003 flood of the Muda River reached 1,340?m3/s at Ladang Victoria and adversely impacted 45,000 people in Malaysia. A flood control remediation plan proposed a levee height based on a 50-year discharge of 1,815?m3/s obtained from hydrologic models. This design discharge falls outside the 95% confidence intervals of the flood frequency analysis based on field measurements. Instream sand and gravel mining operations also caused excessive riverbed degradation, which largely off sets apparent benefits for flood control. Pumping stations have been systematically required at irrigation canal intakes. Several bridge piers have also been severely undermined and emergency abutment protection works were needed in several places. Instream sand and gravel mining activities should be replaced with offstream mining in the future.     

Role of Artificial Dissipation Scaling and Multigrid Acceleration in Numerical Solutions of the Depth-Averaged Free-Surface Flow Equations

A numerical model is developed for solving the depth-averaged, open-channel flow equations in generalized curvilinear coordinates. The equations are discretized in space in strong conservation form using a space-centered, second-order accurate finite-volume method. A nonlinear blend of first- and third-order accurate artificial dissipation terms is introduced into the discrete equations to accurately model all flow regimes. Scalar- and matrix-valued scaling of the artificial dissipation terms are considered and their effect on the accuracy of the solutions is evaluated. The discrete equations are integrated in time using a four-stage explicit Runge–Kutta method. For the steady-state computations, local time stepping, implicit residual smoothing, and multigrid acceleration are used to enhance the efficiency of the scheme. The numerical model is validated by applying it to calculate steady and unsteady open-channel flows. Extensive grid sensitivity studies are carried out and the potential of multigrid acceleration for steady depth-averaged computations is demonstrated.     

3D Unsteady RANS Modeling of Complex Hydraulic Engineering Flows. II: Model Validation and Flow Physics

A chimera overset grid flow solver is developed for solving the unsteady Reynolds-averaged Navier-Stokes (RANS) equations in arbitrarily complex, multiconnected domains. The details of the numerical method were presented in Part I of this paper. In this work, the method is validated and applied to investigate the physics of flow past a real-life bridge foundation mounted on a fixed flat bed. It is shown that the numerical model can reproduce large-scale unsteady vortices that contain a significant portion of the total turbulence kinetic energy. These coherent motions cannot be captured in previous steady three-dimensional (3D) models. To validate the importance of the unsteady motions, experiments are conducted in the Georgia Institute of Technology scour flume facility. The measured mean velocity and turbulence kinetic energy profiles are compared with the numerical simulation results and are shown to be in good agreement with the numerical simulations. A series of numerical tests is carried out to examine the sensitivity of the solutions to grid refinement and investigate the effect of inflow and far-field boundary conditions. As further validation of the numerical results, the sensitivity of the turbulence kinetic energy profiles on either side of the complex pier bent to a slight asymmetry of the approach flow observed in the experiments is reproduced by the numerical model. In addition, the computed flat-bed flow characteristics are analyzed in comparison with the scour patterns observed in the laboratory to identify key flow features responsible for the initiation of scour. Regions of maximum shear velocity are shown to correspond to maximum scour depths in the shear zone to either side of the upstream pier, but numerical values of vertical velocity are found to be very important in explaining scour and deposition patterns immediately upstream and downstream of the pier bent.