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.     

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.     

Head Reconstruction Method to Balance Flux and Source Terms in Shallow Water Equations

This paper presents the head reconstruction method (HRM), a new technique that can be used within the finite volume framework to make shallow water models well balanced, i.e., to correct the imbalance that exists between flux and source terms in the equation discretization in the case of irregular bathymetry thus providing unphysical solutions. This technique, based on considering, within each computational cell, the total head of the flow (i.e., the sum of the elevation, pressure and kinetic energies per unit weight of the fluid) as an equilibrium variable, enables the preservation of dynamic equilibria under subcritical, transcritical, and supercritical flow conditions. The new technique is applied to the one-dimensional total variation diminishing (TVD) MUSCL-Hancock scheme and the conservation property is then proven mathematically for this scheme under static equilibrium conditions. Furthermore, the effectiveness of the HRM is tested and compared with two other well balancing techniques based on considering the water elevation as an equilibrium variable in various steady flow case studies. In the end the robustness of the HRM is tested in the simulation of dam-break flow over irregular bathymetry.     

EEG spectral topography in neurology: I. A review of techniques

As researchers investigate methods of automated interpretation of the electroencephalogram (EEG), spectral topography is emerging as an important and popular technique in applied clinical neurophysiology. Several computer-based EEG topography systems have been developed to produce topographic maps showing the spatial distributions of pre-defined frequency bands in the EEG. However, there is ongoing debate as to which technical approaches to EEG topography generate maps that can be most accurately interpreted by clinicians. This paper reviews existing topographic techniques, particularly as they apply to diagnostic neurology, and discusses some of the technical choices that must be addressed by topography users. These choices include the selection of montage, epoch length, interpolation scheme, graphical display method, and artifact removal technique. The points summarised here highlight the general opinion that although EEG topography has many benefits, it should be invoked with care and the user should possess an indepth understanding of the procedures used to produce the topographic maps.     

Case Study: Malpasset Dam-Break Simulation using a Two-Dimensional Finite Volume Method

The accuracy, stability, and reliability of a numerical model based on a Godunov-type scheme are verified in this paper, through a comparison between calculated results and observed data for the Malpasset dam-break event, which occurred in southern France in 1959. This event is an unique opportunity for code validation because of the availability of extensive field data on the flooding wave due to the dam failure. In the code the shallow water equations are discretized using the finite volume method, and the numerical model allows second order accuracy, both in space and time. The classical Godunov approach is used. More specifically, the Harten, Lax, and van Leer Riemann solver is applied. The resulting scheme is of high resolution and satisfies the total variation diminishing condition. For the numerical treatment of source terms relative to the friction slope, a semi-implicit technique is used, while for the source terms relative to the bottom slope a new explicit method is developed and tested.     

Finite Volume Model for Nonlevel Basin Irrigation

A finite volume model for unsteady, two-dimensional, shallow water flow is developed and applied to simulate the advance and infiltration of an irrigation wave in two-dimensional basins of complex topography. The fluxes are computed with Roe's approximate Riemann solver and the monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to eliminate spurious oscillations and the model incorporates an efficient and robust scheme to capture the wetting and drying of the soil. Model predictions are compared with experimental data for one- and two-dimensional problems involving rough, impermeable, and permeable beds, including a poorly leveled basin.     

Numerical Model for Channel Flow and Morphological Change Studies

In this paper a depth-integrated 2D hydrodynamic and sediment transport model, CCHE2D, is presented. It can be used to study steady and unsteady free surface flow, sediment transport, and morphological processes in natural rivers. The efficient element method is applied to discretize the governing equations, and the time marching technique is used for temporal variations. The moving boundaries were treated by locating the wet and dry nodes automatically in the cases of simulating unsteady flows with changing free surface elevation in channels with irregular bed and bank topography. Two eddy viscosity models, a depth-averaged parabolic model and a depth-averaged mixing length model, are used as turbulent closures. Channel morphological changes are computed with considerations of the effects of bed slope and the secondary flow in curved channels. Physical model data have been used to verify this model with satisfactory results. The feasibility studies of simulating morphological formation in meandering channels and flows in natural streams with in-stream structures have been conducted to demonstrate its applicability to hydraulic engineering research∕design studies of stream stabilization and ecological quality among other problems.     

Frequency Modeling of Open-Channel Flow

A good model is necessary to design automatic controllers for water distribution in an open-channel system. The frequency response of a canal governed by the Saint-Venant equations can be easily obtained in the uniform case. However, in realistic situations, open-channel systems are usually far from the uniform regime. This paper provides a new computational method to obtain a frequency domain model of the Saint-Venant equations linearized around any stationary regime (including backwater curves). The method computes the frequency response of the Saint-Venant transfer matrix, which can be used to design controllers with classical automatic control techniques. The precision and numerical efficiency of the proposed method compare favorably with classical numerical schemes (e.g., Runge–Kutta). The model is compared in nonuniform situations to the one given by a finite difference scheme applied to Saint-Venant equations (Preissmann scheme), first in the frequency domain, then in the time domain. The proposed scheme can be used, e.g., to validate finite difference schemes in the frequency domain.     

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.     

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.     

Two-Dimensional Numerical Model of Spillway Flow

A numerical model using the finite-element and finite-volume methods is developed for the resolution of two-dimensional free-surface flow equations including air entrainment and applied to calculation of the flow in a spillway. The model is implemented on an unstructured triangular mesh where the standard Galerkin scheme and an upwind finite-volume scheme are developed to solve the continuity and the conservative momentum equations, respectively. The time integration is performed using the fourth-order-accurate Runge-Kutta method with time steps that satisfy the Courant-Friedrichs-Lewy condition. An artificial dispersion term is introduced to eliminate spurious oscillations. A test problem in a spillway is solved to verify the applicability of the model to practical design. A physically realistic solution is obtained that represents a series of flow state alternations from supercritical to subcritical and vice versa, as well as the surface level increase due to the entrained air. The temporal mean of the calculated solution is compared with experimental temporal mean data and examined by posteriorly evaluating the residual term due to vertical nonhomogeneity of velocity. The investigations prove that the model is valid as a primary analysis tool for hydraulic design of spillways.     

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.     

Application of Three-Dimensional Hydrodynamic Model for Lake Okeechobee

A 3D hydrodynamic and heat transport model was developed for Lake Okeechobee. Continuity, momentum, and temperature transport equations were solved. Dynamically coupled transport equations for turbulent kinetic energy and turbulent scale also were solved. The numerical scheme used spatial finite differencing and a three-time-level, external-internal mode splitting procedure. A 28-day calibration was conducted, using measured bathymetry, rainfall, relative humidity, total solar radiation, wind velocity, inflow, and outflow data. During the calibration period, little rainfall occurred, and lake water levels receded. Water surface elevation, horizontal velocities, and temperature were computed. Agreement between observed and simulated values was based on graphical comparisons, minimizing mean absolute and root-mean-square errors, and spectral analysis. Comparisons showed that the model reproduced general observed trends and short-term fluctuations. The model's heat transport and turbulence closure schemes behaved as expected with regard to water column stratification and mixing. Simulation accuracy may potentially be improved by adding wind-wave and vegetation resistance algorithms to the model.     

Ellipsoidal fitting of corneal topography data after arcuate keratotomies with compression sutures

BACKGROUND AND OBJECTIVE: After paired arcuate keratotomies and compression sutures (AK) for treatment of high postkeratoplasty astigmatism, corneal topography tends to be irregular. The purpose of this study was to demonstrate a mathematical method for approximation of discrete corneal topography power data with an ellipsoid for better appreciation of the clinical outcome after AK. PATIENTS AND METHODS: Thirty-one eyes of 28 consecutive patient who underwent AK for excessive postkeratoplasty astigmatism were studied. Regular keratometry, corneal topography (TMS-1), subjective refraction, and best-corrected visual acuity (VA) were assessed preoperatively and at 1 week and 1 year postoperatively. A simplex algorithm was applied for fitting an ellipsoidal surface to raw corneal topography power data. A set of parameters (meridional power, axis, and asphericity) were calculated. The cylinder of subjective refraction was correlated with the keratometric readings, the simulated keratometry (SimK) of the topography system, and the respective parameters of the model surface. RESULTS: Keratometric astigmatism and the cylinder of the model surface decreased from 8.1 +/- 3.2 and 7.9 +/- 2.9 D preoperatively to 4.5 +/- 2.1 and 5.3 +/- 2.0 D after 1 year, respectively. The asphericity in both meridional cross sections changed from a prolate ellipse preoperatively to an ablate ellipse at the early postoperative follow-up stage. Regarding the cylinder axis, there was a significant correlation of the model surface with the refractive cylinder at all examinations (P < .05), whereas there was no significant correlation of the SimK axis and the refractive cylinder axis. CONCLUSION: The approximation of corneal topography power data with an ellipsoidal model surface renders reconstruction of clinically relevant corneal topography parameters, including corneal asphericity with a marked data compression. Even in markedly irregular corneal surfaces, such as after AK, the correlation of amount/axis of refractive cylinder with the model surface parameters is more accurate than it is with respective SimK values of corneal topography analysis.     

Low-Tension Cable Dynamics: Numerical and Experimental Studies

An efficient and robust numerical method is presented for the dynamic analysis of low-tension cables. The numerical solution strategy is based on finite-difference approximations of differential equations. In a scheme used by other researchers, known as the box scheme, the trapezoidal method is employed in both space and time domains. This scheme, however, gives rise to spurious high-frequency oscillations in cable tension response, as discovered in the research work reported herein. A modified box scheme is proposed to eliminate the problem. To improve computational efficiency, an iterative procedure is used to solve the resulting nonlinear simultaneous equations. A “free-fall” problem of cable dynamics involving low tension and large displacement motion is studied numerically. An experimental program is carried out to verify the accuracy of the numerical solution with regards to cable tension response.     

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.     

High-Resolution Method for Modeling Hydraulic Regime Changes at Canal Gate Structures

A method for modeling flow regime changes at gate structures in canal reaches is presented. The methodology consists of using an approximate Riemann solver at the internal computational nodes, along with the simultaneous solution of the characteristic equations with a gate structure equation at the upstream and downstream boundaries of each reach. The conservative form of the unsteady shallow-water equations is solved in the one-dimensional form using an explicit second-order weighted-average—flux upwind total variation diminishing (TVD) method and a Preissmann implicit scheme method. Four types of TVD limiters are integrated into the explicit solution of the governing hydraulic equations, and the results of the different schemes were compared. Twelve possible cases of flow regime change in a two-reach canal with a gate downstream of the first reach and a weir downstream of the second reach, were considered. While the implicit method gave smoother results, the high-resolution scheme—characteristic method coupling approach at the gate structure was found to be robust in terms of minimizing oscillations generated during changing flow regimes. The complete method developed in this study was able to successfully resolve numerical instabilities due to intersecting shock waves.     

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 Analytic Model for Flow and Transport in Unsaturated Zone

The finite analytic method is employed to solve the vertical, two-dimensional subsurface flow and transport equations in an unsaturated zone. The finite analytic method treats the nonlinear coefficient terms of the governing equations as constants in the element so that linearized partial differential equations can be obtained and solved in each element. The accuracy and limitations of the numerical method are systematically explored. The flow and transport simulations are examined using a one-dimensional laboratory infiltration test and an analytical solution of a two-dimensional subsurface transport problem, respectively. In the advection-dominant, vertical, one-dimensional infiltration problem, nine spatial weighting schemes are proposed to evaluate the averaged unsaturated hydraulic conductivity in a discretized element. Among them, the geometric mean weighting scheme provides the most accurate results as compared with the infiltration data. In verification of the two-dimensional solute transport problem, the nine-node elements are placed in the interior domain, and different layers of five-node elements are placed at the boundaries to investigate if the numerical experiment setup was proper and the algorithm was accurate. The developed numerical model is then applied to an irregular-domain landfill leaching problem to reveal the features of subsurface transport in unsaturated zone. Numerical aspects to be further explored are suggested.     

Simple, Robust, and Efficient Algorithm for Gradually Varied Subcritical Flow Simulation in General Channel Networks

This paper details the development of a method for subcritical flow modeling in channel networks by using the implicit finite-difference method. The method treats backwater effects at the junction points on the basis of junction-point water stage prediction and correction (JPWSPC). It is applicable to flows in both looped and nonlooped channel networks and has no requirement on the flow directions. The method is implemented in a numerical model, in which the Saint-Venant equations are discretized by using the four-point implicit Preissmann scheme, and the resulting nonlinear system of equations is solved by using the Newton-Raphson method. With the help of JPWSPC, each branch is computed independently. This guarantees the simplicity, efficiency, and robustness of the numerical model. The model is applied to two hypothetic channel networks and a real-life river network in South China. All the networks contain both branched and looped structures. The simulated results compare well with the results from literature or the measurements.