Conservative Scheme for Numerical Modeling of Flow in Natural Geometry

A numerical model is proposed to compute one-dimensional open channel flows in natural streams involving steep, nonrectangular, and nonprismatic channels and including subcritical, supercritical, and transcritical flows. The Saint-Venant equations, written in a conservative form, are solved by employing a predictor-corrector finite volume method. A recently proposed reformulation of the source terms related to the channel topography allows the mass and momentum fluxes to be precisely balanced. Conceptually and algorithmically simple, the present model requires neither the solution of the Riemann problem at each cell interface nor any special additional correction to capture discontinuities in the solution such as artificial viscosity or shock-capturing techniques. The resulting scheme has been extensively tested under steady and unsteady flow conditions by reproducing various open channel geometries, both ideal and real, with nonuniform grids and without any interpolation of topographic survey data. The proposed model provides a versatile, stable, and robust tool for simulating transcritical sections and conserving mass.     

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.     

Analysis of Debris Wave Development with One-Dimensional Shallow-Water Equations

The objective of this contribution is to analyze the formation of debris waves in natural channels. Numerical simulations are carried out with a 1D code, based on shallow-water equations and on the weighted averaged flux method. The numerical code represents the incised channel geometry with a power-law relation between local width and flow depth and accounts for all source terms in the momentum equation. The debris mixture is treated as a homogeneous fluid over a fixed bottom, whose rheological behavior alternatively follows Herschel-Bulkley, Bingham, or generalized viscoplastic models. The code is first validated by applying it to dam-break tests on mudflows down a laboratory chute and verifying its efficiency in the simulation of rapid transients. Then, following the analytical method developed by Trowbridge, the stability of a uniform flow for a generalized viscoplastic fluid is examined, showing that debris flows become unstable for Froude numbers well below 1. Applications of the code to real debris flow events in the Cortina d’Ampezzo area (Dolomites) are presented and compared with available measured hydrographs. A statistical analysis of debris waves shows that a good representation of wave statistics can be obtained with a proper calibration of rheological parameters. Finally, it is shown that a minimum duration of debris event and channel length are required for waves showing up, and an explanation, confirmed both by field data and numerical simulations, is provided.     

Implicit Bidiagonal Scheme for Depth-Averaged Free-Surface Flow Equations

A general fast implicit bidiagonal numerical scheme, based on the MacCormack's predictor-corrector technique requiring the inversion of only block bidiagonal matrices, has been developed and subsequently applied for subcritical and supercritical free-surface flow calculations. The model has been applied to depth-averaged steady flows. There are two main advantages of the proposed method: the technique has fast convergence and utilizes a body fitted nonorthogonal local coordinate system to simulate irregular geometry flows. The model is used to analyze a wide variety of hydraulic engineering problems including flows in a converging-diverging subcritical flume, supercritical expansions at various Froude numbers, and supercritical converging chutes. For each of these test cases, the calculated results are compared with experimental data. The comparisons with measurements as well as with other numerical solutions show that the proposed method is comparatively accurate, fast, and reliable.     

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

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

Lagrangian Simulation of One-Dimensional Dam-Break Flow

The note demonstrates the application of a pure Lagrangian numerical method to dam-break flows by solving the St. Venant equations. The method is developed based on the smoothed particle hydrodynamics. It is easy to apply and is shown to be capable of providing accurate simulations for mixed flow regimes with strong shocks.     

Two-Dimensional Depth-Averaged Flow Modeling with an Unstructured Hybrid Mesh

An unstructured hybrid mesh numerical method is developed to simulate open channel flows. The method is applicable to arbitrarily shaped mesh cells and offers a framework to unify many mesh topologies into a single formulation. A finite-volume discretization is applied to the two-dimensional depth-averaged equations such that mass conservation is satisfied both locally and globally. An automatic wetting-drying procedure is incorporated in conjunction with a segregated solution procedure that chooses the water surface elevation as the main variable. The method is applicable to both steady and unsteady flows and covers the entire flow range: subcritical, transcritical, and supercritical. The proposed numerical method is well suited to natural river flows with a combination of main channels, side channels, bars, floodplains, and in-stream structures. Technical details of the method are presented, verification studies are performed using a number of simple flows, and a practical natural river is modeled to illustrate issues of calibration and validation.     

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

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

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.     

Numerical Modeling of Basin Irrigation with an Upwind Scheme

In recent years, upwind techniques have been successfully applied in hydrology to simulate two-dimensional free surface flows. Basin irrigation is a surface irrigation system characterized by its potential to use water very efficiently. In basin irrigation, the field is leveled to zero slope and flooded from a point source. The quality of land leveling has been shown to influence irrigation performance drastically. Recently, two-dimensional numerical models have been developed as tools to design and manage basin irrigation systems. In this work, a finite volume-based upwind scheme is used to build a simulation model considering differences in bottom level. The discretization is made on triangular or quadrilateral unstructured grids and the source terms of the equations are given a special treatment. The model is applied to the simulation of two field experiments. Simulation results resulted in a clear improvement over previous simulation efforts and in a close agreement with experimental data. The proposed model has proved its ability to simulate overland flow in the presence of undulated bottom elevations, inflow hydrographs, and colliding fronts.     

平展流燃烧器内湍流输运过程的数值模拟

采用任意拉格朗日一欧拉(ALE)框架下的有限体积法对平展流湍流输运特性进行了数值模拟研究。根据平展流特征对湍流模拟的κ-ε模型进行了修正，并对计算网格的生成进行了探讨，在此基础上分析了旋流强度对平展流输运特性的影响以及两个平行平展流间的相互作用。     

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.     

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.     

High-Order Compact Difference Scheme for Convection–Diffusion Problems on Nonuniform Grids

In this study, a high-order compact (HOC) scheme for solving the convection–diffusion equation (CDE) under a nonuniform grid setting is developed. To eliminate the difficulty in dealing with convection terms through traditional numerical methods, an upwind function is provided to turn the steady CDE into its equivalent diffusion equation (DE). After obtaining the HOC scheme for this DE through an extension of the optimal difference method to a nonuniform grid, the corresponding HOC scheme for the steady CDE is derived through converse transformation. The proposed scheme is of the upwind feature related to the convection–diffusion phenomena, where the convective–diffusion flux in the upstream has larger contributions than that in the downstream. Such a feature can help eliminate nonphysical oscillations that may often occur when dealing with convection terms through traditional numerical methods. Two examples have been presented to test performance of the proposed scheme. Under the same grid settings, the proposed scheme can produce more accurate results than the upwind-difference, central-difference, and perturbational schemes. The proposed scheme is suitable for solving both convection- and the diffusion-dominated flow problems. In addition, it can be extended for solving unsteady CDE. It is also revealed that efforts in optimizing the grid configuration and allocation can help improve solution accuracy and efficiency. Consequently, with the proposed method, solutions under nonuniform grid settings would be more accurate than those under uniform manipulations, given the same number of grid points.     

Simulation of Flow past an Open-Channel Floor Slot

In the past, solutions to the problem of flow past a floor slot in a rectangular open channel used to divert flow from one stream to another were obtained mainly on the basis of model tests or through the development of simplified theoretical expressions. In the present study, the free-surface turbulence model is applied to obtain the flow parameters such as pressure head distribution, velocity distribution, and water surface profile. The predictions of the proposed numerical model are validated using previous experimental data. In particular, the model predictions agree well with the test data related to flow parameters. The study indicates that the free-surface turbulence model developed is an efficient and useful tool for predicting characteristics of free surface flows such as flow past a floor slot. For flow past an open-channel floor slot, a model that is properly validated can be used to predict the flow characteristics under various flow configurations encountered in the field, without resorting to expensive experimental procedures.     

Dam Break in Channels with 90° Bend

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

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.     

Analysis of the Friction Term in the One-Dimensional Shallow-Water Model

The numerical simulation of unsteady open channel flows is very commonly performed using the one-dimensional shallow-water model. Friction is one of the relevant forces included in the momentum equation. In this work, a generalization of the Gauckler-Manning friction model is proposed to improve the modeling approach in cases of dominant roughness, unsteady flow, and distorted cross-sectional shapes. The numerical stability conditions are revisited in cases of dominant friction terms and a new condition, complementary to the basic Courant-Friedrichs-Lewy condition, is proposed. Some test cases with measured data are used to validate the quality of the approaches.     

Flow Characteristics of Skimming Flows in Stepped Channels

Skimming flows in stepped channels are systematically investigated under a wide range of channel slopes (5.7°?θ?55°). The flow conditions of skimming flows are classified into two flow regimes, and the hydraulic conditions required to form a quasi-uniform flow are determined. An aerated flow depth of a skimming flow is estimated from the assumption that the residual energy at the end of a stepped channel coincides with the energy at the toe of the jump formed immediately downstream of the stepped channel. In a quasi-uniform flow region, the friction factor of skimming flows is represented by the relative step height and the channel slope. The friction factor for the channel slope of θ=19° appears to have a maximum. The residual energy of skimming flows is formulated for both nonuniform and quasi-uniform flow regions. Further, a hydraulic-design chart for a stepped channel is presented.     

Identification of Manning’s Roughness Coefficients in Shallow Water Flows

A numerical method based on optimal control theories for identifying Manning’s roughness coefficients (Manning’s n) in modeling of shallow water flows is presented. The coefficients are difficult to be determined especially when the spatial variation is significant, and are usually estimated empirically. The present methodology is applied to determine the optimal values of the spatially distributed parameters, which give least overall discrepancies between simulations and measurements. Through a series of systematic studies to identify the n values in both a hypothetical open channel and a natural stream stretch, several identification procedures based on unconstrained and constrained minimizations are analyzed. It is found that the limited-memory quasi-Newton method has the advantages of higher rate of convergence, numerical stability and computational efficiency. Although the identification of Manning’s n is chosen as an example, the identification methods can be applied to numerical simulations of various flow problems.