One-dimensional Open Channel Flow Research Articles

Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography

In this paper, an accurate numerical model is presented for one-dimensional open-channel flows with varying topographies; the model is specifically applied to rectangular channels with variable widths. A pressure term is introduced in the shallow water momentum equation to construct a new conservation term, and the resulting non-conservation term is included in the source term to characterize the actual topography changes in the open-channel flow. Based on a Harten Lax and van Leer (HLL) Riemann solver for a homogeneous system, an upwind scheme is introduced into the model in which the flux is determined via randomly selecting a local Riemann solution state. This two-step random choice method enables the scheme to reach second-order accuracy in space. A Runge–Kutta scheme is introduced into the discretization of the source term of the system to achieve second-order accuracy. The proposed model is validated via a selection of steady and transient hydraulic problems with reference solutions. When compared with published experimental results, the predictions of the proposed model show a high degree of accuracy.

Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method

A computationally efficient hybrid finite-volume/finite-difference method is proposed for the numerical solution of Saint-Venant equations in one-dimensional open channel flows. The method adopts a mass-conservative finite volume discretization for the continuity equation and a semi-implicit finite difference discretization for the dynamic-wave momentum equation. The spatial discretization of the convective flux term in the momentum equation employs an upwind scheme and the water-surface gradient term is discretized using three different schemes. The performance of the numerical method is investigated in terms of efficiency and accuracy using various examples, including steady flow over a bump, dam-break flow over wet and dry downstream channels, wetting and drying in a parabolic bowl, and dam-break floods in laboratory physical models. Numerical solutions from the hybrid method are compared with solutions from a finite volume method along with analytic solutions or experimental measurements. Comparisons demonstrates that the hybrid method is efficient, accurate, and robust in modeling various flow scenarios, including subcritical, supercritical, and transcritical flows. In this method, the QUICK scheme for the surface slope discretization is more accurate and less diffusive than the center difference and the weighted average schemes.

A lattice Boltzmann model for solute transport in open channel flow

A lattice Boltzmann model of advection–dispersion problems in one-dimensional (1D) open channel flows is developed for simulation of solute transport and pollutant concentration. The hydrodynamics are calculated based on a previous lattice Boltzmann approach to solving the 1D Saint–Venant equations (LABSVE). The advection–dispersion model is coupled with the LABSVE using the lattice Boltzmann method. Our research recovers the advection–dispersion equations through the Chapman-Enskog expansion of the lattice Boltzmann equation. The model differs from the existing schemes in two points: (1) the lattice Boltzmann numerical method is adopted to solve the advection–dispersion problem by meso-scopic particle distribution; (2) and the model describes the relation between discharge, cross section area and solute concentration, which increases the applicability of the water quality model in practical engineering. The model is verified using three benchmark tests: (1) instantaneous solute transport within a short distance; (2) 1D point source pollution with constant velocity; (3) 1D point source pollution in a dam break flow. The model is then applied to a 50-year flood point source pollution accident on the Yongding River, which showed good agreement with a MIKE 11 solution and gauging data.

Using a characteristic-based particle tracking method to solve one-dimensional fully dynamic wave flow

The theoretical behavior of a one-dimensional (1-D) open-channel flow is embedded in the Saint-Venant equation, which is derived from the Navier–Stokes equations. The flow motion is described by the momentum equations, in which the terms for the inertia, pressure, gravity, and friction loss are retained while all other terms are discarded. Although the problem is valid for most channel-flow scenarios, it is numerically challenging to solve because robust, accurate, and efficient algorithms are critical for models to field applications. The method of characteristics (MOC) is applied to solve the diagonalized Saint-Venant equations. Most importantly, the boundary conditions can be naturally implemented based on the wave directions. This is considered more closely related to realistic flow conditions and sufficiently flexible to handle mixed sub- and supercritical fluid flows in natural rivers. A computer model, WASH1DF, derived from the proposed numerical method, and which differs from other commercial software packages such as HEC-RAS and SOBEK, was developed. To test the accuracy of the proposed method, four benchmark problems were examined. Analytical solutions to these benchmark problems, covering a wide range of cases, were provided by MacDonald et al. (J. Hydrol. Eng. ASCE 123(11), 1041–1045, 1997). The simulations indicate that the proposed method provides accurate results for all benchmark cases, which are valid for all transient flow scenarios. Comparisons of WASH1DF with other commercially available software packages were also conducted under the same simulation conditions. The results indicate that our proposed model demonstrates high accuracy for all problems and achieves the highest simulation precision among all packages tested.

Development of an upwinding kernel in SPH-SWEs model for 1D trans-critical open channel flows

In this study, an upwinding SPH model with a non-symmetric kernel function is proposed to predict one-dimensional open channel flows. Due to the application of non-symmetric kernel function biased in favor of the upstream side, numerical diffusion is intrinsically added into the discretized momentum equation using SPH. The proposed model thus has shown to have good potential to resolve steep gradient or discontinuous solutions without the need of exactly adding artificial viscosity to the discretized equation. Furthermore, an upwinding coefficient for the determination of the degree of upwinding is derived to accommodate the dispersion-relation-preserving (DRP) property. In wave number space, the error between the discretized SPH equations and the original partial differential equations is minimized, thereby yielding the optimized upwinding coefficient. The proposed model has been validated by solving four benchmark problems involving non-rectangular cross section, varying channel width, non-uniform bed slope and hydraulic jump. Comparison of the numerical and exact solutions shows that the proposed model has the ability of accurately predicting various open channel flows involving complicated transcritical flows. The consistency condition of the proposed model is also analyzed theoretically for the sake of completeness.

Development of a river-groundwater interaction model and its application to a catchment in Northwestern China

The river-groundwater interaction is an important component of the hydrological cycle. This study develops an integrated river-GW model that uses a one-dimensional open channel flow model and a three-dimensional saturated GW flow model to describe the dynamic river-GW relationship at the basin scale, as well as groundwater flow and streamflow in arid regions. The model is tested with three cases, and the good agreement between the simulated and observed results demonstrates that the model can be used to simulate river-GW interactions. The integrated river-GW model is applied to the middle reaches of the Heihe River Basin and is calibrated using multi-source field data, including hydraulic heads from observation wells, streamflow, and spring flow. The case studies in the Heihe River Basin find that the following: (1) the river-GW relationships vary seasonally and spatially and depend on many factors, such as the river flow and GW uses; (2) in the middle reaches, the annual mean river–groundwater flux exchange from Yinluoxia to the Heihe Bridge is approximately 17% of the mean streamflow and increases to more than 49% from the Heihe Bridge to Zhengyixia; and (3) after the implementation of the water reallocation plan in 2000, the river-GW relationship in some reaches changed from a gaining stream to a losing stream due to the increase of GW abstraction. These findings suggest that GW pumpage should be controlled rationally and demonstrate that the integrated river-GW model can be used to analyse the temporal-spatial trends of river-groundwater interaction in arid regions.

Effect of channel shape on selection of time marching scheme in the discontinuous Galerkin method for 1-D open channel flow

One-dimensional open channel flows are simulated using the discontinuous Galerkin finite element method. Three different explicit time marching schemes, including multistep/multistage schemes, are evaluated for different channel shapes for accuracy and efficiency. The Forward Euler, second-order Adam-Bashforth (multistep), and second-order total variation diminishing (TVD) Runge-Kutta (multistage) time marching schemes are utilized. The role of monotonized central, minmod, and zero TVD slope limiters for each of the time marching scheme is investigated. The numerical flux is approximated using HLL function. The accuracy and robustness of different time marching schemes are evaluated for steady and unsteady flows using analytical and measured data. The unsteady flows include dam break tests with wet and dry beds downstream of the dam in prismatic (rectangular, trapezoidal, triangular, and parabolic cross-sections) and non-prismatic (natural river) channels. The steady flow test involves simulation of hydraulic jump in a diverging rectangular channel. The various schemes are evaluated by comparing accuracy using statistical measures and efficiency using maximum possible time step size as well as CPU runtime. The second-order Adam-Bashforth time marching scheme is found to have the best accuracy and efficiency among the time stepping schemes tested.

Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. II. Numerical Simulations

AbstractIn this study, finite difference numerical methods, first order accurate in time and second order accurate in space, are proposed to solve the governing equations of the one-dimensional unsteady kinematic and diffusion wave open-channel flow processes in fractional time and fractional space, which were derived in the accompanying paper. Advantages of modeling open-channel flow in a fractional time-space framework over integer time-space framework are threefold. First, the nonlocal phenomena in the open-channel flow process in either space or time can be considered by taking the global correlations into consideration. Second, the proposed governing equations of the open-channel flow process in the fractional order differentiation framework are generalization of the governing equations in the integer order differentiation framework. Third, the physics of the observed heavy tailed distributions of particle displacements in transport processes, as reported in the literature, may be explained by a flow...

Scaling and self-similarity in one-dimensional unsteady open channel flow

The conditions under which the Saint Venant equations system for unsteady open channel flow, as an initial–boundary value problem, becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. One of the advantages of this methodology is that the self-similarity conditions due to the initial and boundary conditions can also be investigated thoroughly in addition to the conditions due to the governing equation. The obtained self-similarity conditions are compared with the scaling relationships that are derived through the Froude similitude. It is shown that the initial–boundary value problem of a one-dimensional unsteady open channel flow process in a prototype domain can be self-similar with that of several different scaled domains. However, the values of all the flow variables (at specified time and space) under different scaled domains can be upscaled to the same values in the prototype domain (at the corresponding time and space), as shown in this study. Distortion in scales of different space dimensions has been implemented extensively in physical hydraulic modelling, mainly because of cost, space and time limitations. Unlike the traditional approach, the distinction is made between the longitudinal–horizontal and transverse–horizontal length scales in this study. The scaled domain obtained by the proposed approach, when scaling ratios of channel width and water depth are equal, is particularly important for the similarity of flow characteristics in a cross-section because the width-to-depth ratio and the inclination angles of the banks are conserved in a cross-section. It is also shown that the scaling ratio of the roughness coefficient under distorted channel conditions depends on that of hydraulic radius and longitudinal length. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal and economical flexibility in setting up physical hydraulic models. Copyright © 2013 John Wiley & Sons, Ltd.

Local sensitivity for uncertainty analysis in one-dimensional open channel flow modelling

Uncertainty analysis may be of high importance in the field of water engineering. Sampling-based techniques, that resort to extensive exploration of the space of possible model inputs are a common approach. When inputs uncertainties can be described statistically by means of probability distribution functions, Monte Carlo simulation yields a statistical description of uncertain model outputs. However, the combination of commonly encountered input space dimensions and model CPU times often makes this approach hardly practicable. We propose to assess the potential of a local sensitivity analysis for uncertainty analysis in the modelling of open channel flows with the one-dimensional shallow water equations, which are generally viewed as being largely non linear. The first-order local sensitivity is simply assessed using the so-called empirical method requiring only two runs of the model (for a single uncertain parameter). The global sampling-based approach consists in running Monte Carlo simulations with the hydrodynamic model, given a statistical distribution of the model input parameters. Monte Carlo results can be analysed through classical statistical estimators, such as mean and standard deviation. It is shown that these characteristics can also be accurately estimated from the local sensitivity and the input distribution. Steady state and highly transient configurations (gate opening and closing) are explored, accounting for uncertainty of the main parameters (bottom slope, friction coefficient, boundary and initial conditions).

A kinetic scheme for the one-dimensional open channel flow equations with applications on networks

In this paper, a kinetic method to compute approximative solutions of the one-dimensional open channel flow equations with a varying cross-sectional area and a varying bottom profile is proposed. The scheme preserves the steady states at rest, keeps the water height non-negative and is able to handle dry channel beds. These three properties are essential indicators for the quality of a numerical scheme. The ability of the scheme to treat varying cross-sectional areas along the channel is an important advancement in comparison to previous papers. Further the same equations are considered on a network by coupling the equations at the nodes. The main difference to the majority of common numerical hydraulic models is the usage of the energy equality as coupling condition which seems to be more realistic than conditions based on the water height. Moreover, we set up a numerical method for computing subcritical flow on such networks. Several examples are treated to illustrate the results.

Auxiliary Variable-based Balancing (AVB) for source term treatment in open channel simulations

Practical engineering applications of open channel flow modelling involve geometric terms arising from variations in channel shape, bottom slope and friction. This paper presents the family of schemes that satisfy the generalised C-property for which static equilibrium is a particular case, in the framework of one-dimensional open channel flows. This approach, named Auxiliary Variable-based Balancing, consists of using an auxiliary variable in place of the flow variables in the diffusive part of the flux estimate. The auxiliary variable is defined so as to achieve a zero gradient under steady-state conditions, whatever the geometry. Many approaches presented in the literature can be viewed as a particular AVB case. Three auxiliary variables are presented in this paper: water elevation, specific force and hydraulic head. The methodology is applied to three classical Riemann solvers: HLL, Roe and the Q-scheme. The results are compared on five test-cases: three steady-state configurations including friction, singular head losses and variations in bottom elevation, channel width and banks slope and two transient test-case (dam-break problems on rectangular and triangular channel). In each case, the auxiliary variable that best preserves the steady-state configuration is the hydraulic head. Besides, using the head as auxiliary variable allows head loss functions due to singularities to be incorporated directly in the governing equations, without the need for internal boundaries. However, it is generally less accurate when sharp transients are involved.

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.

Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows

Several new techniques are proposed to overcome the deficiencies in the conventional formulation of the approximate Riemann solvers for onedimensional open channel flows, which include numerical imbalance and inaccuracy in the solution of discharge. The former arises in the case of irregular geometry and the latter in the presence of a hydraulic jump. These new techniques include: (1) adopting the form of the saint venant equations that include both gravity and pressure in one source term; (2) using water surface level as one of the primitive variables, in stead of cross-sectional area; (3) defining discharge at interface and evaluating it according to the flux obtained by the HLL Riemann solver; and (4) estimating water surface gradient based on piecewise linearly reconstructed variables in the second-order scheme. The performance of the resulting schemes is evaluated by means of theoretical analysis and various test examples, including ideal dam-break flows with dry and wet bed, hydraulic jump, steady flow over bump with hydraulic jump, wave interactions, tidal flow in an open channel, and wave propagation. it is demonstrated that the schemes have excellent numerical balance and mass conservation property and are capable of satisfactorily reproducing various open channel flows.

Tribalj dam-break and flood wave propagation

In this paper we present numerical simulations for the dam-break flood wave propagation from Tribalj accumulation to the town of Crikvenica (Croatia). The mathematical models we used were the one-dimensional open channel flow and the two-dimensional shallow water equations. They were solved with the well-balanced finite volume numerical schemes which additionally include special numerical treatment of the wetting/drying front boundary. These schemes were tested on CADAM test problems. The aim of this study was to assess potential damage in the village of Tribalj and the town of Crikvenica. Results of these simulations were used as the basis for urban planning and micro-zoning of the flood-risk areas. Several different dam-break scenarios were considered, ranging from sudden dam disappearance to partial and dynamic breach formation.

Multilayer Averaged and Moment Equations for One-Dimensional Open-Channel Flows

A model is developed to account for the vertical distribution of velocity and nonhydrostatic pressure in one-dimensional open-channel flows. The model is based on both classical multilayer models and depth-averaged and moment equations. The establishment of its governing equations and the flow simulation are performed over a number of flow layers as in classical multilayer models. However, the model also allows for vertical distributions within a flow layer by including both Boussinesq terms and effective stress terms due to depth-averaging operations. These terms are evaluated on the basis of vertically linearly approximated profiles of velocity and pressure. The resulting additional coefficients can be solved by the moment equations for the relevant layers. Three verifications demonstrate satisfactory simulations for water surface profile, as well as vertical distributions for horizontal velocity, vertical velocity, and nonhydrostatic pressure. Sensitivity analysis shows that the model can be applied with fewer flow layers, more flexibility of layer division, and less computational cost than classical multilayer models, without a remarkable compromise in accuracy.

Boundary control of linearized Saint-Venant equations oscillating modes

The paper investigates the control of oscillating modes occurring in open-channels, due to the reflection of propagating waves on the boundaries. These modes are well represented by linearized Saint-Venant equations, a set of hyperbolic partial differential equations which describe the dynamics of one-dimensional open-channel flow around a given stationary regime. We use a distributed transfer function approach to compute a dynamic boundary controller that cancels the oscillating modes over all the canal pool. This result is recovered with a Riemann invariants approach in the case of a frictionless horizontal canal pool. The effect of a proportional boundary control on the poles of the transfer matrix is then characterized by a root locus, and we derive an asymptotic result for high frequencies closed-loop poles.

Upwind Conservative Scheme for the Saint Venant Equations

An upwind conservative scheme with a weighted average water-surface-gradient approach is proposed to compute one-dimensional open channel flows. The numerical scheme is based on the control volume method. The intercell flux is computed by the one-sided upwind method. The water surface gradient is evaluated by the weighted average of both upwind and downwind gradients. The scheme is tested with various examples, including dam-break problems in channels with rectangular and triangular cross-sections, hydraulic jump, partial dam-break problem, overtopping flow, a steady flow over bump with hydraulic jump, and a dam-break flood case in a natural river valley. Comparisons between numerical and exact solutions or experimental data demonstrated that the proposed scheme is capable of accurately reproducing various open channel flows, including subcritical, supercritical, and transcritical flows. The scheme is inherently robust, stable, and monotone. The scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients or discontinuities in the solution.

Improved treatment of source terms in TVD scheme for shallow water equations

A number of high-resolution schemes have been recently developed to solve the homogeneous form of the shallow water equations. However, most approximate Riemann solvers experience difficulties with natural river applications if the irregular bed topography is not handled correctly. Based on the finite-difference flux-limited total variation diminishing (TVD) scheme, this paper develops a simple approach to handle the source terms for the one-dimensional open channel flow simulation with rapidly varying bed topography. Conclusions on the validity of the operator-splitting approach, the eigenvector-projection approach, and the proposed approach are presented. Analytical solution, experimental data, and available numerical result comparisons are shown to demonstrate the accuracy, robustness, stability, simplicity, and applicability of the proposed model.

Improved treatment of source terms in TVD scheme for shallow water equations

A number of high-resolution schemes have been recently developed to solve the homogeneous form of the shallow water equations. However, most approximate Riemann solvers experience difficulties with natural river applications if the irregular bed topography is not handled correctly. Based on the finite-difference flux-limited total variation diminishing (TVD) scheme, this paper develops a simple approach to handle the source terms for the one-dimensional open channel flow simulation with rapidly varying bed topography. Conclusions on the validity of the operator-splitting approach, the eigenvector-projection approach, and the proposed approach are presented. Analytical solution, experimental data, and available numerical result comparisons are shown to demonstrate the accuracy, robustness, stability, simplicity, and applicability of the proposed model.