Open Channel Flow Equations Research Articles

Conservation equations for open-channel flow: effects of bed roughness and secondary currents

AbstractTime-averaged velocity fields in uniform open-channel flows over rough beds may exhibit spatial heterogeneities due to the effects of bed roughness and secondary currents (SCs). The latter typically originate from the turbulence anisotropy and spatial heterogeneity introduced by the solid and mixed corners (i.e., between sidewalls and water surface), but may also appear due to roughness spanwise heterogeneities, e.g., associated with patchy vegetation distributions or streamwise sediment ridges on the channel bed. In this paper, we propose rigorous conservation equations for momentum, kinetic energy and fluid stresses accounting for the contributions of bed roughness and SCs, separately. Particular attention is given to the terms regulating the energy exchanges between roughness-induced and SC-related motions, which are expected to provide information on the physical mechanisms leading to the generation of roughness-induced SCs. The proposed approach is illustrated using a large-eddy simulation of a rough-bed open-channel flow.

Influence of Multi-Cross Structures on the Flood Discharge Capacity of Mountain Rivers in the Yellow River Basin

This study investigates the impact of cross structures on flood occurrences in mountainous rivers. The governing equations of open channel flow were formulated based on the Saint-Venant equations. The open channel was segmented, and a node equation was established at each section’s connection point. An overflow model of bridges and weir dams was also developed. The physical model of the open channel was simplified and modeled using actual building data and model calculation requirements. The study found that the primary impact of weirs and bridges on the open channel was the backwater effect on the water level. The influence of these structures on the water level in the Huang Stream urban section in the Yellow River Basin was assessed under various working conditions. The results showed that deleting the #1 weir could reduce the maximum backwater height by 1.14 m, and deleting the #2 weir could reduce it by 1.09 m. While reducing the weir height significantly decreased the backwater range and height, it did not enhance the river’s flood discharge capacity. The Huang Stream contains 17 bridges, 13 of which could potentially affect flood discharge. The eight flat slab bridges in the submerged outflow state had a significant impact on flood discharge, with a maximum water level change of 0.51 m. Conversely, the three single-hole flat slab bridges in the free outflow state downstream had a negligible impact on flood discharge. The study found that bridges had a greater influence on flood discharge capacity than weirs. This research provides valuable insights for the reconstruction of cross structures in mountainous rivers and for managing flood discharge capacity and flood control.

On the equations of open channel flow

The traditional equations for describing open channel flow have appeared in the literature for decades and are so ingrained that they might seem to be statements of settled science. Careful derivations that detail assumptions and approximations relied upon in the formulation are mostly absent. We derive mass, momentum, and energy equations by averaging their small-scale counterparts and formulate forms that are Galilean invariant as required by continuum mechanics. Averaging is over a time increment and a spatial region in a single step, clarifying the need for closure relations. The derivation leads to the Boussinesq tensor and Coriolis vector as rigorous generalizations of the Boussinesq and Coriolis coefficients typically proposed. Examples are provided for the computation of these coefficients from published data. The approach employed here can be extended to systems such as pipe flow or shallow water equations, and the Galilean invariant forms are also suitable for entropy generation analyses.

Random-walk-path solution of unsteady flow equations for general channel networks

The random walk path method (RWP) is a stochastic model that transforms the solution of the unsteady open channel flow equations into a probability problem. In this paper, this method is introduced to solve the discretized Saint-Venant equations, with probabilistic representation of the water levels at junctions. Different boundary conditions, which specify either water levels or discharge hydrographs, can be implemented by specifying how walkers react when arriving at the boundaries. The pointwise solution at river junctions can be obtained via random walk simulations. The efficacy of the model was verified against the following test cases: (1) a real-world looped channel example documented in the manual of the software Hec-Ras and (2) a hypothetical channel system consisting of dendritic and divergent networks. The number of random simulations is an important aspect of the RWP method and needs to be chosen by considering the trade-off between precision and computational efficiency. The impact of the boundary water level or discharge on the water levels at internal nodes can be quantitatively evaluated by adopting the terminal weights, which present a distinct advantage of the RWP method. This assessment of water levels can serve as a guide in the operation of hydraulic controls, such as dams, sluices and pumps, to effectively regulate the flow in mitigating flood risks.

Drag Reduction in Aerated Chute Flow: Role of Bottom Air Concentration

Wall shear stress modification in self-aerated flows has been observed in boundary layer–type flows mainly in the form of drag reduction. Due to its complexity, drag reduction has predominantly been studied using experimental techniques and empirical analyses, whereas the development of conceptual approaches is limited. In this study, two-phase open-channel flow equations with variable density are revisited and bottom air concentration is identified as a key parameter for drag reduction in air-water flows. The application of this new theory to smooth and stepped chute flows is introduced, and implications for the determination of friction factors are discussed. A modified version of the Manning-Strickler formula is proposed to model friction factors across a wide range of roughness and bottom air concentration levels.

Flow resistance in mobile bed rills shaped in soils with different texture

AbstractRill erosion is considered one of the most important soil processes because of the large amount of soil loss due to the development of a rill network able to promote an efficient transport of both rill flow‐detached particles and those delivered from the inter‐rill areas. Rill flow experiments are useful to overcome the gap in rill hydraulics knowledge and to test the reliability of currently applied uniform open channel flow equations for mobile bed rills. In this paper the applicability of a theoretical flow resistance equation, based on a power‐velocity profile, to rill channels shaped on soils having different textures was investigated. The measurements (437 runs) of water depth, cross‐section area, wetted perimeter, bed slope and flow velocity carried out in rill reaches shaped on experimental plots, having different slope values (9–26%) and soil texture (clay fraction range of 32.7–73%), were carried out. These measurements were jointly examined with 35 experimental runs carried out by other authors in a rill flume having a slope range of 18–84% and a clay content of 3%. The main aim of this paper is to deduce a relationship between the velocity profile parameter Γ, the channel slope and the flow Froude number using a database characterized by a wide range of slope (9–84%) and textural classes (clay ranging from 3 to 73%). The obtained relationship was also verified using 117 experimental runs carried out in this investigation by sloping plots (slope equal to 9 and 15%) with a soil having a clay content of 32.7%. The analysis demonstrated that the applicability of the theoretical flow resistance law is experimentally supported and that the effect of soil grain‐size distribution is represented by silt and clay fractions which take into account the influence of the detachability and transportability properties of the soil particles.Highlights Rill flow resistance is affected by interaction between flow, bed morphology and sediment transport. A new theoretical flow resistance law is tested for rills shaped on different textured soils. The velocity profile is related to the channel slope, the flow Froude number and the soil texture. The Darcy‐Weisbach friction factor can be accurately estimated by the proposed theoretical approach.

Timescale interpolation and no-neighbour discretization for a 1D finite-volume Saint-Venant solver

ABSTRACT A new finite-volume numerical method for the one-dimensional (1D) Saint-Venant equations for unsteady open-channel flow is developed and tested. The model uses a recently-developed conservative finite-volume formulation that is inherently well-balanced for natural channels. A new timescale interpolation approach provides transition between 1st-order upwind and 2nd-order central interpolation schemes for supercritical and subcritical flow, respectively. This interpolation meets a proposed “no-neighbour” criterion for simplicity in future parallel implementation. Tests with a highly-resolved transitional flow and a coarsely-resolved natural channel show that the method is stable and accurate when applied with a flowrate damping algorithm that limits propagation of energy down to subgrid scales.

Optimal design of interception for flood control: An integrated simulation approach

This study develops a computational framework that integrates a flood simulation model, a response surface method, and an optimization technique for an optimal flood control design. The objective function was the minimization of the difference between simulated flooding depths and pre-specified allowable flooding depths, with simulated flooding depths constrained to less than allowable flooding depths in the control regions. The design variables were the design capacities of hydraulic structures. The flood simulation model was based on two-dimensional shallow water flow equations for open channel flow, rainfall, overland flow flooding, and river interactions. The response surface method was combined with the central composite design method to construct approximate functions for the objective and the constraint functions in terms of the design variables. Sequential quadratic programming was used to obtain the optimal design variables with the minimum value of the approximated objective function. Actual flood data from Typhoon Morakot for the Linbian River was used to verify the flood simulation model. The integrated computational framework was then used to calculate the optimal upstream interception strengths for controlling downstream flooding under hydrologic conditions associated with a 100-year return period storm. Numerical examples demonstrated that the proposed computational framework can efficiently provide optimal designs for practical flood control problems.

Discussion of “Time-Space Fractional Governing Equations of Unsteady Open Channel Flow” by M. L. Kavvas and A. Ercan

Discussion of “Time-Space Fractional Governing Equations of Unsteady Open Channel Flow” by M. L. Kavvas and A. Ercan

Closure to “Time-Space Fractional Governing Equations of Unsteady Open Channel Flow” by M. L. Kavvas and A. Ercan

Closure to “Time-Space Fractional Governing Equations of Unsteady Open Channel Flow” by M. L. Kavvas and A. Ercan

Time-Space Fractional Governing Equations of Unsteady Open Channel Flow

AbstractIn this study, the complete governing equations for unsteady open channel flow in fractional time-space are developed from the fractional continuity equation combined with the fractional motion equation, which is based on Newton’s second law of motion, by accounting for the acceleration terms in order to render physically interpretable hydraulic terms. Then the kinematic wave and diffusion wave approximations to unsteady open channel flow with physically interpretable terms in fractional time-space are developed from the fractional continuity and motion equations. Because the powers of all of the derivatives in the conventional governing equations of unsteady open channel flow in integer time-space are one, these conventional equations are derived as special cases of the fractional governing equations when the fractional powers in the fractional equations approach unity.

Some Solutions of Simplified Open Channel Flow Equations

Open channel flows of shallow depth in hydraulics are characteristically turbulent due to disturbances generated by bed friction and other causes. The governing equations of such flows based on Reynolds Averaged Navier – Stokes (RANS) equations have been developed in recent years in collaborative papers by Bose and Dey, and have found useful applications. In as much as the equations are of nonlinear evolution type, several types of solutions can exist. The present paper develops five types of solutions, based on the similarity method, localized solution method, and the method of separation of variables. The last two methods deliver two solutions in each case. The self-similar solution and the two localized solutions, reduce to solution of ODE’s that are numerically presented. The two separation of variables solutions are obtained in explicit form.

Influence of Bed Roughness and Cross Section Geometry on Medium and Maximum Velocity Ratio in Open-Channel Flow

AbstractThis paper deals with the analytic-theoretical derivation of the relationships between the entropic quantity Φ(M), representing the ratio between the mean and maximum flow velocities, and the relative submergence and aspect ratios, using classical open-channel flow equations. Φ(M) is found to be highly dependent on the relative submergence when large or intermediate roughness scales occur, whereas it might be assumed to be almost constant for a small roughness scale. Furthermore, considering the hydraulic geometry relationships, an attempt is made to relate the relative submergence to the aspect ratio of flow through a log relationship whose coefficients depend on the local bed slope, with an important implication for hydrological practices. Then a practical relation between Φ(M) and the aspect ratio is proposed and validated in an operative chain for discharge assessment that shows high robustness and stability. The proposed model has been applied to a set of experimental velocity data collected ...

SISCO: surface irrigation simulation, calibration and optimisation

A model is described which applies the full one-dimensional version of the Saint–Venant equations for open channel flow to simulate the process of surface irrigation. The resulting software for surface irrigation simulation, calibration and optimisation, abbreviated to SISCO, was developed for use in a standard PC environment. Unlike some other models currently in use, SISCO can accommodate temporal variations in inflow rates and spatial variability in soil infiltration, surface roughness, slope and furrow geometry. The main focus of the paper is in regard to the calibration functionality, whereby it is capable of estimating the soil infiltration characteristic and Manning roughness from various combinations of practically obtainable field measurements.

Calibrating remotely sensed river bathymetry in the absence of field measurements: Flow REsistance Equation‐Based Imaging of River Depths (FREEBIRD)

AbstractRemote sensing could enable high‐resolution mapping of long river segments, but realizing this potential will require new methods for inferring channel bathymetry from passive optical image data without using field measurements for calibration. As an alternative to regression‐based approaches, this study introduces a novel framework for Flow REsistance Equation‐Based Imaging of River Depths (FREEBIRD). This technique allows for depth retrieval in the absence of field data by linking a linear relation between an image‐derived quantity X and depth d to basic equations of open channel flow: continuity and flow resistance. One FREEBIRD algorithm takes as input an estimate of the channel aspect (width/depth) ratio A and a series of cross‐sections extracted from the image and returns the coefficients of the X versus d relation. A second algorithm calibrates this relation so as to match a known discharge Q. As an initial test of FREEBIRD, these procedures were applied to panchromatic satellite imagery and publicly available aerial photography of a clear‐flowing gravel‐bed river. Accuracy assessment based on independent field surveys indicated that depth retrieval performance was comparable to that achieved by direct, field‐based calibration methods. Sensitivity analyses suggested that FREEBIRD output was not heavily influenced by misspecification of A or Q, or by selection of other input parameters. By eliminating the need for simultaneous field data collection, these methods create new possibilities for large‐scale river monitoring and analysis of channel change, subject to the important caveat that the underlying relationship between X and d must be reasonably strong.

Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. I. Development of the Equations

AbstractIn this study the fractional governing equations for diffusion wave and kinematic wave approximations to unsteady open-channel flow in prismatic channels in fractional time-space were developed. The governing fractional equations were developed from the mass and motion conservation equations in order to provide a physical basis to these equations. A fractional form of the resistance formula for open-channel flow was also developed. Detailed dimensional analyses of the derived equations were then performed in order to ensure dimensional consistency of the derivations. It is shown that these fractional equations of unsteady open-channel flow are fundamentally nonlocal in terms of nonlocal fluxes. The derived fractional governing equations of diffusion wave and kinematic wave open-channel flow can accommodate both the long-memory nonlocal behavior of open-channel flow as well as its local, finite memory behavior, as is numerically demonstrated in the accompanying paper by the authors.

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...

Balance errors generated by numerical diffusion in the solution of non-linear open channel flow equations

► The dissipative numerical methods for solution of nonlinear equations are considered. ► We examine the numerical solution of 1D equations in open channel hydraulics. ► The numerical diffusion generated by the method of solution produces a balance error. ► The balance error is connected with an inadequate conservative form of equation. The paper concerns the untypical aspect of application of the dissipative numerical methods to solve non-linear hyperbolic partial differential equations used in open channel hydraulics. It is shown that in some cases the numerical diffusion generated by the applied method of solution produces not only inaccurate solution but as well as a balance error. This error may occur even for an equation written in the conservative form not only in the solution containing a shock wave but also for the smooth solution with strong gradients. In order to explain this problem 1-D kinematic wave equation and 1-D system of shallow water equations, both solved by the finite-difference implicit box scheme, are considered. Accuracy analysis carried out using the modified equation approach revealed why the numerical solutions of these equations written in the conservative forms contain the balance errors. It was shown that the balance error is directly connected with the additional source term generated by the numerical diffusion during the solution of equation written in an inadequate conservative form. Consequently the balance error depends on the values of time step, spatial interval and the weighting parameters involved in the applied method of solution.

Three-Dimensional Numerical Modeling of Cavitation and Aeration System in Dam Outlets

Abstract Preservation of the live storage of reservoirs is a serious challenge for most of the countries that encounter drought. Flushing of deposited sediments through bottom outlets in a dam is one of the suitable means to transport sediments toward downstream of the dam. Due to complicated conditions of flow through bottom outlets, we are facing a lack of accurate information about the various phenomena that occurs in bottom outlets such as cavitation, corrosion, and abrasion. Cavitation is the most important problem that causes remarkable damages on the lining of these tunnels. A numerical model based on the finite volume method for fully three-dimensional (3D) open channel flow equations is incorporated into a modeling cavitation and aeration system along the dam outlets. In this study, complex two-phase turbulent flow is simulated using the K-ɛ model. The pressure and velocity distribution under different conditions of gate opening and reservoir water level are computed along the tunnel and validation of the work has been obtained by comparison with measurements of a laboratory model. For both nonaerated and aerated flow, cavitation index in flow direction have been estimated and compared with each other.

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.