Bernoulli Theorem, Minimum Specific Energy, and Water Wave Celerity in Open-Channel Flow

One basic principle of fluid mechanics used to resolve practical problems in hydraulic engineering is the Bernoulli theorem along a streamline, deduced from the work-energy form of the Euler equation along a streamline. Some confusion exists about the applicability of the Bernoulli theorem and its generalization to open-channel hydraulics. In the present work, a detailed analysis of the Bernoulli theorem and its extension to flow in open channels are developed. The generalized depth-averaged Bernoulli theorem is proposed and it has been proved that the depth-averaged specific energy reaches a minimum in converging accelerating free surface flow over weirs and flumes. Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics.     

VOF Model for Simulation of a Free Overfall in Trapezoidal Channels

In the past, solutions to open channel flow problems involving free surfaces were generally found on the basis of experimental data or through the development of theoretical expressions using simplified assumptions. The volume of fluid (VOF) turbulence model is applied to obtain characteristics of three-dimensional open channel flows involving free surfaces. In particular, the VOF model is used to determine the pressure head distributions, velocity distributions, and water surface profiles for the free overfall in a trapezoidal open channel. The predictions of the proposed model are validated using existing experimental data.     

Backwater Prediction due to the Blockage Caused by a Single, Submerged Spur Dike in an Open Channel

A method is proposed for predicting the backwater effect due to a single, submerged spur dike located within an open channel flow. A theoretical analysis based on the momentum principle relates the backwater effect to the drag force exerted by the spur dike on the flow. Experimental data obtained in laboratory flumes having subcritical flow conditions throughout the flow field have been used in developing predictive relationships for the spur dike drag coefficient, which is found to be strongly correlated to the blockage created by the spur dike within the flow cross section. The predictive relationships provide a means of obtaining a first-level estimate of the backwater effect due to a single, submerged spur dike in an open channel flow.     

Inception Point and Air Concentration in Flows on Stepped Chutes Lined with Wedge-Shaped Concrete Blocks

Stepped channels lined with wedge-shaped concrete blocks are a promising solution to provide overtopping protection for embankment dams if the discharge capacity of existing spillways is not adequate. The paper addresses the characteristics of the two-phase transition and skimming flows in stepped channels lined with this type of block. An experimental setup was developed with two flumes designed with a relative scale of 1:2.5. Air concentration was measured with an optical probe in several cross sections of both flumes. The scale effects are analyzed. An expression for the location of the inception point is proposed. The vertical air concentration profiles and their longitudinal variation are studied, considering data and models proposed by other researchers. The establishment of the uniform flow regime is analyzed.     

Overbank Flow in Compound Channels with Nonprismatic Floodplains

In this paper, the authors present experimental results of overbank flow in compound channels with nonprismatic floodplains and different convergence angles. The depth-averaged velocity, the local velocity distributions, and the boundary shear stress distributions were measured along the converging flume portion for different relative depths. The momentum balance is used to analyze the force acting on the flow in the main channel and for the whole cross section. Using the experimental data, various terms in the momentum equation are also calculated. The apparent shear forces on the vertical interface between the main channel and floodplains are evaluated for compound channels with nonprismatic floodplains, and the results are then compared with the prismatic cases. The energy balance in nonprismatic compound channels is also investigated by using the water surface elevation.     

Numerical Modeling of Bed Evolution in Channel Bends

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data than those obtained by previous numerical models. The model is used to investigate the process of evolution and stability of bed deformation in circular bends.     

Lateral Weir Flow Model using a Curve Fitting Analysis

A numerical approach is considered for flow over side weirs as a substantial part of distribution channels in irrigation systems and treatment units. The model is based on the energy principle and a curve-fitting technique. For this purpose, the side weir was divided into elementary strips to develop generalized equations for discharge and surface profile. The change in water surface elevation towards the weir crest and the inclination of the deflected flow over the weir were also taken into account. Dimensionless parameters were used and the normalized equations solved to obtain the hydraulic parameters of side weirs. The results were plotted to determine general relationships based on the curve-fitting technique. A practical application of the derived equations to obtain hydraulic parameters of side weirs is performed using literature data.     

Modeling Sediment Transport Using Depth-Averaged and Moment Equations

A 1D mathematical model to calculate bed variations in alluvial channels is presented. The model is based on the depth-averaged and moment equations for unsteady flow and sediment transport in open channels. Particularly, the moment equation for suspended sediment transport is originally derived by the assumption of a simple vertical distribution for suspended sediment concentration. By introducing sediment-carrying capacity, suspended sediment concentration can be solved directly from sediment transport and its moment equations. Differential equations are then solved by using the control-volume formulation, which has been proven to have good convergence. Numerical experiments are performed to test the sensitivity of the calibrated coefficients α and k in the modeling of the bed deposition and erosion. Finally, the computed results are compared with available experimental data obtained in laboratory flumes. Comparisons of this model with HEC-6 and other numerical models are also presented. Good agreement is found in the comparisons.     

Effect of Applying Different Distribution Shapes for Velocities and Pressure on Simulation of Curved Open Channels

Most of the computational models of curved open channel flows use the conventional depth averaged De St. Venant equations. De St. Venant equations assume uniform velocity and hydrostatic pressure distributions. They are thus applicable only to cases of meandering rivers and curved open channels where vertical details are not of importance. The two-dimensional vertically averaged and moment equations model, developed by the writers, is used to study the effect of applying different distribution shapes for velocities and pressure on the simulation of curved open channels. Linear and quadratic distribution shapes are proposed for the horizontal velocity components, while a quadratic distribution shape is considered for the vertical velocity. Linear hydrostatic and quadratic nonhydrostatic distribution shapes are proposed for the pressure. The proposed model is applied to problems involved in curved open channels with different degrees of curvature. The implicit Petrov–Galerkin finite element scheme is applied in this study. Computed values for depth averaged longitudinal and transverse velocities across the channel width and vertical profiles of longitudinal and transverse velocities are compared to the observed experimental data. A fairly good agreement is attained. Predictions of overall flow characteristics suggest that the results are not very sensitive to different approximations of the preassumed applied velocity and pressure distribution shapes.     

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.     

Flow-Establishment Length in Rectangular Channels and Ducts

Laboratory experiments related to open channel hydraulics are generally intended to be carried out in the fully developed flow region of a rectangular flume. The developing region, in channels as well as in ducts, is a complex three-dimensional flow region influenced by secondary circulation and free surface effects, and is therefore not amenable to analytical solution. Laboratory data and a numerical model have been used to study the variation of the length of flow establishment in rectangular channels.     

Turbulence Modeling of Flows over Circular Spillways

To predict the characteristics of flows over circular spillways, a turbulence model based on the Reynolds stress model (RSM) is presented. Circular spillways are used to regulate water levels in reservoirs. The flow over the spillway is rapidly varied with highly curvilinear streamlines. The isotropic eddy-viscosity models such as k-ε models are based on the Boussinesq eddy viscosity approximation that assumes the components of the turbulence Reynolds stress tensor linearly vary with the mean rate of strain tensor. Hence, they cannot very precisely predict the characteristics of flows over the spillway. On the other hand, the non-isotropic turbulence models such as the turbulence Reynolds stress models (RSM) that calculate all the components of the Reynolds stress tensor can accurately predict the characteristics of these flows. The k-ε models and RSM were applied in the present study to obtain the flow parameters such as the pressure and velocity distributions as well as water surface profiles. The previously published experimental results were used to validate the simulation predictions. For flow over a circular spillway, RSM appears to properly validate the characteristics of the flow under various conditions in the field, without recourse to expensive experimental procedures.     

Field Testing of a Simple Flume (SMBF) for Flow Measurement in Open Channels

In this paper the stage–discharge relationship of a new flume named SMBF (Samani, Magallanex, Baiamonte, Ferro), originally proposed by Samani and Magallanez and tested by Baiamonte and Ferro, for measuring flow discharge in open channels is reviewed. The flume is obtained inserting two semicylinders in a rectangular cross section. The results of some experimental runs carried out using horizontal flumes characterized by different values of the contraction ratio (ranging from 0.17 to 0.81) are used for determining the two coefficients of the power stage–discharge equation. The stage–discharge equation is tested using flow measurements carried out in the period between December 2004 and March 2006 in the Sicilian experimental SPA1 basin. Field testing of the SMBF flume is developed using discharge measurements carried out by a Khafagi–Venturi flume placed in the field measurement channel.     

Direct Integration of the Equation of Gradually Varied Flow

The steady flow in open channels, when the depth of the flow varies gradually with distance, is governed by the classic gradually-varied-flow equation. The solution of this ordinary differential equation allows the tracing of the longitudinal profiles of the water surface of the flow. In this note, a relation obtained by direct integration is proposed for a wide rectangular channel, when Manning’s formula is used for the computation of the energy slope. Then the profiles for subcritical and supercritical flow in a mild and steep channel are presented and a comparison with the Bresse solution, relative to the same channels, is carried out.     

Investigation on Water Model for Fluid Flow in Slab Continuous Casting Mold With Consideration of Solidified Process

This study aims to propose suitable simulation methods, which enable to reduce the major differences between water model and real caster, such as the gradually decreased flow space, flow mass in the casting direction, and the momentum decay in the mushy zone. With consideration of solidified process, the method is concerned with the change of flow space and flow mass at the casting direction in water model. The level fluctuations, stimulus–response curves, velocities of liquid surface, and distributions of liquid slag have been changed in the water model to study the differences of flow character and the variation of fluid flow in molds. The mold with a solidified shell leads to significant differences in flow behaviors, such as higher level fluctuations, higher surface velocities, and worse liquid slag distributions. Neglecting the solidified shell causes unrealistic lower surface velocities and level fluctuations in water model. The mold with consideration of flow mass balance has higher level fluctuations and surface velocities than the mold without shell, and has lower level fluctuations and surface velocities than that of mold with a shell. The results indicate that it is necessary for water model to take the solidified process into account to acquire more accurate and reliable experiment results, especially for thinner slab.     

Flow and Velocity Distribution in Meandering Compound Channels

An investigation of flow and velocity distribution in meandering compound channels with over bank flow is described. Equations concerning the three-dimensional variation of longitudinal, transverse, and vertical velocity in the main channel and floodplain of compound section in terms of channel parameters are presented. The flow and velocity distributions in meandering compound channels are strongly governed by interaction between flow in the main channel and that in the floodplain. The proposed equations take adequate care of the interaction affect. Results from the formulations, simulating the three-dimensional velocity field in the main channel and in the floodplain of meandering compound channels are compared with their respective experimental channel data obtained from a series of symmetrical and unsymmetrical test channels with smooth and rough sections. The aspect ratio of the test channels varies from two to five. The equations are found to be in good agreement with the experimental data. The formulations are verified against the natural river and other meandering compound channel data. The power laws used for simulating the three-dimensional velocity structure are found to be quite adequate.     

Efficient, Robust Design Tool for Open-Channel Flow

The use of computational fluid dynamics to simulate physical phenomenon, such as flow through open channels, has become more prolific as computational resources and numerical algorithms have improved. With current algorithms, the designer defines the shape of the channel geometry, and the code calculates the fluid dynamic state within the channel. For these computational tools to be effectively used in the design process, the designer should be able to define the desired flow characteristics and the computational tool should generate the channel geometry that yields the closest match. In this paper, a finite-element simulation code, which solves the viscous, 2D, shallow water equations, is modified to become an efficient, robust design tool by inclusion of discrete sensitivity analysis and the Gauss-Newton optimization algorithm. This technique is demonstrated for design of supercritical flow channels with the objective of producing constant depth flow downstream of the channel transitions.     

Identification of Geologic Fault Network Geometry by Using a Grid-Based Ensemble Kalman Filter

Discrete geologic features such as faults and highly permeable embedded channels can significantly affect subsurface flow and transport characteristics. Therefore, they must be properly identified, parameterized, and represented in subsurface simulation models. In this work, we use an improved ensemble Kalman filter (EnKF) for history-matching fault network geometry from production data. EnKF is a sequential Monte Carlo data assimilation method that simultaneously propagates and updates an ensemble of model states, resulting in a set of calibrated model realizations that can be readily used for model prediction and uncertainty analysis. A pattern-based stochastic simulation algorithm was used to generate fault network realizations based on a priori fault trace data. The classic EnKF algorithm was enhanced with a grid-based covariance localization scheme to better handle non-Gaussian permeability distributions resulting from the presence of faults. Numerical experiments indicate that the modified EnKF can be a promising method for uncovering unmapped faults by using production data.     

Diffusive Modeling of Aggradation and Degradation in Artificial Channels

The unsteady flow and solid transport simulation problem in artificial channels is solved using a three-equation model, coupled with a local erosion law. The three equations are the water mass and momentum balance equations, as well as the total solid load balance equation. It is shown that even during severe hydrological events inertial terms can be neglected in the momentum equation without any substantial change in the solution sought. Empirical equilibrium formulas were used to estimate the solid load as a function of the flow variables. Local erosion, due to the scour generated at the jump between two channels connected at different bottom elevations, was estimated adapting a literature formulation. The double order approximation time and space marching scheme, previously proposed for the solution of the unsteady flow problem in the fixed-bed case, is applied to the solution of the new system. The model was validated with both literature and new laboratory experimental data. No parameter calibration was used to fit the computed results to the experimental ones.     

Flume for Teaching Spatially Varied Open-Channel Flow

Spatially varied flow in open channels is a topic that is often included in undergraduate open channel hydraulics courses. Physical and computational models are developed to enhance the presentation of spatially varied flow to engineering students at the late undergraduate or early graduate level. The physical model is inexpensive and easy to build and the computational model is easily developed using commercially available spreadsheet software. The physical model consists of a 30.48 cm nominal-diameter PVC pipe that is 6.1 m in length and has circular orifices approximately 1.40 cm in diameter drilled on 15.24 cm centers along the pipe invert. A relationship between the orifice discharge coefficient and a modification of the Froude number, as measured in the flume upstream of the orifice in question, was developed in repeated trials having varying flume slope, volumetric inflow rate, and end conditions. With this relationship, a stepwise solution to the energy equation is used to predict the water surface profile. Differences between the water surface profiles observed and predicted in repeated trials averaged approximately 2 mm.