Hydraulic Jumps in Sloping Channels: Sequent Depth Ratio

A large variety of hydraulic jumps on horizontal and sloping inverts at the end of an ogee standard weir is investigated. An ogee standard weir was used to create supercritical flow and slopes of 0.0, ?0.025, ?0.05, ?0.075, and ?0.10 were built downstream of the weir. Based on the momentum equation in the horizontal direction, a method to predict the sequent depth ratio is presented. The theory agrees well with the results of the writers and previous investigators. A correlation was developed to predict the minimum Froude number needed to establish jumps on negative slopes. Observations showed that in those cases where the gravity force component in the jump was opposite to the flow direction, the water surface of the surface roller became undular and unstable. The hydraulic jump on an entirely adverse slope was almost impossible to control. The analysis of experimental data showed that the negative slope of the basin reduces the sequent depth ratio, while a positive slope increases the sequent depth ratio.     

Effect of Lateral Water Surface Profile on Side Weir Discharge

The average water surface slope in the lateral direction is taken into account as a local parameter ψ to study flow over a side weir. It was later shown that for smaller side weir lengths and side weir portions with no entrance and exit transition effects, ψ can be obtained from a numerical integral and also from the measurements of water surface elevations in the lateral direction. The effect of elemental weir length was further determined. Dividing the weir length into smaller computational segments has the effect of lowering the water surface to approach the measured profile, the downstream ends being coincident. The model was verified using experimental data.     

Hydraulic Characteristics of Gabion-Stepped Weirs

Experimental studies on the hydraulics of flow through and over gabion-stepped weirs are presented. Two flow components were observed, i.e., base flow through the void between filled stones and overflow on the gabions. The energy loss ratios in the gabion-stepped weirs are greater than those in the corresponding horizontal stepped weirs by approximately 7, 10, and 14% for weir slopes of 30, 45, and 60°, respectively. As a result, the velocity at the outlet is lower. Stone size and shape have little influence on the energy loss and flow velocity as compared to the increasing effect of the weir slope. The pressure acting on the step face for the gabion-stepped weirs is less than that of the horizontal steps owing to the cushioning effect generated by filled stones. To reduce pressure on a step face at a fixed discharge for different weir slopes, the void ratio of the filled stones should be low to allow a small amount of base flow. The pressure distribution pattern on the horizontal face of a step is provided.     

Transition Effects in Flow Over Side Weirs

The effect of water surface slope in the lateral direction on flow over side weirs was studied. Water surface elevation on the weir plane was expressed by a parameter ψ based upon the hydraulic profile on the channel axis. Two different relationships of ψ as a function of the Froude number were used to calculate side weir discharges. Results were compared with the experimental data. It was shown that better results are obtained when transition conditions of ψ = 1 at the ends of the side weirs with no lateral surface slope are taken into account. However the effect of water surface slope in lateral direction is of secondary importance as compared to the angle of the deflected water jet along the side weir.     

Discharge Coefficient for Sharp-Crested Side Weir in Subcritical Flow

To estimate the outflow over a rectangular sharp-crested side weir, the discharge coefficient in the weir equation needs to be known. Although this type of structure has been designed and used extensively by hydraulic engineers, a universally acceptable discharge coefficient does not exist. In this study over 250 laboratory tests were conducted, and the results were analyzed to find the influence of the flow hydraulics and the geometric, channel, and weir shapes on the coefficient. The results show that for subcritical flow the De-Marchi assumption of constant energy is acceptable, and the weir discharge can therefore be used. Furthermore, it was discovered that the De-Marchi coefficient of discharge is a function of the upstream Froude number and the ratios of weir height to upstream depth and weir length to channel width, whereas the channel slope in subcritical flow can be ignored. Hence, an accurate equation for the coefficient of discharge is introduced.     

Inception Point and Air Entrainment on Flows under Macroroughness Condition

Rock chutes or block ramps are fishway passages with low environmental impact. They also contribute to reaeration of rivers with low dissolved oxygen content, owing to the turbulence enhanced by their three-dimensional macroroughness conditions. This paper analyzes the air entrainment inception in flows over beds in macroroughness condition and the self-aerated flow features of the developing flow downstream of the inception point. Air concentration, inception point locations, and water depth elevations have been measured on two different scaled chutes for slopes ranging between 1V:5.88H and 1V:2.17H. Moreover, two different ogee crest lengths have been tested to assess the role of the inlet conditions on the location of the inception point. New equations have been developed to estimate the location of the point of inception and the respective water depth. Longitudinal variations of the mean air concentration downstream from the inception point have been studied and compared with data from the literature. An expression is presented to estimate the optimum length of the block ramp in natural rivers for maximizing air-water mixing.     

Behavior of Submerged Ogee Crest Weir Discharge Coefficients

Weir head-discharge relationships are typically described using the discharge coefficient-dependent standard weir equation. The submerged weir (tailwater exceeds the weir crest elevation) head-discharge relationship can vary from the free-flow head-discharge relationship, particularly at high submergence levels. The accuracy associated with predicting the upstream head or discharge, corresponding to submerged weir flow conditions, is dependent upon the accuracy with which a representative submerged discharge coefficient can be determined. A submerged ogee crest weir discharge coefficient predictive method developed by the U.S. Bureau of Reclamation (USBR) is reviewed and its predictive accuracy compared to laboratory-scale submerged ogee crest weir experimental data associated with a wide range of submerged flow conditions for nine different ogee crest weir geometries. Discussion is presented in an effort to partially explain the relatively poor correlation between the USBR method and the experimental data set. Alternative submerged discharge coefficient relationships are also presented.     

Effect of Weir Face Angles on Circular-Crested Weir Flow

The standard circular-crested weir is often found in engineering applications and is used as a discharge measurement device or as an overflow structure. This research determines the discharge coefficients for ten circular-crested weir configurations with various combinations of up- and downstream angles. Two different weir heights and four different overflow depths are considered for each weir shape. For free overflow, the discharge coefficient is determined experimentally by using the total head of the approach flow. The results indicate that the upstream weir face angle has only a small effect on the discharge coefficient. In contrast, increasing the downstream weir face angle increases the discharge coefficient notably. A new formula for the discharge coefficient is presented, including both the up- and downstream weir face angles. Further, the hydraulic performance of the circular-crested weir, the resulting discharge reduction from tailwater submergence, and transition flow are discussed.     

Lateral Behavior of Vertical Pile Group Embedded in Stabilized Earth Slope

An experimental study of the lateral behavior of vertical pile groups embedded in reinforced and nonreinforced sandy earth slopes was carried out. The model tests include studies of group configurations, pile spacing, embedment length of pile, relative densities of sand, and location of pile groups relative to the slope crest. Several configurations of geogrid reinforcement with different lengths, widths, and number of layers were used to reinforce a sandy slope of 1 (V): 1.5 (H). Pile groups of 2×2 and 3×3 along with center-to-center pile spacing of 2D, 3D, and 4.5D and piles with embedment length to diameter ratios of L/D = 12 and 22 were considered. Based on test results, geogrid parameters that give the maximum lateral capacity improvement are presented and discussed.     

Multilayer Depth-Averaged Flow Model with Implicit Interfaces

Using a depth-averaged model to obtain the velocity and pressure distributions in the vertical direction is difficult. A multilayer model is an option that can be used to improve on the depth-averaged model. However, the unknown flow depth needs to be predicted first and then divided into layers as an input for the multilayer model. An improved multilayer model is proposed here by introducing an implicit layer dividing interfaces that are associated with the flow velocity and pressure distribution. The formulation of interfaces also applies to boundary faces: Free surface and channel beds. Therefore, each flow layer behaves like that in the classical depth-averaged model. Subsequently the governing equations are also simplified due to the vanishing terms related to interfacial flow exchanges. This improved model has been satisfactorily applied to steady flow simulations in three cases: Flow over a slope transition from mild to steep, from steep to mild, and over a trapezoidal weir. The results demonstrate the efficiency and validity of the proposed models to simulate open channel flows with bed slope changes.     

Lateral Resistance of Single Pile Located Near Geosynthetic Reinforced Slope

An extensive program of laboratory tests was carried out to study the effect of reinforcing an earth slope on the lateral behavior of a single vertical pile located near the slope. Layers of geogrid were used to reinforce a sandy slope of 1 (V):1.5 (H) made with sands of three different unit weights representing dense, medium dense, and loose relative densities. Several configurations of geogrid reinforcement with different numbers of layers, vertical spacing, and length were investigated. The experimental program also included studies of the location of pile relative to the slope crest, relative density of sand, and embedment length of pile. The results indicate that stabilizing a soil slope has a significant benefit of improving the lateral load resistance of a vertical pile. The improvement in pile lateral load was found to be strongly dependent on the number of geogrid layers, layer size, and relative density of the sand. It was also found that soil reinforcement is more effective for piles located closer to the slope crest. Based on test results, critical values are discussed and recommended.     

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.     

Head-Discharge Relationships for Submerged Labyrinth Weirs

Low-head labyrinth weir control structures installed on mild sloping channels or where the channel downstream of the weir is constricting and/or heavily vegetated can experience submergence. Weir submergence occurs when the tailwater surpasses the weir crest elevation, causing an increase in the upstream driving head for a given discharge, relative to a free-discharge condition. The most familiar relationship for predicting submerged weir head-discharge relationships is likely that of James R. Villemonte, which he published in 1947 for sharp-crested linear weirs. For lack of a better alternative, Villemonte’s relation has also been applied to predicting submerged labyrinth weir performance. A new dimensionless submerged head relationship developed in this study is presented for submerged labyrinth weirs. A similar relationship is also presented for linear sharp-crested weirs. The accuracy of the submerged linear weir relationship was equivalent to Villemonte’s and is simpler to solve when working in terms of total upstream head. Relative to Villemonte’s relationship applied to labyrinth weirs, the new submerged labyrinth weir relationship reduced the predictive errors from 23 to 3.5% (maximum) and 8.9 to 0.9% (average), relative to the experimental data.     

Skimming Flow in the Nonaerated Region of Stepped Spillways over Embankment Dams

Traditionally, research on stepped spillway hydraulics has been focused on the air-water flow region but for the hydraulic design of small embankment dams experiencing relatively large overtopping flows, the nonaerated region can be very important. Empirical formulas are presented for predicting skimming flow properties upstream of the point of inception of air entrainment for 1V:2H sloping stepped spillways, and the location and flow depth at the point of inception. Particular emphasis is placed on the clear-water depth, velocity distribution, and the energy dissipation characteristics in the developing nonaerated flow region. The velocity distribution is well described by a power law. The normalized clear-water depth and the normalized specific energy varied with the relative distance along the spillway and the effect of the normalized critical depth was negligible. Finally, the rate of energy dissipation was small, which has direct implications for the design of the downstream energy dissipator.     

Method of Solution of Nonuniform Flow with the Presence of Rectangular Side Weir

An iterative step method for solving the nonlinear ordinary differential equation, governing spatially varied flows with decreasing discharge, like the flow over side weirs, is developed. In the procedure, starting at a known flow depth and discharge in the control section, the analytical integration of the dynamic equation with bed and friction slope is carried out. The specific energy, the weir coefficient and the velocity distribution coefficient are considered as local variables, then for the explicit integration, the respective average values along the short side weir elements are assumed. The water surface profiles and the discharges for flow over side weirs, obtained with the proposed relation and valid for rectangular channels, are compared with experimental data for subcritical and supercritical flow conditions. The validation of the method is accomplished by the comparison with the solution obtained by De Marchi’s classical hypothesis, about the specific energy, which is constant along a side weir. In addition, the influence of the coefficient velocity distribution is considered.     

Earth Dam with Toe Drain on an Impervious Base

The required height of a toe drain for a homogeneous earth dam on an impervious base has been determined considering reservoir water level, capillary rise for the embankment soil, free board, top width of the earth dam, embankment slopes, and tail-water position, such that the surface of seepage does not develop on the downstream sloping face of the earth dam and capillary saturation above phreatic line is contained well within the downstream sloping face. Using Kozeny’s analytic function, exact solution to the unconfined flow through an earth dam having parabolic equipotential boundaries on either side has been obtained. For straight toe drain face, and for various positions of tailwater, approximate toe drain heights and heights of surface of seepage have been determined using the Kozeny’s function and the method of fragments. It has been found that for an earth dam with 1/2 upstream slope, ?1/3 downstream slope, no tailwater, and 2?m capillary rise, capillary saturation is contained within the earth dam and the phreatic line is prevented from emerging on the downstream sloping face by providing a toe drain of height equal to 1/3 of the height of water level in the reservoir.     

Two-Phase Flow Characteristics of Stepped Spillways

An experimental study on a large model flume with fiber-optical instrumentation indicated that minimum Reynolds and Weber numbers of about 105 and 100, respectively, are required for viscosity and surface tension effects to become negligible compared to gravitational and inertial forces expressed by Froude similitude. Both the location of and the flow depth at the inception point of air entrainment can be expressed as functions of a so-called roughness Froude number containing the unit discharge, step height and chute angle. The depth-averaged air concentration is found to depend only on a normalized vertical distance from the spillway crest and the chute angle for chute slopes ranging from embankment to gravity dam spillways. Air concentration profiles can be expressed by an air bubble diffusion model. The pseudobottom air concentration allows the assessment of the cavitation risk of stepped chutes and is approximated by a regression function. Finally, a new velocity distribution function is presented consisting of a power law up to 80% of the characteristic nondimensional mixture depth, and a constant value above.     

Seepage Modeling Assisted Optimal Design of a Homogeneous Earth Dam: Procedure Evolution

Presented herein is a procedure for arriving at an optimal design of a homogeneous earth dam laid on an impervious foundation and provided with a drain. The procedure, heavily dependent on variably saturated flow modeling, involves optimizing a multiobjective function comprising a weighted summation of four objective functions, viz., the dam section area, seepage discharge, wetted area of the dam section and the drain area. The design variables considered in the optimization are the upstream and downstream slopes and the drain dimensions. The optimization is carried out subject to the constraints ensuring safe upstream and downstream slopes and sufficient distance between the free surface and the downstream face. Two of the objective functions (viz., the seepage discharge and the wetted area) and the constraints are implicit functions of the design variables. Their values are obtained by employing a numerical model of two-dimensional (vertical plane) variably saturated flow in a homogeneous earth dam. Optimization, conducted by the sequential unconstrained minimization technique procedure is preceded by several runs of the model for various combinations of the design variables. The discrete values of the implicit functions so generated are invoked during optimization to compute the implicit objective functions and constraints. The results are presented in the form of nondimensional design tables/curves. The design procedure is illustrated with the help of few examples.     

Depth-Averaged Model of Open-Channel Flows over an Arbitrary 3D Surface and Its Applications to Analysis of Water Surface Profile

A new set of depth-averaged equations is introduced to study the flow over an arbitrary three-dimensional (3D) surface. These equations are derived based on a generalized curvilinear coordinate system attached to the 3D bed surface, therefore it allows us to include the effect of centrifugal force due to the bottom curvature. These general equations make it possible to analyze flows over complex terrain without the limitation of mild slope assumption used in conventional depth-averaged models. This new model is then applied to calculate the water surface profiles of (1) flow over a cylindrical surface; (2) flow over a circular surface; and (3) flow with an air-core vortex at a vertical intake. A simple hydraulic experiment is conducted in the laboratory to observe the water surface profile of flow over a circular surface. The results obtained from the model are in good agreement with experimental measurements and calculation by an empirical formula. Consequently, it demonstrates the applicability of the model in cases of flow over a highly curved bottom.     

Turbulent Open-Channel Flow in Circular Corrugated Culverts

This paper presents the results of a laboratory study of the velocity field in turbulent open-channel flow in a circular corrugated pipe of diameter D of 0.622 m for three slopes S of 0.55, 1.14, and 2.55% and a range of discharges from 30 to 200 L∕s. The Manning n was found to be equal to 0.023. Velocities were relatively small in some portion of the flow near the boundary of the pipe, and these low velocity regions may be useful for fish passage upstream. In the region of fully developed flow, in the central vertical plane, the longitudinal velocity u was described by the Prandtl equation for rough turbulent flow, with a dip in the velocity profiles near the water surface. The velocity profiles in the noncentral planes were also described by the Prandtl equation for rough turbulent flow, but with a significant dip in the upper part of the flow. An empirical method was devised to describe the geometrical and kinematical properties of this velocity dip. The general findings of this study were also found to be valid for flow in a large corrugated pipe of diameter of 4.27 m with two slopes of 0.14 and 1.42%.