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.     

Discharge Characteristics of Weirs of Finite Crest Length

This technical note presents a critical analysis flow over weirs of finite crest length, with square-edged or rounded entrance, for free-flow conditions. Using the flow equation for the broad-crested weir with parallel flow in the critical state as the basis, we have defined the discharge coefficient Cd, with the head on the weir as the length scale. Based on an extensive analysis of the experimental observations in the literature, we have confirmed the classification of finite crest length weirs into four classes of long-crested, broad-crested, short-crested, and sharp-crested weirs. For the square-edged entrance, we have developed robust correlations for Cd when the Weber number is greater than 1. For weirs with a rounded entrance, for which the data set is not that extensive compared to the square-edged case, we have developed good correlations for Cd.     

Discharge Characteristics of Chimney Weir under Free-Flow Conditions

The assumption of a constant coefficient of discharge in the linear head-discharge relationship of a chimney weir is reinvestigated. Based on dimensional analysis and subsequent experiments conducted with three different chimney weirs at various crest heights and channel widths, it is found that the coefficient of discharge in the linear relationship is not a constant, but is found to vary with the ratio of head to altitude, h/d; half-vertex angle in the form of w/d, w being the half crest width; h/(h+P), P being the crest height; and the channel width contraction ratio, w/C, C being the channel width. A linear regression equation correlating the coefficient of discharge with the above variables is proposed that, along with the linear head-discharge relationship, provides an accurate prediction of free-flow discharge.     

Discharge Capacity of Labyrinth Side Weir Located on a Straight Channel

Side weirs, also known as lateral weirs, are flow diversion devices widely used in irrigation as a head regulator of distributaries and escapes, land drainage, and urban sewage systems. The studies on the subject have been generally focused on rectangular and triangular side weirs located on a straight channel. However, the same is not true for labyrinth side weirs. The current studies deal with sediment transport and scour problems around side weirs and lateral structures. The investigation of the hydraulic effects of labyrinth side weirs to increase discharge capacity of them has been studied in this particular work. In the study, 2,830 laboratory tests were conducted for determining discharge coefficient of labyrinth side weirs, and results were analyzed to find the influence of the dimensionless weir length L/b, the dimensionless effective length L/?, the dimensionless weir height p/h1, triangular labyrinth side weir included angle θ, and upstream Froude number F1 on the discharge coefficient, water surface profile and velocities in the channel along the side weir. It has been found that discharge coefficient of labyrinth side weirs gives pretty higher coefficient value compare to that of classical side weirs and a reliable equation for discharge coefficient of labyrinth weirs is presented. Discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than rectangular side weir.     

Optimal Design of Composite Channels Using Genetic Algorithm

In the past, studies involving optimal design of composite channels have employed Horton’s equivalent roughness coefficient, which uses a lumped approach in assuming constant velocity across a composite channel cross section. In this paper, a new nonlinear optimization program (NLOP) is proposed based on a distributed approach that is equivalent to Lotter’s observations, which allows spatial variations in velocity across a composite channel cross section. The proposed NLOP, which consists of an objective function of minimizing total construction cost per unit length of a channel, is solved using genetic algorithm (GA). Several scenarios are evaluated, including no restrictions, restricted top width, and restricted channel side slopes, to account for certain site conditions. In addition, the proposed NLOP is modified to include constraints on maximum permissible velocities corresponding to different lining materials of the composite channel cross section, probably for the first time. The proposed methodology is applied to trapezoidal and triangular channel cross sections but can be easily extended to other shapes or compound channels. Optimal design graphs are presented to determine the channel dimensions of a composite trapezoidal channel cross section. The results obtained in this study indicate that cost savings up to 35% can be achieved for the unconstrained velocity case and up to 55% for the limiting velocity case when the proposed NLOP is solved using GA as compared with the existing NLOP solved using either the classical optimization solution technique or GA.     

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.     

Hydraulics of Broad-Crested Weirs with Varying Side Slopes

The flow of water over a trapezoidal, broad-crested, or embankment weir with varying upstream and downstream slopes has been investigated. Data are presented comparing the effect of slopes of 2H:1V, 1H:1V and vertical in various combinations on the upstream and downstream faces of the weir. Pressure and surface profiles were self-similar for all cases tested. Increasing the upstream slope to the vertical decreased the height of the surface profile and, hence, the static pressure of the crest. It also reduced the discharge coefficient. The variation in downstream static pressures was negligible though. Varying the downstream slope had a negligible effect on the surface and pressure profiles over the weir. Changes in flow were constrained to the region downstream of the crest. Cavitation could occur at the downstream corner of the weir if the upstream head was sufficiently high and a sloped face was used. This paper presents data that will be of use in the design of hydraulic structures for flow control and measurement.     

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.     

Measurement of Small Discharges in Open Channels by Slit Weir

A rectangular slit weir is designed for measurement of small (<0.005 m3/s) discharges. The discharge coefficient is determined experimentally using the measured discharges and the corresponding heads over the weir. The relationship between the discharge coefficient and all relevant dimensionless parameters is investigated. It is concluded that the discharge coefficient can be represented solely as a function of Reynolds number.     

Design and Calibration of a Compound Sharp-Crested Weir

This paper describes the design and calibration of a compound sharp-crested weir consisting of two triangular parts with different notch angles. This weir provides accurate measurement for a wide range of flows without discontinuities. The lower triangular part of the weir handles the normal range of discharges at the measurement structure while the upper part measures the occasional higher peak flows. This weir will be installed on the crest of small concrete gravity dams for measuring runoff in experimental catchments. The weir was calibrated in the laboratory using a scaled model. Different geometries (combination of notch angles) were tested in order to validate the proposed theoretical discharge equations.     

Optimal Design of Parabolic-Bottomed Triangle Canals

The characteristics of a parabolic-bottomed triangle cross section are introduced. For this geometry the “best” hydraulic cross section is determined by using the undetermined multipliers method of Lagrange. For a given flow, roughness coefficient, and longitudinal slope, the best hydraulic section is the channel section having the least wetted perimeter or cross-sectional area. The cross-sectional parameters of this novel geometry are compared with those of trapezoidal, parabolic, and round-bottomed triangle cross sections. It is shown that for all values of side slopes, the flow area and wetted perimeter in a parabolic-bottomed triangle cross section are less than those of trapezoidal and parabolic cross sections for the same discharge. This indicates that less excavation and linings are involved and therefore implies that the parabolic-bottomed triangle cross section is more economical than trapezoidal and parabolic cross sections.     

Overtopping Breaching of Noncohesive Homogeneous Embankments

Homogeneous small-amplitude embankments were constructed in flumes from a range of uniform noncohesive materials and breached by overtopping flows under constant reservoir level conditions. Embankment erosion evolves from primarily vertical to predominantly lateral in nature. The breach channel initially erodes the downstream face of the embankment with an invert slope parallel to the face, the breach invert slope then progressively flattening to a terminal value by rotating about a fixed pivot point along the base of the embankment, the location of this pivot point being a function of the size of the embankment material. The breach channel is of a curved (hourglass) shape in plan. Below the water line, breach cross-section width B variation with elevation y above the breach invert is nondimensionally described by B? = 2k?y?0.5, where for the breach cross section at the embankment crest k? = ?2.82[ln(Hb?)]+0.351, and Hb is the centerline breach crest elevation. Breach discharge Qb can be nondimensionally expressed as a function of the head hb on the breach-crest centerline and the breach crest length in plan Lb using Qb? = 0.242?Lb?(hb?)1.5. All expressions presented are applicable to full-width breach sections (double the half-breach section tested). The present findings enable prediction of the development with time of breach cross section, breach longitudinal profile, eroded volumes, and breach flows. The findings can be utilized for predictions of erosion and flooding occurring as the result of embankment failure, although in an engineering sense the quantitative findings of the present work await confirmation for larger embankments.     

Semianalytical Model for Shear Stress Distribution in Simple and Compound Open Channels

Semianalytical equations were derived for distribution of shear stress in straight open channels with rectangular, trapezoidal, and compound cross sections. These equations are based on a simplified streamwise vorticity equation that includes secondary Reynolds stresses. Reynolds stresses were then modeled and their different terms were evaluated based on the work of previous researchers and experimental data. Substitution of these terms into the simplified vorticity equation yielded the relative shear stress distribution equation along the width of different channel cross sections. In compound channels the effect of additional secondary flows due to the shear layer between the main channel and the flood plain were also considered. Comparisons between predictions of the model and experimental data, predictions of other analytical or three dimensional numerical models with advanced turbulent closures, were made with good agreement.     

Inception Point Relationship for Flat-Sloped Stepped Spillways

A two-dimensional, physical model was constructed to evaluate the air entrainment inception point location in a 4(H):1(V) stepped spillway. Step heights of 38, 76, and 152 mm were evaluated. The physical model was constructed with a broad-crested weir, and model unit discharges ranging from 0.11?m3/(s?m) to 0.82?m3/(s?m) were tested. Hubert Chanson developed a relationship for predicting the inception point location for primarily steep (θ ≥ 22°) stepped spillways. In this study, Chanson’s relationship effectively predicts the location of the inception point for slopes as flat as 14° for Froude surface roughness values (F?)>10 which, in this study, corresponds to model step heights of 38 and 76 mm. A new relationship for predicting the location of the inception point was developed, applicable for flatter sloped (θ ≤ 22°) stepped spillways with 1相似文献   

Critical Flow over Circular Crested Weirs

The critical flow principle is a useful approach for the hydraulic analysis of round-crested weirs due to their single head-discharge relationships. The hydraulics of circular-crested weirs is examined using simplified models incorporating streamline curvature effects, comparing their predictions with experimental data. A generalized one-dimensional model based on the critical flow in curvilinear motion has been developed. The discharge coefficient increases with the specific energy normalized with the radius of curvature, E/R, when streamline curvature effects are included. The relative flow depth at the crest decreases as E/R increases. The flow at the weir crest is only critical for a normalized specific energy value of E/R ≈ 0.5–0.6. For larger heads, the flow at the weir crest has been found to be supercritical.     

Improved Channel Cross Section with Two-Segment Parabolic Sides and Horizontal Bottom

A channel cross section with parabolic sides and horizontal bottom has been recently published and proved to be more economical (provide lesser construction cost per unit length) than the trapezoidal cross section. This paper presents a new and improved cross section with two-segment parabolic sides and horizontal bottom. Each side of the cross section consists of two parabolic segments smoothly connected. Closed-form relationships for the cross-sectional area and perimeter are developed. For specific parameter conditions, the new cross section produces most of the common cross sections, including the parabolic sides—horizontal bottom and trapezoidal cross sections, as well as new cross-sectional shapes. It provides an additional degree of freedom in determining the optimal cross-sectional design. A spreadsheet-based optimization model for the new cross section that minimizes the total construction cost (excavation and composite linings) is developed. The constraints of the model include channel discharge and physical requirements, such as flow depth, top width, and side slope with fixed or depth-dependent freeboard. The model was validated and the cross-sectional performance was evaluated using different design scenarios. The optimization results show that the new cross section is more economical and more flexible than a cross section with (one-segment) parabolic sides. As such, it should be of interest to the irrigation and drainage engineers.     

Flow Interaction of Meandering River with Floodplains

A series of laboratory test results are presented concerning the boundary shear stress, shear force, and discharge characteristics of compound meandering river sections composed of a rectangular main channel and one or two floodplains disposed off to its sides. Five dimensionless parameters are used to form equations representing the total shear force percentage carried by floodplains. A set of smooth and rough sections is studied with an aspect ratio varying from 2 to 5. Apparent shear forces on the assumed vertical, diagonal, and horizontal interface plains are found to be different from zero at low depths of flow and change sign with an increase in depth over the floodplain. A variable-inclined interface is proposed for which apparent shear force is calculated as zero. Equations are presented giving proportion of discharge carried by the main channel and floodplain. The equations agree well with experimental and river discharge data. Using the variable-inclined interface, the error between the measured and calculated discharges for the meandering compound sections is found to be the minimum when compared with that using other interfaces.     

Quick Method for Open-Channel Discharge Measurements Using Air Bubbles

An economical methodology is proposed by which distinct air bubbles released at the bottom of a channel may be utilized for determining the local flow discharge q per unit width. Simple theoretical analysis shows that q is linearly dependent on the rise length L of bubbles released at the bottom. This length is the horizontal displacement of the bubbles between the release cross section and the cross section where they emerge. The theoretical findings were compared with measurements in three laboratory flumes and in an irrigation canal. Based on the above, a relationship between L and q has been established. The empirically proposed relationship is very useful for fast discharge measurements in channels and natural streams.     

EDR in Circular Channels

This paper presents a simplified approach to determine the end depth of a free overfall in horizontal or mildly sloping circular channels. The flow over a free overfall in a circular channel is simulated by that over a sharp-crested weir to calculate the end-depth ratio (EDR). The theoretical model is calibrated by the experimental data, making the coefficient of velocity as a free parameter. The EDR, related to the critical depth, varies almost linearly from 0.72 to 0.74 for a critical depth-diameter ratio up to 0.86. The method of estimation of discharge from the known end depth is also presented.     

Supercritical Flow Measurement Using a Small Parshall Flume

An experimental program was conducted to determine if a Parshall flume, developed to accurately measure open-channel subcritical flow, could also be used to measure discharge in a supercritical flow regime. Fifteen experimental configurations were tested using two small Parshall flumes [6-in. (15.2-cm) and 9-in. (22.9-cm) crest width] with varying approach channel slopes, approach channel roughness, and flume convergence. It was determined that a single Parshall flume can be used to measure flow (within ±5%) for both supercritical and subcritical flow regimes for a specified range of flows. The original Parshall flume equation was then modified to incorporate crest width, channel slope, channel roughness, and convergence in the prediction algorithm. Unique expressions were developed for both supercritical and subcritical flow regimes to estimate the discharge. A single expression does not appear feasible for accurate discharge measurement for both flow regimes in a Parshall flume at this time.