Nonlinear PLS Method for Side Weir Flows

Side weirs are flow-regulating devices commonly encountered in hydraulic engineering. In the past, the discharge coefficient for flow past a side weir was investigated experimentally by many researchers. In this study, a modified discharge coefficient Cd for side weirs in rectangular channels and circular channels is defined. The multivariable nonlinear partial least square (PLS) method is used to determine the empirical equations relating Cd with the dimensionless weir parameters F1, S/Y1, and L/D. Compared to the previous studies, the procedures to calculate the discharge of the side weirs is simplified. The discharge predicted using the empirical equations based on the nonlinear PLS method is in good agreement with the measured discharge. The nonlinear PLS method can also be applied to many other hydraulic flow configurations characterized by a large number of variables.     

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.     

Aeration Performance of Triangular Planform Labyrinth Weirs

A high level of dissolved oxygen is vital for the maintenance of healthy streams and rivers. Structures in rivers can increase dissolved oxygen levels by creating turbulent conditions where small air bubbles are carried into the bulk of the flow. Plunging overfall jets from weirs are a particular instance of this, and the aeration properties of such structures have been studied widely in the laboratory and field over a number of years. On the other hand, labyrinth weirs, where the weir sill is cranked in planform thus increasing their length, have received little or no attention in this context. They have a proven hydraulic advantage over straight weirs of increased discharge at the same head for design conditions. However, they also serve to modify the combined overfall jet as individual jets from adjacent sections of the weir collide. This paper describes an experimental investigation into the nature of these jets and how they affect the aeration performance of a triangular plan labyrinth weir. It is demonstrated that the aeration efficiency of these labyrinth weirs generally is better than their equivalent-length linear weir and that this advantage becomes more pronounced as the weir included angle becomes smaller and also at lower overfall drop heights and higher discharges. These results point to the possible advantage of these type of weir in situations where both hydraulic and aeration performance needs to be optimized.     

Broad-Crested Weirs with Rectangular Compound Cross Sections

A series of laboratory experiments was performed in order to investigate the effects of width of the lower weir crest and step height of broad-crested weirs of rectangular compound cross section on the values of the discharge coefficient, the approach velocity coefficient, and the modular limit. For this purpose, nine different broad-crested weir models with rectangular compound cross sections and a model with a rectangular cross section were tested in a horizontal laboratory flume of 11.0 m length, 0.29 m width, and 0.70 m depth for a wide range of discharges. The compound cross sections were formed by a combination of three sets of step heights and three sets of lower weir crest widths. The sill-referenced heads at the approach channel and at the tailwater channel were measured in each experiment. The dependence of the discharge coefficient, approach velocity coefficient, and modular limit values on model parameters was investigated, and these quantities were compared with those of the broad-crested weir models with a rectangular cross section.     

Aeration Performance of Rectangular Planform Labyrinth Weirs

Maintenance of a high level of dissolved oxygen is important in natural rivers and in some water treatment processes. This can be enhanced by creating turbulence in the water. One method of producing such turbulence is via the overflow jets downstream of weir structures. The aeration characteristics of straight weirs have been measured and analyzed by a number of investigators. However, labyrinth weirs, which are cranked in planform, have received little attention in this respect. The aeration performance of a weir under a particular set of flow conditions is largely determined by the characteristics of the overall jet. The geometry of labyrinth weirs provides increased sill length and often results in the overfall jets colliding with each other, both of which may lead to increased aeration. A series of laboratory experiments were run on rectangular planform labyrinth weirs. These showed that, although detailed geometry of the weir was not important, the labyrinth weirs aerated significantly better than an equivalent straight weir, particularly at low drop heights.     

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.     

Generalized Head-Discharge Equation for Flow over Sharp-Crested Inclined Inverted V-Notch Weir

The sharp-crested weir is the most commonly used device in channels for flow measurement and flow regulation due to its simplicity. Attempts have been made to study in detail the flow over different shapes of normal conventional weirs, side weirs, and oblique weirs. Most of the investigators have expressed the discharge coefficient as a function of various parameters. The concern of this paper is to study flow over an inclined inverted triangular notch weir. Based on experimental work on 0° (normal) and 15, 30, 45, and 60° weirs, the writers have established an equation for discharge in terms of the inclination angle of the weir plane with the plane normal to the flow axis. A new general algebraic optimization procedure has been developed for evolving a linear head-discharge relationship for an inclined inverted V-notch (IIVN), which is relatively superior to similar procedures developed earlier. It has been shown that the IIVN does not lose its near-linear discharging characteristics even when it is inclined to the channel bed. The discharge estimated by the linear discharge-head relationship has been found to be in good agreement with the actual discharges well within a ±5% error. Further, the advantages of the IIVN as a flow measuring and flow control device have been highlighted.     

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.     

Multislit Weir Characteristics

The slit weir has been recently shown to be effective in measuring very small discharge rates. In this study, the slit weir concept is extended to permit one to measure both very low and very high discharge rates accurately. For this purpose, the hydrodynamic concept of images is adopted to form a bank of multislit weirs in which the individual slit weirs have nearly identical flow characteristics. The discharge coefficient of multislit weirs is determined experimentally using the measured discharges and the corresponding heads causing weir flow. The relationship between the discharge coefficient and the main dimensionless parameters is investigated. It is concluded that the discharge coefficient can be represented solely as a function of the Reynolds number.     

Experiments on Discharge Equations of Compound Broad-Crested Weirs

A linear combination of traditional discharge equations for simple rectangular and/or triangular weirs is proposed to describe the discharge equations of compound broad-crested (CBC) weirs. The CBC weirs are composed of rectangular, triangular, and/or truncated triangular weirs. Dimensionless discharge equations have been also derived. Laboratory experiments on discharge relations for flows over four CBC weirs were conducted in this study in order to calibrate the proposed discharge equations. The experiments were carried out under the conditions of the H1/H2-ratio of water heads above upper and lower crests less than 0.54, and a dimensionless discharge less than 2.174. The result shows that the differences between the calculated discharges by the proposed equations and the measured ones are less than 3% for flows over these CBC weirs under the present experimental conditions.     

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.     

Flow over Gabion Weirs

A conventional weir typically consists of an impermeable body constructed of concrete, since its primary functions are to heading up water and efficiently regulate flow. However, an impermeable body prevents the longitudinal movement of aquatic life and transportation of physical and chemical substances in water, eventually having a negative impact on the water environment. One of the advantages of gabions as a building material is that the motion of individual stones comprising the gabion is not of much concern. The wire mesh of the gabion basket serves to restrain any significant movement. Also, gabion weirs offer an alternative design that could be adopted for flash flood mitigation. In this study, a series of laboratory experiments was performed in order to investigate the flow over gabion weirs. For this purpose, two different gabion weir models were tested in two horizontal laboratory flumes of 10-m and 17-m length, 0.3-m width, and 0.3- and 0.5-m depth, respectively, for a wide range of discharge, upstream water depth, downstream water depth, weir height, weir length, and gabion filling gravel material size. The results of the gabion weir were compared with those of experiments carried out on solid weirs having the same dimension and it was found that there is a large deviation when the solid weirs equation is applied to gabion weirs (permeable weirs). So, using one of the existing solid weir flow formulas would lead to an erroneous calculated value. Multiple regression equations based on the dimensional analysis theory were developed for computing the discharge over gabion weirs for both free and submerged flow regimes. Also, equations were introduced for computing the discharge coefficient to be applied with the traditional solid weir equation.     

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.     

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.     

Hydraulic Relations for Clinging Flow of Sharp-Crested Weir

Available discharge coefficient formulas for sharp-crested weirs are only applicable to the free-flow regime. To extend the range of discharge measurement by a rectangular sharp-crested weir, critical heads of the transition flow regime, the head-discharge relation for clinging and free flow, and the discharge coefficient for clinging flow were investigated experimentally based on more than 300 experimental points with head ranging from 0.0048 to 0.0455 m. The results indicate that the transitions from clinging to free flow and vice versa do not occur at the same head. Upper and lower critical heads, Hu,crit and Hl,crit, can be identified at which these transitions occur. For the condition studied, the head relation between clinging and free flow is found to be linearly correlated at the same discharge. Expressions for the discharge coefficient for clinging flow are developed.     

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.     

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.     

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.     

Performance of Slit Weir

Slit weirs designed to measure small discharges in open channel flows are investigated. Experimental data available in the literature are added to study the effect of the slit width. An explicit relation for the discharge coefficient is obtained by a conventional Reynolds number definition. The discharge coefficient is modified to increase the working range of slit weirs for small values of head to slit-width ratios.     

Discharge Coefficient of a Triangular Side-Weir Located on a Curved Channel

This study aims to obtain sharp crested triangular side-weirs discharge coefficients both in the straight channel and along the bend by using a total of 1,735 experiments. It was found that triangular side-weirs discharging coefficients along the bend depend on the upstream Froude number in the main channel (F), the apex angle of side-weir (θ), and the bend angle (α). Because there is much greater intensity of secondary flow created by lateral flow with an increase of the overflow length, sharp crested triangular side-weirs discharge coefficients of the apex angle θ = 120° were achieved more frequently than the others even at the straight channel in subcritical flow conditions. In a curved channel, the path of maximum velocity and the secondary current created by the bend both cause a much greater deviation angle towards the side-weir which is involved within F and L/b. Therefore, triangular side-weirs discharging coefficients along the bend are greater than the values obtained in the straight channel.