Free over-fall in exponential channel cross-sections based on free vortex theory in supercritical flow regimes

Free over-fall can be used as a flow metering hydraulic structure by a single measurement of an end depth. Many theoretical and experimental researches carried out on free over-fall with various approaches for different cross-sectional shapes. This paper presents a theoretical method to compute the end depth ratio (EDR) and the end depth discharge (EDD) relationships in steep sloping channel for the exponential cross-section. The exponential cross-section is a general section which can reduce to rectangular, wide rectangular, parabolic, semi-parabolic, triangular, and semi-triangular channels. Applying the momentum equation based on the free vortex theory, a theoretical approach is presented to obtain the EDR for the exponential channel cross-section in supercritical flow regime. Experimental and theoretical studies are then utilized to verify the proposed EDR and EDD relationships. The computed results are in acceptable agreement with the relevant experimental and theoretical studies. Direct solutions of the discharge for the known end depth for each cross-section are provided in tabular forms where two empirical discharge expressions with their relevant range of applications for each channel cross-section are detailed as the main outcome of this study.     

Free overall in circular channels with flat base: a method of open channel flow measurement

A free overfall at the end of an open channel offers a simple means of measuring flow discharge. In this paper, two methods are presented for the computation of end-depth and discharge of a free overfall from smooth circular channels with flat base. Firstly, applying the momentum equation based on the Boussinesq approximation, the flow upstream of a free overfall is theoretically analyzed to calculate the end-depth-ratio (EDR). This approach eliminates the need of an experimentally determined pressure coefficient. In subcritical flows, the EDR is related to the critical-depth that occurs far upstream. In supercritical flows, the Manning equation is used to express the end-depth as a function of streamwise slope of the channel. Methods to estimate discharge from the end-depth in subcritical and supercritical flows are presented. The upstream flow profiles of a free overfall are computed using the streamline curvature at free surface. Secondly, an alternate method for analyzing free overfall from circular channels with flat base is also presented, where a free overfall in a circular channel with flat base is simulated by the flow over a sharp-crested weir to calculate the EDR. The comparisons of the computed results with the experimental data are satisfactory for subcritical flow and acceptable for supercritical flow.     

Flow measurement using free over-fall in generalized trapezoidal channels based on one velocity point method

A free over-fall offers the possibility of being used as a flow measuring device in hydraulic structures with a single depth measurement of the end section. Due to its practical importance, considerable attention has been paid to investigate free over-falls for different channel cross-sections using various approaches. This paper presents a new theoretical approach for computing the end depth ratio (EDR) relationship for the generalized trapezoidal channel cross-sections at free over-falls in sub critical flow regimes from which the end depth discharge (EDD) can be computed. The generalized trapezoidal channel is a geometric shape that is defined mathematically with a single equation where five widely known prismatic channel cross-sectional shapes can be generated (trapezoidal, inverted triangular (Δ), rectangular, parabolic, and triangular). This suggested theoretical approach uses one velocity point at the geometric center of the end section based on the energy and the continuity equations. Relevant experimental and theoretical results were utilized in order to examine the suggested method through the statistical measuring indices (percentage difference and the correlation coefficient (R²)). The computed results show very close agreements with the earlier works. Furthermore, simple equations are also generated using the regression curve fitting technique in order to estimate the direct discharges (Q) using the end depth (y_e) for each of the above mentioned channel cross-sections.     

Flow measurement by the end-depth method in inverted semicircular channels

A free overfall at the end of an open channel provides a simple means for measuring flow discharge. The paper presents a simplified approach for the computation of end-depth of a free overfall in horizontal or mildly sloping inverted semicircular channels. Using a known end-depth the discharge can be estimated. The flow over a free overfall in an inverted semicircular channel is simulated by that over a sharp-crested weir to calculate the end-depth-ratio (EDR). The mathematical model is calibrated by the experimental data, making the coefficient of velocity a free parameter. The EDR, related to the critical depth, is around 0.705 for a critical depth–diameter ratio up to 0.42. The computed results agree satisfactorily with the experimental data.     

End depth computation in inverted semicircular channels using ANNs

The paper presents the application of artificial neural network (ANN) to determine the end-depth-ratio (EDR) for a smooth inverted semicircular channel in all flow regimes (subcritical and supercritical). The experimental data were used to train and validate the network. In subcritical flow, the end depth is related to the critical depth, and the value of EDR is found to be 0.705 for a critical depth–diameter ratio up to 0.40, which agrees closely with the value of 0.695 given by Dey [Flow Meas. Instrum. 12 (4) (2001) 253]. On the other hand, in supercritical flow, the empirical relationships for EDR and non-dimensional discharge with the non-dimensional streamwise slope of the channel are established.     

Direct solution for discharge in generalized trapezoidal free overfall

In open channels, free overfall can be used as a discharge measuring structure by a single measurement of depth at the end of the channel. If the slope of channel is negative, zero or mild, the flow at upstream of end section will be critical and end depth value depends only on the shape of the approach channel and its critical depth. This research presents a theoretical end depth–discharge (EDD) relationship for free overfall (end section) in a horizontal open channel with generalized trapezoidal section. The generalized trapezoidal shape reduces to the commonly used trapezoidal section as well as to the Δ-shaped section. Two direct discharge equations in terms of end depth for subcritical flows are proposed by simulating free overfall as a weir without crest. Experimental data are then used to verify the proposed EDD relationships. The calculated discharges, using the proposed EDD relationships, show excellent agreement with the experimental data.     

Stage-discharge relationship for slide gates installed in partially full pipes

Sluice/slide gates are widely used for flow depth control and flow discharge measurements in open channels. The hydraulic behavior of the sluice gates located in the rectangular open channels is well documented in the literature. This study reports the results of an investigation conducted to establish the stage-discharge relationship for the sluice gates located in horizontal, circular open channels/pipes under free outflow conditions. Different stage-discharge models were proposed based on the Buckingham's theorem of dimensional analysis and orifice theory. A comprehensive series of laboratory experiments (729 runs) were performed to study the sluice gates located at the middle, and at the end of two circular pipes. Using the data collected from two circular open channels of nominal diameters 20 and 30 cm, the proposed models were calibrated. For the middle slide gates, the experimental results showed that the discharge prediction can be improved by introducing the Reynolds number. For the slide gates located at the middle of the channel, the best proposed model has an average error of 1.40% with a maximum error of 7.12%. For the slide gates located at the end of the channel, the Reynolds number has no significant effect and best proposed model has an average error of 2.47% with a maximum error of 6.59%. The results also showed that the flow discharge of the end slide gate (with unconfined free jet under gravity) is higher than the flow discharge of the middle slide gate for the same gate opening areas and upstream flow depths. The proposed sluice/slide gate for circular open channels offers a simple and reliable discharge measurement approach with acceptable accuracy.     

Hydraulics of free overfall in steeply sloping rough rectangular channel: A general computational approach

All supercritical free overfalls were almost considered as frictionless flow or pseudo-uniform flow before, the effect of friction had not been paid due attention. Currently, a new mathematical model is proposed to analyze supercritical rectangular free overfall by Boussinesq-type energy equation accounting for frictional effect. A computational approach of loop iteration algorithm is utilized to simultaneously calculate the implicit water surface profile by Runge-Kutta method, end-depth-ratio, and discharge. Subsequently, the dependency of discharge on end depth and Nikuradse equivalent sand roughness for different slope is illustrated. Comparison of computed results with experimental observations reveals satisfactory agreement. Additionally, the present energy approach leads to more accurate results than the momentum method for high roughness. Finally, to aid application for engineers, an explicit solution of discharge for practical application in the commonly using range is also proposed. The most important purpose of this study is to present a general computational approach, which could be used for the rough supercritical free overfalls of any other cross-sections.     

Prediction of the end-depth ratio and discharge in semi-circular and circular shaped channels using support vector machines

This paper presents the results of an application of a support vector machine based modelling technique to determine the end-depth ratio and discharge of a free overfall occurring over an inverted smooth semi-circular channel and a circular channel with flat bases. The data used in this study are taken from earlier studies reported in the literature. The results of the study indicate that the support vector machine technique can be used effectively for predicting the end-depth ratio and the discharge for such channels. For subcritical flow, the value of the end-depth ratio is found to be 0.704, which compares favorably to the predicted values obtained by using empirical relations derived in previous studies, while for supercritical flow, the support vector machines perform equally well and are found to work better than the empirical relationship proposed by S. Dey [Free over fall in circular channels with flat base: a method of open channel flow measurement, Flow Meas. Instrum. 13 (2002) 209–221]. The results also suggest the usefulness of support vector machine based modelling techniques in predicting the end-depth ratio and discharge for a semi-circular channel using the model created for the circular channel data, and vice versa, for supercritical flow conditions.     

Discharge prediction in flow measurement flumes with different downstream transition slopes

Experimental testing of 9 different rectangular compound cross-section flow measurement flumes with different downstream slopes was conducted to yield the coefficient of discharge and the approach velocity coefficient. The aim of the experimental research was the determination of stage–discharge relationship in compound cross-section flow measurement flumes with different downstream slopes. One empirical predictive model for each of the coefficient of discharge and the approach velocity coefficient for the 9 cross-sections have been derived using one dimensionless parameter for the coefficient of discharge and another one dimensionless parameter for the approach velocity coefficient as the single independent variable. This approach is preferred as it allows the estimation of discharge by only measuring the water depth at the head measurement section. All obtained predictive models statistics have indicated the high reliability of the derived models in estimating discharge in an open channel flume of a rectangular compound cross-section using the predicted coefficients.     

Horizontal bottomed semi-cubic parabolic channel and best hydraulic section

This paper presents a new channel section having semi-cubic parabolic sides and horizontal bottom. The formulae for calculating the area, wetted perimeter are presented. The best hydraulic section is derived using three variables (water depth, water surface width and horizontal bottom width). Results show the ratios of the water surface width to depth, bottom width to depth and water surface width to bottom are all constant for the best hydraulic section. Explicit equations of the best hydraulic section for design are also deduced. Examples show these explicit equations are convenient for design. This type of best hydraulic section is compared with the trapezoid and classic semi-cubic parabolic sections. Results indicate that the area and wetted perimeter are less than those of trapezoid and classic semi-cubic parabolic sections for a given flow discharge. It means less lining and excavation cost is required for construction.     

Discharge equation of semi-circular side weirs: An experimental study

Side weirs are diverting structures and usually used for diverting and controlling the water flow into the side open channel. The present study deals with an experimental study regarding the hydraulic performance of side weirs with semi-circular vertical sections along the main channel. As flow depth of the main channel increases the top flow width of the semi-circular side weir (SCSW) increases which is an advantage when high discharge enters the main channel and should be immediately diverted for safety reasons. In this study, the flow discharge of semi-circular sharp-crested side weirs and their affecting parameters are investigated. To investigate the hydraulic behavior and geometric characteristics of the SCSWs, a comprehensive laboratory study including 155 tests (for three weir diameters 0.25, 0.30 and 0.40 m) was conducted in a physical model under subcritical flow conditions. Flow discharge of the SCSW was investigated in relation to height, diameter and flow head of side weir, also approach Froude number (Froude number at upstream end of the side weir) and main channel width. Three different discharge models were developed based on; purely dimensional analysis technique, classical weir equation with linear water surface and classical weir equation with horizontal water surface profile (conventional weir theory along with dimensional analysis technique). The presented mathematical discharge models enable estimation of discharge along the SCSW with acceptable accuracy (best model has an average error of 1.87% with a maximum error of 6.31%) compared with the measured data under subcritical flow conditions. Additionally, a relationship was proposed for computing the limiting flow depth at the downstream end of the SCSW. Experimental results confirm that the proposed relationship well explains the behavior of flow over the SCSW regarding the downstream flow conditions.     

Flow structure of the wake behind an elliptic cylinder close to a free surface

The flow fields around an elliptic cylinder of axis ratio AR=2 adjacent to a free surface were investigated experimentally using a water channel. The main objective is to understand the effect of the free surface on the flow structure in the near-wake. The flow fields were measured by varying the depth of cylinder submergence, for each experimental condition, 350 velocity fields were measured using a single-frame PIV system and ensemble-averaged to obtain the spatial distribution of turbulent statistics. For small submergence depths a large-scale eddy structure was observed in the near-wake, causing a reverse How near the free surface, downstream of the cylinder. As the depth of cylinder submergence was increased, the flow speed in the gap region between the upper surface of the cylinder and the free surface increased and formed a substantial jet flow. The general flow structure of the elliptic cylinder is similar to previous results for a circular cylinder submerged near to a free surface. However, the width of the wake and the angle of downward deflection of the shear layer developed from the lower surface of the elliptic cylinder are smaller than those for a circular cylinder.     

Discharging capacity of rectangular side weirs in straight open channels

A side weir is a hydraulic control structure used in irrigation and drainage systems and combined sewer systems. A comprehensive laboratory study, including 843 tests for the discharge coefficient of a sharp-crested rectangular side weir in a straight channel, was conducted in a large physical model under subcritical flow conditions. The discharge coefficient is a function of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth. An equation was developed considering all dimensional parameters for discharge coefficient of the sharp-crested rectangular side weir. The average error of the proposed equation is 4.54%. The present study data were compared with ten different discharge coefficient equations developed by several researchers. The study also presents water surface profile and surface velocity streamlines.     

共面薄膜电极表面放电冷却技术

为研究一种可用于雷达领域的新型冷却技术,本文研究了高电压下共面薄膜电极之间的DC表面放电现象。设计制作了一系列具有不同参数且带有一个阳极针尖以及两个对称布置阴极的样机,通过改变基底表面粗糙度以及不同的结构参数(如凹槽深度、凹槽宽度以及阴极长度等)进行实验测试。结果表明:主要受到基底表面离子迁移率的影响,凹槽深度对于共面薄膜平面电极的表面放电现象影响最大;表面放电的电流稳定性随着深度的增加而增加;而放电起始电压则随着深度的增加而减小;离子与平面基底之间的流体阻力影响相对较小。共面薄膜电极表面放电的研究对于推动电冷却技术在雷达技术领域的应用具有重要意义。     

Experimental analysis of combined side weir-gate located on a straight channel

Flow measurement and control in open channel system for lateral flow is important to support the system management. The flow characteristics of combined side weir-gate are complicated due to changes in flow conditions along the side weir-gate section. Experimental investigation of the flow characteristics of both weir and sluice gates is crucial for predicting the flow through a combined side weir-gate structure. Although weir-gate structures are widely used for frontal flow in hydraulic structures, the same is not true for lateral flows. In this study, 650 laboratory tests were conducted to determine the flow characteristics of combined side weir-gate for subcritical flow, and the experimental results were analyzed to determine the effect of these characteristics and weir-gate geometry on discharge capacity. Interaction factor of combined side weir-gate is a function of the upstream Froude number, the ratio of gate opening to upstream flow depth, the ratio of distance between the top of the sluice gate and the weir crest to gate opening, the ratio of weir and gate length to upstream flow depth, and the ratio of weir and gate length to main channel width. Consequently, more discharge is passed through combined side weir-gate compared to side weir and sluice gates. The empirically derived equations for the discharge of combined side weir-gate show good compatibility with the experimental data.     

Discharge characteristics of sharp-crested circular side orifices in open channels

A side orifice is a flow diversion structure provided in one or both side walls of a channel to spill/divert water from the main channel. It is widely used in irrigation and environmental engineering. Analytical and experimental studies related to the discharge characteristics of sharp-crested circular side orifices in open channels under free flow conditions have been presented in this paper. Considering the side orifice as large, the discharge equation for the side orifice is derived analytically. Experiments were performed to estimate the coefficient of discharge which depends on the approach flow Froude number and ratio of the diameter of the orifice and bed width of the channel. Relationships for the coefficient of discharge, considering the orifice as large and small were developed. Such relationships were used to compute the discharge through the orifice for data not used for proposing such relationships for the coefficient of discharge. The computed discharges were within ±5% of the observed ones. The average percentage error in computation of discharge through the orifice considering it as large and small are, respectively, 1.59% and 1.66% which are practically the same. Therefore, it is recommended that the discharge through the side orifice can be computed considering it as a small orifice within the range of data used in the present study.     

Experimental and theoretical modeling of a quarter-circular gate flow

Gates are important hydraulic structures and used for flow measurement, water delivery, and water level regulation in open channels and irrigation networks. In this study, the quarter-circular gate is introduced and investigated. The cross section of this gate consists of a quarter circular arc and the lip angle of the gate equals to zero. Discharge coefficient, variation of downstream flow depth, and velocity distribution at opening section of gate were experimentally measured. Using potential flow theory supported by dimensional analysis, equations for discharge coefficient and velocity distribution at gate opening section of quarter-circular gate were derived and then validated using experimental data. The mean percentage error (MPE) of obtained equation for discharge coefficient of quarter-circular gate was calculated as 2.24%, indicating the high precision of the proposed theory. Based on obtained results, downstream flow depth of quarter-circular gate is uniform. Also, velocity distribution at gate opening section is nearly uniform. Discharge coefficient of quarter-circular gate was averagely obtained 55% larger than that of sluice gate. It was also obtained larger than that of radial gate. Elimination of contraction section at downstream of gate opening, which is the main source for energy loss and therefore discharge capacity reduction, is the main reason for larger discharge coefficient of quarter-circular gate.     

Study of the hydraulic properties of the cylindrical crested weirs

Since the cylindrical and circular crested weirs are economical and easily made compared to some other weirs, they can be used to measure the flow velocity, the water discharge and flow surface control structures in the canals and reservoirs. In this paper, the hydraulic properties of the cylindrical and circular crested weirs, such as the discharge coefficient, the depth on the crest of the weir and the energy loss in 18 laboratory models categorized in 5 categories have been investigated. The results of the experiments indicated that in both the cylindrical and circular crested weirs by increasing the total partial head, the discharge coefficient increases and any changes in the upstream wall slope has no effect on the discharge coefficient. Also, the partial energy loss more occurs in the cylindrical weirs than the half shaped cylindrical weirs and by increasing the downstream inclination angle, the partial energy loss increases subsequently. The partial flow depth on the crest of both the cylindrical and half shaped cylindrical weirs is equal to 0.7 and by increasing of the upstream inclination angle, the partial flow depth on the crest in the values greater than 0.6, increases slightly and the downstream inclination angle changes of the partial flow depth has no effect on the crest.     

Free overfalls in flat-based circular and U-shaped channels

This paper presents a theoretical model based on the free vortex theorem that is capable of predicting the pressure head distribution at the brink of free overfalls in open channels. This approach is coupled with the momentum equations to obtain the end-depth-ratio (EDR) from which the discharge can be estimated. In order to illustrate its flexibility, the theory is successfully applied to flat-based circular and U-shaped channels. Using previous experimental data, the proposed method is validated.