Discharge coefficient equation for semi-circular sharp-crested side weir in subcritical flow regime

Side or lateral weir is a flow measurement structure that is extensively used in irrigation and drainage networks and sewer systems. The main channel cross-section and side weir shape completely modify the flow conditions. Up to now, many studies have been conducted on side weirs. Most of the previous theoretical analysis and experimental research works are related to the flow over rectangular side weirs. This study focuses on a circular sharp-crested side weir located in a rectangular main channel in subcritical flow regime. This kind of side weir has some advantages, in comparison to other general weir types. Since the height of this side weir varies along its length, it can control flood better than other side weirs in flood conditions. In addition, because of its circular shape, it has a greater discharge coefficient.
The governing equation of the circular sharp-crested side weir has no analytical solution and thus should be solved by numerical methods. In order to simplify the solution, the equation of inline circular weir with discharge coefficient as a calibration parameter is used. The reference flow depth which should be used in this equation is an important point. In this study, three reference depths were considered (average depth 0.5(y1), center depth and initial depth y1).
In this research, the equation of conventional circular weir presented by Vatankhah (2010) is used for circular sharp-crested side weir:where Qa=actual discharge; Cd = discharge coefficient; g = gravitational acceleration; and η=H/D, with H being flow depth above the circular sharp-crested weir of diameter D.
For circular sharp-crested side weir, effective non-dimensional parameters were determined using dimensional analysis and Buckingham's Pi-Theorem. Finally, the following non-dimensional parameters were considered as the most effective ones on the discharge coefficient:where Fr= Froude number and B= main channel width.
The experiments were performed in a rectangular open channel having provisions for a side weir at one side of the channel. The main channel was horizontal with 12 m length, 0.25 m width, and 0.5 m height which was installed on a frame. The lateral channel has a length of 6 m, width of 0.25 m, and height of 1 m that was set up parallel to the main channel; the walls and its bed were made up of Plexiglas plates. The side weir was positioned at a distance of 6 m from the channel’s entrance. A total of 123 experiments on circular sharp-crested side weirs were carried out.
Different depths were considered as the reference depth. According to the experimental analysis, the discharge coefficient is a function of the upstream Froude number and the dimensionless ratio of its diameter to the main channel width.
The proposed empirical equations for discharge are as: (Average depth 0.5(y1) as the reference depth)
(Center depth as the reference depth)
(Initial depth y1 as the reference depth)
The average errors of the proposed equations are less than 2.4%, and only 5.5% of the experimental data have error more than 5% if the averaged and center depths are considered as the reference depth. However, 8.1% of the experimental data have error more than 5% if the initial depth is used as the reference depth.
In this research, the characteristics of the circular sharp-crested side weir overflows in subcritical flow regime are discussed. For this purpose, experimental data related to the water surface profile of the side weir and discharge coefficient were collected and carefully analyzed to develop accurate and simple discharge equation. The results showed that the most efficient section for measuring the water surface profile is located on the center line of the approaching channel. It was also found that for the circular sharp-crested side weir, the discharge coefficient depends on the Froude number and the ratio of weir diameter to the main channel width. In this study, the conventional circular sharp-crested weir theory has been used in order to evaluate the discharge coefficient and provide a side weir discharge equation. For this purpose, three reference flow depths were considered for conventional weir (average depth 0.5(y1), center depth and initial depth y1), and for each flow depth an equation was developed for the discharge coefficient. Comparison between predicted values and experimental data showed that the average and center flow depths result in accurate outcomes for estimating the discharge coefficient. The average value of error for discharge coefficient estimation by the proposed equations is less than 2.4%. Thus, these equations are proposed for practical use.