Weir velocity formulation for sharp-crested rectangular weirs S. Gharahjeh n , I. Aydin, A.B. Altan-Sakarya Department of Civil Engineering, Middle East Technical University, Ankara 06800, Turkey article info Article history: Received 1 October 2013 Received in revised form 24 October 2014 Accepted 29 October 2014 Available online 8 November 2014 Keywords: Flow measurement Discharge Sharp-crested weirs Rectangular weir Open-channel flow abstract Discharge in open channels can be measured by sharp-crested rectangular weirs. Generally, measured head over the weir crest is substituted into an empirical formula derived from energy considerations to calculate the discharge. Assumptions made on the derivation are taken into account by defining a discharge coefficient that fits into the experimental data. In this study, a physical quantity, the average velocity over the weir section defined as ‘weir velocity’ is directly formulated as function of weir geometry and head over the weir. Weir velocity plotted against the weir head has a universal behavior for constant weir width to channel width ratio independent of the weir size. This unique behavior is described in terms of weir parameters to calculate the discharge without involving a discharge coefficient. Combining weir velocity data for variable weir widths provides a basis for direct formulation of discharge. The weir velocity exhibits simpler functional dependency on weir parameters in contrast to the discharge coefficient. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Sharp crested rectangular weirs are designed as control sec- tions to provide a particular link between the discharge and water depth. The depth at a section upstream of the weir is measured and the discharge is calculated. These classical devices are com- monly utilized in laboratories, irrigation practices and industry. Three-dimensional flow phenomena involving significantly curved streamlines and vortex structures can take place around the weir plate. Due to this complex nature of the flow over the weir, any analytical derivation for the flow rate entails simplifying assumptions and requires complementary functions established by utilizing experimental data. Most of the weir formulations involve a discharge coefficient to complement the discharge expression obtained from energy considerations. For a rectangular weir (Fig. 1) the discharge, Q, may be written in terms of the measured head over the weir, h, the weir width, b and the discharge coefficient, C d [6]: Q ¼ 2 3 C d ffiffiffiffiffi 2g p bh 3=2 ð1Þ where g is the gravitational acceleration. The discharge coefficient is assumed to represent all effects due to assumptions made in the derivation of Eq. (1). Therefore, C d may be expected to depend on the channel width, B, the weir height, P and dimensionless parameters such as Reynolds number, R e , Weber number, W e and Froude number, F r . It is possible to set limitations on the working range of Eq. (1) to eliminate such dependencies. Many researchers have studied weirs to determine the dis- charge coefficient, C d . Their findings are either empirical or analytical and sometimes a combination of both. Some of the available C d formulations in the literature are presented in Table 1. One of the oldest experimental researches is traced back to Rehbock [11]. He proposed a relation for C d which is derived with neglecting viscous and capillary effects. Swamee [13] suggested a full-range weir equation by combin- ing the proposed equations of Rehbock [11] and Rouse [12] and fitting the experimental data of Kandaswamy and Rouse [9]. The resulting equation would hold good for extreme variations of head over weir height ratios (h/P). It can be applied to sharp-crested, narrow-crested, broad-crested and long-crested weirs. Another study on sharp crested weirs was presented by Bagheri and Heida- rpour [4] based on integration of velocity due to free-vortex motion assumed between the upper and lower nappe profiles. Form of the discharge coefficient expression was obtained from the best fit approximations of the measured nappe profiles. For the range of 0 oh/P o9, Bagheri and Heidarpour [5] revised their equation by enlarging the data base and extended the validity range to 0 oh/P o10. Aydin et al. [1] introduced the slit weir suitable for measuring small discharges with high accuracy due to increased head over the weir. The most important characteristic of the slit weir is the independency of the discharge coefficient from the channel width, B, because of the completely contracted nature of the flow over the weir. Formulation of the discharge coefficient was improved with increased data range in a later study by Aydin et al. [2]. Although the measuring capacity for large discharges is limited, the precision and reliability achieved by the slit weir is significant when compared to other weir types. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/flowmeasinst Flow Measurement and Instrumentation http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.018 0955-5986/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Flow Measurement and Instrumentation 41 (2015) 50–56