Discharge Coefficient of Circular-Crested Weirs Based on a Combination of Flow around a Cylinder and Circulation

Abdorreza Kabiri-Samani1 and Sara Bagheri2

Abstract: Circular-crested weirs are overflow structures with a reasonably high discharge coefficient, compared with that of sharp- and broad-crested weirs. In this study, combining the potential flow around a circular cylinder and the free vortex flow, to determine the discharge coefficient and velocity distribution simulated flow over circular-crested weirs. Both the potential flow around a fixed-circular cylinder and the free vortex flow independently consider a symmetrical flow field. However, in realized conditions, the inertia force induces deviation of the flow streamlines over the circular cylinder. Therefore, a more comprehensive theory is essential to simulate the asymmetry of the flow streamlines properly. For this purpose, in the present study, both the mentioned theories were combined to simulate the flow around the circular-crested weirs using the potential flow past a circular cylinder with circulation. The analytical results were calibrated and compared with the experimental data of former investigations. Results indicate that the suggested semi-analytical model can predict the discharge coefficient and velocity distribution satisfactorily for the relative crest curvature (H1=Rb) up to 7. In addition, results of the sensitivity analysis demonstrated that H1=Rb is the most prominent parameter influencing the discharge coefficient of circular-crested weirs. DOI: 10.1061/(ASCE)IR.1943-4774.0000695. © 2014 American Society of Civil Engineers.

Author keywords: Circular-crested weir; Circulation; Cylinder flow; Discharge coefficient; Velocity profile.

Introduction

Circular-crested weirs are used for measuring flow discharge and diverting streamflow in water conveyance and distribution systems, and also for controlling the water level in farm ponds and reservoirs. Merits of circular-crested weirs are as follows: stable overflow pattern compared with sharp-crested weirs, simplicity of design compared with the ogee crest design, ease of passing floating debris, and associated lower cost. A circular-crested weir consists of a circular crest of radius Rb set tangentially to the up- and downstream faces having slopes α and β, respectively (Fig. 1).

Other parameters are also shown in Fig. 1. The flow discharge per unit width of the weir crest (q) can be calculated using the following relation (Bos 1976): q ¼ Cd ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2H1 3 3 s ð1Þ where Cd = the discharge coefficient; g = the gravitational acceleration; and H1 = the total head of the approach flow measured above the weir crest level. Ample studies have been conducted to investigate the effects of various parameters on discharge characteristics of circular-crested weirs. Matthew (1963) presented a simple theory to elucidate the effects of surface tension, viscosity, and radius of curvature of the flow streamline over the weir crest on

Cd for H1=Rb less than 1. Cassidy (1965) employed the ideal fluid flow theory to derive an equation for determining the flow discharge over circular-crested weirs. Sarginson (1972) studied the influence of surface tension on discharge coefficient of circularand sharp-crested weirs. He proposed a model to determine the weir discharge coefficient for 2 less than H1=Rb less than 4. Bos (1976) analyzed the existing data of other researchers [e.g., Verwoerd (1941), Escande and Sananes (1959), and Matthew (1963)] to derive a correlation between Cd andH1=Rb for circular-crested weirs.

He stated that Cd increases from 0.64 up to 1.48 when H1=Rb values are between 0.05 and 5.5 and then remains constant in the range 5.5 less than H1=Rb less than or equal to 9.5. Indeed, if the upstream water head is increased up to a certain value (here

H1=Rb ¼ 5.5), the flow momentum increases, resulting in an augmentation of the discharge coefficient Cd. However, for greater values of water head, a circular-crested weir behaves in a way similar to a sharp-crested weir. Consequently, the discharge coefficient remains constant with respect to H1=Rb values.

Hager (1985) extended Matthew’s (1963) theoretical approach and established an equation for estimating the flow discharge coefficient (Cd). Ramamurthy et al. (1992) employed the momentum principle to estimate the pressure distribution over the weir crest.

While investigating the influence of up- and downstream slopes (α and β) on discharge coefficients, Ramamurthy and Vo (1993b) found that the variation of upstream slope does not alter Cd. In addition, for 0 less than H1=Rb less than 3.5 and α ¼ 90°, the weir discharge coefficient does not change with the downstream slope.

Ramamurthy and Vo (1993a) used the velocity profile of a shallow irrotational flow over a curved bed, following the method of

Dressler (1978), to predict the velocity distribution over a cylindrical weir. Ramamurthy et al. (1994) used the principle for irrotational flow to estimate the geometrical characteristics of flow over the crest of circular-crested weirs. Chanson and Montes (1998) studied the flow behavior over circular-crested weirs considering five types of upstream approach flow conditions: partially developed flow, fully developed flow, upstream ramp, upstream normal hydraulic jump, and upstream undular hydraulic jump.

Assuming the potential flow theory around a circular cylinder, 1Associate Professor, Dept. of Civil Engineering, Isfahan Univ. of

Technology, 84156-83111 Isfahan, Iran. E-mail: akabiri@cc.iut.ac.ir 2Ph.D. Candidate, Dept. of Civil Engineering, Isfahan Univ. of

Technology, 84156-83111 Isfahan, Iran (corresponding author). E-mail: sara.bagheri@cv.iut.ic.ir

Note. This manuscript was submitted on August 16, 2013; approved on

November 25, 2013; published online on February 20, 2014. Discussion period open until July 20, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Irrigation and Drainage Engineering, © ASCE, ISSN 0733-9437/04014010(6)/ $25.00. © ASCE 04014010-1 J. Irrig. Drain Eng.