Discharge coefficient of a semi-elliptical side weir in subcritical flow

Abstract

A labyrinth weir is an overflow weir, folded in plan view to provide a longer total effective length for a given overall weir width. The total length of the labyrinth weir is typically three to five times the weir width. In this study, a semi-elliptical labyrinth weir was used as a side weir structure. Rectangular side weirs have attracted considerable research interest. The same, however, is not true for labyrinth side weirs. The present study investigated the hydraulic effects of semi-elliptical side weirs in order to increase their discharge capacity. To estimate the outflow over a semi-elliptical side weir, the discharge coefficient in the side weir equation needs to be determined. A comprehensive laboratory study including 677 tests was conducted to determine the discharge coefficient of the semi-elliptical side weir. The results were analyzed to find the influence of the dimensionless weir length $L/B$, the dimensionless effective length $L/\ell$, the dimensionless weir height $p/h_1$, the dimensionless ellipse radius $b/a$, and upstream Froude number $F_1$ on the discharge coefficient. It was found that the discharge coefficient of semi-elliptical side weirs is higher than that of classical side weirs. Additionally, a reliable equation for calculating the discharge coefficient of semi-elliptical side weirs is presented.

Introduction

Side weirs have been extensively used in hydraulic and environmental engineering applications. They are substantial parts of the distribution channel in irrigation systems and treatment units. In combined sewer systems suitable locations are selected where, during storms, the approach flow can be partitioned so that only a small fraction of water remains in the system and continues towards the sewage treatment station. The remainder is either discharged into a rainwater storage basin or fed directly into a receiving water body. To achieve this, two structures have been favored in recent decades: bottom openings and side weirs. The bottom opening is particularly suitable when the upstream flow is supercritical, whereas the side weir is usually adopted with subcritical upstream flow. The required water for irrigation can also be obtained from any irrigation channel by using a side weir. A side weir is an overflow weir formed at the side of a channel, which allows lateral flow of the water when the surface of the water in the channel rises above the weir crest. The ability to predict the flow that is diverted in this way is useful in the design of diversion structures and in flood alleviation works.

A review of previous studies indicated that rectangular sharp-crested side weirs have been investigated extensively, including work by Ackers [1], Collings [2], Frazer [3], Subramanya and Awasthy [4], El-Khashab and Smith [5], Uyumaz and Muslu [6], Helweg [7] and, Agaccioglu and Yüksel [8]. Borghei et al. [9] studied the discharge coefficient for sharp-crested side weirs in subcritical flow, and developed an equation for the discharge coefficient of sharp-crested rectangular side weirs. Also, in order to study the variation of the discharge coefficient along side weirs, Swamee et al. [10] used an elementary analysis approach to estimate the discharge in smooth side weirs through an elementary strip along the side weir. The hydraulic behavior and the discharge coefficient of different types of weirs have been studied by many researchers, including: Nandesomoorthy and Thomson [11], Singh et al. [12], Yu-tech [13], Cheong [14], and others. Ranga Raju et al. [15] investigated the discharge coefficient of a broad-crested rectangular side weir, based on the width of the main channel, Froude number and head/weir width ratio. Kumar and Pathak [16] investigated the discharge coefficient of sharp and broad-crested triangular side weirs. Ghodsian [17] studied supercritical flow in rectangular side weirs. Khorchani and Blanpain [18] investigated flow over side weirs using video monitoring techniques. Coşar and Agaccioglu [19] studied the discharge coefficient of a triangular side weir both on straight and curved channels. Yüksel [20] modeled the effect on flow of changes in specific energy height along a side weir. Aghayari et al. [21] investigated experimentally the effect of height, width and side weir crest slope on the spatial discharge coefficient over broad-crested inclined side weirs under subcritical flow conditions in a rectangular channel.

The flow over a side weir falls within the category of spatially varied flow. The existing studies deal mainly with the application of the energy principle in the analysis of side weir flow. The concept of constant specific energy [22] is often adopted for studying the flow characteristics of these weirs [23], [12], [14], [16], [15], [4].

There is considerable interest, particularly in rectangular side weirs. De Marchi [22] was one of the first researchers to provide equations for flow over side weirs. Considering the discharge $dQ$ through an elementary strip of length $ds$ along the side weir in a rectangular main channel as a De Marchi equation, one gets

$$q=-\frac{dQ}{ds}=\frac{2}{3}{C}_d\sqrt{2g}{[h-p]}^{3/2}$$

where $Q$ is the discharge in the main channel, $s$ is the distance from the beginning of the side weir, $dQ/ds$ (or $q$) is the spill discharge per unit length of the side opening, $g$ is acceleration due to gravity, $p$ is the crest height of the side weir, $h$ is the depth of flow measured from the channel bottom along the channel centerline, and $C_d$ is the discharge coefficient (De Marchi coefficient) of the side weir. Thus, the side weir discharge equation can be written as:

$$Q_w=\frac{2}{3}{C}_{d}L\sqrt{2g}{[h-p]}^{3/2}$$

in which total flow over side weir $Q_w$ is in $m^3/s$, the discharge coefficient $C_d$ is dimensionless, the width of side weir $L$ is in meters, and $h$ and $p$ are in meters (see Fig. 1). This equation is usually used for flow over flat, broad-crested, quarter-round, half-round, and nappe (ogee) profile weirs.

Some of the proposed formulas for the discharge coefficient of the rectangular side weirs (see Fig. 1(a), (b)) are as follows:

$$C_d=0.864{\left(\frac{1-F_1^2}{2+F_1^2}\right)}^{0.5}\text{Subramanya and Awasthy [4]}$$$$C_d=0.81-0.6F_1\text{Ranga Raju et al. [15]}$$$$C_d=0.485{\left(\frac{2+F_1^2}{2+F_1^2}\right)}^{0.5}\text{for }p=0\text{Hager [24]}$$$$C_d=0.7-0.48F_1-0.3\frac{p}{h_1}+0.06\frac{L}{B}\text{Borghei et al. [9] .}$$

Most of presented equations for $C_d$ depend on the Froude number. Most researchers have concentrated on investigating rectangular and triangular side weirs in straight channels. Kumar and Pathak [16] investigated the variation of discharge coefficient for a sharp-crested triangular side weir having 60°, 90°, and 120° apex angles, and presented equations depending only on the Froude number and the apex angle. Hager [25] studied supercritical flow in circular-shaped side weirs. Oliveto et al. [26] studied the hydraulic characteristics of side weirs in circular channels when flow along the side weir is supercritical. As mentioned above, the most common type of side weir is rectangular. Moreover, triangular and circular types are also used in hydraulic and environmental engineering applications.

Emiroglu et al. [23] studied the discharge coefficient of sharp-crested triangular labyrinth side weirs on a straight channel. Dimensionless parameters for triangular labyrinth side weir discharge coefficient on a straight channel given by them are

$$C_d=f(F_1,L/B,L/\ell ,p/h_1,\theta ,\psi )$$

in which, $F_1$ is the upstream Froude number at the beginning of the side weir in the main channel, $C_d$ is the discharge coefficient (De Marchi coefficient), $p$ is the crest height of the side weir, $L$ is the width (length) of the side weir; $B$ is the width of the main channel; $\ell$ is the overflow length of the side weir, $h_1$ is the depth of flow at the upstream end of the side weir in the main channel centerline. The dimensionless parameters were explained as follows: $L/B$ is the dimensionless weir length (width), $L/\ell$ is the dimensionless effective side weir length, $p/h_1$ is the dimensionless weir crest height, $\theta$ is the included angle of the triangular labyrinth side weir.

The water nape deviation or deflection angle $\psi$ is defined as the deflection of the side weir nape from the water surface toward the weir side, and is given as follows [4]:

$$sin\psi =\sqrt{1-{\left(\frac{V_1}{V_s}\right)}^2}$$

in which, $V_s$ is velocity of flow $d{Q}_s$ over the brink. According to Eq. (7), $\psi$ takes different values for each fluid particle and varies with the Froude number, which changes along the side weir due to spilling over the side weir. The deviation angle increases towards the weir side when the Froude number in the main channel decreases towards the downstream direction. El Khashab [5] also mentioned that the dimensionless length of the side weir $(L/B)$ includes the effect of the deviation angle on the discharge coefficient. Therefore, the deviation angle $\psi$ is not included in the side weir discharge coefficient equations in the literature.

Emiroglu et al. [23] obtained the following results regarding the labyrinth side weir discharge coefficient:

• Discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than that of the rectangular side weir.

• The discharge coefficient $C_d$ increases when the $L/B$ ratio increases. A decrease in the labyrinth weir included angle $\theta$ causes a considerable increase in $C_d$, due to increasing the overflow length. The labyrinth side weir with $\theta =45°$ has the greatest $C_d$ values among the weir included angles that were tested.

• The following proposed equation for $C_d$, the De Marchi coefficient, for subcritical flow can be used reliably:

  $$C_d={[18.6-23.535{\left(\frac{L}{B}\right)}^{0.012}+6.769{\left(\frac{L}{\ell }\right)}^{0.112}-0.502{\left(\frac{p}{h_1}\right)}^{4.024}+0.094.sin\theta -0.393F_1^{2.155}]}^{-1.431}\text{.}$$

An ellipse is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane. It is also the locus of all points of the plane whose distances to two fixed points add to the same constant. By using an appropriate coordinate system, the ellipse can be described by the canonical implicit equation

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

where, $(x,y)$ are the point coordinates in the canonical system. In this system, the center is the origin (0, 0) and the foci are $(-ea,0)$ and $(+ea,0)$. $a$ is called the major radius, and $b$ is the minor radius. The quantity $e=\sqrt{(1-b^2/a^2)}$ is the eccentricity of the ellipse (Fig. 2).

The perimeter of an ellipse can be calculated by using Eq. (13) or Eq. (14). Eq. (13) is an “infinite sum” formula. Eq. (14) was developed by the Indian mathematician Ramanujan.

$$h=\frac{{(a-b)}^2}{{(a+b)}^2}$$$$P=\pi (a+b)\sum _{n=0}^{\infty }{\left(\begin{array}{c}0.5\hfill \\ n\hfill \end{array}\right)}^2{h}^n$$

where, $P$ is the perimeter of the ellipse. Eq. (11) expands to this series of calculations

$$P=\pi (a+b)(1+\frac{1}{4}h+\frac{1}{64}{h}^2+\frac{1}{256}{h}^3+\frac{1}{16384}{h}^4+\cdots )\text{.}$$

Note that when $a=b$, the ellipse is a circle. If $h$ is equal to zero, then the perimeter becomes $2\pi a$. The Ramanujan formula is

$$P\cong \pi (3(a+b)-\sqrt{(3a+b)(a+3b)})=\pi (a+b)(3-\sqrt{4-h})\text{.}$$

A semi-elliptical side weir is defined as a weir crest that is not straight in planform. The increased sill length provided by the semi-elliptical side weirs effectively reduces upstream head to the particular discharge. They can therefore be used to particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. To the best of our knowledge, no previous work has reported on semi-elliptical side weirs. This paper investigates the discharge coefficient of semi-elliptical side weirs for a subcritical flow regime and, in particular, the effect of the Froude number $F_1$, the dimensionless weir crest height $p/h_1$, the dimensionless weir width $L/B$, the dimensionless effective side weir length $L/\ell$ and, the dimensionless ellipse radius $b/a$ on the discharge coefficient.