Analytical and experimental study of flow through elliptical side orifices

Flow Measurement and Instrumentation 72 (2020) 101712

Contents lists available at ScienceDirect

Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst

Analytical and experimental study of flow through elliptical side orifices Ali R. Vatankhah *, F. Rafeifar Irrigation and Reclamation Eng. Dept., University College of Agriculture and Natural Resources, University of Tehran, P. O. Box 4111, Karaj, 31587-77871, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Flow measurement Flow diversion Elliptical side orifice Stage-discharge relationship Discharge coefficient Laboratory study

Discharge estimation is important for water management. Side orifices are commonly used in irrigation and drainage networks for distributing the water. Despite the vast amount of theoretical and experimental studies published, no generally applicable discharge equations are available for elliptical sharp-crested side orifices. When the length (diameter) of the circular side orifice is not sufficient to divert the water, an elliptical side orifice is a good alternative. In this paper, the elliptical sharp-crested side orifices were studied theoretically and experimentally. Several models were developed to predict the discharge coefficient of elliptical side orifices based on the Buckingham’s theorem of dimensional analysis. A series of laboratory runs (588 runs) were con­ ducted for different values of orifice geometry. The main channel discharge used in the tests ranged from 13.8 to 39.6 l/s and the side orifice discharge ranged from 3.66 to 21.4 l/s and upstream Froude number ranged from 0.22 to 0.77. Using measurements obtained by laboratory runs carried out in this investigation the proposed models of elliptical side orifices were calibrated under free outflow conditions. The model that includes the approach Froude number had an average error of 1.74%, while other model that does not include the approach Froude number had an average error about 2.43%. Moreover, suitable models were proposed for design pro­ cedure when measurement data for flow depths above the centroid of the orifice are not available. In this case, the model that includes the approach Froude number had an average error of 1.92%, while other model that does not include the approach Froude number had an average error of 2.24%. It was found that the proposed stagedischarge relationships were in an excellent agreement with the experimental results.

1. Introduction Along with global climate change, water scarcity has become one of the most significant factors for sustainable water management. In recent years, in order to save water and improve irrigation performance, flow measurement structures are often used in irrigation networks. Side ori­ fices are widely used in drainage networks and irrigation systems, urban wastewater treatment plants, and hydroelectric power facilities. Side orifices are generally placed in the sides of an open channel for diverting some of the flow into irrigated areas. Determining the lateral discharge through the side orifices is important for environmental engineering, water pricing and payment, and water management purposes. Flow along a side orifice is considered as spatially varied flow with decreasing flow discharge. Several researchers have studied the hydraulic characteristics of the side orifices with various shapes in open channels. The flow character­ istics of rectangular side orifices were firstly studied experimentally by Ramamurthy et al. [1,2]. They derived a discharge coefficient relation in

terms of the ratio of the orifice length to the main channel width, and the ratio of the average velocity in the approaching channel to the orifice jet velocity. Their experimental data provided a good verification of the proposed discharge coefficient of the orifice when a reduction factor of 0.95 was applied to account the velocity distribution effects of the flow in the main channel. Gill [3] studied the square side orifice as a special case of spatially varied flow by ignoring the frictional head loss in the parent channel over the length of the orifice. Ojha and Subbaiah [4] studied the flow through side rectangular orifice using spatially varied flow equation and elementary discharge coefficient relation. They validated the proposed discharge relation using their collected data. Swamee et al. [5] introduced the concept of the elementary discharge coefficient for flow through an elementary strip along the gate length of a side sluice gate. They related the elementary discharge co­ efficient of the side sluice gates with the main channel depth and gate opening under free outflow conditions. Discharge characteristics of a skew side sluice gate explored using the concept of an elementary discharge coefficient [6]. The hydraulic characteristics of rectangular

* Corresponding author. E-mail address: [email protected] (A.R. Vatankhah). https://doi.org/10.1016/j.flowmeasinst.2020.101712 Received 7 May 2019; Received in revised form 27 December 2019; Accepted 11 February 2020 Available online 5 March 2020 0955-5986/© 2020 Elsevier Ltd. All rights reserved.

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

Fig. 1. Definition sketches of an elliptical side orifice located in a horizontal rectangular channel under subcritical outflow condition (water surface profile increases from upstream to downstream along the elliptical side orifice).

side sluice gates (rectangular side orifices) were also experimentally studied [7]. Based on this study, the discharge coefficient of sluice gates depends on the flow depth in the main channel, approach Froude number, and gate opening for free outflow conditions. The flow char­ acteristics of the sharp-edged side sluice gate were studied by Esmail­ zadeh et al. [8]. They reported that the maximum value of flow diversion angle was observed at the end of the side gate. The flow discharges of the experimental study were obtained using an acoustic Doppler velocity meter. Hussain et al. [9] conducted a series of laboratory tests for rectan­ gular main channels with circular side orifices. They presented discharge coefficient as a function of the Froude number and the ratio of the orifice diameter to the width of the main channel. Hussain et al. [10] also conducted a laboratory study for rectangular main channels with rectangular side orifices and studied the variables that affect the orifice flow discharge. The discharge coefficient expressions proposed by Hussain et al. [9,10] are mainly depend on the upstream Froude number and ratio of the orifice size and the main channel width. Hussain et al. [11] estimated the flow discharge of a rectangular side orifice using analytical solutions and compared it with the experimental data of Hussain et al. [10]. They derived a discharge equation for rectangular side orifices in rectangular open channel with due consideration of ve­ locity direction and area vectors and also polynomial variation of discharge coefficient with velocity ratio. The accuracy of proposed discharge equation was checked using the data of Hussain et al. [10] and showed that their discharge equation predicts the side orifice discharge with an accuracy of �5%. Hussain et al. [12] studied the flow through side circular orifice under free and submerged outflow conditions. They concluded that the coefficient of discharge depends mainly on the up­ stream Froude number and the ratio of diameter of the orifice and channel width under free outflow conditions. The average error in computation of discharge through the circular orifice was 4%. Ebtehaj et al. [13] modeled the flow discharge of rectangular side orifices via the group method of data handling. The parameters affecting the discharge coefficient were considered by introducing Froude number, the ratio of flow depth in main channel to orifice width, the ratio of sill height to orifice width, and the width of main channel to width of rectangular orifice. They showed that Froude number has the most effect on the coefficient of discharge among the presented dimensionless parameters. Vatankhah and Bijankhan [14] used the energy principle for obtaining a theoretical discharge equation which is valid for small and large circular orifices. Vatankhah [15] also developed a combined uni­ fied discharge relation for circular orifices and circular weirs. Azimi et al. [16] assessed the flow discharge coefficients of side rectangular orifices by using the adaptive neuro-fuzzy inference system and genetic algorithm. They simulated the discharge coefficient in terms of the ratio of the main channel width to the side orifice length, the ratio of the side orifice height to the side orifice length, the ratio of the flow

depth in the main channel to the side orifice length, and upstream Froude number. Vatankhah and Mirnia [17] studied sharp-crested side triangular orifices analytically and experimentally. They developed several models for the discharge coefficient based on Buckingham’s theorem of dimensional analysis. Review of the literature concerning side orifices and gates indicates that most of the available studies dealt with the discharge characteristics of non-elliptical side orifices and the elliptical side orifice has not been previously studied. In this investigation, the flow discharge of a sharpedged elliptical side orifice located in a horizontal rectangular open channel under free outflow conditions was studied analytically and experimentally. 2. Orifice discharge equation The present study deals with the side orifices under free outflow condition, for which the discharge equation can be defined in two different ways: as the small orifices or large orifices. When the flow depth behind the orifice exceeds the opening crown, for a low flow head, the pressure profile is not uniform through the orifice section and thus large orifice flow occurs [14]. For large values of water head, the orifice acts as a small orifice. For example, the theoretical discharge computed by large circular orifice equation differs maximally 4% compared to that computed by a small circular orifice equation. The standard equation used to calculate the flow discharge through a small orifice under free outflow condition where the flow depth above the orifice is high compared with the dimensions of the orifice is written based on energy principle as: pffiffiffiffiffiffiffiffiffi Q ¼ Cd A 2ghc (1) where; Cd is the discharge coefficient, A is the orifice flow area, g is the acceleration due to gravity, and hc is the flow depth or head above the centroid of the vertical section of the orifice. Eq. (1) is only valid for a horizontal water surface profile along the orifice (orifices with small length). The discharge coefficient is introduced to correct nonrealistic assumptions such as approach velocity, viscous effects, streamline cur­ vature due to orifice contraction, and non-uniform velocity distribution in the orifice cross-section. Because of the significant variation in head at various points in the vertical section of the large orifices, the velocity distribution varies over the entire cross section of the large orifices. For large vertical orifices, the orifice discharge is determined by integrating the partial discharges through the small elements of the orifice area. Based on spatially varied flow theory [18], in the case of supercritical flow regime, due to decreasing unit discharge in the approach channel and over the side orifice, the water surface profile along the approach channel axis decreases from upstream to downstream. In comparison, in the case of subcritical flow regime, the water surface profile increases 2

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

from upstream to downstream along the side orifice. In practice, usually the flow regime in the main channel is subcritical, and thus the water surface profile increases gradually from upstream to downstream along the side of the elliptical orifice as presented in Fig. 1. In this figure, h1 and h2 are the heights from the water surface to the elliptical orifice edge at the upstream and downstream ends of the elliptical orifice, respec­ tively, and y1 and y2 are the flow depths at the upstream and down­ stream ends of the elliptical orifice, respectively.

Z

þ1 1

2 � �3 6 hc pffiffiffiffiffiffiffiffiffiffi2ffiffi 2 þ 1 t 4 b

2 � �32 6 hc ffi w1 4 þ w2 b

� hc b � hc b

3 3 pffiffiffiffiffiffiffiffiffiffiffiffi �2 7 1 t2 5dt �32 w2

3 7 5

(7)

3. Analytical considerations: discharge relation of elliptical side orifices

where w1 and w2 coefficients to be determined by curve fitting method against accurate numerical integration. when hc/b approaches infinity, Eq. (7) takes the form:

3.1. Small elliptical orifice with horizontal water surface profile along the orifice

� �12 Z hc 3 b

For an elliptical side orifice of horizontal semi-axes, a, and vertical semi-axes, b, located in a horizontal approaching channel with a hori­ zontal water surface profile (h1 ¼ h2), the stage-discharge Eq. (1) takes below form: pffiffiffiffiffiffiffiffiffi Q ¼ Cd π ab 2ghc (2)

thus w1w2 ¼ π/2. Substituting w1 ¼ π/(2w2) into Eq. (7) yields: 2 3 �32 � �32 Z þ1 � p p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi h h c 6 c 7 þ 1 t2 1 t2 5dt 4 b b 1

1

π 6 hc ffi þ w2 4 2w2

�32

b

� hc b

�32 w2

(8)

3 7 5

(9)

For hc/b � 1, using the Solver toolbox of Microsoft Excel by mini­ mizing the relative errors between the integral values in LHS of Eq. (9) and those computed by the function in RHS of Eq. (9) one gets w2@15/ 17 thus Eq. (9) takes the form: 2 3 �3 � �3 Z þ1 � hc pffiffiffiffiffiffiffiffiffiffi2ffiffi 2 7 6 hc pffiffiffiffiffiffiffiffiffiffi2ffiffi 2 þ 1 t 1 t 4 5dt b b 1

3.2. Large elliptical orifice with horizontal water surface profile along the orifice In Fig. 1, the origin of the coordinate system is located at the centroid of the elliptical orifice section. Considering a small elemental rectangle of sides dx and dy at height y from the centroid of the elliptical side orifice, the partial discharge dQ through this small elemental rectangle is expressed as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dQ ¼ Cd 2gðhc yÞdydx (3)

2 �3 � 17π 6 hc 15 2 ffi þ 4 30 b 17

The pressure distribution across a large orifice cannot be ignored, and thus Eq. (3) should be integrated to compute the total discharge of the large orifice. For a horizontal water surface profile (h1 ¼ h2), Eq. (3) can be written for an elliptical side orifice of horizontal semi-axes, a, and vertical semi-axes, b, as: ffiffiffi2ffi Z a Z þbpffi1ffiffiffixffiffi2ffi=a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ Cd (4) 2gðhc yÞdydx pffiffiffiffiffiffiffiffiffiffiffi b

1

� �2 pffiffiffiffiffiffiffiffiffiffiffiffi hc 1 t2 dt ffi 3w1 w2 b

2 �

The upstream water depth and the downstream water depth are equal in this case (y1 ¼ y2), and the flow depth above the centroid of the orifice section is equal to hc ¼ y1 W b in which W is the orifice crest height.

a

þ1

� hc b

15 17

�32

3 7 5

(10)

Approximate term in Eq. (10) is valid for hc/b � 1 with a maximum relative error less than 0.06%. Substituting Eq. (10) into Eq. (6) yields the following equation for the orifice discharge: 2 3 � �3 � �3 17π pffiffiffiffiffi 32 6 hc 15 2 hc 15 2 7 Q¼ (11) Cd 2gab 4 þ 5 45 b 17 b 17

1 x2 =a2

Integrating Eq. (4) with respect to y yields: 2 3 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �32 � pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �32 2 pffiffiffiffiffi þa 4� Q ¼ Cd 2g hc þ b 1 x2 =a2 hc b 1 x2 =a2 5dx 3 a

An asymptotic solution for Eq. (11), which corresponds to the large values of hc/b, can be analytically obtained as: (5)

Q¼

Assuming x ¼ at, (dx ¼ adt, where a is the horizontal semi-axes and t is a dimensionless variable) Eq. (5) takes the form: 2 3 � �3 � �3 Z 2 pffiffiffiffiffi 32 þ1 6 hc pffiffiffiffiffiffiffiffiffiffi2ffiffi 2 hc pffiffiffiffiffiffiffiffiffiffi2ffiffi 2 7 Q ¼ Cd 2gab (6) þ 1 t 1 t 4 5dt 3 b b 1

� �12 pffiffiffiffiffiffiffiffiffi 17π pffiffiffiffiffi 3 hc 15 ¼ Cd πab 2ghc Cd 2gab2 � 3 45 b 17

(12)

For large values of hc/b, the elliptical side orifice acts as a small side orifice. The relative deviation of Eq. (12) compared with Eq. (11) is small even for the minimum head ratio of hc/b ¼ 1 (about 4%). For hc/b ¼ 1.5, the maximum relative deviation is 1.5% and for hc/b ¼ 1.8, the maximum relative deviation is only 1%. Since agreement between the exact complex solution (11) and the approximate asymptotic solution (12) is very good, Eq. (12) for elliptical side offices with horizontal water surface profiles is used in this research for its simplicity and accuracy.

Analytical integration of Eq. (6) yields complex results in terms of elliptic integrals. A simple and suitable form for integral term in Eq. (6) may be considered using the method of undetermined coefficients [14, 15] as follows:

3.3. Large elliptical orifice with linear water surface profile along the orifice An elliptical side orifice, under a linear water surface profile is shown in Fig. 1. The water surface depth relative to the x-axis is expressed as hc þ mx, in which m is the slope of the linear water surface profile and hc is the flow depth above the centroid of the elliptical orifice section at x ¼ 0. 3

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

Under the linear water surface profile, Eq. (3) can be written for an elliptical side orifice as: ffiffiffi2ffiffi Z þa Z þbpffi1ffiffiffixffiffi2ffi=a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ Cd (13) 2gðhc þ mx yÞdydx pffiffiffiffiffiffiffiffiffiffiffi a

b

1 x2 =a2

Integrating Eq. (13) with respect to y yields: 2 3 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �32 � pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �32 2 pffiffiffiffiffi þa 4� 2 2 2 2 Q¼ Cd 2g hc þmxþb 1 x =a hc þmx b 1 x =a 5dx 3 a (14) Eq. (14) can be written as: 2 3 � �3 � �3 Z 2 pffiffiffiffiffi 3 þa 6 hc mx pffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi2ffi 2 hc mx pffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi2ffi 2 7 Q¼ Cd 2gb2 1 x =a 5dx þ þ 1 x =a þ 4 3 b b b b a

B b

(21)

Π4 ¼

y1 b

(22)

V1 Π 5 ¼ pffiffiffiffiffi gb

(23)

σ ρgb2

(24)

μ

(25)

in which G1 is a functional symbol. In the following, some reasonable simplifying assumptions are made. The effects of the Reynolds number, Re, and the Weber number We, have been found to be negligible for side orifices [1,9–13,16,17]. Neglecting the effects of the Reynolds and Weber numbers the dimensionless Eq. (26) can be written in the form: � � W B B y1 V1 Cd ¼ G2 ; ; ; ; F1 ¼ pffiffiffiffiffiffi (27) b a b b gy1

When m approaches to zero, the stage-discharge Eq. (16) reduces to Eq. (6) for a horizontal water surface profile. Generally, ma/b is less than unity in practical applications. For ma/b ¼ 1, the maximum deviation between Eq. (16) and Eq. (6) is about 2.3% which occurs at hc/b ¼ 1.415. Similarly, for ma/b ¼ 1, the maximum deviation between Eq. (16) and Eq. (2) is about 4% which occurs at hc/b ¼ 1.415. Based on the general analytical analysis done in this work, Eq. (2) can be reliably used for both small and large elliptical side orifices and for both horizontal and linear water surface profiles along the elliptical side orifices.

in which F1 is the Froude number and G2 is a functional symbol. In this study, the following nonlinear regression form is considered for the discharge coefficient Cd of the elliptical side orifices: � �a2 � �a3 � �a4 � �a5 W B B b Cd ¼ a0 þ a1 Fa16 (28) b a b y1

4. Dimensional analysis for variables affecting Cd This study attempted to establish the relationship between the discharge coefficient Cd and other effective variables. The flow discharge of an elliptical side orifice of horizontal semi-axes, a, and vertical semi-axes, b, located in a horizontal main channel with a linear water surface profile, is expressed in terms of the flow depth above the centroid of the orifice section, hc, as Eq. (2). Several different physical and geometrical quantities may influence the value of the discharge coefficient. Variables that are likely to affect the discharge coefficient Cd of an elliptical side orifice are: the horizontal semi-axes, a, the vertical semi-axes, b, the orifice crest height W, the upstream velocity in the main channel V1, the upstream flow depth in the main channel y1, the main channel width B, water density ρ, water viscosity μ, water surface tension σ , and gravitational acceleration g. The discharge coefficient Cd of an elliptical side orifice can be expressed as:

where ai (i ¼ 0,1 to 6) are empirical coefficients and should be deter­ mined using laboratory data. The discharge coefficient of the orifices approaches a constant value as upstream water depth approaches infinity. This physical meaning is considered in Eq. (28). In this equation for positive values of a0, a5 and a6 when y1→∞ (b/y1→0 and F1→0) then Cd→a0. In the following, the effects of the above dimensionless quantities on the discharge coefficient were analyzed using the data collected in this study. 5. Experimental setup The laboratory tests were performed at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran, Iran. The experimental setup contained a pumping station. A horizontal rectangular channel 12 m long, 25 cm wide and 50 cm deep was used for the laboratory tests. Further details on the experimental setup are reported in Refs. [17]. The total discharge upstream of the orifice, Qu and the discharge passing through the side orifice, Qs, was determined by two different triangular and rectangular weirs which were calibrated using an electromagnetic flowmeter with an accuracy of �0⋅5% of the full scale. The elliptical side orifices were made of 1 cm thick Plexiglas sheets with a crest thickness of about 1 mm, and the downstream edge beveled to a 45� angle. The measurements were carried out for two different orifice lengths (a ¼ 15 and 20 cm), three different values of the orifice heights (b ¼ 2, 3 and 4 cm) and two different crest height (W ¼ 5 and 10 cm). A total number of 12 geometric configurations were used for the

(17)

where K1 is a functional symbol. Eq. (17) can be expressed in a dimensionless form as: (18)

where Π1, Π2, Π3, Π4, Π5, Π6 and Π7 are dimensionless groups and K2 is a functional symbol. Using b, g and ρ as repeating variables, the following dimensionless groups can be obtained: a b

Π3 ¼

By combining the original dimensionless groups [Eqs. (19)–(25)], the functional relationship of an elliptical side orifice can be rewritten as: � � W B B y1 V1 ρV1 a σ Cd ¼ G1 ; ; ; ; F1 ¼ pffiffiffiffiffiffi ; Re ¼ ; We ¼ (26) 2 gy1 ρga b a b b μ

(16)

Π1 ¼

(20)

Π 7 ¼ pffiffiffiffiffiffiffi ρ gb3

Assuming x ¼ at, the stage-discharge Eq. (14) takes the form: 2 3 � �3 � �3 Z 2 pffiffiffiffiffi 3 þ1 6 hc mat pffiffiffiffiffiffiffiffiffi2ffiffi 2 hc mat pffiffiffiffiffiffiffiffiffi2ffiffi 2 7 Q ¼ Cd 2gab2 þ 1 t 1 t þ þ 4 5dt 3 b b b b 1

Cd ¼ K2 ðΠ 1 ; Π 2 ; Π 3 ; Π 4 ; Π 5 ; Π 6 ; Π 7 Þ

W b

Π6 ¼

(15)

Cd ¼ K1 ða; b; W; B; y1 ; V1 ; g; σ ; μ; ρÞ

Π2 ¼

(19) 4

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

experimental program (six orifices, each with two crest heights). The elliptical side orifices were installed in the wall of the main channel at 6 m distance from its entrance. Flow depths upstream and downstream of the orifice (y1 and y2), and flow depth in the center of the orifice length, yc, were measured at the centerline of the main channel using a point gauge with an accuracy of 0.1 mm. Fig. 2 shows the experimental setup used in this study. A total number of 588 runs were performed under free outflow conditions. For each experiment, the flow discharges (Qu and Qs) and the flow depths (y1, y2 and yc) were measured. The main channel discharge, Qu, used in the tests ranged from 13.8 to 39.6 l/s and the side orifice discharge, Qs, ranged from 3.66 to 21.4 l/s. The upstream Froude number, F1, ranged from 0.22 to 0.77, centroid flow depth ratio hc/b ranged from 1.82 to 5.25 and orifice shape factor a/b ranged from 3.75 to 10. The experi­ mental data sets of this research under free outflow conditions are presented in Table 1. The experimental discharge coefficients were pffiffiffiffiffiffiffiffiffi computed using Table 1 and equation Cd ¼ Qs =ðπab 2ghc Þ in which hc ¼ yc W b. 6. Results and discussions The empirical coefficients of the discharge coefficient Eq. (28) was determined by minimizing the summation of the absolute relative errors of the flow discharge coefficients. Minimizing the relative errors be­ tween the observed (588 runs) and computed discharge coefficient leads to: � �0:572 � �1:27 � �0:04 � �0:13 W B B b Cd ¼ 0:64 0:1 F0:85 (29) 1 b a b y1

Fig. 2. Physical model of elliptical side orifice in operation.

The average error of Eq. (29) compared with the observed values, Qs/

Fig. 3. Comparison of experimental discharge coefficients with computed ones using proposed Eqs. (29) and (30) for elliptical side orifices, along with their discharge error distributions. 5

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

Fig. 4. Comparison of experimental discharge coefficients with computed ones using proposed Eqs. (31)–(33) for elliptical side orifices, along with their discharge error distributions.

[πab(2ghc)0.5], is 1.73%. Eq. (29) has a maximum error of 7.66%. According to Eq. (29), the influences of b/y1 and B/b are less than the other variables and these two variables may be ignored without any significant loss in accuracy. Considering three variables B/a, W/b and F1, and by minimizing the relative errors between the observed and computed discharge coefficients, the following relationship was obtained: � �1:33 � �0:57 B W Cd ¼ 0:635 0:085 F0:91 (30) 1 a b

within �3%. Eq. (30) is valid for 0.95<(B/a)1.33(W/b)0.57F0.91 1 <2.5. In Eq. (30), a/B has a clear physical meaning (2a/B is the orifice length to the channel width), F1 shows flow regimes, and b/W has a clear geo­ metric meaning (2b/W is the orifice height to the orifice crest height). A comparison between measured discharge coefficients and the ones calculated by Eq. (29), and (30) along with their errors are also illus­ trated in Fig. 3. A perusal of Fig. 3, indicates that the experimental discharge coefficient values can be simply represented by B/a, W/b and F1 without a loss in accuracy. According to Eq. (30), the conjugated influence of B/a, W/b and F1 on Cd is significant. In the following, the accuracy of the models with only two dimensionless variables was also evaluated. Using B/a, W/b and F1, three models with two variables were considered. These models were obtained as:

The average error of Eq. (30) was less than 1.74%. The discharge coefficient Eq. (30) had a maximum error less than 7.8%. Also, for 97% of the measured discharge coefficient values, the relative error was within �5% and for 82.5% of the measured values, the relative error was 6

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

Fig. 5. Comparison of experimental discharge coefficients with computed ones using proposed Eq. (34) for elliptical side orifices, along with their discharge error distributions.

Fig. 6. Comparison of experimental discharge with computed ones using proposed Eq. (35), and (36) for elliptical side orifices, along with their discharge error distributions.

Cd ¼ 0:536

� �2:11 W 0:005 F1:35 1 b

(31)

Cd ¼ 0:578

� �1:47 B 0:039 F0:24 1 a

(32)

Cd ¼ 0:549

� �0:969 � �2:631 W B 0:004 b a

(33)

are considerable compared with the model with three variables pre­ sented by Eq. (30). Calculated discharge coefficients by Eqs. (31)–(33) versus measured discharge coefficients, along with their relative errors are plotted in Fig. 4. This figure shows that the experimental discharge coefficient cannot be accurately represented with only two dimensionless variables. When the upstream discharge of the main channel, Qu, is not known, the upstream Froude number, F1, cannot be computed or considered in Eq. (30). In such a case, the following relationship can be obtained using the collected experimental data:

The average errors of Eqs. (31)–(33) are respectively 3.43%, 3.56% and 3.10% and their maximum errors are respectively 19.13%, 25.54% and 17.34%. As noted, the maximum errors in models with two variables 7

8

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10

13.84 13.84 13.84 13.84 13.84 13.84 13.84 15.19 15.19 15.19 15.19 15.19 15.19 15.19 16.63 16.63 16.63 16.63 16.63 16.63 16.63 18.21 18.21 18.21 18.21 18.21 18.21 18.21 19.51 19.51 19.51 19.51 19.51 19.51 19.51 21.27 21.27 21.27 21.27 21.27 21.27 21.27 22.64 22.64 22.64 22.64 22.64 22.64 22.64 17.13 17.13 17.13 17.13

4.14 4.63 5.09 5.74 6.30 6.90 7.30 4.47 4.84 5.42 5.76 6.09 6.42 7.10 4.10 4.53 4.94 5.40 5.78 6.13 6.41 4.23 4.55 5.08 5.44 5.69 6.17 6.63 4.45 4.76 5.15 5.51 6.06 6.49 6.96 4.29 4.90 5.13 5.58 6.08 6.38 6.79 4.74 5.09 5.56 5.80 6.17 6.54 6.95 3.89 4.38 4.76 5.41

10.86 11.77 12.44 13.52 14.62 15.8 16.61 11.32 11.94 13.02 13.56 14.18 14.89 16.23 11.26 11.95 12.43 13.34 14.1 14.67 15.3 11.41 11.86 12.68 13.35 13.9 14.93 15.88 11.96 12.49 13.24 13.94 14.81 15.8 16.87 11.27 12.32 12.99 13.71 14.75 15.23 16.19 11.88 13.15 14.11 14.65 15.25 16.27 17.06 15.92 16.69 17.25 18.3

11.33 12.13 12.75 13.86 14.87 16.02 16.85 11.88 12.32 13.4 13.85 14.52 15.14 16.48 11.57 12.38 12.99 13.73 14.51 15.03 15.66 11.97 12.43 13.22 13.96 14.36 15.3 16.2 12.36 12.97 13.67 14.53 15.33 16.26 17.25 12.47 12.84 13.46 14.14 15.28 16.01 16.64 13.07 13.45 14.61 15.1 15.67 16.85 17.49 16.16 16.94 17.51 18.47

11.7 12.39 13.02 13.99 14.99 16.09 16.92 12.05 12.67 13.61 14.12 14.71 15.37 16.58 11.95 12.82 13.24 14.07 14.73 15.21 15.87 12.51 12.67 13.67 14.16 14.74 15.49 16.46 13.03 13.22 14.09 14.75 15.71 16.48 17.42 12.9 13.44 13.92 14.33 15.72 16.08 16.94 13.6 14.07 14.84 15.29 16.19 17.12 17.86 16.37 17.07 17.7 18.66

197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10

26.14 26.14 26.14 26.14 26.14 26.14 26.14 27.30 27.30 27.30 27.30 27.30 27.30 27.30 28.30 28.30 28.30 28.30 28.30 28.30 28.30 29.48 29.48 29.48 29.48 29.48 29.48 29.48 30.39 30.39 30.39 30.39 30.39 30.39 30.39 31.72 31.72 31.72 31.72 31.72 31.72 31.72 32.71 32.71 32.71 32.71 32.71 32.71 32.71 27.88 27.88 27.88 27.88

11.62 12.16 12.45 12.66 12.90 13.12 13.27 12.33 12.59 12.88 13.04 13.25 13.47 13.64 11.66 11.93 12.33 12.51 12.82 13.05 13.21 12.32 12.73 13.20 13.44 13.73 13.90 14.14 11.73 12.09 12.53 13.09 13.36 13.53 13.76 12.44 12.73 13.23 13.63 13.89 14.23 14.59 12.69 13.07 13.49 13.85 14.23 14.52 15.08 11.56 11.95 12.33 12.52

15.62 16.28 16.52 16.76 17.02 17.26 17.43 15.91 16.22 16.55 16.79 17 17.22 17.36 15.88 16.31 16.63 16.83 17.12 17.35 17.6 16.35 16.91 17.22 17.3 17.76 18.16 18.27 16.07 16.17 16.32 17.17 17.51 17.7 17.86 16.31 16.77 17.15 17.61 18.2 18.5 18.72 16.96 17.58 17.77 18.01 18.78 19.2 19.74 20.77 21.18 21.51 21.73

16.5 16.8 17 17.1 17.3 17.7 17.7 16.8 16.9 17 17.2 17.4 17.6 17.8 17 17.2 17.6 17.7 17.8 18 18.1 17.5 17.7 18 18.4 18.5 18.6 18.9 17.2 17.3 17.9 18.1 18.3 18.7 18.8 17.7 17.9 18.5 18.8 19.1 19.3 19.7 18.3 18.7 19.2 19.6 19.8 20.1 20.5 21.3 21.7 22 22.2

17 17 18 18 18 18 18 17 18 18 18 18 18 18 18 18 18 18 18 19 19 18 18 19 19 19 19 20 18 18 18 19 19 19 19 19 19 19 19 20 20 20 19 19 20 20 20 21 21 22 22 22 23

393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10

26.80 26.80 26.80 26.80 26.80 26.80 26.80 27.88 27.88 27.88 27.88 27.88 27.88 27.88 28.95 28.95 28.95 28.95 28.95 28.95 28.95 30.02 30.02 30.02 30.02 30.02 30.02 30.02 31.11 31.11 31.11 31.11 31.11 31.11 31.11 32.21 32.21 32.21 32.21 32.21 32.21 32.21 33.32 33.32 33.32 33.32 33.32 33.32 33.32 27.00 27.00 27.00 27.00

11.78 12.19 12.38 12.56 12.84 13.05 13.28 11.63 11.87 12.21 12.72 13.10 13.49 13.72 11.55 12.05 12.38 12.67 13.12 13.42 13.87 11.73 12.09 12.50 12.92 13.23 13.55 13.92 11.51 11.93 12.50 12.92 13.53 13.65 14.16 11.90 12.43 12.90 13.45 14.02 14.33 14.73 12.32 12.74 13.22 13.76 14.31 14.92 15.50 10.53 11.15 11.42 11.83

13.7 14 14.5 14.4 14.4 14.7 15.2 13.8 13.9 14.3 14.9 15.2 15.3 15.8 14.2 14.4 14.6 14.7 15.3 15.8 16 14.4 14.5 14.9 15.2 15.3 15.7 16.3 14.2 14.8 15.2 15.4 15.6 15.9 16.6 14.5 15.1 15.5 15.9 16.2 16.5 17.1 14.7 15.2 15.7 16 16.6 17.1 17.9 18.1 18.7 18.9 19.2

15.3 15.38 15.82 16.13 16.21 16.18 16.4 15.56 15.85 16.05 16.31 16.77 16.95 17.09 15.27 15.95 16.47 16.55 16.8 17.36 17.65 15.16 15.74 16.45 16.86 17.19 17.34 17.91 14.69 15.45 16.63 17.06 17.51 17.7 18.21 14.8 15.69 16.85 17.51 18.01 18.31 18.68 15.06 15.59 16.48 17.71 18.39 18.71 19.52 18.93 19.44 19.61 20.09

15.77 15.86 16.16 16.23 16.5 16.48 16.73 16.03 16.22 16.3 16.62 16.88 17.16 17.43 16.05 16.51 16.61 16.95 17.23 17.44 17.83 16.34 16.67 16.88 17.21 17.52 17.71 18.04 16.43 16.74 17.28 17.56 17.8 18.15 18.46 16.71 16.89 17.37 18.14 18.34 18.57 18.95 16.91 17.14 17.91 18.21 18.69 18.99 19.39 19.13 19.72 19.82 20.2

(continued on next page)

Flow Measurement and Instrumentation 72 (2020) 101712

Row

A.R. Vatankhah and F. Rafeifar

Table 1 Collected experimental data in this study for elliptical side orifices located in a horizontal rectangular main channel (B ¼ 25 cm).

9

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5 5 5 5 5 5 5 5

17.13 17.13 17.13 19.20 19.20 19.20 19.20 19.20 19.20 19.20 21.06 21.06 21.06 21.06 21.06 21.06 21.06 22.31 22.31 22.31 22.31 23.30 22.31 22.31 22.31 23.30 23.30 23.30 23.30 23.30 23.30 24.23 24.23 24.23 24.23 24.23 24.23 24.23 24.82 24.82 24.82 24.82 24.82 24.82 24.82 19.22 19.22 19.22 19.22 19.22 19.22 19.22 20.73 20.73

6.07 6.45 7.07 3.77 4.24 4.82 5.31 5.60 5.95 6.59 3.79 4.22 4.70 5.09 5.53 6.11 6.56 3.80 4.25 4.58 5.06 5.26 5.62 6.04 6.42 4.01 4.50 4.86 5.68 6.03 6.40 3.66 3.88 4.18 4.66 5.06 5.42 5.83 3.98 4.31 4.72 5.05 5.48 5.88 6.40 7.72 8.05 8.34 8.88 9.29 9.56 9.96 7.68 8.36

19.54 20.35 21.54 16.08 16.71 17.41 18.43 19.16 19.77 21.1 16.11 16.82 17.71 18.4 19.27 20.4 21.26 16.21 16.98 17.58 18.57 18.45 19.6 20.39 21.46 16.31 17.01 17.68 19.23 20.04 20.78 16.48 16.8 17.3 18.02 18.75 19.62 20.58 16.51 17.12 17.66 18.44 19.17 20.08 21.17 13.55 13.89 13.99 14.61 15.14 15.56 16.16 13.6 14.41

19.76 20.55 21.86 16.36 16.98 17.74 18.76 19.43 20.04 21.34 16.54 17.2 18.06 18.7 19.54 20.67 21.54 16.71 17.47 17.96 18.88 18.76 19.92 20.68 21.75 16.73 17.54 18.13 19.58 20.41 21.1 16.67 17.2 17.76 18.54 19.23 19.95 20.95 16.79 17.48 18.1 18.96 19.65 20.42 21.49 13.97 14.4 14.63 15.06 15.7 16.03 16.49 13.95 14.73

19.88 20.67 21.95 16.58 17.17 17.92 18.85 19.59 20.14 21.45 16.68 17.47 18.25 18.94 19.7 20.83 21.63 17 17.63 18.21 19.09 19.07 20.09 20.8 21.92 17.06 17.73 18.36 19.78 20.6 21.25 17 17.55 18.06 18.76 19.46 20.29 21.16 17.04 17.71 18.39 19.13 19.87 20.65 21.63 14.39 14.65 14.89 15.41 15.96 16.15 16.75 14.58 15.16

250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5 5 5 5 5 5 5 5

27.88 27.88 27.88 28.83 28.83 28.83 28.83 28.83 28.83 28.83 29.84 29.84 29.84 29.84 29.84 29.84 29.84 31.17 31.17 31.17 31.17 31.17 31.17 31.17 32.15 32.15 32.15 32.15 32.15 32.15 32.15 33.64 33.64 33.64 33.64 33.64 33.64 33.64 34.51 34.51 34.51 34.51 34.51 34.51 34.51 17.49 17.49 17.49 17.49 17.49 17.49 17.49 18.88 18.88

12.86 13.06 13.33 11.36 11.88 12.37 12.62 13.04 13.48 13.95 11.00 11.43 11.83 12.19 12.59 13.19 13.52 11.37 11.77 12.29 12.61 13.11 13.64 13.99 11.58 11.94 12.50 12.93 13.22 13.75 14.33 12.24 12.87 13.41 14.14 14.81 15.10 15.72 12.86 13.31 13.86 14.69 15.16 15.63 16.37 6.51 6.72 7.02 7.50 7.68 8.17 8.38 6.73 7.00

21.99 22.29 22.58 20.69 21.1 21.54 21.72 22.35 22.8 23.31 20.82 21.06 21.49 21.81 22.26 22.93 23.4 20.99 21.24 21.68 21.98 22.45 22.92 23.32 21.51 21.73 22.13 22.51 22.96 23.42 24.29 21.78 22.45 22.93 23.71 24.39 24.73 25.56 22.61 23.05 23.79 24.41 25.07 25.89 26.75 11.06 11.43 11.86 12.68 12.85 13.54 13.84 11.31 11.79

22.5 22.8 23.1 21.3 21.8 22.1 22.4 22.8 23.2 23.8 21.4 21.7 22.2 22.4 23 23.4 23.9 21.4 21.7 22.2 22.6 23.1 23.6 23.9 21.9 22.1 22.6 23.3 23.7 24.2 24.7 22.3 23 23.7 24.3 25 25.4 26 23.1 23.7 24.4 25.2 25.8 26.5 27.3 12 12.3 12.7 13.5 13.6 14.1 14.4 12.4 12.9

23 23 23 22 22 23 23 23 24 24 22 22 22 23 23 24 24 22 22 23 23 23 24 24 22 23 23 24 24 24 25 23 24 24 25 25 26 26 24 24 25 25 26 27 28 13 13 13 14 14 14 15 13 13

446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 5 5 5 5 5 5 5 5 5

27.00 27.00 27.00 28.00 28.00 28.00 28.00 28.00 28.00 28.00 29.00 29.00 29.00 29.00 29.00 29.00 29.00 30.08 30.08 30.08 30.08 30.08 30.08 30.08 31.11 31.11 31.11 31.11 31.11 31.11 31.11 32.03 32.03 32.03 32.03 32.03 32.03 32.03 33.01 33.01 33.01 33.01 33.01 33.01 33.01 36.22 36.22 36.22 36.22 36.22 36.22 36.22 36.66 36.66

12.43 12.88 13.48 10.52 10.92 11.32 11.72 12.05 12.46 13.06 10.57 10.99 11.31 11.61 12.10 12.46 12.96 10.90 11.34 11.70 12.11 12.49 12.88 13.29 10.45 10.95 11.12 11.57 11.91 12.32 12.77 10.91 11.25 11.62 11.96 12.47 12.83 13.30 11.27 11.75 12.14 12.78 13.23 13.69 14.07 17.28 17.53 17.83 18.71 19.21 19.48 19.91 17.63 18.04

19.8 20.2 20.8 18 18.7 18.9 19.2 19.6 19.9 20.6 18.3 18.5 18.8 19.3 19.7 20.1 20.7 18.2 18.6 19 19.4 19.9 20.3 20.7 18.2 18.4 18.6 19.3 19.5 19.9 20.4 18.3 18.5 18.7 19.2 19.7 20 20.7 18.6 19.2 19.4 20.2 20.9 21.3 21.9 17 16.6 16.8 17.4 17.9 17.8 18.1 16.8 16.9

20.4 20.75 21.41 18.74 19.2 19.61 19.92 20.31 20.63 21.22 18.99 19.44 19.69 20.1 20.54 20.85 21.45 19.01 19.47 19.8 20.15 20.72 21.06 21.55 19.2 19.47 19.59 19.96 20.23 20.64 21.21 19.19 19.63 19.93 19.98 20.62 20.92 21.4 19.68 20.33 20.52 21.14 21.64 22.21 22.85 18.33 17.37 18.25 18.71 19.01 19.17 19.71 17.52 17.86

20.8 21.14 21.7 19.39 19.68 19.92 20.27 20.45 20.9 21.5 19.77 19.97 20.07 20.47 20.85 21.17 21.7 19.79 20.12 20.33 20.63 20.97 21.3 21.71 19.74 19.87 20.13 20.66 20.93 21.21 21.53 19.69 20.09 20.32 20.73 21.14 21.53 21.78 20.23 20.78 21.19 21.77 22.12 22.67 22.89 19.22 18.81 19.04 19.44 19.7 19.88 20.12 18.9 19.27

(continued on next page)

Flow Measurement and Instrumentation 72 (2020) 101712

Row

A.R. Vatankhah and F. Rafeifar

Table 1 (continued )

10

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10

20.73 20.73 20.73 20.73 20.73 22.14 22.14 22.14 22.14 22.14 22.14 22.14 23.64 23.64 23.64 23.64 23.64 23.64 23.64 25.19 25.19 25.19 25.19 25.19 25.19 25.19 26.69 26.69 26.69 26.69 26.69 26.69 26.69 28.12 28.12 28.12 28.12 28.12 28.12 28.12 20.20 20.20 20.20 20.20 20.20 20.20 20.20 21.22 21.22 21.22 21.22 21.22 21.22 21.22

8.58 8.94 9.37 9.64 10.29 7.88 8.32 8.65 9.04 9.36 9.60 9.95 7.73 8.11 8.41 8.83 9.20 9.47 9.94 8.25 8.52 8.87 9.34 9.66 10.04 10.38 8.28 8.74 9.03 9.29 9.56 10.03 10.46 8.41 8.75 9.07 9.52 9.73 10.00 10.33 7.34 7.78 8.03 8.53 8.98 9.41 9.96 7.45 7.77 8.21 8.67 9.01 9.70 10.04

14.57 14.98 15.33 15.8 16.64 13.63 14.2 14.62 15.23 15.5 15.82 16.09 13.57 13.95 14.4 14.98 15.46 15.94 16.46 13.57 14.15 14.8 15.34 15.98 16.35 16.82 13.89 14.22 14.81 15.4 15.75 16.73 17.18 14.33 14.66 15.27 16.19 16.49 16.72 17.52 18.33 18.77 18.98 19.65 20.15 20.72 21.48 18.28 18.6 19.11 19.66 20.09 20.95 21.39

15.14 15.58 15.87 16.22 17.03 14.14 14.61 14.99 15.46 16.12 16.47 16.83 14.18 14.73 15.08 15.54 15.87 16.43 17.12 15.01 15.09 15.47 15.98 16.31 16.75 17.43 15.15 15.61 15.95 16.02 16.69 17.11 17.68 15.82 16.04 16.51 17 17.22 17.65 18 18.6 19.06 19.37 19.96 20.49 20.99 21.72 18.63 18.91 19.45 19.94 20.41 21.25 21.66

15.42 15.81 16.23 16.6 17.36 14.77 15.19 15.5 16.07 16.37 16.66 17.04 15.34 15.44 15.6 15.94 16.32 16.94 17.33 15.7 15.97 16.13 16.42 16.72 17.21 17.73 16 16.18 16.62 16.65 17.04 17.47 18.1 16.44 16.68 17.2 17.44 17.5 17.77 18.33 18.92 19.27 19.58 20.17 20.63 21.22 21.87 18.9 19.18 19.71 20.2 20.59 21.4 21.8

304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10

18.88 18.88 18.88 18.88 18.88 20.04 20.04 20.04 20.04 20.04 20.04 20.04 21.27 21.27 21.27 21.27 21.27 21.27 21.27 22.64 22.64 22.64 22.64 22.64 22.64 22.64 23.70 23.70 23.70 23.70 23.70 23.70 23.70 25.11 25.11 25.11 25.11 25.11 25.11 25.11 19.25 19.25 19.25 19.25 19.25 19.25 19.25 20.57 20.57 20.57 20.57 20.57 20.57 20.57

7.28 7.61 7.85 8.28 8.67 6.49 6.79 7.12 7.42 7.63 8.06 8.28 6.57 6.90 7.17 7.62 7.81 8.03 8.26 6.77 7.07 7.39 7.75 7.98 8.23 8.64 6.69 7.11 7.61 7.88 8.06 8.43 8.67 6.85 7.25 7.73 8.01 8.42 8.83 9.21 5.62 5.85 6.18 6.50 7.05 7.52 7.68 5.38 5.60 6.11 6.32 6.76 7.16 7.70

12.12 12.68 13.16 13.72 14.31 11.16 11.64 11.77 12.31 12.63 13.39 13.82 11.1 11.64 12.27 12.51 13.11 13.37 13.77 11.62 11.99 12.3 12.99 13.19 13.87 14.46 11.99 12.78 13.06 13.87 14.01 14.54 15.09 12.45 12.39 12.96 13.57 14.25 14.65 15.79 15.58 15.78 16.21 16.59 17.24 17.78 18.11 15.75 15.99 16.35 16.72 17.18 17.79 18.53

13.1 13.6 14 14.5 14.8 12.3 12.7 13.1 13.5 13.6 14.2 14.7 13 13 13.4 13.8 14.8 14.5 14.8 13.4 13.8 14 14.2 14.6 15.1 15.4 14 14.2 14.7 15.2 15.4 15.7 16 13.2 14.1 14.7 15.3 15.4 16.1 16.7 16.1 16.3 16.8 17 17.6 18.2 18.5 16.2 16.5 17 17.3 17.8 18.2 19

13 14 14 14 15 13 13 14 14 14 14 15 13 14 14 14 15 15 15 14 14 15 15 15 15 16 14 15 15 16 15 16 16 14 15 15 16 16 16 17 16 17 17 17 18 18 19 16 17 17 17 18 18 19

500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10

36.66 36.66 36.66 36.66 36.66 37.18 37.18 37.18 37.18 37.18 37.18 37.18 37.69 37.69 37.69 37.69 37.69 37.69 37.69 38.34 38.34 38.34 38.34 38.34 38.34 38.34 39.05 39.05 39.05 39.05 39.05 39.05 39.05 39.64 39.64 39.64 39.64 39.64 39.64 39.64 34.76 34.76 34.76 34.76 34.76 34.76 34.76 35.46 35.46 35.46 35.46 35.46 35.46 35.46

18.53 18.94 19.21 19.81 20.10 17.62 17.96 18.40 18.74 19.04 19.57 20.13 17.64 18.01 18.37 18.79 19.35 19.66 20.18 17.69 18.05 18.43 18.89 19.32 19.81 20.22 17.48 17.90 18.24 18.37 18.95 19.22 19.67 17.74 18.27 18.78 19.37 20.04 20.80 21.41 16.28 16.70 17.21 17.75 18.12 18.85 19.41 16.89 17.33 17.67 18.12 18.50 19.26 19.78

17 17.4 17.6 18.2 18.4 16.6 16.9 17.2 17.5 17.7 18 18.4 17 17.2 17.4 17.8 18.2 18.3 18.6 17.2 17.1 17.2 17.6 18.1 18.4 18.6 16.7 17.2 17.5 17.4 17.8 18.1 18.6 16.2 17.6 17.8 17.9 18.8 19.1 19.4 20.4 20.6 21.1 21.8 21.9 22.5 23.1 20.8 21.1 21.1 21.6 22.2 22.7 23.2

18.41 18.52 19.01 19.46 19.6 17.59 18 18.02 18.82 18.87 19.6 19.93 17.43 17.93 18.5 18.6 19.47 19.71 20.15 17.71 17.86 18.55 18.84 19.42 19.97 20.24 17.34 17.6 17.9 18.31 18.79 19.12 19.78 17.69 17.9 18.56 19.35 19.8 20.89 21.19 21.56 21.65 22.01 22.43 22.78 23.26 24.01 21.78 22.09 22.37 22.63 22.93 23.51 24.09

19.27 19.47 19.62 20.16 20.22 19.11 19.11 19.55 19.51 19.75 20.21 20.37 19.21 19.41 19.62 19.89 20.16 20.37 20.8 19.43 19.42 19.58 19.91 20.26 20.41 20.56 19.06 19.51 19.48 19.77 20.05 20.36 20.45 19.11 19.47 19.84 20.3 20.61 21.24 21.63 22.16 22.4 22.57 22.91 23.3 23.69 24.17 22.26 22.44 22.73 23.21 23.35 23.98 24.27

(continued on next page)

Flow Measurement and Instrumentation 72 (2020) 101712

Row

A.R. Vatankhah and F. Rafeifar

Table 1 (continued )

A.R. Vatankhah and F. Rafeifar

Table 1 (continued )

11

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

Row

a (cm)

b (cm)

W (cm)

Qu (l/s)

Qs (l/s)

y1 (cm)

yc (cm)

y2 (cm)

162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

22.31 22.31 22.31 22.31 22.31 22.31 22.31 23.42 23.42 23.42 23.42 23.42 23.42 23.42 24.54 24.54 24.54 24.54 24.54 24.54 24.54 25.85 25.85 25.85 25.85 25.85 25.85 25.85 27.12 27.12 27.12 27.12 27.12 27.12 27.12

7.18 7.55 8.02 8.49 8.87 9.24 9.57 7.67 8.08 8.45 8.89 9.38 9.73 10.21 7.57 7.83 8.09 8.44 8.86 9.36 9.64 7.40 7.67 8.04 8.40 8.74 9.21 9.54 7.70 8.07 8.51 8.87 9.34 9.67 10.00

18.17 18.61 19.2 19.62 20.1 20.67 21.09 18.44 18.86 19.33 19.9 20.47 21.05 21.55 18.53 18.78 19.23 19.6 20.16 20.8 21.32 18.53 18.87 19.27 19.66 20.24 20.81 21.44 18.74 19.11 19.59 20.07 20.55 21.09 21.74

18.53 19.03 19.52 19.96 20.43 20.99 21.49 18.84 19.22 19.76 20.31 20.83 21.27 21.96 18.99 19.3 19.64 20.01 20.61 21.25 21.65 19.01 19.47 19.81 20.18 20.65 21.18 21.81 19.19 19.7 20.15 20.64 21.13 21.56 22.25

18.81 19.18 19.75 20.24 20.75 21.21 21.69 19.21 19.55 20.04 20.52 21.09 21.53 22.13 19.35 19.73 20.02 20.26 20.83 21.53 21.91 19.48 19.76 20.21 20.49 21.1 21.52 22.11 19.62 20.07 20.32 20.95 21.45 21.9 22.47

358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

21.93 21.93 21.93 21.93 21.93 21.93 21.93 23.14 23.14 23.14 23.14 23.14 23.14 23.14 24.54 24.54 24.54 24.54 24.54 24.54 24.54 25.51 25.51 25.51 25.51 25.51 25.51 25.51 26.54 26.54 26.54 26.54 26.54 26.54 26.54

5.79 6.18 6.44 6.88 7.14 7.41 7.98 5.79 6.29 6.75 6.97 7.38 7.81 8.18 5.60 6.00 6.45 6.84 7.32 7.80 8.33 5.86 6.37 6.89 7.38 7.63 7.95 8.37 5.73 6.26 6.94 7.36 7.85 8.29 8.93

15.85 16.38 16.7 17.14 17.42 17.78 18.57 16.14 16.75 17.23 17.49 18.04 18.68 19.15 15.81 16.34 17.04 17.7 18.27 18.96 19.63 16.08 16.74 17.56 18.3 18.6 19.18 19.65 15.84 16.78 17.88 18.47 19.27 19.83 20.65

16.5 16.9 17.2 17.7 18 18.4 19.1 16.7 17.3 17.9 18.1 18.6 19.3 19.7 16.5 16.9 17.6 18.3 18.9 19.4 20.3 16.8 17.4 18.2 18.8 19.2 19.7 20.2 16.8 17.5 18.5 19.2 19.8 20.4 21.3

17 17 17 18 18 19 19 17 17 18 18 19 20 20 17 18 18 18 19 20 20 17 18 18 19 19 20 21 17 18 19 19 20 21 22

554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

36.03 36.03 36.03 36.03 36.03 36.03 36.03 36.66 36.66 36.66 36.66 36.66 36.66 36.66 37.24 37.24 37.24 37.24 37.24 37.24 37.24 38.01 38.01 38.01 38.01 38.01 38.01 38.01 38.63 38.63 38.63 38.63 38.63 38.63 38.63

16.50 17.13 17.73 18.17 18.34 19.06 19.69 16.49 16.93 17.31 17.95 18.16 18.72 19.28 16.41 16.94 17.45 18.11 18.52 19.04 19.41 16.82 17.46 17.84 18.43 18.81 19.30 19.82 17.45 17.98 18.69 19.14 19.76 20.08 20.70

20.5 21 21.2 21.9 22 22.4 23 20.5 20.8 21.3 21.7 21.8 22.3 22.9 20.6 20.9 21.5 21.8 22.3 22.7 23.1 20.6 20.8 21.1 21.8 22.2 22.4 22.9 20.9 21.2 21.9 22.3 22.9 23.1 23.8

21.59 22.09 22.37 22.7 22.74 23.29 23.97 21.96 22.03 22.34 22.89 22.76 23.21 23.75 22.13 22.13 22.51 23.02 23.29 23.81 24.02 21.95 22.24 22.51 22.7 23.23 23.41 23.96 22.25 22.71 23.2 23.47 23.97 24.2 24.68

22.05 22.57 22.97 23.2 23.44 23.71 24.28 22.29 22.68 22.88 23.21 23.27 23.69 24.07 22.59 22.68 23.21 23.58 23.93 24.13 24.45 22.41 22.76 23.04 23.31 23.88 23.96 24.31 22.81 23.32 23.6 23.85 24.41 24.57 25.26

Flow Measurement and Instrumentation 72 (2020) 101712

Row

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

Fig. 7. Comparison between discharge coefficients of the elliptical side orifices and the equilateral triangular side orifices (left figures W ¼ 5 cm and right figures W ¼ 10 cm).

Cd ¼ 0:543

� �1:6 � �5:4 � �2:6 W B a 0:0006 b a y1

Fig. 5. As seen, the experimental discharge coefficient values can be accurately represented by Eq. (34), which does not depend on the up­ stream Froude number. In conclusion, the stage-discharge relationship of an elliptical side orifices, having known B/a, W/b and F1 values, is:

(34)

The average and maximum errors of Eq. (34) are respectively as 2.43% and 12.15%. Calculated discharge coefficients by Eq. (34) versus measured discharge coefficients, along with their relative errors are plotted in 12

A.R. Vatankhah and F. Rafeifar

� Q ¼ πab 0:635

� �1:33 � �0:57 � pffiffiffiffiffiffiffiffiffi B W 0:085 F0:91 2ghc 1 a b

Flow Measurement and Instrumentation 72 (2020) 101712

measured discharge values, the relative error is less than �5% and for 75% of the measured discharge values, the relative error is less than �3%. If the upstream flow discharge is unknown, F1 cannot be evaluated in Eq. (40). In such a case, this study proposes the following equation for Cd: � � 0:06 � �a� 0:014 � y 1 Cd ¼ 2 1 0:936 1:06 b W ffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� (41) 0:53 � �1:06 � � 1:2 a a y1 1 þ 8:45 y1 B W

(35)

and similarly when the upstream discharge of the main channel is un­ known, the stage-discharge relationship of an elliptical side orifices, having known B/a, W/b and a/y1 values, can be expressed as: � � �1:6 � �5:4 � �2:6 � pffiffiffiffiffiffiffiffiffi W B a Q ¼ πab 0:543 0:0006 2ghc (36) b a y1 Estimated discharge values by Eq. (35), and (36) versus measured discharge values, along with their relative errors are plotted in Fig. 6. This figure shows that the experimental discharge values can be accu­ rately represented (estimated) with three dimensionless variables. As a high correlation exists between the estimated and observed discharge values when the discharge value is estimated as a function of B/a, W/b and F1 (or B/a, W/b and a/y1) these dimensionless quantities are considered as predominant parameters affecting the flow discharge values of elliptical side orifices. For an equilateral triangular side orifice located in a horizontal channel, the stage-discharge relation (Eq. (1)) can be expressed in terms of the flow depth above the centroid of the orifice section, hc, as [17]: Q ¼ Cd

LH pffiffiffiffiffiffiffiffiffi 2ghc 2

The discharge computed by Eqs. (39) and (41) has an average error of 2.24% with a maximum error less than 9.90%. Also, for 92% of the measured discharge values, the relative error is less than �5% and for 74% of the measured discharge values, the relative error is less than �3%. 8. Conclusions This study showed that Eq. (2) can be reliably used for both small and large side orifices and for both horizontal and linear water surface profiles along the elliptical side orifices. Then this equation was used for any nonlinear water surface profiles with small variation along the side orifice (small values of y2 y1). A comprehensive experimental study was carried out to determine the discharge coefficient for sharp-crested elliptical side orifices under free outflow conditions. The applicability of the discharge relationship (Eq. (2)) was realized analytically while the discharge coefficient relationships was deduced using dimensional analysis and experimental data collected in this study. The laboratory data gathered in this study were used to develop different discharge coefficient relationships with various accuracies. When the upstream discharge of the main channel is known, Eq. (35) which depends on the approaching flow Froude number may be used to obtain the greatest accuracy. When the upstream discharge of the main channel is not known, the discharge coefficient Eq. (36), which does not depend on the upstream Froude number, can be used to accurately predict the flow discharge of the sharp-crested elliptical side orifices under free outflow conditions. Moreover, the proposed model by Eqs. (39) and (40) is suitable when measurement data for hc are not available. The proposed model by Eqs. (39) and (41) is also useful when the upstream discharge and upstream Froude number are unknown. In the future, the authors intend to build prediction models through employing water surface profile data measured along the side orifice.

(37)

in which L is the orifice length and H is the orifice height. Similar experimental study on the flow characteristics of equilateral triangular side orifice under free outflow conditions is available in Ref. [17] which can be used for a correct comparison with the elliptical side orifices. In this study using similar laboratory setup, below discharge coefficient relation was developed for the triangular side orifices: � �0:86 � �0:35 B W Cd ¼ 0:75 0:52 F0:56 (38) 1 L H in which B (¼25 cm) is the main channel width, L (¼30 and 40 cm) is the orifice length, H (¼4, 7 and 10 cm) is the orifice height, W (¼5 and 10 cm) is the orifice crest height, and F1 is the upstream Froude number. In Fig. 7, the discharge coefficient for the elliptical side orifice located in a horizontal rectangular channel (Eq. (30)) is compared with that of the equilateral triangular side orifice under free outflow condi­ tions (Eq. (38)). Both discharge coefficient relations are obtained using the theoretical orifice discharge concept (Eq. (1)). According to Fig. 7, the discharge coefficient of the elliptical side orifice is greater than that of the equilateral triangular side orifice.

References

7. Discharge coefficient for unknown hc

[1] A.S. Ramamurthy, S.T. Udoyara, S. Serraf, Rectangular lateral orifices in open channel, J. Environ. Eng. (1986) 292–300, https://doi.org/10.1061/(ASCE)07339372(1986)112:2(292). [2] A.S. Ramamurthy, S.T. Udoyara, M.V.J. Rao, Weir orifice units for uniform flow distribution, J. Environ. Eng. (1987) 155–166, https://doi.org/10.1061/(ASCE) 0733-9372(1987)113:1(155). [3] M.A. Gill, Flow through side slots, J. Environ. Eng. 113 (5) (1987) 1047–1057. [4] C.S.P. Ojha, D. Subbaiah, Analysis of flow through lateral slot, J. Irrigat. Drain. Eng. (1997), https://doi.org/10.1061/(ASCE)0733-9437(1997)123:5(402)405. [5] P.K. Swamee, S.K. Pathak, M.S. Ali, Weir orifice units for uniform flow distribution, J. Irrigat. Drain. Eng. 119 (6) (1993) 1026–1035. [6] P.K. Swamee, S.K. Pathak, T. Mansoor, C.S.P. Ojha, Discharge characteristics of skew sluice gates, J. Irrigat. Drain. Eng. (2000), https://doi.org/10.1061/(ASCE) 0733-9437(2000)126:5328-334. [7] M. Ghodsian, Flow through side sluice gate, J. Irrigat. Drain. Eng. (2003) 458–463, https://doi.org/10.1061/(ASCE)0733-9437(2003)129:6(458). [8] M. Esmailzadeh, M. Heidarpour, S.S. Eslamian, Flow characteristics of a sharpcrested side sluice gate, J. Irrigat. Drain. Eng. 141 (7) (2015), 06014007. [9] A. Hussain, Z. Ahmad, G.L. Asawa, Discharge characteristics of sharp-crested circular side orifices in open channels, Flow Meas. Instrum. 21 (3) (2010) 418–424. [10] A. Hussain, Z. Ahmad, G.L. Asawa, Flow through sharp-crested rectangular side orifices under free flow condition in open channels, Agric. Water Manag. 98 (10) (2011) 1536–1544. [11] A. Hussein, Z. Ahmad, C.S.P. Ojha, Analysis of flow through lateral rectangular orifices in open channels, Flow Meas. Instrum. 36 (2014) 32–35.

In preceding considerations, hc is considered as a known independent variable (measurement variable). For design procedure of the side ori­ fices, hc is not a predetermined variable and thus should be considered as a dependent variable [19]. In such a case, the flow discharge of an elliptical side orifice can be expressed in terms of y1 as: pffiffiffiffiffiffiffiffiffi Q ¼ Cd π ab 2gy1 (39) Different discharge coefficient models can be constructed based on the dimensional analysis, intuition, theory, experience as well as an analysis of previous studies. Using the data of this study, the following equation is proposed for Cd: ffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 0:63 � �3:8 � � 0:84 � 0:057 �y a a y1 1 0:035 F1 Cd ¼ 2 1 0:873 1 þ 1:8 1:23 y1 B W W (40) The discharge computed by Eqs. (39) and (40) has an average error of 1.92% with a maximum error less than 8.19%. Also, for 96% of the 13

A.R. Vatankhah and F. Rafeifar

Flow Measurement and Instrumentation 72 (2020) 101712

[12] A. Hussain, Z. Ahmad, C.S.P. Ojha, Flow through lateral circular orifice under free and submerged flow conditions, Flow Meas. Instrum. 52 (2016) 57–66. [13] I. Ebtehaj, H. Bonakdari, F. Khoshbin, H. Azimi, Pareto genetic design of group method of data handling type neural network for prediction discharge coefficient in rectangular side orifices, Flow Meas. Instrum. 41 (2015) 67–74. [14] A.R. Vatankhah, M. Bijankhan, “Discussion of “New method for modeling thin walled orifice flow under partially submerged conditions” by David Brandes and William T. Barlow, J. Irrigat. Drain. Eng. (2013) 789–793, https://doi.org/ 10.1061/(ASCE)IR.1943-4774.0000584. [15] A.R. Vatankhah, Discussion of “stage-discharge models for concrete orifices: impact on estimating detention basin drawdown time” by WT Barlow and D. Brandes, J. Irrigat. Drain. Eng. 142 (11) (2016), https://doi.org/10.1061/(ASCE) IR.1943-4774.0001102.

[16] H. Azimi, S. Shabanlou, I. Ebtehaj, H. Bonakdari, S. Kardar, Combination of computational fluid dynamics, adaptive neuro-fuzzy inference system, and genetic algorithm for predicting discharge coefficient of rectangular side orifices, J. Irrigat. Drain. Eng. 143 (7) (2017), 04017015. [17] A.R. Vatankhah, S.H. Mirnia, Predicting discharge coefficient of triangular side orifice under free flow conditions, J. Irrigat. Drain. Eng. 144 (10) (2018), 04018030. [18] S.C. Jain, Open-channel Flow, John Wiley & Sons, 2000. [19] A.R. Vatankhah, S.H. Mirnia, Closure to “predicting discharge coefficient of triangular side orifice under free flow conditions” by Ali R. Vatankhah and SH Mirnia, J. Irrigat. Drain. Eng. 145 (8) (2019), 07019009.

14