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Flow through sharp-crested rectangular side orifices under free flow condition in open channels
Agricultural Water Management 98 (2011) 1536–1544
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Flow through sharp-crested rectangular side orifices under free flow condition in open channels A. Hussain a , Z. Ahmad a,∗ , G.L. Asawa b a b
Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India GLA University, Mathura 281406, UP, India
a r t i c l e
i n f o
Article history: Received 11 January 2011 Accepted 9 May 2011 Available online 16 June 2011 Keywords: Open channel Flow measurement Flow diversion Side orifice Coefficient of discharge Froude number
a b s t r a c t Side orifices are widely used in irrigation and environmental engineering to spill or divert water from a channel. Flow characteristic of sharp-crested rectangular orifices under free flow condition in open channels is studied in the present paper. Existing discharge equations are checked for their accuracy using the data collected in the present study and available data in the literature and a new discharge equation has been proposed. The coefficient of discharge mainly depends on the approach flow Froude number and ratio of the size of orifice and bed width of the channel. Relationships for the coefficient of discharge, treating the orifice as large and small were developed. The computed discharges using these relationships were within ±5% of the observed ones. Measurement of three-dimensional velocities and visualisation of streamlines in a horizontal plane at the centerline of the orifice indicates that for the flow of low Froude number, almost all the streamlines divert towards the orifice. However, in the case of high Froude number flows, only those streamlines which are close to the side orifice are diverted towards the orifice. The opposite side of the boundary has significant effect on the diverted discharge of low Froude number compared to the flow of high Froude number. Circular orifice has been found to be more efficient compared to the rectangular orifice of the same opening area. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Side sluice gates, side weirs and side orifices are commonly used flow diversion structures provided in the side of a channel to spill or divert water from the main channel. They are widely used in irrigation engineering, wastewater treatment plants, flocculation basins, sedimentation tanks, aeration basins, etc. Open channels, with flow diversion structures, are commonly used in irrigation systems, both for conveying water from the source to the irrigated areas, and also for distributing the water within the irrigated area. It is necessary to have knowledge of the volumes of water actually being applied through irrigation systems to agricultural areas for better crop production and optimum use of water. Several investigators like Ghodsian (2003), Ojha and Subbaiah (1997), Tanwar (1984), Gill (1987), Ramamurthy et al. (1986), Ramamurthy et al. (1987), Panda (1981), Kra and Merkley (2004) and many others have extensively studied the discharge characteristics of diversion structures. Amaral et al. (2005) presented a comparative study on the auto-regulator flow control device with sluice gate flow and concluded that auto-regulator is more effective in controlling the
∗ Corresponding author. Tel.: +91 1332285423; fax: +91 1332275568. E-mail addresses: [email protected] (A. Hussain), zulfi[email protected], zulfi[email protected] (Z. Ahmad), [email protected] (G.L. Asawa). 0378-3774/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2011.05.004
distribution of flow from the main channel to the sub channel. For zero sill height, the rectangular orifice behaves like sluice gate and once the upper crest of the rectangular orifice is above the free surface, it corresponds to a side weir. Ghodsian (2003) and Swamee et al. (1993) introduced the concept of the elementary coefficient of discharge for flow through an elementary strip along the gate length of a side sluice gate. They found that, for a sluice gate, the elementary discharge coefficient depends on the depth of flow in the main channel, gate opening for free flow conditions, and approach flow Froude number. For the sharp-crested side weirs, however, the approach flow Froude number, ratio of the depth of flow in the main channel, sill height, and ratio of the length of weir and width of channel affect the coefficient of discharge (Borghei et al., 1999). Side orifices have been studied by Gill (1987), Ramamurthy et al. (1986), Ojha and Subbaiah (1997), and Hussain et al. (2010). Gill (1987) studied on relatively short square side orifice (as in sewers) as a special case of spatially varied flow in the open channel and pressure flow. Lewis et al. (2011) studied the bottom slot in a free flow pipe as a sewer diversion structure and observed that even for subcritical approaching flow, supercritical flow exists along the slot length. Oliveto et al. (1997) studied flow through the bottom slot in a free flowing supercritical flow in a pipe functioning as a sewer diversion structure. Considering the velocity of jet Vj issued from the rectangular side orifice equal to the resultant of the velocity in the main channel V1
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Nomenclature B Cd Ce D Fr g Ho H2 H1 L b Q Qm Re V1 W Ym ε
width of the main channel (m) coefficient of discharge elementary discharge coefficient diameter of orifice (m) Froude number acceleration due to gravity (m/s2 ) head of water above the centerline of the orifice (m) head over the upper crest of the orifice (m) head over the lower crest of the orifice (m) width of rectangular orifice (m) height of rectangular orifice (m) discharge through orifice (m3 /s) discharge in the main channel (m3 /s) Reynolds number velocity in the main channel (m/s) sill height (m) depth of approach flow in the main channel (m) mass density (kg/m3 ) viscosity (N s/m2 ) average percentage error
and the velocity due to differential pressure across the orifice Vn , as shown in Fig. 1, Ramamurthy et al. (1986) derived the following discharge equation for the orifice: Q = 0.95
V13 L
f
g
1 ,
in which
f
L 1 , B
f
2 ,
1 =
L B
L B
=
(1 − 31 )
= (1 − 31 )
1/2
V12 V12 + 2gH1
C1 = 0.254
L B
−f
2 ,
L B
0.611 C3 + 3 331 C3 0.611 + 3 332 and
2 =
(1)
+ (1 − 1 )
+ (1 − 2 )
V12 V12 + 2gH2
− 0.538; C2 = 0.234
+ 0.058; C3 = −0.489
L B
C
1
1
C
1
2
+ C2
+ C2
(2)
(3)
1/2 .
(4)
L
where L is the width of rectangular orifice; B is the width of the main channel; V1 is the velocity in the approach channel; H1 is the head of water above the lower crest of the orifice; H2 is the head of water above the upper crest of the orifice; g is the acceleration due to gravity. One can compute discharge through a side rectangular orifice using Eqs. (1)–(5) for known V1 , L, B, H1 and H2 . Ojha and Subbaiah (1997) considered the flow in the main channel provided with a side orifice as spatially varied flow and proposed the following equation for the diverted discharge per unit length of the orifice: 2
dQ 1.5 = Ce 2g[(y − W )1.5 − (y − W − b) ] 3 dx
(6)
The elementary discharge coefficient Ce , mainly dependent on the orifice geometry and sill height, and is given by Ce = 0.55 − 0.07 − 0.22 2
(7)
where = b/(b + W); b is the height of rectangular orifice; W is the sill height; and y is the depth of flow in the main channel. Eq. (6) and the equation for spatially varied flow are to be solved jointly to compute the diverted discharge through the orifice. Prohaska et al. (2010) studied the flow through orifices in riser pipe and investigated the various parameters affecting the discharge coefficient of an orifice. Recently, Hussain et al. (2010) studied the discharge characteristics of sharp-crested circular side orifices in open channels through analytical and experimental considerations. They developed relationships for the coefficient of discharge, treating the orifice as large and small and concluded that the discharge through an orifice can be computed treating it as small orifice within the range of ratio of head of water above the centerline of the orifice to the diameter of the orifice, i.e., 0.75–7.86. The computed discharges using the developed relationships were within ±5% of the observed ones. Flow characteristics of sharp-crested rectangular orifices under free flow condition in open channels have been studied in the present investigation. Existing discharge equations have been checked for their accuracy using the data collected in the present study and available data in the literature and a new discharge equation has been proposed. 2. Theoretical considerations
B
− 0.129
1537
(5)
Considering varying pressure head over the flow area of a rectangular orifice of width L and height b fitted in the side of an open channel at sill height W, discharge through the orifice under free flow condition is (Doughlas et al., 2005; Chadwick et al., 2004). Q = Cd
2 3/2 3/2 2g L(H1 − H2 ) 3
(8)
However, for a small rectangular orifice with constant pressure distribution over the flow area, the discharge equation is (Ojha and Subbaiah, 1997)
Q = Cd Lb
2gHo
(9)
Ho is head of water above the centerline of the orifice. As the side sluice gate and side weir in open channel are special cases of side orifice, it is likely that variables affecting the coefficient of discharge of side sluice gate and side weir would also affect the coefficient of discharge of a side orifice. The probable variables affecting the coefficient of discharge Cd for rectangular side orifice are L, b, B, W, V1 , depth of approach flow Ym , mass density , viscosity , and acceleration due to gravity g. Thus, Fig. 1. Rectangular side orifice in an open channel.
Cd = f1 (L, b, B, W, V1 , Ym , , , g)
(10)
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A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Fig. 2. Layout of the Experimental set-up.
Dimensional analysis yields
Cd = f2
centre of the orifice. The range of data collected in the present study is given in Table 1 and a complete set of data is given in Table 2 .
V1 L B W Ym , , , Re = , Fr = L L L
V
1
(11)
gYm 3.1. Observations during experimentations
Here Re and Fr are the Reynolds number and Froude number, respectively. Data collected in the present experimental study were analyzed to investigate the effect of the above dimensionless parameters on Cd .
3. Experimental program The experiments were performed in a rectangular channel of 9.15 m length, 0.50 m width and 0.60 m depth. The channel was fitted with a sluice gate at the end of the channel to regulate the depth of flow, and a square orifice in the left side of the channel (Fig. 2). Diverted discharge through the orifice was passed into a diversion channel and, then, to a return channel. Two rectangular sharp-crested weirs, namely, Weir-A and Weir-B, calibrated using ultrasonic flow meter, were provided at the end of the diversion channel and return channel, respectively. The return channel carries the discharge from the main channel too. Experiments were performed for three sizes of square orifice equal to 0.044, 0.089, and 0.133 m and crest height W = 0.05, 0.10, 0.15 and 0.20 m under free flow conditions. For each size of orifice and crest height, three to four discharges (Qm ), in the main channel, were flown and for each Qm , different depths of flow were maintained in the main channel by the sluice gate. Water level in the main channel upstream of the orifice and head over the crests of weir-A and weir-B were measured by digital point gauge of least count 0.1 mm for each run. For selected runs, streamlines in the main channel were visualised by injecting food dye in the flow and three-dimensional velocities were measured using Acoustic Doppler Velocimeter (ADV, Sontek, 1997) in the main channel in the vicinity of the orifice in a horizontal plane passing through the
A vortex was noticed in the main channel in the vicinity of the orifice during the experimentation when the head of water above the orifice was relatively less or velocity in the main channel was low. However, no run for discharge through the orifice was taken under this condition. A small decrease in water level in the main channel was observed downstream of orifice for all other runs; however, the specific energy in the main channel was constant. The shape of the jet issuing out from the orifice was distorted downstream of the orifice. The centreline of water jet was having inclination with normal to flow in the main channel. Such inclination increases with the increase in the Froude number of the main channel as shown in Fig. 3(a) and (b). Fig. 3(a) shows emerging jet for Froude number = 0.45 while Fig. 3(b) is for comparatively low Froude number = 0.18. Large inclination of jet for higher Froude number is due to high momentum of flow in the main channel compared to the flow of low Froude number.
Table 1 Range of data collected in the present study. Parameters
Unit
Qm Q L Ym V1 W Fr
m3 /s m3 /s m m m/s m Dimensionless
Range of data Min.
Max.
0.0281 0.0009 0.044 0.154 0.12 0.05 0.05
0.1467 0.0288 0.133 0.590 0.79 0.20 0.48
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544 Table 2 (Continued )
Table 2 Data collected in the present study for square orifice. Run no. 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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
L (cm) 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 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 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9
Q (m3 /s) 0.00273 0.00287 0.00302 0.00341 0.00366 0.00213 0.00233 0.00259 0.00301 0.00336 0.00366 0.00172 0.00195 0.00262 0.00333 0.00369 0.00239 0.00259 0.00273 0.00294 0.00327 0.00350 0.00131 0.00175 0.00238 0.00294 0.00353 0.00173 0.00202 0.00238 0.00299 0.00317 0.00338 0.00252 0.00263 0.00289 0.00315 0.00343 0.00356 0.00134 0.00180 0.00206 0.00239 0.00273 0.00315 0.00343 0.00149 0.00194 0.00230 0.00285 0.00337 0.00200 0.00196 0.00220 0.00253 0.00291 0.00316 0.00142 0.00173 0.00226 0.00269 0.00302 0.00099 0.00157 0.00233 0.00902 0.01065 0.01130 0.01200 0.01465 0.00790 0.00915 0.01069 0.01381 0.00849 0.00983
Qm (m3 /s) 0.10532 0.10532 0.10532 0.10532 0.10532 0.05985 0.05985 0.05985 0.05985 0.05985 0.05985 0.02819 0.02819 0.02819 0.02819 0.02819 0.10625 0.10625 0.10625 0.10625 0.10625 0.10625 0.03189 0.03189 0.03189 0.03189 0.03189 0.06522 0.06522 0.06522 0.06522 0.06522 0.06522 0.11018 0.11018 0.11018 0.11018 0.11018 0.11018 0.07132 0.07132 0.07132 0.07132 0.07132 0.07132 0.07132 0.04215 0.04215 0.04215 0.04215 0.04215 0.03262 0.11647 0.11647 0.11647 0.11647 0.11647 0.07530 0.07530 0.07530 0.07530 0.07530 0.04202 0.04202 0.04202 0.10578 0.10578 0.10578 0.10578 0.10578 0.07212 0.07212 0.07212 0.07212 0.08582 0.08582
1539
Ym (cm) 30.94 33.63 36.89 44.90 50.65 22.57 25.88 29.56 36.85 43.79 50.55 15.38 18.27 27.29 40.38 49.35 29.50 32.62 35.00 38.88 44.41 49.61 18.04 21.95 31.71 38.65 50.74 21.24 25.27 29.88 40.50 45.34 50.93 35.69 37.59 41.15 47.00 52.65 56.16 22.79 26.17 28.91 33.56 38.93 46.69 52.64 22.96 27.07 31.94 40.49 50.46 27.94 34.39 36.86 40.78 46.72 54.49 27.64 29.87 36.56 43.41 50.39 25.92 32.30 48.34 27.05 32.63 35.16 38.54 51.58 23.35 27.18 32.46 46.99 25.13 29.55
W (cm) 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 10 10 10 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 20 20 20 20 5 5 5 5 5 5 5 5 5 5 5
Run no.
L (cm)
Q (m3 /s)
Qm (m3 /s)
Ym (cm)
W (cm)
77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153
8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3
0.01189 0.01314 0.01470 0.00760 0.00865 0.00995 0.01187 0.01445 0.00770 0.00917 0.01285 0.01431 0.00896 0.00993 0.01174 0.01422 0.01028 0.01123 0.01185 0.01305 0.01334 0.00777 0.00915 0.01052 0.01194 0.00733 0.00991 0.01196 0.01325 0.00631 0.00733 0.00898 0.01044 0.01246 0.00921 0.01105 0.01246 0.00689 0.00902 0.01038 0.00857 0.01059 0.01334 0.00631 0.00803 0.00983 0.01134 0.01233 0.00720 0.00935 0.02178 0.02367 0.02580 0.02827 0.01931 0.02035 0.02173 0.02367 0.02596 0.02876 0.01906 0.02160 0.02859 0.02830 0.01709 0.01926 0.02217 0.02594 0.01789 0.01999 0.02146 0.02451 0.02621 0.02842 0.01873 0.02391 0.02647
0.08582 0.08582 0.08582 0.08384 0.08384 0.08384 0.08384 0.08384 0.06553 0.06553 0.06553 0.06553 0.10934 0.10934 0.10934 0.10934 0.04738 0.04738 0.04738 0.04738 0.04738 0.11198 0.11198 0.11198 0.11198 0.07497 0.07497 0.07497 0.07497 0.07140 0.07140 0.07140 0.07140 0.07140 0.05435 0.05435 0.05435 0.04600 0.04600 0.04600 0.08737 0.08737 0.08737 0.07720 0.07720 0.07720 0.07720 0.07720 0.08920 0.08920 0.13163 0.13163 0.13163 0.13163 0.10830 0.10830 0.10830 0.10830 0.10830 0.10830 0.06545 0.06545 0.06545 0.06545 0.07148 0.07148 0.07148 0.07148 0.10784 0.10784 0.10784 0.10784 0.10784 0.10784 0.07555 0.07555 0.07555
37.83 44.00 51.40 27.15 29.59 33.55 41.48 54.26 27.02 31.29 45.72 52.80 30.71 34.07 40.87 53.22 34.64 38.06 40.79 46.59 48.10 32.90 37.14 41.69 48.00 31.61 39.08 47.07 53.73 28.89 31.09 35.85 41.04 49.64 36.35 43.18 49.46 34.00 40.69 44.30 38.90 45.90 57.67 32.86 36.99 43.02 48.76 53.51 34.88 40.95 33.90 37.45 41.90 48.20 29.61 31.42 33.80 37.40 42.44 50.05 28.76 33.35 41.95 48.32 30.67 34.15 39.20 47.29 32.14 35.33 37.89 44.09 48.19 53.22 33.15 43.24 48.45
5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 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 20 20 20 20 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
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Table 2 (Continued ) Run no.
L (cm)
Q (m3 /s)
Qm (m3 /s)
Ym (cm)
W (cm)
154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173
13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3
0.01770 0.02017 0.02284 0.02585 0.01959 0.02126 0.02268 0.02437 0.02300 0.02541 0.02771 0.02233 0.01697 0.01794 0.01974 0.02284 0.02605 0.02120 0.02260 0.02549
0.08436 0.08436 0.08436 0.08436 0.11580 0.11580 0.11580 0.11580 0.09718 0.09718 0.09718 0.14672 0.12723 0.12723 0.12723 0.12723 0.12723 0.08659 0.08659 0.08659
34.71 38.63 43.77 50.03 37.90 40.73 42.80 46.98 43.40 49.75 54.63 42.80 40.15 41.49 44.44 49.91 59.00 46.71 49.44 56.09
15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20
3.2. Flow pattern near the orifice Three-dimensional velocities were measured using Acoustic Doppler Velocimeter (ADV) (Sontek, 1997) with down looking probe in a horizontal plane at the centreline of the orifice from 0.4 m upstream of orifice to 0.30 m downstream and full width of the main channel. Velocities were measured at all grid points spaced 0.10 m in the longitudinal direction and 0.05 m in the transverse direction. The vector plot of velocity in the horizontal plane for two Froude numbers viz., 0.18 and 0.45 are shown in Fig. 4(a)
and (b). These figures depict that for low Froude number almost all the streamlines divert towards the orifice while in the case of high Froude number; only those streamlines which are very near to the side orifice are diverted towards the orifice. Thus, it may be concluded that the other side of the boundary has significant effect on the diverted discharge at low Froude number. For low Froude number, a reverse flow was also observed as shown in Fig. 4(a). Velocity distribution in a vertical plane passing through centreline of orifice indicates convergence of flow towards the orifice opening. Food dye was also injected in the upstream of the orifice at the level of the horizontal plane and the streamlines were captured with high resolution camera. Such streamlines are shown in Fig. 5(a) and (b) for two Froude numbers i.e., 0.18 and 0.45. These observed streamlines clearly indicate the flow pattern and also comply with the observed velocity vectors shown in Fig. 4(a) and (b). For low Froude number, the streamlines returned to the side orifice while moving downstream of the orifice, which indicates reverse flow. As the velocity in the main channel increases, the reverse flow diminishes. The strength of the reverse flow depends especially on the Froude number at the upstream side of the orifice in the main channel in addition to the dimensions of the side orifice. 4. Checking the accuracy of the existing discharge equations Data collected in the present study and those from Ojha and Subbaiah (1997) were used to check the accuracy of the discharge equations proposed by Ramamurthy et al. (1986) and Ojha and Subbaiah (1997). Fig. 6(a) shows that the computed discharge using Ramamurthy et al. (1986) discharge equation for the present study data is well within ±15% of the observed ones. It may be noted that Ramamurthy et al. (1986) equation underestimates the discharge. However, the computed discharge values for the data of Ojha and Subbaiah (1997) are beyond 1.15 times of the observed ones except for a few data as shown in Fig. 6(b). Therefore, the reliability of Ojha and Subbaiah (1997) data is doubtful. Ojha and Subbaiah (1997) used their own experimental data to verify their Eq. (6). Therefore, their data were not used to check the accuracy of their discharge equation. The computed discharges for the present study data with the observed ones are shown in Fig. 7, which indicates that the computed discharges are much lower than the observed ones. It is noted that Ojha and Subbaiah (1997) equation is much less accurate. Underestimation of the discharge by Ramamurthy et al. (1986) is likely due to arbitrarily adoption of flow reduction factor 0.95 in Eq. (1) to account for the velocity distribution in the main channel. Therefore, it is proposed to develop a new discharge equation using the data collected in this study. 5. Proposed equation for the coefficient of discharge
Fig. 3. View of the water jet issuing out from the orifice for (a) Fr = 0.45 and (b) Fr = 0.18.
Treating the orifice as large, the coefficient of discharge is computed using Eq. (8) for each data set collected in the present study for known values of L, H1 , H2 , and Q. The effect of the dimensionless parameters B/L, W/L, Ym /L, Re and Fr on the coefficient of discharge was examined. A thorough data analysis reveals that Fr and B/L are, indeed, the predominant parameters which affect the value of Cd . Variation of Cd with Fr is shown in Fig. 8, which clearly indicates decrease of Cd with increase in Froude number. It should be noted that the approach Froude number is an important parameter in the flow over a side weir and under a side sluice gate. Therefore, its effect on Cd for side orifice is apparent. For the range of data used in the present study, Cd is unaffected by
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
1541
Fig. 4. Velocity vector in horizontal plane for (a) Fr = 0.18 and (b) Fr = 0.45.
the parameters W/L, Ym /L and Re. Further, in the present experimental work, b/L = 1. The value of Cd for B/L = 3.72, 5.56, and 11.25 are 0.60, 0.63, and 0.66, respectively for Qm = 0.108 m3 /s, Ym = 0.35 m and W = 0.1 m. This indicates that Cd increases with the increase in B/L. Therefore, it may be concluded that the discharge coefficient decreases with the increase in the size of the orifice. Generally, 70–80% of the total data are used for the calibration of any equation and 20–30% for its validation (Kumar et al., 2010; Bowden et al., 2005; Hammerstrom, 1993). Therefore, in the present study, 75% data have been used for the calibration of the equation for Cd and remaining 25% for its validation. Using the least square technique and 130 data sets selected randomly out of 173
data sets of the present study, the following equation is obtained for Cd : Cd = 0.714 − 0.062Fr − 0.347
L B
(12)
This equation for Cd is validated using the remaining 43 data sets for the computation of discharge through the orifice. A graphical comparison between the observed and computed values of discharge through side orifice using Eqs. (8) and (12) for the test data is shown in Fig. 9. It is apparent from Fig. 9 that the computed discharge is within ±5% of the observed ones. The prediction of discharge through the orifice, using Eqs. (8) and (12) may, therefore, be considered as satisfactory.
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A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Fig. 5. Observed streamlines in a horizontal plane for (a) Fr = 0.18 and (b) Fr = 0.45.
Similarly, treating the used orifices as small, the following equation for Cd is obtained using 130 observed data sets and invoking least squares technique: Cd = 0.714 − 0.066Fr − 0.354
6. Efficiency of rectangular and circular orifices
L B
(13)
The computed discharges using Eqs. (9) and (13) for 43 data sets that were not used for obtaining Eq. (13) is compared with the observed ones in Fig. 10 to validate Eq. (13). It is, therefore, obvious that even from the concept of small orifice, the computed discharges are within ±5% of the observed ones. For the quantitative measure of selecting best-fit equation that can represent the agreement between the observed and computed values, an average percentage error term ε was defined as (Ghodsian, 2003)
An orifice would be more efficient if it diverts more discharge through it compared to the orifices having the same opening area and under the same geometry and flow characteristics in the main channel. Coefficient of discharge is, obviously, an indicator of the efficiency of the orifice. Thus, an orifice having more Cd will be
100 Q (computed) − Q (observed) N Q (observed) N
ε=
Fig. 6. Checking the accuracy of Ramamurthy et al. (1986) equation using (a) present data and (b) Ojha and Subbaiah (1997).
(14)
i=1
The average percentage error in computation of discharge through the orifice treating it as large and small are 2.52% and 2.53%, respectively. Likewise, the values of R2 for large and small orifices are 0.998 and 0.997, respectively; and mean root square error are 2.56 × 10−5 and 2.58 × 10−5 , respectively. Thus, discharge through a side orifice can be computed treating it as small orifice within range of Ho /L = 0.84–9.88. When the head over a normal orifice is less than five times the height of the orifice, it is referred to as a large orifice (Gupta, 1989). However, as inferred from the results of the present investigation, the side orifice behaves like a small orifice even for Ho = 0.84L.
Fig. 7. Checking the accuracy of Ojha and Subbaiah (1997) equation using present study data.
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Fig. 8. Variation of Cd with Froude number.
1543
Fig. 11. Variation of coefficient of discharge for circular and square orifices with L/B or D/B.
considered more efficient and vice versa. Hussain et al. (2010) proposed the following relationship for coefficient of discharge Cd for a circular side orifice: Cd = 0.670 − 0.076Fr − 0.136
Fig. 9. Comparison of the computed discharge through orifice using Eqs. (8) and (12) with the observed ones treating orifice as large.
D B
(15)
Here D is the diameter of the orifice. Variations of Cd for square and circular orifices with L/B or D/B for different Froude numbers are shown in Fig. 11. For L/B or D/B < 0.2, Cd for square orifice is more than the circular orifice while for L/B or D/B > 0.2, Cd of circular orifice is more than the square orifice. Further, Cd is more for low Froude number for a given L/B or D/B. Due to the boundary effect of the opposite wall of the channel, Cd decreases with increase in L/B or D/B for a given Froude number. Normally, pressure in the water jet at vena contracta is higher than the atmospheric pressure (Ahmad, 2001). The inward force due to pressure at vena-contracta is likely to be more for the square orifice than for the circular orifice due to symmetry of the circular jet, which results in higher Cd for the circular orifice. 7. Conclusions
Fig. 10. Comparison of the computed discharge through orifice using Eqs. (9) and (13) with the observed ones treating orifice as small.
It is concluded from the present study on discharge characteristics of sharp-crested square side orifices in open channels that the coefficient of discharge depends mainly on the approach flow Froude number and ratio of width of orifice and bed width of the channel. The computed discharges using the proposed relationships, by treating the orifice as large and small, were within ±5% of the observed ones. The average percentage error in computation of discharge through the orifice treating it as large and small are 2.52% and 2.53%, respectively, which are practically the same. Therefore, discharge through the orifice can be computed treating it as small orifice within range of Ho /L = 0.84–9.88. Measurement of three-dimensional velocities and visualisation of streamlines in a horizontal plane at the centreline of the orifice indicates that for flow of low Froude number, almost all the streamlines divert towards the orifice while in the case of high Froude number, only those streamlines which are close to the side orifice are diverted towards the orifice. The opposite side of the boundary has significant effect on the diverted discharge at low Froude number compared to flow at high Froude number. Circular orifice diverts more discharge through it compared to the square orifice having the same opening area. Therefore, it is recommended that the cir-
1544
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
cular orifice be used in place of rectangular orifice to divert flow through an open channel when L/B or D/B > 0.2. References Ahmad, Z., 2001. Flow measurement with trapezoidal free overfall. Journal of Hydraulic Engineering 7 (2), 32–44. Amaral, L.G.H., Righes, A.A., Filho, P.S.S., Costa, R.D., 2005. Automatic regulator for channel flow control on flooded rice. Agricultural Water Management 75 (3), 184–193. Borghei, M., Jalili, M.R., Ghodsian, M., 1999. Discharge coefficient for sharpcrested side weir in subcritical flow. Journal of Hydraulic Engineering 125 (1), 1051–1056. Bowden, G.J., Maier, H.R., Dandy, G.C., 2005. Input determination for neural network models in water resources applications. Part 2. Case study: forecasting salinity in a river. Journal of Hydrology 301, 93–107. Chadwick, A., Morfett, J., Borthwick, M., 2004. Hydraulics of Civil and Environmental Engineering, 4th ed. Spon Press, New York. Doughlas, J.F., Gasoriek, J.M., Swaffield, J., Jack, L., 2005. Fluid Mechanics, 5th ed. Prentice Hall, New York. Ghodsian, M., 2003. Flow through side sluice gate. Journal of Irrigation and Drainage Engineering 129 (6), 458–462. Gill, M.A., 1987. Flow through side slots. Journal of Environmental Engineering 135 (21874), 1047–1057. Gupta, R.S., 1989. Hydrology and Hydraulic Systems. Prentice-Hall, Inc., Englewood Cliffs, NJ. Hammerstrom, D., 1993. Working with neural networks. IEEE Spectrum 30 (7), 46–53.
Hussain, A., Ahmad, Z., Asawa, G.L., 2010. Discharge characteristics of sharp-crested circular side orifices in open channels. Journal of Flow Measurement and Instrumentation 21 (3), 418–424. Kra, E.Y., Merkley, G.P., 2004. Mathematical modelling of open-channel velocity profiles for float method calibration. Agricultural Water Management 70 (3), 229–244. Kumar, S., Ahmad, Z., Kothyari, U.C., Mittal, M.K., 2010. Discharge characteristics of a trench weir. Flow Measurement and Instrumentation 21 (2), 80–87. Lewis, J.W., Wright, S.J., Pribak, M., Sherill, J., 2011. Bottom slot discharge outlet for combined sewer diversion structure. Journal of Irrigation and Drainage Engineering 137 (2), 248–253. Ojha, C.S.P., Subbaiah, D., 1997. Analysis of flow through lateral slot. Journal of Irrigation and Drainage Engineering 123 (5), 402–405. Oliveto, G., Biggiero, V., Hager, W.H., 1997. Bottom outlet for sewers. Journal of Irrigation and Drainage Engineering 123 (4), 246–252. Panda, S., 1981. Characteristics of side sluice flow. ME Thesis, University of Roorkee, Roorkee, India. Prohaska, P.D., Khan, A.A., Kaye, N.B., 2010. Investigation of flow through orifices in riser pipes. Journal of Irrigation and Drainage Engineering 136 (5). Ramamurthy, A.S., Udoyara, S.T., Serraf, S., 1986. Square lateral orifices in open channel. Journal of Environmental Engineering 135 (5), 292–298. Ramamurthy, A.S., Udoyara, S.T., Rao, M.V.J., 1987. Weir orifice units for uniform flow distribution. Journal of Environmental Engineering 113 (1), 155–166. Swamee, P.K., Pathak, S.K., Ali, M.S., 1993. Weir orifice units for uniform flow distribution. Journal of Irrigation and Drainage Engineering 119 (6), 1026–1035. Tanwar, M.P.S., 1984. Flow through side sluice. ME Thesis, University of Roorkee, Roorkee, India.
Contents lists available at ScienceDirect
Agricultural Water Management journal homepage: www.elsevier.com/locate/agwat
Flow through sharp-crested rectangular side orifices under free flow condition in open channels A. Hussain a , Z. Ahmad a,∗ , G.L. Asawa b a b
Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India GLA University, Mathura 281406, UP, India
a r t i c l e
i n f o
Article history: Received 11 January 2011 Accepted 9 May 2011 Available online 16 June 2011 Keywords: Open channel Flow measurement Flow diversion Side orifice Coefficient of discharge Froude number
a b s t r a c t Side orifices are widely used in irrigation and environmental engineering to spill or divert water from a channel. Flow characteristic of sharp-crested rectangular orifices under free flow condition in open channels is studied in the present paper. Existing discharge equations are checked for their accuracy using the data collected in the present study and available data in the literature and a new discharge equation has been proposed. The coefficient of discharge mainly depends on the approach flow Froude number and ratio of the size of orifice and bed width of the channel. Relationships for the coefficient of discharge, treating the orifice as large and small were developed. The computed discharges using these relationships were within ±5% of the observed ones. Measurement of three-dimensional velocities and visualisation of streamlines in a horizontal plane at the centerline of the orifice indicates that for the flow of low Froude number, almost all the streamlines divert towards the orifice. However, in the case of high Froude number flows, only those streamlines which are close to the side orifice are diverted towards the orifice. The opposite side of the boundary has significant effect on the diverted discharge of low Froude number compared to the flow of high Froude number. Circular orifice has been found to be more efficient compared to the rectangular orifice of the same opening area. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Side sluice gates, side weirs and side orifices are commonly used flow diversion structures provided in the side of a channel to spill or divert water from the main channel. They are widely used in irrigation engineering, wastewater treatment plants, flocculation basins, sedimentation tanks, aeration basins, etc. Open channels, with flow diversion structures, are commonly used in irrigation systems, both for conveying water from the source to the irrigated areas, and also for distributing the water within the irrigated area. It is necessary to have knowledge of the volumes of water actually being applied through irrigation systems to agricultural areas for better crop production and optimum use of water. Several investigators like Ghodsian (2003), Ojha and Subbaiah (1997), Tanwar (1984), Gill (1987), Ramamurthy et al. (1986), Ramamurthy et al. (1987), Panda (1981), Kra and Merkley (2004) and many others have extensively studied the discharge characteristics of diversion structures. Amaral et al. (2005) presented a comparative study on the auto-regulator flow control device with sluice gate flow and concluded that auto-regulator is more effective in controlling the
∗ Corresponding author. Tel.: +91 1332285423; fax: +91 1332275568. E-mail addresses: [email protected] (A. Hussain), zulfi[email protected], zulfi[email protected] (Z. Ahmad), [email protected] (G.L. Asawa). 0378-3774/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2011.05.004
distribution of flow from the main channel to the sub channel. For zero sill height, the rectangular orifice behaves like sluice gate and once the upper crest of the rectangular orifice is above the free surface, it corresponds to a side weir. Ghodsian (2003) and Swamee et al. (1993) introduced the concept of the elementary coefficient of discharge for flow through an elementary strip along the gate length of a side sluice gate. They found that, for a sluice gate, the elementary discharge coefficient depends on the depth of flow in the main channel, gate opening for free flow conditions, and approach flow Froude number. For the sharp-crested side weirs, however, the approach flow Froude number, ratio of the depth of flow in the main channel, sill height, and ratio of the length of weir and width of channel affect the coefficient of discharge (Borghei et al., 1999). Side orifices have been studied by Gill (1987), Ramamurthy et al. (1986), Ojha and Subbaiah (1997), and Hussain et al. (2010). Gill (1987) studied on relatively short square side orifice (as in sewers) as a special case of spatially varied flow in the open channel and pressure flow. Lewis et al. (2011) studied the bottom slot in a free flow pipe as a sewer diversion structure and observed that even for subcritical approaching flow, supercritical flow exists along the slot length. Oliveto et al. (1997) studied flow through the bottom slot in a free flowing supercritical flow in a pipe functioning as a sewer diversion structure. Considering the velocity of jet Vj issued from the rectangular side orifice equal to the resultant of the velocity in the main channel V1
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Nomenclature B Cd Ce D Fr g Ho H2 H1 L b Q Qm Re V1 W Ym ε
width of the main channel (m) coefficient of discharge elementary discharge coefficient diameter of orifice (m) Froude number acceleration due to gravity (m/s2 ) head of water above the centerline of the orifice (m) head over the upper crest of the orifice (m) head over the lower crest of the orifice (m) width of rectangular orifice (m) height of rectangular orifice (m) discharge through orifice (m3 /s) discharge in the main channel (m3 /s) Reynolds number velocity in the main channel (m/s) sill height (m) depth of approach flow in the main channel (m) mass density (kg/m3 ) viscosity (N s/m2 ) average percentage error
and the velocity due to differential pressure across the orifice Vn , as shown in Fig. 1, Ramamurthy et al. (1986) derived the following discharge equation for the orifice: Q = 0.95
V13 L
f
g
1 ,
in which
f
L 1 , B
f
2 ,
1 =
L B
L B
=
(1 − 31 )
= (1 − 31 )
1/2
V12 V12 + 2gH1
C1 = 0.254
L B
−f
2 ,
L B
0.611 C3 + 3 331 C3 0.611 + 3 332 and
2 =
(1)
+ (1 − 1 )
+ (1 − 2 )
V12 V12 + 2gH2
− 0.538; C2 = 0.234
+ 0.058; C3 = −0.489
L B
C
1
1
C
1
2
+ C2
+ C2
(2)
(3)
1/2 .
(4)
L
where L is the width of rectangular orifice; B is the width of the main channel; V1 is the velocity in the approach channel; H1 is the head of water above the lower crest of the orifice; H2 is the head of water above the upper crest of the orifice; g is the acceleration due to gravity. One can compute discharge through a side rectangular orifice using Eqs. (1)–(5) for known V1 , L, B, H1 and H2 . Ojha and Subbaiah (1997) considered the flow in the main channel provided with a side orifice as spatially varied flow and proposed the following equation for the diverted discharge per unit length of the orifice: 2
dQ 1.5 = Ce 2g[(y − W )1.5 − (y − W − b) ] 3 dx
(6)
The elementary discharge coefficient Ce , mainly dependent on the orifice geometry and sill height, and is given by Ce = 0.55 − 0.07 − 0.22 2
(7)
where = b/(b + W); b is the height of rectangular orifice; W is the sill height; and y is the depth of flow in the main channel. Eq. (6) and the equation for spatially varied flow are to be solved jointly to compute the diverted discharge through the orifice. Prohaska et al. (2010) studied the flow through orifices in riser pipe and investigated the various parameters affecting the discharge coefficient of an orifice. Recently, Hussain et al. (2010) studied the discharge characteristics of sharp-crested circular side orifices in open channels through analytical and experimental considerations. They developed relationships for the coefficient of discharge, treating the orifice as large and small and concluded that the discharge through an orifice can be computed treating it as small orifice within the range of ratio of head of water above the centerline of the orifice to the diameter of the orifice, i.e., 0.75–7.86. The computed discharges using the developed relationships were within ±5% of the observed ones. Flow characteristics of sharp-crested rectangular orifices under free flow condition in open channels have been studied in the present investigation. Existing discharge equations have been checked for their accuracy using the data collected in the present study and available data in the literature and a new discharge equation has been proposed. 2. Theoretical considerations
B
− 0.129
1537
(5)
Considering varying pressure head over the flow area of a rectangular orifice of width L and height b fitted in the side of an open channel at sill height W, discharge through the orifice under free flow condition is (Doughlas et al., 2005; Chadwick et al., 2004). Q = Cd
2 3/2 3/2 2g L(H1 − H2 ) 3
(8)
However, for a small rectangular orifice with constant pressure distribution over the flow area, the discharge equation is (Ojha and Subbaiah, 1997)
Q = Cd Lb
2gHo
(9)
Ho is head of water above the centerline of the orifice. As the side sluice gate and side weir in open channel are special cases of side orifice, it is likely that variables affecting the coefficient of discharge of side sluice gate and side weir would also affect the coefficient of discharge of a side orifice. The probable variables affecting the coefficient of discharge Cd for rectangular side orifice are L, b, B, W, V1 , depth of approach flow Ym , mass density , viscosity , and acceleration due to gravity g. Thus, Fig. 1. Rectangular side orifice in an open channel.
Cd = f1 (L, b, B, W, V1 , Ym , , , g)
(10)
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A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Fig. 2. Layout of the Experimental set-up.
Dimensional analysis yields
Cd = f2
centre of the orifice. The range of data collected in the present study is given in Table 1 and a complete set of data is given in Table 2 .
V1 L B W Ym , , , Re = , Fr = L L L
V
1
(11)
gYm 3.1. Observations during experimentations
Here Re and Fr are the Reynolds number and Froude number, respectively. Data collected in the present experimental study were analyzed to investigate the effect of the above dimensionless parameters on Cd .
3. Experimental program The experiments were performed in a rectangular channel of 9.15 m length, 0.50 m width and 0.60 m depth. The channel was fitted with a sluice gate at the end of the channel to regulate the depth of flow, and a square orifice in the left side of the channel (Fig. 2). Diverted discharge through the orifice was passed into a diversion channel and, then, to a return channel. Two rectangular sharp-crested weirs, namely, Weir-A and Weir-B, calibrated using ultrasonic flow meter, were provided at the end of the diversion channel and return channel, respectively. The return channel carries the discharge from the main channel too. Experiments were performed for three sizes of square orifice equal to 0.044, 0.089, and 0.133 m and crest height W = 0.05, 0.10, 0.15 and 0.20 m under free flow conditions. For each size of orifice and crest height, three to four discharges (Qm ), in the main channel, were flown and for each Qm , different depths of flow were maintained in the main channel by the sluice gate. Water level in the main channel upstream of the orifice and head over the crests of weir-A and weir-B were measured by digital point gauge of least count 0.1 mm for each run. For selected runs, streamlines in the main channel were visualised by injecting food dye in the flow and three-dimensional velocities were measured using Acoustic Doppler Velocimeter (ADV, Sontek, 1997) in the main channel in the vicinity of the orifice in a horizontal plane passing through the
A vortex was noticed in the main channel in the vicinity of the orifice during the experimentation when the head of water above the orifice was relatively less or velocity in the main channel was low. However, no run for discharge through the orifice was taken under this condition. A small decrease in water level in the main channel was observed downstream of orifice for all other runs; however, the specific energy in the main channel was constant. The shape of the jet issuing out from the orifice was distorted downstream of the orifice. The centreline of water jet was having inclination with normal to flow in the main channel. Such inclination increases with the increase in the Froude number of the main channel as shown in Fig. 3(a) and (b). Fig. 3(a) shows emerging jet for Froude number = 0.45 while Fig. 3(b) is for comparatively low Froude number = 0.18. Large inclination of jet for higher Froude number is due to high momentum of flow in the main channel compared to the flow of low Froude number.
Table 1 Range of data collected in the present study. Parameters
Unit
Qm Q L Ym V1 W Fr
m3 /s m3 /s m m m/s m Dimensionless
Range of data Min.
Max.
0.0281 0.0009 0.044 0.154 0.12 0.05 0.05
0.1467 0.0288 0.133 0.590 0.79 0.20 0.48
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544 Table 2 (Continued )
Table 2 Data collected in the present study for square orifice. Run no. 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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
L (cm) 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 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 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9
Q (m3 /s) 0.00273 0.00287 0.00302 0.00341 0.00366 0.00213 0.00233 0.00259 0.00301 0.00336 0.00366 0.00172 0.00195 0.00262 0.00333 0.00369 0.00239 0.00259 0.00273 0.00294 0.00327 0.00350 0.00131 0.00175 0.00238 0.00294 0.00353 0.00173 0.00202 0.00238 0.00299 0.00317 0.00338 0.00252 0.00263 0.00289 0.00315 0.00343 0.00356 0.00134 0.00180 0.00206 0.00239 0.00273 0.00315 0.00343 0.00149 0.00194 0.00230 0.00285 0.00337 0.00200 0.00196 0.00220 0.00253 0.00291 0.00316 0.00142 0.00173 0.00226 0.00269 0.00302 0.00099 0.00157 0.00233 0.00902 0.01065 0.01130 0.01200 0.01465 0.00790 0.00915 0.01069 0.01381 0.00849 0.00983
Qm (m3 /s) 0.10532 0.10532 0.10532 0.10532 0.10532 0.05985 0.05985 0.05985 0.05985 0.05985 0.05985 0.02819 0.02819 0.02819 0.02819 0.02819 0.10625 0.10625 0.10625 0.10625 0.10625 0.10625 0.03189 0.03189 0.03189 0.03189 0.03189 0.06522 0.06522 0.06522 0.06522 0.06522 0.06522 0.11018 0.11018 0.11018 0.11018 0.11018 0.11018 0.07132 0.07132 0.07132 0.07132 0.07132 0.07132 0.07132 0.04215 0.04215 0.04215 0.04215 0.04215 0.03262 0.11647 0.11647 0.11647 0.11647 0.11647 0.07530 0.07530 0.07530 0.07530 0.07530 0.04202 0.04202 0.04202 0.10578 0.10578 0.10578 0.10578 0.10578 0.07212 0.07212 0.07212 0.07212 0.08582 0.08582
1539
Ym (cm) 30.94 33.63 36.89 44.90 50.65 22.57 25.88 29.56 36.85 43.79 50.55 15.38 18.27 27.29 40.38 49.35 29.50 32.62 35.00 38.88 44.41 49.61 18.04 21.95 31.71 38.65 50.74 21.24 25.27 29.88 40.50 45.34 50.93 35.69 37.59 41.15 47.00 52.65 56.16 22.79 26.17 28.91 33.56 38.93 46.69 52.64 22.96 27.07 31.94 40.49 50.46 27.94 34.39 36.86 40.78 46.72 54.49 27.64 29.87 36.56 43.41 50.39 25.92 32.30 48.34 27.05 32.63 35.16 38.54 51.58 23.35 27.18 32.46 46.99 25.13 29.55
W (cm) 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 10 10 10 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 20 20 20 20 5 5 5 5 5 5 5 5 5 5 5
Run no.
L (cm)
Q (m3 /s)
Qm (m3 /s)
Ym (cm)
W (cm)
77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153
8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3
0.01189 0.01314 0.01470 0.00760 0.00865 0.00995 0.01187 0.01445 0.00770 0.00917 0.01285 0.01431 0.00896 0.00993 0.01174 0.01422 0.01028 0.01123 0.01185 0.01305 0.01334 0.00777 0.00915 0.01052 0.01194 0.00733 0.00991 0.01196 0.01325 0.00631 0.00733 0.00898 0.01044 0.01246 0.00921 0.01105 0.01246 0.00689 0.00902 0.01038 0.00857 0.01059 0.01334 0.00631 0.00803 0.00983 0.01134 0.01233 0.00720 0.00935 0.02178 0.02367 0.02580 0.02827 0.01931 0.02035 0.02173 0.02367 0.02596 0.02876 0.01906 0.02160 0.02859 0.02830 0.01709 0.01926 0.02217 0.02594 0.01789 0.01999 0.02146 0.02451 0.02621 0.02842 0.01873 0.02391 0.02647
0.08582 0.08582 0.08582 0.08384 0.08384 0.08384 0.08384 0.08384 0.06553 0.06553 0.06553 0.06553 0.10934 0.10934 0.10934 0.10934 0.04738 0.04738 0.04738 0.04738 0.04738 0.11198 0.11198 0.11198 0.11198 0.07497 0.07497 0.07497 0.07497 0.07140 0.07140 0.07140 0.07140 0.07140 0.05435 0.05435 0.05435 0.04600 0.04600 0.04600 0.08737 0.08737 0.08737 0.07720 0.07720 0.07720 0.07720 0.07720 0.08920 0.08920 0.13163 0.13163 0.13163 0.13163 0.10830 0.10830 0.10830 0.10830 0.10830 0.10830 0.06545 0.06545 0.06545 0.06545 0.07148 0.07148 0.07148 0.07148 0.10784 0.10784 0.10784 0.10784 0.10784 0.10784 0.07555 0.07555 0.07555
37.83 44.00 51.40 27.15 29.59 33.55 41.48 54.26 27.02 31.29 45.72 52.80 30.71 34.07 40.87 53.22 34.64 38.06 40.79 46.59 48.10 32.90 37.14 41.69 48.00 31.61 39.08 47.07 53.73 28.89 31.09 35.85 41.04 49.64 36.35 43.18 49.46 34.00 40.69 44.30 38.90 45.90 57.67 32.86 36.99 43.02 48.76 53.51 34.88 40.95 33.90 37.45 41.90 48.20 29.61 31.42 33.80 37.40 42.44 50.05 28.76 33.35 41.95 48.32 30.67 34.15 39.20 47.29 32.14 35.33 37.89 44.09 48.19 53.22 33.15 43.24 48.45
5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 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 20 20 20 20 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
1540
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Table 2 (Continued ) Run no.
L (cm)
Q (m3 /s)
Qm (m3 /s)
Ym (cm)
W (cm)
154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173
13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3
0.01770 0.02017 0.02284 0.02585 0.01959 0.02126 0.02268 0.02437 0.02300 0.02541 0.02771 0.02233 0.01697 0.01794 0.01974 0.02284 0.02605 0.02120 0.02260 0.02549
0.08436 0.08436 0.08436 0.08436 0.11580 0.11580 0.11580 0.11580 0.09718 0.09718 0.09718 0.14672 0.12723 0.12723 0.12723 0.12723 0.12723 0.08659 0.08659 0.08659
34.71 38.63 43.77 50.03 37.90 40.73 42.80 46.98 43.40 49.75 54.63 42.80 40.15 41.49 44.44 49.91 59.00 46.71 49.44 56.09
15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20
3.2. Flow pattern near the orifice Three-dimensional velocities were measured using Acoustic Doppler Velocimeter (ADV) (Sontek, 1997) with down looking probe in a horizontal plane at the centreline of the orifice from 0.4 m upstream of orifice to 0.30 m downstream and full width of the main channel. Velocities were measured at all grid points spaced 0.10 m in the longitudinal direction and 0.05 m in the transverse direction. The vector plot of velocity in the horizontal plane for two Froude numbers viz., 0.18 and 0.45 are shown in Fig. 4(a)
and (b). These figures depict that for low Froude number almost all the streamlines divert towards the orifice while in the case of high Froude number; only those streamlines which are very near to the side orifice are diverted towards the orifice. Thus, it may be concluded that the other side of the boundary has significant effect on the diverted discharge at low Froude number. For low Froude number, a reverse flow was also observed as shown in Fig. 4(a). Velocity distribution in a vertical plane passing through centreline of orifice indicates convergence of flow towards the orifice opening. Food dye was also injected in the upstream of the orifice at the level of the horizontal plane and the streamlines were captured with high resolution camera. Such streamlines are shown in Fig. 5(a) and (b) for two Froude numbers i.e., 0.18 and 0.45. These observed streamlines clearly indicate the flow pattern and also comply with the observed velocity vectors shown in Fig. 4(a) and (b). For low Froude number, the streamlines returned to the side orifice while moving downstream of the orifice, which indicates reverse flow. As the velocity in the main channel increases, the reverse flow diminishes. The strength of the reverse flow depends especially on the Froude number at the upstream side of the orifice in the main channel in addition to the dimensions of the side orifice. 4. Checking the accuracy of the existing discharge equations Data collected in the present study and those from Ojha and Subbaiah (1997) were used to check the accuracy of the discharge equations proposed by Ramamurthy et al. (1986) and Ojha and Subbaiah (1997). Fig. 6(a) shows that the computed discharge using Ramamurthy et al. (1986) discharge equation for the present study data is well within ±15% of the observed ones. It may be noted that Ramamurthy et al. (1986) equation underestimates the discharge. However, the computed discharge values for the data of Ojha and Subbaiah (1997) are beyond 1.15 times of the observed ones except for a few data as shown in Fig. 6(b). Therefore, the reliability of Ojha and Subbaiah (1997) data is doubtful. Ojha and Subbaiah (1997) used their own experimental data to verify their Eq. (6). Therefore, their data were not used to check the accuracy of their discharge equation. The computed discharges for the present study data with the observed ones are shown in Fig. 7, which indicates that the computed discharges are much lower than the observed ones. It is noted that Ojha and Subbaiah (1997) equation is much less accurate. Underestimation of the discharge by Ramamurthy et al. (1986) is likely due to arbitrarily adoption of flow reduction factor 0.95 in Eq. (1) to account for the velocity distribution in the main channel. Therefore, it is proposed to develop a new discharge equation using the data collected in this study. 5. Proposed equation for the coefficient of discharge
Fig. 3. View of the water jet issuing out from the orifice for (a) Fr = 0.45 and (b) Fr = 0.18.
Treating the orifice as large, the coefficient of discharge is computed using Eq. (8) for each data set collected in the present study for known values of L, H1 , H2 , and Q. The effect of the dimensionless parameters B/L, W/L, Ym /L, Re and Fr on the coefficient of discharge was examined. A thorough data analysis reveals that Fr and B/L are, indeed, the predominant parameters which affect the value of Cd . Variation of Cd with Fr is shown in Fig. 8, which clearly indicates decrease of Cd with increase in Froude number. It should be noted that the approach Froude number is an important parameter in the flow over a side weir and under a side sluice gate. Therefore, its effect on Cd for side orifice is apparent. For the range of data used in the present study, Cd is unaffected by
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Fig. 4. Velocity vector in horizontal plane for (a) Fr = 0.18 and (b) Fr = 0.45.
the parameters W/L, Ym /L and Re. Further, in the present experimental work, b/L = 1. The value of Cd for B/L = 3.72, 5.56, and 11.25 are 0.60, 0.63, and 0.66, respectively for Qm = 0.108 m3 /s, Ym = 0.35 m and W = 0.1 m. This indicates that Cd increases with the increase in B/L. Therefore, it may be concluded that the discharge coefficient decreases with the increase in the size of the orifice. Generally, 70–80% of the total data are used for the calibration of any equation and 20–30% for its validation (Kumar et al., 2010; Bowden et al., 2005; Hammerstrom, 1993). Therefore, in the present study, 75% data have been used for the calibration of the equation for Cd and remaining 25% for its validation. Using the least square technique and 130 data sets selected randomly out of 173
data sets of the present study, the following equation is obtained for Cd : Cd = 0.714 − 0.062Fr − 0.347
L B
(12)
This equation for Cd is validated using the remaining 43 data sets for the computation of discharge through the orifice. A graphical comparison between the observed and computed values of discharge through side orifice using Eqs. (8) and (12) for the test data is shown in Fig. 9. It is apparent from Fig. 9 that the computed discharge is within ±5% of the observed ones. The prediction of discharge through the orifice, using Eqs. (8) and (12) may, therefore, be considered as satisfactory.
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Fig. 5. Observed streamlines in a horizontal plane for (a) Fr = 0.18 and (b) Fr = 0.45.
Similarly, treating the used orifices as small, the following equation for Cd is obtained using 130 observed data sets and invoking least squares technique: Cd = 0.714 − 0.066Fr − 0.354
6. Efficiency of rectangular and circular orifices
L B
(13)
The computed discharges using Eqs. (9) and (13) for 43 data sets that were not used for obtaining Eq. (13) is compared with the observed ones in Fig. 10 to validate Eq. (13). It is, therefore, obvious that even from the concept of small orifice, the computed discharges are within ±5% of the observed ones. For the quantitative measure of selecting best-fit equation that can represent the agreement between the observed and computed values, an average percentage error term ε was defined as (Ghodsian, 2003)
An orifice would be more efficient if it diverts more discharge through it compared to the orifices having the same opening area and under the same geometry and flow characteristics in the main channel. Coefficient of discharge is, obviously, an indicator of the efficiency of the orifice. Thus, an orifice having more Cd will be
100 Q (computed) − Q (observed) N Q (observed) N
ε=
Fig. 6. Checking the accuracy of Ramamurthy et al. (1986) equation using (a) present data and (b) Ojha and Subbaiah (1997).
(14)
i=1
The average percentage error in computation of discharge through the orifice treating it as large and small are 2.52% and 2.53%, respectively. Likewise, the values of R2 for large and small orifices are 0.998 and 0.997, respectively; and mean root square error are 2.56 × 10−5 and 2.58 × 10−5 , respectively. Thus, discharge through a side orifice can be computed treating it as small orifice within range of Ho /L = 0.84–9.88. When the head over a normal orifice is less than five times the height of the orifice, it is referred to as a large orifice (Gupta, 1989). However, as inferred from the results of the present investigation, the side orifice behaves like a small orifice even for Ho = 0.84L.
Fig. 7. Checking the accuracy of Ojha and Subbaiah (1997) equation using present study data.
A. Hussain et al. / Agricultural Water Management 98 (2011) 1536–1544
Fig. 8. Variation of Cd with Froude number.
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Fig. 11. Variation of coefficient of discharge for circular and square orifices with L/B or D/B.
considered more efficient and vice versa. Hussain et al. (2010) proposed the following relationship for coefficient of discharge Cd for a circular side orifice: Cd = 0.670 − 0.076Fr − 0.136
Fig. 9. Comparison of the computed discharge through orifice using Eqs. (8) and (12) with the observed ones treating orifice as large.
D B
(15)
Here D is the diameter of the orifice. Variations of Cd for square and circular orifices with L/B or D/B for different Froude numbers are shown in Fig. 11. For L/B or D/B < 0.2, Cd for square orifice is more than the circular orifice while for L/B or D/B > 0.2, Cd of circular orifice is more than the square orifice. Further, Cd is more for low Froude number for a given L/B or D/B. Due to the boundary effect of the opposite wall of the channel, Cd decreases with increase in L/B or D/B for a given Froude number. Normally, pressure in the water jet at vena contracta is higher than the atmospheric pressure (Ahmad, 2001). The inward force due to pressure at vena-contracta is likely to be more for the square orifice than for the circular orifice due to symmetry of the circular jet, which results in higher Cd for the circular orifice. 7. Conclusions
Fig. 10. Comparison of the computed discharge through orifice using Eqs. (9) and (13) with the observed ones treating orifice as small.
It is concluded from the present study on discharge characteristics of sharp-crested square side orifices in open channels that the coefficient of discharge depends mainly on the approach flow Froude number and ratio of width of orifice and bed width of the channel. The computed discharges using the proposed relationships, by treating the orifice as large and small, were within ±5% of the observed ones. The average percentage error in computation of discharge through the orifice treating it as large and small are 2.52% and 2.53%, respectively, which are practically the same. Therefore, discharge through the orifice can be computed treating it as small orifice within range of Ho /L = 0.84–9.88. Measurement of three-dimensional velocities and visualisation of streamlines in a horizontal plane at the centreline of the orifice indicates that for flow of low Froude number, almost all the streamlines divert towards the orifice while in the case of high Froude number, only those streamlines which are close to the side orifice are diverted towards the orifice. The opposite side of the boundary has significant effect on the diverted discharge at low Froude number compared to flow at high Froude number. Circular orifice diverts more discharge through it compared to the square orifice having the same opening area. Therefore, it is recommended that the cir-
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