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Hydraulic characteristics of a new weir entitled of quarter-circular crested weir
Flow Measurement and Instrumentation 33 (2013) 168–178
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Hydraulic characteristics of a new weir entitled of quarter-circular crested weir J. Mohammadzadeh-Habili n, M. Heidarpour, H. Afzalimehr Department of Water Engineering, College of Agriculture, Isfahan University of Technology, Isfahan 84156-83111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2012 Received in revised form 2 June 2013 Accepted 15 July 2013 Available online 20 July 2013
Weirs are used for flow measurement, flood control in reservoirs and water level control in irrigation systems. In this study, a new weir entitled of quarter-circular crested weir is investigated. This weir is geometrically consisted of a quarter-circular crest of radius R, upstream slope α and vertical downstream face. The downstream face of the weir must be ventilated. Discharge coefficient, crest section velocity and pressure profiles, pressure distribution on the crest surface and upper and lower nappe profiles of flow over the quarter-circular crested weir were experimentally investigated. Results indicated that discharge coefficient of the weir is a constant value and equals to 1.261. In the range of H/Ro 1.5, it is more than the discharge coefficient of circular crested weir. The lower nappe profile of free jet over the weir can also be considered as the ogee shape of the proposed weir. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Quarter-circular crested weir Discharge coefficient Open channel Circular crested weir Ogee weir
1. Introduction Weirs are of the most important parts of hydraulic structures and are used for flow measurement, flood control in reservoirs and water level control in irrigation systems. Discharge coefficient is the most important hydraulic parameter of the weirs. A weir with the large coefficient of discharge is usually preferred for practical purposes. The most common types of weirs are the sharp crested weir, the circular crested weir and the ogee weir. The sharp crested weir is the simplest form of the weirs and serves as a simple and accurate device for flow measurement in open channels. It can also enable one to control and regulate the open channel flows. The hydraulic characteristics of sharp crested weir have been investigated by Kandaswamy and Rouse [1], Rajaratnam and Muralidhar [2], Ramamurthy et al. [3], Afzalimehr and Bagheri [4], Bagheri and Heidarpour [5], Ferrari [6] and Lv et al. [7]. The lower nappe profile of free jet over the sharp crested weir is considered as the shape of ogee weir profile [8]. The circular crested weir is geometrically consisted of a circular crest of radius R, upstream slope α and downstream slope β (Fig. 1). Advantages of circular crested weir include the stable overflow pattern, the ease to pass floating debris, the simplicity of design compared to ogee crest design and the associated lower costs [9].
n
Corresponding author. Tel.: +98 916 363 1625. E-mail addresses: [email protected] (J. Mohammadzadeh-Habili), [email protected] (M. Heidarpour), [email protected] (H. Afzalimehr). 0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.07.002
For steady two-dimensional flow past a circular crested weir, the general discharge equation can be expressed as [10], Q¼
2 C LH 3 d
rffiffiffiffiffiffiffiffiffiffi 2 gH 3
ð1Þ
where Cd is the discharge coefficient, Q is the discharge, L is the weir width, H is the total head above the weir crest and g is the acceleration due to gravity. Discharge coefficient of circular crested weir was experimentally obtained as a function of H/R [10–15]. The hydraulic characteristics of this weir have been investigated by Cassidy [16], Ramamurthy et al. [11], Ramamurthy and Vo [12,13], Heidarpour and Chamani [14], Heidarpour et al. [15], Castro-Orgaz et al. [17], Tadayon and Ramamurthy [18], Bagheri and Heidarpour [19] and Schmocker et al. [20]. Based on experimental results of Bos [10] and Ramamurthy et al. [11], the crest pressure of circular crested weir is reduced with increasing H/R. For 1.35 oH/R, the negative pressure occurs on the crest surface [11]. Negative pressure increases the discharge coefficient but it causes cavitation on the weir surface and therefore weir damage. For practical purposes, the occurring of negative pressure in the lower ranges of H/R limits the application range of circular crested weir. In this study, to increase the discharge coefficient and to delay the occurrence of negative pressure to the higher ranges of H/R, the geometric form of circular-crested weir is changed and a new weir entitled of quarter-circular crested weir is introduced. To determine the hydraulic characteristics of the weir, experiments
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Notations Cd d dd g h H Hd k1 k2 L p q Q R u ucr us U V
discharge coefficient; crest flow depth (m); design crest flow depth (m); acceleration due to gravity (m/s2); water head over the weir (m); total upstream head (m); design total upstream head (m); a constant coefficient; a constant coefficient; weir crest width (m); weir height (m); and pressure (kg/m s2); unit discharge (m2/s); discharge (m3/s); radius of the weir crest (m); horizontal component of the velocity vector (m/s); horizontal component of the crest streamline velocity (m/s); horizontal component of the surface streamline velocity (m/s); average velocity of the approach flow (m/s); magnitude of the velocity vector (m/s);
Vcr Vs y yp α β γ ρ s μ θ
169
crest streamline velocity (m/s); surface streamline velocity (m/s); vertical distance from the weir crest (m); pool water depth (m); upstream slope (1); downstream slope (1); specific weight of water (Kg/m2s2); water density (Kg/m3); water surface tension (kg/s2); water viscosity (kg/m s); clockwise angle relative to the horizontal line passed from the weir crest center (1); inclination angle of streamline relative to the horizontal (1); depth of the stilling basin (m).
φ Δz
Subscripts cr d p s
crest; discharge; and design; pool; surface.
over the quarter-circular crested weir is aerated and pressure on this surface is atmospheric. While, the lower surface of flow over the circular crested weir is matched on the weir surface and pressure on this surface is not atmospheric. Fig. 2 shows the hydraulic parameters and the geometric form of quarter-circular crested weir. Referring to Fig. 2, a functional relationship linking the main variables of flow over the quarter-circular crested weir can be expressed as, ΦðQ ; H; R; p; L; α; g; μ; s; ρÞ ¼ 0
Fig. 1. Flow past a circular crested weir.
were conducted on six suppressed models of quarter-circular crested weir. 2. Material and methods 2.1. Geometry and theory of the weir To increase the discharge coefficient and to delay the occurrence of negative pressure to the higher ranges of H/R, the geometric form of circular crested weir is changed and a new weir is introduced. This new weir is geometrically consisted of a quarter-circular crest of radius R, upstream slope α and vertical downstream face. Due to the quarter-circular shape of the weir crest, this new weir is named quarter-circular crested weir. The same as the sharp-crested weir, the air cavity of quarter-circular crested weir must be ventilated to circulate the air under the lower nappe. If the nappe is not adequately ventilated, the negative pressure occurs under the nappe and drags the lower surface of the nappe towards the downstream face of the weir. This results the weir vibration, cavitation on the weir downstream face and finally weir damage. Geometry of quartercircular crested weir is the same as upper half (above the weir crest section) of circular crested weir. For 0ox, the lower surface of flow
ð2Þ
in which Φ is a functional symbol, p is the weir height, μ is the water viscosity, s is the water surface tension and ρ is the water density. Eq. (2) represents a physical phenomenon. Based on the Buckingham Π theorem, this equation can be expressed in a dimensionless form as, Π 1 ¼ φðΠ 2 ; Π 3 ; Π 4 ; Π 5 ; Π 6 ; Π 7 Þ
ð3Þ
in which Π1, Π2, Π3, Π4, Π5, Π6 and Π7 are the dimensionless groups and φ is a functional symbol. Considering H, Q and ρ as dimensional independent variables, the dimensionless groups were obtained as, Π1 ¼
gH 5 Q
2
; Π2 ¼
R p L μH sH 3 ; Π 3 ¼ ; Π 4 ¼ ; Π 5 ¼ α; Π 6 ¼ and Π 7 ¼ : H H H ρQ ρQ 2
ð4Þ
Taking into account that some groups should be combined to deduce the dimensionless variables commonly used in hydraulics, Eq. (3) is expressed in the following form, pffiffiffi 3 3 1 1 1 Π7 : ð5Þ ¼ f ; ; Π ; ; 5 Π2 Π3 Π6 Π1 2Π 4 ð2Π 1 Þ1=2 Replacing Π1 and Π2, Π3, Π4, Π5, Π6 and Π7 from Eq. (4) into Eq. (5) leads, ! Q H H ρQ s qffiffiffiffiffiffiffiffiffi ¼ f ; ; α; ; : R p μH ρgH 2 2 2 3 LH 3 gH
ð6Þ
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Fig. 2. The hydraulic parameters and the geometric form of the quarter-circular crested weir.
The left-hand side of Eq. (6) shows the weir discharge coefficient. The fourth and the fifth terms on the right-hand side represent the Reynolds number R and the Weber number W, respectively. The effects of the Reynolds number and the Weber number can be negligible except for very low values of the measured water head h over the weir [21,22]. If the effects of the Reynolds number and the Weber number can be considered negligible, the discharge coefficient equation of flow over the quarter-circular crested weir is, H H Cd ¼ F ; ; α : ð7Þ R p Further than discharge coefficient, the crest velocity Vcr and the surface streamline velocity Vs can be related to the total upstream head H. Based on the Bernoulli equation, Vcr is the maximum velocity at the weir crest section and equals, H¼
pffiffiffiffiffiffiffiffiffi V 2cr ⇒V cr ¼ 2gH 2g
ð8Þ
also, Vs is the minimum velocity at the weir crest section and equals, H¼dþ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2s ⇒V s ¼ 2gðHdÞ: 2g
ð9Þ
The horizontal component of the velocity vector at the weir crest section is, u ¼ V cos φ:
ð10Þ
The inclination angle of streamlines φ increases from 01 at y¼ 0 to a maximum value of φs at y¼d. 3. Experiments Experiments were conducted in a smooth rectangular horizontal flume 0.20 m wide, 0.60 m high and 4.00 m long. The flume sidewalls and its floor were made of transparent Plexiglas. Before the test section, sufficient stilling measures were provided to obtain the smooth flow. Fig. 3 shows a simple sketch of the experimental setup. Six suppressed models of quarter-circular crested weir (models A–F) were tested. The models were made of transparent acrylic plate of 2.5 mm thickness. To determine the scale effect on the obtained results, two models of suppressed circular crested weir (models G and H) were also tested. The effects of the Reynolds number and the Weber number on discharge coefficient were
Fig. 3. Simple sketch of the experimental setup.
eliminated to a large extent by keeping 4.65 cm oh. Table 1 presents a summary of the test conditions and the model characteristics. The water surface profile and the lower nappe profile were measured with a point gage to 71.0 mm reading accuracy. The discharge was measured using an electromagnetic flow meter with an accuracy of 70.5%. Velocity profile at the weir crest section was measured with a Prandtl–Pitot tube. To determine the pressure profile at the weir crest section, manometers were spaced at 20 mm centers aside of the flume wall along the weir crest section and were then connected to a manometer board aside of the flume wall. Also, to determine the longitude pressure distribution on the weir crest surface, manometers were spaced along the flow direction on the center line of the weir crest surface and were then connected to the manometers board. Present method was also used by Naghavi et al. [23] to measure the pressure distribution on the circular weir surface. The flow cavity was well aerated. Measurements of velocity profiles on 1.00 m and 2.00 m upstream of the weir crest section were conducted with an ADV to make sure that the approach flow was fully developed. Results indicated that for the each test, the measured velocity profiles were fitted together and were shown that the approach flow was fully developed.
4. Results and discussion The crest flow depth d is shown against the total upstream head H in Fig. 4. A linear equation is fitted to all the data and a very good agreement is observed (R2 ¼ 0.993), d ¼ 0:721H:
ð11Þ
Based Eq. (1), if measured discharge data is shown against pon ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2=3ÞLH ð2=3ÞgH , the slope of the fitted line will then equal to Cd.
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Table 1 A summary of the test conditions and the model characteristics Models
R (cm)
p (cm)
α (1)
Δz (cm)
q (m3/s.m)
H/R
Remark
A B C D E F G H
5.10 12.50 5.20 7.60 7.30 5.10 7.50 7.00
11.20 12.50 20.40 23.10 29.50 30.60 15.00 21.60
63 – 90 90 59 90 90 90
0.00 8.50 8.50 8.50 0.00 0.00 8.50 0.00
0.0215–0.0732 0.0453–0.1897 0.0361–0.1860 0.0430–0.1667 0.0510–0.1730 0.0397–0.1141 0.0126–0.2249 0.0198–0.1421
0.930–2.043 0.621–1.591 1.268–3.746 0.987–2.391 1.143–2.551 1.374–2.753 0.483–2.779 0.676–2.226
– Quarter-circular model – – – – Circular crested weir (β ¼ 451) Circular crested weir (β ¼ 751)
Fig. 4. The crest flow depth d against the total upstream head H.
Fig. 5. The measured discharge against ð2=3ÞLH
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=3ÞgH .
For all the conducted tests on the quarter-circular crested weir pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi models, Q is shown against ð2=3ÞLH ð2=3ÞgH in Fig. 5. Analyzing pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the results indicated that Q is closely related to ð2=3ÞLH ð2=3ÞgH 2 by a linear equation (R ¼0.999), rffiffiffiffiffiffiffiffiffiffi! 2 2 LH gH : ð12Þ Q ¼ 1:261 3 3 Comparison of Eq. (12) with Eq. (1) results that discharge coefficient of quarter circular crested weir is a constant value and equals to 1.261. Estimated discharge coefficient (Cd ¼1.261) of quarter-circular crested weir is shown against H/R in Fig. 6 and compared with the discharge coefficient data of circular crested weir from present study (models G and H), Bos [10] and Ramamurthy and Vo [13].
Fig. 6. Comparison of the discharge coefficient of quarter-circular crested weir with circular crested weir (Cd versus H/R).
Fig. 7. Comparison of the discharge coefficient of quarter-circular crested weir with sharp crested weir (Cd versus H/p).
As seen from Fig. 6, discharge coefficient of circular crested weir is increased with increasing H/R. In the range of H/Ro 1.5, discharge coefficient of quarter-circular crested weir is larger. Reduction of the crest pressure with increasing H/R [10,11] and suction the lower nappe profile by the weir downstream face increase the discharge coefficient of circular crested weir. The pressure on the lower nappe surface of quarter-circular crested weir is atmospheric and it causes constant discharge coefficient of this weir. For 1.5oH/R, the crest pressure of circular crested weir is negative and it causes larger discharge coefficient of this weir. Dimensions (R, α and p) of circular crested weir models (models G and H) are in the range of quarter-circular crested weir models dimensions. The measured discharge coefficient data of these two models are fitted on the discharge coefficient data of circular
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Fig. 8. Unit discharge of quarter-circular crested weir against the crest flow depth.
crested weir from literature. It shows that obtained results are not affected by scale effect. Discharge coefficient data of quarter-circular crested weir models is presented against H/p in Fig. 7 and compared with the discharge coefficient of sharp-crested weir from Afzalimehr and Bagheri [4] and Johnson [24]. It follows that the discharge coefficient of proposed weir is significantly larger than the discharge coefficient of sharp crested weir. It can also be resulted from Figs. 6 and 7 that discharge coefficient of quarter-circular crested weir is a constant value (Cd ¼ 1.261) based on H/R or H/p. Also, it is not affected by the weir upstream slope. This result shows that the discharge coefficient of quarter-circular crested weir can be expressed as a function of H/R or H/p. Replacing H from Eq. (11) into Eq. (12) and dividing both sides of Eq. (12) by L, the unit discharge q over the quarter-circular
Fig. 9. The normalized water surface profile data (y/H versus x/H) over the quarter-circular crested weir.
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crested weir is related to the crest flow depth d as, 3=2
q ¼ 3:512d
Fig. 10. Estimated values of cos φs against H/R.
:
ð13Þ
The measured unit discharge data of quarter-circular crested weir models is shown against the crest flow depth in Fig. 8. The curve of Eq. (13) is also added and a very good agreement is observed. It shows that the discharge over the proposed weir can directly be measured from the crest flow depth with good precision. Using this interesting characteristic, the quarter-circular crested weir can serve as an accurate device for flow measurement in open channels. Using the boundary condition of ucr ¼ Vcr cos 01¼√2gH, a general equation for crest velocity profile of quarter-circular
Fig. 11. The normalized velocity distribution [(y/d) versus (u/√2gH)] at the weir crest section.
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Combining Eqs. (12) and (15) and replacing d/H from Eq. (11) yields,
crested weir can be expressed as, pffiffiffiffiffiffiffiffiffi yk2 u ¼ 2gH 1 þ k1 d
ð14Þ
where k1 and k2 are constant coefficients. A relationship between k1 and k2 can be derived by integration of Eq. (14) as, Z Q ¼L
d 0
u dy ¼
pffiffiffiffiffiffiffiffiffi i Ld 2gH h ð1 þ k1 Þk2 þ1 1 : k1 ðk2 þ 1Þ
ð15Þ
1:261
2 LH 3
rffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffi i 2 Ld 2gH h ð1 þ k1 Þk2 þ1 1 ð1 þ k1 Þk2 þ1 1 ⇒0:673 ¼ : gH ¼ k1 ðk2 þ 1Þ 3 k1 ðk2 þ 1Þ
ð16Þ
Another relationship between k1 and k2 can be derived from the surface streamline velocity. Using Eq. (10), the horizontal component of the surface streamline velocity is: us ¼ V s cos φs :
Fig. 12. The normalized pressure distribution [(y/d) versus (p/γd)] at the weir crest section.
ð17Þ
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Replacing Vs and us from Eqs. (9) and (14) into Eq. (17) and considering d/H ¼0.721 gives, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi lnð0:528 cos φs Þ : 2gH ð1 þ k1 Þk2 ¼ 2gðHdÞ cos φs ⇒k2 ¼ lnð1 þ k1 Þ
ð18Þ
To estimate k1 and k2 from Eqs. (16) and (18), φs should first be estimated. φs can be estimated from the fitted equation to the water surface profile data. The normalized water surface profile
175
data (y/H versus x/H) over the quarter-circular crested weir models is shown in Fig. 9. For each test, a polynomial equation is fitted to the water surface profile data and φs is then estimated as, dy dy ⇒φs ¼ tan 1 : ð19Þ tan φs ¼ dx x ¼ 0 dx x ¼ 0 In Fig. 9, a single approximate line is also fitted to all the data of the each model. Estimated values of cos φs are shown against H/R in
Fig. 13. The normalized longitude pressure distribution [p/[γ(H+R)] versus θ/901] on the center line of the crest surface of quarter-circular crested weir.
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Fig. 10. The line of cos φs ¼0.941 is closely fitted to all the data and a good agreement is observed. Using cos φs ¼0.941, φs is equal to cos 1 0.941¼19.781. Replacing cos φs ¼ 0.941 into Eq. (18) gives, k2 ¼
0:699 : lnð1 þ k1 Þ
ð20Þ
Combining Eqs. (16) and (20) yields, k1 ¼ 2:184 and k2 ¼ 0:604:
ð21Þ
Replacing the estimated values of k1 and k2 into Eq. (14), the normalized crest velocity equation of quarter-circular crested weir is, u y0:604 pffiffiffiffiffiffiffiffiffi ¼ 1 þ 2:184 : ð22Þ d 2gH The normalized crest velocity data [(y/d) versus (u/√2gH)] of quarter-circular crested weir models is presented in Fig. 11. The curve of Eq. (22) is also added and a very good agreement is observed. Using the Bernoulli equation, the crest pressure profile can be expressed as, p V2 u2 ¼ Hy ¼ Hy γ 2g 2g cos 2 φ
ð23Þ
φ increases from 01 at y¼0 to 19.781 at y¼ d. Assuming the linear variation of φ with y yields, φ¼
y y φ ¼ ð19:78o Þ: d s d
ð24Þ
Replacing H, u and φ from Eqs. (11), (22) and (24) into Eq. (23) and simplifying, the dimensionless crest pressure profile is, p y 1:387 y1:208 y ¼ 1:387 : ð25Þ o 1 þ 2:184 2 γd d cos d ð19:78 Þ d The normalized crest pressure data [(y/d) versus (p/γd)] of quartercircular crested weir models is shown in Fig. 12. The curve of Eq. (25) and the hydrostatic distribution case are plotted for comparison. As seen from Fig. 12, the dimensionless pressure data has a good agreement with the curve of Eq. (25). Also, the crest pressure distribution of quarter-circular crested weir is not hydrostatic. Both the applying force from the water pressure and the occurring range of negative pressure on the weir crest surface can be estimated from the longitude pressure distribution on the weir crest surface. The normalized longitude pressure distribution data [p/[γ(H+R)] versus θ/901] on the center line of the crest surface of quarter-circular crested weir models is presented in Fig. 13. where θ is the clockwise angle relative to the horizontal line passed from the weir crest center. It can be seen from Fig. 13 that the maximum pressure on the weir crest surface occurs at θ/901¼ 0 and nearly equals to γ(H+R). Due to the lower nappe aeration, the pressure at θ/901¼ 1 equals to zero. With increasing H/R, the normalized pressure on the weir crest surface is reduced and pressure profiles are closed to the θ/901 axe. Some of the normalized pressure profiles of models C, D, E, and F are also intersected with the θ/901 axe. The part of the normalized pressure profile placed under the θ/901 axe shows the negative pressure region or cavitation region on the weir crest surface. To avoid the cavitation, the pressure profile on the crest surface should be placed upper the θ/901 axe. Depends on the weir geometric, the negative pressure on the crest surface of quartercircular crested weir models occurs in the range of 2.13 oH/ Ro2.74. while, the negative pressure on the crest surface of circular crested weir occurs in the range of 1.35 oH/R [11].
It shows that the quarter-circular crested weir can be used for the higher ranges of H/R. Determining the pool water depth yp is important from two aspects. First, the water pool applied a force to the weir body that must be estimated from the pool water depth. Second, the downstream face of the weir becomes submerged for (p+Δz)oyp. To avoid the weir submergence, yp should be less than the p+Δz. The normalized pool water depth yp/(p+Δz) is shown against the normalized total upstream head H/(p+Δz) in Fig. 14. A polynomial equation is fitted to all the data and a reasonable agreement is observed (R2 ¼ 0.821), 2 yp H H : ð26Þ ¼ 0:472 þ 1:343 p þ Δz p þ Δz p þ Δz The same as the sharp crested weir, the lower nappe profile of free jet over the quarter-circular crested weir can be considered as the ogee shape of proposed weir. To this end, the equation of the lower nappe profile of free jet over the quarter-circular crested weir is taken as the shape of the downstream surface profile of the ogee shape of quartercircular crested weir. The upstream surface profile of quarter-circular crested weir is also taken as the upstream surface profile of the ogee shape of proposed weir. Ogee shape of quarter-circular crested weir is a suggestion and it must be tested in the future researches. To derive the equation of the lower nappe profile, the normalized lower nappe profile data (y/H versus x/H) of free jet over the quarter-circular crested weir models is presented in Fig. 15. After analyzing the results, a power equation is fitted to all the data and a very good agreement is observed. The equation of the fitted line is, x 1:696 y ¼ 0:629 : H H
ð27Þ
Ogee shape of quarter-circular crested weir is schematically plotted in Fig. 16. For H¼Hd, the pressure on the downstream face of the weir is atmospheric and discharge coefficient equals to 1.261. If the weir is operating under a head lower than its design head (HoHd), the positive pressure occurs throughout the weir surface and discharge coefficient is reduced in comparison with the quarter-circular crested weir. A greater head (H4Hd) will cause negative pressure on the weir surface and increase the discharge coefficient.
5. Conclusions In present study, a new weir entitled of quarter-circular crested weir is experimentally investigated. The geometric form of this
Fig. 14. The normalized pool water depth yp/(p+Δz) against the normalized total upstream head H/(p+Δz).
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Fig. 15. The normalized lower nappe profile data (y/H versus x/H).
weir is complex in comparison with the sharp crested weir. But, it is simple in comparison with the circular crested weir. Results indicated that discharge coefficient of quarter-circular crested weir is a constant value and equals to 1.261. In the range of H/Ro 1.5, it is more than the discharge coefficient of circular crested weir. Discharge over the quarter-circular crested weir can easily be measured from the crest flow depth with the good precision. Using this interesting characteristic, the quarter-circular crested weir can serve as an accurate device for flow measurement in
open channels. The lower nappe profile of free jet over the quarter-circular crested weir can also be considered as the ogee shape of this weir. In the design procedure of quarter-circular crested weir, two parameters (R and α) should be estimated. While, for circular crested weir, three parameters (R, α and β) should be estimated. For the same weir height, the surface area of quarter-circular crested weir is nearly equal to the half of the surface area of circular crested weir. Also, in the range of H/R o1.5, discharge coefficient of quarter-circular crested weir is more.
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Fig. 16. Ogee shape of quarter-circular crested weir.
Based on Eq. (1), for the same discharge capacity Q and the same total upstream head H, the larger coefficient of discharge causes reduction in the weir crest width L. Reduction in both of the weir surface area and the weir width reduced the required materials for the weir constructing and finally reduced the weir costs. References [1] Kandaswamy PK, Rouse H. Characteristics of flow over terminal weirs and sills. J Hydraul Div 1957;83(4):1–13. [2] Rajaratnam N, Muralidhar D. Pressure and velocity distribution for sharpcrested weirs. J Hydraul Res 1971;9(2):241–8. [3] Ramamurthy AS, Tim US, Rao MVJ. Flow over sharp-crested plate weirs. J Irrig Drain Eng 1987;113(2):163–72. [4] Afzalimehr H, Bagheri S. Discharge coefficient of sharp-crested weirs using potential flow. J. Hydraul Res 2009;47(6):820–3. [5] Bagheri S, Heidarpour M. Flow over rectangular sharp-crested weir. Irrig Sci 2009;28(2):173–9.
[6] Ferrari A. SPH simulation of free surface flow over a sharp-crested weir. Adv Water Resour 2010;33:270–6. [7] Lv X, Zou Q, Reeve D. Numerical simulation of overflow at vertical weirs using a hybrid level set/VOF method. Adv Water Resour 2011;34:1320–34. [8] U.S. Bureau of Reclamation. Design of small dams. 3rd ed. Technical Service Center, Denver, USA; 1987. [9] Chanson H, Montes JS. Over flow characteristics of circular weirs: effects of inflow conditions. J Irrig Drain Eng 1998;124(EM3):152–62. [10] Bos MG. Discharge Measurement Structures. Wageningen, the Netherlands: Institute of Land Reclamation and Improvement ILRI; 1978(Publication 20). [11] Ramamurthy AS, Vo ND, Vera G. Momentum model of flow past weir. J Irrig Drain Eng 1992;118(6):988–94. [12] Ramamurthy AS, Vo ND. Application of Dressler theory to weir flow. J Appl Mech 1993;60:163–6. [13] Ramamurthy AS, Vo ND. Characteristics of circular-crested weir. J Hydraul Eng 1993;119(9):1055–62. [14] Heidarpour M, Chamani MR. Velocity distribution over cylindrical weirs. J Hydraul Res 2006;44(5):708–11. [15] Heidarpour M, Mohammadzadeh Habili J, Haghiabi AH. Application of potential flow to circular-crested weir. J Hydraul Res 2008;46(5):699–702. [16] Cassidy JJ. Irrotational flow over spillways of finite height. J Eng Mech Div 1965;91:155–73. [17] Castro-Orgaz O, Giráldez JV, Ayuso JL. Critical flow over circular crested weirs. J Hydraul Eng 2008;134(11):1661–4. [18] Tadayon R, Ramamurthy AS. Turbulence modeling of flows over circular spillways. J Irrig Drain Eng 2009;135(4):493–8. [19] Bagheri S, Heidarpour M. Overflow characteristics of circular-crested weirs. J Hydraul Res 2010;48(4):515–20. [20] Schmocker S, Halldórsdóttir BR, Hager WH. Effect of weir face angles on circular-crested weir flow. J Hydraul Eng 2011;137(6):637–43. [21] Rao SS, Shukla MK. Characteristics of flow over weirs of finite crest width. J Hydraul Div 1971;97(11):1807–16. [22] Ranga Raju KG, Asawa GL. Viscosity and surface tension effects on weir flow. J Hydraul Div 1977;103(10):1227–31. [23] Naghavi N, Esmaili K, Yazdi J, Koroosh Vahid F. An experimental and numerical study on hydraulic characteristics and theoretical equations of circular weirs. Can J Civ Eng 2011;38(6):1327–34. [24] Johnson MC. Discharge coefficient analysis for flat-topped and sharp-crested weirs. J Irrig Sci 2000;19(3):133–7.
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Hydraulic characteristics of a new weir entitled of quarter-circular crested weir J. Mohammadzadeh-Habili n, M. Heidarpour, H. Afzalimehr Department of Water Engineering, College of Agriculture, Isfahan University of Technology, Isfahan 84156-83111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2012 Received in revised form 2 June 2013 Accepted 15 July 2013 Available online 20 July 2013
Weirs are used for flow measurement, flood control in reservoirs and water level control in irrigation systems. In this study, a new weir entitled of quarter-circular crested weir is investigated. This weir is geometrically consisted of a quarter-circular crest of radius R, upstream slope α and vertical downstream face. The downstream face of the weir must be ventilated. Discharge coefficient, crest section velocity and pressure profiles, pressure distribution on the crest surface and upper and lower nappe profiles of flow over the quarter-circular crested weir were experimentally investigated. Results indicated that discharge coefficient of the weir is a constant value and equals to 1.261. In the range of H/Ro 1.5, it is more than the discharge coefficient of circular crested weir. The lower nappe profile of free jet over the weir can also be considered as the ogee shape of the proposed weir. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Quarter-circular crested weir Discharge coefficient Open channel Circular crested weir Ogee weir
1. Introduction Weirs are of the most important parts of hydraulic structures and are used for flow measurement, flood control in reservoirs and water level control in irrigation systems. Discharge coefficient is the most important hydraulic parameter of the weirs. A weir with the large coefficient of discharge is usually preferred for practical purposes. The most common types of weirs are the sharp crested weir, the circular crested weir and the ogee weir. The sharp crested weir is the simplest form of the weirs and serves as a simple and accurate device for flow measurement in open channels. It can also enable one to control and regulate the open channel flows. The hydraulic characteristics of sharp crested weir have been investigated by Kandaswamy and Rouse [1], Rajaratnam and Muralidhar [2], Ramamurthy et al. [3], Afzalimehr and Bagheri [4], Bagheri and Heidarpour [5], Ferrari [6] and Lv et al. [7]. The lower nappe profile of free jet over the sharp crested weir is considered as the shape of ogee weir profile [8]. The circular crested weir is geometrically consisted of a circular crest of radius R, upstream slope α and downstream slope β (Fig. 1). Advantages of circular crested weir include the stable overflow pattern, the ease to pass floating debris, the simplicity of design compared to ogee crest design and the associated lower costs [9].
n
Corresponding author. Tel.: +98 916 363 1625. E-mail addresses: [email protected] (J. Mohammadzadeh-Habili), [email protected] (M. Heidarpour), [email protected] (H. Afzalimehr). 0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.07.002
For steady two-dimensional flow past a circular crested weir, the general discharge equation can be expressed as [10], Q¼
2 C LH 3 d
rffiffiffiffiffiffiffiffiffiffi 2 gH 3
ð1Þ
where Cd is the discharge coefficient, Q is the discharge, L is the weir width, H is the total head above the weir crest and g is the acceleration due to gravity. Discharge coefficient of circular crested weir was experimentally obtained as a function of H/R [10–15]. The hydraulic characteristics of this weir have been investigated by Cassidy [16], Ramamurthy et al. [11], Ramamurthy and Vo [12,13], Heidarpour and Chamani [14], Heidarpour et al. [15], Castro-Orgaz et al. [17], Tadayon and Ramamurthy [18], Bagheri and Heidarpour [19] and Schmocker et al. [20]. Based on experimental results of Bos [10] and Ramamurthy et al. [11], the crest pressure of circular crested weir is reduced with increasing H/R. For 1.35 oH/R, the negative pressure occurs on the crest surface [11]. Negative pressure increases the discharge coefficient but it causes cavitation on the weir surface and therefore weir damage. For practical purposes, the occurring of negative pressure in the lower ranges of H/R limits the application range of circular crested weir. In this study, to increase the discharge coefficient and to delay the occurrence of negative pressure to the higher ranges of H/R, the geometric form of circular-crested weir is changed and a new weir entitled of quarter-circular crested weir is introduced. To determine the hydraulic characteristics of the weir, experiments
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Notations Cd d dd g h H Hd k1 k2 L p q Q R u ucr us U V
discharge coefficient; crest flow depth (m); design crest flow depth (m); acceleration due to gravity (m/s2); water head over the weir (m); total upstream head (m); design total upstream head (m); a constant coefficient; a constant coefficient; weir crest width (m); weir height (m); and pressure (kg/m s2); unit discharge (m2/s); discharge (m3/s); radius of the weir crest (m); horizontal component of the velocity vector (m/s); horizontal component of the crest streamline velocity (m/s); horizontal component of the surface streamline velocity (m/s); average velocity of the approach flow (m/s); magnitude of the velocity vector (m/s);
Vcr Vs y yp α β γ ρ s μ θ
169
crest streamline velocity (m/s); surface streamline velocity (m/s); vertical distance from the weir crest (m); pool water depth (m); upstream slope (1); downstream slope (1); specific weight of water (Kg/m2s2); water density (Kg/m3); water surface tension (kg/s2); water viscosity (kg/m s); clockwise angle relative to the horizontal line passed from the weir crest center (1); inclination angle of streamline relative to the horizontal (1); depth of the stilling basin (m).
φ Δz
Subscripts cr d p s
crest; discharge; and design; pool; surface.
over the quarter-circular crested weir is aerated and pressure on this surface is atmospheric. While, the lower surface of flow over the circular crested weir is matched on the weir surface and pressure on this surface is not atmospheric. Fig. 2 shows the hydraulic parameters and the geometric form of quarter-circular crested weir. Referring to Fig. 2, a functional relationship linking the main variables of flow over the quarter-circular crested weir can be expressed as, ΦðQ ; H; R; p; L; α; g; μ; s; ρÞ ¼ 0
Fig. 1. Flow past a circular crested weir.
were conducted on six suppressed models of quarter-circular crested weir. 2. Material and methods 2.1. Geometry and theory of the weir To increase the discharge coefficient and to delay the occurrence of negative pressure to the higher ranges of H/R, the geometric form of circular crested weir is changed and a new weir is introduced. This new weir is geometrically consisted of a quarter-circular crest of radius R, upstream slope α and vertical downstream face. Due to the quarter-circular shape of the weir crest, this new weir is named quarter-circular crested weir. The same as the sharp-crested weir, the air cavity of quarter-circular crested weir must be ventilated to circulate the air under the lower nappe. If the nappe is not adequately ventilated, the negative pressure occurs under the nappe and drags the lower surface of the nappe towards the downstream face of the weir. This results the weir vibration, cavitation on the weir downstream face and finally weir damage. Geometry of quartercircular crested weir is the same as upper half (above the weir crest section) of circular crested weir. For 0ox, the lower surface of flow
ð2Þ
in which Φ is a functional symbol, p is the weir height, μ is the water viscosity, s is the water surface tension and ρ is the water density. Eq. (2) represents a physical phenomenon. Based on the Buckingham Π theorem, this equation can be expressed in a dimensionless form as, Π 1 ¼ φðΠ 2 ; Π 3 ; Π 4 ; Π 5 ; Π 6 ; Π 7 Þ
ð3Þ
in which Π1, Π2, Π3, Π4, Π5, Π6 and Π7 are the dimensionless groups and φ is a functional symbol. Considering H, Q and ρ as dimensional independent variables, the dimensionless groups were obtained as, Π1 ¼
gH 5 Q
2
; Π2 ¼
R p L μH sH 3 ; Π 3 ¼ ; Π 4 ¼ ; Π 5 ¼ α; Π 6 ¼ and Π 7 ¼ : H H H ρQ ρQ 2
ð4Þ
Taking into account that some groups should be combined to deduce the dimensionless variables commonly used in hydraulics, Eq. (3) is expressed in the following form, pffiffiffi 3 3 1 1 1 Π7 : ð5Þ ¼ f ; ; Π ; ; 5 Π2 Π3 Π6 Π1 2Π 4 ð2Π 1 Þ1=2 Replacing Π1 and Π2, Π3, Π4, Π5, Π6 and Π7 from Eq. (4) into Eq. (5) leads, ! Q H H ρQ s qffiffiffiffiffiffiffiffiffi ¼ f ; ; α; ; : R p μH ρgH 2 2 2 3 LH 3 gH
ð6Þ
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Fig. 2. The hydraulic parameters and the geometric form of the quarter-circular crested weir.
The left-hand side of Eq. (6) shows the weir discharge coefficient. The fourth and the fifth terms on the right-hand side represent the Reynolds number R and the Weber number W, respectively. The effects of the Reynolds number and the Weber number can be negligible except for very low values of the measured water head h over the weir [21,22]. If the effects of the Reynolds number and the Weber number can be considered negligible, the discharge coefficient equation of flow over the quarter-circular crested weir is, H H Cd ¼ F ; ; α : ð7Þ R p Further than discharge coefficient, the crest velocity Vcr and the surface streamline velocity Vs can be related to the total upstream head H. Based on the Bernoulli equation, Vcr is the maximum velocity at the weir crest section and equals, H¼
pffiffiffiffiffiffiffiffiffi V 2cr ⇒V cr ¼ 2gH 2g
ð8Þ
also, Vs is the minimum velocity at the weir crest section and equals, H¼dþ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2s ⇒V s ¼ 2gðHdÞ: 2g
ð9Þ
The horizontal component of the velocity vector at the weir crest section is, u ¼ V cos φ:
ð10Þ
The inclination angle of streamlines φ increases from 01 at y¼ 0 to a maximum value of φs at y¼d. 3. Experiments Experiments were conducted in a smooth rectangular horizontal flume 0.20 m wide, 0.60 m high and 4.00 m long. The flume sidewalls and its floor were made of transparent Plexiglas. Before the test section, sufficient stilling measures were provided to obtain the smooth flow. Fig. 3 shows a simple sketch of the experimental setup. Six suppressed models of quarter-circular crested weir (models A–F) were tested. The models were made of transparent acrylic plate of 2.5 mm thickness. To determine the scale effect on the obtained results, two models of suppressed circular crested weir (models G and H) were also tested. The effects of the Reynolds number and the Weber number on discharge coefficient were
Fig. 3. Simple sketch of the experimental setup.
eliminated to a large extent by keeping 4.65 cm oh. Table 1 presents a summary of the test conditions and the model characteristics. The water surface profile and the lower nappe profile were measured with a point gage to 71.0 mm reading accuracy. The discharge was measured using an electromagnetic flow meter with an accuracy of 70.5%. Velocity profile at the weir crest section was measured with a Prandtl–Pitot tube. To determine the pressure profile at the weir crest section, manometers were spaced at 20 mm centers aside of the flume wall along the weir crest section and were then connected to a manometer board aside of the flume wall. Also, to determine the longitude pressure distribution on the weir crest surface, manometers were spaced along the flow direction on the center line of the weir crest surface and were then connected to the manometers board. Present method was also used by Naghavi et al. [23] to measure the pressure distribution on the circular weir surface. The flow cavity was well aerated. Measurements of velocity profiles on 1.00 m and 2.00 m upstream of the weir crest section were conducted with an ADV to make sure that the approach flow was fully developed. Results indicated that for the each test, the measured velocity profiles were fitted together and were shown that the approach flow was fully developed.
4. Results and discussion The crest flow depth d is shown against the total upstream head H in Fig. 4. A linear equation is fitted to all the data and a very good agreement is observed (R2 ¼ 0.993), d ¼ 0:721H:
ð11Þ
Based Eq. (1), if measured discharge data is shown against pon ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2=3ÞLH ð2=3ÞgH , the slope of the fitted line will then equal to Cd.
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Table 1 A summary of the test conditions and the model characteristics Models
R (cm)
p (cm)
α (1)
Δz (cm)
q (m3/s.m)
H/R
Remark
A B C D E F G H
5.10 12.50 5.20 7.60 7.30 5.10 7.50 7.00
11.20 12.50 20.40 23.10 29.50 30.60 15.00 21.60
63 – 90 90 59 90 90 90
0.00 8.50 8.50 8.50 0.00 0.00 8.50 0.00
0.0215–0.0732 0.0453–0.1897 0.0361–0.1860 0.0430–0.1667 0.0510–0.1730 0.0397–0.1141 0.0126–0.2249 0.0198–0.1421
0.930–2.043 0.621–1.591 1.268–3.746 0.987–2.391 1.143–2.551 1.374–2.753 0.483–2.779 0.676–2.226
– Quarter-circular model – – – – Circular crested weir (β ¼ 451) Circular crested weir (β ¼ 751)
Fig. 4. The crest flow depth d against the total upstream head H.
Fig. 5. The measured discharge against ð2=3ÞLH
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=3ÞgH .
For all the conducted tests on the quarter-circular crested weir pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi models, Q is shown against ð2=3ÞLH ð2=3ÞgH in Fig. 5. Analyzing pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the results indicated that Q is closely related to ð2=3ÞLH ð2=3ÞgH 2 by a linear equation (R ¼0.999), rffiffiffiffiffiffiffiffiffiffi! 2 2 LH gH : ð12Þ Q ¼ 1:261 3 3 Comparison of Eq. (12) with Eq. (1) results that discharge coefficient of quarter circular crested weir is a constant value and equals to 1.261. Estimated discharge coefficient (Cd ¼1.261) of quarter-circular crested weir is shown against H/R in Fig. 6 and compared with the discharge coefficient data of circular crested weir from present study (models G and H), Bos [10] and Ramamurthy and Vo [13].
Fig. 6. Comparison of the discharge coefficient of quarter-circular crested weir with circular crested weir (Cd versus H/R).
Fig. 7. Comparison of the discharge coefficient of quarter-circular crested weir with sharp crested weir (Cd versus H/p).
As seen from Fig. 6, discharge coefficient of circular crested weir is increased with increasing H/R. In the range of H/Ro 1.5, discharge coefficient of quarter-circular crested weir is larger. Reduction of the crest pressure with increasing H/R [10,11] and suction the lower nappe profile by the weir downstream face increase the discharge coefficient of circular crested weir. The pressure on the lower nappe surface of quarter-circular crested weir is atmospheric and it causes constant discharge coefficient of this weir. For 1.5oH/R, the crest pressure of circular crested weir is negative and it causes larger discharge coefficient of this weir. Dimensions (R, α and p) of circular crested weir models (models G and H) are in the range of quarter-circular crested weir models dimensions. The measured discharge coefficient data of these two models are fitted on the discharge coefficient data of circular
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Fig. 8. Unit discharge of quarter-circular crested weir against the crest flow depth.
crested weir from literature. It shows that obtained results are not affected by scale effect. Discharge coefficient data of quarter-circular crested weir models is presented against H/p in Fig. 7 and compared with the discharge coefficient of sharp-crested weir from Afzalimehr and Bagheri [4] and Johnson [24]. It follows that the discharge coefficient of proposed weir is significantly larger than the discharge coefficient of sharp crested weir. It can also be resulted from Figs. 6 and 7 that discharge coefficient of quarter-circular crested weir is a constant value (Cd ¼ 1.261) based on H/R or H/p. Also, it is not affected by the weir upstream slope. This result shows that the discharge coefficient of quarter-circular crested weir can be expressed as a function of H/R or H/p. Replacing H from Eq. (11) into Eq. (12) and dividing both sides of Eq. (12) by L, the unit discharge q over the quarter-circular
Fig. 9. The normalized water surface profile data (y/H versus x/H) over the quarter-circular crested weir.
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173
crested weir is related to the crest flow depth d as, 3=2
q ¼ 3:512d
Fig. 10. Estimated values of cos φs against H/R.
:
ð13Þ
The measured unit discharge data of quarter-circular crested weir models is shown against the crest flow depth in Fig. 8. The curve of Eq. (13) is also added and a very good agreement is observed. It shows that the discharge over the proposed weir can directly be measured from the crest flow depth with good precision. Using this interesting characteristic, the quarter-circular crested weir can serve as an accurate device for flow measurement in open channels. Using the boundary condition of ucr ¼ Vcr cos 01¼√2gH, a general equation for crest velocity profile of quarter-circular
Fig. 11. The normalized velocity distribution [(y/d) versus (u/√2gH)] at the weir crest section.
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Combining Eqs. (12) and (15) and replacing d/H from Eq. (11) yields,
crested weir can be expressed as, pffiffiffiffiffiffiffiffiffi yk2 u ¼ 2gH 1 þ k1 d
ð14Þ
where k1 and k2 are constant coefficients. A relationship between k1 and k2 can be derived by integration of Eq. (14) as, Z Q ¼L
d 0
u dy ¼
pffiffiffiffiffiffiffiffiffi i Ld 2gH h ð1 þ k1 Þk2 þ1 1 : k1 ðk2 þ 1Þ
ð15Þ
1:261
2 LH 3
rffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffi i 2 Ld 2gH h ð1 þ k1 Þk2 þ1 1 ð1 þ k1 Þk2 þ1 1 ⇒0:673 ¼ : gH ¼ k1 ðk2 þ 1Þ 3 k1 ðk2 þ 1Þ
ð16Þ
Another relationship between k1 and k2 can be derived from the surface streamline velocity. Using Eq. (10), the horizontal component of the surface streamline velocity is: us ¼ V s cos φs :
Fig. 12. The normalized pressure distribution [(y/d) versus (p/γd)] at the weir crest section.
ð17Þ
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Replacing Vs and us from Eqs. (9) and (14) into Eq. (17) and considering d/H ¼0.721 gives, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi lnð0:528 cos φs Þ : 2gH ð1 þ k1 Þk2 ¼ 2gðHdÞ cos φs ⇒k2 ¼ lnð1 þ k1 Þ
ð18Þ
To estimate k1 and k2 from Eqs. (16) and (18), φs should first be estimated. φs can be estimated from the fitted equation to the water surface profile data. The normalized water surface profile
175
data (y/H versus x/H) over the quarter-circular crested weir models is shown in Fig. 9. For each test, a polynomial equation is fitted to the water surface profile data and φs is then estimated as, dy dy ⇒φs ¼ tan 1 : ð19Þ tan φs ¼ dx x ¼ 0 dx x ¼ 0 In Fig. 9, a single approximate line is also fitted to all the data of the each model. Estimated values of cos φs are shown against H/R in
Fig. 13. The normalized longitude pressure distribution [p/[γ(H+R)] versus θ/901] on the center line of the crest surface of quarter-circular crested weir.
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Fig. 10. The line of cos φs ¼0.941 is closely fitted to all the data and a good agreement is observed. Using cos φs ¼0.941, φs is equal to cos 1 0.941¼19.781. Replacing cos φs ¼ 0.941 into Eq. (18) gives, k2 ¼
0:699 : lnð1 þ k1 Þ
ð20Þ
Combining Eqs. (16) and (20) yields, k1 ¼ 2:184 and k2 ¼ 0:604:
ð21Þ
Replacing the estimated values of k1 and k2 into Eq. (14), the normalized crest velocity equation of quarter-circular crested weir is, u y0:604 pffiffiffiffiffiffiffiffiffi ¼ 1 þ 2:184 : ð22Þ d 2gH The normalized crest velocity data [(y/d) versus (u/√2gH)] of quarter-circular crested weir models is presented in Fig. 11. The curve of Eq. (22) is also added and a very good agreement is observed. Using the Bernoulli equation, the crest pressure profile can be expressed as, p V2 u2 ¼ Hy ¼ Hy γ 2g 2g cos 2 φ
ð23Þ
φ increases from 01 at y¼0 to 19.781 at y¼ d. Assuming the linear variation of φ with y yields, φ¼
y y φ ¼ ð19:78o Þ: d s d
ð24Þ
Replacing H, u and φ from Eqs. (11), (22) and (24) into Eq. (23) and simplifying, the dimensionless crest pressure profile is, p y 1:387 y1:208 y ¼ 1:387 : ð25Þ o 1 þ 2:184 2 γd d cos d ð19:78 Þ d The normalized crest pressure data [(y/d) versus (p/γd)] of quartercircular crested weir models is shown in Fig. 12. The curve of Eq. (25) and the hydrostatic distribution case are plotted for comparison. As seen from Fig. 12, the dimensionless pressure data has a good agreement with the curve of Eq. (25). Also, the crest pressure distribution of quarter-circular crested weir is not hydrostatic. Both the applying force from the water pressure and the occurring range of negative pressure on the weir crest surface can be estimated from the longitude pressure distribution on the weir crest surface. The normalized longitude pressure distribution data [p/[γ(H+R)] versus θ/901] on the center line of the crest surface of quarter-circular crested weir models is presented in Fig. 13. where θ is the clockwise angle relative to the horizontal line passed from the weir crest center. It can be seen from Fig. 13 that the maximum pressure on the weir crest surface occurs at θ/901¼ 0 and nearly equals to γ(H+R). Due to the lower nappe aeration, the pressure at θ/901¼ 1 equals to zero. With increasing H/R, the normalized pressure on the weir crest surface is reduced and pressure profiles are closed to the θ/901 axe. Some of the normalized pressure profiles of models C, D, E, and F are also intersected with the θ/901 axe. The part of the normalized pressure profile placed under the θ/901 axe shows the negative pressure region or cavitation region on the weir crest surface. To avoid the cavitation, the pressure profile on the crest surface should be placed upper the θ/901 axe. Depends on the weir geometric, the negative pressure on the crest surface of quartercircular crested weir models occurs in the range of 2.13 oH/ Ro2.74. while, the negative pressure on the crest surface of circular crested weir occurs in the range of 1.35 oH/R [11].
It shows that the quarter-circular crested weir can be used for the higher ranges of H/R. Determining the pool water depth yp is important from two aspects. First, the water pool applied a force to the weir body that must be estimated from the pool water depth. Second, the downstream face of the weir becomes submerged for (p+Δz)oyp. To avoid the weir submergence, yp should be less than the p+Δz. The normalized pool water depth yp/(p+Δz) is shown against the normalized total upstream head H/(p+Δz) in Fig. 14. A polynomial equation is fitted to all the data and a reasonable agreement is observed (R2 ¼ 0.821), 2 yp H H : ð26Þ ¼ 0:472 þ 1:343 p þ Δz p þ Δz p þ Δz The same as the sharp crested weir, the lower nappe profile of free jet over the quarter-circular crested weir can be considered as the ogee shape of proposed weir. To this end, the equation of the lower nappe profile of free jet over the quarter-circular crested weir is taken as the shape of the downstream surface profile of the ogee shape of quartercircular crested weir. The upstream surface profile of quarter-circular crested weir is also taken as the upstream surface profile of the ogee shape of proposed weir. Ogee shape of quarter-circular crested weir is a suggestion and it must be tested in the future researches. To derive the equation of the lower nappe profile, the normalized lower nappe profile data (y/H versus x/H) of free jet over the quarter-circular crested weir models is presented in Fig. 15. After analyzing the results, a power equation is fitted to all the data and a very good agreement is observed. The equation of the fitted line is, x 1:696 y ¼ 0:629 : H H
ð27Þ
Ogee shape of quarter-circular crested weir is schematically plotted in Fig. 16. For H¼Hd, the pressure on the downstream face of the weir is atmospheric and discharge coefficient equals to 1.261. If the weir is operating under a head lower than its design head (HoHd), the positive pressure occurs throughout the weir surface and discharge coefficient is reduced in comparison with the quarter-circular crested weir. A greater head (H4Hd) will cause negative pressure on the weir surface and increase the discharge coefficient.
5. Conclusions In present study, a new weir entitled of quarter-circular crested weir is experimentally investigated. The geometric form of this
Fig. 14. The normalized pool water depth yp/(p+Δz) against the normalized total upstream head H/(p+Δz).
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Fig. 15. The normalized lower nappe profile data (y/H versus x/H).
weir is complex in comparison with the sharp crested weir. But, it is simple in comparison with the circular crested weir. Results indicated that discharge coefficient of quarter-circular crested weir is a constant value and equals to 1.261. In the range of H/Ro 1.5, it is more than the discharge coefficient of circular crested weir. Discharge over the quarter-circular crested weir can easily be measured from the crest flow depth with the good precision. Using this interesting characteristic, the quarter-circular crested weir can serve as an accurate device for flow measurement in
open channels. The lower nappe profile of free jet over the quarter-circular crested weir can also be considered as the ogee shape of this weir. In the design procedure of quarter-circular crested weir, two parameters (R and α) should be estimated. While, for circular crested weir, three parameters (R, α and β) should be estimated. For the same weir height, the surface area of quarter-circular crested weir is nearly equal to the half of the surface area of circular crested weir. Also, in the range of H/R o1.5, discharge coefficient of quarter-circular crested weir is more.
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Fig. 16. Ogee shape of quarter-circular crested weir.
Based on Eq. (1), for the same discharge capacity Q and the same total upstream head H, the larger coefficient of discharge causes reduction in the weir crest width L. Reduction in both of the weir surface area and the weir width reduced the required materials for the weir constructing and finally reduced the weir costs. References [1] Kandaswamy PK, Rouse H. Characteristics of flow over terminal weirs and sills. J Hydraul Div 1957;83(4):1–13. [2] Rajaratnam N, Muralidhar D. Pressure and velocity distribution for sharpcrested weirs. J Hydraul Res 1971;9(2):241–8. [3] Ramamurthy AS, Tim US, Rao MVJ. Flow over sharp-crested plate weirs. J Irrig Drain Eng 1987;113(2):163–72. [4] Afzalimehr H, Bagheri S. Discharge coefficient of sharp-crested weirs using potential flow. J. Hydraul Res 2009;47(6):820–3. [5] Bagheri S, Heidarpour M. Flow over rectangular sharp-crested weir. Irrig Sci 2009;28(2):173–9.
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