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The hydraulics of parallel sluice gates under low flow delivery condition
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The hydraulics of parallel sluice gates under low flow delivery condition M. Bijankhan, S. Kouchakzadeh
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S0955-5986(14)00136-8 http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.017 JFMI1011
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Received date: 10 July 2013 Revised date: 8 October 2014 Accepted date: 30 October 2014 Cite this article as: M. Bijankhan, S. Kouchakzadeh, The hydraulics of parallel sluice gates under low flow delivery condition, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The hydraulics of Parallel sluice gates under low flow delivery condition By: Bijankhan M., and Kouchakzadeh, S Mohammad Bijankhan, PhD Candidate, Irrigation and Reclamation Engrg. Dept., University of Tehran, Iran, and Visiting Researcher UMR G-EAU, SupAgro, place Pierre Viala, 34060 Montpellier Cedex 1, France. Email: [email protected], [email protected] (Corresponding Author) Salah Kouchakzadeh, Prof., Irrigation and Reclamation Engrg. Dept., University of Tehran, P.O. Box 31587-4111, Karaj, 31587-77871, Iran. Email: [email protected] ABSTRACT In this paper the flow through parallel sluice gates under low flow conditions and with some of the gates closed resulting in symmetrical or asymmetrical gate installations were studied experimentally. The current stage-discharge formula for single sluice gates cannot be used for either free flowing or submerged parallel sluice gates. Then, on the basis of experimental observations, the effect of the closed gates was considered to develop a submergence distinguishing condition curve formula. For both free and submerged regimes, the Π-theorem along with the incomplete self-similarity concept were used to develop head-discharge formulas for symmetrical and asymmetrical gate installations. The proposed formulas were then calibrated using the compiled experimental data. The new approach is shown to be applicable within the entire range of operation, i.e. from free to submerged flow regimes as well as the transition zone.
Keywords: free flow, submerged flow, parallel sluice gates, incomplete self-similarity, distinguishing condition curve 1. Introduction Gates have been widely used in irrigation networks for regulating water surface elevation or measuring flow rates. In this regard a head discharge formula would be a key issue for the network designers. Usually the gate width is almost equal to the channel width. Such a gate installation is termed “single gate installation”, however, in wide channels this could be impractical due to operation and maintenance difficulties. In wide canals usually two or more gates are installed in parallel and called “parallel gate installation”. According to gate openings the operating conditions of parallel gates might be classified as follows:
(1) Gates with same opening sizes, which is not a preferable operation condition. Since, in most cases it might not be easy to manually open the gates so that the openings
have identical dimensions and for low discharge delivery such a case requires small gate openings resulting in trapping debris [1]. (2) Only some gates are in operation, others are closed. The open gates ordinations can be divided into symmetrical and asymmetrical cases. This operation condition is applied in low flow delivery conditions. (3) Gates with different opening sizes. Depending on the tail water depth and the amount that the gate is opened, all cases might experience free or submerged flow conditions. Although a sound background can be found in the literature for both free and submerged single sluice/radial gates [e.g. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] very limited studies on sluice or radial gates installed in parallel exist. Nago and Furukawa [12] conducted some experiments in a channel having a sudden expansion with negative vertical step, lateral expansion, and both lateral expansions and negative step. They proposed many graphical solutions in order to find both free and submerged discharge coefficients. Employing the energy and momentum equations, Clemmens et al. [13] indicated that the downstream flow conditions would significantly affect the distinguishing condition curve. They showed that for a radial gate having a smaller width than its channel width, the available submerged head-discharge methods are inappropriate for the case of parallel sluice gates with closed gates at least near the transition from free to submerged flow conditions. Clemmens [1] indicated that the submergence transition zone should be avoided in parallel radial gate operations to achieve reliable water measurement. In this regard he evaluated the performance of Energy-Momentum (EM) method proposed by Clemmens et al. [13] to distinguish the flow condition of each radial gate. However, he did not propose any headdischarge formulas for parallel sluice gates. On the basis of the mentioned EM method and by incorporating the effect of the downstream channel width, Bijankhan et al. [14] developed a theoretical distinguishing condition curve permitting to calculate the maximum tailwater depth for which the free flow regime happened. In this paper the flow through parallel sluice gates with both symmetrically and asymmetrically closed gates will be analyzed theoretically and experimentally for free and submerged flow conditions. This is a very practical case existing for either parallel gates with closed sides or the gates installed in a channel with sudden expansion. It is shown that the current head discharge formulas used for single sluice gates cannot be employed to calibrate parallel sluice gates cases. On the basis of the experimental observations, a new approach is presented to distinguish the hydraulic jump type occurred in a rectangular channel
downstream of parallel sluice gates with closed ones. Furthermore, a distinguishing condition curve formula including the effect of the closed gates is developed. Afterward, employing the Π-theorem and the incomplete self-similarity concept a generalized head-discharge formula was proposed and calibrated using the compiled experimental data for both free and submerged regimes. The proposed formula is applicable within the entire range of operation, i.e. from free to submerged flow regimes as well as the transition zone. 2. Experimental setup The experiments were performed in a 1.179m wide, 1m high and 7m long Plexiglas flume located at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran. The flume was supplied by an elevated constant head tank and an electromagnetic flow-meter with the accuracy of ±0.5% was installed on the feeding pipe to measure the flow rate. Water depths were measured using point gages with accuracy of 0.1 mm. The tail water depth, y3, was adjusted using a tail gate installed at the downstream end.
The flow through parallel sluice gates with both symmetrical and asymmetrical closed gates are simplified to the flow in an open channel through a cross-section that is blocked except for a symmetrical or asymmetrical gap of dimensions b×w located at the bottom of the channel as shown in Fig. 1 which also indicates the parameter used. Geometric characteristics of the parallel sluice gates used in the present study are summarized in Table 1. Table 1. Geometric characteristics of the parallel sluice gates y1/w
Q (l/s)
y3/w
Free
Submerged
Symmetrical
0.31 15.2-19.4
15.2-18.9
22-34.6
23.8-37.6
−
2.3-8.9
0.48 21.7-31.6
17-28.6
17.9-36
12.3-37.1
−
2.1-14.8
0.65 27.9-45.6
21.6-40.8
15.7-38.2
11-40.4
−
2.47-12.1
0.48 21.4-28.7
21.5-27
19-33.9
20.5-41
−
2.7-13
0.57 23.5-36.2
18-33.5
17-38.9
11.7-41.9
−
2.7-12.8
0.65 28.9-42.4
29.1-37.7
18.4-38.9
19.2-40.5
−
2.6-12.4
38.8-59.6
11.8-33
13.1-37.9
−
5.2-14.8
gate
Free
Single
Submerged
Asymmetrical
b/B
1
38-64.6
Free Submerged
First the free flow condition was established in each test for a given for a given discharge and the flow depth upstream of the gate was measured. Then, the tail gate was adjusted incrementally until the upstream water depth begun to respond to the downstream variation. This condition was taken as the beginning of the transition zone from free to submerged flow condition. Submerged flow regime was occurred with further increase in the tail gate height in which both flow depths upstream and downstream the gates were recorded. 3. Free flow condition For a single sluice gate the dimensionless head-discharge formula can be written as [15, 16]: b
K §y · = a¨ 1 ¸ w ©w ¹
(1)
where w is the gate opening K=[Q2/(b2g)]1/3, Q is the discharge, b is gate width, g is the acceleration due to gravity, y1 is the upstream depth, a and b are empirical coefficients that should be determined experimentally. According to Fig. 2 current experimental data of the single sluice gate are in a very good agreement with the experimentations carried out by Rajaratnam and Subramanya [4], Ferro [15], and Cassan and Belaud [17]. Also, the coefficients a and b were obtained as 0.8749 and 0.3455 respectively. If the head-discharge equation for a single sluice gate is employed to calculate the discharge through parallel sluice gates with closed gates, it is necessary to evaluate the performance of the current head-discharge equations to estimate the flow rate through parallel sluice gates. Accordingly, Eq. (1) was depicted along with the experimental data associated with the parallel sluice gates with symmetrical/asymmetrical closed gates in Fig. 3. As indicated in the figure for b/B<1 the overall trend of the data points associated with both symmetrical and asymmetrical cases lie lower than the head-discharge relationship of the single sluice gate. Consequently, the free flow formula of the single sluice gate is not able to accurately determine the discharge through parallel sluice gates including closed gates. It is interesting to note that, while the asymmetrical installation collapses all the compiled data very well on a single trend, slight deviation in the symmetrical installation data trends is observed (Fig. 3). The free flow condition through parallel sluice gates with closed gates can be expressed by the following functional relationship:
F1 ( Q, y1 , B, b, w, g , µ, ρ ) = 0
(2)
Where, F1 is a functional symbol, B is channel width, µ is the water viscosity, and ρ is water density.
Considering µ, w, and g, as dimensional independent parameters, the following five dimensionless variables were obtained: Π1 =
y1 ,Π w
2=
Q2 w 3/2 g 1/2 ρ B b , , , Π = Π = Π = 3 4 5 gw5 w w µ
Rearranging the dimensionless parameters yield:
Π 21/3 Q 2/3 K = 2/3 1/3 = 2/3 b g w w Π4 in which K is critical depth under the gate opening and
Π 5 Π 21/ 2 ρ Q = = Re Π3 µB in which Re is the Reynolds number Consequently, neglecting the effect of Reynolds number, the functional relationship, Eq. (2), takes the following form:
K §y b B · = F2 ¨ 1 , , ¸ w ©w w w ¹
(3)
The incomplete self-similarity theory (ISS) can be used to deduce the mathematical shape of the functional relationship (Eq. 3). For a given dimensionless group, Πn, a phenomenon is defined as self-similar when the functional relationship Π1 = ϕ1 ( Π 2 , Π 3 ,........., Π n ) is independent of Πn. In such a case, the behavior of ϕ1 is studied for Π n → 0 or Π n → ∞ . When the function ϕ1 has a limit equal to 0 or ∞, the phenomenon is expressed by the following functional relationship [18, 19]:
Π1 = Π εn ϕ2 ( Π 2 , Π 3 ,........., Π n −1 )
(4)
Considering Eq. (4), when y1 / w → 0 then K / w → 0 , and when y1 / w → ∞ then
K / w → ∞ , allowing the ISS to be used so that, Eq. (3) can be written as: b
f K §y · = Af ¨ 1 ¸ w ©w ¹
(5)
Where Af and bf are constant coefficients that should be obtained experimentally. It should be noted that Af should be considered as a function of b/w and B/w. Therefore the following expression was proposed: b
f K §y · = af ¨ 1 ¸ w ©w ¹
c
df
f §b · §B · ¨ ¸ ¨ ¸ ©w ¹ ©w ¹
(6)
Employing the experimental data compiled in the current study the empirical coefficients were obtained for both symmetrical and asymmetrical gate installations, i.e. b/B<1. It was found that the best solution would be obtained by considering d f = −c f , i.e., channel contraction ratio due to the closed gates (b/B) plays an important role. The results are given in Table 2. Table 2. Empirical coefficients of Eq. (6) Type
af
bf
cf
Symmetrical
0.8457 0.358 0.0415
Asymmetrical
0.821
0.346
0
The estimated values of K/w, obtained on the basis of employing Eq. (6) were compared to those
observed
experimentally
(Ke/w)
and
the
associated
relative
errors,
i.e.
R(%)=(K−Ke)/Ke×100, were calculated and depicted in Fig. 4. For the symmetrical installation case the maximum and mean absolute relative errors were obtained as 2.5% and 0.48% respectively. These values were 1.43% and 0.36% for the asymmetrical condition. It should be mentioned that no significant differences were observed between the empirical coefficients calculated for symmetrical and asymmetrical cases. While for the symmetrical case K/w relates to b/B with the power of 0.0415, no correlation was observed between K/w and b/B for the asymmetrical case. However, for the asymmetrical case it is anticipated that as b/B tends to unity the associated head discharge formula should tend to the single sluice gate curve. Consequently, more experimental study should be carried out for 0.65
§ K y1 · , , DRF , Re ¸ = 0 ©w w ¹
ψ¨
(7)
DRF is a Discharge Reduction Function defined by the following formula [9, 16]:
DRF =
(y 1 − y 3) λ1 ( y 3 − y * ) + ( y 1 − y 3 )
(8)
In which λ1 is a numerical constant and y* is the conjugated depth obtained by the classical hydraulic jump formula [20, 21]:
· y * Cc § y = ¨¨ 1 + 16( 1 − 1) − 1¸¸ w 2 © C cw ¹
(9)
Where Cc is the contraction coefficient. Neglecting the effect of Reynolds number and considering the ISS concept Eq. (7) reduces to: b
K §y · = a ¨ 1 ¸ DRF c w ©w ¹
(10)
In which a, b, and c are empirical coefficients. Employing the experimental data obtained in this study, Eq. (11) was developed for the single sluice gate:
K §y · = 0.77 ¨ 1 ¸ w ©w ¹
0.374
§ · (y 1 − y 3) ¨ ¸ © 3.506( y 3 − y * ) + ( y 1 − y 3 ) ¹
0.171
(11)
Based on Eq. (11) any parameters affecting the maximum tailwater depth for which the free flow condition occurs, i.e. y*, would affect the head-discharge curve and should be considered. Consequently, investigating the distinguishing condition curve of parallel sluice gates flow is necessary for obtaining appropriate water measurements. Therefore, experiments were carried out in order to clarify the case. In this regard, for a given parallel sluice gate dimensions, the free flow condition was established first. Then, in order to determine the threshold between free and submerged flow conditions, the tail water depth was increased incrementally until maximum change of 3 to 5% was observed in the upstream flow
depth. The distinguishing condition curves associated with different gate widths, including both symmetrical and asymmetrical cases, were compared with Eq. (9) (Fig. 5). The figure indicated that the distinguishing condition curve obtained based on the classical hydraulic jump (i.e. Eq. 9) accurately determines the threshold between free and submerged flow regimes for the single sluice gate. However, parallel sluice gates with closed gates tend to become submerged for smaller tail water depth. Consequently, Eq. (9) cannot be used to distinguish the flow condition through parallel sluice gates. The reason might be attributed to the fact that the momentum flux associated with the contracted section is less than that of the case with no contraction, i.e. single sluice gate condition. Accordingly, for the parallel sluice gates the gate width, b, should appear in the functional relationship of the distinguishing condition curve. Therefore, the following functional relationship was considered to describe the distinguishing condition curve of the parallel sluice gates:
y*p = f1 ( y j , b, B, Q, w, g , ρ, µ)
(12)
Where, f1 is a functional symbol, y*p is the maximum downstream depth for which the free flow condition occurs through parallel sluice gates, and yj(=Ccw) is the jet thickness. Considering µ, w, and g as dimensional independent parameters, the functional relationship, Eq. (12), takes the following form:
y *p w
= f 2(
yj b B K , , , , Re) w w w w
(13)
Neglecting the effect of the Reynolds number and considering the critical depth as a function of y1, and considering that the effects of b/w and B/w can be represented by b/B (as described for the free flow condition) Eq. (13) reduces to:
y *p yj
= f 3(
b y1 , ) B w
(14)
Considering the ISS hypothesis the mathematical form of Eq. (14) takes the following form: β
§y · §b · = αC c ¨ 1 ¸ ¨ ¸ w ©w ¹ ©B ¹
y *p
γ
(15)
In which, α, β, and γ are constant parameters. It is necessary to examine the value of the contraction coefficient for the case of parallel sluice gate with closed gates. As indicated in Fig. 6 an average value of Cc=0.61 was obtained which is equal to the common agreed value of 0.61 for single sluice gate [4, 22]. Considering Cc=0.61 and employing the experimental data of different values of b/B, the empirical coefficients of Eq. (15), i.e. α, β, and γ becomes as shown in Table 3. Eq. (15) is only valid for the experimental data ranges listed in Table 1. It must be recalibrated for any other data points which are out of the calibration ranges Table 3. The empirical coefficients of Eq. (15) Type
α
β
γ
Symmetrical
0.645 0.69 0.55
Asymmetrical
1.54
0.34 0.38
Tail water depth that exceeds the value of y*p obtained by Eq. (15) generates submerged flow condition. Plotting the experimental data on the distinguishing curve would greatly assist identifying the submergence level [23]. Experimental data points which are very close to the
distinguishing condition curve fall into low submergence zone. In other words the submergence ratio, (y3−y*p)/w, tends to zero for very low submerged conditions. As indicated in Figs. 7 and 8 for both symmetrical and asymmetrical cases, the experimental data cover the entire range of the submergence ratios including the low submergence zone. 4.1. Categorizing the downstream flow conditions for different tailwater levels Symmetrical installation By increasing the tailwater depth incrementally three submerged flow regimes were observed and identified (Fig. 9): (1)- For the tailwater depth slightly greater than y*p an oblique hydraulic jump was formed and the flow downstream the gate became asymmetrical having a reverse flow near the channel wall (Fig. 9a). Graber [24] was also reported the asymmetric flow in symmetric supercritical expansions. (2)- As shown in Figs. 7 and 8, for b/B<1 it was not possible to record any data points in a specific range. This range was termed “blank data zone”. That is, for a given value of y1/w, which was called y*L, the tailwater depth was suddenly increased. Exactly at this point it was observed that the oblique hydraulic jump started converting to a symmetrical submerged jump. However, the reverse flow was still intermittently observed. (3)- Beyond a certain value of the downstream depth called y*U, the submerged flow condition was always symmetrical. In such a case, two steady vortexes were formed just downstream the gate. Also, two symmetrical steady reverse flows were observed near the channel wall (Fig. 9b). The lower and upper limits of the blank data zone, i.e. y*L and y*U, were formulated and the following relationships were obtained:
y *L §y · = 3.35 ¨ 1 ¸ w ©w ¹
0.162
y *U §y · = 3.75 ¨ 1 ¸ w ©w ¹
0.238
§b · ¨ ¸ ©B ¹
0.376
§b · ¨ ¸ ©B ¹
(16) 0.445
(17)
According to the above discussion two different submerged flow zones were identified: a. Experimental data located between the distinguishing condition curve and the lower limit of the “blank data zone” which could be considered as low submerged range. b. Experimental data having the tail water depth of greater than y*U is called high submerged zone.
Asymmetrical installation Fig. (10) indicates the flow characteristics downstream an asymmetrical installation of parallel sluice gates. According to the figure for low submergence condition (Fig. 10a) an oblique like hydraulic jump was formed with a reverse flow near the channel wall. Similar to the symmetrical case the blank data zone was observed and beyond it the submerged hydraulic jumps became as that indicated in Fig.10b. A single large vortex was formed just after the gate with the reverse flow formed at the tailwater section. For the asymmetrical parallel sluice gate installation the lower and upper limits of the blank data zone, i.e. y*L and y*U, were formulated and the following relationships were obtained:
y *L §y · = 1.685 ¨ 1 ¸ w ©w ¹ y *U §y · = 2.26 ¨ 1 ¸ w ©w ¹
0.415
(18)
0.405
(19)
It should be noted that for the tested experimental data of the asymmetrical parallel sluice gate installation, the lower and upper limits of the blank data zone was independent of the variable b/B. However more experimental study should be carried out for 0.65
F ( Q , y1 , y3 , y* p , B , b, w, g , µ, ρ ) = 0
(20)
Considering the variables w, g and µ as dimensional independent parameters, the following seven dimensionless variables were obtained: Π1 =
y1 ,Π w
2
=
y *p y3 Q2 w 3/2 g 1/2 ρ B b , , , Π = , Π = , Π = Π = Π = 6 7 3 4 5 w w gw5 w w µ
Rearranging the dimensionless parameters yield:
Π 21/3 Q 2/3 K = = ; 2/3 2/3 1/3 Π4 b g w w Π1 − Π 6 =
y1 − y 3 ; w
λ1 ( Π 6 − Π 7 ) + ( Π1 − Π 6 ) = λ1
y 3 − y *p w
+
y1 − y 3 ; w
in which K is critical depth under the gate opening and
Π 5 Π 21/ 2 ρ Q = = Re Π3 µB in which Re is the Reynolds number Neglecting the effect of Reynolds number the dimensionless form of Eq. (20) would be: f
e1 b1 c1 d1 1 K § y · § b · § B · § y − y 3 · § y 3 − y *p y 1 − y 3 · = a1 ¨ 1 ¸ ¨ ¸ ¨ ¸ ¨ 1 + λ ¸ ¸ ¨ 1 w w w © w ¹ ©w ¹ ©w ¹ © w ¹ © ¹
(21)
There are some physical constrains that must be taken into account before calibrating Eq. (21): 1- For fully submerged flow condition, when y1 tends to y3, K/w must tend to zero; 2- For very low submergence condition, when y3 tends to y*p, Eq. (21) should converge to the free flow formula, i.e. Eq. (6). In order to apply the second constrain the coefficient d1 must be equal to −c1 and also f1=−e1. Consequently Eq. (21) takes the following form: e1
b1 c1 · y1 − y 3 K § y1 · § b · § = a1 ¨ ¸ ¨ ¸ ¨ ¸¸ ¨ w © w ¹ © B ¹ © λ1 ( y 3 − y * p ) + ( y 1 − y 3 ) ¹
(22)
According to Eq. (22) a very interesting dimensionless coefficient called Discharge Reduction Factor, DRF, was identified which can be written as:
DRFp =
(y1 − y 3) λ1 ( y 3 − y * p ) + ( y 1 − y 3 )
(23)
For the free flow condition, i.e. y3=y*p, DRF is unity. Also, increasing the tailwater level and making the gate to be submerged it would be decreased, so that for the fully submerged flow condition it would be zero. Consequently DRF is a coefficient ranging from 0 to 1 and introduces the level of submergence. It should be mentioned that to achieve a continuous head-discharge formula, the submerged and free flow data points were used all together to calibrate Eq. (22). In this regard, DRFp was considered to be unity for free flow data points. The coefficients λ1, a1, b1, c1, and e1 were listed in Table 4.
Table 4. The empirical coefficients of Eq. (22) for both symmetrical and asymmetrical gate configurations λ1
Type
a1
c1
b1
e1
Symmetrical
0.1127 0.838 0.349 0.018 2.18
Asymmetrical
0.161
0.885 0.332 0.045 3.02
Eq. (22) could be used for both free and submerged flow (y*p≤y3≤y*L) conditions. The estimated values of K/w, obtained by using Eq. (22) were compared to those observed experimentally (Ke/w) and the associated relative errors, i.e. R(%)=(K−Ke)/Ke×100, were calculated and depicted in Fig. 12 for free and low submerged data of both symmetrical and asymmetrical cases. For y3≥y*U the functional relationship takes the following form:
F ( Q, y1 , y3 , y*U , B, b, w, g , µ, ρ ) = 0
(24)
Similarly, neglecting the effect of Reynolds number and considering the ISS concept Eq. (24) reduces to: b
c
2 2 K §y · §b · = a2 ¨ 1 ¸ ¨ ¸ DRFU e 2 w ©w ¹ © B ¹
(25)
In this case the DRF was considered as the following form:
DRFU =
(y1 − y 3) λ2 ( y 3 − y *U ) + ( y 1 − y 3 )
(26)
Using the complied data the coefficients λ2, a2, b2, c2, and e2 were calculated (Table 5). Table 5. The empirical coefficients of Eq. (25) for both symmetrical and asymmetrical gate configurations Type
λ2
a2
Symmetrical
0.408
0.64
Asymmetrical 0.795 0.568
b2
c2
e2
0.414 0.003 0.695 0.43
0.027 0.458
The relative error associated with Eq. (26) compared to the experimental data of the high submergence zone were calculated and depicted in Fig. 13.
5. Conclusion The flow characteristics of flow through parallel sluice gates with closed gates were studied under both free and submerged flow regimes. The threshold between free and submerged
flow conditions, i.e. the distinguishing condition curve, was formulated for both symmetrical and asymmetrical parallel sluice gates’ configurations. Then, the Π-theorem and the incomplete self-similarity concept were employed to develop stage-discharge relationships. Finally, the proposed formulas were calibrated using the experimental data obtained in the current study. 6. Acknowledgements The research was supported by the “Center of Excellence for Evaluation and Rehabilitation of Irrigation and Drainage Network” and the University of Tehran grant No. 7102021/1/06 . Both supports are highly appreciated. 7. Notation a1, b1, c1, d1, λ2, a2, b2, c2, and d2 = empirical coefficients; af, and bf = constant parameters B = channel width; b = gate width; Cc = contraction coefficient; DRF = Discharge Reduction Function; f1 = a functional symbol; F1 = functional symbol; g = acceleration due to gravity; K = critical depth; Ke/w = experimental values of K/w Q = discharge; Re = Reynolds number; Relative error (R) =(K−Ke)/Ke×100, y* = maximum downstream flow depth for which the free flow condition occurs; y*L and y*U = The lower and upper limits of the blank data zone;
y*p = maximum downstream depth for which the free flow condition occurs through parallel sluice gates; y1 = the upstream depth; yj(=Ccw) = jet thickness; ψ = a functional symbol; ρ = water density; µ = water viscosity; α, β, and γ = constant parameters; 8. References [1] Clemmens, A. (2004) Avoiding Submergence Transition Zone for Radial Gates in Parallel. Critical Transitions in Water and Environmental Resources Management: pp. 1-10. [2] Henry H. Discussion: diffusion of submerged jet. Transactions of the American Society of Civil Engineers 1950;115:687–97. [3] Swamee PK. Sluice-gate discharge equations. Journal of Irrigation and Drainage Engineering 1992;118(1):56–60. [4] Rajaratnam N, Subramanya K. Flow equation for the sluice gate. Journal of Irrigation and Drainage Engineering 1967;93(3):167–86. [5] Rajaratnam N, and Subramanya K.. Flow immediately below submerged sluice gate. J. Hydr. Div., 1967;93(HY4), 57–77. [6] Sepulveda C, Gomez M, Rodellar J. Benchmark of discharge calibration methods for submerged
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2009;135(5):676–82. [7] Lozano D, Mateos L, Merkley GP, Clemmens AJ. Field calibration of submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2009;135(6):763–72 [8] Castro-Orgaz O, Lozano D, Mateos L. Energy and momentum velocity coefficients for calibrating submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2010;136(9):610–6. [9] Bijankhan M, Ferro V, and Kouchakzadeh S. New Stage-Discharge Relationships for Radial Gates. J. Irrig. Drain Eng., 2013;139(5), 378–387.
[10] Khalili Shayan H, and Farhoudi J. Effective parameters for calculating discharge coefficient of sluice gates. J. Flow measurement and instrumentation., 2013;33, 96-105. [11] Castro-Orgaz O, Mateos L, and Dey S. Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions. J. Irrig. Drain Eng., 2013;139(4), 325–335. [12] Nago H, and Furukawa S. Discharge coefficient of a sluice gate placed at sudden expansion of open channel. Memoirs of the School of Engineering,Okayama University 1979; 14(1), 127–138. [13] Clemmens AJ, Strelkoff TS, and Replogle JA. Calibration of Submerged Radial Gates. J. Hydraul. Eng., 2003;129(9), 680-687. [14] Bijankhan M, Kouchakzadeh S, Bayat E. Distinguishing condition curve for radial gates. Journal of flow measurment and instrumentation 2011;22(5): 500–6. [15] Ferro V. Simultaneous flow over and under a gate. Journal of Irrigation and Drainage Engineering 2000;126(3):190–3. [16] Bijankhan M, Ferro V, and Kouchakzadeh S. New stage-discharge relationships for free and submerged sluice gates. J. Flow measurement and instrumentation 2012; 28, 50-56. [17] Cassan L, Belaud G. Experimental and numerical investigation of flow under sluice gates. Journal of Hydraulic Engineering 2012;138(4):367–73. [18] Barenblatt GI. Similarity, self-similarity and intermediate asymptotics. New York: Consultants Bureau; 1979. [19] Barenblatt GI. Dimensional analysis. Amsterdam: Gordon and Breach, Science Publishers Inc.; 1987. [20] Bos MG. Discharge measurement structures. Wageningen, the Netherlands: ILRI; 1989. [21] Lin CH, Yen JF, Tsai CT. Influence of sluice gate contraction coefficient on distinguishing condition. Journal of Irrigation and Drainage Engineering 2002;128(4):249–52. [22] Henderson, F.M. Open Channel Flow, Macmillan, New York. 1966. [23] Bijankhan M, Darvishi E, and Kouchakzadeh S. Discussion of “Energy and Momentum Velocity Coefficients for Calibrating Submerged Sluice Gates in Irrigation Canals” by Oscar Castro-Orgaz, David Lozano, and Luciano Mateos. J. Irrig. Drain Eng., 2012;138(9), 852–854. [24] Graber, S. ”Asymmetric Flow in Symmetric Supercritical Expansions.” J. Hydraul. Eng., 2006;132(2), 207–213.
Figure Captions Fig. 1. Schematic sketch of the parallel sluice gates configuration under free and submerged flow conditions of low flow delivery condition Fig. 2. K/w versus y1/w associated with the single sluice gate of the current study in comparison with the other experimental data Fig.
3.
K/w
versus
y1/w
associated
with
parallel
sluice
gates
including
symmetrical/asymmetrical closed gates Fig. 4. The relative error distribution associated with symmetrical and asymmetrical gate installations Fig.5. distinguishing condition curves associated with different gate widths, including both symmetrical and asymmetrical cases Fig. 6. The contraction coefficient versus y1/w for symmetrical and asymmetrical parallel sluice gate installations Fig. 7. Experimental data points plotted on the distinguishing condition curves for symmetrical case Fig. 8. Experimental data points plotted on the distinguishing condition curves for asymmetrical case Fig. 9. Flow characteristics downstream a symmetrical installation of parallel sluice gates Fig. 10. Flow characteristics downstream an asymmetrical installation of parallel sluice gates Fig. 11. Absolute relative error distribution to predict the discharge through parallel sluice gates by employing the head-discharge relationship of a single sluice gate, a)Symmetrical, b)Asymmetrical Fig. 12. Relative error distribution associated with Eq. (23) for determining the discharge through parallel sluice gates, a) symmetrical b) asymmetrical Fig. 13. Relative error distribution associated with Eq. (25) for determining the discharge through parallel sluice gates
9. Research highlights ¾ The flow through submerged and free parallel sluice gate is considered under low flow delivery condition. ¾ The head-discharge curve of the single sluice gate cannot be used for the parallel sluice gates. ¾ New distinguishing condition curves were developed to identify the flow regime through parallel gates. ¾ Buckingham theorem and Incomplete Self Similarity concepts were used to develop a new head-discharge relationship.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
The hydraulics of parallel sluice gates under low flow delivery condition M. Bijankhan, S. Kouchakzadeh
www.elsevier.com/locate/flowmea-
PII: DOI: Reference:
S0955-5986(14)00136-8 http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.017 JFMI1011
To appear in:
Flow Measurement and Instrumentation
sinst
Received date: 10 July 2013 Revised date: 8 October 2014 Accepted date: 30 October 2014 Cite this article as: M. Bijankhan, S. Kouchakzadeh, The hydraulics of parallel sluice gates under low flow delivery condition, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The hydraulics of Parallel sluice gates under low flow delivery condition By: Bijankhan M., and Kouchakzadeh, S Mohammad Bijankhan, PhD Candidate, Irrigation and Reclamation Engrg. Dept., University of Tehran, Iran, and Visiting Researcher UMR G-EAU, SupAgro, place Pierre Viala, 34060 Montpellier Cedex 1, France. Email: [email protected], [email protected] (Corresponding Author) Salah Kouchakzadeh, Prof., Irrigation and Reclamation Engrg. Dept., University of Tehran, P.O. Box 31587-4111, Karaj, 31587-77871, Iran. Email: [email protected] ABSTRACT In this paper the flow through parallel sluice gates under low flow conditions and with some of the gates closed resulting in symmetrical or asymmetrical gate installations were studied experimentally. The current stage-discharge formula for single sluice gates cannot be used for either free flowing or submerged parallel sluice gates. Then, on the basis of experimental observations, the effect of the closed gates was considered to develop a submergence distinguishing condition curve formula. For both free and submerged regimes, the Π-theorem along with the incomplete self-similarity concept were used to develop head-discharge formulas for symmetrical and asymmetrical gate installations. The proposed formulas were then calibrated using the compiled experimental data. The new approach is shown to be applicable within the entire range of operation, i.e. from free to submerged flow regimes as well as the transition zone.
Keywords: free flow, submerged flow, parallel sluice gates, incomplete self-similarity, distinguishing condition curve 1. Introduction Gates have been widely used in irrigation networks for regulating water surface elevation or measuring flow rates. In this regard a head discharge formula would be a key issue for the network designers. Usually the gate width is almost equal to the channel width. Such a gate installation is termed “single gate installation”, however, in wide channels this could be impractical due to operation and maintenance difficulties. In wide canals usually two or more gates are installed in parallel and called “parallel gate installation”. According to gate openings the operating conditions of parallel gates might be classified as follows:
(1) Gates with same opening sizes, which is not a preferable operation condition. Since, in most cases it might not be easy to manually open the gates so that the openings
have identical dimensions and for low discharge delivery such a case requires small gate openings resulting in trapping debris [1]. (2) Only some gates are in operation, others are closed. The open gates ordinations can be divided into symmetrical and asymmetrical cases. This operation condition is applied in low flow delivery conditions. (3) Gates with different opening sizes. Depending on the tail water depth and the amount that the gate is opened, all cases might experience free or submerged flow conditions. Although a sound background can be found in the literature for both free and submerged single sluice/radial gates [e.g. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] very limited studies on sluice or radial gates installed in parallel exist. Nago and Furukawa [12] conducted some experiments in a channel having a sudden expansion with negative vertical step, lateral expansion, and both lateral expansions and negative step. They proposed many graphical solutions in order to find both free and submerged discharge coefficients. Employing the energy and momentum equations, Clemmens et al. [13] indicated that the downstream flow conditions would significantly affect the distinguishing condition curve. They showed that for a radial gate having a smaller width than its channel width, the available submerged head-discharge methods are inappropriate for the case of parallel sluice gates with closed gates at least near the transition from free to submerged flow conditions. Clemmens [1] indicated that the submergence transition zone should be avoided in parallel radial gate operations to achieve reliable water measurement. In this regard he evaluated the performance of Energy-Momentum (EM) method proposed by Clemmens et al. [13] to distinguish the flow condition of each radial gate. However, he did not propose any headdischarge formulas for parallel sluice gates. On the basis of the mentioned EM method and by incorporating the effect of the downstream channel width, Bijankhan et al. [14] developed a theoretical distinguishing condition curve permitting to calculate the maximum tailwater depth for which the free flow regime happened. In this paper the flow through parallel sluice gates with both symmetrically and asymmetrically closed gates will be analyzed theoretically and experimentally for free and submerged flow conditions. This is a very practical case existing for either parallel gates with closed sides or the gates installed in a channel with sudden expansion. It is shown that the current head discharge formulas used for single sluice gates cannot be employed to calibrate parallel sluice gates cases. On the basis of the experimental observations, a new approach is presented to distinguish the hydraulic jump type occurred in a rectangular channel
downstream of parallel sluice gates with closed ones. Furthermore, a distinguishing condition curve formula including the effect of the closed gates is developed. Afterward, employing the Π-theorem and the incomplete self-similarity concept a generalized head-discharge formula was proposed and calibrated using the compiled experimental data for both free and submerged regimes. The proposed formula is applicable within the entire range of operation, i.e. from free to submerged flow regimes as well as the transition zone. 2. Experimental setup The experiments were performed in a 1.179m wide, 1m high and 7m long Plexiglas flume located at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran. The flume was supplied by an elevated constant head tank and an electromagnetic flow-meter with the accuracy of ±0.5% was installed on the feeding pipe to measure the flow rate. Water depths were measured using point gages with accuracy of 0.1 mm. The tail water depth, y3, was adjusted using a tail gate installed at the downstream end.
The flow through parallel sluice gates with both symmetrical and asymmetrical closed gates are simplified to the flow in an open channel through a cross-section that is blocked except for a symmetrical or asymmetrical gap of dimensions b×w located at the bottom of the channel as shown in Fig. 1 which also indicates the parameter used. Geometric characteristics of the parallel sluice gates used in the present study are summarized in Table 1. Table 1. Geometric characteristics of the parallel sluice gates y1/w
Q (l/s)
y3/w
Free
Submerged
Symmetrical
0.31 15.2-19.4
15.2-18.9
22-34.6
23.8-37.6
−
2.3-8.9
0.48 21.7-31.6
17-28.6
17.9-36
12.3-37.1
−
2.1-14.8
0.65 27.9-45.6
21.6-40.8
15.7-38.2
11-40.4
−
2.47-12.1
0.48 21.4-28.7
21.5-27
19-33.9
20.5-41
−
2.7-13
0.57 23.5-36.2
18-33.5
17-38.9
11.7-41.9
−
2.7-12.8
0.65 28.9-42.4
29.1-37.7
18.4-38.9
19.2-40.5
−
2.6-12.4
38.8-59.6
11.8-33
13.1-37.9
−
5.2-14.8
gate
Free
Single
Submerged
Asymmetrical
b/B
1
38-64.6
Free Submerged
First the free flow condition was established in each test for a given for a given discharge and the flow depth upstream of the gate was measured. Then, the tail gate was adjusted incrementally until the upstream water depth begun to respond to the downstream variation. This condition was taken as the beginning of the transition zone from free to submerged flow condition. Submerged flow regime was occurred with further increase in the tail gate height in which both flow depths upstream and downstream the gates were recorded. 3. Free flow condition For a single sluice gate the dimensionless head-discharge formula can be written as [15, 16]: b
K §y · = a¨ 1 ¸ w ©w ¹
(1)
where w is the gate opening K=[Q2/(b2g)]1/3, Q is the discharge, b is gate width, g is the acceleration due to gravity, y1 is the upstream depth, a and b are empirical coefficients that should be determined experimentally. According to Fig. 2 current experimental data of the single sluice gate are in a very good agreement with the experimentations carried out by Rajaratnam and Subramanya [4], Ferro [15], and Cassan and Belaud [17]. Also, the coefficients a and b were obtained as 0.8749 and 0.3455 respectively. If the head-discharge equation for a single sluice gate is employed to calculate the discharge through parallel sluice gates with closed gates, it is necessary to evaluate the performance of the current head-discharge equations to estimate the flow rate through parallel sluice gates. Accordingly, Eq. (1) was depicted along with the experimental data associated with the parallel sluice gates with symmetrical/asymmetrical closed gates in Fig. 3. As indicated in the figure for b/B<1 the overall trend of the data points associated with both symmetrical and asymmetrical cases lie lower than the head-discharge relationship of the single sluice gate. Consequently, the free flow formula of the single sluice gate is not able to accurately determine the discharge through parallel sluice gates including closed gates. It is interesting to note that, while the asymmetrical installation collapses all the compiled data very well on a single trend, slight deviation in the symmetrical installation data trends is observed (Fig. 3). The free flow condition through parallel sluice gates with closed gates can be expressed by the following functional relationship:
F1 ( Q, y1 , B, b, w, g , µ, ρ ) = 0
(2)
Where, F1 is a functional symbol, B is channel width, µ is the water viscosity, and ρ is water density.
Considering µ, w, and g, as dimensional independent parameters, the following five dimensionless variables were obtained: Π1 =
y1 ,Π w
2=
Q2 w 3/2 g 1/2 ρ B b , , , Π = Π = Π = 3 4 5 gw5 w w µ
Rearranging the dimensionless parameters yield:
Π 21/3 Q 2/3 K = 2/3 1/3 = 2/3 b g w w Π4 in which K is critical depth under the gate opening and
Π 5 Π 21/ 2 ρ Q = = Re Π3 µB in which Re is the Reynolds number Consequently, neglecting the effect of Reynolds number, the functional relationship, Eq. (2), takes the following form:
K §y b B · = F2 ¨ 1 , , ¸ w ©w w w ¹
(3)
The incomplete self-similarity theory (ISS) can be used to deduce the mathematical shape of the functional relationship (Eq. 3). For a given dimensionless group, Πn, a phenomenon is defined as self-similar when the functional relationship Π1 = ϕ1 ( Π 2 , Π 3 ,........., Π n ) is independent of Πn. In such a case, the behavior of ϕ1 is studied for Π n → 0 or Π n → ∞ . When the function ϕ1 has a limit equal to 0 or ∞, the phenomenon is expressed by the following functional relationship [18, 19]:
Π1 = Π εn ϕ2 ( Π 2 , Π 3 ,........., Π n −1 )
(4)
Considering Eq. (4), when y1 / w → 0 then K / w → 0 , and when y1 / w → ∞ then
K / w → ∞ , allowing the ISS to be used so that, Eq. (3) can be written as: b
f K §y · = Af ¨ 1 ¸ w ©w ¹
(5)
Where Af and bf are constant coefficients that should be obtained experimentally. It should be noted that Af should be considered as a function of b/w and B/w. Therefore the following expression was proposed: b
f K §y · = af ¨ 1 ¸ w ©w ¹
c
df
f §b · §B · ¨ ¸ ¨ ¸ ©w ¹ ©w ¹
(6)
Employing the experimental data compiled in the current study the empirical coefficients were obtained for both symmetrical and asymmetrical gate installations, i.e. b/B<1. It was found that the best solution would be obtained by considering d f = −c f , i.e., channel contraction ratio due to the closed gates (b/B) plays an important role. The results are given in Table 2. Table 2. Empirical coefficients of Eq. (6) Type
af
bf
cf
Symmetrical
0.8457 0.358 0.0415
Asymmetrical
0.821
0.346
0
The estimated values of K/w, obtained on the basis of employing Eq. (6) were compared to those
observed
experimentally
(Ke/w)
and
the
associated
relative
errors,
i.e.
R(%)=(K−Ke)/Ke×100, were calculated and depicted in Fig. 4. For the symmetrical installation case the maximum and mean absolute relative errors were obtained as 2.5% and 0.48% respectively. These values were 1.43% and 0.36% for the asymmetrical condition. It should be mentioned that no significant differences were observed between the empirical coefficients calculated for symmetrical and asymmetrical cases. While for the symmetrical case K/w relates to b/B with the power of 0.0415, no correlation was observed between K/w and b/B for the asymmetrical case. However, for the asymmetrical case it is anticipated that as b/B tends to unity the associated head discharge formula should tend to the single sluice gate curve. Consequently, more experimental study should be carried out for 0.65
§ K y1 · , , DRF , Re ¸ = 0 ©w w ¹
ψ¨
(7)
DRF is a Discharge Reduction Function defined by the following formula [9, 16]:
DRF =
(y 1 − y 3) λ1 ( y 3 − y * ) + ( y 1 − y 3 )
(8)
In which λ1 is a numerical constant and y* is the conjugated depth obtained by the classical hydraulic jump formula [20, 21]:
· y * Cc § y = ¨¨ 1 + 16( 1 − 1) − 1¸¸ w 2 © C cw ¹
(9)
Where Cc is the contraction coefficient. Neglecting the effect of Reynolds number and considering the ISS concept Eq. (7) reduces to: b
K §y · = a ¨ 1 ¸ DRF c w ©w ¹
(10)
In which a, b, and c are empirical coefficients. Employing the experimental data obtained in this study, Eq. (11) was developed for the single sluice gate:
K §y · = 0.77 ¨ 1 ¸ w ©w ¹
0.374
§ · (y 1 − y 3) ¨ ¸ © 3.506( y 3 − y * ) + ( y 1 − y 3 ) ¹
0.171
(11)
Based on Eq. (11) any parameters affecting the maximum tailwater depth for which the free flow condition occurs, i.e. y*, would affect the head-discharge curve and should be considered. Consequently, investigating the distinguishing condition curve of parallel sluice gates flow is necessary for obtaining appropriate water measurements. Therefore, experiments were carried out in order to clarify the case. In this regard, for a given parallel sluice gate dimensions, the free flow condition was established first. Then, in order to determine the threshold between free and submerged flow conditions, the tail water depth was increased incrementally until maximum change of 3 to 5% was observed in the upstream flow
depth. The distinguishing condition curves associated with different gate widths, including both symmetrical and asymmetrical cases, were compared with Eq. (9) (Fig. 5). The figure indicated that the distinguishing condition curve obtained based on the classical hydraulic jump (i.e. Eq. 9) accurately determines the threshold between free and submerged flow regimes for the single sluice gate. However, parallel sluice gates with closed gates tend to become submerged for smaller tail water depth. Consequently, Eq. (9) cannot be used to distinguish the flow condition through parallel sluice gates. The reason might be attributed to the fact that the momentum flux associated with the contracted section is less than that of the case with no contraction, i.e. single sluice gate condition. Accordingly, for the parallel sluice gates the gate width, b, should appear in the functional relationship of the distinguishing condition curve. Therefore, the following functional relationship was considered to describe the distinguishing condition curve of the parallel sluice gates:
y*p = f1 ( y j , b, B, Q, w, g , ρ, µ)
(12)
Where, f1 is a functional symbol, y*p is the maximum downstream depth for which the free flow condition occurs through parallel sluice gates, and yj(=Ccw) is the jet thickness. Considering µ, w, and g as dimensional independent parameters, the functional relationship, Eq. (12), takes the following form:
y *p w
= f 2(
yj b B K , , , , Re) w w w w
(13)
Neglecting the effect of the Reynolds number and considering the critical depth as a function of y1, and considering that the effects of b/w and B/w can be represented by b/B (as described for the free flow condition) Eq. (13) reduces to:
y *p yj
= f 3(
b y1 , ) B w
(14)
Considering the ISS hypothesis the mathematical form of Eq. (14) takes the following form: β
§y · §b · = αC c ¨ 1 ¸ ¨ ¸ w ©w ¹ ©B ¹
y *p
γ
(15)
In which, α, β, and γ are constant parameters. It is necessary to examine the value of the contraction coefficient for the case of parallel sluice gate with closed gates. As indicated in Fig. 6 an average value of Cc=0.61 was obtained which is equal to the common agreed value of 0.61 for single sluice gate [4, 22]. Considering Cc=0.61 and employing the experimental data of different values of b/B, the empirical coefficients of Eq. (15), i.e. α, β, and γ becomes as shown in Table 3. Eq. (15) is only valid for the experimental data ranges listed in Table 1. It must be recalibrated for any other data points which are out of the calibration ranges Table 3. The empirical coefficients of Eq. (15) Type
α
β
γ
Symmetrical
0.645 0.69 0.55
Asymmetrical
1.54
0.34 0.38
Tail water depth that exceeds the value of y*p obtained by Eq. (15) generates submerged flow condition. Plotting the experimental data on the distinguishing curve would greatly assist identifying the submergence level [23]. Experimental data points which are very close to the
distinguishing condition curve fall into low submergence zone. In other words the submergence ratio, (y3−y*p)/w, tends to zero for very low submerged conditions. As indicated in Figs. 7 and 8 for both symmetrical and asymmetrical cases, the experimental data cover the entire range of the submergence ratios including the low submergence zone. 4.1. Categorizing the downstream flow conditions for different tailwater levels Symmetrical installation By increasing the tailwater depth incrementally three submerged flow regimes were observed and identified (Fig. 9): (1)- For the tailwater depth slightly greater than y*p an oblique hydraulic jump was formed and the flow downstream the gate became asymmetrical having a reverse flow near the channel wall (Fig. 9a). Graber [24] was also reported the asymmetric flow in symmetric supercritical expansions. (2)- As shown in Figs. 7 and 8, for b/B<1 it was not possible to record any data points in a specific range. This range was termed “blank data zone”. That is, for a given value of y1/w, which was called y*L, the tailwater depth was suddenly increased. Exactly at this point it was observed that the oblique hydraulic jump started converting to a symmetrical submerged jump. However, the reverse flow was still intermittently observed. (3)- Beyond a certain value of the downstream depth called y*U, the submerged flow condition was always symmetrical. In such a case, two steady vortexes were formed just downstream the gate. Also, two symmetrical steady reverse flows were observed near the channel wall (Fig. 9b). The lower and upper limits of the blank data zone, i.e. y*L and y*U, were formulated and the following relationships were obtained:
y *L §y · = 3.35 ¨ 1 ¸ w ©w ¹
0.162
y *U §y · = 3.75 ¨ 1 ¸ w ©w ¹
0.238
§b · ¨ ¸ ©B ¹
0.376
§b · ¨ ¸ ©B ¹
(16) 0.445
(17)
According to the above discussion two different submerged flow zones were identified: a. Experimental data located between the distinguishing condition curve and the lower limit of the “blank data zone” which could be considered as low submerged range. b. Experimental data having the tail water depth of greater than y*U is called high submerged zone.
Asymmetrical installation Fig. (10) indicates the flow characteristics downstream an asymmetrical installation of parallel sluice gates. According to the figure for low submergence condition (Fig. 10a) an oblique like hydraulic jump was formed with a reverse flow near the channel wall. Similar to the symmetrical case the blank data zone was observed and beyond it the submerged hydraulic jumps became as that indicated in Fig.10b. A single large vortex was formed just after the gate with the reverse flow formed at the tailwater section. For the asymmetrical parallel sluice gate installation the lower and upper limits of the blank data zone, i.e. y*L and y*U, were formulated and the following relationships were obtained:
y *L §y · = 1.685 ¨ 1 ¸ w ©w ¹ y *U §y · = 2.26 ¨ 1 ¸ w ©w ¹
0.415
(18)
0.405
(19)
It should be noted that for the tested experimental data of the asymmetrical parallel sluice gate installation, the lower and upper limits of the blank data zone was independent of the variable b/B. However more experimental study should be carried out for 0.65
F ( Q , y1 , y3 , y* p , B , b, w, g , µ, ρ ) = 0
(20)
Considering the variables w, g and µ as dimensional independent parameters, the following seven dimensionless variables were obtained: Π1 =
y1 ,Π w
2
=
y *p y3 Q2 w 3/2 g 1/2 ρ B b , , , Π = , Π = , Π = Π = Π = 6 7 3 4 5 w w gw5 w w µ
Rearranging the dimensionless parameters yield:
Π 21/3 Q 2/3 K = = ; 2/3 2/3 1/3 Π4 b g w w Π1 − Π 6 =
y1 − y 3 ; w
λ1 ( Π 6 − Π 7 ) + ( Π1 − Π 6 ) = λ1
y 3 − y *p w
+
y1 − y 3 ; w
in which K is critical depth under the gate opening and
Π 5 Π 21/ 2 ρ Q = = Re Π3 µB in which Re is the Reynolds number Neglecting the effect of Reynolds number the dimensionless form of Eq. (20) would be: f
e1 b1 c1 d1 1 K § y · § b · § B · § y − y 3 · § y 3 − y *p y 1 − y 3 · = a1 ¨ 1 ¸ ¨ ¸ ¨ ¸ ¨ 1 + λ ¸ ¸ ¨ 1 w w w © w ¹ ©w ¹ ©w ¹ © w ¹ © ¹
(21)
There are some physical constrains that must be taken into account before calibrating Eq. (21): 1- For fully submerged flow condition, when y1 tends to y3, K/w must tend to zero; 2- For very low submergence condition, when y3 tends to y*p, Eq. (21) should converge to the free flow formula, i.e. Eq. (6). In order to apply the second constrain the coefficient d1 must be equal to −c1 and also f1=−e1. Consequently Eq. (21) takes the following form: e1
b1 c1 · y1 − y 3 K § y1 · § b · § = a1 ¨ ¸ ¨ ¸ ¨ ¸¸ ¨ w © w ¹ © B ¹ © λ1 ( y 3 − y * p ) + ( y 1 − y 3 ) ¹
(22)
According to Eq. (22) a very interesting dimensionless coefficient called Discharge Reduction Factor, DRF, was identified which can be written as:
DRFp =
(y1 − y 3) λ1 ( y 3 − y * p ) + ( y 1 − y 3 )
(23)
For the free flow condition, i.e. y3=y*p, DRF is unity. Also, increasing the tailwater level and making the gate to be submerged it would be decreased, so that for the fully submerged flow condition it would be zero. Consequently DRF is a coefficient ranging from 0 to 1 and introduces the level of submergence. It should be mentioned that to achieve a continuous head-discharge formula, the submerged and free flow data points were used all together to calibrate Eq. (22). In this regard, DRFp was considered to be unity for free flow data points. The coefficients λ1, a1, b1, c1, and e1 were listed in Table 4.
Table 4. The empirical coefficients of Eq. (22) for both symmetrical and asymmetrical gate configurations λ1
Type
a1
c1
b1
e1
Symmetrical
0.1127 0.838 0.349 0.018 2.18
Asymmetrical
0.161
0.885 0.332 0.045 3.02
Eq. (22) could be used for both free and submerged flow (y*p≤y3≤y*L) conditions. The estimated values of K/w, obtained by using Eq. (22) were compared to those observed experimentally (Ke/w) and the associated relative errors, i.e. R(%)=(K−Ke)/Ke×100, were calculated and depicted in Fig. 12 for free and low submerged data of both symmetrical and asymmetrical cases. For y3≥y*U the functional relationship takes the following form:
F ( Q, y1 , y3 , y*U , B, b, w, g , µ, ρ ) = 0
(24)
Similarly, neglecting the effect of Reynolds number and considering the ISS concept Eq. (24) reduces to: b
c
2 2 K §y · §b · = a2 ¨ 1 ¸ ¨ ¸ DRFU e 2 w ©w ¹ © B ¹
(25)
In this case the DRF was considered as the following form:
DRFU =
(y1 − y 3) λ2 ( y 3 − y *U ) + ( y 1 − y 3 )
(26)
Using the complied data the coefficients λ2, a2, b2, c2, and e2 were calculated (Table 5). Table 5. The empirical coefficients of Eq. (25) for both symmetrical and asymmetrical gate configurations Type
λ2
a2
Symmetrical
0.408
0.64
Asymmetrical 0.795 0.568
b2
c2
e2
0.414 0.003 0.695 0.43
0.027 0.458
The relative error associated with Eq. (26) compared to the experimental data of the high submergence zone were calculated and depicted in Fig. 13.
5. Conclusion The flow characteristics of flow through parallel sluice gates with closed gates were studied under both free and submerged flow regimes. The threshold between free and submerged
flow conditions, i.e. the distinguishing condition curve, was formulated for both symmetrical and asymmetrical parallel sluice gates’ configurations. Then, the Π-theorem and the incomplete self-similarity concept were employed to develop stage-discharge relationships. Finally, the proposed formulas were calibrated using the experimental data obtained in the current study. 6. Acknowledgements The research was supported by the “Center of Excellence for Evaluation and Rehabilitation of Irrigation and Drainage Network” and the University of Tehran grant No. 7102021/1/06 . Both supports are highly appreciated. 7. Notation a1, b1, c1, d1, λ2, a2, b2, c2, and d2 = empirical coefficients; af, and bf = constant parameters B = channel width; b = gate width; Cc = contraction coefficient; DRF = Discharge Reduction Function; f1 = a functional symbol; F1 = functional symbol; g = acceleration due to gravity; K = critical depth; Ke/w = experimental values of K/w Q = discharge; Re = Reynolds number; Relative error (R) =(K−Ke)/Ke×100, y* = maximum downstream flow depth for which the free flow condition occurs; y*L and y*U = The lower and upper limits of the blank data zone;
y*p = maximum downstream depth for which the free flow condition occurs through parallel sluice gates; y1 = the upstream depth; yj(=Ccw) = jet thickness; ψ = a functional symbol; ρ = water density; µ = water viscosity; α, β, and γ = constant parameters; 8. References [1] Clemmens, A. (2004) Avoiding Submergence Transition Zone for Radial Gates in Parallel. Critical Transitions in Water and Environmental Resources Management: pp. 1-10. [2] Henry H. Discussion: diffusion of submerged jet. Transactions of the American Society of Civil Engineers 1950;115:687–97. [3] Swamee PK. Sluice-gate discharge equations. Journal of Irrigation and Drainage Engineering 1992;118(1):56–60. [4] Rajaratnam N, Subramanya K. Flow equation for the sluice gate. Journal of Irrigation and Drainage Engineering 1967;93(3):167–86. [5] Rajaratnam N, and Subramanya K.. Flow immediately below submerged sluice gate. J. Hydr. Div., 1967;93(HY4), 57–77. [6] Sepulveda C, Gomez M, Rodellar J. Benchmark of discharge calibration methods for submerged
sluice
gates.
Journal
of
Irrigation
and
Drainage
Engineering
2009;135(5):676–82. [7] Lozano D, Mateos L, Merkley GP, Clemmens AJ. Field calibration of submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2009;135(6):763–72 [8] Castro-Orgaz O, Lozano D, Mateos L. Energy and momentum velocity coefficients for calibrating submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2010;136(9):610–6. [9] Bijankhan M, Ferro V, and Kouchakzadeh S. New Stage-Discharge Relationships for Radial Gates. J. Irrig. Drain Eng., 2013;139(5), 378–387.
[10] Khalili Shayan H, and Farhoudi J. Effective parameters for calculating discharge coefficient of sluice gates. J. Flow measurement and instrumentation., 2013;33, 96-105. [11] Castro-Orgaz O, Mateos L, and Dey S. Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions. J. Irrig. Drain Eng., 2013;139(4), 325–335. [12] Nago H, and Furukawa S. Discharge coefficient of a sluice gate placed at sudden expansion of open channel. Memoirs of the School of Engineering,Okayama University 1979; 14(1), 127–138. [13] Clemmens AJ, Strelkoff TS, and Replogle JA. Calibration of Submerged Radial Gates. J. Hydraul. Eng., 2003;129(9), 680-687. [14] Bijankhan M, Kouchakzadeh S, Bayat E. Distinguishing condition curve for radial gates. Journal of flow measurment and instrumentation 2011;22(5): 500–6. [15] Ferro V. Simultaneous flow over and under a gate. Journal of Irrigation and Drainage Engineering 2000;126(3):190–3. [16] Bijankhan M, Ferro V, and Kouchakzadeh S. New stage-discharge relationships for free and submerged sluice gates. J. Flow measurement and instrumentation 2012; 28, 50-56. [17] Cassan L, Belaud G. Experimental and numerical investigation of flow under sluice gates. Journal of Hydraulic Engineering 2012;138(4):367–73. [18] Barenblatt GI. Similarity, self-similarity and intermediate asymptotics. New York: Consultants Bureau; 1979. [19] Barenblatt GI. Dimensional analysis. Amsterdam: Gordon and Breach, Science Publishers Inc.; 1987. [20] Bos MG. Discharge measurement structures. Wageningen, the Netherlands: ILRI; 1989. [21] Lin CH, Yen JF, Tsai CT. Influence of sluice gate contraction coefficient on distinguishing condition. Journal of Irrigation and Drainage Engineering 2002;128(4):249–52. [22] Henderson, F.M. Open Channel Flow, Macmillan, New York. 1966. [23] Bijankhan M, Darvishi E, and Kouchakzadeh S. Discussion of “Energy and Momentum Velocity Coefficients for Calibrating Submerged Sluice Gates in Irrigation Canals” by Oscar Castro-Orgaz, David Lozano, and Luciano Mateos. J. Irrig. Drain Eng., 2012;138(9), 852–854. [24] Graber, S. ”Asymmetric Flow in Symmetric Supercritical Expansions.” J. Hydraul. Eng., 2006;132(2), 207–213.
Figure Captions Fig. 1. Schematic sketch of the parallel sluice gates configuration under free and submerged flow conditions of low flow delivery condition Fig. 2. K/w versus y1/w associated with the single sluice gate of the current study in comparison with the other experimental data Fig.
3.
K/w
versus
y1/w
associated
with
parallel
sluice
gates
including
symmetrical/asymmetrical closed gates Fig. 4. The relative error distribution associated with symmetrical and asymmetrical gate installations Fig.5. distinguishing condition curves associated with different gate widths, including both symmetrical and asymmetrical cases Fig. 6. The contraction coefficient versus y1/w for symmetrical and asymmetrical parallel sluice gate installations Fig. 7. Experimental data points plotted on the distinguishing condition curves for symmetrical case Fig. 8. Experimental data points plotted on the distinguishing condition curves for asymmetrical case Fig. 9. Flow characteristics downstream a symmetrical installation of parallel sluice gates Fig. 10. Flow characteristics downstream an asymmetrical installation of parallel sluice gates Fig. 11. Absolute relative error distribution to predict the discharge through parallel sluice gates by employing the head-discharge relationship of a single sluice gate, a)Symmetrical, b)Asymmetrical Fig. 12. Relative error distribution associated with Eq. (23) for determining the discharge through parallel sluice gates, a) symmetrical b) asymmetrical Fig. 13. Relative error distribution associated with Eq. (25) for determining the discharge through parallel sluice gates
9. Research highlights ¾ The flow through submerged and free parallel sluice gate is considered under low flow delivery condition. ¾ The head-discharge curve of the single sluice gate cannot be used for the parallel sluice gates. ¾ New distinguishing condition curves were developed to identify the flow regime through parallel gates. ¾ Buckingham theorem and Incomplete Self Similarity concepts were used to develop a new head-discharge relationship.
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Figure 13