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Application of the submerged experimental velocity profiles for the sluice gate's stage-discharge relationship
Author’s Accepted Manuscript Application of the submerged experimental velocity profiles for the sluice gate's stage-discharge relationship Mohammad Bijankhan, Salah Kouchakzadeh, Gilles Belaud www.elsevier.com/locate/flowmeasinst
PII: DOI: Reference:
S0955-5986(16)30264-3 http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.009 JFMI1293
To appear in: Flow Measurement and Instrumentation Received date: 22 July 2016 Revised date: 27 November 2016 Accepted date: 30 November 2016 Cite this article as: Mohammad Bijankhan, Salah Kouchakzadeh and Gilles Belaud, Application of the submerged experimental velocity profiles for the sluice gate's stage-discharge relationship, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.009 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.
Application of the submerged experimental velocity profiles for the sluice gate’s stage-discharge relationship Mohammad Bijankhana, Salah Kouchakzadehb, Gilles Belaudc a
Department of Water Engineering, Faculty of Engineering and Technology, Imam Khomeini International University,
Qazvin, Iran. b
c
Irrigation and Reclamation Engrg. Dept., University of Tehran, P.O. Box 31587-4111, Karaj, 31587-77871, Iran.
UMR G-EAU, SupAgro, place Pierre Viala, 34060 Montpellier Cedex 1, France.
Email: [email protected] Email: [email protected] Email: [email protected]
Abstract Sluice gates have been widely used and intensively studied, however their submerged flow conditions still call for in depth attention. A large scale experimental setup equipped with Acoustic Doppler Velocimetry, ADV, and electromagnetic flow-meter was used to thoroughly investigate various aspects of the hydraulics of submerged sluice gate. In this study, new experimental data sets are provided, that help better understand and quantify the flow features for submerged sluice gates. According to the experimental data generic fitting are provided for the velocity profiles from which the velocity correction factors can be obtained. Then, the experimentally obtained submerged head loss coefficient is presented and discussed. The results of this study showed that current classical Energy-Momentum methods (EM) failed to accurately determine the flow rate for the cases of highly submergences, while employing the interaction of the energy correction factors and head loss values in the EM model would result in more accurate head-discharge estimation. The new data set provided in this work can be used effectively for the validation of numerical modeling of submerged sluice gates. 1
Keywords: sluice gate, velocity profile, submerged flow, energy and momentum equations, flow measurement. Notations Cc=contraction coefficient; d1 (L)= flow depth with the associated velocity of um/2; d3 (L)= the depth of the reverse flow region; g (Ls-2)= the acceleration due to gravity; k = experimental coefficient; K=energy loss coefficient; MARE = Mean Absolute Relative Errors; Q (L3s-1) = flow rate; q (L2s-1) =discharge per unit width; qB (L2s-1) = backwater flow rate; qF (L2s-1) = forward discharge; Qm (L3s-1)= the measured discharges; RMSE = Root Mean Square Errors; SR= Submergence Ratio, (y1 y3)/w; T = z/d1; U = q/y; um (Ls-1) = maximum velocity; ur (Ls-1) = roller flow velocity;
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us (Ls-1) = surface velocity; Vj (Ls-1) = jet velocity; w (L) =gate opening; x (L) =horizontal distance; y (L) =the submerged flow depth just after the gate; yj (L) = jet thickness; y1 (L) = upstream water depth; y3 (L) = tail water depth; yc (L) =critical depth; z (L) = distance from the channel bottom; zu (L) = thickness of the potential velocity core;
=Boussinesq coefficient; =the Coriolis coefficient; H (L) = head loss;
Introduction Gates are widely used in open channel flows as water regulator and flow measurement devices. According to the tailwater depth value and gate opening the flow conditions through a vertical sluice gate can be classified as: 1- Free flow condition; which occurs when a free hydraulic jump formed downstream the gate. 2- Submerged flow condition (Fig. 1); which takes place when the tailwater depth increases beyond the conjugated depth. 3
The classical energy and momentum (EM) methods could be used to determine the discharge through free or submerged flow conditions [1]. According to the studies of Henry [2] and Sepulveda et al. [3] the conventional EM method can be used accurately for flow measuring through a vertical sluice gate. While, Lozano et al. [4] observed significant deviations between the measured discharges and calculated values by applying the EM method to calibrate vertical sluice gates of irrigation networks. They considered the contraction coefficient, Cc, as a calibrating parameter to improve the performance of the conventional EM method. Later, Castro-Orgaz et al. [5] indicated that calibration of the contraction coefficient indirectly incorporates the effects of non-uniform velocity distributions into the energy and momentum fluxes. Consequently, they included the kinetic and momentum fluxes for a total depth of y, into the classical EM method and incorporated the Coriolis, , and Boussinesq, , coefficients to account for the highly non-uniform velocity profiles of the roller zone. Castro-Orgaz et al. [6] discussed that the roller momentum flux and energy loss are also important and cannot be ignored. They considered a constant vorticity velocity profile in the roller, thereby obtaining analytical solutions for the momentum correction coefficient and the head loss through vertical sluice gates. They evaluated their method using experimental data available in the literature and reported satisfactory results. Belaud et al. [7] indicated also that the method of Castro Orgaz et al. [6] neglected the friction forces, and assumed that contraction coefficient (Cc) is the same in submerged flow as in free flow. However, this assumption has been questioned by Belaud et al. [8]. The submerged contracted coefficient is a key parameter in most of the energy and momentum methods used for estimating the submerged discharge values. There are however different kinds of assumptions in the literature for considering the amount of the jet thickness at the contracted area. Woycicki [9] indicated that Cc is directly related to the ratio of gate opening to the upstream flow depth, w/y1, for the submerged flow condition. Rajaratnam and Subramanya [10] performed four experimental runs and collected only the forward flow velocity profiles for 0.167
constant value of 0.61 for the submerged contraction coefficient. Using field data and applying energy and momentum equations, Lozano et al. [4] indicated that considering Cc-values between 0.629 and 0.652 would improve the accuracy of discharge estimation. Applying the energy equation and employing the experimental data of Fardjeli [11], Belaud et al. [8] calculated submerged Ccvalues indirectly. They concluded that the submerged Cc can reach values as high as 0.8 and the energy and momentum balance can be used to predict Cc. However, Habibzadeh et al. [12] considered a constant value of 0.611 for the submerged contraction coefficient and investigated the role of energy loss on discharge characteristics of the submerged sluice gates. Cassan and Belaud [13] studied the flow characteristics upstream and downstream of a submerged sluice gate experimentally and numerically. Using ADV measurements, they indicated that the flow condition can be considered two dimensional and applied a 2D- numerical model to determine the velocity fields and the associated Cc-values. The results were also consistent with the energy and momentum balance proposed by Belaud et al. [8]. As mentioned by Belaud et al. [7], Castro Orgaz et al. [6], and Belaud et al. [8] taking Cc as a constant while calibrating the head-discharge formula can be counterbalanced by some other simplifying assumptions or by changing other experimental coefficients like energy and momentum correction factors, friction, and head loss. This means that the EM method, although it may appear as physically-based, remains an empirical method, so fitted relationships may not apply universally [7]. The lack of the knowledge from the velocity distribution of the flow just after the gate would make the researchers consider many hypotheses to solve the energy and momentum formulas. In this paper, the velocity distribution profiles for the submerged flow downstream the gate were studied experimentally. Based on the real velocity profile functions the energy, , and momentum, , correction coefficients were determined. In addition, the compiled data were used to evaluate and report the submerged head-loss coefficients. Note that the proposed energy and momentum sets of
5
equations are independent of the submerged contraction coefficient values which are associated with many uncertainties. Finally, it was indicated that the available classical EM method showed more than 60% relative error in discharge prediction for the highly submerged flow condition, the proposed herein EM method was developed and proposed in this paper. Experimental setup To evaluate different submerged jet types and the performance of different EM methods it was necessary to compile reliable experimental data set covering submerged wall jet from low to high submerged flow conditions, given that it is scarcely available. The experiments were performed in a 1.179 m wide, 1 m height and 7 m long Plexiglas flume located at the Central Hydraulic Laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran (Fig. 2). The flume was supplied by an elevated constant head tank and an electromagnetic flow-meter with an accuracy of ±0.5% of the full scale was installed on the feeding pipe to accurately 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 flume downstream end. Gate openings, w, of 46.5 and 68.3 mm were considered. In order to accurately set the gate opening, prefabricated elements with an accuracy of 0.1 mm were located below the gate, and removed after the gate was positioned [14]. Considering the gate openings of w=46.5 and 68.3 mm, and installing the ADV instrument just after the gate at x/w2 where the hydrostatic pressure distribution occurs [10], the velocity profiles associated with the vena contracta were collected. The velocity profiles were also measured at the tailwater section located just after the roller zone where the release of bubbles was minimized. These experimental data were then used to calculate the velocity correction factors associated with the tailwater section. ADV measurements
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A vectrino+ Nortek Acoustic Doppler Velocity meter system, ADV, equipped with a side looking 3D probe and sampling rate of 200 Hz was used to measure the velocity profiles. Considering different recording times of 15, 30, 45, 90, and 180 seconds, it was found that the mean velocity values associated with a specific node remain almost constant for the recording time of greater than 30 seconds. Consequently, data recording time of 30 seconds was considered for compiling the data and for each node almost 12000 samples were collected. According to Wahl [15], and introducing the so-called WinADV software, for a given data set the results were filtered by considering the average signal-to-noise ratio (SNR) and correlations of 15 and 70 respectively. Nominal velocity range of 4 m/s, transmit length of 1.8 mm, and sampling volume of 7 mm were found suitable to have more than 90% of the data fall within the minimum SNR and correlation of 15 and 70 respectively. However, the minimum correlations were observed at the shear layer between the jet and its above roller zone. One problem with ADV measurements is spikes caused by aliasing of the Doppler signal. The effect of spikes would be highlighted when the flow velocities exceed the preset velocity value or the bed roughness affects the flow near the bed [16]. Considering the velocity range of 4 m/s and the smooth channel bed the measured velocities did not contain many spikes. Such a condition was also reported by Dey and Sarkar [17] for the submerged jumps. However, in order to ensure the measuring accuracies especially near the bed the despiking method of Goring and Nikora [16] which is included in WinADV software was used. Considering z as the distance from the channel invert, the side-looking 3D probe of the ADV instrument allowed velocity measurement in the range of 5z(mm)y210. The depth interval of 5 and 10 mm were considered for 0z (mm)50, and 50
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The experimental velocity profiles obtained in this study are provided in the Appendix 1. Nice to mention that, there is no comprehensive source in the literature to find such dataset of the submerged velocity profiles, and consequently, the original dataset presented in this work can be used effectively to increase the current knowledge about the submerged flow through vertical sluice gates. Developing the velocity profile formula at the contracted section Forward Flow Following the study of Schwarz and Cosart [18], Rajaratnam [19] investigated the submerged hydraulic jump on the basis of wall jet theory. He proposed a general definition of the submerged velocity profile by dividing the submerged flow into forward and backward flows (Fig. 3). Defining the flow depth d1, as the depth with associated velocity of um/2 (um is the maximum velocity); plotting u/um versus z/d1 would result in a general definition for the forward flow [19]. The experimental velocity profiles associated with the vena contracta section collected in this work are plotted in normalized coordinates in Fig. 4. As revealed, all velocity profile data associated with the forward flows collapse to a single curve. Also, the velocity profile at the vena contracta section includes a potential core, i.e. a region of ideal fluid flow, and a shear layer above. As indicated in Fig. 4 the velocity profile associated with the vena contracta section collected by Rajaratnam and Subramanya [10] agrees remarkably well with the current experimental data. However, significant difference between the new velocity profiles and the wall jet theory proposed by Rajaratnam [19] was observed. The wall jet theory is applicable for the fully established flow, i.e. x/w15 [19], and therefore it cannot be used to study the flow at the vena contracta section. The thickness of the potential velocity core, zu, (Fig. 4) is approximately zu/d1=0.66. Neglecting the bottom boundary layer thickness and considering no bed friction, the following empirical equation was then found to describe the non-dimensional forward flow velocity profile at the vena contracta:
8
1 u 2 z z u m 0.6375 2.5972 2.45 d1 d1
0 z / d 1 0.66 0.66 z / d 1 1.483
(1)
According to the experimental data and regression analysis, the flow velocity in the roller zone for the value of z/d1 equal to 1.483 was considered to be zero. Consequently, Eq. (1) should be applied in the range of 0z/d11.483 and for z/d1>1.483 the negative velocity values would be observed. Plotting um versus q/w in Fig. 5, indicates that the observed data could be described by the following relationship:
um 1.48
q w
(2)
According to Eq. (2), due to the jet contraction, the maximum velocity associated with the vena contracta section is averagely 1.48 times greater than the mean velocity of the emerging jet. Also, the normalized depth d1/w, is depicted versus y/w in Fig. 6 which shows that an average value of 0.78 can be considered for d1/w. Backward Flow There are very limited experimental data associated with the backward flow through a submerged hydraulic jump. Rajaratnam [19] employed the experimental data of Liu [20] to investigate the backward velocity distribution. Castro Orgaz et al. [6] considered the following formula for the velocity profile within the eddy:
z C cw u u r 1 2 y C cw
(3)
ur (=0.5q/Ccw) is the roller flow velocity. Note that the coefficient 0.5 was considered as a calibration coefficient to have the best fit of the theoretical Boussinesq coefficient to the experimental data [6]. Belaud et al. [7] indicated that the roller flow velocity should be considered as a function of the submergence ratio. They reported also that the coefficient 0.5 (ur =0.5q/Ccw) 9
could not be always correct and very small values were reported especially for highly submerged flow conditions. Taking a linear velocity profile for the backwater flow rate, qB, the backwater velocity distribution associated with 1.483d1z
u
u r z 1.483d 1
(4)
y 1.483d 1
Integrating Eqs. (1) and (4) the forward and backward flow rates, i.e. qF and qB respectively, would take the following forms:
q F 1.018d 1u m qB
0 z 1.483d 1
( y 1.483d 1 ) u r 1.483d 1 z y 2
(5)
Substituting q, qF, and qB from Eqs. (2) and (5) into q=qF+qB, the ratio ur/um would be: u r 1.307 2.036d 1 / w u m y / w 1.483d 1 / w
(6)
Note that ur-values are always negative. Also for the current experimental data the ratio |ur/um| varies from 0.017 for highly submerged flow condition to 0.25 for low submergence regime. In other words the reverse flow through the roller zone associated with a low submerged flow is significantly higher than that of highly submerged regime. This phenomenon however cannot be described by the fixed coefficient of 0.5 proposed by Castro Orgaz et al. [6]. Coriolis and Boussinesq Coefficients As a result of non-uniform velocity distribution the true velocity head and momentum of a section would be αV2/2g and βq2/gy respectively, where α and β are the energy and momentum correction coefficients. The fundamental definitions of the energy and momentum correction coefficients are [21]:
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y
u dz 3
0
(7)
V3y y
u dz 2
0
(8)
V2y
In order to formulate the velocity correction factors, the proposed velocity distributions at the vena section, i.e. Eqs. (1) and (4), were then used. Accordingly, the following one dimensional predictors of and were found for the vena contracta section:
0.89d 1u m2 u r2 / 3 y 1.483d 1
(9)
q2 / y 0.83d 1u m3 u r3 / 4 y 1.483d 1
(10)
q3 / y 2
Considering um=1.48q/w and simplifying Eqs. (9) and (10), yield:
1.52
y 0.057 y / w w y / w 1.157
(11)
2
0.018( y / w)2 y 2.1 2 w (y/ w 1.157)
(12)
Considering Cc=0.61 and a linear velocity distribution in the roller, with a maximum velocity set at the half of the jet velocity, Castro Orgaz et al. [6] obtained the following relationships for and at the vena contracta section respectively:
y C cw
2
(13)
y 1 y 1 1 C cw 12 C cw
(14)
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The velocity correction factors, α and β were plotted in Fig. 7 versus y/w for the method of CastroOrgaz et al. [6] and the results of the current study. According to the figure the velocity correction factors obtained by Eqs. (13) and (14) are always greater than the experimentally obtained values. This is consistent with the remarks formulated by Belaud et al. [7], based on experimental and numerical results: it seems that the values chosen by Castro-Orgaz et al. [6] to initiate the linear velocity profiles in the roller are usually over-estimated; secondly, the hypothesis of having the fixed values of Cc=0.61 for the submerged flow conditions in Eqs. (13) and (14) is not always true [8]. However, Eqs. (11) and (12) which are based on the real velocity profiles can be used more accurately to calculate the velocity correction factors. Due to the inherent uncertainties of calculating the real Cc-values its use may be avoided in the velocity distribution for the whole section, i.e. jet plus roller zone. This would highlight again the necessity of using the real velocity profiles to calculate the velocity correction factors being independent of the contraction coefficient. Head loss For the submerged gates, the turbulence due to the recirculation zone as well as the shear layer between the jet and the roller zone are key parameters which are responsible for the energy loss. The energy balance between the upstream and vena contracta sections can be written as [5,6]: 2
Vj q2 q2 y1 y K 2 2 2 gy 1 2 gy 2g
(15)
where Vj= [2g(y1−y)]0.5. The head loss coefficients were depicted versus the submergence ratio, SR=(y1 y3)/w in Fig. 8. The figure clearly shows that the head loss coefficient increases significantly as the submergence ratio increases. The following explicit empirical formula was then developed to calculate head loss coefficients:
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1.48 0.51SR 2.33 K 13.5 SR 2.33
(16)
Tailwater section velocity profiles The velocity profiles at the end of the roller zone, where the jet thickness consists more than 95% of the flow depth, were considered for the tailwater section. It was also observed that beyond this point the water depth remained constant. The normalized velocity profiles of different submergence conditions were depicted in Fig. 9. The velocity correction coefficients associated with the tailwater section, i.e. 3 and 3, were then calculated and average values of 3= 2.7 and 3=1.61 were considered. To develop a new energy-momentum model Eqs. (17) and (18) were used to apply the results of this study. Eqs. (11) and (12) were employed to calculate and respectively. The submerged head loss values were also obtained by Eq. (16) and an average value of 3=1.61 was considered. Note that due to using the real velocity profiles the proposed energy and momentum method is independent of the contraction coefficient.
y1
q2 q2 y H 2 gy 12 2 gy 2
y2 q 2 y 32 q2 3 2 gy 2 gy 3
(17)
(18)
Assessment of the available EM methods Review of current EM formulations Applying the energy principle between sections 1 and 2 and the momentum equation to sections 2 and 3 (Fig. 1), Henderson [1] proposed the following set of equations which could be solved for the unknowns y and q:
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q2 q2 y1 y 2 gy 12 2 gC c2w 2
(19)
y 32 q 2 y2 q2 2 gC cw 2 gy 3
(20)
where, q is the discharge per unit width, g is the acceleration due to gravity, and y is the flow depth just after the gate. Henderson’s [1] EM method is based on a static roller assumption with Cc=0.61. The effects of the non-uniform velocity profile and the head loss through the submerged sluice gate were neglected [6]. Considering the role of the energy loss Habibzadeh et al. [12] proposed the following energy and momentum formulas:
y1
q2 q2 y (1 K ) L 2 gy 12 2 gC c2w 2
y 2 q2 y2 q2 3 2 gC cw 2 gy 3
(21)
(22)
where KL is the energy loss coefficient. Employing the experimental data of Rajaratnam and Subramanya [22] they proposed KL=0.088 for the submerged flow condition. Note that KL would also include the effect of non-uniform velocity, which is considered in α in Eq. (15). Therefore it is not exactly the same K as in Eq. (15). Castro Orgaz et al. [5] questioned the validity of considering static roller concept. According to the velocity profiles available in the literature they indicated that the energy and momentum velocity correction coefficients, i.e. and respectively, are significantly greater than unity. Consequently, they considered the general energy and momentum relationships:
q2 q2 y1 y 2 gy 12 2 gy 2
(23)
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y2 q 2 y 32 q 2 2 gy 2 gy 3
(24)
Castro Orgaz et al. [5] proposed =1.7 and =k(y3/w). According to the data obtained from four gates operating in an irrigation canal in Southern Spain, Castro Orgaz et al. [5] found that k=2 to 2.45. It should be mentioned that the proposed values of the exponent of 1.7 in the equation for α=α(β) is valid only for highly submerged flows. The parameter k is the result of a local calibration and is only valid for the gates investigated in Castro-Orgaz et al. [5]. Castro Orgaz et al. [6] applied the energy and momentum equations in a general integral form, resulting for rating vertical sluice gates under submerged flow condition:
q2 q2 y1 y H 2 gy 12 2 gy 2 y2 q 2 y 32 q 2 2 gy 2 gy 3
(25)
(26)
in which H is the head loss, α and β is obtained by Eqs. (13) and (14). The corresponding analytical expression for the head-loss coefficient is [6]:
H K
V j2
where, K
(27)
2g
y y 1
0.5
10 2 gC c
.
Experimental assessment of EM formulations There are rare experimental works in order to compare the performance of different EM methods for low to fully submerged flow regimes. In order to evaluate the accuracy of the current energy momentum methods, 71 experimental runs were considered as listed in Appendix 2. Note that the
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velocity correction factors and head loss coefficient were found by 17 data points and other 54 runs were not used before to find the required parameters of the presented energy momentum model. According to the experimental observations, the measured discharges (Qm) were compared to those obtained by the EM methods (Q) and the relative errors ([Q-Qm]/Qm100) were depicted versus the submergence ratio, SR, Fig. 10. For each method, the values of Mean Absolute Relative Errors, MARE (Eq. 28), and absolute maximum errors were calculated and listed in Table 1.
n MARE Q Q m / Q m / n 100 i 1
(28)
For the low submerged jets (SR≥0.5), the method of Henderson [1], i.e. classical EM model, is as accurate as the model proposed by Castro Orgaz et al. [6] which is a modified version of Castro Orgaz et al. [5]. The associated relative errors for both methods are bounded to 6%. The method proposed by Habibzadeh et al. [12] underestimated the discharges for all the current experimental data with the relative errors ranging from 0.6% to 13.5%. Their model was based on calibrating the head loss coefficient for only one experimental dataset. The recalibrated model might result in better performance. Also, the method of Castro-Orgaz et al. [5] was a calibration for a field channel. For the highly submerged flow condition (SR<0.5), the values of relative errors listed in Table 1 indicated that the absolute maximum relative error associated with the EM models proposed by Henderson [1], Habibzadeh et al. [12], Castro Orgaz et al. [5], Castro Orgaz et al. [6] and the current study were 61.2%, 46.5%, 46.2%, 24.1% and 21% respectively. Consequently, the EM models proposed by Henderson [1], Habibzadeh et al. [12], Castro Orgaz et al. [5], for the highly submerged flow condition, are far away from acceptable accuracies. However the methods of Castro Orgaz et al. [6] and current study are the best, among the available EM models. Note that, as indicated by Henry [2] the head discharge curve of a submerged sluice gate is very sensitive for highly submerged flow conditions. That is, a very small change in the influencing parameters would 16
significantly affect the submerged discharge coefficient and consequently it is inherently difficult to get a better accuracy for largely submerged jets, i.e. SR<0.5. Therefore, the performances of the EM models used to rate the submerged vertical sluice gates depended significantly on the submergence level. In summary, the current study demonstrated that the available submerged Cc-values are difficult to evaluate, but its use can be avoided with a correct consideration of velocity profiles. Consequently, employing the dimensionless form of the submerged velocity profiles, the experimental coefficients such as velocity correction factors (α and β) and head loss can be directly obtained. These results are less dependent on the empirical coefficients and can be considered as universal stage-discharge relationships. Conclusion The submerged jets emerging below a vertical sluice gate is considered in the current study. ADV measurements were used to develop a new dimensionless formula describing the velocity profile at the vena contracta section of low to high submerged flow conditions. The energy and momentum correction coefficients were then determined using the real experimental velocity profile distributions. Experimental observations indicated that the energy loss coefficient and submerged contraction coefficient for highly submerged jets are significantly greater than that of low submerged flow conditions. Employing new experimental data obtained in this work, it was also indicated that the available EM models failed to accurately determine the discharge for fully submerged jets. Finally, the results of the current study were incorporated in the energy and momentum formulas to calculate the submerged discharges. Due to the uncertainties attributed to the calculation of the submerged contraction coefficient, the proposed energy momentum model was developed to be independent of submerged Cc values. Comparing with the current experimental data it was revealed that the results of this study can effectively be used to improve the performance of the classical EM methods especially for the fully submerged wall jets.
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Acknowledgement The authors gratefully acknowledge the financial support (Grant No. 7102021/01/07) and the research facility provided by the University of Tehran and the Center of Excellence for Evaluation and Rehabilitation of Irrigation and Drainage networks. Also, Valuable comments and ideas given by Dr. Oscar Castro-Orgaz and Dr. Luciano Mateos are highly appreciated.
References [1] Henderson F.M. Open channel flow, Macmillan, New York, 1966. [2] Henry H. Discussion: diffusion of submerged jet. Transactions of the American Society of Civil Engineers 1950;115:687–97. [3] 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. [4] Lozano D, Mateos L, Merkley G, and Clemmens A. Field Calibration of Submerged Sluice Gates in Irrigation Canals. Journal of Irrigation and Drainage Engineering 2009;135(6):763– 772. [5] Castro-Orgaz O, Lozano D, Mateos L. Energy and momentum velocity coefficients for calibration of submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2010;136(9):610-616. [6] Castro-Orgaz O, Mateos L, and Dey S. Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions. Journal of Irrigation and Drainage Engineering 2013;139(4):325–335. [7] Belaud G, Cassan L, and Baume J. Discussion of “Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions” by Oscar Castro-Orgaz, Luciano Mateos, and Subhasish Dey. Journal of Irrigation and Drainage Engineering 2014;140(7):07014019.
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[8] Belaud G, Cassan L, Baume J.P. Calculation of contraction coefficient under sluice gates and application to discharge measurement”. Journal of Hydraulic Engineering 2009;135(12):10861091. [9] Woycicki K. Wassersprung, Deckwalze und Ausfluss unter einer Schütze (Hydraulic jump, roller and outflow from below a gate). Dissertation 639. ETH Zurich, Zürich, 1931 [in German]. [10] Rajaratnam N, and Subramanya K. Flow immediately below a submerged sluice gate. Journal of Hydraulic Division 1967;93(HY4):57–77. [11] Fardjeli N. Modélisation d’ouvrages de régulation pour l’aide à la gestion des canaux. MS thesis, Montpellier Supagro, France, 57, 2007 (in French). [12] Habibzadeh A, Vatankhah Ali R, Rajaratnam N. Role of energy loss on discharge characteristics of sluice gates. Journal of Hydraulic Engineering 2011;137(9):1079–84. [13] Cassan L, and Belaud G. Experimental and Numerical Investigation of Flow under Sluice Gates. Journal of Hydraulic Engineering 2012;138(4):367–373. [14] Roth A, and Hager W. Underflow of standard sluice gate” Journal of experiments in fluids, 1999;27(4):339-350. [15] Wahl TL. Analyzing ADV Data Using WinADV. Joint Conference on Water Resources Engineering and Water Resources Planning & Management, American Society of Civil Engineers, 2000, Minneapolis, Minnesota. [16] Goring DG, and Nikora VI. Despiking acoustic Doppler velocimeter data. Journal of Hydraulic Engineering 2002;128(1):117–126. [17] Dey S, Sarkar A. Characteristics of turbulent flow in submerged jumps on rough beds. Journal of Engineering Mechanics 2008;134(1):49-59. [18] Schwarz WH, and Cosart WP. The two-dimensional turbulent wall-jet. Journal of Fluid Mechanics 1961;10:481–495.
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[19] Rajaratnam N. Submerged hydraulic jump. Journal of Hydraulic Division 1965;91(HY4):7196. [20] Liu HK, Diffusion of Flow from a Submerged Sluice Gate. MSc. Thesis, State University of Iowa 1949. [21] Chow VT. Open channel hydraulics. McGraw-Hill, New York 1959. [22] Rajaratnam N, and Subramanya K. Flow equation for the sluice gate. Journal of Irrigation and Drainage Engineering 1967;93(3):167–186.
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Fig. 1. Schematic of flow through a submerged vertical sluice gate Fig. 2. a) Plan and side view of the experimental setup b) free flow condition through the tested sluice gate Fig. 3. Schematic velocity profile of a submerged hydraulic jump Fig. 4. u/um versus z/d1 Fig. 5. um in terms of q/w Fig. 6. d1/w versus y/w Fig. 7. Variations of α and β versus y/w for the methods of Castro-Orgaz et al. [6] and the results of the current study Fig. 8. Head loss coefficient versus (y1-y3)/w Fig. 9. Velocity profiles associated with the end of the roller flow zone Fig. 10. Relative error distribution versus SR for different EM methods; (a) Henderson [1]; (b) Habibzadeh et al. [12]; (c) Castro Orgaz et al. [5] and Castro Orgaz et al. [6]; and (d) Current study
21
Table. 1. Detailed error analysis of the EM models Submerged jet type Submerged flow SR≥0.5
Highly submerged SR<0.5
Method
Absolute maximum error (%)
MARE (%)
Henderson [1] Castro Orgaz et al. [5] Habibzadeh et al. [12] Castro Orgaz et al. [6] Current study Henderson [1] Castro Orgaz et al. [5] Habibzadeh et al. [12] Castro Orgaz et al. [6] Current study
6.02 39.7 13.4 8.3 8 61.2 46.2 46.5 24.1 21
2.8 19 6.9 3.45 3.1 22.3 18.3 14.8 7.8 8.1
22
¶ Appendix 1 Flow description Q(l/s)=61 y1 (mm)= 251.5 y3 (mm)= 156.5 w (mm)= 46.5
z (mm) 0 5 9.8 14.8 19.8 24.8 29.8 34.8 44.8 54.8 64.8 74.8 84.8 100.8
u (cm/s) 167.24 167.24 168.43 168.05 162.30 151.81 130.04 96.98 40.08 4.10 -27.48 -28.06 -39.82 -39.82
Flow description Q(l/s)= 51.1 y1 (mm)= 219.3 y3 (mm)= 151.8 w (mm)= 46.5
Flow description Q(l/s)= 21.6 y1 (mm)= 118.7 y3 (mm)= 103.8 w (mm)= 46.5
z (mm) 0 5 12.3 17.3 22.3 27.3 32.3 37.3 42.3 47.3 57.3 67.3 77.3 87.3 97.6
u (cm/s) 58.60 58.60 59.73 59.07 57.36 48.68 39.32 31.18 16.05 15.29 1.36 -3.48 -10.70 -13.07 -13.07
Flow description Q(l/s)=21.6 y1 (mm)= 187.7 y3 (mm)= 169.2 w (mm)= 46.5
Flow description Q(l/s)= 30.3 y1 (mm)= 196.2 y3 (mm)= 117.7 w (mm)= 46.5
z (mm) 0 5 12 17 22 27 32 37 42 47 57 67 77 87 97 107 117 127 137 147
u (cm/s) 82.82 82.82 85.54 84.70 80.91 75.33 67.30 52.75 37.34 21.81 4.17 -1.69 -7.72 -6.54 -6.47 -6.87 -4.62 -6.21 -1.94 -6.63
Flow description Q(l/s)= 40.1 y1 (mm)= 165.2 y3 (mm)= 123.8 w (mm)= 46.5
156.5
-6.63
z (mm) 0 5 11.5 16.5 21.5 26.5 31.5 36.5 41.5 46.5 56.5 66.5 76.5 86.5 96.5 113.5 z (mm) 0 5 12 17 22 27 32 37 42 47 57 67 77 87 97 107 117 127 137 147 157 165.3 z (mm) 0 5 12.2 17.2 22.2 27.2 32.2 37.2 42.2 47.2 57.2 67.2 77.2 87.2 98.6
u (cm/s) 138.41 138.41 138.11 137.76 134.94 123.89 97.40 79.54 38.74 17.18 -2.56 -8.38 -12.62 -9.74 -15.72 -15.72 u (cm/s) 59.53 59.53 60.64 60.74 59.96 54.26 43.35 36.86 24.01 20.38 5.12 -2.73 -2.59 -3.75 -3.12 -4.68 -3.58 -2.20 -4.26 -3.57 -1.16 -1.16 u (cm/s) 107.03 107.03 108.86 109.02 107.85 97.02 83.19 66.77 33.32 18.17 -4.04 -6.04 -11.03 -7.04 -7.04
Flow description Q(l/s)= 16.3 y1 (mm)= 114.7 y3 (mm)= 104 w (mm)= 46.5
Flow description Q(l/s)=30.3 y1 (mm)= 143.2 y3 (mm)= 117.7 w (mm)= 46.5
Flow description Q(l/s)= 12 y1 (mm)= 139.7 y3 (mm)= 130.4 w (mm)= 46.5
z (mm) 0 5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 100.2 z (mm) 0 5 11.8 16.8 21.8 26.8 31.8 36.8 41.8 46.8 56.8 66.8 76.8 86.8 96.8 106.3
u (cm/s) 43.31 43.31 43.67 44.12 43.60 37.44 26.07 14.60 4.38 -0.42 -2.97 -4.04 -1.99 -2.51 0.86 0.86 u (cm/s) 82.92 82.92 84.03 83.93 80.11 73.35 63.03 45.29 33.56 22.03 5.71 -3.88 -12.04 -13.32 -12.86 -12.86
z (mm) 0 5 11.7 16.7 21.7 26.7 31.7 36.7 41.7 46.7 56.7 66.7 76.7 86.7 96.7 106.7 116.7 126.7 130.7
u (cm/s) 30.60 30.60 33.48 31.72 30.71 28.58 24.35 20.48 12.99 12.45 2.28 -0.13 -0.03 -3.41 -4.05 -7.52 -6.69 -7.31 -7.31
Appendix 1 (continued) 23
Flow description Q(l/s)=80.4 y1 (mm)= 242.7 y3 (mm)= 175.7 w (mm)= 68.3
z (mm) 0 5 11.5 16.5 21.5 26.5 31.5 36.5 41.5 46.5 56.5 66.5 76.5 86.5 96.5 106.5 123
u (cm/s) 146.35 146.35 147.66 147.22 148.84 148.37 145.32 136.78 130.85 111.21 73.43 27.77 -1.87 -20.17 -29.54 -28.87 -28.87
Flow description Q(l/s)=65.35 y1 (mm)= 206 y3 (mm)= 162.4 w (mm)= 68.3
z (mm) 0 5 11.6 16.6 21.6 26.6 31.6 36.6 41.6 46.6 56.6 66.6 76.6 86.6 96.6 106.6 116.6 126.6
u (cm/s) 116.49 116.49 118.98 117.50 118.57 119.07 115.42 110.21 104.53 88.30 41.30 14.23 -2.90 -10.25 -14.94 -15.30 -16.48 -16.48
Flow description Q(l/s)=65.6 y1 (mm)= 283.2 y3 (mm)= 227.3 w (mm)= 68.3
Flow description Q(l/s)= 71.7 y1 (mm)= 302.7 y3 (mm)= 235.1 w (mm)= 68.3
z (mm) 0 5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 50.5 60.5 70.5 80.5 90.5 100.5 110.5 120.5 130.5 140.5 150.5 160.5 170.5 180.5 190.5 203.5
u (cm/s) 134.54 134.54 133.51 134.50 133.84 135.65 134.39 131.93 122.27 96.13 60.39 14.79 2.78 -2.66 1.35 -2.58 -3.30 -8.07 -8.17 -4.57 -1.91 -10.71 -9.54 -11.24 -8.49 -8.49
Flow description Q(l/s)=71 y1 (mm)= 212.7 y3 (mm)= 162.7 w (mm)= 68.3
z (mm) 0 5 14 19 24 29 34 39 44 49 59 69 79 89 99 109 122.8
u (cm/s) 126.55 126.55 128.64 127.82 128.05 124.98 115.88 110.51 93.22 77.50 38.87 20.01 -6.65 -23.16 -35.37 -23.71 -23.71
Flow description Q(l/s)=30 y1 (mm)= 125.1 y3 (mm)= 115 w (mm)= 68.3
z (mm) 0 5 12 17 22 27 32 37 42 47 52 62 72 82 92 102 112 122 132 142 152 162 172 182 192 201.5 z (mm) 0 5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 106.8
u (cm/s) 124.77 124.77 126.18 125.32 125.23 124.90 120.68 112.70 94.61 81.66 62.63 26.77 6.61 4.62 -3.91 -2.59 -1.18 -9.77 -11.42 -13.44 -13.86 -23.36 -8.93 -14.44 -8.84 -8.84 u (cm/s) 50.47 50.47 51.54 52.17 52.74 52.34 50.91 50.55 49.05 41.07 23.65 3.25 -0.39 -3.20 -10.81 -10.81
Appendix 1 (continued) Flow description
z (mm)
u (cm/s)
Flow description
z (mm)
u (cm/s)
24
Q(l/s)=81 y1 (mm)= 343.7 y3 (mm)= 246.8 w (mm)= 68.3
0 5 8.5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 108.5 118.5 128.5 138.5 148.5 158.5 168.5 178.5 188.5 198.5 209.5
157.37 157.37 158.67 158.68 158.82 158.86 158.01 154.55 143.70 121.80 79.77 27.14 1.57 -4.87 -6.20 -10.06 -8.65 -2.92 -11.66 -7.54 -14.63 -19.32 -19.00 -18.95 -13.18 -13.22 -13.22
Q(l/s)=91 y1 (mm)= 369.5 y3 (mm)= 256.5 w (mm)= 68.3
0 5 8.2 13.2 18.2 23.2 28.2 33.2 38.2 43.2 48.2 58.2 68.2 78.2 88.2 98.2 108.2 118.2 128.2 138.2 148.2 158.2 168.2 178.2 188.2 198.2 207.6
169.07 169.07 170.73 170.17 170.79 170.67 170.98 168.86 166.50 153.75 89.22 19.41 1.66 -6.98 -3.33 -7.43 -3.38 -14.09 -13.01 -13.08 -13.69 -12.48 -12.45 -11.68 -9.24 -10.47 -10.47
25
Appendix 2 w (mm)
Q (l/s)
y1 (mm)
y3 (mm)
w (mm)
Q (l/s)
y1 (mm)
y3 (mm)
46.5
12.04
92
84.5
46.5
19.76
213
195.7
46.5
13.5
99.1
90
46.5
19.77
233.5
216
46.5
13.85
96.4
88
46.5
19.73
252.1
235
46.5
15.1
164.7
153
46.5
61.3
251.5
156.5
46.5
15.2
99.9
90.3
46.5
61.3
316.7
198.6
46.5
15.25
202.7
190.8
46.5
61.3
395.3
258
46.5
15.3
133.4
122.6
46.5
51
341.1
248.4
46.5
16.1
100
92.3
46.5
51
298.3
211.7
46.5
18.5
173.3
158.7
46.5
51
219.3
151.8
46.5
18.5
108.4
96.4
46.5
16.3
114.7
104
46.5
18.55
213.3
197.5
46.5
16.3
178.7
166.4
46.5
18.7
252
236
46.5
16.3
229.3
216.1
46.5
18.7
237.2
221
46.5
21.6
118.7
103.8
46.5
20.44
210.6
192.6
46.5
21.6
187.7
169.2
46.5
20.44
243.8
225.4
46.5
30.3
143.2
117.7
46.5
20.6
156.8
141
46.5
30.3
196.2
167.4
46.5
20.65
113
100
46.5
40.3
228.9
179.2
46.5
25.3
125.3
108.6
46.5
40.3
165.2
123.8
46.5
25.3
124.8
107
46.5
12
139.7
130.4
46.5
25.4
192.3
170.5
68.3
80.4
242.5
175.7
46.5
30.4
195
165
68.3
50.5
173.8
145.6
46.5
30.6
311
274.5
68.3
65.6
283
227.3
46.5
30.6
278.2
242
68.3
65.3
205.8
162.4
46.5
30.6
239.2
204.3
68.3
71.7
302.7
235.1
46.5
34.1
323.6
279.4
68.3
71
212.5
162.7
46.5
34.35
238.1
195.8
68.3
40.15
144.9
128.6
46.5
35.35
144.8
113.6
68.3
30
125.1
115
46.5
39.93
245
193.4
68.3
81
343.5
246.8
46.5
40.07
162.8
122.2
68.3
91
369.5
256.5
46.5
13.4
96.5
91.5
46.5
13.4
116.1
108
46.5
13.66
134.1
126
46.5
13.78
151.1
140.7
46.5
13.85
167.3
155.5
46.5
13.94
185
174
46.5
13.95
201.5
190.3
46.5
19.9
142.1
126
46.5
19.98
114.1
102
46.5
19.95
127
112.6
46.5
19.9
160.6
144.7
46.5
19.9
179.1
162.5
46.5
19.89
195.1
177.4
highlights 26
The flow through submerged vertical sluice gates is investigated experimentally. Original experimental data of the velocity profiles at vena contracta are presented. The real velocity profiles were used to find the velocity correction factors and submerged head loss coefficient. The results were implemented in the energy and momentum formulas to find the headdischarge relationship of the submerged sluice gates.
27
28
29
PII: DOI: Reference:
S0955-5986(16)30264-3 http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.009 JFMI1293
To appear in: Flow Measurement and Instrumentation Received date: 22 July 2016 Revised date: 27 November 2016 Accepted date: 30 November 2016 Cite this article as: Mohammad Bijankhan, Salah Kouchakzadeh and Gilles Belaud, Application of the submerged experimental velocity profiles for the sluice gate's stage-discharge relationship, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.009 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.
Application of the submerged experimental velocity profiles for the sluice gate’s stage-discharge relationship Mohammad Bijankhana, Salah Kouchakzadehb, Gilles Belaudc a
Department of Water Engineering, Faculty of Engineering and Technology, Imam Khomeini International University,
Qazvin, Iran. b
c
Irrigation and Reclamation Engrg. Dept., University of Tehran, P.O. Box 31587-4111, Karaj, 31587-77871, Iran.
UMR G-EAU, SupAgro, place Pierre Viala, 34060 Montpellier Cedex 1, France.
Email: [email protected] Email: [email protected] Email: [email protected]
Abstract Sluice gates have been widely used and intensively studied, however their submerged flow conditions still call for in depth attention. A large scale experimental setup equipped with Acoustic Doppler Velocimetry, ADV, and electromagnetic flow-meter was used to thoroughly investigate various aspects of the hydraulics of submerged sluice gate. In this study, new experimental data sets are provided, that help better understand and quantify the flow features for submerged sluice gates. According to the experimental data generic fitting are provided for the velocity profiles from which the velocity correction factors can be obtained. Then, the experimentally obtained submerged head loss coefficient is presented and discussed. The results of this study showed that current classical Energy-Momentum methods (EM) failed to accurately determine the flow rate for the cases of highly submergences, while employing the interaction of the energy correction factors and head loss values in the EM model would result in more accurate head-discharge estimation. The new data set provided in this work can be used effectively for the validation of numerical modeling of submerged sluice gates. 1
Keywords: sluice gate, velocity profile, submerged flow, energy and momentum equations, flow measurement. Notations Cc=contraction coefficient; d1 (L)= flow depth with the associated velocity of um/2; d3 (L)= the depth of the reverse flow region; g (Ls-2)= the acceleration due to gravity; k = experimental coefficient; K=energy loss coefficient; MARE = Mean Absolute Relative Errors; Q (L3s-1) = flow rate; q (L2s-1) =discharge per unit width; qB (L2s-1) = backwater flow rate; qF (L2s-1) = forward discharge; Qm (L3s-1)= the measured discharges; RMSE = Root Mean Square Errors; SR= Submergence Ratio, (y1 y3)/w; T = z/d1; U = q/y; um (Ls-1) = maximum velocity; ur (Ls-1) = roller flow velocity;
2
us (Ls-1) = surface velocity; Vj (Ls-1) = jet velocity; w (L) =gate opening; x (L) =horizontal distance; y (L) =the submerged flow depth just after the gate; yj (L) = jet thickness; y1 (L) = upstream water depth; y3 (L) = tail water depth; yc (L) =critical depth; z (L) = distance from the channel bottom; zu (L) = thickness of the potential velocity core;
=Boussinesq coefficient; =the Coriolis coefficient; H (L) = head loss;
Introduction Gates are widely used in open channel flows as water regulator and flow measurement devices. According to the tailwater depth value and gate opening the flow conditions through a vertical sluice gate can be classified as: 1- Free flow condition; which occurs when a free hydraulic jump formed downstream the gate. 2- Submerged flow condition (Fig. 1); which takes place when the tailwater depth increases beyond the conjugated depth. 3
The classical energy and momentum (EM) methods could be used to determine the discharge through free or submerged flow conditions [1]. According to the studies of Henry [2] and Sepulveda et al. [3] the conventional EM method can be used accurately for flow measuring through a vertical sluice gate. While, Lozano et al. [4] observed significant deviations between the measured discharges and calculated values by applying the EM method to calibrate vertical sluice gates of irrigation networks. They considered the contraction coefficient, Cc, as a calibrating parameter to improve the performance of the conventional EM method. Later, Castro-Orgaz et al. [5] indicated that calibration of the contraction coefficient indirectly incorporates the effects of non-uniform velocity distributions into the energy and momentum fluxes. Consequently, they included the kinetic and momentum fluxes for a total depth of y, into the classical EM method and incorporated the Coriolis, , and Boussinesq, , coefficients to account for the highly non-uniform velocity profiles of the roller zone. Castro-Orgaz et al. [6] discussed that the roller momentum flux and energy loss are also important and cannot be ignored. They considered a constant vorticity velocity profile in the roller, thereby obtaining analytical solutions for the momentum correction coefficient and the head loss through vertical sluice gates. They evaluated their method using experimental data available in the literature and reported satisfactory results. Belaud et al. [7] indicated also that the method of Castro Orgaz et al. [6] neglected the friction forces, and assumed that contraction coefficient (Cc) is the same in submerged flow as in free flow. However, this assumption has been questioned by Belaud et al. [8]. The submerged contracted coefficient is a key parameter in most of the energy and momentum methods used for estimating the submerged discharge values. There are however different kinds of assumptions in the literature for considering the amount of the jet thickness at the contracted area. Woycicki [9] indicated that Cc is directly related to the ratio of gate opening to the upstream flow depth, w/y1, for the submerged flow condition. Rajaratnam and Subramanya [10] performed four experimental runs and collected only the forward flow velocity profiles for 0.167
constant value of 0.61 for the submerged contraction coefficient. Using field data and applying energy and momentum equations, Lozano et al. [4] indicated that considering Cc-values between 0.629 and 0.652 would improve the accuracy of discharge estimation. Applying the energy equation and employing the experimental data of Fardjeli [11], Belaud et al. [8] calculated submerged Ccvalues indirectly. They concluded that the submerged Cc can reach values as high as 0.8 and the energy and momentum balance can be used to predict Cc. However, Habibzadeh et al. [12] considered a constant value of 0.611 for the submerged contraction coefficient and investigated the role of energy loss on discharge characteristics of the submerged sluice gates. Cassan and Belaud [13] studied the flow characteristics upstream and downstream of a submerged sluice gate experimentally and numerically. Using ADV measurements, they indicated that the flow condition can be considered two dimensional and applied a 2D- numerical model to determine the velocity fields and the associated Cc-values. The results were also consistent with the energy and momentum balance proposed by Belaud et al. [8]. As mentioned by Belaud et al. [7], Castro Orgaz et al. [6], and Belaud et al. [8] taking Cc as a constant while calibrating the head-discharge formula can be counterbalanced by some other simplifying assumptions or by changing other experimental coefficients like energy and momentum correction factors, friction, and head loss. This means that the EM method, although it may appear as physically-based, remains an empirical method, so fitted relationships may not apply universally [7]. The lack of the knowledge from the velocity distribution of the flow just after the gate would make the researchers consider many hypotheses to solve the energy and momentum formulas. In this paper, the velocity distribution profiles for the submerged flow downstream the gate were studied experimentally. Based on the real velocity profile functions the energy, , and momentum, , correction coefficients were determined. In addition, the compiled data were used to evaluate and report the submerged head-loss coefficients. Note that the proposed energy and momentum sets of
5
equations are independent of the submerged contraction coefficient values which are associated with many uncertainties. Finally, it was indicated that the available classical EM method showed more than 60% relative error in discharge prediction for the highly submerged flow condition, the proposed herein EM method was developed and proposed in this paper. Experimental setup To evaluate different submerged jet types and the performance of different EM methods it was necessary to compile reliable experimental data set covering submerged wall jet from low to high submerged flow conditions, given that it is scarcely available. The experiments were performed in a 1.179 m wide, 1 m height and 7 m long Plexiglas flume located at the Central Hydraulic Laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran (Fig. 2). The flume was supplied by an elevated constant head tank and an electromagnetic flow-meter with an accuracy of ±0.5% of the full scale was installed on the feeding pipe to accurately 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 flume downstream end. Gate openings, w, of 46.5 and 68.3 mm were considered. In order to accurately set the gate opening, prefabricated elements with an accuracy of 0.1 mm were located below the gate, and removed after the gate was positioned [14]. Considering the gate openings of w=46.5 and 68.3 mm, and installing the ADV instrument just after the gate at x/w2 where the hydrostatic pressure distribution occurs [10], the velocity profiles associated with the vena contracta were collected. The velocity profiles were also measured at the tailwater section located just after the roller zone where the release of bubbles was minimized. These experimental data were then used to calculate the velocity correction factors associated with the tailwater section. ADV measurements
6
A vectrino+ Nortek Acoustic Doppler Velocity meter system, ADV, equipped with a side looking 3D probe and sampling rate of 200 Hz was used to measure the velocity profiles. Considering different recording times of 15, 30, 45, 90, and 180 seconds, it was found that the mean velocity values associated with a specific node remain almost constant for the recording time of greater than 30 seconds. Consequently, data recording time of 30 seconds was considered for compiling the data and for each node almost 12000 samples were collected. According to Wahl [15], and introducing the so-called WinADV software, for a given data set the results were filtered by considering the average signal-to-noise ratio (SNR) and correlations of 15 and 70 respectively. Nominal velocity range of 4 m/s, transmit length of 1.8 mm, and sampling volume of 7 mm were found suitable to have more than 90% of the data fall within the minimum SNR and correlation of 15 and 70 respectively. However, the minimum correlations were observed at the shear layer between the jet and its above roller zone. One problem with ADV measurements is spikes caused by aliasing of the Doppler signal. The effect of spikes would be highlighted when the flow velocities exceed the preset velocity value or the bed roughness affects the flow near the bed [16]. Considering the velocity range of 4 m/s and the smooth channel bed the measured velocities did not contain many spikes. Such a condition was also reported by Dey and Sarkar [17] for the submerged jumps. However, in order to ensure the measuring accuracies especially near the bed the despiking method of Goring and Nikora [16] which is included in WinADV software was used. Considering z as the distance from the channel invert, the side-looking 3D probe of the ADV instrument allowed velocity measurement in the range of 5z(mm)y210. The depth interval of 5 and 10 mm were considered for 0z (mm)50, and 50
7
The experimental velocity profiles obtained in this study are provided in the Appendix 1. Nice to mention that, there is no comprehensive source in the literature to find such dataset of the submerged velocity profiles, and consequently, the original dataset presented in this work can be used effectively to increase the current knowledge about the submerged flow through vertical sluice gates. Developing the velocity profile formula at the contracted section Forward Flow Following the study of Schwarz and Cosart [18], Rajaratnam [19] investigated the submerged hydraulic jump on the basis of wall jet theory. He proposed a general definition of the submerged velocity profile by dividing the submerged flow into forward and backward flows (Fig. 3). Defining the flow depth d1, as the depth with associated velocity of um/2 (um is the maximum velocity); plotting u/um versus z/d1 would result in a general definition for the forward flow [19]. The experimental velocity profiles associated with the vena contracta section collected in this work are plotted in normalized coordinates in Fig. 4. As revealed, all velocity profile data associated with the forward flows collapse to a single curve. Also, the velocity profile at the vena contracta section includes a potential core, i.e. a region of ideal fluid flow, and a shear layer above. As indicated in Fig. 4 the velocity profile associated with the vena contracta section collected by Rajaratnam and Subramanya [10] agrees remarkably well with the current experimental data. However, significant difference between the new velocity profiles and the wall jet theory proposed by Rajaratnam [19] was observed. The wall jet theory is applicable for the fully established flow, i.e. x/w15 [19], and therefore it cannot be used to study the flow at the vena contracta section. The thickness of the potential velocity core, zu, (Fig. 4) is approximately zu/d1=0.66. Neglecting the bottom boundary layer thickness and considering no bed friction, the following empirical equation was then found to describe the non-dimensional forward flow velocity profile at the vena contracta:
8
1 u 2 z z u m 0.6375 2.5972 2.45 d1 d1
0 z / d 1 0.66 0.66 z / d 1 1.483
(1)
According to the experimental data and regression analysis, the flow velocity in the roller zone for the value of z/d1 equal to 1.483 was considered to be zero. Consequently, Eq. (1) should be applied in the range of 0z/d11.483 and for z/d1>1.483 the negative velocity values would be observed. Plotting um versus q/w in Fig. 5, indicates that the observed data could be described by the following relationship:
um 1.48
q w
(2)
According to Eq. (2), due to the jet contraction, the maximum velocity associated with the vena contracta section is averagely 1.48 times greater than the mean velocity of the emerging jet. Also, the normalized depth d1/w, is depicted versus y/w in Fig. 6 which shows that an average value of 0.78 can be considered for d1/w. Backward Flow There are very limited experimental data associated with the backward flow through a submerged hydraulic jump. Rajaratnam [19] employed the experimental data of Liu [20] to investigate the backward velocity distribution. Castro Orgaz et al. [6] considered the following formula for the velocity profile within the eddy:
z C cw u u r 1 2 y C cw
(3)
ur (=0.5q/Ccw) is the roller flow velocity. Note that the coefficient 0.5 was considered as a calibration coefficient to have the best fit of the theoretical Boussinesq coefficient to the experimental data [6]. Belaud et al. [7] indicated that the roller flow velocity should be considered as a function of the submergence ratio. They reported also that the coefficient 0.5 (ur =0.5q/Ccw) 9
could not be always correct and very small values were reported especially for highly submerged flow conditions. Taking a linear velocity profile for the backwater flow rate, qB, the backwater velocity distribution associated with 1.483d1z
u
u r z 1.483d 1
(4)
y 1.483d 1
Integrating Eqs. (1) and (4) the forward and backward flow rates, i.e. qF and qB respectively, would take the following forms:
q F 1.018d 1u m qB
0 z 1.483d 1
( y 1.483d 1 ) u r 1.483d 1 z y 2
(5)
Substituting q, qF, and qB from Eqs. (2) and (5) into q=qF+qB, the ratio ur/um would be: u r 1.307 2.036d 1 / w u m y / w 1.483d 1 / w
(6)
Note that ur-values are always negative. Also for the current experimental data the ratio |ur/um| varies from 0.017 for highly submerged flow condition to 0.25 for low submergence regime. In other words the reverse flow through the roller zone associated with a low submerged flow is significantly higher than that of highly submerged regime. This phenomenon however cannot be described by the fixed coefficient of 0.5 proposed by Castro Orgaz et al. [6]. Coriolis and Boussinesq Coefficients As a result of non-uniform velocity distribution the true velocity head and momentum of a section would be αV2/2g and βq2/gy respectively, where α and β are the energy and momentum correction coefficients. The fundamental definitions of the energy and momentum correction coefficients are [21]:
10
y
u dz 3
0
(7)
V3y y
u dz 2
0
(8)
V2y
In order to formulate the velocity correction factors, the proposed velocity distributions at the vena section, i.e. Eqs. (1) and (4), were then used. Accordingly, the following one dimensional predictors of and were found for the vena contracta section:
0.89d 1u m2 u r2 / 3 y 1.483d 1
(9)
q2 / y 0.83d 1u m3 u r3 / 4 y 1.483d 1
(10)
q3 / y 2
Considering um=1.48q/w and simplifying Eqs. (9) and (10), yield:
1.52
y 0.057 y / w w y / w 1.157
(11)
2
0.018( y / w)2 y 2.1 2 w (y/ w 1.157)
(12)
Considering Cc=0.61 and a linear velocity distribution in the roller, with a maximum velocity set at the half of the jet velocity, Castro Orgaz et al. [6] obtained the following relationships for and at the vena contracta section respectively:
y C cw
2
(13)
y 1 y 1 1 C cw 12 C cw
(14)
11
The velocity correction factors, α and β were plotted in Fig. 7 versus y/w for the method of CastroOrgaz et al. [6] and the results of the current study. According to the figure the velocity correction factors obtained by Eqs. (13) and (14) are always greater than the experimentally obtained values. This is consistent with the remarks formulated by Belaud et al. [7], based on experimental and numerical results: it seems that the values chosen by Castro-Orgaz et al. [6] to initiate the linear velocity profiles in the roller are usually over-estimated; secondly, the hypothesis of having the fixed values of Cc=0.61 for the submerged flow conditions in Eqs. (13) and (14) is not always true [8]. However, Eqs. (11) and (12) which are based on the real velocity profiles can be used more accurately to calculate the velocity correction factors. Due to the inherent uncertainties of calculating the real Cc-values its use may be avoided in the velocity distribution for the whole section, i.e. jet plus roller zone. This would highlight again the necessity of using the real velocity profiles to calculate the velocity correction factors being independent of the contraction coefficient. Head loss For the submerged gates, the turbulence due to the recirculation zone as well as the shear layer between the jet and the roller zone are key parameters which are responsible for the energy loss. The energy balance between the upstream and vena contracta sections can be written as [5,6]: 2
Vj q2 q2 y1 y K 2 2 2 gy 1 2 gy 2g
(15)
where Vj= [2g(y1−y)]0.5. The head loss coefficients were depicted versus the submergence ratio, SR=(y1 y3)/w in Fig. 8. The figure clearly shows that the head loss coefficient increases significantly as the submergence ratio increases. The following explicit empirical formula was then developed to calculate head loss coefficients:
12
1.48 0.51SR 2.33 K 13.5 SR 2.33
(16)
Tailwater section velocity profiles The velocity profiles at the end of the roller zone, where the jet thickness consists more than 95% of the flow depth, were considered for the tailwater section. It was also observed that beyond this point the water depth remained constant. The normalized velocity profiles of different submergence conditions were depicted in Fig. 9. The velocity correction coefficients associated with the tailwater section, i.e. 3 and 3, were then calculated and average values of 3= 2.7 and 3=1.61 were considered. To develop a new energy-momentum model Eqs. (17) and (18) were used to apply the results of this study. Eqs. (11) and (12) were employed to calculate and respectively. The submerged head loss values were also obtained by Eq. (16) and an average value of 3=1.61 was considered. Note that due to using the real velocity profiles the proposed energy and momentum method is independent of the contraction coefficient.
y1
q2 q2 y H 2 gy 12 2 gy 2
y2 q 2 y 32 q2 3 2 gy 2 gy 3
(17)
(18)
Assessment of the available EM methods Review of current EM formulations Applying the energy principle between sections 1 and 2 and the momentum equation to sections 2 and 3 (Fig. 1), Henderson [1] proposed the following set of equations which could be solved for the unknowns y and q:
13
q2 q2 y1 y 2 gy 12 2 gC c2w 2
(19)
y 32 q 2 y2 q2 2 gC cw 2 gy 3
(20)
where, q is the discharge per unit width, g is the acceleration due to gravity, and y is the flow depth just after the gate. Henderson’s [1] EM method is based on a static roller assumption with Cc=0.61. The effects of the non-uniform velocity profile and the head loss through the submerged sluice gate were neglected [6]. Considering the role of the energy loss Habibzadeh et al. [12] proposed the following energy and momentum formulas:
y1
q2 q2 y (1 K ) L 2 gy 12 2 gC c2w 2
y 2 q2 y2 q2 3 2 gC cw 2 gy 3
(21)
(22)
where KL is the energy loss coefficient. Employing the experimental data of Rajaratnam and Subramanya [22] they proposed KL=0.088 for the submerged flow condition. Note that KL would also include the effect of non-uniform velocity, which is considered in α in Eq. (15). Therefore it is not exactly the same K as in Eq. (15). Castro Orgaz et al. [5] questioned the validity of considering static roller concept. According to the velocity profiles available in the literature they indicated that the energy and momentum velocity correction coefficients, i.e. and respectively, are significantly greater than unity. Consequently, they considered the general energy and momentum relationships:
q2 q2 y1 y 2 gy 12 2 gy 2
(23)
14
y2 q 2 y 32 q 2 2 gy 2 gy 3
(24)
Castro Orgaz et al. [5] proposed =1.7 and =k(y3/w). According to the data obtained from four gates operating in an irrigation canal in Southern Spain, Castro Orgaz et al. [5] found that k=2 to 2.45. It should be mentioned that the proposed values of the exponent of 1.7 in the equation for α=α(β) is valid only for highly submerged flows. The parameter k is the result of a local calibration and is only valid for the gates investigated in Castro-Orgaz et al. [5]. Castro Orgaz et al. [6] applied the energy and momentum equations in a general integral form, resulting for rating vertical sluice gates under submerged flow condition:
q2 q2 y1 y H 2 gy 12 2 gy 2 y2 q 2 y 32 q 2 2 gy 2 gy 3
(25)
(26)
in which H is the head loss, α and β is obtained by Eqs. (13) and (14). The corresponding analytical expression for the head-loss coefficient is [6]:
H K
V j2
where, K
(27)
2g
y y 1
0.5
10 2 gC c
.
Experimental assessment of EM formulations There are rare experimental works in order to compare the performance of different EM methods for low to fully submerged flow regimes. In order to evaluate the accuracy of the current energy momentum methods, 71 experimental runs were considered as listed in Appendix 2. Note that the
15
velocity correction factors and head loss coefficient were found by 17 data points and other 54 runs were not used before to find the required parameters of the presented energy momentum model. According to the experimental observations, the measured discharges (Qm) were compared to those obtained by the EM methods (Q) and the relative errors ([Q-Qm]/Qm100) were depicted versus the submergence ratio, SR, Fig. 10. For each method, the values of Mean Absolute Relative Errors, MARE (Eq. 28), and absolute maximum errors were calculated and listed in Table 1.
n MARE Q Q m / Q m / n 100 i 1
(28)
For the low submerged jets (SR≥0.5), the method of Henderson [1], i.e. classical EM model, is as accurate as the model proposed by Castro Orgaz et al. [6] which is a modified version of Castro Orgaz et al. [5]. The associated relative errors for both methods are bounded to 6%. The method proposed by Habibzadeh et al. [12] underestimated the discharges for all the current experimental data with the relative errors ranging from 0.6% to 13.5%. Their model was based on calibrating the head loss coefficient for only one experimental dataset. The recalibrated model might result in better performance. Also, the method of Castro-Orgaz et al. [5] was a calibration for a field channel. For the highly submerged flow condition (SR<0.5), the values of relative errors listed in Table 1 indicated that the absolute maximum relative error associated with the EM models proposed by Henderson [1], Habibzadeh et al. [12], Castro Orgaz et al. [5], Castro Orgaz et al. [6] and the current study were 61.2%, 46.5%, 46.2%, 24.1% and 21% respectively. Consequently, the EM models proposed by Henderson [1], Habibzadeh et al. [12], Castro Orgaz et al. [5], for the highly submerged flow condition, are far away from acceptable accuracies. However the methods of Castro Orgaz et al. [6] and current study are the best, among the available EM models. Note that, as indicated by Henry [2] the head discharge curve of a submerged sluice gate is very sensitive for highly submerged flow conditions. That is, a very small change in the influencing parameters would 16
significantly affect the submerged discharge coefficient and consequently it is inherently difficult to get a better accuracy for largely submerged jets, i.e. SR<0.5. Therefore, the performances of the EM models used to rate the submerged vertical sluice gates depended significantly on the submergence level. In summary, the current study demonstrated that the available submerged Cc-values are difficult to evaluate, but its use can be avoided with a correct consideration of velocity profiles. Consequently, employing the dimensionless form of the submerged velocity profiles, the experimental coefficients such as velocity correction factors (α and β) and head loss can be directly obtained. These results are less dependent on the empirical coefficients and can be considered as universal stage-discharge relationships. Conclusion The submerged jets emerging below a vertical sluice gate is considered in the current study. ADV measurements were used to develop a new dimensionless formula describing the velocity profile at the vena contracta section of low to high submerged flow conditions. The energy and momentum correction coefficients were then determined using the real experimental velocity profile distributions. Experimental observations indicated that the energy loss coefficient and submerged contraction coefficient for highly submerged jets are significantly greater than that of low submerged flow conditions. Employing new experimental data obtained in this work, it was also indicated that the available EM models failed to accurately determine the discharge for fully submerged jets. Finally, the results of the current study were incorporated in the energy and momentum formulas to calculate the submerged discharges. Due to the uncertainties attributed to the calculation of the submerged contraction coefficient, the proposed energy momentum model was developed to be independent of submerged Cc values. Comparing with the current experimental data it was revealed that the results of this study can effectively be used to improve the performance of the classical EM methods especially for the fully submerged wall jets.
17
Acknowledgement The authors gratefully acknowledge the financial support (Grant No. 7102021/01/07) and the research facility provided by the University of Tehran and the Center of Excellence for Evaluation and Rehabilitation of Irrigation and Drainage networks. Also, Valuable comments and ideas given by Dr. Oscar Castro-Orgaz and Dr. Luciano Mateos are highly appreciated.
References [1] Henderson F.M. Open channel flow, Macmillan, New York, 1966. [2] Henry H. Discussion: diffusion of submerged jet. Transactions of the American Society of Civil Engineers 1950;115:687–97. [3] 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. [4] Lozano D, Mateos L, Merkley G, and Clemmens A. Field Calibration of Submerged Sluice Gates in Irrigation Canals. Journal of Irrigation and Drainage Engineering 2009;135(6):763– 772. [5] Castro-Orgaz O, Lozano D, Mateos L. Energy and momentum velocity coefficients for calibration of submerged sluice gates in irrigation canals. Journal of Irrigation and Drainage Engineering 2010;136(9):610-616. [6] Castro-Orgaz O, Mateos L, and Dey S. Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions. Journal of Irrigation and Drainage Engineering 2013;139(4):325–335. [7] Belaud G, Cassan L, and Baume J. Discussion of “Revisiting the Energy-Momentum Method for Rating Vertical Sluice Gates under Submerged Flow Conditions” by Oscar Castro-Orgaz, Luciano Mateos, and Subhasish Dey. Journal of Irrigation and Drainage Engineering 2014;140(7):07014019.
18
[8] Belaud G, Cassan L, Baume J.P. Calculation of contraction coefficient under sluice gates and application to discharge measurement”. Journal of Hydraulic Engineering 2009;135(12):10861091. [9] Woycicki K. Wassersprung, Deckwalze und Ausfluss unter einer Schütze (Hydraulic jump, roller and outflow from below a gate). Dissertation 639. ETH Zurich, Zürich, 1931 [in German]. [10] Rajaratnam N, and Subramanya K. Flow immediately below a submerged sluice gate. Journal of Hydraulic Division 1967;93(HY4):57–77. [11] Fardjeli N. Modélisation d’ouvrages de régulation pour l’aide à la gestion des canaux. MS thesis, Montpellier Supagro, France, 57, 2007 (in French). [12] Habibzadeh A, Vatankhah Ali R, Rajaratnam N. Role of energy loss on discharge characteristics of sluice gates. Journal of Hydraulic Engineering 2011;137(9):1079–84. [13] Cassan L, and Belaud G. Experimental and Numerical Investigation of Flow under Sluice Gates. Journal of Hydraulic Engineering 2012;138(4):367–373. [14] Roth A, and Hager W. Underflow of standard sluice gate” Journal of experiments in fluids, 1999;27(4):339-350. [15] Wahl TL. Analyzing ADV Data Using WinADV. Joint Conference on Water Resources Engineering and Water Resources Planning & Management, American Society of Civil Engineers, 2000, Minneapolis, Minnesota. [16] Goring DG, and Nikora VI. Despiking acoustic Doppler velocimeter data. Journal of Hydraulic Engineering 2002;128(1):117–126. [17] Dey S, Sarkar A. Characteristics of turbulent flow in submerged jumps on rough beds. Journal of Engineering Mechanics 2008;134(1):49-59. [18] Schwarz WH, and Cosart WP. The two-dimensional turbulent wall-jet. Journal of Fluid Mechanics 1961;10:481–495.
19
[19] Rajaratnam N. Submerged hydraulic jump. Journal of Hydraulic Division 1965;91(HY4):7196. [20] Liu HK, Diffusion of Flow from a Submerged Sluice Gate. MSc. Thesis, State University of Iowa 1949. [21] Chow VT. Open channel hydraulics. McGraw-Hill, New York 1959. [22] Rajaratnam N, and Subramanya K. Flow equation for the sluice gate. Journal of Irrigation and Drainage Engineering 1967;93(3):167–186.
20
Fig. 1. Schematic of flow through a submerged vertical sluice gate Fig. 2. a) Plan and side view of the experimental setup b) free flow condition through the tested sluice gate Fig. 3. Schematic velocity profile of a submerged hydraulic jump Fig. 4. u/um versus z/d1 Fig. 5. um in terms of q/w Fig. 6. d1/w versus y/w Fig. 7. Variations of α and β versus y/w for the methods of Castro-Orgaz et al. [6] and the results of the current study Fig. 8. Head loss coefficient versus (y1-y3)/w Fig. 9. Velocity profiles associated with the end of the roller flow zone Fig. 10. Relative error distribution versus SR for different EM methods; (a) Henderson [1]; (b) Habibzadeh et al. [12]; (c) Castro Orgaz et al. [5] and Castro Orgaz et al. [6]; and (d) Current study
21
Table. 1. Detailed error analysis of the EM models Submerged jet type Submerged flow SR≥0.5
Highly submerged SR<0.5
Method
Absolute maximum error (%)
MARE (%)
Henderson [1] Castro Orgaz et al. [5] Habibzadeh et al. [12] Castro Orgaz et al. [6] Current study Henderson [1] Castro Orgaz et al. [5] Habibzadeh et al. [12] Castro Orgaz et al. [6] Current study
6.02 39.7 13.4 8.3 8 61.2 46.2 46.5 24.1 21
2.8 19 6.9 3.45 3.1 22.3 18.3 14.8 7.8 8.1
22
¶ Appendix 1 Flow description Q(l/s)=61 y1 (mm)= 251.5 y3 (mm)= 156.5 w (mm)= 46.5
z (mm) 0 5 9.8 14.8 19.8 24.8 29.8 34.8 44.8 54.8 64.8 74.8 84.8 100.8
u (cm/s) 167.24 167.24 168.43 168.05 162.30 151.81 130.04 96.98 40.08 4.10 -27.48 -28.06 -39.82 -39.82
Flow description Q(l/s)= 51.1 y1 (mm)= 219.3 y3 (mm)= 151.8 w (mm)= 46.5
Flow description Q(l/s)= 21.6 y1 (mm)= 118.7 y3 (mm)= 103.8 w (mm)= 46.5
z (mm) 0 5 12.3 17.3 22.3 27.3 32.3 37.3 42.3 47.3 57.3 67.3 77.3 87.3 97.6
u (cm/s) 58.60 58.60 59.73 59.07 57.36 48.68 39.32 31.18 16.05 15.29 1.36 -3.48 -10.70 -13.07 -13.07
Flow description Q(l/s)=21.6 y1 (mm)= 187.7 y3 (mm)= 169.2 w (mm)= 46.5
Flow description Q(l/s)= 30.3 y1 (mm)= 196.2 y3 (mm)= 117.7 w (mm)= 46.5
z (mm) 0 5 12 17 22 27 32 37 42 47 57 67 77 87 97 107 117 127 137 147
u (cm/s) 82.82 82.82 85.54 84.70 80.91 75.33 67.30 52.75 37.34 21.81 4.17 -1.69 -7.72 -6.54 -6.47 -6.87 -4.62 -6.21 -1.94 -6.63
Flow description Q(l/s)= 40.1 y1 (mm)= 165.2 y3 (mm)= 123.8 w (mm)= 46.5
156.5
-6.63
z (mm) 0 5 11.5 16.5 21.5 26.5 31.5 36.5 41.5 46.5 56.5 66.5 76.5 86.5 96.5 113.5 z (mm) 0 5 12 17 22 27 32 37 42 47 57 67 77 87 97 107 117 127 137 147 157 165.3 z (mm) 0 5 12.2 17.2 22.2 27.2 32.2 37.2 42.2 47.2 57.2 67.2 77.2 87.2 98.6
u (cm/s) 138.41 138.41 138.11 137.76 134.94 123.89 97.40 79.54 38.74 17.18 -2.56 -8.38 -12.62 -9.74 -15.72 -15.72 u (cm/s) 59.53 59.53 60.64 60.74 59.96 54.26 43.35 36.86 24.01 20.38 5.12 -2.73 -2.59 -3.75 -3.12 -4.68 -3.58 -2.20 -4.26 -3.57 -1.16 -1.16 u (cm/s) 107.03 107.03 108.86 109.02 107.85 97.02 83.19 66.77 33.32 18.17 -4.04 -6.04 -11.03 -7.04 -7.04
Flow description Q(l/s)= 16.3 y1 (mm)= 114.7 y3 (mm)= 104 w (mm)= 46.5
Flow description Q(l/s)=30.3 y1 (mm)= 143.2 y3 (mm)= 117.7 w (mm)= 46.5
Flow description Q(l/s)= 12 y1 (mm)= 139.7 y3 (mm)= 130.4 w (mm)= 46.5
z (mm) 0 5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 100.2 z (mm) 0 5 11.8 16.8 21.8 26.8 31.8 36.8 41.8 46.8 56.8 66.8 76.8 86.8 96.8 106.3
u (cm/s) 43.31 43.31 43.67 44.12 43.60 37.44 26.07 14.60 4.38 -0.42 -2.97 -4.04 -1.99 -2.51 0.86 0.86 u (cm/s) 82.92 82.92 84.03 83.93 80.11 73.35 63.03 45.29 33.56 22.03 5.71 -3.88 -12.04 -13.32 -12.86 -12.86
z (mm) 0 5 11.7 16.7 21.7 26.7 31.7 36.7 41.7 46.7 56.7 66.7 76.7 86.7 96.7 106.7 116.7 126.7 130.7
u (cm/s) 30.60 30.60 33.48 31.72 30.71 28.58 24.35 20.48 12.99 12.45 2.28 -0.13 -0.03 -3.41 -4.05 -7.52 -6.69 -7.31 -7.31
Appendix 1 (continued) 23
Flow description Q(l/s)=80.4 y1 (mm)= 242.7 y3 (mm)= 175.7 w (mm)= 68.3
z (mm) 0 5 11.5 16.5 21.5 26.5 31.5 36.5 41.5 46.5 56.5 66.5 76.5 86.5 96.5 106.5 123
u (cm/s) 146.35 146.35 147.66 147.22 148.84 148.37 145.32 136.78 130.85 111.21 73.43 27.77 -1.87 -20.17 -29.54 -28.87 -28.87
Flow description Q(l/s)=65.35 y1 (mm)= 206 y3 (mm)= 162.4 w (mm)= 68.3
z (mm) 0 5 11.6 16.6 21.6 26.6 31.6 36.6 41.6 46.6 56.6 66.6 76.6 86.6 96.6 106.6 116.6 126.6
u (cm/s) 116.49 116.49 118.98 117.50 118.57 119.07 115.42 110.21 104.53 88.30 41.30 14.23 -2.90 -10.25 -14.94 -15.30 -16.48 -16.48
Flow description Q(l/s)=65.6 y1 (mm)= 283.2 y3 (mm)= 227.3 w (mm)= 68.3
Flow description Q(l/s)= 71.7 y1 (mm)= 302.7 y3 (mm)= 235.1 w (mm)= 68.3
z (mm) 0 5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 50.5 60.5 70.5 80.5 90.5 100.5 110.5 120.5 130.5 140.5 150.5 160.5 170.5 180.5 190.5 203.5
u (cm/s) 134.54 134.54 133.51 134.50 133.84 135.65 134.39 131.93 122.27 96.13 60.39 14.79 2.78 -2.66 1.35 -2.58 -3.30 -8.07 -8.17 -4.57 -1.91 -10.71 -9.54 -11.24 -8.49 -8.49
Flow description Q(l/s)=71 y1 (mm)= 212.7 y3 (mm)= 162.7 w (mm)= 68.3
z (mm) 0 5 14 19 24 29 34 39 44 49 59 69 79 89 99 109 122.8
u (cm/s) 126.55 126.55 128.64 127.82 128.05 124.98 115.88 110.51 93.22 77.50 38.87 20.01 -6.65 -23.16 -35.37 -23.71 -23.71
Flow description Q(l/s)=30 y1 (mm)= 125.1 y3 (mm)= 115 w (mm)= 68.3
z (mm) 0 5 12 17 22 27 32 37 42 47 52 62 72 82 92 102 112 122 132 142 152 162 172 182 192 201.5 z (mm) 0 5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 106.8
u (cm/s) 124.77 124.77 126.18 125.32 125.23 124.90 120.68 112.70 94.61 81.66 62.63 26.77 6.61 4.62 -3.91 -2.59 -1.18 -9.77 -11.42 -13.44 -13.86 -23.36 -8.93 -14.44 -8.84 -8.84 u (cm/s) 50.47 50.47 51.54 52.17 52.74 52.34 50.91 50.55 49.05 41.07 23.65 3.25 -0.39 -3.20 -10.81 -10.81
Appendix 1 (continued) Flow description
z (mm)
u (cm/s)
Flow description
z (mm)
u (cm/s)
24
Q(l/s)=81 y1 (mm)= 343.7 y3 (mm)= 246.8 w (mm)= 68.3
0 5 8.5 13.5 18.5 23.5 28.5 33.5 38.5 43.5 48.5 58.5 68.5 78.5 88.5 98.5 108.5 118.5 128.5 138.5 148.5 158.5 168.5 178.5 188.5 198.5 209.5
157.37 157.37 158.67 158.68 158.82 158.86 158.01 154.55 143.70 121.80 79.77 27.14 1.57 -4.87 -6.20 -10.06 -8.65 -2.92 -11.66 -7.54 -14.63 -19.32 -19.00 -18.95 -13.18 -13.22 -13.22
Q(l/s)=91 y1 (mm)= 369.5 y3 (mm)= 256.5 w (mm)= 68.3
0 5 8.2 13.2 18.2 23.2 28.2 33.2 38.2 43.2 48.2 58.2 68.2 78.2 88.2 98.2 108.2 118.2 128.2 138.2 148.2 158.2 168.2 178.2 188.2 198.2 207.6
169.07 169.07 170.73 170.17 170.79 170.67 170.98 168.86 166.50 153.75 89.22 19.41 1.66 -6.98 -3.33 -7.43 -3.38 -14.09 -13.01 -13.08 -13.69 -12.48 -12.45 -11.68 -9.24 -10.47 -10.47
25
Appendix 2 w (mm)
Q (l/s)
y1 (mm)
y3 (mm)
w (mm)
Q (l/s)
y1 (mm)
y3 (mm)
46.5
12.04
92
84.5
46.5
19.76
213
195.7
46.5
13.5
99.1
90
46.5
19.77
233.5
216
46.5
13.85
96.4
88
46.5
19.73
252.1
235
46.5
15.1
164.7
153
46.5
61.3
251.5
156.5
46.5
15.2
99.9
90.3
46.5
61.3
316.7
198.6
46.5
15.25
202.7
190.8
46.5
61.3
395.3
258
46.5
15.3
133.4
122.6
46.5
51
341.1
248.4
46.5
16.1
100
92.3
46.5
51
298.3
211.7
46.5
18.5
173.3
158.7
46.5
51
219.3
151.8
46.5
18.5
108.4
96.4
46.5
16.3
114.7
104
46.5
18.55
213.3
197.5
46.5
16.3
178.7
166.4
46.5
18.7
252
236
46.5
16.3
229.3
216.1
46.5
18.7
237.2
221
46.5
21.6
118.7
103.8
46.5
20.44
210.6
192.6
46.5
21.6
187.7
169.2
46.5
20.44
243.8
225.4
46.5
30.3
143.2
117.7
46.5
20.6
156.8
141
46.5
30.3
196.2
167.4
46.5
20.65
113
100
46.5
40.3
228.9
179.2
46.5
25.3
125.3
108.6
46.5
40.3
165.2
123.8
46.5
25.3
124.8
107
46.5
12
139.7
130.4
46.5
25.4
192.3
170.5
68.3
80.4
242.5
175.7
46.5
30.4
195
165
68.3
50.5
173.8
145.6
46.5
30.6
311
274.5
68.3
65.6
283
227.3
46.5
30.6
278.2
242
68.3
65.3
205.8
162.4
46.5
30.6
239.2
204.3
68.3
71.7
302.7
235.1
46.5
34.1
323.6
279.4
68.3
71
212.5
162.7
46.5
34.35
238.1
195.8
68.3
40.15
144.9
128.6
46.5
35.35
144.8
113.6
68.3
30
125.1
115
46.5
39.93
245
193.4
68.3
81
343.5
246.8
46.5
40.07
162.8
122.2
68.3
91
369.5
256.5
46.5
13.4
96.5
91.5
46.5
13.4
116.1
108
46.5
13.66
134.1
126
46.5
13.78
151.1
140.7
46.5
13.85
167.3
155.5
46.5
13.94
185
174
46.5
13.95
201.5
190.3
46.5
19.9
142.1
126
46.5
19.98
114.1
102
46.5
19.95
127
112.6
46.5
19.9
160.6
144.7
46.5
19.9
179.1
162.5
46.5
19.89
195.1
177.4
highlights 26
The flow through submerged vertical sluice gates is investigated experimentally. Original experimental data of the velocity profiles at vena contracta are presented. The real velocity profiles were used to find the velocity correction factors and submerged head loss coefficient. The results were implemented in the energy and momentum formulas to find the headdischarge relationship of the submerged sluice gates.
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