Subcritical 90° Equal-Width Open-Channel Dividing

Subcritical 90° Equal-Width Open-Channel Dividing Flow Chung-Chieh Hsu, A.M.ASCE1; Chii-Jau Tang2; Wen-Jung Lee3; and Mon-Yi Shieh4 Abstract: Based on experimental observations, for a subcritical, right-angled, equal-width, open-channel dividing flow over a horizontal bed, the contraction coefficient at the maximum width-contracted section in the recirculation region is almost inversely related to the main channel upstream-to-downstream discharge ratio. The energy heads upstream and downstream of the division in the main channel are found to be almost equal. Under the assumption that the velocities are nearly uniformly distributed at the considered boundaries, the depth-discharge relationship follows the commonly used energy equation. The predicted results correlate fairly with the experimental data from this and other studies. The energy-loss coefficient of a division is expressed in terms of discharge ratio, upstream Froude number, and depth ratio. An expression for practical engineering applications is to determine the maximum possible branch-channel discharge at a given upstream discharge with a prescribed downstream Froude number or the maximum possible downstream Froude number if both branch- and main-channel discharges are prescribed. DOI: 10.1061/共ASCE兲0733-9429共2002兲128:7共716兲 CE Database keywords: Open channel flow; Water discharge; Contraction; Froude number.

Introduction Open-channel dividing flow is characterized by the inflow and outflow discharges, the upstream and downstream water depths; and the recirculation region in the branch channel. Taylor 共1944兲 examined the right-angled dividing flow in a 4-in. wide flume with conclusions that the dividing flow cannot be solved analytically from the momentum equation. He presented his experimental data in graphical form with four dimensionless parameters. With the aid of both downstream rating curves, he was able to solve the flow conditions. Grace and Priest 共1958兲 observed the flow division from a 5-in wide main-channel flume for various junction angles and channel-width ratios and presented their data in graphical form using dimensionless parameters. They divided the flows into two categories with and without the appearance of local standing waves near the division region. Without the appearance of standing waves, the downstream-to-upstream depth ratio and the branch-to-downstream depth ratio are nearly equal to unity and the ratios mildly decrease while the branch-to-mainchannel upstream discharge ratio increases. For a right-angled dividing flow, Law and Reynolds 共1966兲 concluded that the momentum equation, with a proper assumption of hydraulic pressure 1

Professor, Dept. of Water Resources and Environmental Engineering, Tamkang Univ., Taipei, Taiwan, 251, Republic of China. 2 Associate Professor, Dept. of Hydraulics and Ocean Engineering, National Cheng-Kung Univ., Tainan, Taiwan, 701, Republic of China. 3 Former Graduate Student, Dept. of Water Resources and Environmental Engineering, Tamkang Univ., Taipei, Taiwan, 251, Republic of China. 4 Former Graduate Student, Dept. of Water Resources and Environmental Engineering, Tamkang Univ., Taipei, Taiwan, 251, Republic of China. Note. Discussion open until December 1, 2002. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on January 27, 1999; approved on October 31, 2001. This technical note is part of the Journal of Hydraulic Engineering, Vol. 128, No. 7, July 1, 2002. ©ASCE, ISSN 0733-9429/2002/7716 –720/$8.00⫹$.50 per page.

force on the depth-averaged stagnation dividing streamline surface, and the energy consideration both describe the onedimensional flow characteristics. Ramamurthy and Satish 共1988兲 found that, for a short branch channel with the downstream Froude number exceeding a threshold value, the branch flow exhibits a unsubmerged recirculation region. They assumed critical flow at the maximum width-contracted section of the recirculation region to derive a relationship for downstream-to-upstream depth ratio Y d , downstream-to-upstream discharge ratio Q d , and upstream Froude number Fu . They estimated that the contraction coefficient was similar to the empirical procedure of twodimensional lateral conduit flow with a barrier outlet. However, Ingle and Mahankal 共1990兲 noted that the prediction cannot be used for small Fu and recommended that the assumption of critical flow at the maximum width-contracted section does not fit for all dividing flows. Ramamurthy et al. 共1990兲, skipping the assumption of critical condition, assumed no energy loss along the depth-averaged stagnation dividing streamline surface to obtain an expression for the momentum transfer rate from the main to the branch channel. They proposed a model for Fu ⬍0.75 which relates Q d to Fu and Y d . Their predictions agree well with their data and those of Sridharan 共1966兲. Hager 共1992兲 derived an expression for the energy-loss coefficient across a division. He assumed critical flow at the maximum width-contracted section and concluded that the branch discharge coefficient is simply a function of Fu and Q d . Recently, Neary and Odgaard 共1993兲 examined the effects of bed roughness on the three-dimensional structure of a dividing flow and considered the flow being similar to a bend flow. The primary objective of this paper is to propose a depthdischarge relationship and energy-loss coefficient for a subcritical, equal-width, right-angled dividing flow over a horizontal bed in a narrow aspect ratio channel. With known upstream discharge, prescribed branch flow discharge and downstream depth, the upstream depth is determined from energy considerations. The energy-loss coefficient due to the flow division is expressed as a function of Fu , Q d , and upstream-to-downstream depth ratio ¯Y . The contraction coefficient C c , in the recirculation region is de-

716 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2002

Downloaded 29 Oct 2011 to 131.170.6.51. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

¯ ,Fd ,Fb ,A u 兲 ⫽0 F共 Q d ,Y

(1)

where Q d ⫽Q d /Q u ⭐1; ¯Y ⫽Y u /Y d ; Fd ⫽Q d /(WY d 冑gY d ) ⫽Froude number in the downstream channel; Fb ⫽Q b /(WY b 冑gY b )⫽Froude number in the branch channel; and A u ⫽W/Y u ⫽upstream aspect ratio in the main channel. Eq. 共1兲 states that subcritical, equal-width, right-angled, open-channel dividing flow over a horizontal bed can be characterized by Q d , ¯Y , Fd , Fb , and A u . A schematic layout of the experimental flume and the sections considered is shown in Fig. 1. The main and branch flumes were 12 and 4.0 m long. Both channels were 14.7 cm wide. The division corner to the branch channel was sharp edged and located 5.35 m downstream from the main-channel inlet. An upstream sluice gate and the measured volume-change rate at each channel end regulated the discharge. Water depths and velocities were measured at several cross sections 共Fig. 1兲 with nine vertical profiles in each cross section. An adjusted weir was installed at each channel end to ensure that the downstream flows were subcritical. The upstream and downstream sections, AB and CD, of the main channel were taken at a distance of four channel widths from point E and F, respectively, while the branch section, GH was set at six to ten channel widths from the branch entrance, section EF. The flow at sections AB, CD, and GH was considered fully developed with velocities nearly uniformly distributed across the section for all runs in this study. An ALEC ACM-250 twocomponent electromagnetic current meter was used for the velocity measurements. Due to the ALEC ACM-250 sensor size limit, measurements were restricted to the mid-depth zone, 14 mm below the water surface and 16 mm above the channel bed in each profile. The maximum width-contracted section in the recirculation region was observed using dye trajectories. The contraction coefficient was computed once the velocity-integrated discharge equaled the branch discharge. The upstream inflow discharges were 3.02–5.37 L/s. The downstream-to-upstream discharge ratios ranged from 0.409 to 0.875. The Froude numbers at the upstream and downstream sections of the main channel were between 0.330 and 0.770 and 0.140 and 0.560, respectively. The Froude numbers in the branch channel were between 0.090 and 0.230. The upstream aspect ratio

Fig. 1. Schematic layout of experimental flume and considered sections

termined using velocity measurements. An expression for practical engineering applications is given to determine the maximum possible branch-channel discharge at a given upstream discharge and a prescribed downstream Froude number. This expression can also be used to determine the maximum possible downstream Froude number as well as the minimum possible downstream water depth if both branch-and main-channel discharges are prescribed.

Experimental Setup and Parameters A subcritical, equal-width, right-angled dividing open-channel flow over a horizontal bed is characterized by the upstream discharge Q u , the downstream discharge Q d , the upstream flow depth Y u , and the downstream flow depth Y d , of the main channel, the branch flow depth Y b , the channel width W, and the gravitational acceleration, g. Applying dimensional analysis yields

Table 1. Experimental Parameters

Discharge ¯ ratio Q 0.875 0.871 0.833 0.826 0.692 0.674 0.613 0.604 0.503 0.496 0.411 0.409

Mainchannel inflow Qu 共L/s兲

Branchchannel outflow Qb 共L/s兲

Mainchannel outflow Qd 共L/s兲

Mainchannel up-stream depth Yu 共cm兲

Branchchannel down-stream depth Yb 共cm兲

Mainchannel down-stream depth Yd 共cm兲

Mainchannel up-stream Froude number Fu

Branchchannel down-stream Froude number Fb

Mainchannel down-stream Froude number Fd

3.920 3.020 3.540 3.050 5.070 5.370 5.060 4.570 5.370 4.590 4.630 4.920

0.490 0.390 0.590 0.530 1.560 1.750 1.960 1.810 2.670 2.315 2.725 2.910

3.430 2.630 2.950 2.520 3.510 3.620 3.100 2.760 2.700 2.275 1.905 2.010

5.340 4.590 4.650 4.260 8.850 9.020 8.360 8.080 10.230 9.790 9.260 9.520

5.330 4.580 4.570 4.240 8.720 8.910 8.200 8.030 10.130 9.760 9.170 9.340

5.770 4.910 5.140 4.790 9.240 9.490 8.900 8.530 10.660 10.170 9.750 10.020

0.700 0.670 0.770 0.760 0.420 0.440 0.460 0.440 0.360 0.330 0.360 0.370

0.090 0.090 0.140 0.140 0.140 0.150 0.200 0.180 0.180 0.170 0.220 0.230

0.540 0.530 0.560 0.530 0.280 0.280 0.260 0.250 0.180 0.160 0.140 0.140

JOURNAL OF HYDRAULIC ENGINEERING / JULY 2002 / 717

Downloaded 29 Oct 2011 to 131.170.6.51. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

Fig. 3. Width ratio W ud /W of dividing streamline at Section AB

Fig. 2. Contraction coefficient in branch channel

Depth Ratio of the main channel was restricted from 1.44 to 3.45. The downstream aspect ratios of the main and branch channel were 1.38 – 3.07 and 1.45–3.47, respectively. It must be noted that due to the narrow range of the aspect ratio, this paper was not capable of depicting the aspect ratio effects on the dividing flow. The experimental parameters are listed in Table 1.

Results and Discussion Contraction Coefficient The location of the contracted section in the recirculation region was first determined using dye trajectories. The velocity vectors at the predetermined section and the nearby sections were then measured. For the effective width C c W, the velocity-integrated discharge from left bank Q I , equals Q b , where C c ⫽the contraction coefficient, and Q I was determined as m

Q I⫽

n

兺 兺 u i, j ⌬Y i, j ⌬W i, j

(2)

i⫽1 j⫽1

where u i, j ⫽longitudinal velocity over the elemental height ⌬Y i, j ; width ⌬W i, j in profile i; n⫽number of elements in each profile; and m⫽number of profiles in a cross section. The velocity was adjusted using a ratio of the measured discharge to the velocity-integrated discharge. A linear interpolation for determining the dividing width was necessary. Fig. 2 shows that for this paper the contraction coefficient C c , increases linearly as Q b increases. This indicates that for a small branch discharge Q b , the contraction results in a small effective width in the recirculation region of the branch channel.

Dividing Streamline The width of the depth-averaged stagnation dividing streamline four-channel widths upstream from point E 共Point I at section AB in Fig. 1兲 was determined for the velocity-integrated discharge from left bank Q I , equal to the downstream discharge Q d . Let the subscript ud refer to section AI with a discharge Q d across section AB, Fig. 3 shows that W ud /W⫽Q d /Q u at a 5% significance level with a correlation coefficient of 0.999. This figure also indicates that the assumption of fully developed velocity uniformly distributed at four-channel widths upstream from point E is reasonably acceptable.

For 0.41⭐Q d ⭐0.88, the energy heads H ud at section AI, H u at sections AB, and H d at section CD were found practically equal. The energy head was calculated as H⫽Y ⫹Q 2 /(2gW 2 Y 2 ), where the energy correction coefficient is assumed equal to unity. Thus H u ⬵H d

(3)

Essentially, Eq. 共3兲 implies that, instead of using the momentum equation with an assumption of the momentum transfer rate through the interface or of the pressure force on the depthaveraged stagnation dividing streamline surface, the commonly used energy equation, without losing generality, can be applied to model the one-dimensional dividing flow. Eq. 共3兲 can be written in a polynomial form as

冋

¯Y 3 ⫺ 1⫹

1 2

册

F2d ¯Y 2 ⫹

1 2Q d

2

F2d ⫽0

(4)

where Fd ⫽ 冑Q 2d /(gY 3d W 2 ). Eq. 共4兲 states that, for an equal-width, right-angled, dividing open-channel subcritical flow over a horizontal bed, ¯Y can be solved using a third-degree polynomial, provided that Q d and Fd are known. Eq. 共4兲 contains two unequal positive roots and one negative root if the discriminant D(Fd ,Q d )⫽Q d ⫺ 兵 F2d / 关 (2⫹F2d )/3兴 3 其 1/2⬎0. One multiple positive root and one negative root can be obtained if D⫽0. The negative root and the smaller positive root, representing Fu ⬎1 and Fd ⬍1, are of no interest in this paper. The predicted values from Eq. 共4兲 together with the measured data of this paper, Ramamurthy et al. 共1990兲, and Sridharan 共1966兲 are plotted in Fig. 4. The downstream Froude numbers of this paper were determined with the calculated momentum corrections while those of other studies were determined with momentum correction coefficient equal to unity. It is recalled that for the control volume AIDC the water depth increases in the downstream direction with increasing channel width at a nearly constant specific energy ¯Y ⭐1. Therefore, one data point from Sridharan 共1966兲 with Y u ⬎Y d was excluded. The agreement between the measured data and ¯Y predicted by Eq. 共4兲 is fairly good, although the predictions deviate for some of the measured data of Sridharan 共1966兲 and Ramamurthy et al. 共1990兲. The sections considered in the experiments by Ramamurthy et al. 共1990兲 were closer to the junction than those in this paper and the flow at the downstream section in their experiments tended to be nonuniform for Fu ⭓0.75, as mentioned in their paper. Thus, the discrepancies may result from: 共1兲 different observation locations, and 共2兲 difficulties in water depth measurements. The ranges of aspect ratio

718 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2002

Downloaded 29 Oct 2011 to 131.170.6.51. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

Fig. 6. Experimental and predicted K e values for various Q d and Fu

Fig. 4. Measured and predicted ¯Y for various Q d and Fd

of Ramamurthy et al. 共1990兲 between 1.420 and 3.30 was almost the same as in this paper. From this figure, ¯Y ⫽Y u /Y d increases with increasing Q d but with decreasing Fd . For Q d ⫽1.0 results ¯Y ⫽1.0, that is Y u ⫽Y d in the main channel.

Energy-Loss Coefficient For the control volume ABEGHFDCA 共Fig. 1兲, conservation of energy implies Q u H u 共 1⫺K e 兲 ⫽Q d H d ⫹Q b H b

(5)

where K e ⫽energy-loss coefficient across the flow division. Since H u ⬵H d and with Y b ⫽Y b /Y u , Eq. 共5兲 can be written as

冋

K e ⫽ 共 1⫺Q d 兲 1⫺

3

2Y b ⫹ 共 1⫺Q d 兲 2 F2u 2

Y b 共 2⫹F2u 兲

册

(6)

Eq. 共6兲 shows that the energy-loss coefficient K e , can be expressed in terms of Q d , Fu , and Y b . From these observations, it was noted that in the branch channel the fully developed region tends to migrate downstream as Q d increases. Thus, in this paper the downstream sections were set at six to ten channel widths from section EF. The measured Y b ⫽Y b /Y u in this paper and the

data of Grace and Priest 共1958兲 without the appearance of standing waves were plotted in Fig. 5. While the data of Grace and Priest 共1958兲 mildly increases with Q d , the mean value for Y b of this paper equals 0.990 with a correlation coefficient of 0.999 at a 5% significance level. The data from Taylor 共1944兲 were excluded since they were determined from an empirical curve. No measurements for Y b were available in the study by Ramamurthy et al. 共1990兲 and Sridharan 共1966兲. The predicted K e value using Eq. 共6兲 with Y b ⫽0.990, and the data from this paper and from Grace and Priest 共1958兲 are shown in Fig. 6. From this figure, K e increases with increasing Fu and Q d which indicates that a relatively higher-branch discharge results in a relatively lowerenergy-loss coefficient across the division. The prediction is somewhat smaller than those according to Grace and Priest 共1958兲. The discrepancies result from the energy correction coefficients taken unity in calculating the data of Grace and Priest 共1958兲.

Practical Applications Approximation Solution ¯ 2 ) can The approximation solution of Eq. 共4兲 with error of O(⌬Q be written in a difference form as ¯Y 共 Q ¯ ⫺⌬Q ¯ 兲 ⫽Y ¯ 共Q ¯ 兲⫹

F2d ¯ 3 关 3Y ¯ 2 ⫺Y ¯ 共 2⫹F2 兲兴 Q d

¯ ⌬Q

(7)

¯ Eq. 共7兲 states that with a division flow the solution ¯Y at Q ¯ ¯ ⫺⌬Q can be determined from the solution at Q for a given Fd . Thus, for subcritical flow throughout the division the depth ratio ¯ can be obtained step by step with the initial condition for any Q ¯Y ⫽1 and 0⬍⌬Q ¯ Ⰶ1. If a simple hydraulic jump condition, ¯Y 2 ⫽( 冑1⫹8Fd ⫺1)/2, is used as the initial condition, the solution is for a dividing flow with upstream supercritical, which we are not interested in for this paper.

Limitations of Frd and Q b Fig. 5. Branch-channel downstream-to-main-channel upstream depth ratio

Let Q b ⫽Q b /Q u ⫽1⫺Q d , the discriminant of Eq. 共4兲 can be expressed as Q b ⭐G(Fd )⫽1⫺ 兵 F2d / 关 (2⫹F2d )/3兴 3 其 1/2 for a given Fd . JOURNAL OF HYDRAULIC ENGINEERING / JULY 2002 / 719

Downloaded 29 Oct 2011 to 131.170.6.51. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

Q u with prescribed Fd and Fd at a given Q u with prescribed Q b are presented.

Acknowledgments The National Science Council, Taipei, Taiwan, under Grant No. NSC-88-2218-E-032-007, and the Tamkang University funded this study.

Notations

Fig. 7. Limitations of Fd and Q b

This inequality indicates that for dividing flow the limitations of Fd as well as Q b exist at a given condition. The experimental data from this paper, Taylor 共1944兲, Grace and Priest 共1958兲, and Sridharan 共1966兲 Ramamurthy et al. 共1990兲, with Y u ⬎Y d being excluded were also plotted. It was found that Q b is always less than G(Fd ) with the exceptions of a few data points being greater than G(Fd ). The three data points from Ramamurthy et al. 共1990兲 and one data point from Sridharan 共1966兲 had an upstream Froude number exceeding 0.89 and that Y d from Taylor 共1944兲 were determined from his empirical curve. Thus, for a given upstream discharge Q u and a prescribed downstream Froude number Frd , the available branch-channel discharge is calculated as Q b,max⫽Q u 兵 1⫺ 兵 F2d / 关共 2⫹F2d 兲 /3兴 3 其 1/2其

再

6 共 1⫺Q b 兲

冋

cos

1 3

cos⫺1 共 Q b ⫺1 兲 ⫹

4␲ 3

Subscripts b ⫽ downstream section in branch channel; d ⫽ downstream section in main channel; and u ⫽ upstream section in main channel.

(8)

The maximum possible downstream Froude number Fd,max for a given upstream discharge at a prescribed branch-channel discharge can be expressed as Fd,max⫽

The following symbols are used in this paper: A u ⫽ main-channel upstream aspect ratio⫽W/Y u ; C c ⫽ contraction coefficient; F ⫽ 冑Q 2 /(gW 2 Y 3 ); g ⫽ gravitational acceleration; H ⫽ energy head; K e ⫽ energy-loss coefficient; Q ⫽ flow discharge; Q b ⫽ branch-channel downstream-to-main-channel upstream discharge ratio⫽Q b /Q u ; Q d ⫽ main-channel upstream-to-downstream discharge ratio⫽Q d /Q u ; W ⫽ channel width; Y ⫽ water depth; ¯Y ⫽ main-channel upstream-to-downstream depth ratio ⫽Y u /Y d ; Y b ⫽ branch-channel downstream depth to main-channel upstream depth⫽Y b /Y u ; and Y d ⫽ main-channel downstream-to-upstream depth ratio ¯. ⫽Y d /Y u ⫽1/Y

册 冎

1/2

⫺2

(9)

As seen in Fig. 7, while the prescribed Q b ⫽0.45, the maximum possible downstream Froude number obtained from Eq. 共9兲 is 0.324.

Conclusions Using the energy considerations, the depth ratio and the total energy-loss coefficient across a division may be predicted. The depth ratio ¯Y and the total energy-loss coefficient K e were found in agreement with the observations. It was found that ¯Y increases with increasing Q d and with decreasing Fd and that K e increases with increasing Fu and Q d . For engineering applications, a difference equation for solving ¯Y and the limitations of Q b at a given

References Grace, J. L., and Priest, M. S. 共1958兲. ‘‘Division of flow in open channel junctions.’’ Bulletin No. 31, Engineering Experimental Station, Alabama Polytechnic Institute. Hager, W. H. 共1992兲. ‘‘Discussion of ‘Dividing flow in open channels.’ by A. S. Ramamurthy, D. M. Tran, and L. B. Carballada.’’ J. Hydraul. Eng., 118共4兲, 634 – 637. Ingle, R. N., and Mahankal, A. M. 共1990兲. ‘‘Discussion of ‘Division of flow in short open channel branches.’ by A. S. Ramamurthy and M. G. Satish.’’ J. Hydraul. Eng., 116共2兲, 289–291. Law, S. W., and Reynolds, A. J., 共1966兲. ‘‘Dividing flow in open channel.’’ J. Hydraul. Div., Am. Soc. Civ. Eng., 92共2兲, 207–231. Neary, V. S., and Odgaard, A. J., 共1993兲. ‘‘Three-dimensional flow structure at open channel diversions.’’ J. Hydraul. Eng., 119共11兲, 1223– 1230. Ramamurthy, A. S., and Satish, M. G., 共1988兲. ‘‘Division of flow in short open channel branches.’’ J. Hydraul. Eng., 114共4兲, 428 – 438. Ramamurthy, A. S., Tran, D. M., and Carballada, L. B., 共1990兲. ‘‘Dividing flow in open channels.’’ J. Hydraul. Eng., 116共3兲, 449– 455. Sridharan, K. 共1966兲. ‘‘Division of flow in open channels.’’ thesis, Indian Institute of Science, Bangalore, India. Taylor, E. H., 共1944兲. ‘‘Flow characteristics at rectangular open-channel junctions.’’ Trans. Am. Soc. Civ. Eng., 109, 893–912.

720 / JOURNAL OF HYDRAULIC ENGINEERING / JULY 2002

Downloaded 29 Oct 2011 to 131.170.6.51. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org