Numerical and Experimental Study of DividingOpen-Channel Flows
A. S. Ramamurthy1; Junying Qu2; and Diep Vo3
Abstract: Dividing flows in open channels are commonly encountered in hydraulic engineering systems. They are inherently three-dimensional �3D� in character. Past experimental studies were mostly limited to the collection of test data on the assumption that the flowwas 1D or 2D. In the present experimental study, the flow is treated as 3D and test results are obtained for the flow characteristics ofdividing flows in a 90°, sharp-edged, rectangular open-channel junction formed by channels of equal width. Depth measurements aremade using point gauges, while velocity measurements are obtained using a Dantec laser Doppler anemometer over grids definedthroughout the junction region. A 3D turbulence model is also developed to investigate the dividing open-channel flow characteristics. Thepredicted flow characteristics are validated using experimental data. Following proper model validation, the numerical model developedcan yield design data pertaining to flow characteristics for different discharge and area ratios for other dividing flow configurationsencountered in engineering practice. Energy and momentum coefficients based on the present 3D model yield more realistic energy lossesand momentum transfers for dividing flow configurations. Data related to secondary flows provide information vital to bank stability, ifthe branch channel sides are erodible.
DOI: 10.1061/�ASCE�0733-9429�2007�133:10�1135�
CE Database subject headings: Open channel flow; Experimentation; Three-dimensional models; Numerical models; Simulation.
Introduction
In hydraulic and environmental engineering, one commonlycomes across branching channel flows. Some of the distinctivecharacteristics of a dividing flow in an open channel are illus-trated in Fig. 1. They include a zone of separation immediatelynear the entrance of the branch channel, a contracted flow regionin the branch channel, and a stagnation point near the downstreamcorner of the junction. In the region downstream of the junction,along the continuous far wall, separation due to flow expansionmay occur.
A great number of experimental and analytical studies dealwith dividing flows. Taylor �1944� conducted the first detailedexperimental study in an open channel and proposed a graphicalsolution, which included a trial-and-error procedure. Grace andPriest �1958� presented experimental results for the division offlow at different width ratios of the branch channel orientation tothe main channel. They also classified the division of flow intotwo regimes, with and without the appearance of local standingwaves near the branch. The regime without waves correspondedto the case where the Froude numbers were relatively small, and
1Professor, Dept. of Civil Engineering, Concordia Univ., 1455 deMaisonneuve W., Montreal, Quebec, Canada H3G 1M8. E-mail:[email protected]
3Research Associate, Concordia Univ., Montreal, Canada.Note. Discussion open until March 1, 2008. Separate discussions must
be submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on June 20, 2005; approved on November 30, 2006. Thispaper is part of the Journal of Hydraulic Engineering, Vol. 133, No. 10,
the regime with the waves corresponded to the free over-fall con-ditions at sections downstream of the junction.
Milne-Thomson �1949�, Tanaka �1957�, and Murota �1958�solved the problem of branch channels analytically using confor-mal transformation based on the assumption of flow depth beingconstant in all the channels. Law and Reynolds �1966� investi-gated the problem of dividing flows using analytical and experi-mental methods. Hager �1984� proposed a simplified model forevaluating the energy loss coefficient for flow through branches,neglecting variation of velocity at the entrance to the branch.
Existing results �Krishnappa and Seetharamiah 1963; Law1965; Sridharan 1966; Ramamurthy and Satish 1988� indicatethat the flow condition at the entrance to the branch is generallyunsubmerged when the Froude number Fr in the branch is greaterthan a threshold value �0.3 or 0.35�. Ramamurthy et al. �1990�obtained an expression for the momentum transfer rate from themain to the branch channel. Hager �1992� derived an expressionfor the energy-loss coefficient for branch channel flows. He as-sumed critical flow at the contracted section and expressed thebranch discharge coefficient as a function of Fu and Qrd. Nearyand Odgaard �1993� examined the effects of bed roughness on thethree-dimensional �3D� structure of dividing flows. For lowFroude number flows, they presented detailed velocity-vector andparticle-trace plots in the initial part of the separation zone. Theirmeasurements indicated no depth variations in the junction�Neary and Odgaard 1995�. Further, Neary et al. �1999� numeri-cally investigated the lateral-intake inflows using 3D two-equation turbulence models without considering the water surfaceeffects.
Weber et al. �2001� performed an extensive experimental studyof combining flows in a 90° open channel for the purpose ofproviding a very broad data set comprising three velocity compo-nents, turbulence stresses, and water surface mappings. Huang
et al. �2002� provided a comprehensive numerical study of com-
bining flows in open-channel junctions using the 3D turbulencemodel and validated the model by using the detailed test data ofWeber et al. �2001�. Recently, assuming the velocities to be nearlyuniform, Hsu et al. �2002� presented a depth-discharge relation-ship and energy-loss coefficient for a subcritical, equal-width,right-angled dividing flow over a horizontal bed in a narrow as-pect ratio channel.
Until now, experimental data related to 3D mean velocity com-ponents and water surface profiles for dividing flows in openchannels were not available. A primary objective of the presentstudy is to obtain these flow characteristics of dividing flows in a90°, sharp-edged, rectangular open-channel junction formed bychannels of equal width. The data set presented in this paper iscomposed of water surface mappings and 3D velocity distribu-tions in the vicinity of the channel junction region. The otherprimary objective of this study is to develop a 3D numericalfree-surface turbulence model to predict the characteristics of di-viding flows in open channels. The volume of fluid �VOF�scheme is used to capture the free surface. The model is validatedusing the data obtained from the experimental studies.
Experimental Studies
Test Setup
The experiments were performed in 90° horizontal dividing flowchannels �Fig. 2�. The main channel is 6.198 m long, the branchchannel is 2.794 m long, and the main channel and the branchchannel are 0.305 m high and 0.610 m wide and are made of12.7 mm Plexiglas plates. The branch channel is positioned at thedistance of 2.794 m from the channel entrance �Fig. 2�. The flowenters the main channel after passing through a transition section
Fig. 1. Flow characteristics of a dividing flow in open channels
containing screens and honeycombs to ensure properly developedflow with low turbulence in the flume. At the end of the mainchannel and branch channel, control gates regulate the flow depth.
The discharge from the main flume or the branch channel ismeasured using 60° standard V-shaped weirs. The maximum errorin the discharge measurement is estimated to be 3%. The pointgauge used to measure the water surface in the channels canrecord depths to the nearest 0.1 mm. A 2D Dantec laser Doppleranemometer �LDA� is used to measure the mean velocity compo-nents of the flow fields. The error of velocity measurement isestimated to be 1%. The water surface profile for a given flowcondition is completely and continuously mapped in one day.
The coordinate system used to record data is shown in Fig. 2.All distances are nondimensionalized by the channel width�B=0.610 m�. The nondimensionalized coordinates are X*, Y*,and Z* for X /B, Y /B, and Z /B, respectively. All measurementsare made with reference to the floor of the upstream lower cornerO of the branch channel where X*=Y*=Z*=0 �Fig. 3�. The veloc-ity components in the X-, Y-, and Z-directions are defined as u, v,and w, respectively. The velocities u*, v*, and w* are nondimen-sionalized by the critical velocity Vc= ��Qu
2� /gB2�1/3 in the up-stream main channel, where Qu denotes the upstream discharge.
Measurements are made only after steady conditions areattained. The measuring locations are shown in Fig. 3. Test con-ditions ensure the Reynolds number of the flow R=VR /� is gen-erally high for both the main and the branch channels. Here R, V,and � denote hydraulic radius, mean velocity, and kinematic vis-cosity, respectively.
The discharges of downstream main and branch channel flowsare denoted as Qd and Qb, respectively. The discharge ratio Qr
denotes the ratio of Qb to Qu �Table 1�. For brevity, only data forsome Qr and selected locations are shown in the discussion ofresults. Detailed results pertaining to all the discharge ratios �Qr�
Fig. 3. Cross sections for measurements of the flow velocity
are provided by Qu �2005�.
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Discussion of Experimental Results
Velocity Component v*
Fig. 4 shows contour plots for the velocity component v* near thewater surface �Z*=0.27� for Qr=0.838 in the junction region. Inthis sketch, the narrow area of return flow near the entrance to thebranch denotes the separation zone that envelopes the recirculat-ing flow. For Qr=0.838, the largest velocities of v* occur justdownstream of the branch near Y*=−1.0 �Fig. 4�. Here, the maxi-mum flow contraction occurs. In fact, the largest velocities of v*
are seen just downstream of the branch near Y*=−1.0 for all fivedischarge ratios tested. Downstream of Y*=−0.1.0 for Qr=0.838,the separation zone begins to shrink. Further, the separation re-gion ends at Y* close to −2.3 for Qr=0.838 and Z*=0.27. Experi-mental results show that the recirculation region becomes smallernear the channel bottom, corresponding to the lower values of Z*
�Figs. 5�a and b��.For the section at Y*=−1.0, Figs. 5�a and b� show the contour
plot of the v* component for Qr=0.149 and 0.838, respectively.Here, the view shown is for an observer looking toward the posi-tive Y*-direction. The increase in the width of the separation withdecreased values Qr is clearly seen in these sketches. It suggeststhat the separation zone increases both in width and length as thedischarge ratios decrease.
For combining flows in rectangular channels, at a fixed dis-charge ratio, it was noted �Weber et al. 2001; Ramamurthy andZhu 1997� that higher angles of entry of the lateral flow into themain channel resulted in the larger separation zones. In thepresent case of the dividing flows, the mean exit angle of thestreamlines for flow entering the branch increases with an in-crease in Z* �surface streamlines� compared to the exit angles of
Table 1. Experimental Flow Conditions
Qu
�m3/s�Qb
�m3/s� QrVc=� Qu
2
gB2�1/3
�m/s�
0.047 0.007 0.149 0.914
0.046 0.014 0.308 0.906
0.046 0.019 0.409 0.907
0.047 0.032 0.672 0.910
0.046 0.038 0.838 0.903
Fig. 4. Contours for velocity v* at the plane Z*=0.27 forQr=0.838
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the streamlines located at the lower Z* �bottom streamlines�. Fur-ther, for a fixed Z*, the exit angle of the flow decreases when thedischarge ratio Qr increases.
Vector Field u*–v*
For Qr=0.838, Fig. 6 shows plots for the velocity components u*
and v* at Z*=0.270 and Z*=0.033 in the junction region. As men-tioned earlier, the wider and longer separation zone occurs athigher Z* value �Figs. 6�a and b��. Further, when Qr decreases, thewidth of the separation zone increases �Figs. 5�a and b��.
A second separation zone �Fig. 1� may occur in the section ofthe main channel downstream of the junction because of flowexpansion in this region. The width and length of the separationzone increase when Qr increases. When Qr is very small, theseparation zone would not exist. As before, the separation zonenear the surface layers is wider �Figs. 6�a and b�� than the sepa-ration zone near the channel bottom. The stagnation zone �Fig. 1�occurs at the downstream corner of the junction. The highest ve-locity u* occurs at the upstream corner of the junction.
Secondary FlowsSecondary flows are very important in 3D experimental investi-gation of open-channel flows. For Qr=0.838, Fig. 7 shows vectorplots for the velocity components u*–w* in the X*–Z* plane atvarious locations, Y*=−0.29 to Y*=−2.5, in the branch channel.The existence of a strong secondary current in the contractingregion is clearly seen in Fig. 7�a�. The spiral motion in the anti-clockwise direction weakens as the flow moves downstream but isstill present at Y*=−2.5, as indicated by significant u* magni-tudes. As one would expect in the recirculation zone, the second-ary flow is quite weak, as shown by the short vectors in this zone.It is also observed that the secondary flow occurs even in themain channel, as shown in the v*–w* vector plot of Fig. 8. Thestrongest spiral motion occurs near the downstream corner of thejunction.
The main objective of the study was the details of the charac-teristics of dividing flows in the junction region. However, to getthe true picture of the secondary flow, the length of the main
Fig. 5. Contours for velocity v* at cross section Y*=−1.0: �a�Qr=0.149; �b� Qr=0.838
channel and the branch channel downstream of the junction
should be much longer. In the present case, the length of thebranch channel was of the order of 5B and hence is consideredshort.
Water Surface ProfilesThe water surface profile is shown for Qr=0.409 and Qr=0.838 inFig. 9. It shows that the flow depth in the main channel is higherthan the flow depth in the branch channel. The lowest flow depthoccurs at the contracted zone in the branch channel. Furtherdownstream of this zone, the flow depth increases when the widthof the separation zone decreases. As the speeds of the curvedstreamlines separating from the upstream corner b �Fig. 1� arevery high, the corresponding flow depths are relatively smallernear b than the depth of the approaching main flow.
For a low value of Qr=0.409, Fig. 9�a� shows that the flow
Fig. 6. Velocity u*–v* vector plot for Qr=0.838 at different levels:�a� Z*=0.27; �b� Z*=0.033
depth at the stagnation zone around h �Fig. 1� is higher than the
depths at other regions in the junction. However, for a highervalue of Qr=0.838, the recirculation zone existed in the far wallregion of the main channel just downstream of the junction �Fig.6�. Consequently, the flow depth in this region rst �Fig. 1� isslightly higher ��2% � than the depth at the stagnation zone.
For all flow conditions, generally the flow depth Yd in the mainchannel downstream of the junction was higher than the flowdepth Yu in the approaching flow. Also, both the flow depths Yu
and Yd were higher than the flow depth Yb in the branch channel.
Fig. 7. Velocity u*–w* vector plot for Qr=0.838 at various locations�looking upstream�: �a� Y*=−0.29; �b� Y*=−0.73; �c� Y*=−1.0; and�d� Y*=−2.50
Fig. 8. Velocity v*–w* vector plot for Qr=0.838 at X*=−1.7
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The flow depth ratio Yu /Yd and Yb /Yd decreases while the dis-charge ratio Qr increases. In Fig. 10, the experimental values ofYu /Yd are plotted against the values of Yu /Yd predicted by thetheoretical relation �Eq. �1�� suggested by Hsu et al. �2002�. Theagreement appears to be reasonable
�Yu
Yd�3
− �1 +1
2Fd��Yu
Yd�2
+1
2�1 − Qr�2Fd2 = 0 �1�
Here, Fd denotes the Froude number at the downstream section ofthe main channel.
Three-Dimensional Numerical Simulation
Governing Equations
In the Reynolds-averaged approach to turbulence, all the un-steadiness is averaged out; in fact all unsteadiness is regarded aspart of turbulence. Eq. �2� denotes the Reynolds-averaged Navier-
Fig. 9. Contours of water surface profiles: �a� Qr=0.409; �b�Qr=0.838
Stokes equation �RANS�, �i=1, 2, 3 and j=1, 2, 3�
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��Ui
�t+ �Uj
�Ui
�xj= −
�p
�xi+
�
�xj�2�Sji − �uj�ui�� �2�
where ��density of water; g�acceleration due to gravity;p�pressure; t�time; and Ui�mean velocity in the xi-direction�i=1,2 ,3�. Also, the continuity equation is
�Ui
�xi= 0 �3�
The two-equation turbulence models are based on the Bouss-inesq eddy-viscosity approximation, which assumes that the prin-cipal axes of the Reynolds stress tensor �ij are coincident withthose of the mean strain-rate tensor Sij at all points in a turbulentflow. This is analogous to the Stokes postulate for laminar flows.The coefficient of proportionality between �ij and Sij�eddy vis-cosity �T.
�ij = − ui�uj� = �TSij −2
3k�ij �4�
The two-equation models are the most popular turbulencemodels in numerical simulation of turbulent flows. In two-equation models, the two most widely used, models�k–� modeland the k–� model. Here the k–� model �Wilcox 2000� is pre-ferred and adopted. Kinetic eddy viscosity is
�T = k/� �5�
Turbulent kinetic energy:
�k
�t+ Uj
�k
�xj= �ij
�Ui
�xj− �*k� +
�
�xj��� + *�T�
�k
�xj� �6�
Specific dissipation rate:
��
�t+ Uj
��
�xj=
�
k�ij
�Ui
�xj− ��2 +
�
�xj��� + �T�
��
�xj� �7�
Closure coefficients and auxiliary relations suggested byWilcox �2000� are used.
Boundary Condition and Initial Conditions
At the free surface, the VOF scheme �Ferziger and Peric 2002� iswidely used. For the VOF scheme, one has to introduce a newvariable c, which is called the void fraction. The void fraction c is
Fig. 10. Comparison of the flow depth ratio Yu /Yd
defined by the ratio of water to air in a cell. Generally, c=1 when
the cell is full of water and c=0 when the cell is full of air. Thegoverning equation of c�following �Ferziger and Peric 2002�:
�c
�t+
��ujc��xj
= 0 �8�
Also, c must satisfy the following condition:
0 � c � 1 �9�
For free-surface problems, one has to solve the equation forthe void fraction along with the conservation equation for massand momentum. Alternatively, near the free-surface boundary, onecan treat both fluids as a single fluid whose properties vary inspace according to the volume fraction of each phase, as follows:
� = �1c + �2�1 − c�; � = �1c + �2�1 − c� �10�
where ��density and ��molecular viscosity. Subscripts 1 and 2denote the two fluids �e.g., water and air�.
At the wall boundary, the wall-function approach proposed byLaunder and Spalding �1972� is used. At the inlet, the gradients ofvelocity quantities, turbulent quantities, and void fractions in theaxial direction are prescribed as zero. The total flow rate Qu isknown. Hence, assuming an initial depth at the inlet yields theinitial inlet mean velocity. As calculation progresses, both thedepth and the velocity at the inlet get altered. At the end of eachiteration, only the new mean velocity at the inlet is adjusted toprovide the same flow rate Qu. The pressure heads at the inlet areobtained using linear extrapolation methods from the interior ofthe solution domain. At the start of calculations, hydrostatic pres-sure distribution is assumed in the solution domain.
At the outlets of the main channel and the branch channel, thewater surfaces are prescribed �given data based on the ratingcurves of the two channel outlet controls�. The velocities andturbulent quantities at both outlets are obtained using the linearextrapolation method from the interior of the solution domain. Atthe end of each iteration, the new mean velocities at the twooutlets are adjusted to ensure the prescribed flow rates Qd in themain and Qb in the branch. All the air boundaries are defined aszero-pressure boundaries. The pressure heads at both outlets areobtained using linear extrapolation methods from the interior ofthe solution domain.
Numerical Algorithm
The proposed numerical model solves the standard 3D RANSequations for turbulent unsteady flow. The control volume tech-nique is used to convert the governing equations to algebraicequations that can be solved numerically. It employs a collocatedgrid. The approximation of the convection term is handled by thedeferred correction scheme �DCS�, which �Ferziger and Peric2002� combines the advantages of both the upwind differencescheme �UDS� and the central difference scheme �CDS�. Thepressure-velocity coupling is achieved using the SIMPLE algo-rithm, and the discretized equations are solved with a Stone-basedtridiagonal solver.
Solution Procedures
The computational domain is shown in Fig. 11. The width B ofthe channel is equal to 0.610 m, and the height H of the channelis equal to 0.305 m. To apply the uniform inlet flow condition tobegin with, the length of the main channel upstream of the junc-tion is extended to 6.1 m �=10B�. However, the length of the
main channel downstream of junction and the length of the
branch channel are 2.44 m �=4B� since the main interest is tostudy the junction flow characteristics. This configuration corre-sponds to the channel length in the experimental setup. Normally,it is desirable to have a channel much longer than 4B in theexperimental setup.
The multiblock algorithm �Ferziger and Peric 2002� is devel-oped for grid generation and computation. The flow domain ismeshed with a power law function that generates a fine mesh inthe vicinity of the channel boundary �Fig. 12�. The grid cells nextto the boundary are constructed well within the turbulent region.
Discussion of Simulation Results
The coordinate system used to record data is shown in Fig. 12.Similar to the experimental result analysis, all distances are non-dimensionalized by the channel width B=0.610 m. For brevity,only the experimental data for Qr=0.838 are considered for com-paring the predicted results with the test data obtained.
Velocity Component v*
For the model predictions, Fig. 13 shows the contour plots of thevelocity component v* near the water surface �Z*=0.270� in thejunction region. The corresponding experimental data are illus-trated in Fig. 4. Both the numerical and experimental results�Figs. 13 and 4� show the separation zone in the branch channel.The maximum flow contraction occurs near Y*=−1.0 in thebranch; further, the recirculation region becomes smaller near thebottom, corresponding to lower values of Z* �not shown�.
Fig. 11. Computational domain
Fig. 12. Grid geometry near the junction region
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The variation of v* at Y*=−1.0 in the branch channel is shownin Fig. 14 for the numerical results and in Fig. 5�b� for theexperimental results. The view shown is for an observer lookingtoward the positive Y*-direction. The agreement between theexperimental results and the numerical results is reasonable.
Vector Field u*–v*
Fig. 15 shows vector plots for the predicted velocity componentu*–v* at Z*=0.270 in the junction region. It shows that there aretwo separation zones in the junction region. One occurs in thebranch channel and the reattachment point is near Y*=−2.0 forthe region close to the water surface. The other separation zonecaused by the flow expansion is in the main channel downstreamof the junction. The stagnation zone occurs near the downstreamcorner of the junction. As one would expect, the highest velocityu* occurs at the upstream corner of the junction. The predictedresults and the test data are in good agreement �Figs. 15 and 6�a��.
For a more quantitative comparison, test data and predicteddata for the velocity component v* at a few locations in thebranch channel are shown in Fig. 16. The agreement between thetwo is reasonable. In the branch, for X*=−0.098 at the locationY*=−0.29 �Fig. 16�a�� and Y*=−1.0 �Fig. 16�b��, the values of v*
are positive, indicating that these points are in the flow separationzone. However, for X*=−0.098 and Y*=−1.62 �Fig. 16�c��, v* isnegative near the channel bottom and positive near the watersurface �high Z*�. This shows that the separation zone is wider atthe top, as mentioned before. For X*=−0.787 at Y*=−1.0, denot-
Fig. 14. Contours for predicted velocity v* at cross sectionY*=−1.0
Fig. 13. Contours for predicted velocity v* at the plane Z*=0.27
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ing the maximum contracting zone �Fig. 16�b��, v* is higher thanthe value v* at an upstream section Y*=−0.29 �Fig. 16�a�� andY*=−1.62 �Fig. 16�c��.
For sections X*=0.0 and −1.70 in the main channel, Fig. 17compares u* based on test data and predicted data. At the sectionX*=0.0, u* is quite high at the location Y*=0.098, which is nearthe upstream corner of the junction. The positive values of u* atY*=0.787 and X*=−1.7 �Fig. 17�b�� again indicate a separationzone in the main channel.
Fig. 15. Predicted velocity u*–v* vector plots at Z*=0.27
Fig. 16. Comparisons of velocity v* in the branch channel���experimental data, —�numerical data�
Secondary FlowsFigs. 18 and 19 show the predicted secondary flows at Y*=−0.29 and −0.73, respectively. Compared to the correspondingexperimental results �Figs. 7�a and b��, they show that the two-equation turbulence model can capture the secondary flow. How-ever, the vortex size of the secondary flow in the prediction issmaller than the vortex size in the experiment. Huang et al. �2002�remark rightly that higher-order turbulence models are needed tocapture the secondary flow characteristics more faithfully.
Water Surface ProfilesFig. 20 displays the water surface profiles in the junction regionin order to compare them with the experimental results �Fig.9�b��. For Qr=0.838, the water surface level in the branch channelappears to be lower than the water surface level in the mainchannel. There is a very sharp drop of the water surface at theupstream corner of the junction �Fig. 20�. This is also observedduring the test �Fig. 9�b��. The flow depth at the downstreamcorner of the junction denoting the stagnation point is higher thanthe other neighboring regions �Fig. 20�. Since Qr=0.838, the axialvelocity u* is reduced considerably as the flow crosses the junc-tion. This results in a large separation zone in the main channel;hence, the variation of the flow depth in this region is very small�Fig. 9�b��. The maximum difference between the predicted re-
Fig. 18. Predicted velocity u*–w* vector plots at Y*=−0.29 �lookingupstream�
Fig. 17. Comparisons of velocity u* in the main channel���experimental data; —�numerical data�
sults and the test data related to the flow depths in the junctionregion is less than 5%. The overall agreement of the water surfaceprofiles is reasonable �Fig. 21�.
Summary and Conclusions
The three mean velocity components and water surface profilesfor dividing flows in a 90°, sharp-edged, rectangular open-channel junction formed by channels of equal width are obtainedon the basis of experimental and numerical studies. Test dataindicate that the width and length of the separation zone in thebranch channel decrease when the discharge ratio Qr increases.The separation zone is smaller near the channel bottom corre-sponding to lower values of Z* than near the surface �highervalues of Z*�. The mean exit angle of the streamlines for flowentering the branch increases with an increase in Z* �surfacestreamlines� compared to the mean exit angle of the streamlineslocated at the lower Z* �bottom streamlines�. Further, for a fixedZ*, the exit angle of the flow decreases when the discharge ratioQr increases.
The largest velocities of v* appear to occur just downstream ofthe branch near Y*=−1.0 for all five discharge ratios tested, wherethe maximum flow contraction occurs. A separation zone mayalso occur in the main channel downstream of the junction, be-cause of the flow expansion. The width and length of this sepa-ration zone increase with the increase in the discharge ratio Qr.The secondary flow �spiral motion in the anticlockwise direction�in the branch channel is considerable in the contracted flow re-
Fig. 19. Predicted velocity u*–w* vector plots at Y*=−0.73 �lookingupstream�
Fig. 20. Contours of predicted water surface profiles
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gion. For subcritical flows, the depth Yd is always higher than Yu.The depths Yu and Yd are always higher than Yb, while the depthratios Yu /Yd and Yb /Yd decrease when the discharge ratio Qr in-creases. The lowest flow depth occurs in the region of flow con-traction in the branch channel.
For dividing flows in 90° rectangular open-channel junctions,the 3D two-equation turbulence model faithfully reproduces themean flow characteristics such as velocity profiles, water surfaceprofiles, and mean flow patterns. In particular, the separation zonenear the entrance to the branch channel and the separation zone inthe main channel downstream of the junction are also predictedby the model, and the predictions qualitatively agree with experi-mental data. The predicted reattachment length on the branch wallof the separated flow is close to the test results. The overall vali-dation of the model is reasonable. Since numerical modeling isrelatively less expensive, a well-validated model can be usedto predict flow behavior in various situations encountered inpractice.
Notation
The following symbols are used in this paper:B � channel width;c � 2 3 2
Fig. 21. Comparisons of water surface profiles at several sections�� experimental data; —�numerical data�
� volume fraction� Qd / gYdW
JOURNA
J. Hydraul. Eng. 2007.1
Fr � Froude number;Fu � Froude number at upstream section of the
main channel;g � acceleration due to gravity;k � turbulent kinetic energy;p � pressure;
Qb � discharge of the branch channel;Qd � discharge of the downstream main channel;Qr � discharge ratio Qb /Qu;
Qrd � main channel upstream-to-downstreamdischarge ratio Qd /Qu;
Qu � discharge of the upstream main channel;R � Reynolds number �=VR /��;R � hydraulic radius;t � time;
Ui � velocity in xi-direction �i=1,2 ,3�;u ,v, and w � velocity components in the x-, y-, and
z-directions, respectively;u*, v* and w* � dimensionless velocity components in the
x*-, y*-, and z*-directions, respectively;V � mean velocity;
Vc � �Qu2 /gB2�1/3;
X, Y, and Z � cordinates of three dimensions;X*, Y*, and Z* � coordinates of three dimensions in
dimensionless directions;Yb � flow depth in the branch channel;Yd � flow depth in the downstream of the main
channel;Yu � flow depth in the upstream of the main
channel;� � dynamic viscosity;� � kinematic viscosity.� � density of water; and� � specific dissipation rate.
Subscripts
i � 1, 2, 3 �x-, y-, and z-directions, respectively�;and
j � 1, 2, 3 �x-, y-, and z-directions, respectively�.
Superscripts
Prime ��� � fluctuating �turbulent� component.
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