TECHNICAL NOTE / NOTE TECHNIQUE

Numerical simulation of sharp-crested weir flows

J. Qu, A.S. Ramamurthy, R. Tadayon, and Z. Chen

Abstract: The sharp-crested weir in a rectangular open channel can be used as a simple and accurate device for flowmeasurement and control in open channels. However, in the past, the solution to this problem was found mainly on the ba-sis of experimental data or through the development of simplified theoretical expressions. In the present study, k-3 turbu-lence model is applied to obtain the flow parameters such as pressure head distributions, velocity distributions, and watersurface profiles. The predictions of the proposed numerical model are validated using existing experimental data. The k-3turbulence model developed is used to predict the characteristics of a sharp-crested weir in a rectangular open channel.The volume of fluid (VOF) scheme is used to find the shape of the free surface. A properly validated model permits oneto obtain the flow characteristics of the sharp-crested weir for a wide range of weir and hydraulic parameters without re-course to expensive and more time consuming experimental methods. Further, the model permits one to incorporate smallchanges in the geometric parameters involving small changes in inlet and outlet conditions and study their impact on theweir flow characteristics.

Key words: k-3 turbulence model, VOF model, sharp-crested weir, flow measurement, open channel flows.

Resume : Les deversoirs a paroi mince dans un canal ouvert rectangulaire peuvent servir de dispositif simple et precis demesure du debit et de controle de l’ecoulement dans les canaux a surface libre. Toutefois, anterieurement, la solution a ceprobleme se trouvait principalement dans les donnees experimentales ou dans le developpement d’expressions theoriquessimplifiees. Dans la presente etude, le modele de turbulence k-3 est utilise pour obtenir les parametres de l’ecoulement,tels que les repartitions de la hauteur piezometrique, les repartitions de vitesse et les profils de la surface de l’eau. Les pre-visions du modele numerique propose sont validees en utilisant des donnees experimentales existantes. Le modele de tur-bulence k-3 developpe est utilise pour predire les caracteristiques d’un deversoir a paroi mince dans un canal ouvertrectangulaire. Le volume de fluide est utilise pour trouver la forme de la surface libre. Un modele correctement valide per-met d’obtenir les caracteristiques de l’ecoulement du deversoir a paroi mince pour une grande gamme de deversoirs et deparametres hydrauliques sans avoir recours a des methodes experimentales dispendieuses et demandant plus de temps. Deplus, le modele permet d’incorporer de petits changements dans les parametres geometriques impliquant de petits change-ments dans les conditions d’entree et de sortie et d’etudier leur impact sur les caracteristiques de l’ecoulement dans le de-versoir.

Mots-cles : modele de turbulence k-3, modele « VOF », deversoir a paroi mince, mesure de debit, ecoulement dans canauxa surface libre.

[Traduit par la Redaction]

IntroductionThe sharp-crested weir in a rectangular open channel

(Fig. 1) serves as a simple and accurate device for flow

measurement in open channels. It also enables one to controland regulate open channel flows. Further, the lower nappeprofile of the weir is often considered as the shape of spill-way profiles. A large number of theoretical and experimen-tal studies have been carried out to determine the weircharacteristics. Rouse and Reid (1935) made an analyticalinvestigation of the design of spillway crests based on theinvestigation of sharp-crested weir flow characteristics. Kan-daswamy and Rouse (1957) experimentally investigated theweir discharge coefficient Cd as a function of H/P, whereH is the driving head and P is the height of sharp-crestedweir. Kindsvater and Carter (1957) presented a comprehen-sive solution for the weir discharge characteristics based onexperimental results and dimensional analysis. Rajaratnamand Muralidhar (1971) experimentally determined the de-tailed distributions of velocity and pressure in the region ofthe weir crest. Han and Chow (1981) used ideal flow theory

Received 16 September 2008. Revision accepted 20 April 2009.Published on the NRC Research Press Web site at cjce.nrc.ca on6 October 2009.

J. Qu. Hydraulic Engineer, KGS Group, 865 Waverly Street,Winnipeg, MB R3T 5P4, Canada.A.S. Ramamurthy,1 R. Tadayon, and Z. Chen. ConcordiaUniversity, 1455 de Maisonneuve W, Montreal, QCH3G 1M8,Canada.

Written discussion of this technical note is welcomed and willbe received by the Editor until 31 January 2010.

1Corresponding author (e-mail: ram@civil.concordia.ca).

1530

Can. J. Civ. Eng. 36: 1530–1534 (2009) doi:10.1139/L09-067 Published by NRC Research Press

and developed a hodograph model to get some gross charac-teristics of the flow. Based on experimental results and sim-plified theoretical considerations, a general relationshipbetween the weir discharge coefficient Cd and the parameterH/P was determined by Ramamurthy et al. (1987). Recently,Khan and Steffler (1996) predicted the water surface profilesfor sharp-crested weirs with sloping upstream faces, usingtwo-dimensional finite element model involving verticallyaveraged continuity, longitudinal momentum and verticalmomentum equations. For weir slopes up to 278 with thehorizontal, their computed results for weir with sloping up-stream faces agreed well with test data. For larger upstreamweir slopes, numerical instability was encountered. Wu andRajaratnam (1996) experimentally determined the reductionfactor for flow over sharp-crested weirs due to submersion.Martinez et al. (2005) presented the characteristics of com-pound sharp-crested weirs.

The Froude number (Fr), the Reynolds number (Re), andthe Weber number (W) are the key flow parameters for thesestudies. In this study, the Froude number is in a very narrowrange. The Reynolds number is pretty high to reduce viscouseffects to negligible values. Also, the Weber number is quitehigh as the investigators have taken precautions to have Hmore than 6.5 cm. Therefore, H/P is the only significant pa-rameter characterizing the flow.

In the present study, the Reynolds Averaged Navier-Stokes (RANS) equations are applied to solve the problemof flow past a sharp-crested weir in a rectangular open chan-nel. The two-dimensional two-equation k-3 turbulence modelis adopted for the numerical simulation. The fractional vol-ume of fluid (VOF) method (Ferziger and Peric 2002) isused, which is an efficient method for treating the compli-cated free-surface problem (Mohapatra et al. 2001; Maron-nier et al. 2003). The results of simulation are validatedusing the experimental (Rajaratnam and Muralidhar 1971;Ramamurthy et al. 1987) data pertaining to surface profilesand the distributions of velocities and pressure heads.

Numerical methodThe proposed model solves the standard two-dimensional

Reynolds averaged continuity and Navier-Stokes equationsfor turbulent unsteady flow based on the two-equation k-3model (Wilcox 2000). For brevity, details of familiar equa-tions (Wilcox 2000) are not included. The control volume

technique is used to convert the governing equations to alge-braic equations that can be solved numerically. It employsthe collocated-grid approach. The pressure–velocity cou-pling is achieved using the pressure-implicit with splittingof operators (PISO) algorithm (Issa 1986). The discretizedequations are solved with a Stone-based tri-diagonal solver.

The volume of fluid (VOF) scheme is used to find theshape of the free surface. For the VOF scheme, one has tointroduce a new variable c, which is called the void fraction.The void fraction c is defined by the quantity ratio of waterto air in a cell. Generally, c = 1 when the cell is filled fullyby water and c = 0 when the cell is filled fully by air. Thegoverning equation (Ferziger and Peric 2002; Chen et al.2002) for c is given by eq. [1]:

½1� @c

@tþ @ðujcÞ

@xj

¼ 0; 0 � c � 1

Here, uj denotes the average flow velocity (j = 1, 2 in twodimensions), xj denotes Cartesian coordinates (j = 1, 2 intwo dimensions) and t represents time. Therefore, for freesurface problems, one has to solve the equation for the voidfraction besides the conservation equation for mass and mo-mentum.

Alternatively, near the free surface boundary, one cantreat both fluids as a single fluid, whose properties vary inspace according to the volume fraction of each phase (Fer-ziger and Peric 2002), i.e.:

½2� r ¼ r1cþ r2ð1� cÞ; m ¼ m1cþ m2ð1� cÞ

Here, r and m denote density and dynamic viscosity, re-spectively. Subscripts 1 and 2 represent the two fluids.

For the velocity components near the free surface boun-dary, it is assumed that the velocity of air is equal to the ve-locity of water. Far from the free surface boundary, thevelocity is equal to zero in the air. The pressure near thefree surface boundary is obtained using linear extrapolationmethod from the interior of the water domain.

Boundary conditionsAt the wall boundary, the wall-function approach pro-

posed by Launder and Spalding (1974) is used. The univer-sal logarithmic law of the wall, which is applicable to thefully turbulent region outside the viscous sub-layer, is ex-pressed as

½3� u

ut¼ 1

kln

utY

nþ C

Here, u = velocity parallel to the wall at the first cell, ut =resultant friction velocity, k (kappa) = 0.41, Y = normal dis-tance to the wall, n (nu) = kinematic viscosity, and C = 5.0for smooth surfaces. The near wall values of the turbulentkinetic energy k and the dissipation rate 3 are specified byassuming local equilibrium of turbulence (Wilcox 2000):

½4� k ¼ ut2

ffiffiffiffiffib�o

p ; 3 ¼ ðb�oÞ3=4k3=2

kYwhere b�0 ¼

9

100

At the inlet ‘an’ in Fig. 1, uniform velocity was assumed.The prescribed turbulent quantities at the inlet were of the

a f

s'c

db

k h

n s

m

P

HFlowt

y

x

Fig. 1. Computation domain for flow past sharp-crested weirs(slope ‘ab’ = 0).

Qu et al. 1531

Published by NRC Research Press

order of 2% of the mean velocity. The section ‘fh’ fardownstream from the weir crest (Fig. 1) is defined as a pres-sure boundary. This allows the water to flow out freely. Allthe air boundaries are defined as pressure boundaries withzero pressure.

Solution procedure

The computational domain is shown in Fig. 1. The chan-nel upstream of the weir ‘ab’ is 6.0 m long in the x-direc-tion. The channel downstream of the weir ‘df’ is 1.0 mlong. The weir is 0.297 m high ‘bc’. Body fitted coordinatesare used in the Cartesian frame. The flow domain is meshed

with a power law function that generates a fine mesh in thevicinity of the channel boundary. The grid cells next to theboundary are constructed well within the turbulent region.There are 19 456 grid cells in total. The program executiontime was of the order of 10 h for the two-dimensional modelsimulation on a personal computer with 800 MHz CPU and256 MB RAM. The results were checked for grid independ-ence using a coarser grid with double the grid spacing aswell as a finer grid with half the grid spacing. The resultsof the coarser grid size were in less agreement than the re-

0.0

0.5

1.0

0.0 0.5 1.0

p/(γYe)

y/Y

e

0.0

0.5

1.0

0.0 0.5 1.0

u/(U0)

y/Y

e

0.0

0.5

1.0

-60 0 60

φ

y/Y

e

Exp. data (Rajaratnam and

Muralidhar 1971)

Exp. data (Ramamurthy et

al. 1987)

Present simulation (H/P = 0.625)

(a) (b) (c)

Fig. 2. Flow characteristics at section c–t: (a) pressure distribution, (b) velocity distribution, and (c) angle of velocity.

0.0

0.5

1.0

-60 0 60

φ

y/Y

0

0.0

0.5

1.0

0.0 0.6 1.2

u/(U0)

y/Y

0

Exp. data (Rajaratnam and

Muralidhar 1971)

Exp. data (Ramamurthy et

al. 1987)

Present simulation (H/P = 0.625) Hydrostatic pressure line

0.0

0.5

1.0

0.0 0.5 1.0

p/(γY0)

y/Y

0

(a) (b) (c)

Fig. 3. Flow characteristics at section s–s’: (a) pressure distribution, (b) velocity distribution, and (c) angle of velocity.

-2

-1

0

1

2

-5 -4 -3 -2 -1 0 1 2

(Rajaratnam and Muralidhar 1971)

O Exp. data at H/P = 0.625

Δ Exp. data at H/P = 6

― Present simulation at H/P = 0.625

― Present simulation at H/P = 6

(Rajaratnam and Muralidhar 1971)

y/H

x/H

Fig. 4. Water surface profiles near the nappe region.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fra

P/(

H+

P)

Theoretical curve (Han and Chow 1981)

Exp. data (Kandaswamy and Rouse 1957)

Exp. data (U.S.B.R. 1948)

Exp. data (Ramamurthy et al. 1987)

Present simulation

Fig. 5. Variation of weir parameter P/Ya with Froude number Fra.

1532 Can. J. Civ. Eng. Vol. 36, 2009

Published by NRC Research Press

sult using the final grid. Further, the results for the finer gridwere almost the same as the case of the final grid chosen.The deviations of parameters, velocities and pressures, weregenerally much less than 1% between the results obtainedfrom the final grid and the finer grid.

Through the time-dependent simulation (for the specifiedinlet and outlet conditions), the water flows in the openchannel and constitutes the free surface between air andwater. In the simulation, two different values of H/P (0.625and 6.000) were chosen. These are the same flow parametersas in experiment number A1 of Rajaratnam and Muralidhar(1971).

Discussion of results

Velocity and pressure head distribution in the napperegion

The model predictions were validated for one full set oftest data related to H/P = 0.625. This involved verificationsfor the water surface profile, pressure profiles, and velocitydistributions at two different sections. Model velocity datain curvilinear flow should normally be compared with theirdata collected by Laser Doppler Velocimetry (LDV). This isdue to the fact that curvature effects do not affect LDV data.Presently, LDV data for the sharp-crested weirs are notavailable. As such, the available velocity databased on Pitottube measurement are used for the model validation.

At the crest section c–t, the finite size of the static anddynamic pressure probes can be expected to cause some in-terference effects while measuring pressure and velocity.Further, the curvature of the flow near the crest c is high.This may prevent very accurate pressure and velocity meas-urements there. For flow past a two-dimensional sharp-crested weir at the location of the weir crest ‘ct’ (Fig. 1),Fig. 2 shows the distributions of pressure and velocity aswell as the velocity angle 4 ( = tan–1v/u). Figure 2a showsthe variation of the distribution of the non-dimensional pres-sure head p/gYe with y/Ye at the crest section. Here, Ye isnappe thickness at crest ‘c’ (Fig. 1) and y is distance abovethe crest. In Fig. 2a, the pressure distribution obtained bythe present simulation is compared with the experimentaldata of Rajaratnam and Muralidhar (1971) and Ramamurthyet al. (1987). In Fig. 2b, the non-dimensional axial velocityu/U0 is plotted against the non-dimensional flow depthabove the sharp crest y/Ye. Here, the velocity U0 ¼

ffiffiffiffiffiffiffiffiffi2gH

p.

The results of the simulation are in generally good agree-ment with the existing test results. The predicted relation be-tween the velocity angle 4 and y/Ye based on the test dataalso appears to follow the trend of the earlier test data.

Figure 3 shows the distributions of pressure and velocityas well as the velocity angle 4 at section s–s’ (Fig. 1). As inthe previous case, the predicted values of flow parametersare close to the test data.

Surface profiles and other characteristicsFigure 4 shows the predicted non-dimensional flow pro-

files denoting y/H as a function of the non-dimensional dis-tance x/H for subcritical approach flows. In Fig. 4, thesurface profiles of the simulation are compared with the ex-perimental profiles (Rajaratnam and Muralidhar 1971). Thetwo profiles agree well with the present simulated profiles.

Figure 5 shows the variation of the weir parameter P/Yawith the Froude number Fra in the approach channel. It in-cludes a few points related to the present simulation and theprevious experimental studies. The agreement between thetest data, simulation results and theoretical predictions basedon ideal flow theory (Han and Chow 1981) are reasonable(Fig. 5).

ConclusionsThe two-dimensional two-equation k-3 turbulence model

together with the VOF method reproduces faithfully thecharacteristics of flow past a sharp-crested weir in a rectan-gular open channel. The predictions of the numerical modelagree well with the existing experimental and theoretical re-sults related to water surface profiles and distributions of thepressure head and velocity components. Due to the lowertime demand and lower cost of numerical methods com-pared to experimental methods in predicting the flow char-acteristics, simulation of the sharp-crested weir flows basedon a properly validated model provides the weir flow char-acteristics for various flow configurations encountered in thefield.

ReferencesBureau of Reclamation. 1948. Studies of crests for overall dams.

U.S.B.R. Bulletin 3, Part IV, Boulder Canyon Project Final Re-ports. Boulder Canyon.

Chen, Q., Dai, G., and Liu, H. 2002. Volume of fluid model forturbulence numerical simulation of stepped spillway overflowof plane free overfall. Journal of Hydraulic Engineering, 128(7):683–688. doi:10.1061/(ASCE)0733-9429(2002)128:7(683).

Ferziger, J.H., and Peric, M. 2002. Computational method for fluiddynamics. 3rd ed. Springer-Verlag Berlin Heidelbeg, New York.

Han, T.Y., and Chow, W.L. 1981. The study of sluice gate andsharp-crested weir through hodograph transformations. Journalof Applied Mechanics, ASCE, 48(6): 229–238.

Issa, R.I. 1986. Solution of implicitly discretized fluid flow equa-tions by operator splitting. Journal of Computational Physics,62(1): 40–65. doi:10.1016/0021-9991(86)90099-9.

Kandaswamy, A.M., and Rouse, H. 1957. Characteristics of flowover terminal weirs and sills. Journal of the Hydraulics Division,83(4): 1–13.

Khan, A.A., and Steffler, P.M. 1996. Modeling overfalls using ver-tically averaged and moment equations. Journal of HydraulicEngineering, 122(7): 397–402. doi:10.1061/(ASCE)0733-9429(1996)122:7(397).

Kindsvater, C.E., and Carter, R. 1957. Discharge characteristics ofrectangular thin-plate weirs. Journal of the Hydraulics Division,83(3): 1–36.

Launder, B.E., and Spalding, D.B. 1974. The numerical computa-tion of turbulent flows. Computer Methods in Applied Me-chanics and Engineering, 3(2): 269–289. doi:10.1016/0045-7825(74)90029-2.

Maronnier, V., Picasso, M., and Rappaz, J. 2003. Numerical simu-lation of three-dimensional free surface flows. InternationalJournal for Numerical Methods in Fluids, 42(7): 697–716.doi:10.1002/fld.532.

Martınez, J., Reca, J., Morillas, M.T., and Lopez, J.G. 2005. De-sign and calibration of a compound sharp-crested weir. Journalof Hydraulic Engineering, 131(2): 112–116. doi:10.1061/(ASCE)0733-9429(2005)131:2(112).

Mohapatra, P.K., Murty Bhallamudi, S., and Eswaran, V. 2001.

Qu et al. 1533

Published by NRC Research Press

Numerical study of flows with multiple free surfaces. Interna-tional Journal for Numerical Methods in Fluids, 36(2): 165–184. doi:10.1002/fld.126.

Rajaratnam, N., and Muralidhar, D. 1971. Pressure and velocitydistribution for sharp-crested weirs. Journal of Hydraulic Re-search, IAHR, 9(2): 241–248.

Ramamurthy, A.S., Tim, U.S., and Rao, M.V. 1987. Flow oversharp-crested weirs. Journal of Irrigation and Drainage Engi-neering, 113(2): 163–172. doi:10.1061/(ASCE)0733-9437(1987)113:2(163).

Rouse, H., and Reid, L. 1935. Model research on spillway crests.Civil Engineering, ASCE, 5(1): 10–15.

Wilcox, D.C. 2000. Turbulence Modeling for CFD. 2nd ed., and2nd printing. DCW Industries, Inc., La Canada, Calif.

Wu, S., and Rajaratnam, N. 1996. Submerged flow regimes ofrectangular sharp-crested weirs. Journal of Hydraulic Engineer-ing, 122(7): 412–414. doi:10.1061/(ASCE)0733-9429(1996)122:7(412).

List of symbols

C dimensionless constant = 5.0Cd Weir discharge coefficient

c void fractionFra Froude number for the approach flow

g acceleration due to gravityH water head upstream of weirk turbulent kinetic energyP Weir height

p pressureq discharge of approach flow per unit widtht time

U0 reference velocity ¼ffiffiffiffiffiffiffiffiffi2gH

p;

Ue average horizontal velocity at the brinku axial mean velocity component (parallel to the

horizontal wall)uj average flow velocity (j = 1, 2 in two dimension)ut resultant friction velocityv vertical mean velocity component (perpendicular to

the horizontal wall)x axial distance from c (Fig. 1) along channel wallxj Cartesian coordinate (j = 1, 2 in two dimensions)Y normal distance from the wall

Ya water depth in the approach channelYe flow depth at section c–t (Fig. 1)Y0 flow depth at section s–s’ (Fig. 1)y vertical distance from c (Fig. 1) perpendicular to

channel wall (floor)b�0 dimensionless coefficient = 9/100g specific weight3 dissipation ratek Karman’s constant = 0.41m dynamic viscosityn kinetic viscosityr density4 angle of the velocity vector with the horizontal

( = tan–1v/u)

1534 Can. J. Civ. Eng. Vol. 36, 2009

Published by NRC Research Press