Scientia Iranica A (2015) 22(6), 1994{2000

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Numerical analysis of the ow over weirs and labyrinthside-weirs

M. Nabatiana;�, M. Amiria, M.R. Hashemib and N. Talebbeydokhtia

a. Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran.b. Department of Water Engineering, Shiraz University, Shiraz, Iran.

Received 20 January 2014; received in revised form 27 September 2014; accepted 30 June 2015

KEYWORDSLabyrinth side-weir;Lateral ow;Spatially varied ow;Di�erentialquadrature;Unsteady ow.

Abstract. The study is undertaken on Di�erential Quadrature Method (DQM) with theaim of developing a 1-D numerical analysis of the ow over side-weirs. The numerical resultswere compared with relevant experimental data for both the simple and labyrinth side-weirs.The results showed that the numerical methodology can e�ectively predict the dischargeand ow pro�le associated with labyrinth side-weirs. Deviation (from experimental data)was found not exceeding 4.5%. Furthermore, the contributions of di�erent terms of thegoverning equation were assessed through a comprehensive sensitivity analysis. The resultsshow that in order to simplify the governing equation, the channel slope and the frictionslope can be eliminated.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Side weirs are traditionally adopted to protect hy-draulic structures and sometimes function as an outletin irrigation canals. Computation of the water surfacepro�le along a side-weir and its outlet discharge are themain data required for design of this structure.

Flow over a side-weir is a Spatially Varied Flow(SVF) with decreasing discharge. Due to nonlinearityof the governing equation, a closed form solution isnot available for the general case [1] and numericalmethods alongside experimental data are used to quan-tify the ow characteristics [2-7]. In particular, manyresearchers have tried to provide empirical relations forthe weir coe�cient [4,5,8-14].

When there is a limitation on the length of a side-weir, the Labyrinth Side Weir (LSW) is recommended,because it allows for a greater discharge by increasing

*. Corresponding author.E-mail addresses: nabatianmohamad@gmail.com (M.Nabatian); mamiri@shirazu.ac.ir (M. Amiri);Hashemi@shirazu.ac.ir (M.R. Hashemi); taleb@shirazu.ac.ir(N. Talebbeydokhti)

the e�ective length of weir. Some researchers haveclaimed that the outlet discharge of labyrinth side-weir can be four times as much as that of the side-weirs [15]. In terms of the previous research, most ofthe studies have focused on quanti�cation of the LSWcoe�cient [15-21] and less attention has been paid tonumerical solution of the water surface pro�le over thistype of weir.

The results of this method are veri�ed by exper-imental data. Many numerical techniques are usedfor analyzing steady SVF, such as Finite Di�erence(FD) and Boundary Element (BE). DQM can getmore accurate results, because the grids are directlycalculated and also it can match with a di�erent bound-ary zone easier than other methods; also, it requiresless formulations and calculations in comparison withother methods [22-27]. DQM was used for modelinggradually varied ow [26] and in this study, it is used formodeling SVF on prismatic and non-prismatic canals.

In this study, the numerical solution of ow overLSW has been investigated using DQM. In terms ofthe weir coe�cient, several formulae were comparedagainst the experimental data [8] and the formulawhich had the most e�cient parameters was proposed.

M. Nabatian et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 1994{2000 1995

2. Material and methods

In this part, numerical methodology of the model isintroduced. After introducing the governing equations,the solution procedure, in numerical aspect, has beenexplained in detail.

2.1. Governing equationsConservations of mass, momentum, and energy in the uid are the most utilized principles for mathematicalmodeling of the hydraulic phenomena. That which ofthe equations are suitable for a certain problem mainlydepends on its physical features as well as the abilityof each equation set. In view of the fact that thelongitudinal velocity of lateral out ow in the SVF withdecreasing discharge over side-weirs cannot be easilyestimated employing momentum equation [28], the setof energy and continuity equations is more appropriateand thus it is proposed as the mathematical tool. How-ever, it should be noted that in derivation of the energyequation, it is assumed that longitudinal velocity isequal to the cross-sectional mean ow velocity which isa reasonable hypothesis [29]. The set can be presentedas follows [29]:

@Q@x

+@A@t

+ ql = 0; (1)

1g@V@t

+@H�

@x+v2l � �V 2

2gQql =

�" Q

+(�� �)V

2gA@A@t

� V2g��@x

; (2)

where A is cross section; Q is discharge; V is velocity;Vl is velocity of lateral ow; ql is discharge per unitchannel width; � is energy coe�cient; H� is total head;�" is rate of work done by shear force per unit channellength; g is acceleration due to gravity; y is waterdepth; x is distance measured along the channel; and tis time.

After some manipulations, the governing equationfor non-prismatic cross section channels can be writtenas:

dydx

=S0 � Sf � �Q

gA2dQdx + �Q2@A

gA3@x jy=cte

1� F 2�

; (3)

where S0 is channel slope; Sf is friction slope; and F 2� =

2�Q2

A3dAdy

dydx . The friction slope (Sf ) can be estimated by

Manning equation which reads [29]:

Sf =n2Q2P 4=3

A10=3 ; (4)

where n is manning coe�cient, and P is wettedperimeter. Eq. (3) can be reduced to the following

form for prismatic channels:

dydx

=S0 � Sf � �Q

gA2dQdx

1� F 2�

: (5)

For side-weir channels, the rate of ow varies withdistance along the main channel. The discharge perunit length (dQdx ) can be computed as [29]:

dQdx

=23Cwp

2g(y � p)3=2; (6)

in which ow discharges over a side-weir, p=weirheight; Cw=side-weir coe�cient. This equation wasobtained by [30].

2.2. Numerical solutionWith reference to Eqs. (3) and (5), a non-linear ordi-nary di�erential equation should be solved to obtainthe ow �eld ow over a side-weir. The analyticalsolutions for these equations exist only for special cases,but have no generality. To overcome this shortage,a numerical solution can be utilized. Typical solu-tion strategies include single-step, predictor-correctoror modi�ed Hinds methods. In this investigation,DQM is employed as an in uential e�cient numericalscheme. The concept of DQM, as a numerical tool forsolving di�erential equations, was emerged from theintegral quadrature technique. The de�nite integral ofa typical function f(x) over a speci�c interval can beapproximated as [26]:Z b

af(x)dx =

NX1

wif(xi); (7)

where wi is weighting coe�cients, and f(xi) is thefunction value at xi. In the same manner, the mthorder di�erentiation of the function with respect tothe independent variable can be estimated applying alinear weighted sum of the function values [26]:

@mf@xm

jx=xi =NXj=1

wmij f(xi); (8)

where wmij is weighting coe�cients; and N is number ofgrid points. The weighting coe�cients can be expressedas [26]:

wij =1

xj � xi�Nk=1;k 6=i

xi � xkxj � xk for j 6= i; (9)

wij =NX

k=1;k 6=1

1xi � xk for j = i: (10)

Substitution of dydx from Eq. (13) into the governing

equation gives [1]:

w(i; 1)y1 +NXj=2

wijyj = f(xi; yi): (11)

1996 M. Nabatian et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 1994{2000

In these expressions, y1 is known from the speci�edboundary condition. Therefore, the governing equationbecomes simpli�ed into a set of N � 1 algebraicnonlinear equations to be solved for y2, y3,:::yn. This isaccomplished by the well-known Newton's method [1]:

Uiter+1 = Uiter + �Uriter; (12)

Jiter�Uiter = �(F )iter; (13)

where U is vector of unknowns; F is equation vector;and J is Jacobian matrix as given by [1]:

Jmn =@Fm@Un

: (14)

Choosing a correct distribution of grids could assist theaccuracy of modeling. Cosine distribution of grids wasadopted herein [1]:

x(i� 1)� x(i) = 0:5�1� cos

�(i� 1) ��

n

���0:5

�1� cos

�(i) ��n

��; (15)

where x is the position of each grid. Schematicpresentation of the scheme is depicted in Figure 1.

3. Results and discussion

In this section, comparing with experimental data,results of the foregoing model are evaluated. There areseveral approaches concerning the formula of dischargecoe�cient; the most suitable one is determined in the�rst numerical test. On this basis, the ow over side-weir was modeled by DQM with di�erent dischargecoe�cients; thereby the one producing less errors wasobtained. The labyrinth side-weir was considered inthe second test. The purpose of this test is to examinehow one-dimensional dynamic equation can model the ow �eld and to assess which terms in the equationare more in uential. The latter is investigated via thesensitivity analysis.

Figure 1. Cosine distribution of grid points.

3.1. Rectangular side-weirThe ow �eld associated with side-weirs of di�erentheights was numerically captured by DQM and veri�edagainst available data from literature [8]. Di�erent weircoe�cients were examined and the most appropriateone was determined.

The ow occurs in a rectangular channel with awidth of b = 30 cm and a length of 570 cm (Figure 2).The height of side-weir is p = 15 cm and the channelconveys a discharge of Q = 0:03888 m3/s beforereaching the side-weir.

Several proposed formulae for discharge coe�-cients are summarized in Table 1, where F1 is Froudenumber of the approach ow; z = h=H; h is waterdepth at the channel; H is speci�c energy (assumedconstant); = 1 � KF ; F is Froude number; K isfunction of L=b; L is length of side-weir; b is channelwidth for L=b=0.402; = 1 � 0:07F for L=b = 0:605and K = 1� 0:087F [7].

The same test, but now with p = 30 cm andQ = 0:0375 m3/s was also considered. The measuredand computed water pro�les are given in Figures 3

Figure 2. (a) Plan view, and (b) longitudinal crosssection of the channel in the �rst case.

Figure 3. Water pro�le over side-weirs was obtained inlaboratory and modeled with DQM with di�erent weircoe�cients and Muslu model.

M. Nabatian et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 1994{2000 1997

Table 1. Existing formulae for discharge coe�cient.

Researcher Weir coe�cient

Subramanya and Awasty (1972) Cm = 0:864h

1�F21

2+F21

i0:5Nandesamoorthy and Thomson (1972) Cm = 0:288

h2�F2

11+2F2

1

i0:5Ranga Ruja et al. (1979) Cm = 0:54� 0:4F1

Singh et al. (1994) Cm = 0:33� 0:18F1 + 0:49 ph1

Jalil and Borghei (1987) Cm = 0:71� 0:41F1 � 0:21 ph1

Borghei et al. (1987) Cm = 0:7� 0:48F1 � 0:3 ph1

+ :06LbHager (1987) Cm = 0:485

h2+F2

12+3F2

1

i0:5Muslu (2001) Cm = 0:611

p3z � 2

Table 2. Estimating the discharge of lateral ow.

n Q(L/s)

y1

(cm)V1

(m/s)F1

P(cm)

L(cm)

Qwobserved

(L/s)

QwestimatedEq. (3)(L/s)

QwestimatedEq. (17)

(L/s)

ErrorEq. (3)

ErrorEq. (17)

1 15 14,155 0,211939 0,179855 12 75 9.11 9.00 9.00 1.2075 1.20752 25 15,391 0,324865 0,264384 12 75 18.62 19.20 19.40 3.1149 4.18903 35 16,554 0,422859 0,331825 12 75 28.33 27.40 27.40 3.2827 3.28274 45 14,831 0,606837 0,503098 12 75 16.43 17.10 17.20 4.0779 4.68655 15,4 18,007 0,171045 0,128693 16 75 8.34 8.60 8.70 3.1175 4.31656 25,1 19,269 0,260522 0,189487 16 75 17.70 17.60 17.70 0.5650 0.00007 35 18,498 0,378419 0,280916 16 75 12.50 12.80 13.00 2.4000 4.00008 45 21,542 0,417789 0,287395 16 75 36.37 37.80 38.00 3.9318 4.4817

Figure 4. Water pro�le over side-weirs was obtained inlaboratory and modeled with DQM and Muslu model.

and 4. From Figure 3, it is evident that the mostsuitable weir coe�cients are those suggested by Hagerand Subramanya. As Figure 4 indicates, a relativelyless deviation from the experiments was obtained whenincorporating Muslu's formula into DQM. Referring toEq. (16), the sum of Relative Errors (RE) for DQMcombined with Hager discharge coe�cient was 1.6%,while it was reported as 5.63% by Muslu [10].

RE =nXi=1

jyobs � yprejyobs

� 100; (16)

Figure 5. Plan of the channel and weir in Sample (2).

where yobs is the observed data; ypre is the predicteddata; and n is the number of grid.

3.2. Labyrinth side-weirIn the second sample, discharge of lateral owover labyrinth side-weir was estimated by the one-dimensional formulation. For further veri�cation, theexperimental data recorded by Emiroglu (2010) wasalso utilized (Figure 5). Experimental discharge beforeside-weir and weir heights were shown in Table 2.Besides, the water pro�le over labyrinth side-weirs areestimated for two samples (Q = 0:02446 m3/s andthe heights for Samples (2-1) and (2-2) are 12 and 16cm, respectively). Figure 3 shows the water surface

1998 M. Nabatian et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 1994{2000

Figure 6. Water surface over labyrinth side-weir obtainedin laboratory and modeled by DQM.

pro�le over labyrinth side-weir in Samples (2-2) withfull equation Eq. (3).

In order to conduct sensitivity analysis on eachterm in Eq. (3), ow over labyrinth side-weir wasmodeled by Eqs. (5), (17), and (18). Root MeanSquare Errors (RMSE) of each model were calculatedby Eq. (19) as shown in Table 2.

dydx

=aqgA2

dQdx + aQ2

gA3@A@x jy=cte

1� F 2�

; (17)

dydx

=S0 � Sf + aQ2

gA3@A@x jy=cte

1� F 2�

; (18)

RMSE =nXi=1

r(yobsi � yprei)2

n: (19)

Data from experimental studies was applied for veri�-cation of the model based on one-dimensional dynamicequation and DQM (Figure 6). Comparison betweennumerical and experimental results shows RE=1.95%for Sample (2-1) and RE=3.16% for Sample (2-2).These negligible errors illustrate that DQM, whencombined with the one-dimensional dynamic equation,provides an e�cient tool to appropriately model the ow over labyrinth side-weirs and estimate the gener-ated water surface pro�le.

Table 2 shows the contribution of each term inEq. (3). In both of the samples, the term �Q2

gA3@A@x jy=cte,

which represents the non-prismatic e�ects, has themost in uence on accuracy of the numerical predic-tions. The term �Q

gA2dQdx , which accounts for spatial

variability of ow discharge, also contributes signi�-cantly to estimate water pro�le. However, the bedslope and the friction slope appear to have the leaste�ect (less than 3% change in RMSE). So as thesesamples demonstrate, it seems reasonable to ignore thee�ects of bed and friction slopes in favor of a simpli�ednumerical model. Consequently, Eq. (22) is o�ered asan appropriate mathematical model governing the owover labyrinth side-weir.

4. Conclusions

The numerical model carried out by DQM on weirsand labyrinth side-weirs was veri�ed by several experi-mental models and other numerical models; the resultsshow that this model has less errors. Furthermore,DQM is less intricate and needs less points to reach theappropriate results. Comparison between numericalresults and experimental data shows that relative errorfor the model on weirs is less than 2 percent, and it isless than 3.5 percent on labyrinth side-weir.

The results of sensitivity analysis and analysis ofcontribution of each parameter show that the channelslope and the friction slope are not requisite parametersand they can be eliminated to simplify the mathemat-ical representation of problem.

References

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2. Khorchani, M. and Blanpain, O. \Free surface mea-surement of ow over side weirs using video monitoringconcept", Flow Meas. Instrum., 15, pp. 11-117 (2004).

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4. Nandesamoorthy, T. and Thomason, A. \Discussionof spatially varied ow over side weirs", J. Hydr Div.,ASCE, 98(12), pp. 2234-2235 (1972).

5. Ranga Ruja, K.G., Grupta, S.K. and Prasad, B. \Sideweir in rectangular channel", J. Hydr Div., ASCE, 98,pp. 1-10 (1979).

6. Venutelli, M. \Method of solution of nonuniform owwith the presence of rectangular side weir". J. Irrigat.Drain Eng., ASCE, 134, pp. 840-846 (2008).

7. Muslu, Y. and Tozluk, H. \A Numerical Approach forSide Weir Discharge Using the Parameters of WaterSurface Pro�le", [Yam savaklarda akinim su yuzupro�lini etkileyen parametrelerle hesabi icin numerikbir yaklasim], Proc., 8th Engineering Conference.,Muhendislik Haftast, S. Demirel Universitesi, Isparta,Turkey (in Turkish) (1994).

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11. Uyumaz, A. and Muslu, Y. \Flow over side weirs incircular channels", J. Hydraul Eng., ASCE, 111, pp.144-160 (1985).

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12. Jalili, M.R. and Borghei, S.M. \Discharge coe�cient ofrectangular side weirs", J. Irrigat Drain Eng., ASCE,122(2), pp. 132-132 (1996).

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16. Emiroglu, M.E., Kisi, O. and Bilhan, O. \Predictingdischarge capacity of triangular labyrinth side weirlocated on a straight channel by using an adaptiveneuro-fuzzy technique", Adv. Eng. Softw., 41(2), pp.154-160 (2010).

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Biographies

Mohamad Nabatian received the BS degree in CivilEngineering from Persian Gulf University and the MSdegree in Hydraulic Structure from Shiraz University.His research concentrates on the numerical and physicalmodeling of hydraulic structures. He is interestedin Di�erential Quadrature Method and CharacteristicMethod. He worked in 2012-2014 as a researcher andstructure designer for Tir and Seton Pars Companyplus Tarh and Sazeh Katibe Company where he was incharge of the calculation department.

Mehrab Amiri received BS and MSc degrees inHydraulic Structures in 2004 and 2007, respectively,from Shiraz University, Iran, where he is currentlyserving as a faculty member at the Department of Civiland Environmental Engineering. His research interestsinclude: computational hydraulics, morphodynamicsand analysis of hydrological time series. He has alsopublished and presented various papers in journals andat conferences in these �elds.

Reza Hashemi is a research fellow at the Centre forApplied Marine Sciences (CAMS), School of OceanSciences. He was awarded BEng (1997) and MSc(1999) in Civil Engineering from Shiraz University.His MSc thesis was about automatic calibration ofa numerical model (i.e. Princeton Transport Model)using nonlinear programming techniques. His PhD(2006) concerned numerical modelling of the free sur-face ow and shallow water equations using Di�er-ential Quadrature Method. During his PhD studies,he went to Bangor University and collaborated inresearch projects in CAMS, relating to tide and surgemodelling in the Bristol Channel and morphodynamicmodelling in the Irish Sea. He worked as an academic

2000 M. Nabatian et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 1994{2000

sta� member in Shiraz University (2006-2011) andcontinued to collaborate with CAMS in that period inareas like applications of Arti�cial Neural Network andSmoothed Particle Hydrodynamics in hydrodynamicand morphodynamic modelling. He was appointed tothis post in December 2011.

Nasser Talebbeydokhti is a professor in Civil and

Environmental Engineering at the School of Engineer-ing, Shiraz University. He is a member of The Academyof Sciences of Iran and also editor-in-chief of IranianJournal of Science and Technology Transaction of CivilEngineering. His major areas of specialty are hydraulicengineering, sediment transport, river engineering, andenvironmental engineering. He has published morethan 40 journal papers.