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Discharge control by a side weir in a triangular main channel Contrle des dbits par un dversoir d'orage latral sur un canal triangulaire

Ali Uyumaz

Department of Civil Engineering, Istanbul Technical University, Maslak, 34469, Istanbul, TURKEY. E-mail uyumaz@itu.edu.tr

RESUME Les dversoirs dorage latraux ont t utiliss de manire extensive pour contrler le niveau deau pour lirrigation et les systmes de canal de drainage, afin de dvier les excs deau dans les canaux des ouvrages de protection contre les crues. Ils ont galement t utiliss comme dversoirs dorage pour les rseaux dassainissement urbain. Une solution analytique complte pour les quations qui rgulent les dbits dans les dversoirs dorages latraux nest pas possible. Jusqu peu, des mthodes approximatives fondes sur des exprimentations ralises partir dun ventail limit des nombreuses variables impliques ont t usits. Dans de nombreux cas, lusage de ce type de mthode a engendr des erreurs dans les dversements calculs. Le dbit dun dversoir dorage latral sur un canal rectangulaire a fait lobjet de nombreuses tudes, mais il ny a pas dtudes sur les dversoirs dorages latraux sur un canal triangulaire dans la littrature. Cette tude fait tat des tudes numriques de dbit dans les dversoirs dorages latraux sur des canaux triangulaires. Les modles numriques sont obtenus partir de principes nergtiques sur la base dune hypothse dnergie constante et solutionns par la mthode de diffrence finie. Les rsultats sont prsents sous forme graphique. Ltude recouvre les rgimes de dbits sub-critiques et super-critiques. Les expressions drives sont compares avec les rsultats exprimentaux pour le dversoir dorage latral et pour les profils de surface deaux pour ses rgimes.

ABSTRACT Side weirs have been used extensively for water-level control in irrigation and drainage canal systems, as a means of diverting excess water into relief channels for flood protection works, and as storm overflows from urban sewage systems. A complete analytical solution of the equations governing the flow in the side weir channels is not possible. Until quite recently, approximate methods based on experiments conducted over a limited range of the many variables involved have been used. In many cases, the use of such methods caused very substantial errors in the calculated spill discharge. The flow over a side weir in a rectangular channel has been the subject of many investigations but there are no more studies about side weirs in a triangular channel reported in the literature. In this study, numerical investigations of flow over side weirs in triangular channels are reported. Numerical models are obtained from energy principles on the basis of a constant energy assumption and solved by a finite difference method. The results are presented in graphical form. The study covers both sub- and supercritical flow regimes. Derived expressions are compared with experimental results for the side weir discharge and water surface profiles for these regimes.

KEYWORDS Side weir; discharge; discharge capacity; water surface; triangular channel.

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INTRODUCTION Side weirs have been used extensively for water-level control in irrigation and drainage canal systems, as a means of diverting excess water into relief channels for flood protection works, and as storm overflows from urban sewage systems. A complete analytical solution of the equations governing the flow in side weir channels is not possible. Until quite recently, approximate methods have been used based on experiments conducted over a limited range of the many variables involved. In many cases, the use of such methods cause very substantial errors in the calculated spill discharge. The flow over a side weir in a rectangular channel has been the subject of many investigations (Allen, 1957; Coleman and Smith, 1923; Collinge, 1957; Engels, 1920; Forchheimer, 1930; Frazer, 1954; Kumar, and Pathak, 1987; Ranga, et al 1979; Tyler, et al 1929). Probably the first theoretical approach to the hydraulics of a side weir in a rectangular channel was reported by De Marchi (1934). Theoretical and experimental studies for a side weir in a circular channel were reported in the literature include Uyumaz, (1982); Uyumaz, and Muslu (1985, 1987); Hager (1987); Uyumaz, and Smith (1981). Practically all the experimental work and theoretical analysis are confined to the flow over side weirs in rectangular and circular channels. There are no studies about side weirs in a triangular channel reported in the literature. Methods of analyzing spatially varied flow in a channel with a side weir have recently been developed to give accurate computations for certain cases, which include subcritical flow along the weir, and supercritical flow both in the upstream channel and along the weir (De Marchi,1934; El-Khashab, 1975; El-Khashab, and Smith, 1976). In side weir channels, the flow is not affected by curvature and the rate of depth varies with distance along the main channel. In the case of subcritical flow, the water surface profile at the main channel axis rises from the upstream to downstream, whereas the opposite occurs in supercritical flow (El-Khashab, 1975; El-Khashab and Smith, 1976). However, when the upstream and along the weir flows are supercritical, the depth remains below the critical depth just before the start of the weir. Also, the water surface is appreciably curved to the extent that it affects the distribution of pressure in the flow (Smith, 1973). In the present study, general expressions for the surface profile along the side weirs in triangular channel are derived on the basis of a constant specific energy assumption.

FLOW EQUATION FOR SIDE WEIRS The specific energy of the flow in the triangular main channel is assumed to remain constant. Referring to Fig.1, the specific energy at any point is

g2vhH2

+= (1) in which H is a specific energy; h is the depth of water; v is the mean velocity of flow in the main channel; and g is the gravitation constant.

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Figure 1. Definition sketch for triangular channel with discharge over side weir

If a is the cross sectional area of flow and Q is a discharge at any point in the main channel, then

2

2

ga2QhH += (2)

and

2tanghQhH 242

+= (3)

in which is the bottom angle of the triangular cross-section. Eq.3 can be rewritten as:

)hH(gh2

tanQ 2 = (4) General assumptions are that the flow in the side weir channel is approximately two-dimensional, and pressure in the channel is approximately hydrostatic despite some curvature and irregularity on the water surface. Although the flow over the side weir makes a considerable angle with the direction of the weir, a conventional weir equation for discharge per unit length is assumed as,

)ph()ph(g2mqdsdQ == (5)

where m is the side weir discharge coefficient; p is the weir height; and s is the distance measured along the channel. Provided that the proportion of the flow over the side weir is not excessive, these assumptions are approximately valid (El-Khashab, 1975; El-Khashab and Smith, 1976; Smith, 1973, 1974). The change of discharge per unit length of a triangular channel due to the weir can be obtained by taking the derivative of Eq.4 with respect to s and substituting it into Eq.5. The results are given below:

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)ph()ph(g2mdsdh)

hH2hhHh2(

2gtan

dsdQ 2 =

=(6) and

)ph(phdsdh)

hH2hhHh2(

m2tan 2 =

(7)

Eq.7 can be changed into dimensionless from through the following parameters that are obtained as a result of dividing the variables by the specific energy:

;zHh = x

Hs = (8)

These dimensionless parameters are then substituted into Eq.7 to obtain a differential equation for the water surface profile along the side weir. The results is given below:

)z12

zz1z2(m2

tan

)Hpz(

Hpz

dxdz

2

= (9)

On the other hand by considering the following functions

)z12

zz1z2()z(f2

1 = (10) and

)Hpz(

Hpz)z(f2 = (11)

One can obtain from the rearrangement of Eq.9, that

dz)z(f)z(f

m2tandx

2

1

= (12) Making use of the forementioned the dimensionless variables it is possible to see that

dxHdsx

Hs == (13)

Hence, its substitution into Eq.12 leads to (14)

dz)z(f)z(f

m2Htands

2

1

=

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Furthermore, integration both sides yields

dz)z(f)z(f

m2Htands

2z

1z 2

12s

1s = (15)

Which can be defined as

2z

1z)z(

m2HtanL = (16)

or

2z

1z)z(

m2HtanL = (17)

in which L is the length of the side weir, and (z) and (z) are integral areas. In the derivation of the foregoing equations it is assumed that the width of the water surface in the triangular channel is small compared with the length of the side weir. For a side weir conveying flood waters from a natural channel, this condition may not be satisfied and it may be necessary to consider the flow in the channel as three-dimensional, requiring a different analysis, or model study. Eq.17 gives the length of the side weir for a triangular channel when the water depth varies from h1 to h2 along the side weir at the main channel. Eq.14 is solved by using the finite difference method and the results are shown graphically in Fig.2 for practical purposes of uses. DISCHARGE COEFFICIENT FOR SIDE WEIR In Eq.17; the parameter m presents the discharge coefficient of the side weir. In this approach the discharge coefficient for a side weir in a rectangular channel is modified for the discharge coefficient for a side weir within a triangular channel, where the equivalent rectangular channel width is calculated as the ratio of the wet cross-sectional area to water depth. A detailed procedure for the determination of the discharge coefficient for sub- and supercritical flow regimes for a rectangular channel is presented by Subramanya and Awasthy (1972). The discharge coefficient for subcritical regimes for Froude number F12 as:

)F008.036.0(32m 1= (19)

Herein, F1 is the Froude number at the beginning of the side weir on the axis of the main channel. Derived expressions for side weir water surface profiles, and discharge for subcritical and supercritical regimes have been compared with experimental results. The analysis indicated errors of less than 12% as shown in Figs.3-6 (Uyumaz, 1982; 1989 and 1991).

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However, these equations will need further extensive checking from laboratory and field data to verify their applicability for design purposes.

Figure 2. Numerical solution of differential equation

. Figure 3. Water surface profile

Figure 4. Water surface profile

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Figure 5. For F11, discharge of the side weir

CONCLUSIONS The theoretical procedures have been presented for calculating flow profiles and discharges along a side weir in a triangular shaped channel. In the case of the subcritical regimes, the water profile at the main channel axis rises from upstream to downstream whereas the opposite occurs in the supercritical regime. Nevertheless, computation techniques based on the constant energy solution give results that are usually considerably more correct than empirical curves or equations based on limited range experiments. However, in this study the specific energy in a triangular side weir channel remains constant.

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REFERENCES Allen, J.W. (1957). The discharge of water over side weirs in circular pipes. Proc. of the I.C.E., London, Vol.6, Feb. p. 270-287. Coleman, G.S., and Smith, D.(1923). The discharging capacity of side weirs. Instn. Civ. Engrg., Selected Engineering Paper No.6. Collinge, V.K.(1957). The discharge capacity of side weirs. Proc. of the I.C.E., London, Vol.6, Feb. p. 288-304. De Marchi, G. (1934). Saggio di teotia de funzionamenta degli stramazzi laterali. LEnergia Electtricia, Milano, Vol.11, Nov. p. 849-860. El-Khashab, A.M.M. (1975). Hydraulics of flow over side weirs. Thesis presented to the University of Southampton, at Southampton, in partial fulfilment of the requirements for the degree of Doctor of Philosophy, England. El-Khashab, A.M.M., and Smith, K.V.H.(1976). Experimental investigation of flow over side weirs. J. Hydraulic Engineering, ASCE, 102(9), p. 1255-1268. Engels, H.(1920). Mitteilungen Aus Der Dresdener Flussbaulaboratorium. Weiten, Jan., p. 362-365 ; 387-390. Forchheimer, P. (1930). Hydraulic., Teubner Verlagsgesellschaft, Leibzig, Berlin, 3rd. Ed., Germany, p. 406-408. Frazer, W. (1954). The behaviour of side weirs in prismatic rectangular channels. Thesis presented to Glasgow University, in partial fulfilment for the degree of Doctor of Philosophy, England. Hager, W.H.(1987). Flow over side weir in circular channels. Discussion, J. Hydraulic Engineering, ASCE, 113(5), p. 685-688. Kumar, C.P., and Pathak, S.K. (1987). Triangular side weir. J. Irrigation and Drainage Engineering, ASCE, 113(2), p. 98-105. Ranga Raju, K.G., Parasad, B., and Gupta, S.K. (1979). Side weir in rectangular channel. J. Hydraulic Engineering, ASCE, 105(5), p. 547-554. Smith, K.V.H. (1973). Computer programming for flow over side weirs. J.Hydraulic Engineering, ASCE, 99(3), p. 495-508. Smith, K.V.H.(1974). Computer programming for flow over side weirs. Closure, J.Hydraulic Engineering, ASCE, 100(11), p. 1722-1723. Subramanya, K., and Awasthy, S.C. (1972). Spatially varied flow over side weirs. J.Hydraulic Engineering, ASCE, 98(1), p 1-10. Tyler, R.C., Carollo, J.A., and Steyskal, N.A.(1929). Discharge over side weirs with and without baffles. J.Boston Soc. Civ. Engrg., 16, p.118. Uyumaz, A.(1982). Yan savaklardaki akmn teorik ve deneysel incelenmesi. Thesis Presented to Istanbul Technical University, at Istanbul, in Partial Fulfilment of the requirements to the degree of Doctor of Philosophy, Turkey. Uyumaz, A., and Muslu, Y. (1985). Flow over side weirs in circular channels. J. Hydraulic Engineering, ASCE, 111(1), p. 144-160. Uyumaz, A., and Muslu Y.(1987). Flow over side weirs in circular channels. J. Hydraulic Engineering, ASCE, 113(5), p. 688-690. Uyumaz, A.(1989). Flow over side weirs in rectangular and circular channels. Water Science and Technology, 21, p. 739-746. Uyumaz, A., and Smith, R.H.(1991). Design procedure for flow over side weirs. J. Irrigation and Drainage Engineering, ASCE, 117(1), p. 79-90.

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