Uyumaz A.

I-Revues

SESSION 6.2 
NOVATECH 2007  1251 
Discharge control by a side weir in a triangular 
main channel  
Contrôle des débits par un déversoir d'orage latéral 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 déversoirs d’orage latéraux ont été utilisés de manière extensive pour contrôler le 
niveau d’eau pour l’irrigation et les systèmes de canal de drainage, afin de dévier les 
excès d’eau dans les canaux des ouvrages de protection contre les crues. Ils ont 
également été utilisés comme déversoirs d’orage pour les réseaux d’assainissement 
urbain. Une solution analytique complète pour les équations qui régulent les débits 
dans les déversoirs d’orages latéraux n’est pas possible. Jusqu’à peu, des méthodes 
approximatives fondées sur des expérimentations réalisées à partir d’un éventail 
limité des nombreuses variables impliquées ont été usités. Dans de nombreux cas, 
l’usage de ce type de méthode a engendré des erreurs dans les déversements 
calculés. Le débit d’un déversoir d’orage latéral sur un canal rectangulaire a fait 
l’objet de nombreuses études, mais il n’y a pas d’études sur les déversoirs d’orages 
latéraux sur un canal triangulaire dans la littérature.  
Cette étude fait état des études numériques de débit dans les déversoirs d’orages 
latéraux sur des canaux triangulaires. Les modèles numériques sont obtenus à partir 
de principes énergétiques sur la base d’une hypothèse d’énergie constante et 
solutionnés par la méthode de différence finie. Les résultats sont présents sous forme 
graphique. L’étude recouvre les régimes de débits sub-critiques et super-critiques. 
Les expressions dérivées sont comparées avec les résultats expérimentaux pour le 
déversoir d’orage latéral et pour les profils de surface d’eaux pour ses régimes.  
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. 
SESSION 6.2 
1252 NOVATECH 2007  
 
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  
g2
vhH
2
+=         (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.  
SESSION 6.2 
NOVATECH 2007  1253 
 
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
ga2
QhH +=       (2) 
and  
2
tangh
QhH 24
2
α
+=       (3) 
in which “α” is the bottom angle of the triangular cross-section. Eq.3  can be rewritten 
as:  
)hH(gh
2
tanQ 2 −α=      (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(g2mq
ds
dQ
−−==−     (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: 
SESSION 6.2 
1254 NOVATECH 2007  
)ph()ph(g2m
ds
dh)
hH2
hhHh2(
2
gtan
ds
dQ 2
−−−=
−
−−
α
=
(6) 
and  
)ph(ph
ds
dh)
hH2
hhHh2(
m2
tan 2
−−−=
−
−−
α
  (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: 
;z
H
h
= x
H
s
=        (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
)
H
pz(
H
pz
dx
dz
2
−
−−
α
−−
−=      (9) 
On the other hand by considering the following functions             
)
z12
zz1z2()z(f
2
1
−
−−=      (10) 
and 
)
H
pz(
H
pz)z(f2 −−=       (11) 
One can obtain from the rearrangement of Eq.9, that 
dz
)z(f
)z(f
m2
tandx
2
1
⎥
⎦
⎤
⎢
⎣
⎡
⎥⎦
⎤
⎢⎣
⎡ α−=       (12) 
Making use of the  forementioned the dimensionless variables it is possible to see 
that  
dx
H
dsx
H
s
=→=        (13) 
Hence, its substitution into Eq.12  leads to   
 
    (14) 
 
 
−
dz
)z(f
)z(f
m2
Htands
2
1
⎥
⎦
⎤
⎢
⎣
⎡
⎥⎦
⎤
⎢⎣
⎡ α−=
SESSION 6.2 
NOVATECH 2007  1255 
Furthermore, integration both sides yields 
 
dz
)z(f
)z(f
m2
Htands
2z
1z 2
1
2s
1s
∫∫
α
−=       (15) 
 
Which can be defined as 
2z
1z
)z(
m2
HtanL φ−α=       (16) 
or 
2z
1z
)z(
m2
HtanL ϕα=        (17) 
in which “L”  is the length of the side weir, and “φ(z)” and “Ψ(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 F1<1 was expressed as: 
)
2F
F3(1611.0
3
2m 2
1
2
1
+
−=       (18) 
and for supercritical regimes where  F1>2  as: 
)F008.036.0(
3
2m 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).  
SESSION 6.2 
1256 NOVATECH 2007  
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 
SESSION 6.2 
NOVATECH 2007  1257 
 
        Figure 5. For F1<1, discharge of the side weir 
 
           Figure 6. For F1>1, 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.  
SESSION 6.2 
1258 NOVATECH 2007  
REFERENCES  
 
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De Marchi, G. (1934). Saggio di teotia de funzionamenta degli stramazzi laterali. L’Energia  
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Contrôle des débits par un déversoir d'orage latéral sur un canal triangulaire

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Contrôle des débits par un déversoir d'orage latéral sur un canal triangulaire

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