Availability and Conditions of Agricultural Machinery in Public and Private Sectors of Borno State, Nigeria


Arid Zone Journal of Engineering, Technology and Environment. October, 2007; Vol.5, 26-36
Copyright© Faculty of Engineering, University of Maiduguri, Nigeria.
Print ISSN: 1596-2490, Electronic ISSN: 2545-5818
www.azojete.com.ng

LABORATORY EVALUATION OF FREE OVERFALL EQUATIONS
FOR PREDICTINGWATER PROFILE OVERWEIR

M. K. Othman1, S. Z. Abubakar1, X. Litrico2 and G. Belaud3

Abstract

Laboratory evaluation of three free overfall equations for predicting water profile over hydraulic structure – weir
was conducted in Irrigation laboratory of Higher National College of Agriculture of Montpellier (ENSAM) ,
France. A broad-crested weir was constructed as a reduced physical model to simulate the gated-structure
(Begemann gate) for upstream water control in the Main Canal of Hadejia Valley Irrigation Project (HVIP),
Northern Nigeria. Instrumentation in the ENSAM laboratory allows precise variation of channel flows with
incremental changes of 5 l/s. Three recently developed models in a progressive succession were used for
comparison between measured and predicted data to establish degree of accuracy in the prediction of water
profile passing over weir in open channel.

Results of the evaluation show that the three equations performed differently in predicting water profile over
weir at the selected channel flows of 60 l/s, 30 l/s and 10 l/s respectively. These stream sizes were chosen
because they were the maximum, medium and minimum for the canal capacity. Although the stream size
directly influenced the water profile as a result of internal pressure due to convergence of stream filaments
between upper and lower laminar flow. Longer velocity jumps were recorded at the higher stream size (60 l/s) of
35 cm from weir crest end while less than 25 cm distance was covered by the least stream size of 10 l/s.
Generally, based on the R2 values and percentage errors for the three models at all the three selected stream sizes,
Cavailhé model predicted the water profile with high degree of accuracy followed by that of Davis and lastly the
Hager model. The same order was observed on ease of manipulation and user friendliness. The choice of water
profile prediction model with high degree of accuracy is important in the design of hydraulic structures for water
conveyance, diversion and measurements in irrigation canal network.

1. Introduction

In the design of water control and measurement structures, the concept of creating critical
flow by inducing a reduction of specific energy through raising of channel bed or constricting
the canal cross sectional area, thereby increasing the discharge per unit width, or combination
of the two, are mostly adopted. This is done to make the discharge a single valued function of
the up stream stage because it was found that when a structure is introduced into a sub-critical
flow causing it to pass through the critical to the super critical, the stage of upstream is
independent of the down stream stage (Withers, 1974). This concept gives birth to the creation
of critical-depth meters such as broad-crested weir, venturimeter, spillway and recently
automatic control gates that have combination of broad-crested weir and gate structure. The
hydraulic problems of the free overfall structures are concerned with characteristics of the
control and dissipation of flow in downstream basin. Ordinarily, flow over the structure is free
discharging where air is admitted to the under side of the flow nappe to avoid the jet being
depressed by reduced underneath pressure. Dissipation of the flow in the downstream basin
may be obtained by hydraulic jump, impact and turbulence induced in the basin where erosion
and down stream flood may be the results. To address this issue, USDBR (1977) developed a
relationship between drop distance Y, unit discharge passing over the crest Q and critical

1 National Agricultural Extension and Research Liaison Services, Ahmadu Bello University, Zaria (e-mail :
mkothman@yahoo.co.uk)

2CEMAGREF (Centre National du Machinisme Agricole, du Génie Rural, des Eaux et des Forêts) Montpellier,
France (xavier.litrico.@montpellier.cemagref.fr)

3ENSAM (Ecole Nationale Supérieur d’Agronomie de Montpellier), France

mailto:mkothman@yahoo.co.uk
mailto:xavier.litrico.@montpellier.cemagref.fr
http://www.azojete.com.ng


Laboratory Evaluation of Free Overfall Equations for Predicting Water Profile Over Weir

27depth hc. To reinforce this effort, several researchers attempted to determine the point of
critical depth where a single measurement at established channel section might permit a
direct computation of the discharge passing over the structure (Rouse, 1936; Hager, 1983;
Davis et al., 1999). Part of this attempt was the development of mathematical equations to
predict water flow profile falling over the hydraulic structures. Among these equations were
those of Hager, Davis and Cavailhé. The three equations developed by these researchers could
apply to both sub and super critical flows conditions. In using these equations, the designers
may have difficulty of choosing the most efficient equation that can give a better prediction of
the water profile at the same time links the water depth and discharge at each section of the
overfall structure most accurately. It is pertinent to note that the overfall condition is of
distinct importance in hydraulic and irrigation engineering for design and use of the structures
to control or divert water in irrigation system. Beside its close relation to broad crested weir, it
forms the starting point in computations of the surface curve in non-uniform channel flow in
which the discharge spill into an open reservoir at the downstream end. Similarly, the
equation of such condition is needed in the design of automatic control gates such as
Begemann and Vlughter gates for determining the depth needed to be maintained at the
upstream level. This paper presents the laboratory work and result of comparable accuracies
of three selected equations in predicting the flow profile over the broad crested weir. The
paper further highlighted the applicability of such principle in real-life situation where such
weirs play dual functions of water diversion and upstream control/measurement structure as in
the case of North Main Canal of Hadejia Valley Irrigation Project (Othman et al., 2003).

2. Background study of free overfall equations

There is limited effort on hydraulic studies of free overfall structures adaptable to open canal
irrigation system, the world over. However, few structures (weir, flumes, orifice, inverted
siphon and gates) attracted researchers’ attention for more than seven decades across Europe,
America, Asia and North Africa. The work on hydraulics of free overfall structure was one of
such studies which could be dated back to sixty years ago. The applicability of weir under
wide range of field situations made it very attractive and a focus of research efforts especially
where computer-based models are used to support the operation and management of canal
irrigation. Starting from the work of Rouse (1936), the relationship between depth at the brink
“he” and critical flow depth upstream “hc”, for a smooth rectangular horizontal channel, the
ratio he/hc was obtained as 0.715. Similarly, Markland (1965), as reported by Davis et al.
(1999), used a relaxation method to integrate the potential flow equations and obtained
solutions for the nappe and upstream profiles in a rectangular free overfall. Strelkoff and
Moayeri (1970) used numerical integration and potential flow theory to compute free overfall
conditions in rectangular, triangular and parabolic channel sections respectively. Hager (1983)
used an analytical approach to solve the conservation of energy equation (Bernoulli equation),
extended to take account of curvature effects, and applied it to rectangular free overfall
conditions. Marchi (1993) also solved the same equations analytically by expanding the
stream function in a power series. With the exception of Hager (1983) and Marchi (1993), all
the approaches are rigorous, requiring numerical solutions (Davis et al., 1999). They all
applied to critical and super-critical conditions and do not apply to sub-critical condition.
Davis et al. (1999) modified the method of Hager (1983) and used parabola method which
assumes two-dimensional flow (a prismatic rectangular channel with a confined nappe). It
also assumed that a particle in a streamline in the nappe follows a free fall parabolic path,
once the overfall structure is passed. That is, it accelerates downward in the vertical direction
under the influence of gravity and possesses no acceleration in the horizontal direction.
Progressively, Cavailhé (2001) further improved the work of Davis et al. (1999) and came out



AZOJETE Vol. 5 2007

28with a slightly different equation which predicts the water profile. The three equations of
Cavailhé, Hager and Davis were developed to predict water profile over hydraulic structure

of weir in this study.

2.1 Presentation of Water Surface Profile Equations
The basic principles of free-overfall equations can be understood from Figure 1 which
illustrates a typical situation of water profile and dynamics of forces and other relevant
parameters.

Figure 1: Schematic diagram of free overfall water profile passing over hydraulic
structure

The three selected equations developed by Cavailhé (2001), Davis et al. (1999) and Hager
(1983) under evaluation were as follows:

2

cos2
1tan

3
2










x
y

xyY
c

c (1a)

2

cos2
tan 













V
xgxhY e (2a)

2

)1cos(2
)1tan( 













Vu
XgXheY (3a)

Where
yc = critical depth, the upstream depth above the weir
It is given by Cavailhé, (2001) as

3
1

2

2











a
c gL

Qy (1b)

Q = discharge passing over the weir, m3/s
g = acceleration due to gravity, m/s2
La= width of the canal, m



Laboratory Evaluation of Free Overfall Equations for Predicting Water Profile Over Weir

29














 

L

yc3
1

tan 1 Refer to Figure 1

L = length of the weir, m
ho = yc

5
23

2
2

2



fr

frh
yh o
ce (2b)

Where, fr = Froude Number

1 : angle = 2

2
2 1)(3

fr
h
he

h
hefr

oo











180 (3b)

2 : angle = (20.555 - 5.686fr)
180
 (2c)

Vu : flow velocity = )(2 heEg 

E : energy at upstream =
g

Voho 2

2



Vo: average flow velocity of the discharge upstream
Vudav : flow velocity on the crest = )(2 heEdavg 

Edav : upstream energy as modified by David =
g

VoCho 2

2



C=
fr
264.2 (Davis et al., 1999).

Detailed procedures of determining these equations could be obtained from Davis et al.
(1999), Hager (1983) and Cavailhe (2001) respectively. However, Figure 1 could assist in
placing all the parameters that appear in the equations.

Figure 2: The experimental set-up
.



AZOJETE Vol. 5 2007

30
3. Materials and methods

A Laboratory experiment was conducted to evaluate the three equations of water profile
passing over crest of hydraulic structure as an overfall. The experiment took place at the
Irrigation Laboratory of Ecole Nationale Supérieur d’Agronomie de Montpellier (ENSAM,)
Montpellier, France in 2002. A regulator was installed at the upstream of the system, which
supplies the canal with water for a desired discharge from 10 l/s up to maximum of 60l/s at
interval of 5 l/s. The reference was taken at the upstream, 30 cm from the terminal end of the
crest and the distance coincided with the thickness of the broad-crested weir used for the
experiment. The broad-crested weir used for the experiment was constructed as a reduced
model of the component of a gated-structure (Begemann gate) for upstream water control in
the North Main Canal (NMC) of Hadejia Valley Irrigation Project (HVIP), Nigeria.
Consequently, the dimensions of the weir were determined using the principle of dynamic
similarity which implies the identity of all the dimensionless laws governing the model and
prototype phenomena. Thus, a comparison of froude numbers of the two structures (the one at
HVIP and the prototype model) at a critical depth with maximum canal capacity of NMC at
HVIP and the prototype model in the Irrigation Laboratory canal of ENSAM was done using
the following relations (Yalin, 1971):

32

2

3

2
2

hgb
Q

gS
bQfr  (4)

3222 hbgfrQ  (4a)
if h'=h (5)
and b’=b (6)

32252' hbgfrQ  (6a)

Where Q’ discharge of the model, Q discharge of the gated-structure in the field (HVIP), b’
and h’ width and breath of the model while b and h are width and breath of gated-structure in
the field. Combining (4a) and (6a) yields (7a):

5
2

Q
'Q








 (7a)

 can be found by using the maximum discharge of the laboratory canal of 0.06 m3/s and that
of the gate in the field (HVIP) which was found from the design value to be 0.91 m3/s

337.0
91.0
06.0 5

2







 (7b)

h and b measured in the field were 93.5 cm and 112.8 cm, respectively and the height and
width of 31.5 and 38 cm respectively were computed and used for the construction of the
prototype model . Small alloy rods installed into a plastic frame following the pattern of the
water trajectory was used to measure the water profile passing over the structure for each of
the desired discharge supplied from upstream (see Figure 2). The water depth was taken from
one-end of the crest length up to distance of 65 cm at interval of 5 cm for flow rates of 10 l/s
to 60 l/s at interval of 5 l/s. The measurements were done with meter rule following the length
of the alloy rod from its end to where it touches the water profile. The measurements were
repeated four times at each point and the average was determined. Thereafter, the three



Laboratory Evaluation of Free Overfall Equations for Predicting Water Profile Over Weir

31

equations were used for the prediction of the profiles using the different discharges as used
during the experiment. This was done using spreadsheet while substituting all the known
parameters and then determining the profile depth at each point for the different discharges
used. The profile predicted by each of the three equations was compared with the measured
data using quantitative analysis. Similarly, regression analysis and percentage error were
conducted between the predicted and measured data for three selected flows in which R2
values were determined to measure the level of accuracy for each of the equations. The
percentage error was computed using the equation ( Dt-Dp )/Dt), where Dt is measured depth
and Dp predicted depth.

4. Result and discussion

4.1 Measured depths of water profile
Table 1 shows the result of the surface water profile measurements for the different discharges
supplied from 0 to 60 l/s respectively. The values in Table 1 represent the profile of the upper
nappe of the flowing water passing over crest with a shape of a trajectory of free overfall. This
means that the flow accelerates downward in the vertical direction under the influence of
gravity and no acceleration in the horizontal direction. As the acceleration in the horizontal
direction is small and hence negligible, the thickness of the nappe could be determined using
relationship given by Rousse (1943). From the table and the graphs plotted, the results show
that the influence of the upstream curvature started before the thickness of the crest, where
there exist hydrostatic pressure condition which agrees with the reasoning of Davis et al.
(1999) that the curvature or downstream effects never reach upstream more than a distance
equal to four times the upstream depth.

Table 1: Average water profile measured depths (cm) at different points and flows
passing over broad-crested weir

Discharge (l/s) in the upstream supplied from the source with sub critical flow

Distance
(cm)

60 55 50 45 40 35 30 25 20 15 10

0
5
10
15
20
25
30
35
40
45
50
55
60
65

Water depths from the bottom of the down stream bed (cm)
69
68,3
66,8
65,5
63,7
61,8
60,2
58,5
56,1
53,2
49,3
44,5
38,5
31,5

68,3
67,1
66,4
64,7
62,6
61,2
59,5
58
55,8
52,5
48,6
44,2
37,9
30

67,3
66,8
65,2
63,7
62
60,4
59
57,4
55
52
47,9
43
37,3
29,6

66,4
65,2
64,3
62,7
61,2
59,7
58,5
56,8
54,3
51,3
47,4
42,1
35,7
26,5

65,2
64,4
63,2
61,7
60
58,7
57,6
56,3
55,7
50,9
46,6
40,7
34,5

64,2
63,3
61,9
60,3
59
58
56,7
55,5
53,2
49,6
45,4
39,6
28,6

62,9
61,8
60,5
59,1
57,8
57
56,2
55
52,5
48,5
44,7
35,7
26,5

61,4
60,6
59,3
57,9
56,8
56,2
55,4
54,4
51,6
47,7
42,7
35,1
22,7

60,2
59,1
57,9
56,4
55,7
55,5
54,7
53,4
50,7
46,2
39,8
30,5

58,9
57,5
56
55,2
54,7
54,4
53,8
52,7
49
43,4
38,7
27,3

56,7
55,6
54,2
53,9
53,6
53,5
52,8
51,4
47,2
40,2
33
21,9



AZOJETE Vol. 5 2007

324.2 Models Comparison with Experimental Data
Figure3 shows the comparison of the data between the predicted values by the three models

and measured values at channel flow of 60 l/s. From the Figure, the Cavailhé and Hager
Models tend to obey the law where the acceleration in the horizontal direction is the lowest.
On the other hand, Davis model seem to obey the law with lesser precision indicative of lower
slope. Similarly, Figures 4 and 5 show the comparison at 30 and 10 l/s flows respectively. The
models predictions commence from where the nappe start. At this point, there is a complete
aeration below and above the nappe, allowing the pressure to be atmospheric. Under this
condition hydrostatic pressure could be assumed to be zero. However, there is still exist
considerable internal pressure due to convergence of the stream filaments between upper and
lower surfaces of the flow. This internal pressure is dependent on the stream size, the higher
the stream size, the higher the internal pressure and the higher the velocity of the jump.
Consequently, the shape of each channel flow is different as shown by the three figures. The
slope for the curves at 60 l/s is lower as also shown in the figures, covering more than 35 cm
horizontal distance from crest end. The slope increases for the curves at 30 l/s covering less
than 30 cm horizontal distance from crest (Figure 4) while the distance covered by 10 l/s was
less than 25 cm from crest with the highest slope. Again, from the models, the profile depth is
dependent on the relationship between variable channel flow (flow velocity) and a constant
geometrical form of the flow changes. Therefore, the complete geometrical and dynamic
similarities of any two different flows are hardly attainable as pointed out by Rouse, (1936,
1943) and Davis et al. (1999). Thus, the points on the curves from the experimental data and
those obtained from the models predictions could attest to this point (Figures 3, 4 and 5).



Laboratory Evaluation of Free Overfall Equations for Predicting Water Profile Over Weir

33

Table 2: Statistical comparisons of measured values with the three models at the maximum, minimum and medium flows (l/s)
M = measured values, P = predicted values, %E = percentage error

Water profile depths (cm) in Y-direction
Discharge of 60 l/s Discharge of 30 l/s Discharge of 10 l/s

Distance
(x-
direction)

Models
Cavailhe Davis Hager

Models
Cavailhe Davis Hager

Models
Cavailhe Davis Hager

M P %E P %E P %E M P %E P %E P %E M P %E P %E P %E

0
5
10
15
20
25
30
35

60.2
58.5
56.1
53.2
49.3
44.5
38.5
31.5

64.1
63.1
60.9
57.6
53.0
47.3
40.4
32.2

6.5
7.9
8.6
8.2
7.6
6.3
4.8
2.3

59.7
59.4
58.4
56.8
54.5
51.6
48.1
43.9

0.8
1.5
4.1
6.8
10.6
16.0
24.9
39.4

59.7
59.1
57.4
54.4
50.3
45.1
38.6
31.0

0.8
1.1
2.3
2.3
2.1
1.3
0.3
1.5

56.2
55.0
52.5
48.5
44.7
35.7
26.5

58.9
57.6
54.3
49.2
43.2
33.3
22.6

4.8
4.6
3.5
1.5
5.5
6.6
14.4

56.1
55.6
54.1
51.5
47.9
43.3
37.7

0.1
1.1
3.0
6.2
7.2
21.4
42.4

56.1
55.1
52.4
47.7
41.2
32.9
22.7

0.1
0.3
0.2
1.6
7.7
7.9
14.4

52.8
51.4
47.2
40.2
33.0
21.9

54.3
51.9
45.6
35.4
21.3

0.0
0.0
0.1
0.4
0.5

53.0
51.9
48.7
43.4
36.0

0.0
0.0
0.1
0.4
0.5

53.0
51.0
45.2
35.6
22.1

0.0
0.0
0.0
0.1
10.3

R2Values 0.9983 0.9962 0.9961 0.9981 0.9956 0.9957 0.9958 0.9948 0.9945
Average %
Error

6.5 12.8 1.5 15.9 11.6 14.6 0.1 0.0 0.1



AZOJETE Vol. 5 2007

34

0

10
20

30
40

50

60
70

80

0 20 40 60 80

Distance from 30 cm from crest end

W
at

er
 d

ep
th

 fr
om

 d
ow

ns
tr

ea
m

 
bo

tt
om

 (c
m

)

measured values cavailehé Davis

Hager Crest

Figure 3: Graph of comparison between the three equations and measurement at 60 l/s

Figure 4: Graph of comparison between the three equations and measurement at 30 l/s

Figure 5: Graph of comparison between the three equations and measurement at 10 l/s



Laboratory evaluation of free overfall equations

35Table 2 shows the statistical comparisons of the models predictions with the experimental
data at the channel flows of 60 l/s, 30 l/s and 10 l/s respectively.

From the table, the following can be deduced:
 Cavailhé model predict better than the other two models irrespective of the channel
flows where R2 values of 0.9983, 0.9981 and 0.9958 were obtained at 60 l/s, 30 l/s and
10 l/s channel flows respectively. Davis model follows in performance at 60 l/s and 10
l/s and performed least at 30 l/s while Hager Model shows the least prediction accuracy
at 60 l/s and 10 l/s and performed better at 30 l/s.

 The trend of percentage error recorded by the three models to predict water profile
over weir has no consistent pattern. (Table 2). The highest percentage errors were
recorded by all the three models at 30 l/s channel flow while the least errors occurred at
10 l/s for all the three models. At highest channel flow (60 l/s), Hager model had the
least percentage of error (1.5) followed by Cavailhé model (6.5) where Davis model
recorded 12.8 % error.

 Comparing the R2 values and percentage error for all the three models at the three
selected channel flows, Cavailhé model predict water profile with highest degree of
accuracy followed by that of Davis while Hager model predict with least degree of
accuracy. It is logical that the progressive improvement incorporated by Cavailhé in
Davis work, who in turn, improved Hager work are empirically justified in this study.

4. Conclusion and recommendations

The laboratory evaluation of the three free overfall models for predicting water profile over
weir bring out the differentiation among the most recent models developed. The experimental
set-up represent a real life situation where managers of irrigation schemes are confronted with
decision on whether to use weirs as solely diversion structures or use them for dual functions
of diversion and flow measurements as well as upstream control hydraulic structures. All the
three models were found to predict fairly accurately the water profile depths flowing over
broad crested-weir in an open channel. The choice among the three depends on the level of
precision required for the type function the structure is intended to serve. The study would
also guide in the choice of which model to use by irrigation engineers for design of water
control and flow measurement hydraulic structures in canal network. Similarly, the choice of
which model can predict most efficiently is also useful in selection and in-corporation into a
canal irrigation simulation packages. Finally, the ease of manipulations of these models is also
instructive of their accuracy in prediction and user friendliness.

Acknowledgement

The first author is grateful to French Embassy in Nigeria for the award of scholarship to do
this study in France. Our thanks go to Management of ENSAM in particular Monsieur Paul
for his assistance in the laboratory work.



AZOJETE Vol. 5 2007

36References

Cavailhé, M. (2001). Modélisation d'une vanne amont constant de type Begemann, Mémoire
de 3° cycle, CEMAGREF, Montpellier, France

Davis, C. A.; P.J. Richard and G.S.E. Brian (1999). Estimating Trajectory of Free Overfall
Nappe, Journal of Hydraulic Engineering, ASCE Jan.P. 97-82

Hager, W. H. (1983). Hydraulics of plane free overfall, J.Hydr. Div. ASCE 109(12) 1683-
1697.

Marchi, E. (1993). On Free-Overfall. J. Hydr. Res., Delft, The Netherlands, 31(6)
Markland, E. 1965. Calculation of Flow at a Free-overfall by Relaxation Method. Proc. ICE,

London, P.31.
Othman, M.K.; X. Litrico, J. P. Baume, and S.Z. Abubakar, (2003). Hydraulic Modeling of

Begemann Gate for Effective Water Management System in HVIP, Paper presented in
the 4th International Conference of Nigerian Institute of Agricultural Engineers (NIAE)
and 24th AGM, held in Damaturu, Yobe State, Nigeria

Rouse, H. (1936). Discharge Characteristics of the Free Overfall. Civ.Engrg., ASCE, 6(4)
Rouse, H. (1943). Discussion to Energy Loss at the Base of Free Overfall. Trans. ASCE, 108
Strelkoff, T. and M.S. Moayeri (1970). Pattern of Potential Flow in Free Overfall. J.Hydr.Div.

ASCE, 96(4), 879-901
USDBR (1977). Design of Small Dams. United States Department of the Interior Bureau of

Reclamation, Water Resources Technical Publication. United State Government
Printing Office, Washington

Withers, B. and S. Vipond (1974). Irrigation: Design and Practice. B. T. Batsford Ltd,
London, UK

Yalin, M. S. (1971). Theory of Hydraulic Models. The McMillan Press LTD, London, UK


	Abstract
	1. Introduction
	Figure 1: Schematic diagram of free overfall water
	Distance
	4.2Models Comparison with Experimental Data


	Models
	Acknowledgement 

	References