Behavior of Submerged Ogee Crest Weir Discharge Coefficients

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Technical Note
Behavior of Submerged Ogee Crest
Weir Discharge Coefficients
B. P. Tullis, M.ASCE1
Abstract: Weir head-discharge relationships are typically described using the discharge coefficient-dependent standard weir equation. The
submerged weir (tailwater exceeds the weir crest elevation) head-discharge relationship can vary from the free-flow head-discharge relation-
ship, particularly at high submergence levels. The accuracy associated with predicting the upstream head or discharge, corresponding to
submerged weir flow conditions, is dependent upon the accuracy with which a representative submerged discharge coefficient can be de-
termined. A submerged ogee crest weir discharge coefficient predictive method developed by the U.S. Bureau of Reclamation (USBR) is
reviewed and its predictive accuracy compared to laboratory-scale submerged ogee crest weir experimental data associated with a wide range
of submerged flow conditions for nine different ogee crest weir geometries. Discussion is presented in an effort to partially explain the
relatively poor correlation between the USBR method and the experimental data set. Alternative submerged discharge coefficient relation-
ships are also presented. DOI: 10.1061/(ASCE)IR.1943-4774.0000330. © 2011 American Society of Civil Engineers.
CE Database subject headings: Wave crest; Submerging; Weirs; Coefficients; Water discharge; Irrigation.
Author keywords: Ogee crest; Submergence; Discharge coefficient; Modular submergence.
Introduction
In practice, weirs typically have four different functions: head-
discharge control structure for spillways, flowmeasurement structure
in canals, flow diversion structure in canals, and/or grade-control
structure in river and stream applications (e.g., lock and dam).
An ogee crest weir is a common weir type whose cross-sectional
profile corresponds to the shape of the underside of a sharp-crested
weir nappe. Just as the shape or trajectory of the sharp-crested weir
nappe changes with flow rate, so does the ogee crest profile. Con-
sequently, a specific ogee crest profile is based on a design flow
rate (Qdesign) or corresponding design head (Ho). The U.S. Bureau
of Reclamation (USBR) and U.S. Army Corps of Engineering
(USACE) have developed procedures for determining ogee crest
profiles based on Ho and free-flow head-discharge relationships
with provisions that account for the influences of piers in the flow,
abutment wall geometry, upstream total head variations, and sloping
upstream face influences (USBR 1987; USACE 1990).
Both ogee crest design procedures use the following weir head-
discharge equation:
Q ¼ Cf LH3=2 ð1Þ
where Q = discharge; Cf = dimensional free-flow discharge coef-
ficient; L = effective weir length (actual weir length adjusted for any
pier and/or abutment wall effects); and H = free-flow total head
upstream of the weir measured relative to the crest elevation.
When an ogee crest weir operates in a submerged condition
(i.e., when the tailwater exceeds the weir crest elevation), the
head-discharge relationship can deviate (increased upstream head
for the same Q) from the free-flow condition. In this study, submer-
gence (S) is quantified as the ratio of the downstream total head
(Hd) over the submerged weir upstream total head (H�):
S ¼ Hd=H�; S ≥ 0 ð2Þ
Notation specific to free-flow and submerged weir flow is pre-
sented in Fig. 1.
At low submergence values, the tailwater is above the weir crest
but not of sufficient depth to affect the upstream flow depth, and the
head-discharge relationship of the submerged weir remains the
same as the free-flow head-discharge relationship. This condition
is referred to as modular submergence. As the tailwater depth
increases, the upstream flow depth (h�) will eventually begin to
increase, relative to the free-flow upstream depth (h), although
Q remains unchanged. The submergence condition where the tail-
water depth first begins to influence the upstream flow depth is
referred to as the modular submergence limit. Above the modular
submergence limit, H� increases relative to H as S increases. When
S exceeds the modular limit, Eq. (1) can still be used, provided that
Cf is replaced with an appropriate weir-specific submerged dis-
charge coefficient (Cs) and H� replaces H.
An evaluation of published submerged head-discharge relation-
ships for ogee crest weirs (Cox 1928; Skogerboe et al. 1967;
Varshney and Mohanty 1973; USBR 1987) by Tullis and Neilson
(2008) found relatively poor correlations between predicted and ex-
perimental data. The objective of this study was to examine the
behavior of Cs as a function of ogee crest upstream and down-
stream weir heights (P and Pd) and Q. Of the submerged ogee crest
weir head-discharge relationships evaluated by Tullis and Nielsen
(2008), only the USBR (1987) and USACE (1990) methods uti-
lized Eq. (1) with Cs replacing Cf . Because the USBR and USACE
free-flow and submerged ogee crest head-discharge predictive
methods are effectively the same, the results of this study will only
be compared with the USBR method for convenience.
The USBR (1987) submerged ogee crest head-discharge predic-
tive method is based on a study by Bradley (1945). Bradley tested a
1Associate Professor, Utah Water Research Laboratory, Dept. of Civil
and Environmental Engineering, Utah State Univ., 8200 Old Main Hill,
Logan, UT 84322-8200. E-mail: blake.tullis@usu.edu
Note. This manuscript was submitted on June 16, 2010; approved on
December 9, 2010; published online on December 11, 2010. Discussion
period open until March 1, 2012; separate discussions must be submitted
for individual papers. This technical note is part of the Journal of Irriga-
tion and Drainage Engineering, Vol. 137, No. 10, October 1, 2011.
©ASCE, ISSN 0733-9437/2011/10-677–681/$25.00.
JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / OCTOBER 2011 / 677
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http://dx.doi.org/10.1061/(ASCE)IR.1943-4774.0000330
http://dx.doi.org/10.1061/(ASCE)IR.1943-4774.0000330
http://dx.doi.org/10.1061/(ASCE)IR.1943-4774.0000330
http://dx.doi.org/10.1061/(ASCE)IR.1943-4774.0000330
lab-scale ogee crest weir, referred to as Dam B, to investigate
the influence of S and Pd on Cs. P was held constant and Pd
was varied (Table 1). Bradley’s results are presented dimension-
lessly as Cs=Cf , where Cs and Cf were calculated via Eq. (1) using
the appropriate upstream head values [i.e., submerged (H�) or free-
flow (H)] and the measured Q. Bradley concluded that Cs was in-
dependent of P; variations in P were properly accounted for
through Cf . Bradley also concluded that, owing to a lack of geo-
metric similitude among ogee crest weirs (i.e., ogee crest profiles
are geometrically similar but P and Pd are unconstrained), a single
Cs=Cf relationship for ogee crest weirs would not be sufficient to
account for the effects of both submergence and weir geometry.
Consequently, Bradley’s results were presented graphically, similar
to Fig. 2, where the submergence-induced percent reduction in Cf
is correlated with a downstream apron parameter [ðPd þ H�Þ=H�]
and a submergence parameter [ðH� � hdÞ=H�]. Bradley’s dimen-
sionless parameters have been redefined using notation from the
current study. A comparison between Fig. 2 and Bradley’s Dam
B submergence data produced variations of �30% in Cs values.
The primary objective of this study was to evaluate the accuracy
with which the USBR (1987) predictive method for determining Cs
(Fig. 2) correlated with experimental data for a range of ogee crest
weir geometries and submergence conditions. Cs data were col-
lected for nine different ogee crest weir geometries with varying
values of P and Pd. New Cs=Cf relationships based on S, Q, P,
and Pd were also explored.Experimental Method
To generate free-flow and submerged ogee crest discharge coeffi-
cient data (Cf and Cs, respectively), a 510.9-mm tall ogee crest weir
constructed with Ho ¼ 233:5 mm and a vertical upstream face was
tested at the Utah Water Research Laboratory at Utah State
University. The crest profile was determined using the compound
curve method (USBR 1987) and constructed using 32-mm thick
high-density polyethylene (HDPE) ribs skinned with sheet metal.
The weir was tested in a 1.2-m wide by 0.9-m deep by 14.6-m
long rectangular flume (Fig. 3). Q was measured using calibrated
orifice flow meters located in the supply piping. The flow depths h
and hd were measured using point gauges (readable to �0:3 mm)
installed in stilling wells hydraulically connected to pressure taps in
the flume sidewall, approximately 5Ho upstream and 11Ho down-
stream of the weir crest, as shown in Fig. 3. hd was varied using an
adjustable gate at the downstream end of the flume. V , V�, and Vd
represent the average cross-sectional flow velocities at the corre-
sponding pressure tap location.
The influences of P and Pd on submerged ogee crest head-
discharge relationships were evaluated by testing weir configura-
tions featuring nine different combinations of P and Pd . P and
Pd were varied by installing false floors of differing heights
Fig. 1. Notation for free-flow and submerged ogee crest discharge
Table 1. Experimental, Lab-Scale Ogee Crest Weir Geometric Parameter
Summary
Study Ho (mm) P (mm) Pd (mm) L (mm) Ho=P Ho=Pd
Bradley (1945)
Dam B
353.6 1,079.0 0–1,079.0 594.4 0.33 0:33�∞
Current study 233.5 121.6 121.6 1,219.2 1.92 1.92
233.5 121.6 252.2 1,219.2 1.92 0.93
233.5 121.6 510.9 1,219.2 1.92 0.46
233.5 252.2 121.6 1,219.2 0.93 1.92
233.5 252.2 252.2 1,219.2 0.93 0.93
233.5 252.2 510.9 1,219.2 0.93 0.46
233.5 510.9 121.6 1,219.2 0.46 1.92
233.5 510.9 252.2 1,219.2 0.46 0.93
233.5 510.9 510.9 1,219.2 0.46 0.46
Fig. 2. Cs graphical predictive method from USBR (1987) and Bradley
(1945) [based on Figs. 9–26 in USBR (1987)]
Fig. 3. Overview of experimental setup
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immediately up and downstream of the weir. The upstream and
downstream weir heights were identified as the dimensionless ra-
tios Ho=P and Ho=Pd, with Ho=P and Ho=Pd equal to 0.46, 0.93,
and 1.92, as summarized in Table 1. Submergence tests were con-
ducted for each weir configuration at two different flow rates,
Q ¼ 50% and 100% Qdesign. All weir configurations were tested
under free-flow and a variety of submergence conditions up
to S≈ 0:97.
Data for each ogee crest weir configuration were collected as
follows. Free-flow head-discharge data were collected for a range
of H and Q values. With Q ¼ 100%Qdesign, the effects of submer-
gence were evaluated by determining H� for a range of Hd values
(S ¼ 0 to ∼0:97). Similar submerged ogee crest data were collected
at Q ¼ 50%Qdesign. Using the experimental data, Cf and Cs were
calculated using Eq. (1) for the free-flow and submerged flow con-
ditions, respectively.
Experimental Results
The free-flow head-discharge relationships for all ogee crest weir
geometries tested are plotted in Fig. 4 along with the predicted
head-discharge relationship for the tallest experimental ogee crest
weir (Ho=P ¼ Ho=Pd ¼ 0:46, Ho ¼ 233:5 mm) determined using
the USBR free-flow ogee crest predictive method (1987). The good
correlation between the USBR predicted and experimental data for
data sets (average and maximum errors were 1.0% and 2.8%, re-
spectively) shows reasonable consistency between the experimental
protocol of the current study and the USBR method. The small var-
iations in the experimental data sets also suggest that the range of P
and Pd tested had measurable though minor effects on the free-flow
head-discharge relationships. In Fig. 4, the weir efficiency increases
with decreasing Ho=P.
The experimental Cs=Cf versus S data from the current study are
presented in Figs. 5 and 6 for Q ¼ 100% and 50% Qdesign, respec-
tively. The data trends in Figs. 5 and 6 agree with Bradley’s asser-
tion that both S and Pd have an influence on Cs; however, they also
contradict Bradley’s conclusion that Cs is independent of P. With
the exception of the Ho=Pd ¼ 1:92 with Ho=P ¼ 1:92 and 0.93
weir geometries, each ogee crest weir geometry tested produced
a unique Cs=Cf curve. All of the weir geometry-specific data plot-
ted in Figs. 5 and 6 include a modular submergence region
(i.e., Cs=Cf ¼ 1:0). All of the Cs=Cf curves transition from a hori-
zontal line (modular submergence) to a mild sloping line (small
submergence effects) to a relatively steep sloping curve (large
submergence effects). Related to the behavior of the Cs=Cf curves
and consistent with the observations reported by Bradley (1945),
the submergence-influenced flow conditions passing over the ogee
crest weir progress from (1) a clinging, free-flow jet, to (2) a sub-
merged jet that remains attached (fully or partially) to the down-
stream weir profile, to (3) a detached surface jet as S increases.
The variation in the Cs=Cf curves withQ for the taller ogee crest
weir (Ho=P ¼ Ho=Pd ¼ 0:46) in Figs. 5 and 6, for example, sug-
gests that the momentum of the discharge passing over the crest,
which varies with Q, influences the point at which the jet detaches
from the crest profile (flow separation) and the head-discharge (or
Cs=Cf ) relationship. The influence of S on Cs=Cf is slightly more
significant for Q ¼ Qdesign than Q ¼ 50%Qdesign. The increased
momentum of the Qdesign, relative to the 50% Qdesign, will cause
the jet to detach at a point farther upstream on the crest profile,
increasing the value ofH� and decreasing Cs=Cf . The submergence
range associated with the transition between the mild and steep
sloping curves varies significantly with P, Pd and Q.
Comparing the curves for common ogee crest weir geometries
among Figs. 5 and 6 shows that the influence of Q on Cs=Cf de-
creases as Pd decreases (i.e., the shape of the Ho=Pd ¼ 1:92 curves
vary little between Figs. 5 and 6). Fig. 5 also shows that at the
smallest Pd value tested (Ho=Pd ¼ 1:92), the influence of P on
Cs=Cf diminishes as P decreases (i.e., the Ho=P ¼ 0:93 and
1.92 data sets, with Ho=Pd ¼ 1:92, essentially follow a common
trend line). For the taller Pd weirs, Cs=Cf is influenced by both
Q and P. In other words, as the weir heights decrease (P and
Pd) for ogee crest weirs with common Ho values, the Cs=Cf rela-
tionship becomes solely a function of S.
Fig. 4. Experimental and predicted (USBR) free-flow head-discharge
data
Fig. 5. Cs=Cf versus S for various combinations of P and Pd at
Q ¼ Qdesign
Fig. 6. Cs=Cf versus S for various combinations of P and Pd at
Q ¼ 50%Qdesign
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A comparison of the experimentally determined Cs values
with USBR (1987) predicted Cs values (Fig. 2), referred to as
Cs-experimental and Cs-predicted, respectively, versus S are presented
in Fig. 7. The data in Fig. 7 show that the USBR (1987) and
Bradley (1945) method for predicting Cs significantly underesti-
mates the effects of submergence on H� for a given value of Q
(i.e., Cs-predicted values, based on Fig. 2, were higher than the
Cs-experimental values). Although the reason for the poor correlation
is not well understood, it may be attributable in part to the location
ofthe hd measurement. Bradley (1945) measured hd at 4Ho down-
stream from the crest. In the current study, hd was measured farther
downstream (11Ho from the crest); the corresponding hd value ap-
peared to better represent the downstream tailwater condition rather
than the local flow depths specific to the weir discharge. When sub-
mergence levels are sufficient to cause the flow passing over the
weir to detach from the weir boundary, thereby creating a surface
jet, the water surface condition downstream of the weir experiences
significant wave action. For submergence conditions at and just
beyond the detached surface jet initiation, the downstream water
surface features a very prominent standing wave. Consequently,
the nature of the water surface profile immediately downstream
of the submerged ogee crest weir and the close relative proximity
of the tailwater measurement location to the weir crest may have
influenced Bradley’s experimental results.
A single-sample uncertainty analysis of the experimental data
from the current study, performed per Kline and McClintock
(1953), produced a data uncertainty of less than 1% based on in-
strument readability. Although not quantified, the phenomenologi-
cal uncertainty associated with determining parameters such as a
representative elevation of a dynamic water surface (i.e., wave
activity) is likely higher than the uncertainty associated with the
instrumentation.
Conclusions
The behavior of the submerged ogee crest discharge coefficient was
evaluated relative to variations in upstream and downstream weir
heights and discharge. An experimental data set was produced by
testing nine different laboratory-scale ogee crest geometries (varied
P and Pd values) with a common design head and then compared
with the USBR (1987) Cs predictive method, which was developed
by Bradley (1945). Based on the findings of this study, the follow-
ing conclusions are made:
1. Fig. 2 shows that the USBR dimensionless relationships devel-
oped by Bradley (1945) [i.e., Cs=Cf versus ðPd þ H�Þ=H�
(downstream weir height effects) and ðH� � hdÞ=H�
(submergence effects)], underestimate the effect of S on Cs re-
lative to the experimental data associated with the nine ogee
crest weir geometries tested in the current study. The predicted
Cs values per Fig. 2 exceeded the experimental values by as
much as 8 times.
2. The poor correlation between the experimental and USBR
(1987) predicted Cs values (Fig. 7) might be attributable,
in part, to differences between hd measurement locations
[Bradley (1945) measured hd significantly closer to the weir
than in the current study]. The fact that the variations in P
are not explicitly accounted for in the USBR submergence
method, a finding contrary to the assumption inherent in the
USBR method, may also be a factor.
3. The experimental data in Figs. 5 and 6 suggest that Cs=Cf is a
function of P;Pd, Q and S. With the exception of the shorter
ogee crest weirs (i.e., Ho=P ¼ 1:92 and Ho=Pd ¼ 1:92 and
0.93), each ogee crest weir geometry and flow rate tested
produced a unique Cs=Cf versus S relationship. This suggests
that the probability of developing a general Cs versus S rela-
tionship applicable to all ogee crest geometries and flow rates
is unlikely.
4. For the ogee crest geometries tested, the upper bound of the
modular submergence range, which appears to be principally
influenced by Ho=P and less so by Ho=Pd, varied with ogee
crest geometry, ranging from S ¼ ∼0:3 to ∼0:67. The shorter
Pd weirs have the higher upper limit of modular submergence.
The primary reason for this is that as Pd decreases, a higher
tailwater is required to overcome the momentum of the super-
critical flow section downstream of the ogee crest weir and
submerge the weir.
5. For ogee crest weirs with common Ho values and Ho=P
and Ho=Pd ≤ 0:93, Cs=Cf is dependent upon P;Pd, Q, and
S. As the weir heights decrease (i.e., Ho=P ≥ 1:92 and
Ho=Pd ≥ 0:93), Cs=Cf appears to be independent of P
and Q½Cs=Cf ¼ f ðSÞ�.
The results of this study show that the submerged head-
discharge relationship is dependent upon the specific weir geom-
etry. This study was limited to two discharge conditions (e.g.,
Q ¼ 50% and 100% Qdesign). The Cs=Cf data presented in Figs. 5
and 6 are recommended over the USBR method (Fig. 2), where
applicable for predicting submerged ogee crest discharge coeffi-
cients. In future submerged ogee crest research, the number and
variety of ogee crest weir geometries should be increased along
with the range of discharges evaluated.
Acknowledgments
Funding for this study was provided by the State of Utah and the
Utah Water Research Laboratory, Utah State University.
Notation
The following symbols are used in this paper:
Cf = free-flow weir discharge coefficient (L1=2=t);
Cs = submerged weir discharge coefficient (L1=2=t);
Cs-experimental = experimentally determined discharge coefficient
(L1=2=t);
Fig. 7. Cs�predicted=Cs�experimental versus S (Cs�predicted was determined
by evaluating the current study experimental data set using Fig. 2)
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Cs-predicted = discharge coefficient predicted using Fig. 2
(L1=2=t);
Cs=Cf = ratio of submerged and free-flow discharge
coefficients for a common discharge;
H = upstream, unsubmerged total head measured
relative to the weir crest (L);
Hd = downstream total head measured relative to the
weir crest (L);
Ho = ogee-crest design head (total head) measured
relative to the weir crest (L);
H� = upstream submerged total head measured relative
to the weir crest (L);
h = upstream unsubmerged flow depth measured
relative to the weir crest (L);
hd = downstream flow depth measured relative to the
weir crest (L);
h� = upstream submerged flow depth measured relative
to the weir crest (L);
L = weir length (L);
P = upstream weir height (vertical distance from the
upstream apron to the weir crest) (L);
Pd = downstream weir height (vertical distance from the
downstream apron to the weir crest) (L);
Q = discharge over the ogee crest weir (L3=t);
Qdesign = free-flow (unsubmerged) discharge associated with
the Ho (L3=t);
Qs = submerged weir discharge (L3=t);
S = submergence ratio equal to total Hd=H�;
Vo = upstream cross-sectional average velocity under
free-flow conditions;
Vd = downstream cross-sectional average velocity under
submerged flow conditions; and
V� = upstream cross-sectional average velocity under
submerged-flow conditions.
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