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5th International Junior Researcher and Engineer Workshop Spa, Belgium, 28-30 August 2014
on Hydraulic Structures
IN SKIMMING FLOW ON STEPPED SPILLWAYS: ENTRAINMENTAIR THE EFFECT OF AN ABRUPT SLOPE CHANGE
M.J. Ostad Mirza1,2 , M. Pfister1 , A.J. Schleiss1, and J. Matos2
1Laboratory of Hydraulic Constructions (LCH),
Ecole Polytechnique Fédérale de Lausanne (EPFL),
CH - Lausanne 1015
SWITZERLAND
2 Department of Civil Engineering, Architecture and Georesources
Instituto Superior Técnico, University of Lisbon
Lisbon, 1049-001
PORTUGAL
E-mail: mohammadjavad.ostadmirza@epfl.ch
Abstract: Numerous stepped spillways were built during the last decades. In particular, a stepped
spillway may be integrated economically into the downstream face of a RCC gravity dam, or on valley
flanks besides embankment or rockfill dams, where slope changes may naturally be implemented due
to topography and economic reasons. This paper presents and discusses preliminary results on the air
entrainment in the vicinity of an abrupt change chute slope, namely the air concentration distribution
and the mean air concentration. A significant influence was observed on the air entrainment pattern,
with a decrease of the mean air concentration immediately upstream of the slope change, followed by
a marked increase immediately downstream, and a subsequent decrease further down the flatter
chute, approaching a practically constant value. Considerable larger air entrainment was observed
shortly downstream of the slope change cross-section, in comparison with that found upstream, in the
quasi-uniform flow.
Keywords: Stepped spillways, Slope change, Skimming flow, Air entrainment.
1. INTRODUCTION
A significant number of stepped spillways were built during the last decades, in particular linked to the
application of the roller compacted concrete (RCC) dam construction technique. A stepped spillway
may be integrated economically into the downstream face of a RCC gravity dam. In combination with
embankment and rockfill dams, stepped spillways have been built on valley flanks besides the dam,
where slope changes may naturally occur due to topography and economic reasons.
For a given stepped chute geometry, the general behavior of the flow may be characterized by three
different regimes, namely nappe, transition and skimming flow (e.g., Othsu and Yasuda, 1997
Chanson, 2002). Nappe flow occurs at low flows and can be defined as a succession of free-falling
nappes. In skimming flow, the water or air-water flows as a coherent stream over the pseudo-bottom
formed by the outer step edges; beneath it three-dimensional vortices occur. Between the upper limit
of nappe flow and the lower limit of skimming flow, a transition flow takes place. For typical hydraulic
design of dam stepped spillways, a skimming flow regime occurs (Matos, 2000, Boes and Hager
2003a).
In the last couple of decades, a significant number of physical model studies were conducted on the
hydraulics of skimming flow over constant sloping stepped spillways (e.g., Chamani and Rajaratnam
1999, Pegram et al, 1999, Sánchez-Juny et al, 2000, Matos 2000, Chanson, 2002, Boes and Hager
2003a,b, Frizell, 2006, Amador et al, 2009, Meireles et al., 2012, Pfister and Hager, 2011, Bombardelli
et al., 2011, Bung, 2011, Felder and Chanson, 2009, Felder, 2013). In addition to the hydraulics of
conventional stepped spillways, a variety of experimental studies have also been carried out on non-
conventional geometries, such as stepped spillways with macro-roughness (e.g., André, 2004, André
et al, 2004, Gonzalez et al, 2008, Bung and Schlenkhoff, 2010), or with non-uniform step heights
(Felder and Chanson, 2011).
Despite some few exceptions, such as the Upper Stillwater dam in the USA (Houston, 1987) and
lower Siah-Bishe dam in Iran (Baumann et al., 2006), most stepped spillways have been designed for
a constant chute slope. Hence there is presently insufficient information available on the flow
behaviour on abrupt slope changes on stepped spillways. The present study was conducted under
different geometric and flow conditions in order to investigate the flow properties in the slope change
region, in particular the air entrainment.
2. EXPERIMENTAL SETUP
A steep channel with variable slope, equipped with steps of constant height, was assembled at the
Laboratory of Hydraulic Constructions (LCH) of the Ecole Polytechnique Fédérale de Lausanne
(EPFL). It consists of four 2 m long and 0.5 m wide modules, with a 0.6 m high transparent sidewall to
allow for flow observation. Since the present study focuses on slope changes, the channel was divided
in two separated parts, each of those including two modules of 4 m length. Each part was of different
slope, with the upstream slope being steeper than the downstream slope (Figure 1a). The bottom
slope of the upstream chute (i.e. pseudo-bottom angle) was set to φ1=50º (1V:0.84H), while the
downstream slope was set to φ2=18.6º (1V:3H).
The flow rate was measured with an electromagnetic flow meter. The maximum unit discharge which
could be provided is approximately 0.46 m2/s. To allow for an independent variation of the inflow depth
(d0) and Froude number (Fr0 = qw/(gd03)1/2; qw is the unit discharge and g is the gravitational
acceleration), the flume inflow device consisted of a jet-box with a maximum opening of 12 cm.
Applying this device, the pressurized pipe approach flow is transformed into a free surface flow. Thus
the location of the inception of air entrainment is shifted upstream and the developing region of the
flow is shorten, such that quasi-uniform flow conditions are reached on the upstream slope, for all step
geometries and discharges. A dual fiber-optical probe developed by RBI Instrumentation, France, was
mounted on an automatic positioning system for measuring the air concentration and velocity.
(a)
(b) (c)
Figure 1 – Physical model of the stepped spillway with an abrupt slope change assembled at LCH-
EPFL: a) General view; b) Initial reach of the chute and jet box, c) Dual fiber-optical probe in
operation.
A series of observations and measurements were conducted in the skimming flow regime for unit
discharges ranging between 0.35 and 0.46 m2/s, and relative critical depths dc/h (dc is the critical
depth and h is the step height) ranging between 3.8 and 4.6. That range corresponds to Reynolds
numbers (Re = qw/υ) varying between 3.4 and 4.6×105 and inflow Weber number at the exit of the jet-
box (We0 = Vm0/(σ sinφ/ ρh)1/2) between 124 and 189, where υ is the kinematic viscosity of water, Vm0
is the inflow depth averaged velocity at the exit of the jet-box (Vm0 = qw/d0), σ is the interfacial surface
tension, and ρ is the water density (Table 1).The air–water flow properties were measured at 20
streamwise cross-sections along the stepped spillway, namely in 5 step edges upstream and 15 step
edges downstream of the slope change, from step number -9 to +15 (Figure 2). The measurements in
each cross-section consisted of 30 points from about 0.005 m distance to the step edge, and
subsequently increasing by 0.01 m.
Table 1 – Chute geometry and hydraulic conditions.
Parameters
Min.
Max.
φ(º)
18.6(1)
50(2)
h (m)
0.06
qw (m2/s)
0.35
0.46
dc/h (-)
3.8
4.6
d0 (m)
0.082
0.093
Fr0 (-)
4.0
6.4
Re (-)ˣ105
3.4
4.6
We0 (-)
124
189
(1) Downstream chute slope, (2) Upstream chute slope.
(a) (b)
Figure 2 – a) Physical model of the stepped spillway with an abrupt slope change assembled at LCH-
EPFL (50º to 18.6º, h = 0.06 m, qw = 0.46 m2/s, dc/h =4.6); step numbers used in the following are
indicated, b) sketch of slope change region.
AIR-WATER FLOW PROPERTIES ON THE SLOPE CHANGE REGION
2.1. Definitions
The local air concentration C is defined as the time-averaged value of the volume of air per unit
volume of air and water. The mean (depth-averaged) air concentration is defined as
90
0
90
Y
dyC
C
Y
mean
(1)
where y is measured perpendicular to the pseudo-bottom formed by the step edges and Y90 is the
depth where the air concentration is 90 %.
2.2. Air concentration distribution
Various air concentration profiles were acquired in skimming flow upstream and downstream of the
slope change, as presented in Figure 3. The air concentration distribution varies significantly along the
slope change region. Four main sub-regions may be identified: sub-region I, characterized by a
decrease in the local air concentration (for identical distance to the pseudo-bottom) within the flow
when approaching the slope change cross-section (Figure 3a), sub-region II, characterized by a
sharp increase in the air concentration within the flow near the slope change cross-section, reaching
maximum values shortly downstream (Figure 3b); sub-region III, where the air concentration
decreases rapidly again and approaches to values close to uniform flow condition for the second slope
(Figure 3c); and sub-region IV, where the air concentration continues to exhibit a decreasing trend,
eventually approaching an almost constant value, hence leading to similar air concentration profiles
(Figure 3d).
In the reaches not affected by the slope change, namely, steps -9 to -3 (Figure 3a) and +12 to +15
(Figure 3d), the air concentration distribution exhibits a S-shape profile, similarly as obtained in other
experimental studies for constant chute slope under uniform flow condition, as well as well described
by the advection-diffusion model for the air bubbles (e.g., Chanson, 1997; Chanson and Toombes,
2002).
Figure 3 - Air concentration distribution upstream, downstream and along the slope change region (“-”
and “+” signs represent the steps upstream and downstream of the slope change region, respectively
(step numbers as per Figure 2 ): (a) sub-region I; (b) sub-region II; (c) sub-region III; (d) sub-region
(IV): dc/h =4.6. “C theory” was obtained from Chanson (1997), for uniform flow on a similar sloping
chute, assuming Cmean equal to 0.6 and 0.3 for 50º and 18.6º slopes, respectively.
2.3. Mean air concentration and characteristic flow depth
The development of the mean air concentration along the chute (obtained from Eq. (1)) is plotted in
Figure 4, where x is the streamwise coordinate from the jetbox. A comparison of the experimental data
against empirical formulae developed for estimating the mean air concentration in uniform flow on 50º
and 18.6 º sloping chutes, either stepped (e.g., Boes, 2000, Takahashi and Ohtsu, 2012) or smooth
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:-09
Step Number:-05
Step Number:-03
Step Number:-02
Step Number:-01
Step Number:+01
C theory(50º)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+02
Step Number:+03
Step Number:+04
Step Number:+05
Step Number:+06
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+07
Step Number:+08
Step Number:+09
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+10
Step Number:+11
Step Number:+12
Step Number:+13
Step Number:+14
Step Number:+15
C theory(18.6º)
235-8.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:-09
Step Number:-05
Step Number:-03
Step Number:-02
Step Number:-01
Step Number:+01
C theory(50º)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+02
Step Number:+03
Step Number:+04
Step Number:+05
Step Number:+06
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+07
Step Number:+08
Step Number:+09
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+10
Step Number:+11
Step Number:+12
Step Number:+13
Step Number:+14
Step Number:+15
C theory(18.6º)
235-8.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:-09
Step Number:-05
Step Number:-03
Step Number:-02
Step Number:-01
Step Number:+01
C theory(50º)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+02
Step Number:+03
Step Number:+04
Step Number:+05
Step Number:+06
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+07
Step Number:+08
Step Number:+09
Step Number:+10
Step Number:+11
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+12
Step Number:+13
Step Number:+14
Step Number:+15
C theory(18.6º)
235-8.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:-09
Step Number:-05
Step Number:-03
Step Number:-02
Step Number:-01
Step Number:+01
C theory(50º)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+02
Step Number:+03
Step Number:+04
Step Number:+05
Step Number:+06
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+07
Step Number:+08
Step Number:+09
Step Number:+10
Step Number:+11
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.4
0.8
1.2
1.6
2.0
C
y/y90
Step Number:+12
Step Number:+13
Step Number:+14
Step Number:+15
C theory(18.6º)
235-8.2
(a)
(b)
(c)
(d)
(e.g., Wood, 1991, Hager, 1991, Chanson 1997, Matos, 1999) indicates that flow conditions not
substantially dissimilar from quasi-uniform flow were observed at far upstream (x/h ~ 52) and far
downstream (x/h ~ 101) of the slope change cross-section, where Cmean u approaches 0.6 and 0.3,
respectively. The influence of the slope change on the mean air concentration is noticeable slightly
upstream of the slope change cross-section, where Cmean decreases considerably, which is judged to
be due to the flow curvature and higher pressures near the pseudo-bottom, in such short region.
As one can see from Figure 5, the streamwise distribution of the characteristic flow depth normalized
by the critical depth (Y90/dc) follows a similar overall trend as that obtained for the mean air
concentration, except in the vicinity of the slope change cross-section.
Figure 4 – Streamwise development of the mean air concentration for two unit discharges of 0.35 m2/s
(dc/h =3.8) and 0.46 m2/s (dc/h = 4.6).
Figure 5 – Streamwise development of the normalized characteristic depth, for two unit discharges of
0.35 m2/s (dc/h =3.8) and 0.46 m2/s (dc/h = 4.6).
In Figure 6, the mean air concentration normalized by the uniform mean air concentration values for
50º and 18.6 º chute slopes (after Boes, 2000, and Takahashi and Ohtsu, 2012, respectively) is
plotted as a function of the normalized distance xoc/Li. Therein the experimental data of xoc and Li are
estimated after modifying the origin for an uncontrolled ogee crest, following an approach similar to
that applied by Boes and Hager (2003b) where Li = (5.9dc6/5)/(sinφ7/5h1/5). The difference between the
calculated Li from an uncontrolled ogee crest and the observed inception point length from the jet-box
has been used to estimate the distance from an uncontrolled ogee crest (xoc). The application of
Pfister and Hager (2011) formula is also included in Figure 6, strengthen the conclusion that quasi-
uniform flow condition was attained on the upstream chute (Cmean/Cmean u ~1).
The sub-regions previously referred in section 3.2 apply to the mean air concentration, including a
decrease in the mean air concentration when approaching the slope change cross-section, followed
by its sharp increase, eventually reaching a peak, and decreasing further downstream, approaching a
practically constant value (Figures 4 and 6). However, uniform flow conditions were likely not reached
in the 18.6º chute, because the mean air concentration is larger than those corresponding to the
uniform flow for an identical slope on stepped (e.g., Takahashi and Ohtsu, 2012) or smooth spillway
chutes (Figure 4), Cmean u ~ 0.3.
Figure 6 – Streamwise development of the mean air concentration ratio for two unit discharges of 0.35
m2/s (dc/h =3.8) and 0.46 m2/s (dc/h = 4.6). The experimental data is normalized by Cmean u, upstream
and downstream of the slope change (after Boes, 2000, and Takahashi and Ohtsu, 2012,
respectively). At the slope change, xoc/Li ~ 2 for both discharges.
3. CONCLUSION
The effect of a 50º to 18.6º abrupt slope change on the air entrainment on stepped chutes was
analysed from data gathered on an experimental facility assembled at the Laboratory of Hydraulic
Constructions (LCH) of EPFL. Measurements of air concentration profiles were conducted and the
extracted data are discussed.
The results demonstrate that abrupt slope changes on stepped chutes have a major effect on the air
entrainment and flow bulking in the vicinity of the respective transition region. Four main sub-regions
were identified, with a decrease in the air concentration when approaching the slope change cross-
section, followed by its sharp increase immediately downstream, reaching a peak, and decreasing
further downstream towards an almost constant value. The peak mean air concentration as observed
downstream of the slope change cross-section may be considerably larger than that corresponding to
the uniform flow condition for the upstream chute, whereas the minimum mean air concentration
downstream of the slope change is larger than that estimated for uniform flow on a similar sloping
chute, possibly due to the limited length of the chute.
4. ACKNOWLEDGMENTS
The study was carried out in the framework of the IST-EPFL Joint Doctoral Initiative. The PhD
research has been granted by the Fundação para a Ciência e a Tecnologia (FCT), Portugal under
grant SFRH/BD/51527/2011, and the Laboratory of Hydraulic Constructions (LCH) of EPFL,
Switzerland.
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