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Cavitation Potential of Flow on Stepped Spillways
Article in Journal of Hydraulic Engineering · June 2013
DOI: 10.1061/(ASCE)HY.1943-7900.0000715
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Cavitation Potential of Flow on Stepped Spillways
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K. Warren Frizell, M.ASCE 1; Floriana M. Renna 2; and Jorge Matos 3
Abstract: Cavitation on spillways has proven to be an undesirable condition. The formation of vapor-filled bubbles and cavities and their
eminent collapse has led to significant damage to major spillway components and appurtenant structures worldwide. Although stepped
spillways are thought to be less prone to cavitation damage than smooth spillways, designers continue conservative practices regarding
specifying stepped spillways at many sites. Using laboratory experiments in a specialized reduced ambient pressure chamber, cavitation
was shown to form on stepped geometries that are representative of typical stepped spillways currently in service. Experiments in a nonaerated closed conduit revealed the strength and extents of the highly intense shear layer that forms above the step tips, and the friction
characteristics were determined and compared with results from previous researchers. Advanced techniques for detecting cavitation characteristics along with high-speed videography have given additional insight into the flow features that drive the formation of cavitation. Finally,
a correlation between the critical cavitation index and the common friction factor is shown, extending the data that also includes shear
layers resulting from uniformly distributed roughnesses, jets, and wakes. DOI: 10.1061/(ASCE)HY.1943-7900.0000715. © 2013 American
Society of Civil Engineers.
CE Database subject headings: Cavitation; Friction; Roughness; Spillways; Channel flow.
Author keywords: Cavitation inception; Friction factor; Macro-roughness; Stepped spillways.
Introduction
Stepped spillways have long been thought to be less prone to
cavitation damage than their more conventional smooth-surfaced
counterparts; however, uncertainty in this notion has perpetuated
conservative design practices. Skimming flow on a stepped channel
forms a highly intense shear layer along the line connecting successive step tips (also called the pseudobottom). Flow structure
within this shear layer supports formation of cavitation along secondary flow features with many similarities to other types of cavitating shear flows, including plane shear layers (O’Hern 1987;
Baur and Köngeter 1998; Iyer and Ceccio 2002) and submerged
jets (Arndt and Taghavi 1984). For the stepped spillway designer,
some important questions are at what flow conditions is cavitation
possible, and can traditional spillway design principles be applied?
The development of roller-compacted concrete (RCC) in the
1980s and its use in dam construction spurred a resurgence in
the application of stepped chutes as spillways. This renewed interest in stepped channels resulted in designs that featured a variety
of slopes and occasionally high hydraulic heads with high specific
discharges. There has been considerable uncertainty in how to
address cavitation on stepped spillways. Matos et al. (2000a) estimated the critical cavitation index at the inception point for air
1
Retired, Research Hydraulic Engineer, Bureau of Reclamation, Denver,
CO 80225 (corresponding author). E-mail: warren.frizell@gmail.com
2
ATB Riva Calzoni SpA, 25030 Roncadelle, BS, Italy; formerly,
Research Assistant, Technical Univ. of Bari, 70126 Bari, Italy. E-mail:
floriana.renna@atbrivacalzoni.com
3
Dept. of Civil Engineering, Architecture, and Georesources, IST,
Technical Univ. of Lisbon, 1049-001 Lisbon, Portugal. E-mail:
jm@civil.ist.utl.pt
Note. This manuscript was submitted on March 16, 2012; approved on
December 4, 2012; published online on December 6, 2012. Discussion period open until November 1, 2013; separate discussions must be submitted
for individual papers. This paper is part of the Journal of Hydraulic Engineering, Vol. 139, No. 6, June 1, 2013. © ASCE, ISSN 0733-9429/2013/
6-630-636/$25.00.
entrainment on steeply sloping chutes based on the relation proposed by Arndt and Ippen (1968) for a uniformly distributed roughness, along with the reanalysis of friction factor data of Tozzi
(1992). The resulting empirically based formula estimated the cavitation index at the inception point as a function of the step height
and specific discharge. Boes and Hager (2003) reported a critical
velocity for cavitation inception in the flow prior to air entrainment
of approximately 20 m=s. Based on the possibility that this velocity
could be reached prior to the initiation of air entrainment for a range
of slopes and step heights, they recommended limiting design specific discharges to ∼25 m2 =s. Pfister et al. (2006) proposed treating
each step as a singular bottom irregularity with risk of cavitation.
Based on this treatment (σc ≈ 1.0 0.1) and pressure measurements on the first step, they presented scaled prototype (h ¼ 1.2 m)
results showing the cavitation index is below 0.9 on the first step
when q > 30 m2 =s. Amador et al. (2009) recommended a mean
velocity of 15 m=s at the inception point based on a 0.1% probability of extreme negative pressures measured near the edge of the
vertical step faces on a steeply sloping chute. A similar negative
pressures approach was adopted by Gomes (2006), on the basis
of which a pseudocritical cavitation index was estimated as a function of the distance along the spillway. Critical unit discharges and
mean velocities were also proposed for typical stepped spillway
geometries (Gomes 2006; Gomes et al. 2007).
This resulted in recommendations to limit the specific discharge
from 11.5 m2 =s to 14 m2 =s for a steep (51.3°) stepped spillway
with step heights of 0.6 m and 1.2 m, respectively. As motivation
for the present studies, a recently modeled stepped spillway featured specific discharges ranging from about 3 to 15 times these
recommended values. The use of specific discharge as a design
recommendation also seems a bit misguided without knowing the
actual conditions of when and if cavitation will form.
In addition to theoretical considerations and reanalysis of literature data, results from physical modeling of representative stepped
geometries are presented. Reduced ambient pressure modeling
provides the first reports of measured critical cavitation indices over
a wide variation of stepped spillway slopes for two step heights.
630 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2013
J. Hydraul. Eng. 2013.139:630-636.
Additional measurements provide insight into the physical processes involved with cavitation formation and reveal a correlation
with the friction factor supporting similar findings shown for uniformly distributed roughness in boundary layers and several types
of free shear flows. Finally, the results will be used to discuss differences between the models tested and a typical stepped spillway
installation, including effects of air entrainment and possibilities
of damage, giving a suggested procedure for including cavitation
potential in the design process for stepped spillways.
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Experimental Program
Experiments were conducted in a specialized low-ambient pressure
chamber (LAPC) at the laboratory of the Bureau of Reclamation in
Denver, Colorado. This is a closed-loop facility capable of circulating up to 0.3 m3 =s with a split case centrifugal pump, while the
ambient pressure within the facility can be maintained between
atmospheric conditions and roughly 7,000 kPa absolute, using a
vacuum pump. Flow rates are measured with a permanent magnetic
flowmeter (0.002 m3 =s), and chamber pressures are measured
with a high-precision pressure/vacuum gauge (50 Pa). Atmospheric pressure is measured adjacent to the facility with a NovaLynx Model 230-7410 Fortin-type barometer (0.25 mm-Hg).
A rectangular closed-channel with machined anodized aluminum steps on the invert and smooth acrylic sidewalls and lid
was placed within the chamber, with water entering the conduit
from a pressure tank with flow straightening vanes and exiting
submerged into a small free-surface reservoir, allowing the chamber pressure to act on the fluid. The conduit dimensions were
148 mm high (pseudobottom to lid) by 203.2 mm wide by
2.21 m long. The conduit orientation was horizontal, with the angle
of the step treads above the horizontal (α) representing the spillway
slope. Two slopes were tested, 21.8° and 68.2°, accomplished
by reversing the step orientation to the flow direction. The step
height perpendicular to the slope, k, was adjusted from 50.8 mm
to 25.4 mm by adding small rectangular blocks at each larger
step. The vertical step height is defined as h. Piezometer taps
(1.6-mm-dia) were installed along the centerline of the channel
on the lid at a spacing of 150 mm in order to measure the pressure
gradient along the conduit. Fig. 1 shows the major geometric
parameters of the stepped channel.
While operating at atmospheric conditions, specific discharge
was incrementally increased from 0.418 to 1.255 m2 =s, corresponding to mean velocities varying from 2.83 to 8.5 m=s. The
energy gradient was measured using water manometers connected
to the lid piezometer-taps for each flow condition. In addition, a
Piezometer taps
Lid
FLOW - A
A&B
FLOW C&D
pseudo bottom
h
A&B
k
Ste
p In
t
ser
C&D
Fig. 1. Sketch showing main geometric components of closed conduit
used in present study
two-dimensional particle image velocimetry (PIV) system manufactured by Dantec Dynamics was used to acquire localized flow
field images on the conduit centerline plane at three locations along
the length corresponding to roughly the beginning, midpoint, and
end of the conduit. The PIV system included a dual cavity Nd:YAG
laser, a FlowSense 4M CCD camera (2,048 × 2,048 pixels), and
Dantec’s FlowManager software to control, collect, and analyze
the images.
Operations at reduced ambient pressures were completed over
a similar range of discharges, first filling the test circuit with tap
water and circulating it slowly for several hours under vacuum
to pull dissolved gases from the water. The degassing of the water
was done to allow operation of the facility with slightly undersaturated fluid at the reduced ambient working pressure. At each flow
condition, an acoustic emissions (AE) sensor was used to indicate
cavitation activity. The AE sensor was a DECI Model SE9125-M, a
mass-loaded transducer that is equally sensitive to both extensional
and flexural waves. The sensor was attached to the exterior sidewall
of the conduit, centered at the level of the step tips and 100 mm
upstream from the end of the test section. A DECI AE1000 signal
conditioner was used to provide excitation and output from the
sensor. Signals in a frequency range of 20 kHz–70 kHz were
recorded, as formation and collapse of cavitation bubbles result
in flexural waves in the conduit and are typically found in this frequency band (Dunegan 1997). A counting technique was used to
quantify the cavitation activity by noting the number of times the
AE signal peaks exceeded a threshold value in a 30-s time period.
High-speed digital video was also collected with a Vision Research,
Inc. Phantom v4.2 camera and associated software. A macro/
zoom lens allowed close-up imaging from outside the LAPC
through the acrylic windows. All videos were collected at a rate
of 2,000 frames=s.
Table 1 summarizes the experimental conditions, where
flow conditions were repeated for each of four basic data sets:
Q = discharge; V o = mean reference velocity; Po = local reference
Table 1. Experimental Conditions for Data Sets A–D
αð°Þ
k (mm)
Q ðm3 =sÞ
V o ðm=sÞ
J (m/m)
σatm
σrap
21.8
21.8
21.8
21.8
21.8
21.8
21.8
21.8
21.8
21.8
21.8
68.2
68.2
68.2
68.2
68.2
68.2
68.2
68.2
68.2
68.2
50.8
50.8
50.8
50.8
50.8
50.8
50.8
25.4
25.4
25.4
25.4
50.8
50.8
50.8
50.8
50.8
25.4
25.4
25.4
25.4
25.4
0.085
0.113
0.142
0.17
0.198
0.227
0.255
0.057
0.113
0.17
0.227
0.113
0.142
0.17
0.198
0.227
0.113
0.142
0.17
0.198
0.227
2.83
3.78
4.72
5.66
6.61
7.55
8.5
1.89
3.78
5.66
7.55
3.78
4.72
5.66
6.61
7.55
3.78
4.72
5.66
6.61
7.55
0.111
0.189
0.289
0.469
0.569
0.725
0.839
0.022
0.145
0.314
0.561
0.328
0.503
0.735
1.062
1.314
0.283
0.452
0.646
0.982
1.163
20.1
11.34
7.29
5.08
3.75
2.88
2.29
44.67
11.23
4.99
2.85
11.24
7.25
5.1
3.79
2.96
11.14
7.19
5.03
3.74
2.9
1.75
1.02
0.69
0.49
0.38
0.3
0.25
4.17
1.02
0.49
0.31
0.97
0.68
0.54
0.44
0.39
0.87
0.62
0.47
0.39
0.33
Note: h = vertical step height; α = angle of slope to horizontal;
k ¼ h cos α–roughness height perpendicular to slope; J = energy loss
per unit length; σatm = cavitation index at atmosphere conditions; σrap =
cavitation index at reduced ambient pressure conditions; reference
pressure = ambient pressure plus local mean pressure at last step.
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2013 / 631
J. Hydraul. Eng. 2013.139:630-636.
pressure; Pv = vapor pressure; ρ = water density; and σ = cavitation
index. The cavitation index is given by
σ¼
ðPo − Pv Þ
1
2
2 ρV o
ð1Þ
Experimental Results and Discussion
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Shear-Layer Development
Flow characteristics over stepped spillways have received much
attention over the last 40 years. A variety of means have been used
to quantify flow parameters, such as the mean velocity, pressures,
turbulence quantities, and air entrainment. The high-intensity shear
layer that develops along the pseudobottom is thought to be the
most important feature affecting energy exchange and dissipation
and, in turn, should influence the inception and development of
cavitation. This region of high shear, where vorticity is produced,
results in secondary streamwise vortical structures, and as the pressure within the cores of these vortices drops, they become probable
sites for cavitation formation. Past measurements have usually been
performed with instruments that measure velocity and air concentration at a point in the flow (e.g., Chamani and Rajaratnam 1999;
Matos 1999; Matos et al. 2000b; Chanson 2002; Boes and Hager
2003). Using vertical profiles of velocity and related turbulence
quantities, it was possible to sketch out the extents of the shear
(or mixing) layer above the pseudobottom. Amador et al. (2004)
were the first to use PIV to characterize the nonaerated flow region
of a typical steep stepped spillway with 53.1° slope, 50-mm step
height, and 0.11-m2 =s specific discharge. They showed peaks in
vorticity and mean shear strain along the pseudobottom just downstream from each step tip in the range of 45 to 50 s−1 for a velocity
of 3.2 to 3.6 m=s. Fig. 2 shows PIV measurements from the current
tests at a flow with mean velocity of 3.6 m=s where the average
shear strain rate is computed using
1 dV x dV y
εxy ¼
þ
ð2Þ
dx
2 dy
Magnitudes of the shear strain rate compare well with data of
Amador et al. (2004), taken at similar mean velocities on the nonaerated portion of an open channel stepped spillway flow. In general,
the strength of the shear layer is slightly higher for the 21.8° slope,
Fig. 2. (Color) Shear strain rate from PIV measurements at last step for each data set; mean velocity 3.6 m=s
632 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2013
J. Hydraul. Eng. 2013.139:630-636.
and the thickness perpendicular to the slope is greater for the larger
roughness height (50.8 mm). At mild slopes, the wake-step interference greatly affects the secondary flows and the mixing layer
thickness. At steep slopes, the interference appears to be less or
nonexistent. Similar observations have been made by Ohtsu et al.
(2004) for stepped spillways ranging from 5.7° to 55°.
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Friction Factor
The steps can be characterized as macroroughness elements where
losses include traditional skin friction and a substantial contribution
from form drag. Additional complicating factors with stepped spillways are that the roughness elements are generally only present on
the invert and not the side walls, and significant air entrainment
causes increased bulked depths, drag reduction—resulting in lower
friction factors as compared to nonaerated flow—and increased
velocities.
Many researchers have previously used the Darcy-Weisbach
friction factor, f, to define flow characteristics on stepped spillways. Several approaches can be used to estimate the friction factor
in the closed stepped channel, from a simple measurement of the
pressure drop along the channel to a more rigorous treatment of the
use of measured velocity profiles acquired through particle image
velocimetry measurements. The friction factor can be defined as
f¼
8Rh gJ
U2
ð3Þ
Experimental data were used to calculate f using Eq. (3). The
head loss/unit length was measured for each flow and roughness
condition. In addition, we defined a virtual centerline or characteristic depth, d, suggested by Tozzi (1992) and based on the measured (PIV) velocity profiles in the conduit. The hydraulic
radius was then defined as Rh ¼ ðWdÞ=ðW þ 2dÞ, where W =
width of the channel, and d = characteristic depth. In order to
use the majority of published equations for friction factor, a shape
factor proposed by Marchi (1961) was used to apply the equations
from circular to square geometries. This factor (ω ¼ 1.177) was
based on his numerous experimental data of turbulent flow in a
conduit of square dimension and is applied to the hydraulic radius.
Using the shape factor of Marchi (1961) and the sidewall correction
method of Schröder (1990) (in Boes and Hager 2003), the overall
Darcy-Weisbach friction factor was adjusted to better represent the
friction factor of a channel with roughness only on the bottom
(more typical of a stepped chute), f b (Table 2). Fig. 3 shows a comparison of friction factor versus k=d, for the present study, the data
presented by Tozzi (1992), and the formulae by Tozzi (1992),
Chamani (1997), and Matos (1999). The data of Tozzi (1992) include closed conduit and open channel (53°-sloping) or closed
conduit (27°-sloping), the Chamani (1997) formula was derived
from some data on pipe flow (Tozzi 1994) as well as the skimming
flow in stepped chutes, and Matos (1999) has presented a reanalysis
of Tozzi’s closed conduit data for the 53°-sloping chute.
Fig. 3. Friction factor as function of step height and characteristic
depth for data from closed conduits and open channels with steps; present study data is fb calculated with Rh ¼ d
Cavitation Potential
The high-intensity shear just above the step tips is likely the most
probable location for cavitation to form; however, it could be argued that cavitation should appear first just below the step tips
on the vertical riser where the lowest measured pressures have been
found (Amador et al. 2009). Based on other related shear flows,
streamwise vortex structures will likely be present and could actually produce the lowest pressures in the flow field. Fig. 2 highlights the location of the high-intensity shear layer and the
interaction with the step tips.
Cavitation detection measurements are presented in Fig. 4 for all
conditions tested in the LAPC under reduced ambient pressures.
The data represents counts that have exceeded a nominal threshold,
normalized by the maximum number of counts recorded in a
particular data set, plotted against the cavitation index [Eq. (1)].
Different techniques have historically been used to determine critical cavitation conditions; ranging from visual and audible to instrumented detection of acoustic pressures, vibrations, and acoustic
emissions. The resulting noise and vibrations associated with cavitation bubble formation and collapse are typically high-frequency
(>10 kHz) events. Acoustic pressures have been measured directly
using hydrophone technology, while accelerometers and acoustic
Table 2. Data from Present Experiments
Dataset
k (m)
d (m)
fb
σc
A
B
C
D
0.051
0.025
0.051
0.025
0.114
0.106
0.086
0.082
0.127
0.089
0.167
0.160
0.404
0.308
0.640
0.621
Note: Characteristic depth and friction factor from Eq. (3), adjusted using
Marchi’s (1961) shape factor and Schröder’s (1990) side-wall correction (in
Boes and Hager 2003); critical cavitation index from the acoustic emission
counts during reduced ambient pressure testing.
Fig. 4. (Color) Normalized acoustic emission threshold counts as function of cavitation index
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2013 / 633
J. Hydraul. Eng. 2013.139:630-636.
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emission sensors can be mounted externally. In the present tests,
critical indices are noted by sudden changes in the rate of counts
of acoustic emissions with decrease in the cavitation index. The
critical index in this case is different from inception. The first
occurrences of acoustic emissions are the indicators of inception
and occur at higher σ values; however, the critical index represents
the point of the largest increase in the rate of cavitation activity
(noted by the arrows on Fig. 4), a point where performance could
possibly be affected (Table 2).
In addition to the acoustic detection, high-speed digital video
was collected during each test. Single-frame images are presented
in Fig. 5, showing typical cavitation patterns for the stepped geometries tested near these critical indices. The secondary streamwise
vortex structures appear to be where cavitation first forms. Nuclei
tend to migrate to the cores of these vortices, providing an ample
source for cavitation inception. Arndt and Ippen (1968) measured
the inception pressure in the cores of vortical structures within
turbulent boundary layers by observing the rate of bubble growth
during inception using high-speed cinematography and discovered
instantaneous pressure drops on the order of 10 times the RMS
(root-mean square) wall pressure. Similarly, Ooi and Acosta
(1984) and Ran and Katz (1994) observed the growth of injected
air bubbles in a jet and found measured negative peaks in pressure
greater than 10 times the RMS pressure. So, while the pressures on
the step risers just below the step tips show the lowest surface pressures, these vortical structures within the shear layer above the
pseudobottom likely have the lowest peak pressures and, as reinforced by the visual evidence (Fig. 5), are most likely the locations
of cavitation inception.
At a given slope, spillways featuring larger step heights are
slightly more prone to cavitation than smaller steps. The effect of
chute angle (spillway slope) is important. Based on Figs. 2 and 5,
Fig. 5. Single frames from high-speed digital video collected at
2,000 frames=s at critical cavitation indices for each of four test
conditions: (a) k ¼ 50.8 mm, σ ¼ 0.39 (set A); (b) k ¼ 25.4 mm, σ ¼
0.37 (set B); (c) k ¼ 50.8 mm, σ ¼ 0.67 (set C); (d) k ¼ 25.4 mm,
σ ¼ 0.63 (set D)
the 21.8° slope shows an impact of the skimming flow about twothirds of the way along the length of the step tread, resulting in
increased local pressures along the flow boundary and a redirection
of the flow. The steeper slope (68.2°) shows the high-intensity shear
layer mostly above the step tips (pseudobottom) with little if any
impingement on the step tread. These slope-related observations
become a significant indicator of the damage potential to the
stepped surfaces as well, even though the steep slope has a larger
measured critical cavitation index than the mild slope.
The critical cavitation index measured in the reduced ambient
pressure tests and the adjusted friction factor calculated from the
measured J and d for that flow condition result in a correlation
between sc and f b , shown on Fig. 6. Arndt and Ippen (1968) first
presented this correlation for uniformly distributed roughnesses in
turbulent boundary layers. They also found that the correlation was
maintained for a variety of other types of shear flows, including
wakes and jets. Including Cp in the expression allows for the cases
where the local mean static pressure is less than the free stream
pressure, such as the wake behind a disk.
The spillway designer needs only to specify basic parameters for
the spillway, such as slope and step height, calculate a friction factor using any of a number of methods in the literature, then apply
the correlation, σc ¼ 16Cf ¼ 4f, to predict the critical cavitation
index. The cavitation indices of the flow can be computed along the
spillway length, namely in the nonaerated region (using empirical
models or numerical simulations) and compared to the calculated
critical index in order to evaluate areas of concern. In addition, the
length down the steps to the inception of air entrainment (Li )
should be estimated using available methods (e.g., Chanson
2002; Hunt and Kadavy 2010; Meireles et al. 2012). Once this
length is determined, the designer can evaluate the air entrainment
effects on the spillway and the possibility of cavitation damage. In
general, for designs of relatively small specific discharges, cavitating conditions will likely not be present on the portion of the chute
prior to the inception of air entrainment, and if the critical index is
exceeded only on the aerated portion of the chute, cavitation damage is unlikely due to the effect of aeration on cavitation (Peterka
1953). Recent studies (using PIV) on microbubble drag reduction
have shown decreases in the production of vorticity and the intensity of the fluctuating strain rate by up to 35%, with a local void
fraction of only 5% (Gutierrez Torres et al. 2008), giving further
evidence of the actual physical effects of air entrainment on the
flow field.
Fig. 6. Correlation of coefficient of friction (Cf ¼ f=4) as function of
critical cavitation index; current data are shown with other types of
wall-bounded and free shear flows
634 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2013
J. Hydraul. Eng. 2013.139:630-636.
Evaluating the exact conditions when cavitation damage will
occur on a stepped spillway is still an unknown. Smooth spillways
have provided numerous prototype examples of damage, resulting
in the ability to predict critical conditions for damage with some
confidence. Stepped spillways have no documented damage. There
is some anecdotal evidence that spillway slope could play a major
role in possible damage from cavitation to stepped channels, and
this remains a topic for future research.
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Summary and Conclusions
Laboratory tests in a specialized facility used to study the formation
of cavitation at reduced ambient pressures have shown that cavitation can form on stepped geometries typical of those found on
stepped spillways of varying slopes. Measurements of flow features
using particle image velocimetry have helped to delineate the
highly intense shear layer that forms just above the step tips along
the pseudobottom. These measurements along with energy losses
and velocity profiles recorded along the conduit were used to calculate friction factors for the four different data sets tested. Testing
at reduced ambient pressures allowed detection of critical cavitation
features using acoustic emissions technology. High-speed digital
video verified locations and patterns of the cavitation that occurs
along the steps of a stepped spillway.
No evidence of cavitation damage has been reported on a
stepped spillway in service; however, design practices have been
conservative, yielding to the uncertainty surrounding this topic.
In addition, most of the larger stepped spillways constructed have
been relatively steep slopes, which, according to all indications,
will be less susceptible to damage that might occur, due to the
slope-dependent observations of the cavitation in the model related
to the shear-layer properties and lack of major impingement of flow
on the step treads. The spillway designer can now, with some confidence, surpass these limiting specific discharge conditions by using the correlation between friction factor and the critical cavitation
index presented within. In addition, the designer has more freedom
in addressing the method in which flow enters a stepped spillway,
easily allowing for consideration of gated inlet structures in addition to the typical free-overfall crests. While naturally induced air
entrainment on stepped spillways has likely protected the stepped
surfaces from cavitation damage previously, this new design information can be used in conjunction with constructed aeration ramps/
slots (commonly used on smooth spillways) in order to protect a
whole new class of stepped spillways.
Acknowledgments
The funding for these and related studies was provided by the
U.S. Army Corps of Engineers, Sacramento District, and the U.S.
Bureau of Reclamation Dam Safety Office’s Technology Development Program and Science and Technology Program. Dr. Renna’s
support during her stay at the Reclamation Laboratory was provided through the Technical University of Bari, Italy. Prof. Matos’s
support was provided through the Portuguese Foundation for
Science and Technology-FCT (Project PTDC/ECM/108128/2008).
Notation
The following symbols are used in this paper:
A = area of flow;
Cf = coefficient of friction;
Cp = coefficient of pressure;
d = characteristic depth (in present study, pseudobottom to
virtual centerline);
f = Darcy-Weisbach friction factor;
f b = friction factor adjusted for shape and sidewall;
g = gravitational constant;
h = vertical step height;
k = roughness height perpendicular to slope (h cos α);
J = energy loss/unit length;
Li = length down chute to beginning of air entrainment;
Po = reference pressure;
Pv = vapor pressure of water;
Q = discharge;
q = specific discharge;
Rh = hydraulic radius (cross-sectional area/wetted perimeter);
u = streamwise point velocity;
u 0 = streamwise fluctuating velocity;
V o = mean streamwise velocity—reference velocity;
V x = velocity component–streamwise;
V y = velocity component–normal;
v 0 = normal fluctuating velocity;
W = width of channel;
x = coordinate of length along channel;
y = coordinate of depth ρ—density;
α = angle of chute to horizontal;
εxy = shear strain rate;
ρ = water density;
σ = cavitation parameter, subscript c indicates critical value;
τ = shear stress; and
ω = Marchi shape factor.
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