Acta Geophysica
vol. 59, no. 2, Apr. 2011, pp. 296-316
DOI: 10.2478/s11600-010-0048-z
Expanding Pools Morphology
in Live-Bed Conditions
Stefano PAGLIARA, Michele PALERMO
and Iacopo CARNACINA
Department of Civil Engineering, University of Pisa, Pisa, Italy
e-mails: [email protected] (corresponding author),
[email protected], [email protected]
Abstract
Sediment load plays a fundamental role in natural river morphology
evolution. Therefore, the correct assessment of the role of the sediment
load on natural or anthropic pools morphology downstream of river
grade control structures, such as rock chute or block ramps, is of fundamental interest for preserving the fish habitat and the river morphology.
This work presents an experimental study on the sediment load influence
on rectangular expanding pools downstream of block ramps in live-bed
conditions. Several longitudinal and transversal expanding ratios have
been tested. Ramp slopes were varied between 0.083 and 0.25. The effect
of the pool geometry and the sediment load on hydraulic jump downstream of block ramp as well as scour morphologies and flow patterns
have been analyzed. Equations were derived to evaluate the maximum
scour hole depth, the longitudinal distance of the section in which it
occurs, and the maximum water elevations both in the pool and in the
downstream contraction.
Key words: block ramp, expanding pool, hydraulic jump, live-bed, scour
morphology.
________________________________________________
© 2010 Institute of Geophysics, Polish Academy of Sciences
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1.
297
INTRODUCTION
Natural river’s erosion features as well as anthropological activities may
change the natural river system sediment convey from the river basin to the
sea (Phillips 2010). Amongst the vast process of sediment erosion from a basin surface, the sediment bed load transport has been found of particular
influence on rivers morphology and on river bank stability (Whitaker and
Potts 2007). An analysis conducted by Lisle (1982) on natural gravel channels in northwestern California after the 1964 flood proved that both bar profiles and bed load transport deeply influence pools and riffles morphology.
Very often the natural evolution of river bodies has to be regulated using
grade control structures which can modify the sediment transport and the
thalweg configuration of both mountain creeks and rivers (Lenzi et al. 2003,
Marion et al. 2004, Martín-Vide and Andreatta 2009). In this perspective,
low environmental impact river training structures (such as block ramps, or
block sills) have been widely studied as they have the advantage to be
extremely eco-friendly and can be even used as fishways.
In order to maintain the natural river evolution and to obtain suitable step
and pool configurations, either unprotected natural or anthropic pools can be
inserted downstream of block ramps. Ideally, the natural fishway flow configuration should present a succession of low velocity zones which constitute
a waiting area for fish migration, and high velocity zones which attract them
(Yasuda and Ohnishi 2009). Therefore, expanding anthropic pools, which
are characterized by a complex three-dimensional flow configuration with
dead water zones (Bremen and Hager 1993), could represent an ideal compromise between energy dissipation and river training structure’s impact on
natural habitat. Moreover, they simulate in an optimal way the natural pools.
However, the presence of sediment load coming from upstream has a big
impact on river morphology evolution. Namely, it reduces the maximum
scour depth and influences the hydraulic jump location.
This paper aims to assess an experimental study on sediment load effect
in the presence of an expanding pool in live-bed conditions, i.e., when sediments are continuously supplied and the equilibrium depth condition is
reached when there is a balance between the sediment supply and sediment
transported out of the hole (Breusers and Raudkivi 1991). The phenomena
will be qualitatively analyzed in terms of flow pattern, maximum water elevation and pool morphology, including the maximum scour hole depth,
length and width, as well as the longitudinal location of the cross-section in
which the maximum scour depth occurs. Data will be analyzed in order to
provide equations based on the main parameters involved in the erosive
process, i.e., geometrical (ramp slope, pool width and length), hydraulic
parameters (Froude number, tailwater level), granulometric characteristics
(specific weight, average diameter) and sediment load.
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2.
S. PAGLIARA et al.
LITERATURE REVIEW
In the present paper, a rectangular expansion basin was analyzed (Fig. 1).
The rapid transition from super- to subcritical flow, which occurs in the
pool, evolves in high turbulence, shear stress and energy dissipation and
causes large morphological variations downstream of the block ramp. The
erosion process in the expanding pool downstream of a block ramp is influenced by several factors, i.e., hydraulic features and sediment load.
Little literature is available dealing with the spatial hydraulic jump and is
mainly based on fixed bed channel experiments, a condition rather far from
that usually present in natural rivers. Rajaratnam and Subramanya (1968),
Herbrand (1973), and Smith (1989) analyzed hydraulic jumps below abrupt
indefinite symmetrical or lateral expansions for a fixed and horizontal channel bed. They described the hydraulic phenomenon including the tailwater
effect. Several typologies of hydraulic jumps were distinguished, depending
on the configuration assumed by the hydraulic jump front. A more detailed
analysis of hydraulic jumps in abruptly expanding channels was provided by
Bremen and Hager (1993). They furnished a classification of jumps based on
the approach flow conditions: Repelled jump (R-jump), Spatial jump
(S-jump), Transitional jump (T-jump) and classical hydraulic jump. Experiments were conducted both in symmetric and asymmetric expansions in
horizontal and rectangular channels with expansions ratio up to 3 for the
symmetric case. It was experimentally proved that an abruptly expanding
channel is generally not appropriate as an efficient energy dissipator.
Amongst the jump typologies, T-jump performs better than any of the other
types in terms of energy dissipation, although it reduces if the expansion
ratio increases.
Another study was carried out by Ram and Prasad (1998). They analyzed
the hydraulic jump in stilling basins with an abrupt drop and an expansion.
Ohtsu et al. (1999) conducted experiments on submerged hydraulic jumps
below abrupt expansions. The hydraulic conditions for their formation were
clarified, and an expression for the transition between symmetric and asymmetric flows was developed. No studies are currently available in which the
hydraulic jump occurs in a symmetrically expanding pools for mobile channel beds, except some tests conducted by Veronese (1937), even if mobile
expanding pools are generally present in natural rivers. Moreover, previous
studies were conducted in horizontal channels and the effect of bed slope on
the phenomenon was not taken into account.
Recently, Pagliara et al. (2009) analyzed both the scour geometry and
hydraulic jump downstream of block ramps in enlarged pools in different
hydraulic and geometric conditions and for both uniform and non uniform
stilling bed granulometry. Different typologies of hydraulic jumps were clasUnauthenticated
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sified, based on the nature of the hydraulic jump behavior, namely toe and
repelled jump, either symmetric or oscillatory.
Erosive processes downstream of hydraulic structures have received
a systematic investigation. Most of the studies were conducted in clear-water
condition, for which no general bed movement and no supply of sediment to
the scour hole from upstream occurs (Veronese 1937, Hassan and Narayanan
1985, Mason and Arumugam 1985, Bormann and Julien 1991, Breusers and
Raudkivi 1991, Hoffmans and Verheij 1997, D’Agostino and Ferro 2004,
Marion et al. 2004, Dey and Raikar 2005, Dey and Sarkar 2006, Pagliara
2007, Pagliara and Palermo 2008).
Conversely, relatively few studies dealing with the sediment load effect
on river morphology downstream of structures such as block ramp, sills or
eventually check dams are present in literature. Melville (1984), Chiew and
Lim (2000), and Sheppard and Miller (2006) conducted live-bed scour tests
with singular circular piles furnishing a significant quantity of scour data and
information about bed form in different hydraulic conditions, observing that
maximum scour hole depth generally occurred at the threshold condition,
while in live-bed condition scour tended to be reduced as a dynamic equilibrium was achieved, i.e., when sediment is continuously supplied and removed from the groove at the pier section. Dey and Raikar (2006) analyzed
the scour process occurring in long contraction in live-bed conditions. They
developed an analytical model to evaluate the live-bed scour depth which
well predicts the experimental data. Moreover, they compared different empirical equations present in literature in order to find the best predictor. In
particular, Marion et al. (2006) experimentally studied the effect of sediment
load on scour downstream of a series of sills in high gradient river. Primarily
the authors observed that the normalized scour, i.e., the ratio between the
maximum observed scour and the energy height over the sills, depends on
a morphological slope. This parameter takes into account the original river
slope, the equilibrium slope assumed by the mobile bed and the distance between the sills. They developed a new experimental equation to predict the
scour depth based on previous clear-water tests. However, the equilibrium
slope analysis has not been performed and only data obtained under the
experimental condition were analyzed. Martín-Vide and Andreatta (2006)
conducted live-bed experiments on steep bed slopes in order to investigate
the equilibrium slope. They also tested different bed sill settings in order to
compare the bed degradation both in the presence and absence of sills.
Bhuiyan et al. (2007) analyzed the interaction between sediment transport
and a W-weir located immediately downstream of a riffle section. They analyzed the scour and bed form in correspondence to the structure and they
observed that the scour phenomenon tends to progress in stages. They found
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S. PAGLIARA et al.
that the scour process starts rapidly, then it develops and eventually reaches
the equilibrium condition.
3.
EXPERIMENTAL APPARATUS AND METHOD
About 70 tests have been carried out at the PITLAB research center of the
University of Pisa, Pisa, Italy. The experimental apparatus consisted of a 6 m
long, 0.50 m wide, 0.60 m deep glass walled flume. Water was supplied in
the inlet ramp section by means of a broad crested weir 0.50 m long, where
the critical condition was achieved. Ramp material was directly glued on
a 2 mm thick, and 0.80 m long stainless steel sheet. The supercritical flow
entered in a movable stilling basin which is confined downstream by a contraction (see Fig. 1b). Transported sediments were collected by means of
a sediment trap located immediately downstream of the contraction. The
sediment trap was shaped in order to avoid backwater effects. The water
level in the pool was regulated by means of a sluice gate located at the model
end section. Previous studies of the same authors (Pagliara 2007 and
Pagliara et al. 2009) allow to state that both the scale effects and the ramp
material size are negligible in terms of scour features in the tested range of
parameters. Figure 1 shows a sketch of the experimental apparatus, in which
both the geometric and hydraulic parameters are represented.
Pools of different longitudinal and transversal expansions were obtained
by means of Plexiglas movable walls which allowed for sediment and flow
visualization. For each series of tests, pool sidewalls were located at B/2
from the channel axis while the pool contraction was located at A from the
ramp toe (Fig. 1b).
Fig. 1. Diagram sketch: (a) longitudinal view and (b) plane view.
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In Figure 1, kc = [(Q/b)2/g]1/3 is the critical depth, where Q [m3/s] is the
water discharge and g is the gravitational acceleration, b is both the ramp
and the downstream contraction width, Qs-in [m3/s] is the inlet sediment load,
Qs-out [m3/s] is the outlet sediment load, h1 is the water depth at the ramp toe
(measured adopting the methodology specified in Pagliara 2007), h0 is the
tailwater level, measured from the original bed level downstream of the dune
or of the scoured zone, and S is the ramp slope. The scour features are: zmax –
the maximum scour hole depth, ls – longitudinal distance of the maximum
scour hole depth from the ramp toe, l0 – the scour hole length, x – longitudinal coordinate, y – transversal coordinate, z – vertical coordinate, and 0 –
origin of the coordinate system.
The three different tested ramp slopes were: 1V:12H, 1V:8H, 1V:4H.
Two uniform materials, m1 and r1 , were used for stilling basin and ramp bed,
respectively. Their granulometric characteristics are listed in Table 1, where
dxx and Dxx is the diameter of the stilling basin and of the ramp bed, respectively, for which the xx% in weight of sediment is finer, σ = (d84/d16)0.5 and
σ = (D84/D16)0.5 are the non uniformity parameters relative to stilling basin
and ramp materials, respectively, Δ = (ρs − ρ)/ρ is the relative sediment density, ρs and ρ are the sediment density and the water density, respectively.
Water discharges were measured by means of an electromagnetic
KROHNE® flow meter of 0.01 l/s accuracy. Water elevations, as well as
dynamic equilibrium pool elevations and scour lengths, were surveyed by
means of a point gauge of 0.1 mm accuracy.
Before each run, the material was carefully leveled and measured in
order to obtain a horizontal bed. After the suitable flow discharge was reached,
the inlet sediment load Qs-in, having the same granulometric characteristics of
the stilling basin, was directly supplied at the entering ramp section using an
hopper located before the ramp inlet (see Fig. 1a). Hence, according to the
desired concentration, prefixed volumes of sediments were prepared and
supplied continuously each 1 minute in order to guaranty an homogenous
sediment load. With the same frequency, the transported sediments collected
by the downstream trap were removed and the volume computed. When the
T ab l e 1
Ramp and pool materials characteristics
Materials
d50, D50
[mm]
d90, D90
[mm]
σ
Δ
m1
r1
3.39
24.84
4.00
28.38
1.19
1.10
1.58
1.54
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Fig. 2. Evolution of: (a) the sediment load and (b) scour depth.
volumes of the supplied and collected sediment were equal, the equilibrium
conditions were supposed to be reached. Consequently, the outlet sediment
loads Qs-out were recorded. Examples of the temporal evolutions of the sediment outlet loads are shown in Fig. 2. The figure shows three typical temporal sediment load evolutions for different inlet sediment concentrations,
ranging between 0.143 < C < 0.574, where C = (Qs-in/Q)×1000.
According to Marion et al. (2006) and Melville and Chiew (1999), livebed scour depth needs less time than clear-water scour depth to reach its
equilibrium value. The equilibrium value is reached when Qs-in ≈ Qs-out and
the variation of zmax/h1 becomes negligible. Figure 2a shows a typical temporal evolution of the ratios Qs-out /Qs-in plotted versus the non dimensional
temporal parameter T = [(g′d50)0.5/h1]t, where t is the time, whereas Fig. 2b
shows the respective temporal evolution of the maximum scour depth which
increases up to reaching the asymptotic equilibrium. The parameter T was
introduced by Oliveto and Hager (2002). In the present paper, the water
depth h1 was assumed as the reference length. According to the value of Qs-in
and varying the hydraulic conditions, the dynamic equilibrium condition
(Qs-out /Qs-in ≈ 1) can be reached in three different ways, as shown in Fig. 2a.
For low C, Qs-out decreased with T, until it became equal to Qs-in. For intermediate C, the equilibrium was reached after few minutes. A further increase
of Qs-in caused an increase of Qs-out up to reaching the dynamic equilibrium
condition. Thus, the time to reach the equilibrium conditions depends also on
C, but in the tested range of parameters it was always obtained for T < 4000
(about 40 minutes in model scale). When the sediment load equilibrium condition was reached, the scour hole shape was surveyed. After the equilibrium
of both the sediment load and the scour hole morphology was attainted, the
water surface and the bed morphology were carefully surveyed.
Table 2 shows the tested experimental ranges for each configuration.
In Table 2 Fd 90 = V1/(g′d90)0.5 is the densimetric Froude number, in which
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T ab l e 2
Experimental tested range
Test
series
E
λ
S
C
Fd 90
h0/h1
Re×10–3
C1-12
1
2
1.8
2.8
0.083
0.143
0.574
4.81
6.24
1.29
1.69
165.0
290.0
min
max
C1-8
1
2
1.8
2.8
0.125
0.143
0.574
5.65
6.54
1.53
1.72
175.0
267.0
min
max
C1-4
1
2
1.8
2.8
0.25
0.143
0.574
6.63
7.16
1.65
2.06
184.0
263.0
min
max
Fig. 3. Hydraulic jumps classification: (a) R-OE=∞, (b) R-SF-J , and (c) R-OI-J.
g′ = gΔ is the reduced gravitational acceleration (Dey and Sarkar 2006), and
V1 = Q/(bh1) is the average velocity at the ramp toe; Re = 4V1R/ν is the
Reynolds number, where ν is the kinematic viscosity and R is the hydraulic
radius. λ = B/b is the transversal expansion ratio and E = A/B is the longitudinal expansion ratio of the pool, where A and B are the length and the
width of the pool, respectively (Fig. 1). Note that E = ∞ for an indefinite
longitudinal expansion (Fig. 3a).
4.
RESULTS AND DISCUSSION
Hydraulic jump features
According to Pagliara et al. (2009), the flow field in presence of an indefinite abrupt expansion (i.e., E = ∞) in clear-water conditions results in flow
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net concentration at the ramp toe, with the presence of strong recirculation
eddies on both pool sides. They found that in clear-water condition, the
peculiar flow patterns resulted in a deeper maximum scour hole as far as λ
increases. In addition, four main hydraulic jumps were classified, namely toe
and repelled jumps, both either symmetric or oscillating.
In the present study, preliminary tests were carried out in live-bed condition for E = ∞ in order to understand the hydraulic qualitative behavior.
It was observed that, in the tested range of parameters and live-bed conditions, only one hydraulic jump typology takes place, namely the repelled oscillatory hydraulic jump (R-OE=∞, Fig. 3a), whereas four hydraulic jump
typologies occurred in clear-water conditions (Pagliara et al. 2009). This is
mostly due to the fact that both the presence of bed load and the higher
tested Fd 90 influence the hydraulic jump behavior, as the scour shape varies
in respect to the clear-water case. The hydraulic jump is not forced to occur
downstream of the ramp toe by a prominent dune and the high Fd 90 causes
periodical deflections towards both stilling basin sides. Moreover, in this
Fig. 4. Flow pattern side view: lateral section (on the left) and axial section (on the
right) for: (a) R-OE=∞, (b) R-SF-J , and (c) R-OI-J.
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Fig. 5. Sediment and flow patterns for R-OI-J, S = 0.25, C = 0.574, Fd 90 = 7.16,
λ = 2.8 and E = 1 (test C1-4-1778-20-2.8-1): (a) plan, and (b) side view. Flow from
the left.
geometric configuration, downstream of the scoured region, the flow nets
were no longer influenced by the pool expansion and shockwaves.
In the presence of a downstream contraction, two main hydraulic jump
typologies can be distinguished, namely Repelled Symmetric Free Jump
(R-SF-J) and Repelled Oscillatory Impact Jump (R-OI-J). Type R-SF-J is
characterized by a flow recirculation in the lateral sides of the pool (Fig. 3b).
The hydraulic jump is entirely located in the pool. It mainly occurs for E = 2
and/or both low Fd90 and C. In Fig. 4 longitudinal schematic representations
of flow pattern in both lateral and axial sections are reported for R-OE=∞
(Fig. 4a), R-SF-J (Fig. 4b), and R-OI-J (Fig. 4c).
In presence of a R-SF-J hydraulic jump type, the maximum scour generally develops in the centre of the pool (Fig. 4b), as observed also for E = ∞
(Fig. 4a), and the dynamic recirculation of sediment in correspondence with
the pool transversal walls is negligible. Conversely, the R-OI-J hydraulic
jump type occurs mainly for E = 1 and/or both high Fd 90 and C. It is characterized by a hydraulic jump whose front directly impacts on the downstream
transversal walls and enters in the contraction. Moreover, a strong lateral
recirculation (Figs. 3c and 5a) in correspondence with the downstream contraction occurs, causing a strong dynamic recirculation of bed sediment
(Fig. 5b), which is less evident in case of R-SF-J hydraulic jump type. The
maximum scour depth can occur in different places, but mostly close to the
pool side walls (Fig. 4c).
Final dry scour morphology
Figure 6 reports typical equilibrium scour morphologies in dry conditions for
several tests carried out with a wide range of variables. Scour morphologies
have been generated by means of MATLAB software based on surveyed
data. Contour lines have been plotted as a function of the non-dimensional
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Fig. 6. Examples of non-dimensional equilibrium scour contour lines in dry conditions (contour line step height is equal to 0.25 z/h1, ● stands for maximum scour
depth location).
coordinates x/h1, y/h1 , and z/h1. Accordingly, the scour morphology depends
on several geometric parameters (λ, E, S), hydraulic parameters (h1, h0, Fd 90)
and sediment load, C. In general, for R-SF-J type, the maximum scour has
been observed in the center of the pool, presenting either a single scour lobe
or two symmetric scour lobes (Fig. 6d, e, respectively). It is worth observing
that, due to the lateral flow recirculation, for R-SF-J the maximum scour hole
depth can also be observed downstream of the ramp toe in correspondence
with the pool side walls (Fig. 6b), even if z/h1 at the center of the pool is
slightly lower than zmax/h1. The maximum scour depth for R-OI-J generally
occurs close to the lateral walls either in correspondence with the downstream transversal walls (Fig. 6a) or in the channel contraction itself (Fig. 6c,
f). As it can be inferred from Fig. 6c and d, for constant S, C, λ and Fd 90, but
for different E (and consequently different hydraulic jump type), the parameter zmax/h1 assumes comparable values.
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(a) 0
1.5
0
Gaudio and Marion (2003)
Pagliara et al (2009)
C1-4-267-12-2.8-1
C1-12-2400-27-1.8-2
C1-12-1200-27-1.8-2
C1-12-600-27-1.8-2
C1-4-533-12-2.8-1
C1-4-1067-12-2.8-1
C1-4-1778-20-2.8-1
0.6
3
x /l s
(b) -0.5
0
0
C1-4-267-12-2.8-1
C1-12-1200-27-1.8-2
C1-4-533-12-2.8-1
C1-4-1778-20-2.8-1
0.2
307
0.5
C1-12-2400-27-1.8-2
C1-12-600-27-1.8-2
C1-4-1067-12-2.8-1
C1-12-2400-27-2.8-1
y /B
0.4
0.6
0.8
1
z /z max
1.2
z /z max
1.2
Fig. 7. Non-dimensional equilibrium (a) longitudinal and (b) transversal scour profiles for R-SF-J (filled symbols), and R-OI-J (empty symbols).
Figure 7 shows the non-dimensional longitudinal and transversal equilibrium profiles through the point of maximum scour. They are obtained normalizing the coordinates x and y by ls and B respectively, for either R-SF-J
(filled symbols) or R-OI-J (empty symbols).
Longitudinal scour profiles have been compared with both the average
longitudinal scour profile deduced by Gaudio and Marion (2003), which are
relative to the scour downstream of sills for prismatic channels in absence of
supplied sediment, and the average longitudinal profile deduced by Pagliara
et al. (2009) valid for clear-water conditions. For x/ls < 1 and R-SF-J , the
scour longitudinal profiles are characterized by the same shape of Gaudio
and Marion (2003) as shown in Fig. 7a. In addition, the transversal profiles
present self-similar and symmetric shape (Fig. 7b). Similarly, the upstream
non-dimensional profile (for x/ls < 1) observed by Pagliara et al. (2009) well
agrees with the latter two. Whereas for x/ls > 1, it is completely different, as
it presents always a concave shape. In addition, the non-dimensional profile
is less extended downstream if compared with those observed in live-bed
conditions and by Gaudio and Marion (2003). This occurrence is mainly due
to the larger dune accumulation downstream of the scour hole. Conversely,
in the presence of a R-OI-J jump type, the longitudinal profiles relative to the
live-bed condition case are characterized by a completely different shape,
if the maximum scour depth occurs close to the lateral pool walls. In this
case, the profiles are convex and can be plotted only for x/ls < 1. In addition,
also transversal profiles are characterized by a self-similar convex shape
(Fig. 7b). When the maximum scour depth occurs in the contraction, the longitudinal profile is similar to the cases relative to R-SF-J, as shown in Fig. 7a.
The respective transversal profile assumes a more defined two-dimensional
shape than the previous cases. It is worth noticing the particular case in
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which the maximum scour hole depth is located laterally and just downstream of the ramp toe (empty triangle symbol in Fig. 7b). As for this case
ls = 0, the non-dimensional longitudinal profile cannot be plotted in the proposed normalization.
Maximum scour depth
The qualitative analysis of both the scour morphology and the hydraulic
behavior proposed in the previous section showed that the phenomenon is
complex and it is influenced by several parameters. The quantitative analysis
of the data confirmed that the scour mechanism cannot be easily described
by analytical or experimental relationships by which one can evaluate the
main characteristic lengths. The analysis conducted by Pagliara et al. (2009)
in clear-water condition for an indefinite abrupt expansion (i.e., E = ∞)
showed that the parameter (zmax + h0) /h1 can be expressed as a function of the
densimetric Froude number, Fd 90, block ramp slope, S, and λ = B/b. In the
case in which the pool is both longitudinally and transversally expanded (see
Fig. 1b) and in live-bed conditions, two other parameters have to be taken
into account. Namely, both sediment concentration, C, and E = A/B influence the phenomenon. Thus the functional relationship becomes
( zmax + h0 )
h1 = f (Fd 90 , S , C , λ , E ) .
(1)
The analysis of experimental data was conducted in steps. First, the
dependent variable (zmax + h0) /h1 was plotted versus Fd 90 for selected values
of λ and S varying the parameters E and C. This preliminary analysis allowed
to state that, adopting this parameterization, the experimental data do not
exhibit a clear and unique trend. Figure 8a, b shows two graphs in which
(zmax + h0) /h1 is plotted versus Fd 90 for λ = 2.8 and S = 0.083 (see Fig. 8a)
and for λ = 2.8 and S = 0.25 (see Fig. 8b), and for all the tested values of E
and C. It can be easily observed that the trend of data is not detectable. The
influence of the several parameters is extremely variable and it depends on
both hydraulic and geometric conditions. This is mainly due to the fact that
the presence of a downstream contraction influences the flow pattern which
contributes to modify the scour morphology, especially in those cases in
which the jump is not confined in the pool. Based on the previous observations, a new parameterization was introduced. Namely, the parameter
ln[(zmax + h0) /h1] /C was plotted versus Fd 90, for all the ramp slopes and all λ,
E and C values tested. Adopting this parameterization the data exhibits
a clear trend and can be easily estimated by a unique and simple analytical
expression. Moreover, using this parameterization, the dependence of the
variable ln[(zmax + h0) /h1] /C on the parameters E and λ can be neglected as
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6.0
6.0
(z max +h 0)/h 1
λ =2.8, S =0.083
E =1, C=0.144
E =1, C=0.287
E =1, C=0.574
5.0
(z max +h 0)/h 1
E=2, C=0.144
E=2, C=0.287
E=2, C=0.574
5.0
4.0
4.0
3.0
3.0
E =1,C =0.144
E =1,C =0.287
E =1,C =0.574
Fd 90
(a)
2.0
6.0
8.0
15.0
Fd 90
4.0
5.0
6.0
7.0
8.0
7.0
ln[(z max +h 0)/h 1]/C
[(z max +h 0)/h 1]meas
C
C
C
C
C
C
C
C
10.0
=0.144, S =0.083
=0.287, S =0.083
=0.574, S =0.083
=0.144, S =0.25
=0.287, S =0.25
=0.574, S =0.25
=0.144, S =0.125
=0.574, S =0.125
Eq. (2) for S =0.125
Eq. (2) for S =0.25
Eq. (2) for S =0.083
3.5
S =0.083
S =0.25
S =0.125
perfect agreement
20% deviation
5.0
0.0
E =2,C =0.144
E =2,C =0.287
E =2,C =0.574
(b)
2.0
4.0
λ =2.8, S =0.25
4.0
(d)
Fd 90
(c)
[(z max +h 0)/h 1]calc
0.0
6.0
8.0
10.0
0.0
3.5
7.0
Fig. 8. Normalized maximum scour depth (zmax + h0) /h1 versus Fd 90 for λ = 2.8 and
all tested C and E values in the case in which ramp slope, S, is (a) 0.083 and
(b) 0.25; (c) Plot of ln[(zmax + h0) /h1] /C versus Fd 90 for all λ, E, S and C tested;
(d) Comparison between measured and calculated (with eq. 2) values of the variable
(zmax + h0) /h1 .
it does not seem to substantially influence the dependent variable. Figure 8c
shows all the experimental data for different slopes and different C values in
a graph ln[(zmax + h0) /h1] /C versus Fd 90. The data appear to be well grouped
and the influence of both ramp slope, S, and concentration, C, is clearly
evident. The dependent variable results to be a decreasing function of C and
an increasing function of the slope S. The experimental data were interpolated by linear functions and one main simple governing equation was found
which, considering the complexity of the phenomenon, well estimated the
dependent variable, namely
ln ⎡⎣( zmax + h0 ) h1 ⎤⎦
= ⎡⎣( −3C 2 + 2.7C − 0.45 ) Fd 90 + ( 63.6C 2 − 61.5C + 15 ) ⎤⎦
C
× ( −33.6S 2 + 15.6 S − 0.03) .
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(2)
310
S. PAGLIARA et al.
From eq. (2) one can easily derive the estimation of the parameter
(zmax + h0) /h1 knowing the geometric characteristic of the ramp slope, S, the
hydraulic conditions, Fd 90 , and the sediment load concentration, C. Note that
eq. (2) is valid in the tested ranges of parameters specified in Table 2. The
comparison between measured and calculated values of the variable
(zmax + h0) /h1 is shown in Fig. 8d. It can be noted that the proposed relationship furnishes a good estimation of the experimental data, and it has the
advantage to be easily applicable.
Maximum scour hole position
Another important parameter is the longitudinal distance of the transversal
section in which the maximum scour depth occurs. The longitudinal location
of this transversal section can be useful in practical applications, as it consents to establish if the maximum scour depth will happen in the pool or in
the downstream contraction. It was experimentally proved that the nondimensional longitudinal distance of the maximum scour hole depth from the
ramp toe, Ls = ls/h1 , can be satisfactorily expressed as a function of the
maximum scour depth. This experimental evidence is also confirmed by
Breusers and Raudkivi (1991) in the case of clear-water condition. For all
the ramp slopes and all the hydraulic and geometric conditions tested, the
experimental data exhibit a clear increasing trend as shown in the Fig. 9a.
For all the tested λ and E values, ln(Ls) was plotted versus the nondimensional maximum scour hole depth, Zmax = zmax /h1 , and the following
general relationship valid in the tested range of parameters was deduced. It is
worth noting that Ls can be expressed just in function of Zmax , thus neglecting all the other geometric and hydraulic parameters, as Zmax already depends
on h0, Fd 90, E, λ, S, and C.
20.0
6.0
ln(L s )
(L s )meas
S =0.083
S =0.25
S =0.125
Eq. (3)
10.0
3.0
data
perfect agreement
30% deviation
(b)
Z max
(a)
(L s )calc
0.0
0.0
0.0
2.5
5.0
0.0
10.0
20.0
Fig. 9: (a) Plot of ln(Ls) versus Zmax for all tested conditions; (b) Comparison
between measured and calculated (with eq. 3) values of the variable Ls.
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EXPANDING POOLS MORPHOLOGY IN LIVE-BED CONDITIONS
311
ln ( LS ) = 0.286 Z max + 1.45 .
(3)
In Figure 9b, the comparison between measured and calculated values of
the variable Ls is reported and it can be seen that eq. (3) satisfactorily predicts Ls, considering the complexity of the phenomenon. Moreover, it has the
advantage to be analytically simple and extremely easily applicable.
Maximum surface water elevation
It is also useful to know what will be the maximum water elevation which
can occur in both pool and contracted channel. According to different
hydraulic and geometric conditions, the maximum water elevation can
occur either in the pool or in the contracted channel. Thus, two different nondimensional parameters were estimated, namely HB = hB/h1 and HR = hR/h1,
in which hB is maximum water elevation in the pool and hR is maximum
water elevation in the contraction (Fig. 10).
In general, when the jump is confined in the pool, the maximum water
elevation occurs inside the pool itself. Moreover, it is generally localized in
correspondence with the downstream transversal walls which delimitate the
pool itself. Conversely, when the front of the hydraulic jump hits the downstream transversal walls, the maximum water elevation mainly occurs in the
contraction. For each slope and for all the tested λ and E values, the parameters HB and HR were plotted versus Fd 90. In Figure 11a, c, and e, a graph
HB(Fd 90) is reported for S = 0.083, 0.25 and 0.125, respectively, and all λ and
E tested. It can be seen that the effect of λ is not detectable whereas the experimental data relative to E = 2 have always higher HB values than those
relative to E = 1. This is mainly due to the fact that the distance of the downstream contraction strongly influences the water elevation in the pool, especially in the case in which E = 2, as for the case in which E = 1 the front of
the hydraulic jump generally hits the downstream transversal walls and eventually enters the contraction. Thus two limiting lines were distinguished for
each tested slope. Namely, the limiting lines were established in such a way
Fig. 10. Diagram sketch with the indication of the water elevations hB and hR .
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S. PAGLIARA et al.
4.0
4.0
HB
HR
S =0.083, λ =1.8, E =1
S =0.083, λ =1.8, E =2
S =0.083, λ =2.8, E =1
S =0.083, λ =2.8, E =2
limiting line for E =2
limiting line for E =1
S =0.083, λ =1.8, E =1
S =0.083, λ =1.8, E =2
S =0.083, λ =2.8, E =1
S =0.083, λ =2.8, E =2
limiting line
2.0
2.0
Fd 90
0.0
Fd 90
(a)
4.0
0.0
6.0
4.0
HB
8.0
(b)
4.0
4.0
S =0.25, λ =2.8, E =1
S =0.25, λ =2.8, E =2
S =0.25, λ =1.8, E =1
S =0.25, λ =1.8, E =2
limiting line for E =2
limiting line for E =1
HR
6.0
S =0.25, λ =2.8, E =1
S =0.25, λ =2.8, E =2
S =0.25, λ =1.8, E =1
S =0.25, λ =1.8, E =2
limiting line
2.0
2.0
Fd 90
0.0
Fd 90
(c)
4.0
0.0
6.0
8.0
(d)
4.0
6.0
8.0
4.0
4.0
HB
HR
S =0.125, λ =1.8, E =1
S =0.125, λ =1.8, E =2
S =0.125, λ =2.8, E =1
limiting line for E =2
limiting line for E =1
S =0.125, λ =1.8, E =1
S =0.125, λ =1.8, E =2
S =0.125, λ =2.8, E =1
limiting line
2.0
2.0
Fd 90
0.0
8.0
Fd 90
(e)
4.0
0.0
6.0
8.0
(f)
4.0
6.0
8.0
Fig. 11. Non-dimensional maximum elevation in the pool HB versus Fd 90 for S equal
to: (a) 0.083, (c) 0.25, (e) 0.125; and non-dimensional maximum elevation in the
contraction HR versus Fd 90 for S equal to: (b) 0.083, (d) 0.25, and (f) 0.125.
that all the experimental data relative to E = 1 and E = 2 are below the
respective line. Likewise the following simple linear equation is also proposed
in order to evaluate the limiting HB values in the tested range of parameters
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EXPANDING POOLS MORPHOLOGY IN LIVE-BED CONDITIONS
313
H B = α ( E , S ) Fd 90 + β ( E , S ) ,
(4)
where the coefficients α(E, S) and β(E, S) are functions of parameters E
and S and their expressions are reported in Table 3.
The same analysis was conducted for HR. Also in this case, the experimental data were distinguished for different ramp slopes, E and λ values.
As shown in Fig. 11b, d, and f, the effect of both parameters, E and λ, cannot
be distinguished, thus one unique limiting line for each slope was plotted. As
for HB, for HR the limiting lines were established in such a way that all the
experimental data relative to all tested conditions are below the respective
line. The following equation can be used to estimate the limiting values of
HR in the tested range of parameters
H R = α1 ( S ) Fd 90 + β1 ( S ) ,
(5)
where the coefficients α1(S) and β1(S) are functions of parameter S and their
expressions are reported in Table 3.
T ab l e 3
Coefficients of eqs. (4) and (5)
5.
Variable
S
α
HB
0.083
0.125
0.25
Variable
S
α1
β1
HR
0.083
0.125
0.25
–0.175
–0.21
–0.49
2.76
3.4
5.82
0.06E-0.24
0.22E-0.71
0.34E-1.1
β
2.52-0.06E
5.48-0.94E
8.89-1.79E
CONCLUSIONS
This research analyzes the sediment load effects in live-bed conditions on
the scour features and on the flow patterns downstream of a block ramp in
the presence of an expanding pool. This configuration can be usually found
also in natural rivers in which a sloped rough bed can be followed by an
enlarged pool. Three ramp slopes, two pool longitudinal and transversal
expanding ratios and wide ranges of both sediment concentrations and densimetric Froude numbers were tested.
Experimental tests showed that both the scour morphology and the flow
patterns mainly depend on geometric and hydraulic parameters. Two main
hydraulic jump types occurred in the tested range of parameters, namely
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S. PAGLIARA et al.
Repelled Symmetric Free Jump (R-SF-J) and Repelled Oscillatory Impact
Jump (R-OI-J). The first typology is characterized by a flow recirculation in
the lateral sides of the pool and the hydraulic jump is entirely located in it.
Conversely, the R-OI-J hydraulic jump type is characterized by a hydraulic
jump whose front directly impacts on the downstream transversal walls and
enters in the contraction.
The effect of sediment load concentration on the scour mechanism is
prominent. It was experimentally proved that scour depth is reduced if the
sediment load concentration increases and it influences the longitudinal position of the section of maximum scour. Simple equations were derived in
order to estimate the main scour lengths.
In addition, an analysis of the water elevations both in the pool and in the
downstream contraction was conducted. In this case as well, some simple
analytical relationships were deducted in order to evaluate the maximum
water elevations in the tested range of parameters.
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Received 15 September 2010
Received in revised form 16 October 2010
Accepted 29 October 2010
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