expanding pools morphology in live-bed conditions

Acta Geophysica vol. 59, no. 2, Apr. 2011, pp. 296-316

DOI: 10.2478/s11600-010-0048-z

________________________________________________ © 2010 Institute of Geophysics, Polish Academy of Sciences

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]

A b s t r a c t

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 funda-mental 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 down-stream 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.

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1. 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 ba-sin 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 chan-nels in northwestern California after the 1964 flood proved that both bar pro-files 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 confi-guration 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 com-promise 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 sedi-ments 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 ele-vation 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. 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 influ-enced 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 chan-nel 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. Expe-riments 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 asym-metric flows was developed. No studies are currently available in which the hydraulic jump occurs in a symmetrically expanding pools for mobile chan-nel 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 clas-


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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 equili-brium was achieved, i.e., when sediment is continuously supplied and re-moved 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 em-pirical 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 be-tween 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 ana-lyzed 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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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 con-traction (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 – longitu-dinal 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, respec-tively, 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 den-sity, 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

Table 1 Ramp and pool materials characteristics

Materials d50, D50 [mm]

d90, D90 [mm]

σ

Δ

m1 3.39 4.00 1.19 1.58 r1 24.84 28.38 1.10 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 sedi-ment outlet loads are shown in Fig. 2. The figure shows three typical tempo-ral 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), live-bed 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 tem-poral 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 inter-mediate 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 con-dition 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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Table 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

165.0290.0

min max

C1-8 1 2

1.8 2.8 0.125 0.143

0.574 5.65 6.54

1.53 1.72

175.0267.0

min max

C1-4 1 2

1.8 2.8 0.25 0.143

0.574 6.63 7.16

1.65 2.06

184.0263.0

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 longi-tudinal 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 indefi-nite 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 condi-tion for E = ∞ in order to understand the hydraulic qualitative behavior. It was observed that, in the tested range of parameters and live-bed condi-tions, only one hydraulic jump typology takes place, namely the repelled os-cillatory 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 gener-ally 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 charac-terized 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 con-traction 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 condi-tions (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 down-stream 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 parame-ter zmax/h1 assumes comparable values.


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0

0.6

1.2

0 1.5 3

x /l s

z /z max

Gaudio and Marion (2003)Pagliara et al (2009)C1-4-267-12-2.8-1C1-12-2400-27-1.8-2C1-12-1200-27-1.8-2C1-12-600-27-1.8-2C1-4-533-12-2.8-1C1-4-1067-12-2.8-1C1-4-1778-20-2.8-1

(a)

0

0.2

0.4

0.6

0.8

1

1.2

-0.5 0 0.5

y /B

z /z max

C1-4-267-12-2.8-1 C1-12-2400-27-1.8-2C1-12-1200-27-1.8-2 C1-12-600-27-1.8-2C1-4-533-12-2.8-1 C1-4-1067-12-2.8-1C1-4-1778-20-2.8-1 C1-12-2400-27-2.8-1

(b)

Fig. 7. Non-dimensional equilibrium (a) longitudinal and (b) transversal scour pro-files for R-SF-J (filled symbols), and R-OI-J (empty symbols).

Figure 7 shows the non-dimensional longitudinal and transversal equili-brium profiles through the point of maximum scour. They are obtained nor-malizing 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 lon-gitudinal 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 down-stream 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 pro-posed 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 influ-ence the phenomenon. Thus the functional relationship becomes

( ) ( )max 0 1 90 , , , , .dz h h f S C Eλ+ = F (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 observa-tions, 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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λ=2.8, S =0.083

2.0

3.0

4.0

5.0

6.0

4.0 6.0 8.0

Fd 90

(z max +h 0)/h 1

=1, =0.144 =2, =0.144 =1, =0.287 =2, =0.287 =1, =0.574 =2, =0.574

(a)

EEE

EEE

CCC

CCC

λ=2.8, S =0.25

2.0

3.0

4.0

5.0

6.0

4.0 5.0 6.0 7.0 8.0

Fd 90

(z max +h 0)/h 1

=1, =0.144 =2, =0.144 =1, =0.287 =2, =0.287 =1, =0.574 =2, =0.574

(b)

EEE

EEE

CCC

CCC

0.0

5.0

10.0

15.0

4.0 6.0 8.0 10.0

Fd 90

=0.144, =0.083 =0.287, =0.083 =0.574, =0.083 =0.144, =0.25 =0.287, =0.25 =0.574, =0.25 =0.144, =0.125 =0.574, =0.125Eq. (2) for =0.125Eq. (2) for =0.25Eq. (2) for =0.083

(c)

ln[(z max +h 0)/h 1]/C

CCCCCCCC

SSSSSSSS

SSS

0.0

3.5

7.0

0.0 3.5 7.0

=0.083 =0.25 =0.125perfect agreement20% deviation

[(z max +h 0)/h 1]meas

[(z max +h 0)/h 1]calc

(d)

SSS

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 interpo-lated by linear functions and one main simple governing equation was found which, considering the complexity of the phenomenon, well estimated the dependent variable, namely

( ) ( ) ( )

( )

max 0 1 2 290

2

ln3 2.7 0.45 F 63.6 61.5 15

33.6 15.6 0.03 .

d

z h hC C C C

CS S

+⎡ ⎤⎣ ⎦ ⎡ ⎤= − + − + − +⎣ ⎦

× − + − (2)

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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 relation-ship 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 con-sents to establish if the maximum scour depth will happen in the pool or in the downstream contraction. It was experimentally proved that the non-dimensional 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 non-dimensional 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 neglect-ing all the other geometric and hydraulic parameters, as Zmax already depends on h0, Fd 90, E, λ, S, and C.

0.0

3.0

6.0

0.0 2.5 5.0

Z max

ln(L s ) =0.083 =0.25 =0.125Eq. (3)

(a)0.0

10.0

20.0

0.0 10.0 20.0

(L s)calc

(L s )meas

dataperfect agreement30% deviation

(b)

SSS

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.


311

( ) maxln 0.286 1.45 .SL Z= + (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 pre-dicts 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 non-dimensional 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 down-stream transversal walls, the maximum water elevation mainly occurs in the contraction. For each slope and for all the tested λ and E values, the parame-ters 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 expe-rimental 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 down-stream contraction strongly influences the water elevation in the pool, espe-cially 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 even-tually 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 .

S. PAGLIARA et al.

312

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H B

=0.083, =1.8, =1 =0.083, =1.8, =2 =0.083, =2.8, =1 =0.083, =2.8, =2limiting line for =2limiting line for =1

(a)(a)

λλλλ

EEEE

EE

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H R

=0.083, =1.8, =1 =0.083, =1.8, =2 =0.083, =2.8, =1 =0.083, =2.8, =2limiting line

(b)(b)

SSSS

λλλλ

EEEE

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H B =0.25, =2.8, =1 =0.25, =2.8, =2 =0.25, =1.8, =1 =0.25, =1.8, =2limiting line for =2limiting line for =1

(c)(c)

SSSS

λλλλ

EEEE

EE

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H R =0.25, =2.8, =1 =0.25, =2.8, =2 =0.25, =1.8, =1 =0.25, =1.8, =2limiting line

(d)(d)

SSSS

λλλλ

EEEE

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H B =0.125, =1.8, =1 =0.125, =1.8, =2 =0.125, =2.8, =1limiting line for =2limiting line for =1

(e)(e)

SSS

λλλ

EEE

EE

0.0

2.0

4.0

4.0 6.0 8.0

Fd 90

H R

=0.125, =1.8, =1 =0.125, =1.8, =2 =0.125, =2.8, =1limiting line

(f)(f)

SSS

λλλ

EEE

SSSS

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


313

( ) ( )90, , ,B dH E S E Sα β= +F (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 experi-mental 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

( ) ( )1 90 1 ,R dH S Sα β= +F (5)

where the coefficients α1(S) and β1(S) are functions of parameter S and their expressions are reported in Table 3.

Table 3 Coefficients of eqs. (4) and (5)

Variable S α β

HB 0.083 0.125 0.25

0.06E-0.24 0.22E-0.71 0.34E-1.1

2.52-0.06E 5.48-0.94E 8.89-1.79E

Variable S α1 β1

HR 0.083 0.125 0.25

–0.175 –0.21 –0.49

2.76 3.4 5.82

5. 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 den-simetric 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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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 posi-tion 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

expanding pools morphology in live-bed conditions

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