Scour holes downstream of bed sills in low-gradient channels
Affouillement local à l’aval des seuils dans des canaux à pente faibleM. BEN MEFTAH, PhD Student,DIAC, Technical University of Bari, Via E. Orabona, 4—70125 Bari, Italy. Tel.: +39 080 5963288;fax: +39 080 5963414; e-mail: [email protected]
M. MOSSA, PhD, Professor, IAHR Member,A.M. ASCE, DIASS, Technical University of Bari, Via E. Orabona, 4—70125 Bari, Italy.Tel.: +39 080 5963289; fax: +39 5963414; e-mail: [email protected] (author for correspondence)
ABSTRACTAn experimental study on long local scouring downstream of bed sills in a monogranular sand bed was carried out in the hydraulic laboratory flume atthe Mediterranean Agronomic Institute of Bari (Italy). The main objectives of this study were to determine scour hole dimension, with its maximumscour depth as a function of time and at the equilibrium stage, the scour hole shapes and the investigation of the influence of sills on the distributionof the three-velocity components through the scour hole at the same stage. Four experimental configurations were tested, the main difference betweenthem being the distance between sills. Based on experimental data, the classical dimensional analysis of the variables that influence the developmentof the scour hole has been carried out in order to obtain two empirical formulas predicting the maximum scour depth and the length of the scour holeat the equilibrium stage. Moreover, it was observed that the distance between sills influences the scour hole dimension and shape. The three-velocitycomponents of the flow, measured with an acoustic Doppler velocimeter, show that in the scour hole, at the equilibrium stage, the three componentsof the flow turbulence intensities are very high. Near-bed flow vortexes in addition to secondary currents are also observed.
RÉSUMÉUne étude expérimentale sur l’affouillement local à long terme à l’aval des seuils le long d’un lit à sable mono-granulaire a été élaborée dans uncanal d’essai au laboratoire d’hydraulique de l’Institut Agronomique Méditerranéen de Bari (Italie). Les objectifs principaux de cette étude sont; ladétermination des dimensions de l’affouillement, à savoir, l’évolution de sa profondeur maximale en fonction du temps et à la phase d’équilibre, sa formeet l’analyse de l’influence du seuil sur la distribution des trois composantes de la vitesse d’écoulement lors de la même phase. Quatre configurationsexpérimentales ont été réalisées, la différence entre ces dernières étant l’écartement entre les seuils. Sur la base des résultats expérimentaux, l’analysedimensionnelle classique des variables influençant le développement de l’affouillement a induit à l’obtention de deux formules empiriques permettantla détermination de la profondeur maximale et de la longueur d’affouillement à l’équilibre. En outre, il a été observé que l’écartement entre les seuilsprésente une influence aussi bien sur les dimensions que sur la forme de l’affouillement. Les mesures de la vitesse d’écoulement tridimensionnelle parle biais d’un Acoustic Doppler Velocimeter (ADV) a montré que dans l’affouillement, durant la phase d’équilibre, les trois composantes de l’intensitéde la turbulence de l’écoulement sont très élevées. Une circulation de fluide en vortex à proximité du lit ainsi qu’un développement d’écoulementsecondaire ont été observés.
Keywords: Bed sills, local scouring, time evolution, scour shapes, secondary currents, turbulence.
1 Introduction
The bed is generally protected against scouring in direct prox-imity to hydraulic structures. The length of the bed protectiondepends on the tolerable scour (maximum scour depth andupstream scour slope). When the bed length protection isincreased, the scour process is less intense due to the decayof turbulence energy and the adaptation of the velocity pro-file downstream of the hydraulic structure. For a designer, themost important scour parameter is the maximum scour depthin the equilibrium phase (defined as the time when the scourhole and the bed profile spatial characteristics along the flumedo not change any further under steady flow and sediment inputconditions). The extent of the scour hole is strongly dependent
Revision received August 30, 2005/Open for discussion until August 31, 2007.
497
on time. Initially, the scour development over time is rapid, thenit decreases gradually to reach the equilibrium stage after a longperiod. The magnitude of the maximum scour depth dependson the bed-shear stress, the turbulence condition near the bed,and the sediment characteristic (density of the bed material,sediment-size distribution, porosity, cohesive or non-cohesivebed material, etc.).
In gravel bed rivers, bed sills are used to limit bed degrada-tion and to control erosion in the proximity of bridge piers orin downstream channels of dam stilling basins. There is exten-sive literature on scour (Bormann and Julien, 1991; Chatterjeeet al., 1994; Habibet al., 1994; Hoffmans and Pilarczyk, 1995;Hoffmans and Verheij, 1997; Hoffmans, 1998; Gaudioet al.,2000; Gaudio and Marion, 2003; Marionet al., 2004). Lenzi
498 Meftah and Mossa
et al. (2002) investigated the main characteristics of local scour-ing downstream of bed sills, forming a staircase-like system inhigh-gradient streams with non-uniform alluvium. Lenziet al.(2003a) surveyed scour holes below 73 grade-control structures(check dams and bed sills) in six mountain rivers located in theeastern Italian Alps. Lenziet al. (2003b) analyzed the localscouring downstream of bed sills forming a sequence for bedstabilization in steep channels, confirming self-affinity of scourholes. D’Agostino and Ferro (2004) described an approach forpredicting local scour downstream of grade-control structures.
In addition, the bed sill has effects on the flow velocity, and theimpacts from the sill can vary spatially. Furthermore, the pres-ence of the bed sill causes a redistribution of the flow velocity. Itcan decrease or increase the flow turbulence, create vortex phe-nomenon and/or secondary current velocity. Nezu and Nakagawa(1993) observed that many literature data of turbulence in rivershave been obtained in well-controlled laboratory flumes. Almostall of the numerical calculations of open-channel turbulence havealso been conducted for simple boundaries such as uniform orgradually-varied flows. Therefore, in literature there is lack ofdata on flow turbulence in scour holes. Yokoshi (1967) carriedout turbulence measurements in a river by use of a propeller-typecurrent meter, but only the large-scale turbulence in the rivercould be measured. Also significant is McQuivey’s (1973) paper,who conducted hot-film measurements in rivers and conveyancechannels. Nezu and Nakagawa (1993) did a brief review on fieldmeasurements of river turbulence.
This study is concerned with an experimental investigation ofthe scour hole phenomenon due to a current flowing over sillsin an erodible bed of sand particles without upstream sedimentsupply. Scour hole variation with time, the water and bed profilesat the equilibrium phase, and three-flow-velocity components atdifferent sections along the channel were assessed for each con-figuration. Two different scour hole topologies were observed,depending on the distance between sills.
2 Theoretical analysis
Local scour hole is typical of river beds, downstream of the bedsills. After a long period of time, the scour hole reaches an equi-librium state. At equilibrium, the geometry of the system underconsideration is shown in Fig. 1 (Gaudioet al., 2000). Based onthis geometric shape of the bed profile, the maximum scour depthym, or the scour lengthls, can be expressed as:
ym or ls = f(g, ν, ρw, ρs, q, hu, d50, L, a) (1)
whereg = gravity acceleration,ν = kinematic water viscosity,ρw = water density,ρs = sediment density,q = discharge perunit width,hu = measured flow depth over the sill,d50 = grainsize for which 50% of the total weight of the sediment is finer,L = distance between sills, anda = difference between the levelof two successive sills, which is calculated as:
a = L · S0 (2)
whereS0 is the initial bed slope.
ym
L
S0
Seq
a
ls
Figure 1 Sketch of the scour hole.
The application of the�-theorem to Eq. (1) leads to:
ym
huor
ls
hu= f
(q2
gh3u
,q
ν,�d50
hu,
L
hu,�a
hu
)(3)
where:
� = ρs − ρw
ρw. (4)
Equation (3) assumes implicitly thatym andls are not correlated.Even if this assumption is not true, the present study highlightsthe variation ofym andls with the dimensionless parameters ofEq. (3). Theq2/(gh3
u) ratio is the square of the Froude num-ber, and theq/ν ratio is the Reynolds number. Generally, inopen channels, the flow is almost always fully turbulent. Thus,the dependence upon the Reynolds number could be neglected.Since the parametera is determined as the product of the distancebetween sillsL and the initial bed slopeS0, the ratioL/hu canbe removed from Eq. (3). Furthermore, the effect of the channelwidth is here implicitly neglected by choosing variables per unitwidth. In addition, in the present study the same sand grains havebeen used, with a constant value of the relative submerged par-ticle density,� andd50. Because of this constancy, the effect of� andd50 is not shown in the present study. Equation (3) is thefunction of the total drop between two sills rather than the actualenergy drop. This dimensional analysis is preferred in order toenable the possible user to use the formula in easier way becausethe total drop between two sills is known, being a constructiveparameter. Another possible approach is that proposed by Gaudioet al. (2000).
According to these simplifying assumptions, Eq. (3) can bereduced to:
ym
huor
ls
hu= f
(q2
gh3u
,�LS0
hu
)(5)
The determination of Eq. (5) is one of the main objectives of thepresent paper.
Scour holes downstream of bed sills in low-gradient channels 499
3 Experimental set-up
The experimental work was carried out in a horizontal flumein the laboratory of the Mediterranean Agronomic Institute ofBari (Italy). This flume is 7.72 m long, 0.30 m wide, and has adepth of 0.40 m. The flume’s floor is constructed with Plexiglasand the lateral walls are made of glass, which allows better sideviewing of the flow. Water is fed in from an upstream reservoirwith a maximum water level of 54 cm equipped with stilling gridand lateral weir, which maintains a constant head upstream ofa movable gate constructed at the upstream end of the flume.This gate is made of Plexiglas and allows the passage of differentdischarges with different corresponding channel flow depths. Theflume is supplied with a pump giving a maximum discharge of24 l/s through a steel pipe. To create a smooth flow transitionfrom the upstream reservoir to the flume, a wooden ramp wasplaced at the inlet of the flume; the wooden ramp is 1.55 m long,0.15 m thick and has the same width of the channel cross-section(Fig. 2). Only a 0.5-m long end of the aforementioned woodenplate is positioned downstream of the channel’s upstream gate.
At the downstream end of the flume, water is intercepted bya stilling reservoir, equipped with three vertical grids to stabi-lize water, and a triangular weir (V-notch sharp crested weir) tomeasure the flow discharge.
In order to protect the flume’s sand bed against erosion, con-trol sills were placed along the channel. Four sets of tests wereperformed during the experimental work, the difference betweenthem being the distance (L) between sills (set 1 withL = 1 m,set 2 withL = 2 m, set 3 withL = 4 m, and set 4 withL = 3 m).The sill level decreases progressively going from the upstreamsection to the downstream section of the channel, respecting aconstant initial slopeS0 fixed at the value of 0.0086. The sillsused in the experiments consisted of PVC plates 0.30 m wideand 0.01 m thick (Figs 1 and 2).
The flume bottom is covered with an erodible bed materiallayer consisting of sand particles with mean average size (d50)of 1.8 mm and density of 2650 kg/m3. The grain-size distributioncurve of the sand is illustrated in Fig. 3. The uniformity coefficientCu = d60/d10 was equal to 1.6, whered60 = grain size forwhich 60% of the total weight of the sediment is finer andd10 =
Figure 2 Laboratory flume.
Figure 3 Grain-size distribution curve.
grain size for which 10% of the total weight of the sedimentis finer. Thus, the sediment can be considered as well sorted.Along the channel, the sand layer decreases progressively fromthe upstream sections to the downstream sections, respecting theinitial slope predetermined by the sills.
During the experiments, the bed profile along the channel wasmarked on the flume’s glass side-walls by means of differentcolors at various times. At the equilibrium stage, the water levelprofile along the centerline was measured using an electricalhydrometer with an accuracy of 1/10 of millimeter. At the samestage, the bed profiles along the centerline and near the two side-walls of the channel were determined using a point gage with anaccuracy of 1/10 of millimeter.
In order to study the effects of the sill upon the flow velocityat equilibrium, the measurements of the three-velocity compo-nents along the flume at different positions have been recordedduring this experimental work, using a Nortek acoustic Dopplervelocimeter (ADV). The ADV was used with a velocity rangeequal to±0.30 m/s, a velocity accuracy of±1%, a samplingrate of 25 Hz, a random noise approximately equal to 1% of thevelocity range, and a sampling volume<0.25 cm3. Hydrody-namic measurements through the scour hole were made for themajority of tests. Measurements were taken when the scour holereached its equilibrium stage. Each vector velocity of the presentstudy is the average of a minimum acquisition of 7000 instan-taneous flow velocity values. The measurements were taken inboth longitudinal and transversal planes.
The main parameters of the tests are illustrated in Table 1,whereQ = flow discharge through the channel,td = total dura-tion of each test, when the equilibrium stage was reached, andhc = critical flow depth. The flow depth over a sill was measuredas the vertical distance between the top of the sill and the waterprofile. The maximum scour depth was determined as the verticaldistance between the initial bed profile and the center of the scourhole at the equilibrium stage. The scour length was determinedin a cell, being a cell the space between two consecutive sills, asthe distance between the upstream sill of the cell and the pointdownstream of the scour hole where the slope reaches a constantvalue (Fig. 1). For tests T04 to T08, the scour hole occupied all ofthe space between the sills and, therefore,ls was not determined.
500 Meftah and Mossa
Table 1 Main parameters of each test
No. Test Q (m3/s) q (m2/s) L (m) td (h) ym (m) ls (m) hu (m) hc (m)
It is important to highlight that the variables of Table 1 refer onlyto the cell whose upstream sill is positioned at 2 m downstreamof the wooden plate’s end, i.e. the third cell for set 1, and thesecond cell for sets 2, 3 and 4. The reason is due to the fact thatat only about 2 m downstream from the wooden plate’s end, thecells show a behavior similar to that of a sequence of infinitecells. This reason will be better clarified in Section 4.1.
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Longitudinal distance [m]
0
5
10
15
20
25
Dis
tanc
e fr
om th
e ch
anne
l flo
or [c
m]
23:30 h
16:30 h
4:30 h
1:30 h
0:30 h
Figure 4 An example of the time evolution of the scour hole profiles (left side) with the water surface profiles at the scouring equilibrium stage(test T10).
4 Scour results
4.1 Scour hole evolution
Examining all the tests, it has been shown that the extent of thescour hole is strongly dependent on time. It was observed thatthere are three stages of local scour hole development (Fig. 4
Scour holes downstream of bed sills in low-gradient channels 501
shows an example). Initially, the scour hole development overtime is rapid, and this is due to the high rate of solid transportachieved on the downstream end of each sill. The high rate ofthe solid transport is a consequence of the high forces of thebed-shear stresses exerted over the sand bed at the initial time.To study the different stages of development of the scour holeas a function of the time for each test, and their similarity, thevalues ofyt/ym have been plotted againstt/td for tests T04 toT17 (whereyt is the scour hole depth at timet, andtd, as written,is the duration of each experiment, i.e., the time when the scourhole reached the equilibrium phase).
It is important to take into account that the first cell of thechannel (i.e., the cell downstream of the wooden bed sill) ischaracterized by clear water, while at the beginning of each runall the other cells are characterized by an inflow and outflow ofbed particles with a net sediment erosion. Looking at the tem-poral development, scour holes at the upstream flume end aredeeper than those downstream, but at the equilibrium clear waterflows on all the cells. This means that scour should be virtuallyidentical. But it does not occur because the scour development ofthe first upstream cell was different during the first time period,because of the clear water, and scour also remains different whenclear water is present in the whole channel (i.e., the major effectstaking place at the beginning of the tests are not compensatedsuccessively). This means that the first cells closer to the channelupstream gate will be characterized by a greaterym than those ofthe other downstream cells. Another reason for the different scourhole depths of the first cells are due to the flow inlet effects. Thistypical phenomenon is reported in Fig. 5, which shows an exam-ple of the maximum scour depths in each cell for test T05, i.e.one of those characterized by a greater number of cells. Figure 5highlights that the maximum scour depths are always greater inthe first two cells and roughly equal in the other ones. The figureshows that at each time measurements of the maximum scourdepths are greater in the first cell of the channel. This is the rea-son why afterwards in the present paper the values ofym andls
are referred to at the cell positioned 2 m downstream of the endof the wooden plate.
Figure 6 shows that the scour development as a function oftime presents three different phases. As previously written, the
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6
Cell number
Max
imu
m s
cou
r d
epth
[m
m] At t = 20 min
At t = 50 min
At t = 110 min
At t = 240 min
Figure 5 Example of the maximum scour depth in each cell for test T05.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
t/td
yt/y
m
T04
T05
T06
T07
T08
T09
T10
T11
T12
T13
T14
T15
T16
T17
Figure 6 Similarity of scour holes evolution.
quantities refer only to the cell at 2 m downstream of the woodenplate’s end. Figure 6 shows, as well, that at the initial stage thescour hole depth for each test reaches about 65% of the maximumscour depth during a time<10% of the duration after which theequilibrium stage was reached. A second stage is characterizedby an increased rate of scour development much slower than thefirst stage. Figure 6 also shows that during this stage, the depthof the scour hole increases globally with a percentage of 90% ofthe maximum scour hole through a time estimated of 30% of theduration after which the equilibrium stage was reached. As wellknown, the increased degree ofyt/ym depends on the magnitudeof the bed-shear stress. Therefore, it can be said that the shearstress acting over the bed reduces with the increase of the scourhole depth. A final slow stage is that in which the scour achievesequilibrium after a long period of time. The equilibrium phase isassumed to be reached when no transport of sediment particlesis observed along the channel. Figure 6 indicates that the beddeformation or the scour hole evolution is extremely slow duringthis stage. It can be observed that during a time period longerthan 60% of the duration after which the equilibrium stage wasreached, the scour depth increased only with a value around 10%of the maximum scour depth. Examining all the experimentalruns, the effect of time on the scour hole is visualized as a veryimportant variable. Therefore, the bed profile measurement atvarious times was one of the purposes of the present experimentalwork.
As shown in Fig. 4, the upper part of the upstream scourslope is in equilibrium during the whole period of the scourhole processes’ development, whereas, the lower part is stilldeveloping.
Figure 7 shows the plot ofyt/ym versust/td (when the timeratio is ≤ 15%) for test T04 to test T17 and those by Gaudioand Marion (2003) at time 25 h (i.e., when the time magnitude iscomparable withtd of the present study). The figure indicates thatthe initial stages of scouring are characterized by very scattereddata. The linear regression of these points leads to the followingequation (derived using the present data and those by Gaudio andMarion, 2003) with a correlation coefficientR2 = 0.66:
yt
ym= 5.7005
t
td+ 0.1303 (6)
502 Meftah and Mossa
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15
t/td
y t/ym
T04 to T17 Gaudio and Marion (2003) yt/ym=5.7005t/td+0.1303
Figure 7 Similarity of scour hole evolution fort/td < 0.15.
The same work has been performed when the time ratio is>15%as shown in Fig. 8, where Gaudio and Marion’s (2003) data havebeen added to those of the present study. Figure 8 highlights thatthe relationship betweenyt/ys andt/td (derived using the presentdata and those by Gaudio and Marion, 2003) is a logarithmicfunction with a correlation coefficientR2 = 0.32:
yt
ym= 0.098 ln
(t
td
)+ 0.9828 (7)
The analysis of Figs 7 and 8 shows larger data scattering in therange oft/td between 0.1 and 0.4.
4.2 Scour hole shape
According to the experimental results, it was observed that thescour hole shape depends strongly on the distance between sills.
0
0.2
0.4
0.6
0.8
1
1.2
0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05
t/td
y t/ym
T04 to T17 Gaudio and Marion (2003) yt/ym=0.098*ln(t/td)+0.9828
Figure 8 Similarity of scour hole evolution fort/td ≥ 0.15.
It was seen that during the first set of tests with an interval lengthequal to 1 m, the scour hole shape is quasi-parabolic. This con-figuration is characterized by a scour hole which occupies all ofthe space between sills. It was also observed that the scour holedimensions were particularly influenced by the proximity of sillswhen the flow discharge is high.
Furthermore, when the interval length between sills is largesuch as in the cases of the second, third and fourth sets of tests,which have an interval of 2, 4 and 3 m, respectively, the scourhole shape is similar to a “spoon” profile.
To study the self-affinity of the scour profile, the values ofyx/ym at the equilibrium stage have been plotted againstx/ls,as shown in Fig. 9, whereyx is the depth of the scour holeat the longitudinal distancex (x = 0 m at the upstream sillposition).
Figure 9 indicates that the scour profiles for test T09 to test T22are affine in nature. This result is in agreement with the resultsof other authors (Gaudioet al., 2000; Lenziet al., 2003b). Fur-thermore, according to the experimental results, it can be notedthat the scour hole profiles for the first set of tests (with 1 m dis-tance between sills) are not affine in nature. This is due to thestrong influence of the distance between sills for this configura-tion. Thus, it should be noted that when the length of the scourhole is comparable to the distance between sills, the sills interferewith the development of the scour.
4.3 Scour hole dimensions at equilibrium stage
Recall that over a sill with a small thickness, the depth of theflowing water fluctuates around the critical flow depth. A com-parative study between the flow depth over sills,hu, measuredduring the experimental work and the critical flow depth,hc, cal-culated for each test is presented in Table 1. It can be seen that themaximum rate of variation between the measured and calculatedflow depth [calculated as the absolute value of(hu − hc)/hu] is16.7% and the minimum is 0.00%. The average variation for alltests is 8.48%, which is considered small. Thus, the flow depthover sill in the present study can be estimated with the formula
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.20.0 0.2 0.4 0.6 0.8 1.0
x/ls
y x/y
m
T09
T10
T11
T12
T13
T14
T15
T16
T17
T18
T19
T20
T21
T22
Figure 9 Affinity of the scour profiles.
Scour holes downstream of bed sills in low-gradient channels 503
of the critical flow depth,hc:
hu ≈ hc =(
q2
g
)1/3
(8)
According to these simplifying assumptions, Eq. (5) can bereduced to:
ym
hcor
ls
hc= f
(�LS0
hc
)(9)
where the magnitude ofhu could be obtained from Eq. (8).Figure 10 showsym/hc as a function of�LS0/hc of the exper-
imental data of the present paper (withd50 = 0.0018 m) withthose of Gaudioet al. (2000) withd50 equal to 0.0085 m (D1in the figure) and 0.0042 m (D2 in the figure), Gaudio and Mar-ion (2003) withd50 equal to 0.0018 m (D3 in the figure), andLenzi et al. (2003b) withd50 equal to 0.0085 m (D4 in the fig-ure). Figure 11 shows the same results as Fig. 10 but only forsmaller values of�LS0/hc (until 3.0) with the regression lines.From Fig. 11 it is possible to observe the tendency that all theexperimental results are dependent on the sand diameter. In fact,the greater the sand diameter the lower the scour depth.
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
0.00 5.00 10.00 15.00 20.00
∆LS0/hc
y m/h
c
Gaudio et al. (2000) D1 Gaudio et al. (2000) D2Gaudio and Marion (2003) D3 Results of the present paperLenzi et al. (2003b) D4
Figure 10 Correlation of theym/hc ratio with�LS0/hc ratio.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00
∆LS0/hc
ym/h
c
Gaudio et al. (2000) D1 Gaudio et al. (2000) D2 Gaudio and Marion (2003) D3Lenzi et al. (2003b) D4 Results of the present paper
Figure 11 Correlation of theym/hc ratio with �LS0/hc ratio (until�LS0/hc equal to 3.0) with the regression lines.
The plot ofym/hc versus�LS0/hc for the tests of the scourhole of the present study which are affine in nature, as shownin Fig. 11, indicates that all the points of the present paper fallapproximately on a single line. Therefore, the linear regressionof these points leads to the first of Eq. (9) becoming:
ym
hc= 0.592
�LS0
hc+ 1.1582 (10)
This equation is valid for tests with “spoon” scour holes, with acorrelation coefficientR2 = 0.65.
Figure 12 shows the plot ofls/hc against�LS0/hc of theexperimental data of the present study (withd50 = 0.0018 m)with those of Gaudioet al. (2000) withd50 equal to 0.0085 m(D1 in the figure) and 0.0042 m (D2 in the figure), Gaudio andMarion (2003) withd50 equal to 0.0018 m (D3 in the figure), andLenziet al. (2003b) withd50 equal to 0.0085 m (D4 in the figure).It is possible to see the interesting and expected result, i.e., thegreater the sand diameter, the lower the scour length. Figure 13shows the same results as Fig. 12 with�LS0/hc until 3.0 andthe addition of the regression line. Particularly, the plot ofls/hc
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
0.00 5.00 10.00 15.00 20.00
∆LS0/hc
ls/h
c
Gaudio et al. (2000) D1 Gaudio et al. (2000) D2Gaudio and Marion (2003) D3 Lenzi et al. (2003b) D4Results of the present paper
Figure 12 Correlation of thels/hc with �LS0/hc.
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
∆LS0/hc
ls/h
c
Gaudio et al. (2000) D1 Gaudio et al. (2000) D2Gaudio and Marion (2003) D3 Lenzi et al. (2003b) D4Results of the present paper
Figure 13 Correlation of thels/hc with �LS0/hc (until �LS0/hc
equal to 3.0) with the regression lines.
504 Meftah and Mossa
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Longitudinal distance [m]
0
5
10
15
20
25
Dis
tanc
e fr
om th
ech
anne
l flo
or [c
m]
= 20 cm/s
(a) Test T07
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Longitudinal distance [m]
0
5
10
15
20
25
Dis
tanc
e fr
om th
ech
anne
l flo
or [c
m]
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3
Longitudinal distance [m]
0
2
4
6
8
10
12
14
16
18
20
Dis
tanc
e fr
om th
e ch
anne
l flo
or [c
m]
= 20 cm/s
=10cm/s
(b) Test T16
Figure 14 Time-averaged velocities in the central longitudinal profile of the channel at the scouring equilibrium stage.
Scour holes downstream of bed sills in low-gradient channels 505
against�LS0/hc of the present experimental results leads to thefollowing relationship:
ls
hc= 4.6083
�LS0
hc+ 31.006 (11)
This equation is valid for tests with “spoon” scour holes, with acorrelation coefficientR2 = 0.34.
Equations (10) and (11) have been obtained with�d50/hc
in the range 0.049–0.103. They do not contain the sedimentdiameterd50, because, as written, in the present study only onediameter has been used. Nevertheless, it does not mean that ingeneral the results do not depend on the sediment parameter (seeEq. (3) and the limitations of the parameter�d50/hc previouslyreported). Equations (10) and (11) fit only the data withL >
1 m, because in the former case the scour holes do not presentself-affinity.
5 Flow field
5.1 Time-averaged velocities
As indicated in the foregoing paragraphs, the local scour holedownstream of the sills is a consequence of the local increase ofthe flow velocity caused by the plunging jet stream generated atthe crest of the sill. Thus, the flow velocity distribution in thescour hole region has a very large effect upon the developmentprocesses. Therefore, it is a very important flow characteristic tostudy.
Figure 14 shows two examples of the resulting mean velocitiesof the vertical and stream-wise velocity components. Examiningthese figures, it can be seen that the flow velocity vectors in thescour hole decrease as the flow depth increases. In addition, inthe vertical sections of the aforementioned figures, the formationof a vortex-dominated flow is observed. The large turbulence inthis region explains well the dissipation of a big rate of the flowenergy by the mixing process of the momentum jet associatedwith the flow over the crest of the sill and the cross flow in thescour hole.
Figure 15 shows a vector map of velocity distribution intransversal cross-sections of the channel. Vectors are determinedas the mean velocities of the span-wise and vertical velocitycomponents. It has long been established in scientific literaturethat flow in alluvial rivers is strongly three-dimensional (Petersand Goldberg, 1989). The secondary currents were originallydefined by Prandtl (1952) as currents occurring in the planenormal to the axis of the primary flow. They originate frominteractions between the primary flow and channel features. Themeasurements of the flow velocity along different cross-sectionsof the channel were determined during this experimental work.Figure 15 shows examples of the resulting mean velocities ofthe span-wise and vertical velocities along the transversal center-line of the scour hole. Particularly, Fig. 15 clearly indicates thatthere is a development of cells of secondary currents through thecentral cross-section of the scour hole. The secondary current,as well shown in Fig. 15, consisted of two symmetrical cells ofhelical rotation located near the sidewalls of the channel. These
-15 -10 -5 0 5 10 15
Transversal distance [cm]
0
2
4
6
8
10
12
14
16
18
20
Dis
tanc
e fr
om th
e ch
anne
l flo
or [c
m]
= 1 cm/s
Bed profile
Figure 15 Typical time-averaged velocities in the cross-sections of thechannel with the cross-sections of the sand bed profiles at the scouringequilibrium stage (front view of test T13 at 73 cm from the end of thewooden plate).
secondary currents have a direct effect upon the scour hole char-acteristics and the flow features. Figure 15 indicates also thatthere is a formation of a sand ribbon along the centerline axis ofthe scour hole, which is a result of the redistribution of primaryvelocities by the secondary currents.
5.2 Turbulence characteristics
During all the experimental runs, it was observed that at theequilibrium stage, the sediment particles through the bed scourhole move in a random fashion without leaving it. In order tostudy the causes of this observation, calculation of the root meansquare (rms) value of the three components of velocity fluctua-tion through the scour hole were determined. Figure 16 shows anexample of the distribution of the rms(u′), whereu′ is the longi-tudinal turbulent velocity flow component (the results are fromthe second cell T16 test). These experimental results indicatethat the velocity fluctuation is very large through the scour holefor the three components. This result explains well the formationof the turbulent eddies and vortex indicated in Figs 14 and 15.
=15cm/srms(u’)
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3
Longitudinal distance [m]
2
4
6
8
10
12
14
16
18
20
Dis
tanc
e fr
om th
e be
d of
the
chan
nel[c
m]
Figure 16 An example of the turbulence intensity rms(u′) (test T16).
506 Meftah and Mossa
Furthermore, it is seen that the large scale of velocity fluctua-tion is always localized in the upstream side of the scour hole.The rms of the velocity fluctuation decreases gradually with theincrease of the longitudinal position of each cell, until reachinga quasi uniform value. The difference between the velocity fluc-tuation in the upstream and downstream sides is also the resultof the large rate of the stream energy dissipation in the scourhole. As the experimental results show, the rms of the verticalvelocity component fluctuation rms(w′) is smaller than rms(u′)and rms(v′). This result well highlights the movement of the sed-iment particles in a random fashion without leaving the scourhole at the equilibrium stage, because the small vertical velocityfluctuation makes the flow incapable of picking up particles fromthe bed. The large value of rms(v′) confirms the development ofthe secondary current in the scour hole.
Figure 17 shows an example of the distribution of the Reynoldsshear stresses, apart from the water density (the results are alwaysfrom the second cell T16 test). The quantityu′w′, presented in
Figure 17 An example of the cross-correlationu′w′ (test T16).
this figure, is the time-averaged streamwise-vertical Reynoldsshear stresses apart from the water density. From the experi-mental results, it is possible to highlight thatu′v′, u′w′ andv′w′ Reynolds shear stresses present high absolute values in thearea localized in the upstream side of the scour hole, wherethe velocity fluctuation is also very large. Recalling that turbu-lence diffusion is associated with the evolution of the Reynoldsstresses, it is possible to conclude that the turbulence along theflow, in the scour hole, decreases gradually with the increaseof the longitudinal distance from the sill. This result confirmswell the large rate of the flow energy dissipation through thescour hole.
Furthermore, one can see, as indicated in Fig. 17, thatu′w′Reynolds stresses are very large and they present negative valuesin the upstream side. It is possible to conjecture that there isa zone of turbulent transport of the flow momentum localizedin the upstream side of the scour hole which is governing themaximum scour. Near the bottom and in the downstream side of
Scour holes downstream of bed sills in low-gradient channels 507
0.01
0.10
1.00
10.00
0.01 0.10 1.00
z/h
rms(
u’)
/U
T05
T07
T08
T11
T12
T13
T15
T16
T21
A-1 Nezu (1977)
A-3 Nezu (1977)
A-5 Nezu (1977)
A-6 Nezu (1977)
1 - Laufer (1951)
4 - Blinco et al. (1971)
Figure 18 Variation of relative turbulence intensity rms(u′)/U with z/h.
the scour hole a zone of low turbulent transport can be identifiedwhich explains the limit ability of the stream to erode the bedsediment particles in the equilibrium stage.
Figure 18 shows the relative turbulence intensity using uni-versal expressions proposed by Nezu (1977), Laufer (1951), andBlinco and Partheniades (1971) (see Nezu and Nakagawa, 1993).The relative turbulence intensity in the longitudinal direction,rms(u′), was measured in earlier research work in open channelflow as shown in Fig. 18. This figure shows a plot of rms(u′)/U
of the present study with those proposed in literature in the caseof uniform flow, whereU is the time-averaged longitudinal flowvelocity in each measurement point. It is possible to highlightthat the relative turbulence intensity is greater in the case of thepresent study, with erodible flume bottom than in the case ofuniform flow in a fixed plane bottom channel.
6 Conclusions
Four sets of tests were conducted in the hydraulic laboratory ofthe Mediterranean Agronomic Institute of Bari (Italy) to predictlong-term local scour at bed sills. One uniform sediment was usedrespecting a constant initial bed slope of 0.0086. The experimentsshow the following main results.
The extent of the scour hole is strongly dependent on time.Three stages of the local scour hole development were observed:an initial rapid stage, a progressive stage, and a final deceleratedstage.
The scour hole shape depends strongly upon the distancebetween sills. Two scour hole forms were discovered. A quasi-parabolic shape occurred with a small distance between the sills,while a “spoon” shape profile occurred with large distances. Thescour profiles are similar for the cases of large distances betweensills (“spoon” profiles), but not similar for a small distance(quasi-parabolic profiles).
Using the dimensional analysis, the maximum scour depth andthe length of the scour hole at the equilibrium stage were articu-lated as a function of non-dimensional parameters. According tosome logical simplifying assumptions, two empirical formulaswere determined to predict the maximum scour depth,ym, andthe length of the scour hole,ls, respectively. The maximum scourdepth and the length of the scour hole depend strongly on thedistance between sills.
In order to study the effects of the sill upon the flow veloc-ity at the equilibrium, the measurements of the three-velocitycomponents along the flume at different positions have beenrecorded during this experimental work, using a Nortek ADV.Hydrodynamic measurements through the scour hole were madefor the majority of tests. Measurements were taken when thescour hole reached its equilibrium stage. Each vector velocityof the present study is the average of a minimum acquisitionof 7000 instantaneous flow velocity values. The measurementswere taken in both longitudinal and transversal planes. The bedsills lead to a redistribution of the flow velocity and a flow rota-tionality. Intensive flow turbulence and vortex trends occurredin addition to secondary currents in the center of the scour hole.The measurements of the flow velocity along different cross-sections of the channel were determined during this experimentalwork. The secondary current consisted of two symmetrical cellsof helical rotation located near the sidewalls of the channel. Thesesecondary currents had a direct effect upon the scour hole char-acteristics and the flow features. The comparison between therelative intensities of the longitudinal velocities in uniform flowsand those of the present study highlighted that the latter are higherthan the former.
Acknowledgments
The research described and the results presented herein wereobtained from an experimental study on long local scouring
508 Meftah and Mossa
downstream of bed sills in monogranular sand bed conductedin the hydraulic laboratory of the Mediterranean AgronomicInstitute of Bari in 2002.
Notation
a = Difference between the levels of twosuccessive sills (m)
Cu = Uniformity coefficientd10 = Grain size for which 10% of the total
weight of the sediment is finer (m)d50 = Grain size for which 50% of the total
weight of the sediment is finer (m)d60 = Grain size for which 60% of the total
weight of the sediment is finer (m)g = Gravity acceleration (m/s2)h = Flow depth (m)
hc = Critical flow depth (m)hu = Measured flow depth over sills of the cell
at 2 m downstream of the end of thewooden plate (m)
L = Distance between sills (m)ls = Scour length of the cell at 2 m
downstream of the end of the woodenplate (m)
Q = Water discharge (m3/s)q = Water discharge per unit width (m2/s)
S0 = Initial sand bed slopeSeq = Sand bed slope at the equilibrium stage
t = Time (h)td = Duration of each experiment, i.e., the
time when the scour hole reached theequilibrium phase (h)
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