Evolution of scouring process

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papausuow passes through a GCS, a critical ow condition may befound immediately upstream of the GCS, whereas a

based on a ume study. Themain parameters in his equationerittodrsd7)ordtonsemoor

plunging jet, if the channel bed slope is verymild, eventually

HYDROLOGICAL PROCESSESHydrol. Process. (2012)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/hyp.9318supercritical ow condition can be found on the ramp ofthe GCS. Downstream of the GCS, an impinging/plungingjet diffuses the energy into the pool. Far downstream of thepool, a uniform ow may occur (Lenzi et al., 2003a).Scour downstream of a GCS/drop has attracted many

researchers to study. Initially, they focused on the geometryof the scour hole, especially for the equilibrium maximum

a deposit mound will occur downstream of the scour holeand becomes a boundary condition to arrest the developmentof the scour hole (Chiew and Lim, 1996).Considering the effect of channel slope on the scouring

processes downstream of bed sills, Gaudio et al. (2000)proposed formulae to describe the maximum scour depthand the length of scour hole based on a morphologicallength and a median particle size. Lenzi et al. (2002)extended the research of Gaudio et al. (2000) andproposed improved relationships to predict the scour*CNaE-m

Colacement costs. Many civil engineering structures, suchbridges and levees, have been extensively affected byannel incision in Taiwan and also throughout the world.e of the most widely used remedial measures is to installde-control structures (GCSs) in the river to limit furthergradation upstream of the GCSs. However, the GCSsnot completely solve the incision problem in case ofiment starvation because edge failure may occurwnstream of the GCSs and nally damage the GCSsough head cutting.After a GCS is constructed, initially, the coarser sedimentrticles are trapped upstream of the GCS, whereas the nerrticles may pass through it during oods. The GCSs areally 1.52m higher than the original riverbed. When the

are the unit discharge, the height between head and tailwatlevel, and D90. Schoklitsch (1932) considered the undischarge and the height difference as parameters relatedthe impact forces on the bed, andD90 as a parameter relateto the resistance. After Schoklitsch (1932),many researchepaid a lot of attention to the scouring issue. Mason anArumugam (1985) and Hoffmans and Verheij (199summarized a wide range of empirical equations fpredicting the equilibrium maximum scour depths induceby plunging jets. Bormann and Julien (1991) took inaccount the effect of the jet angle near the water surface othe scour depth downstream of a GCS. However, in theprevious studies, a deposit mound was formed downstreaof the scour hole because of the horizontal channel slope. Nmatter whether it is an impinging jet, horizontal jetstructures under stead

Jau-Yau Lu,1* Jian-Hao Hong,11 Department of Civil Engineering, Nationa

2 Department of Statistics and Informatics Sci

Abs

For many incised channels, one of the most common strategies(GCSs), in the riverbed to resist further incision. In this study, aconditions, were conducted to investigate the scouring process ddeveloping and equilibrium phases, during the evolution of scouproposed to predict the equilibrium scour-hole prole for the scthe experimental and simulated results are reasonably consistentcaused by oods are limited, the effect of ood hydrograph shapsupply was also investigated experimentally in this study. Basedmethodology is proposed to simulate the time evolution of theMoreover, the proposed scheme predicts reasonably well the temwith both single and multiple peak. Copyright 2012 John W

KEY WORDS grade-control structure; scour; sediment transpor

Received 3 March 2011; Accepted 16 March 2012

INTRODUCTION

Channel incision receives public attention as a result of theorrespondence to: Jau-Yau Lu, Department of Civil Engineering,tional Chung-Hsing University, Taichung, Taiwan 40227.ail: jylu@mail.nchu.edu.tw

pyright 2012 John Wiley & Sons, Ltd.downstream of grade-controly and unsteady ows

ai-Po Chang1 and Tai-Fang Lu2

hung Hsing University, Taichung, Taiwance, Providence University, Taichung, Taiwan

act:

to install some hard structures, such as grade-control structuresseries of experiments, including both steady and unsteady owwnstream of a GCS. Three distinct phases, including the initial,holes were identied. In addition, a semi-empirical method wasr countermeasure design. In general, the comparisons betweenAs the studies on temporal variation of the scour depth at GCSss on the scour downstream of GCSs without upstream sedimenton the dimensional analysis and the concept of superposition, aaximum scour depth downstream of a GCS for steady ows.oral variations of the maximum scour depth for unsteady owsley & Sons, Ltd.

unsteady ows

scour depth. Schoklitsch (1932) proposed one of the earliestformulae to predict the scour depth for the ow over a dropdepth and length. In the studies of Gaudio et al. (2000)and Lenzi et al. (2002), the distance between two bed sills

shows the schematic diagram of experimental arrangement

J.-Y. LU ET AL.is long enough for the development of the scour hole.Moreover, Marion et al. (2004) investigated the effect ofthe sill spacing and sediment grading on the potentialerosion by jets formed over the sills. They found thatdecreasing the distance between two sills may lead to thereduction of the scour hole as compared with its potentialsize. Similar results can also be found in Meftah andMossa (2006). Both Marion et al. (2004) and Pagliara(2007) studied the effect of sediment gradation on scourdepth and pointed out that the effect is pronounced fornon-uniform sediment.Many investigations on scour proles induced by jets

have been carried out over the past decades. Chatterjee et al.(1994) and Dey and Sarkar (2006) investigated the scourdownstream of an apron because of submerged horizontaljets and found that the scour proles are similar in nature.Pagliara (2007) also proposed a polynomial equation todescribe themeasured scour proles.However,Gaudio et al.(2000), Lenzi et al. (2003a,b), Meftah and Mossa (2006)and Tregnaghi et al. (2007) pointed out that scour prolesat equilibrium are afne. To reduce the scour depth andto prevent the failure of the GCS, the hydraulic engineersusually launch riprap or gabions downstream of the GCS.Although it is very useful to know the scale of the scourhole for the adoption of the proper scour countermeasure,only little attention has been placed on the scour-holeproles downstream of the GCSs under sloping channelconditions.In natural rivers, severe riverbed scour usually occurs

during oods. The unsteady force acting on the riverbedis different from that acted by a steady ow. Breusers andRaudkivi (1991), Hoffmans and Verheij (1997) andDargahi (2003) showed that many scour phenomena aretime dependent and should be investigated in advance.To date, the time variations in bridge-scour depths have

attracted the attention of many researchers (Kothyari et al.,1992; Cardoso and Bettess, 1999; Melville and Chiew,1999; Mia and Nago, 2003; Chang et al., 2004; Oliveto andHager, 2005). Oliveto andHager (2005) also examined theircalculating scheme under a multiple-peak hydrograph.With consideration of the primary horseshoe vortex,

sediment pick rate and superposition concepts, Kothyariet al. (1992) proposed a semi-empirical model to calculatethe time evolution of the maximum pier-scour depth understeady and unsteady ow conditions. Lu et al. (2008)modied the method by Kothyari et al. (1992) to simulatethe variation of pier-scour depths in the eld and obtainedgood agreement with the measured values. Recently, asimilar concept was also adopted for the simulation of thetemporal variations of scour at non-uniform piers underunsteady ows (Lu et al., 2011).In contrast to pier scour, little research has been conducted

regarding the time evolution of the maximum scour depthdownstream of a sill/GCS. Gaudio and Marion (2003) andDargahi (2003) proposed different time scales for thesimulation of the temporal variations of the maximum scourdepth. Tregnaghi et al. (2009; 2010) investigated the effectof ash oods on the scour at sequence sills with

symmetrically triangular-shaped and long ood recession

Copyright 2012 John Wiley & Sons, Ltd.under equilibrium scour conditions. The working length ofthe ume L that represents the sediment recess was 3.5m,which was chosen to be large enough to guarantee full scourdevelopment with no geometrical interference (Marionet al., 2004). Two types of perspexmaking the GCSmodels,with different ramp slopes (Srm=0.25, 0.143) and twouniform sediment sizes (D50 = 2.7mm, 3.5mm), were usedfor the scouring tests. The head of GCSmodelsP (total dropheight) was kept constant (P= 45mm). Eighteen sets ofexperimental runs under steady ow conditions (Table I),and 32 sets of experiments under unsteady ow conditions(Table II) were performed in this study. As shown in Table I,all the ows belong to subcritical ow. The Froude numbersrange from 0.36 to 0.93. The stepwise hydrographs forthe unsteady ow conditions, including the advanced,symmetrical and delayed hydrographs, are shown inFigure 2. No sediment was supplied upstream during theexperiments. A video camera was attached to the glasssidewall to record the temporal variations of the bed proles.The video records were then analysed using a digital imageprocessing technique with a precision of 0.3mm developedby the authors. It had also been tested against point gaugemeasurements. Reasonably close results were obtainedusing the two methods. As shown in Figure 1, point Bhydrographs, respectively. They provided a methodology topredict the nal scour depth downstream of a sill after theoccurrence of a ood. Recently, Tregnaghi et al. (2011)assumed that at any time, the scour depth evolves at the samerate as in an equivalent steady ow. Based on thisassumption, they proposed a model for the simulation oftime-varying scouring at bed sills under single-peakhydrograph conditions. However, the scouring processunder a multiple-peak hydrograph was not considered.Furthermore, the effect of incision-related undermining

of GCSs tends to occur during high-ow conditions.Based on the eld observations and investigations (WaterResources Planning Institute, 2009), it was found thatmost GCS failures occurred during oods. More researchinto scour downstream of a GCS is required, especiallyunder unsteady ow conditions.The present study aims to investigate the development

of scour hole in non-cohesive sediment downstream of aGCS. The experimental results are used to develop asemi-empirical model for predicting the equilibrium scourprole. An attempt is also made to calculate the temporalvariation of the maximum scour depth downstream of aGCS under both steady and unsteady ows.

EXPERIMENTAL SETUP AND PROCEDURE

The experiments were carried out in a re-circulating ume17.5-m long, 0.6-m wide, and 0.6-m deep with glasssidewalls. The ow discharge was controlled by an inletvalve and calibrated by a water tank downstream of theume. The linear correlation coefcient between the valveopening reading and the ow discharge was 0.998. Figure 1represents the location (xs,me) of the maximum equilibrium

Hydrol. Process. (2012)

Figure 1. Schematic diagram of scour downstre

Table I. Experimental results

Run Sb Srm D50 (mm) q (m2/s)

S-M1R4-5 0.01 0.25 3.5 0.0167S-M1R4-8 0.01 0.25 3.5 0.0475S-M1R7-5 0.01 0.143 3.5 0.0167S-M1R7-8 0.01 0.143 3.5 0.0475S-M2R4-5 0.01 0.25 2.7 0.0167S-M2R4-8 0.01 0.25 2.7 0.0475S-M2R7-5 0.01 0.143 2.7 0.0167S-M2R7-6 0.01 0.143 2.7 0.0225S-M2R7-7 0.01 0.143 2.7 0.0283S-M2R7-8 0.01 0.143 2.7 0.0475S-S1R4-1 0.015 0.25 3.5 0.0167S-S1R4-4 0.015 0.25 3.5 0.0475S-S1R7-1 0.015 0.143 3.5 0.0167S-S1R7-4 0.015 0.143 3.5 0.0475S-S2R4-1 0.015 0.25 2.7 0.0167S-S2R4-4 0.015 0.25 2.7 0.0475S-S2R7-1 0.015 0.143 2.7 0.0167S-S2R7-4 0.015 0.143 2.7 0.0475

Sb = channel bed slope.Srm = ramp slope.D50=median particle size.q= discharge per unit width.Fdd= (q/hd)/(gD50)

0.5, where g= [(rs-r)/r]g, reduced gravitational accelerars= sediment density,r = water density,g= the gravitational acceleration.xm,e = position of maximum scour depth at the equilibrium stage for steadyym,e=maximum scour depth at the equilibrium stage for steady ows.lm,e=maximum scour hole length at the equilibrium stage for steady ows.

Table II. Experimental conditions for unsteady ows

Run Sb Srm D50 (mm)Type of

hydrograph

U-S1R7 0.015 0.143 3.5 (a) ~ (d)U-M1R7 0.01 0.143 3.5 (a) ~ (d)U-S1R4 0.015 0.25 3.5 (a) ~ (d)U-M1R4 0.01 0.25 3.5 (a) ~ (d)U-S2R7 0.015 0.143 2.7 (a) ~ (d)U-M2R7 0.01 0.143 2.7 (a) ~ (d)U-S2R4 0.015 0.25 2.7 (a) ~ (d)U-M2R4 0.01 0.25 2.7 (a) ~ (d)

SCOURING PROCESS OF GCSS DURING FLOODS

Copyright 2012 John Wiley & Sons, Ltd.scour depth. The distance from the original bed to point Brepresents the maximum equilibrium scour depth (ys,me),and the distance between point C and the upstream endcorresponds to the length of the maximum equilibrium localscour hole (ls,me).

RESULTS

Scouring process

Based on the study with clear-water inow, in general,the scouring process downstream of a GCS under steady

am of a grade-control structure (not to scale)

for steady ow conditions

Fdd xm,e (mm) ym,e (mm) lm,e (mm) Fr

3.50 119 59 450 0.814.75 316 136 1450 0.774.12 124 55 470 0.935.87 259 129 1220 0.633.62 150 71 683 0.565.05 604 151 2500 0.584.43 149 74 700 0.854.68 171 95 894 0.645.20 337 104 1087 0.615.98 522 152 1750 0.603.05 213 111 888 0.674.99 445 204 1750 0.863.05 152 92 532 0.544.64 679 241 1950 0.672.85 326 169 857 0.414.73 678 246 2150 0.453.19 335 149 1056 0.364.83 836 300 2150 0.53

tion,

ows.

Hydrol. Process. (2012)

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J.-Y. LU ET AL.ow condition can be identied into three distinct phasesduring the evolution of scour hole, that is, the initial,

Figure 2. Stepwise hydrographs for the tests under unsteady ow conditiowith a lodeveloping, and equilibrium phases. Figure 3 depicts thetypical experimental results for the development of thescour depth as follows:

1. Phase I (initial stage)

The movable bed was scoured by the impinging jetinduced by the GCS, and a small deposit mound wasformed downstream of the scour hole.

2. Phase II (developing stage)

The dimension of the scour hole increased with time. Thesediment particles oscillated along the downstream slope ofthe scour hole. Furthermore, the maximum scour depthoccurred near the sides of the ume because of the formationof secondary ows in the xz plane induced by the hydraulicjump (z is perpendicular to xy plane in Figure 1).

3. Phase III (equilibrium stage)

It can be further identied into three sub-scouringprocesses. First (Phase III(a)), bursting phenomenoncan be found near the location of the maximum scourdepth. The scour hole near the GCS was approximatelytwo-dimensional. Second (Phase III(b)), a depositmound was temporarily formed on the downstreamslope of the scour hole because of the accumulationof the falling sediment particles entrained by the

Copyright 2012 John Wiley & Sons, Ltd.impinging jet. During this period, the hydraulic jumpoccurred very close to the GCS. The impinging jet was

: (a) symmetric; (b) delayed peak; (c) advanced peak; and (d) symmetric,er peaksomehow restrained by the hydraulic jump, and thelocation of the maximum scour depth moved slightlyupstream. Finally (Phase III(c)), the hydraulic jumpgradually moved downstream. The impinging jet wasnot restrained by the hydraulic jump any more, and thedeposit mound vanished gradually. During the equi-librium stage, the scouring process followed thecycle (a)(c) of Phase III. The bursting phenomenondecreased with time, and the scour hole almostremained unchanged. Finally, the maximum scourdepth occurred near the sides of the ume.

Scour hole dimensions at equilibrium stage under steady ows

The placement of an armor layer downstream of aGCS, such as rock riprap and rock gabions, is one of themost widely used countermeasures to protect the structurefrom erosion or edge failure. The estimation of thegeometric parameters of the scour hole, including theposition (xs,me, ys,me) of the maximum equilibrium scourdepth and the length of the scour hole (ls,me), is requiredfor the proper design of the countermeasures. Farhoudiand Smith (1985) and Dey and Sarkar (2006) found thatthe sediment size and tailwater depth ht affect the scourdepth downstream of a spillway apron. Based on theexperimental data, Dey and Sarkar (2006) and Pagliara(2007) reported that the densimetric Froude number hadgreat effect on the scour depth downstream of an apronbecause of submerged horizontal jets and block ramps,respectively. However, the experiments performed by

Hydrol. Process. (2012)

SCOURING PROCESS OF GCSS DURING FLOODSDey and Sarkar (2006) and Pagliara (2007) were underhorizontal bed slope conditions. In this study, theexperiments were conducted with different bed slopes(Table I). With consideration of the ndings by Farhoudiand Smith (1985), Dey and Sarkar (2006) and Pagliara(2007) and incorporating the effect of bed slope,regression analysis for the experimental data (Table I)yielded the following dimensionless equations for thegeometric parameters of the scour hole (xs,me, ys,me, ls,me)

xs;me=P 38:9S1:263b F 0:887dd ht=D50 0:867 (1)

ys;me=P 351:3S1:667b F 0:759dd ht=D50 0:494 (2)

ls;me=P 11:3S0:595b F 0:812dd ht=D50 0:761 (3)

Figure 4 shows the comparisons of the measured andpredicted scour hole parameters. The R2 values forEquations (1) ~ (3) are 0.86, 0.93, and 0.81, respectively,indicating that Equations (1) ~ (3) correspond satisfactorilywith the experimental data. Physically, Equations (1) ~ (3)show that the locus of the maximum scour depth or scourlength increases with an increase in channel slope Sb,densimetric Froude number Fdd, or tailwater depth ht, and adecrease in bed material median size d50. In fact, a

Figure 3. Schematic diagrams of scouring process downstream of

Copyright 2012 John Wiley & Sons, Ltd.dimensional analysis can be performed to prove the validityof the functional forms of Equations (1) ~ (3).A comparison of the scour depths predicted by Lenzi

et al. (2002), Marion et al. (2004) and Equation (2), and ourmeasured data was also made. As expected, Equation (2), in

a grade-control structure under a typical steady ow condition

Figure 4. Comparisons of the measured and computed dimensionlessgeometric parameters (xs,me, ys,me, ls,me) for scour holes

Hydrol. Process. (2012)

Figure 6 shows the comparisons of the results obtained

J.-Y. LU ET AL.general, gave the best predictions. The performance of theequation by Marion et al. (2004) was slightly better thanthat of Lenzi et al. (2002). No empirical equation for theprediction of scour hole length was derived in the study byMarion et al. (2004). A comparison of the scour holelengths predicted by Lenzi et al. (2002) and Equation (3)and our measured data indicated that both equations gavefair predictions.

Scour hole proles at equilibrium stage under steady ows

Dumped riprap and rock gabions are the most commonscour countermeasures to minimize the scour depthdownstream of a GCS. In this section, a semi-empiricalmodel is proposed to predict the scour-hole prole.Accordingly, river engineers can design the scourcountermeasures more economically. Dene X= xs/ls,meand Y= ys/ys,me, where the longitudinal and verticalcoordinates xs and ys for describing the scour hole aremeasured with respect to the origin A in Figure 1. Basedon the analysis of the existing data [Gaudio and Marion,2003; Meftah and Mossa, 2006] and those collected inthis study, the dimensionless scour-hole prole can bedescribed by

ysys;me

C0 exp C1 xsls;me

1 exp C2 xsUn

(4)

where C0, C1, and C2 are empirical coefcients; U ghtSeq

p; g = gravitational acceleration; ht= tailwater

depth; n= kinematic viscousity. The coefcients C0, C1,and C2 are determined using the scour-hole character-istics, which are as follows: (i) atxs = xs,me, ys/ys,me= 1;(ii) atxs= xs,me, dys/dxs= 0; and (iii) atxs= ls,me, ys = a2,where a2= equilibrium general scour depth in reach CD(Figure 1), which can be estimated as follows based onGaudio and Marion (2003):

a2 ht hc nq

6=7

cD50 3=7 q

2

g

1=3; for a2=ht > 0:15 (5a)

q 23

2g

p0:605 0:001

ht a2 0:08ht a2a2

ht a2 3=2;

for a2=ht0:15

(5b)

where n =Mannings roughness coefcient; c =criticalShields mobility parameter =htSeq/(D50) for widechannels; Seq= equilibrium bed slope downstream ofthe maximum scour depth; = relative submergedparticle density = (rsrw)/rw; rs= density of sedimentparticles; and rw =density of water; q=water dischargeper unit width. Using the pre-mentioned three boundaryconditions, one can obtain three equations. The empiricalcoefcients C0, C1, and C2, can then be determined by

solving the simultaneous nonlinear equations.

Copyright 2012 John Wiley & Sons, Ltd.from the proposed model with the experimental data ofMeftah and Mossa (2006). The model predicts thescour-hole proles between the origin and the maximumscour depth very well but slightly overestimates the scourdepths for reach beyond the maximum scour depth.Overall, however, the model predicts the scour-holeproles quite reasonably.

Temporal variation of maximum scour depthsteadyow conditions

The time variation of scour downstream of a GCSbecause of plunging jet was tested for uniform sedimentwith different sizes (Table I). According to the experi-mental observations, the scour depth ys,t at time t can beexpressed as a function of tailwater depth (ht), densimetricFroude number, time, and channel slope as follows:

ys;t=ys;me f Sb;Fdd;T (6)

where T= t/tR; tR= reference time = ht/(gD50)0.5; g=g.

The dimensionless maximum scour depth ys,t/ys,me can beexpressed as an exponential function of T as follows:

ys;t=ys;me 1 exp b0 Tb1

(7)

where bo = 0.0468 and b1 = 0.421 are coefcients deter-mined by using SPSS. The coefcient of determination R2

for Equation (7) is 0.95. As shown in Figure 7, all the datacollapse in a narrow band. Although there are somediscrepancies between the measured and predicted resultsat the initial stage of scour activity, the model, in general,predicts the temporal variations of scour depths down-stream of a bed sill under steady ow conditionssatisfactorily.

Evolution of maximum scour depthsingle-peakhydrograph

Tregnaghi et al. (2009, 2010) had performed a series ofexperiments on scour downstream of uniformly spacedbed sills with uniform sediment under both symmetricaland asymmetrical hydrographs. The time variations of themaximum scour depth during oods were investigatedand tted with polynomial equations. In the present study,a semi-empirical model is proposed for calculating thetime variations of scour at a GCS under unsteady owThe scour-hole proles computed using the presentedmethod are compared with the observed experimentaldata in several typical cases as shown in Figures 5(a)(h).In general, the proposed method predicts the experimen-tal data with bed slope Sb= 0.015 better than those withSb = 0.01. It is because the undulant variations occurredon the lower slope (Sb = 0.01), although this phenomenongradually disappeared on the higher slope (Sb = 0.015),especially with higher ow discharges or smaller mediansediment size, for example, Figure 5(h).conditions based on the concept of superposition. Figure 8

Hydrol. Process. (2012)

SCOURING PROCESS OF GCSS DURING FLOODSshows the schematic diagram of the model. A stepwisehydrograph consisting of the rising (Q1 ~Q3) andrecession (Q4) limbs as depicted in Figure 8(a) is usedto illustrate the procedure.Chatterjee et al. (1994), Dey and Sakar (2006) and

Pagliara (2007) found that the scour hole shapesdownstream of a GCS and cross-river hydraulic structureare similar in nature. Meftah and Mossa (2006) also foundthat if the spacing between two GCSs is long enough, theshapes of scour holes are also similar. In Figure 3, duringthe initial stage (only a few seconds), a deposit moundwas temporarily formed downstream of the GCS andgradually washed away by the ow. However, one cansee in the gure that the shapes of scour holes are fairlysimilar during the developing and equilibrium phases.Based on this nding, we assume that the concept ofsuperposition can be applied to most of the scouringprocesses except for the very initial stage under unsteady

Figure 5. Comparisons of the observed (circle) and predicted (curve) scour-ho(c) M2R4-5, (d) M2R4-8, (e) S1R4-1, (f

Copyright 2012 John Wiley & Sons, Ltd.ow conditions. As the initial stage is very short, it wouldnot affect the calculation signicantly.Figure 8(b) shows the time variation of scour depth at a

GCS for different steady ow rates (Qj, j= 1, 2, 3 and 4).The specic steps for computing the scour depth at a GCSunder an unsteady ow are given below:

1. For the rst ow rate Q1, the time evolution of scourdepth follows the ys curve of Q1 [i.e. OA, as shown inFigure 8(b)]. The corresponding scour depth is ys, 1.

2. As the ow rate increases from Q1 to Q2 at time t1, thetemporal variation of scour depth follows the path A00Bof ys curve corresponding to Q2. Because Q2>Q1, thetime required for the scour depth to reach ys,1 for theow rate Q2 is less than t1. This time is designated as t1in Figure 8(b).

3. When the ow rate increases from Q2 to Q3(>Q2) att t1 t2 t1 , the yscurve follows the pathB00Cof the

le proles under steady ow conditions for runs: (a) M1R4-5, (b) M1R4-8,) S1R4-4, (g) S1R7-1 and (h) S1R7-4

Hydrol. Process. (2012)

J.-Y. LU ET AL.scour depth curve for Q3. Because Q3>Q2, the timerequired for the scour depth to reach ys,2 for the owrate Q3 is less thant2. This time is designated as t2 inFigure 8(b).

4. However, at time t3, as the ow rate decreases from Q3to Q4 corresponding to t t2 t3 t2 , the scourdepth may remain unchanged, that is, ys,3 = ys,4. On theother hand, if the slope of the recession limb is mild,the scouring potential of Q4 may cause further scour,which means ys,4 may be slightly greater than ys,3.Furthermore, for the steep recession limb, the scourdepth remains unchanged because of the reducedstream power. The ys curve may follow the pathC00Dof the scour depth curve for Q4.

Determination of time lag tiTo determine the time lag ti , Equation (7) for the temporal

variation of the scour depth at a sill under steady owconditions is adopted.

t1 ht;2

gD50 1=21

b0 ln ys;me2 ys;me1 ys;me10@24

8Q1),Q1 lasts a period of t1, whereas the subsequent Q2 lasts aperiod of (t2 t1). The time evolution of the scour depthfor Q1 is determined by

ys;m t1 ys;me1

1 exp b0t1 gD50 1=2

ht;1

" #b18