Journal of Hydraulic Research Vol. 41, No. 4 (2003), pp. 417–426

© 2003 International Association of Hydraulic Engineering and Research

Scour development downstream of a spillway

Développement d’un affouillement à l’aval d’un déversoirBIJAN DARGAHI, Assoc. Prof. Div. Hydraulic Engrg., The Royal Institute of Technology, Stockholm, Sweden. E-mail: bijan@kth.se

ABSTRACTThe scouring process downstream of spillways is an important research topic of value in engineering practice. The objectives of the present experi-mental study were to examine the similarity development of scour profiles, the controlling scour mechanism and predictions of the scour geometry.No experimental evidence was found in support of the similarity assumption for the scouring process. Scouring downstream of a spillway is controlledby large secondary flows and a vortex system induced by a hydraulic jump. The main dimensionless parameters controlling the scouring process arethe relative operating head, the relative sediment size, and the relative roughness of the scour protective plate. The maximum scour depth and the rateof sediment transport can be estimated by using simple, power-type equations that relate the scour geometry to the controlling scour parameters.

RÉSUMÉLe processus d’affouillement à l’aval des déversoirs est un sujet de recherche important au service de l’ingénierie pratique. Les objectifs de l’étudeexpérimentale présentée ici étaient d’examiner la similitude des profils, le mécanisme contrôlant l’affouillement et les prévisions de sa géométrie. Onn’a trouvé aucune preuve expérimentale permettant de valider l’hypothèse d’une similitude pour le processus d’affouillement. En aval d’un déversoir,l’affouillement est contrôlé par des écoulements secondaires importants et un système de vortex induit par un ressaut hydraulique. Les principauxparamètres sans dimensions contrôlant le processus d’affouillement sont la hauteur de chute de fonctionnement relative, la taille relative de sédiment,et la rugosité relative de la semelle de protection. La profondeur maximum d’affouillement et le taux de transport de sédiment peuvent être estimés aumoyen d’équations simples de type puissance qui relient la géométrie de l’affouillement aux paramètres de contrôle de celui-ci.

Keywords: Scouring; spillways; scour similarity; scour equations; sediment transport; temporal scouring; hydraulic jumps;flow visualization; secondary flows.

Introduction

Background

Local scour downstream of hydraulic structures such as low headand high head structures, spillways and culverts is an importantresearch field due to its significant practical value. Scour inducedby low head structures such as gates and low dams has beeninvestigated by many researchers. These works can be classifiedinto two groups. One group concerns scour depth equations andthe other group deals with the flow and scour characteristics andthe temporal behaviour of scouring.

There are many scour depth equations that have been devel-oped for different ranges of experimental parameters. Most ofthese equations relate the scour depth to the flow depth, dis-charge and sediment size. One of the earliest equations wasgiven by Schoklitsch (1932). He used model tests to proposetwo scour depth equations for overflow and underflow gates withshort horizontal sills. The equation for the overflow case reads:

Z + Zo = 4.75H0.2q0.57D−0.3290 (1)

Revision received February 17, 2003. Open for discussion till December 31, 2003.

417

In which Z = scour depth, Zo = downstream flow depth,H = difference between upstream and downstream water sur-faces, q = flow discharge per unit width and D90 = sedimentsize in mm (90% passing by weight). Equation (1) is in metricunits and is intended for coarse sediment in the range D90 = 9–36 mm. Jaeger (1939) reanalysed previous experimental data toarrive at a similar form of scour depth equation (2) for plungingjets.

Z + Zo = 6H0.25q0.5

(H

D90

)0.33

(2)

A similar form of scour depth equation was given by Eggenberger(1944) for a gate with underflow and overflow

Z + Zo = CH0.5q0.6D−0.490 (3)

In which C is a constant that depends on the ratio of the underflowto the overflow discharges, being 22.8 when the underflow is zero.In Eq. (3) all the variables are in metres with exception of D90

which is in millimetres. Shalash (1959) extended the previousworks by including the effect of an apron with length L for an

418 Bijan Dargahi

underflow structure. The proposed equation reads:

Z + Zo = 9.65H0.5q0.6D−0.490

(1.5H

L

)0.6

(4)

Equation 4 is based on D50 = 0.7 mm, 1.9 mm and 2.65 mm andq = 0.01 − 0.07 m2/s.

The studies on the flow field and temporal variation of thescour hole are more limited in comparison to the first group.Experiments show (e.g. Breusers (1966) and Dietz (1969)) thatfor a given geometric arrangement, the shape of the scour hole isalmost independent of the flow velocity and bed material at theequivalent values of Zo/Zmx (Zmx = maximum scour depth). Theflow field depends on the type of the flow. In a two-dimensionalscour hole the flow is similar to that of a turbulent mixing layer.The flow character is dominated by vortices in case of a three-dimensional situation.

The temporal variation of the scour depth was investigated byBreusers (1966) and Kotoulas (1967). Breusers (1966) suggesteda power type relationship for computing the temporal variationof scour depth. Kotoulas (1967) found that in case of coarse sandabout 64% of the final scour occurred in the first 20 s, and about97% of scour depth was attained in 2 h. Novak (1955) studied theinfluence of a stilling basin on local scouring. He found that thescour depth could be reduced to 45 to 65% if the basin is longenough to contain the hydraulic jump. In a recent work Farhoudiand Smith (1985) studied the scour profiles downstream of aspillway. They found a similarity relationship for the temporalvariation of the scour profiles.

A detailed review of the foregoing works and further studiesare given by Simons and Sentürk (1977). Further informationcan be found in the recent design manual for hydraulic structuresissued by IAHR (Breusers and Raudkivi (1991)).

Generally, scouring downstream of hydraulic structures isassociated with a jet type flow characterised by high velocity andvelocity decay laws. The physics of jetflow induced scouring isless complex than the scouring developed around hydraulic obsta-cles such as bridge piers and abutments. However, an importantcommon feature in all local scouring problems is the existenceof strong secondary flows. Secondary flows are generated dur-ing flow separation or interaction with flow obstacles such asbridge piers. Secondary flows induce complex vortex structuresthat are the main scouring agents in local scour problems. Exper-iments have served to improve understanding of the physicsand to yield useful relationships to predict the maximum scourdepths. Regarding bridge pier and abutment scouring, the three-dimensional nature of the scouring process has been thoroughlyinvestigated (Dargahi (1990)). In the case of scouring down-stream of hydraulic structures, most investigations have beenlimited to two-dimensional planes.

Objectives

The present study is an attempt to extend the previous experimen-tal works by providing detailed 3-D measurements downstreamof a spillway. Three issues were of major interest: first, the

similarity development of the scour profiles; second, the non-dimensional scour vs. time function; and third, the controllingscour mechanism. Similarity of the scour profiles is an impor-tant consideration, because if it exists it could provide a meansto predict the time factor. Many local scouring results are basedon small scale model tests, which are excellent tools to study thephysics of the problem and to find useful relationships. However,scaling problems occur when one needs to know the temporalscour development in a prototype situation. The difficulty wouldbe overcome if a temporal similarity relationship could be found.Previous research on local scouring has not yet resulted in sucha relationship, although in a few cases where the flow can besimplified a solution is possible. The present study investigatesthe temporal variation of the scour profiles downstream of a spill-way. As will be shown no evidence was found of the existence ofa similarity relationship. In this respect, the results do not confirmthe findings of Farhoudi and Smith (1985).

Dimensionless parameters

The flow field and local scouring downstream of a spillway withan apron (bed protection plate) depends on a large number of flowand sediment variables. These variables can be grouped into thefollowing dimensionless parameters (5) by use of dimensionalanalysis:(

ho

hd,

ks

ho, Fr, Re,

ho

Dm,

Ls

ho,

u∗u∗c

, σ

)(5)

in which h0 = operating head, hd = design head, ks = spillwayroughness, Fr = Froude number, Re = Reynolds number,Dm = characteristic sediment mean size, Ls = length of theprotection plate (apron), u∗ = shear velocity, u∗c = critical sed-iment particle velocity and σ = geometrical standard deviationof sediment sizes. The first four parameters are related to theflow over the spillway and the remaining parameters are relatedto the scouring process. The present study showed that the oper-ating spillway head (h0) and the sediment size (D50) were thedominating variables for local scouring.

Experiments

The experiments were conducted in a 22-m-long, 1.5-m-wide(B = 1.5 m) and 0.65-m-deep flume. A fixed layer of uniformlygraded fine sand covered the channel bed. The dimensionlesswall roughness height defined by Y+ = u∗ks/υ (υ = kinematicviscosity) was approximately 1.8 (ks = 0.36 mm), so that the sur-face was effectively smooth. Scouring tests were carried out in a0.26-m-deep, 4-m-long and 1.5-m-wide sediment recess sectionsituated 16.5 m from the inlet. The flow entering the test sectionwas uniform within 1% of the free stream velocity and the tur-bulence intensity was 5%. An overflow spillway of 0.205 m crestheight was placed at a distance of 16.5 m from the inlet. Thespillway was designed according to the Waterways ExperimentStation (WES) standard (USArmy Corps of Engineers (1952)) fora design head of hd = 0.1 m. A Perspex plate was placed down-stream of the spillway’s toe having dimensions 1.5 × 0.615 m

Scour development downstream of a spillway 419

Scoured surface

hO

sediment recess section

4 m

Protection plate

L = 0.615 mS

Cont

0.26

m

0.20

5mrol gate

Lower basin

Zmx

ZO

16.5 m

Flow

0 XZ

-Z

Figure 1 The details of the test section in a vertical plane i.e. xz-plane (not to scale).

(Fig. 1). To measure the temporal variation of the bed, a speciallydesigned instrument called here a profile-metre was used. Theprofile-metre was mounted on a motorized carriage systems. TheXY-coordinates were measured by two potentiometers attachedto the carriage. The origin of the coordinates was at the end of theplate touching the right flume wall (looking in the flow direction).

The experimental variables were the sediment size, the plateroughness and the flow discharge. Two different bed materialswere used, one fine sand with D50 = 0.36 mm (D90 = 0.50 mm)and the other medium size gravel with D50 = 4.9 mm (D90 =7 mm). Both materials were effectively uniformly graded, thevalues of geometric standard deviations begin less than 1.4. Twoseries of experiments were carried out, one with a smooth plateand the other with additional roughness on the plate, in the formof hexagon nuts glued to the plate. Each nut was 1.1 cm high and2.0 cm wide. They were placed in three staggered rows with aspacing of 7.5 cm between each row and between each nut. Thefirst row was 10 cm downstream of the spillway. The flow dis-charge was varied from 20 to 100 l/s in 20 l/s intervals. Table 1shows the key flow parameters. Each test is characterized by thematerial type, flow discharge, plate roughness and the sectionlocation. For example S20R means, the bed material is sand (S),the flow discharge is 20 l/s, the protection plate has added rough-ness (R). S20R-250 (in case of Fig. 8) refers to S20R results at asection located at y = 250 mm or Yn = y/B = 0.167.

Table 1 Experimental data.

Additional D50 Q ho hn = ho/hd

Test roughness Scour material [mm] [l/s] [mm] (hd = 0.1 m)

S20 No Medium Sand 0.36 20 38.8 0.388S40 No Medium Sand 0.36 40 56.4 0.564S60 No Medium Sand 0.36 60 73 0.73G20 No Fine Gravel 4.90 20 38.5 0.385G40 No Fine Gravel 4.90 40 57 0.57G60 No Fine Gravel 4.90 60 71 0.71G80 No Fine Gravel 4.90 80 83.2 0.832G100 No Fine Gravel 4.90 100 95.8 0.958S20R Yes Medium Sand 0.36 20 38 0.38S60R Yes Medium Sand 0.36 60 72.8 0.73G20R Yes Fine Gravel 4.90 20 37.0 0.37G60R Yes Fine Gravel 4.90 60 70.6 0.706G100R Yes Fine Gravel 4.90 100 96.2 0.962

Each test was run for 8 h, which was sufficient for most of thetests to reach a quasi equilibrium state of scouring. However, intests S40, S60, and G100, 8 h might not have been sufficient toreach equilibrium. This was because the bed sediment thickness(0.26 m) was not sufficient to retain the maximum scour depthsin these tests.

The measurements were computer based and consisted of tem-poral bed profiles, water surfaces and flow visualizations. The bedprofiles were measured continuously during 8 h both in longitu-dinal (X) and spanwise (Y) directions. Each measurement setconsisted of seven longitudinal sections at y = 0.033B, 0.167B,0.333B, 0.5B, 0.66B, 0.833B and 0.967B and ten spanwise sec-tions in the range y = 0.033B to y = 0.967B. The spanwisesections covered the entire scoured area. Water surfaces weremeasured upstream of the spillway, at the toe of the spillway, atthe end of the scour protection plate, and along the longitudinalsections.

To visualize the flow field and the scouring process a potassiumpermanganate solution was added to the flow. A video camerawas used to record the visualizations in both XY (horizontal)and XZ (vertical) planes. The video records were analysed bya digital image processing system developed by the author (fordetails see Dargahi (1997)). The first two hours of each test wererecorded continuously, as the scour process was intensive. Later,10-minute video records were taken at 30 min intervals.At the end

420 Bijan Dargahi

of each experiment, the total amount of eroded material caughtin a downstream basin was measured. The measurements wereused to compute the geometrical properties of the scour cavity,the temporal rate of transport and the total load.

Results and discussions

The flow field

The main flow feature was the rapid development of a hydraulicjump from the initial super-critical condition. The character ofthis hydraulic jump differed from the classical stationary casewhere the bottom (bed) boundary is rigid. The mobile bed causedthe jump to undergo variation both with respect to its kinematicsand dynamic characteristics. The hydraulic jump resembled morea “shock wave” than a travelling wave that is typical of stationaryjumps. The changes in the hydraulic jump were imposed as thesequent depths and the local Froude numbers changed during thescouring process. The hydraulic jump induced large secondaryflows and clockwise vortices. Secondary flow regions resemblinglarge flow circulating zones dominated the flow patterns, both inhorizontal and vertical planes. The development of the scour cav-ity caused the flow to undergo separation at its edge. A clockwisevortex was created during the separation process. In addition, asmall roller flow zone formed within the hydraulic jump.

In contrast to the classic jump, the roller zone occupied asmall portion of the jump, whereas the clockwise vortex has nocounterpart in a stationary jump. To illustrate some of these fea-tures, reference is made to the video clips shown in Fig. 2(a)–(j).The figures show the temporal development of the vortex systemclose to the flume side wall in a vertical plane (XZ) for test G100.The schematic sketches of the corresponding idealised flow linesare also included. A cyclic behaviour was observed that lastedfor about 20 s. Figure 2(a), and (j) mark the start and the endof the cycle, respectively. During the initial stage (b–c) a break-ing tail water wave grew in height and reached the highest pointof the hydraulic jump. The wave moved rapidly upstream andthe approaching supercritical flow was not sufficiently strong topush the rising tailwater downstream. Instead, the approachingflow was pressed down towards the bed, and vortices were created(d) through flow separation. During the first six- second period(a–d), the scoured profile had no inflexion point. A double-slopescour hole was created by the vortex as it interacted with the bedmaterial. Following this period, the vortex attained its maximumstrength and was detached from the flow lines (e). The vortex for-mation was followed by a vortex breakdown stage (f) that createdsmall vortices (f and g). No regular shedding frequency could befound for the vortices. Following the vortex breakdown, the rollerflow grew in strength and finally the cycle was complete (h–j).

The secondary flow system induced a complex flow patternin the horizontal plane (XY) that was composed of several flowcirculation regions. The number of regions depended primarilyupon the flow discharge and the bed material. A two-region cir-culating flow pattern developed in all tests except G20. The twoflow regions were separated by a narrow section referred to hereas the main channel where the flow direction was streamwise. An

Figure 2 Temporal development of the scour profiles and the vortexsystem (test G100), xz-plane (video shot and schematic h : m : s).

Main scoured channel

Scour cavity

Scour cavity

Despositation

Despositation

Protection plate

Flow

Figure 3 Schematic layout of the two scoured regions and the secondaryflows, xy-plane.

example is given in Fig. 3. The number of these regions increasedto four in the gravel test G20. The increase could be due to a lowerflow discharge and a larger particle size.

An interesting aspect of the flow is that each scour hole can beconsidered as a separate flow region having a “separate” vortexsystem and secondary flows. The results also emphasize the need

Scour development downstream of a spillway 421

to consider the 3-D character of the flow field. The author believesthe flow division into several regions is related to flow instabilitiesthat are inherent in the hydraulic jump. Once the instabilitiesinteract with the mobile bed, scouring cavities develop. The flowthen becomes confined to the cavities.

No significant change in the general flow patterns wasobserved when additional roughness was added to the scour pro-tection plate. However, the vortex system became weaker and thehydraulic jump shifted upstream.

Transport mechanism

The main transport agents were the hydraulic jump and theinduced secondary flows. The temporal variation of the bedcaused significant changes in the boundary conditions. Thesechanges in turn led to complex transport patterns that includedall the modes of transport.

During the initial stage, the transport took place randomly.That is to say the intensity and duration of the transport variedirregularly. The irregularity was related to the strong unsteadynature of the hydraulic jump that maintained an oscillating posi-tion for about 30 min. Thereafter, the rate of transport decreasedand the hydraulic jump resembled more the classical case. Thescour cavities illustrated in Fig. 3 developed simultaneously in allthe tests. Each cavity caused a local flow separation that createdclockwise vortices. An example of the final scoured surface fortest S20 is shown in Fig. 4. Within a scour cavity two sedimenttransport regimes existed, one backward due to the reversed flowand the other streamwise. The mean position of the line separat-ing the two regimes for sand tests is shown in Fig. 5. In the caseof the gravel tests, the line moved upstream and the length ratiosof the two regimes were changed from 7/8 to 4/5 and 1/8 to 1/5.The results imply that the larger the sediment size, the greater theportion of the reversed flow in the scour cavity.

The general sediment transport pattern followed closely thetwo flow regimes. Some of the material carried by the reversedflow was deposited near the edge of the plate and some was trans-ported out of the scour hole. The streamwise transport led to theformation of a mound downstream of the scour cavity. In addi-tion to the foregoing transport patterns, a vortex-type transportwas observed. The sediment particles were trapped in the vor-tex circular motion and significant local erosion took place. Themotion was maintained for several minutes until the vortex wasconvected downstream by the main flow. The vortex did not afixed position, and could be convected upstream by the reversedflow or downstream by the main flow.

One issue was the transport mode and its dependence on theflow and sediment variables. The transport mode depended on theflow discharge and the grain size of the bed material. Generally,higher discharges led to a higher proportion of suspension orsaltation. Different modes existed within the scour cavity. In thetests with sand, the sand particles were suspended in the reversedflow region, whereas in tests with gravel, saltation dominated.In both cases, vortex type transport was present. In the lowerpart of the scour cavity (Fig. 5: 1/8 and 1/5), rolling/gliding andsaltation dominated the transport process. With increasing scour

Figure 4 The final scoured surface (tf = 8 h), test S20 (video shot).

78

18

Flow

Figure 5 Schematic of the flow division in the scour cavity ( sand tests),xz-plane.

time, the flow velocities were reduced and the vortex weakened.The transport mode then changed from suspension/saltation torolling/ gliding.

Erosion under the protection plate was initiated by the movingvortex system, that caused scouring near the edge of the plate.Once the scouring started, the secondary side flows in the XYplane (Fig. 3) continued to erode the material under the plate.This process was counteracted by the reversed flow (XZ-plane),that transported eroded material from the scour holes towards theplate. The net effect was to limit scour under the plate. The extentof erosion under the plate depended primary on sediment size. Ingravel tests, little erosion was observed.

The additional plate roughness had a significant influence onthe transport intensity. In all the tests the intensity reduced sharplyand the scour cavities were 18–57% smaller than the case with

422 Bijan Dargahi

no plate roughness (see scour depths in Table 3). The lower rangecorresponded to higher the discharge values.

Temporal scour variation

The temporal variation of scour is an important issue that involvesthree aspects. One is the influence of flow and sediment variablesupon the development of the scour geometry and the maximumscour depths. The second issue is whether the scour profiles fol-low a similarity law. The third issue is whether a general law couldbe found to estimate the maximum scour depths at a given time.To address these issues, large numbers of scour profiles were anal-ysed. Two representative longitudinal sections were selected, onecentre line section (Yn = 0.5) and one side section (Yn = 0.167).The choice of the sections was related to the scour patterns shown

0 20 40 60Xn

Zn

S20–250 hn=0.38

0 20 40 60Xn

Zn

S40–250 hn=0.56

0 20 40 60Xn

Zn

S60–250 hn=0.73

0 20 40 60Xn

Zn

G20–250 hn=0.38

0 20 40 60Xn

Zn

G40–250 hn=0.57

0 20 40 60Xn

Zn

G60–250 hn=0.73

0 20 40 60Xn

Zn

G80–250 hn=0.83

0 20 40 60Xn

Zn

G100–250 hn=0.958

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

Figure 6 Temporal scour profiles normalized with the spillway operating head at section Yn = 0.167, tf = 8 h.

in Fig. 3. In addition, there were significant differences betweenthe two sections regarding the development of the profiles. Thescour profiles at Yn = 0.167 had one minimum point, in contrastto section Yn = 0.5 which had several minima. The latter resem-bled large anti-dunes. Some examples are given in Fig. 6, whichshows a set of scour profile measurements at the side sectionYn = 0.167. The XZ-coordinates in these figures are normal-ized using the corresponding operating heads. The plots showthe influence of the operating head and the bed material upon thetemporal development of the profiles. With increasing operatinghead (ho) or discharge, the scour cavity increased in size and itsshape changed (decreasing values of Zn and Xn). In the case ofthe sand tests, the shape of the cavity changed from an invertedcone into an inverted frustum for ho > 0.4hd. A similar changewas observed for the gravel tests at section Yn = 0.5.

Scour development downstream of a spillway 423

The scour geometry varied considerably as the bed materialwas changed from sand to gravel. In gravel tests, the scour cavitybecame smaller and the slope angles were reduced by 10 to 20%.A common feature in all tests was that 40% of the final scourdepth was reached after about 20 min or 4% of the test duration.One important issue was where in the scoured area the maxi-mum erosion depth occurred. The maximum scour depths werefound at the side section within a distance range 10Xn − 13Xn

(Xn = x/ho). The range remained more or less independent ofthe flow and sediment variables. Regarding the sand test, slightlyhigher scour depths (5%) were recorded at the centre line sectioncompared to the side section.

The bed profiles were characterized by upstream and down-stream slopes given by angles α and β, respectively. An examplefor the gravel tests is given in Fig. 7. The live-bed transport condi-tions caused a strong periodic variation of the angles. The anglesreached at the end of the test durations are summarized in Table 2.For the sand tests, the values of α and β decreased as the oper-ating head increased, although the upstream slope was steeperthan the downstream slope. This result agrees with the positionsof the secondary flow field in the upper region of the scour cav-ity, where secondary flow could maintain a steeper scour profile.For the gravel tests, the angles were reduced but the downstreamslopes were steeper than the upstream slopes.

Similarity among temporal developments of the scour profilesimplies that the profiles could be collapsed into a single curve,if normalized using appropriate variables. A common procedureis to use the temporal maximum scour depths to normalize boththe lengths and depths. Using this approach, the temporal scour

0

5

10

15

20

25

30

0.0 0.1 0. 2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 1.0

t / tf

Slo

pe a

ngle

α ,

β in

deg

rees

G20, α G20, β G40, α G40, β G60, αG60, β G80,α G80, β G100, α G100,β

Figure 7 Temporal development of the upper and lower angles of the scour cavity-gravel tests.

Table 2 Scour cavity slope angles in degrees.

Test S20 S40 S60 G20 G40 G60 G80 G100 S20R S60R G20R G60R G100R

α 27 24 19 11 10 13 13 16 21 23 8 14 17β 20 23 17 19 18 22 21 22 13 17 7 18 14

profiles were normalized to investigate the validity of a similarityrelationship. Some examples are given in Fig. 8(a)–(e) (sectionYn = 0.167). There is a considerable variation between theshapes of the profiles with respect to time, operating head and bedmaterial size. The use of other variables such as the equilibriumscour depths also did not produce the desired results. In con-clusion, the author could not find any experimental evidence insupport of the similarity assumption. This conclusion contradictsthe findings of Farhoudi and Smith. According to their study, thescour profiles downstream of a spillway are geometrically similarif normalized with the temporal maximum scour depths. Theirresults were based only on scour profile measurements close tothe flume wall, whereas the present study considered the entirescoured area. In addition, the use of special instrumentation madeit possible to collect data with high spatial and time resolutions.It can be speculated that the Farhoudi and Smith results are relatedto mean behaviour.

In the opinion of the author it is unlikely that a similarity lawcan be found for problems involving more than one set of timeand length scales. Despite the lack of similarity, useful resultscan be found that describes some mean behaviour. The temporalvariation of the maximum scour depth (Zmx) and the volume rateof transport are two examples. Figure 9 shows the variation ofthe maximum scour, where the coordinates were normalized withthe maximum scour depth (Zf) and time (tf) reached at the endof the experiments. The data can be described with a power typeequation as follows:

Zmx

Zf= A

(tmx

tf

)n

(6)

424 Bijan Dargahi

0 5 10 15

x/Zmx, t

a: S20-250

z/Z

mx,

t

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0 5 10 15

x/Zmx,t

z/Z

mx,

t

b: S40-250

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

c: S60–250

0 5 10 15

x/Zmx, t

z/Z

mx,

t

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.40 5 10 15

x/Zmx, t

d: S60R–250

z/Z

mx,

tt

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.40 5 10 15

x/Zmx, t

z/Z

mx,

t

e: S60–750

0 5 10 15

x/Zmx, t

z/Z

mx,

t

f: G20–250

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

Figure 8 Temporal development of the scour profiles at section Yn = 0.167, normalized with maximum scour depth at time t (tf = 8 h).

in which the exponent n is a variable. Analysis showed that A ≈ 1and that n can be given as a logarithmic function of the relativeoperating head ho/hd

n = −0.17 ln

(ho

hd

)+ 0.04 (7)

A similar power type equation was found for the volume of theeroded bed material

Vt

Vf= A

(t

tf

)n

(8)

in which, Vt = volume of the transported material at timet, Vf = volume of the initial bed material and tf = total

scour time. The coefficient A and exponent n were found to beexpressible as logarithmic functions of the relative sediment sizeho/D50

A = 0.029 ln

(ho

D50

)− 0.028 (9)

n = 0.115 ln

(ho

D50

)− 0.178 (10)

The correlation coefficients (r) for Eqs. (6)–(10) were in the range0.75–0.9.

Scour development downstream of a spillway 425

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1tmx/tf

Z mx/Z

f

Figure 9 Temporal development of the maximum scour depth mea-sured, section Yn = 0.167. � S20 � S4 × S60 � G20 � G40 • G60◦ G80 + G100 – S20R — S60R.

Scour depth equations

The maximum scour depth and its location downstream of a spill-way are important design factors. To the knowledge of the author,there are no equations that deal with scouring depths downstreamof a WES-type spillway. However, scour depth equations for var-ious types of weirs, plunging jets, and stilling basins are availablein the literature (see Breusers and Raudkivi, 1991). For compar-ison purposes some of these Eqs. (1)–(4) were used to calculatethe maximum erosion depths in the present experiments. Of theseequations, the Jaeger (1939) equation gave a good estimate of themeasured scour depths (Table 3).

To develop scour depth equations, one customary approachis to use non-linear multi-regression analysis to fit equationsto experimental data. The method has the disadvantage that theresulting equation normally assumes a complex form and is diffi-cult to use. Furthermore, the use of such an equation may requireknowledge of a large number of variables. In the present work,first the main parameters that control the scouring process were

Table 3 Comparison of maximum scour depths at section Yn = 0.5.

Predicted using literature equations [m]

Present study Stocholitsch (1932) Jaeger (1939) Eggenberg (1944) Shalash (1959)Test identity [m] Eq. (1) Eq. (2) Eq. (3) Eq. (4)

S20 0.2 0.33 0.16 0.99 0.25S40 >0.26 (0.3: Eq. (13)) 0.49 0.26 1.48 0.37S60 >0.26 (0.4: Eq. (13)) 0.61 0.36 1.87 0.45G20 0.07 0.12 0.04 0.31 0.06G40 0.11 0.19 0.08 0.51 0.12G60 0.16 0.23 0.11 0.63 0.13G80 0.2 0.26 0.13 0.73 0.14G100 >0.26 (0.3: Eq. (13)) 0.29 0.15 0.82 0.15S20R 0.1 0.33 0.16 0.99 0.25S60R >0.26 0.61 0.36 1.87 0.45G20R 0.03 0.12 0.04 0.31 0.06G60R 0.13 0.23 0.11 0.63 0.13G100R 0.19 0.29 0.15 0.82 0.15

found, then a simple set of dimensionless equations were fitto the data. The equations are expressed in terms of two mainflow and scouring parameters; i.e. h0/hd, ho/D50. The maxi-mum scour depths (Zmx) and its location (Xmx) are given in Eqs.(11)–(14). Equations (11) and (12) are valid for the side sectionYn = 0.167 and Eqs. (13) and (14) are valid for the centre linesection Yn = 0.5

Zmx

h0=

(h0

D50

)1/3

(11)

Xmx

h0= 10

(ho

D50

)1/12

(12)

Zmx

ho= 1.7

(h0

D50

)1/4.5

(13)

Xmx

h0= 5

(ho

D50

)1/3

(14)

Table 3 also gives the values of the maximum scour depths pre-dicted by Eq. (13) for tests S40, S60 and G100. For these test,the maximum scour depths could not be recorded.

One other issue is to predict the time needed to reach a certainscour depth. In this study the scour depth equal to the operatinghead is chosen. The time Ts required to reach a scour depth equalto the operating head, could be estimated by use of Eq. (15).

Tsu∗c

ho= 8 × 103

(ho

D50

)−1/1.25

(15)

Conclusion

The experimental study of scouring downstream of a spillwaywith an apron (a protective plate) led to the following conclusions.

1. The main flow feature was the rapid development of ahydraulic jump from the initial super-critical conditions. The

426 Bijan Dargahi

jump induced large secondary flows and clockwise vortices.The vortices were created during the flow separation process.

2. The secondary flow system caused several scour cavities todevelop. The numbers of cavities depended upon the bedmaterial and the flow discharge.

3. The main transport agents were the hydraulic jump and theinduced secondary flows. The temporal variation of the bedcaused significant changes in the boundary conditions. Thesechanges in turn led to complex transport patterns that includedall the modes of transport.

4. No significant changes in the general flow patterns wereobserved when additional roughness was added to the scourprotection plate. However, the transport intensities werereduced sharply and the scour cavities became 18–57%smaller than in the case with no plate roughness.

5. No experimental evidence was found in support of a similarityassumption for temporal development of the scouring process.

6. The main dimensionless parameters controlling the scouringprocess are the relative operating head, the relative sedimentsize and the relative roughness of the protective plate.

7. Power type equations to predict the scour geometry werefound in terms of the controlling scour parameters.

Acknowledgments

The first series of the experiments were carried out by MatthiasBartzsch and Insa Will and the second series by Michael Bach.The reviewers provided useful advice and suggestions to improvethe manuscript.

Notations

B = flume widthDm = sediment characteristic sizeD50 = sediment size (50% finer)D90 = sediment size (90% finer)

Fr = Froude numberh0 = operation headhd = spillway design headhn = ho/hd

ks = spillway surface roughnessLs = length of the scour protection plateRe = Reynolds number

t = timetf = duration of the experiments

tmx = scour time corresponded to maximum erosion depthTs = time to reach a scour depth = ho

u∗ = shear velocityu∗c = critical shear velocity according to ShieldsVf = volume of the initial bed materialVt = volume of transported material in time t

XYZ = coordinate system in streamwise, spanwise and verticaldirections

Xn = x/ho

Xmx = x-position of Zmx

Yn = y/BZ = scour depth

Zn = Z/ho

Zmx = maximum scour depthZmx,t = maximum scour depth at time t

Zf = scour depth at time tfσ = geometrical standard deviation of sediment sizesυ = kinematic viscosity

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