Nr. 73 Mitteilungen der Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie

an der Eidgenossischen Technischen Hochschule Zurich Herausgegeben von Prof. Dr. D. Vischer

Scour Related to Energy Dissipaters for High Head Stmctures

Jeffrey G. Whittaker

Anton Schleiss

Ziirich, 1984

P r e f a c e

The fo l l owing communication d e a l s w i t h scour problems a t t h e

t o e o f dams and w e i r s and g i v e s a g e n e r a l view of t h e pos s i -

b i l i t i e s of p r e d i c t i n g t h e f i n a l dep th and form of s c o u r s

u s i n g e m p i r i c a l l y e s t a b l i s h e d fo rmulas and h y d r a u l i c model

tes ts .

Thus t h e a u t h o r s , D r . J . G . Wh i t t ake r and A. S c h l e i s s , pro-

v i d e h y d r a u l i c e n g i n e e r s w i t h a v e r y v a l u a b l e s t a t e - o f - t h e -

a r t r e p o r t and c o n t r i b u t e t o a n i n c r e a s e i n t h e s a f e t y of

s t r u c t u r e s endangered by s c o u r .

P r o f . D r . D . V i scher

- 4 -

CONTENTS Page

Abstract

1, I NTRODUCT I ON

2, BACKGROUND

2.1 Jet Behaviour in Air

2.2 Jet Behaviour in Plunge Pool

2.3 Hydraulic Jump Behaviour

3 , MODEL T E S T S

3.1 Grain Size Effects

4, SCOUR B Y H O R I Z O N T A L J E T S

4.1 Scour Following a Horizontal Apron

4.2 Scour Following a Stilling Basin

5 , SCOUR B Y P L U N G I N G J E T S 38

5.1 Empirical Equations of General Applicability 38

5.2 Semi-empirical Equations of General Applicability 42

5.3 Empirical Equations Specific to Ski-Jump Spill- 45 ways

5.4 General Comments 51

6, APPLICATION OF THE PLUNGING JET SCOUR FORMULAE 51

6.1 Cabora-Bassa 51

6.2 Kariba 54

7 , SCOUR CONTROL - PRACTICAL MEASURES

7.1 Scour from Plunging Jets

7.2 Scour from Horizontal Jets

9 , REFERENCES 65

10, ANNEX - SOME SCOUR FORMULAE 73

- 5 -

Abstract

The provision of means for spilling excess water from

reservoirs created by hydraulic structures has long been

recognised as a problem by engineers. The difficulty does

not so much lie in conveying the water to the downstream

river bed. Rather, it lies in being able to do this in

such a way that catastrophic scour does not occur down-

stream of the structure. Consequently, it is necessary

for the engineerto be able to predict the extent and lo-

cation of the scour downstream of hydraulic structures,

particuliarly high head structures, for a variety of

spillway and energy dissipator types. This report is

addressed to this problem.

Background theory is presented on predicting jet tra-

jectories and behaviour in air, as well as on the cha-

racteristics of a plunging jet in water. The role of mo-

del tests in predicting scour is discussed, and some

difficulties relating to grain size effects noted. Pre-

dicting scour caused by horizontal jets issuing from

energy dissipation basins and by plunging jets from free

overfall, pressure outlet or ski-jump spillways is then

covered in some depth. A large number of different for-

mulae are presented. The accuracy of a number of these

is checked in an application to two prototype scour si-

tuations - namely the Cabora-Bassa and Kariba dams. Some recommendations as to which formulae to use in specific

situations are given, as well as some general recomrnen-

dations for reducing or preventing scour.

Die Beherrschung von energiereichen Hochwasserabflussen bei

Talsperren und Stauwehren stellt oft ein Schlusselproblem

hinsichtlich Sicherheit der Gesamtanlage dar. Problematisch

ist dabei nicht nur die Hochwasserableitung uber das Bauwerk

selbst; die Schwierigkeit besteht vor allem darin, das Hoch-

wasser ohne starke, lokale Erosion (Kolk) ins Flussbett zu-

ruckzufuhren. Die Kenntnis von Ort und Ausmass dieser Kolke

ist fur den Ingenieur bei der Wahl der Hochwasserentlastungs-

anlage und im Hinblick auf konstruktive Massnahmen im Unter-

wasser von entscheidender Bedeutung. Der vorliegende Bericht

befasst sich mit dieser Kolkproblematik.

Der erste Abschnitt behandelt den theoretischen Hintergrund

fur das Verhalten eines frei fallenden Strahles in der Luft

und beim Eintauchen in ein Wasserpolster, sowie die Besonder-

heiten des horizontal abfliessenden Strahles im Wassersprung.

Die Rolle von Modellversuchen bei Kolkprognosen wird anhand

der Fragen, wie Wahl der Korngrosse (Massstabseffekt) und

Simulation von bindigem oder felsigem Untergrund diskutiert.

Die Prasentation einer Vielzahl von Kolkformeln soll es dem

Ingenieur ermoglichen, die Kolkentwicklung fur folgende Falle

abzuschatzen: Horizontal abfliessende Strahlen bei unter-

stromten Schutzen, tiefliegenden Auslassen und nach Wechsel-

sprungbecken; Entlastungsstrahlen bei freien Ueberfallen,

Mauerdurchlassen und Sprungschanzen. Die Anwendung einiger

Formeln auf die aktuelle Kolksituation der Bogenmauern Cabora-

Bassa und Kariba soll deren Schwankungsbereich und die Grenzen

der Anwendbarkeit verdeutlichen. Verschiedene Empfehlungen

erleichtern zudem die Wahl der besten Kolkformel fur konkrete

Fragestellungen. Abschliessend enthalt der Bericht auch einige

praktische Vorschlage zur Begrenzung und Verhinderung von

Kolken.

SCOUR RELATED TO ENERGY DISSIPATORS FOR H I G H HEAD STRUCTURES

1, I NTRODUCT I ON

Scour associated with energy dissipators of high head struc-

tures can be caused by two different flow situations, namely

- vertical or oblique free jets impinging on an erodible

bed,

- horizontal flow eroding bed material immediately down-

stream of a structure such as a stilling basin.

The material eroded may be rock, cohesive material or non-

cohesive material.

Vertical or oblique jets are obtained with the spillway

types shown in figure 1.

Free overfalls and high and low level outlets are usually

used as spillway options only in connection with arch dams.

Jet range increases as the level of the outlet is lowered. If

the energy of the jet is not dissipated mechanically at the

point of impact with the downstream river channel, scour of

large proportions can occur.

The erosion process of a rocky river bed under the action

of free jets is very complex. The resultant scour depends on

the interaction of hydraulic factors, hydrologic factors and

morphological (considering the rather complex structural pat-

terns of the scouring rock) factors. It must be remembered

that scouring is a dynamic process, and so magnitudes, frequen-

cies and durations of spilled discharges need to be taken into

consideration.

If the rock bed on which the jet impacts is fissured, tre-

mendous forces can be created within the fissures by the dyna-

C L A S S I C A L O V E R F A L L OUTFLOW UNDER PRESSURE

Small throw distance

e.g. Kariba

OUTFLOW UNDER PRESSURE

e. g. Sainte -Croix, Cabora - Bassa

S K I - J U M P S P I L L W A Y

e. g. Bort, Aigle e. g . Tarbela - -- _ - - ~ _ _ _ _ - _ _ _ _ _ ~ - - - - - - - -

Figure 1 Spillway types.

mic pressure of the plunging jet and so break up the rock ma-

trix. These forces are to some extent dependent on the angle

of the fissures. Consequently, scour may occur in some condi-

tions to depths consistent with the end of the plunging jet.

The magnitude of scour decreases with a decrease in the ratio

of jet velocity to fall velocity of the disintegrated material

(Doddiah et al. [13 1 ) . Lencastre [ 401 and Martins [ 44, 451 also

state that scour increases with increasing tailwater depth to

a critical value, and then decreases as tailwater depth in-

creases beyond this value.

With stilling basins located at the end of a spillway,

scour occurs at or near the end of the basin structure and is

caused by excess energy in the horizontal jet.

The scouring process can have two major effects:

- The stability of part or whole of the hydraulic struc-

ture(s)may be threatened. This does not necessarily have

to be caused by direct structural failure. In some cases

a scour hole downstream of a stilling basin increases

the seepage gradient beneath the structure, leading to

instability.

- The stability of the downstream channel and side slopes

may be threatened. The failure or collapse of an energy

dissipation device may aggravate this severely.

Ramos [561 mentions that hillside streams may result from '

the mixture of air and water created as a free jet tra-

vels through the air, and these could aggravate side

slope erosion.

The actual development of a scour hole depends on two rela-

ted steps [191.

- Disintegration and/or entrainment of base material,

- Evacuation of the material from the scour hole.

Entrained material removed from the scour hole may be trans-

ported downstream as bed load, or form a mound immediately at

the downstream margin of the scour hole. This mound may limit

the depth of scour [15,161, but may also raise the tailwater

to a level at which it interferes with the operation of bottom

outlets. If the mound does limit the depth of scour, the scour

is considered to have attained a dynamic limit. However, if

the mound is removed and the scour proceeds to a maximum pos-

sible extent, it is considered to have attained the ultimate

static limit [161.

2 , BACKGROUND

2.1 Jet Behaviour in Air

2.1.1 Range of J e t

In evaluating the scour caused by free jets, it is first

necessary to predict the jet trajectory so that the location

of the scour hole is known.

For the situation shown in figure 2, a kinematic theory of

free jets gives the expression

Figure 2

From this, the travel length LT of the jet can be evaluated

for the situation shown in figure 3. This is given by the ex-

pression

LT = ZO sin.20 + 2 cos O \I- (2)

Figure 3

Jet trajectory parameters.

assuming no energy loss, the median velocity vo at the exit

of the outlet being given by

Equation (2) can be transformed to give

LT ZO - = - sin 20 + 2 cos - (?l2 cos20 h h

Martins [ 4 7 ] gives graphical solutions to this equation.

The angle of incidence 0' of the jet with the downstream

river bed or water surface can be evaluated from equations(1)

and (2) ;

tan 0' = ---- I. \/sin2o + zl/z0 cos 0

Again, Martins [ 4 7 ] gives a graphical solution.to this equa-

tion. The free jet will penetrate a downstream pool at this

angle 0'.

The equations presented above predict the behaviour of an

ideal jet. Effects such as air retardation, disintegration of

the jet in flight and flow aeration (if the jet is derived

from a ski jump at the end of a long spillway) are neglected.

A number of researchers have developed equations to predict

jet behaviour accounting for these effects.

Gun'ko et al. give an equation for LT that encompasses

energy losses on the spillway. I t

Symbols are as defined in figure 3, except

Ah difference between lip elevation and bucket invert elevation (Ah - R (1 - cos 0))

q h b = -c"

q 1 $ a coefficient characterising

vb 0 J2g (z2 - hb) energy losses on the spillway

$I can be determined graphically from figure 4 (given in Gun'ko

et al. [ 2 2 ] ) .

Figure 4

Graphical solution for determination of spillway loss 0 co-efficient. 1.00 140 180 220 260 300

(after [ 2 2 ] ) . Spil lway length [m]

Figure 5 gives the ratio of actual distance traveled L to

the theoretical determined from equation 6 plotted against the

kinetic flow factor (~r?) for conditions at the lip of the

flip bucket. Figure 5 was prepared from experimental observa-

Figure 5 Jet travel length.

tions, and includes results from tests in which the spillway

flow was aerated by up to - 50 %. Lencastre 1401 concludes that

this is valid for two dimensional jets if the following cri-

terion is satisfied:

Figure 5 also contains the results of Taraimovich 1711 from

the observation of several prototype structures.

Kamenev [36] gives the theoretical jet range as

in which ho flow depth a t l i p of f l i p bucket,

ZO difference i n elevation between the axis of the f ree j e t a t the e x i t point and the f ree surface,

Z3 difference i n elevation between the l i p of the f l i p bucket and the f ree surface on which the j e t impinges downstream,

a loss coeff ic ient as defined above.

It can be seen that equation (8) can be derived from equa-

tion (2) by substituting 0 =O. Thus Kamenev's method is only

valid for horizontal ski jumps. Further, validity is restric-

ted to Fro2 < 47.

Figure 6 10

Jet travel length. 0

( a f t e r Kamenev [ 36 1 J .

Kamenev g i v e s a g r a p h i c a l s o l u t i o n f o r L/LT (see f i g u r e 6)

t h a t i s v a l i d f o r t h e i n t e r v a l s

2 0.57 < $ < 0.84 a n d 35 < Fr < 47

0 . 6 7 < 0 < 0 . 7 5 and 1 3 < ~ ? < 4 7

T h i s method assumes t h a t t h e j e t h a s a p a r a b o l i c form, and

i n c l u d e s t h e e f fec t of a i r r e s i s t a n c e i n f l i g h t . The r a n g e of

t h e je t i s g i v e n by

i n which

L = - l n ( l + Z k ~ h 6 ' ) 9 k2

(valid for Z1=O)

k = a dimensional coefficient of air resistance (L-1 T) ,

vh = horizontal velocity component of vo

6 ' (in radians) = tan-l (k vv)

in which vv = vertical component of vo.

k i s d e f i n e d g r a p h i c a l l y i n f i g u r e 7 . LT c a n b e e v a l u a t e d f rom

e q u a t i o n ( 2 ) . I n t e r e s t i n g l y , f o r vo 2 1 3 m / s , o n e a t t a i n s t h e

t h e o r e t i c a l l e n g t h . T h i s i s e q u i v a l e n t t o Gunko ' s c r i t e r i o n 2 (Fr. < 30) g i v e n ho - 0.6 m [47 1 . L/LT i s a g a i n d e f i n e d g r a p h i -

c a l l y , as shown below i n f i g u r e 8 .

F i g u r e 7 F i g u r e 8

A i r r e s i s t a n c e co- e f f i c i e n t as- a func - t i o n of v e l o c i t y .

R a t i o o f a c t u a l t ra jec- t o r y l e n g t h t o t h e o r e - t i c a l as a f u n c t i o n of v e l o c i t y .

Zvorykin et a1.[82] present an empirical expression for

calculating the effective maximum range L measured in relation

to the downstream end of the impact zone. The difference bet-

ween this and L for the middle of the jet is - 1 to 8 % , with

a median of about 4 %.

L = 0.59 (1.53) logq Z2 sin 20 + 1.3 Z3 + 16 (10)

Z2 = difference i n elevation between the f r ee surface and the l i p of the bucket.

Parameters are valid in the ranges

2.1.2 Applicabil i ty of Cited Methods

A comparison of the above methods (excluding that of Tarai-

movich [71]) was made by Martins [47] using 27 conceptual situ-

ations, and parameters as defined by Zvorykin et al. [82]. Fi-

gure 15 of [751 was used to evaluate vo. Martins [47] recom-

mends the methods of Kawakami[37] and Zvorykin et al.[82];

the results of Gun'ko et al. showed considerable deviation

from those evaluated by the other methods.

Tangent to the free surface /

Figure 9

Definition sketch for downward oriented jet.

Tangent to the lip

For a free overfall jet situation as shown in figure 9,

Martins [ 4 7 ] recommends using 0 in equation (2), where

Of course 0 is a negative quantity.

2.1.3 Transverse Cross-Section

Strict Froude similarity modelling of the effect of air on

the evolution of a jet is not possible. Consequently, a study

of the transverse characteristics of a jet in flight can only

properly be performed with prototype structures.

Taraimovich [ 7 1 ] measured the characteristics of various

jets issuing from flip buckets. Figure 10 shows the variation

in cross-section of the jet during flight. (Ro is the cross-

section property of the jet as it leaves the bucket, and R

represents the cross-section property at some distance L 1 < L.

Curve 1 refers to the total thickness of the jet and curve 2

to the thickness of the core, both measured vertically).

-

Figure 10

Curves giving change in jet parameters with flight distance.

U.S.B.R. [ 7 5 ] gives two figures (also quoted by Martins [ 4 7 ] )

for lateral divergence of a jet following two types of bucket

shape at the end of tunnel spillways.

Gun'ko et al. [22] give a formula for the lateral angle of

jet expansion B .

in which vbk = t r a n s v e r s e component of t h e v e l o c i t y i n t h e f l i p bucket .

(Note, the assumption behind.this equation is that the flow

is constrained by ribs on the spillway surface but begins to

spread laterally at or just before the flip bucket).

2.2 Jet Behaviour in Plunge Pool

Several studies have used the behaviour of a plunging jet

to derive the possible extent of scour caused by a free falling

jet 125, 28, 48, 49, 70, 791.

Tests performed with submerged jets of air and water (in

air and water respectively) have been observed to conform clo-

sely to equations developed from diffusion and turbulence theo-

ry [2, 26, 27, 60, 721. Because of the applicability of the theory

to both horizontal and vertical jets, Cola [9] states that sub-

merged jet behaviour is not influenced by gravity. Of course

this is not true for density currents or plumes diffusing in

a basin of fluid of different density, and so a jet that is

considerably aerated may in fact be influenced by gravity.

As the jet plunges into the pool, it diffuses almost line-

arly. Water from the pool is entrained at the boundary of the

jet. Plunging jet behaviour may be approximated as shown in

figure 11 (see also table 1 on page 19).

Hartung and Hausler [25] give the following information:

- y = y k at -5(2~,) or 5(2Ru)

I n t h i s zone ( 0 < y < y k ) ; vmax = vu i n t h e whole co re region.

vu is considered to act uniformly over the whole entry

section.

iure

Figure 11

Plunging jet parameters.

- at y = Y ~ I Ejet = 80 % E jet at entry (rectangular)

Ejet = 70 % E jet at entry (round) . - If the jet hits base material, part of the flow energy

builds up as dynamic pressure. At the jet centre this is

equal to the available energy head.

- Dynamic pressure reduces to zero at a distance of about

x =y/3 from the jet axis.

- For practical purposes, the end of the jet may be conside-

red be (rectangular) y - 40 (2 Bu) E - 30 % Eu

(round) y - 20 ( 2 ~ ~ ) E - 15 % Eu.

For the round jet, a plot of P,/Pu v, Ru/y confirms this

[29] by showing that the data points asymptotically ap-

proach a line parallel to the zZ/Pu axis (decreasing), at

RU/y - 0.022.

Table 1 Jet behaviour characteristics.

The development presented above assumes that the angle ai

characteristic of the reduction of the core is constant. In

fact Cii is dependent onReynolds number [4], decreasing with

C i r c u l a r j e t

1

1

1+0.507 y/yk+o. 5oo(y /yk) 2

1-0.550 y /yk+o.21 7 ( y / yk ) 2

-I12 (l+r/Ru.yk/y-yk/y) 2 e

-v2 (r/RU) 2

e

yk / y

(Y ~IY)

2 ~ / ~

0.667 yk/y

2 e -112 (r/Q-yk/y)

2 e -114 (r/%-yk/y)

increasing Re.Characteristic values of ai for submerged jets

are 40-'6O [4], although Homma's [32] data indicates a rever-

sal in the trend of yk with Re for free falling jets with en-

Rectangular j e t

1

1

1 +0.414 y / y k

1 -0.184 y/yk

- ~ / 8 (l+x/~u- ~ / y ~ - ~ ~ / y ) 2 e

-TI16 (x/BU) 2 e

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FZ-

Y ~IY

1.414 \rx 0 . 8 1 6 \ r x

-~/8 (x/B;- yk/y) 2 e

2 e -~/16 (x/BU.yk/y)

Y S Y k

Y L y k

vz - v u

Pz P u

Q - Qu

E - E u

v - Vz

P - P z

v z - v u

P z - P u

Q - Q u

- E u

v - z

P - P z

trained air and shows an ai value of lo0. Further, Holdhusen

[31] notes that a velocity distribution at the orifice of a

nozzle corresponding to a normal turbulent profile might cause

a very significant shortening of yk.

The difficulty thus arises of accounting for effects such

as aeration of the jet in flight when evaluating dispersion

parameters. Jet aeration is likely to be considerable for a

jet originating at a flip bucket, and this complicates selec-

tion of entry velocities and characteristics jet dimensions.

In the free overfall jet, Hausler [29] asserts that although

aeration occurs, a core region in the jet will nearly always

remain during the drop until the tailwater level. He recommends

ignoring aeration for this situation, or considering it by a

careful reduction of the jet impact width. Such a reduction

may be estimated from the similar behaviour of a water jet in

air (see Eck [14 1 ) .

2.3 Hydraulic Jump Behaviour

Scour occurs in alluvium downstream of a stilling basin

even with good hydraulic jump formation in the basin. This

scour is caused by excess energy that is not dissipated within

the jump.

The loss of energy in an hydraulic jump is equal to the

difference in specific energies before and after the jump. The

theoretical energy loss EL in an hydraulic jump on a horizon-

tal floor (a,B assumed = 1.0; a =Coriolis coefficient and 6 =

Boussinesq coefficient) is

where hl and h2 are the hydraulic jump conjugate depths.

However, the velocity distribution downstream of an hydraulic

jump is generally quite non-uniform and high velocity fila-

ments concentrate near the channel bed. Thus, a, f3 # 1.0, and

so the actual energy loss is <EL. The excess energy can be

called macroturbulent energy, and is given by [I81

in which vt2 - -,I is the velocity head immediately Ly downstream of the jump.

Thus the actual energy loss is EL where

The efficiency of the jump can be written

at - a2 v2L Q = (1- -) 100

EL 2g

Figure 12 Definition sketch: Hydraulic jump.

Garg and Sharma [I81 showed that Q = 100 for Fr- > 4.5, but

found that scour occurred up to FrlZ 6. This is because scour

occurs not only because of excess velocities in the transition

region downstream of the jump (i.e. where a < a2) but also be-

cause of turbulence features C21, 23, 41, 581. It has been found

that macroturbulence decays at a slower rate than velocity

distribution non-uniformities, and so scour is observed even

when the velocity distribution has become uniform. Velocity

pulsations in the flow immediately downstream of an hydraulic

jump have the structure

where is the root mean square value of the pulsating velocity component [ 2 3 , 791.

The intensity of pulsations increases with the non-unifor-

mity of the velocity distribution [23].

The structure of the velocity distribution and macroturbu-

lence immediately following an hydraulic jump in a stilling

basin depend on the form of the stilling basin as well as the

incoming flow characteristics. Thus determination of the length -9 \

required for the turbulence intensity to decay to non-erodible

values must be determined for each particular case being in-

vestigated.

3, MODEL TESTS

Most prototype high head structures are modelled before

construction. Model scour depths are then used to predict ex-

pected prototype scour depths. Such predictions can be very

incorrect. For example, initial model tests (more were subse-

quently performed) predicted a scour depth of 30 m below the

original rock surface for the high level outlet spillway of

the Kariba Dam in Zimbabwe [29, 781. By 1979 the scour depth

was 85 m below the original rock surface, and Hausler [29]

predicts this will reach 100m. In order to use model data for

predicting scour depths associated with a stilling basin or

plunge pool, the model bed material type and size must be

chosen carefully to allow scaling.

For free jets impinging on rock underlying a plunge pool

(or for a horizontal jet issuing from a stilling basin onto

rock) a difficulty arises as to how to choose a material that

will behave dynamically in the model as fissured rock does in

the prototype. In most models the disintegration process is

assumed to have taken place, removing the need to model the

dynamic pressures in the fissures and the resistance of the

rock to disintegration. This means only the entrainment and

transport of material from the scour hole needs to be modelled.

Reasonable results are obtained if fissured rock is modelled

by appropriately shaped concrete elements [45, 791. However,

both Ramos [56] and Yuditskii [81] note that the ejection of

blocks is more intense in the model over the initial stages

of scour development than in the prototype. The large number

of blocks ejected lose speed and accumulate to form a bar at

the downstream end of the scour hole. The slower rate of ejec-

tion, combined with wearing down of material within the scour

hole, result in a lower prototype bar height. This in turn

will result in a realised prototype scour depth greater than

that predicted from the model.

If the bed material is chosen carefully, good predictive

results for scour depth can be obtained by using non-cohesive

material. However, the main disadvantage with using non-cohe-

sive material is that while the scour depth may be correct,

the extent of the scour hole is much greater than would occur

in rock. For flood discharges structures located in narrow

gorges, this can be overcome to an extent by considering the

banks to be rigid, only the bed being simulated by means of a

loose granular material [56].

Steep slopes similar to those found in rock and a more re-

presentative shape of scour hole are obtained in tests with

slightly cohesive material [19, 351. Because the eroding jet

is more confined than in the non-cohesive case, cohesive mate-

rial scours more deeply.

In choosing the sediment size for the cohesive mixture a

larger sediment size may have to be used in the model than in-

dicated by scaling prototype block sizes. The model scour depth

should then be adjusted using a formula such as Kotoulas [38].

The next sub-section indicates some difficulties involved with

grain size effects.

In contrast, Yuditskii [81] considers that considerably

more accuracy is needed in modelling prototype conditions.

Block sizes, orientations and the binding effects of the fil-

ler material between blocks were modelled for an investi-

gation of scour below the Mogelev-Podol'sk spillway dam. Big-

ger cracks were left between blocks and layers. This was be-

cause it was realised that although the gradual removal (in a

way analogous to prototype behaviour) of interstitial material

is possible in the model, the weathering of rocks to a size

allowing them to be expelled from the scour hole and entrained

is not possible. Slightly smaller blocks renders this possible

once the binding material is removed.

It is possible to calibrate, in some circumstances, model

scour with scour resulting from first operational experiences

with the prototype. Eventual constructive measures can also

then be tested [24, 421.

Model tests can also be used to evaluate or choose an appro-

priate stilling basin location and geometry [591. The Conowingo

(USA) model tests [59] were subsequently validated by proto-

type behaviour.

3.1 Grain Size Effects

Care must be taken in scaling scour values obtained in mo-

del tests with non-cohesive material to prototype scales.

First, some scour formulae that could be used are dimensional-

ly incorrect (e.g. the equations of Veronese and Schoklitsch).

These will result in incorrect prototype scour values if pro-

totype variables are used. However, using model scale variab-

les (of the same range for which the equations were derived)

and then scaling the result to prototype scale should give

more correct results.

Secondly, there are two grain size limitations that affect

scour, one relative and the other absolute. Conceptually,

scour formula fall into two groups: those that consider such

grain size limitations, and those that do not.

Veronese [77lfound (for the situation shown in figure 13)

that the measured scour with a bed material size of 4 mm was

smaller than that expected from the trend given by the larger

sediment sizes. For his second series of tests reported in

Figure 13

[771 (figure 14), Veronese anticipated a similar trend for

1st #

grain sizes smaller than 5 mm. Consequently, Veronese altered

Veronese / test series 1.

the equation derived for the second series of tests, viz

to indicate that for grain sizes smaller than 5 mm, a scour

depth independent of grain size would result. The scour depth

is then given by the formula

This is suggested by the U.S.B.R. [741 as defining a limiting

. . . . - - . -

. . . . . . . . . . . . . . . Figure 14

Veronese test series 2.

scour depth. This reflects the fact that plunging jets reach

an effective scouring limit that is much more dependent on jet

parameters than on bed material size.

Machado [43] also gives an equation for scour that is in-

dependent of grain size. Mirtskhulava et al. [49] commented on

a limiting grain size effect. They found their equation over-

estimated scour (at model scale) for grain sizes < 2mm. It can

thus be expected that if a prototype has a head/grain size

ratio (or perhaps a dimensionless ratio involving discharge

and grain size) corresponding to the limiting zones of Vero-

nese [77] or Mirtskhulava et al. [49], the same limiting of

scour depth will occur.

Breusers [5] also suggests that scour depth will become in-

dependent of grain size in the range O.lmrn < d < 0.5 mm, but

seems to infer that this is an absolute rather than a relative

(e.g. to head) feature. He supports this by showing that cri-

tical velocity (assumed to be the most relevant characteristic

of the sediment when analysing scour) becomes independent of

the grain size in that range.

The following example illustrates some of the points men-

tioned above. This example is based on a model test described

by Mikhalev [48].

ExumpLe: Have an overfall scour with the following parameters:

assuming a prototype of scale 50 x the model

q = 0.011m3/ms q = 3.88m3/ms (assume d50 - 1.0 mm,

h = 0.19 m h = 9.5 m i.e. 0.05m prototype

h2 = 0.040 m h2 = 2.0 m scale)

The model test gave a final scour depth of t+h2 = 0.25 m

(12.5 m at hypothetical prototype scale). The predictions of

various formulae are listed in table 2 (note: a list of the

respective formulae can be found in Annex 1).

FORMULA SCOUR DEPTH PREDICTED t + h2 [ml -.

Eva lua ted s c a l e

Model t e s t r e s u 1 t C481

Veronese A [ 77 1

Veronese B (limiting eqn. )

[771

Schokl i t s c h 1641

W Y W [481

Smol j a n i n o v [671

Patrashew 1481

I Tschopp-Bisaz 1731

Machado B (limiting eqn.)

[43 1

Table 2 Scour predicted by various formulae - Mikhalev example.

This model test was run with a head/grain size ratio of

126.667. Veronese [77] postulated that the limiting grain size

effect would begin with a grain size of about 5 mrn, which for

his tests corresponds to a head/grain size ratio of 200.00.

Further, the grain size of 1.5 mm employed by Mikhalev is lar-

ger than that indicated by Breusers [51 as giving an absolute

grain size effect.

From the table it can be seen that the Kotoulas formulae

is still accurate at this head/grain size ratio (126.667),

even though it lies well outside the test range of Kotoulas.

The erroneous values predicted at prototype scale (from proto-

type scale variables) by dimensionally incorrect formula are

clearly seen in the last column of the table. Figure 15 illu-

strates the trends in some of the different formulae (at model

scale) for the example just discussed, if a varying grain size

is assumed.

Sediment size ( m m ) -- - .-

Figure 15 Trends in scour formulae with changing grain size.

As can be seen, the scour formulae reflect either of two

forms for small grain sizes. The equations of Mikhalev, Kotou-

las and Veronese A continue the trend given by larger grain

sizes. However, Veronese B and - Tschopp-Bisaz 1 7 3 1 (derived

from fitting an equation of different form to the Kotoulas

data) attempt to reflect the limiting by grain size commented

on above. It should be noted that a limiting of scour depth

with small grain sizes is largely an anticipated trend with

little data to substantiate it.

The difficulty of scaling results from models requiring

very small grain sizes is illustrated by the following example:

- 29 - Example :

Assume the prototype situation from the previous example

(taken from Mikhalev [481) must be modelled at 1:50, but with

dgo (prototype) = 0.02 m.

This gives dgo (model) = 0.4 mm.

A check on whether the model size selected is appropriate

can be performed using the calculation sequence given by Yalin

[80j With some assumptions, this indicates that for the given

grain size, flow in the model will only be rough turbulent if

the model is constructed bigger than -1:18. (A prototype

grain size of 0.075 m would allow the model to be constructed

at - 1:50). However, supposing the model was constructed at 1:50 and

d = 1.5 mm (0.075 m.prototype) was used. Then the scour depth

(prototype) for the dgo = 0.02 m material (prototype) could

be calculated with the Kotoulas formula:

But, it must be noted that the h/dgo value is greater than

300. Thus, a relative limiting effect may occur in the proto-

type, meaning that scaling using the Kotoulas formula may give

an excessive value. Conversely, if dgO = 0.4 mrn had been used

in the model (with a consequent lessening in scour depth as

anticipated by Breusers [5]), then the result scaled from the

model would be smaller than realised in the prototype.

4, SCOUR BY H O R I Z O N T A L J E T S

4.1 Scour Following a Horizontal Apron

In this subsection, supercritical flow is assumed on the

apron, and the hydraulic jump (either submerged or non-submer-

ged) is assumed to form over the erodible bed downstream of

the apron. The supercritical flow may result from flow down

a spillway face or under gates from medium to lower head

Form 1

v - - -

Form 2

Form 3

v - - - Wavy water, surface

Form 4 v - -

v - - - Smooth water surface

Form 5 v - -

v - - - Smooth water surface v - - Form 6

Figure 16 Effect of submergence on form of jet.

structures. The form of the scour after a horizontal apron

depends on a number of factors such as submergence, degree of

dissipation of the jet energy, level of the bed relative to

the apron etc.

Scour following an apron may be modelled by the scour re-

sulting from flow under a sluice gate. The influence of sub-

mergence on the jet form can be seen in figure 16 (after Mul-

ler [161).

In the case of the non-submerged jump, the ultimate static

limit of scour (the mound having been removed as per the

Eggenberger method [15]) is given by the following diagram

(figure 17).

8 Figure 17

Scour as predicted by Valentin [76].

The equation shown by the line in figure 17 is

This situation could result from a low tail water condition

on the apron. However, a high tail water is no guarantee that

the submerged jet will dissipate a significant amount of energy

by the end of apron, as the jet persists for a considerable

distance.

Several researchers have investigated the scour caused by

a submerged horizontal jet over an erodible bed.

Egg~nbmgm [ 7 51 performed tests with combined flow over a

weif and flow under the weir acting as a sluicegate. If the

overflow is zero, the scour resulting from the submerged hori-

zontal jet is h0.5 0.6

t+h2 = 7 .255 q (dgo in mm) (2 0 d9 OOa4O

This refers to an ultimate static limit of scour, where the

mound has been removed. In the prototype this would correspond

to a situation in which the lower than scour forming flows

would remove the mound by higher velocities due to a much lo-

wer tail water level.

MWm [ I 6 1 defined the total scour depth t +h2 for two of

the wave forms shown in figure 16. Using the head behind the

weir To,

w = 6-70 Type 4 and w = 1 0 . 2 0 Type3 (Ultimate s t a t i c l i m i t )

while for

w = 8.80 Type 4 and w = 1 3 . 1 0 Type 3 (Ultimate s t a t i c l i m i t )

The position of the scour hole for Miillers' tests is given

S h d a h [63 ] gives the depth of scour resulting from flow

under gates onto an apron with no end sill (see figure 18) as

in which 2 = length of apron bin = 1.5 h

dgo is defined i n mm.

crr

Figure 18 Scour following an apron (after Shalash [63]).

- % ----- e - - --

Fixed bed

I . . I

It is not clear whether the hydraulic jump (submerged or

otherwise) forms on the apron or over the erodible bed.

For the situation shown in figure 19, Shalash developed the

equation

&I * Figure 19 Scour following a low apron

(after Shalash [ 63 1 ) .

.

Moveable bed

where smin = 0.2 Rmin = 0.3 h

This gives

Wisner et al. [79] found a (shorter) countersloping apron

reduced scour from that obtained with a horizontal apon and a

sloping end sill.

The case where the hydraulic jump does not form over the

erodible bed is covered in the next subsection, where it is

assumed that the hydraulic jump always forms in the stilling

basin.

4.2 Scour Followins a Stillins Basin

The following discussion concerns scour following an hydrau-

lic jump in a stilling basin, irrespective of whether the in-

coming flow is from a spillway or a free overfall jet.

As an approximate guideline, Novak [53] states that stilling

basins decrease scour to about 50 % of the average of the re-

sults (at model scale) according to Veronese [77], Jaeger [33],

Smol janinov [67] and Schoklitsch [641, and to about 12% of the

value according to Eggenberger [15]. (Note, all these formulae

are for plunging jet scour).

In a later paper [54], Novak gives the scour after a stil-

ling basin as (after Jaeger [33])

where k = 0.45 - 0.65 for submergence 0 of the jump of cl = 1.6 -t 1.0 respectively.

Novak [53] cautions that scour must not be allowed to reduce

the tail water level to the point where the hydraulic jump

leaves the stilling basin. However, he also states that deep-

ening the stilling basin beyond a depth of approximately 1.05

to 1.10 times the conjugate hydraulic jump depth is unneces-

sary, and that the depth of scour is practically independent

of the dimensions of the stilling basin as far as it fulfills

the condition of holding the hydraulic jump. The passage of

bed load decreases scour markedly [54].

Catakli et al. [71 give a formula for scour at the end of

a stillina basin as

without a s i l l k = 1.62 with a s i l l k = 1.42-1.53 depending 'on t h e form.

They found that lateral beams set in the stilling basin (but

above the floor) did not decrease scour because, while dissi-

pating some flow energy, they also increased bottom velocities.

Schoklitsch [65!, 661 gives a formula

where - gives t h e r e l a t i v e proport ion of t h e weir c r e s t B2 used a s spillway ( including p i e r s ) t o the down-

stream channel width

6 r e f l e c t s t h e discharge management when more than one ga te i s ava i l ab le

a r e f l e c t s t h e s t i l l i n g bas in and hydraulic s t ruc- t u r e form (0.12 < a < 0.36)

( t a b l e s of a and B a r e given a s examples below)

and z i s a time i n hours f o r any p a r t i c u l a r q

(see Figure 20 on next page)

First, it should be noted that the formula is dimensionally

incorrect, and will only be valid at model scale. Secondly,

sediment size was found to be so poorly correlated that it was

not included in the formula. As can be seen from the formula,

scour is minimised as a' + 0.

Energy line

- - - - - - - - - - - - -

Floor of weir

1

Figu re 20 Scour fo l l owing a s t i l l i n g b a s i n ( a f t e r S c h o k l i t s c h [ 6 5 , 66 1 ) .

T h i r d l y , t h e t i . m e f a c t o r r e s u l t s i n t o o g r e a t s cou r v a l u e s

f o r ve ry long l e n g t h s of t i m e .

Table 3 Table of v a l u e s of a

Stilling basin f o r m

-- - - - - -. - - - - - - - - - - - - - - I - - - H

-- - - - - - - - - - - - - - - - - - -

----f--- H

- p- --

R - H

1.5

2.5

2.5

2.5

2.5

- - - - - - - - - - - - - --- -

- - - - - f - - - - angl e

H 1 :28.5

h' - H

-

-

- - -

1 h,,,,, I I . " 1 :19 I - I m 1 :14.3

/ I -- --

a

0.36

0.30

0.26

0.26

0.28

- 37 - C o n t i n u a t i o n Table 3

S t i l l i n g b a s i n f o r m

- - - - - - - - - - - f --- - - - - - - - - -

d ) c!xqpy '.. .. . . /

5 - L

( w i t h Rehbock den ta ted s i 11 )

- - - - - - - - - - - - A - - - - - - - - f - -

I"'

( w i t h Rehbock den ta ted s i l l )

- - - - - - - - - - - - - - - - - - - - - -

1 - --

- - - - - - - - - - I - - - - - -- -- H

I - 1 4 1

- - - - - ~ - - - I - - - - - - -- - - --

Z = l . O H i = O . l 5 H

I Z = 1 . 5 H

i=0.275 H --

b

R - H

2.5 2.5 2.5 2.5 2.5 2.5 2.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

h ' - H

0.037 0.049 0.061 0.076 0.092 0.107 0.122

0.039

0.057

0.057

0.029

0.057

0.086

0.029

0.057

0.086

0.114

-

-

ci

0.25 0.22 0.21 0.20 0.19 0.18 0.17

0.30

0.23

0.18

0.35

0.28

0.24

0.32

0.27

0.20

0.12

0.30

0.04

Discharge management L e f t I r r~ r r~ed ia te ly bank downstream o f

end o f apron

A1 1 t h r e e bays d i scha rg i ng 1 0.85 1 0 - 0.21 ( R i g h t and l e f t bays 1 0.75 1 0.75 I M idd le and r i g h t bays / 0.70 1 1.0 I M idd le and l e f t bays 1 1 . 0 1 1.0 I M i dd l e bay o n l y 1 0.85 1 0.80 I R i g h t bay o n l y 1 0.85 / 0.95 1 L e f t bay o n l y / 0.95 ( 0.95 1

R i gh t Deepest scour

Table 4 Values of f3 for a weir with three equivalent bays.

Hay and White [30] show that aeration of the flow reduces

scour. For a stilling basin with only an end sill, a bulk air

concentration of 15 -20 % reduces scour by 5 to 10 %. However,

as appurtenances are added to the stilling basin, the effect

is reduced. With a complicated,basin, scour is reduced with

or without air entrainment in the spillway flow.

5, SCOUR BY PLUNGING JETS

A number of empirical and semi-empirical equations have been

developed for predicting the scour resulting from plunging jets.

Some of these are of general applicability. Others are specific

to ski-jump spillways. The different formulae can be classified

as follows in Table 5 (see next page).

5.1 Empirical Equations of General Applicability

K u X u u R a [ 381

The Kotoulas [38] formula is

h0.35 qo.7 t + h2 = 0.78 (dgO d e f i n e d i n m)

0.4 (31)

d9 0

(Symbols are as d e f i n e d i n f i g u r e 21 b e l o w ) .

Table 5 Classification of plunging jet scour formulae.

E m p i r i c a l

Semi - ernpi r i c a l

-

Figure 21 Free overfall jet scour.

I General

a p p l i c a b i l i t y

Ko tou l as [381

Veronese A,B [77]

Schokl i t s c h L64.1

W ~ s g o [481

Smol jan inov [671

P a t r a s hew 1481

Jaeger [331

Tschopp-Bi saz [73]

S t u d e n i c h i kov [69]

M a r t i n s A [44,45 I

Machado A,B [43 I

Mi kha l ev [48 I

M i r t s k h u l a v a A,B,C [49]

Z v o r y k i n e t a l . [821

S p e c i f i c t o sk i - j ump s p i l l w a y

M a r t i n s B [461

Chian [ 8 1

R u b i n s t e i n [62]

Tara imov ich [70]

MPIRI [521

This equation was developed for a free overfall jet scour-

ing a non-cohesive bed. The final scour length &was evalua-

ted to be

and the distance of the point of maximum scour from the free

overfall as

The equation of Studenichikov [ 6 9 ] is

k = 0 . 1 f o r B 2 > 2 . 5 B o

= 0.2 f o r B2 = Bo

where Bo = width of flow on the spillway c r e s t and B2 = width of the downstream bed

hc = c r i t i c a l depth of the j e t

n i s a f ac to r allowing f o r a i r entrainment and dis- in tegra t ion of the jet. n should be > 0.7 and = 1.0 i f the j e t i s compact

where q = s p e c i f i c discharge a t sec t ion of impact and q, = i n i t i a l s p e c i f i c discharge of the j e t

dm = median diameter of bed mater ia l ,

The formula is valid for the ranges

It is intersting that this equation accounts not only for

the reduction in scour depth due to lateral speading of the

jet, but also for the reduction in scour depth that occurs

when the width of jet impact is smaller than the bed width.

Martins [ 4 4 ] notes that small material was used by Studeni-

chikov [69] in his model tests. The maximum dm diameter was

= 16 mrn, and some tests were performed with dm = 0.2 mrn.

M~~ A C44, 451

Martins gives a formula for scour in a bed of rock cubes

(assuming that in the prototype any cohesion is quickly de-

stroyed but yet no fragmentation or abrasion of rocks occurs).

The equation is 0.73 h22

t = 0.14 N + 0.7 h2 - N

where

where a = dimension of one edge of a cube.

Differentiation of equation (5) indicates that scour depth

will become a maximum at a tail water h2 value of

h2 = 0.48 N (37)

This agrees with the value derived by Martins in [44], but

disagrees with the value of h2 = 0.2 N given in [45].

Machado [43]

In reference [43] Machado gives two equations for scour of

rocky beds by jets. The first is

(dgO def ined i n m)

in which c, i s a c o e f f i c i e n t r e f l e c t i n g a e r a t i o n

o f t h e j e t i n f l i g h t .

The other equation is a limiting form of equation (38),

No explanation seems to be given as to the origin of the

two equations. However, they are quoted in a paper dealing

with a dam with a mid-level outlet. The applicability of equa-

tion (38) seems a little doubtful, as can be seen from table 1.

However, equation (39) predicts a reasonable value of scour

depth for Mikhalevls example (see section 3.1).

5.2 Semi-Empirical Equations of General Applicability

The following equations are based on a semi-empirical ana-

lysis of flow behaviour within the scour hole. The basic as-

sumption is that scour caused by an impinging jet will cease

developing when the flow is no longer able to carry entrained

material beyond the mound at the downstream end of the scour

hole. This of course depends on the horizontal velocity com-

ponents of the flow within the scour hole, and so the angle of

impingement of the jet is important.

Using empirical relations for the change in flow velocity

along y and z (see figure 22), Mirtskhulava et al. [ 4 9 ] deve-

loped an equation for the depth of scour in non-cohesive ma-

terial: 30 vu (2 BU)

t+h2=( - 7*5 (2k)) 1 - 0.175 sin 0' cot0' +0.25 h2 (40)

in which ~l = value of instantaneous maximum ve loc i t i e s r e l a t i v e t o the average ve loc i t i e s

q = 2.0 f o r prototypes a .nd0 = 1.5 fo r models

w = f a l l veloci ty of pa r t i c l e s , and may be calcula ted from

1.75 y

y s = speci£ic gravi ty of p a r t i c l e s

y = spec i f ic gravi ty of water/air mixture

For natural conditions, Mirtskhulava et al. note that the

entrance width of the jet is often

vU can be calculated from

where in many cases $I can be set equal to unity. To evaluate

y allowing for some air entrainment effects,

Figure 22

Definition diagram for scour parameters of Mirtskhulava et al. [49].

Equation (40) is valid in the range 5 < vu < 25 m/s, and for

dgo > 2 mrn. For smaller diameters dgO, ( ( 3 ~ 7 v,(2 B,) )/w - 7.5 (2 B 4 must be multiplied by a factor nl (evaluated by Mirtskhulava

et al. [491 experimentally) and which is given by figure 23.

Over the range of sedi-

ment sizes given in figure 2,

nl can be determined by the

1 equation [ 4 4 1

n1 = 0.42 \I= (45)

(dgo in mm)

0 0.5 1 . 0 1 .5 2.0

Sediment size [mml Mirtskhulava et al. [491

further present an equation Figure 23 Correction factor nl for scour of rock beds. This

8.3 vu (2 Bu) sin 0' t+h2 = + 0.25h2 (46)

1-0.175 cot0'

in which Rf = f a t ique s t r eng th t o rupture . (This i s determined i n r e l a t i o n t o t h e s t a t i s t i c a l l i m i t of compression s t r eng th [ 4 9 ] . ~ l l o w i n g f o r t h e f a c t o r s ou t l ined above regarding t h e e f f e c t of j e t s on a f r ac tu red rocky bed, -

- - - - - - -

Rf can be s e t = 0,

n = q 2 = 4 f o r f i e l d s i t u a t i o n s and - 2 .25 f o r laboratory

experiments,

m = c o l l o i d a l sediment inf luence on the flow eroding ca- pac i ty ,

m = 1.0 f o r no sediment i n flow,

m = 1.6 f o r sediment i n flow,

a ,b , c = longi tudinal , l a t e r a l and v e r t i c a l block dimensions respect ive ly .

Martins 1 4 4 1 quotes equation (46) from [50] in a slightly

different form as

4- 1

8-3 u vu (2Bu) sin 0' - 7.5 (2 Bu) +0.25h2 (47) 1-0.175 cot 0'

y sin 0' (0.6b2+0.2c2)

From 150) Martins notes that Mirtskhulava admits the possibi-

lity of quantifying the influence of a non-horizontal bed

downstream. To do this the following expression can be substi-

tuted for the numerator inside the square root part of equa-

tion (47), i.e.

\ 2 mg b c b (ys -y) cos 6 2 3c ys sin 6 ) (48)

in which 6 = angle t h e plane of t h e blocks makes with t h e hor izonta l .

In [49] Mirtskhulava et al. also give an equation for scour

in cohesiye bed material. It is similar in form to those listed

above, but contains some undefined factors. For this reason it

is not listed here.

The following figure from Martins [44] enables the correct

values of ys to be chosen for use in the formulae of Mirtskhu-

lava et al.

Figure 24 Specific masses of different rock types.

'-

$

Mikhalev used a similar approach to that employed by Mirts-

khulava et al. to derive the following equation describing

scour in beds downstream of high head structures.

1 sin 0'

a A ndesite

1 x 2 I

- IU1 I 1- 0.215 cot 0'

Dolomite

An example given by Mikhalev has already been discussed in

section 3.1.

5.3 Empirical Equations Specific to Ski-Jump Spillways

Limestone

The situation to which the equations presented in this sub-

Granite

Marble

Argill ite schists

section refer is shown in figure 25.

Rhyolite

Figure 25 Scour following a ski-jump spillway.

Sandstone

U .- .- ~rys ta l l i ne o -

J schists Q Q

Basalt Gneiss Gabbro

R u b i ~ t c L n [ 6 2 1

For a two dimensional problem, the following equations give

the dimensions of scour (quoted by Gunko et al. [22] from Ru-

binstein [62] )

The length of scour RSc is given by

D = diameter of a sphere with volume equal t o t h a t of a jo int ing block.

The coefficients E and X (from equations (50) and (51) respec-

tively) are products of a number of various factors:

and

Values of ~i and Xi are given in table 6.

Equations (50) and (51) are only valid in the range

where

Zvmykin eX d . [ti21

Zvorykin et al. [82] included in the development of their

equation an empirical determination of the distance travelled

by the plunging jet. Their equation is

in which va = admissable (non-erosive) velocity,

a = angle of internal friction, and

C = turbulence constant = 0.22.

Table 6 Coefficients for Rubinstein's equation.

- Cond i t i ons

30° - 700 en t rance ang le o f j e t

j e t non aera ted

j e t ae ra ted

Block dimensions: cub i c

1 : 1 . 5 : 2 . 0 (N1)

1 : 5.0 : 5.0 (N2)

1 : 2.75: 6.5 (N3)

Almost h o r i z o n t a l bed

D ip o f bed a t l a r g e angle , and w i t h b l ocks N1

N2

I13

The difficulty of course lies in determining Va. The equa-

tion (in this form) is insoluble if va can't be determined. -

However, Zvorykin et al. C821give

where

E i

€1 = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

€2 = 0.8

€2 = 0 . 5 - 0 . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~3 = 1.0

&3 = 1 . 0 .

&3 = 0.8

~3 = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~4 = 1.0

~4 = 0.8 -1 .35

~4 = 0.9 - 1.30

€4 = 0.7 - 1 .O

x can be substituted into equation (56), and then the equation

solved by trial and error approximation for t.

Xi

X i = 1 .8 c o s 0 '

X2 = 1.0

A2 = 1.0

X3 = 1.0

A3 = 1.0

A3 = 1.1

X3 = 1.1

Xq = 1.0

A4 = 0.8 - 1.1

X4 = 0.65 -1 .0

X4 = 0.65 - 1.0

Tahaimvvich [70]

Taraimovich [ 7 0 ] states that the time for formation of the

maximum scouring depth during construction and operation of

spillways ranges from two to seven seasons of passage of maxi-

mum discharges. The maximum scouring depth during a season

varies from 27 % to 65% of the total scour depth.

The length of the scour hole Rsc is given as

where

Rsc = (11 - 12) hc

hc = critical depth of the flow.

Taraimovich then uses this to establish a stability criterion

for safety of the flip bucket structure. Stability is ensured

The maximum scour depth below the original bed level is

t = (5.5 -6.0)hc tan @, (61)

where @, is the upstream angle of the scour hole side.

A further expression is given for establishing the total scour

depth t + h2 as

in which kr = coefficient of strength of the rock, and

ri ' = coefficient of transition from average and maximum bottom velocities to velocities on the ski jump.

In some examples cited by Taraimovich

0.9 < q'/kr w < 1.08

and so this factor can probably be treated as equal to unity.

The following empirical equations are much simpler in form

from those of Rubinstein, Taraimovich and Zvorykin et al. In

fact they reflect the form of equations such as that of Kotou-

las [38] developed for plunging jet scour.

Matim B 1461

Martins [46] evaluated the following empirical equation from

prototype observations. The equation is

0.6 0.1 t + h 2 = 1.5q Z2 (64)

Plotted points are shown in figure 26. '

Figure 26 Prototype scour depths (after Martins [461) . ( Martins1 data)

The equation of Chian for scour below ski-jump spillways is

It should be noted that, apart from the coefficient 1.18,

this is very similar to the limiting equation of Veronese [74].

Equation (65) is also similar to the Martins equation (64),

although Chian uses h instead of Z2., (The small power exponent

minimises the difference due to the use of these two different

parameters). With this in mind, data from prototype observa-

tions of Chian [8] and from some other prototype structures

were re-evaluated with the equation of Martins (equation (64))

and the limiting equation of Veronese [74], assuming that the

error arising from equating Z2 and h is small. Results are

listed in table 7. Appropriate points are plotted in figure

26, and show good agreement.

Martins1 data [46] was re-evaluated and plotted, along with

the data from table 7, in figure 27.

It can be seen that the limiting equation of Veronese pro-

Table 7 Pro to type obse rva t ions of scour .

F igure 27 Pro to type scours .

v ides a reasonable upper envelope f o r t h e sma l l e r p ro to type

scour v a l u e s observed.

90.6 ~ 2 0 . 1

[341

28.76

13.0

9 .73

24.33

11.09

10.998

9

[m2/sl

113.6

40.0

25.0

95.2

32.0

31.4 .

Equat ions (64) and (65) both n e g l e c t t h e inEluence of- se -

diment on t h e scour p rocess . However, Akhmeuov [ l ] conunents

t h a t f r a c t u r e d rocks d i s i n t e g r a t e w i th in t h e scour ho le due

t o flow a c t i o n . Thus t h e scour ing - - process could be l i kened to

t h a t i n non-cohesive m a t e r i a l , w i th t h e a p p r o p r i a t e l i m i t i n g

s i z e a s p e c t s noted i n sub-sec t ion 3.1.

t + h 2

[ m l

43.2

19.7

18 .9

30.1

17.5

1 5 . 0

q0.54 h0.225

[ 581

41.432

16.21

12.32

32.77

16.628

16.496

Z 2 , h

[ml

180

34

31

9 7

26.1

27.0

Refe- rence

C8 I [81

[81

[ 81

[82,591

[82,171

The following relationship is presented in reference [ll] for

estimating the probable depth of scour below a ski-jump bucket.

5.4 General Comments

If the jet expands in plan during flight from width Bo on

the dam crest to width Bdown at the jet entry point to the

downstream water surface, then the scour depth is reduced ac-

cording to

(Gunko et al. [22], after Solov'eva [681)

where max indicates the depth of scour in the absence of la-

teral jet expansion.

The scouring characteristics of submerged flip buckets

have been investigated by Doddiah [12]. Important parameters

are given by him in dimensionless graphical form.

6, APPLICATION OF THE PLUNGING J E T SCOUR FORMULAE

The formulae given in the previous section will now be

applied to two examples.

6.1 Cabora Bassa (Mozambique)

The Cabora-Bassa Dam (see figure 28) has a middle-level

outlet. The outlet consists of eight sluices, with the outlet

section of each being 6 x7.80 m2. The maximum discharge (at

reservoir level 326m a.s.R.) through these 8 sluices is 13 100

m3/s, and the downstream water level is at 225.10 m a.s.2.

The lip of the spillway sluices is located at elevation 244.3

m a.s.2. The following are the rele-

I rn k C r d - rd 2 4 k , 0 E C N

-4 I C Cn -4 rn Cn

E 4 C rd o n -4 a r n w a rn 0 3 . u urn rn 3 m -4 0 Cn a h4

rd - G I -

H U C H 0

6 - ad rd E O d h U 4 r n H Q) 4 J 0 - c u m -4 C a, 3 H b O k

a, 0 k S U a, 4J 4J 3 w w 0 rdOk V

03 CV

a, k 3 b -4 F=l

vant parameters:

In the model tests (perfor-

med at a scale of 1:75), the

bed was modelled as moveable,

with characteristic diameters

dg5, d50 and d15 of 35, 28 and

13 mrn respectively [55]. The

bed was weakly aggregated with

aluminous cement. Assume, then,

the following prototype dimen-

sions :

The modelled scour depth for

all eight sluices discharging

was t +h2 = 75m [55]. In Fek-

ruary 1982, t + h2, was measured to be approximately t + h2 = 68m.

The values of scour depth predicted by various formulae are listed in table 8.

Table 8

The following comments may be made regarding these results:

Pred i c t e d scour depth

t + h2 [ m l

One s l u i c e : 53

Two 5 6

58

58

68

68

84 149

89

117

136

163' 304

170

- The values predicted by the equation of Martins (eqn. (35)

are a little low. This seems to confirm the fears of Yudits-

kii [81] that scour is limited by the prematurely quick de-

velopment of the mound when using beds such as employed by

Martins in his tests. His formula then probably reflects

this limitation.

Comments

S t r i c t l y , j e t wrong shape f o r a p p l i c a t i o n . Consider scour f rom one and two s l u i c e s r e - s p e c t i v e l y

Assume a, = 30'

General equa t i on (eqn. (38) ) L i m i t i n g equa t i on (equ. (39))

Both v a l i d i t y c r i t e r i o n s a t i s f i e d

A e r a t i on cons idered negl i - I g i b l e f o r j e t i n f l i g h t

1 Non-cohesive (eqn . (40) ) Rock-scour (eqn. (46))

Assume D = 2.76 m and E = 0.8

Formula

- The formulae developed for plunging jet scour (e-g-those of

Mikhalev, Mirtskhulava et al., Kotoulas) over-estimate the

scour depth. They should not be used for middle to low level

pressure outlet jets.

Eqn. No.

M a r t i n s A (35)

MPI R I (66)

Chian (65

M a r t i n s B (64

Tara imovich (62)

Machado

S tuden i ch i kov (34)

M i kha l ev

Kotou l as (31 )

(49 )

M i r t s k h u l a v a e t a l .

Rub ins te i n (50)

b

- Rubinsteins equation (eqn. (50) should only be used for

scour caused by jets from ski-jumps located at the end of

long spillway chutes.

- The empirical power formulae developed for ski-jump jet

scour give the best predictions.

6.2 Kariba (Zimbabwe)

The Kariba Dam (Zimbabwe) has a high level outlet spillway.

The structure is shown in figure 29, together with measured

scour depths.

The following are the relevant parameters:

Brighetti [6] noted fractured blocks of 0.5 m size at the

prototype. It can be assumed, then, that the dgO size is appro-

ximately 0.3 - 0.5 m.

The measured scour depth to 1979 was

The values of scour depths predicted by various formulae are

listed in table 9.

The following comments may be made regarding these results:

- Hartung and Hausler assumed the jet to be circular at the

point of impact with the downstream water pool, with a dia-

meter of 6.9 m. The jet at the discharge point is in fact

rectangular. However, allowing for distortion of the jet in

flight, and for the shortening of yk as noted by Homma [32]

and Holdhusen [31], the plunging length evaluated by Hartung

and Hausler for the circular jet may be considered to be a

reasonable approximation for the Kariba situation.

- The equation of Mirtshkulava (eqn.(46)) should not be used for -- - .- - -- - - - - --

"-

'l-

811

$@

lay

"-'PP 'xe"-'

--

--

=

"-' L'S

059

7saJ3 40

y3

6u

a~

Table 9

cases such as described here. - - - - - - -. - - -

- Veronese's limiting equation should not be used for predict-

Predicted scour depth

t + h 2 [ml

129

4 6

7 1 112

7 8

82 84 86

165 180 2 03 238

9 4

51 1 576 833

138

Formul a

ing a limiting scour depth as suggested by USSR [ 741 .

- The jet based evaluations of Hartung and Hausler, and

Eqn. No.

Mikhalev give excellent results.

Comments

Mi khal ev (49)

Studenichikov (34)

Machado

Veronese B (18)

Martins A (35)

Kotoul as (31)

Taraimovich (62)

Mirtskhulava (46) e t a l .

Hartung and Hausl e r

\

As a general comment, it should be noted that the mound

formed by scour in rock beds does not seem to be removed by the

flow in many cases (e.g. the Ricota Dam, see Cunha and Lencastre

General equation (eqn. ( 3 8 ) ) Limiting equation (eqn. (39))

Cube s i z e = 0.5 m = 0.4 m = 0.3 m

dgO = 0.5 m = 0.4 m = 0.3 m = 0.2 m

au = 45O

Rf assumed = 0

Cube s i z e = 0.5 m = 0.4 m = 0.2 m

Evaluated from j e t theory. Assuming scour develops u n t i l P - r O

[lo]). Thus the equation of Eggenberger [15] will not be able

to be used for the prediction of scour depths in such situa-

tions. Also, water cushions are relatively ineffective in dis-

sipating jet energy, unless very deep.

7, SCOUR CONTROL - P R A C T I C A L MEASURES

To avoid scour damage, two options are available:

- avoid scour formation completely

- limit the scour location and extent.

Because of cost usually only the latter is feasible. Ramos

[561 notes that structures for scour control are usually un-

economic.

7.1 Scour from Plunging Jets

One way to control scour from jets is to have them discharge

into a very deep water pool (which may be excavated or formed

by building a small downstream dam). As noted above, water

cushions are not amazingly effective in terms of dissipating

jet energy. However, if the jet is aerated (50 % by volume) the

depth of tail water required for no scour is reduced to half

that required for the solid (or dispersed, but with no air

entrainment) jet [34]. Or, in the absence of sufficient cu-

shioning, the final scour depth can be reduced by 25 % for to-

tal air entrainment, and by 10 % for partial air entrainment

[62]. An example of a deep plunge pool is shown in figure 30.

It should be noted that in view of the potential jet pene-

tration, the pool shown is still not deep enough to prevent

scour. It appears that the grouted base rock is covered by a

concrete apron to protect the bed. Ramos [56] states that such

apron structures should always be model tested to evaluate up-

lift forces that will occur.

Figure 30 Arch Dam Vouglans (after [20]) .

If this solution is chosen, a danger exists if the main dam

is completed while the downstream dam is not. Over a duration

of approximately 20 days, the Calderwood Dam (USA) was forced

to spill flows of up to 10000 cusecs before the downstream

dam had been completed. With a fall of about 56 m to the base

material, this event scoured a hole 15 m deep at about 23 m

out from the toe of the dam. This depth of scour extended to

the depth of the foundation of the dam [31.

Another alternative to control scour is to fabricate a huge

prestressed and anchored slab at the point of jet impact.

Hartung and Hausler [25] illustrate this solution for the Ka-

riba Dam in figure 29. The slab should be of large enough ex-

tent to cover all points of impact for any spillway management

policy, and contain the hydraulic jump formed.

7.2 Scour from Horizontal Jets

As noted above, appurtenances in the stilling basin reduce

scour, but a similar effect can be achieved by aerating the

spillway flow. An optimum solution could be evaluated in terms

of the cost of providing for aeration of the spillway flow as

opposed to the cost of basin appurtenances.

An alternative solution is to design a particular stilling

basin, then use a rigid boundary model to determine how far

downstream of the basin the macroturbulence is still erosive.

Rand [57] proposes on the basis of tests that additional pro-

tection given to a length LE downstream of a stilling basin

will prevent scour. He found LE/LUN = 1.15 (at any scale)

where LUN = length required from the beginning of the hydrau-

lic jump for the establishment of uniform flow (see figure 31).

-1 h'and htd

Entrance Exit section, Section nonerodable bottom

Continuous sill or dentated sill @

I

Figure 31 Flow transition with erosion (after 1571).

Ribeiro [61] used a rigid bed model to determine (with la-

ser Doppler anemometry) the distribution of macro-turbulence

downstream of the stilling basin. An appropriate rip-rap blan-

ket was then designed to resist erosion.

a Sluice gate opening / dimension of one edge of cube

a,b,c Longitudinal, lateral and vertical block dimensions respectively

a ' Difference in height between original bed level and stilling basin outlet height

b Flow width of spillway crest (including piers). [Flow discharging through more than one bay]

d Sediment size

g Acceleration due to gravity

h Difference in height between upstream and downstream water levels / [with subscriptldepth of flow

Ah Height of flip bucket lip above invert

h' Height of end sill above stilling basin floor

i Thickness of riprap following stilling basin

k Aeration coefficient /Coefficient

Coefficient of rock strength

Length of apron or stilling basin

Length of scour hole

Colloidal sediment influence (eqns. 46, 47)

Factor allowing for disintegration of jet in flight

Sediment size (d X 2 m m ) adjustment coefficient

Pressure

Specific discharge

Drop in height from bottom of flow outlet section to stilling basin or apron

Maximum depth of scour below original bed level

Velocity

Pulsating component of velocity

F a l l v e l o c i t y / C o e f f i c i e n t o f form (eqns . 2 1 , 2 2 )

x d i r e c t i o n ( h o r i z o n t a l )

Dis tance from o u t l e t of f low t o s t a r t o f s cou r h o l e

Dis tance from o u t l e t of f low t o p o i n t of maximum scour

Dis tance from o u t l e t of f low t o end p o i n t o f s cou r (i.e. where downstream end o f scour i n t e r s e c t s o r i g i n a l bed l e v e l )

Dis tance from o u t l e t of f low t o t o p of mound downstream of scour h o l e

y d i r e c t i o n ( v e r t i c a l ) /Descending l e n g t h of p lunging j e t t o bottom of s cou r h o l e

Core l e n g t h o f j e t

Ascending l e n g t h of j e t from bottom of scour ho l e t o wa te r s u r f a c e / T i m e /Length of r i p r a p beyond end of s t i l l i n g b a s i n

B T o t a l crest width of s p i l l w a y

Bdown J e t width a t e n t r y p o i n t t o downstream plunge pool

B~ J e t width on sp i l lway

B2 Width o f downstream bed

2Bu Je t t h i c k n e s s of r e c t a n g u l a r je t a t e n t r y p o i n t t o downstream plunge pool

Cv Turbulence c o n s t a n t

C r F a c t o r f o r r e f l e c t i n g a e r a t i o n of j e t i n f l i g h t

D Diameter of a sphe re wi th volume equa l t o t h a t o f a j o i n t i n g block

E Energy /Width between d e n t a t e s i n a d e n t a t e d s i l l

E~ Energy l o s s

F r Froude number ( v / a )

H Dis tance from wate r l e v e l upstream t o s t i l l i n g b a s i n f l o o r

L Actua l je t range

Jet travel distance (I L)

Distance from start of hydraulic jump to end of scour downstream of stilling basin

Theoretical jet range

Distance from start of hydraulic jump to establishment of uniform flow conditions downstream of stilling basin

Factor of Martins

Limiting variable of Rubinstein

Discharge

Radius of flip bucket

Diameter of circular jet at entry point to downstream plunge pool

Spillway length

Depth of water above bed level upstream of a dam struc- ture

Width of dentates in a dentated sill

Difference between upstream water level and mid point of jet at exit from flip bucket

Difference between downstream water level and mid point of jet at exit from flip bucket

Difference between upstream water level and lip of flip bucket

Difference between downstream water level and lip of flip bucket

Angle of spread of plunging jet /Angle of internal fric- tion / A coefficient

Angle of reduction in core of plunging jet

A coefficient

Specific weight of water

Specific weight of sediment

Angle of dip of bed

E Coefficient of Rubinstein

rl Efficiency of hydraulic jump /Value of instantaneous maximum velocities to average velocities

' Coefficient of transition from average and maximum bot- tom velocities to velocities on the ski jump

O Angle of flip bucket, and of jet at flip bucket exit

O' Angle of jet at entry point to downstream plunge pool

X Coefficient of Rubinstein

0 Submergence of hydraulic jump

@ Energy loss coefficient/Angle of scour hole sides

52 Cross-sectional characteristics of the jet in flight

Ro Cross-sectional characteristics of jet at exit from flip bucket

0 At exit from flip bucket

1 At section 1

2 At section 2

a Admissable

b At invert of flip bucket

c Critical

h Horizontal

R Lateral

m Mean

t Excess

u Jet entry conditions to plunge pool/Upstream

v Vertical

z Along axis of plunging jet

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10, ANNEX

SOME SCOUR FORMULAE

All the following formulae have been developed for the plunging jet scour case. h and q are defined in m and m2/s, respectively, and g in m/s2.

(Kotoulas [38] incorrectly gives d as dgo for Veronese A and ~aeger).

limiting equation: t + h2 = 1.9 h0-225 9 0.54

k defined in a table in a reference given in Mikhalev [48]

dgO [m]; for 0' > 60°, k " 1