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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
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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 -- - .- - -- - - - - --
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