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
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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
- 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-
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-
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. -
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]