Fig. 1. Schematic of
Scour in rock riverbeds
downstream of large dams
F.E.Fahlbusch, Consultant
Because of the potential risk of undermining foundations, adequate control of scour
is an integral part of dam safety for both new and existing dams. Repair of damage
caused by high energy jets can be extensive and costly, therefore enhancing
understanding of the scour mechanism and improved methods for predicting plunge
pool size is of considerable practical interest. A simple expression is proposed here
for the ultimate depth of scour below high dams in rock riverbeds.
S
cour holes or plunge pools caused by free-trajectory high energy jets associated withflip buckets, ski jumps and free overfalls are advantageous and inexpensive energy dissipators only if they occur at safe distances from valley slopes and dam foundations, where undermining is of no concern.
Controlling scour by structural means is gener-ally difficult and costly, and the difficulties and costs seem to increase with the height of dams. In very narrow valleys with limited space for the con-struction of stilling basins, it can be more economical to limit the dam height than to provide structures to keep scour within harmless limits.
Scour may impose severe limits on the safety improvement of existing dams. Increasing the spillway capacities to meet upgraded flood esti-mates and safety standards may fail to produce the desired result if the scour associated with the re-vised discharge cannot be controlled adequately.
The repair of damage caused by high energy jets is expensive, as has been shown by Mason [19841], and losses caused by a temporary interruption to the functioning of the dams and reservoirs may be even higher.
The ultimate scour depth is defined here as the depth from the surface of the tailwater to the lowest point of the scour.
1.
Theoretical considerations
Understanding of the development of scour and the interaction of the various agents involved in the process is still incomplete, and an exact ana-lytical solution to the problem still seems to be rather a long way off.
It would seem [Hartung and Hausler, 19732] that the development of scour comprises three phases:
□ initial break up of the solid rock mass as a result of dynamic pressure fluctuation in fissures and cracks induced by the impinging jets;
Control volume
1 Section
q)
Section--- - ---7
~-~-;..-::-;.:=-"!-~
-+-+==:m
<b
C
□ fragmentation of the blocks by the turbulent water; and,
□ removal of the fragments from the pool.
Furthermore, observations indicate rapid scouring at the beginning of the development of the plunge pool, with the time taken to reach the ultimate depth being shorter in soft rather than hard rock. The correlation between the ultimate depth c)f scour below the riverbed and rock type, however, is not clearly corroborated by the data base used here.
Because of the partial understanding of the phenomenon and the difficulties of using anal yti-cal solutions, model tests and prototype data have been the primary means to study the phenomenon and to derive working relationships for hydraulic design [Mason and Arumugam, 19853
]. Dimen-sional analysis is one of the most common means to evaluate experimental data and to determine the relationship between the main variables. An alternative possibility is to assume a structure that appears to be the most reasonable, such as Eq. (1), and to determine the undefined exponents and coefficients by least square, trial and error and similar mathematical methods.
qi hj
s = k - - ... (1)
gm dn
By contrast, in the approach adopted here, the basic relationship between the main variable is derived from the momentum principle, combined with a few simplifying assumptions. Underlying the adopted approach is the reasoning that how-ever intricate the scouring process and the development of the plunge pool may be, in the final state, the hydraulic and all other forces acting on a control volume shown in Figure 1 must be in equilibrium and must comply with the momen-tum equation.
Considering a plane rectangular air/water jet of total thickness d and width ofunity, and with Va'=V
(the density of air Pa being negligible compared with the density of water p), the momentum equa -tion applied in the horizontal direction to Sections 1 and 2 of the control volume [Martin, 19814] is:
1 ((1-E)pqu,+pd) sin a= F+
2
yy2 +pqu2 ... (2) where q = discharge per unit width, u = water velocity, ua=
air velocity, p=
average pressure in the jet, a=
angle of impact, D=
s +y=
the ulti-mate scour depth, F=
the force imparted by the scour wall on the control volume and £ = the av-a jet av-and plunge pool. ' - - - -- - - -- -- - ~erage volumetric concentration of the air over the
total depth of the air/water mixture.
The unknown force F depends
to
some extenton the water pressure distribution along the
plunge pool walls and on the strength of the rock
mass. Since there is no information regarding the
pressure, a linear pressure distribution is assumed
as a first approximation, giving for F the expres
-sion:
... (3)
in which the correction factor
P
is included to ac-count for the unknown rock strength and other
uncertainties. Ignoring the water pressure within
the jet and substituting Eq. (3) into Eq. (2) gives,
after some algebraic manipulation, Eq. (4):
l
qu
1 sin au
2D=s+y= - - - 1 "
-i
P
~ g u1 sina(l- E)... (4) from which the expression for the scour depth D
is eventually obtained by dropping the third term on the right of Eq. (4), since v2 <v1:
~ F . u s i n a
D=s+y= /;
-1
P
,
g\_.--'
Data reference (see figure 2)
... (5)
Source Data identification
Doddiah6 3 M X F Rajaratnam7 4 M + F Wu8 5 p ■
s
p ■s
p ■s
p ■s
p ■s
M □s
Martins9 6 p ♦ -Author's file 7 p•
s
\ p•
F p•
s
p•
s
p•
s
p•
s
M 0 F Zvoryk.inlO 8 p T p T p T Taraimovich 11 9 p...
s
p...
s
p...
s
p...
s
, p...
s
p...
s
p...
F p...
F p...
FHydropower & Dams July 1994
-The final equation, Eq. (5), with u1 replaced by
u, is a dimensional homogeneous expression
re-flecting a grouping of the key variables in
accordance with the physical process. An advan
-tage of Eq. (5) is that neither the structure or the exponents are subject to data errors, generally a
potential source ofbias in strictly empirical
formu-lae.
Eq. (5) predicts a reduction of the scour depth
brought about by disintegrntion of the jet by air or
other means. But it also shows that a notable
re-duction of the depth seems to require a substantial
·dissipation of the jet.
2
.
Evaluation
Model and prototype data published in the sources
listed in the Table were used to compare Eq. (5)
with observations. The data set contains 104
indi-vidual observations or measurements ranging
from scour depths of a few centimetres measured
in model tests to the 80 m-deep scour observed at
the lnguri dam. The largest recorded flows were
used for q in Eq. (5) and the jet velocity at the
impact on the water surface (see Figure 1) was
estimated from ✓2gH, where H is the height
be-tween the head and tail water levels. If not given,
the angles of impact were taken from photographs
or diagrams, or were estimated from hydraulic
considerations. In the absence of pertinent
infor-mation on £, the effect of aeration was ignored .
Project name Geology
(model tests) -(model tests) Kukuan -Wuchien -Tienlung -Houlung -Shimen Tachien
-Santa Rosa Rhyolite
Kariba Gneiss
Tarbela Limestone
Itaipu Basalt Ukai Basalt Picote Granite Brown Canyon -Conovogo Granite Elmali Granite Kondopoga Granite
Dnieper Granite/gneiss
Bratsk Diabase
Farkhad Sandstone
Ust-Ilim Diabase
Sayano-Shushensk Schist
· Krasnoyarsk Granite
Chicoasen Limestone
Inguri Limestone
Toktogul Limestone
> + " 0 100 10 ' JI 11: •• ~ II -.1:J>,.f ---;, 0. I I 0.0 0.001 + Fig. 2. Plot of ultimate scour depth, D, versus
(uq sin ex) lg.
32 0.01 # •
.,
a • ••....
t":.
.
. •
•
0~ ♦.,~
:..-··
•
♦ ♦~·
Model data ' ' Ooddiah ''
•
Rajaratnam ~ I :-
~ 0 Files:
0 Wu Prototype data •-•
Files ~..
Zvorykin.
Taraimovich•
Wu ♦ Martins 7 .. 0.1 10 100 1000 vqsina /gIn accordance with Eq. (5), model and proto-type scour depths are plotted as a function of
(vq sin a)/g in Figure 2; this shows principal agree-ment between theory and observation over the whole range of data. The average value of the
coefficient k
=
[2(1-e)/p]0·5 was found to be 2. 79,which is identical to k
=
2.83 implicit in Veronese'sformula published more than half a century ago [Veronese, 19375). The coefficient according to the
upper data envelope was determined to be 3.92. While the agreement between Eq. (5) with
k
=
2.79 and Veronese's formula is not coinc iden-tal, the potential underestimation of the truescour depth by 40 per cent suggests that caution
should be used in the indiscriminate use of the results of Veronese and of this study. Unfortu-nately there is no indication of whether the
calculated depths are on the conservative side or not.
The uncertainty reflected in the rather wide
scatter in Figure 2, and the fact that the upper
envelope contains prototype observations, would
suggest that until more information is available
about
p
and e, the use of any value lower than k=
3.92 must be accompanied by a careful assess-ment of the possible consequences. In cases where credible information regarding e is available, the maximum value for k may be scaled down by the ratio ( 1-e)0·5.It should be noted that the two data points
listed under the heading "Model data, files and Wu8" are based on model tests from real projects
and are plotted in prototype scales.
3. Conclusion
The momentum equation seems to offer a rational point of departure for the evaluation of the scour
depth and for unifying model tests and prototype
performance data. The wide scatter reflected in Figure 2, however, clearly reflects large gaps in the current state of knowledge of the scouring process.
Because of the scatter, the prediction of individ-ual incidences of scour based on the average performance of model and prototype data may be as much as 40 per cent away from the true value.
However, no data manipulation, however elabo-rate, would seem to be able to reduce this inherent uncertainty. While
P
can be expected to increaseand k to decrease with the quality of the rock mass,
Notation
q discharge per unit width
p density of water
y unit weight of water
u flow velocities
p average pressure in the jet
d thickness of the jet ex
=
the angle of impacts scour depth below river bed level
y tailwater depth
D ultimate scour depth
F force of the scour wall on the control volume
H head drop from reservoir to tailwater level
the data used in this paper do not lend support tc
this hypothesis.
Enhancement of knowledge on the scom mechanism and reduction of the scatter in Fig-ure 2, as well as clarification of the parameteri
P
and e, would seem to require large scale modetests. (
References I '
1. Mason, P.J., "Erosion of plunge pools downstream o
dams due to the action of free-trajectory jets", Proceed
ings, Institution of Civil Engineers, UK; May 1984.
2. Hartung, F. and Hausler, E., "Scours, stilling basin:
and downstream protection under free overflow jets a dams", Transactions, (Q41, Vol II) 11th ICOLD Con gress, Madrid, Spain; 1973.
3. Mason, P.J. and Arumugam, K., "Free jet scour: below dams and flip buckets", Journal of Hydrauli,
Engineering, ASCE, Vol. 111; February 1985. 4. Martin, C.S., "Two-phase flows" in 'Closed Condui
Flow' edited by M.H.Chaudhary and V.Yevjevich, Wate
Resources Publications, Littleton, Colorado, USA; 1981 5. Veronese, A., "Erosioni di fondo a valle di uno scarico'
Annali dei Lauori Publicci, Vol. 75, No. 9; Septembe
1937.
6. Doddiah, D, et al., "Scours from jets", 5th IAHI
Congress, Minneapolis, USA; 1953.
7. Rajaratnam, N., "Erosion byunsubmergedplanewate jets", Proceedings, ASCE Conference: Applying RE search to Hydraulic Practice; August 1982.
8. Wu, C.M., "Scour at downstream end of dams in Taiwan
IAHR Symposium on River Mechanics, Bangkok, Thai
land; January 1973.
9. Martins, R.B.F., "Scouring of rocky river beds by fre
jet spillways", Water Power & Dam Construction, Apr:
1975.
10.Zvorykin, K.A. et al., "Scour of rock bed by a jet fror a deflecting bucket of an overflow dam" 16th IAHI
Congress, Sii.o Paulo, Brazil; June 1975.
11.Taraimovich, I.I., "Deformation of channels belo•
high head spillways on rock foundation", Hydrotechnicc
Construction, No.9; September 1978.
Dr Ing Friedrich E fahlbusch is a consulting engineer with more than 30 years' international
experience in engineering and project management in
the field ofhydropower, dam engineering and other water resources projects. He is a graduate of the University of Karlsruhe and obtained his PhD in civil
engineering at the Technical University of Berlin. Currently he serves as an Alternate Delegate, Lesotho Delegation - Joint Permanent Technical Commission
(JPTC) at the Lesotho Highlands Water Project, under contract with the German Agency for Technical Cooperation (GTZ).
Joint Permanent Technical Commission, Lesotho
Highlands Water Project, Private Bag A 165, Lesotho Bank Tower, Maseru 100, Lesotho, Southern Africa.