ANALYTICAL AND EXPERIMENTAL STUDY
OF BED LOAD DISTRIBUTION
AT ALLUVIAL DIVERSIONS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAG-NIFICUS DR. R. KRONlG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUUR-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 14 JUNI 1961 DES NAMIDDAGS TE 2 UUR
DOOR
KAMAL RIAD
GEBOREN TE MEHALL EL KOBRA, EGYPTEDIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. IR. J. TH. THIJSSE
CONTENTS
Chapter
Abstract . . Introduction
I. Movement of sediment as bed load in open channels 11. Previous studies of the problem of canal bifurcation 111. Analytical study of the problem of canal bifurcation
A. Flow in open channel bends. B. Flow at bifurcation
C. Analytical approach
IV. I. Description of installation and measuring devices .
Page 7 8 9 18
26
26
32
35 39 11. Calibration of measuring devices and determination of theroughness of the walls . . . 45 V. Experiments on the ra te ofbed load movement in the main flume 53 VI. Experiments on sediment distribution at the bifurcation
A. Experimental procedure
B. Analysis of experimental data and observations I. Experiments Ce C l5 and Dl . . . .
Relation between streamlines and the statie head 11. Experiments EeE4o, F cF 4' GeGa and HeHlO Conclusions Summary in Dutch Main symbols References Appendix 53r 58
62
62
76 81 97 99 100 103 105ACKNOWLEDGEMENT
This study has been carried out in the Delft Hydrau-lies Laboratory, with the permission of the Board of the Foundation of Hydraulic Engineering Laboratories and
under the supervision of Prof. Ir.
J.
Th. Thijsse. Thewriter would like to express his deep gratitude to those
who helped to make this study possible through their
encouragement, instructive criticism, and support.
The writer wtmld also like to thank Ir. H.
J.
Schoe-maker, director of the laboratory, and the stafffor their cooperation in making available the excellent facilitiesABSTRACT
It has long been observed that at most canal bifurcations the water diverted to the branch does not carry sediment in direct proportion to the rate of flow. Usually, the major part of sediment reaching a bifurcation is diverted into the small branches.
This phenomenon has always bothered engineers responsible for the main-tenance of irrigation and navigation canals which branch off relatively large alluvial streams.
Experimental studies of this problem have usually been limited to the use of fixed bed flumes in which the velo city of flow was measured at different sections in the vicinity of the bifurcation.
The distribution of the velocity both vertically and horizontally were then determined and considered as the basis of comparison between different cases. Some investigators studied the pattern of flow ne ar the bed either by the introduction of sediment particles or pottasium permanganate crystals.
In the present experimental study, sand was used as bed material and measurements in any run were only taken af ter the sand move ment had reached equilibrium, when the rate of sediment feeding was equal to the sum of the rates of sediment being trapped at the end of main and branch channels. The experimental set-up consisted of a straight flume 20 m long and 0.80 m wide which represented the main canal and a 10 m. X 0.50 m flume which branched off the main flume at 45 degrees, 8.20 m. from the upstream end and which represented the branch canal.
At first a series of tests was carried out without a sand bed in order to study the wall roughness. Then the sand bed was introduced and a series of tests was carried out to determine the effect of the ratio between branch and main canal dis charges up on the sediment behaviour at the bifurcation.
In order to con trol the rate of sediment diversion into a branch, some artificial means have to be applied. In this respect the writer has experimented with the application of dividing walls which direct the bottom flow and guide vanes which direct the surf ace flow. In general and within the scope of the experiments, the guide vanes gave the better results. Hence, tests were concentrated on the determination of the best location and direction for such vanes, and the results of these experiments led to the recommendations described on fig. 50.
INTRODUCTION
In hydraulic engineering as weIl as in geology several problems of sediment transportation are usually found. The design of irrigation canals, navigation canals, river training works, docks, harbours, and similar works, all necessitate a knowledge of the existing conditions of sedimentation and scour and require prediction of the effe cts likely to be produced by the proposed works. This is usually quite difficult, because of the complex nature of the phenomena.
A common problem in sediment transportation is the behaviour of sediment at the bifurcation of an alluvial channel. Usually, sediment tends to enter the branch at a higher rate than its carrying capacity and causes sedimentation in the branch intake as weIl as a reduction of its efficiency in conveying irrigation waters. In Egypt, for instance, a great deal of money is annually spent for the maintenance of irrigation canals.
This study is an attempt towards a better understanding of the mechanics of sediment move ment at the branching of alluvial channels and the use of artificial means to control it.
Numerous investigations have been made with the object of defining 'ind evaluating the parameters governing sediment transport in channels. Before proceeding to the specific object of the present investigation, a brief review of the results of previous work carried out in the general field of sediment transportation is presented in Chapter I and a summary of previous investiga-tions in the particular field of canal bifurcation is given in Chapter 11.
CHAPTER I
MOVE MENT OF SEDIMENT AS BED LOAD IN OPEN CHANNELS
In the last century, Du Boys [1] introduced the concept that bed load movement is due to a tractive force T which is the shearing stress exerted by the flow on its boundary. The average tractive stress in uniform flow is the component - along the flow - of the weight of water divided by the area of the wetted element of the boundary. The value of the tractive stress which is required to start the general movement of the bed particles is called the critical tractive stress Tc.
Du Boys gave the following equation for the ra te of bed load move ment in a wide open channel:
qs = CsT(T - Tc) . . . (1-1) where qs = ra te of transportation of bed material in volume per unit width, Cs
=
a coefficient depending upon sediment characteristics, T=
tractive stress and Tc = critical tractive stress.STRAUB [2] summarized the results of various investigations, and gave the average magnitudes of Cs and Tc for different sediment sizes with specific gravity 2.65 as tabulated below:
size of sediment (mm) '/8 '/
.
'j" 2 4 T {Ibjsq.ft 0,016 0,017 0,022 0,032 0.051 0.09 c kg/sq.m 0.078 0.083 0.108 0.156 0.250 0.44 C {ftGjlb2 sec 0.81 0.48 0.29 0.17 0.10 0.06 s m6/kg" sec 0.0032 0.0019 0.0011 0.00067 0.00039 0.00023He has also shown that by introducing the Manning formula, equation (1-1) can be written in the following form:
y2S1.4qO.6
q s -- C s (1.49/n)1.2 (qO.6 - q 0.6) (1 2)
C . . .
-where, y = specific weight of fluid, S = slope of energy line, q = fluid dis-charge per unit width, qc = fluid discharge per unit width at critical conditions of sediment movement, i.e. at the beginning of bed load move ment, and
CHANG'S [3] data for bed load movement in an experimental flume led him to the conclusion that Du Boys equation was valid, and his proposed formula:
gs = C'nT(T- Tc)/T} . . . . . . . (1-3) is almost identical to equation (1-1) of Du Boys. In equation (1-3) C' is a coefficient depending upon sediment characteristics and gs
=
rate of transpor-tation of bed material in dry weight per unit width.The main objections to the Du Boys equation, however, is the use of certain assumptions in its derivation which are far from reality. These assumptions may be summarized as follows:
1. the bed moves by the sliding of top layers,
2. the velocity distribution in these layers is linear, and 3. the friction factor between these layers is constant.
O'BRIEN & RINDLAUB [4] tried to minimise the effect of the constant friction assumption by considering the friction to vary exponentially with the depth below the bed surface. The same application could be used for the variation of the velocity in order to remedy the second assumption. Assuming that in the case of dynamic equilibrium the shearing force must be constant on planes parallel to the bottom, and that the sand has a static friction which depends upon the weight of the material above any layer, they developed the following equation:
gS= K2(T- Tc)m . . . (1-4) This expression has been used by the U.S. WATERWAYS EXPERIMENT STATION [5] with a value of m = 1.5 to 1.8 and K2 = Kl/n, where Kl depends up on sediment characteristics.
O'Brien & Rindlaub also tested the validity of the equations which had been given for the critical tractive force when applied to the experimental results on 39 sands. They found that to obtain the best correlation of the data, a simple equation of the form:
Tc = 3.5 d . . . (1-5)
could be used, where d
=
mean diameter of the sand bed material in ft, and Tc in lbs/ft2• For din metres and Tc in kgs/m2, this equation becomes:Tc = 600 d. . . (1-5) SHIELDS [6] postulated from a particle entrainment analysis backed by a wide range of experiments that the force exerted upon a sediment particle could be expressed in terms of the usual drag relationship as follows:
FD = Cyd2v2/2g = q;(abvd/v)yd2v2/2g
In which al = a shape factor, v = characteristic velocity of the flow at a height a2d above the bed, d = mean diameter of the bed material, a2 = a
dimensionless coefficient, v
=
kinematic viscosity of the Huid, C=
a dimen-sionless coefficient depending upon al & vd/v, v =V'
T/e[5. 75 log a 2+
+
f{JI (dV'T/e/v)]=
V'T/e f{J2(a 2, dv'Tfe/v) and e=
Huid density.The drag force was then expressed by the equation :
FD
=
Td2f{J3(afl a 2, dv'gDS/v), where D = depth of How.The resistance of the particle to motion, FR, was assumed to depend only upon the bed form and the submerged weight of the particle.
FR
=
a 3(ys-y)d3, where a3=
a dimensionless coefficient.At the beginning of bed load movement the boundary shear should be just enough to overcome the resistance to motion and therefore:
Tcd2f{J4(
afl a 2, dV'gDS/v)
=
a3(ys-y)d3•Shields further simplified this relation and introduced the following equation for the critical stress for a level bed of uniform size sand:
Tc/(Ys-y)d =j(dV'gDS/v) . . . (1-6) Since the thickness of the laminar boundary layer b is equal to II.6v/
v'
gDS, equation (1-6) can be written in the form:Tc/(Ys-y)d = h (d/b) (1-7)
For the ra te of sediment transportation he gave the following empirical equation:
gs/yqS = 1O( T - Tc) / (Ys-y)d . . . (1-8) Shields gave the relation described byequation (1-7) in the form ofa diagram. An interesting analysis of the validity of many of the numerous empirical equations which had appeared in the literature, was made by J OHNSON [7] in his
discussion of Chang's paper. Johnson compared a nu mb er of these equations by applying them to the data taken from the U.S. Waterways Experiment Station [5]. By means of statistical analysis of various plotted data, he found that any of the equations fitted the data as weIl as any other. From this, he concluded th at the choice of an empirical equation for bed load movement could be made on the basis of the convenience in measuring the variables involved.
MEYER-PETER & MÜLLER -[8], in-1948 published 'an empirical formula for
bed-load transportation, based upon the work that had been carried out at the laboratory of Hydraulic Research In Zurich in 1934. _
The original formula which was based on experiments with material of uniform size and wi th specific gra vi ty y s = 2.68 can be exptessed ~s follows:
.:/\
where qb'
=
~b
= specific discharge, Q;b is the part of total discharge whose energy is converted into eddying on the bed and causes the movement of bed particles, B is the bed width and a &f3
are coefficients (considered to be constants).For materials with uniform size and specific gravity other than 2.68, a &
f3
we re found to be variables depending on the specific gravity.Tests with mixtures yielded unsatisfactory results and the deviation from equation (1-9) was substantial.
Concerning the beginning of bed load transportation, some special runs were carried out with material consisting of uniform size particles. The authors stated th at these runs showed that the beginning of movement definitely depended upon the energy gradient such th at :
(qlb·
I•S) , (qlb'I'S'I')
-d- 0 =11SI• or - -d- 0
=
const.=11·
Applying Strickler's formula for the mean velocity V = KbRb·I·S'/. where
c
Kb
=
roughness coefficient of the bed=
'-1 and d=
diameter of the particled '
. d · h - h . I Q,'b D V h 1· .. h .
In m, an wIt t e equatlOn qb =
B
=
Yw· . , t e lmltlng s eanng stress Tc was defined by the-empirical equation::c
= K3(:J
/
...
(1-10) 'Ij;In equation (1-10), the constant K3 = Yw.I' 1 and has the dimensions of c •
specific gravity. Introducing the effect of bed profile, the energy conversion was divided into two types, one due to 10ss occuring through the unevenness of the bed (ripples and dunes), and the other to the loss due to particle roughness.
The energy gradient corresponding to the latter, was called the frictiona1 fall Sr:
Sr =
S(~:r
...
(1-11)The pure frictional fall Sr was then introduced into the bed-load trans-portation equation, and the following empirical formula was developed:
(Q;b) (Kb)'I' DS _
I I g/I.
1'10
Q,
Kr dm - a+f3
dm . • • • . . . (1-12)in submerged weight per unit width and dm = effective diameter of sediment. Experimental runs for the determination of Kr led to the equation:
_ 26 'I. Kr - - )(d '1 m Isec.
90 •
(1-13)
where d90 was the diameter at 90% finer in metres.
The final form of equation (1-12) for bed load of specific gravity 2.65 was given as follows:
Yw
(~) (~:
r
DS = 0.047 (Ys-y)dm+0.25(Yg
w
)"'gIB·
I• • • • (1-14) WHITE [9], by means of laboratory experiments, determined the critical conditions under which the movement of a sand bed began. He distinguished between two cases of the beginning of movement. The first is that of slow speed .and small grains, in which the force applied is th at of viscous stress acting tangentially and where the pressure at the front of the grain does not appreciably exceed that at the rear. The second case is that of high speed and large grains where the pressure difference between the front and the rear of the grain is the dominating force, while the horizontal drag is unim-portant. Assuming th at grains having a diameter d will occupy more area than d2, he introduced a packing coefficient rJ' = d2N, where N = number ofgrains per unit area. If T = the drag exerted by the flow per unit area, then
TI N = Td2/rj'. The resultant of the submerged weight and the drag should,
for equilibrium, fall below the value of the angle of repose cp, and therefore: Tc
=
a"rJ'(nI6) (Ys-y)d tan cp • • • • • • • • • • • • • (1-15) a" was introduced to allow for the difference in level between the drag force and the centroid. For V*dlv more than 3.5, a" = 0.5, and for V*dlv less than 3.5, a"=
1V* is the friction velocity and equals
V T
Ie.V*dlv is called the Reynolds number for the grains
Rg.
TISON [9] indicated that there is no abrupt change in the value of a" when Rg exceeds 3.5, but there is a transition zone which extends up to Rg = 70 in which a" is smaller than unity.
LELIAVSKY [11] mentioned th at there also appears to be two more question-able factors in White's work. The first is that of the angle of repose of the bed material. Secondly, these experiments we re carried out in a very small exper-imental flume where the secondary currents influence may have been negligible. Applying their results to fuH size rivers, Leliavsky stated, would probably yield to misleading conclusions.
The above mentioned authors based their formulae for bed-load movement upon the criterion that there is a shearing stress which the flow has to exert
upon the bed sediment in order to start its move ment as bed load. The writer
would like to stress the fact that all dimensionally correct equations' which
have been given for this critical shear stress can be put in the form:
Tc/(Ys-y)d = C. . . (1-16)
in which C was reported either a constant, depending upon particle shape
and size distribution, or function of dlb where b is the thickness of the laminar
boundary layer at the beginning of move ment. The latter case is according
to Shields, and is in a way a little misleading. Since b at the beginning of
move-ment is equal to II.6vl V*, the result is that once dis known, and for a certain
water viscosity, Tc can be determined by trial. It would have been easier in
application had Shields given a table or a curve for the variation of C with
the diameter rather than with dlb which seems to have been introduced for
the only purpose ofdimensionless plot. Of course the approximation would
have required the assumption of an ave rage viscosity for the water.
On the other hand, EINSTEIN [12] criticised the use of the term "critical
tractive force", stating that a distinct condition for the beginning of bed load
move ment does not exist. He defined the bed load movement as a slow
down-stream motion of a certain top layer of the bed which takes place in quick
steps with comparatively long intermediate periods of rest. Together with the
concept th at bed load movement is governed by the laws of probability, he
adopted a statistical appr.oach to the probelm. Assuming that Ps is the
prob-ability that a particle will start moving in a given unit of time, the rate of
transport was expressed by the following eq ua tion :
or
g'sl(es-e)g = LA 2d3PsIA
ld2
g's AI/(es-e)gd2A2 = ÀoPs .
( 1-1 7)
(1-17)'
where Al & A2 are dimensionless constants such that: Ald2
=
area which thegrain covers in the bed & A 2d3 = volume of the particle, Ào = Lid, and L
is the length of one step, d = diameter at 40% finer.
The time til required to move a particle from its place was assumed to be
proportional to the time t required for a particle to settle in the Huid a distance
equal to its own diameter i.e. til = A3t.
I
t = dl V, =
F
V
delg(es-e) (1-18)in which: F = V,I vgd(es-e)le
= v2J3+36fl2Igd3e(es-e) - v36fl2Jgd3e(es-e).
If
P
=Pst",
equation (1-17) could be put in the form:A A {I g'
(e
)'
/2
1 }The same probability
p
was also expressed in terms of the probability that the hydraulic lift is about to overcome the weight of the particle.p =!
(A2d3(es-e)g)(A4d2V2e) (1-20)
Equations (1-19) & (1-20) led to the introduction ofthe CP-lP relation:
cP =!(lP), where: }
cP
=
Qse,/,/vg(es- e)Fd1.5 & . . . (1-21)lP
=
d(es-e)/SRecP includes the rate of transportation and the size and settling velo city of the particle, while lP describes the ratio' of forces acting upon the particle. The
interrelation between these two functions was supposed to express the law
of transportation.
In order to isolate the effect of different systems of roughness in an alluvial channel, EINSTEIN [13] later divided the bed roughness into that caused by
sediment grains of the surface as a rough wall, and that caused by the
devel-opment of the bars. Again, he divided the cross sectional area A into two parts, A' contributing the shear which is transmitted to the boundary along the
roughness of the grainy sand surface, and AU contributing to the shear trans-mitted to the boundary in the form of normal pressures at the different sides
ofthe bars. Two hydraulic radii were then defined as R' = A'/B & RU = AU/B. N aturally R'
+
RU = Rb the hydraulic radius for the bed.According to KEULEGAN [14]:
V/V~
=
3.25+
5.75 log (R'V:/v) . . . (1-22)for a hydraulicallY smooth bed, and
V/V~ = 6.25
+
5.75 log (R'/ks) ( 1-23)for a hydraulically rough bed wh ere ks = absolute roughness &
V:
= vgR'S.The entire transition between the two cases inclusive of the extremes was th en expressed as fOllOWS:
V/V~ = 5.75 log (12.27 R'x/ks) . . . (1-24)
where x is a function of ks/(l/ & (J' = II.6v/ V~.
The exchange probability was expressed in the same way. Using the normal
error law the probability
p
for motion was given as the probability integralwith limits ±B*lP* -1/'Y/o, and the final bed load function was derived:
B.'P.- 1/TJo
p
= I - I_fe-t ' dl = A*IP*/(I- A*cp*)vn
-B.'P.-1/TJ. . . . (1-25)where A* and B* are universal constants, 'Y/o is the standard deviation of 'Y/
Although d40 was Einstein's choice for sediment characteristic size in deriving
his bed 10ad equation in his earlier paper, he and ELSAMNI [15] proposed the use of d65 to represent the absolute roughness of bed material. Since the
roughness of the bed corresponds more to the particles which are 1eft behind than to those set in motion, the choice of coarser diameter for the roughness seems logical.
KALINSKE [16] based his ana1ysis ofbed movement on two concepts, namely: that there is a minimum Tc needed to set the bed in motion, and th at the force
acting on a particle is not constant but Huctuates about some mean va1ue. He adopted White's method for the determination of the critical shear stress.
Tc = 2Jaa"p'(Ys-y)d tan cp • • • • • • • • • • • • • • (1-26)
in which
p'
=
a factor indicating the proportion of the bed taking the shear = 0.35 (according to White).If a" = 0.5 for Rg
>
3.5 and tan cp = 1.Tc = 0.12(Ys-y)d • (1-26)
In an earlier investigation by Kalinske [17], he showed that the turbulent velocity HtrctjlaÜon near the bottom of a river is such th at the maximum velocity is 1. 75
io
2 times its mean va1ue. Since the shear stress approximately varies as the square of the velocity, the mean value of the tractive stress required to set individual grains in motion can be one fourth to one third that given by equation (1-26).For the estimation of the average amount of transportation, the average velocity of the individual grains Ug was first expressed by the following equation:
Ug = C(U- Uc) . . . • . . • ( 1-27)
where U = the instantaneous Huid velocity at grain level, Uc = the Huid velocity at critical conditions and C is a factor equal to unity.
The ave rage transport:
p'
ng.
=
nd2J4·6 d3Ug=
2Jap'dUg (1-28) Kalinske then developed his equation :(1-29) where
p
'
=
the proportion of the bed that moves,=
about 0.35 according to White, C=
a constant of about 7.3, 8=
a certain function which was defined by the analysis of collected data.Müher's formulae for bed load move ment agree quite well, and that they may be approximated by the following relatively simple formula:
q8 = 5e-O.27.tJdmll"RS
dm"/gf-l'RS
where f-l'is a coefficient that indicates the part of the mean shear stress To that should be used for calculating the bed-Ioad (called a ripple factor) and
A' h l ' d . f h . d {!8 -Q
LJ IS tere atlve enslty 0 t e gralns un er water,
=
- -
.
(!
Although Frijlink's study constitutes a good step towards a dependable formula for the rate of bed load movement, yet it should be kept in mind that all the formulae which he combined contained some empirical assump-tions in their derivation. While Einstein and Kalinske had both used some theoretical approach, they have had to turn to empiricism in defining their proposed functions. Meyer-Peter and Müller applied empiricism from the start.
A good summary of the progress of research in sediment transportation was given by NING CHIEN [19] in 1956.
In studying the subject of sediment transportation, one finds that every investigator was able to show that his suggested formula agreed with some collected data. This may not be as surprising as it sounds if one realises that most of the formulae are plotted logarithmically, where a large error may not be apparent.
BROOKS [20], impressed by this fact, carried out several experiments in a glass flume and suggested th at a good correlation between the sediment movement and the hydraulic conditions was not possible. However, in their discussion, of Brook's paper, CARLSON and MOSTAFA [21] have thrown some doubt upon this conclusion.
It seems logical th at the hydraulic conditions existing in an alluvial channel, namely: the slope, the depth and the water section, should somehow be correlated to the rate of sediment move ment and also to the bed configuration at equilibrium. Since sediment transportation by flowing water involves not only fluid mechanics but also the mechanics of movement of solid particles, a purely mathematical solution to the problem has so far not been successful. It is still hoped that sediment transportation may some day be rationally expressed in an equation which will be acceptable to all specialists in this field. The writer has therefore been convinced that in studying a more particular subject in the field of sediment transportation, a subject which involves not only the move ment of bed "material, but also its scour and deposition, it was advisable to rely upon proper experimental investigation.
CHAPTER II
PREVIOUS STUDIES OF THE PROBLEM
OF CANAL BIFURCATIONS
BULLE [22] stated that the first experiments known to have been carried out for determining the movement of bed load at a diversion point were conducted by H. THOMA in connection with a sedimentation problem at a power plant on the Mittleren Isar river. In his study at the Technical Univer-sity of Karlsruhe, Bulle used an experimental flume rectangular in cross section, 20 cm wide and with a bottom slope of 0.003 for the main canal and for each of the two forks. The total length of the flume representing the main canal was 6.43 meters. At 2.5 meters from the upstream side a 3.93 meters long branch channel, took off. Adjustable weirs were used as tail-gates at the downstream end of both canals.
He investigated the effects of diversion angle which was varied between 30° and 150° degrees, streamlining the corners and the transverse surface slope on the bed load distribution at the diversion. Fig. 1, which is reproduced from his publication, shows the percentage distribution of discharge and of sedimentary material for different
angles of diversion with sharp 100"., - - , - - - - , - - - , - - - , - -, - - - , - - - , - - - ,
edges at the diversion point. He
I
I
I
I
I
also tested the effect of stream
-lining the corners at the branch entrance for 30°, 60° and 90° degrees. His results can be sum-marized as follows:
1.
J
ust inside the branch canal and opposite to the point of diversion an eddy was formed. lts size varied with the angle and form of diversion. This eddy reduced the effective sec-tion of the branch and in-creased slopes at its mouth. 2. For smaller discharges andwith the tail-gates set at the same elevation, the discharge flowing in each branch was
90 '0 70 60 50 t--+---...,...----~ - - - - --40 30 I ~ 4 I x I l's I E w I o~ I ~ I 20 I
"
I I"
, :;.N1 T 10 "0 J)..0·
90·,,..
,...
,
...
Fig. 1. Distribution of sedimentary material in a branching canal for different angles of diversion.
about the same. For greater discharges, the main canal carried more than 50% of the discharge, and the least discharge was carried by the branch
when the diversion angle was 90° degrees.
3. The least percentage of deposits in the branch canal was for a diversion
angle of 120° degrees.
4. Bed material showed almost no tendency to continue in the straight
channel past the diversion point when the angle of diversion was 30°
degrees and tail-gates at the same level. By increasing the discharge in the main canal while decreasing it in the branch more material could be'
made to move down the straight channel.
5. The existance of rounded corners tended to decrease the branch con-traction produced by the eddy, and also reduced the transverse slope at the point of diversion. More water, but less sediment, was carried into the side channel as a result of the streamlined corners.
6. With equal discharges in the two channels, a decrease of one half in the width of the side channel did not produce a proportionate decrease in the bed material diverted into it.
VOGEL [23] reported the results of experimental study carried out at the U.S. Waterways Experiment Station, Vicksburg (U.S.W.E.S.) following the
same general lines of Bulle's study. The first set of experiments were carried
out in a rectangular flume 2 ft wide. The angle of diversion of the branch was 30° and its width was the same as the main channel.
The discharge above the diversion was held practically constant, while varying the percentage diverted to the branch. Two kinds of sediment were used: loess, which was very fine and largely carried in suspension, and Red River sand, which was not as fine. The results are shown in the following tabie: Flow in branch as
percentage of total flow 65 49.4 48.8 34.9 30.2 29.9 24.2 23.3 15.9 Deposits in branch as
percentage of total deposits 85.1 75.9 75.2 63.6 45.3 51.3 37.1 38.2 17.9 Bulle's results showed that the percentage of sediment diverted in the case where the angle of diversion was 30° and with 50% of the discharge flowing in the branch was 97.3%.
The difference may be due to either or both of the two following reasons : 1. Red River sand which was used in these experiments, and which was reported to be rather fine, may have been partially carried in suspension. The suspended material, which would be more or less divided in proportion to the flow, might have been deposited below the diversion where it was measured. In this case, the recorded quantity of deposits in the branch would have been less than if it we re transported entirely as bed load.
2. Sand traps were not provided, and later experiments showed that sand
was carried past the end of the branch before it reached the end of the
main canal. Such lóss, if was accounted for, would probably have in-creased the percentage deposited in the branch.
The second set of experiments was performed in a semicircular flume
which had a radius of 2 ft for the main canal and one ft for the branch. The bottom of both flumes were in the same horizontal plane and the angle of diversion was 30° degrees. In this set of experiments two kinds of sand we re tested namely: Red River sand and Poll Creek sand. Although their distribu-tion curves were not given by V ógel, yet it was stated that the first one was rather fine while the second was coarse.
The stabilized condition showed the branch channel to discharge about
38% of the total flow and 52% of the total bed load when the fine sand was used and about 36% of the flow and 65% of the total bed load in the case of coarse sand.
These conclusions also differ from those ofBulle, but it should be remembered
that the experimental flumes were different in shape and size in both cases.
In the German experiments, the two flumes had rectangular sections of equal
areas. The discharges were likewise maintained equal and each test duration
was about one hour. The U.S.W.E.S. flume was semicircular, with the side
channel given two thirds the area of the main channel. The flow in the side
channel was less than 50% and the duration was about 70 hours.
In addition to the above mentioned experiments LINDNER [24] reported
others, summarized their results and discussed the factors governing the
diversion ofbed load into the branch canals. He mentioned further experiments
which were carried out in Vicksburg for different angles of diversion. Using
a model of a Mississippi bend, three diversion angles were tested. The results of these experiments were found to confirm those of Bulle.
In Lindner's opinion, this agreement was because the branches we re
excavated in a location most favorable for the diversion of bed load. Had these
diversions been excavated on the concave bank far from the bars, the percent-age of diverted bed load would have been less.
Experiments concerning the particle size, at the Iowa State College [25] revealed without exception that as the particle size of a given fraction of the
sediment is reduced, the percentage of this fraction diverted to the branch
is reduced.
The anaylis of the factors governing the diversion of bed load into branch
canals led Lindner to the following conclusions :
1. Downstream of the point of diversion, the average velocity is less than that in the upstream and therefore the tractive force is rather abruptly reduced below its value in the main stream above the diversion. As aresult, material
in, and the branch is th us afforded a better opportunity to withdraw
bed load.
2. The top 1ayers which do not carry bed load have the highest velo city and
are therefore more difficult to deflect. As a result a large portion of this
part of flow continues downstream in the main channel. A substantial
portion of the flow in the branch canal is drawn from the bottom layers
which carry the bed load.
3. The material moving along the bed of a stream with noticeable velo city
will be easily diverted through a small angle and with less force than
required for a large angle, since the particles approaching a divers ion
with a small angle will have a velocity component in the direction of the
branch while for a 90° degrees diversion, the entire movement must be
caused by diversion forces.
4. Any tendency to create a meandering pattern will make the bed load pass
towards the inside of the bend at the upstream side of the bifurcation and
into the branch canal.
5. Experiments conducted by the U.S.W.E.S. to investigate the effect of the
location of the diversion showed that the proximity of the diversion
en-trance to the sand path was of major importance and that a diversion
should be constructed in the concave bank as far as possible from the
sand path so as to ensure that the amount of sediment withdrawn in the
branch canal is minimum
TISON [26] carried out some experiments to investigate the effect of the
angle of diversion, streamlining of the corners and the proximity of the intake
to the sand path. His experiments verified the tendency of bed currents to
enter the branch and the tendency of surface currents to continue their path
in the main canal. He first assumed that the flow was in layers parallel to
the bed, and by applying the equations of rectilinear flow, he showed that
this assumption was not valid. The curvature of the streamlines caused an
ascending motion in the flow portion which is diverted towards the branch
and a descending motion in the portion flowing along the main channel.
Different runs with angles of diversion smaller and greater than 60° degrees
showed that the proportion of solid material diverted to the branch was
reduced as the angle was reduced and also the ascending tendency of the bed
currents was reduced.
He also showed that a reduction in the percentage of solid matter diverted to the branch could be obtained by streamlining of the corners. About 55%
to 60% of solid matter was diverted in the latter case instead of 90% when
the corners had sharp edges. Using a curved approach to the bifurcation he
observed that the proportion of solid matter passing in the convex side varied
between 80-100% depending on the magnitude of the radius of curvature.
convex side occured at a short distance downstream of the apex of the bend. When an experiment was carried out with the intake at the concave side no
solid discharge was diverted to the branch canal.
ISMAIL [27] divided the parameters of this problem into 3 groups:
I. Those related to the shape of the canal.
a. Straightness or curvature of both main and branch canals, which can be represented by the radius of curvature of canal divided by
r its width i.e.
B .
B
b. Relative width of branch to main canal, represented by _ b •
c. Angle of diversion between the two canals, (). Bm
11. Those related to the dynamics of water flow; these can be represented by Reynolds number, Re, Froude's number, Fr, and the ratio ofthe mean velocity in the branch canal to the mean velocity in the main canal, Vb •
V
m111. The third group covered the parameters of sediment transportation both in suspension and as bed load. They were represented by the concentra-tion at the bed Co, relative roughness represented by the sand diameter divided by the depth of flow,
~,
mean velocity of the flow to mean fallV D
velocity ofthe sediment, --, the geometric standard deviation of the fall V,
velocity from its mean a V, (this representing the size distribution of the sediment) and V,t', where t' is the duration of the test and a is a certain
a
length representing the field of the test.
The total amount of sediment diverted into a branch should be a function of all previoüsly mentioned parameters i.e.
(r Bb Vb d V VIt')
Cs = g; ,B' Bm' (), Re, Fr, V
m' Co, D' V,'
av"
----;;-
.
He carried out some experiments on a fixed bed flume keeping the width of the main channel constant while the width and the angle of diversion of the branch canal were varied. By studying the velo city distribution while varying the ratio between the mean velocities in the main and the branch
channels he concluded that:
V
1. The velocity ratio ~ is the main independent variabie.
~ V
2. The strength of the spi ral motion is a direct function of V~. The bigger this ratio, the larger the quantity of sediment entring thebranch canal.
V
3. The smallest possible ~ is recommended for the diversion operation in Vb
order to get minimum separation and minimum secondary currents. LELIAVSKY [28] believes that the angle of diversion (twist) determines the radius of curvature of the streamlines at the entrance of the diversion, which in turn, determines the intensity of the centrifugal force produced by this curvature. As the centrifugal force is the main local factor governing the behaviour of sediment at the bifurcation point, the sediment distribution between main and branch should accordingly be a function of this angle.
He gave the following formula, which was based on experiments and ob-servations, for the ave rage radius of curvature of the filaments diverted into the branch canal at the point where the flow changed its direction.
Bb n-8
r = - t a n -
-2 2 (2-1)
where Bb = width of the branch canal and 8 = angle of diversion (twist).
Assuming that the water surface in a curve or in a vortex was normal to
WV2
the resultant RF of the weight Wand the centrifugal force - - he' obtained
gr
a formula for the average transverse slope of the water surface Sy:
V2 Sy= gr 2 V2 2 V2 8 - - -- - - = - -tan -n-8 Bbg
2
Bbg tan - 2 -(2-2)He suggested that in order to counteract the effect of assymmetry of the flow and the vortex action at the entrance of the branch a sand screen may be used whose sill should have a slope equal to and opposite in direction to the slope Sy given byequation (2-2).
A similar solution was proposed by other Egyptian engineers. U sing the
average slope of the eroded bed S as a parameter, yielded the following formula: S = 8/50 n. . . . (2-3)
An experimental study for the relation between the water depth in each
tributary channel upstream from a bifurcation and their corresponding discharge ratio was carried out by TAYLOR [29] on a small fixed bed flume set-up. Although the author was ab Ie to make reasonable assumptions in the application of the momentum pressure theory for the case of combining flow, he could not find any sensible ones for the case of divided flow and therefore had to depend entirely upon his experimental investigation.
Following the same line as Ismail, his student ELGHAMRY [30] continued the study of canal intakes on the same fixed bed flume. He found th at if a
submerged sill were to be used in
order to decrease the sediment inflow
to the branch, its location and
dimen-sions should be as shown in fig. 2.
The idea of dividing the bottom
flow was first introduced by TISON
[26], who proposed the construction
of a divide wall parallel to the
direc-tion of the main canal and located
near the intake of the branche canal.
MARTIN and CARLSON [31] later
h=O.5D m
Fig. 2. Location and dimensions of sub-merged sili.
improved the design of divide walls by making its upstream part so rounded
as to minimise the bottom flow towards the branch.
TULTS [32] also recommended the use of guide walls with curved upstream
end, together with a gradual reduction of the radius of curvature from surface
to bottom of the upstream transition
between main and branch canals as
shown in fig. 3.
The general practice in Egypt has
for a long time been to construct a
sand screen at the intake of branche
canals, somewhat si mil ar to th at
de-scribed by Leliavsky [28]. It is the
writer's experience that most sand screens of this type have not been
successful. Since the bed material in
Egyptian canals is composed of fine
sand, the bottom currents which carry
this material into the branch can very
easily carry it in suspension over the
sill, into the branch. We re the bed
material composed of large particles,
Main channtl
~~==G=Uid~.~W~.I=I=========~
-
--Fig. 3. Curved guide wal1 and upstream embankment with reduced radius towards the bottom.
they would probably tend to form a ramp, which the particles would then
cross over into the branch canal.
A divide wall, on the other hand, limits the area from which the branch
can get its bed load and hence can be so designed as to render the diversion
of bed material corresponding to the hydraulic conditions existing in the
branch.
Although El Ghamry gave some recommendations for the design of a
submerged type of divide walls, yet his conclusions were based upon experiments
with a fixed bed and have not been verified by other experiments with a
had he continued his work using a movable bed set-up since the moving
sediment and the pres en ce of ripples and bars affect the velocity distribution
and the flow pattern.
Finally, it is worthy,to note that the question of sediment distribution in a water course which divides into naturalor artificial branches was discussed at the l8th INTERNATIONAL NAVIGATION CONGRESS [33] held in Rome in 1953. The recommendations regarding this problem were as follows:
1. It is possible to forecast, approximately, the amount of sediment diverted from a water course to one of its naturalor artificial branches either by direct observations or by model studies.
2. To reduce as much as possible, the amount of solid material diverted to
a branch or a navigable canal and the sedimentation which is liable to occur at their extremities it is necessary that:
a. The branch or the canal should preferably take off the water
course slightly downstream of the apex of a curve on the outer side.
b. At this point, the angle which the branch or canal makes with the
tangent to the curve should be rather small.
c. The sill at the entrance should be designed in such a way as to blo ck the path of bed currents which are particularly laden with solid matter.
d. Eventually, the bottom currents should be diverted to the opposite
CHAPTER III
ANAL YTICAL STUDY OF THE PROBLEM OF CANAL BIFURCATIONS
A. Flow in open channel bends
At fiTst consideration, a branch canal take off seems to cause a part of the flow in the main canal to follow a curved path into the branch. Another part ofthe flow, away from the intake would continue its norm al path. The remain-der, not so close to the intake, would take a hesitant path and end by partly mixing with the remaining flow of the main canal downstream of the take off and partly by entering the branch at the downstream side of the intake. However, the problem is not so simpie, especially due to the presence of spiral flow as a result of the centrifugal action of the diversion curve. It is further complicated by the presence of sediment in transport at the bottorn layers of the flow. While the flow characteristics govern the rate of sediment
movement and the bed configuration in both main and branch canals, the rate of movement and the configurations control the roughness, which in turn controls the flow. The behaviour of sediment particles near to the intake is greatly affected by the secondary currents of the spi ral flow which is created
by the bifurcation curve.
Starting with the simple case of a curved fixed bed channel without any
bifurcation and treating the problem in a one dimensional approach based upon the assumption of constant velo city and curvature for the entire stream, the increase of pressure fJp, which is caused by the centrifugal force, may be
computed as follows:
v2
fJp = Be - . . . (3-1)
r
where fJp = pressure difference between inside and outside of curve, r = mean radius of curvature.
The difference in surface elevation between outside and inside of the curve becomes:
fJp v2
fJD
=
-
=
B - . . . (3-2)y gr
To get the equation of flow in the simple case described byequations (3-1)
v2
dp
=
(2 - dr . . . (3-3)r
where v is the tangential velocity of the particle Applying Bernoulli's energy equation for frictonless flow and differentiating:
dp
+
vdv = 0 . . . . . . . . . (3-4)y g
Combining equations (3-3) & (3-4) we get:
or v2 dp _vdv - dr= - = -gr y g dv dr
- + -
=0
v r . . . (3-5)which is the differential equation of the flow for a particle in a free vortex.
The solution of equation (3-5) is:
vr
=
C=
Constant . . . (3-6)The fact that the pressure gradient varies inversly as the radius of curvature
of the streamline can be shown by applying the general equation ofirrotational.
unsteady flow deduced by Euler.
Assuming a steady concentric flow will lead to equation (3-6).
Combining equation (3-3) & (3-6) we get:
dp C2
- = (2
-dr r3
The total radial pressure difference between the side walls becomes:
. (3-7)
. . . (3-8)
where ri
=
radius of curvature of the inside wall, ro=
radius of curvatureof the outside wall.
The integration of equation (3-8) along one radius gives
• • • • • • • 0 '. • • • • • • • • • (3-9)
Following an assumption similar to WOODWARD & POSEY'S [34] that V is
equation (3-2) would mean that the radial profile ofthe surface is a straight line. On the other hand, GRASHOF [35] showed that:
dD dr
g r
W gr
dD.
In this equation - IS the ra te change of depth with distance from center
dr
of curvature
=
transverse slope, W=
force due to weight of the water. N cglecting the variation in v and by integration:V2 [ Jro V2 r
L1D
=
-
In r=
-
In ~ .g ri g ri
. . . (3-10)
The difference between equation (3-9) & (3-10) is that the latter was aresuit of an approximation in the assumption of constant velocity while in the former the inverse proportionality of a particle velocity and its radius of curvature was taken into consideration.
Again SHUKRY [36] found that the water surface in a 1800 bend did not follow either equations (3-2) or (3-10). The difference was attributed to the
effect of the velocity distribution which is controlled by the friction of walls
and the bottom, since it was not included in either equation.
Woodward and Posey in trying to include the effect of the velocity dis-tribution assumed a parabolic distribution of the velocity with a zero va1ue at both boundaries and a maximum va1ue V m at the center and arrived at
the following equation:
V,
~
[20 r 16r3 (4r2 ) 2r+BJL1D = - - -
+
-
-
1 In - - . . . (3-11)g 3 B B3 B2 2r-B
In developing this equation, the authors also ignored the variation in the radius of curvature across the flow, i.e. B was assumed too small relative to r.
RAMPONI [37], found from his measurements that the value of L1D shou1d
be increased by 50 per cent over the va1ues computed by equation (3-2). However, ROUSE [38] stated that in extreme cases the increase is not likely to be more than 20 per cent above the values computed from equation (3-2), and superelevations usually remain small for velocities and curvatures
normally encountered or employed in subcritical flow.
VAN BENDE GOM [39] used a mathematical approach af ter some
simplifica-tions of the differential equations of flow and arrived at expressions for the transverse water slope and the deviation of the surface velocity and the
bottom shear from the average. Ris solution was based upon the following
assumptions:
1. Constant roughness coefficient of DE CHEZY. 2. No influence of bank roughness.
3. No ripples or eddies present.
4. Sand transport porportional to the fourth power of velocity. Van Bendegom's conclusions can be summarized as follows:
1. The transverse water surface gradient at a certain point in the cross
V 2
section is Sy, = l.06 _a
gr
2. The deviation of the surface velocity from the average direction of the flow
Vy Va2
-
,
=
0,025 - -S-V", g·r· '"
3. The deviation of the tractive force on the bottom caused by the spiral flow
Fy Va2
- = tan À = 0 04
-F",' , g·r·S",
where Va
=
average velocity in the vertical under consideration, r=
averageradius of curvature of streamlines in the vertical under consideration,
Vy
=
surface velocity in the lateral direction, V",=
surface velocity in thedirection of flow, S'" = longitudinal slope of the water surface, F", = tractive
force in the direction of the flow, Fy = tractive force in the lateral direction.
We can relate the mean tangential velocity with the value of C in the equa
-tion ur = C. For any particle:
'0 '0
V
=
J
-
Udr=
JC
-
dr=
- -
C In -roB rB ro-ri ri
Ti Ti
Applying the equation of continuity Q
=
VAwhere A = Dm (rO-ri) . . . .
The specific energy equation can be written in the form:
V2 H = D
+-2g '0(3-12)
(
3-13)
(
3-14
)
f(
H-~)
dr . r2• 2gThe average depth Dm
=
'iC2 Dm= H - -2g·rO·ri Substituting in equation (3-14):
A
=(H
-
~)
(rO-ri) 2grori(3-15
)
(3-16)
Substituting in equation (3-13) for A from equation (3-16) and for V from eq uation (3-12) Shukry derived the following equation:
Q,=
~
ln
~(
H
-~
)
(rO-ri)ro-ri ri 2grori
=
C In~
(H
-~)
. .
.
. . .
ri 2grori (3-17)
From equation (3-17) and knowing the discharge Q, and the specific energy H
for any given section in a bend one can always compute C. Thus, the super
-elevation iJD =
~
p
for any given point in the bend can be determined byy
the application of equation (3-9).
He stated that for the practical application of the above equations, the position of the point of maximum surface depression d, should be first deter-mined. Since this position was slightly affected by varying the parameter
D
I
B,the table given for the location of this point with respect to other variabie parameters and constant
D
I
B = 1 could be used.The specific energy he ad for a section A in the straight approach can be computed by:
V2A
HA = DA
+
a-2g
in which a, the velocity head corrective factor may be assumed equal to unity without appreciable error. In order to estimate the specific energy head H at the radial section passing through d, he suggested the following equation:
H = HA-Lhf-'Y)hb • • • • • • • • • • • • • • • • • • (3-18)
where L is the distance between the two sections, hf
=
energy lost by frictionper unit length of the straight channel, hb
=
energy lost due to the bend res ist-ance and 'Y) is a coefficient which was found to be practically constant for any curve which = 0.4By applying equation (3-6) we can find the tangential velocity at any radius rand the dep th of flow can then be computed by subtracting the kinetic energy head from the specific he ad H.
Earlier, MACKMORE [40], treated the problem of open channel bends in a three dimensional way. He made the following assumpitions:
If Vn is the velocity component in the radial direction, Vy is the component
in the vertical direction and v is the tangential component, then Vn was assumed
to have a parabolic distribution in a horizontal plane with Vn
=
0 at both walls. It had a linear distribution in the vertical plane, such th at Vn became a maximum at top and bottom (with opposite signs) and zero at mid depth.maximum at the walls, with opposite signs, and zero at the center line, while it had a parabolic variation in the vertical plane, with its maximum at mid depth and starting with zero at top and bottom.
v had a parabolic distribution in the vertical with its maximum at the surface and becoming zero at the bottom, while it had a parabolic distribution in the horizontal plane, being zero at both walls and maximum at the center line. Following these assumptions, the velocities in the three directions were put
10 the following forms:
vy = kyx(yD - f) . . . (3-20)
& v = k
(~2
_
x2) (2yD-y2) • • • • • . . . • • • • • • (3-21) vn and x are in the direction of radius, Vy and y are in the vertical direction, v is in the direction of the tangent and the axes origin being the middle of the channel bottom.Mackmore then derived the stream lines function for the
xr
&zr
planes and gave a plotting of the stream lines in section, plan and elevation for flow around bends. He also derived expressions for the acceleration a." ay & az in the x, y, and z-directions.Equation (3-6) means th at the velocity is more for the inner than for the outer particles. At the beginning of a curve, the velocity will consequently tend to increase and the pressure to decrease at the inner wall. Near the end of the curve, the process will be reversed in order to turn back to norm al. A rapid increase of pressure at the downstream vicinity of the curve is therefore expected and may cause separation. Naturally, separation occurs when the traction between the flow and the boundary becomes zero, or, in other words, when the ra te of change of velocity
along the radius becomes zero. This is approached at high rates of pressure lncrease.
In eXplaining the phenomena of secondary currents at the bends, the different authors usually assumed that the pressure difference resulting from the centrifugal action is the same on any particle in a vertical, and as the water particles near the surface will
- - - SURFACE-CURRENTS
- - - - 9OTTOH- CURRENTS
Fig. 4. Diagrammatic plan showing flow lines.
have greater velocities than those close to the bottom, it follows that the radius of curvature of the path of the bottom particles will be smaller than that of the surface ones as shown in fig. 4, and spiral motion inevitably develops.
The writer would like to mention in this respect th at the actual pressure
measurements carried out in some experiments did not confirm this assumption.
As will be shown later in chapter VI the pressure near the bottom was
usually less than that near the surface towards the outside of the diversion,
while the reverse occured at the inside. The difference was too small to
com-pensate for the lower values of P near the bottom and therefore helicoidal motion still developed.
B. Flow at Bifurcations
It is evident from the preceding discussion that the curved flow into a
branch canal causes the bottom currents, which carry the sediment, to deflect towards the diversion. Consequently a great portion of the bed load of the main canal is diverted into the branch. In most cases, the branch bed load
capacity is less than the diverted load. The excess load is gradually deposited
in the branch, starting from the intake, thus reducing its capacity to carry
water and eventually choking the branch unless the accumulated sediment is
periodically dredged out.
Now, considering the parent alluvial channel separately, its bed-load
movement, upstream of the intake, should correspond to its hydraulic con
-ditions provided it is neither subject to degradation or aggradation. If at
some section this channel has to divide its water discharge and its sediment
load among a diverted branch and its downstream, and since the laws go
v-erning the discharge of water and the rate of sediment transportation are
different, there has to be some upsetting of the regime of either or both forks unless their sections and slopes are so designed as to conform with both laws,
which is rarely the case.
The branch taking off at an angle will get more sediment than it needs and that forming an extension of the parent channel may get less than its
share. The angle of twist, the radius of curvature of the transition and the
cross sectional dimensions of the branch will, for a certain dis charge ratio
be the most important factors affecting the diversion of sediment into the branch. This is naturally due to the fact that they are the factors affecting the
centrifugal force which causes the spiral flow which in turn causes the deflection
of bottom currents towards the branch.
Let us consider the conditions prevailing in the branch canal separately
i.e. as an ordinary alluvial channel whose slope is usually dictated by the
natural slope of the site. The hydraulic conditions of the branch correspond
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