GOVERNMENT OF THE REPUBlIC OF INDONESIA
MINISTRY OF PUBLIC WORKS • DIRECTORATE GENERAL OF WATER RESOURCES DEVELOPMENT
PROGRAMME Of ASSISTANCE fOR THE IMPROVEMENT Of HYDROLOGIC DATA COLLECTION. PROCESSING AND EVALUATION IN INDONESIA
BED- MATERIAL LDAD (EINSTEIN'S METHOOJ
by
M TRAVAGLIO
po ~ SOCIETE CENTRALE ~~ POUR L" EQUIPEMENT DU TERRITOIRE INIERNATIONAt INTERNATIONAL BANDUNG MARCH 1981 f
Bed-Materia1 Load (Einstein's Methed)
by
M. TRAVAGLIO
Bandung, March 1981 Taole of Contents
Page
List of symbols • r
Introduction 1
Einstein's Procedure 2
1. Hydraulic.Calculations 2
1.1 Test Reach • 2 1.2 Surface Drag and Bedform Drag (or Bar Resistance) 3 1.3 Mean velocity 4 a. Manning-Stricler's Equation 4 b. Logarithmic Type Formula ••• 5 1.4 step by Step Procedure for Hydraulic Calculations 6
2. Bed-Material Load Calculation . 8
2.1 Rouse Equation for Vertical Distribution of Suspended Matter 8 2. 2 Suspended Load Equation la 2.3 Einstein's Bed-Load Formula 11 2.4 Bed-Material Load Equation •• 13
3. Example of Bed-Material Load Calculation 21
Concluding Remarks 27
Annex 1 · 28 Annex 2 · 30 Annex 3 · . 31 Annex 4 · 33 Annex 5 35
References 37 l
LIST OF SYMBOLS
A cross-sectional area diameter of particle. In a mixture d d or median diameter d = 50 D depth of flow 2 g gravitational constant, mean value 9.81 rn/s gs bedload rate in weight per unit time and unit width gss suspended.load rate in weight per unit tirne and unit width gst bed-material load rate in weight per unit time and unit width G bedload rate in weight per unit time S G suspended load rate in weight per unit time SS bed-material loadrate in weight per unit time Gst n Manning roughness value p fraction of bed rnaterial in a given grain size p wetted perimeter 3 water discharge (m /s) A hydraulic radius ~ = p
S channel slope u fluid velocity shear or friction velocity v settling velocity of particle 0 3 density of fluide For water at 20 C l = 1000 kg/m 3 density of particle.Usually taken as 2650 kg/m when the actual value is unknown 0 3 fluid specif~c weight. Water at 20 C = 1000 kgtm .; ... 3 partiele specifie weight. Taken usually as 2650 kgf/m when actual value is unknown o· kinernatic viscosity of fluide For water at 20 C. -2 2 = 10 cm /s shear stress or friction force per unit area exerted by the fluid at a depth y above the bed shear stress at the bottom 'Ï"'" = y RS or 1: = "(DS \0'0 H 0 other symbols are defined in due course in the following sections. l
INTRODUCTION
The bed-material load is made up of only those particles consisting of grain sizes represented in the bed.
In theory the bed-material load can be predicted with the hydraulic
knowledge of the streamJ that is,
velocity bed composition and configuration shape of the measuring section water temperature concentration of fine sediment
Therefore the problem at issue is to determine the relationship between the bed-material load and the prevailing hydraulic conditions such a problem has proved to be a difficult task and is not yet completely solved.
50 far comparisons of measured and calculated bed-material loads exhibit discrepancies which lead to think that first the problem o~ sediment transport is not fully understood and second great care must be taken in using bed-material load formulae.
As pointed out by GRAF (see references at the end) "Einstein's method represents the most detailed and comprehensive treatment, from the point of fluid mechanics, that is presently available". This method is described in the following paragraphs.
. Nota We prefer the name "Bed-material load" to the name "Total load" since the so-called "washload" is not taken into account when one speaks of bed-material load. 2
EINSTEIN'S PROCEDURE
Introduction
The bed-materi~l load is divided in two parts according to the mode of transport. In the immediate vicinity of the bed in the so-called bed layer takes place the bedload whereas the suspended-load takes place above the bed layer where the particle's weight is supported by the surrounding fluid and thus the particles move with the flow at the same average velocity:
Some researchers think the division of the bed-material load in two fractions is questionable. Actually such a division is rather artificial particularly when it comes to define a zone of demarcation between bed-load and suspended-load, nevertheless it is often convenient for the sake of clarity to distinguish these two modes of transport.
Nota Figures number 2 ta number 9 are grouped fr.om page 15 to page 20.
1. HYDRAULIC C.l\LCULATIONS
1.1 Test Reach
To calculate or measure the flow and the sediment transport in a stream, a test reach has to be selected first, the following requirements have to be fulfilled, the better they are the more reliable the results.
It should be sufficiently long to determine rather accurately the slope of the channel
It should have a fairly uniform and stable channel geometry with uniform flow conditions and bed material composition
It should have a minimum of outside effects such as strong bends, islands, sills or excessive vegetation
No tributaries should join the river within ~r immediatly above the test reach.
It is worth noting that the foregoing requirements are those usually . sought-for to set up a gauging station. 3
1.2 Surface Drag and Bed-Form Drag (or Bar Resistance)
To take into account the contribution the bedforms make to the channel roughness it was proposed that both the cross section area, denoted A, and the hydraulic radius, denoted ~, be di~ided into two parts: one related to the surface drag or grain roughness designated by A' and R' , the other related H to the bedform drag designated by A" and R~ respectively.
In terms of hydraulic radii we have
+ R" = H
It follows that both shear stress and friction velocity are in turn divided since:
~= = Y(RH ~ R")S and (1)
= = (2) so we have:
a. In terms of shear stresses
't" + 't;' (3) = o 0
b. in terms of friction velocities
= (4)
the "prime", 1 , used in the notation pertains to the surface àrag whereas the "double prime", fi , pertains to the bedform drag.
Einstein and Barbarossa derived a curve fram data of river measurements which relates the "flow intensity" denoted y 35 and defined as 4
= is the bed sediment size forwhich (5 ) RES 35% of the material is finer)
u to the ratio of the me an stream velocity, denoted u, to the friction u" * velocity due to the bar resistance denoted u;. This curve which has come to
be known as "bar resistance curve" is shown in fig. 3.
Nota: Different bedform shapes are sketched in Annex 1
1.3 Mean Velocity
DePending on the surface roughness, Einstein and Barbarossa recommended use of either the Manning-Strickles equation or a"logarithmic type formula.
a. Manning-Strickler's equation
Is defined as
u 7.66 (RH )i/6 (6 ) = d u' * 65 where d is the bed sediment size for which 65% of the bed material is 65 finer.
The well-known Manning fonnula is defined as
1 1 2 u = R 3/2 5 / (7) n H 5
Let us assume firstly the velocity would be the same with a fIat bed and secondly the bedform would affect both the roughness coefficient n and the hydraulic radius i1i Be n' and ~ the values when no bedform exists. 50 we have
l 3/2 u = R' si/2 {8} n' H
By combin~ng {7} and {8} we get:
n' ~ { }3/2 {9} n ~
and by combining· {6} and {8} we get
d 1/6 n' 65 . {10} 24
Equations {9} and (lO) enable to ascertain whether there is a bedform drag or not and ta calculate ~ if need be. This is the case when direct measurement were made of the mean velocities for examp1e at a permanent gauging station.
b. Logarithrnic Type Formula
Einstein and Barbarossa chose the fo110wing equation which was derived from Nikuradse's experirnents by Keulegan.
12.27 ~ x U 2.3 log { } {11} u k d .*' 65 where k is the Prandtl - Von Karman coefficient equal to 0.4 for clear fluid and, x , is a correction factor for the transition from hydraulically (see AnneA 2 for a discussion about k) "rough to hydraulically srnooth surface, 6
d 65 the roughness being in turn related to the ratio T' where ~ is the thick- ness of the so-called laminar sublayer and is defined as
11.6 J) (J,I kinematic viscosity of the fluid) (12) u~
In figure 2, the factor x is given as a function of
Use of Manning-Strickler's formula is recommended when the grain rough ness produces a hydraulically rough surface, i.e. when d65 is more than d 65 ~ about 5. Whereas use of a logarithmic formula when r ~s less than about 5 (see fig. 2).
In case direct measurements of velocities are made, a trial and error procedure is used to determine R' and x. The chosen values have not only H to verify equation (11) but to verify both the 2 functions depicted by the curves given in figures 2 and 3.
1.4 Step by Step Procedure for Hydraulic Calculations
Once· a test reach has been selected, the following informations are needed.
l. Slope
2. Description of the cros~ section, that is,
2.1 Curve of ~ versus D A Cross section area 2.2 Curve of A versus DD Depth or stage 2.3 Curve of p versus D P Wetted perimeter
3. Bed sediment distribution curve 7
The determination of the depth (or stage) - dischargerelation proceeds as follows:
l. Select a value of ~ 2. Calculate u' and ~ through * equations (2) and (12) respectively 3. Determine x frcm fig. 2 4. Calculate u through equation (6) or equation (11) (5 ) 5. Calculate y 35 fram equation b . u 6. o ta~n-;;-- frcm fig. 3 then calculate un and RH u * * 7. Calculate ~ =~ +~ 8. Determine A and D through the description of the cross section 9. Calculate Q = u A
Remark
In flume experiments a side-wall correction is introduced to take into account differences in roughness between the sand-coverad bed and the flume walls. In most natural streams such a correction neednot be applied. 8
2. BED-MATERIAL LOAD CALCULATION
The bed-material transport is calculated in its two modes, namely, bed-load and suspended-load for each grain fraction of the bed at each given flow depth.
The procedure used to compute the suspended-load is based on the so-called Rouse equation which is in turn an application of the diffusion dispersion model.
The Einstein's bedload-function is used to calculate the bedload rate. Sorne theoretical considerations are in place here to shed some light on the procedure.
2.1 Rouse Equation for vertical Distribution of Suspended Matter
Let us consider particles of uniform shape, size and density in a two dimensional, uniform,' turbulent flow.
Since the particle continuously settles with its settling velocity in relation to the surrounding fluid an equilibrium suspension is possible only if the flow provides a countermotion with an equal velocity. This.upward movement is due to the turbulence of the flow, which turbulence results fram eddies that are bei~g formed continuously and are swirling in an irregular manner as they are carried along by the flow.
The diffusion-dispersion theory states that the settling rate due to gravity per unit area is balanced by the upward movement due to diffusion. This can be expressed by the following·equilibrium equation
vc = _E .2s. (13) s dy where v is the settling velocity of the given particle and c the concentration at the height y above the bed. v is given with fig. 4 as a function of the particle diameter, the curve due to Rubey will roughly describe the sedi ment of most streams. 9
E being a function of y which has been found to be proPQrtional to the s diffusion coefficient E so we have: m
E E (14) s. = @ m
In most applications the ft factor is taken as unity. "Though experiments have shown that f3 decreases when both the diameter d and the sediment concentration increase such changes are small in comparison with the changes observed in k.
Furthermore, the local shear stress, that is, the shear stress at the height y above the bottan can be expressed as:
CE ~ (15) , m dy
Assuming the Karman-Prandtl velocity law valid, that is,
u-u 2.3 y max -~ log 0 (16)
we finally get the so-called Rouse equation (see Annex 3 for the derivation of this equation).
c = (17) c a
The quantity ~"ku. is often denoted z. It has been found that the dis- crepancies observed between theoretical values of z and the ones based on experiments are chiefly .due to variations of the k factor. So taking ~ as unity as wel~ as using for v the settling velocity in clear, still water do not seriously change the z values. (See Annex 2). 10 D
-----.... fla,,",
J
•
Figure l
50 relation (5) may be used to calculate the concentration, c , of a given grain size whose diameter is, d , at a distance, y , above the bed provided that the concentration, c , at a distance, a , above the bed is a available. 5ee fig. 1.
2.2 5uspended Load Equation
To obtain the suspended load rate in weight per unit time and unit width, denoted g , we have to integrate the product of the velocity and the ss concentration over the part of the vertical concerned with suspended load, say from a to D.
= [ cudy (18)
This time, we use for the velocity distribution the following relation due to Keulegan which relates the velocity not only to the depth y but to d as well. 65 11
u 2.3 30.2 yx l og' (19) \T k d * 65
Substituting the Rouse equation (17) for c and cquation (19) for u into (18) we get: (see Annex 4 for the derivation)
_1..:1 c u' (20) gss - k a *
a where :: D
According to equation (20) when y approaches zero the concentration becomes infinite,obviously this is not true. In fact the sediment distribution does not apply right at the bed because the concept of suspension, that is, solid particles being continuously surrounded by the fluid fails and so the proclem is to determine the thickness of the layer above which suspension is possible and under which takes place the so-called bedload which is actually the source of the suspended load.
2.3 Einstein's Bed-Load Formula
For mixtures with small size spread the total bedload transport of the mixture can be determined directly by using d as the effective dia 35 meter, that is the case when only the bulk rate is needed to predict scour or deposition or when the suspended load is negligeable. The case was dealt with in a previous note entitled "Bedload measurement and sampling."
A few more parameters come up when transport rates of each size fraction have to he computed, mainly to take into account the fact that particles of different sizes in a mixture have not the same behaviour as uniform bed materials.
In that case, the "intensity of bed. load transport" , ~ *' and "flow intensity" , y *' are expressed respectively by: 12
l (21) P
r being the fraction ~f bed material in the given grain size who se repre sentative diameter is d.
. 2 ~og ~ = j y 10.6 (fs - r) _.d_ (22) Y. log 10.6 Xx P RH S d65 '
X is defined as a characteristic grain size of the mixture computed as follows
d d 65 X O.7·7~ if ;> 1.80b or (23) x x
d 65 X = 1.39 ~ if c::: 1.80~ (23 ) x '
We recall that ~ laminar sublayer is equal to
~= 11. 6 V --ur-•
Two correction factors are introduced namely ; and Y. S or "hiding" factor takes into account the fact that srnall particles seems to hide between larger ones. Fig. 5 depicts the relation between.5 and the d ratio 65 . X
y takes into account changes of the lift coefficient in mixtures with various roughness. Fig. 4 depicts the relation between Y and d65 • S Once ~.is deterrnined, we get ~. through figure 6 which depicts the Einstein's bedload function, namely,
1/7Y.(-2) 2 l r -t 43.5 ~ * l = e dt = (24) 1+43. S9i. ff J-1/7'r.(-2) J3
2.4 Bed-Material Load Equation
For a given vertical, it is logical to think that the summation of the bed-load and the suspended load leads to the determination of the bed-material load. In order to relate the concentration c to the bed-load, Einstein a introduced the notion of bed-layer whose depth is equal to 2d and stated that suspension is possible only above this layer. Assuming a bed-load move ment in the bed layer he derived the reference concentration at 2d fram the bed as (see Annex 5 for the derivation)
l c with a 2d (25) 11.6 a =
Introducing relation (25) into the suspended load equation (20) we get
gs z-l 30.2 Dx A (~)z. dy + Az-l (l-Y)z gss = n r lny dY}26) 0.216 d t 65 (l-A) z 1: Y (l-A)~ A Y
The bed-material load denoted gst is given by
= (27)
Substituting (26) into (27) we obtain
= (27' )
30.2 Dx where ln (28) FE = d 65
z-l ~ Z dy (29) Il 0.216· z (!:.:lé) (l-~) rAY E z-l 1 = 0.216' ( ). Iny dy (30 ) 2 Z (l-A"" ) Z r!=lAY EE 14
The two integrals are not expressible in closed form in terms of elementary functions.
and I are graphically depicted in figures 8 and 9 respectively 2 for various and z values.
Equation (27') gives a stream's capacity as to how much bed material load it can transport under uniform.and steady flow conditions; washload is not included in Equation (27'). In applying the methodfor a particular water course, Einstein (1950) stresses the following points:
(1) The length of a uniform reach should be such that the slope 5 may be determined accurately; (2) the channel geometry, the sediment composition, and aIl other factors influencing the roughness velue n, such as vegetation, etc., should be uniform, so that an average representative cross section may be selected.
50 Einstein's (1950) method of computing the bed-material load 15 elegant and allows the calculation without measuring either the suspended or the bedload matter. 15
FIGURES
Fil. 3 Flow rcsisl~nce due 10 bedforms. [Afra EI:-.;snIN f!t al. {/952}.J 16
200 ~EDIMENTATION ENGINEERING I.e ~;-..;.=--...:.l __~2.:...- ,;.;,"_.:;;;.•:.-.:a~'....;;;ar;..,..;•.;.:.4_.;;,"'1> 0.9' _, 1 0.8; /_" _ O.T ~;_: _., . '/ i i).. . f . ~-- 0.&, ~ 1 -_. _.. - ." 1.-{ . ~., -r oS> " ~ ....f------fT 1" i 15 0.41 ~..,!--'o-+-.;__I_---_+...,.I-+-i---t--I~1,,"";"'--1 ~ O.3~_~.~~~;__;.. ~.~.:~.~:-j-Ë; E··~- ~-;;B'~Ë·~1§ll'g~ O.Z:~§§~§§§ -5 4 3 2 l 0.8 0.6 0.5 0.4 0.3 VII"'~·." --- FIG. 4.-Fac:tor Y ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of . d."j
200.---,------...... _--......
• 1 . 1 ·1 r- "0 ! ~ >
5 4 3 2 l 0.8 0.6 0.4 0.3 0.2 0.1 VI"" 01 \+ •.-- • AG. 5 -Fact« ~ ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of d./X -t++++1ti 10 ~ mtllmttttl!l1t: 1 --f 5
j : .~ l, '1 I , 1t l IJ 1 .!t l '. Il ,1 1 1 : i rH-Wt++ltt-H-f'ld+H 1.0
IlH-H-I+HII++HtH:!
1 tIlH+1-"l+#q:++f+'ftH1 l' li;qll 1 l '
5 fi 7 89 0.1 10 18
~ ~ . e ~ .. ~ ~~ ~ 0.1 l' 45 I.() 11.0 100 1100 10 1000
; , ':, ",Ii Iii l'i!:! 1.0 100
sr·~Fc. Wc.:,.\.11 ~ ~'5~/e...'!>
Wf11 t-r te""l"er..""n: , 1 . ' '.: 1 , 1 -': l', 1 ': ! 1 l, ' "Ii 1 ' ',1:: 1 ! i ~ j 1 l'OC Il,I 1 '1 1' 1 ![j'II 1 1 il::' 1 il 10·) 10·' .. il ! j 1iilll Il
,, .. , ' 1 " . 1 ;" ; , j'II' 1 i i " 1 , 1 Il , ; : ~ I! li;11 i I!;I! 1 i;Pi 1 1 , : Jill 10" 10'2 / i 1 •• j
: ", :!l , :,., . ,,' 1 'I: il:! Il'! 1 1111'1 Il Iii l' l i 1 : [1 !lli 1 ilili 1 1!lIi 11111, lili QOl 0.1 1 J. 1.0 10 100 ',000 Groin sile. mm- FIG. 7 . Seulini velocit)' ~. for quartz iraïna of various aizes according to Ruhey [lOt.
,.. 19
....
l '
Fi,. 8 FunClion JI in terms of AI for values of =. [Afur E/:-;STEIS (/950).) 20
! l' l 'i 1 j 1 Il r il ~: ::-~~':~}1??~;:·:i\:.p~:;:~~,,~\ ! !! Iii ; ~: IOZrooo... ~ . ,'." .. :1:,"~ .' ~:·~"~".N . .~~.~~ , 1 h--'. ~... '-'-;::-i .
!- i _ ,- ., l , ~----::-:••: ..:}:N-U ! iii:!I' : -lilil: -.~: , tO"~-~ê'3---~"~--~'.~~~:~-~.~;~'~~'~:~~;~~~~~~~~~-~--i i ~--l--+---+--4 ~---++t+-++';---I-++_":-_-";""'-+-+~";'+~ ,1-:-7TI l , Il!
t~rms Fil. 9 FunClion 1, in of A E for values of :. [Afler EI:-;STEl:-O (/950,.] (I:l i~ .. ,G. t";",~ ) 21
3. EXAMPLE OF BED-MATERIAL LOAn CALCULATION (After GRAF'Hydraulics of Sediment Transport*p.222)
A test reach, representative of the watercourse to be investigated, has been selected. It was concluded that thé channel can be represented by a trapezoidal cross section with bank slopes of 1:1 and a bottom width of 91.45 m. The channel slope was determined and given by S = 0.0007.
Five samples, taken down to a depth of approximately 2 ft, were.collected to obtain information on the grain size distribution of the entire wetted perimeter. The average values of the five samples are given in table 1.
Table 1
Grain Size Average Grain Size Distribution, mm mm Peroentage
d > 0.589 2.4 0.589 > d > 0.417 0.495 17.8 0.417 > d > 0.295 0.351 40.2 0.295 > d > 0.208 0.248 32.0 0.208 > d > 0.147 0.175 5.8 0.147 > d 1.8
The average grain size is the geometric mean between the upper and the lower limits of each division, i.e. 0.495 "0.589 x 0.417 .
The grain size distribution curve is given in fig. 10.
Description of cross section is given in fig. Il.
Hydraulic calculations are presented in Table 2 and bed material load in table 3. The table heading, its meaning and caleulation are explained with footnotes. 22 t.O 0.9 1 , 0.8 , i 1 0.7 1 1 1 J 1 ! 0.6 1 ;-. 1 , ~ , 0.5 , , ; , i ~ ! 1 i ::0.4 1 , -ct,~- - -.--~ , : i, j .. r i 1 1 r-' 1 1 ..,-----.;;;- , 1 1 0'), 1 1 I~1. ' i 1l' : . , , 0.2 , 1 1 1 i 1 l' i 1 ! 1 1 1 1 i 1 1 1 !1 1 0.1 il 95 90 80 70 605040 30 20 10 5 2 Plrelnl finer
FI,_ 10 Grain size distribution of bed material. 23
Table 2 Hydraulic calculation for sample problem
.' 3 u" 103S" x -u ü/u: u" ~ * d65/ r 10 d 65/ x Y35 * R}i 1 2 3 4 5 6 7 8 9 10 11
0.61 0.0647 0.179 1.96 1.40 0.25 1.745 1.12 34 0.51 0.379
.' (i'ft)
m mis m m mis mis m
(1) Values of ~ are assumed
friction velocity due to grain roughness
11.6 (3) V 0 = u' laminar sub layer. V (kinematic viscosity) at 20 C * V = 10-2cm2/s = 10-6m2/s •
(4) d65/ r
(5 ) x = fct given with fig. 2 (d65/S" Correction factor for roughness transition.
(6) apparent roughness diamet~r d65/ x 12.27' Rif (7) u- u~ 5.75 log = d 65 d 35 (8) flow intensity with d ,as representative diameter y 35 = ~s 35
(9) --lL- = fct given with fig. 3 u" Y35 *
(10) u" = (l/u~ ) u friction velocity due to bedform drag * * ,,2 (11) R" =_u_ gS hydraulic radius due to bedform drag
(12) ~ = RH + RH hydraulic radius 24
Table 2 (Continued)
2 3 0 A P 10 X y \109 10.6) ~ u. Q 0"- PE '=<
12 13 14 15 lG 17 18 19 20 21 22
0.99 0.U83 1.02 94 94.3 164 0.249 O.GO 1.024 1.003 11.72
2 3 m mis m m m m Is m
(13) u. = Jg~S friction velocity
(14) D ::: fct(i1:I) given with fig. 11 Depth
(15) A ::: fct (0) given with fig. 11 Cross Section Area
(16) P ::: fct(O) given with fig. 11 Wetted perimeter
(17) Q ::z uA \'later discharge
·d d 65 (18) X ::: 0.77 -2.i if 1.80 Characteristic grain size x x > d 65 or X = 1.39 i' if < 1.80 x
(19) y ::: given with fig. 4 Pressure correction term
10.6 XX . (20)' 0<. log d 65
(22) ::: 25
Table 3 Bed material load. calculations for sample problem
3 R' 10 d p diX ~* H § y* gs Gs ZG s
1 2 3 4 5 6 7 8 9 10
0.61 0.495 0.1.78 1.99 1.00 1.15 6.7 0.140 13.202 13.202 0.351 0.402 1.25 1.01 0.82 9.6 0.271 25.555 38.757 0.248 0.320 1.00 1.13 0.65 12.2 0.160 15.088 54.637 0.175 0.058 0.70 1.60 0.65 12.2 0.018 1.697 56.334
. m m kg/m-sec kg/sec kg/sec
(1) RH
(2) d taken fram fig. 10 and Table 1 grain size diameter
(3) p taken fram table 1 fraction of bed material whose diameter is d
(4) ..s!. X
(5) 5 = fct (d/X) given "in fig .' 5 hiding factor
j y [lOg 10.61 2 ( ~ s - r) d (6) Y * = (RHs) flow intensity on individual ,0< r grain size
(7) §. = fct (Y*) given in fig. 6 intensity of transport for indi ~ vidual grain size P !p. '(s Jt f r;. p bedload rate in weight per unit time and width for a size fraction
(9) G = Pgs s bedload rate ln weight per unit time for a size fraction for the entire cross-section
bedload rate in weight per unit timefor all size fractions for entire cross-section
50 according to Einstein's procedure the bedload rate is in the region of 56 kg/s. 26
Table 3 (Continued)
103A v z -I P I +I +1 G ~Gst E Il 2 E 1 2 gst st
11 12 13 14 15 16 17 18 19
0.97 0.063 2.43 0.15 0.95 1.760 0.246 23.198 23.198 0.61 0.045 1. 74 0.27 1.80 2.36 0.640 60.352 83.550 0.49 0.035 1.35 0.51 3.00 3.98 0.636 59.975 143.525 0.34 0.022 0.85 2.70 10.0 22.64 0.396 37.362 180.887
mis kg/rn-sec kg/sec kg/3ec
2d (11) ratio of bed layer to water depth ~ = D
(12) v = fct(d) given with fig. 7 Sett1ing velocity
v (13) z = 0.4 u~
(14) Il = f(~, z) given with fig. 8
(15) I f(~, z) given with fig. 9 2 =
(16) P I +I +1 E 1 2
(17) gs(P I +I +1) bed rnaterial rate in weight per unit time gst E l 2 and width for a size fraction
(18) G Pg bed rnaterial rate in weight per unit time st = st for a'size fraction for the entire cross . (P : wetted perimeter) section
(19) bed rnateria1 rate in weight per unit time Z. Gst for aIl size fractions for the entire cross section
Obviously the digits (given by using a calculator) after the decima1 point in colurnn 19 are not significant/at best the number of significant figures is 3. 50 according to Einstein's pxocedure the·bed rnaterial 10ad rate is in the reqion of 180 kg/s. 27
CONCLUDING REMARKS
Several items in Einstein's method were questioned. For instance to use u' instead of in calculating z in the suspended load equa- * tion may seem inapprop~iate b~cause the diffusion coefficient E ' upon m which the equation is based is likely to depend on the total shear stress 1; and not only on \:' , let alone that taking 0.4 for k is also o 0 questionable.
Anyway any method has its own limitations and is at best for the time being a mere estimate even though aIl pertinent variables are taken into account to set it up as it is the case in the Einstein's method.
In the foregoing chapt~rs it was assumed that at any time the sedi ment bed could afford a continuous and full availability of its particles to be transported under any likely hydraulic conditions, if not/that i~if the supply were partially exhausted the stream would obviously transport less material and a bed material load equation which is supposed to give the maximum capacity (load capacity) would fail.
Last but not least, wherever washload plays an essential role the bed material equations.are merely helpful for the understanding of the problem but cannot give correct results since not only such equations are of no help to determine the washload rat~ but the parameters used to derive them are most likely to undergo drastic changes due to the very presence of the wash~ load (i.e. the factor k which is no longer equals to 0.4 when heavy sediment laden flows are considered). 28
Annex l
The following table shows that in the lower regime the values of RH are likely to be high as the form roughness predominates whereas in the upper regime when grain roughness predominates RH is often negligeable and R' H
CI~ssificatjon of bedforms ~nd other inform~tion (ufrer SI'10:-.s "t ul. l/CJ(>5) und 5"10:-;5 et al, (/966 JI
Bed lIlureriul .\tud~ lJI clJnc<'ntratilJlIs. s~dil/li!nr T.r~uI . R(}/'3hn~ss .. FllJ'" regilll<' Be"J"rm . PP"' transport rlJughll<'ss .("\,:; l
Io-~OO 7.8-1~.4 Rippk'S . Form Rippll:S ,ln 100-1.200 Discrl:ll: L•.m.:r regiml: roughncss dun~s sleps predominales Dun.:s 200-2.000 7.0-13.~
Transilion Washed·,lul 1.000-3.000 ; Variable 7.0-:0.0 dunes
: Plane b<:tfs 2,<>00-6.000 16.3-:0 Grain Antidunes 2,000 - 10.8 ·:0 Upp.:r rcgimc: Conlinuous roughnl:SS ChutC'i anJ 2.000 - 9A-10.i pr.:dominalcs : pools
A useful flow regime criterion is the Froude number denoted NF and defined·as :
_u_ JgO • where u is the stream mean velocity and 5 the mean depth over the entire cross-section.
A ~ classification is as follows:
tranquil (streaming) flow lower regime
= l critical flow transition regime
rapid (shooting) flow upper regime 29
Annex l (Continued)
Sketches of various bedforms are shown in the following figure •
..,..
Ct'l Plane Dea
_-----c.:!!.~':. ':~::___----
lD: ::>unes ... fft flCDles 5yD@fOOStd C9a,'..
lc) Dunes ..,.. . ~- /.\""'- tI', ~Poo' Ct'u!e '
(d) WO$h~d-ouT dunes or tranSITion (/Il ChuTes and POOlS
Idcalized bcdrorms in alluvial channcls. [Afte, SIMO~S ~I al. (196/).)
It is worth noting that should the bedforrn change for the same depth (or stage) bath .the velocity and the water discharge would in turn do, sornetirnes discontinuous rating curves or rating curves with loops may be . interpreted in this way.
To explain the fact that in the upper regime the depth-discharge relation is reasonably stable we will quote Einstein and al.
The effect of irregularities (bedforrns) is to distort the flow pattern. When the discharge is least, the distortion of the flow pattern is greatest; as witness the meandering of natural streams at low flows. As the discharge increases and hence the sediment transport along the bed also increases, the distortion of the flow pattern becomes less and less because the alinement of flow becomes progressively straighter. Consequently, one rnay expect that the additional friction loss, u~ , dirninishes as the discharge increases. 30
Annex 2 variations of k
The value of k is approximately 0.4 for clear fluids, but it has been observed to diminish to as low as 0.2 in flows with high concentration of suspended material. The following figure shows that the logarithmic velocity distribution law holds true but with different values of k according to the mean concentration.
THE SUSPENOEO LOAO
1 O,..------"'r"'--..... lÇ~=_~~ O~=== 0.9 >------f----'f-i 0.6~,------+ 0.8 !------t--~t-. O~'-·--- C.4 ------+---,... 0.71---- 03 ----~-_,,I-- 0.6,...... ----- y 02 ~---+___"+--____. 15 0.51------t-4-.-J~- y [) o 4 :------+---+-~ O' _ ~.O 0.3 r------J'--~---; 0.C8-~~-j~~~~~~ 0.2'-':,------j,-~--__I 0.06-= 1 0-0295 ft: o.c~ D·".2?: •• ! -s.o.ooas ~ 0.C4.:o' ;----s' ·)0025 1 1-- -.::::-...... J1 O.O~ l '. 1.0 2.;) 3.0 40 '0 2 C 30 4.0 Veloe' ~ y Il. ~cs .. !IC-C, ~., J. 'C\
VelOl:llY protil~." fvr ,lear-\\;lIer and >.:Jiment-Iaden I1v\\ .. [Afra \'.~:\u:-"I ,'1 ul, (/Y6UI.)
It has been suggested that a reduction of k means that mixing is less effective and that the presence of sediment suppresses or damps the turbulence.
Anyhow drastic changes may arise in the veloèity distribution when high concentrations take place but in that case it is likely that the bulk of the ~otal load is made up of particles finer than the bed material ones and 50 wash-Ioad is the predomlnant forro of transport. 31
Annex 3
Derivation of the Rouse Equation
We have the following set of equations
vc ~. (1) Equilibrium equation = - Es dy
du E (2) E diffusion coefficient in the diffusion theory 1:y = m dy m
(3) ~ Es = ~ Em constant
= OSD (4) Bottom shear stress, often simply called shear stress t 0
'Ç' D-y ~-= (5) Ratio of the local shear stress to the bottom l'V D \"0 shear stress u... = ft (6) Shear stress velocity or friction velocity u-u max 2.3 1 Y -- 09- (7) Karman-Von Prandtl law u ... k D
Let's take the derivatille in equation (7) noting that 2.3 logL= ln:L DD we get
du = dy ky (8)
Let's' express 1i in terms of ~o in equation (2) by means of equation (5) we get :
(D-Y) 't' (9) D 0 32
Annex 3 (Continued)
/'\,. Substituting equation (8) into equation (9) and expressing ~ in o terms of u* by means of equation (6) we get :
D- y Em D) u* = ky (10)
Combining equation (3) and (10) Es can be expressed by
(11)
Substituting equation (11) into equation (1) and separating the variables we get :
v Ddy ~ = (12) c --- y(D -y
Let us assume that the concentration of suspended sediment at a point a is c . Then integrating (12) from a to y we get : a
Ya _~D...;;d:.l.Y__ [ln(-.:L..)] Y -:L- log a (D-y) = y(l~ - y) ~ku* j L D-y a Y (D-a) and taking the antilogarithms v
c = [a(D-Y)] Pku* c y (D-a) a
v The quant±ty ~ is. often ca11ed z. \Jku* 33
Annex 4
Derivation of the Suspended Load Equation
We have the following three relations
1 5.75 log 30.2 xy ln 30.2 xy (1) d 0.4 d 65 65
y c = [a (D-Y)] Z (2) ca y (D-a)
c u dy (3) gss = y Y
Substituting (1) and (2) into (3) we get
30.2 x c a introduce =- then we have Let us ~ D l a (D-Y)] Z (~f (l-~J (5) Y(D-a) l-A •Y E-D Let us take as new variable u=:L then we have D dy = D du and the new limits of integration are u = Ae: and u = 1 for y = a and y = D respectively. 34 Annex 4 (Continued) Consequently we get A Z D Ga (D-y) ] D(_E_)Z and (6) y(D-a) dy l-A Ja E ~ l-u Z D [a (D-yil log Y dy (l-u) Z log u du ~ (7) y(D-a) u + log D l (u) du Ja J J Substituting (6) and (7) into (4) we finally get: 30.2 Dx l-u)z l-u Z )% ~Og (- du (-) log u (8) d u u 65 or taking the Naperian logarithms 30.2 Dx 1 (l-u)z ln u (8 ) d du + 65 u . r~ 35 Annex 5 Derivation of the Be~-Material Load Equation Einstein foun& that in the so-called laminar sub-layer whose depth is the bottom velocity, u ' is related to the shear stress velocity by B 50 assuming that the particles in the sublayer move with an average velocity equal to Ua' the bed..load per unit width g 5 may be considered as the product of the concentration c and the discharge per unit width, a 50 we can write: :: C a u or 9 s a B = c a 11.6 u. 9 s a and with a = 2 d we get gs = (1) Ca Il. 6 u. 2d Let us resume the suspended load equation [Annex 4, equation (8 ' i] (l~Y)Z J~ (l~y)Z (2) dy + ln y dyJ which may be rewritten as follows: z-l l + ~ (l-y)zln y dyJ (3) --u Oc rI 0.4 * a (l-A )z JA y EE 36 Annex 5 (Continued) Substituting (1) into (3) and noting that a = 2d and consequently a 2d ~=D = 0 we get 1 1 2d gs g =- u. D (4) ss 0.4 Il.6 o [ .••] = 0.216 [- ... -] Finally we get for the bed-material 10ad gst gs + g9S gg (PE Il + I 2 + 1) where ln (30.2 Dx) PE "" d 65 z-l ~ (l-Y) z Il = 0.216 dy (l_~)z [ y \:z-l 1 = 0.216 (l-Y) z Iny dy 2 (l_~)Z J: y 37 REFERENCES Hydraulics of Sediment Transport. GRAF, W. H., MacGraw Hill. Sedimentation Engineering. American Society of Civil Engineers. River Sedimentation. EINSTEIN, H. A. in Handbook of Applied Hydrology{VEN TE CHOW~ These books are available at the DPMA library.