Calculation of the Natural Pavement in Canals in Sands

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CALCULATION OF THE NATURAL PAVEMENT IN CANALS IN SANDS 
S. Ya. Pavlov UDC 626.821.3:624.131.21 
In [i] the calculation of the initial stretch of a canal situated under conditions of 
general erosion is based on the assumption that a pavement formed by an accumulation of the 
coarsest sediments in the deforming channel develops if the current velocity u does not ex- 
ceed the noneroding velocity for particles whose size (do) lies in the 75-85Z range of 
the grading curve. It is suggested to calculate the velocity uo by known formulas pertain- 
ing to streams whose bed is characterized only by granular (sand) roughness, whereas a channel 
has a ripple structure, which should lead to a change in the velocity uo owing to a change 
in the resistance to the flow. So far experiments to study a pavement have been conducted 
with rather coarse-grained material, where quite gently sloping ripples having little ef- 
fect on the kinematics of the flow occurred. 
A completely different picture occurs in the case of the affect of a flow on a channel 
composed of fine- or medium-grained soils. If the flow velocity u exceeds the noneroding 
velocity calculated for conditions of a flat bottom (Uo) by a factor of 1.1-1.5, then the 
bottom of the stream has a ripple (under natural conditions, a microripple) structure. Ac- 
cording to the data of on-site [2] and laboratory [3] investigations, ripples have a three- 
dimensional structure and stability ratio Zr/hr-10 (Z r and h r are, respectively, thelength 
and height of the ripple); they have a considerable effect on the hydraulic regime of the 
flow and wholly determine the channel roughness. 
The question of the possibility of formation of a pavement under such conditions has 
r~,nined largely unclear so far. An attempt to solve this problem was made at the depart- 
mant of engineering hydrology of the M. I. Leningrad Polytechnic Institute. 
The investigations ware based on the following consideration: a natural pavement can 
develop in a channel if the composition of the sand characterized by an average diameter do 
includes particles sufficiently large so that they remain at rest on ripples whose size corre- 
sponds to a given ratio u/uo [3]. 
The state of limit equilibrium of particles of a particular size can be characterized 
by a parameter called the "noneroding velocity on ripples" u r. This concept was introduced 
for the first time in [4], where it was assumed without any grounds that the ratio uo/u r 
1.17 and does not depend on the size and shape of the ripples. 
To determine the structure of the equation associating the value of u r with the flow 
parameters, we turn to an examination of the relation for determining uo [5]: 
= .7 -~- . , - - -~ I /T (1) 
, . aK - i - . ,p~a~ =a. i , - i =-, 
where 0' is the relative density of the suspended sand; ~, coefficient of hydraulic friction 
of the bottom with a granular roughness which depends, as shown in [6], on the hydraulic 
radius R, average diameter do, and diameter corresponding to 95Z with respect to the grad- 
ing curve dc; u, a dimensionless parameter which depends only on do and the physical proper- 
ties of the sediment material and llquid [6]; uo,, friction velocity corresponding to the 
start of the flow. 
To determine u r we can use a relation of analogous type, substituting in place of A the 
coefficient of hydraulic friction corresponding to the ripple structure of the bottom (At). 
To check this assumption, a series of experiments was conducted in a flume 12 m long 
and 0.48 m wide to study the stability of sediment particles on the surface of the ripples. 
Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 9, pp. 29-31, September, 1979. 
890 0018-8220/79/0009-0890507.50 �9 1980 Plenum Publishing Corporation 
NATURAL PAVEMENT IN CANALS IN SANDS 891 
As the eroded material from which ripples were formed we used sand whose granulometric com- 
position was the following: 
d, mm I--0,5 0,5--0,2 0,2--0, ! 0,1 
p, e/~ 8.40 45.5 33 2 12.9 
The average d iameter of the sand do •0.280 mm; dc/do =2.65. 
The exper iments were car r ied out by the fo l low ing method. A f te r s tab i l i za t ion of r ip - 
p le fo rmat ion caused by the e f fec t of the f low of a g iven regime, a cont ro l sect ion wi th a 
length of 5-6 r ipp les was covered w i th mater ia l invest igated fo r mob i l i ty . As such mater ia l 
we used sands of f rac t ions 1 -0 .5 mm (do •0.75 mm), 0 .5 -0 .25 .- . (do •0.375 mm), and the main 
mater ia l be ing eroded. To e l iminate the e f fec t of the in le t cond i t ions , the cont ro l sec - 
t ion was located a t a d i s tance equal to 30 depths from the s tar t of the f lume ( the d i s tance 
from the sur face of the f low to the mid l ine of the r ipp le bottom was taken as the depth of 
the f low/ . In the exper iments w i th do =0.75 mm and do =0.375 mm the r ipp le bottom above the 
cont ro l sect ion was secured wi th cement grout . The cement c rus t p revented the f low of smal l 
par t i c les from the upstream sect ion to the cont ro l r ipp les (covered w i th less mobi le mater i - 
a l than the main mater ia l / . The exper iments began wi th ve loc i ty u =O.5uo and ended when the 
ve loc i ty reached a va lue cor respond ing to the ent ra inment of par t i c les over the ent i re length 
of the r ipp le . The t imes of the s tar t of movement of the par t i c les in the t roughs and along 
the c res ts of the r ipp les were recorded . 
Accord ing to the method [7] , the noneroding ve loc i ty on r ipp les was determined as the 
ar i thmet ic mean between the ve loc i ty cor respond ing to the f i r s t movements of the par t i c les 
in the r ipp le t roughs and the ve loc i ty a t which the sand gra ins began to move a long the r ip - 
p le c res ts . The parameter a in the equat ion fo r the noneroding ve loc i ty in the case of a 
bottom having a r ipp le s t ruc ture was ca lcu la ted by Eq. (11 wi th subst i tu t ion in to i t of the 
coef f i c ient I r determined accord ing to [5] and the ve loc i ty u r obta ined as a resu l t of the 
exper iment . 
We see from the exper imenta l data that the f r i c t ion ve loc i ty of the s tar t of f low fo r 
par t i c les of a g iven d iameter does not depend on the channel forms. Thus the noneroding 
ve loc i ty on the r ipp les can be ca lcu la ted by Eq. (11 i f the coef f i c ient t r i s subst i tu ted 
ins tead of A in i t . Let us examine the inverse problem: we w i l l f ind the s i ze of the par - 
t i c les that a re s tab le on r ipp les formed by a f low with a g iven ra t io u /uo . The re la t ion 
descr ib ing the s ta te of equ i l ib r ium of par t i c les wi th d iameter dpa on r ipp les has the form 
u -~ , ~- . _ _ (2) 
It follows from the preceding relation that the flow velocity exceeding the noneroding 
velocity for particles with diameter do lying on a bottom with a granular roughness should 
be equal to the noneroding velocity for particles dpa located on the surface of the ripples. 
As was stated above, the size of the material forming the pavement should correspond to 
75-85% with respect to the grading curve, consequently, with particular error we can assume 
the average diameter of the particles of the pavement dpa to be equal to d c to which 95% 
corresponds with respect to this curve. 
In view of the fact that the average diameter of the eroded material being investigated 
by us does not exceed 1 mm, according to the recommendations [7] we can consider that a flow 
with a bottom formed by granular roughness belongs to the smooth (when do <0.25 mml or to 
the transition region of resistance (when 0.25 ~do < 1.0 mm). To calculate parametera 
equations are recommended in [7] for the smooth region 
0.34 
a=~, (3) L~ #d 
892 S. Ya. PAVLOV 
TABLE 1 
.q, m 
1,0 
5,0 
I0,0 
200 
=/.0 
I,I 
1,4 
I,I 
1,4 
I,I 
1,4 
I,I 
1,4 
do~O,I mm 
' ! t �9 ~-x =oo ~xloo "h~<l~176 -Pa 
! ,04 
1,66 
1,28 
2,07 
7,07 1,05 
15,18 1,27 
6,61 1,03 
14,19 1,25 
4,63 1,03 
13,80 1,24 
6,26 1,02 
13,43 1,24 
60,82 
118,7 
(dcldo) nat= 8,3 
Hax 
76,52 
150,09 
85,34 
185,41 
03,19 
184,43 
d,,=O,~ mm 
1,53 
2,40 
i,90 
3,06 
2,07 
3,36 
1,40 
2,28 
1.50 
2,47 
2,26 
3,68 
7,36 
15,81 
6,86 
14,73 
6,07 
14,32 
6,49 
13,92 
1,06 82,40 
1,29 124,60 
1,04 98,10 
1,27 161,86 
1,04 128,61 
1,26 179,52 
i ,03 126,55 
1,25 224,65 
4=0~$ n lm 
. kr'~lu~ de' m/ 'Xl= I ..... [ ' 
7,60 1,07 107,91 
16,32 1,30 108,73 
7,07 1,05 134,40 
15,61 1,27 196,20 
6,86 1,04 140,28 
14,73 1,27 230,53 
6,67 1,04 154,00 
14,32 1.26 254,08 
2,03 
3,19 
2,55 
3,80 
2,80 
4,50 
3,05 
4,96 
(dc/d.)rmt~ 7,4 (de~de) mr== 6,4 
MaX V~X 
for the transition region 
0.26 
a'= R- -~, (4) 
where Re, d =uo,d /v . 
In [6] it is shown by an analysis of dimensionless numbers and criteria for the condi- 
tion of the start of flow that when 0.I0 mm< do< 0.25 mm 
�9 ~e..,==o.5o.~.=~, (5) 
and when 0.25 ~do <1-2 mm 
Re*d=O'314At~ (6) 
where Ar =,--1- - - ! i s the Archimedes number determined wi th respect to d iameter do. 
The va lue of A in the t rans i t ion of res i s tance i s ca lcu la ted by the equat ion [6] 
~41 lg -~ + I 1.28 -- 1-4 Ig Arc (7) 
where Ar c is the Archimedes number determined with respect to diameter d c. 
When 0.i mr, <do <0.25 mm we can calculate Aby the equation 
6.0435~","' 
x - o.oooe + (-~/~p ~, , ,~o, , , =o. (8 ) 
Substituting Eqs. (3)-(8) and the equations in [3] for determining the relatlve dimen- 
sions of the ripples and X r into Eq. (2) (taking into account that in the range u/uo =i.i- 
1.5 when do •0.7 mm the ratio of the ripple height to its length is approximately equal to 
i0), we obtain a relation giving in implicit form the relation between the sought quantity 
d c and the given parameters U/Uo, do, R. This relation was solved by the trlal-and-error 
method on a Nalrl-2 computer. The results of the calculation 0' =Px/P -- 1 •1.7 and v = 
0.0125 cm=/sec are given in Table 1, which also gives the values of the allowable shear 
stresses on the bottom of the canal z b corresponding to the limit equilibrium of the parti- 
NATURAL PAVEMENT IN CANALS IN SANDS 893 
cles of the pavement on ripples having height hr/R and creating a resistance characteristic 
At. The same table presents data [8] on the maximum possible ratios dc/do for natural bot- 
tom deposits (dc/de)ma x- 
For practical purposes the procedure of using the materials presented in Table 1 is the 
following. On the basis of the known value of dc/de and given ratio u/ue we find the hy- 
draulic radius of the canal for which the formation of a pavement is possible. Further cal- 
culation of the canal parameters is carried out with respect to u, R, and discharge Q by 
known hydraulic methods. If for some reason the value of the hydraulic radius must be as- 
sumed greater than that recommended by Table i, then the formation of a pavement is impossi- 
ble and the canal must be calculated by the method given in [i]. Stability o f the canal 
in the latter case in the section of general erosion can be provided only by special mea- 
sures ensuring through-flow of sediments within this section or the formation of a pavement. 
In conclusion we can note that the development of a natural pavement in a canal is pos- 
sible only if the flow velocity do@s not exceed the noneroding velocity on ripples for par- 
ticles having a diameter greater than dpa. A natural pavement develops only in flows hav- 
ing either a small hydraulic radius or considerable inequigranularity of the material be- 
ing eroded. 
LITERATURE CITED 
i. M.A. Mikhalev, "Hydraulic design of unlined large-capacity canals in coheslonless 
soils," Gidrotekh. Strolt., No. 4 (1976). 
2. D. Ya. Ratkovlch, "On-slte investigations of the ripple movement of sediments," Tr. 
GGI, No. 132 (1966). 
3. V .S . Knoroz, "Effect of microroughness-of a channel on its hydraulic resistance," 
Izv. VNIIG, 62 (1959). 
4. V .M. Lyatkher and A. M. Prudovskil, Investigations of Open-Channel Flows on Pressure 
Models [in Russian], Energiya, Moscow (1971). 
5. I . I . Levi, Dynamics of Stream Channels [in Russian], Gosenergoizdat, Moscow-Leningrad 
(1957). 
6. M.A. Mikhalev, "Data on modeling certain types of flow of a viscous fluid," Izv. VNIIG, 
108 (1975). 
7. V.S. Knoroz, "Nonerodlng velocity for cohesionless soils and factors determining it," 
Izv. VNIIG, 59 (1958). 
8. G. I . Shamov, "Granulometrlc composition of sediments in rivers of the USSR," Tr. GGI, 
No. 18(72) (1951).