A knowledge of the distribution of the boundary shear stress in open channeIs is necessary not onlly for the economical designing of water courses but also for understanding the mechanics of turbulent f1<nv in these channels. Leighly [1] and Lane [2, :3] obtainec1 the shear stress distribution for certain channel shapes by dividing the area into sub-areas bounded by orthogonals to the isovels.
Lundgren and .Jonsson [4] developed a method for shallow channels with rough beds. Ippen et al. [5]
measured the stress distribution in trapezoidal channels, in the course of theiT work on flow in channell bends, using the Preston tube [6]. Enger [7] computed the shear stress distribution for some trapezoidal sections with erodible boundaries assum- ing a certain velocity distribution equation.
H.ecently Crufl' [8] has obtained considerable experimentaI data for smooth rectangular ehan- neIs. In some cases, the shear stress was measur- cd ,,'Ïth a Preston tube and in the remaining cases they were computed using the weIl known PrandtI- Karman logarithmic velocity distribution equation.
H.eplogle and Chow [9] measured the shear stress distribution in circulaI' channels using the Pres- ton method. Some shear stress measurements for rectangular channels have also heen reported by Davidian and Cahal [10].
This paper presents some more measurements on the shear stress diskibution in smooth rectangular channe'ls. Further the maximum stresses on the
• N. HAJARATNAM, Associate Professor; D. j\IURALIDHAH, He- search Associate. Department of Civil Engineering, Univer- sity of Alberta, Edmonton, Alberta, Canada.
of the boundary layer theory. A relation has been obtained connecting the maximum wall shear stress with the bed shear stress in an infinitely wide channel. Further the boundary layers on the bed and sides have been studied regardÏ'ng the velocity distribution. Some interesting results have been obtained regarding the veloeity dip at the free surface and the discharge distribution. Only su- percritical flows have been considered in this work.
The Froude number of the flows varied from about 1.1 to about 2.4. In the entire range of Froude numbers studied, there ,vas no perceptible forma- tion of roll waves [11]. The subcritical studies have not been successful so far due to difficulties with the velocity and shear measuring instruments for low velocities.
Experimental arrangements and experiments
The experiments were done in two rectangular ehannels. The first channel was a smooth chan- nel with the bed and sides made of plexiglass. It had a width of nine inch es, depth of eight inc1Ies and length of :32 feet with an adjustable slope.
'Vatel' entered the flume from a constant head tank provided with suitable screens. The discharge Q was measured by means of an orifice-meter located in the supply line to the constant head tank.
The water surface profile in the flume was mea- sured by means of pi'ezometers located in the bed of the flume. The test section was located at a distance of about 24 feet from the entrance section.
Velocities were measured using a 3 mm external dia- 603
10' ESSAI - RUN 28
QZ=0.00inch )..;;" 0.00
• '0.50 '0.33
o 'LOO '0.67
"'125 '0.83
( b)
~::5.7510gY~+5.50 YQb'1.58 ESSAI -RUN . 2A
n "0.00Inch ).,0.00
• '0.50 '0.33
o 'LOO '0.67
" '1.25 '0.83 25.0
25.0
15.0
-':'.. ( a)
u. 20.0
~.=5.75 log
Y7
+5.5015.0 .LY 083
o .
20 102 103
yU.
11
u
u. 20.0
in the velocity distribution in the neighbourhood of the free surface. This aspecL is discussed Iater on.
Figure 2 shows a typical boundary shear stress distribution on the bed and side waNs as obtained by the Preston tube technique. From a study of Figure 2 and the resuIts for the other experiments, it was found that in general, the boundary shear stress on the bed 1:/) has a InaxinHlnl value of1:/)n! on the centerline of the bed and decreases as the \vaH is approached. On the side wall, the shear stress
1:," increases with the distance y from the bed up
to SOUle value of y, then remains constant at a maxi- mum value of 1:'Vn! for a certain depth and exhibits a decrease as the free surface is approached.
For the second series of LwoeX!periments, the width of the plexiglass channel was reduced to 3.0 inches by Lhe introduction of a box-like unit wiLh a smooth plywoodside. In these Lwo series, the aspect ratio b/yo has been varied l'rom 0.8:~
Lo 10.82.
The thircl series of experiments were done in a large rectangular channel of width of 2.94 feet, depth of 2.50 feet and length of 120 feet with smooth si des and hottom. In aIl these l'uns, the distribution of Lhe bed shear stress 1:/1 and waU shea'r stress T",and cenLerplane velocity distribution were obtained whereas only in some of the l'uns detailed velocity distributions were obtained. Due to cerLain difficu'lties with the orifice-meter, the discharge was not directly measured and in the experimeuLs with detailed velocity measurements, it was computed by graphical integration. The aspect ratio b/yo was varied from 6.79 to 20.28.
yu.
11
3/Velocity distributions for various aspect ratios (a to f).
Répartition des vitesses en fonction de l'allonaement
(a il f).
aux
FREE SURFACE SURFACE LiBRE
RUN ID ESSAI Twm ' 0.045 Ib./sq.ft.
, (ft.)
... 1
t
o '-- ---L_ _- ' -_ _
o 0.05 0.10 Tw (Ibs/sq.ft)
1
0.375 0.30 0.20 0.10 0
O r - - - , - - - , - - - , - - - . - - 0.15
0.10
0.05 Y (ft) 6.0
5.0
~
'.0
3.0
( ftlsee.1
~
2.0 "
inch ~ 000
, 15 033
1.0 '30
0&7
'35 078
,40 089
:A5 , 0
00 0.001 0,01 Dl 1.0
Y (ft)
'bm =0.055 Ibs/sQ.ft.
0.20
0.05~....,---...--~
Tb 0.10
(Ibs/sq.ft) 0.15
2/
Typical boundary ,shear stress di-stdhutions.Répartition-type de la contrainte de cisaillement limites.
meLer Prandtl-type PitoL static tube. The same Prandtl tube was also llsed as a Preston tube [12, 13] .
Six experiments (1 A 1 F) were done in this channe1. If V is the mean veloeity of fiow and Yo is the normal depth, then the Froude number, defin- ed as §7= V/Vgyo, 'where g is the acceleration due Lo gravity, was varied from 1.25 to 2.31. Velocity and the boundary shear stress distributions were obtained in one half of the channel. Figure1shows a typical velocity distribution vvhere II is the velo- city at a norma'l distance of y above the bed and
z
denotes the transverse distance from the center- plane of the channel. If),
=
z/(b/2), wherebis the breadth of the channel, for À from zero up to about 0.89, II varies linearly with log y and for À = 1.0, the variation becomes curviIinear. (No corrections have been made to the velocity lueasurements for the Pitot displacement, turbulence and other ef- fects.) Further, Figure 1shO'\vs that there is a dip1/
Typical velocity distribution.Répartition-type des vitesses.
604
yU.
V
\ u yu.
U.~5 75 log - +5 50
J?..~7.01 '0
~-;. 5.7 5 logy~.+5.50
~ ~10.82 '0
ESSAI -RUN le
" z~0.0 inch >.~0.00
• ~1.5 ~0.3 3
o ~3.0 ~0.67
v ~3.5 ~0.78
... ~4.0 ~0.8 9 ESSAI -RUN 18
(iz:;0.0inch À:;0.00
• ~1.5 ~0.33
o ~3.0 ~0.67
V ~3.5 ~0.78
... ~40 ~0.89
25.0 25.0
U 20.0 (.)
U.
15.0
10.0 10
U 20.0 (f)
u.
15.0
10.0
\0
3 hl
ESSAI.- RUN 1D -0z:O.O inch >"=0.00
• ~1.5 ~0.33
o ~3.0 ~0.67
V ~3.5 ~0.78
... ~4.0 ~0.89
25.0
U Ici
U. 20.0
15.0 ~:; 5.75 logy~.l'5.50
J?..~332
'0
10.0
10 102 103 10'
yU.
V 30.0
ESSAI -RUN lE '0 z:;0.0inch >.~O.OO
•
~1.5 ~0.3325.0 0 ~3.0 ~0.67
V ~3.5 ~0.78
... ~4.0 ~0.89
U 20.0 1d)
U.
15.0 ~:;5.75 logy~.+5.50
~~4'0 .75
10.0
10 102 103 10'
yU.
V
3
al
Ibl
0"
.n."' ' C
•'0 ,. lE 030
'00
• SERifS 1 AND 2 stRIES1CT 2
b
Y,
[~]:: --~
~
oo O~
Velocity distribution The basic experimentall data for aIl the three series are given in Table 1. In Table 1, So is the slope of the channel, y is the specific weight of water and v is its coefficient of kinemalic viscosity. U is the maximum velocity in the centerplane. In the l'un 3 T, "b was measured for the full bed width and
"10
was measured on both the side walls and theresults were the same for either side of the cen- teI'plane.
even for values of À as large as 0.89. It was also noticed that near the free-surface the!re is a dip in the veloci ty profile.
The vclocity distribution obtained on the center- line of the channel (ie. À = 0) as welil as for sec- tions with non-zero values of À were plotted with Il/Il. versus log (Yll./V) where Il. is the shear velo- city, equal to 'l'tb/P, at the foot of the correspond- ing normal line, and Pis the mass density of water.
A few typical plots are shown in Figure 3 for the aspect ratio b/yo equal to 0.83, 1.58, 3.32, 4.75, 7.01 and 10.82 respectively. From Figure 3 and the other related plots, it was found that the velocity distribution agrees weIl with the weIl known Kar- man-Prandtl equation [14]:
Il - - - 1 /Ill.
+ -
r;0-- =
a.';) og -,~ a.aIl. v (1)
41
Velocity clip at the free surface.Chule de vitesse il la surface libre.
Concerning the dip in the velocity profile, it is generally believed that it is more due to the pre- sence of secondarv currents rather than the air drag [15,16]. In' this paper, without going into a discussion of these above two causes, a prelimi- nary mcthod is suggested for correlating the dip in the velocity at the free surface with the aspect ratio.
\Vith reference to Figure 4a, let (All)o be the difference between the actual velocity and that given by Eq. Cl) at the free surface. Itshould be mention- cd that in this investigation, the Pitot tube was placed as close as possible to the fI' ce-surface and the surface velocity was obtained by a small extra-
, /
~~
= 4.15 log[~8 VC;] +
:UH) (4)51
Velocity di'stribution in the side wall houndary layer.Répartition des vitesses dans la couche limite de la paroi latérale.
Eqs.
(in
and (4) are shown plotted in Figure 7. The experimental results for cfli are plotted in Figure 7 and they are somewhat higher than the Clausel' and Karman curves. Further comments on this comparison will be made later artel' analysing the maximum shear stress on the side walls.olA
• 1B
1>IC .. ID viE .. 1 F
-z u.v
15.0
~ 20.0 25.0
~:5.75 log z'v.+5.50
u. •
polation of the measured velocity profile. In a few cases, this predicted vailue of surface velocity ,vas checked and found to agree with the observations on small surface floats. If [(~Il)/U.Jo is the di- mensiol1'less veilocity dip at the free-surface, its va- riation with b/yo is shown in Figure 4b for the first and second series. Il is seen that [(~u)/Il.Jo decreases rapidly from about 6.0 at b/yo = 1.0 to zero at an aspect ratio of about 7.0. The experi- ments in the bigger channel did not show any appre- ciable value for [(~Il)/Il.Jo'
Further the experiments in the smaller channel showed tITat for any given experiment with an aspect ratio less than about 7.0, the dimensionless velocity dip at the surface [(~Il)/Il.J increases as the side wall is approached. This effect is shown in Fi- gure 4 c. Only the experiment with the smallest aspect ratio shows an opposite trend. Il is not known definitely as to whether this reversaI at smaN aspect ratios is due to the change in the pat- tern of the secondary currents.
Considering the side wall, taking the portion on which"Cwwas essentially constant,ifZ'is the normal distance from the wall, the velocity distribution nor- mal to the ,vaU is studied in Figure 5 by plotting Il/Il. versus log (z'Il.)/u where Il.
=
V"C,v/p. Il is interesting to find that the data for the first series agree weU with the Karman-Prandn equation. Data of the other experiments also gave the same results.Maximum bed shear stress
1 . 2 r - - - , - - , - - - - , - - - , . . - - - , -
..
60
50
b Yo
61
Studv of the maximum hed shear stress.Etucle de la contrainte de cisaillement maximale au fond.
o 0
• SERIES 1
<Il " 2
0 " 3
..
CRUFF10 15 20 25
\.0
0.8
Tbm 0.6 )'Yoso
0.4
0.2
0 0
If "Clim is the maximum value of the bed shear
stress, occurring at the center of the channel, for channels of very large aspect ratio, "C1i'" should be equal to Y!JoSo. The variation of "Clim/Y!JOSo with the aspect ratio is studied in Figure 6 using the present experimental results n'long with those of Crufl'. Even though there is some scatter, a mean curve could be drawn, which increases from about 0.3 at b/yo= 1.0 to unity at b/!Jo = 15. Renee for the centerline bed shear stress to be equal to Y!JoSo , the aspect ralio should be greater than about 15.
Considering a rectangular channel of large aspect ratio with an uniform depth of !Jo, at least the cen- terplane 110w could be considered as a turbulent boundary layer of a thickness 8
=
Yo. Il is a fuUy grown boundary layer and the channel bed is under constant pressure. If U is the maximum observed velocity in the centerplane, then U corresponds to the free stream vei}ocity of the boundary layer. A skin-friction coeflicient cfI) is defined in the conven- tional manner:Karman [18J gave the equation for
Cf
with pipe flow constants as:For the plane turbulent boundary layer growing under constant pressure, on a flat plate the skin- friction coeflicient has been given by a number of investigators. The relation given by Clausel' [17J can be written as:
't'bm Cfl'
=
pU2/2\/+ Cf
= :J.96IogI-~ _
u-JCfJ..
_ 4.15(2)
(3)
Cfxl 03 .-
30
VALEURS DE:'\0'" -.
OH/;c;'::%Z.p~:s~'3 EIIFONCTlO~'Dl
2..0 L'AllONGEMENT!"~ 0.83 __\5~
T...DATA ., 300_ .... 00 1,2t.3SH,H. ~OO__ \OOO (,QOUP!08Y D 1000 1.'.000
10 A5P(CIR.UIQ • 1500 2000
•"'~.. ()AIA11,2 AJ'5UIE5) V"U:URS [JE 'bm (StRfES " 2,f JI
01L,0''--~~~I--;O'~~~~I--;O'~:'''-~~10-;:'~~~~'07,~~...J,O.
RS:~
71 Skin friction eoefficient of the hed and side wall.
Coefficient de frottement superficielsurle fond et la paroi latérale.
Maximum side wall shear stress
Considering a rectangular channel of small aspect ratio, leaving a smal'l layer near the bed, the flow could be imagined ta be made of two fully-grown boundary Iayers of thickness /)
=
b/2 on the si de walls. Defining the skin friction coefficient c fw as:(6)
't'11)11/.
Cfw = pU2/2 (5)
where U is the maximum centerplane velocity, the variation of cfll; with 6t.a= (Ub/2u) is shawn in Fi- gure 7. The data for cfw varies in the same manner
asc fb , thereby showing that the bed boundary layer
and side wall boundary layers are aIl governed by the general flat plate turbulent boundary layer theo- l'Y. A curve is drawn through the experimcntal points and this curve cOl1'ld be expressed by the equation:
Il [- u - ] -
V 0
= 4.4 log DO \Icf+
1.60o. lb)
~
~.'O.0.6 ':'5~104
O.
in which, for the bed boundary layer cf
=
c fb and/) =
Yo and for the sicle wall boundary layercf=
cfwand /)= b/2. The interaction of the bed and side boundary layers results in a variation of the hed as weIl as wall shear stresses which is considered in a later section.
0'
8/
Study of the maximum bed and side wall shear stress.Etude de la contrainte de cisaillement nW;l;Ïl1lale sur le fond et en ]Jaroi latérale.
oO!:---:--::,0---;-'':-'--::'0"---:':::-'---;-3':-0---',,:----:':.O--b--..J
r;
(7)
(8)
(9) 6t.= UYo
u
"t'lem == .!!...tU)
't'bill Cfb
. /-i- V =
4.4 log ,-.6t. \/Cfb-,+
1.60CflJ _ __
'
/_1_ = 4.4 'log
,---c:;- J!.-- VC
rlV-1 +
1.60CflV _ ~ Yo _
Using Eqns. (7) ta (H), il could be shawn that If:
It is possible ta establish a relation between 't'bm
and 't'Will and the aspect ratio of the channel. One
could write:
Relation between
'7bmand
'7wm'0r-~rAr-...~
o. "(RUH'S Mf AN (URvt FOR
/:ouRBE,I(Orw/~:: ~. ~\
"•••16.;.È.. .; iC ";"~
06 TO ~
DE CRUFF POUR 1C<-f;.-;' 20 • o3R ~']02&
_3D10,1960 r.3~; , 1&9 7
"3l .IS6] 10'1 v J I <1771 ::::31: '1606
ooC---'''---:':,0--''--'---"O--:':"-b----'
r;
{a) PRtsENn truDe
oPRESENT WOIlI(
... (RUFF 15
0'
" a
"
'.
'O. 811°~
..
~,0
Tb
.
T~
O.
.!:.-'.68 CI<
010 1°'332
>1 [ '.175
,.,
C·1A '!5&
CSJ C
0 Dl
"
Tb.,
9/Bed shear stress distribution.
Répartition de III eontrainte de cisaillement Sllr le fond.
6t.
=_U~Jo l
(10)T'Cllle =
t
1. -]) ,'t lnn _!Jo
Typical variation of 't'IJ and 't'w have been shawn in Figure 2. An attempt was made ta give these distributions in groups, each fora certain range of the aspect ratio. This is done in Figure 9 and 10.
Bed and side wall shear stress distributions
Eq. (l0) was evalllated for three values of 6t.
=
5 X 104, 10° and lOG and the variation of't''':III/'t'/)/ll with b/yo for these tIll'ee Heynolds numbers
is shown in Figure 8 a. Il is seen that in the range of HeynO'lds number considered (vl'llÏch is typical of laboratory investigations) the effect of 6t. is ne- gligible only for b/yo less than about tlll'ee. The experimental results of the authors as weIl as those of Cruff shown in Figure 8a, agree generaUy weIl with the above calculations. For the range of 6t.
bel\veen 5 X 104 and lOG using the results of Fi- gure 6 and Figure 8a, the variation of "w",/rYQSO with b/yo was computed and the results are shawn in Figure 8 b. It is interesting ta see that for b/yo greater than about three the effect of the variation of the Heynolds nllmber is appreciabIe.
"OtN
."o
"'.
": .0
~
...
ft'
(b)
"
·0· ': .. .
A ,"0
0".
, ,,1 15<~qO ~
Yo ~,
cl'
\.006 .. " ~'756
".'8 YO'701 .J: '679
"H ,7J5
"3G '793
:JJH '911
.3A '9,U
'862
"
c 3e 3i
ç3C " iJO;
Ç,3F
"3"'. '12:5 :·3J " J 6 !
"Je 'I~77
o o0
b 0
- ' 1 5 8 '0
\0 O~O'-r: C:>T~ <»
COURBE MOYENNE i-
DE CR!JFF POUR' . (RUfF'5 MEAN CuRvE FOR
t1~<..!..< :J 5 '0
'.0 02-
0'
y Yo
10/ Sille wall sheal' stress distribution.
Répartilion de la contrainte de cisaillement en paroi latérale.
"-"-q 1.10
LOS
..
1.00
0 10
"
20 25Y,b
11/ Distribution of the discharge intensity.
Répartition de l'intensité du débit.
Table 1 Basic eJ.:perimental data
Yo 1bm F = -'L ~ x 10-4 Ub 10-4 1WIll
ExpL110. ( ft)b 50 ( ft) b!Yo Q - x
(cfs) 1bs!sq.fL a
.m
v 2v 1bs!sq.ft.lA Q.75 0.0197 0.160 4.68 0.574 0.1155 2.10 8.05* 18.9 0.0810
lB 0.75 0.0197 0.107 7.01 0.344 0.OB75 2.31 4.85 17.0 0.0720
lC 0.75 0.0197 0.069 10.82 0.172 0.0735 2.24 2.59 14.1 0.0450
10 o75 0.00656 0.226 3.32 0.574 0.0550 1.25 6.95 13.2 0.0450
lE 0.75 0.00656 0.153 4.75 0.344 0.0425 1.28 4.B6 11.5 0.0370
1F 0.75 0.00656 o099 7.56 0.172 0.0315 1.30 2.59 9.81 0.0180
2A 0.25 0.0193 0.158 1.58 0.143 0.0720 1.60 6.36 5.15 0.0763
2B 0.25 o0193 0.301 0.33 0.330 0.0874 1.41 1.36 5.70 0.08BO
3A 2.94 0.0096 0.289 9.84 5.472* 0.1344 2.11 17.6 89.3 0.0940
3B 2.94 o0096 0.341 3.62 6.B72* 0.1492 2.07 22.7 97.8 0.1160
3C 2 94 0.0096 0.258 11.39 4.664* 0.1236 2.13 15.2 87.1 0.0880
30 2.94 0.0096 0.150 19.60 2.168* 0.0338 2.33 6.67 65.5 0.0550
3E 2.94 0.00636 0.400 7.35 7.128* 0.1126 1.69 23.8 87.9 0.1040
3F 2.94 0.00636 0.246 11.94 3.520* 0.0864 1. 73 11 .8 70.8 0.0650
35 2.94 0.00636 0.371 7.93 6.496* 0.1233 1.72 21.6 85.5 0.0940
311 2.94 0.00636 0.317 9.28 - 0.1138 - 17.3 80.3 0.0800
31 2.94 0.00636 0.166 17.71 - 0.0738 6.65 59.0 0.0430
3J 2.94 0.00636 0.215 13.68 - 0.0887 - 9.55 65.6 0.0550
3K 2.94 0.00636 0.183 16.06 - 0.0802 - 7.67 61.6 0.0470
3L 2.94 0.00636 0.158 18.60 - 0.0781 - 6.24 58.1 0.0560
31·1 2.94 0.00636 0.242 12.15
-
0.0864-
11.6 70.9 0.06303N 2.94 0 .. 00636 0.155 18.97 - O.OllO
-
5.96 56.5 0.045030 2.94 0.00636 0.199 14.77
-
0.0806 - 8.54 63.2 0.06203P 2.94 0.00636 0.167 17.61 - 0.0706 - 6. II 59.3 0.0530
3Q 2.94 0.00636 0.260 11 .31 - 0.0958 - 12.8 73.0 0.0690
3R 2.94 0.00636 0.145 20.23 - 0.0597 - 5.31 54.0 0.0390
35 2.94 0.00636 0 .. 187 15.72 - 0.0696
-
7.84 61.6 0.048031 2.94 0.0053 0.433 6.79 7.16* 0.1100 1.506 24.4 82.8 0.0820
1
* Intcgratcd discharge
Using Figure 9 it could be saitl in general that as the aspect ratio increases the bed shear stress dis- tribution tends to become more uniform. There is considerable but no sys tematic scatter and the present data agree fairly ,vell with the correspond- ing average Cluves of CrufL Figure 9d for b/yo 16.1 to 20.0 exhibits some contradiction to the above general remarle Figure 10 shows the correspond- ing distributions for the side wall shear stress. Here it is even more difllcult to discern clearly the efl'ect of the aspect ratio.
50
me supplementary resultsIf
q
is the average discharge intensity, equal to Q/b and q,n is the maximum discharge intensity, which was aI\Vays found to occur on the center- plane, the ratio of q"Jq is ploUed in Figure 11 agai'11st the aspect ratio b/Uo' It is interesting to see that for small values of b/yo, q",/q is much la l'gel' than unity. It could also be seen from Fi- gureIl, tat as b/Uo increases from 1.0 to 20, q",/q decreases t'rom about 1.20 to about 1.01.Conclusions
Based on the experimenta'l results for supercriti- cal 110ws with the Froude number varying from 1.1 to about 2.4, the following conclusions could be drawn.
1. In the ,vhole range of the aspect ratio studied, ie. 0.83 to about 20, the veilocity distribution normal to the bed agrees well with the weIl known Karman-
60S
Prandtl logarithmic equation for values of À up to about 0.89. The velocity distribution normal to the side walls also agrees with the same equation, but this aspect has not been extensively studied.
2. The maximum boundary shear stress on the bed and si de walls has been correlated on the same basis as the fiat pIate con~tant pressure turbulent boundary 'layer. Some difference has been found to exist between the skin friction coeflicient obtain- cd in this work and that given by Clausel' for the fiat plate boundary layer.
3. H has been found that the ratio of the maxi- mum side waH stress and the maximum bed shear stress varies with the aspect ratio as weIl as the Heynolds number. These remarks apply also to the ratio of the maximum wall shear stress and the maximum bed stress in an infinitely wide channel.
The variation of the ratio of the maximum bed shear stress to that in an infinitely wide channel, with the aspect ratio has also been found. In addition, the distribution of the bed and wall shear stresses has also been obtained.
4. A preliminary analysis has been made regard- ing the dip in the velocity profile at the frce surface.
Acknowledgements
The research work reported in this paper \Vas done in the Civil Engineering Department of the University of Alberta, Edmonton. The authors are thankful to Mr. H. Tape for doing the experiments in the smaller channel. The authors are grateful to the National Hesearch Council of Canada for the financiaI assistance provided.
Notations References
The following symbols have been used in this paper:
b : breath of the rectangular channel, also suffix for bed;
cf: coefficient of skin friction;
cfli skin friction coefficient, with respect to maximum bed stress;
cf",: skin friction coefficient, with respect to maximum 'wall stress;
f: function;
§i : Fronde number;
g: acceleration due to gravity;
q : discharge intensity;
q
average discharge intensity;q", maximum discharge intensity;
Q discharge;
,-=
uuYO ']al Reynolds number So bed slope;
Zl point velocity;
LI. shear velocity;
U maximum centerplane velocity;
V: average velocity;
y distance normal to the bed;
Yo normal depth of flow;
z: transverse distance from the centerplane;
z': normal distance from the side wall;
y: specific weight of water;
o
bonndary layer thiclmess;À non-dimensional transverse distance;
u kinematic viscosity of water:
p mass densi ty of water;
"t'o houndary shear stress;
"t'Ii hed shear stress;
"t'", \vall shear stress;
"t'IJln rllaxinlllIll value of "t'Ii;
"t'Will rllaxirllum value of "t'",;
1::.LI/ LI.
[(I::.LI)/LI·]O:
velocity dip at the free surface;
velocüy dip at the free surface centre-plane.
for the
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