a w l LFL" 91 W O R L D H E A L T H ORGANISATION MONDIALE O R G A N I Z A T I O N &>I u>& &I DE LA SANTE REGIONAL OFFICE FOR THE EASTERN MEDITERRANEAN

Mghth Session

EM/k8fiecht1)isc*/3 30 July 1958

BILHARZZASB m ITS CONTROL

STUDY OF WATER FLOW VELOCITIES m ZRRIGaTION C A N U S - IN IRAQ AN0 THEIR MATHEMATICAL ANALYSIS

Joaquin de Araoz4+, C.E., M.S.S.E. WHO Public Health %Leer

Bllharziasis Control Project Iraq 15

Ecologists have found that water velocity in streams has a great bearing

on the breeding and developmmt of b i h r z i a s i s vector snails. C a r e f u l

lzlvestigat ion on this subject has been hampered by the difficulty of d e t e m b d n g

velocities near the periphery of the watercourse, t h e usual habitat of these

diseasb +vechrs.

U r n revision of hydraulic treatises, it was %md that t h e -subject of

water velocity lin Canals is d a l t with mainly from the englneerhg viewpoht,

emphasis is h 3 d on the determfnation of the mean velocity for calculating the

rate of water flow. The distribution of water velocities within t h e cross-

s e c t i ' d area of the canal f s considered of a rather academic hterest;. There

is a c e r k k diversity of opinion amongst authors concerning the distribution

of water ttelocities, and thei r statements regarding &cities near the periphery

of the ad. lack tbe required precision f o r ecological studies of the vector ~13aXk. S a m of these authorities will be quoted later, when the results of

this study are compared with available material on thf s subJectr

* The.:w&uable a e s b t b c e of lkg- al-, b i a t r y of Health, % undertook most of the field work for t H s dudy*, is here* achowl- w i t h gmteful t-ks,

The undertaking of a study of water f l o w in canals by the Bilhsrziasis

Control Pro.ject mo/'Iraq 15 has a twofold purpose: (a) to determine the water velocity at any po'ht of the cross-sectbn

of the canal3

(b) to d d s e a pract ical and accurate method of determining the rate of

water flow in canals, of wide application in mollusicidd work.

These two objectives are direct ly connected with the shape of the cross-

section of the canal; therefore, t h i s study includes the d~.%emknation of the

shape of the contour of . the ~ m s s ~ s e c t i o n , essenthl when dealing with earth

canals which, through erosion and sflt deposition, are allowed to shape freely

t h e i r channels until a state of stability is reached.

It is obvious that such study of water f l o w has a wide application, The

h y d r a a c engineer will prof i t from a simplified method of measuring the amount

of water flowing in canals, either for power, i r r iga t ion or drainage purposes,

The design engineer w i l l be able to draw cross-sections of canals adjusted ta

a more stabilized condition which reduces .erosive action and silt deposition on

the stream bed. 3hintenance work, mch as clearance and chemical destruut3on

of vegetation will a l s o be assisted.

~anoix''' lists a series of "straightforward questions which an i r r iga t ion

or pubUc health engineer might address to a group of e~o log i s te '~3 among theae,

question 4 is What are the r n i n h m "averagetf, %argllna17' and nbttomn velocfties

w h i c f i w i l l discourage the m l t i p l i c a t i o ~ of varsous disease-mrrying e;naUs in

earth canals?. . .I1 The ecologist in his turn may ask the engineer '%Jar can I d e t e m h e navetagem, lbrginaLv and "bottomn velocAties?" Bn attenpt is made in

this paper. to give a proper answer to the Zattek question,

A U material for this study was obtained from 20 measufing stations on 14.

canals of t h e d i s t r i c t of Tanniya and adjacent lands, area of operations of

the Bilharz iasb Control Pm ject Iraq 15.

These canals are located north of Baghdad at a distance of 20 to 50 h.

from this city. They are not part of ah i r r i g a t i o n system, but are independent

and wlthout inter-connections, most b e k g In fested w%m t h e s n a i l vector of bilkarziasb

The canals originate at the western bank of the Tigris River, f r o m which

they are fed by means of mohr punrps that lift water to the head of the canal

at a height of 5 to 7 m, above the normal water lev& of the river. Bmater

pump-statiana are installed on the longer canals so as to overcome the. gradual

upward stops of the land away from the river course. The conUnuous operation

of these pumps gives a uniform and constant flow regime to the canals

Poor engineering s W is shown in the construction of these canals, as

most of them were dug by the farmers themselves without professional advice.

Unnecessaxy curves and irregularities in the cross-section are frequently

observed, In general, canals are only partially dug in the s o u , as embank-

m e n t s are built From the excavated material to complete t h e i r required

capacity* Periodha1 silt clearance is carried out without attemptlng to

correct ikeguladt ies in the cross-s ection and caurse of the canal, the md

dug out' from the stream bed is piled on top of the embahents, r e d t i n g i n

a gradual increase of their hetght and consequently of their water-cawing

capadty. Thus water in the canals f l o w s at a level higher than that of the

&tivat& fidds, and irrigation water can be drawn out by gravity.

me canals run along the alluvial Mesopotamian plain, and the mterial

which constitutes their beds is a s i l ty loam, r ich in &ay with a Inw ~tand

content, This msterial is compact amd hardwhen dry, but whenuhderwater,

it becomes smooth and of plastic consistency, of a grayish brown colour, and

pliant to water action,

The Wer in the canals is turbid, o r d h a w the current met= used for m e a m r i n g water velocit ies could not be seen at water depths below 25 cm,

The water velocity i s vexy low and eddies, turbulence and cross-currents were

only observed f o r a short distance downstream of the pumping stations anl in

place where the canal course varies abruptly, Submerged vegetation is scarce,

but grass is usua l ly found along the water edge.

Careful at tent ion was paid to the selection of measuring stations. Long straight reaches were alwaya chosen and the measuring post was fixed on t h e

mid portion, Bridges and culverts were avoided. The presence of aquatic

vegetation or of grass and s h b s on t h e Mnks were reasons for discarding a whole section of a canal, Another condition sought waa that the water flow

should be as m o t h d streamlined as it cbuld be vismlly appraisd.

Particular care was taken to prevent disturbmg the strean bed and water flow

during measuri33g operations,

Despite these precautions, eymmetrical reauings were seldom idmticalj

howevep, wihh regard to water veloc i t ies ) the diversion from the meari value of

sytnmetrfcalw located measurements was less than 5$ In about half of the t o t a l

rmmber of r-gs.

The general characteristics of the canals at t he measuring s tat ions are

summarized in Table I.

TABLF, f

General Chrac tc r i s t i c s of thc! 1.rrna2s at the measuring stations:

Station Water Width Wat ex Depth Wat er Vela ci ty Number at at at surface level

surface level centre l ine on centre line ( c b t ~ e t r e s ) (centirnetres) (cm. per second) -

1 372 76 26 2 344 69 20 3 290 84 U 4 351 64 16 5 30 5 4-6 16 6 317 30 17 7 240 96 24 8 318 45 13 9 6 58 120 22 10 271 46 16 3.l 281 80 15 12 239 58 19 13 24.0 72 46 U. 18 5 L;z 24 15 210 62 18 l6 192 44 19 17 209 58 32 v 193 55 27 19 197 64 32 20 2% 74 30

Mean 263 # 50 64.25 22.24

For the study of the shape of the contour of the cross-section, measurements of water depths were taken at five equidkstant p o h t s across the canal. TO

obtain more accurate results a special staff was designed. Ris staff is

provided with, an articulated base that prevents its s iddng in the soft m d of

the stream bed while reading water depths, The ar t icu la t ion p e d t s the

rotation of the base and i ts adjustmat to t h e slope of t h e bottom and sides of

the canal. A rectangular cut in the base permits, t h e f i t t i n g ; of the connect-

Fng hinge t o the underside of t h e base, thus corrections of w a t e r depths when

the base is set at an inclined pos i t ion are obviated, A plummet and a

sliding guide attached to the staff assure i t s verticality during measurements.

The staff is graduated every LO crn., measurements were taken with a s t e e l

tape. Readings were recorded with an approximation of half cen tbe t re . A

drawing of this staff is shown in Plate I.

For the study of the distribution of w a t e v d o c i t i e s , within the cross-

sect ional area of the canal, water velocity measurements were taken along the

vert icals of the points where water depths were measured, at depth htemala of

10 cm. starting from the water surface. A Watts a. IV Current Metre, of the bucket-wheel. type, attached to an elec t r ic revolution counter, was used

throughout. Ow- to the dow water velocity, im d l cases less than 40

revolutions per minute, more accurate results were obtained by measuring the

reciprocal of the velocity, 1.8. the time required for a fixed number of

revolutions, In geaeral, the basis used for water vslocit'y readings was the

number of seconds required fop completing ten revolutions. Time was measured

with a stop watch and readings were recorded to the nearest half second.

The measuring procedure was as follows r two wooden planks were laid. across

the canal above the water surface a& perpendicular to the course, one of these

was used as a 'b~idge ahd working platform, and t h e other as a horizontal ruler

for measurements. The two water edges of the canal were referred to this plank

by means of a plummet; by measuring the distance between these two marks the

water width of t h e canal was determir~ed, This width was divided into six

equal parts; obtaining, besides the two edge-points, flve divis ion points, i,e.

the centre l f i e of the canal and two equidistant intermediate points on dther

side of the centre line, ft was at these points that water depths and water

velocities were measured as described above. The sinker weight of the current

metre assured the verticality of the suspension cable, which was not affected

by the alow water current, but t h i s sinker prevented the measuremat of water

velocities under 15 cm. f rom t h e stream bed. The slope of the sides of the

canals; Itn some cases, and the presmce of grass, in others, did not allow

readings at the water edge; .in ordy nine out of twenty measuring stations,

watw edgereadings wsre possible. The number of current metre readings per

station, depending on the w a t e r depth of the cakial, varied f r o m thir teen to

tbirty-eight, wAth an average of twmty-five readings. The position of these

meamring, pohta form a reticular pattern w i t h i n the cross-sectional area of

the canal,

The record form fo r registering f i e l d data is shown in Appendix A *

PIXrrTING OF DATA FOR ANALYSIS

Field measurements were plotted for each of the twenty measuring stations,

and graphs of the shape of the cross-section of t he canal and of the variation

of water velocity-on vertical and horizontal planes, were draw,,, It wag

aasluned that the cross-section and the distributi an of vdocit ies should be

symmetrical with respect to the centre l i n e of the canal, mean values of

readings symmetrically located were used for plot t ing and drawing graphs.

To reduce a11 measurements to a common scale, th i s study is based on

re la t ive values of lengths and veloci t ies and not on their ac tua l dimensional

values, Thus all water velocfties are related t o the velocity occurring on

the centre l h e of t h e canal at thesurface water l eve l , which is considered

as un i t . For horizontal lengths, me uni t of measuremerit' chosen is the distance between the centre line and t h e water edge (half-width of the canal),

and for vertical lengths, the unit . $s the water depth on the centre line of the

canal.

In drawing the curve of the contour of the.cross-section, t h e following procedure was employed. Relative water depths were determined from the mean

values of symmetrically located measurements, approximteb t o the nearest

centirnetre. It was assumed that a max3m.m error of one centimetre in excess

or deficit could be ascribed t o these calculated means, this tolerance covered

instrumental and operationAl errors as well as those resulting f r o m approxima-

tion, According to t h i s acceptable ermr, m3dmum and minimwn values of the

relative water depth3 were calculated. When plotting the relative water

depkhs, this allowed difference was taken into consideration by -king,

instead of points, segments of l ines ' indicating the accepted range between the

two U m i t h g values of the water depth, A smooth continuous curve was drawn

so as t o cmss all the plotted segments of lines. From th i s curve relatlve

water depths were measured at four int m e d i a t e equidistant verticals between the centre f i e and the water edge,

A similar procedure was fol lowed f o r drawing the curves of relative water

velocity distribution on vert ical and horizontal planes. Measurements w e r e

approxhated to the whole second, and the allowed error was plus or minus one

second. The drawing of velocity curves on v e r t i a and horizontal planes

was carried out simultaneously so as to verify t he accuracy of t h g traced

curves. From these graphs relative water velocities were measured at d e w 1

intervals of the relative water depth at the centre l b e of the canal.

As an illustration of t h e procedure, p la te 2 shows the case of measuring

s t a t i on No ,15.

V STUDY OF THE CONTOUR OF THE CBOSS-SECTXON W EARTH C A N D

The shape of the contour of t h e cross-section of each of the twenty

measuring stations was drawn following the procedure %xpMned in section 4 ,

The r e su l t i ng curves are referred t o a coordinate system having its origin at

the bitersection of the centre-line of the canal with the waterr surface. The

abacissss-axia of t U s systag l i e s on t he horizonbal plane a t water surface

lael, and the ordhates-axis coincides with the ver-bical plane an the &re

line of the canal.

Relative water depths were measured from the p lo t ted graphs at four equf-

distant verticals between the centre line, where the relative water depth is one,

and the water edge, where t h e re la t ive water depth is eero. These measurements

W e ahown Fn Table III mean values of the relative water depkhs of the twenty

measuring, rntions,. t h 6 standard deviation and standard error of these meana appear on the lower part of the table.

TABLE II

Relative water depths at four equidistant poin t s between the centre line and the water edge at twenty measuring stations :

Station Relative distance f r g m the c a r e Une Number 0.20 O e 4 0 0.60 0.80

Mean Standard deviation Standard error

It was considered that the law of variation of the relative water depth

with respect to the relatfve distance from the centre l i n e could be express&

by an equation of second degree of t he form:

Ay2 + ~e~ + 2Fa + C = 0

in w h i c h y is t h e relative distance f r o m the centre l ine of the canal

(distance from the centre line to t h e water edge, y 1.W) and z is the

rdatfve water depth (water depth at the centre lhe, z = 1.00). water

depths are measured downwards, and therefore are considered negative.

The coefficients of the second degree equation were determined so as to

fit closely with the means of the relative depths of the twenty measuring

scations , The f oUowing equati,on was obtained :

The f itncss of t . h i ~ equa+ion is +own in Table If 1, in nh,ich a comparison

is made between the , relat ive water depth values computsd from the equation and

the means of the twenty measuring stations,

TABLE I11

Camparisan of the values of r h t i v e waher depths obtained f r o m the equation with the mean values of twenty meawing stations:

Relative distance f r o m the centre ling, 0 0.20 0.40 0,60 0.80 1.04

Mean -1,0000 4,9750 -0.8975 -0.7620 -0.5290 0 Equation -1.0000 ~0.97% ,0,9000 -0.7613 -0.264 0 Discrepancy 0 +0.0008 +O,OO25 -0.0007 4,0026 Y3

Sum of discr4pancies 0 Sum of the sGuare of discrepancies : 0,0000U

The graphic representation o f the equation re la t ing the water depth at any

point with i t s distance from the centre l ine, is an e l l i p s e w i t h i ts mjor d s

coinciding with the vertical plane on the centre lfne of the canal. W s

ellipse has the f o U d n g characterist2cs:

Coordinates of ,the Centre, yo = 0 zo = - 0.249159

Semi-axes. Major (vertical) a = 1.249159 Vj,n_q~ (hn*!.n.mt,l h a 1,020506

&centricity c = 0.576702

Thp relakive area of the cross-section of ' the canal was determined by

integration .of $he equation of t h e ellipse,

Relative Area: a 1.497266 zy,, f o r z and y equal 1,000

The ac tua l area of t h e cross-section is obtained by substitutfng z f o r H,

water depth on the centre l b e of the canal, and y f o r ~ 1 2 , half-width of t he

canal at water surface level,

Actual Area : A = 0.748633 HW

T h i s means that the area of the cross-section of an ePbh canal in which

erosion and silt deposition have reached a s t a t e of stabili-by, is very close

to the three-fourths of t he circumscribd rectangle,

The degree of confidence that may be expected f rom th i s equation is plus

or minus 3.96$ of t h e computed d u e . %is degree of confidence was d e t e m h e d

considering that in 95 cases out of 100, the true value does not l i e more than

two standard errors of the mean away.from ,the average value, -and that .the area

is directly proportional to t h e sum of i t s ord%tes,

%e contour of the cross-section, with lines jndicating its uppar and lower

confidmce U t s , is dmm in Plate 3,

By applyihq the equation of the ellipse, reh t i t re distances from the centre

Ilne to the contour of the cross-section were calculated at ten equidistant

level-s, These valuea, whlch w i l l be referred to later when d d h g with

periphertd. vel~cit ies , are Indicated in mate 3 .

VI S W Y OF VARIATION OF THE WATER VELOCITY ON 3 VEKCICAL PUNB FAMlDL TO THE WATHl F r n

Thq , c u r ent metre reaungs showed clearly that the m i m u m water velocity

occurs below the water surface. Out of twenty measuring stat-, the

maximum velocity was recorded at the water surf ace in five s t a t i ons f o r the

measurements of the cedre line, in three statiom for measurements on the

~ertical located at one-third of the distance from the centre Line to the wa$er

edge,, 'Bnd'in twb stations for measurements on the vertical at two-thf rds of

this same dbtance. mis fa& makes invalid t he assumption that the variation

of the fdativerwater velocity codd 'be representd by a logarithmic o~ semi- , .

logarithmic curve, as the maximum value in these two types of curves would be

at waterhsurface level ,

The.occurrence of 'a regyes&on In ' h e values of relative wa$er velocities, i ,e, an h 5 t i a l incrsase up to a m m value f ol lmed by a gradual decrease,

as the relative water depth increases f r o m the water surface to the steam bed, indicates that on either s ide of the depth at which the maximum takes place,

the same vrglatiw velodty occbrs at two different levels, from this it may

be inferred that the c u p e of water velocit ies is of second degre'e.

Table IV shows, Por.each measurhg stat ion and for the t h r e e w r t i c h l

planes on- which currerk'metre f e a d b g s were taken, the values of relative

hter velocities : at sq@dist&i levels hetws m the water sjrf ace and the stream bed, These relative water velocities were obtained from measurements on the

blotted graphs as explahed in section 4 . The means of t h e relative water

relocit les of the twenty measuring stations, the standard deviation and

tandard error of these mans were calculated, r e su l t s are shown in Table V.

EM/h~8fis&.~isb /3 page 10

TABLE IV

Relative water v e l o c i t i e s on vertical. planes p a r a l l e l t o t h e flow:

a ) On.the centre line of the canal. Stat ion Relative Water h p t h Number r.10 G,20 '0,SO 0.40 0,50 0,bo 0.70 0,gQ 0.90 1.00

. . -. - . . . - . - - - - - . . -

b) At 1/3 of the distance from the centre l ine to the water edge.

Stat ion Relative.Water Depth Number 0.00 0,10 020 0.30 0,40 0,50 0,60 0.70 %90:

c) At 2/3 of the distance f r o m the centre line t o the water edge.

Station Relative - Wat ex Depth Number b.oo 0.10 0.20 0.30 0.40 0.50 0.60 0.70

TABLE V

b a n relative velocity, standard deviation and standard error of the manr Computed fmm the values of Table fV:

a) On the centre h e of the cand-,

Relative Mean Standard standard Depth Deviation Error

4 1 0 1.9345 0,0291 0.006 5 -0.20 1.0430 0,0612 0.0137

1,0285 0. oeql 0,0188 4 4 . 0 0.9945 o*lO% O*oZ36 * b T O 0.9433 0.1184 0.0265 -0.60 0,8700 0.1325 0.0296 4.70 0.7830 0.193 0.0336 -0, $0 0,6730 0.1734. 0,0388 -0.90 O* 5300 0 a1913 0.0428 -1.00 0.3595 0,2057 0 04.m

b) At 1/3 of the distance f m m centre ljne to waterr edge, Relative Mean Standard Standard D&h Devhtion Error

o m 0.8830 0;0615 0,0138 9,30 Q;4210 0,0688 o.oXU. 4 2 0 0,1932 5 0.W10 0 -01 59 4;30 0 9215 0.0894 0,0200 lo& Q;8915 0,1067 0.0239

9 0 8430 o a 5 5 0.02% *0;# 0.7765 0.1659 0.0326 3OmW 0,6890 0.1724 0,0385 4.80 0,9730 0.1855 O * W 5 4.W 0 . U 5 0 1922 0.0430

c) At 2/3 of the distance from centre line to water edge,

Relative Mean Standard Standard

As an h i t i a l asscuaptian, it was considered that the dagram of

distribution of relative water velocities. on vert ical planes aligned t o the

direction of the w a t e r f l o w in the canal, cou ld he closely f i t t e d to a second

degree curve (conic section), with i ts axes parallel to' the coordinate axes

of the diagram. This curve, which. could be either a circle , an el l ipse , a

hyperbola or a parabola, is.expressed by an equation of'the form:

Ax2 + I3e2 + 2Gx + 2Fs + C a 0

in which x is the r e t i v e water velocity (water 4velocity on the cent6e l h e

at water surface level, x = 1.000) and z f s the relat ive water .depth measured

from the water surf ace (water depth a t the centre m e , ,z = 4.000), negative

as it is measured downwards,

W t l p l e trials for d e t e W g the values of the coefficimts of the

equations so as to obtain close fitness between the two curves, t he curve of

means of, the twenty stations and t h e curve expressed by the equation, gave

unsatisfactory results, as the discrepancies were found to be of the order of

O.m, which were considered insuf f ic icnt ly approximate . A further attempt t o fit an equation t o the mean curve of relative watw

velocities was carried out assuming that t h e curve of second degree had i t s

axes jhclined with respect t o the coordinate axes of the diagram. This

assumptfon gave mch better results, as discrepancies were more evenly

distributed along t h e curve and thei r values were, in general, about ten

times smaller than those obtained from t h e previous assumption.

The case of inclined axes i s expressed by t h e equation of second degree

between two variables i n its most general form:

~x~ + 2 h z 1x. +s2 + 2Gx + 2Fs + C = O

where x and z have t he same meaning as in the previous equation.

Upon determination of suitable coefficients, t h e foUming equations

were obtained.

Quatiions of the q mi at ion of relative water velocities on three

vertical planes parallel to the water flow:

(a) On the centre line of the c a m 1

(b) A t 1/3 of the distance f r o m centre line to water edge

( c ) At 2/3 of the distance from centre line to water edge

'?he fLtness of these equations is shown ,in Table VI, in which the values

of the mean curves of relative water velocity distribution are compared with

the values obtained f r o m 'the equations

Comparisbn of the values of relative water velocity obtained from the equations with the mean values of twenty measuring stations:

Relative On At 1/3 off At 2/3 off Depth the cern9r.e Line the centre line the centre line

Mean Equation Mean Mean Equation -

Swn of the discrepancies

+ 0.0031 * 0.0120 * 0.oW Sum of the square of the discrepancies

0,000278 0.0001U 0,000~60

The graphic representation of the equation of the relat ive water vdoci ty

at any pojnt of the three vertical planes under study with respect to the relative depth of the point in question, is m ellipse whose axes are not

parallel to the c o o r ~ n a t e axes of the diagram,

The characteristics of these three el l ipses are shown on Plate 4.. The

diagrams on this plate also show the posit ion and value of the maxiraum relative

water velocity as obtained by differentiation of t h e three equations. Upper and lower confidmce M t s of the relative water velocities are indicated.

The degree of confidence t h a t can be expected f r o m the det eminat ion of

areas of t h e water velocity curves of the mean values of twenty measuring -+ + stations, is - 6.05% for the diagram on the centre l k e , - 6.90$ for the diagram

+ at 1/3 of t h e distance f rom the centre l i ne t o t h e water edge, and - 10.33% for the diagram at 2/3 of t he same distance. The overall degree of confidence

expected from the caXcuZation of the volume de l imi t ed by the vertical, curves

of t h e water velocity distribution is 2 7.63%. ' The determination of these

degrees of confidence is based on the same considerations as established for

the ares of the cross-section, i.e. in 95 cases out of100 the t rue d u e of

the relative water velocity does not l i e more than two standard errors of the

mean away f r o m the average value found, and the areas of the dlagrarns are

directly proportional to t h e sum of t h e i r ordinates.

This study is based on the values of relative water velocities obtained

from the three eguations of the velocity curves on vertical planes parallel to

the direction of the water flow, whose study is t h e subject of sec t ion 6.

Ten equidistant horizontal planes, between t he water surface level a d the

level at relative depth -0.90, were studied, and equations of the relative

water velocity a t any point on each of- these planes wi th respect to t he distance

of the point in question, measked from the ver t i ca l ~ h n e of t he centre line,

were determined,

The diagram of the distribution of relative water vehc i t i e s on k o r i z ~ n t a l

planes should be symmetrical with respect to the vert ical plane on the centre

line of the canal; therefore, the ax is a$ t h e curve should be contained in

this vertical plane. It was considered that a curve of second degree (conic

section), with i t s axes paral le l to t he . axes of the diagram, c o d d ' b e closely

f i t t e d to the curve represenbing t h e variat ion of the relative water vdoc i ty

on a horizontal plane, These conditions are expressed by an equation of

the form:

+ B X ~ + 2Fx + C = 0 in which y is t h e relative distance from t h e centre m e (for distance f r o m

the centre line to the water edge at surface level, y = 1.00) and x is the

relative water v&ocity (fox water velocity on t he centre Line at water

surPace level, x c. 1.00).

Coefficimts for the general equation were d e t w e d so as to obtain

exact coincidence ( t o the fourth' decimal cipher) between the relative

velocities determined f r o m the equations of t h e v e r t i c a l velocity curves

and those obtajned f r o m t h e equations representing t h e variation of

relative water velocity on horizontal planes, The following tm equations

were deterrmlnd:

Equation8 of t h e variation of the relative water veloci-by on horizontal planes at ten equidistant levels.

2 At water surface : y2 - 1.177793~ + 3.094769~ - 1.916976 At depth -0,10 : y2 - 1,022605~~ * 2.99148% - 1.991882

-0,20 . : y2 - 0.959353x2 4 2.93852% - 2.015U8 -0J0 : y2 - 0.8853756? + 2.803937~ - 3.940764 -O&o : y2 - 0.771104~~ + 2.559553 - 1.777372

2 4,s : y2 - 0.591042~ + 2.188397~ -1.533875 2 -0.60 : y - 0.358167~~ + 1,744901x- 1.248761

-0.70 : y2 - 0 . ~ 3 4 2 0 ~ ~ + 1,319809~ - 0.966500

Analysis of these equations shows that the graphic representation of the

variation of reLative water velocities on a horizontal plane between t he

water surface level and t he relat ive d q t h - 0.755 is a hyperbola of decreas- ing sxcentriclty, at relative dep%h - 0,755 .the curve turns i n t o a parabola, and below this depth the curve beaomes an ellipse. At relat ive depth - 1.000 (stream bottom) the relative water velocity is represented by a pint ( x - 0,3618) as determined f r o m the equation of the variation of relative water

velocit ies on the ve r t i ca l plane at t h e centre l i n e , See Plate 5.

The characteristics of the curves representing the va~ation* of water

veloc i t ies on the t e n horizontal planes a r e s h m in Table Vf 1.

TABU VII

Charasteristics of the curves of relative water velocity distribution on horisonLa1 planes.

1.- Hyperbolas. Relative Coordinate of centre Semi-axes &centricity Depth P xo Transverse Conjugate

EM/k~8/CTech .DISC page 16

2.- Ellipses.

Relative Coordinate of centre Smi-es &centricity Depth yo xo mYb jor Maor

-0.80 O -'l .19 5607 7.872607 2,081 527 0.964413 -0.90 0 -2,821012 3,361812 1*23@43 0 * 9 3 W

The overall degree of confidence, based on t h e same considerations as

established in section 6 , that can be expected Fn determining the areas of

the diagram of water velocity dis t r ibu t ion an horizontal planes, is equal to

the degree of confidence w i t h respect b t h e diagrams on.vertica1 planes, + i , e . - 7.68%.

V I I T DFfEFWNATION OF R E L A T m WATER VELOCITIES ON TIlE PlBIFFSW OF THE CROSSSETION OF THE CANAL

Relative water velocities on the periheter of the 'cross-section .of the

canal were determined at ten equidistant levels f r o m the equations %fv

relative water distr ibut ion on horizontal 'planes. Values of re la t ive

water veloci t ies were calculated for rela€ive distances from t h e centre l i n e

equal to t he abscissas of t h e perimeter of the cross-section as determined

in t h e study of i t s shape as explained in section 5 and indicated on. Plate 3.

The calculated relat ive water velocities on the periphery of the cross-

section are shown in Table VIII.

TABLE VIfX

Relative water velocities on the periphery of t h e canal.

Relative Relative Velocity Depth on t he periphery

0;m oi34.04 -0.10 0.3993 4 2 0 0.4405 -0.30 ok4.591 -0.40 0.4616 -0.50 0~4.93 -0;bo 0.4331 -0.70 O.G241 4.80 0.a33 4.90 0.3595 -1.W 0,3618

As mentioned in section 3, - i t was possible to- take measurements of water velocity at t h e water ed'ge of the canal in nine o u t of twmty skations. The

analysis of. these data shows d o s e correspondence w i t h the value of the

peripheral velocity obtained f r o m the equation of re la t ive water distribution on the horizontal plane at the water surface level. T.he study of the

water velocity at the water edge is -summarized in Table n.

T A B U III

Relative water velocity at the water edge on the surface level,

Stat ion Current metre readings Relative water velocity Number (seconds per 10 -rev*)

Centre l i ne Water edge - Mean, Nin. k* - -

Avexa ge s Value obtained from the equation

The detendGtion of the relative water velocities on t h e perimeter of

the canal by applying the equations found f o r the velocity distribution on

horizontal planes impliw the acceptance of the hypothesis that the law of

variation found for the central portion of t he canal is applicable t o the

whole width of t h e stream, In reality, however, many factors of very

complex nature, such as the increased f r i c t i o n between t h e water particles

and the material of the stream bed, the disturbance produced by the unevenness

of the sides and bot tom that deflect the water' current towards the centre of

the canal, the accumul.ation of sugpended matter where t h e mter velocity is

low, e t c ., may have considerable influence on 'the variation of water velocities near the periphery of t h e c b a l , With the means in hand, it is practically

impossible la determine the f i e l d of action of these disturbing factors;

however, it may be considered that at the boundsrgr line,,just above the sad-

fluid layers of sediment on t he bottom and sides of t he canal, the mean

relative water velocity along the perimeter 'of the oanal. is about O.LO. and

that this value fluctuates within 2 12%.

The relative rate of flow, or water discharge of the canal, is expressed

by the volume delimited by the hcrrixorrtal plane of the water surface, the

vertical plane across the section of t h e canal perpendicular to the direct ion

of the water current, and the curves representing the varia$ion of relative

water velocfty. For the d e t e r m i ~ t i o n of t h i s volume the velocity-curves

on the ta equidistant horizontal planes were considered,

-8hech.~isc*/3 page 18

The areas of the diagrams of relative water distribution on horisontal

planes were determined by integration of their respective equations between the

IfmitSng value^ of the relative peripheral velocity, The resulting areas are

indicated in P l a t e 5.

The cur= representing the variation of the areas of t he diagrams of

relative water distr ibut ion on- horizontal planes is shown on P l a t e 6 , For the

stu* of t h i s acume, i* was considered as being formed by five consecutive

parabolic segmmts, the i r ends heeting at points of junction f ixed at relative

depths -02, -0.4, -0.6'and 4.8. These segments are expressed by an equation

of t h e form:

in which a is the area-of the diagram of relative water distribution on

horizontal planes and z .is t h e relative depth of the horizontal plane under

consideration.

Coefficients for t he general equation were determined so as to reach

exact coincidence (to the sixth significant figure) between the c alculated

values of the areas of the diagrams of horizontal distribution of relative

velocities and tho se of the equation.

Zquations of the curve of t he diagrams of areas.

e Prom 0 to -0.2: a = -2.807550s2 - 0.930675s + 1.462037

'she area of the diagram,of:$&xss of t h e curves of relative wa, ter distri-

bution on horizontal planes was .de temhed by integration of the above . '.

equations, between the i r h i t i n g relative depths. The t o t a l area is equal

to t h e volume d e l h i t e d by the vertical plane of t h e cross-section, the

horizontal plane of the water surface and the el l ipsoidal surface that contains

all the points representing relative water velocities, This volume, Jn i t e

turn, represents the relat ive rate of water flow h the canal.

The areas of the five parabolic segments that constitute the c m e of

variation of t he areas of the diagrams of relative water dis t r ibut ion on

horizontal planes are Fndicat ed in P l a t e 6. Their s m be- t A 1,096232.

Thus the rehtive mte of flow is expressed by q " 1.09623-s in which

x i s the relative water velocity on the centre l ine at the water surface

1.00, y is the relative distance from the centre l ine to the water edge of

t h e canal = 1.00, and z is the relative water depth at the centre line ' 1.00..

The actual rate of flow is obtained by substituting these relative values

by their actual djmensional values, thus x is substituted by v,, actual water

velocity at the centre l i n e on the water surface, y is substituted by W/Z,

actual half-width of the canal on the water surface and z is substituted brg H,

actual water depth of the canal at t h e centre line,

Therefore, the expression of t h e actual rate of flaw is:

Q is expressed h the name units for measuring t h e velocity and dimensions of

the canal. ,nus, Q w i l l be im cubic metres per second if the velocity is measured in metres per second and the width ard depth of the canal are

measured in metres.

A simplified procedure f o r measuring the rate of flow in a canal, w i t h i n

a degree of confidence of 2 lo$, w h i c h is the degree of confidence that can be expected from more elaborate methqds, consists in determining the water

velocity on the centre l i n e at a relative depth f r o m the water surface of

0.544 (where the relative w a t er velocity is 0.9122), and t he dimensions of the

width of the canal on the water surface and the depth at the centre l lne .

The half-product of these three measurements w i l l give the rate ~f flow, or

Q O C S ~ . ~

X D E T ~ l I N A T I O N OF THE MEAN VELOCITY &iD ITS POSITION IM THE CAPI3AL

The mean velocity pf water flawing through a canal is determined by

dividing the rate of water flow by the area of the cross-section. Thus,

The relative depth a t which 'this water velocity occurs is determined by

substituting i t s value In the equatian of the variation of relative water

velocity on the vertical plane at t h e centre Ibe.

Relative water d@h of the mean vslocityt -0,752665,

mhtc8/)Pech . ~ i s c , /3 page 20

Therefore, it may be concluded that t he mean water vdoci ty in a canal of

similar charac te r i s t ics and dimensions t o the canas covered by this study, can

be determined by measuring t he water velocity on t h e centre Une at a deptb from

the water szr~face equal t o the three-fourths of t h e water depth of the canal at

i t s centre Une. It may also be concluded that t h e mean velocity of the

water f lowind through the canal is 734% of the water velocity on the water

surface at (tihe ~oeribre line of the canal,

XI GRAPHIC rn U ~ E W T I C A L D E T ~ A T I ~ OF THE R ~ T ~ V E WATER vaocIm AT ANY POINT QF THE CROSS-SECTION OF. THL CANAL

The ten equations of the relat ive water v d o c i t y distribution on horizontal

planas, and t h e equation of vertical dis tr ibut ion of relative water velocit ies

on the plane at the centre line of the canal, Were employed for calcuiating'

relative dlstmces from the centre l ine and relative depths, respectively, f o r

t h i r t een values of , relat ive water veloci ty varying at 0,5 htervab from 1,00

to 0.40.

The values thus fourpd were plotted on t he drawing of t h e shape crf the

cross-section (mat& 3 ) . AU points of a q d v d o c i t y were c~nnected by smooth continuous c w e s r These equal velocity lines cari be compared with

the topographic contour lines used on maps fa r representing points of

equal alt Ltude . Plate 7 shows the resul t ing diagram of 'equal *velocity curves w i t U t h e

cross-sectional area of t he canalj the accompanyinp: table summariaes the

r e su l t s of calculations.

The r e l a t i v e water velocity at any point of the cross-section can be

determined ' graphically by int erpolatlon between the two curves aY! jacenb to

the point in question.

The mathematical determination of t h e relative water velocity at any point

of t he cross-sectional area is based on the formulation of an equation

expressing. t h e relationship of the three variables. x (relative water velocity),

y (relative distance from the cent re line) and z (relative depth from the

water surface) ,

A satisfactory, although not a simple, equation found is:

+(lr132817e2 - 0.828486~ - 2.148601) = 0 in which, following the nornenclatnrc used throughout this paper, y is the

relat ive horizontal distance, perpendicular to t he watercourse, measured

EM/Rc8/Slech.~is c. /3 page 21

from the vertical plane at the centre line (y = 1.00, for t he distance from

the centre.-line t o the water edge at t h e water surface),

z is t he relative water depth measured downwards from thewater surface, and

always negative (z -1.00, for t h e water depth at the centre l i n e of the

x is, th.e relattve, water velocity (x = 1.00, fo r the water velocity on the water

surface at the centre line).

The, fitness of this equation is shown in Table X, in which t h e mean

relative water velocit ies of twen ty measuring s t a t i ons are compared w i t h t h e

computed values from this general equation,

TABLE X

Comparison of t h e values of relative water velocities obtained from the general equation with t h e mean values of twenty stations,

Belat ive On ~t l/3 off Depth the centre line the centre l ine

'Mean Equation - Mean Equation - At 2/3 off

the centre h e Mean Equation -

0,6430 0.6617 0 *673 5 0.6769 0,689 0.6791 0,6735 0,6642 0.6420 0.6366 0,5905 0.5882 0,965 04977 0.4145 0.4J-86

Sum of the discrepancies - 0.0004 - O.OW1 - 0.0041

Sum of the squares of the discrepancies 0,000608 0,000132 0,000492

XI1 EFFECT OF THE ACTUAL WATER VEtOCIT'Y AND THE S W E OF THE CROSS-SECTION ON THE VARIATION OF W A T ~ VSLOCITIES OM VERTICAL AND HORIZONTAL PLANS

A correlation analysis was carried out to determine the influence that

the actual w a t e r velocity (in metres per second) and t h e shape of the cross-

sectional area of t h e canal may exert on the curves of relative water

velocity distribution on vert ical and horizontal planes,

In this study, the actual w a t e r velocity in the canal is represerrted by

the water velocity occurring on the water surface at the c a t r e line; t h e

shape of t he cross-section is represented by t h e relat ion water depth - water width, and t he area of the segmmt delimited by the chord subtended

EM/Rc8/~ech.D.i sc. /3 page 22

betwet::, -1':s x5rzmt. poj l~?,s c;P t h e half~per-heter of the canal and the contour

of the cross-sectlcn, thr'.s a r e s is proportlocal t o the curtatwe of the cross-

section and ewresx? t h e relat,irc c o ~ v ~ c i t y of t he periphery,

Thc c-~rve:: of rc la tL-~e water distrlbukAon selected for analysis are those

contained in ' - 3 pr:incipai. plan&?, t h e -?erkiczl plane at t h e centre Zine, where

*.he mn-+mwn ~-elocit;,- O C C U ~ : ~ , arxl the hor i zm:z l p h n e at re la t ive depth -0.20,

which is the d e c h z l plane nearest to t i e po in t of rn- velocity ( m a n of

t-denty fieasurj ng stzLions). It was considered t hah these cunres could be

repres cnted r:ith suT_"ici~:t,. apprcximath:nn bj two rrzriables that fix the i r

posi t ion and con~-~:Y,yt

(2) Th!; nlop. ~f t h e chord. esubt.,:~,ded between %he velocity-pohts on the

water cZgo a d t h e sJ;rea:,i 5oLtc,r ( f o r t h e ve?-tical plane), and between the . " velocity poir::;r: a t Ll;e ccx'3::e 2 ne e vl at 213 of ";he distance from the centre

].he tc t h eva t& edze (:'or th,? l :or 'r . ;~~ltal lane); the l a t t e r point had to be

s d e c t e d as ib was not ~sss!.blr! Lo $ete?"mine the velocity on the water edge in

all t he measrlrhg s t a t i o n s ,r.s expIa5_r,ed i n sect ion 3.

(b) The area of the segmext delimited b, t h i s chord and the curire of r e l a t k e ~ra-ter v d o city .distributions

Swnma~i ~iilll; , the f ollowfng variables weye taken in to consideration for the

correlat :.on analysis :

V: actual water vdoci,tj. on t h e yater surfa,ce at the centre line.

R: r e l a t i on of t l ~ e w a t c r d e p t h to the?-ater 'widt ,h o f ' t h e canal.

A: area ~f tk s e g m ~ ~ t of t he c ~ o s s - ~ B c ~ ; ~ ~ ~ de-ted by t he chord subtmded across t h e smi-;erime'cer and t h e conkow of t h e canal.

m,: slope of t L c c!l~rd. su': .,t.ndFtc? bcLxea 5he rx+,reme p o b t s of the curve of

water vdo; i ty d i s t r i l l ~ t l o s c : ~ -6112 -7ertical pime a t the centre l i n e *

: area of t l ~ e seg,c:,-it; dd:i:&dr ? :y %he chord of slope q and the velockty curve on t h e 1:e1-;;ical p i w e c ~ t tile ce:l'ure Line,

% : slop& of -!,he ck jrd s ~ b ~ , ~ , ; ~ d b :?+rem the veLocity-points at the cent re

l i n e 2n.i - ' " -, c l Yi;e r2,1s1;:2.cc 2 ~ c r l thz+ c e ~ t r e If.na to the water edge,

ah: area -. of t:?c seg;,l~:;t dz:l.h?ii';efl by -21~ c!iorc? of slope rrl, and t he odocity

cur rc on, the h o r i ~ , c ~ t z ~ - plan^ of r d a t i v e dep5h -4.20.

The values of there variable2 f o r each measudng station are shown in

Table XI.

*oduct m o m a t correlation ca efficient s were calculat ad for each variable

representhg the actual water velocity and t h e shape of the cross-section of

the canal with each of t h e variables representing the curves of the relative

water distribution on the two p r b c i p a l planes selected,

TABLE XI

Variables representing the actua l water velocity, the shape of the cros s-section and t h e variation of relative wat& vdoc i t i es , used for the correla t ion analysis.

Station V 11 A % a, mV &v Number

-- --

&an 0,224 Q;239 0.193 0*6&t 0.157 0,555 0.066 St. dev. 4.148 0,072 0,053 00,206 0,057 0,174 0,042

To h o w whether t h e correlation coefficients oould have arisen by chance,

~tudmt'!s "ttt values were determined and results t e s t e d by comparison w i t h

values. of the respective table, The correla t ion analysis was carried further

t o d e t e m e if a linear correlat ion exis ted betwem any two of the variables

representing the cumes of the variation of relative w a t e r velocity,

Results of these calculations are summarized in Table 113. Values of

Student's *tn for different levels of sigrrificance are included.

TABLE X I 1

Product bment Correlation Coefficisnts for the variables of'Table XI.

Between V R A mv % mh

and ra, * 0.004 -3;058 .+0.295 a~ - 0.0'77 +Om333 4 , 1 3 1 +O.28? mh +a001 -0,353 -0.208 G.261 4.068 ah - Or012 4.287 -0.283 -0,220 -0aLJ-2 +0*294

m~c8/~;eCh*~h c*/3 page 24

Studenrt Is "t" values f o r the variables of Table XI

Between V R A mv av mh

and %. 0,037 0.245 1.308 aV 0.326 1,495' 0.563 1,270 ~h 0,001 1.604 0.9C2 1,160 0 291 "h 0.0% 1,271 1,252 0.9% 1.918 1,307

Level 0,001 9PPy+ 0,@ q.85 0.20 0.20 0.30 O;4O O i $ "t" 1.922 ?re8 2,332, 2,101 1,734 '1,330 1;067 0.862 0;6$8

~ t a t i s t i c a ~ . y sigmiiitant

Camparing the ,.calciXlat kd valuee of Student s l T t 1 l . w i t h those for .diff ermt

degrees of s i r n d a n c e , i$.may be, qeen that the calculated values are not

statistically 8 i~ r l i f i can t ; ' 'therefdre., it may b e adsum& thata the correlatlo~,

coefficients codldchave arisen by chmce,

It may be t cobcluded I $bat, w i t hi& the 'ral-yge of w$t er velocity and variation

in shape of the cfoss-section of the canals covered ty th i s study, the difference

of water velocity .distr ibution on vgrtical and horizpn-bal pl&es, o b s w e d

amongst the twenty m a s u m g stations, cannot be attributecf either to the

actual velocity or t o t h e shape of the crcss-section of t h e canals at the

measuring stattons. A l s q , it may be concluded Lhat'there ia no l b & r correlation between any two of thevariables chosen to represent the curves

of water velocity distribution on ver t ica l and' horfzohtal plaries.

X I I T UALYSTS OF M A OBTAINED FROM BOOKS CONSULTED AND COMPARISON Wf TH &SULTS OF THIS STUDY

Books dealing with t h e subject under study were consulted for comparison

Turposes. See reference section at the end of this report . Authors are

freely T ~ o t ed . Shape of t h e cro ss-sect f on.

stbates t h a t the, parabolic section approximates the f o m assmed

by many natural streams and o ld earth canals.

WFUiams, in b r i m a n and Wiggin s ~a~dbook'~~' asserts tMtt ' th 8 cross-

section of natura l channels, pgr mlarly h, so f'b matwiala, .4ppy.xhat ea to

one or two pwabom sqments. vigleyrs eqwtidns -as g u o t ~ W ' ~ ~ i a m s ,

show that f o r streams where the ehanrre;l- occupies the fuU width of the

waterway and f o r s t raight reaches of t h e canal, t h e cross-section is a

parabola of vertical axts. According to this equation the mean relative

depth of the canal is 0.692.

EMh~8flech,msc ./3 page 25

The present study leads t o the conclusion that the shape of the mean

cross-section determined* f r m .twenty measuring stations, is a segment o f an

eUpser with one of its axes coinckding w i t h t h e cehtre l ine of the canKL,

and its centre located abmeL.the water mrface at a distance equal to one-fourth

of the water depth a t the - centre l h e , The mean relative wat er depth of the can&, as detennfned :from the equation of the shap0 of the cross-section, is

0,749, or about 8% greater than that obtained from Wleyts equation,

Shape of the curve of water vs;tocity distribution .on vert ical pblies. >

(16) '9' ~ o f i , ~rover(ll)-, ha'%', Merrimn , hover and Harrington , (20) Russell , ~-rd'g' agree with Bazin, who was t h e f i rs t to investigate

systsrmatf c a w the distribution of. water v&city op vertical planes, in adopt-

ing the paraboLa as the b e s t f i t t i n g curve to t h e &agrw of variation of water

velocity,

WiUiams in Ww*n h d W g g h t s Handbook (17.) states that the curve

approximates an ell ipse tangental t o the bobtidm and with Ats a x i s 3n or near ~ h e

surface of the stream, Wave action, eddies, etc. cause the velocities near the

surface to be decreased, this effect may extend to nearly ha33 the depth. Thus

the actual curve follows an eUipse halfway to t h e surface,. where f t leaves , the

ellipse and baea~es.ssmewha% flattened.

H i n d s in Da9ists andb book'^) considers t h a t the *riation of velocity depends on the conditions of flow ( s t r U n e d or f urI5i~Xerrb) and that the eff ec€

of this variation is us- hidden in empiri6 coefficients oi f o d a s r

Dodge end horn pa on'?^ afFlna that the curve is t o o complex to admit m a t h a t i c a l andysis .

~ larr ia ( lO ' cbnsiderk that bed f r i c t i o n greatly influences the vertical

distribution of water velocities, distort ing t h e pattern and producing a curve

of uncertain equation.

is also sceptical. He affirm that the velocit ies in the cross- section vary between wide limits, frequently in a most irregular and unaccount-

able manner; that minor disturbances in a stream w t U often produce comparatLvely great changes Fn the velocity distribution.

The present study leads to the conclusion, based on the mean curves at

three vertical planes, that t h e curve of water velocity distribution is an

ellipse whose axes are inclined with respect to the vertical and horizontal

coordinate axes, The position of the centre and the angle of i n e lha t i on of

t h e axes of this eUipse vary for each vertical plane.

-8/~aob ,Dbsct* k3 page 26

Relative water depth of the mean velocity,

~ddison"', ~aughert~" I, Harris'"', Hoyt and Gmver (w9 Hughes a d Safford (12) Lea(u)J L ~ t " 9 Me- and Wig gin f s Handbook (17)

(I8 Parker 'I9 ' , us s ell ' , Tennard -in , (21) give t he position of the meal water velocity, f o r any ver t ica l section parallel to the f l o w , at a relative

depth hear 0.6 from the water surface, Willi&s in &rriman and Wigghls

Handbook sets the posi t ion of the mean velocity at aWut 0.577, and Russell

fixes the U t s of. 0.58 and 0.65 between which he mgan velocity may occur.

However, most of t h e e authoes do not consider very accurate the procedure

of detemwng t h e mean water velocity by: measuring t h e velocity a t this depth,.

nesults €mmmthe-present study lead ;to the cottc2uafoh that t h e posrtfon of

t h i m h i rdoef ty vdries-a'ccarding to the locat~on of t he vertical 'plane mi

which it is determined. From the equations of vel*tmical distribution of

velocities, qeqn velocities and their p s i t i o n with respect .ko #e a r gepth

at each of . the .verbica planes were calculated with the fo l lowbg results:

Pbsiti'm, of 6he mean v d ~ e i t y ' . o n the~cantre l ine - - - - - - - - - -: 4 . 6 9 ~opl3i ion .o~ %he mean velokity on,the vePcica1 plane at -1/3 of the distance . . from the centre line tq the'watw edge - - - - - - - - 2 -0,632 Position o f %he mean velocity on the vertical plane at 213 of the distance from the centre l i n e to the water edge - - - - - - - - t -0.691

%st authors remmnd as a very re l iable method of determin* the mean

velocity on a vertical plane, the measurment of water velocities at relative

water depths 4,2 and -0.8, the arit-hmetlc mean of these two fleasuraents L6

very nearw'equal to'the memv&aditg, Awlyhg this principle to t ;Re

values calculated fmm t h e equation of water distribution on t h e centre l h e

the results are:

Arithmetic mma of 11-zelative water velocity value - - - - - - : O , w 2 A r i t h e t i c mean of relative water velocity at re la t ive depths 4 . 2 m d -0.8 - - - - - - - - - - - - - - - - - - - - - - : 0.8368 tion on between the two arithmetic means - - - - - - - - - - - : 0 . 9 8 . ~ Relative water depth of the .rnxhmxn velocity,

The position of the.mxSmum velocity, according to 'the authoPs consulted,

varies from the Hater surface to the mid-depth of thc stream.

Oreeley end StariLer in Davis's andb book'^' places %he velocity 3n ther-upper'one-third of the depth. kdge and ~ h ~ s o n ' ~ ' locate it

ebout 0-25 of th@ d@th, Hughes and Saf ford '12) say it us- appears

s~meubara betvem sdd-hpth-ad WateT surface. Lea "4''places it ~

the upper two-teartha. b M n '18' considers that the position of the

velocity is affected ,by the roughness 011' the bed, under general conditions

its re la t ive dmth 5s 0,25, for c a d s and smooth flowing rivers, is U.20.

Rusa e l l (20) states that the position varies considerably, sometimes being close

or at the surface, .but more generally a t 2110 or 3m of t he depth, V-d (21) places it at rda t i ve depths betwem 0.25 and 0.50.

Results from the' preseflk study lead to the conclusion that the position

of the m a x h u m velncitv varies according t o the location of the vertical plane

on which it is gwisured, The positions of the na;yrjrmun values obtained by

different iat ion of the equations of vertical distribution of velocities

referred to the water depth at each of the vertical planes are the followhgt

Position of the mximm veloci ty on the centre l b e : 0,176

Position of the e m u r n vdoc i t y on the vert ical plane a t 113 of the distance from the centre l h e to the water edge : 0,210

Position of the maximum velocity on the vertical p h q e at 213 of the distance Srom the centre l ine to the water edge : 0,338

Relation of velocity to m e a n v d o c i t y .

The rehtive value of the maxirrmm water v d o d i t y with respect to t h e mean

velocity is about ;L.2,5, accordmg t o GTeeley and Stmley in ]3avisYs Handbook (61,

not very different f r o m 1,10 for uide streams, according t o Lga M, .aa

nopmally about 1.11, according t o H W s in Merrfman and Wiggin's Handbdbk (17)

Result a f r o m the present study lead to the cdnclusion that t h e relation

of the m a x h m ~elocity to the mean velocity varies accqxding t o the location of the ver t ica l plane on which it is d e t d n e d , Frqm t he equatsons of

vertical distribution of velocities the following values were obtained:

Relation l h x h u & h n water velocf?ty on the centre li&-------: 1*233 Ralation &/hean water *velocity on the vertical plane at 1J3 of the distance from the centre line to the water edge---- : 1,189 Relation w a t e r velocity on t h e v e r t k d p ~ a n e a t 2/3 of the distance from the.'.centre line to the water edge-----: '1.134

R a t i o n of surface velocity to the mean veloci*.

The relative value of t he surface velocity wi th respect to the mean (3) water veocity &I, according to Box ,1,20. On the graph presented by

Cox and ~ermano'~) this relation is about 1.15. Harris 'lo) states tnat

the surface velocity ie usually less' than the meJdmum, except in the case of

downstreamwind. Hoyt and &rover ("' consider that t h i s relation l e

influenced- by the depth of the canal and the water velocity, va- hetween

1.02 and 1.28. k-g'l3' gives l u t i n g values between 1.06 and. 1.25, with an

average. of 1.U. Merriman 'I6' states that the mPXinaun veloci ty i a usually

about 1.17 the rneanvelocity. tlilliaas in Merrhm and wigg&js Handbook (17)

gives the value of 1.25, within an error of 108. Parker '19) l i m i t 6 its value

between 1.12 and 1-23, Vennard (21) fixes the value at 1.18, Wilson (22 )

considers that the relation of surface velocity to mean velocity varies w i t h t h e

shape of the .section, the roughness of the bed and the depth of the canal, from

wew*3meslts quoted it s e a s that the relation varies between 1.10 and 1.20.

Results from the present study lead to the conclusion that, based on

calculated values from the equations of vertical distribution of velocities,

the r e l a t i on of the surface velocity to the mean velocity has t he fo&wing

values t

RelatAon h f a c e h water velocity on the centre l i n e ----------- -1 1.m Relation Surfaceban t r a t q .velocity on the v e r t i c a l plane at 1/3 of the distance from the centre I h e to the water edge -------- : 1,117 Relatton ~urface/Meaa water . d o c i t y on the v e f i i d plane at 213 of t f ie d i s u c e froi.the centre line to the water edge -------- t 1.032 Relation of bgttom velocity to mean velocity.

The relative value of the bottom velocity with respect to the moan water (2) velocity,, accardlng to, Addison , is 0,70. Fmm Cox and G e m o Is graph ( 4 )

,,I

it sews t o beabout.0,52. Greeleyand Stanley in Davists andb book'^' state - thaf tha.relation'my possibly-be 0*50. Harris (10) considers that it is Less

t h h h&U the rrmchm ve2ocit;y + Hughes and Saff ord (12' explain that the

velocity is least at the perimeter, and .increases towards the centre f r o m .the

sf des, and f mm t h e bottom towards the b p . W i l l i a m s in Herriman and Wiggin's

Handbook (I7) says that the velacity in the semi-fluid iayere'of sediment next

to the bottom is relatively very low, whereas that in the mter jus t above

my be considerable. Parker fixes the re la t ion of the bottom velocity

between 0.9 and 0.52 the mean vdoci ty ,

Results fmm t h e presen'c study lead t o the conclus5on that, based on calculated values fram the equations of vertical distribution of velocit ies ,

the relation of t h e velocity at the bottom to the mean nates velocity has

t h e following values r

Relation ~ottom/rlllean water velocity an the centre kine ------------- : 0&0 Relation Batto& - water velocity. on the vertical plane at I/-3 of the distance From the centre Una t o the water edge ------- : 0-4.38 Relation Botto&ean water velocity on the verti,cal plane at 213 of the distance from t h e cen t re l ine to the water edge ------- : 0.689

EMh~8bech .Disc * /3 page 29

X I V SUMMhRY AND CONCLUSIONS

1, The Bilharziasis Control Project WH0Araq 15 undertook a study of water flow in earth canals w3th two main purposes:

(a) to determine the water velocity at any point of t&e cross-sectional

area;

(b) to devise a pract5cal and accurate method of measuring the rate of

f l o w ,

The study of the contour and area of the cross-secthn was essential for

attaining the two objectives,

2. Twenty measuring stations in fourteen canals were selected f o r carxyhg

out . t h i s study, their general characteristics .are descdbed,

3. The study of t h e shape of t h e cross-section is based on 100 water depth

measurements. The analys;is of the distrfbution of water velocities within

the cross-sectional area is based on TOO water veldcity readings. The

procedure followed for collecting t h i s information is described,

4.. F i e l d e t a were plotted and graphs of the shape of the cross-section and

velocity distribbkion were drawn for each measuring station. The tR êhnique

used b explained.

5 , The study of pe rhh t er of the cross~section, based on t he mean of twmty

measuring stations, shows that th i s curve appro*.t;es an e l l t p t f a l segment

of vertical and horizontal axea; the centre of this ellipse be- on the

vertical plane of 'the centr'e line a t a. 'di scance above the water surf ace equal t o one-fdurth of t h e maan water depth, The area of this e l l i p t i c a l

segment is three-fourbhs of t he circumscribed rectangle,

5. b e diagram of water velocity distribution on a vertical pUne approximates

an e l l i p t i c a l s e ~ e n t ~ i t s axes being inclined with respwt t o the horizontal

and vertical coordinate axes. The characteristics of the ellipse vary

according to the location of the vertical plane under consideration. me m a x h water velocity throughout the ~ana.1 is 1.037 the vddcity on t h ~

water surface at the centre line, This w u m velocity takes place on the

centre U n e a t a depth measured f m m the water surface equal to 0.176 the

depth.

7. The diagram of water velocity d i s t r ibnt ion on a hor izon ta l plane located

in the upper threequart ers of the stream, approximates a hyperbolic segment,

q m e t r i c a l with respect to the vertical plane at the centre l ine . In the

lower quarter of the stream, the curve becomes an e l l ip se ,

8. Water velocities along the periphery of the cross-section, calculated

rmm the equations of horioontal distribution of water velocities, have a

m m value of 0.40 the velocity on the water surface at the centre line.

Peripheral velocities from the water edge to the centre point on the bottom

vary within ;L2$ of the m e a n value.

9r The rate of water f l o w is represented by the volume delimited by the

vertical plane of the mss-section, the horizontal plane of t he water surface

and the ellipsoidal surface that corrbaba a l l the points representing the

values of water velociw, This volume, calculated by double integration,

gives the following expression for the rate af flow in a canal:

b w h i c h W is the width of the canal a t water surface lev&, H is the water

depth at the centre l f n e and v, is t h e water veXocity measured on the water

surface at the centre lbe.

When the water velocity is measured an the centre l ine at a depth from

the water surface equal to 0 . 5 ~ the to* depth a t the centre the above

expression becomei :

10. The mean water velocity, as detemnined by dividing the rate of flow. by

the area of the cros&section, is 0.732 the velocity on the water surface at

t h e centre lisle, b w a t e r velocity of a p o h t located on t h e centre line

at a depth f r o m the water surface aqua1 to 0,753 the t o t a l depth at t h e

centre, has t h e same value as t h e mean 'velocity of the water flowing throughout

the canal,

11. A diagram of equal' velocity curves and a general equation are provided

f o r the determination of the water velocity a t any point of the cross-

sectionKl area of' a d,

IZ. Correlation studies show that the actual water velocity 3m the canal

and t h e shape of its cross~section have no statisticaUy significant

influence on the distribution of water velocities on vertical and horizon~l

planes.

3 In comparing the results of t h i s study with amailable literature dealing with the same subject, no w m contradictory csonclusbns w e r e bmd.

E@l~8flech ,~iac r 13 page 33,

(1) LANOIX, J .N, ,ReXation b_etween ir riaation sngineerfnp; and 4$&harej;aax8., World Health Or-'--'aixi eatLon- h cuyent , WHO/B~~. ~onP.A1 Add.1, Geneva (19'56)

(2) ADDISQM, H. A TW-Book of Applied Hydraulics. Chapman and Hall, London (1934) po65 HydraulLc hsuremats. Wiley, N,y. (1943) pa213

(3) BOX, T, Practical w a u l i c a . Sponc London 11943) p.63

( l e ) GOg, Q.N. and Q-0, .F,J,, F h i d Mechanics. Van Nostrand, Hay. (1946) 'p.226

( 5 ) DAUOHERTY, R.Lh Hydraulics. McGraw-Hill, N,Y, (1925) p,176

(61 DAVIS, CbV,, editor Handbook of Applied Hydraulics, %hawsi l l , N,Y, (1952) p 4 6 and pr1045

(7) DODGE, %A. and THOMPSON, Me J,, FLuid &chanic;s. McGrawd111, Nay. (19371 p.239

(8) ELLIS, W*M, Imitation. Government Press, Mras (1926) p.35

(9) N,C, and HARRmTON, A&?., Stml3.111 F'fow- Wfley, NmY* (1943) p.240

(Y) HOYT, JrC. and GROVER, N.G., River Dieohar e. W i l e y , NIYo 7Fsyz&

(12) HUGHES, H .J. and SAF'FORD, A.T ., A, Treatise, on ifydrau33c.s. 2(acUlan, N,Y, (19263 ppv201 ff, and 264

(13) KING, H,W,, Handbook of Hydraulics. McGraw-Hill, N,Y, (19%) Sec, 7 pp,5 and 9, secb 9 p a l l

(ul Each Hydraulics, Arnold, London (1919) pp.210 - 213

(16)=1W,M, RsmentsofHfiraulics. Wiley, N.Y. (1912) pel23

(17) m I W , T. and WIWm, T.H. (editors) American Civll Engineers' Handbook. Wiley, HiY, (1945) pp.1343,'-1347, 1351, U64

(18) mIRIN, E 4 R . Practical River- and - C a d &ygwdpinG. Griffin, Lomion (1920) p.37

(19) PAR-, ' The Control of Water. Routledge, London (192 51 p. 53

(21) m, J,& ETamtary muid Mechanics. Wiley, N.Y. 11954) pa325

m) WILSON, K.M. & u a l o f ' I r r i ~ a ~ o n k f n e a r l n g . Wilw, NeY. (1893) ~ 9 5 2

A P P r n D I x A

Sample of record f om fo r measurmmts of section and water veloc%ty,

STUDY OF WATER FLQW IEJ M T H CANALS.

llessurhg Station: x u ,

Wfdth on the water surface metres

Water depth at Cerltre 2 b s metres

CROSS-SXTIm Distance f r o m c.i,

Water 2/3 ~ / 3 2fi Wa$sr Wac, he Edge

Water Depth

WATER V D C I T Y

Current metre At Readings ~ e ~ t h

(seconds per 10 r ewlut ions 1 0

RF,MARaS:- Windr

Vegetation: S W t u r e :,

Date :

. + . , - . * q hL,