Research Collection

Doctoral Thesis

Mean roughness coefficient in open channels with differentroughnesses of bed and side walls

Author(s): Yassin, Ahmed Mostafa

Publication Date: 1953

Permanent Link: https://doi.org/10.3929/ethz-a-000099176

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ETH Library

Prom. No. 2267

Mean Roughness Coefficient

in Open Channels with Different Roughnessesof Red and Side Walls

THESIS

PRESENTED TO

THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

FOR THE DEGREE OF

DOCTOR OF TECHNICAL SCIENCES

BY

Ahmed Mostafa Yassin, M. E.

of Cairo, Egypt

Accepted on the recommendation of

Prof. Dr. E. Meyer-Peter and Prof. Dr. R. Miiller

1Zurich 1953

Dissertationsdruckerei Leemann AG

Erscheint als Mitteilung Nr. 27

der Versuchsanstalt fiir Wasserbau und Erdbau

an der Eidgenossischen Technischen Hoohschule in Zurich

Herausgegeben von Prof. G. Schnitter

Verlag Leemann Zurich

TO MY WIPE

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Acknowledgement

The writer wishes to express his thanks and sincere gratitudeto Prof. Dr. E. Meyer-Peter, for his valuable help and advice duringthe preparation of this thesis.

The writer is also grateful to Dr. A. Preissmann for his help.

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Index

Symbols and abbreviations 7

I. Introduction 9

II. The model and the measuring apparatus 10

1. The model 10

2. The measuring apparatus 13

III. Fundamental equations for turbulent flow and uniform roughnessof bed and side walls 16

1. General equation of Darcy 16

2. Strickler'a equation 17

IV. The procedure of work 18

1. Experimental details and velocity distribution curves .... 18

2. The value of a 38

3. The energy line gradient Je 40

V. Mean roughness coefficient at different roughnesses of bed and

side walls 42

1. H. A. Einstein's equation based on Strickler's formula....

42

2. Another equation based on Darcy'a formula 44

3. A second hm equation 44

VI. Results and discussion 45

VII. Determination of the shearing stress on side walls and bed from

the velocity distribution 69

VIII. Summary 89

References 90

5

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Symbols and Abbreviations

h depthb width

A area

P wetted perimeter, for a rectangular cross section = b + 2h

R Chezy's hydraulic radius =

-p

v velocityto written as index at the right hand side of a symbol, as vm or

symbolize an average value within a regionA ze loss of energy head

A zw drop in water level

y specific weight, weight of a cubic unit of fluid

p density, mass per cubic unit of fluid

g acceleration of gravity

7} viscosityv kinematic viscosity-n -r, it t

velocity X diameter of pipe vDR„ Reynolds number = ^—^—-.—;

,^

^ ^=—

e J kinematic viscosity v

r, , t-x , -, , velocity X hydraulic radius v RR. Reynolds number = r^—f-.—; ^

=—

6 J kinematic viscosity v

I, A I lengthQ discharge, volume of bulk flow per unit time

y vertical distance from the bed

x horizontal distance from the side wall

Log decimal logarithmD the diameter of a pipeh StricMer's roughness coefficient

hw StricMer's, roughness coefficient of the side walls

kft StricMer's, roughness coefficient of the bed of the channel

A Darcy's roughness coefficient in the general formula

Je energy line gradient

Jw water surface gradient

Js bed gradienta a correction factor to compensate for the use of the mean

velocity in calculating the velocity head

£ the ratio of the discharge measured at the flow nozzle to that

calculated from the mean velocity distribution curve

P correction factor for flow nozzle measurement

s the roughness height parameter, absolute roughness

drelative roughness

t0 shearing stress between the wall and the fluid

t0s shearing stress between the bed and the fluid

t0 shearing stress between the side walls and the fluid

v# the Prandtl "friction velocity" = l/—

u symbol refers to upstreamd symbol refers to downstream

T temperature of water

Abbreviations

SBZ. Schweizerische BauzeitungS.I.A. Schweizerischer Ingenieur- und Architektenverein

VDI Verein Deutscher Ingenieure

8

I. Introduction

All the old laws of friction put forward by Chezy, Darcy, Strickler

and other hydraulic engineers were given for uniform roughness of

the whole surface of closed or open conduits. But, as far as I know,

nobody had tried to carry out experiments on this special problemof getting the mean roughness coefficient in open channels in case

bed and side walls are of different roughnesses. H. A. Einstein

determined an equation [l]1) concerning this problem based on

Strickler's formula [2] making some special assumptions, but, no¬

body had tried to verify this equation.The principal aim of this work was to verify the equation of

Einstein or to find another equation which may be used with

sufficient accuracy in both field and laboratory experiments. As the

ratio between the depth and width of the open channels differs

widely in both cases, as it is small in most of the existing rivers and

canals, and relatively big in many experiments, three channels

were examined:

Channel I

100 cm wide and 35 cm deep. The measured

water depth varied between 4.99 cm and 32.20 cm

discharge varied between 17.22 lit/sec and 319.80 lit/sec

Reynolds number i?/ between 13430 and 174700 (RJ = ^-\Perforated steel plates were used as artificial roughness.

Channel II

50 cm wide and 35 cm deep. The measured

water depth varied between 4.81 cm and 33.64 cm

x) Figures between brackets [ ] indicate references.

9

discharge varied between 7.56 lit/sec and 153.50 lit/sec

Reynolds number Re' between 10810 and 114470

Perforated steel plates were used as artificial roughness.

Channel III

60 cm wide and 60 cm deep. The measured

water depth varied between 4.71 cm and 57.72 cm

discharge varied between 6.90 lit/sec and 388.70 lit/sec

Reynolds number Re' between 8000 and 191550

Rounded gravel having diameter = 4.0 mm to 11.7 mm fixed in

place with cement was used as artificial roughness.

In each channel the following cases were examined:

a) (la, IIa, Ilia) both bed and side walls smooth,

b) (lb, lib, IIIb) bed rough and side walls smooth,

c) (Ic, lie, IIIc) both bed and side walls rough.In channel III the following case was also examined,

d) (Hid) bed smooth and side walls rough.In channels I and II the bed and side walls were made smooth

by the help of white putty, but in channel III only cement was

used because of the cold weather.

It may be important to note here that the flow was always

streaming flow as the water depth was always greater than the

critical depth. The flow was practically uniform as the difference

in depth upstream and downstream was not greater than 0.2 mm.

Also the flow was turbulent. Supposing that according to Krey [3]we leave the transition region at RJ ^ 1500 to 6000, and seeing that

the minimum Re' calculated in our measurements was 8000, we can

say that our flow was always fully turbulent.

II. The Model and the Measuring Apparatus

1. The Model

It consists of three main parts:A. the inlet, B. the channel, C. the outlet.

10

modifications.

its

and

model

The

1.

Fig.

2m

I0

OS

IC-C

Section

mj[c

hann

elPlan

Part

D-0

Section

—-^^^^^S*«H

Rete

inin

q'

.—r

wall

Concrete

hosm

os.

m>.

/]

:„

,.

plates

Concrete

8-B

Section

n)(C

hann

elPlan

Part

Profit!

e7

"

-'hi—-.--4"--"I-,

om—

--39

us-—i

(.mati.-

A. The inlet

A basin of a rectangular cross section 1.12 m clear width, 2.25 m

depth, and 3.5 m length, Fig. 1. The water is introduced by two pipesof 25 cm diameter each. Each pipe has 17 cm diameter flow nozzle

connected to a differential manometer to allow the measurement

of the discharge which is regulated by a special valve.

To ensure smooth water surface in the channel the followingprecautions were used:

a) a half, brick wall spaced 80 cm from the inlet, 75 cm high to

diminish the energy of the entering water,

b) a horizontal steel screen at the top of this wall,

c) a vertical steel screen fixed at this wall,

d) a vertical steel screen at about the middle of the inlet basin,

e) a sloped steel screen just before the channel,

and at the entrance of the channel,

f) a sloped steel screen which was made free to move at first, but

later on, it was found more convenient to fix it to get better

velocity distribution in profile 7 (7 m from the beginning of the

channel),

g) a floating timber screen 1 m long in channel I, and 2 m long in

the two other channels.

Two 5 cm diameter pipes were provided to help to empty the

inlet basin.

B. The channel

It consists of 11 separate rectangular part channels, each 4 m

long, 1 m clear width, 0.40 m depth, giving a total length of 44 m.

Each channel was supported at the joints on broad flanged I beams

imbedded in concrete foundations spaced 4 m from each other. The

joints allow the expansion or contraction of the channels due to

change of temperature and were made water tight.After fixing the channel straight and horizontal it was given a

slope of 1 °/00 using cement mortar. This 1 °/00 slope made a loss of

more than 5 cm in the usefull depth of the channel.

When the bed surface was smoothened by cement it was found

that the bed was smoother than the side walls, so both side walls

12

and bed were covered with white putty to have them equallysmooth.

To facilitate measurements of velocity, water and bed levels,

three portions of rails were set up, the first between 4 metres from

the inlet and metre 11, the second from metre 19 to 27, and the

third from metre 36 to 41. Further, a small carriage rolled on each

portion. The brass rails of rectangular cross section 5x15 mm were

fixed on small brass supports (of a T section with a groove in the

middle just bigger than the width of the rail) imbedded in cement and

spaced 30 cm from each other. The rails were free to move as a

precaution against the change of temperature. An attempt was made

to fix all the rails horizontally at the same level. The differences in

level between the separate portions were exactly measured. Each

of the three carriages rolled on three rollers the diameter of which

were adapted till each carriage was exactly horizontal and all

carriages were at the same level.

C. The outlet

A basin 5.05 m in length and 1.12 m clear width. A sluice gate

having a vertical scale was placed 4 m from the end of the channel

to allow the regulation of the water level in the channel. The water

flows only over the gate.

2. The Measuring Apparatus

a) Discharge

The discharge was measured by the help of two flow nozzles,

each in one of the two inlet conduits, one connected to an air-water

differential manometer, and the other connected to a mercury-

water differential manometer, both of the [7-tube type.The two flow nozzles used were constructed according to the

VDI specifications [4]. A length of at least 50 D upstream and 10 D

downstream from the nozzle"must be kept free from disturbances.

We can get the discharge by applying the formula:

Q = pA2i2gTAh (1)

13

in which Q is the discharge, /? is the discharge coefficient (taken

according to the VDI specifications), A2 is the area of the nozzle

mouth, and Ah is the difference in pressure head in the conduit

immediately upstream and downstream from the nozzle.

Each inlet conduit can supply about 200 lit/sec, but, if the

required discharge was less than 200 lit/sec, the conduit connected

to the air-water manometer was always used as the reading of A h

is much easier due to the higher water columns.

b) Velocity distribution

It was measured by the Pitot-tube, Prandtl type, Fig. 2 and 3 [5].Prandtl devised an instrument [6] in which the pressure openingsare so located that the slight decrease in pressure due to the highervelocity around the nose of the tube is exactly offset by the rise in

pressure caused by the reduction in velocity ahead of the stem of

the instrument; thus the instrument is given a coefficient of unity.The diameter D of the instrument used in channels I and II is

0.92 cm, with a total height of 60 cm. As it was not sufficient for

channel III (the water depth was much bigger), another instrument

pt d izzz p*

:o.so

a-b t>i

Fig. 2. The Pitot-tube, Prandtl type.

14

with a height of 96 cm and D = 1.24 cm was used. The instrument

was connected to a special E7-type differential manometer havingtwo sliding indicators with mirrors. One side of the manometer

indicates the static pressure ps and the other the dynamic pressure,

or let us say, the total pressure pt. The velocity can be calculated

from the formula.

t, = 2flr"|MzZk (2)

Fig. 3. The measuring apparatus, the Pitot-tube, the point gauge and the

hook-gauge well.

15

Both pressures ps and pt change with the increase of the angle 6

which the stem of the instrument may make with the direction of

flow. Prandtl found that the difference pt—ps practically remains

constant till 0 = 17°.

c) Bed and water levels

Bed level was measured by an ordinary point gauge, Fig. 3,

supported on the sliding carriage, and could be transported from

carriage to carriage. The gauge was mounted on a graduated rod

actuated by a slow-motion screw equipped with a vernier for

accurate readings to 0.1 mm. At the beginning, this point gauge

was also used to measure the water level, but later, it was found

inconvenient for big water depths due to the big oscillation of the

water surface. To overcome this difficulty, two side hook gauge

wells were constructed at both measuring profiles. Each was connec¬

ted with a short pipe 5 mm clear diameter to the axis of the bed of

the channel, and had a hook gauge with a vernier for accurate

readings to 0.1 mm, with the same mechanism as the point gauge.

The oscillation was then much smaller and slower in the wells which

gave much more accurate readings of the water level. Each time we

have to move the hook gauge with the oscillation of the water level

in the well and take x maximum readings and x minimum readings,(x^lO), then compute the average value of these 2x readings.The zero point for each well was accurately fixed.

III. Fundamental Equations for Turbulent Flow and Uniform

Roughness of bed and side walls

1. General equation of Darcy

In 1858 Darcy published the following equation which was very

useful to hydraulics:

Je = -yr— for circular pipes (3)

16

for other cross sections we have to replace the term D (the diameter

of the pipe) with the hydraulic radius, and so we come to the form:

J°=ll^ (4)

or A = i?Je|f (5)

From dimensional considerations it is known that the dimensionless

coefficient A is only a function of the Reynolds number Re' and of

the relative roughness p.For high values of Reynolds number and

rough flow it was found by experiment that A depends only on the

relative roughness. So, A is not a constant characteristic of the

surface.

2. StricMer's Equation

In 1923 A. Strickler from Switzerland, independently from

Manning, published what he called a practical and convenient

equation for the mean flow velocity [2]:

vm = kiri.je1i' (6)

This equation was given for fully turbulent rough flow with the

assumption that k is a function of the absolute roughness of the

surface of the walls of pipe or channel, independent of the Reynoldsnumber.

,const

, . ._.

k = ^^- |*| =m (7)

is

In the metric system the dimension of \k\ is m!'jsec.It was found by experiment that this equation did not act

totally right in our case. In Fig. 26, 27 and 28, in which k is drawn

against h the curves rise to a certain limit and then begin to fall

again instead of either going further up or keeping horizontal. It is

possible that we are in the transition zone to rough flow because k

was shown as function of the Reynolds number at least for small

depth.

17

IV. The Procedure of Work

1. Experimental details and velocity distribution curves

(Fig. 5-17)

The first step was to get a smooth water surface for big discharges.That was attained by using a group of screens as mentioned before.

The arrangements of the screens had a big influence on the velocitydistribution near the inlet. This can be easily seen from curves

3 and 4 in Pig. 9. Curve 4 was measured when the steel screen at the

entrance of the channel was free to move and the timber floating

Fig. 4. Velocity distribution in 6 horizontal sections, profile 39, case la,

h= 10.18 cm.

18

screen was only one meter long. After fixing the steel screen and

replacing the floating screen by another two metres long the

velocity distribution was measured again (curve 3) and found to be

much better.

To choose the measuring profiles, the velocity distribution was

measured in different cross sections. Each cross section was divided

into 6, 7, 8 or 9 horizontal sections (depending on the water depth)and for each horizontal section the velocity was measured usingthe Pitot-tube for channel I, in 6 points chosen according to the

S.I.A. specifications [7] (11 points in channel II and 13 points in

channel III) as shown in Fig. 4. From the horizontal velocitydistribution curves we can draw the mean vertical velocity distri¬

bution curve, the area of which if multiplied by the channel's width

will give the discharge. The ratio of the discharge measured at the

flow nozzle to that calculated from the mean velocity distribution

curve, which we call £, gives an idea of the accuracy of Pitot-

tube and flow nozzle measurements. The deviation of this ratio

from unity was generally found to be smaller than 1%.The mean velocity distribution curves in different profiles for

channel I can be seen in Fig. 5 and 6. For each curve h, Q, a. 2), £ and

Re' are given. As it is better to have a long measuring length, the

first measuring profile must be as near as possible to the inlet and

the other as near as possible to the outlet, and in the same time both

profiles must have a normal velocity distribution curve and a

reasonable <x value. It was found that the most convenient measuring

profiles were profiles 7 and 39 (9 and 39 in channel III).In each of the ten cases the velocity distribution was measured

in profile 39 at three different depths. The velocity distribution

near the inlet (profile 7 or 9) was measured at three different depths,

only for the case of totally smooth channel (la, IIa and Ilia). The

mean velocity distribution curve at the inlet is more or less vertical,

but in profile 39 tends to be curved. The character of the velocitydistribution can be judged on the basis of the correction factor a

(eqn. 13). The value of a upstream is smaller than that downstream.

2) a = correction factor to compensate for the use of the mean velocityin calculating the velocity head, see page 38.

19

0.25 0.50 0.75 vm in m/sec

Case la

Curve Profilecm

h

cm

a

Lit/seca. I *;

1 7 99 95 10.06 ess J. 038 0.999 S5 370

2 7 99 95 20.18 lee.e 1.016 1.002 lit ISO

) 6 99.86 20.3? 187.7 1 035 0.998 133 S90

« — 19 99.99 10M 188.3 1025 1.008 133 700

5 26 100.2 20.32 188.7 i.oa 0.8S8 133 980

Fig. 5. Mean velocity distribution curves in different profiles, case la.

20

25

20

10

-A2

/ / /

0.25 0.50 0.75 I'm in m/sec 1.0 1.15

Case la Curve h

cm

Q

Lit/seca i r>i

Profile 39 1 W.IB 67.2 I. Oil I.OOS IS 860

1 4 = 99.95 2 20.fi 187.7 1.038 1.006 133 320

3 29.7S 319.3 1.026 0.989 I7t 0*0

Fig. 6. Three mean velocity distribution curves in profile 39, case la.

21

3

i

\

J j11

0 0.25 0.50 0.75 vm in m/sec 1.0

Case lbCurve h

cm

Q

Lit/sectt 4 R>

Profile 39 1 9.97 52.40 ;.<?*« 0.985 17120

1 b' 98.95 2 19.51 ISO.oo 1.0*2 0.999 919W

S 29.91 273 SO l.03li 0.994 US 300

Fig. 7. Three mean velocity distribution curves in profile 39, case lb.

22

0.25 0.50 0.75 vminm/sec loo

Case IcProfile 59

|| ..«, ||

Curve h

cm

Q

Lit/seca I He

1 9.95 52.10 Wit 0.96S 37IS0

2 20M IU9.00 I.0S1 0.9)7 S0S70

3 29.S7 259.30 t.ou 0.868 13$ 270

Fig. 8. Three mean velocity distribution curves in profile 39, case Ic.

23

4

3 %

• 1

-

2

1

/ M

/ P *

——-*r^L-ca3-*'

P •

/ /

/ /

/ /

0 0.25 0.50 0.75 vmi„m/sec 1.0

Case Ha Curvehcm

aLit/sec

a. i Re'

Profile 7 l 10.1$ 10.92 I.01S 0 978 leoso

2 20.43 80.80 1.0it 0.990 77 no

16=50.263 3S.S6 150.70 1.022 0.99S III $30

4 11.72 ISO AO 1.02k 0.977 III 700

Fig. 9. Four mean velocity distribution curves in profile 7, case II a.

J

\\

2

/

0 0.25 0.50 0.75 vm inm/sec 1.0

Case IlaProfile 39

1 b=t9.82 1

Curve h

cm

a

Lit/seca. I Hi

1 I0.1S iO.Bi I.0S6 0.995 38130

2 20S2 80.80 1.051 1.0)5 77330

i ii.62 ISI.VO 1.01,1 0.991 112690

Fig. 10. Three mean velocity distribution curves in profile 39, case Ila.

30

25

20

IS

10

0.25 0.50 0.75 vm in m/iec loo

Case Eh Curve hcm

QOf/sec

a t *i

Profile 391 10.29 26.28 1070 0.972 12110

1 b-if9.82 2 2OA0 71 90 I.0S7 0 936 68 280

3 33.63 HO. 70 1 Oil 0.991 101420

Fig. 11. Three mean velocity distribution curves in profile 39, case lib.

30

S

c

25

20

15

10

0.25 0.50 0.75 vminm/sec loo

Case lie Cum hon

aLit/sec

a. i *.'

Profile 3s 1 10.33 2S.*0> 1.077 0.SB3 30 770

|';6=4S.«jl! 1

Z 20.17 ee.io LOSS 0.592 62*20

3 S0A8 113.to 1.057 0.954 671.20

Fig. 12. Three mean velocity distribution curves in profile 39, case lie.

27

3

1

2

i

\

/

>

) //JJJ

0.25 0.50 0.75 vm in m/sec l.oo

Case Ma Curveft

cm

Q

Lit/sec

a. I Re

Profile 9 1 2012 101 6 1.022 1012 83920

1 0 = 6015 Z 40 36 2S4.3 1.027 l.02i ISH70

3 S7 2S iBSS I 022 1.012 191620

Fig. 13. Three mean velocity distribution curves in profile 9, case Ilia.

28

3

2

/ I /

0.25 0.50 0.75 vm in m/sec loo

CaseHa Curveh

cm

a

Lit/seca 4 «e

Profile 39I 20.12 102.10 1.027 1.009 88(50

1 b=60.0S2 41.00 252.60 1.017 1.017 151170

3 57.50 SBS.oo I.03S 1.021 ISt 250

Fig. 14. Three mean velocity distribution curves in profile 39, case Ilia.

29

60

%

SO

to

10

20

10

0.15 050 0 75 vm in m/sec l.oo

CaseMt Curveh

cm

a

Lit/seca. £ Be

Profile 391 10.14 63.3 1.071 /.CO .5 seioo

1 b'60.0S 2 2./S J04.6 i.oes 0.995 IIS 700

3 53.98 2$l S I.0S7 1.009 itetto

Fig. 15. Three mean velocity distribution curves in profile 39, case Illb.

30

3

\

2

\

\/

0 0.25 0.50'

0.75 vm in m/sec 1.0

Case McProFile 39

I b'602 I

Curve hcm

a

Lit/seca I Re

1 70.28 eo.ss I.I09 0.993 46980

2 iO.tl 111.to 1.097 1.009 84/80

3 5*88 zn.ro 1.097 0.997 96400

Fig. 16. Three mean velocity distribution curves in profile 39, case IIIc.

31

3

\\

2

\

1

0 0.25 0.50 0.75 vminmfiec loo

CaseMdProfile 59

I b'60.2 a

I 1

Curvehcm

QLit/sec

a I «i

1 20.27 80.00 1068 0.398 6*040

2 40.22 I7B.S 1.070 I.OOS WZISO

3 S87Z 262.25 1.076 0.937 117730

Fig. 17. Three mean velocity distribution curves in profile 39, case Hid.

32

Fig. 18. General view of channel I, taken from the downstream, case lb.

In channel I, Pig. 18, the existing channel was used with | = 0.35

In channel II, Fig. 19, a concrete slab was built in the middle of the

channel making ^= 0.7. In channel III, Fig. 20, two concrete

slabs were built 60 cm apart and 60 cm high (-=- = l.o). In channels

I and II perforated steel plates 2 mm thick, with perforations

33

Fig. 19. General view of channel II, taken from the upstream, case lie.

5 mm diameter spaced 7 mm centre to centre, Fig. 21 and 22, were

used as artificial roughness3). In channel III rounded gravel, Fig. 23,

the grain-size distribution curve of which is given in Fig. 24, was

used as artificial roughness. The gravel was cemented to one side of

each of the 44 concrete slabs each 2 m long, with which the side

walls of channel III were built. For bed roughness the gravel was

cemented to concrete slabs 59 X 59.5 cm and 3 cm thick.

s) For further details see pages 9 and 10 and Fig. 1.

34

Fig. 20. General view of channel III, taken from the upstream, case Hid.

To calculate the roughness coefficient the following data were

required: the discharge Q, the water depth at both measuring

profiles hu and hd and A zw. These data were measured at different

depths and repeated at least four times for each depth taking into

consideration that we must have in each case uniform flow, by the

help of the sluice gate at the outlet. As it was very difficult to have

the water depth in the two profiles exactly the same (the difference

35

Fig. 21. The perforated steel plates used as artificial roughness in channels

I and II.

was not bigger than 0.2 mm), it was necessary to apply mean values

for v, h, B, P and A, namely,

depth: m 2 upstream

wetted perimeter:p

Pu + Pd symbol d refers to

m 2 downstream

wetted area:a Au +Adm 2

Hydraulic radius: p _

^w + ^dm p 1 p

ru^~rd

mean velocity:Q

Vm =aT

36

Fig. 22. The channel in case Ic, the carriage and the point gauge.

And so both fundamental equations of Darcy and Strickler must be

written in the following form:

Darcy: X=RmJe^ (8)m

Strickler: k = -^—^ (9)

and as the depth and the velocity are not the same upstreamand downstream we have to make correction for the energy line

gradient.

37

:~*T»mm**L^k «&#-*» t.«mm,w wv„ *jff~» *? -*t wr- vs^r°P^r^Fj[TI^3L**»*&m»>*rj**. #W1^% i^»»Wl \^P

Fig. 23. The gravel used as artificial roughness in channel III.

2. The value of a

As mentioned before a is a factor to compensate for the use of

the mean velocity in calculating the velocity head. The value of a

depends on the shape of the velocity distribution curve. If the latter

is known a can be calculated as follows:

The kinetic energy

Ek = lPjj*dA = «±v3mA (10)

A

thus a = -3^- I I v3dA (11)

A

38

too

.5>

80

60

10

20

by the square-mesh sieve with /

openings of 10mm /

96.66 % passed through /3.34 % retained on the sieve /

100.0 %

8 10

o d in mm (square - mtsh eiere)

12

Fig. 24. Grain-size distribution curve of the rounded gravel used as artificial

roughness in channel III.

at any point, v = vm + Sv, where 8v is the deviation from the mean

value vm [8], so,

uvlA =HvldA + 3Hv*m8vdA + 3$jvm8^dA+ttWdAA A A A

but, according to the definition of the mean value j$8vdA = 0 so that,A

39

AAA

if St; is small, JfSt;3^ will be a very small term that could beA

neglected, and it follows that,

•2dA (12)

for practical purposes, with a good approximation, referring to

equation 11, we can say that,

Zv3AA(13)

3. The Energy Line Gradient Je

According to Bernoulli, see Fig. 25,

*« + *« +- + «»!£ = ** + ha +^ + «*£ + A*e (14)y zg y zg

where p0 is the atmospheric pressure which can be assumed equalin both sections, and a is a correction factor already defined depend¬

ing on the shape of the velocity distribution.

vu = the mean velocity in profile I—I

vd = the mean velocity in profile II—II

Fig. 25.

40

but,

thus,

Aze = Azw +-~ KvJ-ocdv/)

and 4^= J„Aze_ r

Jp. = t4> +2gAl

Al

(«u*\.a-*dt>«ia)

(15)

(16)

As is was found by experiment that the difference between ocu and

ocd is very small, it may be assumed that au = <xd = 1, in such a

case (see the tables in fig. 5 to 17 giving the computed values of a),

Je, — Jw +2gAl

(17)

To get an idea of the percentage error due to this assumption

we have to take two numerical examples, one having the biggestcalculated values of <xM and ad, and the other having the biggestcalculated difference between ad and au, as the error in calculating

Je depends mostly on xd—

<xu and not on the value of <xu or otd itself.

Example 1: Case Ila:

Xu = 1.044 hu = 19.99

ad =1.051 hd = 20.00

cm bu = 50.26

cm bd = 49.82

cm Q =

cm Jw =

78.40 lit/sec1-034 %0

considering a values:

neglecting a values:

with an error of:

Je

1.010%0

1.018%0+ 0.8%

A

0.01434

0.01446

+ 0-8%

k

106.64

106.22

-0.4%

Example 2: Case la:

a„ = 1.016 hu = 20.31

a.d = 1.038 hd = 20.30

cm bu = 99.95

cm bd = 99.95

cm Q =

cm Jw =

189.0 lit/sec1.046 "/oo

considering a values:

neglecting a values:

with an error of:

Je

1.014%

1.046%0+ 3.1%

A

0.01325

0.01366

+ 3.0%

k

106.27

104.66

-1.5%

This shows clearly that the biggest error comes from biggestdifference between <xd and <xu even when their values are small. The

41

error increases also with the increase of the mean velocity. And as

the error will be of the same order of magnitude in the three or four

different cases of roughness arrangements, then the total error in

km or Xjn will be so small that it can be safely neglected.

V. Mean Roughness Coefficient at Different Roughnesses of

Bed and Side Walls

1. H. A. Einstein Equation based on Strickler'a Formula

H. A. Einstein [1] tried to apply Stickler's formula also for the

case where the roughness coefficient along the wetted perimeter is

not constant. He divided the wetted perimeter into n parts Pv P2,... ... Pn having roughness coefficients kv k2, kn and dividingthe whole cross section into n parts with areas Alt Az, An.He assumed that each part has the same mean velocity

vi — which at the same time will be equal to the mean velocity of

the whole cross section vm — and the same energy line gradientJe. Applying the Strickler'B law of friction we get for each part:

Vt =vm = ktRt'i*j;L (18)

Further, vm = kmRi.j;i. for the whole cross section

so,

or.

kmIP' = ktBti'

k '/>R. = R m lift)

We can get now another equation by assuming that the area of each

part At equals the wetted perimeter Pi multiplied by the hydraulicradius Rt,

as A = EAi (20)

and as R =-=-, or, A = PR

Thus, PR = SPt Bt = Rk'i>ZA

42

for a rectangular open channel,

fcm —•*» + *«

K1' K'u

(22)

where Pb — the wetted perimeter of the bed = 6

and Pw = the wetted perimeter of the side walls = 2h

To apply this equation for a case of the bed rough and side walls

smooth at any certain depth h we have to apply kb the value found

by experiment at the same depth h for the case of totally roughchannel, and kw the value found by experiment at the same depth h

for the case of totally smooth channel. The km value computed from

this equation km (theoretical) must be compared with the experi¬mental value km found by experiment at the same depth h for

the case of the channel having bed rough and side walls smooth.

The deviation

equation.

K tc~

K„

gives an idea of the accuracy of the

Example:

For channel III, case bed rough and side walls smooth,

(case IIIc) = 61.826 = 60.1 cm at h =

table 13

Applying the equation,

50 cm,

K„

(case Ilia) = 104.73

(case IIIb) = 82.66

k —

60.1+2X50

60.1+

100

61.82'/. 104.733/'

81.69

deviation81.69-82.66

8X66X 100 =-1.17%

the minus sign means that the theoretical value is smaller than the

experimental value.

43

2. Another equation based on Darcy's formula

As in Darcy's formula the value of A is a function of the relative

roughness and Reynolds number we cannot apply the assumptionof Einstein as we shall have a different relative roughness for each

separate part of the cross section. Any how we can try the following

empirical formula:

K =^ (23)

for a rectangular open channel.

the application is the same as by Einstein's formula.

Example:

For case Illb as above, table 13,

at h = 40 cm A6 (case IIIc) = 0.0366

Xw (case Ilia) = 0.0125

Xmxp (case Illb) - 0.0221

Applying the equation,

. 0.0366X60.1 + 0.0125X2X40nooQ

'W =

lioj= 0-0228

j .-0.0228-0.0221

1AA „ noo.deviation = X 100 =

+3.03%

3. A second Jcm equation

We can also try an empirical equation based on Stickler's

formula with the same form as the above Am equation, namely,

K =^ (25)

for a rectangular open channel,

t. &ft "ft + ^w "w /9R\

44

Example:

For case Illb as above, table 13,

at h = 30 cm kb (case IIIc) = 62.87

kw (case Ilia) = 107.18

kmexp (caselllb)= 76.15

Applying the equation,

. 62.87X60.1 + 107.18X2X30OK nl

.

*W =

12^= 85-01

deviation = 85,°!~~^6,15 x 100 = +11.63%

VI. Results and Discussion

All the experimental datas for the three cases are given in tables

1 to 10. For each channel the following curves are given:

k in function of hm, Fig. 26, 27 and 28,k in function of Bm, Fig. 29, 30 and 31,

A in function of hm, Fig. 32, 33 and 34,

A in function of Bm, Fig. 35, 36 and 37, and

Log (100 A) in function of Log Be', Fig. 38, 39 and 40.

In the last curve both Blasius and Nikuradse,'& curves for smooth [9]and rough pipes [10] are drawn in dotted lines after converting the

values of Log Be to the corresponding values of Log Be'. As for

circular sections

Be' = -5, then Log Re' = Log Re-Log 4 = Log Re-0.6.

In fact we cannot compare between Nikuradse'a curve for rough

pipes and our case as each line of Nikuradse's curve is given for a

constant relative roughness, but in open channels the relative

roughness changes with the change of the hydraulic radius. The

interesting case is that of total smooth channel. In all three cases,

la, IIa and Ilia, our curves go exactly parallel to Blasius line, and

45

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0 5/0/5 20 25 JO hm in an

Fig. 26. Strickler's roughness coeff. k in function of depth h, for 3 cases la,lb and Ic. Tables 1, 2 and 3.

„

b-j^.

•

•

_3—

—^_

0 5/0/5 20 25 30 hmincm

Fig. 27. Strickler's roughness coeff. k in function of depth h, for 3 cases

IIa, lib and lie. Tables 4, 5 and 6.

56

no

k

100

90

80

10

60

SO

'/~nia

%

/•

20 10 iO SO hm in cm

Fig. 28. k in function of the depth h for 4 cases Ilia, Illb, IIIc and Illd.

Tables 7 to 10.

k

W5

95

90

85

SO

°

/;

a

• ^^

h

I«,

"""*""""^h

0J05 0.05 0.07 009 0.11 015 0.15 0J7 Rm in m

Fig. 29. Strickler's roughness coeff. k in function of hydraulic radius R, for

3 cases la, lb and Ic. Tables 1, 2 and 3.

0.14 Rminm

Fig. 30. Strickler's roughness coeff. k in function of hydraulic radius B, for

3 cases IIa, lib and lie. Tables 4, 5 and 6.

0.20 Rminm

Fig. 31. k in function of the hydraulic radius B for 4 cases Ilia, Illb, IIIc

and Hid. Tables 7 to 10.

0.016

\

0.032

0028

0.024

0.020

0.016

0.012

\\

fc

—>-4

20 25 50 hm in cm

Fig. 32. A in function of the depth h, for 3 cases la, lb and Ic. Tables 1, 2

and 3.

0.016

\

0.012

0.028

0.024

0020

0.016

0.012

\

Bc

~J$__"~~

*.

a.

10 20 21 10 hmincm

Fig. 33. A in function of the depth h, for 3 cases Ila, lib and lie. Tables 4,

5 and 6.

59

0.068

X

0.060

0.052

aou

0.036

\

-Me

0.028-Ed

0.020

-Mb

20 50 to 50 hm in cm

Fig. 34. A in function of the depth h for 4 cases Ilia, Illb, IIIc and Hid.

Tables 7 to 10.

0.036

A

0.032

0.028

0.021,

0.020

0.016

0.01!

4

0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 Rmmm

Fig. 35. A in function of the hydraulic radius R, for 3 cases, la, lb and Ic.

Tables 1, 2 and 3.

60

0.036

A

0.032

0.028

0.02i

0.016

0.01}

;

He

0.08 0.10 0.12 OM Rm in m

Fig. 36. A in function of the hydraulic radius R, for 3 cases Ha, lib and He.

Tables 4, 5 and 6.

0.068

0.060

0.052

0.0U

0.036

0.028

0.020

0.012

Ec

:-***.

%

0.0% 0.06 0.08 0.12 0.16 0.18 0.20 Rm in m

Fig. 37. A in function of the hydraulic radius R for 4 cases Ilia, Illb, IIIc

and Hid. Tables 7 to 10.

61

0.6

0.5

04

03

0.2

0.1

J,,"'"

4-/5.5

30

.---*

^-_63

\

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~^ -lb

NIKURkDSE.^^""1

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3.6 3.8 4.0 4.2 4.4 4.6 4.3 5.0 5.2 log Ri

Fig. 38. Log (100 A) in function of Log Be', for 3 cases la, lb and Ic. Dotted

lines represent Nikuradse's curve for rough and smooth pipes. Tables

1, 2 and 3.

0.6

0.5

0.4

0.3

0.2

0.1

.iL*M.

,.--"

\»

30

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BUSIUS

NIKURAD.

— "V^*

lib

3.6 3.8 4.2 4.4 4.6 4.8 5.0 5.2 log Hi

Fig. 39. Log (100 A) in function of Log Re', for 3 cases Ha, lib and lie.

Dotted lines represent Nikuradse's curve for rough and smooth pipes. Tables

4, 5 and 6.

62

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l.:i5.3...

.,-'''

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l!B

2035

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NIKURAD.

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36 38 io 41 ti 46 «S 5o 52 %/?i

Fig. 40. Log (100 A) in function of Log Be', for 4 cases Ilia, Illb, IIIc and

Hid. Dotted lines represent Nikuradse's curve for rough and smooth pipes.Tables 7 to 10.

a little higher, at the beginning and then curved more or less to

Nikuradse's line. This shows that our case was totally smooth with

a possible little waviness in the channel. It shows also that the

equations for smooth surfaces of pipes can be also applied for

smooth open channels.

In tables 11 to 14, the percentage deviation of the theoretical

values of the roughness coefficient in equations 22, 24 and 26 from

the experimental values is given. It is clear that the deviation at

small depths is relatively big and that can be explained by the fact

that the shearing stress between the side walls and the fluid, t0b, is

smaller than that against the bottom for small depths as will be

mentioned later.

The third channel shows clearly that equation 26 cannot be

applied as the deviation is big.

63

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65

It appears that the deviation in eqn. 24 is bigger than that of

eqn. 22. To find the reason for that we have to discuss the origin of

both equations. When Strickhr wrote his law of friction he assumed

that the value of k depends only on the state of roughness of the

surface, or assumed that k does not vary with the variation of the

depth or hydraulic radius. That was found to be not exactly true.

The value of k increases with the increase of the depth till a certain

limit then begins to decrease again, Fig. 26, 27 and 28. Any how

the difference between the maximum and the minimum computedvalues of k is not bigger than 5 % in all our cases, except in case III c

(10 %). This shows that the assumption of Strickhr is a good approxi¬mation for practical purposes.

On the other hand, the factor A in the equation of Darcy dependson the Reynolds number and the relative roughness for small

Reynolds numbers, and only on the relative roughness for bigReynolds numbers. For this reason, the value of A, in some cases,

for small depths was found to be about the double of A at biggerdepths. For example, in case IIIc, Fig. 34, at 5 cm depth A = 0.0673

and at 60 cm depth = 0.0357. In case lie, Fig. 33, A = 0.0329 at

5 cm depth and 0.0193 at 34 cm depth. And as A depends to a certain

limit on the Reynolds number, a sure mistake will result in applying

eqn. 24 at a certain depth, as at this depth the Reynolds number

in case of totally rough is not the same as of totally smooth channel.

For example, in channel III, at 5 cm depth Se' = 8000 for totally

rough channel and 14000 for totally smooth channel. For 10 cm

depth Ee' = 22000 and 40000 respectively, and for 20 cm depthRe' = 48000 and 85000 respectively. This shows that eqn. 24 can

not be expected to be totally correct.

v^

v ^A *)On the other hand, as Jp =

7„!"

„and J, =

„

"

', an error

**JBW*'. 8giJmof a % in measuring Je will give an error of a % in A and about <r%

in k. So the percentage deviation in A equation ought to be about

the double of the deviation in the km equation. But, as the deviation

in the AOT equation is generally smaller than the double of the

deviation in the km equation, we can say that — in spite of the

*) dX _dJe dk_ 1 dJe

66

above explanation — the Am equation acts still as a rough approxi¬mation but not so well as km equation (eqn. 22).

Any how, as the percentage deviation in both equations is not

very big, and within the possible errors in measurements, both

equations can be accepted for laboratory and field purposes.

The deviation—~ '^ changes in signwith the increase ofdepth

and that is due to the change of the ratio —- with the change of

depth. As has to be expected the deviation in the Am equation and

the deviation in the hm equation have generally different signs.

It was found also that for totally smooth channel, Prandtl-

Karman's formula 1

— = ALogRelf\+B (27)

can be applied for open channels. According to Nihuradse [11]A = 2 and B= —0.8. As for circular sections i?e = 4i?e', the

equation must be written in the form,

-^ = 2 Log 4 Be'iX- 0.8 (28)

As an example: for case Ilia, at h = 57.72 cm, Re' = 191550,A = 0.0126, l/A~= 0.112.

Log 4 Rji\ = Log 4x191550x0.112 = Log 85810 = 4.9336,

the right hand side of the equation =2 X 4.9336-0.8 = 9.067,

the left hand side of the equation = jr^r^= 8.929

with a deviation of j^X 100 = 1.5%.

At h = 40.46 cm, Re' = 156560, A = 0.0124, yX= 0.111

the r.h.s. = 8.884, the l.h.s. = 9.000 with a deviation of 1.3%.

Concerning the mean velocity distribution curves (cal¬culated with mean values over horizontal sections), the followingmay be noticed, see Fig. 5 to 17 and table 15.

a) The maximum velocity is below the water surface and the

vratio -5^ does not change too much and decreases with the increase

of depth.

67

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68

b) For the same case of roughness at the same distance from the

bed the mean velocity increases with the increase of the total water

depth, except in case Hid, i. e. with bed smooth and side walls

rough, as in this case for small water depth the bed has a biggerinfluence on the velocity distribution than the side walls, see Fig. 17.

c) The shape of the mean velocity distribution curve is more

or less the same in profile 39 independent of the case of roughness.

d) The value of a, calculated according to eqn. 13 and not from

the mean velocity distribution curve, increases with the increase of

roughness and its value at the inlet is smaller than that at the

outlet. (Further see page 41.)

VII. Determination of the shearing stress on side walls and

bed from the velocity distribution

1. Introduction

All the developments above are based on the concept of the

hydraulic radius introduced by Chezy for pipes of non-circular

cross section or for open channels.

It may be of interest to recall the statement which leads to the

concept of the hydraulic radius in the case of the same uniform

roughness of bed and side walls for the normal flow in an open

channel.

Fig. 41

1 Chezy assumed that:

a) The shearing stress t0 is proportional to the square of the velocity,

b) the shearing stress is uniformly distributed along side walls and

bed.

69

A simple consideration of equilibrium leads then to the conclusion:

r0 = ev*

AlPr0=AlAyJe

A

To = YyJe = V*

v~fRJe with -B =-p-

One can ask if the second assumption of the uniform distribution

of t0 proves satisfactory if the width is much greater than the depth.A first attempt to answer this question would be to compare the

results of measurements of energy losses for two channels with the

same roughness but with different shapes at the same hydraulicradius, table 16.

But it would be even more interesting to measure the values of

the shearing stress at different points of the wetted perimeter in

order to have more detailed informations on the mechanism of

fluid friction in open channels. As direct measurements of the very

small shearing stresses by means of some mechanical device seems

practically impossible, we must use some indirect method of esti¬

mating the shearing stress. Fortunately the introduction of the

universal velocity distribution formula in the neighborhood of a

wall by Prandtl allows such an estimation on the basis of careful

measurements of the velocity distribution.

It must be emphasised that the formula of H. A. Einstein is

entirely based on the concept of the hydraulic radius, as it is based

on the Strickler's formula where the coefficient k is assumed to be

the same for all parts of the wetted perimeter.

2. The universal velocity distribution formula of Prandtl

The introduction by Prandtl and his school of the mixing lengthin the theory of turbulent fluid motion leads to a velocity distri¬

bution formula in the neighborhood of a wall of the following form 5):

5) Concerning all the equations used in this part, see the comprehensivebook of Bakhmeteff on turbulent flow [12],

70

0.0190

0.0198

0.0207

0.0219

0.0246

0.0299

0.0211

0.0218

0.0226

0.0237

0.0264

0.0305

87.70

88.35

88.98

89.50

88.30

85.04

83.64

84.45

85.22

85.35

84.44

83.43

0.0137

0.0139

0.0144

0.0154

0.0168

0.0195

0.0135

0.0142

0.0150

0.0158

0.0170

0.0203

104.27

105.53

106.45

106.80

'

106.45

104.75

104.27

104.84

105.65

105.77

104.30

102.20

0.15

0.13

0.11

0.09

0.07

0.05

II

channel

Ichannel

II

channel

Ichannel

II

channel

Ichannel

II

channel

Ichannel

36

Fig.

A35

Fig.

30

Fig.

k29

Fig.

36

Fig.

A35

Fig.

30

Fig.

k29

Fig.

channel

rough

Totally

channel

smooth

Totally

mRm

cm

35

=h

cm,

50

=b

with

II

channel

and

cm

35

=h

cm,

100

=b

with

IChannel

roughness.

same

the

and

forms

diff

eren

tof

channels

two

for

radius

hydraulic

the

offunction

in

coef

fici

ent

roughness

The

16.

Table

^ = C1 + C2Logy (29)

where v is the velocity at a distance y from the wall, v* the so called' 'friction velocity'', i—

v* = y^ (so)

02 a universal constant = 5.75, which is a characteristic of turbu¬

lence,

Ct a constant characteristic of the surface which may be called the

"surface characteristic".

On the basis of the development of theory and of experiments

by Nikuradse a distinction must be made between the smooth flow

and the rough flow according to the nature of the walls. For these

two cases the formula for the velocity distribution in best agree¬

ment with his experiments were:

— = 5.5 +5.75 Log^ for smooth flow (31)

— = 8.48 + 5.75 Log -^ for rough flow (32)v% s

where s is the roughness elevation measured by the mean diameter

of the sand attached to the inner surface of the pipes experimented

by Nikuradse [9 and 10].

Eqn. 31 could be applied when the number — is smaller than

5, eqn. 32 when this number is greater than 70. For 5 < — < 70

(the transition zone) neither eqn. 31 nor eqn. 32 is applicable. The

logarithmic law 29 is however valid in this case with the value

<72 = 5.75.

The universal validity of the logarithmic distribution law allows

to compute the values of the factors v* and C1 in any special case

by comparison with an actual distribution of the velocities in the

vicinity of the wall. From the value of v* = ]/— we can get the

value of t0 = p v*2 = ^-JL-.

Although the formulas were deduced only for the vicinity of the

wall, the experiments of Nikuradse on pipes with circular cross

section showed that formula 29 remains valid with good accuracy

over the whole cross section.

72

3. First attempt to determine the shearing stress z0

The velocity of flow was measured at distances 1, 2, 3, 5 resp.

7 cm from the bed and from the side walls of the channels, tables

17 to 28. The first idea was to compare the measured velocitydistribution with that according to eqn. 29 and to fix the values

of the two factors v* and Cx in order to get the best agreement.The values of v% and C1 were determined in accordance to the

method of least squares: Iiy1,y2 yn were the distances from

the wall at which the velocities vvv2 vn were measured, the

difference between the measured velocity at yi and the computed

velocity according to eqn. 29 is:

Vi - Ci v* - 5.75 v* Log yt

the terms G1 v* and 5.75 v* are then to be chosen so that the sum

of the squares of the differences is a minimum, or,

n

2 K-C>* -5.75 V* Log%)2 = min. (33)

The conditions for the minimum lead to the two equations:

n n

»C>* + 5.75 v* £ Log^ = X vi (34)

n n n

Ci*>* 2 Log& +5.75«* 2 (Logy,)2 = 2 ^LogSfc (35)t=l i=l t=l

The values of Cx and v* can easily be computed from these two

linear equations.In principle the method could be applied for the determination

of the local shearing stress at any point of the wetted perimeter.However the unavoidable errors in measuring the velocity make

it impossible to determine the local shearing stress with reason¬

able accuracy. As the velocity does not change very much from

point to point on the same horizontal near the bed it was found

more convenient to introduce "an average velocity distribution in

the vicinity of the bed", and similarly "an average velocity distri¬

bution in the vicinity of the side walls" by taking the average value

of the velocities at a certain distance from the bed respectively

73

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82

x. I 3 7 14 22 30 22 li 7 3 1

Fig. 42. Velocity measuring points, channel III, h = 60 cm.

from the side walls. The points in the vicinity of the corners were

neglected as the velocity in these points is influenced by both bed

and side walls. The points from which the average velocity distri¬

bution was computed are located in the shaded area of Fig. 42.

The results of the calculations for the cases Ilia, Illb, IIIc and Hid

are summarized in table 29.

The values of the factors C1 and t>% are also given as computed

by the above described method. From v% the value of t0 =^-*-

can

be determined. It was assumed that t0 is constant along the bed

(t0j) and constant along the side walls (t0J. A simple consideration

of equilibrium gives the relation:

br0b + 2hr.=yAJe (36)

The relative percentage difference between the 1. h. s. of this equation

computed from the velocity distribution and the r. h. s. computedfrom the measured gradient is also noted on table 29. In order to

check the validity of equations 31 and 32 the values of Gx were also

83

V*computed and compared with the value 5.5 + 5.75 Log — in the

smooth case. In the rough case the "equivalent sand roughness of

Nikuradse" was deduced from the term

or Logs =

'

M.

If the difference in the values of the 1. h. s. of eqn. 36 computedfrom the velocity distribution and of the r. h. s. computed from the

measured gradient is taken as a measure for the suitability of the

method, it can be seen that we obtained good results for roughflow, but for smooth flow (case Ilia) the differences are so big that

in this case no definitive judgement can be made. The differences

between the value of Cx as calculated from the actual velocitydistribution and from eqn. 32 are very appreciable too.

It must be emphasised that the determination of r0 is very sensi¬

tive to small errors in the measurements of velocities. As an examplethe velocity distribution in the vicinity of the side walls for case Illb,

at h = 20.34 cm (table 22), was measured twice and we obtained:

Distance from mean velocity mean velocitythe wall, x 1 "* measurement 2 nd measurement

1 cm 47.850 cm/sec 47.900 cm/sec2 51.250 50.750

3 53.450 53.550

5 56.817 56.967

209.367 cm/seo 209.167 cm/sec

It seems to be a good agreement between the first and the

second measurements. The computed value of r0a was however in

the first case 0.0050 gm/cm2 and in the second case 0.0053 gm/cm2,i. e. with a difference of 6 %, which is bigger than for the velocity.

For the rough case the real value of s "equivalent sand rough¬ness", seems to be somewhere between 0.5 and 0.6 cm.

The mean diameter of the rounded gravel used as artificial

roughness sm as computed from Pig. 24 = 0.82 cm. The difference

between sm and s is due to the fact that a part of the gravel was

imbedded in the cement used in fixing it to the concrete slabs, and

to the manner of rolling of the gravel surface.

84

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-§ 57.50 40.37 20.12 53.98 42.16 20.34 54.88 40.43 20.28 58.72 40.22 20.27

Case Ilia Illb IIIc Hid

85

4. Second attempt to determine the shearing stress t0

As the determination of the shearing stress by the method

described above requires a great number of velocity measurements,

and as the results were very sensitive to errors in measuring the

velocity, an attempt was made to find a method which would re¬

quire the measuring of the velocity in few points only.The simplest method is to assume the validity of eqn. 31 and 32

and to introduce in these equations the value of the velocity mea¬

sured at the centre of the channel. As the equations do not contain

any unknown except v*, its value could be determined.

Example:

Case Ilia, h = 20.12 cm, v = 0.01156 cm2/sec,

v in the centre of the channel at a distance of 10 cm from the bed and

30 cm from the side walls is 97.5 cm/sec.

Introducing these values into eqn. 31 assumed valid for the velo¬

city distribution in the vicinity of bed and side walls we get:

5.5 + 5.75 Log Q0115*6 for v* on bed,

30 Xv5.5 + 5.75 Log

* for v% on side walls.

From these two equations the value of v* could be computed bythe method of successive approximation, and from v* the value

of t0 is computed with eqn. 30.

For the rough case we computed the value of v% for the two

assumptions s = 0.5 cm and s = 0.6 cm. The values of t0 computedin this way are given in table 30, also the relative difference between

the two sides of eqn. 36. The errors are not so great for the roughcase, but for smooth case the computation of t0 by this attempt

gave consistently too low values compared with the first attempt(20 to 30%). This can be explained by the fact that the velocitydistribution as given by equation 31 does not remain valid at

great distances from the walls of the channel.

97.5

97.5

86

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< a 57.50 40.37 20.12 53.98 42.16 20.34 54.88 40.43 20.28 58.72 40.22 20.27

Case Ilia Illb IIIc PHI

87

5. Third attempt to determine the shearing stress z0

As it was not sure that the theoretical velocity distribution

formula remains valid at relatively great distances from the walls,

and as on the other hand the measurements of the velocity at

distances 1 or 2 cm from the solid boundary is possibly disturbed

from the effect of the velocity measuring device (Pitot-tube), an

attempt was made to determine the shearing stress from the mean

velocity measured in one line at a distance of 7 or 5 cm from the side

walls and the bed by an analogical computation as in the second

attempt. At 20 cm depth t0 was determined from the mean velocity

at a distance of 3 cm from the walls. The value of the velocity at

7 or 5 or 3 cm distances from the wall were the mean values as

explained in the first attempt. The computed values of t0 are

summarized on table 30.

6. Results

1. It was found that at 20 cm depth the shearing stress on the

bed t0j is bigger than that on the side walls t0 in both cases totallysmooth and totally rough channel. For 40 and 60 cm depths we can

not give definitive judgements as the difference between t0j and t0w

is small and within the possible error in measurements speciallybecause we have found that the value of t0 is very sensitive to any

small error in the velocity measurement.

2. We can say that eqn. 32 for rough flow can be applied for open

channels as well as for pipes. But, concerning eqn. 31 for smooth

flow we can not give a definitive opinion about it as the difference

between the 1. h. s. and r. h. s. of eqn. 36 was very big in case Ilia.

That may result from the inaccuracy of our measurements of velo¬

city with ordinary mechanical devices or from the equation itself.

3. To get an exact value for t0 we have to choose sufficient lines

parallel to the solid boundary and measure the velocity in many

points in each line (as in first attempt), and it would be necessary

to repeat the measurements in the same profile then go to another

profile and do the same and so on till we are certain of the value

of t0 computed. As equations 29, 31 and 32 were written for the

vicinity of the wall we are not supposed to go far from the wall but

88

to be as near as possible to it. On the other hand, we cannot be sure

of the velocity measured near the wall as the measuring device

causes some disturbances near the wall giving a bigger velocityhead than would be expected. The fixing of the safe allowable

nearest point to the wall depends on the dimensions of the velocity

measuring device and must be determined by comparison with

different devices. We have to notice here that the factor of time is

a very important factor by using a Pitot-tube as we may have to

wait at least 10 minutes in each point till the oscillations in the

manometer connected to the Pitot-tube disappear, to get exact

readings of the velocity head.

4. If we want to get a rough value of t0 for any surface we have

to measure the mean velocity on a line spaced 3 to 7 cm from the

surface and parallel to it and then apply the method explained in

the third attempt. For rough surfaces, if we don't know the approxi¬mate value of s we have to measure the mean velocity at least in

two lines parallel to the surface.

5. This point of the shearing stress t0 is very interesting and very

important and requires further experimental investigations.

VIII. Summary

1. The principal aim of this work was to study the problem of

the mean roughness coefficient in open channels, in the case that

the bed and side walls have different roughnesses.2. Three channels with different shapes and different roughness

arrangements were studied. The dimensions of the three channels

were: 100X 35 cm, 50 X 35 cm and 60X 60 cm.

3. It was found that both equations:

2/3^m —

P.Einstein

act satisfactorily enough for both field and laboratory purposes.

89

4. An attempt was also made to determine the shearing stress

between the wall and the fluid r0 from the velocity distribution.

5. It was found that at 20 cm depth (6 = 60 cm), r0j is biggerthan t0k. That shows that the definition of the hydraulic radius

introduced by Chezy represents only a mean value of t0. At bigger

depths no definitive judgement could be given. It seems that the

difference between t0j and t0w is small.

6. The problem of the shearing stress t0 requires further experi¬mental investigations which may lead to a new definition of the

hydraulic radius, based on the separate values of r0b and t0w.

References

1. H. A. Einstein, Der hydraulische oder Profil-Radius. SBZ. Band 103,

No. 8, 24. 2. 1934.

2. Strichler, SBZ. Band 83, No. 23, 7. 6. 1924.

3. Schiller, Eisener, Handbuch der Experimentalphysik, IV. Teil, page 297.

4. VTJI-DurchflufimeBregeln, Regeln fur die Durchflufimessung mit genorm-ten Diisen, Blenden und Venturidiisen.

5. Prandtl-Tietjens, Hydro- und Aeromechanik, II. Band.

6. Hunter Rouse, Engineering hydraulics.7. Normen fur Wassermessungen, SIA. No. 109, 1924.

8. Charles Jaeger, Technische Hydraulik.9. Nikuradse, VDI-Forschungsheft 356, GesetzmaBigkeiten der turbulenten

Stromung in glatten Rohren.

10. Nikuradse, VDI-Forschungsheft 361, Stromungsgesetze in rauhen

Rohren.

11. Erwin Hoeck, Druckverluste in Druckleitungen groBer Kraftwerke.

12. B. Bakhmeteff, The mechanics of turbulent flow.

90

Curriculum vitae

I was born on January 1st, 1923 in a small village near El-Mansura,

Egypt. I completed my primary education in 1935 and my secondaryeducation in 1940. I joined the Faculty of Engineering of Fouad 1st

University and graduated in 1945 with the degree of Bachelor of

Civil Engineering (B. E.). In the same year I joined the EgyptianState Railways as an Engineer where I spent 8 months. On March

20th, 1946, I was transferred to the post of Demonstrator in the

Civil Engineering Department of Faculty of Engineering, Fouad 1st

University, where I succeeded to get the degree of Master of

Engineering (M. E.). My thesis was accepted on the recommendation

of Prof. M. Hafez, Cairo, and Prof. W. W. Hay, University of

Illinois, U.S.A.

In December 1949 I came to Switzerland in a mission from the

Egyptian Government to study for a Doctorate in HydraulicResearches. In April 1950 I joined the Swiss Federal Institute of

Technology as a hearer. After passing a special admission examina¬

tion in October 1951, I began my doctor work under the super¬

vision of Prof. Dr. E. Meyer-Peter.

Zurich, May 1953.