Resistance Coefficients 

and Velocity Distribution 

Smooth Rectangular 

Channel

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1592-A

Property of

U.S. GEGi«AL SURVEY

WATER RESOURCES DIVISION 

GROUND WATER BRANCH
Resistance Coefficients 

and Velocity Distribution 

Smooth Rectangular 

Channel

By H. J. TRACY and C. M. LESTER

LABORATORY STUDIES OF OPEN-CHANNEL FLOW

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1592-A

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C.
CONTENTS

	Page 

Symbols.... ____________________________________________ iv

Abstract.._______________________________________________ A-1

Introduction- ______________-__---__-___----_-______ 1

Theoretical considerations__________________________________________ 2

Apparatus and procedure_..._________--_-__-___-___-_-___-__- 5

Results________________________________._____ 8

Variation of the friction factor in subcritical flow-__---_--__--_-__- 9

Variation of the friction factor in supercritical flow.________-_--_-_ 9

Velocity distribution......_____________________________________ 13

References cited_______-__-____-__-__________-_--__-___-_--_--__- 18

ILLUSTRATIONS

Page 

FIGURE 1. Average configuration of bottom and sides of flume _ _________ A-7

2. Flume deflection under load_____________________________ 7

3. Typical surface profiles.-.-.-.-________________._-__-- 9

4. Variation of the friction factor in subcritical flow.___________ 11

5. Variation of the friction factor in supercritical flow_________ 12

6. Surf ace-wave amplitude relative to flow depth_____________ 13

7. Typical vertical-velocity distribution in the central region of

flow__________________________________. 14

8. Width-depth ratio of wall region as function of channel aspect

ratio.__________________________________________________ 15

9. Point velocity in central region related to local wall shear.____ 15

10. Point velocity in central region related to average wall shear. _ 16

11. Dimensionless velocity distribution_______________________ 17

TABLES

Page 

TABLE 1. Data summary, subcritical flow. ___________________________ A-10

2. Data summary, supercritical flow_______________--.-___- __- 10

in
SYMBOLS

A Area, of channel cross-section 

b Width of rectangular flume 

C A constant

e A constant; natural logarithm base 

/ Friction factor, SRSoQ/CP 

F Froude number, UJ^fgy 

g Acceleration caused by gravity 

k Universal constant 

I Mixing length 

In Logarithm to the base e 

log Logarithm to the base 10 

P Wetted perimeter 

R Hydraulic radius, AfP 

R Reynolds number, 4RU/v 

r0 Pipe radius 

S0 Hydraulic gradient 

u, v, w Time-averaged velocity components in the x, y, z directions,

respectively 

u', v', wr Fluctuating components of velocity in the x, y, z directions,

respectively

U Average velocity in flume cross-section 

Uc Average velocity in central region of flow 

u* Average "shear" velocity, 

u* "Local" shear velocity, 

V Resultant point velocity

x> y> z Cartesian coordinates. Distances in the x-direction are 

measured along the longitudinal axis of the flume, the 

y-direction is normal to the flume floor, the z-direction is 

normal to the flume sidewalk. 

ym Distance from wall at which u=Q 

2/o Uniform depth of flow 

Z Width of wall region 

p Mass density 

v Kinematic viscosity 

n Dynamic viscosity 

t\ Eddy viscosity 

r Shearing stress

IV
LABORATORY STUDIES OF OPEN-CHANNEL FLOW

RESISTANCE COEFFICIENTS AND VELOCITY DISTRIBUTION, SMOOTH RECTANGULAR CHANNEL

By H. J. TRACT and C. M. LESTER

ABSTRACT

Measurements were made in a smooth rectangular channel to establish the 

effect of the side walls of the channel on velocity distribution and resistance to 

flow. The velocity distribution in the central region of a section normal to the 

channel followed the logarithmetic law. In the vicinity of each wall, at a point 

in the vertical which varied with distance from the wall, the slope of the vertical 

velocity curve reversed. The velocity distribution over the entire section has 

been defined in dimensionless terms. As well, the width-depth ratio for the wall 

region is expressed as a function of a similar ratio for the channel.

The variation of a resistance coefficient computed from the measurements is 

shown to be a function of the flow Reynolds number. The resistance coefficient, 

for the same Reynolds number, is slightly greater than for smooth pipe or a 

smooth channel of infinite width.

INTRODUCTION

Except for the effects of the free surface, the laws which govern the 

motion of fluids in closed conduits are equally applicable to uniform 

open channel flow. Rapid advances have been made in recent years 

on the problem of flow in circular closed conduits. A similar progress 

has not been made in the field of open channel flow, because of the 

additional complexities introduced by the presence of the free surface 

and by the usually asymmetrical cross section. Comparatively few 

systematic investigations have been made in open channels since the 

monumental work of Bazin over a century ago.

This is a report of the progress, to date, on a study of the uniform 

flow of water in rigid open channels. The project is a part of the 

research program of the U.S. Geological Survey. Experimental work 

is being conducted in the hydraulics laboratory of the Georgia Institute of Technology, under a cooperative agreement with the Georgia 

Tech Research Institute. C. E. Kindsvater, Regents Professor of 

Civil Engineering at Georgia Tech, is consultant to the research 

program.

The initial phases of the study have been concerned with the determination of the relationship between the uniform flow depth, disA-l
A-2 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

charge, and slope in a rectangular, smooth channel, and with the 

measurement of time-averaged point velocities in the flume cross 

section at various depths of flow and discharges. Both subcritical and 

supercritical flows have been considered.

THEORETICAL CONSIDERATIONS

The Navier-Stokes equations for turbulent flow may be derived 

from those for the laminar condition by the substitution for each 

velocity and pressure term a mean value and the instantaneous 

deviation therefrom. The time average of the resulting expressions 

differ from the original equations only by the addition of terms 

involving the time-averaged velocity fluctuations. The terms thus 

added are commonly referred to as "apparent" or Keynolds stresses.

The modified Navier-Stokes equations, together with the differential equation of continuity, are a starting point for the calculation 

of turbulent flows. There are many more unknowns than equations, 

however. The direct measurement or prediction of the quantities 

which involve the turbulent fluctuations has proven to be an almost 

insurmountable obstacle because of the random and extremely complicated nature of the turbulence. During the last decade, the 

development of the hot-wire anemometer has permitted the measurement of a number of the statistical properties of fluid turbulence to 

be made in air flow. A comparable instrument has not yet been 

developed for use in water.

The determination of the mean flow properties of the fluid are 

usually the eventual objective of the worker concerned with turbulent 

flow. Because of this, and because of the difficulties encountered in 

the direct solution of the Navier-Stokes equations, attempts have 

been made in the past to simplify the equations by the development 

of empirical relationships between the Keynolds stresses and the 

mean values of the velocity components. Some success in this direction has been attained.

Boussinesq (1897) apparently was among the first to use this approach. Except that in the one case, mixing takes place on a molecular scale, while in the other, masses of fluid are interchanged between 

neighboring flow regions, the shear processes in laminar and turbulent 

flow are approximately similar. Boussinesq suggested that the shear 

stress in turbulent flow could be represented as the sum of a stress due 

to purely molecular viscosity and a stress due to the turbulent transfer of momentum, thus:

du du --
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~3

In this equation, the factor q, by analogy with the molecular viscosity p, has the significance of a viscosity coefficient; and has been 

called the "virtual" or "apparent" viscosity. It is not, unfortunately, 

like /it, a property of the fluid itself, but instead varies with the magnitude of the turbulence fluctuations.

Prandtl (1925) carried this approach a step further. By assuming 

that the momentum of each fluid particle remains unchanged during 

movement from one region to another, he postulated, for a twodimensional flow:

in which Z is the mixing length; the average migration distance of 

particles of fluid starting from a given point. This statement is 

equivalent to an assumption regarding the value of the apparent 

viscosity, thus:

By assuming that the mixing length is proportional to the distance 

from the wall, y, and that, near the wall, r is equal to TO , the mixing 

length hypothesis becomes :

which may be integrated to give the logarithmic vertical-velocity 

distribution,

(1)

in which k is a constant. Because of the assumption regarding TO , 

the integrated expression is applicable only to the wall region. Experimental data, however, confirm the fact that it is a better description of the distribution of velocity away from the wall region than 

it is near the wall. Rouse (1959) suggests that it is just as reasonable 

to assume that,

wnich eliminates the inconsistency, since T TO ( 1 ) \ yo/
A-4 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

Taylor (1935), using a vorticity-transport concept, derived an expression applicable to free-turbulence flow almost identical to that 

given by Prandtl for turbulent flow near a fixed boundary. Von 

Karman (1931) basing his deductions upon the experimentally confirmed assumption that the correlation between the fluctuating components of flow in pipes remains constant almost from the center of 

the pipe to the region of the wall, concluded that the turbulent 

mechanism over this region is similar, and further, that it differs only 

in the time and length scales from the turbulence in any other pipe, 

regardless of size and boundary roughness. From this, he developed 

a relationship between the mixing length and certain characteristics 

of the mean flow. His expression is more complicated than that of 

Prandtl; when integrated, however, it leads to the logarithmic distribution under the assumption that r is constant and equal to r0.

The mixing-length theories have lately fallen into disrepute. They 

contain the implicit assumption that what is locally produced is 

locally dissipated, whereas in actuality the production of turbulent 

energy and its ultimate conversion into molecular heat are the initial 

and the final phase of a complex process, and do not coincide spatially. 

The mixing-length concept has lost much of its importance because 

the mean-velocity distribution may also be obtained from simple 

dimensional considerations. Probably no completely adequate theory 

will be forthcoming until more is known about the origin and structure of turbulence. Until such time, however, these concepts will 

continue to be useful and convenient tools.

The constant of integration of equation (1) may be obtained from 

the condition that u=Q at a certain distance from the wall, say at 

y=yn. Then

u=-~(lny lnym)

For flow near smooth walls, the motion pattern is determined by 

v and u*. It may be concluded, therefore, that ym is proportional to 

the ratio of U* and v, which has the dimension of a length. Then

or

u

the constants k and <72 have been determined by Nikuradse (1932) 

from experiments on smooth pipes. Substituting the experimentally
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A-5

determined values into the equation, and converting to common 

logarithms, the expression becomes

^=5.75 log »+5.5 (2)

Equation (2) may be integrated across a normal pipe section to 

determine the rate of flow. The mean velocity, U Q/A, is

or, in terms of the hydraulic radius, R=AjP=rol2

3.50 (3)

or

4== 2.03 log #V/-0.91 (4) v/

where R=2UrJv=±RUIv.

A similar integration over the half-section of an infinitely wide 

conduit having parallel, smooth walls leads to the relationship :

 =2.03 log JBV7- 1-08 (5) v/

In equation (5), the Reynolds number, to be consistent with that for 

flow in pipes, is equal to 4RU/v=4yQU/v, where y$ is the half -width of 

the section.

In following sections of this report, the equations developed from 

the integration of the Prandtl mixing length hypothesis will be compared with the results obtained in the smooth, rectangular open test 

channel, in order to ascertain the combined effects of the free surface 

and the secondary motions which exist in the open channel.

APPARATUS AND PROCEDURE

The laboratory tests were made in a steel flume 3^ feet wide, 18 

inches deep, and 80 feet long. The flume is supported at its midpoint on a fixed pivot and at the two ends by pairs of screw jacks. 

An electric motor at the pivot drives the two pairs of jacks simultaneously through torque tubes of equal length. The drive mechanism also furnishes a method for the determination of the flume slope.

585225 -61-
A-6 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

The number of turns made by the torque tubes from the level flume 

position is indicated by a mechanical counting device. The vertical 

displacement of the two ends of the flume for various counter readings 

has been measured by precise levels. A calibration of counter reading 

against flume slope has been prepared from this data and the measurement of the distance between the two end points.

Two longitudinal, variable-depth, built-up steel beams furnish the 

principal support for the flume box. Two 4-inch channels placed 

back-to-back are welded in place on top and across the beams at 

3-foot intervals. Five lengths of threaded %-inch rod, inserted vertically between each channel pair and spaced at equal intervals, are 

held in place by nuts bearing on the top and bottom of the channels. 

The rods project to a height of about 3 inches above the top of the 

channels, and form bearing supports for five small car channels which 

extend the length of the flume. The flume floor is bolted directly to 

the car channels. Small irregularities in the flume floor as well as the 

curvature due to the dead weight deflection of the supporting beams 

may be removed by adjustments in the height of the tops of the 

threaded rods. The side walls are supported by channel lengths 

welded to the angles that form the bottom support for the side plates. 

The channel lengths are vertical, and on 3 foot centers. Bolts tapped 

through the channels permit slight adjustments to be made in the 

position of the side walls. The inner flume surfaces are covered by 

one seal coat and two coats of synthetic enamel, applied by brush.

Before tests were made in the flume, the bottom and sides of the 

flume were adjusted to make them as plane as possible. Precise 

levels were run to determine the variation of the bottom and where 

necessary, the bottom elevation adjusted by means of the supporting 

rods under the flume floor. Figure 1 shows the average configuration 

of the bottom and sides of the flume after the adjustments were made.

The deflection characteristics of the flume under various depths of 

flow are shown in figure 2. The deflection is inconsiderable except at 

the extreme downstream end of the flume at depths greater than 

one-half foot of water. Tests thus far in the flume have been limited 

to depths equal to or less than one-half foot.

Water surface levels are read by point gages from a carriage which 

moves on rails mounted on the top of the sidewalls of the flume. 

Piezometer openings have also been installed in the flume floor.

Water is supplied to the flume by the laboratory recirculating system 

through either a 12-inch or a 6-inch line. Discharges are measured 

by a venturi meter located in the 12-inch line, or by an orifice meter 

located in the 6-inch line. Both meters were calibrated gravimetrically.
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~7 

3.490 rv

s 3.485

I

«I 3.480

0 10 20 30 40 50 60 70 

Distance along flume, feet

80

.002

o

CO

Each point is average of 

5 transverse readings 

 Aug. 1957, ---Mar. 1958

o -.001

-.002 10 20 30 40 50 60 

Distance along flume, feet

70 80

FIGURE 1. Average configuration of bottom and sides of flume.

The depth of flow in the flume is regulated by an adjustable tailgate, 

or by a sluice gate at the entrance to the flume. Point velocities have 

been measured with small pitot tubes, or by stagnation tubes used in 

conjunction with floor piezometers.

Near the normal depth, steady flow in uniform channels varies 

gradually from section to section. The change may be almost imperceptible over short distances as, for example, the 80 feet of length of 

the experimental flume. For mild channel slopes, the normal surface-

-1 

« -

o

x 1

c ^

o

4> O

Q 4 

«;

.. *"

 ^'

Depth

 -

of water, i

.._._._. (

....... (

_ TL i

............. (

 

\

n feet

).2IO

).328

).500 

).7IO

.010

'''"'' "\.

^ ^ -..

\

"^^

" - -.... \^'S"ZJ

^^

X\

\ ' -.

\ 

\

10 20 30 40 50 60

Distance along flume, feet 

FIGTTRE 2. Flume deflection under load.

70 80
A-8 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

profile is the upstream asymptote to the Ml and the M2 backwater 

curves. For steep slopes, the normal profile is the downstream asymptote to the $2 and the 83 curves. In each case, the normal line lies 

between the two curves. The method used to determine the normal 

flow depth consisted of the measurement of the surface profile at 

depths just greater than the normal, and at depths just smaller than 

the normal. The normal depth was then interpolated between the 

two curves. Point gage readings to the nearest 1-1000th of a foot 

were made at five points in cross sections at one-foot intervals for the 

length of the flume. The point-gage readings were compared with 

the piezometric level at selected points in the flume. The normal 

flow depth has been referenced to the piezometric level for tests in 

which the flow was subcritical. For the supercritical flow tests, in 

which the velocities were generally considerably greater than those in 

the subcritical range, the piezometric height is more difficult to 

measure accurately. For these runs, a mean depth based upon pointgage readings was computed. This will be discussed in a later section.

The normal depth of flow for a zero slope is infinite. Small slope 

changes at slopes close to the horizontal correspond to large changes 

in normal depth. Because of this, an apparently normal flow profile 

may be obtained over a comparatively large change in depth at the 

smaller slopes (see figure 3a). At these slopes, therefore, the interpolated value of the normal depth is somewhat less reliable than at 

greater slopes.

On the other hand, at very steep slopes and at small depths, very 

large changes in slope are required to cause perceptible changes in 

depth. Errors in the measurement of depth assume a relatively 

greater importance at steep slopes. At these depths and slopes, 

therefore, it becomes important to observe $2 and S3 curves as close 

as possible to the normal profile in order that the error in estimating 

the normal depth be minimized. A typical set of measurements of 

flow on a steep slope is illustrated in figure 3B.

RESULTS

In all, 49 uniform flow tests have been made in the rectangular 

smooth flume. Twenty-four of the tests were in the subcritical flow 

range, and 25 in the supercritical flow range. The depth of flow was 

varied from 0.1 to 0.5 feet, the discharge from 0.3 cfs to 7.3 cfs. The 

principal results from the tests are shown in tables 1 and 2.

The flow Reynolds number, 4RU/v, is approximately proportional 

to the discharge per foot of width of the flume. The Reynolds 

number, for the discharge range tested, varied from about 35,000 to 

about 700,000.
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~9

VARIATION OF THE FRICTION FACTOR IN SUBCRITICAL FLOW

The results of the twenty-four tests made in the subcritical flow 

range are shown on figure 4. For comparative purposes, equation (4), 

for turbulent flow in smooth pipes; and equation (5), for flow near 

smooth, parallel walls are also shown. The line drawn through the 

open-channel flow-data is described by the equation

=2.03 log yf - 1-30 (6)

which differs from the closed-conduit relationship only in the magnitude of the constant. Equation (6) may also be written:

£=2.40+5.75 log & (7)

in which u* is the average "shear" velocity, ^Tolp = -^RS0g; R is the 

hydraulic radius, A/P; and U is the average velocity in the flume 

cross-section. The average deviation, in terms of /, of the plotted 

points from the line defined by equation (6) is less than one percent.

VARIATION OF THE FRICTION FACTOR IN SUPERCRITICAL FLOW

The typical free surface in supercritical flow is characterized by 

highly agitated transverse waves of short length. The amplitude

.340

.320

.300

10

Estimot 

r.ormol

^--M 1 curve

. .in. u O-*O~Q O*O**(

R = 51,500 F= 0.156 S0 =7.I63 

I 1 1

i-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-j

eddeoth^ ° ° H-0 ° 9 ° o-l-o o o o

 -- I J i-

.^M2 curve0 

 = o-o-o-o-o-< 

1

7 " t ~ T

< IO'5

o 

~o 

20 30 50 60 70

Distance along flume, feet 

A

80

..300

-c".260

.220

S2 curve v

- Ooooo-0l0.c 

Normal depth-^.

O _ o- Or O-(

'-O-O-O-O-,;

t=^

-0-0-0-0-C

>-0-0-0-o-<

J-O-o- o-O-C

>-o-o-a-o_c

>-o-o-°-o-c

R = 580, 000 

F = 2.05 ._ _ 

S0=9.03 X IO' 3

)-0_Q.p_.0.^_0.0^

-Q.O-o^O^cr «-L o_

10 20 30 40 50 60 70

Distance along flume, feet 

B

FIGURE 3. Typical surface profiles.

80
A-10 LABORATORY STUDIES OP OPEN-CHANNEL FLOW 

TABLE 1. Summary of data, subcritical flow

Run

1 

Z.. ..............

^.. ..............

4 .

5

6...-..- - -

8

Q

10

11

12 

24 . 

30 

34. 

40 .

41

42

43. 

45 .

46- 

47 

48 

50 

(cfe)

1.21

2.98

1.99

2.01

2.00

1.99

.498

1.50

2.00

2.60

3.54

4.01

1.03

5.69

3.50

1.98

1.46

1.00

1.00

.370

.372

.400

.700

.398

Ho 

(feet)

0.300

.503

.267

.303

.353

.399

.313

.407

.401

.402

.413

.418

429

.454

.496

.503

.206

.146

.215

.107

.223

.100

.148

.148

SbXlO*

3.799

4.680

13.21

9.195

6.057

4.295

.7158

2.423

4.188

6.277

11.07

13.16

1.101

19.22

6.497

2.147

15.64

21.14

6.938

9.532

1.157

12.95

11.02

4.132

T

74

73

67

67

65

65

77

72

69

69

70

70

70

78

78

79

79

79

79

72

70

75

77

77

»X10+« 

(ft2 per sec.)

1.005

1.018

1.104

1.104

1.134

1.134

.967

1.032

1.074

1.074

1.059

1.059

1.059

.954

.954

.942

.942

.942

.942

1.032

1.059

.991

.967

.967

/

0.0188

.0163

.0172

.0169

.0172

.0176

.0235

.0184

.0172

.0166

.0167

.0151

.0206

.0142

.0168

.0170

.0180

.0191

.0192

.0251

.0257

.0242

.0208

.0242

R

117,800

260,700

179,300

177,900

172,400

163,800

51, 510

134,700

173, 700

217,000

310, 100

350,400

89,520

533,400

327,600

187, 100

158,900

112,300

108,100

38,730

35,720

43,770

76,540

43,510

F

0.372

.423

.730

.610

.483

.399

.144

.292

.398

.497

.677

.760

.186

.922

.607

.281

.787

.903

.507

.535

.179

.636

.625

.355

TABLE 2. Summary of data, supercritical flow

Run

13

14..

15 

19..- .- .

20 

21 

22 

23 

26 

27 . 

28 

29

31

33 

35 _.. . 

36 

37- 

38 

39 

a

51 

52 

53 

54.. _ _ .

55. 

(cfs)

7.33

2.30

1.78

0.796

0.597

5.00

4.50

3.98

7.18

7.18

7.01

5.69

5.48

4.52

3.51

3.01

3.01

3.01

2.01

1.00

.917

1.207

1.643

.974

.917

Vv (feet)

0.341

.0952

.0867

.0993

.0935

.186

109

.195

456

OCJ

.260

oen

.258

.279

11A

.172

.136

.201

AQ1Q

no ATI

.124

.103

.124

.104

.101

SoXlO"

66.12

330.5

276.5

46.80

33.04

195.6

148 7

115.6

90 ftd

CQ 4/5

141.5

OQ QO

90.30

51.04

261.1

95.81

190.5

62.49

298.0

74.33

29.37

82.65

82.65

55.10

53.45

T

(degrees 

F)

77

77

77

74

74

81

73

74

74

74

76

76

79

80

7Q

79

on

79

79

79

83

81

81

81

83

»X10+« 

(ft* per sec.)

0.967

.967

.967

1.005

1.006

.919

1.018

1.005

1.005

1.005

.979

.979

.942

.930

.954

.942

Q<?n

049

O49

Q49

.897

.919

.919

.919

.897

/

0.0128

.0160

.0169

.0214

.0223

.0141

.0146

.0151

.0136

.0133

.0137

.0142

.0141

.0146

.0153

.0153

ni *vi

.0157

.0170

.0195

.0197

.0182

.0171

.0191

.0194

R

726,900

258,800

201,300

85,970

64,850

564,000

456,700

408,600

649,500

RRn son

714,900

652,600

681,300

480,500

391, 400

333,600

344,200

OOQ KAA

232,500

115,400

109,400

142,200

191,500

114,700

110,800

F

1.86

8.96

3.66

1.29

1.06

3.17

2.70

2.34

1.18

1.72

2.68

1.34

2.20

1.55

2.14

3.03

1.69

3.64

1.70

1.06

1.85

1.90

1.48

1.45
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~ll

7T

= C, + 2.03 log R/T

Smooth pipe, C. = -0.91

Half - section, smooth parallel walls 

C, = -l.

O

8.5

8.0

7.5

7.0

6.5

'0.5 0.6 0.70.8 1.0 2 345678

R/T x ICT 4

FIGURE 1 Variation of the friction factor in subcritical flow.

and the period of the waves varies with the Froude number. The 

determination of the flow depth by means of point gages is influenced 

to some extent by the judgment of the observer. In this connection, 

also, the pressure indications given by floor piezometers are extremely 

sensitive to small imperfections in construction and alignment. 

Piezometric errors vary directly with the velocity head of the flow.

The mean depth for each of the test runs in supercritical flow has 

been experimentally established by first measuring the elevation of 

the crests, and next the elevation of the valleys of the wave pattern 

with the point gage. The mean depth has been computed as the 

average of the two levels thus observed. This procedure eliminates 

much of the judgment required in the location of the mean depth by 

a single reading of the point gage. It does not, however, give adequate consideration to the fact that the location of the true depth line 

is a function of the shape and the spacing of the waves.

The piezometric depth was measured from four floor piezometers 

in a section normal to the flow seventy feet from the flume entrance. 

Because of the high flow velocities of many of the supercritical tests, 

and the consequent possibility of error, the piezometric measurements 

were not used. It is of interest to note, however, that at all veloci-
A-12 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

ties, the depth indicated by three of the four piezometers agreed with 

that computed from the point-gage readings within one percent. The 

fourth gage read consistently low by amounts up to two percent of the 

point-gage depth.

The friction factors for the supercritical flow tests have been computed using the point-gage mean depth, which was determined as the 

mean of the elevations of the peaks and valleys of the waves. The 

results are shown in figure 5. The average results are described by 

the same function as for the subcritical tests. The mean depth is 

listed in table 2.

An idea of the magnitude of the wave heights for the various supercritical flow runs may be gained from figure 6. In this figure, the 

mean water-surface levels computed from the point-gage readings 

have been subtracted from the point-gage levels for the corresponding 

wave peaks. The differences, or wave amplitudes, have been divided 

by the mean depth and the resulting ratios shown as a function of the 

Froude number.

/F

y.u 

8.5 

8.0 

7.5 

7.0

6'5o

X

o

1 j

/r

s

Xo

30+2.0

O

b°

0'°'

x

x^

X"

)

V^ Q>o 

b 

X^

)

/#

c

0

/^ 

0

8 1.0 2 3 456789IC

IR/Tx icf 4

FIGURE 5. Variation of the friction factor in supercritical flow.
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A-13

5.0

1.00 1.50 2.00 2.50 3.00 3.50 4.00

Froude number 

FIGURE 6. Surface-wave amplitude relative to flow depth.

VELOCITY DISTRIBUTION

Point velocity measurements were made over a section normal to the 

flow in the region of fully established motion at the downstream end 

of the flume for twenty-four of the experimental runs. The measurements were made for runs in both the supercritical and subcritical 

flow range, using either a small pitot-static tube (0.0058 ft outside 

diameter), or a stagnation tube (0.0040 ft outside diameter) in conjunction with floor piezometers. Differential pressures were measured 

to the nearest 0.001 inch of water. The following general conclusions 

have been drawn from the measurements:

1. The flow is symmetrical about the vertical plane equidistant from 

the two sidewalls of the flume. Each half-section may be divided 

into two regions; one in the central portion of the channel, and the 

other in the vicinity of the wall.

2. Each vertical-velocity curve in the central region of flow is logarithmic from the surface to a point very close to the floor. The 

velocity distribution may be expected to deviate from the logarithmic law near the laminar sublayer as purely viscous stresses become 

of the same order of magnitude as those due to momentum exchange. Because of the relatively large diameter of the instruments 

that were used, measurements in the laminar zone were not obtained. 

The measurements in the floor vicinity also display a greater random 

variation than those away from the floor, because of the relatively 

greater difficulties encountered in the measurement of very low 

velocities.

Within the limits of experimental accuracy, the vertical-velocity 

curves appear to be identical over the width of the central region.

At a given distance from the floor, the velocity gradient ^ is thus

equal to or very close to zero. A typical curve for the central 

region is shown in figure 7.
A-14

9

, 8

o 

u

0>

»

0> r 7 i

>s ! 

 ?

: LABOEATOEY STUDIES OF OPEN-CHANNEL FLOW

Sym. 3 ,ft

A 1.30 

O 1.50

V 1.70 

X 1.90

 + 2.10 

Y 2.30 

A 2.50

^

^

/

/

s

/

s

s

^s

S*

\s 9 

ISs

'1 s

s

\- Q'**w

IO' 3 ID'2 

Distance above flume floor, feet

ICT

FIGURE 7. Typical vertical-velocity distribution in the central region of flow.

3. In the vicinity of the sidewalls, the maximum velocity in each 

vertical is depressed below the surface.

4. In the wall region, the magnitude of the mean velocity in the 

vertical increases with distance from the wall.

The limits of the central region of flow have been defined as the 

vertical-velocity curves on either side which first deviate from the 

characteristic logarithmic profile. The distance from each of the 

boundaries to the nearest sidewall has been called Z. The distance Z 

is a function of the depth-width ratio of the channel. The relationship between the channel-aspect ratio, yjb, and the width-depth 

ratio of each of the wall regions, Z/y0 , is shown in figure 8. The precise 

location of the central region boundaries are difficult to ascertain 

experimentally, which accounts for the scatter of the points about 

the curve of figure 8.

The velocity distribution in the central region is shown in dimensionless form in figure 9. The ordinate scale is the ratio of point velocity 

to local shear velocity, u/u%; the abscissa scale is the dimensionless

tyl/tl ___

depth, . The test data involves both supercritical and subcritical

flow. The uniform flow depth, ym ranges from 0.100 to 0.456 ft. 

The mean depth has been used in the computations for the supercritical-flow tests.

Lines have not been drawn through the points shown on figure 9. 

Each set of data, however, appears to define a straight line; each of 

which may be represented by an equation of the slope-intercept form.
RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~15

0 0.05 0.10 0.15

w

Ratio of flow depth to flume width, r 

FIGURE 8. Width-depth ratio of wall region as function of channel aspect ratio.

£8

26 

24 

22 

20 

18 

16 

14 

12 

10 

a

\ 1

Run No.

6 

12 

2\ 

22 

26 

29 

43 

47 

48 

50

x

i ii

Symbol 

O 

A 

X 

A 

A 

V

a

Y

'

x

1

Depth, fe 

0.399 

.418 

.186 

.192 

.456 

.359 

.215 

.100 

.148 

.148

Y

X

/tp

/> y

+

- III u

A

x;j(Y I

5.5 + 5.75 log -^

^

7

1 4

£ *

X

7

A

 - ,.

a »

- v

X*

 o^ °

'/I

]f

; 

'tj

* * 1 7 7' 1

«

*v

10 I02 I03 I04

FIGURE 9. Point velocity in central region related to local wall shear.
A-16 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

The lines do not have a common slope. At the smaller depths, a 

limiting slope of 5.75 is approached, which is the same as that of the 

Karman-Prandtl relationship, equation (2), for flow in an infinitely 

wide channel. At the greater depths, farther from the infinitely wide 

case, the slopes become smaller.

By relating the point velocities to the average wall shear stress 

over the flume perimeter, the data of figure 9 may be plotted upon the 

single logarithmic curve of figure 10. The function shown upon the 

ordmate scale of the figure is the ratio of point velocity to average 

shear velocity, u^=^RSog '> the abscissa function is the same as for 

figure 10. The line through the points is described by the equation

=5.50+5.75 log (8)

The constants of equation (8) are numerically equal to those of the 

Karman-Prandtl expression, equation (2).

Equation (8) may be integrated over the central region to obtain an 

expression for the mean velocity in the region. This gives:

(9)

The distribution of mean velocity in the rectangular flume section 

is shown in dimensionless form in figure 11, in terms of the ratio

4U

28 

26 

24 

22 

20 

18 

16 

14 

12 

in

i i

Run No.

 

 

6

12 

21 

22 

26 

29 

43 

47 

48 

50

1 1 1 1 1 1 1 

Symbol Depth, feet

O

* 

X 

A 

A 

V

a

Y

,

0.399 

.418 

.186 

.192 

.456 

.359 

.215 

.100 

.148 

.148

V

s

' '

t

s

V

mill

-=-.5.5+ 5.75 

a*

*'

V

**

I '

log

V

 7

V

i

{

-.

$I>B &

f

*

X^.U 

10 I02 I03 10"

FIGUEE 10. Point velocity in central region related to average wall shear.
z/Z

.10

o

.10

\

/

.50\

T 

.70\.90//I.O

RESISTANCE COEFFICIENTS, SMOOTH RECTANGULAR CHANNEL A~17 

1.00 

.90 

.80 

.70 

.60

.50 

»|=S

.40

.30 

.20

-12 -II -10 -9 -8 -7 -6 -5 -4 -3-2-1 0 I 2 3

u-Uc 

u*

FIC.URE 11. Dimensionless velocity distribution.

y/y0 and the ratio (% Ue}/u#. The curves are for various values of 

the ratio z/Z, where z is distance from the sidewall, and Z is the distance from the sidewall to the central region boundary. The curves 

were interpolated from the point-velocity observations; the coordinates of the points at which the data were taken did not always correspond to the coordinates of the points through which the dimensionless curves are drawn. The data may not therefore be conveniently 

shown.

The curve for which 2/Z=1.0 corresponds to equation (8). The 

algebraic expression for the curve may be obtained by subtracting 

equation (9) from equation (8). The resulting equation is:

^-'=2.50+5.75 log (10)

The reversal of the curves near the surface in the wall region is an 

indication of transverse motion which proceeds outward from the 

vicinity of the wall. Visual observation of small objects immersed in 

the flow at various points in the wall region lead to no very definite 

conclusions concerning the pattern of the secondary flow.
A-18 LABORATORY STUDIES OF OPEN-CHANNEL FLOW

REFERENCES CITED

Boussinesq, T. V., 1877, Theorie de 1'e'coulement tourbillant, Me"m. prese'ntes

par divers savant Academic des sciences, v. XXIII, Paris. 

Prandtl, L., 1925, tlber die ausgebildete Turbulenz, Zeitschr. Angew. Math.

u. Mech., v. 5, p. 136. 

Taylor, G. I., 1935, Statistical theory of turbulence, Proc. Roy. Soc. London A,

v. 151, No. 183. 

Von Karman, Th., 1931, Mechanische Ahnlichkeit und Turbulenz, NACA Tech.

Memo. No. 611. 

Nikuradse, J., 1932, Gesetzmassigkeiten der Turbulenten Stromung in glatten

Rohren, Forschg. -Arb. Ing.-Wesen. No. 356. 

Rouse, Hunter, ed., 1959, Advanced mechanics of fluids, New York, John Wiley

and Sons.

o