The Structure of a Turbulent Flow in a 

Channel of Complex Shape

GEOLOGICAL SURVEY PROFESSIONAL PAPER 983

The Structure of a Turbulent Flow in a 

Channel of Complex Shape

By H. J. TRACY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 983

UNITED STATES GOVERNMENT OFFICE, W A S H I N G T O N : 1 9 7 6
UNITED STATES DEPARTMENT OF THE INTERIOR 

THOMAS S. KLEPPE, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Tracy, Hubert Jerome, 1918-

The structure of a turbulent flow in a channel of complex shape.

(Geological Survey Professional Paper 983)

Bibliography: p. 24.

Supt.ofDocs.no.: I 19.16:983

1. Turbidity. 2. Channels (Hydraulic engineering) I. Title. II. Series: United States Geological Survey Professional Paper 983.

QE75.P9 no. 983 [GB665] 557.3'08s [532'.517] 76-608249

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

Washington, D.C. 20402

Stock Number 024-001-02875-1
CONTENTS

Page

Abstract ____________________________________________________ 1

Introduction ________________________________________________ 1

Acknowledgments.-___-___________________________________-___ 1

Preliminary considerations _________________________-________--_ 2

Equipment __________________________________________-________ 2

Procedure ________________________________________-___ 3

Results_______-__________-____-___________________________ 5

Distribution of longitudinal mean velocity _____________________________ _ 5

Distribution of normal turbulent stresses ______________________--___-- 7

Distribution of turbulent shear stress ________________________-_ 7

Secondary motion pattern _____________________________________ __ 15

Origin of the secondary motions ________________________________ 16

Role of the secondary motions ____________________________________ __ 19

Conclusions ____________________________________________ 23

References cited _______________________________________________ 24

ILLUSTRATIONS

Page

1. Schematic drawing showing components of turbulent stress on an elementary particle _ _ _ _ ____ ___ _ ______ 1

2. Schematic drawing of test conduit ____ _ ___ ___ _ _ _ _ _ _ _ __ _ _ _ __ _ ____ _ __ _ _ __ __ ___ __ ___ 2

3. Schematic drawing of channel section _ ____ _ _ ____ _ __ _ ____ __ __ __ _ __ ___ __ ___ _ __ _ __ ___ ___ 4

4. Graph showing mean pressure along channel _ __ _ _ ___ ____ _ _ __ ___ _ _ ___ _ _ _ __ _ __ _ ___ 4

5. Graph showing distribution of longitudinal velocity near wall _ __ _ _ _ ___ __ _ __ ___ _ __ ___ _ 5

6. Graph showing distribution of longitudinal velocity away from wall _ __ ___ __ ___ _ __ __ _ __ 6

7.-12. Graphs showing distributions of turbulence quantities:

7. u'lU ____________ _ _______ _ _____ __ __________________ __ _______________________________ _ - __ _______ 8

8. v'lU ______________________________________________________________________ 9

9.

10.

w'/U 10

11. Uw/U* 2 ____________________________________________________________________ 12

1 O IT 79 1O 12. vw/U* _____________________________________________________________________________________________________ io

13. Graph showing comparison of velocity gradient, turbulent shear-stress, and correlation coefficient ____ -___ __ 15

14. Graph showing distributions of correlation coefficients uv/u'v' _____________-______ __ - 16

15. Graph showing distributions of correlation coefficients uwlu'w' ______________________________ 17

16. Chart showing secondary motions in channel _____________________________________ ___ 18

17. Graph showing distribution of i; 2 _i£2 ___________ _______________________________________ 20

s\9. /To _ .o\

18. Graph showing variation of dy dz

19. Graph showing balance of first equation of motion aty/d=O.OS______________________ ___ __ __-_ 22

20. Graph showing distribution of sum of turbulent shear farces __________________________ 23

SYMBOLS

C=constant used to describe velocity variation

d=half height of channel, 6 in. (15.2 cm)

D= height of channel

p=piezometric pressure; Pm is pressure of measuring station

#=hydraulic radius, ratio of cross-sectional area to section perimeter

£=time

U, V, W=mean velocity parallel tox, y, z directions, respectively

[7o=maximum value of longitudinal mean velocity in the channel

[7* = "friction" velocity, VWp

u, v, w= instantaneous values of velocity fluctuations parallel to the 

x, y, z directions, respectively

u', v', w'=root mean square of velocity fluctuations u, v, w, respectively

x=longitudinal coordinate along channel, x=0 correspondents to test 

section

y=vertical coordinate, y=0 corresponds to channel floor

z=lateral coordinate, z=0 corresponds to channel side wall

£=rotation iny-z plane

(JL= dynamic viscosity

p=density

TO=average shear stress on boundary

t>=kinematic viscosity, n,lp

ill

THE STRUCTURE OF A TURBULENT FLOW 

IN A CHANNEL OF COMPLEX SHAPE

By H. J. TRACY

ABSTRACT

Measurements of the Reynolds stresses and the mean motion pattern were made in a uniform turbulent motion in a conduit consisting of a large, nearly square section joined by a smaller rectangular 

section. The results indicate that the boundary shearing stress is 

nearly constant over large segments of the boundaries. It is shown 

that the magnitudes of the lateral and the vertical components of 

turbulence are not the same near a boundary and that the component normal to the boundary is smaller than the component parallel 

to the boundary. The difference in the two components in the corner 

regions of the channel produces secondary mean motions in the plane 

of the channel section. The strength of the motion depends upon the 

angle subtended by the corner.

The magnitude of a component of turbulent shearing stress is a 

function of boundary proximity. In the corner regions, the effect of 

the combination of the two walls is to create large shear stress gradients, or forces. The forces are frequently larger than those of the 

pressure. Depending upon the variation of the mean streamwise velocity, the shear force may either oppose or act in the same direction 

as the pressure force.

A principal function of the secondary motions is to transfer 

momentum into the corner regions and, elsewhere, to compensate for 

the excess force due to the shear gradients. In the absence of the 

secondary motions, the fluid must stagnate and separate from the 

boundaries in certain regions and be greatly accelerated in others.

The secondary motions are conventionally described in terms of 

symmetrical rotations in cells bounded by the corner bisectors. The 

measured motion pattern is at variance with this view, unless the 

symmetry is confined to a very local region.

INTRODUCTION

The mean flow characteristics are usually the ultimate concern of those who deal with the movement of 

fluids in open channels and in closed conduits. Often, 

engineering solutions of satisfactory accuracy may be 

obtained from equations of the Manning or Chezy type, 

or from one-dimensional energy and momentum considerations. As the body of experimental information 

grows, however, it becomes increasingly evident that 

the rational interpretation of many of the phenomena 

that are encountered depends upon a better knowledge 

of the internal structure of the flow than is presently 

available. In particular, much of the current work in 

this field is devoted to an understanding of the role of 

the turbulent fluctuations in shaping the mean flow 

field.

Presently, comprehensive data are limited to simple

examples of turbulent flow. These include (1) a twodimensional flow (Laufer, 1951), (2) an axially symmetrical flow (Laufer, 1954), and flow in (3) a square 

conduit (Brundrett and Baines, 1964), (4) a rectangular conduit (Tracy, 1965), and in (5) a trapezoidal conduit (Rodet, 1960); all examples are for a uniform flow 

condition. The present study was carried out in a 

channel having a boundary form more complex in configuration than any of those listed above. Its principal 

added feature is a corner subtending an angle greater 

than 180 degrees.

In section, the test channel consisted of a larger rectangular channel joined by a smaller rectangular 

overbank channel. The selection of this arrangement 

was influenced by the procedures commonly used in 

natural stream channel flow computations. The section 

of natural stream channels is frequently subdivided, 

and each subchannel is subsequently treated as an independent entity for flow computation purposes. The 

subdivisions are usually made at major changes in 

boundary alignment and at points on the boundary at 

which the boundary roughness changes significantly. 

The test data, though restricted in application by the 

somewhat idealized shape and uniformity of the laboratory channel, and by its smallness, should nevertheless yield information relevant to the validity of the 

subdivision procedure.

The objective of the investigation was a study of the 

turbulent and the secondary mean motions generated 

in the larger of the two channels, and an evaluation of 

the effect of the various corners of the section in terms 

of the momentum transport engendered by the motions. To this purpose, measurements were made of the 

mean motions, both primary and secondary, and of the 

relevant statistical quantities of the turbulence over 

the section. These, when substituted into the equations 

of motion, can be expected to show the manner in 

which a given flow adjusts to the configuration of its 

boundaries, even though many of the mechanisms by 

which this is accomplished are yet unknown.

ACKNOWLEDGMENTS

The study was carried out in a test facility located in
STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

the hydraulics laboratory of the Georgia Institute of 

Technology, under a cooperative agreement with the 

Geological Survey, U.S. Department of the Interior. 

D. W. Hicks assisted in the collection of the test data.

PRELIMINARY CONSIDERATIONS

The continuity equation and the Reynolds equation 

in rectangular coordinates for incompressible flow may 

be written as follows:

-

ay

DU 

Dt

Dt dy

__ duv duw ~dx~ +~dy~ +~dz~'

cduv d> 

dx~ ~d>

DW _ dp

Dt dz

dz

duw dvw dw2 T; t-~ l--r 

dx dy dz

(D

(2)

(3)

(4)

where the superscript bar is used to denote a time average; DlDt refers to the total derivative

D _ d rrd d d 

Dt ~ dt + ~dx + ~dy +W ~(te' 

and V 2 to the operator

^2 /)2 Q2 V 2 =

dx2

_ dz2

U, V, and W are the components of mean velocity in 

the x, y, and z directions, respectively, and u, v, and w 

are the correspondingly directed values of the turbulent velocity fluctuations. The mean pressure at a point 

is p, and p and ^ are the density and_dynamic viscosity 

of the fluid. The normal stresses pu2 , pv 2 , and pur^and 

the tangential stresses puv, puw, and pmv are orientated as shown in figure 1.

In a steady, uniform motion, the velocity field is independent of the longitudinal coordinate Gc) and of 

time. The equations therefore become:

dV dW n J, r~z = 0

dy dz

dU dU duv. duw.

(5)

(6)

9^

dw1*. dz2 '

The abbreviated equations describe the motion to be 

considered here and, except possibly for the terms involving the viscosity, may not be further simplified. 

The viscous terms are of relative significance in a fully

puv

dy

pvw

puw

pwr

puv

puw

pu'

dx

FIGURE 1. Components of turbulent stress on an elementary

particle.

developed turbulent flow only in regions very close to a 

boundary.

EQUIPMENT

Test Conduit. The experimental measurements 

were made in the closed-circuit air tunnel shown 

schematically in figure 2. A cross section of the test 

channel is shown in figure 3.

The test channel was constructed of commercial 

aluminum plate and standard aluminum structural 

shapes. The transition section at the entrance to the 

channel and at its end were formed from aluminum 

sheet. The return section and the upstream and 

downstream turn boxes were constructed of interior 

grade plywood. The measurement cross section was located 84 ft (25.6 m) downstream from the end of the 

upstream transition section.

Air was supplied to the tunnel from a centrifugal 

blower powered by a 15 horsepower (11.2 kW) 

variable-speed direct-current motor. Blower speed, 

over a continuous range, was controlled by variations 

in the current supplied to the motor. The variable 

speed feature of the motor and blower was utilized for 

inplace calibration of the hot-wire probes used during 

the investigation.

Hot-wire equipment. A commercially j(&Vailable 

two-channel DISA Electronics1 constant-temperature 

hot-wire instrument was used during the study. Each 

channel consisted of a 55 D 01 anemometer unit, a 55 D 

10 linearizer, and a 55 D 25 auxiliary unit. Outputs 

were measured with a 55 D 30 DC meter and a 55 D 35 

RMS meter. A model 823 Fluke voltmeter, in conjunction with a wave generator, was used as a voltage

'The use of brand names in this report is for identification purposes only and does not 

imply endorsement by the U.S. Geological Survey.
PROCEDURE

FIGURE 2. Schematic of test conduit.

standard for the calibration of both the AC and the DC 

DISA meters.

The hot wires were either 0.0001 in. (0.0025 mm) or 

0.0002 in. (0.0051 mm) diameter platinum wire and 

were used either singly or in x arrays depending upon 

the variable to be measured. Wire length for the 

single-wire probes was 0.030 in. (0.76 mm), and for the 

x arrays, 0.042 in. (1.06 mm). The x array wires subtended a central angle of 90 degrees and were separated by a distance of about 0.020 in. (0.51 mm). The 

wires, in both cases, were supported by fine sewing 

needles projecting from one end of short lengths of 

small diameter brass tubing.

Traversing device. The probes were traversed over 

the channel section by independent vertical and horizontal mechanisms. Vertical movement was registered 

on scales attached to rack-and-pinion mounted plates 

inserted into the two vertical faces of a transparent 

working section on one side of the channel at the 

measurement station. Horizontal movement was 

measured on a sliding scale extending through and 

perpendicular to the plates. The probe support was fastened to the end of the sliding scale.

Zero readings of the traverse scales were found by 

placing the probe tip arbitrarily close to the vertical 

and horizontal walls and by then measuring the distance between the tip and its image in the appropriate 

wall on an ocular micrometer. The traversing scale 

reading corresponding to one-half of the observed distance marked the location of the wall.

PROCEDURE

The test channel was constructed so that its axis of 

symmetry was vertical, with the smaller channel 

above the larger. Cartesian coordinates are used for 

point location and are orientated with x along the

length of the channel, y in the vertical direction, and 2 

horizontally directed. For economy in the required 

number of figures and for ease of comparison, the 

measurements in the upper and in the lower halves of 

the larger channel are often shown together. For this 

purpose only, the origin of coordinates fory and 2 for all 

data in the upper half of the channel is located at the 

upper left corner of the channel. For data in the lower 

half, the origin fory and 2 is the lower right corner. For 

any other purpose, only the lower origin is used, and y 

increases continuously from the lower to the upper 

boundary. For x, the origin is at the measuring station 

and is positive in the downstream direction. The 

root-mean-square values of the turbulent components 

in the x, y, and 2 directions are denoted by u', v', and 

w'. The height of the lower channel is D. The halfheight, d, is used as a reference length to nondimensionalize the location of the points at which data were 

taken.

The channel was sufficiently long (LID greater than 

80) for full development of the flow at its downstream 

end. Velocity measurements were made on each side of 

the axis of symmetry to verify that the motion was 

symmetrical with respect to the axis. All tests were 

made with the maximum channel velocity maintained 

constant at 75 ft/s (22.9 m/s), corresponding to a 

Reynolds number of about 440,000, using D as the appropriate length.

The longitudinal mean pressure was measured from 

piezometer taps located every 3 ft (0.91 m) along the 

channel for the last 45 ft (13.7 m) of length. The results 

are shown in figure 4. The pressures are referenced to 

the pressure at the measuring station, pm.

Mean velocities were measured by small diameter 

total head tubes used in conjunction with wall 

piezometers and by hot wires. The total head tubes
STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

150

 VB" aluminum plate

I

4"x 158" x'/s" 

aluminum channel

Dimensions are in feet and 

inches-1 inch=25.4 mm

_iy2 " x r/2" x'/s"

aluminum angle

FIGURE 3. Channel section.

were either 0.035 in. (0.89 mm) or 0.020 in. (0.51 mm) 

in diameter. The smaller diameter tubes were used for 

measurements in the vicinity of the wall.

The longitudinal mean velocity measurements were 

corrected for the effect of the turbulent velocity fluctuations by the relation

U (corrected) = U (measured)

U2 (measured)

Hot-wire measurements, except those for V and W, 

were obtained by standard techniques described by 

Schubauer and Klebanoff (1946), Hinze (1954), and

100

-10 -20 -30 -40 -50

x/2d

FIGURE 4. Mean pressure along channel.

Rodet (1960). The measurement of V and W follows 

from the geometric relationships between U, V, and W 

and their relative magnitudes and from certain 

characteristics of the hot wire. These are outlined 

below.

The secondary velocities are at right angles to the 

velocity, U, and are small relative to U. Their effect is 

a change in the local direction of the mean flow from 

the direction parallel to U. The angular deviation, in 

radian measure, from the U direction is V/U in the x-y 

plane and W/U in the x-z plane.

The heat loss from a hot wire is a function of the 

magnitude of the component of velocity normal to the 

wire. The heat loss from each of two wires which are 

arranged in an x configuration and which are independently maintained at a constant temperature thus 

depends upon the degree to which each wire is normal 

to the flow, or upon the angle which each subtends to 

the flow. Since the position of each wire relative to the 

other is fixed, a rotation of the array to increase the 

exposure of one of the wires will decrease the exposure 

of the other and thus change the heat-loss characteristics of each wire. Correspondingly, a change in the direction of the flow with the array in an unchanged 

angular position will have the same effect. Hence, a 

given array may be calibrated to sense a change in the 

angular direction of the flow by a procedure in which 

the heat loss difference between the two wires is determined as a function of flow angle and velocity. After 

calibration in this manner, the probe may be inserted 

into the flow and traversed throughout the section, 

maintaining the array in an invariant angular orientation. The difference between the losses can be related 

to V/U for the array placed in the x-y plane and to W/U 

for the array in the x-z plane by the results of the calibration.

On the other hand, the directional sensitivity of the* 

array probe type was a source of considerable incon-
RESULTS

venience during the measurement of the turbulent 

shearing stresses puv, puw, and pvw. The measurements require that each wire of the array subtend the 

same angle with respect to the mean flow direction at 

each measurement point, which necessitated a pointto-point alinement of the angular position of the array 

to correspond to the variations in mean flow direction 

due to the superposed secondary motions. Equal voltages through each of the two matched wires indicated 

the proper setting to place each wire at the same angle 

to the mean flow. The response of the arrays to the 

normal stresses pv 2 andpTP is less critically dependent 

upon probe orientation.

Hot wires placed close to a boundary are affected by 

heat transfer into the boundary. The transfer is variable, depending upon boundary distance. Some compensation for the effect is provided by the constanttemperature feature of the mode of operation, which 

allows the transfer, and therefore the slope of the 

voltage-velocity relationships upon which the turbulence computations depend, to remain somewhat constant at a given wall distance, irrespective of velocity 

variation. Also, the large mean velocity gradients near 

a boundary are also a source of error. When parallel to 

the boundary, the wires of a two-wire array are exposed to velocity differences which may be quite large 

for wires separated by even small distances. When 

normal to a wall, the mean velocity distribution along 

the wires of single and two-wire arrays alike is 

nonuniform.

Adjustments have not been made to the hot-wire 

data to account for the boundary effects. The data 

closer to the wall than about 3 mm must thus be considered less reliable than elsewhere.

RESULTS

DISTRIBUTION OF LONGITUDINAL MEAN VELOCITY

The variation of the mean longitudinal component of 

velocity over the left one-half of the nearly rectangular 

portion of the test channel is shown in figures 5 and 6. 

Each curve represents a traverse horizontally outward 

from the vertical wall. Velocity measurements close to 

the wall are shown in figure 5 and the remainder of the 

traverse to the channel centerline is shown in figure 6. 

The vertical location (yld) of each traverse with respect 

to distance from the upper horizontal boundary or from 

the lower horizontal boundary is shown in the legend. 

Velocities in the lower quadrant of the channel are 

identified by open symbols. A slash mark through the 

symbol designates an observation in the upper quadrant. The half-channel height, d, is 6 in. (152.4 mm).

The distributions shown in figure 5 in the two parts 

of the channel superpose on the basis of yld, except

0 0.050 

Q 0.0333 

0.0167 

0 0.00833 

O 0.00500 

Q 0.00291

O 1.0

D 0.833

A 0.500 

0.333 

0.167

O 0.0833

Open symbol-origin at lower channel corner 

Line symbol-origin at upper channel corner

0.01 0.02 0.03

VALUE OF z/d

FIGURE 5. Distribution of mean velocity near wall.

near y/d = 1.0. The variation, to a good approximation, 

can be represented by the equation

UU-= C (*-)*> 0 C( d }

with n equal to about 7.5. The curves in the two quadrants of the channel diverge farther from the wall 

(fig. 6). The relatively greater number of observations 

near the horizontal boundaries tend to emphasize the 

differences in the flow in the two quadrants, whereas 

the differences are actually minimal outside a limited
STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

1.0

0.90

0.80

0.70

0.60

0.50

0.40

Projection of upper corner-i

y/d 

o i.oo a o.ie? a o.oosss

O 0.833 O 0.0833 <a 0.00500

O 0.667 0 0.0500 o 0.00291

A 0.500 0 0.0333

4 0.333 O 0.0167

Open symbol-origin at lower channel corner 

Line symbol-origin at upper channel corner

I I I

0.2 0.4 0.6 0.8 1.0

VALUE OF z/d 

FIGURE 6. Distribution of mean velocity away from wall.
RESULTS

region (y/d<0.17). The effect of the upper adjoining 

channel on the mean flow is essentially restricted to 

the vicinity of the junction between the two channels. 

The differences outside the limited region are primarily related to differences in the secondary motion 

patterns of the two quadrants.

The secondary motions are a system of mean circulations in the plane of the channel section, which transfer mass from one region to another region of the flow. 

If the velocities in the two regions are not the same, 

momentum is either gained or lost by the migrating 

mass, and a corresponding retarding or accelerating 

force is exerted on the flow during the course of the 

movement.

The decrease in velocity at the outer ends of many of 

the curves of figure 6 is an example of the effect of the 

circulations. The decrease can be explained by an upward motion (and the measurements of the motion to 

be shown later indicate this to be true) near z/d = 1.0 

near the lower horizontal boundary and by a downward motion in a corresponding location in the upper 

half of the channel. In each case, the motion is from a 

region of a lower velocity to one of a higher velocity, 

which retards the flow along its path. The motions, and 

their effects, are discussed in a later section of the report.

Although the larger channel is nearly square in 

shape, the coordinates y and z are not interchangable, 

even in the lower quadrant of the channel which is the 

most removed from the effects of the smaller channel. 

The characteristic decrease in velocity at the outer 

ends of many of the U/Uo versus z/d curves is not typical of the variation of U/Uo versus yld. The velocity 

increases in a continuous manner in a direction vertically outward from each of the horizontal boundaries to 

the effective horizontal centerline of the channel.

The channel section is not suitably proportioned to 

establish a region of two-dimensional flow. Nevertheless, except for some flattening at the outer end where 

dU/dz is zero, the velocity curve for yld = 1.0 approximates a logarithmic distribution of the form

U /-,/-,!

where [7* is the "friction" velocity, Vro/p, and v is the 

kinematic viscosity. The average shear stress at the 

channel boundaries, TO, was computed from the mear\

sured longitudinal pressure gradient, , and from the

ox

geometric properties of the flow section as follows:

OP T, TO = ~ R, 

dx

where R is the ratio of cross-sectional area to section 

perimeter. The constants Cl and C2 are 4.5 and 6.6, 

respectively, which agree only approximately with 

those obtained by others in a two-dimensional flow. 

The differences are functions of channel aspect ratio 

(ratio of width to height) and boundary roughness.

DISTRIBUTION OF NORMAL TURBULENT STRESSES

The distributions of the measured values of u'lU, 

v'/U, and w'/U over the test channel are shown in 

figures 7, 8, and 9. As for the velocity measurements, 

the traverses at a common distance from the upper and 

from the lower horizontal boundaries are designated 

by the same symbol. The traverses in the upper half of 

the channel are distinguished from those in the lower 

half by a slash mark through the symbol. The u'lU 

values were computed from the response of a linearized 

anemometer, and the values of v'/U and w'/U from a 

non-linear signal.

The measurements represent the standard deviations of the velocity fluctuations in the three directions 

divided, in each case, by the mean longitudinal point 

velocity. The ratio is defined as the relative intensity of 

turbulence. The standard deviation is a measure of the 

spread or dispersion of the component about its 

mean the greater the value, the larger the dispersion. 

Although they are a convenient and useful device, the 

measurements do not, obviously, completely describe 

the eddy formations.

The measured distributions show that the turbulence is nearly isotropic in the center of the channel 

and has its smallest values there. All three components become larger in any direction toward a boundary. The longitudinal component increases at a faster 

rate than the others and at the boundary attains 

roughly twice the magnitude of the larger of the remaining two.

A comparison of figures 8 and 9 indicates that v'/U 

increases at a faster rate than does w'/U near a vertical 

boundary and that the reverse is true near a horizontal 

boundary. A boundary acts, apparently, to restrain the 

eddy in the direction normal to the boundary, whereas 

the eddy is not restrained in the direction parallel to 

the boundary.

The results of Laufer (1951, 1954), Brundrett and 

Baines, (1964), and Tracy (1965) confirm these generalizations to a good approximation over a Reynolds 

number range of about 12,000 to nearly 500,000.

DISTRIBUTION OF TURBULENT SHEAR STRESS

The turbulent cross-products uv, uw, and vw, after 

division by U* 2 , are shown in figures 10, 11, and 12, 

respectively. They are referenced to coordinate origins
STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

0.20

0.15 -

30

0.10

0.05

O 1.0 A 0.333 O 0.0167

D 0.833 Q 0.167 Q 0.0083

O 0.667 O 0.0833 0 0.0050

0.500 0 0.0417 O 0.0033

i-Projection of upper corner Open symbol-origin at lower channel corner 

Line symbol-origin at upper channel corner

VALUE OF z/d 

FIGURE 7. Distribution of u 717.
RESULTS

0.09

-Projection of upper corner

y/d

C

0.08

O 1.0 

0 0.833 

O 0.667 

A 0.500 

A 0.333

0 0.167 

t. 0.125 

<? 0.0833 

0 0.0417 

Q 0.0167

Open symbol-origin at lower channel corner 

Line symbol-origin at upper channel corner

0.07

0.06

0.05

0.04

0.03

VALUE OF z/d 

FIGURE 8. Distribution ofv'IU.

at the upper and lower corners, as for figures 5-8. The 

signs of the stresses, where they depend upon the vertical coordinate, are assigned on the basis of the lower 

origin, with y increasing continuously from the lower 

to the upper boundary. 

The quantities puv,puw, and pvw are physically significant as tangential, or shearing, stresses, which result from the turbulent interchanges of mass and 

momentum between neighboring flow regions. The 

concept, and its details, are well documented in hydraulic literature. The exchanges, in all directions, 

lead to the stress system shown in figure 1. The sign of
10 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

Projection of upper corner

o 1.0 a cue/

D 0.833 0 0.125 

O 0.667 O 0.0833 

0.500 0 0.0417 

A 0.333 O 0.0167 

0.0083

Open symbol-origin at lower channel corner 

Line symbol-origin at upper channel corner

0.04

0.03

0.6

VALUE OF z/d 

FIGURE 9. Distribution ofw'/U.
RESULTS 11

-1.6

-1.2

fell)* -0.4

upper corner I

o 1.0

D 0.833 

O 0.667 

A 0.500

0.333 

Zl 0.250

0.167 

O 0.0833 

0 0.0417 

O 0.0167 

0 0.000

At0.0167d7 

U above junction

Lower half of channel

+0.4

+0.8

+1.2

VALUE OF z/d 

FIGURE 10. Distribution ofuv/U* 2 .
12 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

fe*

Projection of upper corner 

-0.4

1.0

VALUE OF z/d 

FIGURE 11. Distribution ofuw/U* 2 .
RESULTS 13

0.2

0.1

-0.1

-0.2

£

a&

V

0

V

I

t?

xy

f

v

Ji

Q

z

c

' y/d

0 1 .0 ^3 0.667 d 0.333 Q 0.0833

D 0.833 A 0.500 V 0.167 o 0.0417

Open symbol-origin at lower channel corner

Line symbol-origin at upper channel corner

3

:

 - /, 

0

a

&»

3?

??

A

A

A

D

0

4d

?

J

9^

^

01

/n

a 

fci

0

4?

a

^. 0

^

1

J^

R^l

0.2 0.4 0.6 0.8 1.0

VALUE OF z/d 

FIGURE 12. Distribution of uw/U* 2 .

the stress is a function of the mean velocity gradient in 

the direction of the transfer. The nature of the transfer 

process is to produce a negative correlation between 

the two members of the ishearing-stress terms for a 

positive gradient that is, a velocity increasing in a 

positive direction outward from a boundary.

The signs of the tangential stresses are related, also, 

to the rotational aspects of the stresses. Thus, each of 

the tangential stresses and as well, the normal 

stresses is opposed by its counterpart on the opposite, 

parallel face of figure 1. The tangential components in 

opposite directions along parallel faces are stress 

couples which tend to rotate the particle. The sign of 

the stress component indicates the angular direction of 

the rotational tendency, which can be shown to be a 

function of the velocity gradient normal to the stress 

couple. Because the stresses are negative for a positive 

gradient, they are shown with negative signs on 

figures 10, 11, and 12.

The surface force due to a stress is the product of the 

stress intensity and the area of the face over which it 

acts. The resultant surface force on the particle of 

figure 1 due to a given stress component is the difference between the oppositely directed forces on the opposite, parallel faces, and is conveniently expressed in 

gradient form. The resultant surface force, per unit of 

volume, of a tangential stress is its gradient normal to 

the stress couple. For a normal stress, the resultant, 

also per unit of volume, is the gradient of the stress in 

the direction of the stress. The resultant of all surface 

forces due to the turbulent exchanges, per unit of volume, in the x, y, and z directions are the turbulence 

terms on the right side of equations (2), (3), and (4). For 

the case of a uniform motion, they are given by the 

right side of equations (6), (7), and (8).

The total intensity of shear is the sum of the turbulent stress and that due to direct viscous action. The 

second is small relative to the first, except at locations 

where the mean velocity gradients are large. In the 

test channel, the gradients are large near the boundaries. Elsewhere, the turbulent stresses are a close approximation to the total stress. By ignoring the decreasing portion of the turbulent stress distributions 

near the boundaries, the distributions of figures 10 and 

11 may be projected to the boundaries to obtain an 

estimate of the value of the total stress at the boundary. The value ofuw/U* 2 projected in this manner to 

the vertical boundary of the larger section is remarkably constant over almost the entire length of the 

boundary (the immediate corner regions are the exception) and equal to 1.13. A corresponding projection 

of uv/U* 2 to the upper and lower boundaries is similarly constant and equal to 1.17 and 1.11, respectively. 

Measurements made in the upper channel (not shown) 

yield boundary values smaller than these, equal to 

0.61 on the horizontal boundary and 0.76 on the vertical wall. An average, obtained by weighing the boundary stresses for the large and small channel against 

peripheral length, is 1.01, which is a favorable comparison between the average boundary stress computed from the hot-wire measurements and from the 

piezometric grade line. The comparison serves as a
14 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

general check on the hot-wire techniques used during 

the study and on the magnitude of the stress terms. It 

is suggested, moreover, that the apparent constancy of 

the boundary shear stress over large segments of the 

test channel periphery may be typical of flow in more 

complex channels. This observation is substantiated by 

the successful use of equations of the Manning or 

Chezy type in the large variety of channels to which 

they have been applied.

Because of the dependency of the sign of the stresses 

upon a mean velocity gradient and because of the analogy with laminar flow, in which the stress is directly 

proportional to the gradient, attempts have been made 

to relate the magnitude of the turbulent stresses to the 

magnitude of the velocity gradients, without complete 

success. An example is shown in figure 13, on_which is 

plotted the variation of uv/U* 2, dU/dy, and uv/u'v' for 

yld = 0.167 in the upper half of the channel. The variables are imperfectly correlated, and over a part of the 

length of the curves, dU/dy actually is decreasing while 

uv is increasing.

In explanation, Schubauer and Klebanoff (1951), 

point out that a fundamental difference between laminar and turbulent flow is the scale of the mixing, that 

for turbulent flow being much larger. Their measurements of correlation distance show that the large-scale 

components of the eddies occupy a large portion of the 

width of the channel. Their simplified model describes 

a transfer of energy from larger to successively smaller 

eddies, the action of viscosity becoming progressively 

greater with diminishing size. Because the largest eddies thus make the greatest contribution to the average shearing stress, they suggest that the stress should 

be expected to depend upon a velocity gradient averaged over a distance, rather than upon the local gradient at a point.

While the foregoing is reasonable, the selection of an 

averaging distance to reconcile the velocity gradients 

with observed values of the turbulent stress is not obvious. According to figure 13, the distance is evidently 

variable, depending upon location. It is also frequently

r\ -T T

zero. In figure 11 and figure 6, for example, uw and -*-

are perfectly correlated in the numerous instances 

where both are equal to zero (in addition to the obvious 

case at zld = 1.0) at identical locations. It thus appears 

that a simple method of predicting the magnitude of 

the turbulent stress from the mean velocity variation 

is not at hand.

The variation of uw and uv over the section creates 

force components that frequently act in the same direction as the pressure force that is, a "negative" shearing force. Reversals in the direction of the forces occur

duv duw

at changes in the sign of the gradients "^77 and dz

Figure 11 shows many such locations for duw 

dz for

duv 0.7<z/d<0.9. The reversals in the sign of ^ (fig. 10)

_cty

are less evident, because the variation of uv with y is 

not shown. The gradient does change sign, however, at 

many locations in the vicinity of the 260-degree corner. 

An inspection of figures 10 and 11 also shows that 

the stress gradients become large in all corner regions.

The combination { T - ^ ) of stress gradients produces large positive forces (in opposition to the pressure force) in the 90-degree corner regions and large 

negative forces in the 270-degree corner regions, and 

at locations near the vertical axis of symmetry of the 

channel. A numerical evaluation of the forces in the 

upper half of the channel is given in the last section of 

the report.

The shear-stress variation at the junction of the 

larger and the smaller channel is pertinent to the 

channel subdivision procedures commonly used in engineering practice. The procedure does not consider the 

junction to be a shear surface and hence omits its 

length from the "wetted" perimeter of each subsection, 

which is to say that the shear-stress component normal 

to the junction (puv) is assumed to be zero everywhere 

along the junction.

A reference to figure 10 indicates that puv is not zero 

at the junction; in fact, it is large over the portion adjacent to the apex of the corner. Figure 10 also indicates 

that the stress decreases rapidly with distance above 

the junction, however, and at yld=0.0167 above the 

junction it is almost insignificant. The large stress values at the corner are probably due to locally large valf\TT

ues of at the corner. If the locally large values of the

dy

stress are discounted, the zero stress assumption is 

probably more valid than one in which the junction is 

treated as a solid boundary.

The values of vw/U* 2 of figure 12 depend upon the 

separation of a small difference between two large signals, each of which is characterized by some degree of 

uncertainty. The individual measurements must 

therefore be accepted with reserve. The measurements, 

together, probably can serve to indicate the order of 

magnitude of the stress with more confidence, and the 

figure is shown primarily for this purpose. No average 

lines have been drawn through the points.

The Uw component of stress is a momentum exchange which depends upon the intensity of its turbulent 

components and upon some function of the gradients 

dV/dz and dW/dy. The gradients may be appreciable, even 

though V and W are small relative to U.

The correlation coefficients uv/u'v' and uw/u'w' are 

shown in figures 14 and 15. Maximum values of the
RESULTS 15

100

1.0

VALUE OF z/d

FIGURE 13. Comparison of velocity gradient, turbulent shear-stress, and correlation coefficient at yld = 0.167 in

upper half of channel.

coefficients are dependent upon Reynolds number 

(Laufer, 1951). The maximum values in figures 14 and 

15 (about 0.4) are close to those obtained by Laufer at a 

similar Reynolds number in a one-dimensional flow, 

except at the 270-degree corner, where they are greater in the case of uv/u'v'.

SECONDARY MOTION PATTERN

The secondary motions in the section of the nearly 

square lower channel are shown in figure 16. The vectors of the figure are the combination of separate 

evaluations of V and W. The untipped end of each vector represents the point at which the measurements 

were made, and the length of the vector is proportional 

to the velocity. A reference vector equal to 0.0133 Uo 

(1 ft/s, 0.305 m/s) is shown on the figure.

The secondary velocity components are difficult to 

measure accurately because of their small magnitude. 

On the other hand, the measurements are relatively 

easy to obtain, and so each was repeated several times 

to minimize error sources, such as voltage drift.

The void space in the center of the figure is an 

acknowledgment of a lower limit to the velocity signal 

separable from instrument noise and normal voltage 

drift. Although they probably are not zero, the motions 

in the region are small enough to be inconsequential. Elsewhere, the components were compared on 

the basis of continuity, using the relationship 

8V/dy--dW/dz. From this standpoint, the measured 

motion pattern was reasonable everywhere except at 

the upper 90-degree corner, where the motions are 

small, and probably wall influenced, resulting in poor 

definition in this area.

The secondary motions consist of one large cell in the 

upper and in the lower part of the channel and a smaller cell alongside each of the larger eddies. The adjacent 

cells are counterrotating. The magnitude of the motions in the lower half of the channel are comparable to 

those of an earlier study (Tracy, 1965) in a narrow, 

rectangular channel at a Reynolds number of 96,000 

and to those in a square channel at a Reynolds number 

of 83,000 (Brundrett and Baines, 1964). The Reynolds 

number of the present study is about 450,000.

From this data, and the fact that the motions in the 

upper large cell near the 270-degree corner are larger 

than those in the lower cell by a factor of almost three, 

it appears that the boundary configuration in the vicinity of the motion is a more decisive influence upon the 

strength of the motion than is Reynolds number. It 

may be conjectured, also, that the effects of a given 

boundary feature are not entirely local. The differences 

between the secondary motions in the upper and lower 

90-degree corner regions are probably an indication 

that the strong motions in the vicinity of the 270- 

degree corner affect those of the upper 90-degree 

corner.

In any event, the measured pattern is at variance 

with the conventional view held of the motions. They 

are usually described in terms of symmetrical rotations in cells bounded by the corner bisectors, the rotation having the opposite sense in adjacent cells. The 

measured motions are symmetrical only in a local region about the bisector of the lower corner, and because 

of the poor definition at the corresponding upper 

corner, they may also be locally symmetrical there. 

The motions at the 270-degree corner are divided into
16 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

y/d

O 1.0 Q 0.167 

D 0.833 O 0.0833 

O 0.667 0 0.0417 

A 0.500 Q 0.0167 

A 0.333

-0.5

1.0

VALUE OF z/d 

FIGURE 14. Correlation coefficient uv/u'v'.

two distinct circulations; interestingly, they are directed toward the corner along the boundaries and 

outward elsewhere, in contrast to the motions at the 

interior corners. The bisector, however, only approximately divides the cells.

The peculiarities in the distributions of longitudinal 

velocity of figure 6 are consistent with the measured 

secondary motion pattern.

The exchange of fluid between the two parts of the 

channel of their junction is not zero. Fluid is transported out of the larger channel at the center, and an 

equal amount is returned near the boundary.

ORIGIN OF THE SECONDARY MOTIONS

The secondary motions are believed to have their 

origin in the characteristic behavior of the normal 

stress components 17* and H^ in the corner regions of 

noncircular conduits. The typical unbalance of the 

components near a boundary has been outlined in an 

earlier section of the report.

A connection between the normal stresses and the 

secondary motions can be established by a minipulation and combination of equations (7) and (8), which

describe the forces and the resultant accelerations in 

the plane of the secondary motions. The pressure terms 

may be eliminated by the differentiation of each equation with respect to the direction of action of the other 

and by the subsequent subtraction of one from the 

.other. A further use of equation (5) produces the following result:

_ a2

dy'- pvw +dydz

' + ), (9) ' dy 2 dz2

where£ = -J -jand is a measure of the rotation of a

* dy \dz '

fluid particle about an axis normal to the y z plane.

Equation (9) may be obtained in an alternative fashion. Brundrett and Baines (1964) suggest that it can be 

derived directly by equating the rate of change of angular momentum of a volume of fluid the terms on 

the left side to the moments of the surface forces on 

the volume, which are the terms on the right side.

A laminar, uniform state of motion is described by
ORIGIN OF THE SECONDARY MOTIONS 17

0.4

0.3

0.2

0.1

-0.1

-0.2

-0.1

-0.3

y/d

O 1.0 o 0.250

D 0.833 Q 0.167

O 0.667 O 0.0833

A 0.500 0 0.0417

A 0.333 Q 0.0167

VALUE OF z/d 

FIGURE 15. Correlation coefficient uw/u'w'.

the accelerative terms on the left of equation (9) and by 

the last term on the right. Using the laminar condition 

as a starting point, Maslen (1958) demonstrated mathematically that if the secondary components are zero 

anywhere in a viscous motion, they are zero 

everywhere. Because they are zero at a boundary, the 

transverse velocities must vanish in a uniform, laminar flow. The terms involving the viscosity are thus 

counterproductive to the generation of the secondary 

motions.

The relative importance of the terms containing 

vw the first and second terms on the right of equation 

(9) are difficult to assess experimentally because of

the small magnitude of vw and the probable measurement error. For the purpose of eliminating vw as a 

possible cause of the secondary motions, it probably 

suffices to remark that the stress depends upon the 

existence of V and W (or, more strictly, upon their gradients) and not the reverse. It is to be noted, moreover, 

that the vw terms of equation (9) are the turbulent 

counterparts of the viscous terms and may reasonably 

be expected to behave in a parallel manner that is, to 

retard the rotation of the fluid particles.

To the extent that the latter conclusion is justified, 

the remaining term on the right side of equation (9), 

which is descriptive of the action due to the forces fJ2
18 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

EXPLANATION

Length of vector equal to 0.01 33 U0 

(1 ft/s or 0.305 m/s )

0.

t '

, ,

* '

, v I 1

,

- \ \

0.4 -

' / 1

0.6 -*

0.8 -

u. 

O 

uj 1.0'

D

0.8

0.6

0.4

0.2

0

0

'<*.,,,,-.<* t

| ^ ' ' f

x x ^ » - - * x x ' f

y / y / * f

/X '-- -~ ^ , f

x ' ' ' * _^__-.

0.2 0.4 0.6 0.8 1.0

VALUE OF z/d 

FIGURE 16. Secondary motions in channel.
ROLE OF THE SECONDARY MOTIONS 19

and w2 , is evidently responsible for the existence of 

nonzero values of the accelerative terms on the left and 

thus for nonzero values of V and W. The magnitude of 

U2 and w2 is a function of location, Reynolds number, 

and boundary configuration and roughness. It is remarked that the structure of _ dydz

__

(v 2 -w2 ) (or of its

equivalent in cylindrical coordinates) precludes a nonzero value of the term in an axially symmetric or twodimensional flow. The observation that secondary motions are not present in these types of flow is support 

for the conclusion just drawn.

The variation of v2 -w2 over the section of the large 

channel is shown in figure 17. The measurements of u 2 

aty/d=Q.QQ16 in both the upper and the lower half of 

the channel appear erroneous, and the probable value 

of v2 w2 at these locations is indicated as a dashed 

line. The principle function of figure 17 is to show that 

the velocity fluctuations in the plane of the channel 

section v and w vary in a consistent manner near a 

boundary, with the component of turbulence normal to 

the wall always being smaller than the component 

which is parallel to the wall, and that the typical variation will always produce a nonzero value of

d2 r^L ^-

 (v* w*) in a corner region. dydz

The result of the combination of boundary effects on

a2 - 

the variation of dydz (v2 w2) is shown in figure 18 for

the upper one-half of the larger channel. The correlation of the magnitude of the derivative with the magnitude of the secondary motions is good (see fig. 16).

ROLE OF THE SECONDARY MOTIONS

The turbulent shear gradients, and thus the turbulent shear forces, become very large in the corner regions of the channel. Without the secondary motions, 

which compensate for the force excess, the motion pattern in the channel would be greatly different from 

that which actually exists. The role of the motions is 

thus an essential one. It is described by equation (6) 

which expresses the longitudinal force balance in the 

channel:

dy dz dy

dy dz

The terms on the left, excluding p, are the longitudinal acceleration of a volume of fluid. The terms on the 

right, in order, are the forces due to pressure, viscosity,

and turbulent shear, all per unit of volume. The negative pressure gradient indicates a decreasing pressure 

in the downstream direction. The negative turbulent 

shear terms are in recognition of the forementioned 

fact that the turbulence components normally correlate in the negative sense.

Despite their accelerative significance, and like the 

turbulent products, the terms on the left side of equation (6) can also be interpreted as forces due to momenr\ -T T

turn transport. Consider, for example, the term p V -^ ,

in which V is a volume rate of vertical flow per unit of 

area. As the volume is carried from one region to 

another the U velocity generally being different in 

the two regions it undergoes a momentum change per

r\T T

unit of volume equal to p dy in the distance dy. In

dy

contrast to a turbulent fluctuation, which is assumed 

to retain its U velocity throughout its migration, the 

volume transported by V undergoes a continuous 

change in U. The integral of the product of V and

£\T T

p dy is thus the rate of transport of momentum per dy

unit of area as a result of the secondary motion V and 

is equivalent to the stress intensity due to the momentum exchange. The resultant surface force per unit of 

volume is the differential, with respect toy, of the inteftTT gral, which is pV and which is the first term of 

dy

equation (6).

The viscous forces are small relative to the other 

forces except near a boundary where the velocity derivatives upon which they depend are large. At a 

boundary distance equal to 0.07d, the viscous forces 

are typically less than 3 percent of the pressure force. 

The viscous forces have been omitted from consideration at greater distances.

Thus, not too close to the boundaries, equation (6) is 

a balance between the momentum transfer due to turbulence, the transfer due to the secondary motions, and 

the pressure force. Because the pressure force is 

everywhere constant over the section, a variation in 

one of the transfer forces must be accompanied by an 

opposite variation in the other.

The forces have been evaluated from experimental 

data at yld =0.0833 in the upper and in the lower parts 

of the section and are shown in figure 19. To facilitate 

comparison, the negative of the secondary motion 

trasnport forces have been plotted in the figure. (To be 

noted, also, is the division of each term of equation (6) 

by. the mass density, p, so that the force significance of 

each is lost. The result, however, is proportional to 

force.)
20 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

y/d

O 1.00 

D 0.833 

A 0.500 

0.333 

0.167 

O 0.083 

Q 0.0417 

0.0167

-1.0

0.1

VALUE OF z/d 

FIGURE 17. Distribution of ~v*-~w*.
ROLE OF THE SECONDARY MOTIONS 21

400

-400

-800

-1200

-1600

-1700

y/d

O 1.0 

O 0.667 

A 0.500 

0.333 

A 0.250 

D 0.167 

O 0.0833 

ft 0.0417

FIGURE 18. Variation of

VALUE OF z/d

d2 ~r
22 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

A Near upper boundary

0 Near lower boundary

y/d=0.0833

0.9 1.0 1.1

VALUE OF z/d 

FIGURE 19. Balance of first equation of motion aty/d=0.0833.

The force variation shown in figure 19 is a function 

of the details of the motion pattern in the vicinity of 

the variation. The motion pattern is described primarily by boundary geometry, with Reynolds number of 

secondary importance. The extremes of variation of 

figure 19 occur near points of change in boundary 

alignment, or corners. To the extent that the corners 

are isolated from the direct influence of other corners, 

the motions and the resulting forces near each are 

"mostly" representative of the corner, although they 

are affected to some degree by every element of the 

boundary, however distant.

The 90-degree corners, for example, are characterized by large positive values of turbulent shear 

force, which are opposed by the pressure and the secondary motion transport forces. The large shear forces 

are the result of the typical variation of the turbulent 

shear stress components in such regions, illustrated by 

the lowermost of the curves of both parts of figure 11, 

which detail, for example, the variation of uw at small 

values of yld. Near the corner, the magnitude of the 

component of shear stress at a point is a function of the 

relative proximity of the two walls to the point, the

closer wall having the greater effect. The stress at a 

point close to one wall is not greatly affected by the 

opposite wall until the point is located equally as close 

to the latter. As the opposite wall becomes effective, 

the stress changes rapidly. To be noted in figure 11 are 

the large stress gradients near the vertical wall. Similar gradients of uv in the vertical direction account for 

the large shear force values in figure 19. The secondary 

motions, and the forces therefrom, are the consequence 

of the difference in v^ and w2 at the boundaries and the 

reversal in the sign of the difference at the corner.

A "typical" force variation near the 270-degree 

corner is more difficult to describe, because of the influence of the other boundary elements on the more 

complex motion pattern. The effect of such an isolated 

corner may be approximated, however, as the difference between corresponding force distributions in the 

upper and in the lower part of the section at the same 

value ofy/d. This assumes that the force distributions 

in the upper part of the section would be identical to 

those in the lower in the event that the junction between the two channel sections the upper, smaller 

rectangular section and the lower, nearly square
CONCLUSIONS 23

larger section were formed of a solid, smooth material; a further assumption is that the distributions in 

the lower half of the larger channel would not be 

greatly affected by such a procedure. The first assumption is completely reasonable; the second is reasonable 

enough to indicate without serious error the trends and 

magnitudes of the forces for the replaced-boundary 

condition.

When compared in this manner, the trends shown by 

the distributions of figure 19 in the upper part of the 

section are largely unchanged. The effect of the corner 

is thus to cause a large change in the magnitude of 

both sets of the variable forces and to reverse the direction of each from one to the other side of the corner. At 

the left, the shear forces are large and are opposed by 

the pressure force. The momentum transport forces 

due to the secondary motions act in the direction of the 

pressure. Both forces are larger than the pressure 

force. At the right side, the roles are reversed: the 

shear forces now act in the same direction as the pressure, and the secondary motion force opposes each. 

Farther to the right, the forces again reverse.

At increased values ofy/d, the corner influences lessen, and the forces decrease. At y/d = l.Q, a twodimensional flow pattern is approximated, and the 

secondary motions are small or nonexistent. The 

two-dimensional counterpart of figure 19 is a straight

horizontal line representing the value of - ~H±*L anc[ js

dz

located a constant distance equal to -£- above the ^ p dx

line corresponding to a zero force.

The force variation for y/d>0.0833 in the upper half 

of the channel is illustrated in figure 20. The sum of 

the turbulent exchange forces only is shown in the

figure. The sum of V-r and W differ from corres- dy dz

ponding values of these forces by the constant value of 

 ' -3 1, which is equal to 8.6 m/s/s.

Figures 19 and 20 graphically depict the role of the 

secondary motions that of an agency to equalize the 

large turbulent exchange forces in the corner regions. 

The turbulent forces can be much larger than the pressure force. If the motions did not exist, the fluid must 

stagnate and separate from the boundaries in the 90- 

degree corners and would be greatly accelerated in the 

vicinity of the 270-degree corner.

CONCLUSIONS

The experimental work reported herein is devoted to 

the internal details of a fully developed turbulent motion in a conduit of complex boundary form. The principal feature of the boundary configuration is a corner 

subtending an angle greater than 180 degrees. The results are principally concerned with the production 

and effect of the secondary motions in the conduit.

y/d

O 1.00 

D 0.833 

O 0.667 

A 0.500 

0.333 

0.250 

0.167

-20

VALUE OF z/d 

FIGURE 20. Distribution of sum of turbulent shear forces.
24 STRUCTURE OF A TURBULENT FLOW IN A CHANNEL OF COMPLEX SHAPE

The following are the major results:

1. The variation of the turbulence components v and w 

in the corner regions of the conduit is the most 

probable cause of the secondary motions.

2. The magnitude of the secondary motions produced 

in a given corner region is a function of the angle 

subtended by the corner. The function may not be 

defined with the limited data at hand.

3. The secondary motions are an effective mechanism 

for the transfer of momentum in the plane of the 

channel cross section. The forces due to the 

momentum transfer are of the same order of 

magnitude as those due to turbulent shear and to 

pressure.

4. The combination of boundary effects at a corner 

produce turbulent shearing forces which are frequently larger than the force due to pressure. The 

shearing forces may oppose the pressure force or 

act in the direction of the pressure force, depending upon the corner configuration.

5. The secondary motions transfer an appropriately 

directed momentum into the corner regions to 

compensate for the excess of shearing force and, 

thus, to prevent flow separation or acceleration of 

the flow in the regions.

6. The secondary motions also, apparently, act to 

equalize the boundary shear stress over considerable portions of the boundary, accounting in large 

measure for the success of equations of the Manning or Chezy type applied to channels of complex 

form.

7. The transfer of fluid across the line separating the 

larger from the smaller channel is small. The 

shear stress at the line is also small, indicating 

that the channel subdivision practices commonly 

used are probably applicable, at least to the 

channel of this study.

REFERENCES

Brundrett, E., and Baines, W. D., 1964, The production and diffusion 

of vorticity in duct flow: Jour. Fluid Mechanics, v. 19, part 3, 

p.375-392.

Hinze, J. O., 1954, Turbulence: New York, McGraw-Hill, p. 73-119.

Laufer, John, 1951, Investigation of turbulent flow in a twodimensional channel: Natl. Advisory Comm. Aeronautics report 

1053, Washington, B.C.

 1954, The structure of turbulence in fully developed pipe flow: 

Natl. Advisory Comm. Aeronautics report 1174, Washington, 

B.C.

Maslen, S. H., 1958, Transverse velocities in fully developed flows: 

Appl. Mathematics Quart., v. 16, p. 173-175.

Rodet, E., 1960, Etude de 1'ecoulement d'un fluide dans un tunnel 

prismatique de section trapezoidale: Ministere L'Air scientifiques et techniques Pub. no. 369.

Schubauer, G. B., and Klebanoff, P. S., 1946, Theory and application 

of hot-wire instruments in the investigation of turbulent boundary layers: Natl. Advisory Comm. Aeronautics report W-86, 

Washington, B.C.

 1951, Investigation of separation of the turbulent boundary

layer: Natl. Advisory Comm. Aeronautics report 1030,

Washington, B.C. 

Tracy, H. J., 1965, Turbulent flow in a three-dimensional channel:

Am. Soc. Civil Engineers Proc., Jour. Hydraulics Biv., paper

4530, v. 91, no. Hy 6, p. 9-35.