Exploratory Laboratory Study of 

Lateral Turbulent Diffusion at the 

Surface of an Alluvial Channel

x\\\

GEOLOGICAL SURVEY 

CIRCULAR 484
Exploratory Laboratory Study of

Lateral Turbulent Diffusion at the

Surface of an Alluvial Channel

By W. W. Sayre and A. R. Chamberlain

GEOLOGICAL SURVEY CIRCULAR 484

Washington 7964
United States Department of the Interior

STEWART L. UDALL, SECRETARY

Geological Survey

THOMAS B. NOLAN, DIRECTOR

Frw» on application to the U.S. G«oJogkaf Survey, Washington, D. C 20242
CONTENTS

Page Page

Selected list of symbols___________ IV Description of experiment,___________ 6

Abstract ___________________________ 1 Results ____________________________ 8

Introduction.___________________ 1 Application to practical problems ___ 17

Turbulent diffusion in open channels.__ 2 Conclusions ________________________ 17

Theory___________________________ 2 References _________________________ 17

Turbulence measurements _________ 4

Diffusion at the water surface ______ 5

Alluvial channels...___________ 5

ILLUSTRATIONS

Page 

Figure 1. Variance of the lateral distribution, °"2 2 (t), as a function of relative dispersion

time, t/Lt ________________________________________ ___________________ 3

2. Schematic diagram of the flume.___________________________________________ 6

3. Section of compartmented-sieve particle collector ___________________________ 7

4. Cumulative distribution plots for x = 300 centimeters ________________________ 9

5. Variance of lateral distribution as a function of distance downstream from

source ________________________________________________________________ 12

6. Flume bed after the experiment.___________________________________________ 13

7. Intensity and scale of turbulence as determined from limits of dispersion data__ 14

8. Theoretical solution of diffusion equation with

RU,' (a) =e LHt ------- - 15

9. Standard deviation of lateral distribution as a function of dispersion time____ _ 15

10. Lateral diffusion coefficient as a function of distance from source __________ _ 16

TABLE

Page 

Table 1. Lateral diffusion data and computed parameters ______________________________ 10

III
SELECTED LIST OF SYMBOLS

Symbols Definition Units 

K'~ Coefficient of lateral turbulent diffusion.. ______________________________ cm2 / sec

L[ Lagrangian integral time scale of turbulence in the lateral direction sec

RW'(CL) Lagrangian correlation coefficient which correlates the instantaneous 

lateral turbulent velocity components, w' , of a fluid particle, at the 

times t and t + a,

t Dispersion time _____________________________________________________ sec

A t Time interval between release of particles _____________________________ sec

U Mean velocity of flow in the longitudinal direction _______________________ cm/ sec

U s Mean velocity of flow in the longitudinal direction at the water surface ____ cm/ sec

x Longitudinal distance downstream from the source ______________________ cm

XQ The x intercept of the extension of the straight part of the az 2 (x) curve ______ cm

w ' 2 The mean of the squared instantaneous lateral turbulent velocity cm2 / sec 2 

components, w'

z Lateral distance from the source which is located at x = 0, z = 0 __________ cm

a Delay time associated with the Lagrangian correlation coefficient Rw > (o)___ sec

&c Lateral deviation of the center of concentration of a group of dispersed cm 

particles f rom z = 0.

0,2 Variance of the lateral distribution of a group of fluid particles which have cm2 

passed through the point x - 0, z = 0. In the experiments, the variance of 

the lateral particle -concentration -distribution curve was taken as an 

estimate of °~z 2 .

cr_2(tj o-2 2 as a function of t _________________________________________________ cm2

o, 2 (xj crz 2 as a function of x _________________________________________________ cm2

crz 2 The mean variance of concentration distribution curves obtained for cm2 

several values of At, other conditions remaining the same.

o-. The mean standard deviation of concentration distribution curves obtained cm 

for several values of At, other conditions remaining the same.

IV
Exploratory Laboratory Study of Lateral Turbulent Diffusion 

at the Surface of an Alluvial Channel

By W. W. Sayre and A. R. Chamberlain

ABSTRACT

In natural streams turbulent diffusion is one of the principal mechanisms by which liquid and suspended-particulate 

coDtaminants are dispersed in the flow. A knowledge of 

turbulence characteristics is therefore essential in predicting 

the dispersal rates of contaminants in streams.

In this study the theory of diffusion by continuous movements for homogeneous turbulence is applied to lateral diffusion at the surface of an open channel in which there is 

uniform flow. An exploratory laboratory investigation was 

conducted in which the lateral dispersion at the water surface 

of a sand-Led flume was studied by measuring the lateral 

spread from a point source of small floating polyethylene 

particles. The experiment was restricted to a single set of 

flow and channel geometry conditions.

The results of the study indicate that with certain restrictions lateral dispersion in alluvial channels may be successfully described by the theory of diffusion by continuous 

movements. The experiment demonstrates a means for evaluating the lateral diffusion coefficient and also methods for 

quantitatively estimating fundamental turbulence properties, 

such as the intensity and the Lagrangian integral scale of 

turbulence in an alluvial channel.

The experimental results show that with increasing distance from the source the coefficient of lateral turbulent 

diffusion increases initially but tends toward a constant 

limiting value. This result is in accordance with turbulent 

diffusion theory. Indications are that the distance downstream from the source required for the diffusion coefficient 

to reach its limiting value is actually very small when compared to the length scale of most diffusion phenomena in 

natural streams which are of practical interest.

INTRODUCTION

One of the many important functions of 

natural streams and rivers is to transport 

industrial and domestic wastes. The increasing quantities of wastes which must be 

disposed of and the increasing demand on 

available water resources for many competing uses obviously necessitate the imposition of strict controls on the rates at which 

pollutants are released into streams. The 

establishment of realistic control measures 

requires that adequate criteria be available 

for predicting the rates at which pollutants 

are dispersed by natural mechanisms once

they have been discharged into streams. At 

present adequate criteria are not available. 

The need for more knowledge concerning 

dispersion in open-channel flows is especially acute in relation to the disposal of 

low-level liquid radioactive wastes because 

tolerance levels for radioactive contaminants are generally several orders of magnitude lower than tolerance levels for most 

other contaminants. Furthermore, many radioactive contaminants cannot be removed 

from the water by conventional water-treatment practices.

The U.S. Geological Survey, in cooperation 

with the Division of Reactor Development of 

the Atomic Energy Commission, is engaged 

in several research projects which are concerned with the feasibility of using the natural environment for the disposal of certain 

kinds of radioactive wastes. One area of investigation is concerned with developing criteria for predicting the rate of dispersal of 

low-level liquid radioactive wastes in natural waterways.

In streams and rivers the initial dispersion of liquid and suspended-particulate contaminants is accomplished principally by 

turbulence and convection. Turbulent diffusion occurs in all turbulent flows. Common 

examples of convective dispersion in natural 

streams are longitudinal dispersion caused 

by mean velocity gradients and the associated mass transfer and dispersion induced 

by flow around bends.

This paper is restricted to a discussion of 

turbulent diffusion in a broad open channel 

under conditions of steady, uniform flow. 

Experimental results are presented for one 

set of flow conditions in a laboratory-scale
EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

alluvial channel. The experiment was exploratory and conducted for the purpose of 

determining whether certain experimental 

techniques which have been successfully applied to the study of turbulence and diffusion 

in rigid-boundary open channels could be 

similarly applied to alluvial-channel studies.

TURBULENT DIFFUSION IN OPEN CHANNELS

According to Hinze (1959), satisfactory solutions to transport problems in turbulent 

flows depend largely on the adequate determination of Lagrangian statistical functions 

describing turbulent motion. It follows that 

turbulent diffusion can be more effectively 

investigated if the Lagrangian turbulence 

characteristics are also investigated.

In turbulent flow, the relative positions of 

a group of neighboring fluid particles change 

in such a way that the particles tend to 

spread out or disperse with the passage of 

time. Dispersion which is due to the random 

turbulent motion of the fluid particles, as 

opposed to molecular diffusion or convective 

dispersion, is called turbulent diffusion. The 

dispersion of dyes or small suspended particles which have inertial properties identical to those of the fluid particles are visible 

manifestations of turbulent diffusion.

The rate of dispersion due to turbulent diffusion depends on the turbulent motions of the 

fluid particles. In a wide straight channel, 

with uniform flow, the turbulence characteristics depend on the geometrical characteristics of the channel (that is, cross-sectional 

dimensions, boundary roughness, and slope) 

and the flow discharge.

Diffusion theories which are based on the 

idealized concept of homogeneous turbulence 

may be applied to some extent in open channels because in channels such as described 

in the preceding paragraph, the turbulence is 

approximately homogeneous in planes which 

are parallel to the water surface, except 

close to the channel boundaries. Homogeneity means here that the statistical properties 

of the turbulence are independent of position 

in a given plane. Owing to the variation of 

mean shear stress with depth, however, the 

statistical properties of the turbulence do 

vary in the direction normal to the planes.

THEORY

The following discussion is based on Taylor's (1921) theory of diffusion by continuous 

movements. In the discussion the assumption is made that the turbulence field is at 

rest with respect to its coordinate system 

and that the statistical properties of the turbulence do not vary with time. Although the 

discussion is restricted to a consideration 

of turbulent diffusion in the lateral (or z ) direction, the concepts also apply to turbulent 

diffusion in the longitudinal (or x ) direction.

The state of dispersion of a group of fluid 

particles at any given time may be described 

.statistically as the variance, 0-2 ft), of their 

distribution about the center of concentration. In a homogeneous turbulence field the 

fundamental equation of turbulent diffusion 

derived by Taylor (1921) and expressed in 

terms of lateral diffusion in the 2 direction 

is:

0-2

 t a 

= 2w '2 I f R. da (1)

Kamp de Feriet (1939) transformed equation 

1 to a simpler form through integration by 

parts and obtained

__ t 

a, 2 ft) = 2u;' 2 / (t-n), R w > (a; da, (la)

0

in which

°2 2 ft ) = the variance at time t of the 

lateral distribution of a group of 

fluid particles which were located 

at 2 = o at time t = 0;

w' 2 = the mean of the squared instantaneous turbulent velocity components in the 2 direction;

t = the dispersion time;

w'(t)w'(t + a)

Also

; ,which is the

Lagrangian correlation coefficient 

that correlates values of w' for a 

fluid particle at the times t and t

+ 0. ;

T

Lt ~ i R w '(a)dais the Lagrangian integral time scale of turbulence.
TURBULENT DIFFUSION IN OPEN CHANNELS

Exact solutions of equation 1 are possible 

only when the Lagrangian correlation function, Rw'(a), is known; usually the form of 

Rw > (a.) is not known. However, even when the 

form of Rw >(a) is not known, there are some 

approximate solutions of equation 1 which 

apply over certain ranges of dispersion time. 

For example, when the relative dispersion 

time, t/Li, is very small, the correlation coefficient, Rw ' fa) t is nearly one, and equation 

1 reduces to

o-2t"3 T^tz. (2)

When t/Lf is large, equation. 1 reduces to

___ 00

= 2w' z Lt t >'2 J aRw '(a) da. (3)

In a turbulent flow with a given w ' 2 and a 

finite Lagrangian correlation function, the 

the last term in equation 3 is a constant. 

When t/Lt is very large, this constant is 

small relative to the other terms in the 

equation, and equation 1 reduces further to

cr2 2 (t) = 2w' z Lt t. (4)

At intermediate relative dispersion times, 

the variance, Crz 2 f0, depends on the nature 

of the Lagrangian correlation function. The 

general functional form of equation 1 is indicated on figure 1; there may be some variation, depending on the form of the Lagrangian correlation function.

Batchelor (1949) has shown that turbulent 

diffusion can be described by a simple Fickian diffusion equation provided that the onedimensional concentration distribution of a 

contaminant released from a point source 

conforms to the normal distribution law. 

According to Hinze (1959) the experimental 

evidence of many investigators shows that 

for diffusion in .homogeneous turbulence the 

concentration distribution is normal for both 

long and short dispersion times. It is reasonable to assume that this result applies 

for intermediate dispersion times also.

The one-dimensional Fickian 

equation for lateral dispersion is

scl

St-l K2

diffusion

(5)

in which

Kz = 'is the coefficient of lateral turbulent diffusion, and

C « C(z,t) is the amount of contaminant 

per unit, as a function of 2 and t.

Figure 1. Variance of the lateral distribution, ^zz (t), as a funcof relative dispersion time, t/Lf.

When Kz is assumed to be a constant, the 

solution of equation 5 for the case of a quantity of contaminant, q , released from an instantaneous point source at time t = 0 into a 

homogeneous turbulence field is,

--L. z 2 

C-(z.t) = - e 2 (2Kz t)'

which is a normal distribution function with 

variance

o-2 2 =2K2 t.

Equating this variance to equation 4 results in

This relationship which assumes that the 

diffusion coefficient is a constant and states 

that it is equal to the product of the Lagrangian velocity variance and the Lagrangian integral time scale of turbulence 

has been found by many investigators to be 

approximately true when t/Lt » 1. For short 

and intermediate dispersion times the diffusion coefficient, in keeping with the theory
EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

of diffusion by continuous movements, is 

usually defined as

"* 2 dt

and is in general a function of dispersion 

time.

Solutions of equation 5 for this and various other initial and boundary conditions are 

available in the literature (for example, 

Frenkiel, 1953; Hinze, 1959; and Scull and 

Mickelsen, 1957).

The foregoing discussion of turbulent diffusion pertains to a turbulence field which is 

stationary with respect to its coordinate 

system. However, with the transformation

x - Ut

the theory applies also to a homogeneous 

turbulence field which is being transported 

bodily at a constant velocity, U, in the x direction. The. discussion has also been simplified to the extent that complicating factors such as confining boundaries and velocity gradients, which influence the diffusion 

process in open channels, have not been considered. In spite of these simplifications, 

the concepts presented constitute a theoretical framework for studying turbulent diffusion in open channels.

TURBULENCE MEASUREMENTS

In experimental fluid mechanics, studies 

of turbulence in water have lagged considerably behind corresponding studies in air. 

The lag is partially due to the fact that instruments such as the hot-wire anemometer, 

Which has proven satisfactory for turbulence 

measurements in air, have not been developed to the level of adequacy required for 

turbulence measurements in water. Without 

such instruments, measurements of dispersion afford one of the most promising means 

by which fundamental turbulence characteristics can be quantitatively estimated. Specifically, the intensity and the integral scale 

of turbulence can be determined, and the 

Lagrangian correlation coefficient can be 

estimated on the basis of a statistical analysis of dispersion data. Baldwin and Mickelsen (1962) have demonstrated that these Lagrangian properties can be related empirically to the corresponding Eulerian properties which are usually obtained from anemometer measurements.

If the lateral diffusion in the xz plane of a 

contaminant released from a point source in 

a uniform flow has a uniform mean velocity, 

u~ = x/t, the intensity of turbulence is given 

by: __

lim

(6)

The Lagrangian integral time scale of turbulence is given by

Lt = U

(7)

Once u/sT is known, the Lagrangian correlation coefficient can be estimated graphically by successive differentiations of a 

o-2 2 (t)curve, since by applying Leibnitz' rule 

to equation 1 and differentiating twice,

lV(t)=4= ^ az2(t) . (8) 

2w' 2 dt 2

If the form of the Lagrangian correlation 

function is known or assumed in advance, and 

is integrable, the intensity and scale of turbulence can be determined from diffusion 

data obtained at intermediate values of t/Lt , 

say (0.1<t/Lt < 10). This technique is useful 

when, as is often the case, it is difficult or 

impossible to obtain diffusion data at the 

limiting conditions specified by equations 6 

and 7.

There is experimental evidence (Ippen and 

Raichlen, 1957) that at large Reynolds numbers the correlation coefficient may be approximated reasonably well by an equation of 

the form

_ M

!V(a) = e

(9)

=71 -2 * /Lt r'/Lt-«,

By converting equation 1 to the dimensionless form

(10)

and by making the substitution indicated by 

equation 9, Frenkiel (1953) obtained the following approximate solutions to equation 10. 

If 0<t/Lt ^0.030,

(10a)
TURBULENT DIFFUSION IN OPEN CHANNELS

if 0.030 <t/L( ^3.63,

if 3.63 <t/Lt <

^llil = 2 (t/L( -l);and

if 101 <t/Lt,

= 2t/Lt

(10b)

(10c)

(10d)

The intensity and scale of turbulence can 

now be determined graphically by superimposing a cr2 (t) curve, based on experimental 

data and represented on logarithmic coordinates, on the theoretical az / t

curve as determined from equations 10 a d, 

so that the curves match. The ratios of the 

respective ordinates and abscissas then define the ratios t/(t/Lt ) and az (t)

from which Lf and may be evaluated.

The corresponding parameters describing 

the longitudinal turbulence characteristics 

may be obtained in an analogous manner by 

observing the longitudinal diffusion in the xz 

plane of a contaminant released instantaneously from a point source or a line source 

which is parallel to the z axis.

Application of these and similar techniques 

to the determination of turbulence characteristics in open channels has been pioneered 

by Kalinske and Pien (1944) and Orlob (1959). 

Kalinske and Pien (1944) obtained measurements of the lateral diffusion of a hydrochloric acid-alcohol mixture (spgr = l) in a threedimensional turbulence field in a laboratory 

flume. Orlob (1959) studied the lateral diffusion of small polyethylene particles 

(spgr=0.975)in a two-dimensional turbulence 

field defined by the water surface in a laboratory flume.

DIFFUSION AT THE WATER SURFACE

The water surface (or a plane parallel 

thereto) of a wide channel in which there is

uniform flow constitutes a two-dimensional 

nondecaying turbulence field in which the 

turbulence may be assumed to be statistically homogeneous except in regions very 

close to the sidewalls. These conditions 

conform approximately to the restrictions 

by which diffusion theory for homogeneous 

turbulence is limited. Therefore, the water 

surface of a channel such as the one defined 

above is a good medium for the quantitative 

determination of turbulence characteristics 

from dispersion observations. Furthermore, 

measurements of dispersion on the water 

surface of a laboratory flume may be easily 

and accurately obtained, as Orlob (1959) has 

demonstrated.

The extent t«a which the diffusion patterns 

at the water surface reflect the diffusion 

process and turbulence characteristics beneath the surface is questionable, particularly in alluvial channels. The experimental 

results of Kalinske and Pien (1944), however, 

tend to confirm the Reynolds analogy that the 

local diffusion coefficient, K(y), and the kinematic eddy viscosity,e, in the equation

T - ep

du

dy

are virtually equivalent, 

equation

In the preceding

r = rfYl- ] is the mean intensity of the

local shear, where TQ is the mean intensity of shear at the bed and D is 

the mean depth of flow, 

p = the mass density of the fluid, and 

u = the mean velocity at distance, y, above

the bed.

If the Reynolds analogy is indeed a reasonable approximation, measured velocity profiles could be used to predict K(y), and perhaps other turbulence characteristics, as 

functions of y.

ALLUVIAL CHANNELS

In general, diffusion phenomena in open 

alluvial channels should be similar to the 

corresponding phenomena in rigid-boundary 

open channels. Owing to the differences in 

the mechanics of flow associated with the two 

types of channels, however, certain differences are to be expected. For example, in 

alluvial channels there is a mutual interaction between the boundary form and both the
6 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

turbulence characteristics and the velocity 

distribution (the flow characteristics determine the bed form and vice versa);.in rigidboundary channels, the form and roughness 

of the boundary influence the turbulence 

characteristics and the velocity distribution, 

but there is no reciprocal effect. Another 

important difference is that the individual 

ripples and dunes, which usually make up the 

dominant roughness elements in alluvial 

channels, migrate in the direction of flow 

and undergo continuous transformation in 

shape and size, although a certain statistical 

constancy prevails. The extent to which factors such as these affect the diffusion process and the turbulence characteristics in 

open alluvial channels has not been systematically investigated.

DESCRIPTION OF EXPERIMENT

An exploratory experiment was conducted 

in an 8-foot-wide, 150-foot-long, laboratory 

flume at Colorado State University. The bed 

of the flume was covered with sand to a 

depth of about 8 inches. A schematic diagram of the flume circulation system is 

shown in figure 2.

Diffusion data were obtained for one set of 

flow conditions. Data describing the flow 

conditions are as follows:

Discharge, 7.41 cubic feet per second

Average depth, 0.58 feet

Average water-surface and bed slope, 

0.00071

Water temperature, 17.4°C

Total sediment concentration, 63 parts 

per million

Bed configuration, small dunes: 

Average dune length, 2.8 feet 

Average dune height, 0.029 feet 

Average dune velocity, 0.067 feet per

minute

Bed material:

Median fall diameter, 0.93 mm

Gradation coefficient,__ /_5£_ + 

2 Wjq Q = 1.53

The equipment and procedures employed 

in obtaining data on the flow conditions are 

described by Simons, Richardson and Albertson (1961).

The experimental techniques for studying 

lateral diffusion were similar to those employed by Orlob (1959). Polyethylene particles were released one at a time from a 

point source located on the water surface on 

the center line of the flume, 30 feet downstream from the head box. As the particles 

floated down the flume, their paths, which

Figure 2. Schematic diagram of the flume, a, pumping units; 6j, \) z, orifices; C, head-box and diffuser; d, baffles and screens; e, flume 

(8 ft x 2 ft x 150 ft);_f, tailgate; .g, total load sampler; h, tail-box; ,-'. jacks supporting the flume. After Simons and others (1961, fig. 1).
DESCRIPTION OF EXPERIMENT

tended to diverge, defined a lateral dispersion pattern. The dispersion pattern was 

determined by intercepting the particles at 

definite distances downstream from the 

source and noting their lateral positions.

The polyethylene particles were dis-shaped 

and had a diameter of 1/8 inch and a thickness of 1/16 inch. The specific gravity of 

the particles was approximately 0.96. 

Therefore, there was little density effect in 

the inertial response of the particles to turbulent impulses. The particles were sufficiently large to eliminate any appreciable 

effect due to molecular diffusion but probably not too large to trace the motions of the 

larger turbulent eddies at the water surface. 

The effect of particle size was not investigated in this study. However, recent and as 

yet unpublished data obtained by the senior 

author in the same flume and at about the 

same Reynolds number indicate that there is 

little difference in the lateral' dispersion 

patterns of 1/8 inch polyethylene particles 

dispersing on the surface and Rhodamine-B 

dye dispersing beneath the surface. This 

would indicate that the particles were not 

too large.

The particles were released into the flow 

through a funnel which was mounted on a 

point-gage assembly. The 3/16 inch diameter tip of the funnel was located just above 

the water surface. The particles were 

ejected sequentially in lots of 100 into the 

funnel at a constant rate from a manuallyoperated dispenser. In order to detect any 

effect due to periodicity in the turbulence, 

the experiment was repeated for sevferal 

lots at different rates of release by varying 

the time interval, At , from 1 to 5 seconds. 

No periodicity effect was detected. The 

particle-introduction methods simulated 

continuous release from a point source in a 

2-dimensional system or, if diffusion in the 

y direction is neglected, from a vertical line 

source in a 3-dimensional system.

The particles were caught in a compart - 

mented sieve which was inserted in the flow, 

normal to the flow direction. The collector 

extended across the entire width of the 

flume. The collector was divided into 1- 

centimeter compartments as shown in figure 

3, and each compartment was numbered. 

Because only the bottom 1/2 inch, or less, of 

the collector was immersed in the flow, 

disturbance to the flow was assumed to be

Sheet-metal dividers 

Spaced at 1-cm intervals

Window-screen covering on 

bottom and back

Figure 3. Section of compartmented-sieve particle collector.

negligible. The lateral distribution of particle paths for each 100-particle lot was determined by counting and recording the number of particles trapped in each compartment. Lateral distributions of 3 to 5 lots of 

100 particles were determined in this manner for source-to-collector distances of 15, 

30, 48, 60, 120, 240,300,450, 600, 900; 1500, 

1950, 2400, 2850, and 3300 centimeters.

The diffusion theory discussed in the previous section is based on the Lagrangian 

reference system in which attention is focused on the displacement histories of individual particles with respect to time. If the 

data had been obtained in strict accordance 

with the theory, the lateral position of each 

particle at a specified dispersion time, t , 

following release would have been determined, and the extent of dispersion would be 

described as the variance, o-z z (t), of the lateral distribution of particles about the mean 

lateral position at time t. Actually the lateral position of each particle was determined at a specified distance, x, downstream 

from the source rather than at a specified 

time, t, and the" extent of dispersion was described asca-2 2(x j. Because the flow was uniform (x = Ut) and x was large compared to 

the lateral deviations from the mean path, z , 

it seemed reasonable to assume that 

o-z 2(t)|= crz 2(x ),In the analysis which follows,no 

distinction between crz ^(i) and \o~z 2 (x) is made, 

and o-z 2 is substituted foro-^fx^
8 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

The times of transit over various known 

distances was recorded for 27 particles. 

The mean surface velocity computed from 

these data was U s = 62.5 centimeters per 

second. Extrapolation of three measured 

velocity profiles to the water surface indicated a mean surface velocity of 65.8 centimeters per second. The velocity as determined by the particle-timing method was 

used in analyzing the data.

Data on longitudinal diffusion were also 

collected at distances from the source of 

240, 600, and 900 centimeters. Particles 

were released from the source at 10-second 

intervals, and times of transit for each longitudinal distance were determined indirectly by measuring the times between arrival of successive particles at the end station. This procedure was repeated with 100 

or more particles for each of the three longitudinal distances. This method, although 

theoretically sound, did not prove to be sufficiently accurate in practice because operator errors were of the same order of magnitude as the spread about the mean transit 

time. The results were, therefore, inconclusive.

RESULTS

The lateral diffusion data were plotted on 

arithmetic probability paper. A typical set 

of data, for a source-to-collector distance 

of 300 centimeters, is shown in figure 4. 

These results indicate that the distribution 

of particle paths along the z axis followed 

the normal probability law. This was typical 

of all runs. It was also typical that no systematic variation of distribution due to variation of the time interval, At, between particle release could be detected. The center of 

concentration of the particles deviated from 

the center line of the flume by an amount %c , 

In nearly all runs this deviation was to the 

left. The standard deviation, a2 , of the particle distributions was determined from the 

cumulative distribution plots according to 

the relationship

r _ Z 84.1 -"15.9

z - 

from the cumulative distribution plots and 

other experimental data are listed in table 1.

In figure 5, the variance of lateral spread, 

a,2 , is shown as a function of distance downstream from the source, \ . The plotted experimental data follow a peculiar trend in 

that the data fall consistently below the 

curve obtained from equation 10 in the range 

(450 < x < 900) and consistently above the 

curve in the range ( x > 1500) with an apparent 

break occurring somewhere in the range 

(900 <x < 1500). It is believed that the break 

was due to a change in turbulence characteristics induced by a change in bed roughness. This hypothesis is supported by the 

photographs in figure 6 which show the condition of the flume bed after the experiment. 

Both photographs show a path, approximately 

2 feet wide, of comparatively smooth bed 

extending for some distance down the flume. 

This path, which extended from the head box 

to about 9 meters downstream from the 

source, conceivably could have caused a reduction in dispersion in the range (0< x < 900) 

as indicated by the data in figure 5. It is not 

known definitely if the condition of the bed 

indicated by the photographs persisted 

throughout the entire experiment; however, 

previous experience with the same flume 

suggests the possibility. As the distance 

downstream from the source increases, the 

slope of the v: 2 (x) curve becomes constant; 

this fact indicates a constant coefficient of 

lateral diffusion: K2 = 15.1 square centimeters per second.

Figure 7 illustrates the use of equations 6 

and 7 in determining the magnitudes of the 

intensity and integral scale of turbulence 

from experimental dispersion data obtained 

close to and far from the source. Due to the 

apparent shift in turbulence characteristics 

in the range (900 < x < 1500), use of experimental data obtained at the limiting distances 

from the source would in this case require

that estimates of U"s and L t be based

where z 8 4j and Zj 5 9 are the positions along 

the z axis where the 84.1 and 15.9 percentile 

lines intersect the distribution curves. The 

procedure for determining crz is'illustrated 

in figure 4. The parameters determined

on different sets of turbulence characteristics. Also, in view of the limitations of the 

polyethylene-particle techniques, the reliability of the dispersion measurements obtained near the source, where small scale 

motion is important, is questionable. Specifically, the measurements obtained at * =15 

and 30 centimeters are open to question because the diameter of the particles, the compartment spacing, az t and the diameter of
RESULTS

JL_

20 30 40 50 60 70 80 

PERCENT OF PARTICLES ACCUMULATED

90 95 98 99

Figure 4. Cumulative distribution plots for Jt = 300 centimeters. Solid circle indicates centerline of flume.
10 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

Table 1. Lateral diffusion data and computed parameters

Ran

No.

15.1 

1 R 9

1 ^i ^

1 R 4

14.1 

149

1 4 q

14 4

13.1

1 ^ 9

1 ^ ^

1^4

7. la 

7.1b

7 2a

7 9h

7 ^a

7.3b

6.1 

6.2

6.3

5.1 

5 2

^i 3

^1 4

4.1 

4 9

4 *3

4.4

4 ^

2.1

9 9

9 ^

9 4

2

K

3.1

^ 9

3 ^

*3 4

3 ^i

1.1 

1 9

1 q

1.4

X

cm

15

30

48

60

120

240

300

450

600

900

At 

sec

4

q

0

1

4

 3

1

4

Q

o

1

3

Q

0

O

1

1

3 

9

1

5

Q

0

1

5

4

 3

o

1

5 

1

4

q

o

5 

4

q

9

1

5 

4

q

2

t = x/Us 

sec

0.240

.480

.769

.960

1.92

3.84

4.80

7.20

9.60

14.4

az 

cm

0.70 

.77

.77

79

1.50 

1.40

1 Ti

1 49

1.70 

1 75

1.80

1.58

2.38 

9 49

9 1 Q

9 99

1 Q7

9 0^

4.10 

*3 Q9

3.80

7.3 

6.6

fi q

6.7

8.5 

7.8

7.7

8 ^i

8 Q

10.5 

10 3

U Q

1 n Q

10.0

13.8 

13.6

1 9 n

12 7

lo 4

18.4 

17.4

16.3

18.4

2 ^Z

cm2

0.490 

^IQ?

.593

.518

2.25 

1 Qfi

1.82

9 0*3

2.89 

3.06

9 94

9 ^10

5.66 

5.86

4.80

R 94

3.88

4 90

16.8 

1 c; 4

14.4

53.3 

43 fi

47.6

44 Q

72.3 

60.8

59.3

79 2

7Q 9

110.2 

106.1

m 7

118.8

100.0

190.4 

185.0

144.0

161.3

153.8

339 

303

266

339

°~z 

cm

0.74

1.42

1.71

2.22

3.94

6.87

8.28

10.6

12.9

17.0

2 CT,

cm2

0.548

2.02

2.92

4.94

15.5

47.3

68.8

112.6

166.9

289

2 °Z 

X

cm

0.0366

.0673

.0608

.0823

.129

.197

.229

.250

.278

.322

a_z_

X

0.0493

.0473

.0356

.0370

.0328

.0286

.0276

.0236

.0215

.0189

bc 

cm

1.40 

1.45

1.15

1.00

1.70 

1.70

1.90

1.65

1.95 

2.50

1.05

1.85

2.50 

2.45

2.45

1.98

2.00

2.30

3.8 

3.4

3.0

7.2 

6.6

6.3

8.3

8.0 

9.8

6.4

8.8

7.8

3.5

5.4

7.5

11.4

9.0

10.0 

8.8

10.0

6.9

9.4

8.4 

7.7

13.0

6.0

8C 

cm

1.25

1.74

1.84

2.28

3.4

7.1

8.2

7.4

9.0

8.0
RESULTS

Table 1. Lateral diffusion data and computed parameters Continued

11

Run

No.

1.5

8.1

8 9

8

0

9.1

n 9

Q 3

10.1 

i n 9

in s

12.1

1 9 9

1 9 °.

11.1 

i i 2

1 1 °.

X

c.m

1500

1950

2400

2850

3300

At 

sec

1

3 

o

1

3 

9

1

3

9

1

3 

9

1

3 

9

1

sec

24.0

31.2

38.4

45.6

52.8

cm

1 4 9

28.3 

9 ^ n

97 1

28.7

33 0

32.4

Q7 7

37.0

on 9

350

40.6 

38.8

cr, 2

cm2

909

801 

625

7°.4

824 

1030

1089

1050 

1136

1421

1369

1 99 R

1648 

1505

1560

cr,

cm

26.8

31.3

34.6

37.1

39.6

<^ 2 

cm2

723

981

1202

1377

1571

X

cm

0.482

.503

.501

.483

.476

X

0.0179

.0161

.0144

.0130

.0120

cm

4.7

7.6 

10.4

13.7

13.7 

14.3

10.4

13.2 

10.5

-4.0

6.0 

9.0

5.0

8.4 

12.0

8.0

cm

10.6

12.8

6.6

6.7

9.5

the tip of the funnel through which the particles were ejected, were all of the same order of magnitude.

Consequently, the method of estimating 

turbulence characteristics by use of an assumed Lagrangian correlation function is 

considered preferable for interpreting the 

results of this experiment. This method is 

illustrated by figures 8 and 9. The correlation function _io|

R w > (a) = e L t

was assumed, and solutions lOb and lOc of 

equation 10 were used to plot the theoretical

urve in figure 8. In figure 9,

and the ratio of the ordinates defined the intensity of turbulence

1

-(6.5) = 0.037.

2.8 x 62.5

The principal advantage of this method is that 

it involves the entire range of data, not only 

data obtained at limiting distances from the 

source.

With the values of L t and as deterthe experimental data indicating the vz (t) 

relationship obtained by experiment were 

plotted. Figure 9 was superimposed on 

figure 8 in the position which corresponded 

to the, best fit of the entire range of data to 

the theoretical curve. The ratio of the abscissas then defined the integral scale of 

turbulence

(-TJT- = 2.8 seconds,

mined in the preceding paragraph, equations, 

lOa, lOb, and lOc were used to construct the 

theoretical curves shown in figures 5 and 7.

Figure 10 was plotted to indicate the manner in which the coefficient of lateral diffusion, K~ , approaches a constant as the distance from the source increases. The manner of approach depends in part on how Kz is 

defined. The K z (X ) relationship is shown in 

figure 10 for the two alternate definitions

1

K, = -r

dx

and

2 TT

(11)

(lla)
12 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

c 50

u 40 

|

£ 30

z

'N 20

to

10

0

1500

Enlargement of region near source

From eq lOc /

100

= US L, = 175 cm

200 300 

x, IN CENTIMETERS

SYMBOLS

x cr,2 for Af = 1,2,3,4, or 5 sec 

Af

500-

|-See enlargement above

1000 2000 

x, IN CENTIMETERS

3000

Figure 5. Variance of lateral distribution as a function of distance downstream from source.
RESULTS 13

Figure 6. Flume bed after the experiment. Upper photograph: 

View of bed, upstream toward source. Lower photograph: 

View of bed, downstream from source.

The experimental values of Kz corresponding 

to equation 11 were obtained by graphical 

differentiation of the data plotted in figure 5. 

Values of K 2 corresponding to equation lla 

were computed directly from the data. The 

theoretical curves were obtained from equations lOb and lOc by using the computed values of L t = 2.8 seconds and w"2- = 5.38 square 

centimeters per second squared. Although 

the diffusion coefficient as defined inequation 

lla appears frequently in the literature and 

the coefficients defined in equations 11 and 

lla both approach the same limiting value as 

the distance from the source increases, the

definition given in equation 11 is preferable 

because it is consistent with the theory of 

diffusion by continuous movements and because it approaches the limiting value, 

lim K2 = constant, more rapidly.

Semiempirical equations which relate the 

limiting value of K2 to mean flow parameters 

in a broad rigid-boundary open channel have 

been derived by both Orlob (1961) and Elder 

(1959). Or lob's equation (1961), which was 

based on experimental observations of lateral turbulent diffusion at the water surface of 

a laboratory flume with extreme bottom 

roughness is

= 0.085 1 / 3 (L t U) 4/ 3

in which

and _

E = UgSe

In these equations

(12)

(13)

(14)

E = the rate of energy dissipation per

unit mass of fluid in a broad open

channel,

f = the Darcy-We isbach friction factor, 

K = the von Karman turbulence constant 

_ and equals 0.4, 

U = the mean flow velocity, 

3 = the acceleration of gravity, 

S e = the slope of the energy gradient,

and 

U s = the mean flow velocity at the water

surface.

Elder's equation (1959), which was conr 

firmed experimentally in a small flume with 

a smooth boundary for . very small flow

depths, is

(15)

in which D is the mean depth of flow, and 

"\fr^Jp = VgDSc is the shear velocity at the

bed of the channel.

If values of the mean flow parameters determined in this experiment are used, equation 12 yields K2 = 20.4 square centimeters 

per second, and equation 15 yields K 2 = 14.3 

square centimeters per second. Both of these 

values of K z compare reasonably well with 

the value K 2 = 15.1 square centimeters per 

^second determined from the data in this experiment. Equation 12 is based on Kolmogoroff's similarity hypothesis for small-scale 

turbulence components, whereas the extent of

Kz = 0.23
14 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

0.05

0.04

0.03

0.02

ffz

Experimental values of 

100 200

x, IN CENTIMETERS 

A.

300

0.50

0.40

0.30

0.20

0.10 -

From eq 1

t =-^L: lim (2JL) = 2.8 sec '2

O Experimental values of

1000 2000

x, IN CENTIMETERS 

E

3000

Figure 7. Intensity and scale of turbulence as determined from limits of dispersion data. Upper diagram shows intensity and lower

shows integral scale.
RESULTS 15

10

"If1 '0

O.K

t/L

= [2J Wr^)M«)<KT,

I_______I I 1

3.10 1.0 t/L, 10

FigureS. Theoretical solution of diffusion equation with Rj^' (a) - e

60

10 

to

QC.

LU

LU 6.5

1.0

Symbols

for Af = 1,2,3,4, or 5 sec 

Af

i ! i i i i i 111 i i i i 11

0.10 1.0 218 10 

DISPERSION TIME, t, =*IUS , IN SECONDS

Figure 9. Standard deviation of lateral distribution as a function of dispersion time.

100
16 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE OF AN ALLUVIAL CHANNEL

1000 2000 

X, IN CENTIMETERS

3000

Figure 10. Lateral diffusion coefficient as a function of distance from source.
REFERENCES 17

dispersion at large times is generally believed to be controlled mainly by the largescale components. Owing to this apparent 

contradiction, there may be conceptual advantages to a relationship of the form of 

equation 15.

APPLICATION TO PRACTICAL PROBLEMS

In order to predict the lateral diffusion 

pattern in a channel in which the mean flow 

conditions are known, the first and most important requirements is to estimate K2 by 

using a relationship such as equation 12 or 

equation 15. The second requirement is to 

estimate X 0 (see fig. 5), which defines the 

point at which an extension of the straightline portion of the ^Z 2 (x) curve intersects the 

x axis. Next, by using 2KZ /U S as a slope and 

X 0 as an intercept, 0,2 is established as a 

function of x which is valid for large distances from the source. If the Lagrangian 

correlation function is known, crz z (x) can be 

established for the range of values of x where 

K 2 varies. If the correlation function

_|o[

R w > (a) - t L t is assumed, then X 0 = U s L t , K 2 

is constant for x> 6 X 0 , and equations lOa 

and lOb may be used to predict the diffusion 

pattern near the source. When the numerical 

value of K 2 = if' 2 L t is known, application of 

equations lOa and lOb requires knowledge 

of either Lt or * w '2. At present neither Lt 

nor Vu/2 can be determined for alluvial 

channels,, except by experiment. For predicting dispersion at very large distances 

from the source, the error introduced by assuming XQ = 0 would generally be small.

CONCLUSIONS

A review of the theory of turbulent diffusion by continuous movements as applied to 

open channels coupled with the analysis of 

experimental diffusion data for one set of 

flow conditions in a laboratory alluvial channel gives rise to the following conclusions:

1. The statistical theory of diffusion by continuous movements in a homogeneous turbulence field appears to have useful application in the study of lateral turbulent diffusion in alluvial channels.

2. The scale and intensity of the lateral turbulence components at the surface of an 

alluvial channel with uniform flow can be

estimated by analyzing observations of the 

lateral dispersion of small polyethylene 

particles floating on the surface.

3. The coefficient of lateral diffusion in an 

alluvial channel reaches a limiting constant value, in accordance with turbulent 

diffusion theory, as the distance from the 

source is increased. Indications are that 

the distance required for the diffusion coefficient to become constant is quite small 

in comparison to the length scale of most 

diffusion phenomena in natural streams 

which are of practical interest.

4. The assumption that the Lagrangian velocity correlation function is exponential in 

form appears to be a useful first approximation in considering turbulent diffusion in 

an alluvial channel. This assumption permits an approximate prediction of the diffusion pattern near the source.

5. It appears that it will be possible to derive for alluvial channels a relationship 

between the limiting value of the lateral 

diffusion coefficient and mean flow, channel-geometry, and sediment parameters 

which is analogous to the semi-empirical 

relationships for rigid-boundary open 

channels derived by Orlob (1951) and Elder 

(1959).

REFERENCES

1. . Baldwin, L. V. and Mickelsen, W. R., 1962, 

Turbulent diffusion and anemometer 

measurements: Am. Soc. Civil Engineers Proc., Paper 3095, v. 88, no. 

EM 2, p. 37-69.

2. Batchelor, G. K., 1949, Diffusion in a field 

of homogeneous turbulence, I. Eulerian 

analysis: Australian Jour. Sci. Research, v. 2, p. 437-450.

3. Elder, J. W., 1959, The dispersion of 

marked fluid in turbulent shear flow: 

Jour. Fluid Mechanics, v. 5, no. 4, p. 

544-560.

4. Frenkiel, F. N., 1953, Turbulent diffusion: 

Mean concentration distribution in a 

flow field of homogeneous turbulence: 

New York, Acad. Press, Advances Appl. 

Mechanics, v. 3, p. 61 106.

5. Hinze, J. O., 1959, Turbulence: An introduction to its mechanism and theory: 

New York City, MeGraw-Hill Book Co., 

Inc., 586 p.
18 EXPLORATORY LABORATORY STUDY OF LATERAL TURBULENT DIFFUSION AT THE SURFACE ON AN ALLUVIAL CHANNEL

6. Ippen, A. T. and Raichlen, F., 1957, Turbulence in civil engineering: Measurements in free surface streams: Am. 

Soc. Civil Engineers Proc., Paper 1392, 

v. 85, no. HY5, 27 p.

7. Kalinske,A. A. and Pien, C. L., 1944, Eddy 

diffusion: Indus, and Eng. Chemistry, v. 

36, p. 220-223.

8. Kamp'e de Feriet, J., 1939, Les fonctions 

aleatoires stationnaires et la theorie 

statistique de la turbulence homogene: 

Soc. Sci. de Bruxelles Annales, ser. 1, 

v. 59, p. 145-194.

9. Orlob, G. T., 1959, Eddy diffusion in homogeneous turbulence: Am. Soc. Civil 

Engineers Proc., Paper 2950, v. 85, no. 

HY9, p. 75-101.

10. Orlob, G. T., 1961, Eddy diffusion in homogeneous turbulence: Discussion closure: Am. Soc. Civil Engineers, v. 87, 

no. HY1, p. 137-143.

11. Scull, W. E. and Mickelsen, W. R., 1957, 

Flow and mixing processes in combustion chambers: Natl. Advisory Comm. 

Aeronautics TR 1300, chap. 2, p. 32-82.

12. Simons, D. B., Richardson, E. V. and Albertson, M. L., 1961, Flume studies 

using medium sand (0.45 mm): U.S. 

Geol. Survey Water-Supply Paper 

1498-A, 76 p.

13. Taylor, G. I., 1921, Diffusion by continuous movements: London Math. Soc. 

Proc., A 20, p. 196-211.

INT.DUP..D.C.64-