Title: Lagrangian aspects of turbulent transport in pipe flow
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00099398/00001
 Material Information
Title: Lagrangian aspects of turbulent transport in pipe flow
Alternate Title: Turbulent transport in pipe flow, Lagrangian aspects of
Physical Description: xv, 200 leaves : ill. ; 28cm.
Language: English
Creator: Breton, David Lee, 1940-
Copyright Date: 1975
Subject: Flow visualization -- Mathematical models   ( lcsh )
Fluid dynamics   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Statement of Responsibility: by David Lee Breton.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 185-198.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00099398
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000161997
oclc - 02692774
notis - AAS8342


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David Lee Breton




I wish to express my sincere thanks to the members

of my supervisory committee: Dr. Ray W. Fahien, Chairman;

Dr. Dale W. Kirmse, Co-Chairman; Dr. John G. Saw; Dr. Her-

bert E. Schweyer; and Dr. Mack Tyner. Special thanks are

due to Dr. Fahien and Dr. Kirmse for their guidance during

the course of this work.

Thanks are also due to my colleagues and friends

of the Chemical Engineering department. For the assist-

ance in the construction and the maintenance of the experi-

mental equipment, I would like to extend my appreciation to

Jack Kalway and Myron Jones.

I wish to thank my wife, Fran, for her continued

support and aid in organizing and proofing. Many thanks

to Mary Van Meer for the typing of the manuscript. I also

wish to thank David Trissel for his computer programming

assistance in data compilation.

I am very grateful to the College of Engineering

for its financial aid in the form of a graduate assistant-




ACKNOWLEDGMENTS..................... ................ ii

LIST OF TABLES........................... .. .......... v

LIST OF FIGURES .. ............ ....... ............. ...... vi

KEY TO SYMBOLS ............................. ......... x

ABSTRACT ................ ............ ..... ........... xiv

1. INTRODUCTION................................ 1
2. LITERATURE REVIEW........................... 3
2.1 Dispersion of a Scalar Property
in a Turbulent Flow................. ... 7
2.1.1 Isotropic homogeneous flow...... 8
2.1.2 Anisotropic homogeneous flow.... 13 Single-particle
dispersion............. 13 Two-particle
dispersion ............. 20
2.1.3 Shear flow ...................... 22
2.1.4 Interaction of molecular
and turbulent dispersion........ 24
2.2 Lagrangian Characteristics of
Turbulent Flow......................... 26
2.2.1 Lagrangian numerical
simulations ..................... 26
2.2.2 Lagrangian experimental
studies ......................... 31
2.2.3 Eulerian-Lagrangian
transformation .................. 35
2.3 Experimental Determination of
Eddy Diffusivities................ ..... 40
2.3.1 Determination by
Lagrangian analysis............. 41
2.3.2 Determination by
Eulerian analysis............... 45

3. EXPERIMENT AND DATA ANALYSIS ..................
3.1 Experiment ............................
3.1.1 Flow loop .........................
3.1.2 Lighting system....................
3.1.3 Camera ................ ............
3.1.4 Procedure..........................
3.1.5 Processing of
photographic film.................
3.2 Data Analysis ............................
3.2.1 Transformation of
photographic data.................
3.2.2 Computation of particle
3.2.3 Computation of Lagrangian
4. RESULTS AND DISCUSSION............ ............


4.1 Results ..........................
4.1.1 Turbulent intensities and
shear stresses.............
4.1.2 Lagrangian correlations....
4.2 Equipment and Procedure...........
4.3 Data Collection and Analysis......






....... 80

........ 81
........ 81

A. DATA COLLECTION ALGORITHM......................152
EQUATION....................................... 167
PATHS ..................... ................. 171
BIBLIOGRAPHY.......................................... 185
BIOGRAPHICAL SKETCH ..................... ............. 199


Table Pae

4.1 Relative Turbulent Intensities.............. 82

4.2 Relative Turbulent Shear Stresses........... 82

4.3 Typical Particle Path Coordinates ........... 125

4.4 Number of Contributionsto the Correlations.. 133




... .. 55

... .. 58

. . . 6 1

...... 62

... .. 64

...... 69

...... 73

...... 91

...... 92

3.1 Experiment Apparatus..................

3.2 Observation Enclosure and Camera

3.3 Chopping Frequency Detection Circuit.....

3.4 Camera Construction....................

3.5 Film Mounting Arrangement.............

3.6 Geometry of Optics for a Typical
View of the Flow Field................

3.7 Arrangement of Datum Points...........

4.1 <(r'(t))2 /D2 Versus t* for
All Radial Zones................. .....

4.2 /D Uz,MAX Versus t*
for All Radial Zones ..................

4.3 / Versus t*
for All Radial Zones ..................

4.4 <(z'(t))2>/D2 Versus t*...............
4.5 <(r'(t))2>/D2 Versus t*...............
4.56 <(r '(t))2>/D 2 Versus t*.
4.6 <(r o '(t) >)2 /D2 Versus t*.............
4.7 6 z(t)rit)'/D Versus t.

4.8 /D2 Versus t* ............
4.9 /D2 Versus t*...........
4.1 /D 2 Versus t* ...........

4.10 /D Uo z,MAX Versus t*.......

4.11 /D Uz,MAX Versus t*.......

4.12 /DoUz,MAX Versus t*.......

. 93


. 95








Figure Page
4.13 /D Uz,MAX Versus t*............. 103
4.14 /D Uz AX Versus t*............. 104
4.15 /DoUzMAX Versus t*............. 105
4.16 /DoUz MAX Versus t*........... 106
4.17 /DoUz,MAX Versus t*........... 107
4.18 /DoUzMAX Versus t*........... 108
4.19 d /D U ,MAX Versus t*........... 109

4.20 d /DoUzMAX Versus t*.......... 110

4.21 t-/DoUz,MAX Versus t*.......... 111

4.22 /<(V'(t ))2> Versus t* ......... 112
4.23 / Versus t*............ 113
4.24 /<(V (to))2> Versus t* ......... 114

4.25 <(V (to))2 Vr t )>) /2 Versus t*.......... 115
(<(Vz o) 2>

4.26 z 2 1/2 Versus t*....... 116
(<(Vz(t )) 2 ><(V (t ))2 1/2

4.27 2 r Versus t*.......... 117

4.28 Projection of the Particle Paths
onto the r-O Plane........... ................ 124
4.29 Equivalent Convolution Function
for the Least-Square Smoothing.............. 127

Figure Page
4.30 Energy Spectra Function for the
Equivalent Convolution Function............... 129
4.31 Mean Axial Velocity Profile................... 131

4.32 An Estimate of the Statistical
Error for <(z'(t))2>/D2 .......................135

4.33 Time Symmetry in z- for the
<(V (t )) >
Fourth Radial Zone from Center of Pipe........ 136

4.34 Time Symmetry in for the

r o
Fourth Radial Zone from Center of Pipe........ 137

4.35 Time Symmetry in for the
Fourth Radial Zone from Center of Pipe........ 138

4.36 Time Symmetry in --
(<(Vz(to))2> for the Fourth Radial Zone from
Center of Pipe ................................ 139

4.37 Axial Energy Spectra Function................. 141

4.38 Radial Energy Spectra Function................ 142

4.39 Tangential Energy Spectra Function............ 143

4.40 Axial Autocorrelation Function Obtained
from the Axial Energy Spectra Function........ 145

4.41 Radial Autocorrelation Function Obtained
from the Radial Energy Spectra Function....... 146

4.42 Tangential Autocorrelation Function Ob-
tained from the Tangential Energy Spectra
Function .................. ... ............ .. 147

v i i i

Figure Page

A.1 Electrical Skematic of Interface
between the PDP-11 Minicomputer and
the Mosely X-Y Recorder....................... 154

A.2 Flow Diagram for Data Collection
Algorithm.................................... 156




















- initial value of x, L

- parameter of Equation (4.1)

- parameter of Equation (4.1)

- parameter tensor defined by Equations (B.14)
through (B.19)

- proportionality parameter for Equation (2.57)

- coefficient vector of Equation (3.9)

- height of image from photo-enlarger, L

- distance from object point to air-glass interface,

- height of object point, L

- diameter of pipe, L

- relative dispersion tensor for two particles, L

- eddy diffusivity tensor, L2/t

- asymptotic eddy diffusivity tensor, L2/t

- frequency, l/t

- conditional p.d.f. of the separation of two partic

- energy spectra function vector

- turbulent flux; heat, M/t3; mass, M/L t

- equivalent convolution function for the least-squa

Fourier transform of g(t)

image height on film, L

virtual image height at air-glass interface of ob-
servation section, L




i unit vector for radial direction

i unit vector for axial direction

i unit vector for tangential direction

KBW proportionality parameter for Equation (2.69)

m parameter of Equation (4.1)

m parameter vector defined by Equation (B.5)

M magnification factor for photo-enlarger

n refractive index of the glass, glycerin, and

N number of contributions to a statistical sample

P probability distribution function of particle
concentration with displacement and time

Q probability distribution function of particle
displacement with time

r radial distance, L

r starting radial position, L

R(L) Lagrangian velocity correlation coefficient

R(F) autocorrelation function obtained from the energy
spectra function

S time-stationary tensor defined by Equation (2.28),

t time, t

T a specific time, t

T(t) Reynolds stress tensor

u Eulerian velocity, L/t

ub bulk velocity, L/t

u lateral velocity, L/t

u Eulerian acceleration, L/t2

U Eulerian velocity, L/t

U ZAX maximum mean axial velocity which occurs at the
centerline of pipe, L/t

Uz,AVE average axial bulk velocity, L/t

v relative velocity of two particles, L/t

v Lagrangian acceleration, L/t2

V Lagrangian velocity, L/t

x distance, L

X' displacement from origin, L

Y relative displacement of tvo particles, L

z axial distance, L

Greek Letters

B proportionality parameter for Equation (2.68)

r scalar property

S separation distance between velocity vectors, L

6 tangential angle

6. incidence angle of light at air-glass interface
1of observation section

er refacted angle of light at air-glass interface
of observation section

X parameter function defined by Equation (B.6)

A macro integral length scale, L

distance from air-glass interface of observation
section to lense, L

v distance from film to lense, L

elapsed time, t

p density of fluid, M/L3

o statistical variance

T micro time scale, t

a co-factor of determinant

R determinant of displacement-displacement correlation


E Eulerian

i coordinate direction

j coordinate direction

k discrete time increment

L Lagrangian

m view index

o starting value

y lateral coordinate direction in 2-D

z longitudinal coordinate direction in 2-D


fluctuation about the mean

* dimensionless variable

- time average

< > expected value (ensemble average)


Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



David Lee Breton

August, 1975

Chairman: Dr. R. W. Fahien
Co-Chairman: Dr. D. W. Kirmse
Major Department: Chemical Engineering

A Lagrangian analysis of the turbulent dispersion

process was examined by observing the motion of suspended

particles in a turbulent flowing liquid. The liquid, tri-

chloroethylene, was pressure-driven through a 0.0254 meter

(1 inch) I.D. glass pipe. The average velocity was 0.969

meter/sec (3.18 feet/sec) which corresponded to a Reynolds'

number of 110,000. A photographing technique was developed

which enabled the photographing of particle motion over

large distances. Particle positions in time were computed

from the photographic images.

Lagrangian time correlations were computed from the

displacement and velocity deviations obtained from the

particle trajectories. The time dependence of the various

correlations was obvious from an examination of the plotted

results. The effect of a confining wall on the radial spread

of the particles was apparent for large times. It appeared

that the assumption of a constant radial eddy diffusivity

coefficient could be used as a fair approximation of the

time-dependent coefficient. Within the experimental accur-

acy of the data, the resultant correlations agreed well with

the theory and published data.



Turbulence is a common occurrence in both natu-

rally and artificially generated flow systems. The rapid

spread of smoke from chimneys and the vapor trails of

high-altitude jet aircraft are examples of turbulent

effects in the atmosphere. The diffusion of mass and

heat in turbulent fluids in numerous chemical engineering

processes is another example. Few physical phenomena

have attracted more interest in the scientific and en-

gineering fields than turbulence, which attests to its

general occurrence and importance. A great deal of work

has yielded only a little understanding of turbulent fluid

motion, indicating its difficult and complex nature.

The Lagrangian characteristic of turbulent liquid

flow is the object of study in the present work. To in-

vestigate the turbulence, paths of small particles were

observed as approximations of the paths of fluid points.

A distinction is made here between a "particle," a solid,

rigid piece of matter immersed in the continuum, and a

"fluid point," a mathematical point moving with the con-

tinuum. A fluid point is a volume of fluid so small that

in the context of the continuum it may be considered as


an individual point moving along with the fluid. The

system selected for study is fully developed pressure-

driven flow in a pipe. Under these conditions the fluid

is in shear flow, and the turbulence is nonhomogeneous

and anisotropic.



The concept of turbulence is widely discussed,

and in a general sense, its effects and meaning are

understood. Turbulence was discussed as early as the

fifteenth century in the writings of Leonardo da Vinci;

however, it wasn't until the nineteenth century when

quantitative data were reported. The first reporting

on turbulence were attempts to present the characteris-

tics. Hagen and Poiseuille independently and qualita-

tively observed two modes of fluid flow in pipes, non-

turbulent and turbulent (Rouse and Ince, 1963). The

concept of an eddy diffusivity was introduced by Bous-

sinesq (1877) to allow for turbulent effects analogous

to Newton's law of fluid viscosity. With the aid of the

Reynolds number advanced by Reynolds (1883) through an

experimentally determined dimensionless ratio of fluid

and system parameters, it was possible to tell when to

expect turbulence in a flow system. Turbulence is said

to exist in flow systems for which the Reynolds number

has exceeded a critical value. Reynolds (1895) also

recognized the statistical nature of turbulence. By

expressing the dynamic variables of the equation of motion

as a mean value plus a deviation, and by averaging the

terms formed in the equation of motion, the turbulence


T t) = -p (2.1)

was generated. This term can be interpreted as the turbu-

lent stresses on an element of the fluid in addition to

the stresses of pressure and viscous stress. Because

Reynolds was the first to give the equation for turbulent

flow, the turbulence stresses, T.. are often called
I j
Reynolds' stresses.

It was in the early twentieth century that useful

and workable theories and models of turbulence appeared.

It is obvious that, in the early days of turbulence in-

vestigations, when knowledge about and insight into the

mechanism of turbulent fluid flow were rather poor, the

turbulent processes could be studied only in a rather

rough way. To describe the behavior of turbulent motion

of a fluid near a wall, Prandtl (1904) proposed the

boundary layer theory. The flow far from the wall is

assumed uniform. Also among the theories arising from

the early studies of turbulent flow are those based upon

the concept of a "mixing length." These theories have

been the most fruitful, not so much because they describe

the mechanism correctly, but because they have resulted

in useful and practical semiempirical relationships. Taylor

(1915), Schmidt (1917), Prandtl (1925), and Von Karman (1934)

proposed mixing length theories for turbulence. The dif-

fusive action of turbulence is considered to result in an

eddy viscosity or eddy heat conductivity from which the

distribution of mean values can be calculated, just as, in

the kinetic theory of gases, molecular-transport processes

result in a viscosity and heat conductivity. Basically,

it is assumed in the mixing length theories, that each lump

of fluid that is subjected to the turbulent motion may be

considered as an individual entity. The properties of the

lump are conserved during a certain time, i.e. over a cer-

tain distance. At the end of this time, the lump is as-

sumed to mix with the surrounding fluid. These theories

are purely phenomenological. Later studies have shown that

the physical picture based upon the concept of a mixing

length cannot be correct in all details. The mixing length

theories still prove to be very useful to engineers.

The statistical theory of turbulence also began

its existence in the early twentieth century. The concepts

of the correlation coefficient, spectrum function, and

local similarity which have proved so helpful in revealing

the structure of turbulence are all a part of this theory.

Taylor (1921) advanced the concept of the Lagrangian

correlation coefficient which provided a theoretical

basis for turbulent diffusion. Richardson (1920, 1926)

postulated the turbulence consists of a hierarchy of

eddies. The energy is cascaded from the largest eddies

to smaller and smaller eddies until the molecular level

is reached at which point the energy is dissipated by

viscous effects. Taylor (1935a,b,c,d) again considered

the statistical nature of turbulence in a revised form

and defined many of the properties of isotropic turbu-

lence. This was followed by the Von Karman and Howarth

(1938) demonstration of the tensor properties of the

Eulerian correlation coefficient and their formulation

of the relationship between the double and triple coef-

ficients through the Navier-Stokes equations. At about

the same time, Taylor (1938) showed the transform relation-

ship between the correlation coefficient and the energy

spectrum function. The Taylor concept of the energy spec-

trum has dominated turbulence research to the present. The

first theory for the small scale structure of turbulence,

presented by Kolmogorov (1941) and Obukhov (1941), served

as the starting point of many investigations and led to

general theories of spectral similarity such as those by

Von Karman and Lin (1949) and Lin (1948). This theory

predicts for any highly turbulent motion, the small scale

structure is isotropic which was empirically shown

earlier by Richardson (1926).

The following publications, among others, present

turbulence in general or some of its particular aspects:

Prandtl and Tietjens (1934a,b), Goldstein (1938), Agostini

and Bass (1950), Hopf (1952, 1957), Batchelor (1953),

Frenkiel (1953), Sutton (1953), Batchelor and Townsend

(1956), Corcoran et al. (1956), Opfell and Sage (1956),

Townsend (1956), Hinze (1959), Lin (1959), Priestly (1959),

Schlichting (1960), Lin (1961), Schubauer and Tchen (1961),

Tatarski (1961), Favre (1962), Pasquill (1962), Lumley and

Panofski (1964), Brodkey (1967), and Monin and Yaglom (1971).

Papers considered classics on turbulence are collected by

Goering (1958) and Friedlander and Topper (1961). The

textbooks by Bird et al. (1960) and Brodkey (1967) also

present sections on turbulence.

2.1 Dispersion of a Scalar Property in Turbulent Flow

Since turbulent fluid motion has an intrinsic ran-

dom fluid motion, an inherent characteristic of turbulent

flow is its dispersive action; accordingly, the dispersion

process is more active in turbulent than in nonturbulent

flow. It is not unusual then that most of the work on

turbulence centers around its transport or dispersive

characteristic. One of the most important analytical

inroads into the problems of turbulent dispersion was

made by Taylor (1921). Taylor's work not only laid the

groundwork for the Lagrangian study of turbulent disper-

sion but it also represented an initial step in the ideas

essential to the development of a general statistical

theory of turbulence. From this base, Richardson (1926),

Kampd de Fdriet (1939), Batchelor (1949, 1952, 1957), Tay-

lor (1954), and Monin and Yaglom (1971) have successively

expanded and deepened the theoretical foundation for under-

standing the statistical nature of turbulence as well as

its transport processes.

2.1.1 Isotropic homogeneous flow

Consider a condition in which the turbulence in a

fluid with no mean flow is uniformly distributed so that

the average conditions of every point in the fluid are the

same. The above fluid flow condition is that investigated

by Taylor (1921). Within this flow field, Taylor analyzed

the dispersion of a scalar property which may be associated

with all or a fraction of the fluid particles. Because of

the isotropy of the problem, only one direction of disper-

sion need be considered, say the x direction. By defini-

tion, the velocity of a fluid particle at time t, V(t), is

V(t) = x(t) (2.2)

and its position at time t, x(t),

x(t) = a + tV(tl)dtI


where a = position of dispersing particle at t = 0.

Let X'(t) be the displacement of the particle from a, then

X'(t) = x(t) a.


Taylor further supposes that the statistical properties

of V(t) are known, in particular the standard deviation
2 1
of V(t), , and the Langrangian correlation coeffi-

cient, R(L)(t). R(L)(tl) is the correlation coefficient

between the value of V for a particle at any instant, and

the value of V for the same particle after an interval of

time tl, i.e.

R(L (tl )

It is also assumed

time, so that

+-t9>112 (2.5)

the standard deviation is constant with

= = (2.6)

1 he indicates a statistical ensemble average.
The <.> indicates a statistical ensemble average.


R( (tl) 2 (2.7)

One measure of the degree of dispersion is given
by the variance of the displacement, . The rate

of change in the variance can be expressed in terms of

and R(L)(t1) as follows:

d 2 d
First = 2 = 2 (2.8)

then d s 2 0
and since X'(t) = / V(t + t, )dt, (2.9)

then dt< '2(t)> = 2 (2.10)

= 2 fdt1 (2.11)

2 (L)
= 2I R( (tl)dtl. (2.12)

For constant , R(L)(t ) is symmetric about zero, and


d ,2 2 (L
-d- = 2 f R(L)(tl)dtI. (2.13)

Integrating Equation (2.13) yields

= 2 2R(L)(t )dt1dt2. (2.14)

Equation (2.14) reduces the problem of dispersion, in a

simplified type of turbulent motion, to the consideration

of a single quantity, the Langrangian correlation coeffi-

cient, R(L)(t).

As mentioned above, an indication of the rate of

dispersion is the rate of change in the variance. Taylor

(1935a) explicitly recognized as the eddy dif-

fusivity, E(t), a turbulent transport property similar to

the molecular diffusivity. The eddy diffusivity is also

related to the Lagrangian correlation coefficient. From

Equations (2,8) and (2.13)

E(t) = 2 R (L) t )dt (2.15)
o 1 1

Now consider the physical implications of (2.14)

and (2.15). When t is so small that R(L)(t) does not

differ appreciably from 1, then

= t2 (2.16)


E(t) = 2t


Looking further, as t becomes large

R(L)(t) 0 (2.18)

a t (2.19)


E(t) constant. (2.20)

Defining a Lagrangian integral time scale as

TL = L R(L)(t )dt (2.21)

where R(L)(t) = 0 for t > T1, the eddy diffusivity for large

time can be expressed as

E = TL. (2.22)

TL is considered a measure of the average longest time dur-

ing which a particle persists in a motion in a given direc-
tion. Also a Lagrangian integral length scale may be
defined as

AL = 1/2TL (2.23)

which leads to
E_ =

This result is analogous to that obtained from the kinetic

theory of gases for the molecular diffusivity. It is also

consistent with Prandtl's mixing-length theory (Prandtl,

1925); however, it is obtained without having to depend on

the idea of mixture by subdivision and ultimate molecular


A Lagrangian micro time scale may also be defined

(Taylor, 1935a) as

1 1 d2 (L)
2 2 2R (t) (2.25)
T2 dt t = 0

Physically, this time scale is taken as a measure of the

time duration of the smallest eddies which are responsible

for energy dissipation.

2.1.2 Anisotropic homogeneous flow

2.1.2,1 Single-particle dispersion

Batchelor (1949) investigated the three-dimensional

homogeneous turbulent flow field. Even though Batchelor

started with a generalization of Taylor's (1921) single-

particle dispersion which was a Lagrangian analysis,

Batchelor's final results were Eulerian. The Lagrangian

to Eulerian transformation was through probability argu-

ments and some simplifying assumptions about the statistical

nature of homogeneous turbulence.

To generalize Taylor's analysis to three dimensions,

let Xi(t) be the i-th component of the vector displacement

of a fluid particle. The time rate of change of the covari-

ance of two displacement components Xi(t) and X (t) is

written as

d X (t)X.(t)> = + (2.26)


t t
1x(t)X.(t)> = Idtl+ /dtl (2.27)
o 0

Using tensor notation, define the second order tensor S as


Sij(t-t ) = (2.28)
i j 1 1 j

If the process is assumed to be a time-stationary process,

then the elements of S depend only on the elapsed time

= (t tl). Now Equation (2.27) can be written

d- = f (S .(d) + S.j.( ))d (2.29)
dt i 0 ij 31

and when integrated yields

t t
= f f (S ij( ) + Sji ( ))d dt (2.30)
1 J 0 0oo

As shown by Kampd de Fdriet (1939) in a study of stationary

random functions, a change of variable in Equation (2.30)

and a partial integration can be made to give

= (t -
1 J o

The above results can

grangian velocity correlation

generalized coefficient as

S)(Sij( ) + Sji(C))d .(2.31)

be expressed in terms of La-

coefficients. Defining a

R( L)(t t ) 1 i
ij 1 1/21/2
1 1
S..(t t )
1 1 1



and since the turbulence is homogeneous and stationary


then Equations (2.29) and (2.31) become

d 2 1/21/2 (L) + R ())d
d = Ht I j 1 3 1



2 1/2 2 1/2 t (L) (L))dg.
= 1/212 (t )(R () + Ri ))d (2.36)
11 J o

= =

The probability distribution of X.(t) is intimately

connected with the spatial distribution of themean concentra-

tion of the property transported by the fluid particle.

From this viewpoint Batchelor considers a volume of marked

fluid particles with a probability that a fluid point de-

fined by the position vector X lies within the marked fluid

at time t. Note that P(X,O) is unity or zero depending on

whether the point X lies within or outside the initial volume

of marked fluid.

Now by using the probability associated with the

dispersion of a fluid particle, the probability density

function, P(X,t), can be related to its initial p.d.f.,

P(X,O). Each particle diffuses independently of its neigh-

bors. Representing by Q(X,t) the p.d.f. of the vector dis-

placement X of any fluid particle during a time t, then

Q(X,t)dv(X) is the probability that a fluid particle lies

within a volume element dv(X) at time t. Furthermore, the

p.d.f., P(X,t), for a finite volume of marked fluid is

P(X,t) = fP(X',O)Q(X-X',t)dv(X'). (2.37)

If the form of Q(X,t) is known, the solution for

P(X,t) can be obtained from (2.37). The experimental re-

sults of Schubauer (1935), Collis (1948), and Kalinske

and Pien (1944) indicate that the separate p.d.f.'s of the

X's are normal for all values of t. The above data are

not sufficient to define the p.d.f. Q(X,t) in all its

generality. However, in order to make the analysis as

general as possible, Batchelor assumed the p.d.f.'s of

the XVs to be jointly as well as separately normal. In

this case it is known that the appropriate form of Q(X,t)


(x,t) 1/
(8 C)'/2


Q =

w X X
rs r s
exp(- 2Q )

Wrs is the co-factor of the typical element Qrs of the

determinant Q, and repeated suffixes implies summation

over values 1, 2, and 3. Thus Equation (2.37) becomes


P(,t) = P(X", exp(- )(X )(X ))dv(X") .
P(X3t) 1=)I/2 exp(- rr(XrX")(Xs -X))dv(X')
(87r 3 )1/ r rr s

Furthermore, Equation (2.40) is a solution to the diffusion


P E. 32P(X-t) (2.41)
t ij 3X X. '



provided the diffusion coefficient tensor E is such as to

satisfy the equations

E.i. 1 d (2.42)
13 13 2 dt

E..i. c. 1 2 d( rs/()
J1 ir Js 2 dt (2.43

Remembering that by definition

rir r = ij

where 6ij is the delta function, then Equation (2.43) can

be reduced to

E. ij 1 d < Xi(t)X (t)> (2.44)
1 1 d
Ei 2 dt 2 dt (t)(t (2.44)

which is also consistent with (2.42).

Thus the diffusion Equation (2.41) provides a

description of the turbulent dispersion process when

Eulerian analysis is used, provided the p.d.f.'s of X1,

X2, and X3 are jointly as well as separately normal. At

present, sufficient information is not available to sup-

port the above normality hypothesis rigorously; however,

it is a fortunate discovery that a single differential

equation describes the homogeneous turbulent dispersion

process with high accuracy. Even though this hypothesis

is presently (Monin and Yaglom, 1971) thought to be exact

or nearly exact, Equation (2.44) should be treated as a

phenomenological and not a fundamental law of turbulent


It is important to notice that the tensor coeffi-

cient E.. is a function of time, which is in contrast to

the molecular diffusion coefficient, and that Equation

(2.44) is valid at all times. The tensor E is a generali-

zation of the eddy diffusivity coefficient. From Equations

(2.29) and (2.35) the generalized eddy diffusivity coeffi-

cient can be expressed as

E. (t) = (S (C) + S .(M))dS (2.45)
Eij(t) =2o ij

or in terms of the Lagrangian velocity correlation coeffi-


E..(t) = < V 12 /2 2 >1/2
i2 (2.46)

It(R(L) + R())dS.

In addition, a generalized Lagrangian integral time scale

may be defined as


r(R(L.)l ) (L)
I 0 ij j i )) g

In view of (2.46), for large times the generalized eddy

diffusivity coefficient can be expressed

(E ) = 1/2 1/2 (2.48)
j T ij (2.4)

and one may easily generalize Boussinesq's phenomenologi-

cal hypothesis as

F(t) = -E *Vr (2.49)

where r is any scalar and F(t) is the turbulent flux.

The limiting cases for the fluid element displace-

ment covariance and eddy diffusivity are given below. For

small t

= i J I J

and E. (t) is proportional to t. For large t

= 1/2 1/2 T. .t + constant (2.51)
i ij i J J

and E. (t) is constant. Two-particle dispersion

To thoroughly describe turbulent dispersion in a

homogeneous flow field, all n-particle interactions should

be included in the statistical analysis, The single-

particle dispersion problem has been discussed in the

previous section, and the two-particle dispersion is dis-

cussed in this section. Due to increasing mathematical

difficulties, the analysis of the other n-particle inter-

actions have not been made. Even though a complete sta-

tistical determination of turbulent dispersion is desirable,

significant and workable models based on the presently

available works have been obtained.

As in the single-particle studies, the two-particle

analysis was first made for one-dimensional isotropic case.

This work was done by Richardson (1926). Batchelor (1952)

extended the two-particle analysis to the general homo-

geneous situation by investigating the relative displace-

ment and relative velocity of two fluid particles. Batch-

elor shows that even though the flow field may be assumed

stationary, the relative velocity covariance, ,

is not stationary; it depends on the actual elapsed time

t-to. The asymptotic form of the covariance is related to

Eulerian correlations as follows

2, (2.52)

and the relative displacement asymptotic form is related

to the single-particle displacement as

+ 2. (2.53)
1 J 1 J

Furthermore, under the normality assumption for the joint

probability distribution of the components of Y(t), the

conditional p.d.f. of the separation of two fluid particles

satisfies the diffusion equation,

tF(Y,t|Yo o) ij (t) F(Y,t Y to) (2.54)
1 J

where the relative dispersion tensor is determined by

Dij t (< i(t)Y (t)> ) (2.55)

ij(t) = ( vi( v (t )> + )dt (2.56)

Unfortunately, the diffusion equation does not apply to the

p.d.f. of the relative concentration.

2.1.3 Shear flow

The number of contributions in which there is a

theory developed on shear turbulence is very small when

compared to those contributions to isotropic turbulence.

Obviously this is because of the extreme complexity of

the problems encountered. Though the results are still

very meager, a few features of actual shear flow have been

studied and give some understanding of the nature of this

type turbulence.

Batchelor (1957) did some work in the area to free

turbulent shear flows. He extended the single-particle

analysis to the dispersion of a scalar to nonhomogeneous,

self-preserving turbulent flows such as exist in steady

turbulent jets, wakes, and mixing layers. Flow fields,

which are self-preserving, retain their structure with

elapsed time since only changes in the length, time, and

velocity scales occur. Using this property of the flow

field, Batchelor transformed this nonstationary dispersion

problem into a stationary one by an appropriate transfor-

mation of the Lagrangian velocity of the fluid particle.

An important result is that the lateral and longitudinal

dispersions, as functions of time, are proportional to the

average width of the shear layers at identical times. Also,

the time rate of change of the dispersion becomes constant

for large times. No convenient diffusion type equation was

found applicable as for the homogeneous turbulence case.

The study of turbulent shear flow has been mainly

directed at correlations between velocities and their

derivatives. One of the most reported or discussed studies

of this type is that of Laufer (1954). Laufer studied the

fully developed turbulent shear flow in a large pipe. He

not only investigated the second and third order correla-

tions but some fourth order correlations as well. The

importance of a detailed knowledge of flow conditions near

the wall is clearly suggested. The various turbulent

energy rates, such as production, diffusion, and dissipa-

tion, were shown to reach a maximum at the edge of the

viscous sublayer. It was also found that kinetic energy

is transferred away from the edge of the viscous sublayer

while an equally strong movement of pressure energy is

transferred toward the sublayer.

2.1.4 Interaction of molecular and turbulent dispersion

The effects of turbulent dispersion and molecular

diffusion are not easily separated. Taylor (1935) suggested

that the two are statistically independent, so that the vari-

ances due to these two effects are additive. Townsend (1954)

and Batchelor and Townsend (1956) showed that the two effects

are synergistic in a homogeneous turbulent field. They

concluded that for short times, the dispersion of a scalar

is accelerated since the stretching and rotational effects

of the turbulence would enhance the molecular diffusion.

It was decided that the total transport of a scalar depends

on the interaction of the two modes of transport in a de-

pendent fashion. Townsend confirmed his predictions experi-

mentally. He studied the heat transport behind a source in

homogeneous turbulence, and in addition, indicated adjust-

ments were needed on Uberoi and Corrsin's (1953) studies

of a similar transport problem. The adjustments had no

noticeable effect on the asymptotic eddy diffusivity ob-

tained by Uberoi and Corrsin. Mickelsen (1960) also found

the accelerated molecular diffusion is negligible for large

times in his experimental measurements of mass transport in

the flow field.

Later, Saffman (1960, 1962) added to the analysis

of Townsend and Batchelor. He pointed out that the molecu-

lar diffusion reduced the turbulent dispersion since the

scalar within the original fluid particle is diluted by

the molecular diffusion. However the two effects, when

not negligibly small, tend to balance themselves. Saffman

(1960) and Okubo (1967) have shown that the interaction re-

duces the dispersion relative to the origin from the value

it would have had if the two processes had been independent

and additive. Saffman claimed that his predictions were in

qualitative agreement with the measurements of Mickelsen

(1960). Both Saffman and Mickelsen suggested that further

experimentation was needed to resolve this problem for

large times. In general the small absolute value of the

molecular diffusion relative to the turbulent dispersion

allows one to neglect the purely molecular diffusion.

Monin and Yaglom (1971) present an excellent survey plus

supplementary results covering this area of study.

2.2 Lagrangian Characteristics of Turbulent Flow

The fundamental analytical approaches of Taylor

(1921) and Batchelor (1949) are basically derived from

the Lagrangian statistics of a turbulent homogeneous flow

field. Both experimental and numerical efforts have been

made to obtain the Lagrangian characteristics pertinent to

these studies. Ideally, the statistics should be generated

from numerous individual fluid point paths. Numerically,

the generation of fluid point paths is straightforward once

the statistical parameters and model have been decided upon.

However, to experimentally track or follow fluid particles

would be a very difficult if not impossible task. Approxi-

mate Lagrangian statistics have been obtained from experi-

ments which make possible the visualization of the paths of

small particles suspended in a turbulent flow field. These

small particles consist of a separate phase and retain their

identity, and their paths are thought to closely approximate

the paths of fluid points. In addition, attempts have been

made to extract Lagrangian statistics from the concentration

and temperature profiles generated by the turbulent disper-

sion of mass and heat, respectively.

2.2.1 Lagrangian numerical simulations

Several different stochastic models of turbulent

dispersion have been used to obtain the different Lagrangian

statistics which are of interest. Because of the mathe-

matical difficulties and the number of repetitive calcula-

tions involved in a numerical simulation, the digital

computer is required to expedite the solution. The capacity

and speed of the computer used often limits the complexity

and size of the simulation to be made.

Using a Monte Carlo procedure, Kirmse (1964) simu-

lated turbulent dispersion in pressure-driven nonhomogeneous

pipe flow. A proportionality parameter was used to relate

the variance of the Lagrangian acceleration, o to measur-

able Eulerian variances in the expression

0. = Ba u (2.57)

where o = variance of Eulerian velocity

o0 = variance of Eulerian acceleration.

The parameter "B" is similar to Mickelsen's (1955) Eulerian

to Lagrangian transformation parameter. By trial and error

the parameter "B" was adjusted until concentration profiles

obtained in the simulation coincided with available experi-

mental data. From a simple random walk model, Patterson

and Corrsin (1966) simulated a one-dimensional dispersion

process. A homogeneous velocity field was assumed and both

Lagrangian and Eulerian temporal velocity correlations were

obtained. The model generated results which appear to

represent the expected turbulent behavior.

Single-particle paths and their Lagrangian sta-

tistics were simulated by Kraichnan (1970) using a normal

multivariate velocity distribution. An incompressible,

stationary, isotropic flow field was assumed. The velocity

field was generated from a set of Fourier components ob-

tained from Eulerian correlations and spectra. Dispersions,

velocity correlations, and eddy diffusivities were computed

using the simulation. The results were consistent with

Taylor's earlier work.

Deardorff and Peskin (1970) generated single- and

double-particle trajectories using a three-dimensional

numerical model of shear flow within a channel. The study

was limited to regions outside of the viscous sublayer for

a large Reynolds' number. The model used was developed by

Deardorff (1970) and was based on averaging the Navier-

Stokes equations over grid volumes by the method of Reynolds

(Reynolds, 1895; Lamb, 1932). After generating a fully de-

veloped velocity field, forty-eight fluid particles at dif-

ferent initial points within the turbulent core were selected

and followed for two hundred time steps. A new set of fluid

particles was then selected with the same initial starting

points and the procedure repeated. An ensemble of ten such

sets of forty-eight particles was used to compute the

Lagrangian temporal mean square displacements and velocity

autocorrelation coefficients for single-particle tracks.

The decay of the two-particle velocity correlation coef-

ficients toward zero was very slow; and consequently, the

two-particle mean square separation did not reach the

asymptotic value of twice the single-particle mean square

displacement predicted by Batchelor (1952). The Lagrangian

time macroscales of the two-particle correlation were five

to twenty-five times greater than the corresponding single-

particle macroscales.

A numerical stochastic model depending on only

three parameters was proposed by Sullivan (1971) for fully

developed turbulent flow in an open channel. The two-dimen-

sional model uses only the bulk velocity, ub, the local

Eulerian space macroscale, AE, and the standard deviation

of the local lateral velocity fluctuations, ouy With an

initial lateral velocity equal to the local standard devi-

ation,a fluid particle is allowed to retain its velocity

for a time period of

T = (2.58)
mu ( o)

The longitudinal and lateral displacements at the end of

the time period T are

AE Y(tk)+t)
Ax =- ubdy (2.59)
v y(tk)


Ay = + AE- = + AE (2.60)

respectively, where

tk = (k l)T. (2.61)

From the new position, the values of ub, AE, and Uy are

corrected, and the sign of the next lateral displacement

is randomly assigned. By repeating this procedure, particle

trajectories of any time length could be generated. Sulli-

van generated one thousand particle tracks for ten initial

lateral points distributed evenly on the height of the channel.

Ensemble averages of the longitudinal concentration profiles

for given elapsed times and the temporal Lagrangian velocity

autocorrelation coefficients were obtained. In view of the

simplicity of the simulation, the agreement with experimental

data is surprisingly good,

Gielow (1972) numerically studied shear flow between

parallel plates using a stochastic model based on Kinnse's

(1964) model. Also by using the mean Lagrangian accelera-

tion obtained from the Reynolds' stresses, deterministic

velocity and position loci were found. By imposing the

condition that these deterministic loci are closed curves

in a coordinate system connected by the local mean velocity,

relationships were obtained to supplement and to check the

consistency of available experimental data used in the

stochastic model. In the model, the time derivative of

the velocity signal was chosen from a probability distri-

bution and integrated to give the velocity. Statistical

and spectral analyses of the simulated velocity records

were made to show the suitability of the model. Lagrangian

time correlations of the nonhomogeneous pressure-drive flow

were computed and compared to the same correlations for

homogeneous flow computed from the stochastic model. Be-

cause of the confining walls the eddy diffusivity obtained

from the Lagrangian correlations was clearly time-dependent.

The model was also used to simulate the turbulent transport

of a scalar such as heat or mass. The results for the

steady state simulations were consistent with available

experimental data,

2,2.2 Lagrangian experimental studies

The Lagrangian characteristics of fluid flow have

been investigated experimentally by many researchers. The

experiments performed in this area used "tagged" fluid

particles to observe both microscopic and macroscopic

turbulent activity. "Tagging" is accomplished by the

release of a visible foreign material into the flowing

fluid. The bulk of the experimental investigations have

been macroscopic in nature, i.e. due to the type experi-

ment or objectives, the attention was focused on the over-

all effect of the large scale nature of the turbulent

activity. The microscopic type of experiments deal mainly

with the individual fluid particles and their paths through

the flow field,

Hagen, in 1854, appears to be the first to report

the use of this "tagging" technique to study fluid particle

motion (Rouse and Ince, 1963). By observing the behavior

of suspended particles, Hagen discovered the existence of

two different modes of flow. Reynolds (1883) also, in his

well-known experiments using dye injection, demonstrated

vividly the gross Lagrangian characteristic of a flowing

fluid. Prandtl and Tietjens (1934b) discussed and pre-

sented results of their investigations using visual tech-

niques for liquid and gas flows.

In confined flows, the turbulent activity has been

visually observed in all regions of the flow field, the

core region as well as the wall region. Turbulence has

even been observed inside the viscous sublayer using the

particle "tagging" technique. Fage and Townend (1932)

and Fage (1936), using a microscope to view suspended

particles in a confined turbulent fluid, observed signifi-

cant turbulence in the viscous sublayer. In addition to

observing the qualitative nature of the turbulence, Fage

and Townend made quantitative computations of local turbu-

lent intensities. More recently, Corino and Brodkey (1969)

visually observed the turbulent fluid motions very near a

pipe wall, including the viscous sublayer, using a high-

speed movie camera moving with the flow. This study was

made in an effort to establish a physical picture which

related the turbulent activities near the wall to the

generation of turbulence and the turbulent transport pro-

cesses. With an apparent consistent average period, fluid

elements were observed to be ejected outward into the

turbulent core from a thin region adjacent to the viscous

sublayer. These ejections and resulting fluctuations were

the most important feature of the wall region and were

believed to be a factor in the generation and maintenance

of turbulence. Other studies of this type have been done

by Kline and Runstadler (1959), Nedderman (1961), Kim et

al. (1971), and Grass (1971).

Very few Lagrangian experiments of the microscopic

nature have been performed. The reason for the lack of

experimental work in this area is most likely because of

the laborious and tedious chore of following the particles.

Even of the experiments done, the main interest was in the

Lagrangian correlation of the velocity of a solid particle.

The limiting case of the particle velocity correlation is

the Lagrangian correlation function discussed by Taylor

(1921). Studies of the movements of neutrally buoyant

particles were made by Vanoni and Brooks (1955) and later

extended and improved by Frenzen (1963). Droplets of

nitrobenzene and olive oil were injected into the grid-

generated turbulence of a water tunnel. They made mea-

surements of successive displacements of the particles

and adjusted the velocity data to account for decay of

the turbulent intensity. The Lagrangian velocity auto-

correlation function was computed from the corrected data.

Kennedy (1965) measured the dispersion of 1250-micron soap

bubbles and 700-micron and 900-micron polystyrene beads in

the grid-generated turbulence of a vertical wind tunnel.

Kennedy did not measure particle velocities directly, but

inferred them from the initial slope of the dispersion

curve. Snyder and Lumley (1971) made some experimental

measurements of the particle velocity autocorrelation

function. Several types of particles were used in dis-

persion experiments within a wind tunnel designed after

Kennedy's. Within the experimental error, the particle

velocity correlations coincided with the Eulerian spatial

correlations, when the separation was made dimensionless

by dividing by the integral scales. Data generated using

hollow glass beads, were thought to be a good estimate of

the Lagrangian fluid properties. The Lagrangian time

integral scale was found to be approximated byAE/u', where

AE is the Eulerian integral scale and u' is the turbulent


2.2.3 Eulerian-Lagrangian transformation

A fundamentally correct theory of turbulent disper-

sion of fluid points is that of Taylor (1921). The major

reason this theory has not been put to common use is that

it is formulated in Lagrangian terms rather than the much

more easily measured Eulerian variables. As was indicated

in the previous section, quantitative measurements of the

Lagrangian characteristics are laborious and tedious, par-

ticularly since reliable ensemble averages required that

numerous individual particle paths be investigated. Under

these circumstances it is advantageous to develop a the-

oretical, or empirical if need be, relationship between

the Lagrangian variables and the more easily measured

Eulerian variables. As a start in this direction, Taylor

(1938) proposed that the changes at a fixed Eulerian point

are simply due to the passage of an unchanging pattern of

turbulent motion through that point. This concept has been

designated as frozen turbulence or Taylor's hypothesis. An

immediate result of this hypothesis is that time is given

by the displacement divided by the mean velocity, -x/ U

and in derivative form

1 a (2.62)
3x at

Taylor considered the hypothesis applicable only to uniform

low intensity turbulence. Lin (1953) and Uberoi and Corrsin

(1953) tested more fully the conditions under which Taylor's

hypothesis is useful. For isotropic turbulence and large

Reynolds' numbers, Lin found the accuracy of Taylor's hypo-

thesis can be estimated by

dU 2>
dt 5 (2.63)
2 U 2 2
<() >

For shear flow, the validity of the concept is less clear.

Laufer (1954) and others indicated experimentally the hy-

pothesis can be extended to fully developed shear flow

outside the viscous sublayer. An important application of

Taylor's hypothesis permits the analysis of Eulerian data

obtained from a truly Lagrangian type dispersion experiment.

Numerous Lagrangian studies have been made of turbulent

dispersion from continuous sources in uniform turbulence.

By applying Taylor's hypothesis to the above type experi-

ments in which Eulerian measurements are obtained at fixed

points downstream of the source, temporal mean functions

can be approximated. Each fixed position is assumed to

correspond to a certain elapsed time of a fluid particle

flight from the originating source.

Taylor's (1921) theory reduces the dispersion pro-

blem to that of determining the Lagrangian autocorrelation

function. The Lagrangian velocity correlation may be

extracted from the mean square displacement function by a

double differentiation as suggested by Equation (2.14).

In experiments in which the fluid particles were not followed

individually, the autocorrelation function, if obtained,

was found through the questionable step of doubly differen-

tiating the second moment of the distribution of particles.

Graphical differentiation is always difficult, but the

double-differentiation of an empirical curve, even one in

which the scatter of the data is seemingly small, can give

results which vary widely from one analyst to another.

Burgers (1951) proposed that the Lagrangian velocity corre-

lation may be approximated from an easily measured Eulerian

correlation as follows
2V 22 u 2
<( > 2 < ( > (2.64)
at ax

It has been suggested by several authors (Mickelson,

1955; Baldwin and Walsh, 1961; Hay and Pasquill, 1959;

Kirmse, 1964) that Lagrangian and Eulerian velocity corre-

lations are similar in shape and one can be used to generate

the other by scaling their arguments. For homogeneous tur-

bulence, Mickelsen (1955) investigated the similarity between

the Lagrangian velocity autocorrelation function and the

Eulerian longitudinal velocity correlation. Assuming a pro-

portionality between the Lagrangian and the Eulerian integral

length scales; i.e.

AE = BAL (2.65)

and through the definitions of the two length scales, it was

derived that

S = B 1/2 T (2.66)

5 is the separation distance between velocity vectors

in the Eulerian correlation, and

is the elapsed time for the Lagrangian correlation.

Kirmse (1964) further deduced that the above implies

0 = BuO (2.67)

The value of "B" was determined by Mickelsen to be approxi-

mately 0.6 for relative turbulent intensities of about 0.03.

This value of "B" was determined by comparing the doubly

integrated Eulerian longitudinal correlation evaluated from

hot-wire anemometry data with the corresponding value of the

Lagrangian turbulent spreading coefficient evaluated from

data collected on the dispersion of helium into air. Baldwin

and Mickelsen (1963) used the same value of "B" to fit heat

dispersion data. Hay and Pasquill (1959) related the La-

grangian velocity correlation to the Eulerian time correlation

by assuming a proportionality of the time arguments, i.e.

T = Bt (2.68)

where T is the Lagrangian time and

t is the Eulerian time.

The constant was evaluated from experimental data of the

integral of the Eulerian time correlation and the Lagrangian

dispersion in medium scale atmospheric turbulence. At a

relative turbulent intensity of about 0.14, f varied from

1.1 to 8.5 in an uncorrelated fashion. Baldwin and Walsh

(1961) also investigated the similarity in the Eulerian and

Lagrangian correlation coefficients. They assumed the re-

c = K (2.69)

By an incremental integration technique applied to both heat

and mass dispersion data, a twofold variation of the empiri-

cal factor KBW with T and turbulent intensity was observed.

Applying Taylor's hypothesis, can be related to KBW as


=- 12 KBW (2.70)

Philip (1967) theoretically developed a relationship for

the ratio of the Eulerian and Lagrangian integral time

scales for homogeneous, isotropic, stationary turbulence.

The experimental results already mentioned as well as

that of Angell (1964a, b) agree with Philip's results.

A general treatment and review of the theoretical

considerations of the relationship between the Eulerian

and the Lagrangian velocity correlation functionscan be

found in Hinze (1959), Brodkey (1967), and Monin and

Yaglom (1971).

2.3 Experimental Determination of Eddy Diffusivities

From the work of Taylor (1921) and Batchelor (1949),

it was shown that turbulent dispersion has its grass roots

implanted in the Lagrangian aspects of the flow field.

Unfortunately, direct measurement of the physically appeal-

ing Lagrangian characteristics is a laborious and awkward

task. The overwhelming majority of the experimental data

available on turbulent dispersion is of the more easily

measured Eulerian type. As was seen in the previous section

there is no completely satisfactory Eulerian to Lagrangian

transformation scheme. At this juncture there are two princi-

pal methods that may be used to analyze the Eulerian measure-

ments of turbulent dispersion. One method is to assume the

frozen turbulence hypothesis and use Taylor's analysis. The

second method is to retain the Eulerian description of the

phenomenon with the aid of Boussinesq's hypothesis. In

either case the eddy diffusivity seems to be the common

denominator of the two approaches. The concept of the eddy

diffusivity is a natural extension of the more fundamental

concept of the molecular diffusivity and was introduced as

a consequence of the phenomenological approach to the dispersion

of a scalar in turbulent fluids. Boussinesq (1877) pre-

sented an essay on the theory of flowing water, followed

by a book (Boussinesq, 1897), in which he discusses the

flux of momentum in open channels. Generalizing Newton's

law of viscosity, he developed the concept of an effective

momentum diffusivity, later to be referred to as the eddy

diffusivity. Numerous measurements of the eddy diffusivity

for heat and mass transfer have been made for systems hav-

ing various geometries and a wide range of Reynolds numbers.

Subbotin et al. (1966) correlated numerous data on the

radial component of the thermal eddy diffusivity and found

an empirical relationship that fits very well the data for

Prandtl numbers from zero to 10, and for Reynolds numbers

ranging from 10,000 to 500,000. The relationship depends

only on the Reynolds number and the dimensionless radial

position. Simplifications can be imposed for the core

region of the flow. General discussions of the eddy dif-

fusivity can be found in Hinze (1959), Brodkey (1967), and

Monin and Yaglom (1971).

2.3.1 Determination by Lagrangian analysis

Many Lagrangian type experiments have been performed

in which only Eulerian type measurements have been made.

In these experiments heat or mass was dispersed from dif-

ferent source configurations by the turbulent fluid action.

Subsequent temperature and concentration profiles, re-

spectively, were measured at several stations located

downstream from the source. Then by assuming the frozen

turbulence hypothesis and applying Taylor's (1921) La-

grangian analysis to the mean square displacements obtained

from the profiles, certain Lagrangian statistics were com-


The simplest source configuration is of course the

point source. This configuration is most adaptable to

mass transfer experiments. Towle and Sherwood (1939)

studied the turbulent dispersion of carbon dioxide and

hydrogen from an injection point within the turbulent

core of pipe flow using air as the carrier gas. By mea-

suring concentrations at several points downstream from the

injection point, the eddy diffusivities were computed for a

Reynolds' number range from 12,000 to 180,000. The eddy dif-

fusivity was found to increase with Reynolds' number and to

asymptotically approach a constant value. Kalinske and

Pien (1944) obtained eddy diffusivities using Taylor's

analysis. They investigated the dispersion of liquids

within water flowing in an open channel. It was revealed

in this study that the scale of the turbulence enters di-

rectly into the eddy diffusion relationship and it must be

measured or estimated if dispersion in turbulent fluids is

to be predicted accurately. The dispersion from a point

source within packed and fluidized beds has been investi-

gated by Hanratty et al. (1956). These experimenters in-

jected dye into particle beds having various bulk densities

ranging from the pack bed through the fluidized bed condi-

tion to the case where no particles were present. Eddy

diffusivities were computed as functions of elapsed time

using Taylor's analysis. A nonlinear variation of the eddy

diffusivity with particle Reynolds number was observed.

With a constant Reynolds number, the eddy diffusivity was

found to vary also with the particle size. Finally, for a

certain particle size and a constant Reynolds number, the

eddy diffusivity increased with particle density to a maxi-

mum at a 0.35 solid fraction then decreased with particle

density until a fixed bed condition was reached.

Some heat transfer experiments were made using a

line source generated by stretching a wire across the flow

field and heating it by an electrical current. Schubauer

(1935) appears to be first in the use of this technique. He

measured the lateral spread of heat from a line source in

small scale isotropic air flow. Thermal eddy diffusivities

were computed from measured temperature profiles. A thor-

ough investigation of the dispersion of heat from a line

source was made by Uberoi and Corrsin (1953) using grid-

produced isotropic turbulence within a wind tunnel. The

mean temperature distributions were measured for systematic

variations in wind speed, size of turbulence-producing grid,

and downstream location from heat source. Temperatures and

velocity fluctuations were also measured. A comparison of

Lagrangian and Eulerian analysis for dispersion in the

non-decaying turbulence was performed. The ratio of Eulerian

to Lagrangian microscales was determined theoretically and

was shown to roughly agree with the experimental results.

Townsend (1954) carried out a similar experimental study.

Experimental work by Brodkey (1967) supports the Eulerian-

Lagrangian relationships developed by Uberoi and Corrsin.

Crum and Hanratty (1965) measured mean temperature profiles,

temperature fluctuations, and spatial correlations of the

temperature fluctuations in the wake of a heated wire. The

flow system was fully-developed turbulence inside of a pipe.

Traverses of the pipe were made at fixed stations down-

stream from the source to obtain the measurements. The

mean square of the temperature fluctuations was found to

have a relative minimum at the centerline and to have a

maximum at about two thirds'the radius. As the wall is

approached, the mean square fluctuation tends to zero. For

increasing distances downstream of the source, the maximum

of the mean square temperature fluctuation increased to an

apparent asymptotic value at large distances.

Other source configurations have been studied using

Lagrangian analysis. Taylor (1954) investigated the dispersion

of a "plug" of foreign material instantaneously inserted

into a fluid flowing in a pipe. The dispersion of the

"plug" for long elapsed times was the principal point of

interest in his study. Taylor adapted his one-dimensional

single-particle analysis to pipe flow. In spite of the

simplicity of the resulting model, the predictions for

long elapsed times of the longitudinal concentrations were

in excellent agreement with experimental results (Taylor,

1954; Batchelor et al. 1955). Hanratty (1956) theoretically

studied the transfer of heat in isotropic turbulent flow

between a line source-sink pair by superpositioning of

solutions for point source-sink pairs. Using Taylor's

exponential form of the Lagrangian velocity autocorrelation

function and an experimental thermal eddy diffusivity, Han-

ratty calculated temperature profiles between the source

and sink. The results agree well with experimental data

for heat transfer between parallel plates.

2.3.2 Determination by Eulerian analysis

The transformation of Eulerian data to Lagrangian

data is not necessary if a complete Eulerian analysis is

used. In the Eulerian approach it is assumed that the

mean-values of velocity, temperature, and concentration

can be calculated with the aid of a suitably chosen eddy

diffusion coefficient using Boussineq's hypothesis or the

mixing-length theories. In these theories the rate of

transfer of a property is taken to be proportional to the

spatial rate of change of the mean value of the property.

By application of the mass, energy, or Navier-Stokes equa-

tions to experimental data, the eddy diffusivity parameter

can be evaluated and used in numerical solutions to these


As in Lagrangian studies, the point source has also

been used in Eulerian analysis of turbulent dispersion.

Various atmospheric scales of turbulence have been measured

and analyzed using one-dimensional Fickian type models

(Akerblom, 1908; Taylor, 1915; Schmidt, 1917; and Defant,

1921). Richardson (1926) summarized the results of these

and additional workers. Roberts (1923) broadened the analy-

sis to multidimensional problems by using a constant value

of the mass diffusivity for each principal direction. Then

Sutton (1932) allowed for the components to be positionally

dependent as well. This model incorporated the experimental

facts then known. Sutton also proposed functional forms

of the Lagrangian velocity correlation and obtained dif-

fusivities for a nonhomogenous atmosphere which depended

only on the dispersion time. Sutton's model despite its

limited accuracy, is still one of the few theoretical

analyses that accounts for many atmospheric observations

(Monin and Yaglom, 1971). Recently, measurements of medium

and large scale atmospheric turbulence have been made by

Hay and Pasquill (1957, 1959), Smith (1961), and Angell

(1964a,b). Pasquill (1962) discusses the experimental

results up to 1962.

Based on experimental data, Pasquill (1961) and

Gifford (1961) presented semi-empirical nomograms which

could be used to predict scalar dispersion in the atmos-

phere. Meteorological services have made extensive prac-

tical use of the Pasquill-Gifford nomograms (Pasquill,


Further atmospheric studies in the early fifties

by Lettau (1951) and Davies (1954) have measured the com-

plete eddy diffusivity tensor including the off-diagonal

components of the tensor.

Experiments using point sources have also been

performed to investigate turbulent dispersion in nonhomo-

geneous confined flows. These studies have been carried

out in pipe flow as well as in packed bed flow. Bernard

and Wilhelm (1950) studied the dispersion of a tracer in-

jected in a gas flowing through a cylindrical packed bed.

Eddy diffusivities were found by fitting the experimental

data to analytical solutions of the mass equation. The

analytical solution was obtained by assuming the average

axial and radial diffusivity coefficients to be equal and

then solving the system of equations by separation of

variables. The diffusivities were found to increase with

Reynolds' number for the particle Reynolds' number of 5

to 2,400 studied. Considerable work has been done in this

area by Fahien and co-workers using a procedure developed

by Fahien (1954) for a point source. The axial component

of the eddy diffusivity tensor was neglected on grounds of

small gradients in that direction. This assumption enabled

an analytical solution of the mass equation for the axial

direction to be obtained using the separation of variables

technique. The radial component of the eddy diffusivity

tensor is allowed to vary, and the mass equation for the

radial direction is solved numerically.

This solution technique has been applied to both

fully-developed pipe flow and packed bed flow. Fahien

and Smith (1955) studied dispersion in a packed bed for

gas flows by injecting a tracer. The radial dependence

of the radial component of the eddy diffusivity tensor;

Err, was determined for flow conditions having a particle

Reynolds' number ranging from 12 to 15,000. This component

was found to have a relative minimum at the centerline, a

maximum near the wall, and tended toward zero at the wall.

The value of Err observed at the centerline was in good

agreement with the asymptotic values obtained by Hanratty

et al. (1956) for packed beds. Dorweiler and Fahien (1959)

confirmed these results and extended the interpretation of

the data. Using a modification of Fahien's procedure,

Seagrave and Fahien (1961) investigated dye injection

into water flowing in a pipe. They found that the turbu-

lent eddy diffusivity was greater than the molecular

diffusivity by a factor of 105 for Reynolds' numbers of

about 7,500. For the lower Reynolds' numbers, axial dis-

persion appears to increase in its importance. Similar

behavior was reported by Roley and Fahien (1960) for gaseous

flows. Additional data and eddy diffusivity analysis have

been obtained by Frandolig and Fahien (1964) and Konopik

and Fahien (1964).

Eulerian studies of heat transfer between parallel

plates have been conducted by a group of workers between

the years 1947 and 1970 under th leadership of Sage and

Corcoran (Corcoran, 1948; Corcoran et al., 1947, 1952,

1956; Cavers et al., 1953; Schlinger et al., 1953a,b;

Schlinger and Sage, 1953, Hsu et al., 1956; Sage, 1959;

Venezian and Sage, 1961; Chia and Sage, 1970). For Rey-

nolds' numbers ranging from 8,900 to 100,000, the mean

temperature profile of air was measured between a heated

source wall and the parallel sink wall. Only the lateral

heat flux was considered, thereby allowing only the

lateral component of the thermal eddy diffusivity tensor

to be calculated. The dependence of this component on

lateral position was found to be similar to dependence

found for Err in pipe flow by Fahien and co-workers. The

value of the thermal eddy diffusivity increases with in-

creasing Reynolds number for a given lateral position.

It was generally greater than the eddy viscosity. It was

found in the latter phase of the work that the lateral

heat transfer was influenced significantly by changes in

viscous dissipation. Since the work of Venezian and Sage

(1961), the eddy diffusivity results have been corrected.

The viscous dissipation effect (Chia and Sage, 1970) mani-

fests itself as a sigmoid increase of the lateral heat

flux away from the source wall and maximizes at the sink

wall. The correction required increases with Reynolds

number, being about four per cent at a Reynolds number

of 40,000 and thirteen per cent at a Reynolds number of

100,000. When the viscous dissipation effect is accounted

for, the thermal eddy diffusivity is found to be symmetric

about the centerline. Chia and Sage (1970) summarize the

results of this excellent series of works.

Heat transfer within a uniformly heated pipe has

been studied in the fully developed flow of air and water.

The principal investigations of air flow systems were done

by Deissler and Eian (1952), Nunner (1956), Schleicher

(1958), Abbrecht and Churchill (1960), Tanimoto and Han-

ratty (1963), Ibragimov et al. (1969, 1971), and Bourke and

Pulling (1970). Both the thermal entrance region and the


isothermal region have been investigated. Mean tempera-

ture profiles, mean square temperature fluctuations, and

the radial thermal eddy diffusivity have been measured.

Axial heat transfer was neglected, an assumption acceptable

for a Peclet number greater than 100 (Schneider, 1957).

The behavior of the eddy diffusivity profile for the axial

and radial direction was similar to that observed in above

mentioned studies of the diffusivity. In these studies

Reynolds numbers ranged from 7,000 to 71,000, except for

the work of Ibragimov et al.; a Reynolds number of 32,500

and 260,000 was used. The correlation between velocity and

temperature fluctuations was measured by Ibragimov et al.,

and they also determined the axial heat flux. The ratio be-

tween the local radial and axial turbulent flux components

is always smaller than one, with a maximum of about 0.70

midway between the centerline and the wall. Beckwith and

Fahien (1963), Truchasson (1964), Rust and Sesonske (1966),

and Smith et al. (1966) have investigated liquid flow systems.

Reynolds numbers varied from 5,000 to 243,000. All used

water as the liquid except Rust and Sesonske; they used

mercury and ethylene glycol. The corresponding Prandtl

numbers are 0.0018, 10, and 44 for mercury, water, and

ethylene glycol, respectively.

Analytical and seminumerical Eulerian investigations

of the mass and energy equations have been made for fully

developed turbulent flow within a pipe and between parallel

plates. Beckers (1956), Schleicher and Tribus (1957), Spar-

row et al. (1957) studied heat transfer for pipe flow using

empirical correlations for the thermal eddy diffusivity

and considered only the radial dependence. Gielow (1965)

considered fully developed flow in a pipe having a constant

temperature at the wall. He performed a numerical analysis

of the energy equation to obtain local mean temperature pro-

files. Kakac and Paykoc (1968) did a similar study for flow

between parallel plates with constant heat flux and constant

temperature at the walls. Again, since experimental data

are available for only the radial direction, only the radial

dependence was considered. Seminumerical investigations

of mass transfer was made by Russo (1965). Using experi-

mentally available data for various components of the eddy

diffusivity tensor, Russo obtained solutions to the mass




3.1 Experiment

The experiment's objective required the generation

of fully developed turbulent flow with subsequent observa-

tion of individual fluid point paths. To accomplish this

objective, liquid flowing in a glass pipe was chosen as the

flow system. Small, spherical particles were suspended in

the liquid and illuminated so their paths were made visible.

Ideally, the particles should be small in size and of the

same density as the fluid. However, the requirement of

visibility and availability necessitated a compromise. The

particles selected were hollow glass micro spheres with

diameters in the range of 10 to 100 microns. Particle

density was about 0.34 grams per cubic centimeter. These

solid particles were assumed to approximate fluid points.

The apparatus used to photograph the particle motion

consisted of three basic systems; (1) the flow loop in which

the turbulent motion to be observed was generated, (2) a

lighting system for illuminating small suspended particles

depicting the fluid motion, and (3) a unique camera for

photographing the particle motions over extended distances.

Figure 3.1 shows the integrated arrangement of these three


3.1.1 Flow loop

To insure fully developed turbulent flow, several

precautions were taken in the design of the flow loop.

These included (1) a constant head fluid source, (2) a

baffled entrance section to reduce disturbances caused by

piping, and (3) a sufficiently long entrance region to the

test section to allow the flow to develop fully.

A constant head source was necessary to eliminate

the pressure fluctuations created by the pump. Process

fluid, trichloroethylene (TCE), was pumped from a receiv-

ing and storage drum, through a filter, and to an elevated

drum for a constant head supply of fluid. The pump was a

centrifugal type equipped with a mechanical seal which

provided an excellent seal for the TCE and reduced the

potential of contaminating the process fluid with foreign

oils and greases present in other types of sealing arrange-

ments. A pot type filter assembly was used to remove

solids and contained five 0.304 meter (1 foot) long Fram

Corporation, CF10EIH filter elements.l These filter ele-

ments removed all particles above 5 microns in diameter.

Both the receiving and elevated drums were constructed

See Fram Corporation Bulletin No. 171.5.

-Z 0

L <




Li i,




_1 Li1

from standard 55-gallon drums with appropriate pipe coup-

lings welded into their walls to accommodate piping. An

internal standpipe was installed inside the elevated drum.

By pumping the TCE to this drum at a faster rate than was

required for the test section, the excess fluid flowed

down the standpipe maintaining a constant level in the

drum. The receiving drum was vented to the elevated drum

so that vapors entrained in the downcoming fluid would

seek their level in the upper drum without hindering liquid

flow down the standpipe. Sight glasses in both drums pro-

vided a convenient method for examining the quantity of

fluid in each drum.

From the bottom of the elevated drum, the TCE flowed

down a 0,0508 meter (2 inch) diameter aluminum pipe to a

reducing section before flowing into the horizontal test

section, a glass pipe 4.56 meters (15 feet) long with an

inside diameter of 0.0254 meter (1 inch). The reducing

section provided a reduction of pipe size as well as damp-

ing of abnormal fluid motion created by the piping configur-

ation. Internal straightening vanes were installed just

before the reducer. The flow cross section was divided into

axial quadrants by the vanes. This was thought adequate to

dissipate any large scale disturbances generated by the

piping configuration. Two fused glass joints were present

in the test section, one at the 0.304 meter (1 foot) position

and the other at the 1.52 meter (5 foot) position. Obser-

vations were made over 1.52 meters of pipe starting 1.22

meters (4 feet) from the last fused glass joint. This

arrangement provided 48 diameters of straight unobstructed

pipe for flow development. The first 1.52 meters of pipe

also provided a flow developing region even though two

fused glass joints were present. Following the observa-

tion section, the remaining length of pipe reduced the

influence of the exit region. The full length of the glass

pipe was supported at several points to facilitate leveling

and eliminate sagging of the pipe.

To minimize distortion of the particle visualiza-

tion, an observation section was constructed so that the

effects of the pipe's curvature were eliminated. Approxi-

mately 1.83 meters (6 feet) of the glass pipe were enclosed

within two glass plates and a supporting frame. Care was

taken in the construction of the enclosure to make sure the

angle between the glass plates was 90 degrees. The space

within the enclosure surrounding the flow tube was filled

with glycerin to yield two undistorted perpendicular views.

Glycerin has approximately the same refractive index as

glass and TCE, and thereby circumvents the distortion which

is present when viewing a bare circular glass pipe. It is

through the two perpendicular glass plates that the particles

are photographed, See Figure 3.2.

The process fluid was recycled to minimize the

amount of TCE needed for proper circulation in the loop.

The TCE exits the glass section through the side arm of

a tee into an expansion section. The tee allows passage

of fluid through the flow loop while enabling the flow

tube to be illuminated axially through a glass port

mounted to a second arm of the tee. A 0.0508,meter (2-

inch) aluminum pipe was used to return the TCE through a

rotameter to the receiving and storage drum. The rota-

meter was calibrated with TCE and had a capacity of 1.26

liters/sec (20 G.p.m.), yielding a maximum Reynolds number

for the glass pipe of 200,000. The temperature of the pro-

cess fluid was measured just before returning the fluid to

the receiving drum.

A small stream of the process fluid was passed

through a silica gel trap to remove water from the TCE.

This minimized corrosion of iron materials used in the

construction of the flow loop.

3.1.2. Lighting system

The suspended particles were illuminated using a

high intensity carbon arc lamp. Two 0.0634-meter (2.5-inch)

diameter condensing lenses were used to collimate the

light beam and enhance the dark field illumination. By

rotating a four bladed, 0.304-meter (12-inch) diameter disk

perpendicular to the axis of the light path and at the

focal point between the two condensing lenses, the illumi-

nation of the suspended particles was interrupted at a

constant frequency. By interrupting the light, a time

scale was imposed on the photographic images obtained.

The chopping frequency was sensed with the photo-

cell circuit shown in Figure 3.3. When one of the blades

of the disk came between the arc and the photo cell, the

illumination of the cell was reduced and its resistance

to electrical current flow increased. This increase in

resistance caused a corresponding increase in the voltage

drop across the photo cell. The reverse occurred when the

cell was illuminated. The voltage pulses generated in

this manner were counted by a General Radio model 1191

counter and the corresponding period displayed.

3.1.3 Camera

The camera was unique in that it took 80 pictures

simultaneously-- forty pictures for each view, along a

1.52-meter (5-foot) length of the flow tube. This length

corresponded to 60 pipe diameters. The camera construc-

tion is shown in Figure 3.4 and is described below. Forty

0.0159-meter (5/8-inch) holes were drilled and tapped in

two 0.0761-meter (3-inch) wide aluminum channels. Lenses,

0.012 meter in diameter and having a 0.138-meter focal

+15 VDC




Chopping Frequency Detection Circuit

Figure 3.3





Z4 ,


W -,



length, were mounted in copper bushings and inserted in

the drilled and tapped holes. In each lense mount, a

disk with a central 0.00159-meter (1/16-inch) diameter

hole was used as an aperture. This size aperture pro-

vided sufficient light to expose the film and resulted

in good image definition for a field depth of about 0.0254

meter (1 inch). The two aluminum channels were attached

to a 0.127-meter (5-inch) angle iron to establish a right

angle between the two views of the flow tube and give a

rigid camera construction. Each lense and aperture assembly

focused a 0.038-meter (1 1/2-inch) section of the flow tube

onto a corresponding frame of 35-millimeter photographic

film. Each sequential section of the flow tube was photo-

graphed on a corresponding sequential frame of the film.

Partitioned compartments for each lense prevented double

exposure on adjacent frames. The film was mounted on two

0.0761-meter (3-inch) wide aluminum channels, one for each

view and positioned 0.70 meter from the lenses, see Figure

3,5. The distance from the glass-air interface to the

lenses was approximately 0.72-meter. The camera assembly

was positioned over the flow tube and aligned such that

the projected image of the tube was centrally located on

the film. Each viewed direction was at approximately 45

degrees with the vertical, see Figure 3.2,



A dark room, 2.43 meters (8 feet) square by 2.43

meters (8 feet) high was constructed around the camera

area of the flow loop so as to yield a darkened volume

suitable for exposing the film. One shutter was installed

to admit a light beam along the axis of the flow tube into

the darkened room enclosing the camera.

3.1.4 Procedure

The procedure used to photograph the suspended

particles is given below:

(1) The system is charged with sufficient TCE to

avoid cavitation in the pump, approximately

303 liters (80 gallons),

(2) The pump is turned on, and the system is al-

lowed to reach thermal equilibrium at the

desired flow rate.

(3) A small 10-micron filter assembly is charged

with particles by allowing TCE to flow from

the pot filter through the small filter

assembly. In this step, particles collected

in the large pot filter are flushed to the

smaller filter and reused.

(4) By reversing the flow through the 10-micron

filter assembly, particles are entrained and

directed to the glass tube for observation.

The flow rate through the small filter

assembly is adjusted so that approximately

10 to 20 particles are entrained per 10-

second interval,

(5) Photographic film is then clamped into place

on the aluminum channel pieces and inserted

into the camera.

(6) With the shutter closed the carbon arc lamp

and light chopper are turned on. When the

period at which the light beam is interrupted

becomes constant, the shutter is opened for

approximately 7 seconds and closed again.

(7) The flow rate, temperature, and chopping period

are noted and recorded.

(8) The light source is turned off, and light chopper

is turned off.

(9) The film holders are removed from the camera,

and the film removed for processing as described

in the next section.

3.1.5 Processing of photographic film

High speed photographic film was used to photograph

the particles. Kodak, 35-millimeter, 2475 recording film

was selected because of its availability and inexpensive

cost. The ASA number rating for this film is 1600. After

exposure of the film as described earlier, the film was

removed from the film holder and processed in a dark room.

The film was processed for 30 minutes at 65 degrees Fahren-

heit in Kodak DK-50 developer. The developer solution was

prepared in the manner and in the proportions given in the

directions which came with the chemical. A 30-second stop

bath was used to neutralize the developing chemicals. Then

a 10-minute fixer step was used, followed by a 30-minute

water wash step. The fixer solution was prepared as pre-

scribed by Kodak directions. To reduce the possibility

of drying stains, the film was flushed with a solution of

Kodak Photo Flo 200. The film was then allowed to dry for

several hours.

3,2 Data Analysis

Before the Lagrangian correlations could be computed,

two transformations of the data were necessary. First, the

two-dimensional coordinates of two perpendicular views of

the same point in the flow tube were transformed into a

three-dimensional Cartesian coordinate point. Secondly, the

Cartesian coordinates were transformed into cylindrical

coordinates. The resulting particle points were then smoothed

and the Lagrangian correlations computed.

The photographed particle tracks were scanned

using a PDP-11 minicomputer in conjunction with a X-Y

recorder. The track images were projected into the plane

of the X-Y recorder by using a photo enlarger. To obtain

the coordinates of a particle's position, both parallelax

and the refraction at the glass-air interface were consid-

ered. The algorithm used to collect the two-dimensional

coordinates of the perpendicular views is given in Appendix


3.2.1 Transformation of photographic data

Using the laws of geometry and optics as applied to

the photographing system, transformations of the image can

be made to account for parallelax and refraction. In the

analysis that follows, the difference in refractive index

of the glass, glycerin, and the TCE is neglected. Figure

3.6 is used as a guide to the following discussion and a

means of defining the nomenclature. Note that the coordi-

nate perpendicular to D1 and D2 is measured from the center-

line of the pipe in the direction of "d", i.e. perpendicular

to the glass-air interface. For the present time consider

"d" to be the distance from the glass-air interface to the

centerline of the pipe. The analysis is started by first

relating the image in the photographing plane to the virtual

image in the glass-air interface. From similar triangles




3 ..J





< -


U 0

V p

H = G

Also using the law of refraction,

sin 8 (H2 2 )1/2
sin 60 (D-H)

((D-H)2 + d2)1/2

where "n" is the refractive index of the

tion (3,2), the ratio between D and H is

D d
H ((n2 )H2 + n2 )1/2

From Figure 3.6, it is obvious that

D D2 = D
H1 2 DH
H H = (1 +
H 2 H

(3 .1)


glass. From Equa-

found as

d -). (3.3)

((n2 1)H2 + n2p2)1/2

Using Equation (3.1), Equation (3.3) can be written for

coordinate direction m

m = (l +
G v

-((n2 1)G2 + n2v2) 1/2


m_ + d
m ((n2 )G2 + n2 2 17/2 (3.4)
((n )G2 + n v%1"

The length of "G ", i.e. the image height produced on the

film, is projected by the photo enlarger with a magnifica-

tion factor of "M"; therefore, the distance "Gm" on the

film is magnified to a length "Cm" according to

Cm = MGm" (3.5)

Using Equation (3.4) and (3.5), now the coordinate posi-

tions in the flow tube, (D, D2), can be related to the

coordinates measured by the computer in the plane of the

X-Y recorder, (C1,C2), as

Dm + d
Cm Mv y (3.6)
Cm ((n2 1)C2 + n2M2 2)1/2

The analysis given above was concerned with only

one view of the particle position; however, to fix the

three Cartesian coordinates of the particle position, two

such perpendicular views were necessary. Let D1 be the

coordinate direction corresponding the flow direction, i.e.

the common coordinate of two perpendicular views. Introducing

an index "j", j = 1 or 2,

tion (3.6) can be written


to denote the two views, Equa-

for view "j" as

dj D2
2 2 + n2 2 2V2 1/2
((n )C. + n M v )


where the ""


index denotes the complement of view "j".

D 2 1

In order to align the images projected and to evaluate the

terms J and d. of Equation (3.7), several datum points
vj J
were recorded for each view of the flow tube. An alignment

and scaling tape with inch markers on it was also placed

within the enclosure. This tape could be photographed from

both directions and thereby yield an axial scale as well

as known positions within two corresponding views of a tube

section. The two corresponding views of a flow tube sec-

tion were projected simultaneously into the plane of the

X-Y recorder. Figure 3.7 shows the arrangement of the

datum points taken. Adding another index, "k", to Equation

Dj Dl

D11 = D21

012 i D22

(View 1)

Axial Scale Tape

D115 'D 25 (D116 D126)
(1) (5) (6) (4)
(D111 D0121) (D114,D124)

(DI12,0122) (0113,Dl23)
(2) Flow Tube Walls
(1) (4)
(D211,D221) (D214', 224)

(D212 0222) (D213 D223)
(2) (5) (6) (3)

(D215,D225) (D216', 226)

Axial Scale Tape

(View 2)

Figure 3.7 Arrangement of Datum Points

(3.7) to denote the datum point within a view, Equation

(3.7) becomes

Djmk p- d D2k
V 2 2 2,1 /2
Cjmk j ((n2 1)C + n Mj~v )1 2 (3.8)

The value of "n" is known; however, the values of M
dj, and D-2k were evaluated from the datum points recorded

from each view. The method used to compute these constants

is discussed in Appendix B. Using Equation (3.8), all

points on a track image projected in the plane of the X-Y

recorder can be transformed to their corresponding Cartesian

coordinates within the flow tube. The second transformation

from Cartesian to cylindrical coordinates is straightforward

and can be obtained from any elementary algebra text.

3.2.2 Computation of particle velocity

The velocity of the suspended particle was obtained

from smoothed particle position data. A fourth-order poly-

nomial was used to represent the particle's trajectory over

a short distance. This order polynomial allows for second

order variations in the particle's acceleration. The poly-

nomial form is

S+ t + c t + ct
Xi = C1 + c2ti + 3 i + c5t i


where i designates the time coordinate number. Differenti-

ating Equation (3.9) gives the velocity as

2 3
V = c2 + 2c3t + 3c4t2 + 4c5t (3.10)

The coefficients of Equation (3.9) can be evaluated using

a least-squares fit over a short length of the particle

track which spans the point being smoothed. The coeffi-

cients found in this manner can now be used to compute

the velocity via Equation (3.10).

3.2.3 Computation of Lagrangian correlations

The statistical properties of the random motion

of a single fluid particle can be used to effectively

evaluate the dispersion of a scalar property within a

turbulent flow field. Particle position and velocity

with respect to time is a starting point from which cer-

tain statistical correlations can be generated that help

describe turbulent dispersion. The flow field of par-

ticular interest in this study is that of turbulent shear

pipe flow. Consider a fluid particle starting at time t

from a point x within the flow field. At any time during
the particle's flight, its velocity, V(t t ,x ), is equal

to the Eulerian velocity, U(t,x), at the point x which is

the point at which the fluid particle is located at time t.


V(tlt ,x ) = U(t,L)


or, expressing the velocity as a mean plus a deviation

V(t|,t x ) = + u'(t,x).


that the deviation in the fluid particle's velocity

the local mean velocity of the field is

V'(t to,xo)

X'(tlt ,x ) =
-_ 0-

= V(t|t ,xo) = u'(t,x) (3.13)

f V'(t tt oo )dt1
o 1

X'(t t ,x ) = i V(tllto.xo)dt

/ dt1



be the deviation of the fluid particle's position after an

elapsed time t-to from the position it would have had had

it been moved only by the local mean velocity.

The transport of a scalar quantity within a steady

nonhomogeneous flow field has not been theoretically derived



at the present time. However, from the studies of iso-

tropic homogeneous turbulent flow by Taylor (1921) and
Batchelor (1949) respectively, a functional relationship
between the eddy diffusivity tensor and the Lagrangian

correlations apparently exists. From the results of these

studies, it was shown that the eddy diffusivity tensor is

Eij(t) = d

E< (t) = ( + ). (3.17)

From the statistical analysis of single-particle trajectories,
the above Lagrangian correlations can be obtained, i.e.
and . Other correlations such as

and can be computed easily from

the data on the particle's position and velocity at discrete
time intervals.

In the case of cylindrical coordinates, the computa-
tion of displacements and velocities needs further explanation.

The application of Equations (3.13) and (3.15) is straight-
forward for the axial and radial directions, i.e.

V;(t to', ) = Uz(t t ,xo) (3.18)

V'(tl to x ) = V (tlt x ))


t t
z'(t t ,xo) = Uz(t It )dt i dt (3.20)
to to 1 1 d1 (3.20


r'(tltoX o) = V (tl tlo, )dtl. (3.21)
to r -o

The displacement and velocity deviation in the angular

direction needs additional interpretation. In reference

to the starting point or origin, x of a given particle,
the angular velocity is defined as

V(tt ,x ) = r de(tto xo) (3.22)
0to -o o0 dt

and the angular displacement follows from Equation (3.15) as

r e'(t toX ) = i V (tlltoo)dtl (3.23)

where r is the value of the radius at time t = t Note

that if the angular velocity is defined using r(t t ,x ),

then when Equation (3.15) is used to compute the angular

displacement, the displacement would not necessarily have

the proper value of zero for a closed path. Using the

above definitions for displacements and deviations in

velocity, the elements of the eddy diffusivity tensor, E,

are consistent with Equation (2.49) where


Vr i + + (3.24)
=r -r r 36-6 z-z3

for cylindrical coordinates, and

i is the unit vector in the radial direction

S is the unit vector in the angular direction

iz is the unit vector in the axial direction.



Paths of suspended particles were photographed over

large distances, and the turbulent intensities and Lagrangian

correlations were computed from the paths observed in a 0.0254

meter (1 inch) I.D. diameter pipe. The Reynolds number at

which the observations were made was 110,000. Particle paths

were observed for elapsed times as great as 1.2 seconds, cor-

responding to an axial distance of 1.32 meters (4.33 feet).

Seven particle tracks were examined with a total of 1600

time intervals recorded. The time period of the observa-

tions was 6.16 X 103 seconds. The direction of the par-

ticle paths varied very little over this interval of time,

so that the images of the paths were nearly straight lines

with very little curvature.

4.1 Results

The particle track data were used to compute both

turbulent intensities and Lagrangian correlations. These

variables were computed for five radial intervals. The

annular zones formed by the radial intervals are shown in

Figure 4.28. In view of the frequent use of homogeneous

models for turbulent pipe flow and the relatively small

sample size, the overall averages of the correlations for

the five zones were computed.

4.1.1 Turbulent intensities and shear stresses

The computed relative turbulent intensities are

shown in Table 4.1, and the relative shear stresses are

given in Table 4.2. The relative turbulent intensities

agree with those given by Patterson and Zakin (1967) for

organic solvents using hot-film anemometry. However, the

radial variations are not in agreement. This lack of agree-

ment probably stems from the relatively small number of points

available to form ensemble averages for the five radial zones.

The relative radial turbulent intensities are also in fair

agreement with published values by Laufer (1954) for air and

by Cermak and Baldwin (1964) for water. The relative tur-

bulent shear stress, < (t )V (t )>/Uis in good agree-
bulent shear stress ment with published data. The relative shear stress,

/U ,MAX appears to be zero as expected. How-
r z o o z;MAX'
ever, the nonzero results for /U X cannot

be explained. Both shear stresses and

should be zero since there is symmetry in

the tangential direction.

4.1.2 Lagrangian correlations

The computed correlations for fully developed turbu-
lent pipe flow are shown in Figures 4.1 through 4.27. Figures

Table 4.1 Relative Turbulent Intensities


0.0 to 0.2
0.2 to 0.4
0.4 to 0.6
0.6 to 0.8
0.8 to 1.0

Vr2(t )1/2




V 2 (t ) 1/2



Table 4.2 Relative Turbulent Shear Stresses

Interval, V (to) (to) Vz(to)Vg(to) Vr(to)V (to)
2r/Do U2 M U2 U2 A
____z ,MAX z ,MAX z ,MAX

0.0 to 0.2 0.000592 0.000543 -0.000160
0.2 to 0.4 0.000562 -0.000204 0.000240
0.4 to 0.6 0.000537 0.000387 -0.000013
0.6 to 0.8 0.001015 0.000851 0.000044
0.8 to 1.0 0.000709 0.000581 -0.000447
Overall 0.000740 0.000521 -0.000035

4.1 to 4.3 give a representative sample of the results

obtained for the 5 radial zones. However, since the sample

size is relatively small, only the overall correlations

given in Figures 4.4 through 4.27 will be discussed. Speci-

fic observations about the correlations are made below.

The time period between observations is an adjusted

3.08 X 10-3 seconds or 0.145 dimensionless time units. One

dimensionless time unit, t* = 1, corresponds to the time re-

quired to traverse the pipe diameter at a velocity equal to

the maximum centerline velocity.

Displacement-displacement correlations, .

The overall displacement-displacement correlations are given

in Figures 4.4 to 4.9. These correlations are normalized

to the pipe diameter, D The correlations form a symmetric

second-order tensor; therefore, only the six elements pre-

sented need to be considered. The diagonal component for

the axial direction, , is at least an order of mag-

nitude greater than the other components, except for the

diagonal component for the tangential direction, <(r e'(t))2>.

The shape of the correlations of the diagonal com-

ponents of the above tensor are the form expected. The cor-

relation for the spread in the axial direction, Figure 4.4,

does not contain the time linear portion predicted by Equa-

tion (2.51); however, the correlation apparently does tend

towards an inclined straight line for large elapsed times.

The shape of the curve is similar to that reported by

Gielow (1972) for shear flow between parallel plates. For
the radial diagonal correlation, in Figure 4.5,

the effect of the confining pipe wall is evident for the

larger elapsed times. There is an asymptotic limit for

the radial displacement correlation because of the confin-

ing walls of the system. This limit is found by assuming

a random dispersion of the particles from an initial radial

position, ro, i.e. the mean square displacement from ro at

infinite time is

2 (r r dr D2 2D ro

2(_ r) rdr o2 2r0 + r2 (4.1)
Do 8 3 o

27r rdr

For the overall correlation, the particle may start at any

radius; therefore, the asymptotic limit is

2 0 2D r
27r (_ 0 0 + r )rdro D2
8 3 o o

2 (4
2rr f r dr

Since the ordinate of Figure 4.5 is normalized relative to

D2, the asymptotic limit for the ordinate of the figure is

1/36 or 0.0278.

The off-diagonal components of the tensor are given

in Figures 4.7 to 4.9. These correlations are of the same

order of magnitude as the radial diagonal correlation. The

positive value of the correlation between the axial and

radial components of the fluid particle displacement vector,

in Figure 4.7, is to be expected. Due to the

mean velocity profile, if a fluid particle has a positive

displacement in the radial direction, it has moved to a

flow region of lower mean axial velocity. Consequently,

the average effect would be a positive axial deviation in

the particle's displacement. Similarly, a negative radial

displacement leads to a negative axial displacement, on the

average. The product of the two displacement combinations

is positive, thereby resulting in a positive correlation.

The two remaining correlations are expected to be zero be-

cause of angular symmetry in the pipe. This appears to be

the case for the correlation between the radial and tangen-

tial components of the fluid particle displacement vector,

in Figure 4.9. However, the correlation be-

tween the axial and tangential components,

in Figure 4.8, has a significant positive nature. This

probably occurs because of the relatively small number of

samples present in the correlation. If numerous samples

had been taken, the effect of multiple crossings of the

starting angular position could reduce the magnitude of

the correlation.

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