LAGRANGIAN ASPECTS OF
TURBULENT TRANSPORT IN PIPE FLOW
By
David Lee Breton
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
ACKNOWLEDGMENTS
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-
ship.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS..................... ................ ii
LIST OF TABLES........................... .. .......... v
LIST OF FIGURES .. ............ ....... ............. ...... vi
KEY TO SYMBOLS ............................. ......... x
ABSTRACT ................ ............ ..... ........... xiv
CHAPTER
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
2.1.2.1 Single-particle
dispersion............. 13
2.1.2.2 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
velocity...........................
3.2.3 Computation of Lagrangian
correlations......................
4. RESULTS AND DISCUSSION............ ............
5.
APPENDIX
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......
SUMMARY AND CONCLUSIONS................
Page
53
53
54
59
60
65
68
74
75
80
....... 80
........ 81
........ 81
........118
........121
.......150
A. DATA COLLECTION ALGORITHM......................152
B. SOLUTION FOR COEFFICIENT TO X-Y
RECORDER COORDINATE TRANSFORMATION
EQUATION....................................... 167
C. COMPUTER PROGRAM FOR COMPUTING THE
CYLINDRICAL COORDINATES OF PARTICLE
PATHS ..................... ................. 171
D. COMPUTER PROGRAM FOR THE COMPUTATION
OF THE LAGRANGIAN TIME CORRELATIONS........... 177
BIBLIOGRAPHY.......................................... 185
BIOGRAPHICAL SKETCH ..................... ............. 199
LIST OF TABLES
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
LIST OF FIGURES
Figure
Page
... .. 55
... .. 58
. . . 6 1
...... 62
... .. 64
...... 69
...... 73
...... 91
...... 92
3.1 Experiment Apparatus..................
3.2 Observation Enclosure and Camera
Arrangement............................
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
o
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*...............
o
4.5 <(r'(t))2>/D2 Versus t*...............
0
4.56 <(r '(t))2>/D 2 Versus t*.
4.6 <(r o '(t) >)2 /D2 Versus t*.............
2
4.7 6 z(t)rit)'/D Versus t.
4.8 /D2 Versus t* ............
0
4.9 /D2 Versus t*...........
0
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
.94
. 95
96
.97
.98
99
..100
..101
..102
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
<(VN(to))2>
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
KEY TO SYMBOLS
a
a2
a2
A
B
C
c
C
d
D
D
0
D
E
Em
f
F
F
F(t)
F
9
G
G
H
- 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
smoothing
Fourier transform of g(t)
image height on film, L
virtual image height at air-glass interface of ob-
servation section, L
L
les
re
i unit vector for radial direction
--r
i unit vector for axial direction
--z
i unit vector for tangential direction
-9
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
trichloroethylene
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),
L2/t2
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
Subscripts
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
Superscripts
fluctuation about the mean
* dimensionless variable
- time average
< > expected value (ensemble average)
xiii
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
LAGRANGIAN ASPECTS OF
TURBULENT TRANSPORT IN PIPE FLOW
By
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.
CHAPTER 1
INTRODUCTION
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
2
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.
CHAPTER 2
LITERATURE REVIEW
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
term
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)
dt
and its position at time t, x(t),
x(t) = a + tV(tl)dtI
(2.3)
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.
(2.4)
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.
then
(L)
R( (tl) 2 (2.7)
One measure of the degree of dispersion is given
2
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)
-t
0
= 2 fdt1 (2.11)
-t
2 (L)
= 2I R( (tl)dtl. (2.12)
-t
For constant , R(L)(t ) is symmetric about zero, and
therefore
d ,2 2 (L
-d- = 2 f R(L)(tl)dtI. (2.13)
0
Integrating Equation (2.13) yields
= 2 2R(L)(t )dt1dt2. (2.14)
0o
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)
and
E(t) = 2t
(2.17)
Looking further, as t becomes large
R(L)(t) 0 (2.18)
a t (2.19)
and
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
diffusion.
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)
or
t t
1x(t)X.(t)> = Idtl+ /dtl (2.27)
o 0
Using tensor notation, define the second order tensor S as
follows:
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
t
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
= (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 )
1/21/2
1 1 1
(2.32)
(2.33)
and since the turbulence is homogeneous and stationary
(2.34)
then Equations (2.29) and (2.31) become
d 2 1/21/2 (L) + R ())d
d =
Ht I j 1 3 1
(2.35)
and
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)
is
(x,t) 1/
(8 C)'/2
where
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
(2.40)
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
equation,
P E. 32P(X-t) (2.41)
t ij 3X X. '
1]j
(2.38)
(2.39)
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
dispersion.
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-
cient
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
(2.47)
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
13
= 1/2 1/2 T. .t + constant (2.51)
i ij i J J
and E. (t) is constant.
2.1.2.2 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)
or
t
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
AE(Yo)
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)
and
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
intensity.
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
2
dU 2>
dt 5 __ (2.63)__
__2 U 2 ____2__
__<() >__
3x
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)__
where
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-
lationship
__1/2__
c = K (2.69)
KBW
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
follows
=- __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-
puted.
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
equations.
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,
1971).
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
equation.
CHAPTER 3
EXPERIMENT AND DATA ANALYSIS
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
systems.
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
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L <
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Li
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LD
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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
PHOTOCELL VOLTAGE SIGNAL
TO COUNTER
PERIOD
TYPICAL PULSE TRAIN PRODUCED
BY LIGHT CHOPPER
Chopping Frequency Detection Circuit
Figure 3.3
62
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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,
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In
4
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
A.
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
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G H
V p
H = G
V
Also using the law of refraction,
H
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)
(3.2)
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
D
m = (l +
G v
d
-((n2 1)G2 + n2v2) 1/2
(+
or
D
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
D
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
M I
to denote the two views, Equa-
for view "j" as
dj D2
2 2 + n2 2 2V2 1/2
((n )C. + n M v )
J J J
(3.7)
where the ""
Note
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
UPPER VIEW
(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
LOWER VIEW
(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)
ik
The value of "n" is known; however, the values of M
J J
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
(3.9)
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
-0
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.
Therefore
V(tlt ,x ) = U(t,L)
(3.11)
or, expressing the velocity as a mean plus a deviation
V(t|,t x ) = __ + u'(t,x).__
(3.12)
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)__
t
f V'(t tt oo )dt1
o 1
t
X'(t t ,x ) = i V(tllto.xo)dt
t
/ __dt1__
I
(3.14)
(3.15)
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
Note
from
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
or
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 ))
(3.19)
t t
z'(t t ,xo) = Uz(t It )dt i dt (3.20)
to to 1 1 d1 (3.20
and
t
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,
--0
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
t
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
79
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.
-z
CHAPTER 4
RESULTS AND DISCUSSION
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
Radial
Interval,
2r/Do
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
Overall
Axial
Vr2(t )1/2
Uz,MAX
0.0927
0.0799
0.0695
0.0869
0.0879
0.0819
Radial
Vr2(to)1/2
Uz,MAX
0.0295
0.0315
0.0351
0.0397
0.0344
0.0359
Tangential
V 2 (t ) 1/2
Uz,MAX
0.0367
0.0378
0.0400
0.0470
0.0451
0.0429
Table 4.2 Relative Turbulent Shear Stresses
Radial
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
2
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
Do
2 (r r dr D2 2D ro
2
2(_ r) rdr o2 2r0 + r2 (4.1)
Do 8 3 o
27r rdr
0
For the overall correlation, the particle may start at any
radius; therefore, the asymptotic limit is
Do
2 0 2D r
27r (_ 0 0 + r )rdro D2
8 3 o o
2 (4
2rr f r dr
0
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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