Experimental Study of Lagrangian Velocity and Energy Statistics in Inhomogeneous Turbulence
81
Wesleyan University The Honors College Experimental Study of Lagrangian Velocity and Energy Statistics in Inhomogeneous Turbulence by Surendra Bahadur Kunwar Class of 2010 A thesis submitted to the faculty of Wesleyan University in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Departmental Honors in Physics Middletown, Connecticut April, 2010
Experimental Study of Lagrangian Velocity and Energy Statistics in Inhomogeneous Turbulence
Experimental Study of Lagrangian Velocity and Energy Statistics
in
Inhomogeneous Turbulence
faculty of Wesleyan University
Degree of Bachelor of Arts
with Departmental Honors in Physics
Middletown, Connecticut April, 2010
Lagrangian velocity and energy statistics are studied in
inhomogeneous tur-
bulence. A Rλ = 285 flow between two oscillating grids, with
regions of nearly
homogeneous and highly inhomogeneous turbulence, is studied. Large
data sets of
three-dimensional tracer particle velocities have been collected
using stereoscopic
high speed cameras with real-time image compression technology.
Lagrangian
structure functions conditioned on the instantaneous large scale
velocity are mea-
sured in both homogeneous and inhomogeneous regions of the flow to
assess the
effects of the large scales on the small scales in turbulence. At
all scales, the
structure functions depend strongly on the large scale velocity,
the dependence
showing clear signatures of inhomogeneity near the oscillating
grids. But even in
the homogeneous region in the center, a strong dependence on the
large scale ve-
locity remains at all scales. The conditional structure function
measurements are
powerful tools in assessing the effects of inhomogeneity and
intermittency of the
large scales on the small scales in turbulence. There is a bias
present in the mea-
surement of Lagrangian structure functions at all timescales due to
the finiteness
of the measurement volume. A method we developed to estimate such
bias for
different timescales in our data is described. Our method gives an
extrapolation
based on the structure functions for different artificial volumes,
revealing the es-
timates of errors at different timescales. Analysis of structure
functions is limited
to the timescales whose errors do not exceed 17%. Components of the
turbulent
kinetic energy budget are estimated to identify the chief agents of
turbulent energy
transport using Lagrangian energy measurements. Our estimation of
the terms
in the turbulent energy equation identifies pressure transport as
being significant,
along with the velocity transport term, in turbulent energy
transport. These en-
ergy measurements provide a basis for further studying Lagrangian
and Eulerian
energy transport in turbulent systems. Lastly, we study the average
decay of the
kinetic energy of a particle as it enters a measurement volume in
the shape of a
slab or a cube. We see that the bias due to trajectory length
affects the energy
measurement. When this sample bias is removed, particles lose
two-thirds of their
energy during their residence time.
Acknowledgment
At the beginning of my second semester at Wesleyan, when I couldn’t
get into
a class that I wanted to, my faculty advisor Prof. Greg Voth showed
me around
in the lab, and suggested me to do research with him. Having been
fascinated by
the stories of scientists since childhood, I had no hesitation in
trying out research.
More than three years later, I look back and appreciate that Prof.
Voth has
guided me in research, academics, career and life in general. I
will always be
deeply grateful to him for his steady and encouraging efforts to
develop me as a
physicist, both in lab and in classroom. I have no doubts that the
opportunity
that Prof. Voth provided me to be his Research Student for three
summers were
indispensable in the making of this thesis.
All the data that I used for analysis have been produced by Dan
Blum, a
PhD student of Prof. Voth. Without Dan’s diligence and helpfulness,
it would
be unlikely for me to achieve what I have done today. I sincerely
thank him for
everything, including information and nicely formatted figures from
a recently
published paper.
My parents have always strived to give me the best possible
education, and
this has been instrumental in building my academic foundation.
Without my
family’s love, dream and endeavors, I wouldn’t be able to undertake
this thesis
project successfully today.
During my years at Voth lab, I have had the pleasure of interacting
with, or at
least knowing, superb graduate and undergraduate students. Some
were involved
in building components of the experiment that I have studied. Many
contributed
to building the solid tank and oscillating grid system, and the
image compression
circuit is a fruit of the hardwork of a long list of advisees of
Prof. Voth. They all
contributed to my research, and I am extremely thankful for their
efforts. I am
also deeply indebted to the Physics Department at Wesleyan
University for being
warm and family-like in nurturing young students like me. Also, all
my labmates,
my friends and my hallmates have been greatly supportive and
enthusiastic about
my thesis and research. It has been a pleasure talking about my
research with
them, and I have always been positively driven by their interest in
what I am
doing.
Contents
1.1 Fluids and Navier Stokes equation . . . . . . . . . . . . . . .
. . . 2
1.2 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . .
. . . 3
1.5 Kolmogorov Theory . . . . . . . . . . . . . . . . . . . . . . .
. . . 6
1.7 Turbulence and Research . . . . . . . . . . . . . . . . . . . .
. . . 9
2 The Experiment 12
2.2 Demands of Lagrangian statistics . . . . . . . . . . . . . . .
. . . 16
2.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 17
2.3.1 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 17
2.4 Stereomatching: the final step . . . . . . . . . . . . . . . .
. . . . 19
3 Measurement Volume Bias in Lagrangian Statistics 22
3.1 The possible sources of bias . . . . . . . . . . . . . . . . .
. . . . 22
3.2 Estimating the possible bias in structure functions . . . . . .
. . . 24
3.2.1 Structure function vs volume . . . . . . . . . . . . . . . .
25
3.2.2 Particle density vs volume . . . . . . . . . . . . . . . . .
. 27
3.2.3 Structure function vs volume again . . . . . . . . . . . . .
29
3.3 Extrapolating the Structure Function . . . . . . . . . . . . .
. . . 30
3.4 Comparison of errors . . . . . . . . . . . . . . . . . . . . .
. . . . 34
4 Conditional structure functions and their significance 36
4.1 Eulerian Structure functions . . . . . . . . . . . . . . . . .
. . . . 37
4.2 Determining the large and small scale velocities . . . . . . .
. . . 38
4.3 Conditioned structure functions . . . . . . . . . . . . . . . .
. . . 39
4.3.1 Eulerian structure functions . . . . . . . . . . . . . . . .
. 39
4.3.2 Lagrangian structure functions . . . . . . . . . . . . . . .
. 40
4.4 Effects of Inhomogeneity . . . . . . . . . . . . . . . . . . .
. . . . 44
4.5 Other Causes of Dependence . . . . . . . . . . . . . . . . . .
. . . 45
4.5.1 Kinematic Correlation . . . . . . . . . . . . . . . . . . . .
45
4.5.2 Reynolds number . . . . . . . . . . . . . . . . . . . . . . .
46
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 49
5.1 Decomposition of quantities . . . . . . . . . . . . . . . . . .
. . . 52
5.1.1 Velocity decomposition . . . . . . . . . . . . . . . . . . .
. 52
5.2 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . .
. . . 54
5.2.1 Reynolds equation . . . . . . . . . . . . . . . . . . . . . .
54
5.2.4 Energy budget for the center of the tank . . . . . . . . . .
59
5.3 Energy decay . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 60
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 64
6 Conclusion 65
List of Figures
2.1 A diagram of the experiment containing the tank illuminated
by
the laser beam, the cameras and the grids. . . . . . . . . . . . .
. 13
2.2 Flowchart showing the determination of 3D velocities from
camera
images of tracer particles . . . . . . . . . . . . . . . . . . . .
. . . 21
3.1 Second order LVSF vs timescale for different artificial
measurement
volumes, with radii ranging from 0.5 cm to 4.5 cm in increments
of
0.5 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 26
3.2 Comparing the average particle density at each measurement
volume. 28
3.3 Second order LVSF vs timescale for different artificial
measurement
volumes, with radii ranging from 0.8 cm to 2 cm. . . . . . . . . .
30
3.4 Second order LVSF vs measurement volume radius r for
different
timescales up to the inertial subrange . . . . . . . . . . . . . .
. . 31
3.5 Second order LVSF and its functional fit vs measurement
volume
radius r for distinct timescales up to the inertial subrange . . .
. . 33
3.6 Second order LVSF (both the data and the extrapolation) vs
timescale 34
viii
4.1 Eulerian second order conditional structure function versus
large
scale velocity at the center of the tank. . . . . . . . . . . . . .
. . 40
4.2 Lagrangian second order conditional structure function versus
timescale
for different instantaneous velocities at the center of the tank. .
. 41
4.3 Lagrangian second order conditional structure function versus
large
scale velocity, for different timescales at the center of the tank.
. . 43
4.4 Lagrangian second order conditional structure function versus
large
scale velocity, for different timescales in the near grid region. .
. . 44
4.5 Eulerian second order conditional structure function versus
large
scale velocity; comparing the dependence in different
Reynolds
number turbulence. . . . . . . . . . . . . . . . . . . . . . . . .
. . 47
scale velocity magnitude at the center of the tank. . . . . . . . .
. 49
5.1 The axial component of velocity against time on the
centerline
of a turbulent jet (figure taken from the experiment of Tong
and
Warhaft (1995) as published by Pope [1]). . . . . . . . . . . . . .
52
5.2 The mean kinetic energy (KE) of a particle vs time (as a
multiple
of Kolmogorov timescale at different detection volumes (slab
and
cube) in the center of the tank. . . . . . . . . . . . . . . . . .
. . 61
5.3 Mean KE vs time (as a multiple of Kolmogorov timescale)
along
particle trajectory for cubes of different sizes but with a
common
center at the homogeneous region of the flow. . . . . . . . . . . .
63
List of Tables
3.1 Comparison of the error estimates obtained by our method and
the
one Berg et al. developed, for different timescales. . . . . . . .
. . 35
4.1 Comparing the standard deviation of different components of
par-
ticle velocity at the center of the tank. . . . . . . . . . . . . .
. . 48
5.1 The estimated values of the production term, the dissipation
term,
and the velocity transport term in the turbulent kinetic
energy
equation for the center of the tank. . . . . . . . . . . . . . . .
. . 59
x
Theories
Turbulence is a common phenomenon in our everyday life. River water
flowing
at high speed, air in the wake of a speeding car, smoke emitted
from a power plant
chimney are all turbulent. Turbulence can be a nuisance at times.
Turbulence in
the air is a concern during flights, and turbulent water can be
very destructive
during natural calamities like floods. On many occasions,
turbulence is desirable.
Turbulent fluids are very efficient at mixing different components
in a solution.
This mixing property also makes it useful in combustion in engines,
as the fuel
needs to mix well with oxygen. Besides having such industrial
applications, tur-
bulence comes into play in a host of other processes like cloud
formation and
transport of pollutants.
Broadly speaking, the goal of my research is to better understand
turbulence.
1
1.1 Fluids and Navier Stokes equation 2
Before describing turbulence as my field of research and before
explaining my spe-
cific area of interest in turbulence, I will introduce some basic
physics concepts
relevant to the field.
1.1 Fluids and Navier Stokes equation
Before quantifying turbulence, it is important to understand what a
fluid is.
Anything that flows is a fluid. This definition includes all
liquids and gases.
However, not all fluids have the same macroscopic properties of
compressibility
and viscosity. As long as the fluid velocities remain low compared
with the speed
of sound, a fluid is incompressible, so air and water are
incompressible in a wide
range of circumstances. Also, they have constant viscosity and so
belong to the
class of fluids called Newtonian fluids. Let’s denote the velocity
field of a fluid by
U(r, t), where r is the position vector and t is time. Then, the
assumption that
the density of air and water do not change means the divergence of
their velocity
field is zero:
∇ ·U(r, t) = 0 (1.1)
Eq 1.1 is actually a result of the principle of conservation of
mass applied to a
constant density fluid [1]. The behavior of all Newtonian fluids
can be described
by the Navier Stokes equation, which is really a momentum
conservation equation
in fluids:
Dt = ∂U
∂t + (U · ∇) U; p(r, t) is the pressure field;
ρ is the fluid density; and ν is the kinematic viscosity of the
fluid. Sometimes, it
is convenient to write equations in tensor notation. The advantage
of expressing
Eqs 1.1 and 1.2 in tensor notation is clear when one derives other
equations
from them. Below are the tensor versions of the Navier Stokes
equation and the
’incompressibility’ equation:
DUj Dt
∂Ui ∂xi
= 0 (1.4)
Perhaps, it is also insightful to present the non-dimensional
version of Eq 1.2 to
see a powerful property of the Navier Stokes equation. This needs
the introduction
of the Reynolds number first.
1.2 Reynolds number
A widely used characteristic of fluid flows is Reynolds number,
defined as
Re = UL ν
(1.5)
where U and L are the characteristic speed and lengthscale of the
flow respectively,
and ν is the kinematic viscosity of the fluid. Reynolds number is
seen as an
indicator of the intensity of turbulence taking place in a flow-
the higher the
Reynolds number, the more intense the turbulence. Most flows show
an onset
1.3 Lagrangian Statistics and its Importance 4
of turbulence at Reynolds number of a few thousand. It should be
noted that
Reynolds number is dimensionless. Usually, turbulence researchers
use the Taylor-
scale Reynold’s number
15Re (1.6)
With the knowledge of Reynolds number, it is now possible to
understand the
non-dimensional form of Navier Stokes equation (Eq 1.2):
∂U
, p = p
ρU2 0
From Eq 1.7, it is clear that water and air will behave in the same
way if they
have identical Reynolds number and flow geometry. This is a big
advantage as it
makes possible the comparison of experiments using air with those
using water.
1.3 Lagrangian Statistics and its Importance
Before discussing the theories of turbulence, it is important to
understand La-
grangian analysis- the study of the properties of a fluid element
as it moves around
in the fluid. This is different from the Eulerian concept, which
involves monitoring
the behavior of the fluid at fixed positions in space. Temporal
evolution of the
properties of fluid elements could be studied in two ways. One is
keeping track
of a specific fluid element as it moves to different positions with
time. Another
is just recording the events happening at a fixed point in the
fluid with time.
1.3 Lagrangian Statistics and its Importance 5
The second method has a drawback, which will be clear once energy
cascade is
discussed in Section 1.5.
It is important to understand why Lagrangian statistics should be
explored. In
other words, why should it matter what a fluid particle does along
its trajectory
with time. Well, we want to understand whether the properties that
a particle
picks up at a particular locality of the fluid have any correlation
with its properties
some time later during its motion. For instance, our experiment is
a system where
kinetic energy is passed on to the fluid at an inhomogeneous region
of the fluid.
The particles then move to the homogeneous region of the fluid in
the course of
time. We want to understand whether the properties of the particle
at the inho-
mogeneous origin has any influence on its behavior in the
homogeneous region or
vice versa. The right way to do this is by tracking the particle
and studying the
correlation of its properties, say velocity values, at different
times. So the whole
point of Lagrangian analysis of turbulence is to figure out whether
fluid particles
‘remember’ their kinetic properties from some past time during
their motion in a
trajectory.
There are various Lagrangian quantities that can be measured; one
Lagrangian
entity has been widely measured in different experiments. It is
introduced in the
next section.
1.4 LVSF (Lagrangian Velocity Structure Func-
tion)
Suppose we are following a specific fluid element during it’s
motion with ve-
locity U, where U(t) = u(t)i + v(t)j + w(t)k. Then we can define
the pth order
Lagrangian Velocity Structure Function (LVSF) to be
Sp(τ) = (uτ )p = (u(t+ τ)− u(t))p (1.8)
where u(t) is a component of particle velocity at time t, τ is the
time lag, and p
is the order of the function. With this, the second order LVSF (in
x-component
of velocity) is
S2(τ) = ⟨ (u(t+ τ)− u(t))2⟩ (1.9)
It is worth going back to the issue of why Lagrangian study is
important. Using
the second order Lagrangian velocity structure function as an
example, we see that
the structure function is nothing but the ‘correlation’ between the
velocities of an
average particle some timescale τ apart. In other words, it
explores if the velocity
of a particle at a time t is ‘correlated’ with its velocity time t+
τ later.
1.5 Kolmogorov Theory
The first concept to have profound significance in the development
of theories
about turbulence was that of energy cascade by Richardson in 1921
[1]. He pro-
posed the idea that energy introduced through large scale eddies in
turbulence is
1.5 Kolmogorov Theory 7
handed down to smaller and smaller eddies. The cascading of energy
terminates
as the eddies reach the smallest size possible, and energy is
dissipated then. This
powerful idea of cascading of energy and scale of motion inspired
Kolmogorov to
hypothesize three statements about homogeneous and isotropic
turbulence. Col-
lectively called K41, they were:
1. Local Isotropy: In a very high Reynolds number turbulence, all
small scale
motions are statistically isotropic.
2. First Similarity Hypothesis: In a very high Reynolds number
turbulence,
all small scale motions are universal and are determined by the
mean energy
dissipation rate ε and the viscosity ν.
3. Second Similarity Hypothesis: In a very high Reynold’s number
turbu-
lence, the inertial range statistics depend only on the mean energy
dissipa-
tion rate and are independent of the viscosity of the fluid.
Energy cascade dictates that at any given time, eddies of different
scales are
present in a fully developed turbulence. A small scale eddy could
be swept by
a large scale motion at a fixed point in a fluid. If we are keeping
track of the
properties of a fluid at a fixed position, the properties of large
scales motions would
suddenly dominate at that point. This would result in a mix-up of
properties of
different scales. In order to avoid this confusion due to the
sweeping action by
large scales, fluid elements are tracked wherever they go in the
Lagrangian study
of fluids.
1.6 Kolmogorov’s predictions for second order
LVSFs
With the above three hypotheses in mind, and using dimensional
analysis, we
can infer the following behavior of the second order LVSFs in
different length
scales. For a very small time lag τ , it is possible to write Eq
1.9 as
S2(τ) =
)2 ⟩ τ 2 (1.10)
So we expect the second order LVSF to be quadratic with time lag τ
for very
small values of τ . The acceleration variance quantity
⟨( d
has been
determined for fully developed turbulence by Bodenschatz et al.
[2]. In the inertial
subrange of energy cascade, when there is neither production nor
dissipation of
energy, Kolmogorov’s prediction for second order LVSFs using Second
Similarity
Hypothesis and dimensional analysis is
S2(τ) = C(L) p ετ (1.11)
where C (L) p is the Lagrangian Kolmogorov constant (whose value is
estimated to
be about 6 in literature), and ε is the mean energy dissipation
rate of the turbulent
system. Thus, in the inertial range, where the energy is just
passed from large
scale eddies to smaller ones, the second order LVSF is linear with
τ . We can also
1.7 Turbulence and Research 9
expand the second order LVSF algebraically:
S2(τ) = ⟨ (u(t+ τ)− u(t))2⟩
⟩ − 2 u(t+ τ)u(t) (1.12)
Since the flow is considered to be statistically stationary, we can
assume that
u(t + τ) = u(t) for a large time lag τ . Also, at large time lags,
there is no
correlation between the velocities. This makes the correlation term
(the third
term in Eq 1.12) zero. Thus the second order LVSF has a constant
value for large
values of τ :
S2(τ) = 2u2(t) (1.13)
1.7 Turbulence and Research
A clear understanding of the Lagrangian framework is essential to
correctly
model the behaviors of particles in turbulent systems. In the last
decade or so, the
trend in turbulence research has been to explore Lagrangian
statistics in different
flows, which was not possible earlier. As mentioned in section 1.3,
Lagrangian
study of turbulent fluids requires tracking of particles
‘uninterruptedly’. High
speed cameras used to take hundreds of pictures of particles every
second will
then release enormous amount of data, saturating ordinary computer
memories in
few seconds. Fortunately, a system capable of extracting only
useful information
from the camera images (thereby needing significantly less memory)
was devel-
oped at the Voth lab in Wesleyan University. Many graduate students
worked
1.7 Turbulence and Research 10
with Professor Greg Voth (at different times) in building this
image compression
system. This allowed even simple desktop computers to store
information about
the tracked particles continuously for hours, without worrying
about memory lim-
itations.
When I joined the Voth lab in the Spring of 2007, thanks to the
hardwork of
PhD student Dan Blum, particle tracking data from a few runs of the
experiment
were already available. Dan was focussing on the Eulerian
perspective of turbu-
lence (which will be explained in due course). In the beginning, I
worked on 3D
calibration and particle tracking. Later, I focussed on the
Lagrangian analysis of
our data. Since then, I have devoted myself to making various
Lagrangian mea-
surements, primarily the Lagrangian structure functions in our
experiment.
From the very definition of Lagrangian statistics, it is easy to
guess that my
goal is to study whether fluid particles at any given time exhibit
the kinetic prop-
erties they gained at another location in an earlier time. We use a
flow between
two oscillating grids, which is largely inhomogeneous near the
grids and relatively
homogeneous in the center. Particles are set in motion near the
grids, and move
to different parts, including to the center, with time. In such a
non-uniform tur-
bulence system, it is interesting to investigate to what extent a
particle retains
the velocity information from its past along its trajectory.
In looking for the signs of ‘memory’ of particles in our
experiment, I stud-
ied the behavior of Lagrangian structure functions when they are
conditioned on
large scale velocities. The results have already been published in
a paper, and I
consider them my major contribution. Measurement volume bias has
long been
1.7 Turbulence and Research 11
considered an issue with Lagrangian statistics, as we will see in
Chapter 3. I
developed a method to estimate how big the error due to the bias is
at different
timescales in second order LVSFs. Thus we were able to leave out
the results
that would have high percentage of error by our estimation. The
error estimates
that we had already calculated seems to agree roughly with a
recently published
paper by Berg et al. [3]. In the final stages of my research with
Prof. Voth, I
also studied the transport of energy in a Lagrangian context in our
flow. Here
again, my interest was to study what factors caused the transport
of energy from
the inhomogeneous and energetic parts to a quiescent homogenous
region of our
experiment. I suggested that some agents were more important than
others in
transporting turbulent kinetic energy.
My goal will be to convey those three important results, their
consequences
and their accompanying theories in separate chapters. But first, I
will describe
the experiment and calculate few quantities to highlight the
importance of some
special equipments and materials in our experiment. The final
chapters will talk
about the results.
Chapter 2
The Experiment
When I started research in January 2007, the experimental setup was
complete
and it had already gone through few successful runs of experiment.
The basic
premise on which the experiment was built was keeping track of
fluid elements in
a turbulent flow so that important information like the position
and the velocity
of the elements is recorded accurately and in sufficient amount.
From there,
various statistics like the mean velocity and Lagrangian structure
functions are
calculated. This fundamental need to track fluid elements to study
their behavior
as they move around in a fluid is experimentally challenging in
many ways. I will
mainly describe the apparatus that was used to meet those
challenges, sometimes
taking help of dimensional analysis and arithmetic.
12
2.1.1 Sequence of events
Figure 2.1: The experiment containing a tank, two cameras and two
grids (figure
adapted from Blum et al. [4]). The green cylinder is the expanded
laser beam.
The main body of the apparatus consists of a transparent tank
containing wa-
ter with tracer particles in it. Two oscillating grids in the tank
bring turbulent
motion in the water. An expanded laser beam illuminates a region in
the tank.
Two cameras take pictures of the laser-illuminated part of the
tank. The camera
images go through an image compression circuit, which extracts only
useful in-
formation from the images. This information from the circuit is
then sent to the
computer, where the partcles are identified and their 3D positions
and velocities
determined. Data files in hard disks store all the information
about the detected
2.1 Apparatus 14
particles. Computer programs access these files and help understand
the behav-
ior of all detected tracer particles along their trajectories. For
instance, I access
the 3D velocity information from the data files to generate
Lagrangian structure
functions.
2.1.2.1 Tank
We use a large transparent tank to hold water in our experiment. It
has
an octagonal cross-section of width 1 m. The height of the tank is
1.5 m. It
is made of Plexiglas, which is strong and light. Plexiglass is also
unaffected by
moisture, making it further reliable for our experiment. But most
importantly,
it is incredibly transparent, which is a necessity in particle
tracking. The tank
rests on a table about 4 feet high, and it can hold approximately
1100 liters (300
gallons) of water. Due to the big volume of water being used, any
leakage of water
from the tank can cause damage to other equipment in the lab. Thus,
equipment
in the lab is not left on the floor, to keep lab items safe in the
event of flooding.
2.1.2.2 Grids
Two identical grids are used inside the tank to generate turbulence
in the
water. Each grid is octagonal, to match with the cross section of
the tank. The
mesh size of the grids is 8 cm. The grids have 36% solidity. The
distance between
the upper grid and the top part of the tank is the same as that
between the lower
2.1 Apparatus 15
grid and the bottom. The two grids are 56.2 cm apart all the time.
Four rods
pass through both the grids and are eventually connected to a
motor. The 11 kW
motor drives the four rods in an identical manner, causing the
grid-pair to oscillate
in phase. The stroke of the oscillation is 12 cm peak-to-peak. The
oscillation of
the grids can be controlled; while taking the data that I analyzed
for this thesis,
the frequency of the grid’s oscillation was 5 Hz.
2.1.2.3 Water
As it was mentioned in Chapter 1, water is an incompressible fluid.
For this
and a variety of other properties, water was used as the fluid to
be studied in the
experiment. Water in the tank needs to be clear and free of
impurities like foreign
particles that can be mistaken for tracer particles. The presence
of air bubbles in
water creates the same confusion, just on a bigger scale. So water
is filtered and
degassed before being pumped into the tank. About 1100 liters (300
gallons) of
water is contained in the tank during a run of the experiment. In
order to avoid
changes in the viscosity or the index of refraction, water in the
tank is maintained
within ±0.1C of the initial temperature of about 22C.
2.1.2.4 Tracer particles
Tiny polystyrene tracer particles are seeded in the water inside
the tank. Each
such particle is 136 µm in diameter and is neutrally buoyant. The
size and the
buoyancy of the particles make them ideal for the experiment, as
they are being
used to represent fluid elements. The particles were added until an
optimum
2.2 Demands of Lagrangian statistics 16
particle density of about 50 per frame was achieved in each camera.
With this
particle density, the amount of data per frame was maximized while
at the same
time, the error in tracking was minimized.
2.2 Demands of Lagrangian statistics
Here I quantify the smallest length and time scales in our flow and
identify
the camera frame rate necessary to resolve these scales.
Kolmogorov’s First Similarity hypothesis (see Chapter 1) states
that a quantity
should only depend on the mean energy dissipation rate ε and the
kinematic
viscosity of the fluid ν at the smallest scale during energy
cascade. By means
of dimensional analysis, we find that the smallest length scale η
of the flow is
uniquely determined by
= 142µm
where ε = 0.00246 m2s−3 is determined from the Eulerian structure
functions as
described by Blum et al. [4]. ν = 10−6 m2s−1 is a property of
water. We can see
that the diameter of the tracer particles (136 µm) is slightly less
than the smallest
length scale of the flow. This consolidates our belief that the
tracer particles
can represent the motion of fluid elements correctly. Similarly, we
can uniquely
2.3 Detection 17
determine the smallest time scale of our flow using ε and ν:
τη = (ν ε
= 20 milliseconds
This is again a good value, when it is compared with the duration
between two
successive frames of the camera. When our camera runs at a speed of
500 Hz, the
time between two frames is 2 milliseconds. This is a tenth of the
smallest timescale
of the flow (we are resolving up to one tenths of the Kolmogorov
timescale). Thus,
a frame rate of 500 Hz allows us to do particle tracking
comfortably in all time
scales in our experiment.
2.3 Detection
We want to track and record the position and the brightness of
tracer particles
when water is turbulent. This is accomplished using special
equipments like high
speed cameras, laser and an image compression system.
2.3.1 Laser
The tracer particles are so tiny that they have to be illuminated
by laser so
that the cameras can detect them. We used a 532 nm pulsed Nd:YAG
laser for this
purpose. On average, the laser generates 50 W of power with pulses
only 200 ns
in duration. In order to track particles over a reasonable
duration, the detection
volume of cameras cannot be very small. So the laser beam was
expanded to
2.3 Detection 18
create an illumination volume of dimensions 7 cm × 4 cm × 5 cm in
the tank.
The expanded beam can be directed to different places in the tank
depending on
whether we want to study the center or the region near the
grids.
2.3.2 Cameras
Two cameras took pictures of a fraction of the illuminated region
of the tank.
The tracer particles looked bright in the camera images owing to
laser illumina-
tion. Both the cameras were Basler A504K video cameras that could
produce
images with 1280 × 1024 pixel resolution at a speed of 500 frames
per second. As
was shown in Section 2.2, a frame rate of 500 Hz is desirable.
However, it posed a
big problem- data was produced by each camera at the rate of 625
Megabytes per
second. This means the 4 Gigabyte Random Access Memory (RAM) of the
com-
puter connected to the cameras could hold data only for about 7
seconds. After
this, the cameras would have to be stopped for 7 minutes so that
the enormous
amount of data could be downloaded to a computer hard disk from the
RAM.
The larger the number of data points, the better the statistical
analysis. This is
true for Lagrangian statistics too. From the definition of
Lagrangian structure
funcitons, it is easy to see that the more the number of particles
we detect, and
the longer the particle trajectories are, the better the
measurement. Stopping a
run every 7 seconds to download data would only cut short the
trajectories in
view.
2.3.3 Image compression circuit
The computer memory limitation just discussed was tackled using
image com-
pression circuit [4]. The image compression circuit was placed
between each cam-
era and the computer connected to the camera. From each camera
image, the
circuit selects pixels that contain brightness over a certain
threshold level and
ignores the rest of the image in real time. This is done to pick
only useful in-
formation, that is the particle center position and brightness of
every detected
particle. The discarded background information takes up huge amount
of storage
units in computer. With this method, data files were compressed 100
to 1000
times their original size. Thus, it was possible to take data for
hours and store all
the information in an ordinary hard disk without stopping the
experiment.
2.4 Stereomatching: the final step
Data files from the image compression circuit contain the 2D
position (of
the center) and brightness of every particle tracked by each
camera. Using the
information from the image compression circuit, the 3D position of
each particle
is found by the process of stereomatching. The 3D position and the
magnification
of each camera have to be known for stereomatching. We obtained an
accuracy of
about 11 µm in the stereomatching process. To obtain this level of
accuracy, it is
necessary to have very good calibration of camera position
parameters. We take
traditional camera position parameters first. Then using those
parameters, we
stereomatch known pairs of particles from the two cameras. We run a
nonlinear
2.4 Stereomatching: the final step 20
optimization to minimize the error in stereomatching and find the
optimal camera
position parameters.
With the camera positions known, it is possible to calculate the 3D
position of
a particle from the 2D positions as seen from the perspectives of
several cameras.
The pixel coordinates of each particle seen on a camera indicates
that a particle
must lie somewhere along a specific line in 3D space. If the
minimum distance
between lines from multiple cameras is within a certain tolerance,
then we have
the 3D position of a particle. Thus, we obtain a list of the 3D
positions of each
particle that came into the field of view as a function of time.
From position and
time information, the 3D velocity can be found out. This is the
ultimate objective
of particle tracking. We use computer languages (mainly MATLAB) to
use this
data for various analysis. This process is shown in a flowchart in
Figure 2.2.
2.4 Stereomatching: the final step 21
2D Image from Camera A
2D position and brightness of each particle
2D Image from Camera B
2D position and brightness of each particle
Identification of particles + 3D
position
each frame
Image Compression
Experiment
Figure 2.2: Flowchart showing the determination of 3D velocities
from camera images
of tracer particles
3.1 The possible sources of bias
From the description of the experiment in Chapter 2, it is easy to
see that we
use a large volume of water and numerous tracer particles in the
tank. However,
the cameras can only capture a limited region, called the
measurement volume,
of the tank if they are to produce images with desirable
resolution. As a result,
the particles that lie outside the measurement volume cannot be
detected.
Not all tracer particles in the measurement volume have the same
speed. Some
have speeds high enough to move through the measurement volume in a
very brief
duration. Others are slow enough to linger for a relatively long
time. The longest
duration for which a tracer particle has been detected in our
experiment is about
22
3.1 The possible sources of bias 23
3 seconds. When making measurements, we usually take the ensemble
average
of a quantity we measure for all detected particles. Since the slow
particles stay
in the detection volume for longer durations than their fast
counterparts, it is
possible that the data obtained from the cameras is mostly of the
slow particles
and not of the high-speed ones. Hence a possible bias towards slow
moving parti-
cles in the measured quantity. This causes a statistic like the
LVSF (Lagrangian
Velocity Structure Function), which is a velocity ensemble average
quantity, to be
underestimate of the actual value.
However, there is another possibility that we need to consider. The
probability
of a particle entering the the detection volume depends on its
velocity [5]. A high
speed particle is more likely to enter the detection volume than a
slower one. This
should lead to velocity measurements biased towards fast-moving
particles.
Thus we have two hypotheses here, leading to opposite conclusions.
Buchhave
et al. [6] resolved this confusion by showing that as long as
measurement is made
for the entire time the particle is in view, the two velocity bias
effects mentioned
earlier negate each other exactly. Thus, if we had to measure a
single-time ve-
locity statistic like the mean velocity of a particle, we would
simply add up the
velocities of the particle at all times inside the detection volume
(and of course,
divide the sum by total number of data points).
Structure function, however, is not as simple as a single-time
quantity like
the mean velocity. By definition, structure function is a two-time
quantity. Per-
haps, this is best explained by invoking the mathematical
definition of a pth order
3.2 Estimating the possible bias in structure functions 24
structure function presented earlier as Eq 1.8 in Chapter 1:
Sp(τ) = (uτ )p = (u(t+ τ)− u(t))p
where u(t) is a component of particle velocity at time t, τ is the
time lag, and p is
the order of the function. We need velocity information from two
different times,
some timelag apart, for every particle in order to obtain the value
of structure
function for every time lag. For small time lags, it is possible to
use all the
velocity pair-values of particles that lie inside the measurement
volume. However,
it is also possible that the timelag τ is sufficiently large that a
particle can traverse
the observation volume in time t less than τ . In such cases (t
< τ), the particle
in view will not contribute to the measurement of structure
function for timelags
greater than τ . Thus the very nature of the structure function
prevents a bias-
free evaluation; this bias in the measurement of time-delay
statistics is called the
‘measurement volume bias’.
tions
Quantifying the error due to measurement volume bias is very
important as
it might be a significant percentage of the quantity we are
measuring. Data with
very little percentage of this error, if it exists, can be used to
make Lagrangian
analysis reliably. However, if this error is significant,
correction is necessary in our
estimation of quantities. Berg et al. [3] have developed a
theoretical method of
3.2 Estimating the possible bias in structure functions 25
estimating the error in LVSFs due to measurement volume bias,
building on the
ideas first presented by Ott and Mann [7]. With their method, one
can estimate
the percentage of error in a pth order LVSF occurring at any
timescale τ for an
observation sphere of radius r. We use a different approach to
estimating the
bias error in second order LVSFs in our experiment, following a
method that we
independently developed before Berg et al. [3] published their
results. Later we
compare the results from the two different methods and see that
they roughly
agree.
3.2.1 Structure function vs volume
Our first step in studying detection volume effects on second order
LVSFs was
examining how the second order LVSF varied with different
observation volumes.
The detection region for the cameras was kept constant for the
entire run of the
experiment. Instead, we artificially changed the measurement
volume, starting
from a very small sphere (radius 0.5 cm) and going up to a sphere
(radius 4.5 cm)
comparable in size to the actual observation volume of the
experiment. The origin
of the concentric artificial spheres was very close to the center
of the tank. For
each artificial volume, we calculated the second order Structure
function. When
we plotted all the curves on the same axes, Fig 3.1 was
obtained.
The top red dotted curve represents the second order LVSF for all
datapoints in
Fig 3.1. Each of the other 9 solid curves is the structure function
for a particular
volume. One feature of this multiple plot is the coincidence of all
the curves at
very small timelags (τ < 2). This agrees with Buchhave et al.
[6]. For small
3.2 Estimating the possible bias in structure functions 26
0 5 10 15 20 0
0.5
1
1.5
2
2.5
3
.5 to 4.5 cm rad; also the run str func
0.5cm 1.0cm 1.5cm 2.0cm 2.5cm 3.0cm 3.5cm 4.0cm 4.5cm Run
Figure 3.1: Second order LVSF vs timescale for different artificial
measurement vol-
umes. The radii of volumes range from 0.5 cm to 4.5 cm; each volume
is represented
by a color as indicated in the legend (which lists colors and their
corresponding radius
value); the top dotted red curve labeled ‘Run’ is the structure
function obtained from
all trajectories available with no artificial volume
restriction.
timelags, almost all particles in view will be contributing to
structure function;
the residence time of the particles cannot be smaller than a
certain timescale
on average, and so no bias occurs. When the time intervals get
larger, some
particles in the detection volume with small residence time will
not contribute
to the structure function (but this is less likely to happen in a
bigger volume).
Thus, the structure function plots get progressively higher for
bigger measurement
3.2 Estimating the possible bias in structure functions 27
volumes when τ > 2. We also see that the curves for the large
volumes (radius > 3
cm) and the one for the run (all detected trajectories) coincide at
all timescales.
This opens up the possibility that at large volumes, there might
not be many
extra particles (and hence extra velocity information) for each
additional volume.
If we are not tracking more particles in proportion with additional
volume, then
every new plot of the structure function is going to look the same
as the previous
one. Eventually, we will not be able to extrapolate a structure
function in a large
radius limit, and estimation of the finite volume error would not
be possible. So
we investigated the (tracked) particle density at each artificial
volume.
3.2.2 Particle density vs volume
For each artificial volume, we calculated how many particles the
cameras had
detected per frame on average. If a particle appeared in five
consecutive frames
inside an artificial volume, it would be counted as five different
particles. As
long as the number of particles detected in the measurement volume
is increasing
proportionally with the volume itself, we need not worry about
sufficiency of data.
This is exactly what happens in our experiment, as shown by Fig
3.2. Up to a
radius of 2 cm, the slope of the ‘log (Cr3) vs r’ curve (dashed) is
the same as that
of the ‘log(particle number) vs r’ curve (solid). So we can
comfortably claim that
the number of particles detected in a volume is a cubed law of the
radius of the
corresponding volume as long as the volume in consideration is not
more than 2
cm. We observe that the number of particles recorded in a sphere of
radius 2.8
cm is approximately equal to that of the 3 cm radius sphere.
Subsequent (larger)
3.2 Estimating the possible bias in structure functions 28
0.2 0.5 1 2 3 5 102
104
106
108
1010
Number of particles in the volume Cr3
Figure 3.2: The horizontal axis is the radius of a sphere whose
center coincides with
the geometrical center of the tank. The blue line shows the number
of particles for a
volume with radius r. The red line is a function Cr3 with C a
constant; it is proportional
to the volume corresponding to a radius r. Note that both the axes
have been plotted
on logarithmic scales for convenient comparison with power law
behavior.
radii also have the same number of particles. The effective maximum
observation
volume of the experiment must be about 2 cm. Sufficient number of
particles
are being detected up to a sphere of radius 2 cm; a measurement
volume of 2.2
cm or 2.4 cm has all the information that greater volumes have. Now
we look
at the Structure functions within the reliable volume of radius 2
cm and make
extrapolation.
3.2.3 Structure function vs volume again
Each camera is set to capture about 10 cm ×10 cm of 2D plane, so a
measure-
ment volume of about 1000 cm3 is possible at most. In reality, it’s
smaller than
this. We decided to stay within a safe region, which is a shell of
inner radius 0.6
cm and outer radius 2 cm. The inner radius is fixed at 0.6 cm due
to the fact
that the second order LVSF is noisy for smaller volumes. The noise
comes from
the detection of insufficient number of particles and short tracks
for the detected
ones.
Having established 2 cm as the upper limit of the artificial
measurement vol-
ume, and 0.6 cm as the lower limit, we can check how the second
order LVSF
varies for different artificial measurement volumes inside the
tank. The volumes
are all spheres, with radii ranging from 0.8 cm to 2 cm in
increments of 0.2 cm,
and centered at roughly the geometrical center of the tank. The
results in Fig
3.3 show that the second order LVSF actually varies over the volume
intervals
used for not so small timelags. The bigger the observation volume,
the higher the
structure function. This is a clear sign of the presence of
measurement volume
bias, especially when compared with the dotted curve (includes all
the detected
particles in the experiment).
Fig 3.3 shows that the LVSF for the entire data is about 1.5 times
that for
the smallest volume at 7τ/τη. This information is enough to
convince us that
compensation is needed against volume bias. The other implication
of Fig 3.3 is
that we should be able to extrapolate the LVSF for infinitely big
detection vol-
ume, considering the nice pattern of curves for different volumes
in Fig 3.3. The
3.3 Extrapolating the Structure Function 30
0 5 10 15 20 0
0.5
1
1.5
2
2.5
3
0.8cm 1.0cm 1.2cm 1.4cm 1.6cm 1.8cm 2.0cm Run
Figure 3.3: Second order LVSF vs timescale for different artificial
measurement vol-
umes; The radii of volumes range from 0.8 cm to 2 cm; each volume
radius is repre-
sented by a color as indicated in the legend; the top dotted red
curve labeled ‘Run’ is
the structure function obtained from all trajectories available
with no artificial volume
restriction.
extrapolation, in turn, will enable us to estimate the bias error.
The next section
discusses this.
3.3 Extrapolating the Structure Function
Fig 3.3 was a plot of structure function against timescale for
different radii.
There is another way of presenting the same data- plotting
structure function
3.3 Extrapolating the Structure Function 31
against radii for different timescale. The range of timescales that
interests us
mostly is the inertial subrange- from one Kolmogorov unit to the
tenth. There
is a pattern in structure function, which can be described by a
mathematical
function depending on both the radius of sphere and the timescale.
Figure 3.4
below has the details:
We needed a mathematical function of radius r such that it would
yield
0.8 1 1.2 1.4 1.6 1.8 2 0.6
0.8
1
1.2
1.4
1.6
1.8
2
2 )
Figure 3.4: Second order LVSF vs measurement volume radius r; each
blue curve
represents the structure function for a distinct timescale in the
inertial subrange. The
curves get higher for larger timescales.
a limiting value of the structure function at a large radius limit.
The function
should also have an additional responsibility of being a good fit
for a family of
curves (for different timescales), and not just one curve. Also,
there is no well
3.3 Extrapolating the Structure Function 32
known mathematical form that the bias should obey, so we just
looked to find
a function that would fit reasonably well with the data. As it
turned out, the
following functional form satisfied our conditions:
f(r) = A [ 1−Be−CrD
] , (3.1)
where r is the radius of the artificial measurement volume, and A,
B, C and D
are all constants for a given timescale. The values of the
constants A, B, C and
D are all unique for a given timescale in the inertial range. In
the beginning, we
tried to fit a function f(r) with D = 1 but such a function would
not fit well to
the data. In contrast, a function with D as a variable fit
relatively well to the
Structure function. Figure 3.5 gives a picture of how the data
(blue circles) and
the fit (red line) look like.
A quick calculation reveals that the function represented by Eq 3.1
will have
the value A when r tends to ∞. We fit the structure function to the
function
f(r) using a fitting function in MATLAB, and obtain the optimum
values of A
for all timescales in the inertial range (as we are mainly
interested in the inertial
subrange behavior of structure functions). On plotting the optimum
values of A
against corresponding timescale τ , we should get the extrapolation
of the LVSF
based on the pattern in Fig 3.3.
The dotted line in Fig 3.6 represents the extrapolation. The solid
curve is the
LVSF for the entire data that we have. We can clearly see that the
uncompensated
structure function is lower in value than the extrapolated one.
This emphasizes
the need for estimating the error due to measurement volume bias in
Lagrangian
3.3 Extrapolating the Structure Function 33
0.8 1 1.2 1.4 1.6 1.8 2 0.8
1
1.2
1.4
1.6
1.8
2
2.2
2 s− 2 )
Figure 3.5: Each set of blue circles (connected by a red line)
represents ‘Second order
Lagrangian velocity structure function (LVSF) vs ‘measurement
volume radius’ for a
distinct timescale in the inertial subrange .Each set of blue
circles is the same as a blue
curve in Figure 3.4. Each red line joining the blue circles is the
fit of the function in
Equation 3.1 to the blue curves in Figure 3.4.
statistics and making compensation when the error is too high. The
extrapolation
is only reliable between the timescales 4τη and 8τη due to
statistical reasons and
the difficult nature of the family of curves. Below 4τη, the fit
yields unusually large
values for some of the constants due to the flat nature of the
Structure function
at small timescales. At timescales larger than 8τη, the
extrapolation curve takes
off abruptly, as seen in Fig 3.6.
3.4 Comparison of errors 34
4 5 6 7 8 9 10 11 0.5
1
1.5
2
2.5
3
3.5
4
Extrapolation Data
Figure 3.6: Second order LVSF vs timescale; The red dotted line is
the structure
function for all available datapoints; the solid blue curve is our
extrapolation of second
order structure function.
3.4 Comparison of errors
As promised earlier, a comparison will now be made between the
error esti-
mates using our method and the one developed by Berg et al. [3].
Based on the
concept of Greens function, Berg et al. propose a theoretical
method of estimating
the error in pth order Lagrangian structure functions. For a second
order struc-
ture function, the error was roughly proportional to timescale τ ,
with 5% error at
3.3τη a given pair of values. Form there, we extract the percentage
error at any
timescale. The discrepancy seen in Fig 3.6 between the two curves
is the error
that our method estimates. Since the fitting worked well for times
between 4τn
3.4 Comparison of errors 35
and 8τn inclusive, we only look at the error estimates at 4τη, 5τη,
6τη,7τη and
8τη. Table 3.1 lists the error values as percentage of the
uncompensated structure
function. The error values that the two methods estimate are
somewhat close.
Timescale Our error estimate Error from Berg method
4τη 2.9% 6.1%
5τη 8.7% 7.6%
6τη 13.4% 9.1%
7τη 16.5% 10.6%
8τη 16.3% 12.1%
Table 3.1: Comparing the error estimates from two different
methods
Most of the time, our method gives slightly bigger error. With the
knowledge of
the size of error, we now study structure functions at timescales
less than or equal
to 8τη, beginning from the next chapter.
Chapter 4
Conditional structure functions
and their significance
In the Lagrangian study of fluids, a fluid element is followed
along its trajec-
tory and its properties are observed with time. Different from this
is the Eulerian
study, which entails studying the properties of fluids at fixed
points in space. Ob-
viously, in the Eulerian method, we will not be tracking the same
particles all
the time. It is interesting that Lagrangian analysis is a
relatively new field com-
pared to Eulerian study, owing to the experimental difficulties in
particle tracking
until recent technological advances. Many Lagrangian entities like
the structure
function and Kolmogorov’s hypotheses that I have referred to have
Eulerian coun-
terparts.
As mentioned in Chapter 1, Richardson’s idea of energy cascade in
turbulence
envisions the presence of large scale eddies, which break down into
smaller ones.
36
4.1 Eulerian Structure functions 37
The process ends in the smallest scale eddies, also called the
Kolmogorov scale
eddies (1921). Kolmogorov’s hypotheses go further to claim that the
small scale
statistics are universal and independent of the large ones in high
Reynolds number
turbulence (1941). These concepts have been tested for decades by
scientists. The
results are mixed; some small scale properties of the flow show
dependence on the
large scale while others are independent. However, such studies
have been done
in Eulerian perspective only.
My research has centered on the Lagrangian aspect of turbulence. In
fact,
investigating small scale dependence on the large scale in a
Lagrangian perspec-
tive has been a major topic of my research. My results are actually
the very
first ones in the Lagrangian framework to have ever been published.
In the same
paper (by Blum et al. [4]), Dan B Blum reports his Eulerian
measurements of
large scale dependence. Despite having focused on Lagrangian
analysis, I will fol-
low the sequence of history by first introducing Eulerian structure
functions and
then presenting evidence of relation between large scales and small
scales through
Eulerian structure functions. Lagrangian results will follow
them.
4.1 Eulerian Structure functions
Eulerian structure function is obtained from the velocity
difference ur be-
tween two fluid elements r distance apart. The pth order
longitudinal Eulerian
velocity structure function is defined as
Dp(r) = (ur)p = (u(x + r)− u(x))p (4.1)
4.2 Determining the large and small scale velocities 38
where u(x) is the particle velocity at position x, r is the 3D
vector connecting
the two particles, ur = u(x + r)− u(x) is the the projection of the
3D velocity
difference vector onto r and p is the order of the function. With
this definition,
the second order longitudinal Eulerian velocity structure function
is
D2(r) = ⟨ (ur)
= ⟨ (u(x + r)− u(x))2⟩ (4.2)
Like for the Lagrangian structure functions, Kolmogorov’s
hypotheses have pre-
dictions for the Eulerian structure functions. According to K41, in
the inertial
subrange,
ur = C(E) p (εr)p/3, (4.3)
where C (E) p is Eulerian Kolmogorov constant and ε is the mean
energy dissipation
rate. With this brief introduction to longitudinal Eulerian
structure functions, we
now go back to the topic of scale dependence.
4.2 Determining the large and small scale veloc-
ities
The instantaneous velocity of a particle is dominated by the large
scales. So
it is reasonable to represent the large scale velocity by the
instantaneous veloc-
ity. Lagrangian structure function is a two-time quantity- the
calculation of the
function at a particular timescale τ needs the velocities of the
particle at two
times, τ apart, along a trajectory. We average the velocities of
the particle at
these two times, u(t) and u(t+τ), to get the instantaneous velocity
of the particle
4.3 Conditioned structure functions 39
(denoted as Σuz). Thus, the average of the the pair of velocity
values used in
evaluating structure functions is the large scale velocity. The
small scale velocity,
on the other hand, is determined by the difference in the same
velocity pair. In
the Eulerian framework also, the the two velocity values, u(x and
u(x, r), that
are used to calculate the structure function, are averaged to get
the large scale
velocity (also denoted as Σuz) while their difference is the small
scale velocity.
It is worth recalling that by definition, structure function is
just the ensemble
average of some power of the difference between a pair of velocity
values. The im-
plication of this is the dependence of small scales on the large
scales can be studied
by simply conditioning the second order structure function on the
instantaneous
velocity for various timescales. The results of such conditioning
are discussed in
the following sections.
4.3 Conditioned structure functions
4.3.1 Eulerian structure functions
Figure 4.1 is a plot of the second order longitudinal Eulerian
structure functions
conditioned on the vertical component of the large scale velocity.
Different curves
represent different separation distances r/η. The vertical axis has
the conditioned
structure functions (scaled by their value when Σuz = 0) with the
vertical pair
velocity (scaled by √ < u2
z >) on the horizontal axis. The curves show several
remarkable properties. The conditional structure functions are
steep parabolas
when plotted against large scale velocity, and they vary by as much
as a factor
4.3 Conditioned structure functions 40
2 <!ur|!uz>
1.0
1.5
2.0
2.5
3.0
Figure 4.1: Eulerian second order conditional structure function
versus large scale
velocity. Data taken in the center region. Each curve represents
the following separation
distances r/η: + = 0 to 40, ∗ = 40 to 70, = 70 to 110, 4 = 110 to
140, = 300 to
370, × = 370 to 440.
of 2.5, showing strong dependence of the small scales on the large
ones. Also, all
the curves for different separation distances r/η collapse well,
showing that large
scales affect all length scales in the same way.
4.3.2 Lagrangian structure functions
Since the grids inject energy in the vertical direction in our
experiment, its in-
teresting to condition (z-component) Lagrangian structure functions
on Σuz, the
z-component of instantaneous velocity. In Figure 4.2, we plot
several such second
order structure functions conditioned on the vertical component of
instantaneous
4.3 Conditioned structure functions 41
10 1
10 0
10 1
10 2
Figure 4.2: Second order Lagrangian velocity structure function
(conditioned on the
vertical component of instantaneous velocity) vs τ/τη at the center
of the tank. The
colors represent dimensionless vertical velocities, Σuz/ √ u2 z: +
= 3.1 to 1.9, ∗ = 1.9
to 0.62, = 0.62 to -0.62, 4 = -0.62 to -1.9 , = -1.9 to -3.1.
4.3 Conditioned structure functions 42
velocity. We see that the structure functions show a dependence on
Σuz. The
conditional structure functions for different large scale
velocities differ by factor
of about 2, and remain almost parallel for the time range
considered.
It is worth mentioning that measurement volume bias affects
Lagrangian struc-
ture functions. It was discussed in Chapter 3 that this bias
increases with larger
timescales. The bias introduced about 17% error in second order
LVSFs at
τ = 8τη. We have not compensated the structure functions for the
bias as Berg
et al. suggest. However, we will only focus on conditional
structure functions for
timescales τ ≤ 10τη. Eulerian statistics are free of this
mesurement volume bias
as we do not track a particle over some duration to evaluate the
statistics.
A powerful way of looking at Fig 4.2 is by plotting the structure
function
against instantaneous velocity Σuz for different timescales (within
10τη of course).
The structure functions are scaled by their value when Σuz = 0. The
instanta-
neous velocity is scaled by √ < u2
z >. This way of visualizing the results of Fig
4.2 has some advantages, which will be clear in Fig 4.3. In Fig
4.3, we see that
for every timescale, the curve is a parabola. The curvature is a
good indication
that the small scales are being affected by the large scales. The
structure func-
tion has larger values for bigger instantaneous velocity. The graph
gives a hint
that the large scales affect small scales in different ways for
different timescales,
as the curves do not collapse. We see that the value of structure
function differs
by as much as 2.5 times for the different timescales we consider
[4]. Notably, the
curvature is less at all lengthscales for the Eulerian structure
function (Figure
4.1) when compared with the Lagrangian structure function in Figure
4.3; The
4.3 Conditioned structure functions 43
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
5
uz "<uz>
Figure 4.3: Lagrangian second order conditional structure function
vs large scale
velocity for the center of the tank. The symbols represent the
following τ/τη: + = 0.42
, ∗ = 1.3, = 3.5, 4 = 10.
Lagrangian structure functions show stronger dependence on the
large scales than
the Eulerian ones.
We also looked at conditional structure functions in a measurement
volume
half the original volume of the run. Such a change in observation
volume shifts
the parabolic curves down by roughly the deviation between the
curves; the de-
pendence is still retained. This highlights that measurement volume
bias is not a
significant issue for the conditioned structure functions we have
presented.
4.4 Effects of Inhomogeneity 44
4.4 Effects of Inhomogeneity
Figure 4.3 was a result for the center of the tank, which is the
most homogenous
region in the experiment. We now turn our attention to the same
conditional
structure functions but in a more inhomogeneous region of the tank
(near the grid
and below the center). The parabolic curves in Fig 4.4 are a result
of conditioning
-4 -3 -2 -1 0 1 2 3 4
2
3
4
5
6
!uz
Figure 4.4: Lagrangian second order conditional structure function
vs large scale
velocity for the near grid region of the tank. Symbols represent
the following τ/τη: +
= 0.94, ∗ = 2.8, = 8.0.
the Lagrangian structure function on Σuz near the grid. They are
remarkably
asymmetrical compared to the corresponding plot for the center of
the tank (Fig
4.3). This is due to the fact that the bottom grid is providing
kinetic energy to
the fluid particles bound for the detection volume. These upward
moving particles
4.5 Other Causes of Dependence 45
tend to have higher velocities (and higher kinetic energies) than
their downward-
moving counterparts from the more quiescent center of the tank.
This eventually
appears in the structure function calculation as well, as seen by
the leftward shift
of the minimum of the parabolas. The curvature of the curves is
higher towards the
right and lower towards the left, when compared with the
conditioned structure
function at the center. At the center, the incoming fluids (coming
from up and
down) are equally energized by the grids, resulting in symmetrical
curves.
4.5 Other Causes of Dependence
The set of plots we’ve presented so far indicate dependence of
small scales on
the large scales. This leads us to an important question- what
causes the condi-
tional structure functions to behave as they do? We have seen that
inhomogeneity
plays a significant role in the behavior of the structure functions
in the previous
section. There are several other factors that need to be considered
in fully explain-
ing the dependence. One possible concern is that kinematic
correlation might be
behind the behavior of the curves in Figures 4.3-4.4.
4.5.1 Kinematic Correlation
We obtain the large scale (instantaneous) velocity of a particle by
averaging
the pair of velocity values, u(t) and u(t + τ), used in calculating
the structure
function. Interestingly, we find the difference between the same
two values to get
the structure function. This hints at the possibility of kinematic
correlation. A
4.5 Other Causes of Dependence 46
pair of velocity values along a particle trajectory can have a
large sum and still
have a large difference. If this is the case, then the structure
functions conditioned
on the large scale velocities are bound to have parabolic (or near
parabolic) curves,
as shown in Figures 4.3-4.4 above. Several studies have confirmed
that velocity
sums and differences are in fact correlated. However, there are
evidences that
kinematic correlation is only partly responsible for the effects we
have seen. Thus
it is important to determine how much contribution kinematic
correlation has on
the conditional structure functions. Blum et al. [4] have estimated
that more than
70% of the effect is unexplained by kinematic correlation in the
Eulerian structure
functions. The conclusion here is that kinematic correlation might
affect large
scale dependence of small scales significantly but not
completely.
4.5.2 Reynolds number
Another concern when explaining large scale dependence is low
Reynolds num-
ber of the flow. It is often argued that if the Reynolds number of
the flow is not
sufficient enough, the separation of the large scales and the small
scales is not ad-
equate. This will lead to the small scale statistics retaining some
properties of the
large scales. Results published by Blum et al. [4] and Sreenivasan
and Dhruba [8]
together serve to point out that ‘insufficient’ Reynolds number
does not cause
large scale dependence in the Eulerian case. Figure 4.5 compares
the conditional
Eulerian structure functions from the two different experiments-
one with a very
large Reynolds number ( Rλ > 104 ) and the other with Rλ = 300.
The two
experiments have nearly the same dependence. Also, all lengthscales
collapse to
4.5 Other Causes of Dependence 47
-4 -2 0 2 4 0.5
1.0
1.5
2.0
2.5
3.0
Figure 4.5: Eulerian second order conditional structure function
versus large scale
velocity. The thin plots are from atmospheric boundary layer data
[8], r/η: ∗ = 100, 4
= 400, = 1000, × = 1250. The thick line is from Figure 4.1, which
has been overlaid
for comparison, r/η: = 70 to 110.
roughly the same functional form. This suggests the little
significance of the size
of Rλ on the large scale dependence of conditioned Eulerian
structure functions.
4.5.3 Anisotropy
Anisotropy might also be a suspect for causing the dependence of
small scales
on the large scales, especially as our flow is somewhat
anisotropic. Such anisotropy
in our flow can be seen from the difference in the velocity
statistics in different
directions, presented in Table 4.1. The standard deviation of
z-component veloc-
ities is about 1.5 times greater than that of the x- and
y-component velocities.
The magnitude of the velocity is a non-directional quantity, so the
structure
4.5 Other Causes of Dependence 48
Statistic x-direction y-direction z-direction
Standard deviation 0.0106 0.0108 0.0160
Table 4.1: Comparing the standard deviation of different components
of particle ve-
locity for the center of the tank. The values are in units of
cm/frame.
function should not show dependence on the magnitude of the
instantaneous ve-
locity. Our data (Figure 4.6) shows that the structure function
exhibits stronger
dependence on the magnitude of the instantaneous velocity than on
its vertical
component. The same was observed for the conditioned Eulerian
structure func-
tions by Blum et al. [4], and so we conclude that anisotropy is not
a major cause
of the dependence.
4.5.4 Large Scale Intermittency
Our discussion of the various possible causes of the dependence of
small scales
on the large scales concluded that inhomogeneity is an important
factor but it does
the fully explain the magnitude of the dependence. Other causes
like kinematic
correlation, low Reynolds number and anisotropy are not very
significant. This
leaves large scale intermittency as a possible significant
contributor.
Large scale intermittency is not easy to quantify [4]. It is just
the fluctuation
(in time) of the large scales that occur on timescales longer than
the eddy turnover
time. Fernando and De Silva [9] have shown that large scale
intermittency can
occur in an experiment like ours, where oscillating grids create
the flow, depending
4.6 Conclusion 49
2
4
6
8
10
12
<Δ u2 τ| Σ
Figure 4.6: Lagrangian second order conditional structure function
vs large scale
velocity magnitude for the center of the tank. The colors represent
distinct timescale
τ , as shown in the legend.
on the boundary conditions. Blum et al. [4] show that there are
clear signatures
of large scale intermittency in our flow. Large scale intermittency
is a topic that
needs further exploration so that our understanding of the
dependence of small
scales on the large scales is enhanced.
4.6 Conclusion
We see that in a flow like ours, which is not homogeneous
everywhere, the
large scales do influence the small scales in both the Lagrangian
and the Eu-
lerian perspectives. We have also identified that inhomogeneity and
large scale
4.6 Conclusion 50
intermittency are mainly responsible for this behavior of
turbulence. However,
these very first Lagrangian results of large scale dependence
should be examined
in other flows. Our results here also contribute to understanding
what properties
of turbulence are actually universal.
Chapter 5
Energy transport in turbulence
Lagrangian transport of energy in turbulence plays an important
role in pro-
cesses like mixing and dispersion. However, it has been very
difficult to quantify
turbulence transport in complex flows. Traditional measurement
tools have not
been able to capture the full 3D features of the flows, which are
required for
evaluating many quantities. One available model about the dynamics
of energy
transport in a turbulent flow is in a recently published paper by
Berg et al.. The
group claims that the rate of change of kinetic energy is wholly
determined by the
mean energy dissipation rate. Our calculation shows that this claim
is not true.
We know that, on many circumstances, Lagrangian quantities have had
Eulerian
‘ancestors’, the structure functions being very good examples. I
identify what
factors are mainly responsible for transporting energy in the
Eulerian description
of the flow. These first Eulerian transport measurements and their
inferences can
be the bases for the foundation of Lagrangian energy
transport.
51
5.1 Decomposition of quantities
5.1.1 Velocity decomposition
Figure 5.1: The axial component of velocity against time on the
centerline of a tur-
bulent jet (figure taken from the experiment of Tong and Warhaft
(1995) as published
by Pope [1]).
Figure 5.1 is a representation of the axial component of the
velocity of a fluid
element at a fixed point along the centerline of a jet varying with
time. The mean
velocity is constant for the flow, although the fluctuation is
random around the
mean value of the velocity. If the mean velocity were deducted from
the actual
velocity signal, the corresponding plot would still be like Figure
5.1. Only the
curve is shifted down by the mean velocity. Notably, the mean
velocity of the
new curve will be zero. The same idea can be applied to the
velocity field in a
5.1 Decomposition of quantities 53
turbulent flow in general:
U(x, t) = u(x, t) + U(x, t) (5.1)
where, U(x, t) is the actual velocity field, u(x, t) is the
fluctuation velocity field,
and U(x, t) is the mean velocity field. Equation 5.1 is called the
Reynolds de-
composition. So far, vector notation has been used to express
quantities and
equations. However, tensor notation is more convenient in deriving
many equa-
tions. We will be using the tensor notation very frequently in this
chapter. As a
start, we rewrite Equation 5.1 as:
Ui = ui + Ui (5.2)
where, Ui is an actual velocity component, ui is fluctuating
velocity component,
and Ui is the mean velocity component. With Eqn 5.2 in mind, the
velocity
field can now be seen as the sum of the mean velocity field and the
fluctuating
velocity field.
5.1.2 Decomposing kinetic energy
Like the velocity field, the mean of the kinetic energy (per unit
mass) of a
fluid element can also be decomposed. We start with the definition
of the kinetic
energy E(x, t):
2 U ·U (5.3)
Taking the mean of Equation 5.3, substituting U(x, t) with u(x,
t)+U(x, t) and
using the fact that u = 0, we get the following expression for the
mean of the
5.2 Energy Equation 54
kinetic energy E(x, t):
E = E + k, (5.4)
where,
E = 1 2 U ·U is the mean of the kinetic energy,
E = 1 2 U · U is the kinetic energy of the mean flow, and
k = 1 2 u · u is the turbulent kinetic energy.
In tensor notation,
2 uj · uj (5.7)
Both the velocity and the mean of the kinetic energy have been
decomposed
into the mean part and the turbulent part in tensor form. With
this, one can
derive the insightful energy equation.
5.2 Energy Equation
5.2.1 Reynolds equation
We start with the Navier Stoke’s equation in tensor notation (the
vector form
is in Chapter 1):
∂ui ∂xi
= 0 (5.10)
When the Reynolds decomposition is substituted in the Navier Stokes
equation
(Equation 5.8), with some tensor algebra, we get the Reynolds
equation:
DUj Dt
− ∂uiuj ∂xi
DUj Dt
, (5.12)
and p is the mean pressure field. The velocity covariances uiuj are
called
Reynolds stress.
5.2.2 The equation for turbulent kinetic energy
Basically, the derivation of the equation for turbulent kinetic
energy involves
substituting the Reynolds decomposition, the ‘energy’
decomposition, and the
incompressibility condition in the Navier Stokes and Reynolds
equations. The
derivation of this energy equation is done in many steps. Here we
explain the
different terms in the equation and try to estimate them for our
flow. Like the
5.2 Energy Equation 56
evolution equation for any quantity in a continuum field theory,
the turbulent
kinetic energy equation has the flux (transport) part, the source
and the sink:
Dk
5.2.3 Estimating the terms of energy equation
Evaluating any of the terms in the energy equation is very
difficult with tradi-
tional measurement methods. Since we have full 3D measurements, we
can extract
some of the terms.
P = −uiuj ∂Uj ∂xi
(5.14)
The production is usually positive. It can be roughly estimated for
our flow.
First, it is important to understand that the production term has
two distinct
terms that can be evaluated separately. The two terms are uiuj
(Reynolds
stress) and ∂ Uj ∂xi
(mean velocity gradient). In our relatively homogeneous flow,
the off-diagonal Reynolds stresses are much smaller than the
diagonal terms that
represent the kinetic energy in each velocity component. With this
assumption,
we can approximate the production term to be
P ≈ −u2 j ∂Uj ∂xj
(5.15)
The advantage of using only the isotropic parts of the Reynolds
stress is the
production term becomes very easy to evaluate. Along each axis xi,
we divide
5.2 Energy Equation 57
the detection volume into 5 same-sized subvolumes. Each subvolume
is a sphere
of radius 0.5 cm. The distance between two successive subvolumes is
1cm. For
all the particle tracks within each subvolume (for every frame), we
calculate two
terms- u2 j and Uj in each axis. Then we find the ensemble average
of the two
quantities for each subvolume. We choose the value of u2 j of
central subvolume
(very close to the value of other spheres). To find the mean
velocity gradient, we
just use the slope of the plot of Uj against position in each axis.
The product of
the ‘fluctuation velocity component squared’ term and the mean
velocity gradient
gives the production term for each axis. Finally, adding the three
production
values for the three axes gives us the estimate of the
production.
5.2.3.2 Sink
ε ≡ 2νsijsij, (5.16)
The dissipation term itself is always non-negative (sum of
non-negative numbers),
so with the negative sign in front, it acts as the sink.
Dissipation is quite difficult
to measure, but because it is the central quantity in Kolmogorov’s
description of
turbulence, there have been several methods developed for measuring
it. In fact,
the energy dissipation term has even been estimated using a two
dimensional
surrogate, as pointed out by Pope [1].
5.2 Energy Equation 58
5.2.3.3 Flux
∂
+ 2νujsij ] , (5.18)
where, p′ = p − p is the fluctuating pressure field. Unlike the
production and
dissipation, evaluating the transport term has been difficult. One
reason is there
are three distinct terms, and not all of them are easy to
calculate. The first term
can be written as
2 u2 i (5.19)
which is the kinetic energy transported by velocity. This triple
correlation term
can be estimated quite easily, by dividing the observation volume
into multiple
same-sized subvolumes along each axis, like earlier when we
evaluated the pro-
duction term. For each subvolume along the xj axis, we find the
average of the
term xixjxj of all particles lying inside the subvolume for each
frame. A plot of
the triple correlation term versus the position of the subvolume
can be generated.
The gradient of this plot gives us an estimate of the velocity
transport term (and
its error).
The second term of T ′j , the pressure transport term, cannot be
evaluated easily.
In fact, we are trying to estimate roughly how much contribution
this term has on
the energy budget by either evaluating or neglecting other terms in
the equation.
The third transport term, which is essentially stress transport, is
negligible.
5.2 Energy Equation 59
5.2.3.4 Mean substantial derivative
Dt , of the turbulent kinetic energy has two
terms, the time derivative of the turbulent kinetic energy,
dk
dt and the advection
term Uj dk
dxj . The advection term is small owing to the fact that it is a
product
of the mean flow velocity (which is almost zero for our experiment)
and another
term. The time derivative of the kinetic energy is zero on average,
for the turbulent
kinetic energy is oscillating around a mean value of zero over time
at the center.
5.2.4 Energy budget for the center of the tank
At the center of the tank in our flow, with oscillating grids
feeding energy at
the top and bottom of the tank, production of energy is not
significant. Instead,
turbulent kinetic energy is obtained through the velocity transport
of energy by
particles coming from the top and bottom of the tank (near the
grids). Obviously,
the kinetic energy is lost (or rather transformed into heat)
through viscous dis-
sipation. Table 5.1 has the values of these three terms for our
flow. From Table
Quantity Value (×10−5 m2s−3)
P 13.25
ε 246
Velocity transport −179± 40
Table 5.1: The estimated values of different terms in the turbulent
kinetic energy
equation.
5.3 Energy decay 60
5.1, we see that, despite a big error in the transport term, for
the center of our
tank, velocity transport provides the main input of turbulent
kinetic energy for
particles. However, the velocity transport term is only about
three-fourths of the
total amount of energy dissipated. The pressure transport term
likely accounts
for the remaining energy transport into the volume.
5.3 Energy decay
As I mentioned in the very first part of this chapter, it has been
difficult to
quantify Lagrangian energy transport. However, we can still
investigate changes
in the kinetic energy of a particle (on average) as it enters the
center region of the
tank. We see clear effects of sample bias in our study of energy
change with time
for particles.
5.3.1 Single measurement volume
We look at the experimental data on how on average, the kinetic
energy of a
particle entering the measurement volume varies with time. We study
such decay
of energy in cubic and slab-shaped measurement volumes at the
center of the tank.
The slabs have two parallel surfaces, above and below the center of
the tank, with
no boundary on the sides. The cubic volumes have six planes of same
size, and are
also positioned very close to the center of the tank. We chose only
those particles
that entered the slab or the cube through one of the surfaces.
Moreover, those
particles were considered only while they were inside the
measurement volume.
5.3 Energy decay 61
The kinetic energy values at all times are ensemble averages. The
results are
plotted in Figure 5.2.
0.006
0.008
0.01
0.012
0.014
0.016
2 s− 2 )
Slab (independent of sample) All particle average Cube (independent
of sample) Slab Cube
Figure 5.2: The mean kinetic energy (KE) of a particle vs time (as
a multiple of
Kolmogorov timescale) along particle trajectory; The two lower
curves, red and green
in color, are plots obtained by removing the sample length bias of
particle trajectories.
The two higher ones, blue and orange lines, have sample bias in
them. The dotted
line represents the average KE for all particles that were detected
in one run of the
experiment.
We first looked at a slab and a cube, each of height 3.96 cm. Such
a measure-
ment volume just encloses a sphere of radius 2.8 cm, which is about
the largest
volume that has ‘healthy’ particle density, as seen in Chapter 3.
First, we investi-
gate the top two curves. The energy decay curve for particles
entering the slab is
5.3 Energy decay 62
slightly higher than the one for the cube. The slab has bigger
surface areas than
the cube in the xy-plane, which is perpendicular to grid movement.
Also, both
the surfaces of entrance are perpendicular to the vertical axis for
the slab. The
dotted line, which marks the ensemble average of the kinetic energy
of a particle
for the entire run of the experiment, is less than three-fourth of
the initial