Wesleyan University Physics Department
Two Point Correlations Between Velocity Sumsand Differences, and Their Implications for
Large-Small Scale Correlations in FluidTurbulence
by
Nicholas Joseph Rotile
Class of 2012
An honors 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, 2012
Dedication
I dedicate this thesis to my oncological team in pediatrics at Memorial Sloan-Kettering Cancer
Center so that they feel obliged to tack it up on the wall in their office, where I think it will
offer a most humourous contrast to the multitudinous notes and photographs from appreciative
small children.
Abstract
Recent work by Blum, et al has shown the existence of a dependence between large and small
scale statistics in measurements of isotropic fluid turbulence, violating the hypothesized univer-
sality of small scales in fluid turbulence. The authors have argued that that non-ideal effects,
such as inhomogeneity and large scale intermittency are the most likely causes of these depen-
dences. Recent studies of kinematic relations, which seem to imply correlations between large
and small scale statistics, have also been suggested as an explanation for the effects seen by
Blum,et al. This work has focused on measuring the kinematic relation arrived at by Hosokawa.
The first 3–D particle tracking velocimetry measurements of the Hosokawa relation are presented,
as well as a discussion on the sensitivity of the relation to inhomogeneous effects. Ultimately,
the conclusions in Blum et al are supported, as the Hosokawa relation itself is dominated by
inhomogeneity in our flow.
Contents
1 Introduction 1
1.1 A Brief Introduction to the Cascade Model of Fluid Turbulence . . . . . . . . . . 1
1.1.1 Two-Point Velocity Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Longitudinal Velocity Structure Functions . . . . . . . . . . . . . . . . . . 5
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 The Hosokawa Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Experiment 12
2.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Results and Discussion 15
3.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Other Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Conclusions 23
4.1 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
iii
List of Figures
1.1 Second Order Structure Function Plotted Along with the Velocity Sum Analogue
for Comparison. Note that the velocity sum, 〈 Σ u2 〉 in blue, is much larger than
the velocity difference, 〈 ∆ u2 〉 in red, for all values of r . . . . . . . . . . . . . . 6
1.2 Third Order Structure Function normalized by ε r. The 4/5th’s Law is shown
in the inertial range, where the slope of the plot is approximately flat. This was
generated from data taken in the upper atmosphere, where the gas is extremely
turbulent (Rλ ≈ 10,000). Plot from Sreenivasan and Dhruva [1] . . . . . . . . . . 8
1.3 The square of the velocity difference conditioned on the vertical velocity sum,
plotted against the velocity sum. The different colors correspond to different r
ranges. Were ∆ u2 and Σ uz independent, the curves would be flat. Plot from
Blum at al [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Schematic of the turbulence tank. Each of the four cameras point approximately
to the center of the tank, which is illuminated by a 6cm diameter laser beam. . . 14
3.1 Hosokawa Relation. It was expected that 3〈 Σ u2 ∆ u 〉 = -〈 ∆ u3 〉, with 3〈 Σ
u2 ∆ u 〉 in blue and -〈 ∆ u3 〉 in red. As can be seen, they are not equal. . . . . 16
3.2 Full kinematic relation, with all terms included. 〈 ∆ u3 〉 is in red, 〈 Σ u2 ∆ u 〉 is
in blue, 〈 u31 〉 is in brown, and 〈 u32 〉 is in green. 〈 u31 〉 and 〈 u32 〉 were expected
to be negligible, but clearly are not. . . . . . . . . . . . . . . . . . . . . . . . . . 17
iv
3.3 Full kinematic relation. The full kinematic relation is plotted, with the expected
equivalent of 〈 ∆ u3 〉 in solid black, and all other terms the same as in the
previous figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 The terms which the Hosokawa relation predicts to be equal divided by each
other. u+ and u− are simply Σ u and ∆ u, respectively, divided by 2. The value
is 1 where the Hosokawa Relation holds. This data was acquired from a variety
of hot-wire experiments. The data labelled 102, SNM12, and SNM11 correspond
to large masts which are lifted into the atmosphere, while Falcon corresponds to
measurements taken via aircraft, and Jet data is from an indoor jet experiment.
Figure from Kholmyansky and Tsinober [3] . . . . . . . . . . . . . . . . . . . . . 21
Chapter 1Introduction
1.1 A Brief Introduction to the Cascade Model of Fluid
Turbulence
With a wide range of potential applications, from the mixing of fuel with oxygen in a combustion
engine, to oceanic and atmospheric dispersion, to modelling planet formation, the behavior of
fluid turbulence has been an important and puzzling problem for quite some time. In part
because of the many possible boundary conditions in turbulent systems, producing solutions for
and predicting turbulent behavior is a very difficult process. One method of modelling turbulence
has involved utilizing the simple picture of the energetic cascade. This simple cascade model
can be thought of in the following way- imagine rowing a boat. As you move the oar through
the water, little tornado-like vortices follow behind it. Off of those vortices smaller and less
energetic vortices will branch off, and off of each of those still smaller and weaker vortices will
branch off. This continues across a breadth of length scales until the structures become so small
and weak that the motion is simply dissipated into heat. In this general picture, energy is input
at a “large” length scale, from whence it cascades down to smaller and smaller length scales
until it is dissipated as heat.
It has been hypothesized that as the energy transfers downscale, the kinetic motion at each
successive scale becomes more and more randomized to the point that at the smallest scales,
1
Chapter 1 - Introduction 2
it does not retain any of the information about the large scales, such as the details of how the
energy was input to create the turbulence. In this picture, the smallest scales of turbulence
should be universal, depending only upon the viscosity and rate of energy dissipation in the
fluid [4]. Consider the vastly different large scale flows that exist for a wide variety of systems–
water flowing through a stream, turbulence in the wind, the flutterings of a bird in flight, and
even the churning and roiling of a stellar atmosphere– they all require vastly different large
scale descriptions. As such, if a common description for every type of flow could be found, such
a description would be very useful. Should such universality exist, finding solutions to many
aspects of turbulent behavior could be greatly simplified and made more generally applicable.
As such, understanding if and when such universality exists is very important in attempting to
understanding fluid turbulence.
1.1.1 Two-Point Velocity Statistics
In an effort to understand the behavior of this turbulent motion, we must necessarily look to
statistics. Such an approach is necessary because predicting the trajectories of particles of fluid
is impossible, and analytical solutions to turbulent fluid systems do not exist.
In order to probe the various length and energy scales in the energetic cascade, we utilize
two-point measurements of the turbulent velocity. That is, we measure the velocity at two
points, separated by a vector ~r pointing from the first point to the second point. Knowing the
positions of the measurements can be helpful in understanding inhomogeneities, anisotropies,
and boundary condition effects, while the velocity contains directional information about the
flow, and is also related the kinetic energy of the flow.
For our analysis, we focus on the longitudinal component of the velocities. This piece of the
velocity vector tells us whether one particle is going towards or away from the other particle, as
well as how quickly it is doing so. Henceforth, all references to velocities will only be concerning
the longitudinal component of that velocity vector, as defined below.
u1 ≡ ~u(~x) · ~r (1.1)
u2 ≡ ~u(~x+ ~r) · ~r (1.2)
Chapter 1 - Introduction 3
Taking the difference or the sum of the velocity at two points, as defined below, provides infor-
mation on different scales of the flow.
∆u ≡ u2 − u1 (1.3)
Σu ≡ u2 + u1 (1.4)
Taking the velocity difference will largely cancel out effects from structures larger than the
separation distance r. The sum of the velocities, on the other hand, will be dominated by effects
of large scale structures because these larger scale structures are far more energetic than smaller
scale structures. To help visualize this, imagine a large, swirling vortex, like water going down
a drain. Within this large vortex, there are multiple smaller, slower spinning vortices which are
within the larger vortex. Two points within one of these small vortices will be moving along
with not only the small vortex, but also along with the large vortex. Taking the longitudinal
velocity difference of these two points will essentially cancel out any signature of the large vortex,
because both particles will be moving along with the large vortex at essentially the same velocity.
However, effects from the smaller vortex, which significantly changes in structure over smaller
length scales, will leave a signature in the velocity difference. Similarly, the velocity sums will
be dominated by large scale effects because small scale structures, such as these vortices, have
relatively uncorrelated velocities at the two points, whereas the largest scale motions will always
be common to both.
Typically, the separation distance r is normalized by what is called the Kolmogorov scale length,
η. η represents the smallest possible scale of structure before the turbulent motion is entirely
dissipated to heat. This characteristic scale is used to remove individual, specific, large scale
flow details and to normalize the length scales relative to the size of the smallest structures in
a particular flow. For example, the size of the small scales of turbulence in your bathtub and
in astrophysical nebulae are extraordinarily different, but universality demands that they be
related, hence the use of η. In keeping with the universality hypothesis, η is defined in terms of
the fluid viscosity and the energy dissipation rate of that flow, as shown below.
η ≡ (ν3/ε)1/4 (1.5)
Chapter 1 - Introduction 4
Where ν is the kinematic viscosity of the fluid, and ε is the energy dissipation rate per unit
mass.
Though the scales of different flows can be normalized by η to more easily compare relative length
scales, characteristic ranges of those scales do not necessarily end up being the same size. One
such range of interest is the inertial range, which is a sort of middle range. That is, the inertial
range is a range of scales small enough that large scale effects should have become effectively
randomized by the turbulent cascade, but still large enough that dissipation of kinetic energy
into heat remains negligible. The differences in the breadths of these ranges between flows is
largely caused by differences in the intensity of the turbulence. As an example, the inertial range
in our flow scarcely covers one order of magnitude of η, whereas the far more turbulent flows in
the atmosphere have inertial ranges over nearly 3 orders of magnitude in η. One can roughly
picture the sense of this by imagining rowing your boat with a much larger oar. The larger oar
inputs more energy, generating stronger turbulence. The energy has a larger range of length
scales to cascade through. Thus, the large scale effects will be randomized sooner before entering
the dissipative range, allowing for a larger inertial range. This is a considerably simplified picture
considering that the length scale of the energy input is not exclusively tied to the intensity of the
turbulence, but for a simple mental image it gets the point across. For a greater understanding
of what effects the intensity of turbulence, consider the Reynold’s number.
Re =UL
ν(1.6)
Where U and L correspond, respectively, to the characteristic velocity and length scale of the
flow. The Reynold’s number is a dimensionless number which is characteristic of the intensity of
the turbulence in the flow, with a higher Reynolds number indicating more intense turbulence.
The more commonly used quantity, however, is the Taylor-scale Reynolds number, which will
be used whenever the Reynolds number is reported.
Rλ =√
15Re (1.7)
Chapter 1 - Introduction 5
1.1.2 Longitudinal Velocity Structure Functions
The statistics of the longitudinal velocity differences have been widely studied. These statistics
tell us about scales of the flow smaller than the distance, r, between the two points, while the
statistics on velocity sums are related to the large scales of the flow. Within the inertial range,
structure functions scale as shown in equation 1.8.
〈∆un〉 = Cεrn/3 (1.8)
Where C is some constant, n is a positive integer, and again ε is the average energy dissipation
rate of the fluid.
Since the inertial and dissipative ranges are the scales in which large scale effects should not
play a significant role, they are the ranges in which universality should be most detectable. The
inertial range, however, is more often studied because within it, simple power laws exist, such
as velocity structure functions. In the inertial range, these structure functions tell us about how
energy is transferred down from larger to smaller scales, without being affected by non-universal
large scales or viscous dissipation. Because larger length scales have much larger energies, the
velocity differences are of greater magnitude at larger separation distances, as they include
larger, more energetic structures. For any r within the inertial range, most of the contribution
to the structure function is from effects of scales just below r, with negligible contributions from
the considerably less energetic scales well below r. Thus, structure functions are scale local
quantities. That is, they are dominated by contributions from structures of size near r. Take
the second order structure function, for example.
In Figure 1.1 I have plotted the second order structure function, 〈∆u2〉, and 〈Σu2〉 vs. the
separation distance between the particles. ∆u2 is an energy-like quantity characteristic of scales
smaller than r, whereas Σu2 is similar, but is characteristic of scales larger than r. Notice that as
you increase r, ∆u2 increases dramatically due to the inclusion of larger, more energetic scales.
Σu2, on the other hand, has much less r dependence, as it always includes the largest, most
energetic scales, which dominate over the smaller scales.
Another well studied structure function is the third order structure function. This particular
structure function has been derived rigorously from the Navier-Stokes Equation for fluid motion
Chapter 1 - Introduction 6
101
102
103
104
Second Order Structure Function
log(r/η)
log
(u2)
<∆u2>
<Σu2>
Figure 1.1: Second Order Structure Function Plotted Along with the Velocity Sum Analogue for Comparison.
Note that the velocity sum, 〈 Σ u2 〉 in blue, is much larger than the velocity difference, 〈 ∆ u2 〉 in red, for all
values of r
within the inertial range, and the constant C has been analytically found. It is called the 4/5th’s
Law in reference to that constant.
〈∆u3〉 = −4
5εr (1.9)
This relation can be helpful in conceptualizing the energy cascade. For example, 〈∆u3〉 goes like
the third moment of the distribution of velocity differences, or the skewness of ∆u. A negative
∆u indicates the two points are moving closer together, and so the negative skewness indicates
a greater probability that particles coming together will do so more rapidly. The average of
the ∆u distribution must be zero on account of the incompressibility of the fluid, so there must
be a larger number of particles moving apart at slower speeds. This indicates that particles
Chapter 1 - Introduction 7
coming together will tend to be moving more rapidly. than if they were moving apart. This
is precisely due to the energy cascade- as particles move from larger to smaller scales (coming
closer together), they are following the rapid, energetic flow, whereas when they move to larger
scales (move further apart), they are following fluid which has already dissipated its energy into
heat.
This can be visualized as a vortex which is being stretched perpendicular to its rotation. Two
particles, each far from the center of the vortex, will rapidly spiral towards the center of the
vortex, and rapidly approach each other as they do so. As they get closer to the center of the
vortex, however, they will approach each other with a smaller relative velocity. They may spiral
around about the center, but they will have small velocities relative to each other. As the vortex
stretches, the particles will move along with the vortex, most likely in opposite directions, slowly
with the stretched and diminished flow.
Another view might be to look at 〈∆u3〉 as 〈∆u2∆u〉. From this one might consider the quantity
as being something like an energy times a direction. Since a negative ∆u indicates particles
coming closer together, the 4/5th’s Law shows that energy will flow such that particles become
closer. In other words, this relation shows that energy flows downscale.
An example of the third order structure function of an extremely turbulent system is shown in
Figure 1.2. Note that the relation extends over nearly three orders of magnitude of η, whereas
in our flow, which is at much lower Reynolds number (and is thus less intensely turbulent), the
inertial range is relatively small, extending scarcely over one order of magnitude.
1.2 Motivation
Recent work looking into this question of universality, has shown that there exist dependences
between large and small scale quantities [2] [1]. Large scale quantities are non-universal, typi-
cally depending largely upon the boundary conditions of the flow, such as the turbulence driving
mechanism, the geometry of the fluid’s container, etc. Sreenivasan and Dhruva found a depen-
dence between the small scale velocity in the form of 〈∆u2〉 and the instantaneous large scale
velocity u. They attributed this effect to shear in the upper atmosphere, as data from perfectly
isotropic and homogeneous direct numerical simulations and low-shear wind tunnel measure-
ments did not show this dependence. Blum et al, using measurements from a flow between two
Chapter 1 - Introduction 8
Figure 1.2: Third Order Structure Function normalized by ε r. The 4/5th’s Law is shown in the inertial range,
where the slope of the plot is approximately flat. This was generated from data taken in the upper atmosphere,
where the gas is extremely turbulent (Rλ ≈ 10,000). Plot from Sreenivasan and Dhruva [1]
oscillating grids, found that there were dependences between large and small scale quantities,
such as the square of the velocity difference, ∆u2, and the velocity sum, Σu.
The dependence found by Blum et al was extremely similar to that found by Sreenivasan and
Dhruva. As the flow in which Blum et al measured their dependences has very little shear, they
concluded that shear must not be the cause of this dependence, but some other property similar
to the oscillating grid flow and the atmospheric flow. Using measurements at differing distances
from the oscillating grids, Blum et al showed that inhomogeneity is a major contributor to
this dependence. Since the grids input energy, which then essentially cascades away towards
the central region, making measurements at different distances from the grids is analogous to
measuring in more or less homogeneous regions, from which the differing effects of inhomogeneity
could be ascertained. Inhomogeneity only accounted for part of the dependence seen, and Blum
Chapter 1 - Introduction 9
Figure 1.3: The square of the velocity difference conditioned on the vertical velocity sum, plotted against the
velocity sum. The different colors correspond to different r ranges. Were ∆ u2 and Σ uz independent, the curves
would be flat. Plot from Blum at al [2]
et al. argued that the effects of large scale intermittency are most likely the cause. Large scale
intermittency involves fluctuations at the largest length scale, such as when input energy might
temporarily cascade to higher scales instead of lower, for example. In a later paper, they showed
similar large-small scale dependences in a wide variety of flows, indicating that this is not a rare
artefact of a few particular flows [5].
While Blum et al have argued that the measured dependences are the result on non-idealities
in the flows in which they were found, such as inhomogeneity and large scale intermittency, it
has also been suggested that these could instead be signatures of kinematic relations of the kind
arrived at by Hosokawa [6]. Hosokawa found, from the 4/5ths Law, that there must exist some
non-zero correlation between Σu2 and ∆u. These kinematic relations suggest that there must
Chapter 1 - Introduction 10
be certain correlations between velocity sums and differences. My work has been to investigate
these kinematic relations and measure them in our flow.
1.3 The Hosokawa Relation
What is the Hosokawa Relation? In words, it says that, in homogeneous turbulence, the square
of the velocity sums must be correlated with velocity differences. The equation reads
3〈Σu2∆u〉 = −〈∆u3〉
The derivation is quite simple and almost entirely algebraic. To begin, simply recall the defini-
tions of Σu and ∆u from equations 1.3 and 1.4
∆u ≡ u2 − u1
Σu ≡ u2 + u1
Invert them to find u1 and u2 in terms of Σu and ∆u
2u1 = Σu−∆u (1.10)
2u2 = Σu+ ∆u (1.11)
Cube both equations and take the ensemble average
8〈u31〉 = 〈Σu3〉 − 3〈Σu2∆u〉+ 3〈Σu∆u2〉 − 〈∆u3〉 (1.12)
8〈u32〉 = 〈Σu3〉+ 3〈Σu2∆u〉+ 3〈Σu∆u2〉+ 〈∆u3〉 (1.13)
Taking the difference between the two equations, some terms cancel and we are left with...
4(〈u32〉 − 〈u31〉) = 3〈Σu2∆u〉+ 〈∆u3〉 (1.14)
(1.15)
Chapter 1 - Introduction 11
By definition, in a homogeneous flow, the statistics at one point and the statistics at another
are the same, so if we assume homogeneity, then 〈u32〉 = 〈u31〉, and the left side of the equation
drops to zero, resulting in the Hosokawa relation [6].
3〈Σu2∆u〉 = −〈∆u3〉 (1.16)
Substitute in the 4/5th’s Law for ∆u3 (see equation 1.9)
〈Σu2∆u〉 = 415εr (1.17)
Note that this treatment can be used for arbitrary positive integer powers of u1 and u2 to arrive
at a number of kinematic relations. Instead of cubing equations 1.10 and 1.11, raise them to any
power, then subtract the two equations to arrive at a kinematic correlation. The third power
case arrived at by Hosokawa is particular due to the 4/5th’s Law relation involving 〈∆u3〉, which
means that it must be a non-zero correlation for r, ε 6= 0.
At first glance, this suggests that large and small scales are related. Considering that ∆u
is dominated by small scales and Σu is dominated by large scales, it would seem natural that
Σu2∆u being nonzero, as shown by the 4/5th’s law, would be indicative of a correlation between
large and small scales, apparently violating the universality hypothesis. However, recall that the
magnitude of Σu is always much larger than that of ∆u. So the fact that the Hosokawa Relation
says that this correlation is of the order of ∆u3 means it must be a very weak correlation. Were
it a strong correlation, it would be of the order of Σu2√
(∆u2), but ∆u3 is much less than
Σu2√
(∆u2), hence the weakness of the correlation. It is more plausible that scales very near
r in both the velocity sum and velocity difference are correlated, with large scale effects in the
velocity sum cancelling out and leaving a correlation of the order of the velocity difference. The
Hosokawa Relation is not indicative of large small–scale correlation. It is only a correlation of
scales near r in Σu2 with scales near r in ∆u. That is, the Hosokawa relation is a correlation at
scale r.
Chapter 2Experiment
2.1 Apparatus
Our experimental apparatus consists of a 300 gallon, 1 x 1 x 1.5 m3 octagonal tank in which
two grids, equally spaced from the center of the tank, oscillate in phase to generate turbulence.
The tank was designed to produce very nearly homogeneous turbulence in the center region.
For the type of setup used for the data I analysed, the tank is filled with water and seeded with
neutrally buoyant tracer particles, which are then illuminated by an expanded 50W Nd:YaG
pulsed laser through the center, with a diameter of about 6cm. Four cameras then take images
of these particles at about 480Hz with a pixel resolution of 1280 x 1024 each. Such resolution
and imaging frequency would normally produce a data stream that vastly exceeds the rate
at which the data could be written to a hard drive. Real time image compression circuits
reduce the size of the data by a factor on the order of 100, thus allowing essentially endless
data collection. Altering the frequency at which the grids oscillate produces different Reynolds
number turbulence. The grids can be brought to oscillate at 5Hz, but due to uncertainty in the
tanks structural integrity and fear of it shaking itself apart, very little data has been taken at
that high of a frequency.
12
Chapter 2 - Experiment 13
2.2 Data Processing
Once the data has been taken, it is then processed with stereo matching, particle tracking, and
velocimetry algorithms to find very precise particle positions and velocities through time. A
mean subtraction algorithm removes the mean motions of the flow so that only the turbulent
velocities of the particles are used for analysis.
Due to the geometry of the laser beam, particle pairs are more likely to be detected along the
direction of the beam, especially at large separations. To ensure an isotropic sampling of the
data, a rejection method, developed by Susantha Wijesinghe, was applied to guarantee an equal
probability of finding particle pairs oriented in any direction.
The data sets which I have used in my analysis were obtained by Dr. Dan Blum and Susantha
Wijisinghe. Data taken at 1Hz, 2Hz, and 3Hz by Dan Blum and Susantha Wijisinghe. Some
data sets taken at grid frequencies of 4 and 5Hz were considered early in the analysis, but were
soon discarded in favor of the larger 1, 2, and 3Hz data sets.
Chapter 2 - Experiment 14
Figure 2.1: Schematic of the turbulence tank. Each of the four cameras point approximately to the center of
the tank, which is illuminated by a 6cm diameter laser beam.
Chapter 3Results and Discussion
3.1 Experimental Results
In working to quantify the possible effects of kinematic relations such as the Hosokawa relation,
I have analyzed multiple data sets, taken at grid frequencies at 3, 2, and 1Hz. The data sets
used in my analysis are among the largest available to me, having between 1 and 3 million
frames of data, typically with on the order of 109 particle pairs after the use of our anisotropy
rejection algorithm. All plots presented in this chapter were analyzed from data taken at 3Hz
grid frequency by Dan Blum.
In my investigation of the Hosokawa Relation 3〈Σu2∆u〉 = −〈∆u3〉 I have measured the quan-
tities 〈Σu2∆u〉 and 〈∆u3〉 as functions of r. I have plotted in Figure 3.1 those quantities, with
3〈Σu2∆u〉 in blue and −〈∆u3〉 in red.
As is evident in the plot, the two quantities are not equal, and thus do not follow the Hosokawa
relation. Indeed, they are not even close but are nearly opposite one another for all r. Presented
such a plot, once one has satisfied themself that this is not a simple sign error (which it isn’t),
the natural next step is to investigate the reason for this inequality by looking back to any
assumptions that have been made. With this in mind, recall the equation from which the
Hosokawa Relation was extracted, equation 1.14)
15
Chapter 3 - Results and Discussion 16
0 50 100 150 200 250 300 350 400
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
x 104 Hosokawa Relation
r/η
u3 [mm3s−3]
−<∆u3>
3<Σu2∆u>
Figure 3.1: Hosokawa Relation. It was expected that 3〈 Σ u2 ∆ u 〉 = -〈 ∆ u3 〉, with 3〈 Σ u2 ∆ u 〉 in blue
and -〈 ∆ u3 〉 in red. As can be seen, they are not equal.
4(〈u32〉 − 〈u31〉) = 3〈Σu2∆u〉+ 〈∆u3〉
From this equation, making the assumption that 〈u31〉 = 〈u32〉 due to homogeneity leads to the
Hosokawa Relation: 3〈Σu2∆u〉 = −〈∆u3〉. Though our flow is among the more homogeneous
experimental flows, it would seem that this homogeneity assumption cannot be taken for granted,
and the 〈u31〉 and 〈u32〉 terms should be measured.
In figure 3.2 I have plotted 〈u31〉 in brown and 〈u32〉 in green, along with the terms from the
Hosokawa Relation from Figure 3.1 for comparison, 3〈Σu2∆u〉 in blue and −〈∆u3〉 in red. As
can be seen, 〈u31〉 and 〈u32〉 are not equal and are not zero, but are exactly opposite of each other
(again, not a sign error). They look small compared to the third order structure function, but
Chapter 3 - Results and Discussion 17
0 50 100 150 200 250 300 350 400
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
x 104 All Terms of Kinematic Relation
r/η
u3 [mm3s−3]
<u1
3>
<u2
3>
−<∆u3>
3<Σu2∆u>
Figure 3.2: Full kinematic relation, with all terms included. 〈 ∆ u3 〉 is in red, 〈 Σ u2 ∆ u 〉 is in blue, 〈 u31 〉
is in brown, and 〈 u32 〉 is in green. 〈 u3
1 〉 and 〈 u32 〉 were expected to be negligible, but clearly are not.
this is because only a quarter of their contribution is plotted (the u1 and u2 terms are multiplied
by 4 in the kinematic relation). Not only is the inequality of these terms unexpected, but their
non-negligible size is, as well. Our flow is very nearly homogeneous in the center region where
the data was taken. Homogeneity should force the u1 and u2 terms to be equal, yet this is
not so. Furthermore, our flow should be both sufficiently isotropic and homogeneous that those
terms, even if not equal, ought to be nearly zero. It seems intuitive that, with no preferential
direction, u31 should be just as likely to be positive as negative, and if there little or no spatial
dependence on the turbulent motion, u31 in any direction should be very small.
Double checking to ensure that there are no errors that have been left unaccounted for, I have
plotted the full kinematic identity, solving for the third order structure function from equation
1.14 4(〈u32〉 − 〈u31〉) − 3〈Σu2∆u〉 = 〈∆u3〉 in a solid black line along with 3〈Σu2∆u〉 ,−〈∆u3〉,
Chapter 3 - Results and Discussion 18
〈u31〉, and 〈u32〉 in Figure 3.3. As expected, the identity holds true.
0 50 100 150 200 250 300 350 400
−14
−12
−10
−8
−6
−4
−2
0
2
4
6
x 104 All Terms of Kinematic Relation
r/η
u3 [mm3s−3]
<u1
3>
<u2
3>
−<∆u3>
3<Σu2∆u>
3<Σu2∆u>−4(<u
2
3−u
1
3>)
Figure 3.3: Full kinematic relation. The full kinematic relation is plotted, with the expected equivalent of 〈 ∆
u3 〉 in solid black, and all other terms the same as in the previous figure.
These same results show for each data set analyzed. The terms of the Hosokawa relation, instead
of being equal, are nearly opposite each other, and including the u31 and u32 terms always returns
the relation to an exact identity. These results were not expected, and caused quite a bit of
confusion. Our flow is very nearly homogeneous, and we took great effort to ensure that our
data was isotropically sampled, so the overwhelming effect of the u31 and u32 terms, regardless
of data set, seemed to go against intuitive sense. It was only after very careful thinking and
inquiry that the likely cause of this effect was understood.
Chapter 3 - Results and Discussion 19
3.2 Discussion
Considering that our flow is very nearly homogeneous, we are left to consider how the u1 amd u2
terms are not only unequal, but also non-zero and large, entirely opposite of would be expected in
a homogeneous flow. Following many sign checks, I was able to ascertain that part of the answer
may lie in the fact that our data analysis guarantees that the 〈u1〉 and 〈u2〉 measurements are
opposite one other. This happens because, in order to prevent any sort of artificial anisotropy
introduced by the computer’s choice of which particle to call 1 and which particle to call 2, we
measure each particle pair twice, switching the choice of 1 and 2. This is shown below.
For each step in the analysis, you will have two particles to consider. First, pick which particle
is 1 and which particle is 2. This will define the direction of ~r to go from particle 1 to particle
2. Calculate u1 and u2.
u1 = ~u1 · ~r (3.1)
u2 = ~u2 · ~r (3.2)
Then switch which particle is 1 and which particle is 2, and recalculate. u’ and r’ will refer to
the measurements with the switched choice of 1 and 2.
u′1 = ~u2 · ~r′ (3.3)
u′2 = ~u1 · ~r′ (3.4)
Consider the average value of each quantity, ignoring a factor of 2.
〈u1〉 = ~u1 · ~r + ~u2 · ~r′ (3.5)
〈u2〉 = ~u2 · ~r + ~u1 · ~r′ (3.6)
Since ~r′ = −~r,
Chapter 3 - Results and Discussion 20
〈u1〉 = (~u1 − ~u2) · ~r (3.7)
〈u2〉 = (~u2 − ~u1) · ~r (3.8)
〈u1〉 = −〈u2〉 (3.9)
This alone should cause no problems, because u1 being equal to u2 doesnt matter if they are both
zero, as they should be in a homogeneous flow. If, however, this is paired with a small radial
inhomogeneity, which could feasibly occur in our tank as energy passes from the oscillating grids
to the central region, this effect could be observed. Because our observation volume is finite, for
any choice of particle, its pair will tend to be in the direction of the center of the observation
volume, which corresponds to the approximate center of the tank. Imagining a sphere with a
uniform density of particles, it becomes clear that for any particle chosen which is not in the
very center of the volume, the average direction towards another particle will be towards the
center of the volume. If there is a radial inhomogeneity, the 〈u3〉 measurement will measure
some of the skewness of the particle motion. Given some radial inhomogeneity, this motion
towards the center of the detection volume will be, on average, aligning with the movement of
the energy cascade as the energy cascades towards the center of the tank. Particles will thus
tend to go towards the center more rapidly, with many more particles moving away from it more
slowly to maintain incompressibility. Combining the tendency for ~r to point towards the center
of the tank and the radial inhomogeneity, 〈u31〉 will measure the tendency of particles to more
rapidly enter the center of the tank, while 〈u32〉 will measure precisely the same thing, but with
a reversed coordinate system.
This explains why we see non-zero and opposite 〈u31〉 and 〈u32〉 measurements, but not why they
are so large considering that our flow ought to have only a small degree of inhomogeneity. The
reason for the high magnitude of these terms is because, for a highly inhomogeneous flow, 〈u31〉
would be on the order of the large scale motion, which is vastly larger than the small scale
structures being measured by 〈∆u3〉. As such, even a small amount of inhomogeneity is enough
to overwhelm the weakly correlated, relatively low magnitude 〈Σu2∆u〉.
Chapter 3 - Results and Discussion 21
Figure 3.4: The terms which the Hosokawa relation predicts to be equal divided by each other. u+ and u− are
simply Σ u and ∆ u, respectively, divided by 2. The value is 1 where the Hosokawa Relation holds. This data
was acquired from a variety of hot-wire experiments. The data labelled 102, SNM12, and SNM11 correspond to
large masts which are lifted into the atmosphere, while Falcon corresponds to measurements taken via aircraft,
and Jet data is from an indoor jet experiment. Figure from Kholmyansky and Tsinober [3]
3.3 Other Results
It should be noted that the Hosokawa Relation has been experimentally verified in several
flows [3,7]. In Figure 3.4 from [3], the Hosokawa relation has been shown to hold in a variety of
experiments. The plot shows one term from the Hosokawa relation divided by the other. Where
the plot is at unity is when the Hosokawa relation is satisfied.
This is not indicative of inadequacy in our flow or measurements. It is, in fact, not surprising
that these experimental measurements of the Hosokawa Relation have been made. In light of
our understanding of the strong effects of inhomogeneity, it is actually perfectly reasonable that
the experiments carried out by Mouri and Hori [7] and Kholmyansky and Tsinober [3] were able
to measure the relation because they utilized hot-wire measurements. Hot-wire experiments are
performed, as the name implies, by using a hot wire with a current run through it and exposing it
to a flow. As fluid passes the wire, it cools it depending on how quickly the fluid flows past it. The
Chapter 3 - Results and Discussion 22
resistance of the wire changes with temperature, and this is measured in the change in the voltage
across the wire, which is eventually translated into velocity data. Typically, the measurements
utilize Taylors Hypothesis to relate the change in time between measurements to a change
in position in order to calculate two-point statistics. This method of measurement, however,
guarantees homogeneity because the measurements will always be made at a single point. Thus,
it is not unusual that a variety of hot-wire measurements have experimentally confirmed the
Hosokawa Relation, because the inhomogeneous terms which overwhelm our relation are forced
to cancel.
Chapter 4Conclusions
4.1 Conclusions and Future Work
Using 3D particle tracking velocimetry, we have measured the individual components of the
Hosokawa relation and found that the relation does not hold in our flow. Using multiple data
sets, typically with on the order of 109 particles pairs per set, and utilizing a rejection method to
prevent anisotropic sampling, we have found that, in our flow, the correlation found by Hosokawa
is obscured by comparable effects of inhomogeneity. This lends support to the conclusions
arrived at by Blum et al, that the large–small scale correlations they found using conditional
structure functions were not the result of these kinematic correlations, but were more likely due
to inhomogeneous and non-ideal effects.
Our measurements show that inhomogeneity represents a major factor in the Hosokawa relation.
Even though our flow is very nearly homogeneous, the Hosokawa relation is very sensitive to
that small inhomogeneity in our flow. The dominance of this inhomogeneity is understand-
able given the weakness of the correlation implied in the Hosokawa relation. The correlation
3〈Σu2∆u〉 = −〈∆u3〉 is not a correlation of large and small scales, but is a correlation at scale
r. Since the inhomogeneous terms u1 and u2 are dominated by large scale, an inhomogeneity in
our flow will result in large scale effects not cancelling out. Because these large scale effects are
so much stronger than small scale effects, even a small degree of inhomogeneity will be sufficient
23
Chapter 4 - Conclusions 24
to overwhelm the Hosokawa relation. This result lends support to the conclusions of Blum, et
al. that the large-small scale correlations they saw were not due to kinematic relations, but
were rather due to non-idealities in real systems, such as the inhomogeneity encountered in our
measurements. Future work includes finding ways to more precisely quantify and measure this
inhomogeneity which is dominating the Hosokawa relation. Coincidentally, one way to investi-
gate this inhomogeneity is to measure the components of the energy budget of the turbulence,
in particular the energy transport. This was a former project of mine, following on the work of
Surendra Kunwar, which I dropped on account of seemingly unreasonable results and an insuffi-
cient amount of background knowledge to thoroughly investigate the issue. Perhaps, in light of
the picture presented here, future analysis of the energy budget can be better understood.
Acknowledgements
Neither this thesis, nor most of my career at Wesleyan would have been possible without the
help and support I was so fortunate to receive over the past 6 years. I owe a great many thanks
to a great many people.
Most immediately, many thanks go to my parents, Brian and Helen for their unending support,
love, and our lovely nightly tea times.
To my advisor, Greg Voth, for his great kindness, patience, enthusiasm, and for guiding me
through multiple projects, assisting me with all the inevitable problems I ran into along the
way.
To my colleagues, Susantha Wijesinghe, Shima Parsa, Guy Geyer, Sam Kachuck, and Dan Blum
for their assistance in pretty much every aspect of working in a physics lab.
To the rest of my family and relatives, whose constant, immediate support, love and (when
necessary) prodding, helped me to stay focused, upbeat, and generally jovial through all of my
troubles.
To my oncological team at Memorial Sloan-Kettering Hospital– Rosemary, Maura, Katiri, and
Doctors Shukla and Steinherz, as well as all of the nursing and reception staff. They made a
hospital visit for chemotherapy treatment into a trip to look forward to.
Lastly, to my friends, especially my roommates, Tom, Max, Dan, and Jeff, whose company kept
me appropriately (in)sane while writing this thesis, and throughout the year.
25
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