Two Point Correlations Between Velocity Sums and ... · Turbulence by Nicholas Joseph Rotile ... Note that the velocity sum, h u2 iin blue, is much larger than ... and Jet data is

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)


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)


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)


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


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


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


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


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


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)


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


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〉,


〈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.


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,


〈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〉.


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


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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[3] M. Kholmyansky and A. Tsinober. Kolmogorov 4/5 law, nonlocality, and sweeping decorre-

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26

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