DavidRatner_MScThesis_August2010

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

Lagrangian measurements of Kolmogorov size

neutrally buoyant particles in turbulent flow

A thesis submitted toward the degree of

Master of Science in Mechanical Engineering

David Ratner

May 2010

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

Lagrangian measurements of Kolmogorov size

neutrally buoyant particles in turbulent flow

A thesis submitted toward the degree of

Master of Science in Mechanical Engineering

David Ratner

This research was carried out in the School of Mechanical Engineering

Under the supervision of

Dr. Alexander Liberzon

May 2010

ACKNOWLEDGMENT

This work was carried out in the Turbulence Structure Laboratory at the Tel Aviv

University.

I am most thankful to my supervisor Dr. Alex Liberzon, for his support, help and

guidance; without them my work would not have been possible.

I also would like to thank all my colleagues from the laboratory, to Reut Elfassi for her

guidance and help, to Mark Kreizer for his friendship and help, to Dikla Kersh, David

Altura, Oleg Babin and Vitaly Haslavsky for their wonderful company, assistance and

moral support.

My special thanks goes t to all the technical and management staff in the Department of

Fluid Mechanics and Heat Transfer that have made it possible for me to accomplish all

the duties for the M.Sc. degree. They were all kind and helpful.

Also, I want to thank Ira, my family and friends for their support and help; without them

I wouldn’t be here today.

ABSTRACT

There has been increased effort to the develop measurement techniques for

turbulent flows research, both in an attempt to achieve accurate small-scale

measurements as well as to extract the maximum possible information from the

processes observed. Small-scale measurements are especially important in two-phase

flow in which the second phase consists of very small particles (in the range of a tenth

of a millimeter). Despite extensive research, it is difficult to predict the velocity or

distribution in space of the particles in turbulent flow, due to their interaction with the

carrying fluid. Two-phase interaction bears significance in a range of applications

including conceptual understanding of the physics of raindrops, aerosol sprays, and dust

storms, among others.

Measurement problems become far more complex when the flow is turbulent

and the particles in question are on the smaller end of the flow length scales. The Maxey

& Riley (1983) approach and similar, which assume that the particles are point-like, are

no longer applicable to equation derivations for the motion of relatively large particles.

We must ask to what extent such approach is applicable and whether it might help to

adjust the definition and the physical significance of the Stokes number, St τ/τ. The present work deals with the question: if neutrally buoyant particles with a diameter

in the range of the Kolmogorov length scale behave differently than the flow, what is

the main difference and which quantities underlie the physical mechanisms responsible

for the observed differences? Particles movement was measured using a three-

dimensional PTV (particle tracking velocimetry) system that allows for identification

and tracking of particles as they move through a control volume. The system can also

identify the particles trajectory and kinematic properties, such as location, velocity and

acceleration with great precision. Carrying fluid flow properties were measured using

the same technique with neutrally buoyant flow tracers (with a mean diameter of 10

microns). The study was performed in a turbulent LDC (lid-driven cavity) flow at a

Reynolds number of 10,000 based on the cubic cavity width (80 mm) and the velocity

of the lid.

We demonstrate differences between the various quantities we measured among

which the most striking results came from the second order structure—the large

particles do not behave like the flow tracers despite their low Stokes number and in fact

show relatively higher correlation to the velocity-time increments.

ACKNOWLEDGMENT

ABSTRACT…………………………………………………………………………...I

TABLE OF CONTENTS……………………………………………………………..II

LIST OF FIGURES…………………………………………………………………..IV

NOMENCLATURE………………………………………………………………...VIII

1 Introduction .............................................................................................................. 1

2 Literature review ...................................................................................................... 3

2.1 Turbulent particle-laden multiphase flows ........................................................ 3

2.1.1 Basic equations ........................................................................................... 7

2.2 Overview of measurement techniques ............................................................... 8

2.2.1 Three-dimensional particle tracking velocimetry (3D-PTV) ..................... 9

2.2.1.1 Principles of 3D-PTV ........................................................................ 10

2.2.1.2 Tracking algorithm ............................................................................ 11

3 Objectives ............................................................................................................... 12

4 Experimental method .............................................................................................. 12

4.1 Preliminary experiment .................................................................................... 12

4.2 3D-PTV experimental setup ............................................................................ 15

4.2.1 Calibration ................................................................................................ 16

4.3 Methods of analysis ......................................................................................... 17

4.3.1 3-D PTV program processing analysis ..................................................... 17

4.3.2 Matlab analysis ......................................................................................... 19

5 Results and Discussion ........................................................................................... 22

5.1 Preliminary experiment .................................................................................... 22

5.2 3D-PTV experiment ......................................................................................... 23

5.2.1 Eulerian Results ........................................................................................ 24

5.2.1.1 Average velocity maps ...................................................................... 24

5.2.1.2 Root-mean-square velocity contours ................................................. 27

5.2.1.3 Reynolds stresses .............................................................................. 31

5.2.1.4 Summary of Eulerian analysis .......................................................... 31

5.2.2 Lagrangian results .................................................................................... 32

5.2.2.1 Scales evaluation ............................................................................... 33

5.2.2.2 Distribution analysis .......................................................................... 34

5.2.2.3 Estimation of the settling velocity effect .......................................... 36

5.2.2.4 Spatial derivatives ............................................................................. 38

5.2.2.5 Lagrangian correlation of velocity .................................................... 40

5.2.2.6 Autocorrelation of acceleration ......................................................... 42

5.2.2.7 Acceleration analysis ........................................................................ 44

5.2.2.8 Velocity increments .......................................................................... 46

5.2.2.9 Structure function .............................................................................. 47

5.2.2.10 Logarithmic derivatives .................................................................... 48

5.2.2.11 Mixed velocity-acceleration structure function ................................. 50

6 Summary and conclusions ...................................................................................... 52

7 Bibliography ........................................................................................................... 54

APPENDIX A: Uncertainty analysis

LIST OF FIGURES

Figure 1: Basic elements of a PTV system, components of the system are described in

section 4.2 (Dracos 1996). .............................................................................................. 10

Figure 2 : Determination of the observation volume (Dracos 1996). ............................. 11

Figure 3 Scheme of particle trajectory evaluation (Dracos 1996). ................................. 11

Figure 4 Schematic view of the preliminary experiment setup along with combined

images from the experiment. x2-represents measuring volume, x1- represents distance

between measuring volume and water surface, xn- xn-1 –distance between two adjusted

mark lines. ...................................................................................................................... 13

Figure 5 Electronic microscope images of two particles. ............................................... 14

Figure 6 3D-PTV experimental setup ............................................................................. 15

Figure 7 Schematic view of cameras and cavity with axes origin and cavity location

with respect to the origin. ............................................................................................... 16

Figure 8 Isometric view of calibration body of 0.06×0.06×0.006m, the body is 3D

shaped and has channels of 1mm depth on each side with distance between white dots in

x-y plane of 5 mm and of 1 mm in z plane. .................................................................... 17

Figure 9 Four calibration images (taken at the same time by 4 cameras), a-d) are

cameras 1-4, respectively, according to the camera positions. ....................................... 17

Figure 10 Flow chart of the 3D-PTV software procedure (Willneff, 2003)................... 18

Figure 11 Example of an image (captured by camera 1) in tracers and particles

experiment. ..................................................................................................................... 18

Figure 12 Lagrangian (right) to Eulerian (left) transformation of system of coordinates.

........................................................................................................................................ 19

Figure 13 Settling velocities of the particles as function of the depth. .......................... 23

Figure 14 Velocity vector maps. (a) z mid plane map of Ux , Uy , (b) y mid plane map Ux

, Uz , (c) x mid plane map of Uz , Uy . Blue color represents tracer velocity and red color

represents particle velocity, top arrow represent velocity scale. .................................... 25

Figure 15 Average velocity profiles comparison: (a)- Prasad and Koseff (1989) results (

SAR 1:1 Re=10000), tracers velocity profile using the Eulerian analysis, particles

velocity profile using the Eulerian analysis. (b)- particles average profile for two

different voxels sizes. ..................................................................................................... 26

Figure 16 ux r.m.s in z mid-plane (a) – r.m.s velocity of tracers, (b) – r.m.s velocity of

particles, (c) – r.m.s velocity of particles calculated using average velocity of tracers,

(all axes are normalized by cavity width D). .................................................................. 28

Figure 17 uy r.m.s in z mid-plane (a) – r.m.s velocity of tracers, (b) – r.m.s velocity of

particles, (c) – r.m.s velocity of particles calculated using average velocity of tracers (all

axes are normalized by cavity width D). ........................................................................ 29

Figure 18 uz r.m.s in x mid-plane (a) - r.m.s velocity of tracers, (b) - r.m.s velocity of

particles, (c) - r.m.s velocity of particles calculated using average velocity of tracers (all

axes are normalized by cavity width D). ........................................................................ 29

Figure 19 Profiles of r.m.s velocity for middle z plane of the cavity. (1) – uy_r.m.s in

cross-section of z and y middle planes, (2) - ux_r.m.s in cross-section of z and x middle

planes. (a) - Prasad and Koseff (1989) results ( SAR 1:1 Re=10000), (b)- tracers

velocity profile, (c)- particles velocity profile. ............................................................... 30

Figure 20 Reynolds stress contour maps at the z mid-plane (a)- tracers, (b)-particles. . 31

Figure 21 Particles trajectories in 3D cavity (a)- trajectories Longer than 100 frames, (b)

- longest particle trajectories. Red circle represent the end point of trajectory. Cube

denotes the limits of the cavity. ...................................................................................... 33

Figure 22 Particles/tracers histogram of spatial distribution, along (a)- x, (b)- y, (c)- z,

directions, respectively. .................................................................................................. 35

Figure 23 PDF of spatial particle/tracer location. (a)- x location, (b)- y location, (c)- z

location. .......................................................................................................................... 36

Figure 24 PDF of vertical velocity component Uy. (a)- tracers (average value is 0.012

m/s), (b) particles (average value is 0.007 m/s). ............................................................. 37

Figure 25 Number of detected particles as function of frame number, with average

number line (black) of 274 detected particles. ............................................................... 38

Figure 26 (Left) Plot of x component of particle velocity for several insulated

trajectories in time, each blue line represent single trajectory. (Right) Velocity

fluctuation v’ in respect to the linear fit representing average velocity. ........................ 39

Figure 27 Spatial derivatives. (1) Spatial derivative of full velocity, (a)- tracer (average

value is 25.5 1/s2 ), (b)- particle (average value is 25.8 1/s2). (2) Spatial derivatives of

velocity fluctuations. (a)- tracer (average value is 23.9 1/s2 ), (b)- particle (average value

is 24.4 1/s2 ). ................................................................................................................... 40

Figure 28 Particle location and velocity at two different times on its trajectory............ 41

Figure 29 Lagrangian velocity autocorrelation coefficient. (a)- tracer (represented by

solid lines curves) and particle (represented by symbols) autocorrelation coefficient of

Lagrangian velocity vector and its components, (b)- tracer and particle autocorrelation

coefficient of “fluctuating” velocity vector and its components. (full-Rvv , Rxx , Ryy , Rzz

time is normalized by Kolmogorov time scale τ which is equal to 0.1s ) . .................... 42

Figure 30 Sketch of acceleration vector with its components of two points on particles

trajectory. ........................................................................................................................ 43

Figure 31 Acceleration autocorrelation function. (a) Autocorrelation function of full

acceleration vector, Raa (b) autocorrelation function of tangential to trajectory

component of acceleration vector RaTaT. Time is normalized by Kolmogorov time scale

τ which is equal to 0.1s. .................................................................................................. 44

Figure 32 Acceleration in direction of x axis (ax), tracers r.m.s <ax2>0.5=0.16 m/s2,

particles r.m.s <ax2>0.5=0.17 m/s2 . ................................................................................ 45

Figure 33 Acceleration in direction of y axis (ay), tracers r.m.s <ay2>0.5=0.14 m/s2,

particles r.m.s <ay2>0.5=0.17 m/s2 . ................................................................................ 45

Figure 34 Acceleration in direction of z axis (az), tracers r.m.s <az2>0.5=0.3 m/s2,

particles r.m.s <az2>0.5=0.3 m/s2 . .................................................................................. 46

Figure 35 Velocity increments. (a) velocity increments of tracers, (b) velocity

increments of particles. Time lags are 0.07 0.15 0.3 1.5 3.5 τ, and displayed with a

vertical shift from 0.07 (up) to 3.5 (down) for clarity. ................................................... 47

Figure 36 Lagrangian second order structure function of particles and tracers x and y

velocity components, τlag is lag time from minor time interval between adjusted point to

maximum interval between start and the end point of the trajectory, τ is Kolmogorov

time scale of the flow. .................................................................................................... 48

Figure 37 Logarithmic derivatives of order 4. (a)-tracers, (b)-particles, τlag is lag time

and τ is Kolmogorov time scale of the flow. .................................................................. 49

Figure 38 Logarithmic derivatives of order 6. (a)-tracers, (b)-particles, τlag is lag time

and τ is Kolmogorov time scale of the flow. .................................................................. 49

Figure 39 Lagrangian time mixed structure function, τlag – is a lag time and τ is

Kolmogorov time scale of the flow. ............................................................................... 50

Figure 40 Probability density function (F) of the cosine of the angle between ∆τv and

∆τa ................................................................................................................................... 52

NOMENCLATURE

a ax,ay,az particle/tracer acceleration projection on x y z axes respectively

vi Lagrangian velocity i components. vn,vt are normal and tangential projection on

the trajectory , vx vy vz – projection on the Cartesian axes.

ui Eulerian velocity components.ux - x component of velocity, uy – y component of

velocity, uz - z component of velocity

Ui Average velocity component

g Gravitational acceleration

ρ Density [ ]

ρf , ρp – density of the fluid and particles accordingly

ν Kinematic viscosity

µ Dynamic viscosity ·

PIV Particle Image Velocimetry

PTV Particle Tracking Velocimetry

LDA Laser Dopler Anemometry

Re Reynolds number =ν

LV ⋅ (V-velocity, L- length, ν -kinematic viscosity)

Rep , Rel - Reynolds number which length scale is particle diameter and flow

representing length accordingly

St Stokes number (- particle response time, -fluid response time)

2D, 3D 2 and 3 dimensional

l, λ, η Integral , Taylor and Kolmogorov flow length scales accordingly,

Kolmogorov time scale

ε Mean rate of dissipation per unit mass

fps frame per second

dp Particle diameter

Ub Velocity of the leading band

∆τ increment time function defined as ∆f ft τ ft. ∆r increment length function defined as ∆"f fx r fx.

1 Introduction

Development of experimental techniques capable of measuring the smallest scales is

especially important in two-phase flows where the second phase of bubbles, particles

and droplets typically affects the flow on small scales. Extensive research is needed

because we do not have yet the correct prediction of particle position or velocity in

complex and turbulent flows. As of late we have some statistical functions of velocity

and concentration for some cases. We have no general theory for the turbulent flow

itself. Although the equations of the flow are known (i.e., Navier Stokes), it is a non-

linear partial differential equation which has no general solution; this equation is one of

seven “millennium problems” and a million dollar prize is currently being offered for its

solution (Clay Mathematical Institute). In general, solutions are gained from neglecting

the non-linear components of the equation, but this is not possible in turbulent cases.

When dealing with turbulent two-phase flows, we have not only the unknown motion of

the carrier fluid, but also the second phase with its own inertia, density, etc. Interaction

between the two phases is a problem of great interest, since most of the natural and

engineering flows include more than one phase. Such flows we see in aerosols,

raindrops, dust particles in atmosphere, raindrops, plankton in the oceans and much

more. The problem becomes much more complicated when the flow is turbulent where

many factors and scales are involved, and particles are of the size of the flow scales or

larger. It is noteworthy that, in order to derive the equation of motion of particles

significantly larger than the smallest scale of the flow, it is not possible to use the

approach of Maxey & Riley (1983), which is based on the assumption of point-like

particles. It is therefore natural to ask to what extent an extension of the above approach

is possible, and whether it might help to maintain the definition and the physical

significance of the Stokes number. There are many questions we can ask about the

interaction starting from governing equations and continuing with the drag coefficient

where a lot of un-explained phenomena such as drastic reduction in drag coefficient of a

rigid sphere at high Re numbers. Our way of understanding the phenomena is by

measuring velocities, accelerations and locations of bodies in turbulent flow and trying

to find some regularity and statistical order, so that we might say that in certain cases,

flow with bodies will have some same predicted properties, with hope of predicting not

only the same cases but also slightly different ones, trying to match formulas, non-

dimensional numbers like Re, St, Pa (Tanaka and Eaton, 2008). There are many

measurement techniques of measuring flow and bodies in flow, from classical ones such

as anemometers and Pitot tubes, requiring insertion of a probe into the place interested

to measure, to later and more complicated techniques that need not intrude the flow as,

PIV, LDA and PTV (Dracos 1996), but insert into the fluid a second phase, follow after

it with high speed cameras, analyze it's motion and conclude on the flow motion. With

development of advanced techniques, better cameras and strong computers to process

the data, we can now get to smallest scales with miniature particles that follow the

smallest eddies thereby measuring and exploring phenomena that we could only dream

of measuring before. Therefore, the development of these methods, which allow for the

measurement of parameters that could only be estimated in the past, can be extended

further for the exploration of new concepts and ideas.

2 Literature review

The literature review focuses on the studies of turbulent multiphase flows of fluids and

solid particles, which are in the focus of the present study. Recent experimental,

numerical and theoretical studies are reviewed in the following section.

2.1 Turbulent particle-laden multiphase flows

Single-phase turbulent flows of Newtonian fluids are very complicated and require

extensive experimental, theoretical and numerical research in order to obtain relatively

simple empirical models. It is even more difficult when the flows include a second

phase, e.g., particles. The Brennen (2005) textbook, for example, mentions two main

groups of complications related to the particle-laden turbulent flows:

1. Complex unsteady motions of the particles that may result in non-uniform spatial

distribution of the particles and possibly even particle segregation. It can also result in

particle agglomeration or in particle fission, especially if the particles are bubbles or

droplets.

2. Modifications of the turbulence itself caused by the presence and motions of the

particles. Turbulence could be reduced by the presence of particles, or it could be

enhanced by the wakes and other flow disturbances that the motion of the particles may

introduce.

The question that remains asks under which conditions each of these scenarios might

occur and what are the important parameters that dominate the various phenomena.

The effect of particles on the flow was studied by direct numerical simulations (DNS)

as well as through a variety of experimental methods. Since the particles are Lagrangian

objects, Lagrangian properties such as dispersion of particles, their velocity and

acceleration history become of primary interest. The first numerical studies in isotropic

turbulence were performed by Yeung and Pope (1989) for various Reynolds numbers.

The authors found a strong correlation of the acceleration statistics with the Taylor

micro-scale Reynolds number, %&'. The connection between Lagrangian velocity

autocorrelation and Eulerian velocity autocorrelation was examined experimentally by

Shlien and Corrsin (1974). They found that the major difference in Taylor micro-scales

calculated from autocorrelations resulted from the different definitions between

Lagrangian and Eulerian autocorrelations, where Eulerian autocorrelation relies on

constant distances in pre defined control volume, and Lagrangian autocorrelation relies

on distances particle makes on its way (trajectory).

Particle-liquid interaction is the key to understanding major multiphase flows. It

includes different issues, starting from fundamental aspects of modeling dispersion for

numerical studies Shirolkar et al. (1996), which reviewed and summarized leading

works in order to describe the problem, to experimental research of turbulent transport

of particles with different diameters in isotropic turbulent flows; Qureshi et al. (2007)

who found that there is no shape change in acceleration probability density function in

contrast with its variance and chaotic flows; Ouellette et al. 2008, who studied

millimeter-scale particles compared to finite sized particles and found a systematical

difference between their statistical properties.

The effect of particles on the flow was researched also by Gore and Crowe (1989) who

studied the effect of particles on turbulent intensity and claimed that the value of the

ratio d η⁄ (particle diameter to Kolmogorov length scale) can indicate whenever

turbulent intensity is increased or decreased by the particles: where this ratio is larger

than 0.1, turbulent intensity of the carrier fluid (first phase) is increased, and smaller

than 0.1 is decreased.

Measures have been carried out in an attempt to understand the problems theoretically.

Tanaka and Eaton (2008) added new parameters to try to classify turbulence

modification by sphere particles with the novel dimensionless particle moment number

+Pa ../

0120134345 673

8 9:;, which was created in order to simplify the carrier-phase Navier

Stokes equations in the presence of particles, along with Re number. They found a

turbulence attenuation region between augmentations for two critical particle

momentum numbers.

The problem of assessing large and heavy particle distribution and clustering regions in

turbulent flow has been examined by Guala et al. (2008) who found clustering of these

particles in strain-dominating regions. The continuation of this work with heavier-than-

fluid Kolmogorov sized particles at fully developed turbulent flows was performed by

Gibert et al. (2010) who found that the mixed Eulerian structure function of two particle

velocity and the acceleration difference vector =∆"v · ∆"a? increase significantly with

particle inertia, and showed that this increase can be attributed to a preferential

alignment between these dynamical quantities.

One of the major characteristic dimensionless parameter describing particles in flow is

the Stokes number, St, which has many definitions and significant impact on its motion.

The Stokes number is defined as a ratio of particle time scale τp to a representative time

scale of the flow τf , or as viscous drag term in the equation of motion (Maxey and

Riley, 1983).

The time scale of the particle reflects how long it takes the particle to react to flow

conditions. Its value is as follows:

For small particles which Reynolds number based on settling velocity (Vs) is very small

O 6AB·73C 9 D O1, time scale estimation is:

FG./µ (1)

When particle Reynolds number based on settling velocity is of order of one

O 6AB·73C 9 I O1, we include Oseen correction to particle time scale:

FG./µ.J

KLMN (2)

According to Fallon and Rogers (2002), the definition of the fluid time scale is not

always apparent and depends on limiting factors in the flow and there are many ways of

choosing time scales based on different definitions. The same problem exists in

definition of Reynolds number, and there are many definitions of Stokes number based

on many fluid time scales, usually defined with same values as Reynolds number with

characteristic velocity and length scale.

Ouellette et al. (2008) defined Reynolds number as Re=ur.m.s L/ν where ur.m.s is r.m.s

velocity and L is magnet spacing (the flow was induced by a Lorentz force who set into

motion KCL solution), and Stokes number defined as OP Q43R45 67

89Q · Re .

Guala et al. (2008) defined particle time scale as:

τ U3VU5U5 73

./C (3)

(Neglecting the Basset and Faxén forces as well as acceleration of the fluid in particle

location) and τ as Kolmogorov time scale of the flow.

Gibert et al. (2010) defined Stokes number for solid particles with densities varying

between 1 W ρ/ρ W 8, and particle diameters of the order of Kolmogorov length scale

St 73.QZ[ (4)

where β ] 3ρ/2ρ ρis the modified density ratio that takes into account the added

mass effect, and η is the Kolmogorov length scale.

Bourgoin et al. (2009) suggested that Stokes number effects (defining Stokes number as

τp/τD where the particle time scale τp has been corrected in agreement with Schiller and

Nauman (1933) and τD is the flow eddy turnover time at a scale of particle diameter)

cannot be considered alone to properly characterize and model inertial particles.

Optical experiments can supply the direct experimental data of particle velocity location

and acceleration properties. For example, optical particle tracking methods were used

by Sato and Yamamoto (1987) who performed Lagrangian measurements of fluid-

particle motion and determined r.m.s. values of fluctuating velocity and estimated

velocity autocorrelations. More complicated properties such as velocity structure

functions and logarithmic derivatives, which are derivatives of basic properties

(location, velocity and acceleration), are often used to shed light on subjects like physics

of dispersion and mixing, (e.g. Toschi and Bodenschatz 2009).

One of the parameters requiring greater attention is acceleration and its derivatives. The

statistical parameters, such as autocorrelation and mixed structure functions, are

typically used. These provide us clues about the forces particles experience while

moving in flow. For example Calzavarini et al. (2009) demonstrated that point particle

models do not capture the effects of particle finiteness on their statistical properties if

the Faxén forces are not taken into account. The impact of parameters, such as particle

density and diameter on these forces was studied by Bourgoin et al. (2009), who

demonstrated that Stokes number alone is not enough to properly characterize and

model inertial particles, and by Brown et al. (2009), who, in contrary, found little

dependence between particle size and the scaled acceleration probability density

function.

2.1.1 Basic equations

The majority of the research reviewed above describes particle motion according to the

well-known equations of a small rigid sphere proposed by Maxey and Riley (1983). The

equations are written for a small rigid sphere of radius ` which is present in the fluid

flow and which, in absence of the particle, is otherwise defined by the velocity field

ab, P. The presence of the sphere (or particle) and its motion through the fluid will

modify the flow locally and lead to a new velocity field, acb, P.

The authors (Maxey and Riley, 1983) applied the momentum and continuity equations

for incompressible flow of uniform density:

d 6efgeh aciac9 dj ik liQac

i · ac 0

with the following boundary conditions ac n Ω p b qP on the sphere

surface and acx, t ux, t as |x Yt| u ∞

If the fluid stress tensor is defined as:

σxy pδxy µ +|fg|~ |fg

then the equation of motion for the spherical particle is:

m 7fg7 mgx σxynydS.

where surface integral is over the surface of the sphere and is the normal to the

surface direction. The problem is: in order to evaluate the fluid stress tensor on the

sphere one can transform the coordinates to a frame moving with the center of the

sphere with given transformation equations: , , , , . Also, in order to take advantage of small parameters, it is further

convenient to separate the flow field into two parts, q(0) and q(1) ; q(0) being the

undisturbed flow without application of the boundary condition and q(1) is the

disturbance made by the sphere , the two components of the

flow that together satisfy the equation of motion, and each separately satisfies

incompressibility.

The equation for the disturbance flow reduces to

ρ |K| |K

| µ |K| | (9)

provided that ` · q/ν 1 Reynolds number defined on sphere radius and velocity

of undisturbed flow) and `Q/νU/L 1 where L is representative length scale of

the undisturbed flow, and U/L is the scale for the corresponding velocity gradient.

Besides the disturbed flow, there is also a contribution to the fluid force from the

undisturbed flow q, which may be found quite generally and without specific

assumptions about low Reynolds number, such as the conditions used to derive the

previous equation. From the above, the contribution to the fluid force on the particle

from q is F; Fx ny pδxy µ |

| || ¡ (10)

which may be converted to a volume integral and approximately evaluated as

Fx ¢: πa: ¤

| µ || |¥ (11)

under the assumption that the terms in parentheses are nearly uniform over the sphere,

provided that the sphere is sufficiently small. This assumption is valid if the size of the

sphere is small compared to the length scale of variations in the undisturbed flow

(Maxey and Riley, 1983). This is the key assumption, which does not exist in

turbulent flows with particles of the order of scales of the undisturbed flow

(particles of this size are the key interest in this thesis). The smallest scales of the

turbulent flows are Kolmogorov length scales, whose dimensions are of the order of the

particles diameter and even for tracers the difference is not very significant (around

1:50-100), but sufficient for the Maxey and Riley (1983) equations. For flows with large

Reynolds number and not point like particles, flow around the sphere is not uniform

over the sphere. That is the reason why we cannot use analytical approach predicting the

motion of the particles.

2.2 Overview of measurement techniques

There are many techniques that can be utilized for the measurement of flow/particle

velocity. Flow field measurements are mainly based on non-intrusive optical

techniques, among which are particle image velocimetry (PIV), which is designed to

determine the movement of small groups of particles embedded in a fluid element

during a short time as well as its stereo, multi-plane and holographic applications that

allow to measure velocity vectors in three-dimensional space. Particle tracking

velocimetry (PTV) allows for determination of the paths of individual particles inside

an illuminated volume. Laser Doppler anemometry (LDA, or LDV), allows for the

measurement all three components of the velocity at one location. Another technique is

acoustic Doppler in which measurements are limited to particles approximately five

times larger than the Kolmogorov scale. All aforementioned techniques have their

advantages and disadvantages which are reduced with technology advance, (i.e. faster

cameras with higher resolution, smaller sizes of the probes, higher illumination, and

faster and bigger storage devices).

2.2.1 Three-dimensional particle tracking velocimetry (3D-PTV)

Three-dimensional particle tracking velocimetry (3D-PTV) determines the velocity and

acceleration from the displacement of individual particle in moving fluid during a

prescribed time interval. 3D-PTV method has its limitations: the relatively low number

of tracked particles per unit volume (as compared to the PIV, for example), illumination

must be pulsed at high rate or continuous so that cameras can “follow” the particle in

the flow and the optical access requirement adds constraints to the experimental setup.

Moreover experiment duration time is also limited by the capacity of storage devises

and their recording speed.

Snyder and Lumley (1971) provided the first systematic set of particle tracking velocity

measurements from wind-tunnel grid turbulence, and later Sato & Yamamoto (1987)

reported similar measurements in water-tunnel grid turbulence. Three-dimensional

Lagrangian particle tracking using multiple cameras for the stereoscopic reconstruction

of particle tracks was pioneered by Virant and Dracos (1997) and further developed by

Ott and Mann (2000). The first measurements at high Reynolds numbers, using direct

optical imaging were accomplished by means of silicon-strip-detector technology

(Mordant et al. 2004a, b; La Porta et al. 2001; Voth et al. 2001, 2002); however, only

single-particle dynamics could be investigated. Today the technology of complementary

metal-oxide semiconductor (CMOS) cameras allows the tracking of many particles at

rates equivalent to those of the silicon strip detectors, but with only one quarter of the

spatial resolution (Bourgoin et al. 2006). Recently, improved three dimensional particle

tracing velocimetry techniques allowed access to velocity gradients along particle

trajectories (Lüthi et al. 2005). The same technique has been extended to scanning

particle-tracing velocimetry to reach even larger particle densities (Hoyer et al. 2005).

A recent development involves instrumented particles which are promising tools in

exploring Lagrangian turbulence (Gasteuil et al. 2007). These particles can include

sensors capable of measuring local quantities such as temperature or acceleration and

transmitting the signal in real time to receiving stations (current large size of the

particles limit their utility for turbulent flows) (Toschi and Bodenschatz, 2009).

2.2.1.1 Principles of 3D-PTV

3D-PTV tracks the trajectories of individual particles in three-dimensional space.

Particle identification and determination of its position in space is a very important step

in PTV, so in order to achieve the needed accuracy: therefore, the seeding density of

particles is limited.

In order to measure flow velocity, the use of very small and neutrally buoyant particles

(called tracers), which claim to be a perfect trace of the flow and react with its minor

changes is necessary. Using them we can say that their velocity and trajectories are

velocity and streamlines of the flow.

Figure 1: Basic elements of a PTV system, components of the system are described in section 4.2

(Dracos 1996).

Detection of these tracers and bigger particles in the observation volume depends on the

phenomena studied and the equipment used (such as the power of a laser or another

light source, focus range, and quality of CCD cameras), so the images produced are

suitable for later analysis. Examples of an array of this type are shown in Figure 1 and

the observation volume with its dimensions and important parameters in Figure 2.

Figure 2 : Determination of the observation volume (Dracos 1996).

After the tracers/particles are identified and located in the pre-set coordinate system the

second stage analysis role is to find a match between tracers/particles in time and thus to

create trajectories that can later be used to determine velocity, acceleration and other

parameters.

2.2.1.2 Tracking algorithm

Figure 3 Scheme of particle trajectory evaluation (Dracos 1996).

Tracking algorithms requires assuming the maximum velocity of the particle which

determines the searching sphere in which the particle might be found in the next time

step. Knowledge on the previous particle location and velocity provides the algorithm

with possible location where particle may be in the next (and if this analysis goes back

in time then the previous) frame is received. Figure 3 demonstrates a three-frame

algorithm basic, including particle searching spheres and five possible variants of

trajectory path. From frame f to frame f+2, the particles are located inside the searching

sphere and each one could represent the particle for which the trajectory is being built.

The best trajectory is chosen by its smoothness and minor possible changes in velocity

and acceleration. Typically the trajectory is smoothed using a polynomial or another

low-pass filter from which the tracer location, velocity and acceleration are calculated.

3 Objectives

The primary objective of this study is to provide a set of properties that will

unequivocally define whether the solid particle motion is different from the carrying

fluid motion. It is of our interest to compare the Eulerian and Lagrangian approaches to

the 3D data obtained using 3D-PTV experimental method. Specifically, in this work we

intend to established 3D-PTV to acquire the motion of the solid particles in a turbulent

lid-driven cavity flow, attempting to measure turbulent (fluctuating) velocity of the

solid particles and compare it to that of the flow tracers. We use Eulerian ensemble

averaging and directly measured Lagrangian properties to demonstrate main

similarities/differences of the finite size particles and flow tracers.

4 Experimental method

In this section there is a description of the experimental arrangement, and a brief

explanation about the ways of how the experiment was done.

The first set of experiments is designed to measure the settling velocity of the particles

under investigation in order to obtain precise definition of the Stokes number and verify

the values of the particle diameter and density.

Following the precise definition of the particles used in the following study, the PTV

experiments, which include tracers and particles experiments, were designed and

performed as described below.

4.1 Preliminary experiment

The settling velocity experiment, which is represented in Figure 4 along with direct

diameter measurements represented in Table 1, provides us with sufficient information

about particles we used later in 3D-PTV experiments.

Figure 4 represents a schematic view of the experimental setup, showing a glass beaker

aligned with a camera and a combination of two images, where the particle is shown at

the beginning and toward the end of measuring volume (103 frames between those two

images).

Figure 4 Schematic view of the preliminary experiment setup along with combined images from the

experiment. x2-represents measuring volume, x1- represents distance between measuring volume

and water surface, xn- xn-1 –distance between two adjusted mark lines.

Experimental procedure:

Ten particles were released one by one into a tall beaker that has marks at constant and

known distances of 2.7¦0.1 mm, (small unnumbered marks that are too numerous to be

counted in Figure 4). Particle release was made in the center of the vessel (this was

done so influence of the walls would be minimal), and the settling was measured by a

high-speed camera at a speed of 15 frames per second. We used a high-speed camera

because its internal electronic shutter is very precise and we know that the time delay

between the frames is 0.06667 seconds with a negligible error, therefore we know the

exact time that it takes for a particle to fall when thrown a known distance. By filming

the falling of the particle in the beaker and knowing the time passed between each frame

where we see the particle passing the marks, we can manually extract (using Matlab,

detecting manually particle location and calculating distances the particle made between

frames) particle velocity according to the following equation: §¨©gª«

¬ p n 12

Where ®¯°cN is the distance between marking lines, ± is the number of frames, –

frames-per-second (fps) rate and n - settling velocity.

The particles fall almost vertically, for example, in Figure 4, the upper particle x

position and y position is 442 and 78 pixels respectively, x and y position of the same

particle later on (shown as lower particle in Figure 5 ) is 440 and 871 pixels, indicating

that it moved 2 pixels in horizontal position while moving 793 pixels in vertical

position.

The observation volume is 5 cm high and located 20 cm beneath the water level. At this

depth particles already reach constant velocity according to Stokes law. Using the

settling velocity results, we can estimate particle density with high accuracy. If the

particles are falling in the viscous fluid by their own weight, then a terminal velocity,

also known as the settling velocity, is reached when the frictional force combined with

the buoyant force exactly balances the gravitational force. The resulting settling velocity

is given by:

²³ ´µ¶´·¶¸·¹º»¼½ (13)

where dp is the radius of the particle, d, d - is the density of the particle and of the

fluid, respectively and l is dynamic viscosity and g is the acceleration due to the

gravity.

Figure 5 presents electronic microscope images of two particles with measured

diameters. It shows the rigid spherical shape of the particles.

Figure 5 Electronic microscope images of two particles.

In the following Table 1 we summarize the results of the measurements of particle

diameters. Ten randomly selected particles were measured using analog caliper

(measurement error of 0.005 mm) and the results are given in millimeters.

Table 1 Diameter of 10 random particles used in preliminary experiment and the mean diameter

µ¾¶

Particle 1 2 3 4 5 6 7 8 9 10

¿ 0.73 0.68 0.74 0.72 0.79 0.66 0.64 0.72 0.67 0.8

4.2 3D-PTV experimental setup

The main experimental method used in this work is 3D-PTV, providing particle and

tracer location, velocity and acceleration, along with statistical information needed to

fulfill the research objectives. Figure 6 shows a schematic view of the experimental

arrangement, with the main components listed.

Figure 6 3D-PTV experimental setup

We use a glass cubic cavity of 80 x 80 x 85 mm. The cavity is located inside a large

glass tank and positioned using a plastic frame such that its top is parallel to the top

edge of the large glass tank. The main reason of using the small cavity in the larger tank

is to have the lid submersed in water, preventing air entrainment and open surface

disturbances. A flexible smooth plastic belt (total length of 103 cm, 2 mm thick) serves

as a lid in our setup. The belt is driven by a servomotor (Dahaner Motion, Santa

Barbara, CA USA) and the mechanical system designed by IT&ES Engineering (Haifa,

IL) The control is done using Kollmorgen SepLink 2.0.7 software and a SERVOSTAR

controller. The controller adjusts the angular velocity of the driving motor with the

precision of 1 [rev/min]. The speed of the belt is measured manually using the time

lapse for the 10 full cycles of the belt. The experiments shown in this work were

performed using the belt speed 0.125 [m/s]. Laser light provided by the dual pulsed

laser (527 nm, Quantronix, USA) was pumped at 1000 Hz, providing approximately 75

watt of light, illuminating the small cavity.

We used four digital CMOS cameras (1280 x 1024 pixels, 8 bit, Mikrotron, Germany).

The cameras are located as shown in Figure 7, while two tapirs of cameras were facing

the cavity from two sides, pointing toward the center of it. The positions are given in

coordinate system attached to the calibration target as we explain below. The cameras

simultaneously captured and transferred the video using a real-time image streaming

and storage system. The four cameras are synchronized with a maximum possible time

jitter of 1/1000 frame rate. The digital video recording system includes 48 hard drives

and four special frame grabbers (CLFC, IO Industries), each allowing for a 700 Mb/sec

data transfer rate.

In addition to the large particles described above, the flow was seeded by the flow

tracers that are hollow-glass spheres with the density of 1.03 g/cm3 and an average

diameter of 11 microns.

Figure 7 Schematic view of cameras and cavity with axes origin and cavity location with respect to

the origin.

4.2.1 Calibration

The calibration procedure included in the 3D-PTV software provides configuration of

the cameras, consisting of 16 parameters such as focal length, X, Y, Z position of the

camera imaging center, angles of the camera imaging axis in respect to the coordinate

system in the cavity frame of reference, magnification factor and optical distortions. A

detailed description is given in Dracos (1996). The results are the calibration files that

allow direct and inverse projection of every pixel into the 3D space and every point in

3D space onto the 4 imaging planes of the cameras.

The calibration body used in calibration is made of a non reflective black metal alloy

with white light-reflective dots see Figure 8, central dot is defined as axes origin of the

cavity and the whole experimental coordinate system.

Figure 8 Isometric view of calibration body of 0.06×0.06×0.006m, the body is 3D shaped and has

channels of 1mm depth on each side with distance between white dots in x-y plane of 5 mm and of 1

mm in z plane.

The calibration procedure included the following steps:

• The calibration body is securely placed inside the cavity corresponding to the axis

shown in (Figure 7).

• Pictures of the calibration body are taken in the cavity by all four cameras (Shown

in Figure 9).

• Measurement of the approximate camera location and viewing angles with respect

to the calibration body axes.

a) b) c) d)

Figure 9 Four calibration images (taken at the same time by 4 cameras), a-d) are cameras 1-4,

respectively, according to the camera positions.

4.3 Methods of analysis

We used 3D-PTV software developed by ETH Zurich (Swiss Federal Institute of

Technology) and available under a free academic license from http://ptv.origo.ethz.ch.

The output is a set of trajectories organized in ASCII files. The database is processed

then using Matlab software that creates the necessary Lagrangian and Eulerian statistics.

4.3.1 3-D PTV program processing analysis

The software analyzes the images according to the following Figure 10, which

schematically represents the 3D-PTV algorithm.

Figure 10 Flow chart of the 3D-PTV software procedure (Willneff, 2003).

Figure 11 presents an example of an image captured by one of the cameras. The image

shows the laser illuminated cavity with tracers and particles (marked in red). The

particles are clearly large compared to tracers. The software allows discriminating the

particles by size and track separately the large particles and the flow tracers.

Figure 11 Example of an image (captured by camera 1) in tracers and particles experiment.

In this experiment we analyzed 1000 frames of tracers and 1617 frames of particles,

captured at rate of 150 frame per second. The output of the software is positions of the

particles in the cavity at every time instant, linked together into trajectories. Particle

positions in time are used to estimate velocity and acceleration along the trajectory.

During the post-processing stage, the particle positions are smoothed to fit into curve

performing spline functions for parts of the trajectory (detailed explanation on the

process and its errors are given in Lüthi et al., 2005).

4.3.2 Matlab analysis

Data obtained in the experiment is in Lagrangian system of coordinates (meaning that

we were following an individual fluid parcel as it moves through space and time). In

Lagrangian sections, short trajectories are usually negligible (less than 10 frame

trajectories) and long trajectories are the ones that provide us with useful data for a

relatively long period of time and different locations in the cavity. In Eulerian section

(at fixed locations) we use the entire data processed throw the analysis described below

without filtration. The reason for this is that most of the data are in short trajectories

(trajectories that have more than 10 frames are less than 5 percent of all the trajectories)

and the accuracy of statistical processing that describe the Eulerian analysis depends

directly on the amount of data. Figure 12 presents a sketch of Lagrangian to Eulerian

coordinate system transformation (explained below), showing briefly how particles

velocities are transformed to fixed system locations velocity.

Figure 12 Lagrangian (right) to Eulerian (left) transformation of system of coordinates.

Lagrangian analysis

Lagrangian analysis is carried out using the information about the path of particles. It

contains information about evolution of particles in flow. The analysis is pointed toward

particle information along its path, and is a direct continuation of 3-D PTV program

processing, so that no further processing is necessary. Using Matlab functions and self

made codes (like autocorrelation and structure functions) results are achieved in the

form presented in the results section. Lagrangian analysis provides information about

particle trajectories, locations, velocity and acceleration autocorrelations, velocity

structure functions etc.

There are flow parameters that cannot be evaluated using Lagrangian analysis, including

mean and fluctuating velocity analysis, spatial distributions of Reynolds stresses.

Therefore the Lagrangian data has to be interpolated onto a fixed grid to proceed with

the Eulerian analysis.

Eulerian analysis

The Eulerian results we obtain by the interpolation of Lagrangian (sparse and unevenly

distributed in space) to Eulerian (uniformly distributed in space) data. The procedure is

performed in Matlab®, using linear interpolation in three-dimensional space. The cavity

is divided into voxels (3D spaces) of 5×5×5 mm with special attention devoted to the

cavity boundaries. In some cases the reflected light masked the data in near wall

regions. Thus, for example, at the top part we had to remove from the analysis the top 5

mm. Therefore we do not have a direct measure of the lid velocity by the 3D-PTV.

Due to sparse distribution of the particles, there are plenty of empty voxels (usually near

the boundaries of the cavity) at different time steps. This causes an undesired effect of

regions with minor statistics and in result we have windows where the resultant data is

not reliable, for 5202 (18×17×17) voxels we usually have 274 tracers and a quarter of

that amount of particles, therefore although we use the data from these regions in our

Lagrangian analysis part we cannot do that in the Eulerian part.

Error of the interpolation procedure:

Eulerian analysis uses velocity components in order to calculate average and fluctuation

velocity as well as Reynolds stresses, which are derived from fluctuating velocity

components. Therefore it is important to perform error analysis of the interpolation

procedure, in which the main component of uncertainty is velocity due to an error

related to the position of the interpolated point in respect to the sparse data points.

The first issue is related to the number of voxels (3D boxes the center of which we are

the Eulerian grid nodes). In order to evaluate the average analysis error we must

estimate the amount of particles and tracers been in voxel and participated in the

averaging procedure. The average number of flow tracers per voxel is evaluated using

location analysis presented in results section ( 5.2.2.2), from which we get an average of

2000 tracers and 1000 particles per 18 voxels, which gives us around 111 tracers and 55

particles per voxel; average velocity is generated from this amount of tracers/particles

velocity data.

Instant velocity maps are used for averaging and fluctuation analysis. In assembly

average analysis, 3D-PTV error (calculated in appendix) is divided by square root of

average number of tracers/particles that appeared in voxel as a result of average

operation. In order to calculate root mean-square-maps, fluctuation maps are calculated

by subtraction of average velocity map from instantaneous velocity maps. The

uncertainty of the process is the same value as 3D-PTV error. A root-mean-square map

is made by composing averages of squared fluctuation maps and taking square root of

the resulting map. The uncertainty brought by this process is twice as big as the

averaging analysis error. This leads us to the second issue—the evaluation of the error

in instant velocity voxel, for which we established the aforementioned procedures.

The second issue is that in transformation processes there is a significant chance in high

visited areas of the cavity (middle areas) as shown above that there are several

tracers/particles in one voxel. In this case the linear interpolation is made between

velocity components of all the particles/tracers that are in the voxel at the same time.

The error of mentioned interpolation can be estimated according to the linear

interpolation error. This is the algorithm used in the Matlab subroutine (griddata).

Linear interpolation is often used to approximate a value of some function f using two

known values of that function at other points. The error of this approximation is defined

as R(t)=f(x)-p(x) where p denotes the linear interpolation polynomial defined below.

kb b ÀK¸ÀÀK¸À b b It can be proven using Rolle's theorem that if f has

a continuous second derivative, the error is bounded by|%Á| W ÀK¸À/

Q ÂÀÀÃÀÃÀK Ä ′′bÄ. The approximation between two points on a given function becomes even less accurate

with the second derivative of the function that is approximated. That means that in our

case the approximation in high strain regions like near downstream wall is less accurate

then in low strain regions like center of the cavity.

We analyzed squared spatial derivatives ( 5.2.2.4) and assumed that they are of the same

size with second spatial velocity derivative. Since the typical values of the derivatives

are about the same for tracers and particles and equal to 5 [1/s2] we take this value as a

representative value in error analysis.

The maximum distance between particle and center of the voxel to which it is

interpolated is half the size of the voxel and equal to 0.0025m. This provides us with an

error of 0.4·10-5 [m2/s2]. Therefore for typical velocity in voxel of 1 cm/s, we have an

error of square of the error calculated above of 1.1 mm/s.

The resulting error in calculations of instant velocity in a single voxel is 10% of the

measured velocity, both for tracers and for particles.

The error of the averaging procedure of instant velocity maps (as described above) for

tracers is equal to ( .%√... =1%) for tracers and (

.%√ÇÇ =1.35% ) for particles.

The error of root mean square analysis is twice the error of the averaging procedure

(since the average operation in rms procedure is done over squared numbers the

overcool error growth is twice larger then in ensemble average operation), and in it the

particles error stands at 2.7% with tracers error of 2% of the rms velocity values.

The error analysis presented above is an evaluation for most of the cavity regions,

whereas the error in high strain regions (near the corners of the cavity and downstream

wall) is much higher than the evaluated error.

5 Results and Discussion

5.1 Preliminary experiment

In this section, results of the preliminary experiment which setup is explained in the

experimental method section ( 4.1) are presented. Analysis performed to the time data

gathered in the experiment, data measured from 10 particles, is presented as a distance

between marked locations of the particle (see Figure 4 for the sketch of the experimental

setup). The result is the settling velocity n shown in Figure 13. This figure presents

calculated velocities of 10 particles as function of the distance from the first volume

mark line in the control volume (as can be seen as a function of settling time). As we

can see from this figure, the particles have a constant settling velocity, with variation of

about 0.001 m/s, due to the uncertainty of the experiment. The main sources of the

uncertainty are the error in number of frames (maximum error is 1 frame); other error

sources such as fps error (1/1000 of frame rate) and distance measuring errors (0.01mm)

provide negligible errors that do not affect total velocity error.

Figure 13 Settling velocities of the particles as function of the depth.

The average settling velocity is 6.37 ¦ 0.95 mm/s. The uncertainty in velocity

represents the possible deviation of particle settling velocity values from average

settling velocity value (shown as error bars in figure above); the difference in particle

diameters (15% uncertainty, section 4.1) are the main reason for that deviation.

Dependence between settling velocity and density of the settling particle is according to

this equation:

V Q"43¸45ÎRµ ,

from which we estimate the particle density, ρ ρ 0.024 Î"ÐÑ. Thus our particles

density is 1.024 ± 0.004 Î"

ÐÑ. The density difference is small, therefore the Stokes

number (settling) is also small and cannot explain the differences we observe in large

particle behavior in the results of particle tracking experiments, given in the following

section 5.2.

5.2 3D-PTV experiment

This section presents results of the 3D-PTV experiment data analysis done using 3D-

PTV techniques described above. The experiments were carried out according to the

experimental method shown in section 4.2.

Using 3D-PTV techniques two experimental sets were performed:

The first one used flow tracers, and the second one uses large solid particles.

The second set contains 2 experimental runs of about ten seconds each. These

experimental runs were performed for ten minutes, one after another in order to

verify that our ensemble is taken in a steady-state flow.

Both experiments were performed in the cavity under steady conditions of a constant lid

speed (UÒ 0.125 Ñ . The camera frame rates in both sequences of tracers and

particles was 150 fps. The Reynolds number, ReÒ ÓÔ8C 10000, (where L is the side

length of the cavity at 0.08 m and ν is the fluid viscosity, 1 p 10¸Õ Ñ ) in both

experimental sets. The root-mean-square velocity of the velocity fluctuations is

estimated as 0.01 m/s (the distribution is shown below) and the Taylor micro-scale is

estimated as 3.8 mm. Reynolds based Taylor micro-scale is ReÖ × 38.

5.2.1 Eulerian Results

The results presented in this section should be interpreted with care, being interpolated

from the Lagrangian results as described in section 4.3.2. The Eulerian results are

important to understand the overall flow pattern and to enable estimation of fluctuations

in respect to the average flow. The Eulerian voxel maps of instantaneous velocity are

further processed in Matlab® in order to get the results we present below. We present

the mean velocity using the cross-sections through the cavity mid-planes from

orthogonal views, distributions of fluctuating velocity and the Reynolds stress contours.

5.2.1.1 Average velocity maps

The average maps presented below are a product of the ensemble averaging operation

whose procedure and errors are described in section 4.3.2, according to formula

Ø° .¬ ∑ a° (a° is b, Ú, Û component of instant velocity). The maps present cavity mid-

planes from orthogonal views showing each time two average velocity components.

Figure 14 (a) - shows Ux ,Uy (Capital letters denote average) vector field (the scale of

arrows is shown in the figure) in z mid-plane, Figure 14 (b) - shows Ux, Uz vector field

in the y mid plane and Figure 14 (c) - shows Uz, Uy vector field in the x mid plane.

Blue color arrows are tracer average velocity components and red arrows are particle

average velocity.

(a) (b)

Figure 14 Velocity vector maps. (a) z mid plane map of Ux , Uy , (b) y mid plane map Ux , Uz , (c) x

mid plane map of Uz , Uy . Blue color represents tracer velocity and red color represents particle

velocity, top arrow represent velocity scale.

As one can see in Figure 14, the main difference between average velocity of particles

and tracers are in the high shear (near the lid and cavity walls) regions. In the low shear

regions (center of the cavity) the difference is insignificant. The high shear regions are

also regions with fewer particles (as shown below) and therefore the uncertainty is

higher. Also be seen that the main vortex is not exactly in the middle of the cavity but is

located slightly in downstream direction (positive x) of it; this is the reason why we

don’t see zero Uy values in Figure 14. The normalized axes presented in Figure 15 are

from -0.5 to 0.5 of the whole cavity length, but for reasons we’ve discussed above, we

did not present data on the regions near the walls.

In the following, average velocity field in Eulerian setting is compared to the results

available in the literature of average profiles in lid driven cavity flows with the same

geometric and flow conditions. We compare the mean velocity profile at lines of

x= -0.01m and y=0 m for z middle plane to the velocity profiles presented by PK89

(Prasad and Koseff 1989). The comparison has to be taken with care since the precise

horizontal/vertical location of the profile by PK89 is not known.

In Figure 15 (a) we present the mean velocity distributions (velocity profiles) obtained

from the Eulerian results and compare it to the results of Prasad and Koseff (1989),

using the data from flow tracers and particles, and in Figure 16 (b) we made an Eulerian

analysis for the whole cavity height with two different voxel sizes the common used in

the rest of the analysis 5×5×5 mm voxels and a smaller size of 2×2×2 mm voxels, this

analysis is done only to particles x axes velocity in order to see if there is any significant

change in average velocity values between two voxel sizes especially in the region

where our results are different from PK89 data (bottom of the cavity). There are two

pairs of axes in the Figure 16 (a); the inner scale represents the distance normalized by

the cavity side length D and the outer scale represents average velocity normalized

using the lid velocity, UÒ.

(a) (b)

Figure 15 Average velocity profiles comparison: (a)- Prasad and Koseff (1989) results ( SAR 1:1

Re=10000), tracers velocity profile using the Eulerian analysis, particles velocity profile using the

Eulerian analysis. (b)- particles average profile for two different voxels sizes.

From Figure 15 (a) we see that the average velocity profiles are similar for particles and

tracers, though our results are higher than the results of PK89. It is worthy to mention

that in PK89 the two velocity profiles meet in the middle of the cavity presenting the

center of the main vortex. Our expected results do not support this finding. In Figure 14

(a) you can see that the center of the vortex is not located in the middle of the cavity.

From Figure 16 (b) you can see that there is no significant change between two profiles

made using different voxel sizes (2×2×2, and common 5×5×5 mm3). Even through

interpolation the “whole” cavity length we do not get the correct measurements near the

top and bottom regions of the cavity. Therefore we exclude these regions from the

analysis.

5.2.1.2 Root-mean-square velocity contours

Velocity fluctuation maps u′x are estimated by decreasing from instant velocity maps ui

their average velocity maps Ui u′x ux Ux. Root-mean-square (rms) velocity is

calculated from fluctuation velocity as ux ".Ñ. 6.Ü ∑ u′xQ9.Ç

Fields of rms values estimated from fluctuation velocity fields after the Lagrangian to

Eulerian interpolation are described in section 4.3.2. Each figure presented below

shows rms of the velocity components calculated in mid-planes of the cavity in the

same manner as the mean velocity fields. Contours of ux_r.m.s are shown in

Figure 16, uy_r.m.s in Figure 17 and uz_r.m.s in Figure 18 in the corresponding mid-planes.

In addition to the standard method of estimating fluctuating velocity we estimate the

fluctuating particle field as in a one-way coupling manner—by subtracting the mean

flow fields estimated from the tracer motion. In the figures below we demonstrate an

additional contour map of rms velocity component where fluctuating velocity of

particles is calculated by decreasing from instant particle velocity maps tracer average

velocity map: Ý Þ.ß.à áÞÝâãäà å6æ ∑′Ý áÞÝâãäà çÝ ÞâäÞà´9, this was done in

order to see the fluctuation of particles compared to the flow average velocity.

(a) (b)

Figure 16 ux r.m.s in z mid-plane (a) – r.m.s velocity of tracers, (b) – r.m.s velocity of particles, (c) –

r.m.s velocity of particles calculated using average velocity of tracers, (all axes are normalized by

cavity width D).

(a) (b)

Figure 17 uy r.m.s in z mid-plane (a) – r.m.s velocity of tracers, (b) – r.m.s velocity of particles, (c) –

r.m.s velocity of particles calculated using average velocity of tracers (all axes are normalized by

cavity width D).

(a) (b)

Figure 18 uz r.m.s in x mid-plane (a) - r.m.s velocity of tracers, (b) - r.m.s velocity of particles, (c) -

r.m.s velocity of particles calculated using average velocity of tracers (all axes are normalized by

cavity width D).

We see that the structure of r.m.s contours is nearly identical for both particles and

tracers in all velocity components. The r.m.s values of particles are slightly higher at

several regions. This might be due to the fact that we measure particle fluctuations using

its average profile and this does not necessary provide us a true view of the flow pattern

in that region since the “real” particle fluctuation is from the flow field average. In

addition values received in that region are compiled from fewer data sources than in

other regions. Therefore, when we perform an r.m.s velocity analysis of the particles

using tracers average velocity ((c) in all three figures), we observe smaller fluctuation

values. This strengthens the view that the usual way of estimating particle velocity

fluctuations is more sensitive to the low density of the solid particles. The alternative

type of estimating fluctuations of the solid particles seems to be more accurate in this

Similarly to the mean velocity profiles we compare the r.m.s results to those of PK89.

Figure 19 (a) and (b) present profiles ux_r.m.s and uy_r.m.s respectively. The velocity

profiles shown in the figures are of tracers, particles and PK89 results

(1) (2)

Figure 19 Profiles of r.m.s velocity for middle z plane of the cavity. (1) uy_r.m.s in cross-section of z

and y middle planes, (2) ux_r.m.s in cross-section of z and x middle planes. (a) Prasad and Koseff

(1989) results ( SAR 1:1 Re=10000), (b) tracers velocity profile, (c) particles velocity profile.

Velocity r.m.s values are presented in scale of 10:1 for PK89 results and 20:1 to our results,

rescaling was done in order to save the PK89 original scales and to present the results in one figure.

As in section 5.2.1.1, we compare the results of average velocity we calculated in

Eulerian analysis to that of the PK89. We see that although we have the same shape, we

have twice as big values as theirs. The explanation to that may be in differences

between measuring and processing techniques and in understanding the definition of

r.m.s velocity.

5.2.1.3 Reynolds stresses

Reynolds stresses ρ=uèx · uèy? are representation of apparent stress arising from the

fluctuation velocity field and stems from momentum transfer by the fluctuating velocity

field (Pope 2000). Contours of the Reynolds stress fields, represented in Figure 20,

ensemble average of multiplications of the fluctuation velocity component u′x by

component u′y, calculated in z mid-plane of the cavity for tracers and particles.

(a) (b)

Figure 20 Reynolds stress contour maps at the z mid-plane (a) tracers, (b) particles.

High Reynolds stress values (both positive and negative, since sign is according to axes

definition) show us the high shear regions in the cavity for both particles (a) and tracers

(b). We see the same values and shapes, and as we expected the highest fluctuation area

is near the downstream wall, where x velocity from maximum drives to zero, and y

velocity gets its maximum negative values.

5.2.1.4 Summary of Eulerian analysis

Eulerian analysis provided us with information about the general pattern of tracer and

particle flow in the cavity. The information includes: location of central vortex,

presence and location of high shear regions, fluctuating and average velocity values, and

enabled approval of our result by comparing them to the work of Prasad and Koseff

(1989).

Eulerian analysis cannot provide us with a close look into the flow, because our

experimental and data processing methods provide us with indirect Eulerian data and

involve additional processing. Besides the fact that particle fluctuating velocity values

in general are higher than those of the tracers, it does not teach us of the differences

between particles and tracers, which is not surprising, since all the differences are

expected to be in small scales that our Eulerian analyses could not reach. Therefore, we

must proceed to the second part of analyses: the Lagrangian analysis that can provide us

the “closer look” at the smallest scales of the flow and teach us the differences between

particles and tracers.

5.2.2 Lagrangian results

Lagrangian results are derived directly from Lagrangian analysis processing performed

by 3D-PTV software on experiments raw data after the processing described in section

4.3.1. The results presented below include location analysis, settling estimations, spatial

derivatives, autocorrelations of velocity and acceleration, acceleration analysis, velocity

increments and structure functions. All analysis in the Lagrangian results section is

trajectory-based. Results like correlations, acceleration statistics, velocity increments,

structure functions etc. are made from trajectories that last for 10 frames and more. An

example of these trajectories is presented in Figure 21 which shows “long” particle

trajectories (more than 100 frames and 11 longest trajectories), their shape and spatial

location.

(a) (b)

Figure 21 Particles trajectories in 3D cavity (a) trajectories Longer than 100 frames, (b) longest

particle trajectories. Red circle represent the end point of trajectory. Cube denotes the limits of the

cavity.

From figures above we see that we have plenty of long trajectories, which give us

enough data for our analysis. In addition we see that long trajectories are located near

the center of the cavity. This could be explained by the fact that near the center of the

cavity velocity values are relatively small, therefore particles move slower and are

easily located compared to high velocity and strain regions which are located near the

boundaries of the cavity.

5.2.2.1 Scales evaluation

In Lagrangian analysis, absolute values often do not provide us with right view and

understanding of sizes of phenomena observed; therefore most of the values in this part

are presented in non-dimensional.

This part presents a set of calculations performed in order to evaluate several lengths,

time and other scales, which describe the turbulent flow and are being used in figures in

the Lagrangian section for normalization and scaling purposes.

The major value on which most other values evaluations are based is the r.m.s velocity

of the flow. Our method of calculating the r.m.s velocity value is to take it from contour

maps of tracer r.m.s velocity (which represent the flow) in the middle regions of the

cavity (

Figure 16, 18, and 19). The value that was taken represents of the r.m.s velocity

common to most parts of the cavity so it probably could represent the flow. The value

used is 0.01 m/s.

The r.m.s velocity values represent the first velocity scale of the flow (besides the

geometric scales which include belt velocity and geometric properties of the cavity), the

scale which represents the length scale is called turbulent integral length scale. A

theoretical method of estimating turbulent length scale is to take it as something like 1/6

of geometry scale (0.08m) (Pope, 2000); we take integral length scale l as 0.01m.

After evaluating the basic velocity and length scales, we can proceed to the calculation

of one of the energy dissipation rate, on which all other scales (time, velocity and

length) are based. The definition of energy dissipation rate is é ] 2êëìíëìíîîîîîî where

ë°ï ] .Q +ef©′eÀð efð′

eÀ©; ,(Tennekes and Lumley 1972). Since we can't measure sij directly we

use approximation formulas in order to estimate ε, due to the following equations

ε òó.ô.B¯ , where õ is integral length scale. That way we receive an evaluation of energy

dissipation rate: ε .... 1 · 10¸¢ Ñ

. It is important to note that this estimate is

not correct to our case since it was developed for homogeneous flows and the flow in

the experiment is not.

Another length scale which can be estimated is Taylor length scale λ, along with

Reynolds number based on Taylor length scale %&'. We estimate it as:

Öö √15 · Reö¸K

, to be 3.8 mm. Both Reynolds numbers based on Taylor length scale

and on integral length scale make use of velocity r.m.s value and are equal to: 38Re =λ

; Rel=100.

Time scale used widely in analysis of turbulent flow is Kolmogorov time scale which is

the smallest time scale of the flow. Using values evaluated below are 1 · 10¸¢ ,

l=0.01m and a÷.. 0.01 we can evaluate Kolmogorov time scale (τ) defined as:

τ 6Cø9.Ç 0.1 s.

The smallest length scale of the flow is the Kolmogorov length scale defined as

η 6Cø 9.QÇ 0.31mm. Also, adopting the Gibert et al. (2010) definition of Stokes

number for solid particles which densities vary between 1 W ρ/ρ W 8, and particle

diameters of order of Kolmogorov length scale as most appropriate to our particles

(knowing that particle diameter is twice as big as Kolmogorov length scale, we

calculate Stokes number defined in eq. 4 in section 2.1 as 0.34.

5.2.2.2 Distribution analysis

Particle distribution analysis provides us with information about particle and tracer 3D

scatter inside the cavity, and therefore teaches us about the amount of data in every

region. This analysis provides us with estimation of the reliability of our statistical

analysis (it is known that the amount of data processed is the most important factor in

the accuracy of results). This analysis was made using position data disregarding the

sampling time. It is important to know how the homogeneous particle/tracer

distribution, and to find the regions of high/low density. Figure 22 shows the histogram

of particle/tracers distributions in the cavity. In ideal experimental conditions one could

expect a homogeneous, uniform distribution of particles/tracers across the cavity.

(a) (b)

Figure 22 Particles/tracers histogram of spatial distribution, along (a) x, (b) y, (c) z, directions,

respectively.

Figure 22 reflects the larger concentration of tracers as compared to particles, used in

the experiments. On average, there are about 2000 tracers and 1000 particles at every

location in the cavity, with the maximum in the center and slight reduction towards the

edges. Clearly x and y distributions are different from the distribution along z-axis, due

to the optical restrictions of the stereoscopic imaging. (Not uniform Laser beam

diverging by lenses in z direction as well as growing distance from cameras focus. (see

Dracos 1996)).

Although the shapes of the histograms of the particles and of the tracers are similar, in

order to understand if the particles probe the same flow as the tracers we present

“normalized” location distributions in the following Figure 23. It is important to see if

there are some regions that particles sample differently from tracers. Probability density

functions (PDF) allow for the comparative analysis of the distributions.

(a) (b)

Figure 23 PDF of spatial particle/tracer location. (a) x location, (b) y location, (c) z location.

Figure 23 shows that particles and tracers are distributed in a similar manner throughout

the cavity with lower probability near the walls. The main reasons are laser light

reflection from glass walls and optical restrictions described above which caused a

severe reduction in detection in those regions.

5.2.2.3 Estimation of the settling velocity effect

It is important to estimate whether or not the particles we use in our experiment

experience significant influence of gravity forces in the cavity. This is carried out in

addition to the preliminary experiment, comparing vertical components of particle

velocity to that of the tracers. In Figure 24 we present the probability density function of

the particle and tracer Uù velocity component.

Figure 24 PDF of vertical velocity component Uy. (a) tracers (average value is 0.012 m/s), (b)

particles (average value is 0.007 m/s).

As we see from the Figure 24, the shape is almost the same for tracers and particles.

Positive average value of both particles and tracers is caused by detection problem due

to light shading effect in high negative velocity regions near high downstream wall, this

effect is more dominant for tracers than for particles. Although the solid particles

experience lower (on average) vertical velocity due to their settling (negative) velocity,

they do not settle during the experimental run. We show that the number of particles in

the cavity (see Figure 25) per frame is almost constant.

Figure 25 Number of detected particles as function of frame number, with average number line

(black) of 274 detected particles.

5.2.2.4 Spatial derivatives

A spatial derivative is one of the flow parameters, which indicates velocity transport in

the flow. We cannot estimate it directly from knowledge of velocity in a constant

location, since our Eulerian analysis does not gives us enough precision. Therefore we

measure particle velocity derivatives along the trajectory. Normal spatial derivative

6|ú|û9Q

is calculated for “long trajectories” (longer than 10 frames). Derivation tracks the

change in fluctuation velocity as function of distance between two derivate points on the

trajectory. The problem is that average and fluctuation velocity in Lagrangian system of

coordinates is not defined.

This process, which is quite simple in Eulerian coordinates systems, becomes very

complicated in Lagrangian ones, since in the definition of fluctuating velocity presents

an average velocity value which is subtracted from instantaneous velocity. In a system

of coordinates where the origin follows the particle and all that is known is the particle

spatial location and its velocity and acceleration in that location at a specific time, there

is no “simple” definition of local average velocity that can be used to find particle

fluctuations in that location.

Our way of dealing with this problem (since we need to estimate fluctuating velocity in

most sections below) is to define our average velocity in the Lagrangian system.

The definition is as follows: knowing particle trajectory, we define the average velocity

component of a particle as the linear trend of its instantaneous component along the

trajectory. Therefore the fluctuation of particle velocity in time is its components

deviation from the linear trend as shown in Figure 26. A straight line presents linear

trend of velocity component, and a curve represents velocity component evolution along

trajectory in time. The difference of velocity component from trend line is its

fluctuation.

Figure 26 (Left) Plot of x component of particle velocity for several insulated trajectories in time,

each blue line represent single trajectory. (Right) Velocity fluctuation v’ in respect to the linear fit

representing average velocity.

Velocity components for trajectories shorter than 100 frames look like almost straight

lines, without breaking its trend, while velocity of longer trajectories usually experience

more complicated evolution and do not looks like a straight line, and thus linear trend is

not a quite good estimation of their average velocity. That is why statistical data out of

long trajectories for long times (more than 100 frames) received this way should be

trusted carefully. Statistically, these long trajectories do not play a significant role, since

they make up less than 5% out of all trajectories.

Figure 27 (a) shows the derivation of instantaneous particle/tracer velocity and Figure

27 (b) shows the derivation of fluctuating velocity calculated as described above. The

derivation is performed following the presented formula between each set of two

adjusted points on the trajectory :

6|ú|û9Q 6úüJ.¸úü

~J.¸~ , úýJ.¸úýùJ.¸ù , úþJ.¸úþ

J.¸ 9Q. (14)

( is vector tangential to trajectory connecting each set of adjusted points).

(a) (b)

Figure 27 Spatial derivatives. (1) Spatial derivative of full velocity, (a) tracer (average value is 25.5

1/s2 ), (b) particle (average value is 25.8 1/s2). (2) Spatial derivatives of velocity fluctuations. (a)

tracer (average value is 23.9 1/s2 ), (b) particle (average value is 24.4 1/s2 ).

From Figure 27 we see that, as expected, there is no difference between fluctuating and

full velocity component derivatives, neither in values nor in shape, for both tracers and

particles. It shows that the subtraction of linear fit for each component of velocity does

not introduce new errors to the spatial derivatives.

Using the spatial derivatives shown in Figure 27 we can provide another estimate of the

dissipation rate. Based on the known estimate for the isotropic case ε ν ∑ ∂xuy x,y

∂yuxQ I 15 ν 6|ò|~9îîîîîîQ

(Tennekes and Lumley, 1971), and correcting it for the spatial

derivatives 6|ò|û9Qîîîîîîî

, we obtain ε = 4.25 ×10-5 m2/s3 .

5.2.2.5 Lagrangian correlation of velocity

Lagrangian velocity autocorrelation function is for turbulent diffusion study and can be

used to characterize the time and length scales of the flow or of the moving particles.

Figure 28 presents a schematic view of a particle trajectory, showing two points on it

with the velocity components, which is used in the definition of Lagrangian

autocorrelation functions.

Figure 28 Particle location and velocity at two different times on its trajectory.

Figure 29 (a) presents Lagrangian velocity autocorrelation of tracers (solid lines) and

particles (symbol lines). Correlation coefficient of velocity vector is obtained according

to equation

Rúú =úJ·ú?=ú·ú? (15)

where v is the Lagrangian velocity vector and is the lag time (represents a difference

in time between two correlated points).

The correlation coefficient of Lagrangian velocity vector projections on orthogonal axes

is calculated according to formula

Rxxv =úJ·ú?=ú·ú? (16)

where vx is the i component of velocity vector (i = x, y, z).

Figure 29 (b) presents the Lagrangian “fluctuating” velocity autocorrelation of particles

(symbols lines) calculated according to formula

Rxxv′ =ú′ J·ú′ ?=ú′ ·ú′ ? (17)

where v′x is the i component of “velocity vector” obtained with the process described in

section 5.2.2.4 and full present autocorrelation of vector combined from fluctuation

components.

(a) (b)

Figure 29 Lagrangian velocity autocorrelation coefficient (a) tracer (represented by solid lines

curves) and particle (represented by symbols) autocorrelation coefficient of Lagrangian velocity

vector and its components, (b) tracer and particle autocorrelation coefficient of “fluctuating”

velocity vector and its components. (full-Rvv , Rxx , Ryy , Rzz time is normalized by Kolmogorov time

scale τ which is equal to 0.1s ) .

We observe in Figure 29(b) that the correlation time scale of “fluctuating” velocity

components is shorter if compared to the full velocity autocorrelation, shown in Figure

29(a). This analysis demonstrates that most of the correlation is due to the “non-local”

effects and large-scale velocity patterns, imposed on the local variations of the

Lagrangian velocity. This is to emphasize the inability of the undecomposed Lagrangian

correlation analysis to predict dispersion of particles (continuously growing distance

between the neighbor particles on relatively short time scales). The presented

autocorrelation analysis has to be taken with caution, since the flow is not homogeneous

and there is dependence between the values of the autocorrelation and the region in the

flow where the trajectory was detected. Nevertheless, we conclude that in general,

particles have stronger and longer velocity correlations as compared to tracers (almost

twice as long). This can be seen clearly in the example of the correlation of the v

component In Figure 29 (a) and (b).

5.2.2.6 Autocorrelation of acceleration

The acceleration correlation is defined in the same way as the velocity correlation. The

correlation in time of the acceleration vector and its components may provide us with

some understanding of dominant small scale forcing terms, like Bourgoin et al (2009),

who suggest that the dominant small scale forcing term acting on particles is pressure

gradient. From the acceleration correlation we can get parameters such as time scale

based on integration of acceleration autocorrelation coefficient from zero to zero

crossing time. τxû Rsds , where is the first zero crossing time, i.e.

Rτ 0 .

Bourgoin et al (2009) claim that τxû remains of the Kolmogorov dissipation time τ of

the carrier flow for a variety of particle densities and diameters; for which Stokes

number changes by several orders of magnitude from the smallest neutrally buoyant

particles to the largest and heaviest particles, and thus claiming that the Stokes number

effect alone cannot properly characterize and model such inertial particles.

The correlation was calculated for the full acceleration vector and for the tangential to

trajectory component of the acceleration vector (sees Figure 30).

Figure 30 Sketch of acceleration vector with its components of two points on particles trajectory.

R is the full acceleration vector correlation

R =J·?=·? (18)

a - acceleration vector, τöÎ - lag time - a time interval starting from time between two

frames and ending with time interval passed between beginning and the end of the

trajectory.

R is the tangential components correlation of acceleration vector

R =J·û J··û ?=+·û ;··û ? (19)

n - unity vector tangential to trajectory (see Figure 30).

(a) (b)

Figure 31 Acceleration autocorrelation function. (a) Autocorrelation function of full acceleration

vector, Raa (b) autocorrelation function of tangential to trajectory component of acceleration

vector RaTaT. Time is normalized by Kolmogorov time scale τ which is equal to 0.1s.

From Figure 31 (a) we see that there is minor difference in shape and value between

tracers and particles. Also the autocorrelation acceleration of tangential projection on

trajectory did not change significantly in shape and value but there are some

“vibrations” of particle projection autocorrelations which are not presented in tracer

projection autocorrelation ( see Figure 31 (b)), these changes are supposed to be due to

different statistics of “long” trajectories. By calculation, τxû for full acceleration

correlation is 0.35 τ for both tracers and particles, and 0.25 and 0.3 τ for tangential

component acceleration correlation.

5.2.2.7 Acceleration analysis

Some say that acceleration statistics can clearly resolve the transition from tracer

particle to large particle behavior claimed by Brown et al. (2009), while others disagree

claiming that for fixed carrier flow conditions, acceleration statistics of finite size

inertial particles are very robust to size and density variations (Bourgoin et al. 2009). In

particular, the shape of normalized acceleration probability density function remains

unchanged. We checked our particle and tracer acceleration statistics to see which

approach is more correct for our case.

Acceleration values for this analysis are also taken only for those particles/tracers for

which trajectories last for at least 10 frames, since data from these trajectories have been

used in other analyses (see 5.2.2). Figure 32,Figure 33 and Figure 34 present probability

density functions (defined as before) of acceleration components of particles and tracers

of the x, y, z components of acceleration vectors, respectively. In these figures the left

panel (a) represents PDF of the dimensional acceleration, and the right panel (b)

represents the same distribution but of the values normalized with the root mean square

of acceleration component ax/=axQ?.Ç.

(a) (b)

Figure 32 Acceleration in direction of x axis (ax), tracers r.m.s <ax2>0.5=0.16 m/s2, particles r.m.s

<ax2>0.5=0.17 m/s2 .

(a) (b)

Figure 33 Acceleration in direction of y axis (ay), tracers r.m.s <ay2>0.5=0.14 m/s2, particles r.m.s

<ay2>0.5=0.17 m/s2 .

(a) (b)

Figure 34 Acceleration in direction of z axis (az), tracers r.m.s <az2>0.5=0.3 m/s2, particles r.m.s

<az2>0.5=0.3 m/s2 .

Our results confirm the robustness of the acceleration probability density functions

(Bourgoin et al., 2009) – it is similar for the particles and tracers. Therefore, the PDF

cannot be used as a probe for the different behavior of the particles in respect to the

tracers. The single point analysis that combines the statistics that are obtained

disregarding the time history of the trajectory is useless for describing the evolution

(dispersion, diffusion) of particles. Estimates of the dissipation rate using r.m.s values

of acceleration can be done using the formula =axQ?.Ç aε..Çν¸.Ç where a0 is between

1 and 10. We have acceleration r.m.s values of long (more than 10) tracer trajectories

around 0.15 m/s2. Therefore the values of dissipation are in the range of 10-4 m2/s3.

5.2.2.8 Velocity increments

Velocity differences (increments) measured along Lagrangian trajectories at varying

time gaps provide a signal closely related to acceleration, and the change in the shape of

the PDF is typically related to the intermittency of the turbulence (e.g. Toschi and

Bodenschatz 2009). In a view of the previous result it is important that this PDF does

include some “history” effect, as it is taken for two-point statistics at different time

separations. In Figure 35 we present the probability density function of the velocity

increments function defined as ∆v vt τ vt, which is calculated for the same

data as the acceleration statistics. Time lags are dependent of the frame rate and are

calculated for increments of 1, 2, 4, 20 and 50 frames, which are in our experimental

sets (I and II): 0.0067, 0.013, 0.026, 0.133 and 0.333 seconds or 0.07, 0.15, 0.3, 1.5 and

3.5 τ (in Kolmogorov time scale).

(a) (b)

Figure 35 Velocity increments. (a) velocity increments of tracers, (b) velocity increments of

particles. Time lags are 0.07 0.15 0.3 1.5 3.5 τ, and displayed with a vertical shift from 0.07 (up) to

3.5 (down) for clarity.

We observe a continuous change in PDF shape from lines representing the short-time-

lags to lines representing long-time-lags. The shape of short-time-lags curve looks

“sharp” and “thick” compared to long time lags curves. This shape teaches us of a weak

correlation between the velocity values in adjusted times therefore indicating the

strength of the intermittency in our flow.

There is a little difference between Figure 35 (a) and (b) which cannot emphasize some

major difference between particles and tracers as we see also from the acceleration

statistics. Therefore, there is a need to present velocity increment statistics in other more

specific forms to see the dissimilarity between them.

5.2.2.9 Structure function

The velocity increments are shown as probability density functions and we can also

average the differences at different time lags to construct the structure functions. The

Lagrangian structure functions are not directly related to the Eulerian structure

functions, which are used in the core of Kolmogorov theory of turbulence (Kolmogorov

1941). The Lagrangian velocity structure functions are estimated according to:

Sτ =vxt τöÎ vxt? (20)

where i is the orthogonal component of velocity. In Figure 36 we demonstrate structure

functions of the second order with time lags normalized by the Kolmogorov time scale

τ. The convenience of structure functions is that they use the Lagrangian trajectories.

Similar to the Eulerian structure functions, we expect to observe structure functions that

show some asymptotic behavior if normalized with average dissipation. Therefore, we

expect to see a horizontal portion of S2 when plotted versus time lag.

Figure 36 Lagrangian second order structure function of particles and tracers x and y velocity

components, τlag is lag time from minor time interval between adjusted point to maximum interval

between start and the end point of the trajectory, τ is Kolmogorov time scale of the flow.

From Figure 36 we a plateau between the values of

=1 to 7 which reminds us the

asymptotic behavior which present in Eulerian structure functions; it is clearly seen for

tracers and less clear for particles. Also, the velocity structure functions, which are sort

of "correlation of increments", for particles are stronger than for tracers as we can see

from the values in Figure 37 (particles reach 0.8-0.9 and tracers 0.6). The shape is

similar for both tracers and particles, with the same change in dominating component

correlation from x to y, when normalized time lag value is approximately equal to one.

5.2.2.10 Logarithmic derivatives

Logarithmic derivatives of structure functions are used usually in order to emphasize the

intermittency of the turbulence. In our case these, are used to verify the intermittency as

it is shown by the flow tracers and particles. These functions are defined as ζτöÎ 7 öÎ3 7 öÎ , when the results represent some sort of local (at scale τ) scaling exponents.

Figure 37 and Figure 38 show logarithmic derivatives of order 4 (p=4) and 6 (p=6) ,

respectively, as a function of time lag τlag normalized by the Kolmogorov time scale τ,

calculated from structure functions defined and evaluated as in the previous section

(section 5.2.2.9).

(a) (b)

Figure 37 Logarithmic derivatives of order 4. (a)-tracers, (b)-particles, τlag is lag time and τ is

Kolmogorov time scale of the flow.

(a) (b)

Figure 38 Logarithmic derivatives of order 6. (a)-tracers, (b)-particles, τlag is lag time and τ is

Kolmogorov time scale of the flow.

In Figures 37 and 38 above we observe a dip near the scales where the inertial range we

found in the previous section ( 5.2.2.10, 5.2.2.9). The dip represents the border between

dissipative and larger scales of the turbulence (e.g. Toschi and Bodenschatz, 2009), and

we clearly see that the dip in the tracer result is steeper than in the particle data in both

exponents (p=4 and p=6). To the best of our understanding, this is because particles

tend to leave high shear areas faster than the tracers, thus showing less intermittent

behavior, or in other words, “low-pass filtering” of the underlying turbulent flow.

5.2.2.11 Mixed velocity-acceleration structure function

During the analysis of the dynamics of heavy particles, the strong effect of particle

density on mixed velocity acceleration structure function =∆"v p ∆"a? was observed

(Gibert et al. 2010) that can point on preferential alignment between velocity and

acceleration increments. We are interested to test if the mixed structure functions in

time ∆ are also sensitive to the size variations rather than density variations of the

particles. In Figure 39 we present mixed structure function obtained for the particles and

tracers =∆v p ∆a? where: ∆v · ∆a v.t τöÎ v.t · a.t τöÎ a.t vQt τöÎ vQt · aQt τöÎ aQt v:t τöÎ v:t ·a:t τöÎ a:t.

Figure 39 Lagrangian time mixed structure function, τlag – is a lag time and τ is Kolmogorov time

scale of the flow.

In Figure 39 we observe that for longer time lags (¯Â/ 4) the particles show lower

values, pointing towards the lower “dissipation” values, or, in other words, weaker

intermittency and less variations than the tracers. Gibert et al. (2010) showed that for

particles with various densities and the same average diameter, the mixed structure

function equals a constant (approximately -2ε) in the inertial range. The constant value

decreases with density and equals one for neutrally buoyant tracers. If we try to

transform our resulting time scale to length scale, we must expect some constant values

and the same as Gibert et al. (2010) effects in range of ¨

4, but in fact we see a

different result than what we expected: tracers with a small diameter compared to

particles show smaller values than particles, and so only for particles we can assign

some sort of constant value for 5 ¯Â/ 1.

These results could be explained by differences in experimental setups and statistical

processing, or this could be a real difference due to a diameter variation. Both reasons

may be proven through additional experiments with larger variety of particle diameters

and densities or Reynolds and Stokes numbers that are used to define the parameter

space.

It is worthy of mention that there is a clear difference between particles and tracers and

thus the mixed time structure function could be used as an indicator of the different

Lagrangian motion of the particles of varying size is similar to the case of particles of

with different density. In order to see if the effect is seen in the dimensionless units, we

normalize the mixed structure function with the absolute values of the velocity and

acceleration increments and obtain the so-called cosine of the angle between the

velocity-acceleration increments( cos θ ∆ú·∆|∆ú|·|∆| ).

The result is shown in Figure 40. It is shown here that on average, the correlation

between velocity and acceleration is lower for the large particles at extremes, ¦1,

enforcing the ‘filtering’ behavior of larger than Kolmogorov length scale particles in

turbulent flow.

Figure 40 Probability density function (F) of the cosine of the angle between ∆τv and ∆τa

From Figure 40 we see that there is a clear tendency of both particles and tracers toward

negative values, which point out that the two vectors (velocity and acceleration), are

more likely to point in opposite directions. Tracers demonstrate these phenomena

significantly compared to particles, which demonstrate almost no change in angle.

Again, this points out to the filtering effect of large particles, which are less sensitive to

fast changes in velocity vector direction, i.e. strong accelerations.

6 Summary and conclusions

We’ve presented the experimental study of particles and tracers motion in turbulent

flow with emphasis on the difference between them. The experimental technique used is

the three-dimensional particle tracking velocimetry (3D-PTV). Using this technique

with additional processing enables us to obtain particle and tracer Lagrangian

properties, i.e. trajectory along with velocity and acceleration. The experimental volume

is a cubic lid-driven cavity (LDC) flow at Reynolds number 10,000, based on lid

velocity and length of the cavity. Results are presented to provide an insight on (a)

particles and tracers behavior in a turbulent LDC flow, (b) differences in behavior

between large particles and small tracers, and (c) statistical properties that emphasize

these differences.

Summarizing the experimental results we can conclude that:

• Eulerian analysis performed on data transformed from Lagrangian

experimental system provides us with a general view on the flow pattern but

does not emphasize the differences in particle-tracer behavior.

• Acceleration analysis is truly robust to size and density variations.

• Structure functions of velocity components along with mixed acceleration

velocity structure functions best emphasize the difference between particles

and tracers and could become the probe property in experiments dealing with

particles.

• There is are extensive difficulties in working with Lagrangian data received in

experiments with 3D turbulent flow without one main flow direction, as a

result of lack of proper defined fluctuating and average velocity values in

trajectory laden data.

• Evaluation of the dissipation rate value that was done in all three ways (r.m.s

value of velocity components, r.m.s value of acceleration components and

spatial derivation value) provided us with dissipation rate values of the same

order. Therefore evaluation of this flow parameter for the presented

experimental setup can be done with a decent accuracy by any of the three

Concluding our studies we suggest performing a set of experiments using particles with

a variety of diameters and densities in order to see if the conclusions we made are valid.

In addition, we need to extend the resolution of the experimental systems in order to get

a more precise view on the smallest scales along with faster data capturing which will

allow us to perform experiments with much higher velocities.

7 Bibliography

1. Bourgoin M., Qureshi N., Baudet C., Cartellier A and Gagne Y., Lagrangian

statistics of inertial particles in turbulent flow, B.Eckhardt(ed.), Advances in

Turbulence 12 , Springer Proceedings in Physics 132, 2009.

2. Brennen C.E. (ed.), Fundamentals of multiphase flows, Cambridge University

Press, 2005.

3. Brown R.D., Warhaft Z. and Voth G.A., Acceleration Statistics of Neutrally

Buoyant Spherical Particles in Intense Turbulence, Phys. Rev. Lett., 103:194501,

4. Calzavarini E., Volk R., Bourgoin M., Leveque E., Pinton J., and Toschi F.,.,

Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxén

forces, J. Fluid Mech. 630: 179–189, 2009.

5. Clay Mathematical Institute, http://www.claymath.org/millennium/Navier-

stokes_Equations

6. Crowe C.T., Schwarzopf J.D., Sommerfeld M. and Tsuji Y., Multiphase Flows

with Droplets and Particles, CRC Press, 1998.

7. Dracos Th. (ed.), Three-Dimensional Velocity and Vorticity Measurements and

Image Analysis Techniques, Kluwer, 1996.

8. Elfassi R., Experimental study of two- and three-dimensional aspects of lid-driven

cavity turbulent flows, M.Sc. thesis, Tel Aviv University, 2010.

9. Fallon T. and Rogers C.B., Turbulence-induced preferential concentration of solid

particles in microgravity conditions, Exp. Fluids, 33: 233-241, 2002.

10. Gasteuil Y., Shew W.L., Gibert M., Chillá G., Castaing B., and Pinton J.-F.,

Lagrangian Temperature, Velocity, and Local Heat Flux Measurement in Rayleigh-

Bénard Convection, Phys. Rev. Lett. 99, 234302, 2007.

11. Gibert M., Xu H. and Bodenschatz E., Where do heavy particles go in a turbulent

flow, submitted to Phys. Rev. Lett., 2010.

12. Gore R.A. and Crowe C.T., Effect of particle size on modulating turbulent

intensity, Int. J. Multiphase Flow, 15: 279-285, 1989.

13. Guala M., Liberzon A., Hoyer K., Tsinober A. and Kinzelbach W., Experimental

study on clustering of large particles in homogeneous turbulent flow, Journal of

Turbulence, 9: 34 1-20, 2008.

14. Hoyer K., Holzner M., Lüthi B., Guala M., Liberzon A. and Kinzelbach W.,

scanning particle tracing velocimetry, Exp. Fluids, 39(5) 923-934, 2005.

15. Lüthi B., Tsinober A. and Kinzelbach W. Lagrangian measurements of vorticity

dynamics in turbulent flow, J. Fluid Mech, 528: 87-118, 2005.

16. La Porta A.,Voth G.A.,Crawford A.M., Alexander J. and Bodenschatz E., Fluid

particle accelerations in fully developed turbulence, Nature 409, 1017-1019, 2001.

17. Maxey M.R. and Riley J.J., Equation of motion for a small rigid sphere in a non

uniform flow, Phys. Fluids 26(4):883-888, 1983.

18. Mordant N, Crawford A.M., and Bodenschatz E., Three-Dimensional structure of

the Lagrangian acceleration in turbulent flows, Phys. Rev. Lett. 93, 214501, 2004.

19. Ott S. and Mann J., An experimental investigation of the relative diffusion of

particle pairs in three-dimensional turbulent flow, J. Fluid Mech, 422:207-223,

20. Ouellette N.T., O'Malley, P.J.J., Gollub J.P., Transport of finite-sized particles in

chaotic flow, Phys. Rev. Lett., 101, 174504, 2008.

21. Pope S.B. (ed.), Turbulent Flows, Cambridge University Press, 2000.

22. Prasad A.K. and Koseff J.R., Reynolds number and end-wall effects on a lid-driven

cavity flow, Phys. Fluids, 1(2): 208-218, 1989.

23. Qureshi N.M., Bourgoin M., Baudet C., Cartellier A. and Gagne Y., Turbulent

Transport of Material Particles: An Experimental Study of Finite Size Effects,

Phys. Rev. Lett., 99:184502, 2007.

24. Sato Y. and Yamamoto K., Lagrangian measurement of fluid-particle motion in an

isotropic turbulent field, J. Fluid Mech., 176: 183-199, 1987.

25. Shirolkar J.S. Coimbra C.F.M. and McQuay Q., Fundamental aspects of modeling

turbulent particle dispersion in dilute flows, Prog. Energy Combust. Sci., 22: 363-

399, 1996.

26. Shlien D.J. and Corrsin S., Measurement of Lagrangian velocity autocorrelation in

approximately isotropic turbulence, J. Fluid Mech., 62(2) :255-271, 1974.

27. Snyder W.H. and Lumley J.L., Some measurements of particle velocity

autocorrelation functions in a turbulent flow, J. Fluid Mech. 48, 41, 1971.

28. Tanaka T. and Eaton J.K., Classification of turbulence modification by dispersed

spheres using a novel dimensionless number, Phys. Rev. Lett., 101: 114502, 2008.

29. Tennekes H. and Lumley J.L. (ed.), A First Course in Turbulence, MIT Press,

Boston, 1972.

30. Toschi F. and Bodenschatz E., Lagrangian Properties of Particles in Turbulence,

Annu. Rev. Fluid Mech., 41:375-404, 2009.

31. Virant M. and Dracos T., 3D PTV and its application on Lagrangian motion, Meas.

Sci. Technol., 8:1539–1552, 1997.

32. Voth G.A., Bigger B., Buckley M. R., Losert1 W., Brenner M. P., Stone H. A., and

Gollub J. P., Ordered Clusters and Dynamical States of Particles in a Vibrated

Fluid, Phys. Rev. Lett. 88:234301, 2002.

33. Yeung P.K. and Pope S.B., Lagrangian statistics from direct numerical simulations

of isotropic turbulence, J. Fluid Mech., 207: 531-586, 1989.

APPENDIX A: UNCERTAINTY ANALYSIS

We present here the uncertainty analysis of 3D-PTV experiment following the method

presented in Elfassi (2010), who used the same experimental setup, the same calibration

files, and investigated similar Reynolds number flows.

Introduction

The aim of any experiment is to measure some quantities as accurately as possible.

Therefore, the error in the measurement must be estimated to arrive at the conclusion

about correctness of the results obtained. Uncertainty analysis is the procedure used for

quantity data validity and accuracy. Uncertainty NR of the function R is due to the

combined effect of uncertainty intervals ∆b° (b° is the independent parameter in the

experimental measurements). It may be obtained by:

N0 ¦ ¤6~K0

|0|~K n.9Q 6~

0|0|~ nQ9Q . . 6~

0|0|~ n9Q¥.Ç

° ( i=1..k ) are the relative uncertainties of b°. The effect of error in measuring

individual b° on R may be estimated by analogy to the derivative of a function as

follow,

∂Rx |0|~ δxx ; n0 ~

0|0|~ ~

~ ; n~ ~~ .

In the following we estimate uncertainty of the velocity and acceleration in 3D-PTV

experiments. The maximum instantaneous velocity that one expects to capture is

equivalent to the belt velocity 0.125 m/s.

Uncertainty analysis of the 3D-PTV measurements:

In the following we review the main errors that affect the 3D-PTV measurements. The

errors are combined from the optical setup, calibration, experimental errors and data

processing. In each experiment we acquire a sequence of images in the LDC setup and

process it in order to get the velocity and acceleration field. The main error comes from

the calibration process, and it affects the determination of the particle position in time

(Lagrangian data). The measurements of the calibration target play an important role in

the calibration process. The errors in calibration procedure are due to the error in

particle position and image processing. We can estimate the image processing errors

(estimation of the particle center after the high-pass filter) to be of the order of ½ pixel.

The conversion of pixels to millimeters depends on the magnification factor, which in

our setup was 0.1 (mm/pixel). In addition, there are errors from imprecision in the

calibration target production (distance between the points is nominally 5mm±0.1mm)

and errors due to the optical aberrations of the lenses. In the present 3D-PTV system we

use 4 cameras and the particle position in 3D space is estimated using 4 particle

positions in images (x, y in pixels). The stereo-photogrammetric matching of the

particles and reconstruction of the 3D position is the main source of position error. The

software provides the r.m.s estimate of the error using known 3D calibration target

(with multiple dots) which in our case was ±10µm for x and y position and ±30µm for z

position. The time difference between the frames is ∆t (1/frames-per-second) which is

determined by the synchronization unit with very high precision (the error is prescribed

to be 3 orders of magnitude smaller than the interval).

In general, the velocity vector is estimated using the displacement of a given particle

between two frames, divided by the time interval:

ut αxt ∆t xt

And the acceleration is calculated in the similar manner:

at ut ∆t ut∆t

In the next table we summarize the calculations of the errors, and the uncertainties of

the different parameters,

Table A1: The different errors and uncertainties in 3D-PTV process

Parameter xi units ∂ units

x 0.8 mm -150 0.0125 0.01 mm

y 0.8 mm 150 0.0125 0.01 mm

z 0.8 mm 150 0.0375 0.03 mm

∆t 0.006667 s 9000 0.0003 0.000002 s

The error propagation is estimated according to the following equation:

uò +xtu

∂u∂xt u~;Q +xt ∆t

∂xt ∆t u~J∆;Q +∆tu

∂u∂∆t u∆;Q¡

|ò|~ .

∆ |ò

|~J∆ .∆

∂u∂∆t xt ∆t xt

Table A2: Velocity and acceleration values

Parameter xi units

u 120 mm/s

v 120 mm/s

w 120 mm/s

| 120 mm/s

18000 mm/s2

• |a| get of only one component maximum values since there is no place in

the cavity where two velocity components get maximum values.

In summary we can estimate the uncertainty of the velocity and acceleration

measurements (both components and the magnitude of the vector) as:

Table A3: summary of uncertainties of velocity and acceleration

0.01768 0.01768 0.05303 0.05 0.1

And all the other parameters are:

X=0.8 mm ± 1.25% t = 0.00667 ± 0.03%

Y=0.8 mm ± 1.25% U = 120 mm/s ± 5%

Z=0.8 mm ± 3.75% a = 31176 mm/s2 ± 10%

תקציר

בשנים האחרונות הוגבר המאמץ לפיתוח שיטות מדידה במחקר לזרימה טורבולנטית וזאת על מנת

תבסקאלומדידות . להפיק מידע רב ככל הניתן מהתהליך הנצפהו ,הקטנות תבסקאלומדידה לאפשר

מורכבת מחלקיקים קטנים אחת הפאזותפאזיות בהן -קטנות אלה חשובות במיוחד בזרימות הדו

בגלל השפעות הפאזה הנוזלית ולמרות המחקר הרב המושקע . )של עד עשירית מילימטר גדליםב(

להשפעה ההדדית . הפיזור של החלקיקים במרחב עדין קשה מאוד לחזות את המהירות ואת בנושא

, מרססים, מעשיות להבנה בסיסית בזרימת של טיפות הגשם ותחהשלבין שני גורמים אלה יהיו

.סופות חול ועוד

המדידה הופכת להיות מורכבת יותר כאשר הזרימה היא טורבולנטית והחלקיקים נמצאים תבעיי

כי החלקיקים מניחהודומיה ה Maxey & Riley (1983)גישת . קצה התחתון של סקאלת הזרימהב

תם ואינה ישימה לפיתוח משוואות עם חלקיקים גדולים יותר ביחס לא תייםנקודאובייקטים הם

לכן השאלה המתבקשת היא עד כמה גישה זו ישימה ועד כמה עוזרת בהתאמת ההגדרה .סקאלות

.הלמקרה ז Stokesשל מספר סטוקס הפיסיקליתוהחשיבות

נעים Kolmogorovבקוטר של סקאלת בעלי ציפה ניטרליותאם החלקיקים ההמחקר הנוכחי בודק

מנגנוןמהו ההבדל העיקרי ואילו פרמטרים יגדירו את ה ,אם כן. הזורםתנועת באופן שונה מ

.האחראי להבדלים אלה פיסיקליה

המאפשרת particle tracking velocimetry ממדיתתלת תנועת החלקיקים נמדדה בעזרת מערכת

בדיוק רב המערכת מסוגלת לזהות. זיהוי ומעקב אחר החלקיקים בעת זרימתם במרחב נפח הבקרה

.מהירות ותאוצה, מיקום גוןאת מסלול החלקיקים ותכונותיהם כ

התצפיות נעשו . מיקרון 10בקוטר קטניםבאותה שיטה בעזרת חלקיקים הנמדד זורםה נועתת

80על רוחב תא ניסוי של בוססהמ 10,000נדולס של יטורבולנטית במספר רי LDC מסוג בזרימת

.המוביל הסרט מ ומהירות"מ

כאשר התוצאות בין החלקיקים לבין הזורםהעבודה הנוכחית מראה את ההבדלים בתכונות השונות

ובה ,) second order structure function(סדר שני מבנה פונקצית הצגתהבולטות ביותר נתקבלו ב

.נמוך שלהםהחלקיקים גדולים לא התנהגו ככלל הזרם על אף מספר סטוקס

בגודל חלקיקים של ותי'גלגראנ מדידות

טורבולנטית בזרימה קולמוגורוב

בהנדסה מכנית" מוסמך אוניברסיטה"חיבור זה הוגש כעבודת מחקר לקראת התואר

על ידי

רטנר דוד

העבודה נעשתה בבית הספר להנדסה מכנית

ר אלכס ליברזון"בהנחיית ד

2010מאי

בגודל חלקיקים של יות'לגראנג מדידות

טורבולנטית בזרימה קולמוגורוב

בהנדסה מכנית" מוסמך אוניברסיטה"חיבור זה הוגש כעבודת מחקר לקראת התואר

על ידי

רטנר דוד

2010מאי

DavidRatner_MScThesis_August2010

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