lagrangian particle tracking of large-sized particles in...

Master Science de la matière Stage 2009–2010

École Normale Supérieure de Lyon Antoine BérutUniversité Claude Bernard Lyon I M1 Physique

Lagrangian Particle Tracking of large-sizedParticles in Turbulent Flow

Abstract: A turbulent flow can be modified by free particles inside if the density of the particlesis not the same as the density of the fluid or if the particles are big compared to the Kolmogorovscale of the flow. We studied the impact of large-sized particles (i.e. about 200 times bigger thanthe Kolmogorov scale of the flow) with density close to water’s density in a counterrotating vonKàrmàn mixing flow (i.e. a cylindrical fish-tank with two counter-rotating propellers, one at eachside). To follow the translational and rotational motions of the large-sized particles we injectedsome small (i.e. nearly the Kolmogorov scale) fluorescent particles at different positions inside, andtracked them using the Lagrangian Particle Tracking (LPT) a.k.a. Particle Tracking Velocimetry(PTV) method. To characterize the flow around the large-sized particles we added some of thesmall fluorescent particles as free tracers in the flow. To analyze the data we then had to sort thetrajectories of free tracers and the trajectories of tracers inside the large-sized particles, to find thecenter of the large-sized particles and to compare the characteristics (as speed, acceleration, etc.)of the large-sized particles to the characteristics of the surrounding flow.

Keywords : Turbulent flow, Lagrangian Particle Tracking, Large-sized particles comparedto the Kolmogorov scale, counterrotating von Kàrmàn mixing flow.

Internship directed by:Dr. Mathieu Gibert

[email protected] / phone: +49 (0)551-5176-388Max Planck Institute for Dynamics and Self-Organization, Göttingen.Bunsenstrasse 10D-37073 Göttingenhttp://www.ds.mpg.de/

August 22, 2010

mailto:[email protected]

http://www.ds.mpg.de/

Acknowledgements

I want to sincerely thank the following people for their very useful help during this intern-ship: Mathieu Gibert for his disponibility, his numerous advice and his corrections to thisreport, Gaelle Dumas for her English corrections to this report, Simon Klein who workedwith me on this project, who will continue it for his Diploma Thesis and without whomwe couldn’t have results so quickly, and Robert Zimmermann who wrote a nice DiplomaThesis [1] who helped me to understand this project at the beginning and to write thisreport at the end.I would also like to thank Fabio, Sebastian, Yue, Mat, Florian, and all the people in theMax Planck für Dynamik und Selbsorganisation in Göttingen for the very nice atmospherein this lab, the entertaining discussions at lunch and all the cakes for birthdays and othercelebrations.And of course I want to thank the “Ours en JetPack” mailing list for all the stupid linksit provided me, Florent and Clement for allowing me to increase my abilities in Age ofEmpires 2, Sophie for giving me a unique occasion to practice my medical vocabulary inGerman, and all the German people for having forgiven my awful accent.

Lagrangian Particle Tracking of large-sized Particles in Turbulent Flow Antoine Bérut

Contents

Introduction 1

1 Some theoretical backgrounds 21.1 The Kolmogorov scale of one flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Lagrangian Particle Tracking method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Describing large-sized particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Experimental set-up 62.1 The large-sized particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Generating the turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Tracking the particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Data analysis 113.1 Selecting the large-sized particles among the small ones . . . . . . . . . . . . . . . . . . . . . 113.2 Reconnecting the tracks to have long trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Finding the particles’ center and analysing the surrounding flow . . . . . . . . . . . . . . . . . 16

Conclusion and perspectives 20

References 20

Appendix 21

Introduction

From the mixing of milk and tea to motions of smoke clouds produced by an islander volcano, thatforced you to have a cup of tea in the airport instead of taking your plane, turbulence is a very commonphenomenon.

A turbulent flow can be modified by free particles inside if the density of the particles is not the same asthe density of the fluid or if the particles are big compared to the Kolmogorov scale of the flow (which is thesmallest length scale of the turbulent flow). The effect of density differences has already been investigated forparticles small compared to the Kolmogorov scale of the flow (see for example M. Gibert et al [2]). This effectis important to understand phenomena like sedimentation in estuaries and rivers or dust in tornadoes. Butthe effect of large-sized particles with same density as water (such particles are called “buoyant particles”)was never studied. This effect is important to understand phenomena like rain formation in clouds or thecoexistence between several species of plankton.

We developed an experimental protocol and some data analysis algorithms to follow the translationaland rotational motions of large-sized buoyant particles in a counterrotating von Kàrmàn mixing flow (i.e.a cylindrical fish-tank with two counter-rotating propellers, one at each side), and to obtain informationabout the fluid motion around the particles. The first section describes quickly some important theoreticalbackgrounds that are useful to understand what is done after, and gives some references about the descriptionof large-sized particles in turbulent flow. The second section describes the experimental set-up that we usedto obtain the data that are analyzed in the third section.

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1 Some theoretical backgrounds

1.1 The Kolmogorov scale of one flow

In the continuum approach all the physical properties of one flow (as velocity, pressure, density, etc.) aredescribed by a field depending on time and position, and the fields are supposed to be smooth.The motion of an incompressible fluid (i.e. with a constant density), as water, is described by the Navier-Stokes equation:

ρ

[

∂

∂t+ ~u · ~∇

]

~u = −~∇p + η△~u + ~f

where: ρ is the (constant) density of the fluid, ~u its velocity, η its dynamic viscosity, p the pressure and ~fthe external forces.

We call ρ

[

∂

∂t+ ~u · ~∇

]

~u the inertial term and η△~u the viscous term. Since this equation is non-linear, it’s

sensitive to initial and boundary conditions.

We can define the Reynolds number Re as the ratio between the magnitudes of the inertial forces andthe viscous forces. If we call L the length scale of the flow and U its velocity scale, we have:

ρ

[

∂

∂t+ ~u · ~∇

]

~u ∼ ρ × U2

L

η△~u ∼ η × UL2

So:

Re =inertia

viscous=

ULν

where ν = ηρ is the so-called kinematic viscosity.

For small Reynolds numbers, the viscous term dominates, small perturbations are damped out and it’spossible to find steady-state solutions for the Navier-Stokes equation. Then the flow can be pictured asfluid lamina which glide along each other but do not cross. Hence this type of flow is called laminar. Forbigger Reynolds numbers, the system evolves into a spatio-temporal chaotic system. Perturbations arisingfrom the boundaries (for example propellers) or from body forces acting on the flow are not damped outanymore and sum up until the flow breaks up into eddies of different sizes, which are advected by the meanflow. That’s turbulence (see figure 1).

Figure 1: Instantaneous flow past a sphere at Re = 15, 000. The flow is laminar before the sphere and quickly turnsturbulent behind it. ONERA photograph, Werlé 1980, extracted from [3].

It is not possible to predict the behaviour of a turbulent flow because the Navier-Stokes equation ishardly ever analytically solvable. In 1941 Kolmogorov proposed a universal, statistical approach to describeturbulence, using means of distribution and autocorrelation functions [4]. The description is based on theenergy cascade concept, first introduced by Richardson:

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if we start a flow through propellers, it creates big whirls of a size L, and a Reynolds number Re(L). Forsufficiently strong forcing at this scale L, the Reynolds number Re(L), is large enough to neglect viscosityand resulting energy losses. Nevertheless, the injected energy has to be dissipated somehow. FollowingRichardson’s proposal, each big vortex then breaks up into several smaller vortices. The latter would beof a smaller length scale l, resulting in a smaller Reynolds number Re(l) < Re(L). Energy conservationdictates that all energy from the first generation of vortices has to be passed to the next one. The childvortices again split into more vortices of even smaller size l′ with Re(l′) < Re(l). This self-similar process isrepeated until Re ≈ 1, where the viscous dissipation starts to occur.With this idea and some hypothesis introduced by Kolmogorov, the energy conservation assures that therate of energy transferred from a length scale to its child vortices is the same at all scales. So, one candefine the energy dissipation rate (also called energy transfer rate), ε, as the dissipated energy per unitof time and unit of mass. Kolmogorov also postulated that the size of the smallest vortices depends only onthe viscosity of the fluid and the energy transfer rate. This size is the Kolmogorov scale of the flow. It’soften referred as η but here we are going to call it k to avoid confusion with the dynamic viscosity.

Using any characteristic length L and the associated characteristic velocity U , the only possibility toobtain the energy transfer rate is:

ε ∼ U3

LThen, calling uk the smallest velocity scale and k the Kolmogorov scale (which is the smallest length scale),and using the previous hypotheses :

1 = Re =k × uk

ν=

k4/3 × ε1/3

ν

⇒ k = (ν3/ε)1/4

By the same way we can obtain uk:uk = (εν)1/4

At the end we can link these values to the biggest scales of the flow (which are easily measurable), L and U :

k/L ∝ Re−3/4

uk/U ∝ Re−1/4

where: Re = ULν is the Reynolds number of the flow.

Those last equations give us a clear interpretation for the Re in turbulence: the highest the Re, thegreater the range of scale in the flow (and so, the smallest k).

1.2 The Lagrangian Particle Tracking method

There are two frameworks to observe a flow: the Eulerian point of view and the Lagrangian point of view.The first one is based on fixed observations in space: we look at the fluid properties at one or several fixedpoints at different times. The second one describes the flow by trajectories of fluid elements: we follow thevariations of the fluid properties along the trajectories of fluid elements. Even if it’s theoretically possibleto switch from one to the other, it’s experimentally very hard because of the high-resolution (in time andspace) that is then needed.

The Lagrangian Particle Tracking (LPT), also called Particle Tracking Velocimetry (PTV), is used todetermine the three dimensional position of tracer particles in the flow at given times and to build up thepoints into trajectories of particles. So that we can have a Lagrangian view of the flow and then computethe velocity and acceleration from the trajectories. There are two ways to detect the tracer particles: usingthe Doppler shift of a scattered wave in an extended measurement volume, or direct optical detection usingdigital cameras. We use the second one.

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We use 3 high speed cameras that look at the measurement volume from different angles (the actualset-up will be fully described in section 2 ). The cameras take synchronized movies that are analyzed by acomputer. There are three steps in the analysis: find the tracers on each camera, do a 3D-matching andorganize the points as trajectories.

Figure 2: Diagram of the 3D-matching. The line of sight of the point on the first camera cross a point on the secondcamera, the line of sight of the point on the second camera cross a point on the third camera and the line of sight ofthe point on the third camera cross the first point on the first camera. This point is seen by the 3 cameras and wecan find its coordinates in the 3D space.

• The light scattered by the tracers leaves a bright spot on the 2D image of each camera. If the particlesare spherical, the spots can be seen as a 2D Gaussian with noise. The spots are identified using asegmentation technique which considers every pixel with an intensity above a given threshold as partof a spot. Ouellette et al [5] discussed several methods to identify the center of a spot. We use aCenter-of-Mass method: the center of one spot is the center of mass of the pixels in the spot weightedby their intensity.

• The 3D matching uses an algorithm that keeps only the tracers that were seen in the same time byall the cameras and that gives back their coordinates in 3D. We use a camera model proposed by Tsai[6] to have a correspondence between one point on a 2D image of one camera and a direction (lineof sight) in the real 3D space. Of course it’s different for each camera and it needs to be calibrated(the calibration process is briefly described below). The 3D-matching is done at each time position.For each point of the first camera, we calculate the direction that corresponds in the real space. Thenwe calculate the projection of this direction on the image of the second camera and we look if somepoints exist near this direction on the image of the second camera. If there is no point, the secondcamera didn’t see this tracer. If there are some points, for all of them, we calculate the direction thatcorresponds in the real space and we project this direction on the third camera image plane. Thenwe look if some points exist near this direction on the image of the third camera. If there is no point,the third camera didn’t see this tracer. If there are points, for all of them we look if the projectionof their real-space direction on the image of the first camera comes near the first point. If it’s thecase, we have one point, seen by the 3 cameras, and we know the 3 directions that should cross atthe real position of this tracer. Since the 3 directions never cross perfectly, we give to the tracer thecoordinates of the points that is the nearest to the 3 directions (see figure 2).

• Different techniques can be used to connect the points as trajectories (see [5]). We use the so called“Best Estimate” method. For the first points of the trajectory we use the simple nearest neighbourmethod. Then we use 4 frames to calculate speed and acceleration of the tracer and to compute itsestimated position. The next point of the trajectory will be the closest to the estimated position ofthe tracer (if one exists).

The calibration is done with a transparent mask on which a pattern of black dots is drawn. These dotsare aligned and separated by a constant distance (so that we know what is the distance between each of

4


them). Three of the dots are bigger than the others to localize the center of the mask and 2 perpendiculardirections. We put the mask in the middle of the fish-tank and take pictures of the mask with the cameraswhile translating this 2D mask perpendicularly to its plane (its position is measured with a micrometer).Then we use a program to compute the camera-model parameters (which are the link between the blackdots positions in the 2D images of the cameras and their positions in the real 3D space).

At the end of the analysis we can obtain 3D trajectories of particles (see figure 3).

1.3 Describing large-sized particles

To fully describe a spherical particle in motion in a turbulent flow, we need to use the Navier-Stokesequation for the fluid, the Newton’s second law for the particle, non-slip boundary condition at the surfaceof the particles and the conservation of the angular momentum for the particles rotation rate. Solving thissystem is very difficult because it involves a non-linear partial differential equation for the fluid which iscoupled to a moving boundary condition on the particle.

There is a so called “point-particle approximation” that assumes that the perturbation of the flowsurrounding the particles is well described by the Stokes equation (see Maxey et Riley 1983 [7]). It canbe used with Faxén corrections (see Gatignol 1983 [8]) to describe the motion of a particle in a turbulentflow if its diameter is smaller than the Kolmogorov scale of the flow. Homann and Bec (2009) [9] showedwith numerical simulations that Faxén corrections give dominant finite-size corrections to velocity andacceleration fluctuations for particle diameters up to 4 times the Kolmogorov scale. Which means that theseequations can be used for particles slightly bigger than the Kolmogorov scale but are not relevant anymorefor bigger particle because the dynamics is dominated by inertial-range physics.

Since we use particles with a diameter about 200 times bigger than the Kolmogorov scale of the flow,there is no simple model to describe the interaction of the particles with the turbulent flow, and experimentswith direct measurements of the large-sized particles’ trajectories are a first step to develop such a model.

−30−20

−100

10

0

20

40−40

−30

−20

−10

0

10

20

30

40

xy

z

(a) Some trajectories

01020300 10 20 30

−30

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0

10

20

30

40

50

xy

z

(b) Some other trajectories

Figure 3: Example of trajectories obtained with the LPT method. Here the tracers are injected in large-sized particlesthat we let fall in the measurement volume. Distances are in mm.

5


2 Experimental set-up

Our aim was to follow the full motion of large-sized particles with properties very close to those of waterin a turbulent flow and, in the same time, to get some information (as speed and acceleration) about theflow around those particles.

2.1 The large-sized particles

We use “waterballs” as large-sized particles, they are small balls of some super absorbent polymers(usually sodium polyacrylate). Their diameter is about 2.5 mm when they are dry, and then grows up tomore than 1.5 cm when they are put into water. You can see a dry one and a grown up one on figure 4(a).

(a) Waterballs (wet and dry ones). (b) The cube and the syringe.

Figure 4: Pictures of the waterballs in two different states with positions of the injected tracers symbolised by redpoints (a), and picture of the cube with the 6 screws with holes (the top is separated from the rest to allow us to puta waterball inside) and the micro-litre syringe that were used to inject the fluorescent particles in the waterballs (b).

They have several advantages:

• their density is really close to the density of water because they are composed by water for more than99%.

• they are transparent and nearly have the same optical indexes as water, so that we can see throughthem and track particles behind them (for example free tracers in the flow around).

• their diameter is between 10 mm and 20 mm, which is large-sized compare to the Kolmogorov scale ofour turbulent flow (around 50 µm). The waterballs are about 200-300 times bigger than the Kologorovscale.

• they are really cheap and easy to find: it’s possible to buy them in any garden-shop (know as “WaterPearls” and used to supply water to flowers).

• the process to make them ready is really easy: we just have to let them for about 30 minutes in waterso that they can absorb it.

Their only two disadvantages are:

• it’s really hard to modify their size or shape without breaking them.

• since they are nearly invisible when put into water, there is no way to directly follow their motions inthe turbulent flow.

A way to be able to see them is to inject them with some fluorescent particles. Since we want to beable to see their translation and their rotation motions, we need to inject fluorescent particles at differentpositions. The particles are commercial “Red fluorescent polymer microspheres” with a diameter of 107

6


µm and a density of 1.06 (neutrally buoyant). They are the same as those used as tracers to obtain theflow properties with the Lagrangian Particle Tracking method. Even if 4 tracers are enough to measurethe radius and motion of a waterball, we choose to inject 6 particles as close as possible from the externalsurface in each waterball (see figure 4(a)), to have better tracking. For that purpose we use a cube with sixscrews with holes (see figure 4(b)) designed and built before my arrival, and a commercial 10 µL syringe,with a 0.2 mm needle (also seen on figure 4(b)).

The process steps are: the waterball is put in the cube, then fixed by the screws (being careful thatwe don’t screw too much, so that it won’t break), then the cube is closed and the fluorescent particles areinjected through the hole of each screw. The process is neither really practical nor really fast. We triedsome different possibilities for injecting the fluorescent particles as using them dry so that they only stick tothe tip of the syringe by air humidity, or injecting them by air pressure. But we finally found that the moreefficient method was to have them in suspension in water, with a low concentration, so that we can directlysuck them up with the syringe with no risk to block the needle’s hole (that is only 2 times bigger than thefluorescent particles) and inject them as close as possible to the surface of each waterball. The total contentof the syringe is injected in each of the 6 screws of the cube.

Figure 5: Some of prepared waterballs with at least 4 fluorescent particles at different positions in each of them. Oneof the fluorescent particle is illuminated by a small 532 nm LASER (the one that is actually used for checking havea larger beam so that the process can be quicker and easier). Note that the LASER beam is not deflected whiletravelling through the waterballs since their optical indexes are very close to the water ones.

The waterballs need to be checked after being prepared because some of them really don’t get a lot offluorescent particles and we don’t want to have less than 4 of them inside each waterball (to be able tocalculate their radius). So we have to look at them individually with a LASER to determine how manyfluorescent particles were inside (see figure 5)). An easy way to prepare a lot of them is to keep, during thechecking process, all the ones with less than 4 particles inside (that often have 2 or 3) and to re-use them forinjection. Then we have about 100% of success but we lose some information about the disposition of thefluorescent particles in the waterballs (but they still are as close as possible to the surface). A reasonablenumber of waterballs for one measurement would be about one hundred.

2.2 Generating the turbulent flow

The turbulent flow that we use is a von Kàrmàn swirling water flow. It is generated by two counter-rotating propellers submerged in an aquarium confined on top by a removable cover with rubber seals alongthe rim (see figure 6). The propellers are driven by two underwater air-motors so that a highly intenseturbulent flow can be produced in an apparatus with moderate size. The diameter of the propellers is28 cm. The turbulence chamber, shaped as a hexagonal cylinder, measures 40 cm along the axis of thepropellers and 38 cm in both height (vertically) and width (horizontally) in the cross-section. The rotatingaxis of the propellers is in the horizontal direction so that the waterballs (that are still a little bit heavierthan water) settling down towards the bottom of the apparatus are entrained by the strong sweeping ofthe fluid near the bottom surface. In this way the waterballs and the small free fluorescent particles staysuspended in the measurement volume, which is at the center of the apparatus because it’s where the mean

7


fluid velocity is really small. Moreover each propeller has a velocimetry device linked to one computer, sothat we can control their rotation speed and set them equal. (see figure 7 ).

(a) picture (b) view from the side (c) view from above

Figure 6: Picture and diagrams of the fish-tank. The axis of the propeller is called the x axis, the LASER beampropagates along the y axis and the vertical is the z axis. For the picture the grids were removed (they are thedot-lines in diagram (c)) but the mirror that was added just behind the fish-tank can be see on (a).

The small mean fluid velocity is important to allow us to follow one particle for a long time, beforeit leaves the measurement volume, which is nearly a cube with 7 cm edges. We managed to follow someparticles for times approaching 1 s.To avoid that the waterballs get broken by the propellers, we add two grids, as close as possible to thepropellers (see figure 6(c)), to let the water go and generate turbulence, but confine the waterballs in themiddle of the apparatus. A consequence is that we have to increase a little bit the rotation speed of thepropellers to generate the same Reynolds number as without the grids, which is rather a good point becausethe air-motors rotation speed is more stable at higher velocity.

0 200 400 600 800 1000 1200 1400 1600 1800 20001.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

Time (minutes)

Rot

atio

nsp

eed

(rou

nd

per

seco

nd

)

Rotation speeds of the 2 propellers

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000.026

0.028

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0.04

Time (minutes)

ST

Dof

rota

tion

spee

d(r

oun

dp

erse

con

d)

STD of rotation speeds of the 2 propellers

(b)

Figure 7: Speeds and STD of speeds for the two propellers, measured while trying to set them equal (the air-motorsparameters were changed at the 200th minute). The time needed to let them settle down after changing the parametersis really high (because they interact together through the flow), but then we have a stable mode for a very long timeto do the measurements.

The Taylor microscale Reynolds number (Rλ) of our flow is about 600, which correspond to a Re around2400 (Rλ =

√15Re). At this Reynolds number, the Kolmogorov scale of the flow is typically around 50 µm.

8


2.3 Tracking the particles

To track the particles we use 3 high-speed cameras with acquisition speed up to 2900 Hz, and weilluminate the sample with 2 powerful LASERs (wavelength: 532 nm, power up to 100 W for each LASER)in pulsed mode (Q-switch). The LASERs and the cameras are triggered by a function generator, so thatthey are synchronised and the frequency of the LASERs is 3 times bigger than the acquisition frequency ofthe cameras.The material used to lead the LASER beams to the fish-tank were only simple optical devices as mirrors,lenses (to enlarge the beam up to 10 cm) and a diaphragm to have a nice beam through the measurementvolume.

The first configuration was one camera on each side of the fish-tank: one in front of it, one behind andone below (see figure 8(a)). But we then realized that the camera behind, which is the only one that hasthe LASER beam arriving directly to it (camera 2 in figure 8(a)), saw really more particles than the twoothers. We then added a big mirror behind the fish-tank to symmetrise the path of the LASER beam in thewater (see figure 6). With this set-up, the camera below the fish-tank (camera 3 in figure 8(a)) became theonly one to see less particles than the two others. To give an example, for a fish-tank with free fluorescentparticles in water, on a 3 s movie taken with a 2 kHz frequency, the first camera saw in average 244 particlesper frame, the second-one 248, and the third one only 49. Even if the number of particles found by frameis not really important itself because it depends on how much tracers you put in the fish-tank and on howpowerful was your LASER for these movies (the more powerful the LASERs, the more particles the cameraswill see), it’s a problem because we want to find as much 3D matching (which means the number of “real”particles that are found by crossing the data of the 3 cameras) as possible, and it’s clear that this numbercannot be higher than the lowest number of particles found by one of the cameras. For this set-up, the 3Dmatching was in average only 27,4 particles per frame.

(a) first configuration (view from the side) (b) second configuration (view from above)

Figure 8: Diagrams of the two different configurations that were used for the cameras positions.

So we decided to move the cameras to improve the measurements. The new configuration is: one camerain front of the fish-tank (as before) and two behind but with an angle between them (see figure 8(b)). Theangle has to be as small as possible to avoid astigmatism effects due to dioptre air-glass-water at the wallof the fish-tank, that totally disappears only if the camera is looking perpendicularly to the wall. Thisset-up gives a real improvement in term of number of particles seen by each camera. For example: for afish-tank with about 80 waterballs and free fluorescent particles in water, on a 3 s movie taken with a 2.9kHz frequency, the first camera sees in average 197 particles per frame, the second-one 208, and the thirdone 210, for an average number of 3D matching of 93,6 particles per frame. But we then loose some spatialresolution because of the small angle between the two cameras. We then had to calibrate the new set-up forthe Lagrangian Particle Tracking.

9


We made a lot of different movies that were used to test and improve the data analysis processes (whichwill be described in the next section of this report):

• movies with only free fluorescent particles, as references (for example, a good selection algorithm forthe waterballs shouldn’t find anything in such movies).

• movies with only some waterballs falling into water with the propellers off (some nice tracks with thatconfiguration can be seen on figure 3).

• movies with only waterballs in the turbulent flow, also as references.

• movies with both waterballs and free fluorescent tracers in the turbulent flow.

The last movies that are used to obtain some interesting data are:20 movies of 3 s, taken with the 2 LASERs set at 28 A and 8.7 kHz (which gives a power of about 100W per LASER), while the 3 cameras were at 2.9 kHz with a 768 × 768 pixels resolution (knowing that themeasurement volume is approximatively a cube with 70 mm edges, it gives about 0.1×0.1 mm2 for a pixel).The tracks on these movies were in average 107 frames long (which corresponds to 37 ms) as seen on theprobability density function (PDF) of the track lengths figure 9.

0 100 200 300 400 500 600 700 800 900 10000

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pdfmean = 107 frames

Track lengths (number of frames)

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ized

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kle

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PDF of the track lengths

(a)

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mal

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kle

ngt

hs

(in

sem

ilog

)PDF of the track lengths in semilog

(b)

Figure 9: PDF of the track lengths calculated on the 20 movies that are used to obtain the final results with the dataanalysis processing.

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3 Data analysis

In this section we describe how, starting with the raw movies of the particles, we manage to get someinformation about the waterballs trajectories and the surrounding flow.

The first data that we have are the raw movies, which are 3 sets of frames, one set for each camera. All thesets of frames are synchronised so that the first frame of each set corresponds to a same physical instant, andall the frames are taken with the same time between them (which is the acquisition period). So, one framecorresponds to one time position and the number of frames is only the acquisition frequency times the movielength (for example a 3 s movie with 2.9 kHz frequency gives 3 sets of 8700 frames). These raw data aretreated by a LPT program that gives back files that can be read by Matlab and transformed to a structurethat contains all the particles trajectories of the movie, which are calculated by the Lagrangian ParticleTracking method (see 1.2). We call “tracks” these trajectories. Each track contains some informationabout the particle: the numbers (i.e. labels) of the frames where the particle exists, the particle’s positionin 3D for each frame and a interpolation label for each frame, that is “1” if this position was interpolatedfor this frame or “0” if the position is a real position. The number of successive interpolated points cannotbe more than 4. In the following we will use “frame” to designate either the 3D image of all the particlesthat exist together at one particular time, or only a time position.

3.1 Selecting the large-sized particles among the small ones

The first thing to do is to sort the tracks that we have, so that we can know which are the onescorresponding to the fluorescent particles inside the waterballs and which are the ones corresponding to thefree fluorescent particles that follow the turbulent flow.

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Figure 10: Normalized PDF of the distances between all couples of particles, calculated on 9 movies of 4 s taken withthe cameras at 2 kHz (mean = 39.7 mm and STD = 14.9 mm).

The “natural” way to do it, is to think that, if some particles are in one waterball, their relative dis-tance should be a constant. Whereas the free particles should diverge one from each other because of theturbulence. So we made a program that looks at all couples of tracks and keep them only:

• if they exist together for more than a given number of frames (which is useful to eliminate bad tracksor to have tracks long enough to allow us to calculate a derivative and a second derivative).

• if their distance at the first frame where they exist together is in a range that allow us to think thatthey really are in a waterball (for this purpose we chose an estimated diameter size for the waterballsand only the couple with a distance between 0.1 times and 2 times this estimated diameter are kept).

• if the distance between them doesn’t vary more than a given percentage of a reference (we will discussthe choice of this reference below).

11


In the same time, we do a PDF of the selected distances (which means, a PDF done for all the distancesbetween all the selected couples at all the frames where the two particles of this couple exist together), and aPDF of the STD of the selected distances (for all the selected couples, we calculate the STD of the distancesbetween them at the different frames where they exist together, then we do a PDF of all these STD). Thesetwo data give us some useful information to know if the selection process is good or not.Indeed, if all the waterballs have the same diameter and have 6 fluorescent particles injected close to theirsurface as described in 2.1 (see figure 4(a)), and if we see all the fluorescent particles, we should find 2 peaksin the PDF of the kept distances:

• one at a distance that is 2 times the waterball radius.

• one at a distance that is√

2 times the waterball radius.

And the second one should be 4 times bigger than the first one, because if you take one fluorescent particle,it should be coupled with one other at the opposite pole of the waterball and with four others situated onthe equator.To compare the data to a fixed reference, we make a PDF of all the distances between all the particles ofsome movies (see figure 10).

Before adjusting the parameters (the estimated diameter of the waterball, the minimal number of framesof coexistence and the tolerance percentage), we have to choose the reference mentioned before. We had thechoice between:

• an absolute reference: the distance between the 2 particles should never vary more than a givenpercentage of the estimated diameter of the waterball.

• a relative reference: the distance between the 2 particles should never vary more than a given percent-age of the initial distance between the two particles.

As seen on figure 11 the absolute one seems really better because the STD of the distances between theselected couples are smaller and because we don’t favour the couples with bigger distances between them.

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Figure 11: Comparison between the two possible references for the selection of particles that should be in a samewaterball. PDF calculated on 10 movies of 3 s with free tracers and a few waterballs in the fish-tank, cameras at 2kHz. Selection parameters : estimated diameter 10 mm, at least 50 frames of coexistence, tolerance percentage 5%.

Then we need to determine the parameters that give the best results to find good couples:

• The estimated diameter of the waterball is not really important because all the waterballs don’t havethe same size, and because it’s used mainly to eliminate the couples of points that cannot really be ina waterball because they are too distant from each other. And so we keep 10 mm as the diameter.

12


• The minimal number of frames where the particles of a couple exist together is more a practicalparameter than a really relevant one for the selection. Nevertheless, it cannot be too high because, aswe will see in the next section, it can happen that the tracks have a lot of holes (i.e. one “real” trackcan be segmented in a lot of smaller ones), and it shouldn’t be too small because then we cannot reallysee if the distance between the two particles varies or not. Moreover, to be able to find the velocitiesalong the trajectories we need to calculate a derivative, and such derivative is computed by a discreteconvolution between the set of positions and a finite-sized derivative of a Gaussian. Since we chosenot to do derivatives with a finite-sized Gaussian smaller than 6 points (to minimize the influence ofthe noise on the acceleration measurements), we choose a minimal number of coexistence of 25 framesfor the particles in one couple.

• The tolerance percentage is the real parameter for the selection. We first tried it with some movieswhere we had only some waterballs and no free fluorescent particles. With this configuration, it canbe up to 10% and still gives good results (with no significant change for tolerances higher than 5%).But for movies with free particles in the turbulent flow, the tolerance has to be really more drastic. Asseen on figure 12(a) for a tolerance of 5% we still select a lot of free particles: we have peaks but theyare surrounded by a strong background with the same shape as the total PDF (figure 10). It’s onlyfor a tolerance below 0.5% that we can say that the couples selected are mainly couples that shouldbe in a same waterball: we now have only peaks with no background (see figure 12(b)) and the STDsof distances are smaller (see figure 12(c) and (d)).

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Figure 12: Comparison between two tolerance percentages on the same set of data (the last 20 movies we made andthat are described at the end of 2.3 ) for the selection of particles couples. The estimated diameter of the waterball is10 mm and the coexistence is at least for 25 frames.

13


Interestingly we can notice that, even if the waterballs don’t have exactly the same diameter and evenif the fluorescent particles are not exactly injected at the positions were they should be, we find two mainpeaks on figure 12(b). Moreover this two peaks are:

• 14820 occurrences of a couple with a distance of 9 mm between the two particles.

• 4500 occurrences of a couple with a distance of 12.2 mm between the two particles.

And we can notice that 9× 2√2

= 12.7 ≈ 12.2 and that 14820

4= 3705 ≈ 4500. So, even if our waterball are not

perfect, our assumption that we should have two peaks in the PDF of the distances between selected couplesis not completely false: the order of magnitude is correct! We have one peak at the waterball diameter and

one at√

2

2times this diameter, with the second peak 4 times bigger than the first one.

After selecting all the couples of particles that should be in a same waterball, we just have to sort themso that we can have all the particles that should be in the same waterball. To do that, we just put togetherall the tracks that are linked by the coupling selection, and we call “group” the list of all the tracks thatcorrespond to one waterball. For example if the track 6 is selected with the track 8, and if the track 8 isselected with the track 52, then the tracks 6, 8 and 52 will be put in one group.

3.2 Reconnecting the tracks to have long trajectories

As said before, one of the problem we have with the tracks, is that some of them have holes, whichmeans that some tracks that could be really nice and long are segmented into a lot of small different tracks(see figure 14(a)). It’s not really a problem for the selection process (even if it would surely be easier todifferentiate the particles that are in a waterball to the free ones if all the tracks were reconnected andlonger), but it could be disturbing for some of the processes that will be described in the next section(find the center of the particles, calculate the speed of this center, compare it with the velocity of the fluidaround). So we decided to write a program to reconnect the tracks with holes within one group.

(a) Here ~v is the speed at the end of the real track 1. The vector ~d is the distance between the end of track 1 and thebeginning of track 2. The cylinder directed by ~v is represented in green, L is its length and h its radius. To test if thebeginning of the track 2 is in the cylinder, we just use the angle θ that we can know easily by doing a scalar productbetween ~v and ~d. Of course here, the track 1 has only a chance to be reconnect with the track 3, and the track 2 willbe eliminated.

(b) Here ~v1 and ~v2 are the speeds at the end and at the beginning of the real parts of the tracks, and ~v3 and ~v4 arethe speed at the beginning and at the end of the straight line that reconnects the two tracks. If ~v3 is close enough to~v1 and if ~v4 is close enough to ~v2, the reconnection will be kept.

Figure 13: Diagrams of the selection of potential candidates for the reconnection.

14


We don’t reconnect all the tracks of the movies because it takes a really long time and, because of thenumerous free particles, the parameters for the reconnection should be more restrictive to avoid reconnectingtracks that don’t belong to the same particle.

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Figure 14: The same group (in the XZ plane) before and after being treated by the reconnection program. Distancesare in mm.

The method for the reconnection is illustrated on figure 13. For each track, we find all the tracks thatstart after (in time) the end of the considered track. For all these tracks, we look if their beginning pointis in a cylinder directed by the speed at the end of the first track. The cylinder has a radius h and alength L given as parameters. For all the tracks beginnings in the cylinder, we calculate what would be thereconnection between them if it was a straight line: we just add the missing points to obtain a straight linebetween the end of the first track and the beginning of the tracks in the cylinder. Then we calculate thespeed along this straight line and we check if the speed at the beginning of the straight line differs from thespeed at the end of the first track for more than a given percentage of this speed, and if the speed at theend of the straight lines differs from the speed at the beginning of the track that was in the cylinder formore than the same given percentage of this speed. If the speed on the straight line doesn’t vary too much(with regards to the speed on the real parts of the tracks), we decide that the reconnection can be kept. Ifmore than one reconnection are potentially good, we only keep the best one (the one where the speeds onthe straight line are the closer to the speed on the real part of the tracks).We can notice that the reconnection method can be improved by using the tracks filtered by a Gaussian(because sometimes the tracks have kinks that makes the process harder).

Good parameters for the reconnection are: h = 0.4 mm, L = 30 mm, tolerance percentage = 50%. Theresults can be improved by reconnecting a second time with different parameters: h = 2.0 mm, L = 30 mm,tolerance percentage = 10%. With the first set of parameters we only reconnect the tracks that are nearlystraight (in this case, the reconnection by a straight line should be really good). With the second one weallow ourselves to reconnect some curvy tracks (but in this case, we are more strict on the tolerance to avoidreconnecting tracks that shouldn’t be). Some results of the reconnection can be seen on figure 14.Of course, the reconnection process gives us longer tracks, as showed on the PDF of the tracks lengths onfigure 15. Quantitatively, the number of tracks of 500 frames length is multiplied by 10 after this process.It also allows us, assuming that the reconnection was made only on tracks that should be reconnected, to“clean” the groups, by applying once more the couple selection process to eliminate the tracks that shouldn’tbe in the groups (because the first selection is of course never perfect).

15


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Figure 15: PDF of the track lengths calculated on all the groups of the 20 movies that were used to obtain the finalresults, before and after using the reconnection process.

3.3 Finding the particles’ center and analysing the surrounding flow

The next step is to find the waterball center and its radius, knowing only the trajectories of the fluorescentparticles that we injected. We make one assumption that is: all the particles are on the surface of thewaterball (that’s not exactly true in fact, but since they all have been injected with the cube and the samescrews, they should all have nearly the same distance to the center of the waterball). We tried 3 differentmethods to do that:

• for each frame where the group exist, we use the sphere equation (x−xc)2+(y−yc)

2+(z−zc)2−R2 = 0

where (xc, yc, zc) are the coordinates of the center and R the radius of the sphere (that are the 4unknown we are looking for), and we minimize the sum of the residuals using the set of positions asthe (x, y, z).For example if we have n positions (xi, yi, zi) of fluorescent particles, we create a function:

F : (xc, yc, zc, R) →n

∑

i=1

|(xi − xc)2 + (yi − yc)

2 + (zi − zc)2 − R2|

and we minimize it to find the best (xc, yc, zc, R).

• for each frame where the group exists, we find all the sets of 3 points taken into the n fluorescentparticles that we have. For each set of 3 points, we find the center of the only circle that goes throughthem, and the straight lines orthogonal to the plane of the 3 points that passes through this center.Then if we have at least 2 different sets of 3 points (which is achieved with 4 non-coplanar points), wefind the point that is the closest to all the straight lines by minimizing a function that gives the sumof the distances between one point and all the straight lines. This point should be the center of thesphere (because if all the points that we have are on the surface of one sphere, all the straight linesshould cross in the center of the sphere).

• we take all the tracks of the group and we find all the sets of 4 tracks taken into the group. For eachset of 4 tracks we calculate the position of the center and the radius of the sphere that goes throughthem for each frame where they exist all together (there’s only one unique solution that is simply givenby the resolution of a linear system). At the end we take all the frames where the group exists and ifonly one center and radius were find at this frame, we decide they are the good ones. If several oneswere found, we take the average radius and position.

For the minimization we have to give an initial guess of what could be the position of the center and theradius of the sphere. We usually take the barycentre of the positions of the fluorescent particles and the

16


radius found in the PDF of the distances between particles of selected couples (see 3.1 and figure 12). Butthis first method is really sensitive to this guess and often diverges, which is not really practical. Thesecond and the third ones are more stable and give nearly the same results that are fine if we take someprecautions (such as: not taking in account the positions that are too close or too far away from each otherand eliminating the spheres with obviously too big or too small radius). Since the third one was faster(because we have only few tracks in one group, the combinatorial calculus is not so enormous), it’s the onewe keep. One example of the trajectory of one group’s center can be seen in figure 16.

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Figure 16: Two views of the same group with the center and the particles around it:in black the trajectories of the fluorescent particles, in red the trajectory of the center that was found, in green somecircles that symbolise the waterball (they have the same center and radius and are orientated by the speed along thecenter’s trajectory) and in blue the trajectories of the particles that coexist around the waterball (the red point faraway from the group is only the point (0, 0, 0)). Distances are in mm.

After having determined the center of the waterball, we find all the free fluorescent particles that arewithin a sphere of radius 5 times bigger than the radius of the waterball (see figure 16(b)).

Figure 17: diagrams of the XY plane to calculate the coordinates of the free fluorescent particles around the waterballs,assuming a cylindrical geometry around the axis directed by the speed of the waterball’s center:C is the waterball center, P is the free fluorescent particle, ~DCP is the CP vector, ~VC is the speed of the center, ~VP isthe speed of the particle. The XY plane is the plane containing ~DCP and ~VC . Of course, there is no reason why ~VP

should be in the XY plane (it’s only the case on the diagram because it’s more practical to draw).

17


Then we assume a cylindrical geometry around the axis directed by the speed of the waterball center,and going through its center. For each group, each frame and each free fluorescent particle around thewaterball, we find the plane that contains the vector between the waterball center and the free particle, andthe waterball center’s speed (see figure 17). Then we calculate the two relative coordinates of the particlefrom the waterball center, one along the axis directed by the center’s speed and the other orthogonal to thisaxis. We normalize them by the radius of the waterball. We call X the normalized component along theaxis and Y the normalized orthogonal component.Then we calculate the components of the relative particle’s velocity (the particle’s velocity minus the center’sspeed): the one that is parallel to the center speed (longitudinal) and the ones that are orthogonal(transverse and azimuthal) to this speed. Then, using the XY coordinates, we can do a 2D-mapping ofthe values by averaging on all the data that we have, which means on all the groups and all the frames (todo that we assume that the flow is stationary and homogeneous). Of course the mean is significant onlyif we have enough values, so we divided the XY plane in small boxes (the total plane goes from -5 to 5 inX and Y, we divided it into a 65 × 65 matrix) and only kept the boxes were we had more than 100 values(see figure 18). Since we have a lot of values corresponding to a same position in the XY plane, we can alsocalculate a STD and plot the 2D-map of the STD. All the maps are seen on figures 19 to 21. All the mapshave been symmetrised because, as seen in figure 17, if X can be negative, Y can only be positive becauseit’s always the distance between the particle and the axis (we assume then that we should have the samebehaviour for Y and -Y). See the appendix for the same maps without waterballs in the flow.

(a) Direct number of data (b) Normalized number of data

Figure 18: Number of data for each small box in the XY plane. The boxes with less than 100 data inside were alreadysuppressed. The normalization of the number of data is done by dividing the numbers by the theoretical number ofparticles assuming an equipartition around the waterball.

Here the number of data is a bit low to give clear conclusions, but we can detect some tendencies:

• The longitudinal normalized relative velocity of the fluid is mainly negative, which is normal: thewaterball and the fluid should have nearly the same velocity in norm, but the fluid velocity is notsupposed to have the same direction as the waterball speed (see figure 19(a)). We can see that thiscomponent is bigger when we fluid is further from the waterball (there is no more influence of thewaterball). The transverse and azimuthal components of the normalized relative fluid velocity aremainly small (theoretically they should be zero in average). See figures 20(a) and 21(a).

• There is a region around the waterball (and mainly behind the waterball) where the STD of the fluidrelative velocities are smaller, which means that the fluid velocities are more homogeneous than inthe rest of the turbulent flow. This region can be seen as an influence area of the waterball in theturbulent flow. See figures 19(b), 20(b) and 21(b).

18


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19


Conclusion and perspectives

We now have a complete method to follow the translations and rotations of large-sized buoyant particlesin a developed turbulent flow, and to analyze the flow around the particles in the same time.

Some points still need to be improved, in particular:

• the injection of the tracers in the waterballs is not really efficient and we’d like to use a device similarto the one used by diabetic people to test their sugar rate in a blood droplet on their finger.

• we could use a mirror derivation to illuminate all the cameras in the first configuration, to allow us tohave more spatial resolution and keep seeing a lot of particles.

• the reconnection process could be modified to use also the acceleration (and not only the velocity) tolead the search-cylinder and to be more efficient for curvy trajectories.

Moreover some hypotheses should be more investigated, as the cylindrical symmetry that we assumed to doa 2D-mapping of the fluid properties around the waterballs, or the fact that the tracers are at the surfaceof the waterballs. Finally, new movies need to be done in large quantity to have enough data for relevantstatistics.

Nevertheless the first results are promising. They can, by using the rotations of the waterballs, allowus to analyze the turbulence in the non-inertial frame of reference of the waterballs, and maybe help thedevelopment of a simple model to describe the motion of large-sized buoyant particles in turbulent flows.

References

[1] R. Zimmermann, The Lagrangian Exploration Module generation of homogeneous andisotropic turbulence with little mean flow for Lagrangian experiments (Diploma Thesis).Georg-August-Universität Göttingen and Max-Planck-Institut für Dynamik und Selbstorgani-sation, 2008.

[2] M.Gibert, H. Xu and E. Bodenschatz. Inertial effects on two-particle relative dispersion inturbulent flows. EPL, 90 (2010) 64005.

[3] M. Van Dyke. An Album of Fluid Motion. THE PARABOLIC PRESS, Standford, California.

[4] A. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very largeReynolds’ numbers. Dokl. Akad. Nauk SSSR, 30:301-305, 1941.

[5] N.T. Ouellette, H. Xu, and E. Bodenschatz. A quantitative study of three dimensional La-grangian particle tracking algorithms. Experiments in Fluids, 40(2):301-313, 2006.

[6] R. Tsai. A versatile camera calibration technique for high-accuracy 3D machine vision metrol-ogy using off-the-shelf TV cameras and lenses. IEEE Journal of Robotics and Automation,3(4):323-344, 1987.

[7] M. R. Maxey and J. J. Riley. Equation of motion for a small rigid sphere in a non-uniformflow. Phys. Fluids 26, 88-889, 1983.

[8] R. Gatignol. The Faxén formulae for a rigid sphere in an unsteady non-uniform Stokes flow.J. Méc. Théor. Appl. 1, 143-160, 1983.

[9] H. Homann and J. Bec. Finite-size effects in the dynamics of neutrally buoyant particles inturbulent flow. Proposed for Journal of Fluid Mechanic, 30 September 2009.

[10] S. Corrsin. Turbulent Flow. American Scientist, Vol. 49, No. 3, September 1961.

20


Appendix

Here are the 2D-maps of the values calculated in a flow with only free tracers (without waterballs). In this case

we choose 10 tracers randomly in each movie to be the waterballs center, and we choose an arbitrary radius of 6 mm.

Then we do exactly the same process as with the waterballs.

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