AUTOREFERÁT DOKTORSKÉ PRÁCE

Daniel Duda

Kvantová turbulence v supratekutém héliustudovaná vizualiza£ními metodami

Katedra fyziky nízkých teplot

kolitel: Prof. RNDr. Ladislav Skrbek, DrSc.

Studijní program: Fyzika

Studijní obor: Fyzika kond. látek a materiálvý výzkum (F3)

Praha 2017

DOCTORAL THESIS ABSTRACT

Daniel Duda

Quantum turbulence in superuid heliumstudied by particle tracking velocimetry

visualization technique

Department of Low Temperature Physics

Supervisor: Professor RNDr. Ladislav Skrbek, DrSc.

Field of Study: Physics

Specialization: Condensed Matter and Material Research (F3)

Prague 2017

Tato diserta£ní práce byla vypracována v rámci prezen£ního doktorského studia naMatematicko-fyzikální fakult¥ Univerzity Karlovy v letech 2012-2017.

Autor: RNDr. Daniel Duda

Název práce: Quantum turbulence in superuid heliumstudied by particle tracking velocimetry visualization technique

kolitel: prof. RNDr. Ladislav Skrbek, DrSc.

kolící pracovi²t¥: Katedra fyziky nízkých teplotMatematicko-fyzikální fakulta Univerzity KarlovyV Hole²ovi£kách 2, 180 00 Praha 8eská republika

Oponenti: prof. RNDr. Peter Skyba, DrSc.Ústav experimentálnej fyzikySlovenská akadémia viedWatsonova 47040 01, Ko²iceSlovenská republika

doc. ing. Zden¥k Chára, CSc.Ústav pro hydrodynamikuAkademie v¥d eské republikyPod Pa´ankou 30/5166 12, Praha 6eská republika

P°edseda oborové rady: prof. RNDr. Helena t¥pánková, CSc.Katedra fyziky nízkých teplotMatematicko-fyzikální fakulta Univerzity KarlovyV Hole²ovi£kách 2, 180 00 Praha 8eská republika

Obhajoba se koná dne 27. zá°í 2017 v 11:00 p°ed komisí pro obhajoby diser-ta£ních prací oboru F3: Fyzika kondenzovaných látek a materiálový výzkum naMatematicko-fyzikální fakult¥ Univerzity Karlovy, Ke Karlovu 3, Praha 2, v míst-nosti 105 (M252).

S disertací je moºné se seznámit na studijním odd¥lení doktorského studia na MFFUK, Ke Karlovu 3, 121 16, Praha 2.

Preface

This Thesis mainly follows the author's work on the 3-years project of GAUK (Gran-tová Agentura Univerzity Karlovy) number 1968214: Studium oscila£ních proud¥ní

a kavitace v kapalném héliu vizualiza£ními metodami solved during his 4-year doc-toral study at the Faculty of Mathematics and Physics of the Charles University.The Thesis does not track the entire work performed by the author it rather at-tempts to provide a comprehensive report on experimental studies of mechanicallydriven oscillatory ows in both normal viscous liquid 4He and in superuid 4He,focusing on the similarities and dierences between the classical and quantum ows.It is motivated, among others, by questions if macroscopic vortices can exist in thewake, if the mechanically driven coow of both components mimics the classicalturbulence up to such a level that they are indistinguishable and additionally, ifhigher order eects such as viscous streaming can take place in superows.

The presented work is a result of the joint eort of the entire research groupworking on visualization of various types of ows of all forms of cryogenic helium.The author was involved mostly in the technical aspects of the experiments: prepa-ration of the electrical circuits measuring the supporting quantities, programmingthe data acquisition and experiment control in the LabView programming interface.The actual measurements were performed by all members of the research group inthe laboratory under leadership of Marco La Mantia. The following image dataprocessing were performed by various people including the author. The author de-veloped a program used for statistical data analysis, advanced ltering, smoothingand trajectory repairing and he performed the analysis of the presented experiments.Several methods have been suggested and introduced by him (phase averaging, vor-tex proles, PTV-based pseudovorticity and eective bubble radii). The publishedarticles, although on several of them the author of this thesis is written as the rstauthor, were mainly written by Marco La Mantia and Ladislav Skrbek, the authorusually initiated them, took part in designing and preparation of the experiment,data acquisition, subsequent analyzes and discussions and supplied the gures.

1

1. Introduction

Depending on the length-scale, matter can be described by various ways, highlightingthe lack of detailed understanding of this very important part of Nature and hencethe need of further research looking not only for the answers, but also for a newexciting questions. This section freely follows the excellent article of Barenghi,L'vov and Roche [1].

1. Quantum-mechanical description of a macroscopic wave of weakly

interacting bosonic gas1 This approach is represented by the so calledGross-Pitaevskii equation [2] for the macroscopic order parameter Ψ (~x, t).This regime is studied mostly numerically, see e. g. [3], experimental ac-cess to these length-scales is limited only for trapped atomic condensates [4].Proukakis [5] used phenomenological dissipative enhancement of GPE to studythe Kolmogorov spectrum [6, 7] showing that the dissipative GPE can capturekey features relevant to the turbulence [8].

2. Quantized vortices and vortex tangle The quantization of vorticity inBose-Einstein condensate suggests the existence of thin line singularities inthe order parameter Ψ the quantized vortices, whose dynamics is describedby the Biot-Savart Law [9]. Quantized vortices can be directly visualizede. g. by using small particles, see [10] or our results [11]. It is possible todecompose the vortex tangle into the polarized part of vortex line density L‖and the random part L× [12], where just the L‖ seems to be responsible forthe k−5/3 scaling [1], while L× exhibits a dierent k−1 scaling [13].

3. Two-uid model At larger temperatures and/or length-scales, the coarse-grained look on phonons is smoothed creating the so called normal component

of He II described as a continuous uid with non-zero viscosity coexistingwith the superuid component. These two component can move more-or-lessindependently, but their interaction is realized via the quantized vortices, whichis, in the coarse-grained point of view, represented by the mutual friction force,which, at larger length-scales, locks both components together [14]. Indeed, aclear k−5/3 spectrum for both uid components at large scales is found at alltemperatures [1].

To summarize, turbulence in He II can be easier to understand, in such a way thatthe quantized vortices, which the turbulence is built of, are clearly dened objectswith constant circulation and quite simple rules of behavior. On the other hand,at stronger forcing, both components become turbulent inuencing each other viathe still mysterious mutual friction force resulting into one of the most challengingobjects of study ever.

1Note that liquid helium is not weakly interacting gas.

2

2. Experimental setup

Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) arestandard methods in classical hydrodynamic research based on suspending small

particles into the uid and believing that these particles follow the motion of theuid. PIV can estimate the uid velocity in a section of the ow eld, by assuminga single, smoothly varying velocity eld, i. e. the Eulerian point of view is nativefor this technique. PTV allows the measurement of Lagrangean quantities [15], i. e.,the velocity and its derivatives [16], along the individual trajectories of the trackedparticles.

For this purpose we use solid hydrogen and deuterium particles, which are gen-erally of sizes of few µm. To obtain these particles we inject a mixture of helium,hydrogen and/or deuterium gases into the bath at a pressure larger than that ofthe bath, typically 1.5 − 2.0 bar, see gure 2.1. The density of solid hydrogen is88 kg/m3 [17], which is less than that of liquid helium (146 kg/m3 at temperaturesbetween 1.0 and 2.5 K), and solid deuterium has larger density 200 kg/m3 [17].

In the case of He II, which is a superuid quantum liquid containing quantizedvortices and described by the two-uid model, the situation is more complicated [18].In the rst order approach, the particle moves with the viscous normal component,while the superuid component aects the particle motion only via its shear due tothe D'Alembert paradox. Additionally, the particle can be trapped onto a quantizedvortex [19], which allows direct vortex visualization [10], see gure 2.2.

Particles are illuminated by a continuous wave solid state laser with tunablepower RayPower 5000 whose beam is defocussed by using a cylindrical diverginglens into a sheet of approximately 1 mm thickness. The scene is captured by CMOScamera Phantom v12.1 with framerate up to 6273 Hz and resolution 1280× 800 pixof 8-bit gray scale. The camera is situated perpendicularly to the illuminated planeand focused on the latter by using an appropriate telemetrical macro lens, whichcan be enhanced by using a 2× extender. The eld of view has dimensions of about13 × 8.2 mm on the focus plane (these dimensions vary slightly, depending on theactual focusing in the experiment).

The Data acquisition procedure was as follows: rst, there is an injection ofpure helium gas from room temperature under pressure around 1.5 bar by using ourseeding system, gure 2.1. This injection stirs up the deuterium particles seededat the bottom of the cryostat and allows us to perform the measurements. On theother hand, such injection creates a strong residual ow, therefore we have to waituntil this ow is dampened, or, at least, suppressed to the level appreciably weakerthan the studied ow. After this injection we start the driving of the ow by thequartz tuning fork or by the moving obstacle and some reasonable time after theinjection we start acquiring images by the camera at the chosen frequency, see table3.1 for details. The laser illuminating the particles (and the fork as well) is operatedin a pulse regime at the same frequency as the camera, the duration of one shot is∼ 0.4 ms.

3

HeH2D2

V2cleaning pump

HD

fast

PC

-con

trol

ed n

eedl

e va

lve

regu

lation

val

ve

cryo

stat

V1

Figure 2.1: Sketch of the seeding system. The small volume V1 is lled with deuteriumD2 (or with H2 or HD, but these gases are not used in the presented experiments) atchosen pressure (usually 0.75 bar), then it is moved into the 50× bigger volume V2, wherethe deuterium is diluted by helium, in the molar ratio ≈ 1 : 100, i. e. the pressure hasto be 2× higher than in the small volume. This mixture is injected into the cryostat viaregulation valve and fast computer-controlled needle valve with pulse duration of 100 ms.

5

[mm]0 1 2 3

6

0

1

2

3

4

[mm]

Figure 2.2: Left: the photograph of the transparent cylinder used in the experiment[11] taken by a digital camera, the length scale is approximately valid in the focusedplane, temperature 2.16 K, H2 particles, note the laments. Right: negative of image of alament structure, which might be a quantized vortex, temperature 2.15 K, H2 particles.

4

2.1 PTV-based pseudovorticity

The spatial distribution of vorticity ω = ∇ × ~v and its magnitude can be veryinteresting for describing the studied ow, especially, if we believe, there would bemacroscopic vortices. Unfortunately, due to the character of our data, which are inthe form of individual particle trajectories, we are not able to calculate the vorticityω directly, as we do not know the uid velocity everywhere at any time, but onlyalong the trajectories and at times of particle passing. In addition, there is theproblem with the two-uid character of the He II ows.

Hence we introduce an experimentally observable approximation. We do notventure to identify it with the proper vorticity, therefore we call it PTV-based pseu-

dovorticity and we denote it ϑ:

ϑ (~r; t) =

⟨(~ri − ~r)× ~vi|~ri − ~r|2

⟩|t−ti|<T,|~ri−~r|<Rm

, (2.1)

where ~r and t denote the probed point position and time, ~ri and ti stand for theposition and time of i-th trajectory point, while ~vi is its velocity. The vector product× results into a scalar as our vectors are only two-dimensional projections or cuts ofthe real ones. The averaging is calculated only over such points, which are locatedinside a circular area of chosen radius Rm and centered at the probed point ~r. T isthe duration of a time window centered at the probed time t.

In the experiment with large slowly oscillating obstacle, chapter 4, we have useda phase of the obstacle motion instead of time, while in the streaming experiment,chapter 5, we did not care about time anymore, as the time resolution representedby the camera frame-rate was ∼ 10× longer than the oscillation period and thereforewe observed only the steady component of the particle motion.

Pseudovorticity ϑ converges to a regular vorticity ω under three following con-ditions:

1. dense sampling (as dense as Q samples R),

2. ~vi being samples of a continuous eld,

3. Rm → 0.

In practice, none of these conditions is fullled. The sampling is limited by thenumber of tracked particles, which, as we believe, is better to keep lower, because,in the opposite case, serious problems can arise with the trajectory connections.Therefore, in the nal data, there is lot of white places , whose size limits theintegration area size Rm. The violation of the second condition is connected notonly with the two-uid nature of He II, but more signicantly with the behaviorof non-uid particles carried by the ow. But, in the case of classical liquid, wecould, at least, except that for small buoyant particles their velocities would beclose enough to the continuous ow velocity eld. Hence, in the case of He II, wehave to use as an assumption one of our main long-term result, namely, that is thatHe II behaves as a single uid in all cases, where is no reason for dierent forcing ofindividual components (i. e. by viscosity near walls or by heat gradients), and thatthe two-uid behavior is displayed in bulk co-ow only at small length-scales.

5

3. Table of parameters

Table 3.1: Comparison of experimental conditions of three main experiments performedby the author. The viscous penetration depth δ =

√2ν/Ω, where we set ν = κ/6 according

to [20] and Ω = 2πf , where f is frequency. The fork constant is calculated accordingto [21]. The velocity is the root-mean-square-value, therefore, in the case of the Obstacleexperiment, vmax = A · Ωmax/

√2, when the fork is used, this velocity is measured via

the electrical current, v = I/a, hence it is naturally rms; for the sake of clarity the peakmaximum velocity is added too.

· unit Obstacle Streaming CavitationTemperature range T [K] 1.3− 2.18 1.3− 2.22 2.14− 2.24Tracer particles D2 D2 ·Camera frame-rate [Hz] 100 800 400Scale factor [µm/pix] 27.4 10.33 10.56Obstacle shape @A u u visible size [mm] 10× 3 0.9× 0.4 2.2× 19.7 third size [mm] 30 9.0 0.8Obstacle frequency f [Hz] 0.05− 1.25 8298− 8303 4186Amplitude A [mm] 5; 10 up to 0.01 up to 0.05Maximum rms veloc-ity vmax

[mm/s] 55.5 338 1075

Maximum velocityvmax

[mm/s] 78 478 1520

Velocity changed by frequency force forceFork constant a [C/m] · 1.1 · 10−6 3.4 · 10−5

Viscous penetrationdepth δ

[µm] 64− 730 0.8 1.1

The discussed here oscillatory ows have to be characterized by a pair of dimen-sionless parameters: Strouhal number dened as

St =sΩ

v=

s

A, (3.1)

where s is a characteristic size, Ω is the oscillation angular frequency and v is acharacteristic velocity, which is, in the case of ow forced by an oscillating obstacle,v = ΩA, where A is the oscillation amplitude. St was originally dedicated for de-scribing a little bit dierent ow problem: the self-induced oscillations of an obstaclein a stream, therefore, the angular frequency of oscillations was a measurable result,not a control parameter.

Second parameter to make a pair is the Stokes number β dened as

β =fs2

ν, (3.2)

6

10-1

100

101

102

103

104

105

106

107

108

10-1 100 101 102 103 104

Stok

es, β

Strouhal, St

A = 5 mm

A = 10 mm

Streaming

no boundary layer

1 boundary layer

2 boundary layers

laminar

wake

Figure 3.1: Phase diagram of oscillatory ows in the St × β space, Strouhal number

St = s/A, 3.1 and Stokes number β = fs2/ν, 3.2. Regions are denoted according to [22].Green squares and blue circles denote the moving obstacle experiment, see chapter 4, redsquares represent the streaming experiment, see chapter 5. Diagonal line corresponds tostreaming Reynolds number Res ∼ 1 separating region with a single boundary layer fromthat with two boundary layers.

where f is the oscillation frequency and ν is the kinematic viscosity.Figure 3.1 shows two of our visualization experiments in the phase diagram of

St× β together with the regions known from classical experimental and theoreticalworks as summarized in [22] (which presents this diagram in the St×Re axes).

7

4. Oscillating obstacle

The behavior of macroscopic, millimeter-sized vortical structures, generated by abody of rectangular shape oscillating in liquid 4He, has been studied by visual-izing the dynamics of micrometer-sized particles seeding the uid. An obstacleof rectangular cross section (3× 10 mm) performed quasiharmonic oscillations inthe liquid, at various frequencies (0.05− 1.25 Hz) and amplitudes (5 and10 mm),at temperatures between about 1.2 and 3 K, resulting in Reynolds numbers up to105. Consequently, in the present experiments, a direct comparison between similar,mechanically driven ows of He I and He II was carried out.

First of all, our visualization study conrms the existence of large vortical struc-tures shed by the oscillating body, which appear similar in viscous He I and su-peruid He II. Although this result is not surprising (for example, visualization ofmacroscopic vortex rings was reported in [23] and in [24]), our investigation rep-resents, to the best of our knowledge, the rst systematic visualization study ofquantum ows due to an oscillating object. We note, additionally, that in viscousuids vortices are shed by accelerating plates at similar Re as the minimum Reynoldsnumbers investigated here, as reported by Taneda and Honji [25].

At Reynolds numbers lower than approximately 104, a noticeable dierence be-tween the investigated He I and He II ows is observed, see gure 4.2. More pre-cisely, after thorough studies of the quality of the data sets and of the robustness ofthe processing procedure, we found that the probed length-scales are, at these lowReynolds number, smaller than the quantum (dissipative) scale of the ow, deter-mined, in He II, by the average distance between quantized vortices ` and, in He I,by the Kolmogorov dissipative scale η. For Reynolds numbers larger than 104, theeect vanishes and He II behaves similarly to He I, as it is expected for mechanicallydriven coows of superuid 4He at length scales exceeding the average distance be-tween quantized vortices. This suggests that, similarly to thermal counterow [27],both viscous and quantum features can be observed in mechanically driven ows ofHe II, depending on the length scales at which the quantum ow is probed.

The importance of the length-scale ratio R is additionaly illustrated by gure4.3, which shows the normalized distribution of horizontal velocity component forselected movies having dierent R both in He I and in He II. We can easily noticethat all distributions with R > 1 are more or less of a Gaussian shape1, while thosewith R < 1 and in He II are wider, having tails comparable with the v−3 shapeobserved in the thermal counterow experiment, see our publication [28] and theauthor's diploma thesis [29].

Note, however, that the observed eect might also be inuenced by the muchlarger thermal conductivity of He II, compared to that of He I, leading to dierentdissipation mechanisms of the parasitic energy input. Additionally, the obtained〈ϑ2〉 values are aected by the data scatter, which cannot be neglected here. Inorder to clarify the reported behavior, future studies should therefore be devoted toincrease the data set size, as this will likely lead to less uncertainty in the evaluationof various physical quantities of interest.

1Intermittent eects cannot be observable with our statistics.

8

φ

φ = 315°

φ = 0°

φ = 270°φ = 225°

φ = 180°

φ = 135°

0 5 10 15 20

φ = 90°

0 5 10 15 20

φ = 45°

0 5 10 15 20[mm]

θ [s-1]

0

3.5

-3.5-2

2

Figure 4.1: PTV-based pseudovorticity ϑ at dierent phases of the motion in He II attemperature 1.24 K, frequency 0.5 Hz and amplitude 5 mm. The displayed value of theparameter (equation 2.1) Rm = 5.0 mm has been chosen, which corresponds to the half ofthe width of the obstacle, and it is consistent with the choice of Rm in the study of viscousstreaming, chapter 5, in the sense that the more important side is that pushing the uid.Similarly, the width of the phase window has been chosen to 15 as compromise betweenthe preciseness and the statistics. The rectangles represent the actual size of the obstacleand the mask size, respectively. The physical dimension of each panel is 20× 16 mm, oneunit on both axes corresponds to 1 mm. Note also that the motion is not strictly harmonic,see sketch of mechanism in the center.

9

0.1

1

10

1000 10000 100000

He I, N < 40kHe I, N > 40k

He II, N < 40kHe II, N > 40k

θ2 [s-2

]

Re

Figure 4.2: Plot of the dependence of⟨ϑ2⟩on Reynolds number Re, which is in the

case of He II (blue circles) based on the measured eective bulk viscosity tabulated in[26]. Small symbols represent such datasets with less than 40 000 trajectory points (afterltering), while the bigger symbols represent those with more than 40 000 points. Redsquares are the data obtained in He I.

10-3

10-2

10-1

100

101

-15 -10 -5 0 5 10 15

S, 0.05, R 0.07S, 0.1, R 0.14S, 0.2, R 0.36S, 0.5, R 1.3S, 1.0, R 2.8

N, 0.05, R 0.12N, 0.1, R 0.24N, 0.2, R 0.43N, 0.5, R 1.3N, 1.0, R 2.7

ux

PD

F

Figure 4.3: The Probability density function, PDF in percent, of the normalized hori-zontal velocity ux = vx−〈vx〉

σ[vx] of selected datasets with amplitude A = 5 mm and dierentfrequencies (the rst number in the legend) in He II corresponding to dierent ratios R ofthe probed length-scale and the ow small length-scale; current R is the second numberin the legend. Thin black line represents the distribution u−3

x characteristic for particlestrapped on a quantized vortex orbiting around another straight vortex.

10

5. Streaming ow

By using the particle tracking velocimetry technique, we have observed streamingpatterns in the bulk He I and He II surrounding an quartz tuning fork oscillatingwith frequency 8.3 kHz and the rms1 velocity of the fork prongs calculated from thecurrent through the fork reached 340 mm/s (in He II). The movement of micrometer-sized solid D2 particles has been captured at frequency 800 Hz, i. e. 10× lower thanthe oscillation frequency, therefore only the steady component of the motion wasobservable for us.

At the experimentally probed length scales, the observed ow patterns are similarin both uids in the superuid He II, as well as in classical viscous He I and arefound to be consistent with those described in [31].

We have observed 6 cells and 3 outcoming jets around each fork prong, see gure5.2b and c. This can be explained by the large curvature of the prong corners, whichgenerates the steady streaming due to the Reynolds stresses with an outcoming jetpointing along the corner axis. But, this inner (or primary) streaming pattern isinvisible for us as their estimated size [32] δDC = St

√νΩ≈ 26µm for He II and 58µm

for He I is too small (our pixel resolution is 10.33µm, size of the observed cells isaround ∼ 0.7 mm). Therefore we see the complementary incoming jet in the outerstreaming pattern and the complementary outcoming jets located at the centers ofthe sides generated simply due to the mass conservation. As the outcoming jets arealways stronger than the (wider) incoming ones, the rst type dominate the ow,see gure 5.2c.

As our experimental length scale is signicantly larger than the viscous pene-tration depth, any ows inside the Stokes boundary layer are currently invisible tous. At the probed length scale, larger than the quantum length scale the meandistance between quantized vortices, we do not observe any appreciable inuence ofthe quantization of vorticity in He II. We argue that this can be understood basedon the notion that the mechanically driven turbulent coow of He II investigatedhere can be described as if it were a ow of a single quasiclassical uid characterizedby an eective kinematic viscosity, due to the action of the mutual friction forcethat, at scales equal to or exceeding the probed one, couples the motions of the twocomponents of He II [33].

Last but not least, there are practical aspects to our investigation. Althoughquartz tuning forks are employed as convenient probes in both 4He [34] and 3He [35]liquids, the very existence of a streaming ow, generated by their fast oscillations,its extent, and possible inuence on their performance has not yet been investigatedin detail. In both normal He I and in superuid He II we show that, in the range ofinvestigated parameters, a vibrating quartz tuning fork does appreciably inuenceits uid surroundings, at least up to a distance of the order of its cross sectionsize. Future studies could then focus on quantifying how the performance of thisvaluable experimental tool as well as similar oscillating objects used in superuidhydrodynamics is aected by the generated streaming ows, existence of which ourstudy demonstrates.

1Root-Mean-Square,√

2× smaller than the peak value.

11

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

0 2010

|v| [mm/s]

5 15

θ [s-1]0 69.3-69.3 -20 20 40 60-40-60

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

θ [s-1]0 20.8-20.8 -5 5-10-15 1510

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

θ [s-1]0 34.6-34.6 -10 10 20 30-20-30

0 4020 3010

|v| [mm/s]

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

0 1410

|v| [mm/s]

0

1

2

3

4

0 1 2 3 4 5 6 [mm]

He I210 mm/s

He II180 mm/s

He II340 mm/s

Figure 5.1: Left: Trajectories obtained in He II colored via the magnitude of velocity.Only trajectories longer than 10 points are displayed, the trajectories are smoothed byusing the Kolmogorov-Zurbenko linear lter [30] over 3 neighbors and in 3 iterations.Right: PTV-based pseudovorticity ϑ calculated from the trajectory data left, the value ofparameter Rm = 0.4 mm, which corresponds to the thickness of the tuning fork, i. e. itsside which pushes the uid. Top: Data taken in He I, temperature 2.22 K, drive 41µN,achieved fork prong rms velocity 210 mm/s. Middle: the merge of He II data taken atthree dierent temperatures at the lower drive 12µN, velocity 180 mm/s. Bottom: themerge of He II, drive 41µN, velocity 340 mm/s. The gray rectangles show the position ofthe fork prongs of dimensions 0.9× 0.4 mm.

12

inner cell

outer cell

b

1. 2.

3?

4?

5.6.

7.

8.

45°

strong outcoming jets

a

c weaker suction jets

Figure 5.2: Panel Panel a: schematic view of the streaming ow features in the proximityof the fork top left corner (the fork oscillates in the horizontal direction). Locally the uidows past the corner symmetrically with respect to its axis. Panel b: The closeup ofthe Lagrangian pseudovorticity map calculated from all the data obtained in He II, withdriving force F = 41µN. Panel c: corresponding particle trajectories, color coded byvelocity magnitude, see the bottom left panel of gure 5.1. The strong outgoing jets areshown with black arrows, as the weaker incoming jets, orientated along the axes of theprong corners. Each corner generates one incoming jet in the outer streaming pattern and2 outer streaming cells (plus 2 invisible inner cells), in total 8 cells for one prong, althoughthe ones numbered 3 and 4 are barely visible due to the lack of trajectory points in theshadow area.

13

6. Cavitation

The cavitation was produced by decreasing the local pressure by velocity enhance-ment, by passing a moving object a prong of the quartz tuning fork. The cavita-tion is detected optically by camera and illumination laser. The used fork has lengthL = 19.7 mm, thickness T = 0.8 mm and width W = 2.2 mm, the main resonantfrequency is 4186 Hz and the fork constant a = 3.4 · 10−5 C/m.

The velocity of the fork prongs have been measured independently by two ways:rst the standard electrical measurement as described in article [21], second a directoptical measurement of elongation of the projection of small dust particles attachedon the fork surface as visible in gure 6.1. The correlation coecient is 84% withsystematic dependence vel ∼ 0.74vopt. We refrain from declaring which method isbetter.

By using the fast optical camera we have directly observed heterogeneous cav-itation in ow of He I and He II, on both sides of the λ-transition. Typically acluster of small bubbles is produced rather than one bigger hemispherical bubble(gure 6.2), the position, size and shape of such a cluster changes from frame toframe. The produced bubbles are at rst look similar in He I and in He II, however,cavitation threshold is lower in He I than in He II.

Once nucleated, bubbles live longer in He I than in He II and can rise up in thebulk from the place of their nucleation, see gure 6.3. Additionally, under otherwiseidentical conditions the nucleated bubbles brake the motion of the fork's prongsmore in He II than in He I, see the right panel of gure 6.4.

010

020

030

040

050

060

070

0

0 200 400 600 800 1000 1200vopt [mm/s]

vel [mm/s]

cavitation He II

cavitation He I

no-cavitation

y [mm]

0 1 2 3 4 5 60

1

2

3

4

5

x [mm]

Figure 6.1: Left: Photograph of the fork illuminated by the laser sheet from the side,when the fork moves, which elongates the surface features in the direction parallel withthe motion. Right: the correlation of the velocity measured by using the electrical currentthrough the fork vel and the velocity measured optically vopt. The open symbols representcases when a bubble was observed in He I or II respectively, while crosses stand for thosewithout observable cavitation.

14

A

y [m

m]

0 1 2 3 4 5 60

1

2

3

4

5

x [mm]

B

y [m

m]

0 1 2 3 4 5 60

1

2

3

4

5

x [mm]

Figure 6.2: Fork under the same drive 129 Vrms in A: He II, rms velocity of the fork tips576 mm/s, and B in He I, rms velocity 528 mm/s.

A

y [m

m]

0 1 2 3 4 5 60

1

2

3

4

5

x [mm]

B

y [m

m]

0 1 2 3 4 5 60

1

2

3

4

5

x [mm]

Figure 6.3: In He I at highest drives the bubbles live longer than one fork cycle, thesebubbles then rise up. The B frame follows 10 ms after the A frame.

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6 7

v [m

m/s

]

F [mN]

Cavitation, He IICavitation, He I

No-cavitation

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0

100

200

300

400

500

600

700

800

r eff [µm

]

Fb [mN]

He IIHe I

Figure 6.4: Left: Measured velocity of prongs plotted versus applied force. The caseswhen the cavitation did not occur (represented by red crosses) approximately follow theexpected dependence FD ∼ v2 (plotted as a dotted line) valid for the turbulent regime.Right: Eective average radius reff , which the bubble would have, if it were spherical, asa function of the braking force Fb added to the hydrodynamic drag force.

15

7. Small-scale similarity in quantum

turbulence

In this chapter we use the experimental data recorded when studying the oscillatoryows of normal and superuid helium together with the data relevant to our visu-alization studies of thermal counterow characterized by the velocity of normal andsuperuid component vns = vn − vs = q/ρsST (q is the heatux, ρs superuid com-ponent density, S normal component entropy and T temperature) and we analyzethem in order to investigate the idea of small-scale similarity in quantum turbulence.We do not describe in any detail the history of measurements performed in thermalcounterow; the reader is directed to the published paper [36] which the author ofthe thesis coauthored.

The probed length-scale d in our experiment we dene as the mean distancetraveled by the particle between two consecutive particle positions (limited by thecamera frame rate). The relevant ow length scale in He II is the mean intervortexdistance ` estimated, in the case of thermal counterow, as ` ≈ γ−1 (T ) vns

−1, whereγ is known with sucient accuracy [19], while in the case of oscillatory ows we usethe analogy with the Kolmogorov dissipative scale η = ν1/2/ 〈ω2〉1/4 [6], where ν isthe uid kinematic viscosity and ω is the ow vorticity, which we ventured to replacewith ϑ. Then we dene the ratio R = d/` or d/η respectively between the lengthscale probed by the particles and that of the investigated ow.

Figures 7.1 and 4.3 (page 10) show that in He II the velocity PDF is Gaussianor power-law 1/v3 in respect to the ratio R independently on the imposed ow type.The same result is alternatively shown by the velocity atness1 F in gure 7.2, wherewe see that F converges to 3 for R → 1 in He II, while in He I it stays close to 3always.

Based on the analysis of the experimental data we have shown that, similarlyas in classical (viscous) uid turbulence, small-scale universality can be observedin quantum turbulence. Since postulated by Kolmogorov [6, 7], a large number ofrelated studies has been performed in the eld of classical turbulence, including therecent numerical investigations showing that small-scale universality holds alreadyin various ows of moderate Reynolds number [37].

In other words, the small-scale universality we observe can be seen as analogousto that reported to occur in viscous ows, but has also dierent features, as, to date,it is yet to be claried how the Kolmogorov length-scale η can be unequivocally linkedto the intervortex distance `, although it appears that, in most cases, ows of He IIat scales larger than ` behave similarly to viscous ows at scales larger than η.

Nevertheless, our statistical study of the motions of small particles in mechani-cally and thermally generated quantum ows of superuid 4He strongly suggests thatthe concept of small-scale universality is a fruitful one for the deeper understandingof the phenomenon of uid turbulence in general.

14th centered moment of the distribution of the particle velocities. Flatness of Gaussian distri-

bution is equal to 3, while it diverges for the 1/v3 distribution, hence, when calculated numerically

from the real data, it depends on the large magnitude rare events related to small-scales.

16

10-6

10-5

10-4

10-3

10-2

10-1

100

-20 -15 -10 -5 0 5 10 15 20

ux

R 0.09 (H2, t 25)R 0.14 (D2, t 25)R 2.8 (H2, t 50)

R 8.7 (D2, t 250)

Figure 7.1: Probability density function (PDF) of the normalized horizontal velocityux = (vx−〈v〉)/σ [v] of selected measurement of thermal counterow obtained at a relativecounterow velocity of the normal and the superuid component vns = 6.8 mm/s, t is thetime step in ms, the rst data set is obtained with hydrogen particles, the second withdeuterium ones. Compare with the gure 4.3, page 10, where is plotted PDF for oscillatingobstacle experiment, time step there is 10 ms.

0

10

20

30

40

50

0.01 0.1 1 10 100

Fla

tnes

s

R

counterflow, H2counterflow, D2osc. obst., He IIosc. obst., He I

Figure 7.2: Flatness of the normalized horizontal velocity component ux = (vx−〈v〉)/σ [v]in dependence on the ratio R. Horizontal line corresponds to the atness of Gaussian distri-bution. In the case of the counterow data, the ratio R is changed articially by increasingthe time step t [38], while the dierent R in the oscillating obstacle data corresponds todierent experimental conditions.

17

8. Conclusion

The Thesis represents a systematic study of various oscillatory ows of liquid heliumperformed in the Prague Visualization Laboratory - the rst European laboratorycapable of visualizing cryogenic ows of both classical viscous He I and superuidHe II using the methods of particle image velocimetry (PIV) and particle trackingvelocimetry (PTV) utilizing micrometer-sized seeding particles of frozen hydrogenand deuterium. In order to keep the Thesis self contained and comprehensive, theauthor decided to restrict its content to mechanically generated oscillatory owsof cryogenic helium and to a detailed comparison of classical and quantum cases,which is at the same time, the topic of the author's the 3-year GAUK (GrantováAgentura Univerzity Karlovy) project number 1968214 Studium oscila£ních prou-

d¥ní a kavitace v kapalném héliu vizualiza£ními metodami. Additionally, the authorcontributed to studies of the acceleration of small particles in quantum turbulenceas well as visualization studies of steady counterows past a circular cylinder. Heis a coauthor of several papers published in impacted scientic journals - PhysicalReview B, Journal of Fluid Mechanics: Rapid communication and Journal of LowTemperature Physics, see the List of Publications.

The main results described in this Thesis can be characterized as follows:

1. We have observed that, in the case of mechanically driven coow of He II,macroscopic vortical structures form in the wake of an oscillating obstacle ofthe shape of a prism. These structures are similar to those observed in He I,which is a classical viscous liquid. Their size is of the order of the obstaclesize and, additionally, their structure exhibits a turbulent vortex core with aninviscid envelope, similar as in classical liquids [39]. (Chapter 4)

2. The author has introduced the PTV based pseudovorticity, a useful approxi-mation of vorticity calculated from the particle trajectory data, which on onehand, highlights the classical-like vortical structures and, on the other hand,it shows dierent scaling with the Reynolds number, which is dependent onthe ratio of the characteristic lengths probed by the PTV visualization andthe characteristic small length-scales of classical and quantum ows, i. e. theKolmogorov length η and the quantum length scale the intervortex distance`. (Chapter 4)

3. We have observed the viscous streaming (a well known ow in classical hydro-dynamics usually connected with the Reynolds stresses and the curvature ofthe stream) generated in both He I and He II by a vibrating quartz tuningfork, which is a widely used tool for studying quantum turbulence. We haveobserved streaming cells of the order of the fork prong size situated betweenthe corner axes and the side axes; according to their orientation, we have iden-tied them as outer streaming cells, which are classically explained via viscousinduction. (Chapter 5)

4. We have observed cavitation near an oscillating quartz tuning fork in He I andin He II just below the λ-transition, conrming previous results which show

18

the critical cavitation velocity at the λ-transition steeply increases. The newresult is that this heterogeneous cavitation produces a cluster of small bubblesand that these bubbles brake the fork motion more in He II than in He I.(Chapter 6)

5. We have found that in all studied quantum ows of He II, the statisticaldistribution of velocity exhibits a strong dependence on the ratio between thelength-scale probed by the particles, and the ow characteristic length-scale,which is the Kolmogorov length in He I or the quantum length scale the meanintervortex distance in He II. When the experimentally probed length-scale issmaller than the intervortex distance, the probability density distributions ofturbulent velocity contain power-law tails, regardless of the type of large scalemean ow imposed, suggesting small scale similarity in quantum turbulence.At larger probed length-scales, as well as in He I, we have observed classicalGaussian-like distributions. (Chapter 7)

To summarize, the Thesis contributes to the general notion that superuid He II,which is a quantum liquid, behaves as a classical viscous liquid at length-scales largerthan the quantum length scale mean intervortex distance when the individualquantized vortices can be considered as a continuum, while at smaller length-scales,the quantum nature of such turbulence emerges, both in thermally and mechanicallydriven ows. It is shown that both these regimes are experimentally accessible andcan be studied by visualizing suitable tracer particles, although the details of theirinteraction with the normal and superuid components of He II are still poorlyunderstood.

8.1 List of publications

1. M. La Mantia, D. Duda, M. Rotter, and L. Skrbek. Lagrangian accelerationsof particles in superuid turbulence, Journal of Fluid Mechanics, 717 R9, DOI:10.1017/jfm.2013.31, 2013.

2. M. La Mantia, D. Duda, M. Rotter, and L. Skrbek. Lagrangian velocity distributionsin thermal counterow of superuid He-4. Proc. of Conf. on Experimental Fluid

Mechanics (EFM 12), Hradec Králové, NOV 20-23 2012, Book Series: EPJ Web of

Conferences, 45, Article Number UNSP 01005, DOI: 10.1051/epjconf/20134501005,2013.

3. M. La Mantia, D. Duda, M. Rotter, and L. Skrbek. On the Visualization of ThermalCounterow of He II Past a Circular Cylinder. Journal of Low Temp. Physics, 175No1/2 331-338, DOI: 10.1007/s10909-013-0961-z, 2014.

4. M. La Mantia, D. Duda, M. Rotter, and L. Skrbek. Vizualizace obtékání oscilujícíp°ekáºky supratekutým héliem. Proceedings of the 28th Symposium on Anemometry

(Edited by Z. Chára and L. Klaboch, Institute of Hydrodynamics of the Academy ofSciences of the Czech Republic, Prague), pp. 7-13, ISBN 978-80-87117-11-8, 2014.

5. D. Duda, M. La Mantia, and L. Skrbek. Automatized Data Processing in SuperuidFlow Visualization. WDS'14 Proceedings of Contributed Papers - Physics (MAT-FYZPRESS), pp. 73-78, ISBN 978-80-7378-276-4, 2014.

19

6. D. Duda, P. van£ara, M. La Mantia, M. Rotter, and L. Skrbek. Visualization ofviscous and quantum ows of liquid 4He due to an oscillating cylinder of rectangularcross section. Phys. Rev. B 92, 064519, DOI: 10.1103/PhysRevB.92.064519, 2015.

7. D. Duda, P. van£ara, M. La Mantia, M. Rotter, and L. Skrbek. Makroskopické víryv supratekutém héliu. Proceedings of the 29th Symposium on Anemometry (Editedby Z. Chára and L. Klaboch, Institute of Hydrodynamics of the Academy of Sciencesof the Czech Republic, Prague), pp. 7-13, ISBN: 978-80-87117-13-2, 2015.

8. M. La Mantia, P. van£ara, D. Duda, and L. Skrbek. Small-scale universalityof particle dynamics in quantum turbulence. Phys. Rev. B, 94, 184512, DOI:10.1103/PhysRevB.94.184512, 2016.

9. D. Duda, P. van£ara, M. La Mantia, D. Schmoranzer, and L. Skrbek. Pozorováníkavita£ních bublin v kapalném héliu. 30th Symposium on Anemometry proceedings

(Edited by Z. Chára and L. Klaboch, Institute of Hydrodynamics of the Academy ofSciences of the Czech Republic, Prague), pp. 7-14, ISBN: 978-80-87117-14-9, 2016.

10. D. Duda, P. van£ara, M. La Mantia, M. Rotter, D. Schmoranzer, O. Kolosov,and L. Skrbek. Cavitation bubbles generated by vibrating quartz tuning fork inliquid 4He close to the λ-transition. J. Low Temp. Phys., 187 pp. 376-382, DOI:10.1007/s10909-016-1684-8, 2017.

11. D. Duda, M. La Mantia, and L. Skrbek. Visualization of Large-Scale Flow Dueto an Oscillating Tuning Fork in Normal and Superuid Helium. Proceedings Topi-cal Problems of Fluid Mechanics 2017 (Edited by David imurda and Tomá² Bodnár,Prague), pp. 103-112, ISBN: 978-80-87012-61-1 (Print), DOI: 10.14311/TPFM.2017.014,2017

12. D. Duda, P. van£ara, P. Hrubcová, M. La Mantia, and L. Skrbek. Experimentalnístudium stacionárního pohybu £astic v supratekutém héliu v okolí rychle kmitajícíp°ekáºky vliv velikosti oblasti. Proceedings of the 31th Symposium on Anemometry

(Edited by Z. Chára and L. Klaboch, Institute of Hydrodynamics of the Academy ofSciences of the Czech Republic, Prague), pp. 7-13, ISBN:978-80-87117-16-3, 2017.

13. D. Duda, M. La Mantia, and L. Skrbek, Streaming ow due to a quartz tuning forkoscillating in normal and superuid 4He. Phys. Rev. B, 2017, in print

References

[1] C. F. Barenghi, V. S. L'vov, and P. E. Roche. Experimental, numerical, and analyticalvelocity spectra in turbulent quantum uid. P. Natl. Acad. Sci. USA, 111:46834690,2014.

[2] L. Pitaevskii and S. Stringari. Bose-Einstein condensation, volume 116 of Interna-tional series on monographs on physics. Oxford Univ Press, 2003.

[3] N. G. Berlo, M. Brachet, and N. P. Proukakis. Modeling quantum uid dynamics atnonzero temperatures. P. Natl. Acad. Sci. USA, 111:46754682, 2014.

[4] A. C. White, B. P. Anderson, and V. S. Bagnato. Vortices and turbulence in trappedatomic condensates. P. Natl. Acad. Sci. USA, 111:47194726, 2014.

[5] N. P. Proukakis and B. Jackson. Finite temperature models of bose-einstein conden-staion. J Phys B, 41(20):203002, 2008.

20

[6] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous uidfor very large reynolds numbers. Dokl. Akad. Nauk. SSSR, 30:299, 1941.

[7] A. N. Kolmogorov. Dissipation of energy in the locally isotropic turbulence. Dokl.

Akad. Nauk. SSSR, 32:19, 1941.[8] L. Skrbek and K. R. Sreenivasan. Developed quantum turbulence and its decay. Phys.

Fluids, 24(1), 2012.[9] R. Hänninen and A. W. Baggaley. Vortex lament method as a tool for computational

visualization of quantum turbulence. P. Natl. Acad. Sci. USA, 111:46674674, 2014.[10] E. Fonda, D. P. Meichle, N. T. Ouellette, S. Hormoz, and D. P. Lathrop. Direct

observation of kelvin waves excited by quantized vortex reconnection. P. Natl. Acad.Sci. USA, 111:47074710, 2014.

[11] D. Duda, M. La Mantia, M. Rotter, and L. Skrbek. On the visualization of thermalcounterow of he ii past a circular cylinder. J. Low Temp. Phys., 175(1-2):331338,2014.

[12] A. W. Baggaley, J. Laurie, and C. F. Barenghi. Vortex-density uctuations, energyspectra, and vortical regions in superuid turbulence. Phys. Rev. Lett., 109(20), 2012.

[13] P. Walmsley, D. Zmeev, F. Pakpour, and A. Golov. Dynamics of quantum turbulenceof dierent spectra. P. Natl. Acad. Sci. USA, 111:46914698, 2014.

[14] P. Roche, C. F. Barenghi, and E. Leveque. Quantum turbulence at nite temperature:The two-uids cascade. Europhys. Lett., 87(5), 2009.

[15] P. K. Yeung. Lagrangian investigations of turbulence. Annu Rev. Fluid Mech., 34:115142, 2002.

[16] E. Lévêque and A. Nasso. Introduction of longitudinal and transverse lagrangian ve-locity increments in homogenous and isotropic turbulence. Europhys. Lett., 108:54004,2014.

[17] S. C. Bates. Compilation of the engineering properties of solid hydrogen. Glastonbury,2002.

[18] W. Guo, M. La Mantia, D. P. Lathrop, and S. W. Van Sciver. Visualization of two-uid ows of superuid helium-4. P. Natl. Acad. Sci. USA, 111:46534658, 2014.

[19] Y. A. Sergeev, C. F. Barenghi, and D. Kivotides. Motion of micron-size particles inturbulent helium ii. Phys. Rev. B, 74(184506), 2006.

[20] V. S. L'vov, L. Skrbek, and K. R. Sreenivasan. Viscosity of liquid 4He and quantumof circulation: Are they related? Phys. Fluids, 26(041703), 2014.

[21] R. Blaauwegeers, M. Blaºková, M. love£ko, V. B. Eltsov, R. de Graaf, J. Hosio,M. Krusius, D. Schmoranzer, W. Schoepe, L. Skrbek, P. Skyba, R. E. Solntsev, andD. E. Zmeev. Quartz Tuning Fork: Thermometer, Pressure- and Viscometer forHelium Liquids. J. Low Temp. Phys., 146:537, 2007.

[22] C. Y. Wang. On high-frequency oscillatory viscous ows. J.f Fluid Mech., 32(1):5568,1968.

[23] M. Murakami, Hanada M., and T. Yamazaki. Flow visualization study on large-scalevortex ring in he ii. Jpn. J. Appl. Phys., 26:107, 1987.

[24] G. Stamm, F. Bielert, W. Fiszdon, and J. Piechna. Counterow-induced macroscopicvortex rings in superuid helium: visualization and numerical simulation. Physica B,193(2):188 194, 1994.

[25] S. Taneda and H. Honji. Unsteady ow past a at plate normal to the direction ofmotion. J. Phys. Soc. Jpn, 30(1):262272, 1971.

[26] R. J. Donnelly and C. F. Barenghi. The Observed Properties of Liquid Helium at theSaturated Vapor Pressure. J. Phys. Chem. Ref. Data, 27:12171274, 1998.

[27] M. La Mantia and L. Skrbek. Quantum turbulence visualized by particle dynamics.

21

Phys. Rev. B, 90:014519, Jul 2014.[28] M. La Mantia, D. Duda, M. Rotter, and L. Skrbek. Velocity statistics in quantum

turbulence. In Procedia IUTAM, volume 9, pages 7985, 2013.[29] D. Duda. Visualization of selected ows of superuid helium using solid hydrogen

tracer particles. Faculty of Mathematics and Physics, Charles University, 2013.[30] I. Zurbenko. The spectral analysis of time series. North-Holland, 1986.[31] S. K. Kim and A. W. Troesch. Streaming ows generated by high-frequency small-

amplitude oscillations of arbitrarily shaped cylinders. Phys. Fluids, 1(6):975985,1989.

[32] K. Chong, S. D. Kelly, S. Smith, and J. D. Eldredge. Inertial particle trapping inviscous streaming. Phys. Fluids, 25(3):033602, 2013.

[33] W. F. Vinen and L. Skrbek. Quantum turbulence generated by oscillating structures.P. Natl. Acad. Sci. USA, 111:46994706, 2014.

[34] M. Blaºková, M. love£ko, V. B. Eltsov, E. Gaºo, R. de Graaf, J. Hosio, M. Krusius,D. Schmoranzer, W. Schoepe, L. Skrbek, P. Skyba, R. E. Solntsev, and Vinen W.F.Vibrating Quartz Fork A Tool for Cryogenic Helium Research. J. Low Temp. Phys.,150:525, 2008.

[35] M. love£ko, E. Gaºo, M. Kupka, M. Skyba, and P. Skyba. High quality tuning forksin superuid 3he-b below 200 µk. J. Low Temp. Phys., 162(5-6):669677, 2011.

[36] M. La Mantia, P. van£ara, D. Duda, and L. Skrbek. Small-scale universality ofparticle dynamics in quantum turbulence. Phys. Rev. B, 94(18), 2016.

[37] J. Schumacher, J. D. Scheel, D. Krasnov, D. A. Donzis, V. Yakhot, and K. R.Sreenivasan. Small-scale universality in uid turbulence. P. Natl. Acad. Sci. USA,111:1096110965, 2014.

[38] M. La Mantia and L. Skrbek. Quantum, or classical turbulence? Europhys. Lett.,105(4), 2014.

[39] E. Amromin. Analysis of vortex core in steady turbulent ow. Phys. Fluids,19(118108), 2007.

22