An exploration of thermal counterflow in He II using particle tracking velocimetry
aa r X i v : . [ c ond - m a t . o t h e r] M a y An exploration of thermal counterflow in He II using particletracking velocimetry
Brian Mastracci and Wei Guo ∗ National High Magnetic Field Laboratory,1800 E Paul Dirac Dr., Tallahassee, FL 32310, USA andDepartment of Mechanical Engineering, Florida State University,2525 Pottsdamer St., Tallahassee, FL 32310, USA (Dated: October 3, 2018) bstract Flow visualization using PIV (particle image velocimetry) and particularly PTV (particle track-ing velocimetry) has been applied to thermal counterflow in He II for nearly two decades now, butthe results remain difficult to interpret because tracer particle motion can be influenced by boththe normal fluid and superfluid components of He II as well as the quantized vortex tangle. Forinstance, in one early experiment it was observed (using PTV) that tracer particles move at thenormal fluid velocity v n , while in another it was observed (using PIV) that particles move at v n / v n (named “G2” for convenience) whilethe other is centered near v n / v n / ∗ Author to whom correspondence should be addressed. Electronic mail: [email protected] . INTRODUCTION Between the absolute zero temperature and the lambda transition temperature, T λ ≈ .
17 K, He exists in a superfluid phase called He II. Phenomenologically, a two fluid model,in which a superfluid condensate and normal fluid of thermal excitations coexist and arefully miscible [1, 2], provides a useful description of the mechanics of He II. The superfluidcomponent has temperature dependent density ρ s ( T ), and accounts for 100% of the bulkfluid density ρ at absolute zero ( ρ s ( T = 0) = ρ ). It is inviscid, has no entropy, and itscirculation is confined to quantized vortex lines, each with a single quantum of circulation κ ≈ − m /s about a core approximately ξ = 0 . ρ n ( T = T λ ) = ρ ) behaves in the same manneras a classical fluid, though interaction with quantized vortices gives rise to the non-classicalforce of mutual friction [4, 5]. Due to the strong influence of quantum effects, it has becomecommon to refer to turbulence in He II as quantum turbulence [3].Perhaps the most common manifestation of quantum turbulence is thermal counterflow,the mechanism by which He II transports thermal energy. In the presence of a heat source,the normal fluid carries entropy away from the source, with velocity v n , while the superfluidmoves toward it, with velocity v s , such that the overall mass flow is equal to zero, ρ n v n + ρ s v s = 0 [6]. It is important to recall that the two fluid components are fully miscible,such that these two velocity fields are interpenetrating; the counterflow currents are notspatially distinct as in a classical natural convection loop. In a simple one-dimensional case(e.g. counterflow through an insulated channel with a heater at one end), the normal fluidvelocity is related to the magnitude of the heat flux, q , as v n = qρsT (1)where T denotes the fluid temperature and s its specific entropy. The corresponding the-oretical superfluid velocity is easily obtained by conservation of mass, v s = − v n ρ n /ρ s . Asthe heat flux increases, the counterflow velocity, v ns = v n − v s , increases accordingly, andturbulence can develop in both fluid components [7, 8]. Superfluid turbulence manifests asa tangle of quantized vortex lines [5], with the line length per unit volume L approximatedby L = γ ( v ns − v ) (2)3here γ is an experimentally determined temperature dependent parameter [9–14] and v isa small critical counterflow velocity of approximately 2 mm/s [13–15].Flow visualization has become a popular tool for the study of thermal counterflow, withseveral different methods applied in the most recent two decades [16]. Visualization isaccomplished by first seeding the fluid with small tracer particles, illuminating them witha light source (typically a laser beam shaped into a thin sheet), and capturing images oftheir location in the moving fluid [17]. By analyzing particle displacement during the timeinterval between successive images, the flow velocity field can be deduced.Analysis of counterflow visualization is particularly challenging because of the numerousfactors that influence particle motion. Besides interactions with the normal fluid throughviscous forces and the superfluid through inertial and added mass effects [18], particlescan become trapped on quantized vortices [19, 20], which move at some velocity v L = v s . Furthermore, multiple numerical studies have shown that particles are not necessarilystationary on the vortices, but are thought to slide along the core due to a drag forceexerted by the normal fluid [21, 22]. A concrete understanding of particle motion in thermalcounterflow has been the subject of numerous experimental, theoretical, and computationalefforts.The first experiments by Zhang and Van Sciver made use of the particle image velocimetry(PIV) technique [23], in which a pair of images separated by a short time interval aresegmented, and cross-correlation of the segments together with knowledge of the imageseparation time is sufficient to obtain the velocity vector for each segment [17]. Zhangand Van Sciver studied counterflow in a vertical channel generated by a range of heat flux,110 ≤ q ≤ , at a variety of temperatures, 1 . ≤ T ≤ .
00 K. They foundthat for a one-dimensional counterflow, regardless of temperature or applied heat flux, themeasured particle velocity, v p , is approximately half of the theoretical normal fluid velocity: v p ≈ v n /
2. According to the subsequent theory of Sergeev et al., the observed behavior canbe explained by interactions between the particles and quantized vortex lines [24].Other experimental investigations of particle motion in thermal counterflow have em-ployed the particle tracking velocimetry (PTV) technique [25–28], in which individual par-ticle locations are tracked throughout a sequence of images. The results of Paoletti et al.show that some particle tracks correspond to the normal fluid motion, exhibiting relativelystraight trajectories with mean particle velocity v p ≈ v n in the same direction as the heat4urrent, while others show erratic behavior with net motion against the heat current [25].In this experiment the temperature range was 1 . ≤ T ≤ .
15 K and the heat flux rangewas 13 ≤ q ≤
90 mW/cm , an order of magnitude less than that of Zhang and Van Sciver.The numerical work of Kivotides suggests two regimes of particle motion that are separatedby the applied heat flux. The simulations show that when the vortex tangle is relativelydilute, as is the case when the applied heat flux is lower, particles have a relatively largemean free path through the tangle, with some traversing the entire observation volume at v n without interacting with vortices [21]. On the other hand, when the tangle is relativelydense, particles cannot avoid interaction with vortices, and their mean velocity is lower than v n [29].Chagovets and Van Sciver used the PTV method intending to scan a parameter spacecovering that of the PIV experiment by Zhang and Van Sciver as well as the PTV experimentby Paoletti et al., thereby observing the transition between the two proposed flow regimesin a single experiment [26]. However, due to a hardware limitation, the heat flux rangewas limited to 7 ≤ q ≤
100 mW/cm at 1 . ≤ T ≤ .
00 K [26], which does not quiteextend into the region probed by Zhang and Van Sciver. The results were nonethelessinsightful, providing a discussion of the trapping of particles on quantized vortices and theirsubsequent dislocation, which presumably plays a role in the transition between the tworegimes of particle motion [26]. Work has continued on classifying particle motion in thermalcounterflow, with approaches focused on qualitative features of the particle trajectories [27]and analysis of particle motion as a function of their size [28].Another experimental approach to thermal counterflow that makes use of PTV is theanalysis of transverse (i.e., perpendicular to the direction of normal fluid flow) particlevelocity statistics. It has been shown that for both steady state [30] and decaying [31] coun-terflow, the probability density function (PDF) for transverse particle velocity u p exhibitsa Gaussian core with non-classical tails proportional to | u p | − . In some cases, the tails areattributed to the motion of particles trapped on vortices that have just experienced a re-connection event [31]. Others point out that the tails can be predicted from the superfluidvelocity field in the vicinity of a vortex core, without the need to consider vortex reconnec-tion [30, 32]. However, in light of numerical simulations that show particles suitably close tothe vortex core have a tendency to become trapped rather than trace the superfluid velocityfield [33, 34], this explanation is unlikely. Regardless, the tails have been shown to exist5nly when the probing time, t , is smaller than the average travel time between quantizedvortex lines, t = ℓ/ h v p i , where ℓ = L − / represents the mean distance between vortexlines. When the ratio of these times, τ = t /t , exceeds unity, the tails disappear and thePDF assumes the classical Gaussian form [30]. This has been interpreted as an implicationthat counterflow turbulence behaves classically on large length scales [30, 32].It should be mentioned that flow visualization has been applied in He II for many pur-poses other than the study of particle motion. Some other investigations have been focusedon counterflow [35–37] and forced flow [38] around cylinders; velocity profile in mechan-ically driven pipe flow [39]; dynamics of quantized vortices [40–42]; and flow induced byoscillating [43] and towed grids [44]. More recently, a different approach to He II flowvisualization has been introduced, making use of metastable He * molecules as tracer par-ticles [45]. Measurements of the turbulent normal fluid velocity with these particles havelead to non-classical forms of the second order transverse structure function [8], effectivekinematic viscosity in decaying counterflow turbulence [46], and the energy spectrum in asustained thermal counterflow [14]. These measurements are free of the ambiguity associ-ated with PIV and PTV methods since the He * molecules strictly trace the normal fluidfor temperatures above about 1 K [45].In this paper we return to measurement of particle motion in thermal counterflow usingPTV. However, we attempt to remove the particle motion ambiguity by analyzing parti-cles that move with the normal fluid separately from those influenced by vortices. To ourknowledge this approach has not yet been attempted. We were also successful at probing awide range of applied heat flux, overlapping with the PIV experiments of Zhang and VanSciver as well as with the PTV experiments of Paoletti et al. and Chagovets and Van Sciver.In Sect. II, we briefly describe our experimental protocol. The criteria for differentiatingparticle velocity measurements is covered in Sect. III, and we showcase the results obtainedfor streamwise particle motion and transverse velocity statistics, respectively, in terms ofthe separated velocity measurements, in Sect. III and Sect. IV. In Sect. V we offer a briefdiscussion of the physical mechanisms that may be responsible for our observations, andsuggest a number of numerical simulations that may reveal the underlying physics in detail,before concluding in Sect. VI. 6 I. EXPERIMENTAL PROTOCOL
Thermal counterflow is generated and contained inside a vertical flow channel which itselfis immersed in the helium reservoir of a typical research cryostat with optical access. Thechannel, which was designed for visualization of both counterflow and towed grid turbulencein He II (see [44] for details), is constructed from cast acrylic with a square cross section of1.6 cm side length and measures 33 cm long. The bottom end is sealed with an array of evenlyspaced surface mount resistors that occupy about 80% of the channel cross section, such thatthe applied heat flux is distributed nearly uniformly. An illustration of the experimentalapparatus is shown in Fig. 1.For this work we make use of the PTV method. Tracer particles are formed by slowlyintroducing a gas mixture of 5% D gas (balance He) directly into He II [47]. This seed gasis delivered via a tube that passes through the main linear drive shaft for the towed grid LASER SHEETVIEWPORTHEATERPARTICLEDELIVERYTUBEHELIUMRESERVOIRSEED GASINLETMOTORBELLOWSNITROGENRESERVOIRDEWARACRYLICCHANNELIMAGINGREGION DRIVESHAFT
FIG. 1. Simple illustration of the experimental apparatus (not to scale). µ m, as determined from their terminalvelocity in quiescent He II [44]. Since particle size effectively sets the minimum spatialresolution [32], and we anticipate ℓ > µ m, these particles should be suitable for probinglength scales both above and below the mean vortex line spacing. Before measurementsbegin, the particle delivery tube is retracted from the channel by raising the grid driveshaft (the mesh grid itself is removed for counterflow experiments). This prevents any flowstructures that might develop upstream of the tube, a phenomenon known to occur in HeII counterflow [35, 48], from interfering with the velocity field in the region of interest.A continuous wave laser, shaped into a thin sheet of approximately 16 mm height, providesillumination of the imaging plane in the geometric center of the channel. A high speedCMOS camera, triggered at various rates between 60 and 180 frames per second dependingon the anticipated normal fluid velocity, captures sequences of several hundred images of theparticles moving under the influence of counterflowing He II. Tracks are extracted from thesequence of images using an algorithm that is based on the feature point tracking routine ofSbalzarini and Koumoutsakos [49], but that we have tailored for use with solidified tracerparticles in He II. From the tracks, which are essentially lists of spatial coordinates separatedby a known time interval, it is trivial to derive the particle velocity.Using this apparatus we have measured particle motion in steady state thermal counter-flow at three temperatures, and a wide range of heat flux was applied at each temperature:38 ≤ q ≤
215 mW/cm at T = 1 .
70 K, 38 ≤ q ≤
366 mW/cm at T = 1 .
85 K, and17 ≤ q ≤
481 mW/cm at T = 2 .
00 K. This parameter space substantially overlaps thoseof the existing PTV experiments (13 ≤ q ≤
90 mW/cm at 1 . ≤ T ≤ .
15 K [25]and 7 ≤ q ≤
100 mW/cm at 1 . ≤ T ≤ .
00 K [26]) and the original PIV experiment(110 ≤ q ≤ at 1 . ≤ T ≤ .
00 K [23]).
III. STREAMWISE PARTICLE BEHAVIOR
Fig. 2 shows some of the particle tracks observed at 1.85 K as well as the correspondingstreamwise velocity PDFs. Though the particle tracks have the same structure as thoseshown in several previous studies [25–27], and it is well known that streamwise velocityPDFs, at least in the lower heat flux regime, exhibit two peaks [25, 31, 50], this figure8 v p (mm/s) P r( v p ) / v ( s / mm ) v p,m G3 Fit -5 0 5 10 15 v p (mm/s) P r( v p ) / v ( s / mm ) v p,m G1 FitG2 Fit -5 0 5 10 15 v p (mm/s) P r( v p ) / v ( s / mm ) v p,m G1 FitG2 Fit -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Imaging Plane Width (mm) -3-2-10123 I m ag i ng P l ane H e i gh t ( mm ) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Imaging Plane Width (mm) -3-2-10123 I m ag i ng P l ane H e i gh t ( mm ) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Imaging Plane Width (mm) -3-2-10123 I m ag i ng P l ane H e i gh t ( mm ) (a)(b)(c) qqq (d)(e)(f) FIG. 2. (a)-(c): Particle tracks extracted from videos for which T = 1 .
85 K and q = 38, 122 , and320 mW/cm , respectively. (d)-(f): Corresponding particle velocity PDFs (streamwise direction).All six panels are color coded: blue indicates G1, red indicates G2, and black indicates G3. Thedirection of the imposed heat current is shown by the red arrows labeled “ q ”. showcases the novelty of our approach to the data analysis. We use the two-peak structureof the PDFs exemplified in Fig. 2(d) and (e) as the basis for analyzing the motion of particlesmoving with the normal fluid separately from those under the influence of the vortex tangle.Those moving with the normal fluid, whose velocity measurements contribute to the peak9ith higher mean value, we give the name “Group 2” or G2 for short. Those moving withthe vortex tangle, whose velocity measurements contribute to the peak with lower meanvalue, we give the name “Group 1” or G1. For qualitative differentiation, we introduce thefollowing criteria for deciding whether an instantaneous velocity sample represents G1 orG2 behavior. If the instantaneous velocity of a particle satisfies v p < µ − σ , where µ and σ are the mean and standard deviation, respectively, for a Gaussian curve fit to theG2 peak, we assume that it exhibits G1 behavior. Likewise, if v p > µ + 2 σ , we assumethat it exhibits G2 behavior. In cases where µ − µ > σ + 2 σ , i.e. the peaks are wellseparated, the criteria are reversed ( v p < µ + 2 σ counts as G1 and v p > µ − σ counts asG2) to prevent measurements falling in between the two peaks from counting toward bothgroups. As a result, the separation scheme generates ensembles of velocity measurementsthat represent G1 and G2. For brevity in the ensuing discussions, we use these names torefer interchangeably to the entire physical group of particles as well as the representativemeasurement ensemble. We define an additional group, G3, for the high heat flux regime.Since the streamwise velocity PDF exhibits just one peak for higher heat flux, as exemplifiedin Fig. 2(f), all of the measured velocity samples are representative of G3 behavior.The tracks and Gaussian fits of Fig. 2 are color coded: G1 is shown in blue, G2 in red,and G3 in black. In the top row (Fig. 2(a) and (d)), which represents relatively low heat flux( q = 38 mW/cm ), the G2 tracks are long, straight, and oriented in the same direction as theheat current, while the G1 tracks meander and are randomly oriented. The correspondingpeaks in the PDF are well defined (i.e., clearly separated from one another). In the middlerow (Fig. 2(b) and (e)), which represents moderate heat flux ( q = 122 mW/cm ), theG2 tracks are still straight and vertically oriented, but are frequently interrupted by shortG1 segments. This likely represents the trapping of particles by quantized vortices, andthe subsequent dislocation of the particle that stems from the increased normal fluid dragforce [21, 26]. As a whole, the G1 tracks move in the same direction as the heat current,though in a slower and considerably less undeviating fashion than the G2 tracks. A positiveshift in the mean value of both peaks can be observed in Fig. 2(e), and the peaks are lesswell defined, appearing to merge together. Though the evolution of particle velocity as afunction of applied heat flux in this regime has been thoroughly discussed by Chagovetsand Van Sciver, their assumption that the two group behavior continues indefinitely doesnot appear to be correct [26]. The bottom row (Fig. 2(c) and (f)), representing higher heat10urrent ( q = 320 mW/cm ), shows that G3 tracks are all oriented in the same direction asthe heat current but exhibit significant transverse motion, and their PDF exhibits only onepeak.Naturally, a question arises about what causes particles to move under the influence of thenormal fluid or the vortex tangle. Many discussions on the behavior of particles in thermalcounterflow mention particle size [25, 26, 28, 34]. Using our separation scheme, we computedthe PDF for integrated light intensity, or the sum of pixel values in the neighborhood ofthe particle image, which is used as a substitute for particle size since the latter cannot beaccurately measured for a moving particle. Fig. 3 shows that the PDF for G1 and G2 arenearly identical across the full range of observed particle size. This suggests that for solidifiedtracer particles in the size range produced by our seeding system, trapping probability is notinfluenced by particle size, though the observed range is quite small. We do not presentlyhave an explanation for this.An additional consideration is that at the beginning of the image acquisition, particlesare either trapped or untrapped, and whether the G2 particles become trapped during theacquisition period depends primarily on their mean free path through the vortex tangle.As a very simple estimation, we assume that a particle will become trapped if the volumetraversed by its trapping cross section contains a line segment comparable in length to theparticle diameter. We use for the trapping cross section the two-dimensional projection of Integraged Light Intensity (a.u.) P r obab ili t y G1G2
FIG. 3. Probability distribution for the size (indicated by integrated light intensity) of particlescontributing to G1 and G2. The example shown applies to the case where T = 1 .
85 K and q = 38 mW/cm . πd p /
4. The volume traversed by the cross section is then sπd p /
4, where s denotes the mean free path. Multiplication by L gives the vortex line length within thisvolume, and as per our estimation, the result must be less than d p for the particle to remainuntrapped: π d p sL ≤ d p (3)A simple representation for the mean free path is then s < ∼ πd p L (4)As examples we consider the cases shown in Fig. 2(a) and (b) for which we estimate L ≈
743 cm -2 and L ≈ -2 , respectively, using the value for the γ parameter reported byGao et al. [14]. For particles with diameter 4 µ m, the estimated mean free path is about4 cm for case (a). This exceeds the dimensions of the imaging region, and the G2 tracks arequite long and often terminate when the particle leaves the imaging plane instead of with atransition to G1 behavior, which would indicate trapping. For case (b) the mean free path isabout 0.1 cm, and it can be seen the length of many G2 tracks is roughly 1 mm, and the tracksoften terminate in a trapping event. Though this simple estimation is reasonably accurate, aproper determination of the mean free path requires complex numerical simulations, takinginto account the complicated dynamics of He II, such as Kelvin waves on quantized vortices,drag force exerted by the normal fluid, and relative motion of the particles and vortex tangle.Similar simulations by Kivotides indeed show that when the vortex tangle is relatively sparse,particles can move a significant distance (in some cases throughout the entire computationaldomain) without interacting with vortices [21], but when the tangle is relatively dense theparticles experience constant interaction with the tangle [29].For consistency with the existing experimental literature [23, 25, 26, 50] we show v p as afunction of v n for each point in the parameter space in Fig. 4. The top panel shows resultsonly for T = 2 .
00 K and the bottom panel shows results for the other two temperatures. v p is represented by the mean value of Gaussians fit to the streamwise velocity PDFs (asin the examples of Fig. 2(d)-(f)). In all cases, v p for the G2 peak is approximately equalto v n + v slip , where v slip is the velocity offset caused by non-neutral density of the particles.This trend is indicated by the solid black line. Mean velocity of the G1 peak, for very smallheat flux, is similar to the superfluid velocity with the same correction factor, v s + v slip ,indicated by the blue line. This behavior is expected for low counterflow velocities since the12 v n (mm/s) -20246810 v p ( mm / s ) G1, 2.00 KG2, 2.00 KG3, 2.00 Kv p =v n +v slip v p =v n /2+cv p =v s +v slip (a) v n (mm/s) -20246810 v p ( mm / s ) G1, 1.70 KG2, 1.70 KG3, 1.70 KG1, 1.85 KG2, 1.85 KG3, 1.85 Kv p =v n +v slip v p =v n /2+c (b) FIG. 4. Measured particle velocity v p as a function of the theoretical normal fluid velocity v n for(a) T = 2 .
00 K and (b) T = 1 .
85 K and T = 1 .
70 K. superfluid carries the vortex tangle, on average, at v s [51, 52], and it has been demonstratedin recent visualization experiments [25, 26] and numerical simulations [22]. As the heat fluxincreases and mutual friction begins to affect the vortex tangle, the G1 velocity departs from v s + v slip and instead corresponds to v n / c , indicated by the dashed black line, where c is an offset of about 2 mm/s. At 2.00 K, as the heat flux continues to increase, the single13eak PDF structure appears, with the mean value beginning from some value between v n and v n / c and eventually settling at the latter. This transition region occurs betweennormal fluid velocities of roughly 7 and 15 mm/s, as indicated by the vertical dotted lines inFig. 4(a). However, for 1.85 and 1.70 K, this transition appears to be absent, with the meanvalue of G3 PDFs immediately collapsing onto the v n / c trendline when v n exceeds 7 or8 mm/s. We suspect that this results from a limitation of our imaging system; it seems thatwe are not able to resolve particles moving faster than about 9 mm/s. Since the dynamicviscosity of the normal fluid is smaller at these temperatures than at 2.00 K [53], it makessense that the critical drag force preventing particles from remaining trapped on vortices [26]is surpassed at higher values of v n that may be beyond those that we can resolve. In thiscase, the G3 data shown in Fig. 4 would in fact be miscategorized G1 data. As an additionalnote, small values of v ns were not observed at these temperatures, so the transition of G1velocity from v s + v slip to v n / c does not appear either.These observations are consistent with the existing literature on experimental measure-ments of particle motion in thermal counterflow: when the applied heat flux is lower, particlescan be observed moving at approximately v n [25], and when it is higher, particles can beobserved moving at approximately v n / IV. PARTICLE VELOCITY STATISTICS
Statistical analysis of particle motion in thermal counterflow using PTV is typically fo-cused on the evolution of transverse particle velocity or acceleration PDFs with changingtemperature, heat flux, or most commonly, probing time scale [30, 32, 54, 55]. In theseanalyses the statistical sample consists of all of the detected particles. This approach raisessome concern when one considers the vastly different characteristics of the transverse motionexhibited by the G1 and G2 tracks in Fig. 2(a), and to a lesser extent in Fig. 2(b). Ouranalysis of the transverse particle velocity for G1 and G2 shows that some information isindeed missed when the two groups are not considered separately.First we note that, within each group, the streamwise and transverse velocity components14 u p (mm/s) -4 -3 -2 -1 P r( u p ) G1G2G1+G2 (a) -10 -5 0 5 10 u p (mm/s) -5 -4 -3 -2 -1 P r( u p ) G1G2G1+G2 (b) -8 -6 -4 -2 0 2 4 6 8 u p (mm/s) -4 -3 -2 -1 P r( u p ) G1G2G1+G2 (c) -10 -5 0 5 10 u p (mm/s) -5 -4 -3 -2 -1 P r( u p ) G1G2G1+G2 (d)
FIG. 5. Probability distributions for the measured transverse particle velocity at T = 2 .
00 K forthe cases where (a) q = 91 mW/cm and (b) q = 113 mW/cm . The solid lines represent Gaussianfits to the distributions and the dashed line represents a power law curve proportional to | u p | − .Panels (c) and (d) show the same data but the minimum probing length has been changed to 2 ℓ . are uncorrelated. In other words, the samples taken from any slice of the streamwise PDFwill accurately represent the entire transverse distribution (provided the extracted samplesize is large enough), and vice versa. This is important to the success of our separationscheme since the streamwise velocity PDFs for G1 and G2 end up quite lopsided. However,as will be seen in the figures of this section, the transverse velocity PDFs are sufficientlyresolved. 15ig. 5 shows G1, G2, and combined (G1+G2) transverse velocity PDFs. In Fig. 5(a)( q = 91 mW/cm ) the G2 sample size exceeds the G1 sample size. In Fig. 5(b) ( q =113 mW/cm ) the opposite is true. To show the relative contributions of G1 and G2 to thecombined PDF, the normalization is P r g,i = n g,i /N , where n g,i is the number of samples inthe i th bin for group g , and N is the total number of combined G1 and G2 samples. It is clearthat, regardless of the relative sample size, G1 dominates the tail region of the combinedPDF. Though the G2 PDFs appear to have some structure at the ends, it is not coherentand occurs with probability at least an order of magnitude less than the corresponding G1contribution. This is likely due to a small number of misclassified velocity measurements;those with streamwise component more than two standard deviations outside the groupmean can be potentially placed in the wrong group. This effect could be confirmed byinspecting the location of these specific velocity samples in the particle tracks, and judgingbased on the local geometry whether they truly belong to G1 or G2.We also observe that the Gaussian core of the G1 PDF is substantially wider than the G2PDF. This becomes of consequence when the G2 sample size is larger, as in Fig. 5(a). Asa result, the combined PDF may be broken into three regions. The tip region is Gaussianand due primarily to the G2 PDF. The middle region is due to the combined G2 PDF andGaussian core of the G1 PDF, and has a different mathematical description than the tipregion. The tail region exhibits the | u p | − power law behavior due exclusively to the G1PDF tails. If the combined PDF is considered alone, it is possible to mistake the middleregion for the beginning of the power law tails, leading to incorrect conclusions about theparticle velocity statistics.Fig. 5(c) and (d) show the same data as Fig. 5(a) and (b), respectively, except the min-imum probing length scale has been increased to 2 ℓ , twice the mean vortex line spacing.It is important to note that this differs from the approach described in the existing lit-erature [30], which is an adjustment of the probing time scale. The latter is achieved byusing every other, or every third, etc., position measurement along a particle track to calcu-late the velocity, simulating a reduction in the image acquisition rate [30]. Alternatively, atrue adjustment of the minimum probing length can be accomplished by discarding positionmeasurements only if they are not sufficiently separated from the previous location in thetrajectory. Velocity samples are then computed as v i = ( x i + j − x i ) /j ∆ t , where i representsthe i th position along a track and j represents the number of subsequent points to skip such16hat k x i + j − x i k exceeds the desired minimum length scale. Increasing the probing length inthis manner results in a drastic reduction of the number of G1 samples, since the mean G1velocity is small compared to the mean G2 velocity. For a fixed sample size, the PDF tailsare quenched since many of the measurements contributing to them are discarded. Thisis apparent in Fig. 5(c), where the probability of observing a particle with G1 velocity isdrastically reduced, and the extents of the PDF do not resemble the power law curve. Thetails of Fig. 5(d) are more or less eliminated as well, though the effect is not as obvious sincethe G1 sample size for this case was considerably larger.An alternative way to present the data is shown in Fig. 6, which contains several transversevelocity PDFs for each of G1, G2, and G3. Several curves with different values of the non-dimensional time τ are shown in each case. In the same manner as the existing literature, wedefine τ = t /t , where t is the time elapsed between successive images and t = ℓ/ h v p i [30],except that in this case h v p i is computed for each group instead of for all of the detectedparticles. Defined in this way, t represents the average time for a particle of each respectivegroup to traverse the intervortex distance. -8 -6 -4 -2 0 2 4 6 8 u p / u -5 -4 -3 -2 -1 P r( u p / u ) / ( u p / u ) = 0.06= 0.11= 0.12= 0.22 GaussianPower Law (a) G1 -8 -6 -4 -2 0 2 4 6 8 u p / u -5 -4 -3 -2 -1 = 0.31= 0.84= 1.30= 2.02Gaussian (b) G2 -8 -6 -4 -2 0 2 4 6 8 u p / u -5 -4 -3 -2 -1 = 2.97= 3.19= 3.59= 4.14GaussianPower Law (c) G3 FIG. 6. Normalized probability distribution for the measured transverse velocity of particles con-tributing to (a) G1, (b) G2, and (c) G3 at T = 2 .
00 K. Several different values of the ratio τ are shown for each group. Gaussian curves ( A exp (cid:16) − ( u p /σ u ) / (cid:17) ) are fit to the entire PDF andpower law curves ( A | u p /σ u | − ) are fit only to the tail regions, defined as more than 4 σ u from thecenter.
17 Gaussian form is evident in the core of all PDFs, regardless of group or probing timescale, as indicated by the solid black curve in all three panels ( A exp (cid:0) − ( u p /σ u ) (cid:1) , where A is a constant). This is consistent with the existing literature [30, 32]. For G1, shown inFig. 6(a), a power law curve ( A | u p /σ u | − , where A is a constant), indicated by the dashedcurve, can be drawn through the tail region. We define the tails as the data that falls morethan 4 σ u from the center of the PDF. While the size of this data set is not sufficient toresolve extended tails, deviation from the Gaussian profile is clear, and the nondimensionaltime is less than unity for all of the cases shown, indicating that the probing time is smallerthan the average intervortex travel time. According to the existing literature, these are thecorrect conditions for power law tails [30, 32].The G2 PDFs of Fig 6(b) present a different picture: all cases show purely Gaussian form,even though τ < τ > τ > ∼
3) for G3 PDFs, shown in Fig. 6(c),deviation from the Gaussian core can be observed in one case. As with G1, a power lawcurve can be drawn through this tail structure. Previous experimental results suggest thatthe PDF should have Gaussian form if τ >
1, but those investigations did not include thehigh heat flux G3 region [30]. Indeed, little is known about the novel form of turbulencethat exists in this high heat flux region, where both the normal fluid [8] and superfluid canbecome turbulent.We show in Fig. 7 the standard deviation of the Gaussians fit to all streamwise andtransverse velocity data obtained at 2.00 K. As in Sect. III, data for G1 is shown in blue,G2 in red, and G3 in black. At first glance, the figure adds weight to the importance ofthe separation scheme, particularly for the analysis of transverse velocity statistics, since aclear divergence between G1 and G2 transverse velocity fluctuation is evident as the heatflux increases. Closer inspection reveals some additional, more subtle, observations. Themeasured transverse velocity standard deviation σ u for G2 is fairly constant throughout therange of applied heat flux at 2.00 K; the velocity fluctuation does not increase noticeablyuntil the transition to G3. This suggests that the normal fluid may not be turbulent in the18 v n (mm/s) ( mm / s ) G1 TransverseG2 TransverseG3 TransverseG1 StreamwiseG2 StreamwiseG3 Streamwise
FIG. 7. Standard deviation of the measured streamwise and transverse particle velocity for G1,G2, and G3 at T = 2 .
00 K. The solid blue line represents the vortex line velocity fluctuation andwill be discussed in the next section. Note that the probing scale for these data is not constant. two peak region, which is further supported by a brief test of decaying thermal counterflowat 1.70 K in the two-peak region (G1 and G2, q = 50 mW/cm ) and the single peakregion (G3, q = 193 mW/cm ). In the single peak region the line density decays brieflyas L ∝ t − before transitioning to L ∝ t − / , whereas in the two peak region it follows L ∝ t − throughout the entire decay. Gao et al. have shown that the former behaviorcorresponds to decay from a steady state counterflow in which large scale turbulence existsin the normal fluid, while the latter decay behavior occurs when normal fluid turbulence isabsent [56]. It is worthwhile to note that if we scale the normal fluid turbulence transitionheat flux reported by Gao et al. to the wider channel used for our experiment, we obtain aheat flux slightly smaller than that at which the two peak structure disappears. However,the transition to turbulence may be affected by other factors such as the channel materialand surface roughness. Besides the onset of large-scale normal fluid turbulence, increasedfrequency of particle-vortex reconnection may contribute to the larger velocity fluctuations ofG3. Kivotides has shown through numerical simulation that in a dense vortex tangle, suchreconnection events can induce velocity fluctuations of the particles comparable to their19ean velocity [29]. However, directly comparing the results is difficult since the simulationsdid not take into account the turbulent normal fluid. Finally, we note that the streamwisevelocity standard deviation for G2, unlike the transverse, does seem to increase with heatflux. We will carry this observation forward into the following discussion. V. DISCUSSION
There are a couple of potential explanations for the apparent anisotropy of G2 revealedin Fig. 7. This may be an artifact of the acceleration and deceleration of particles at thebeginning and end, respectively, of G2 tracks when they break free from or become trappedon vortex lines. If this is indeed the correct physical interpretation, it implies that particlesinteract with vortices primarily through trapping, as opposed to wide angle scattering, as thelatter would be associated with a significant acceleration, and thus velocity fluctuation, inthe direction normal to the trajectory. It follows that the capture cross section significantlyexceeds the wide angle scattering cross section. Numerical simulations that place a movingparticle on a straight trajectory past a vortex line, with varying distance between the particletrajectory and vortex core, would reveal the trapping and scattering cross sections. Similarwork has already been performed by Kivotides et al. [34, 57] on the results of a direct collisionbetween a moving particle and vortex line at relatively low temperature. Though expansionof this work to study “near misses” would be non-trivial, taking into account the vortexdynamics as well as the normal fluid drag force, it would provide important insight to theparticle velocity statistics in thermal counterflow.A more feasible explanation is that since particles exhibiting G2 behavior move primarilyunder the influence of drag force from the normal fluid, their velocity is subject to localvariations of the normal fluid velocity. As the normal fluid passes across the vortex tangle,wakes can form behind each individual vortex line due to mutual friction; within thesewakes, the normal fluid velocity can vary significantly [58]. It makes sense that the samefluctuations do not appear in the transverse particle velocity since there is no mean flow inthat direction.An additional point of interest is the dynamics of particles trapped in the vortex tangle.Our present work as well as that of Chagovets and Van Sciver [26] has shown that as theheat flux increases, G1 departs from v p ≈ v s behavior and transitions to roughly v n /
2. We20o not have an explanation for this behavior, particularly, why G1 follows the same trend asG3. A similar departure from v p ≈ v s of trapped particles was first observed by Paoletti etal. [25], and has been reproduced in the numerical simulations of Mineda et al. [22], thoughneither presented the particle velocity in terms of v n , and the simulation did not extend tovalues of v n very far beyond the transition point. It would be interesting to know whethera similar simulation, extended to heat currents farther beyond the transition, reveals thesame evolution of mean G1 velocity with v n / R can be written as [59]: v L = κ πR ln (cid:18) Rξ (cid:19) (5)The line velocity fluctuation can then be obtained from h v L i / : (cid:10) v L (cid:11) / = κ π (cid:28) R ln (cid:18) Rξ (cid:19)(cid:29) / (6)Neglecting the slow variation of the natural logarithm with L , we make the substitution R ≈ ℓ and remove the constant ln ( ℓ/ξ ) from the average [60]. The remaining term h /R i / can be replaced by c L / , where c is a temperature dependent parameter [59–61], and theline density can be written in terms of the normal fluid velocity as per (2): (cid:28) R (cid:29) / ≈ c γ (cid:18) ρρ s v n − v (cid:19) (7)The resulting expression for root mean square vortex line velocity fluctuation as a functionof the normal fluid velocity is: (cid:10) v L (cid:11) / ≈ κc γ π ln (cid:18) ℓξ (cid:19) (cid:18) ρρ s v n − v (cid:19) if ρρ s v n > v (cid:10) v L (cid:11) / = 0 if ρρ s v n ≤ v (8)We note that this simple approach yields a linear relationship between h v L i / and v n ,provided the counterflow velocity exceeds v . Using values for c [60] and γ [14] derived fromthe work of Gao et al., and an approximate value of ℓ ≈ µ m, we find the proportionality21onstant to be 0.54. The solid blue line of Fig. 7 represents (8), with a small offset toadjust for environmental noise. It agrees reasonably well with the observed G1 velocityfluctuation, suggesting that the G1 particle velocity fluctuations are, to a good extent,caused by fluctuations of the vortex line velocity. However, it should be kept in mind thatthe particle vortex interaction is quite complicated, and depends also on such factors as therelative motion between the particles, vortices, and normal fluid, and deformation of thetangle due to the presence of particles. The same numerical simulation that predicts themean velocity of trapped particles, suggested above, could produce more detailed informationabout the relationship between G1 and vortex tangle velocity fluctuations.Finally, in the future, we plan to study the structure and scaling laws of PDFs relatedto the particle trajectory geometry, instead of the kinematics. This approach to Lagrangianfluid dynamics has recently emerged in classical fluids, where curvature of the trajectories [62]or the relative angle of velocity vectors as a function of their temporal separation alongthe track [63] are used to characterize the fluid dynamics. Applications to simulations ofclassical turbulent flows have revealed power law scaling of the PDFs for trajectory angle [64],curvature, and torsion [64, 65] that characterize the turbulence. This approach has not yetbeen introduced to quantum turbulence, and may offer an opportunity for quantitativecharacterization of the vortex tangle dynamics in thermal counterflow [66]. This could ofcourse be accomplished using experimental data as well as numerical simulations. VI. CONCLUSIONS
We have performed a systematic study of solidified particle motion in He II thermalcounterflow using the PTV technique. For the first time in a single experiment, the drivingheat flux extends from the low range, previously investigated by PTV, to the high range,previously investigated by PIV. Demonstrating that the streamwise velocity PDFs transformfrom a double peak structure, with one peak centered at v n and one near v n /
2, into asingle peak centered near v n /
2, rectifies the previous experimental observations as well aspredictions obtained through numerical simulations.We have also devised a simple criteria to isolate the normal fluid and vortex tangle velocitystatistics. We apply this separation criteria to show that G1 velocity measurements dominatethe non-classical tail structure of the transverse velocity PDFs, while G2 velocity statistics22xhibit more or less classical behavior. In order to better understand the observed behaviorof each group, we hope that this work will stimulate a number of numerical simulations thatfurther characterize particle motion in He II counterflow.
ACKNOWLEDGMENTS
The authors wish to thank W.F. Vinen, C.F. Barenghi, and M. Tsubota for their valuableinput on the discussion of our results. This work is supported by U.S. Department ofEnergy grant DE-FG02-96ER40952. It was conducted at the National High Magnetic FieldLaboratory, which is supported by NSF DMR-1157490 and the State of Florida. [1] L. Tisza, “Transport phenomena in helium II,” Nature , 913 (1938).[2] L. Landau, “Theory of the superfluidity of helium II,” Phys. Rev. , 356 (1941).[3] W. F. Vinen, “An introduction to quantum turbulence,” J. Low Temp. Phys. , 7 (2006).[4] W. F. Vinen, “Mutual friction in a heat current in liquid helium II. II. Experiments on transienteffects,” Proc. R. Soc. London, Ser. A , 128 (1957).[5] W. F. Vinen, “Mutual friction in a heat current in liquid helium II. III. Theory of the mutualfriction,” Proc. R. Soc. London, Ser. A , 493 (1957).[6] S. W. Van Sciver, Helium Cryogenics (Springer, 2012).[7] W. Guo, S. B. Cahn, J. A. Nikkel, W. F. Vinen, and D. N. McKinsey, “Visu-alization study of counterflow in superfluid He using metastable helium molecules,”Phys. Rev. Lett. , 045301 (2010).[8] A. Marakov, J. Gao, W. Guo, S. W. Van Sciver, G. G. Ihas, D. N. McKinsey, andW. F. Vinen, “Visualization of the normal-fluid turbulence in counterflowing superfluid He,”Phys. Rev. B , 094503 (2015).[9] C. E. Chase, “Thermal conduction in liquid helium II. I. Temperature dependence,”Phys. Rev. , 361 (1962).[10] P. E. Dimotakis and J. E. Broadwell, “Local temperature measurements in supercritical coun-terflow in liquid helium II,” Phys. Fluids , 1787 (1973).[11] R. K. Childers and J. T. Tough, “Helium II thermal counterflow: Temperature- and pressure- ifference data and analysis in terms of the Vinen theory,” Phys. Rev. B , 1040 (1976).[12] K. P. Martin and J. T. Tough, “Evolution of superfluid turbulence in thermal counterflow,”Phys. Rev. B , 2788 (1983).[13] S. Babuin, M. Stammeier, E. Varga, M. Rotter, and L. Skrbek, “Quantum turbulence ofbellows-driven He superflow: Steady state,” Phys. Rev. B , 134515 (2012).[14] J. Gao, E. Varga, W. Guo, and W. F. Vinen, “Energy spectrum of thermal counterflowturbulence in superfluid helium-4,” Phys. Rev. B , 094511 (2017).[15] E. Varga, S. Babuin, and L. Skrbek, “Second-sound studies of coflow and counterflow ofsuperfluid 4He in channels,” Phys. Fluids , 065101 (2015).[16] W. Guo, M. La Mantia, D. P. Lathrop, and S. W. Van Sciver, “Visualization of two-fluidflows of superfluid helium-4,” Proc. Natl. Acad. Sci. USA , 4653 (2014).[17] M. Raffel, C. Willert, and J. Kompenhans, Particle Image Velocimetry (Springer, 1998).[18] Y. A. Sergeev and C. F. Barenghi, “Particles-vortex interactions and flow visualization in4He,” J. Low Temp. Phys. , 429 (2009).[19] P. E. Parks and R. J. Donnelly, “Radii of positive and negative ions in helium II,”Phys. Rev. Lett. , 45 (1966).[20] G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, “Superfluid helium: Visualization ofquantized vortices,” Nature , 588 (2006).[21] D. Kivotides, “Normal-fluid velocity measurement and superfluid vortex detection in thermalcounterflow turbulence,” Phys. Rev. B , 224501 (2008).[22] Y. Mineda, M. Tsubota, Y. A. Sergeev, C. F. Barenghi, and W. F. Vinen,“Velocity distributions of tracer particles in thermal counterflow in superfluid He,”Phys. Rev. B , 174508 (2013).[23] T. Zhang and S. W. Van Sciver, “The motion of micron-sized particles in He II counterflowas observed by the PIV technique,” J. Low Temp. Phys. , 865 (2005).[24] Y. A. Sergeev, C. F. Barenghi, and D. Kivotides, “Motion of micron-size particles in turbulenthelium II,” Phys. Rev. B , 184506 (2006).[25] M. S. Paoletti, R. B. Fiorito, K. R. Sreenivasan, and D. P. Lathrop, “Visualization of super-fluid helium flow,” J. Phys. Soc. Jpn. , 111007 (2008).[26] T. V. Chagovets and S. W. Van Sciver, “A study of thermal counterflow using particle trackingvelocimetry,” Phys. Fluids , 107102 (2011).
27] M. La Mantia, “Particle trajectories in thermal counterflow of superfluid helium in a widechannel of square cross section,” Phys. Fluids , 024102 (2016).[28] W. Kubo and Y. Tsuji, “Lagrangian trajectory of small particles in superfluid He II,”J. Low Temp. Phys. , 611 (2017).[29] D. Kivotides, “Motion of a spherical solid particle in thermal counterflow turbulence,”Phys. Rev. B , 174508 (2008).[30] M. La Mantia and L. Skrbek, “Quantum, or classical turbulence?” Europhys. Lett. , 46002(2014).[31] M. S. Paoletti, M. E. Fisher, K. R. Sreenivasan, and D. P. Lathrop, “Velocity statistics distin-guish quantum turbulence from classical turbulence,” Phys. Rev. Lett. , 154501 (2008).[32] M. La Mantia and L. Skrbek, “Quantum turbulence visualized by particle dynamics,”Phys. Rev. B , 014519 (2014).[33] C. F. Barenghi, D. Kivotides, and Y. A. Sergeev, “Close approach of a spherical particle anda quantised vortex in helium II,” J. Low Temp. Phys. , 293 (2007).[34] D. Kivotides, C. F. Barenghi, and Y. A. Sergeev, “Interactions between particles and quan-tized vortices in superfluid helium,” Phys. Rev. B , 014527 (2008).[35] T. Zhang and S. W. Van Sciver, “Large-scale turbulent flow around a cylinder in counterflowsuperfluid He (He(II)),” Nature Phys. , 36 (2005).[36] T. V. Chagovets and S. W. Van Sciver, “Visualization of He II counterflow around a cylinder,”Phys. Fluids , 105104 (2013).[37] D. Duda, M. La Mantia, M. Rotter, and L. Skrbek, “On the visualization of thermal coun-terflow of He II past a circular cylinder,” J. Low Temp. Phys. , 331 (2014).[38] T. V. Chagovets and S. W. Van Sciver, “Visualization of He II forced flow around a cylinder,”Phys. Fluids , 045111 (2015).[39] T. Xu and S. W. Van Sciver, “Particle image velocimetry measurements of the velocity profilein HeII forced flow,” Phys. Fluids , 071703 (2007).[40] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan, and D. P. Lathrop, “Characterization ofreconnecting vortices in superfluid helium,” Proc. Natl. Acad. Sci. USA , 13707 (2008).[41] M. S. Paoletti, M. E. Fisher, and D. P. Lathrop, “Reconnection dynamics for quantizedvortices,” Physica D , 1367 (2010).[42] E. Fonda, D. P. Meichle, N. T. Ouellette, S. Hormoz, and D. P. Lath- op, “Direct observation of kelvin waves excited by quantized vortex reconnection,”Proc. Natl. Acad. Sci. , 4707 (2014).[43] P. ˇSvanˇcara and M. La Mantia, “Flows of liquid 4He due to oscillating grids,”J. Fluid Mech. , 578 (2017).[44] B. Mastracci and W. Guo, “An apparatus for generation and quantitative measurement ofhomogeneous isotropic turbulence in He II,” Rev. Sci. Instrum. , 015107 (2018).[45] J. Gao, A. Marakov, W. Guo, B. T. Pawlowski, S. W. Van Sciver, G. G. Ihas, D. N. McKinsey,and W. F. Vinen, “Producing and imaging a thin line of He2 molecular tracers in helium-4,”Rev. Sci. Instrum. , 093904 (2015).[46] J. Gao, W. Guo, and W. F. Vinen, “Determination of the effective kinematic viscosity forthe decay of quasiclassical turbulence in superfluid He,” Phys. Rev. B , 094502 (2016).[47] E. Fonda, K. R. Sreenivasan, and D. P. Lathrop, “Sub-micron solid air tracers for quantumvortices and liquid helium flows,” Rev. Sci. Instrum. , 025106 (2016).[48] C. Soulaine, M. Quintard, B. Baudouy, and R. Van Weelderen, “Numerical investigation ofthermal counterflow of He II past cylinders,” Phys. Rev. Lett. , 074506 (2017).[49] I. F. Sbalzarini and P. Koumoutsakos, “Feature point tracking and trajectory analysis forvideo imaging in cell biology,” J. Struct. Biol. , 182 (2005).[50] M. La Mantia, T. V. Chagovets, M. Rotter, and L. Skrbek, “Testing the performance of acryogenic visualization system on thermal counterflow by using hydrogen and deuterium solidtracers,” Rev. Sci. Instrum. , 055109 (2012).[51] R. T. Wang, C. E. Swanson, and R. J. Donnelly, “Anisotropy and drift of a vortex tangle inhelium II,” Phys. Rev. B , 5240 (1987).[52] R. J. Donnelly, Quantized vortices in helium II (Cambridge University Press, 1991).[53] R. J. Donnelly and C. F. Barenghi, “The observed properties of liquid helium at the saturatedvapor pressure,” J. Phys. Chem. Ref. Data , 1217 (1998).[54] M. La Mantia, D. Duda, M. Rotter, and L. Skrbek, “Lagrangian accelerations of particles insuperfluid turbulence,” J. Fluid Mech. , R9 (2013).[55] M. La Mantia, P. ˇSvanˇcara, D. Duda, and L. Skrbek, “Small-scale universality of particledynamics in quantum turbulence,” Phys. Rev. B , 184512 (2016).[56] J. Gao, W. Guo, V.S. L’vov, A. Pomyalov, L. Skrbek, E. Varga, and W.F. Vinen, “The decayof counterflow turbulence in superfluid He,” JETP Lett. , 732 (2016).
57] D. Kivotides, C. F. Barenghi, and Y. A. Sergeev, “Collision of a tracer particle and a quantizedvortex in superfluid helium: Self-consistent calculations,” Phys. Rev. B , 212502 (2007).[58] W. F. Vinen, Private communication.[59] W. F. Vinen and J. J. Niemela, “Quantum turbulence,” J. Low Temp. Phys. , 167 (2002).[60] J. Gao, W. Guo, S. Yui, M. Tsubota, and W. F. Vinen, “Dissipation in quantum turbulencein superfluid He above 1K,” ArXiv:1804.01655 [cond-mat.other].[61] K. W. Schwarz, “Three-dimensional vortex dynamics in superfluid He: Homogeneous super-fluid turbulence,” Phys. Rev. B , 2398 (1988).[62] H. Xu, N. T. Ouellette, and E. Bodenschatz, “Curvature of Lagrangian trajectories in tur-bulence,” Phys. Rev. Lett. , 050201 (2007).[63] S. Burov, S. M. A. Tabei, T. Huynh, M. P. Murrell, L. H. Philipson, S. A. Rice, M. L. Gardel,N. F. Scherer, and A. R. Dinner, “Distribution of directional change as a signature of complexdynamics,” Proc. Natl. Acad. Sci. USA , 19689 (2013).[64] W. J. T. Bos, B. Kadoch, and K. Schneider, “Angular statistics of Lagrangian trajectories inturbulence,” Phys. Rev. Lett. , 214502 (2015).[65] A. Bhatnagar, A. Gupta, D. Mitra, P. Perlekar, M. Wilkinson, and R. Pan-dit, “Deviation-angle and trajectory statistics for inertial particles in turbulence,”Phys. Rev. E , 063112 (2016).[66] C. F. Barenghi, Private communication., 063112 (2016).[66] C. F. Barenghi, Private communication.