Characterizing vortex tangle properties in steady-state He II counterflow using particle tracking velocimetry
aa r X i v : . [ c ond - m a t . o t h e r] O c t Characterizing vortex tangle properties in steady-state He IIcounterflow using particle tracking 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 30, 2018)
Abstract
Historically, there is little faith in particle tracking velocimetry (PTV) as a tool to make quanti-tative measurements of thermal counterflow in He II, since tracer particle motion is complicated byinfluences from the normal fluid, superfluid, and quantized vortex lines, or a combination thereof.Recently, we introduced a scheme for differentiating particles trapped on vortices (G1) from par-ticles entrained by the normal fluid (G2). In this paper, we apply this scheme to demonstratethe utility of PTV for quantitative measurements of vortex dynamics in He II counterflow. Weestimate ℓ , the mean vortex line spacing, using G2 velocity data, and c , a parameter related to themean curvature radius of vortices and energy dissipation in quantum turbulence, using G1 velocitydata. We find that both estimations show good agreement with existing measurements that wereobtained using traditional experimental methods. This is of particular consequence since theseparameters likely vary in space, and PTV offers the advantage of spatial resolution. We also showa direct link between power-law tails in transverse particle velocity probability density functions(PDFs) and reconnection of vortex lines on which G1 particles are trapped. ∗ Author to whom correspondence should be addressed. Electronic mail: [email protected] . INTRODUCTION Whole field flow visualization has become a popular research tool for He II [1], thesuperfluid phase that occurs in He at temperatures below about 2.17 K. One common vi-sualization method, particle tracking velocimetry (PTV), tracks the locations of individualmicron-sized solidified hydrogen or deuterium tracer particles suspended in the flow fieldthroughout a sequence of photographs. It is trivial to obtain an ensemble of velocity mea-surements from these time-resolved sequences of particle locations, which can be used, intheory at least, to characterize quantitatively the fluid behavior.In practice, extraction of reliable, quantitative information from particle velocity mea-surements has been elusive due to the non-classical mechanics of He II. The two fluid modelof Tisza and Landau describes it as two interpenetrating and fully miscible fluid compo-nents [2, 3]. The normal fluid behaves more or less classically, and saturates the He IIsystem at the phase transition temperature T λ ≈ .
17 K. It entrains tracer particles byviscous drag. The superfluid component, which saturates the two-fluid system below about1 K, is inviscid and carries no entropy, but still influences particle motion through inertialand added mass effects [4]. Furthermore, circulation in the superfluid is confined to quan-tized vortex lines, each with a single quantum of circulation κ ≈ − m /s about a core ξ ≈ . v n proportional tothe heat flux q , while the superfluid moves toward it at v s such that there is no net masstransfer. As q increases the two fluids can become independently turbulent [9]. Turbulencein the superfluid manifests as a random tangle of quantized vortex lines [10], and a non-classical form of turbulence arises in the normal fluid [9] due to a force of mutual frictionthat arises from interactions with the vortex tangle [11].Since tracer particles interact with both fluid components, a major challenge when apply-ing PTV to thermal counterflow is determining what influences the motion of an observedparticle at a given time, so that the behavior of the underlying flow field can be interpreted2orrectly [9, 12–14]. Until recently analysis was confined to qualitative characterizations:evolution of particle motion in response to applied heat flux [15], or of statistical distributionsof particle kinematics in response to image acquisition rate [16, 17]. A newer visualizationtechnique employing metastable He * eximers as tracer particles avoids this ambiguity issue,since the eximers are not trapped on vortices above about 1 K [14]. However, as a compro-mise, information about the vortex dynamics cannot be obtained, and thus far this methodyields information about the flow velocity in one dimension only.Recently, we studied particle motion in thermal counterflow across a wide heat flux rangeusing PTV, and found that, indeed, particles moving under the influence of relatively highheat flux, to which we give the name G3, are constantly affected by both the normal fluidand vortex lines. However, for relatively low heat flux, we devised a scheme for analyzingthe kinematics of particles entrained by the normal fluid, to which we give the name G2,separately from those trapped on vortices, which we call G1 [18]. Using this separationscheme, we proposed a simple estimation of the mean free path of G2 particles throughthe vortex tangle, we showed that G1 velocity fluctuations are likely caused by fluctuationsof the local vortex line velocity, and we showed that power-law tails in transverse particlevelocity probability density functions (PDFs) are due entirely to G1. In the present paper, weexpand upon these ideas, with a focus on demonstrating the utility of PTV for quantitativeanalysis of the vortex tangle. After a brief overview of the experimental apparatus and dataanalysis scheme in Sec. II, we motivate, present, and discuss each main result in its ownsection. An experimental estimation of the mean vortex line spacing using flow visualizationis presented in Sect. III. An experimental estimation of c , a parameter related to energydissipation in quantum turbulence [19], is presented in Sect. IV. A direct link between vortexline reconnection and G1 transverse velocity PDF power-law tails is established in Sect. V.We conclude in Sect. VI. II. EXPERIMENTAL PROTOCOL
This work employs the same apparatus, illustrated in Fig. 1, described in our previouspapers [18, 20]. Solidified deuterium tracer particles with mean diameter d p ≈ . µ m aredelivered via stainless steel tube to the center of a 1 . × . ×
33 cm vertical flow channelimmersed in a saturated He II bath. The delivery tube is then retracted by an external3
IG. 1. Simple illustration of the experimental apparatus (not to scale). electric motor, and a 400 Ω planar resistive heater at the bottom of the channel generatesthermal counterflow. A laser beam with cross section approximately 200 µ m thick and9 mm tall illuminates particles as they move through the geometric center of the channel,and a high-speed digital camera captures them on video. A modified feature point trackingalgorithm [21] yields the position of each particle in each video frame, information that canbe readily transformed into velocity measurements for each particle.Our data set covers three temperatures, T = 1 .
70, 1.85, and 2.00 K, with heat currentsranging from 29–481 mW/cm . Fig. 2 shows (a) streamwise and (b) transverse particlevelocity PDFs typical of PTV measurements in thermal counterflow driven by relativelylow heat flux. In the streamwise PDFs, one peak arises from G1, the name we give toparticles trapped on quantized vortices, and the other from G2, the name we give to particlesentrained by the normal fluid. To determine the category to which a velocity measurement v p contributes, we apply the following criteria [18]. If v p < µ − σ , where µ and σ are the mean and standard deviation, respectively, of the G2 peak, then v p exhibits G1behavior. If v p > µ + 2 σ , then v p exhibits G2 behavior. In the event that the peaks are4 v p (mm/s) P r( v p ) / v ( s / mm ) v p G1 FitG2 Fit -6 -4 -2 0 2 4 6 u p / u -4 -3 -2 -1 P r( u p / u ) / ( u p / u ) G1G2GaussianPower Law (a) (b)
FIG. 2. Typical (a) streamwise velocity PDF ( T = 1 .
85 K, q = 38 mW/cm ) and (b) transversevelocity PDFs ( T = 2 .
00 K, q = 113 mW/cm ) obtained from PTV measurements of thermalcounterflow at relatively low heat flux. well separated, i.e., µ − µ > σ + σ ), the criteria are reversed. The separation schemeresults in ensembles of velocity measurements representing G1 and G2, which can be usedfor further analysis, including generation of the transverse PDFs of Fig. 2(b), which arenormalized by standard deviation. It can be seen that a Gaussian curve ( µ = 0, σ = 1),indicated by the solid black line, fits the core of the G1 PDF and the entirety of the G2PDF. Beyond about four standard deviations from the center, a power law curve ( ∝ | u p | − ),indicated by the dashed black line, passes through the tail of the G1 PDF.In addition to PTV, we employ second sound attenuation to measure the average vortexline length per unit volume, or vortex line density L , inside the channel. As a consequenceof the two fluid model, He II supports multiple speeds of sound, including second sound,the wave-like propagation of temperature or entropy. A pair of second sound transducers,as illustrated in Fig. 1, establish a standing second sound wave across the channel that isattenuated in the presence of quantized vortices, and the vortex line density can be obtainedfrom the degree of attenuation [22]. 5 II. MEAN FREE PATH AND VORTEX LINE DENSITY
To explain the underlying mechanism that governs whether particles exhibit G1 or G2behavior, we acknowledged that at the beginning of each video acquisition, some particlesare already trapped on vortices (G1) while others are not (G2). Untrapped particles thenmove over a distance comparable to their mean free path s through the vortex tangle. Weproposed a fairly simple formula to describe the mean free path, s ≤ πd p L , (1)and showed that, qualitatively, the mean free path predicted by this simple model agreeswith the length of observed G2 tracks [18]. To explore the usefulness of this model, we willaccept its validity, and use Eqn. (1) to estimate the mean vortex line spacing ℓ = L − / byusing the length of G2 tracks to represent s .We first recognize that, for 2D planar velocimetry, G2 tracks begin and end for reasonsother than de-trapping or trapping events. Particles tracing the normal fluid are free to enterand leave the imaging plane through the top or bottom of the image as well as by driftingin- or out-of-plane in the direction normal to the camera, leading to many observations oftracks that are shorter than the mean free path. It is therefore inappropriate to assume thatthe mean G2 track length accurately represents s . Alternatively, since it is not possible toobserve a track longer than the mean free path (at least, not much longer), we estimate itwith the mean length of the longest 10% of observed G2 tracks.Fig. 3 shows the mean vortex line spacing as a function of heat flux for (a) 1.70 K,(b) 1.85 K, and (c) 2.00 K. Red markers predict ℓ using Eqn. (1), where the longest 10% of G2tracks observed for each point in the parameter space represent s and d p = 4 . ± . µ m [20].Blue markers represent the line spacing obtained from traditional second sound attenuation.For a simple approximation, the accuracy is remarkable, and suggests that PTV maybe a viable method for estimating vortex line density in steady-state thermal counterflow.However, the assumption that G2 track lengths represent the mean free path should beapproached with caution. It does not account for the possibility of a mean vortex tangledrift, an effect which is difficult to predict due to limited understanding. The true meanfree path might be given as s = s p (1 − C ), where s p represents the observed mean freepath of the particles (i.e., the mean length of the longest 10% of observed G2 tracks), and6
90 120 150
Heat Flux (mW/cm ) -3 -2 -1 V o r t e x L i ne S pa c i ng ( c m ) Track Length MethodSecond Sound Method20 60 100 140 180
Heat Flux (mW/cm ) -3 -2 -1 V o r t e x L i ne S pa c i ng ( c m ) Track Length MethodSecond Sound Method (c)(b)
30 45 60 75
Heat Flux (mW/cm ) -3 -2 -1 V o r t e x L i ne S pa c i ng ( c m ) Track Length MethodSecond Sound Method (a)
FIG. 3. Prediction of mean vortex line spacing using G2 mean free path model and traditionalsecond sound attenuation for (a) T = 1 .
70 K, (b) T = 1 .
85 K, and (c) T = 2 .
00 K. C = v L /v n is a correction factor relating the mean vortex tangle drift velocity v L to thenormal fluid velocity. Several experiments suggest that v L is similar to v s for small enoughheat flux [15, 18, 23, 24]. We therefore constrain C > v s /v n , or equivalently, throughconservation of mass, C > − ρ n /ρ s . It has also been reported that the tangle may drift inthe same direction as the normal fluid when the heat flux is larger [25].7isparity among existing experiments makes it difficult to define a precise value for C ,but we can infer the following picture. For counterflow driven by small heat flux, when v L ≈ v s [15, 18, 23, 24], C ≈ − ρ n /ρ s and a correction factor of ρ/ρ s should be applied toEqn. 1. This may improve agreement between the curves shown in Fig. 3 for low heat flux.For higher heat flux, when v L << v n [23, 26], C ≈
0, which is consistent with the agreementshown in Fig. 3 between line spacing measured by second sound attenuation and the meanfree path model for higher heat flux.Subject to these minor corrections, the mean free path model shows strong validity as analternative to second sound attenuation for estimation of vortex line density in steady-statethermal counterflow. Since PTV provides spatially resolved velocity measurements, this toolmakes localized measurements of vortex line density possible.
IV. EXPERIMENTAL MEASUREMENT OF c In our recent paper, we showed that streamwise and transverse G1 velocity fluctuations σ G as functions of counterflow velocity v ns = | v n | + | v s | can be fit remarkably well by theanticipated root mean square vortex line velocity fluctuations h v L i / [18]. Based on localself-induced vortex motion, h v L i / is given by (cid:10) v L (cid:11) / ≈ κc γ π ln (cid:18) ℓξ (cid:19) (cid:18) ρρ s v n − v (cid:19) (2)provided the counterflow velocity ρv n /ρ s exceeds a small critical velocity v . To illustrateagreement with σ G , we computed h v L i / using values for γ (a temperature-dependentparameter relating L to v ns ), v , and the parameter c reported in the recent work of Gao etal. [27, 28], and we approximated the mean vortex line spacing ℓ = L − / across the entireparameter space. Since the agreement between σ G and h v L i / appeared to be reasonable,we can use measured G1 velocity fluctuations to represent h v L i / , and apply Eqn. (2) toestimate c . This parameter was introduced as a temperature-dependent coefficient relatingvortex line density to local line curvature [29], and has received recent attention for its rolein vortex line dynamics.The parameter c is used to describe both the build-up [30] and decay [19] of quantumturbulence. Recent numerical simulations and experiments suggest that besides tempera-ture, c may depend on the specific flow geometry [28] as well as local vortex line density [30].8ince estimation of spatially-dependent c is still beyond the capability of numerical simula-tions [28], and the traditional second sound method provides averaged information across themeasurement volume, application of PTV to make whole-field measurements of G1 velocityfluctuations offers a unique opportunity to investigate the spatial dependence of c .To demonstrate estimation of c using experimental G1 particle data, we use Eqn. (2)to calculate its average value across the imaging plane. We begin by obtaining values for ℓ , γ , and v using our own apparatus, employing both flow visualization and second soundattenuation according to the procedures outlined by Gao et al. [27]. The results for eachtemperature are tabulated in Table I. It is unclear why v < TABLE I. Measured values for the γ -coefficient and v .T (K) γ (s/cm ) v (cm/s) c . ± . . ± .
135 0 . ± . . ± . . ± .
062 0 . ± . . ± . − . ± .
038 0 . ± . Values for c can then be obtained using the procedure illustrated in Fig. 4. Panels (a)–(c) show σ G ln − ( ℓ/ξ ) as a function of v ns − v for T = 1 .
70, 1.85, and 2.00 K, respectively.The dashed lines represent linear fits for which, according to Eqn. (2), the slope is κγc / π .Values for c that produce the lines are shown in Fig. 4(d) and tabulated in Table I. Theyare slightly less than those reported in existing simulations [28, 29] and experiments [30],but the overall trend, a decrease with increasing temperature, is preserved. Geometricfactors, i.e., the relatively large size of the experimental flow channel, may be partiallyresponsible for the difference. It should also be kept in mind that while fluctuations of thelocal vortex line velocity play a large role in G1 velocity fluctuations [18], they are not solelyresponsible. Other factors, such as drag from the normal fluid, can also affect the G1 particlevelocity [7, 8]. Nonetheless, the results indicate that use of PTV to estimate c is indeedfeasible, implying that the parameter can be spatially resolved by estimating its local valuebased on local G1 velocity fluctuations. 9 v ns - v (cm/s) G l n - ( l / )( c m / s ) TransverseStreamwiseLinear Fit 1.6 1.7 1.8 1.9 2 2.1
T (K) c v ns - v (cm/s) G l n - ( l / )( c m / s ) TransverseStreamwiseLinear Fit0 0.3 0.6 0.9 1.2 1.5 1.8 v ns - v (cm/s) G l n - ( l / )( c m / s ) TransverseStreamwiseLinear Fit (a) (b)(c) (d)
FIG. 4. Linear fit to σ G ln − ( ℓ/ξ ) as a function of v ns − v at (a) T = 1 .
70, (b) 1.85, and (c)2.00 K. (d) Extracted values of c as a function of temperature. V. VORTEX RECONNECTION AND VELOCITY PDF TAILS
Transverse velocity u p PDFs for solidified particles tracing thermal counterflow typicallyexhibit a classical Gaussian core with | u p | − power law tails [16, 31]. Recently, we appliedour separation scheme to reveal that these tails can be attributed to G1 [18]. The tails mightarise from two physical mechanisms, the superfluid velocity field or vortex reconnection, butit is not known conclusively which is responsible.The PDF for a velocity field in the vicinity of a singular vortex is proportional to | v | − for large values of the velocity, i.e., in the tail region [32, 33]. Therefore, the PDF for v s in10he vicinity of a quantized vortex should exhibit the power law tails. This explanation hasbeen invoked to explain the observation of power law tails in transverse particle velocity u p PDFs [16, 17]. We note in passing, however, that solid particles in the vicinity of a vortexline tend to become trapped rather than respond to the superfluid itself [33–35].Alternatively, when two vortices approach, reconnect, and separate from each other, theminimum separation distance δ grows in time as δ ∝ | t − t | / , where t is the time atwhich the reconnection occurs [31, 36, 37]. The separation velocity is then proportional to | t − t | − / , and the PDF should take a form proportional to | v | − . Since particles have atendency to become trapped on vortices, this scaling should be reflected in the observedmotion of trapped particles and their corresponding velocity PDFs. Indeed, Paoletti et al.have shown through visualization of decaying counterflow that particle velocity PDFs takethe form | v | − , and they identified numerous pairs of particles moving away from each otherwith the separation distance growing proportionally to | t − t | / [31]. This is certainly aconvincing link between vortex reconnection and velocity PDF power law tails, but no directlink was established between these pairs of particles and the tail region of the PDF.With the separation scheme, a direct link can be established by analyzing the kinematicsof particles that exhibit G1 behavior and contribute to the transverse PDF tail region. Sinceour data comes from steady-state counterflow, acceleration along the tracks must be con-sidered to remove effects of the mean flow. Based on the δ ∝ | t − t | / scaling, accelerationalong tracks containing a vortrex reconnection should be proportional to | t − t | − / .We first identify G1 tracks containing a segment that contributes to the G1 transversevelocity PDF tail region, which we define as | u p | > µ u p + 4 σ u p (see Fig. 2). Figs. 5(a–c)show an example of these G1 tracks at each temperature, with the first point in each trackindicated by a blue circle. In each of these tracks, the high velocity segment (indicated bythe arrow) is accompanied by a strong acceleration and deceleration as well as a noticeablechange in direction. These characteristics are indicative of vortex reconnection.As a first approximation, we assume that reconnection occurs midway through the tracksegment that contributes to the PDF tail. It follows that the beginning of the identifiedsegment occurs at t = t − dt/
2, and the end of the segment occurs at t = t + dt/
2, where dt is the image acquisition interval. We can then calculate acceleration along each track awayfrom (forward event) and towards (reverse event) [31] the reconnection site as a function ofelapsed time, and fit the acceleration magnitude k a k for each candidate track with a power11
95 500 505 510 515
Image Index (px) I m age I nde x ( p x ) -3 -2 -1 |t-t | (s) -3 -2 -1 || a || ( m / s ) t > t t < t -3 -2 -1 |t-t | (s) -3 -2 -1 || a || ( m / s ) t > t t < t -3 -2 -1 |t-t | (s) -3 -2 -1 || a || ( m / s ) t > t t < t
196 198 200 202 204
Image Index (px) I m age I nde x ( p x )
480 485 490
Image Index (px) I m age I nde x ( p x ) (a)(b)(c) (d)(e)(f) FIG. 5. Selected G1 tracks that contribute to transverse PDF tails at (a) 1.70 K, (b) 1.85 K, and(c) 2.00 K. Blue circles indicate the beginning of each track and black arrows indicate the segmentthat contributes to the transverse velocity PDF tail. (d)–(f) Corresponding acceleration along thetracks. Dashed lines represent Eqn. (3). k a k = C | t − t | − / (3)where C is the fitting parameter. Figs. 5(d–f) show the acceleration magnitudes along eachof the corresponding tracks in Figs. 5(a–c). Forward events are shown in blue and reverseevents in red, and the dashed line represents Eqn. (3). In all three cases, acceleration alongthe track agrees remarkably well with the predicted | t − t | − / scaling. Interestingly, thefitting parameter is approximately the same in all three cases, having an average valueof C ≈ .
25 mm/s independent of temperature. This provides a positive link betweentransverse velocity PDF tails and vortex reconnection, since the G1 tracks that contributeto the tails obey the acceleration scaling extrapolated from the work of Paoletti et al. [31].
VI. CONCLUSION
Our separation scheme for separately analyzing particles entrained by the normal fluidand those trapped on quantized vortices has led to three noteworthy observations. A simplebut remarkably accurate model for the mean free path of particles traveling through thevortex tangle relates G2 track length to mean vortex line spacing, providing a new wayto estimate localized vortex line density in steady-state thermal counterflow. G1 velocityfluctuations have been used to estimate the value of c , an important parameter related todissipation of turbulent energy in He II, using a flow visualization method that allows spatialresolution. Finally, vortex reconnection has been positively linked to particle velocity PDFpower law tails by showing that acceleration along G1 tracks that contribute to the tailsfollows the predicted scaling for vortices accelerating away from (or towards) a reconnectionsite. Together, these observation indicate that with an appropriate approach to data analy-sis, i.e., our separation scheme, PTV is indeed a useful utility for quantifying characteristicsof the vortex tangle in steady thermal counterflow. ACKNOWLEDGMENTS
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