12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

STUDY ON THE CURVATURE OF LAGRANGIAN PARTICLETRAJECTORIES IN QUANTUM TURBULENCE

Naoto Sakaki
Graduate School of Engineering
Nagoya University
Furo-cho, Nagoya-city, 464-8603, Japan
mw.41g.0347@s.thers.ac.jp

Takumi Maruyama
Graduate School of Engineering
Nagoya University
Furo-cho, Nagoya-city, 464-8603, Japan
takumi.maruyama@a.mbox.nagoya-u.ac.jp

Yoshiyuki Tsuji
Graduate School of Engineering
Nagoya University
Furo-cho, Nagoya-city, 464-8603, Japan
c42406a@nucc.cc.nagoya-u.ac.jp

ABSTRACT

thermal counterﬂow. In classical turbulence, particle acceleration and the curvature of Lagrangian trajectory are analyzed by
Xu et al. (2007). In magnetic turbulence, the curvature of themagnetic ﬁeld line is analyzed in a two-dimensional (2D) magneto hydro dynamic (MHD) turbulence (Yang et al. (2019)).
These investigations revealed that the probability density function (PDF) of curvature has an apparent power-law tail shape.
The thermal counterﬂow ﬂow characteristics for helium
are different from classical turbulence and MHD turbulence.
In particular, the PDF varies and often exhibits a bimodal distribution, but the PDF of classical turbulence in homogeneous
isotropic turbulence closely resembles a Gaussian distribution.
It is also assumed that the Lagrangian trajectories of small particles show different characteristics in quantum and classicalturbulence. Trajectory of the particle can rapidly change before and after the particles are trapped by the quantum vortex.
Therefore, characterizing this process by geometrical information of curvature is interesting. In addition, other physicalquantities, such as velocity can be investigated in the region
where the curvature is high for further analysis. Therefore, we
aim to determine the particle trajectory features in quantum
turbulence when they are conditioned by their curvature.
In this study, we analyzed the motion of solid hydrogen particles in thermal counterﬂow using the PTV algorithm.
Here, we do not classify the particle size because the curvature
PDF does not depend on the particle size (Sakaki et al. (2022)),although the PDF of vertical velocity depends on the particle
size (Kubo & Tsuji (2017)). The relation between the particle
velocity and curvature of trajectories is studied by conditional
sampling. Moreover, the particle velocity conditioned by highand low curvature regions is compared with the velocity of thetwo-ﬂuid model. Finally, the interaction between the quantum
vortex and the tracer particles is discussed.

Small particle trajectories are visualized in thermal counterﬂow using the particle tracking velocimetry technique, andthe curvature of two dimensional Lagrangian trajectories arestudied. It is found that the probability density function of
the curvature demonstrates a power-law tail similar to that ofclassical turbulence. The curvature distribution is classiﬁed
into three regions with high, medium, and low values, and theparticle velocity is averaged in each region. Furthermore, theparticle velocity in the low curvature region clearly shows abimodal distribution and agrees with the two-ﬂuid model inthe case of low heat ﬂux. However, in the high curvature region, the particle velocity deviates from the theoretical value
and exhibits a Gaussian distribution.

INTRODUCTION
Quantum turbulence is one of the most important researchﬁelds. Thermal counterﬂow is a simple example of quantum
turbulence. In thermal counterﬂow, the two-ﬂuid velocity ofhelium4 demonstrates the following relation, when considering mass conservation, ρs vs + ρn vn = 0, where ρ and v are ﬂuid
density and velocity, respectively. Moreover, the lower subscript n and s represent the superﬂuid and normal ﬂuid components, respectively. The normal ﬂuid velocity is given by
vn = q/ρ ST(1)with a total density of ρ = ρs + ρn , entropy S, temperature T ,and heat ﬂux q. Additionally, the quantum vortices are generated and tangled in turbulent ﬂow in high q.
In recent years, visualization of thermal counterﬂow has
been conducted by several researchers (Paoletti et al. (2008);
Chagovets & Van Sciver (2011); Kubo & Tsuji (2017); Mastracci & Guo (2018)), using particle tracking velocimetry(PTV). Previous research on thermal counterﬂow visualization mainly focuses on the velocity and acceleration of the
tracer particles in Lagrangian properties. However, the curvature of tracer trajectories has not been studied in detail in

EXPERIMENTS
A schematic of the experimental setup is shown in Fig. 1,
where the acrylic rectangular channel is set inside the cryostat.
The channel cross-section has an area of A = 20 × 20 mm2
and the height is H = 260 mm (Kubo & Tsuji (2017)), and1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
lence (Braun et al. (2006); Xu et al. (2007)). The PDF of
κ (P(κ )) demonstrates the two power-law regions where thehigh and low curvature regions have corresponding PDFs ofP(κ ) ∝ κ −5/2 and P(κ ) ∝ κ , respectively. Xu et al. (2007)
reported that this feature could be understood by the Gaussian property of random variables but not turbulent small-scalestatistics. As is well known, the velocity ﬂuctuation in classical turbulence closely mimics the Gaussian property. In addition, the velocity ﬂuctuations (ux , uy , uz ) in HIT are independent of one another, then the magnitude u2 = u2x + u2y + u2z
should follow a chi-squared distribution with degree of freedom three. When κ → ∞, or u → 0, the PDF of κ follows
the distribution of u−2 with the assumption of ﬁnite an in thislimit, suggesting a power-law with an exponent of −5/2. Similarly, the power-law scaling of κ → 0 is derived. The PDF ofcurvature scales like the PDF of an as an → 0 with the assumption that the components of an are independent Gaussian random variables. Then the PDF of a2n then follows a chi-squared
distribution with a degree of freedom of two. In the case ofκ → 0, the PDF of κ follows the distribution of an as an → 0,then the power-law with an exponent of one is derived by Xu
et al. (2007).
Moreover, the PDF of curvature in 2D case is also discussed. Since the PDF of an near zero can be approximated asa quasi-Gaussian distribution, then the power-law of PDFs arederived as P(κ ) ≈ κ −2 and P(κ ) ≈ κ 0 in κ → ∞ and κ → 0,
respectively, using Taylor’s expansion.

Figure 1. Schematic view of experimental settings.

Table 1. This is an example of a table.

Tq

Number of datasets

1.9 K≈ 800 W/m2

11

2.0 K

≈ 300 W/m2

3

2.0 K

≈ 400 W/m2

5

2.0 K

≈ 800 W/m2

13

2.1 K

≈ 800 W/m2

3
RESULTS AND DISCUSSIONIn Fig. 2, the PDF of the vertical velocity vy in the case of
T = 2 K with q = 310 W/m2 is shown, in which the verticalvelocity is normalized by its standard deviation σ . The PDF
has a bimodal distribution which is similar to previous studies(Paoletti et al. (2008); Mastracci & Guo (2018)). This bimodal
distribution is conﬁrmed in almost all cases. The bimodal distribution is approximated by two Gaussian distributions represented by the blue and red dashed lines in Fig. 2. Their meanvalues are deﬁned as vgauss,s and vgauss,n , respectively. The redline is the positive mean value vgauss,n and the velocity vn iscalculated by the two-ﬂuid model (equation (1)), which is indicated by the solid black line. The bimodal distribution was notobserved in the case of T = 2.1 K; however, the positive peakalways appears. This was also conﬁrmed by previous study
Mastracci et al. (2017) for similar heat ﬂux q ≈ 900 W/m2 .
The mean velocity vgauss,n was plotted against vn in Fig. 3. Different colors represent various experimental conditions. Moreover, the solid line is given by vgauss,n = vn , and the dotted line
is denoted by vgauss,n = vn /2. There have been discussions
on the particle velocity in counterﬂow (Paoletti et al. (2008);
Chagovets & Van Sciver (2011)): the previous studies are summarized in Mastracci & Guo (2018). In the previous studies(Paoletti et al. (2008); Chagovets & Van Sciver (2011); Mastracci & Guo (2018)), the vgauss,n agrees with vn . In this measurement, vgauss,n and vn increase as the heat power becomeslarge, and vgauss,n locates between vn and vn /2.
The example PDF of curvature normalized by the standard deviation σκ is plotted in Fig. 4 in the case of T = 2.0 K
with q = 470 W/m2 . In the larger curvature region, the powerlaw κ −2 is observed. On the other hand, in the smaller curvature region, the power-law κ 0 is shown. The PDFs in othercases have the same trend as the case of T = 2.0 K with
q = 470W/m2 . PDFs of curvature contain no signs of nonclassical behavior attributed to the quantum vortex in superﬂuid. For further analysis, the PDF of the curvature is divided

the bath temperature T is varied from 1.9 to 2.1 K. Furthermore, the plate heater is located at the bottom, and theheat ﬂux q is varied in the range of 300 − 800 W/m2 . The
thermal counterﬂow is generated inside the channel, and theexperimental conditions are listed in Table 1 A high-speed
camera(1024 × 1024 pixels, 8 bit) is used for visualizing thearea of 8.7 × 8.7 mm2 at 250 fps. A continuous laser (wavelength 532 nm, diode-pumped solid-state laser and 1 W at
maximum power) is adopted to make the laser sheet with athickness of about 1 mm. A helium and hydrogen mixing
chamber is designed to vary the mixing ratio and the spouting pressure. The hydrogen particles are generated in the liquid helium. We adopted a mixing ratio of He:H2 = 40:1 anda spouting pressure 20 kPa. Additionally, the injection is carried out just above the λ point, and then the bath temperature
is decreased. In this study, the particle tracking algorithm1 is
employed where the particle size, d, is similar to the previous experiment (Kubo & Tsuji (2019)). It is in the range of5 − 40 µ m (Sakaki et al. (2022)). In this research, we analyze
the motion of all particle sizes, and we do not classify the sizeof the particles.

ANALYSIS METHOD
The instantaneous curvature can be expressed as κ =an /u2 (Braun et al. (2006)) utilizing the Frenet formulas,where an is the magnitude of the normal acceleration, and uis the velocity. The velocity and acceleration are calculated by
the numerical derivative with the central difference method.
The uncertainly is O(∆t) for velocity and O(∆t 2 ) for acceleration.
The distribution of κ has been studied in classical turbu-

1 This was developed by Crocker J. & Weeks E.& Grier D..

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4. PDF of curvature normalized by the standard deviation is shown in the case of T = 2.0 K with q = 470 W/m2 .
The dotted black line represents the κ 0 . The solid black line isgiven by the κ −2 .

Figure 2. PDF of vertical velocity component for the case of
T = 2 K with q = 310 W/m2 . The bimodal PDF is approximated by two Gaussian proﬁles denoted by two dashed lines.
The solid red line is the mean value vy , the mean of the positive Gaussian ﬁtting. The solid black line is the normal ﬂuid
velocity calculated by the two-ﬂuid model.

Figure 5. PDF of curvature is divided into the high, medium,and low regions for the T = 2.0 K with q = 470 W/m2 . Greencircles denote the PDF of curvature, the left solid line is thethreshold between the low and medium region, and the rightsolid line is the threshold between the high and medium region.
Figure 3. Comparison of mean vertical velocity in experiments and the two-ﬂuid model. The red color is T = 1.9 K withq ≈ 800 W/m2 , blue is T = 2.0 K with q ≈ 300 W/m2 , green
is T = 2.0 K with q ≈ 400 W/m2 , black is T = 2.0 K with q ≈800 W/m2 , and magenta is T = 2.1 K with q ≈ 800 W/m2 .
The vertical error bar is the standard deviation in the each data
sets. The horizontal error bar is the range of the theoreticalnormal ﬂuid in the each data sets. The solid line represents
vgauss,n = vn . The dotted line denotes vgauss,n = vn /2.

ical example of the PDF is plotted in Fig. 5, where the low andthe high curvature PDFs are given by κ /σκ < 9.44 × 10−4 and3.80 × 10−3 < κ /σκ , respectively. The histogram of particles
is plotted in Fig. 6 as a function of curvature and vertical velocity. As described in Fig. 5, the curvature is classiﬁed into three
regions according to the deﬁnition. The histogram indicatesthe high probability in the medium curvature region where thepositive vertical velocity is dominant. Around a small vertical velocity of vy ≃ 0, the high curvature motion is indicated.
From these observations, we understand that the vertical velocity has different characteristics depending on the curvature.
Then, the vertical velocity is averaged in each curvature region, and it is usually called conditional sampling.
The vertical velocity is divided into high, medium, andlow curvature regions, and their PDF is shown in Fig. 7 for thecase of T = 2 K with q = 470 W/m2 . The original PDF shows
the bimodal distribution. However, in the velocity conditionedby the low curvature region, there are two more clearly sepa-

into three regions: low, medium, and high curvature regions as
described in Fig. 5. This classiﬁcation is deﬁned as the probability that each region is equal for convenience, therefore it
does not contain any physical signs. In the low curvature region, the PDF shows ﬂat, or the power law is κ 0 , and in thehigh curvature region, the PDF behaves κ −2 . The medium
region is located between these two power-law regions. A typ-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 6. Histogram of particles as a function of y direction velocity and curvature in the case of T = 2.0 K with
q = 470 W/m2 . The lower solid line represents the threshold
between the low and medium regions, and the solid upper line
signiﬁes the threshold between the high and medium regions.
The left dotted line denotes the theoretical superﬂuid velocity
divided by the standard deviation of the vertical velocity andthe right solid line is the theoretical normal ﬂuid velocity divided by the standard deviation of the vertical velocity.

Figure 8. Comparison of mean vertical velocity in experiments and the two-ﬂuid model. The red color is T = 1.9 K withq ≈ 800 W/m2 , blue is T = 2.0 K with q ≈ 300 W/m2 , green
is T = 2.0 K with q ≈ 400 W/m2 , black is T = 2.0 K with q ≈800 W/m2 , and magenta is T = 2.1 K with q ≈ 800 W/m2 .
The vertical error bar is the standard deviation in the each data
sets. The horizontal error bar is the range of the theoreticalnormal ﬂuid in the each data sets. The solid line represents
vgauss,n = vn . The dotted line denotes vgauss,n = vn /2.

velocity in the medium curvature region but is only slightly affected by high curvature regions. The values of the negative
peak vgaussu,s are also different depending on the curvature.
In addition, negative and positive distributions are enhanced
compared with the original PDF because, in the low heat ﬂux,
particles that feel stokes drag do not interact with vortices frequently and sometimes go down with quantum vortices. In
the high heat ﬂux, the PDF of the vertical velocity conditionedby the low curvature region doesn’t have clear bimodal distribution, although the original PDF has a bimodal distribution.
The tracer particles tend to move fast enough to detrap fromthe quantum vortices. Thus, the lower peak is located aroundthe small positive value, and the vertical velocity PDF conditioned by the low curvature region does not have clear bimodal
distribution for the high heat ﬂux.
The vertical velocity calculated by the two-ﬂuid modelis plotted in Fig. 7. Additionally, the normal ﬂuid velocity vgaussu,n agrees with vn . Then the low curvature region
along the Lagrange trajectory represents the laminar ﬂow region where the particles are carried by Stokes drag. The negative particle velocity vgaussu,s is close to vs compared with original PDF, however it does not agree with the two-ﬂuid model.
In Fig. 8, the vertical velocity (vgaussu,n )Low conditionedby the low curvature region is plotted against vn by the ∗ symbol. Furthermore, conditional normal ﬂuid velocities are close
to those of the two-ﬂuid model in the case of vn < 2 mm/s.
However, the conditional velocity still deviates from the original vn in high-velocity regions. In Fig. 9, we calculate thedifference (vgaussu,n )Low − vgauss,n . Note that the data at T =2.1 K with q ≈ 800 W/m2 is not included here because bimodal distribution is not shown in this condition. It is foundthat the difference is large in vn < 2 mm/s but small for

Figure 7. PDF of vertical velocity divided by its standard
deviation in high (top left), medium (top right), low (bottom
left), and all (bottom right) curvature regions at T = 2 K andq = 470 W/m2 . In the high region, the dotted orange line represents the Gaussian ﬁt.

rated peaks. The positive peak is given by vgauss,n /σ ≃ 2.0,
and the negative peak is denoted by vgauss,s /σ ≃ −1.0. In thehigh curvature region, the proﬁle of the PDF is approximatedby the Gaussian distribution. In addition, we found that thehorizontal velocity in high curvature region is also approximated by Gaussian distribution (not shown here). Therefore,
the magnitude of the velocity v2x +v2y follows a chi-squared distribution with a degree of freedom two. When κ → ∞, the PDFof κ follows the distribution of (v2x + v2y )−2 with the assumption of ﬁnite an in this limit. Therefore, the PDF of curvature
has a power-law tail of −2. Similarly, the power-law of κ 0 is
assumed because the normal acceleration could be thought ofas Gaussian, and the velocity is ﬁnite in the limit of κ → 0 or
an → 0.
The positive velocity peak is attributed to the conditional412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 9. Each squares represent difference between themean value of Fig. 3 and Fig. 8. The color is the same withthat of Fig. 3.

Figure 10. An example of high curvature points on the trajectories for the case of T = 2.0 K with q = 470 W/m2 . Red
and blue trajectories represent the upward and downward motion in the average, respectively. Black points indicate the highcurvature region, 7.0 × 10−2 < κ /σκ . Cyan points are the low
curvature region, κ /σκ < 4.0 × 10−5 .

vn > 2 mm/s. Since the quantum vortex density L increases
as the heat ﬂux increases, the interaction between the tracer
particles and quantum vortices frequently occurs. Thus, thezigzag pattern is shown in the particle trajectories. Moreover,the vertical velocity PDF conditioned by the low curvature region tends to include these velocity components. Therefore,
the difference between the PDF of original and low curvature
region is small in the high heat ﬂux.
We plot the examples of the high curvature points on the
trajectories for T = 2.0 K with q = 470 W/m2 in Fig. 10.
In Fig. 10, the red and blue points represent the upward anddownward motion in the average, respectively. Along the
trajectory, the high (7.0 × 10−2 < κ /σκ ) and low (κ /σκ <4.0 × 10−5 ) curvature region are colored in black and cyan,
respectively. As expected, the high curvature region indicates
nonstraight motion or the point where the particle motion suddenly changes its direction, which could be due to the quantum
vortex interaction. On the other hand, the low curvature region
represents the points on almost straight lines.

This work was supported by JSPS KAKENHI Grant
Number 19H00747．The author N. S. would like to takethis opportunity to thank the “Interdisciplinary Frontier NextGeneration Researcher Program of the Tokai Higher Education
and Research System.”

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CONCLUSION
In this study, we calculate the curvature of the 2DLagrangian trajectory in thermal counterﬂow using PTV. The
Lagrange velocities (vertical component vy ) are conditionallyaveraged by curvature. By removing the high curvature region
and conditioning the vertical velocity in the low curvature region, we obtain a bimodal velocity distribution, then the normal ﬂuid velocity is close to the value of the two-ﬂuid model.
Therefore, the quantum vortex which shows tangled complex
interactions are sufﬁciently separated based on the curvaturein the low heat ﬂux case. However, in the high heat ﬂux condition, bimodal distribution does not appear in the low curvature
region although it is seen in the original distribution. Both vy
and vx have a Gaussian distribution with a zero-mean value
in the high curvature region. Based on this experimental evidence, we can understand that the PDF of the curvature has
power-law scaling ≈ κ −2 in the high curvature region, and is
conﬁrmed in the present experiment.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Lu, Q. & Chen, S. 2019 Role of magnetic ﬁeld curvature inmagnetohydrodynamic turbulence. Physics of Plasmas 26,

072306.

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