# On the determination of vortex ring vorticity using Lagrangian particles

O. Outrata1  M. Pavelka1  J. Hron1  M. La Mantia1
<EMAIL_ADDRESS>J. I. Polanco2 and G. Krstulovic2  1Faculty of
Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague,
Czech Republic 2Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, Boulevard de l’Observatoire CS 34229 – F 06304 Nice
Cedex 4, France

###### Abstract

Particles are a widespread tool for obtaining information from fluid flows.
When Eulerian data are unavailable, they may be employed to estimate flow
fields or to identify coherent flow structures. Here we numerically examine
the possibility of using particles to capture the dynamics of isolated vortex
rings propagating in a quiescent fluid. The analysis is performed starting
from numerical simulations of the Navier–Stokes and the
Hall–Vinen–Bekarevich–Khalatnikov equations, respectively describing the
dynamics of a Newtonian fluid and a finite-temperature superfluid. The flow-
induced positions and velocities of particles suspended in the fluid are
specifically used to compute the Lagrangian pseudovorticity field, a proxy for
the Eulerian vorticity field recently employed in the context of superfluid
4He experiments. We show that, when calculated from ideal Lagrangian tracers
or from particles with low inertia, the pseudovorticity field can be
accurately used to estimate the propagation velocity and the growth of
isolated vortex rings, although the quantitative reconstruction of the
corresponding vorticity fields remains challenging. On the other hand,
particles with high inertia tend to preferentially sample specific flow
regions, resulting in biased pseudovorticity fields that pollute the
estimation of the vortex ring properties. Overall, this work neatly
demonstrates that the Lagrangian pseudovorticity is a valuable tool for
estimating the strength of macroscopic vortical structures in the absence of
Eulerian data, which is, for example, the case of superfluid 4He experiments.

###### keywords:

##  1 Introduction

The accurate and well-resolved measurement in space and time of fluid flows
represents one of the main ongoing challenges in experimental fluid dynamics.
A number of methods have been developed to capture the evolution of velocity
fields and other relevant quantities in laboratory flows. A common approach is
to visualize the motion of relatively small particles advected by the flow of
interest, from which properties of the latter can be inferred (see, for
example, Raffel et al., 2018). Various techniques can then be employed to
quantitatively process the obtained images and the choice mainly depends on
the features to be investigated.

The most popular tools are particle image velocimetry (PIV), which allows to
access instantaneous flows fields, i.e. Eulerian data, and particle tracking
velocimetry (PTV), which is a Lagrangian technique, focusing on the temporal
dynamics of the visualized particles. In principle, both methods can be used
to gather complementary information on the same flow but, in practice, this is
not always possible. Leaving aside for a moment the important roles played by
particle size and inertia, addressed below, the main limiting factor is the
particle number density, discussed, for example, by Nobach & Bodenschatz
(2009). Specifically, if the number of particles per unit area is smaller than
about $0.01$ pixel-2, significant errors occur in PIV measurements. It then
follows that, if one cannot seed the flow of interest with enough particles,
PTV is the only choice to get quantitative information from the collected
images.

An exemplary case is represented by superfluid 4He, which is also known as He
II (see, for example, Barenghi et al., 2014; Mongiovì et al., 2018). The
cryogenic flows of this unique liquid are routinely visualized and several PTV
studies have been reported to date – see Švančara et al. (2021) for a recent
example. However, at a given time, one can usually identify the positions of
about 100 particles within a 1 Mpixel image, each particle having an apparent
size of a few pixels. Consequently, the visualization data presently collected
in He II do not match the quality of the PIV data obtained in water or air,
i.e. the former data are more suitable for PTV than for PIV, at least if one
has in mind quantitative studies.

The observed behaviour is related to the available experimental techniques,
which do not allow to seed the flow of interest with a relatively large number
of particles, at a given time. The density of the liquid is equal to about 145
kg m-3 (Donnelly & Barenghi, 1998) and it is consequently challenging to
employ neutrally buoyant particles, as discussed, for example, by Švančara et
al. (2018). Indeed, the mass density of the flow-probing particles should
match that of the liquid but this is yet to be achieved for visualization
experiments in He II. It follows that the particles routinely used – made of
solid hydrogen or deuterium – disappear from the field of view after some
time, of the order of minutes. Additionally, particles are usually injected
into the liquid and, before taking measurements, one must wait for the
injection flow to disappear. Then, as already said, one is left with about 100
particles that can be tracked for some time, of the order of seconds (the
particle size is usually of the order of micrometers).

To illustrate these facts, we display in figure 1 two images obtained from
experimental data discussed by Švančara et al. (2020). Macroscopic vortex
rings propagating in superfluid 4He were studied by using the PTV technique,
i.e. solid deuterium particles were dispersed in the liquid and illuminated by
a planar laser sheet. The ring-induced motions of the particles were collected
by a digital camera, in a plane approximately parallel to the ring propagation
direction, that is, in a plane approximately corresponding to one of the ring
symmetry planes. The white regions correspond to particles reflecting incoming
light and the images were obtained by superimposing hundreds of frames, i.e.
the white regions can be viewed as preliminary indications of the tracks
followed by the particles (heavier than the fluid in this case). The left
panel corresponds to the superposition of movie frames in the laboratory
coordinate system, whereas, in the right one, the white regions have been
centred in the reference system moving with the vortex ring.

Figure 1: Visualisation of particle trajectories induced by a vortex ring in
superfluid 4He at $1.50$ K, with Reynolds number $\Rey\approx 10^{5}$; see
Švančara et al. (2020) for experimental details. The white regions correspond
to particles reflecting incoming light and the images were obtained from
averaging the light intensity reflected by the particles over hundreds of
movie frames. In the left panel – $25$ mm wide and $22$ mm high – the nozzle
ejecting the fluid parcel is located at the bottom centre of the image and the
vortex ring moves upward. The right panel displays (to scale) the same light
intensity values of the left one in the coordinate system moving with the
vortex ring, where the latter time-dependent position and velocity were
computed as discussed by Švančara et al. (2020).

We clearly appreciate the emergence of holes, which would not be observed for
perfect tracers distributed uniformly through the experimental domain. Indeed,
even for relatively small particles, when their action on the flow can be
neglected, particle inertia might lead to the decoupling of the particle
trajectories from the flow streamlines and, in particular, to preferential
concentration (see, for example, Maxey, 1987; Balachandar & Eaton, 2010).

Additionally, in comparison to typical PIV data, very few particles are seen
in figure 1, that is, these data cannot be employed for a robust estimate of
Eulerian quantities, such as velocity and vorticity fields. Indeed, in PTV
experiments, one does not usually have access to the vorticity field because
the latter requires a very large number of particles, vorticity being a small-
scale quantity that can present sharp variations in time and space.
Nevertheless, having access to the vorticity field – or a proxy thereof – can
be of great interest for the description of laboratory flows, e.g. for
quantitative comparisons between particle trajectories obtained in different
experimental conditions.

This research route was specifically followed by Švančara et al. (2020) in
their PTV study on the propagation of macroscopic vortex rings in He II. The
ring-induced motions of the particles were employed to calculate the
Lagrangian pseudovorticity field – a proxy for the Eulerian vorticity field,
which can be used to quantitatively measure the strength of the generated
vortex rings and which can also provide information on the size, position and
velocity of these objects.

Additionally, Švančara et al. (2020) have shown that, in conditions yet to be
met in experiments, the Lagrangian pseudovorticity $\theta$ is equal to half
of the the Eulerian vorticity $\omega$. The present work further clarifies the
relation between $\theta$ and $\omega$, especially in conditions closer to
those encountered in typical experiments. We not only consider in the
following how particle concentration affects this relation but also take into
account the role of particle inertia.

To this end, we numerically study the motion of particles induced by vortex
rings in two different settings. In the first part of the work, we investigate
the behaviour of an isolated vortex ring propagating in a Newtonian fluid by
using the Finite Element Method (FEM) (Brezzi & Fortin, 1991) with the
Incremental Pressure Correction Scheme (Guermond et al., 2006). We then
compute the Lagrangian pseudovorticity field from the positions and velocities
of fluid particles (tracers) advected by the obtained Eulerian flow fields, as
a function of experimentally relevant parameters, such as the number of flow-
probing particles and their spatial distribution.

Secondly, we investigate the effect of particle inertia on the relation
between pseudovorticity and vorticity, in the specific case of a vortex ring
propagating in a superfluid. For this, we perform direct numerical simulations
of the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) model, which is often employed
to describe the large-scale hydrodynamics of He II. Indeed, the dynamics of
relatively small spherical particles in flows of superfluid 4He has been
recently investigated within this theoretical framework (Polanco & Krstulovic,
2020).

In summary, the results presented below reinforce the view that the Lagrangian
pseudovorticity can be used to quantitatively estimate the strength of
relatively large vortical structures in the absence of Eulerian data, which is
specifically the case of experiments in He II. We confirm that the magnitude
of $\theta$ can be appreciably smaller than that of $\omega$, consistently
with previous analytical and experimental estimates (Švančara et al., 2020).
We also show that, in order to get a closer quantitative agreement between the
vorticity and pseudovorticity trends, larger particle concentrations should be
employed compared to those commonly used in experiments. Furthermore, our
results demonstrate that the Lagrangian pseudovorticity can be used to resolve
accurately general features of the observed ring dynamics. Such features
include the ring time-dependent position and propagation velocity, especially
in the case of tracers and particles with sufficiently low inertia. However,
we also report that, at present, $\theta$ does not seem suitable for capturing
the fine details of the vortex ring structure, that is, for reconstructing the
corresponding Eulerian vorticity fields.

##  2 Lagrangian pseudovorticity

The vortices shed at the edges of relatively large objects oscillating in
superfluid 4He have been recently visualised in planes approximately parallel
to the direction of oscillation, by following the flow-induced motions of
small solid particles suspended in the liquid (Duda et al., 2015, 2017). The
strength of these vortices was estimated from the two-dimensional particle
positions and velocities by using a purpose-made scalar measure, introduced to
quantitatively compare vortices obtained in different experimental conditions
because, as already noted, it is in general not possible to obtain flow
vorticity fields from sparse Lagrangian data.

This scalar quantity was named Lagrangian pseudovorticity $\theta$ and, as
mentioned above, it was also used by Švančara et al. (2020) in their
experimental study on macroscopic vortex rings propagating in superfluid 4He.
It is here defined as

$\theta(\boldsymbol{r},t)=\left\langle\frac{\left[\left(\boldsymbol{r}_{i}-\boldsymbol{r}\right)\times\boldsymbol{u}_{i}\right]_{z}}{|\boldsymbol{r}_{i}-\boldsymbol{r}|^{2}}\right\rangle_{\mathcal{M}},$
(1)

where $\boldsymbol{r}_{i}$ and $\boldsymbol{u}_{i}$ denote the particle
positions and velocities in the considered plane, respectively. The angle
brackets indicate the ensemble average within the set $\mathcal{M}$ of
Lagrangian particles, which are captured within a time window centred at time
$t$ and are found within an annular region centered, on a chosen grid, at the
inspection point $\boldsymbol{r}$. The subscript $z$ denotes the axis
perpendicular to the observation plane, i.e. we consider here the only non-
zero component of the vector product (the observed particle tracks occur in a
plane). The size of the annular region was chosen in order to have enough
particles for the calculation of $\theta$ and, at the same time, to exclude
diverging contributions from particles too close to the inspection point
$\boldsymbol{r}$.

At this point, it is useful to note a few properties of the Lagrangian
pseudovorticity $\theta$. As mentioned above, it can be shown analytically
that $\theta$ is closely related to the Eulerian vorticity, in the case of
fluid particles that are homogeneously distributed in space (Švančara et al.,
2020). Furthermore, it was reported that, for a Rankine vortex, the magnitude
of $\theta$ decreases when the area of the annular region used for its
calculation increases, that is, when the outer radius of the region increases,
for given grid and number of fluid particles within the annular region
(Švančara et al., 2020). Additionally, this magnitude can be appreciably
smaller than that of $\omega$ when the area is significantly larger than the
area where the flow vorticity is concentrated. For example, when the region
outer radius is 10 (100) times larger than the Rankine vortex radius, the
maximum pseudovorticity magnitude, located at the vortex centre, is more than
5 (50) times smaller than the corresponding vorticity value. Note in passing
that changing the grid or the number of points within the annular region has
less significant effect on the calculated pseudovorticity magnitude, in
comparison with the just mentioned influence of the annular region size.

A similar outcome – displayed in figure 2 – can be obtained for a Lamb-Oseen
vortex, which is characterised by a vorticity spatial distribution closer to
that observed for vortex rings, see, for example, figure 4 in §3.2, where
relevant numerical results are plotted. It is also apparent that the
pseudovorticity may change sign in the vortex vicinity, similarly to what is
seen in the middle panel of figure 4. Additionally, it is evident from the
inset of figure 2 that the number of fluid particles distributed in the
annular region becomes important solely when it assumes relatively small
values. The same applies to the chosen grid size that affects the computed
pseudovorticity values only for numbers of grid points smaller than those
considered in the figure.

Figure 2: Pseudovorticity for a Lamb-Oseen vortex as a function of the
distance from the vortex axis (radius). Vortex parameters: circulation
$\Gamma=50$ mm2 s-1, kinematic viscosity $\nu=0.01$ mm2 s-1 and time $t=10$ s.
Black circles: vortex velocity $v$; red circles: vorticity $\omega$; open
circles: pseudovorticity $\theta$, the corresponding subscripts in the legend
indicate the radius, in mm, of the area used for the calculation of $\theta$
from (1). 1000 equally spaced grid points were employed for the estimate shown
in the main panel, up to a radius of 10 mm, and 100 fluid particles were
distributed in the annular region, equally spaced along the radial direction
(the pseudovorticity is not computed in the region centre and is set to zero
at the vortex centre). Note that the chosen particle distribution is not
radially symmetric, that is, the particles are clustered in the vortex centre
vicinity. The results plotted in the inset were obtained by using 100 equally
spaced grid points, up to the radius of 10 mm, and the number of equally
spaced fluid particles within the annular region is specified in the legend.
The region outer radius for the data in the inset is 30 mm. The magenta line
indicates the pseudovorticity trend obtained by using 1000 equally spaced grid
points, corresponding to the cyan open circles in the main panel.

Before presenting our numerical results, we also note here that in appendix
§A, additional analytical results on the relation between pseudovorticity and
vorticity are reported. Specifically, we show that, for an isolated vortex
having a purely azimuthal velocity field, the pseudovorticity magnitude
becomes smaller, if the area employed for its estimate increases. We also
suggest an explanation for the sign changes of the pseudovorticity displayed
in figures 2 and 4, that is, the outcome can be explicitly related to the fact
that the corresponding particle distributions are not homogenous nor
isotropic.

##  3 FEM ring

The Finite Element Method (FEM) (Brezzi & Fortin, 1991) was chosen for the
space discretisation of the well-known equations governing the isothermal
dynamics of an incompressible viscous fluid. We specifically employed the
Taylor-Hood mixed FEM by using the FEniCS library (Alnæs et al., 2015). The
discretisation in time was performed by employing the implicit Crank-Nicolson
scheme and the Incremental Pressure Correction Scheme (Guermond et al., 2006)
was used for finding an efficient approximate solution of the coupled
velocity-pressure problem.

Figure 3: Two-dimensional domain used for the FEM simulations, composed of the
grey, light blue and red areas. The yellow area – 5 mm wide and 10 mm high –
represents the nozzle and is not included in the numerical domain (a parcel of
fluid is ejected through the nozzle to generate the vortex ring). The grey
area is 50 mm wide and 110 mm high. In the light blue area – 20 mm wide and 79
mm high – 5676 fluid particles are uniformly distributed at the beginning of
the simulation (_experiment_ particle distribution, labelled E; the area is
located 1 mm above the nozzle and includes the red area). In the red area – 10
mm wide and 5 mm high – 22801 fluid particles are uniformly distributed 1 s
after the beginning of the simulation (_delay_ particle distribution, labelled
D; the area is located 1 mm above the nozzle). The origin of the Cartesian
reference system (indicated by black arrows) is at the bottom centre of the
yellow area.

A schematic view of the numerical domain employed for the simulations in two
dimensions is shown in figure 3. For these simulations we have specifically
chosen boundary conditions close to experimental ones (Švančara et al., 2020).
On all domain boundaries the fluid velocity is set to zero, excluding the top
of the domain, where a traction free boundary condition is imposed, and the
nozzle top, where the vertical fluid velocity has a spatial parabolic profile
during the first second of the simulation (it is zero at later times). There,
the fluid peak velocity linearly increases from 0 to 10 mm s-1 during the
first 0.1 s, remains constant for the following 0.8 s and linearly decreases
from 10 to 0 mm s-1 during the last 0.1 s of the simulation first second.

The latter boundary condition simulates the vortex ring generation process,
which may occur when a parcel of fluid is ejected into a reservoir through a
nozzle. In practice, this is often achieved by using a piston, moving inside
the nozzle with a velocity $U_{p}$, for a stroke length $L$. An adequate
Reynolds number can then be defined as

$\Rey=\frac{U_{p}L}{2\nu}=\frac{\Gamma_{0}}{\nu},$ (2)

where, for the present simulations, the fluid kinematic viscosity is
$\nu=0.01$ mm2 s-1 and $\Gamma_{0}=U_{p}L/2$ indicates the ring nominal
circulation (see, for example, Švančara et al., 2020).

Here, we set $U_{p}$ equal to the mean value of the imposed fluid velocity
during the simulation first second, i.e. $U_{p}=6$ mm s-1, and the stroke
length $L=U_{p}t_{p}=6$ mm, where $t_{p}=1$ s is the imposed duration of the
fluid ejection into the reservoir. It follows that $\Gamma_{0}=18$ mm2 s-1 and
$\Rey=1800$, which means that our FEM ring is most likely in the laminar
regime (Glezer, 1988). Additionally, $L/D=1.2$, where $D=5$ mm is the nozzle
internal diameter, that is, we do not expected a prominent ring wake (Gharib
et al., 1998); note also that the ring nominal impulse, per unit of density,
$I_{0}=\Gamma_{0}\pi(D/2)^{2}=353$ mm4 s-1 (Sullivan et al., 2008).

A mesh independence check was performed on three meshes with different
densities for the space discretisation, resulting in systems with sizes
ranging from $2\times 10^{5}$ to $1.3\times 10^{6}$ degrees of freedom. The
size of the time step ranged from 0.01 to 0.001 s. Full three-dimensional
computations were also performed for the sake of comparison and the obtained
results were not appreciably different from the two-dimensional ones; see
Outrata (2020) for further details. In the following, we discuss the two-
dimensional numerical results obtained by using the most refined mesh, that
with approximately $1.3\times 10^{6}$ degrees of freedom, and with the
smallest time step, equal to 0.001 s (the entire simulation lasts 15 s).

###  3.1 Initial fluid particle distributions

We considered fluid particles in the obtained Eulerian flow fields and tracked
their motions in order to compute – from the particle positions and velocities
– the Lagrangian pseudovorticity, (1). As a first step, we uniformly
distributed, at the beginning of the simulation, approximately 6000 particles
in a relatively large area of the numerical domain, shown as the light blue
region in figure 3, our aim being to replicate actual experimental conditions
(Švančara et al., 2020). We refer to this arrangement as the _experiment_
particle distribution and label it with the letter E. As shown in the
following, its results are characterised by the fact that many particles do
not move, being away from the vortex ring, similarly to what is observed in
experiments.

For our analysis we also employed another particle arrangement and refer to it
as the _delay_ distribution, because the fluid particles were added to the
Eulerian flow fields 1 s after the beginning of the simulation (we label this
distribution with the letter D). Additionally, compared to E, more particles
were distributed in a smaller area, shown as the red region in figure 3, our
main aim being to increase the number of particles significantly contributing
to the pseudovorticity computation, that is, the number of moving particles.

The inner and outer radii of the annular region employed for pseudovorticity
estimates, (1), were set to 0.1 mm and 3 mm, respectively, and the
corresponding time window is 10 ms wide. For both distributions, the chosen
grid is a rectangular mesh of $53\times 126$ inspection points, covering a
20.8 mm wide and 60.0 mm high area of the computational domain, centred just
above the nozzle, see figure 3. Note that on the same grid the obtained flow
vorticity was interpolated and that the fluid particle velocities are known
from the numerical simulation, that is, they are not computed from the
particle positions as in the experiments (Švančara et al., 2020).

###  3.2 Pseudovorticity maps

Having set these conditions, we calculated the Lagrangian pseudovorticity
$\theta$ from the instantaneous positions and velocities of both particle
distributions. In the left and middle panels of figure 4 we display the
$\theta$ maps obtained 10 s after the beginning of the simulation. In the
right panel the same-time flow vorticity is plotted.

Figure 4: Pseudovorticity and vorticity maps obtained 10 s after the beginning
of the FEM simulation. Left and middle panels: pseudovorticity computed from
the _experiment_ (E) and _delay_ (D) particle distributions, respectively.
Right panel: vorticity field. The reference system is that defined in figure
3. Clockwise fluid rotation corresponds to positive values of pseudovorticity
and vorticity (the ring is moving upward). Note that the vorticity and
pseudovorticity values in the nozzle proximity, for vertical positions smaller
than 15 mm, are not taken into account in the other calculations reported
here.

If one compares the left and right panels of figure 4, it is apparent that the
vortex ring vertical position obtained from the _experiment_ particle
distribution is quite close to that resulting from the vorticity map. On the
other hand, the ring spatial extent for the E distribution is appreciably
larger than that apparent from the vorticity map. Additionally, the
_experiment_ pseudovorticity magnitudes are at least 100 times smaller than
the vorticity ones.

Instead, for the _delay_ particle distribution the magnitudes of $\theta$ are
significantly closer to those of $\omega$ – see the middle and right panels of
figure 4. More importantly, the corresponding spatial distributions are quite
similar to each other. This is further confirmed in figure 5, which shows
root-mean-square profiles of $\theta$ and $\omega$, respectively averaged over
the vertical and horizontal directions.

Figure 5: Root-mean-square pseudovorticity (red) and vorticity (black)
profiles obtained 10 s after the beginning of the FEM simulation. The
pseudovorticity is computed from the _delay_ (D) particle distribution.
Profiles are averaged either in the vertical (top panel) or horizontal (bottom
panel) directions. The different scales on the vertical axes are for
pseudovorticity (left axes) and vorticity (right axes). The reference system
is as in figure 4. Note that the vorticity and pseudovorticity values in the
nozzle proximity, for vertical positions smaller than 15 mm, are not taken
into account in the other calculations reported here.

Note in passing that particle positions characterised by velocities smaller
than 0.01 mm s-1 were not taken into account for the present pseudovorticity
estimate because this led to (slightly) smoother $\theta$ maps and (slightly)
larger pseudovorticity magnitudes for the _experiment_ particle distribution.
The effects were, however, absent for the _delay_ particle distribution, due
to the larger velocities of the considered particles. Additionally, for the D
distribution, the vorticity in the nozzle proximity is not seen, because the
particles are distributed 1 s after the beginning of the simulation.

More generally, the results displayed in figures 4 and 5 confirm that the
Lagrangian pseudovorticity can be used to characterize vortex rings’ features
in the absence of Eulerian data (Švančara et al., 2020). Additionally, they
demonstrate that the quantitative agreement between $\theta$ and $\omega$
strongly depends not only on the number of flow-probing particles, but also on
how they are distributed in the region of interest.

###  3.3 Ring position, velocity and radius

In order to further support the previous claims, we calculated other relevant
quantities from the obtained pseudovorticity and vorticity maps, which enable,
for example, the estimation of the vortex ring location and size over time.

As a first step, following Švančara et al. (2020), we chose, for each map, a
positive threshold value $\theta_{0}$ that is employed to identify and track
the vortex ring. The positive (clockwise) part of the ring is then made of
grid points with $\theta>\theta_{0}$, the negative (counter-clockwise) one is
instead described by $\theta<-\theta_{0}$. The chosen values – listed in table
1 – are constant in time and are set to approximately 10 % of the maximum
magnitude at late times.

Once the filtered maps are computed, we can identify the centres of the
positive and negative vortical regions, and the distance between them, which
is known as the ring diameter $2R$ (Gan & Nickels, 2010). Additionally, the
ring position $\boldsymbol{P}$ is defined as the centre of the two vortices
and its velocity is calculated by convolving $\boldsymbol{P}$ with a suitable
Gaussian kernel (Švančara et al., 2020).

Note in passing that the values listed in table 1 are obtained for times
ranging between 3.00 and 14.79 s (the entire simulation lasts 15 s). The upper
limit is due to the Gaussian algorithm chosen for the ring propagation
velocity calculation, while the lower one is imposed to remove the influence
on the results of the fluid vorticity in the nozzle proximity, see figure 4 –
in the present work we decided not to investigate the ring generation process
mainly because this was not accessed in relevant experiments (Švančara et al.,
2020).

data set | $\theta_{0}$ | $v$ | $v/U_{p}$ | $2R$ | $C$ | $A_{C}$
---|---|---|---|---|---|---
vorticity | 0.800 | 2.08 $\pm$ 0.13 | 0.35 | 3.09 $\pm$ 0.14 | 46.21 $\pm$ 2.32 | 13.78 $\pm$ 0.61
tracer D | 0.070 | 2.09 $\pm$ 0.24 | 0.35 | 3.37 $\pm$ 0.09 | 5.37 $\pm$ 0.61 | 16.95 $\pm$ 2.58
tracer E | 0.002 | 2.06 $\pm$ 0.42 | 0.34 | 4.51 $\pm$ 0.23 | 0.28 $\pm$ 0.04 | 35.06 $\pm$ 1.91
Table 1: FEM ring properties. (i) Chosen threshold $\theta_{0}$ for ring
identification in s-1, set to approximately 10 % of the maximum magnitude at
late times. (ii) Ring vertical velocity $v$ in mm s-1; see also the bottom
panel of figure 6. (iii) Ratio between the mean value of the ring vertical
velocity $v$ and the piston velocity $U_{p}$, which is found to be consistent
with values reported in the literature (see, for example, Sullivan et al.,
2008). (iv) Ring diameter $2R$ in mm; see also figure 7. (v) Ring circulation
$C$ in mm2 s-1 from (3); see also figure 8. (vi) Ring area $A_{C}$ in mm2; see
also figure 9. Results obtained by applying the chosen threshold values.
Symbols as in Švančara et al. (2020).

In the top (bottom) panel of figure 6 we plot the position (velocity) of the
vortex ring as a function of the time elapsed from the beginning of the
simulation. It is evident that the ring position is well captured by both
pseudovorticity maps, while the ring velocity obtained from the _experiment_
particle distribution is characterised by a significantly larger standard
deviation compared to that derived from the _delay_ distribution, see also
table 1. On the other hand, the corresponding mean values are quite close to
each other, see again table 1. Note also that, at sufficiently late times, the
ring velocity shows the tendency to decrease in magnitude, which is a
behaviour occurring to vortex rings propagating in a viscous fluid, see, for
example, Maxworthy (1974) and Sullivan et al. (2008).

Figure 6: Ring position (top) and velocity (bottom) as a function of the time
$t$ elapsed from the beginning of the FEM simulation. The reference system is
that defined in figure 3.

Figure 7: Ring radius $R$ as a function of the time $t$ elapsed from the
beginning of the FEM simulation. The reference system is as in figure 6.

One also expects that the decrease of the ring velocity magnitude is
accompanied by the increase of its radius. As shown in figure 7, which
displays the vortex radius as a function of the time elapsed from the
beginning of the simulation, this behaviour is indeed captured by the _delay_
particle distribution. However, this is not the case for the _experiment_
distribution. The outcome can be related to the above mentioned fact that the
ring spatial extent is not well resolved by this particle arrangement, see
again figure 4.

We can therefore say that, by using the Lagrangian pseudovorticity, one can
track rather well the position of a vortex ring and estimate with some
confidence its propagation velocity. On the other hand, in order to get
quantitative information on the ring spatial extent, one should also employ
for the Lagrangian analysis a number of fluid particles significantly larger
than those used to date in experiments (Švančara et al., 2020). Additionally,
these particles should be distributed in an relatively small area, with
dimensions comparable to those of the ring itself.

###  3.4 Ring circulation and area

Following Švančara et al. (2020) we also estimate the ring circulation $C$
from the pseudovorticity (and vorticity) maps. For this, we use the relation

$C=\frac{C_{+}-C_{-}}{2}=\frac{\sum(\theta_{+}a)-\sum(\theta_{-}a)}{2},$ (3)

where the summations only include the grid points characterised by values of
$\theta$ (or $\omega$) passing the threshold $\theta_{0}$ listed in table 1.
The subscript $+$ ($-$) denotes that the pseudovorticity and vorticity values
are positive (negative), and $a$ is the fixed area associated to each grid
point. Since $C_{+}\approx|C_{-}|$, we discuss here only their mean absolute
value $C$.

The temporal evolution of $C$ is represented in the top panel of figure 8
(dashed lines) and, in the same panel, the solid lines correspond to the
actual ring circulation $\Gamma$, obtained using the full maps, that is, with
the threshold $\theta_{0}$ set to zero. We can see that $C$ is consistently
smaller than $\Gamma$, likely because the filtered maps do not capture the
wake shed by the moving ring. The panel also displays the tendency of the ring
circulation to decrease at late times. This is an expected behaviour for both
turbulent and laminar rings, as discussed, for example, by Maxworthy (1974)
and Didden (1979), respectively.

Figure 8: Ring circulation as a function of the time $t$ elapsed from the
beginning of the FEM simulation. In the top panel the solid lines indicate the
ring circulation $\Gamma$ obtained using the full maps, from (3) with
$\theta_{0}=0$, see figure 4. The dashed lines denote the circulation $C$
calculated using the filtered maps, from (3) with the threshold values listed
in table 1. The bottom panel shows $C$ normalised by the ring nominal
circulation $\Gamma_{0}$. Note that in the bottom panel and in all the other
FEM figures – excluding figures 4 and 5 – the chosen pseudovorticity and
vorticity thresholds are taken into account for the estimate of the plotted
quantities.

Additionally, the circulation magnitude derived from the flow vorticity (black
line) is approximately 100 times larger than that obtained from the
_experiment_ particle distribution (blue line). This is also evident from the
bottom panel of figure 8, which displays $C$ normalised by the ring nominal
circulation $\Gamma_{0}$. A similar outcome was reported by Švančara et al.
(2020), who showed that experimentally obtained values of $C$ can be much
smaller than relevant values of $\Gamma_{0}$, up to approximately two orders
of magnitude. Note that for these experiments it was not possible to access
the actual ring circulation $\Gamma$ and that the nominal circulation was
estimated from suitable values of fluid velocity.

From the bottom panel of figure 8 it is also apparent that
$C/\Gamma_{0}\approx 2$, when $C$ is computed from the filtered vorticity maps
(black line). Indeed, measured values of ring circulation were found to be of
the same order of $\Gamma_{0}$ by a few investigators, in various experimental
conditions (see, for example, Maxworthy, 1974; Didden, 1979; Borner & Schmidt,
1985). Note, however, that the ring nominal circulation
$\Gamma_{0}=U_{p}^{2}t_{p}/2$; that is, if we set, for our FEM ring,
$U_{p}=10$ mm s-1 and $t_{p}=0.8$ s, we obtain a nominal circulation more than
two times larger than that employed in the bottom panel of figure 8 – see also
the above discussion on (2). It then follows that the numerical values shown
in the panel should solely be regarded as first-order estimates, considering
also that, as discussed, for example, by Glezer (1988), the nozzle geometry
and the piston velocity time history may have a significant influence on the
resulting ring features.

Following Švančara et al. (2020) we can now estimate the ring area $A_{C}$
from (3), by setting $\theta_{+}=1$ and $\theta_{-}=-1$. We plot the result in
the top panel of figure 9, as a function of the time elapsed from the
beginning of the simulation. Additionally, in the bottom panel, we display the
ratio $A_{C}/A_{R}$, where $A_{R}=\pi R^{2}$ and $R$ is the estimated ring
radius shown in figure 7.

Figure 9: Ring area as a function of the time $t$ elapsed from the beginning
of the FEM simulation.

The figure most prominent feature is likely the peak observed for the _delay_
particle distribution (red lines) at short times. This can be related to the
fact that, at around the same time, the ring wake disappears from the filtered
$\theta$ maps. Note that a similar peak is also observed in the estimated
vertical velocity of the ring, at $t\approx 6$ s (bottom panel of figure 6).

Additionally, from the bottom panel of figure 9, it appears that the ring
radius $R$ can be used as a valuable first-order estimate of the ring spatial
extent. More importantly, we see from both panels that the ring spatial extent
is quite well captured by the _delay_ particle distribution, consistently with
our previous discussion.

In summary, our results confirm that the Lagrangian pseudovorticity can be
used to quantify general features of relatively large vortical structures –
such as their position and propagation velocity – and that this is especially
meaningful when Eulerian data are not available, which is specifically the
case of experiments in superfluid 4He (Švančara et al., 2020). Additionally,
we have shown here that the spatial extent of these vortices can be
quantitatively determined by $\theta$ maps solely in conditions yet to be met
in relevant experiments, that is, only for sufficiently dense particle
distributions, which, by the way, could allow to directly access Eulerian flow
fields.

So far we have considered fluid particles, which by definition exactly follow
the flow, but the particles commonly used in experiments can deviate from
streamlines due to their inertia. In the next section, we then investigate the
effect of particle inertia on the relation between $\theta$ and $\omega$.

##  4 HVBK ring

The Hall–Vinen–Bekarevich–Khalatnikov (HVBK) model is often used to describe
the large-scale hydrodynamics of superfluid 4He. At finite temperatures,
between approximately 1 and 2.2 K, this unique liquid can be seen as a mixture
of two components: the superfluid component, with no viscosity, and the normal
fluid component, with Newtonian viscosity (see, for example, Landau, 1941;
Donnelly, 2009). The HVBK model is valid at scales much larger than the
typical distance between quantised vortices, which are line singularities –
holes – within the superfluid component. The model therefore describes the
coarse-grained dynamics of the two components. Within this framework, their
respective velocity fields $\boldsymbol{v}_{\text{s}}$ and
$\boldsymbol{v}_{\text{n}}$ follow two coupled incompressible Navier–Stokes
equations,

$\displaystyle\frac{\partial\boldsymbol{v}_{\text{n}}}{\partial
t}+(\boldsymbol{v}_{\text{n}}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{\text{n}}$
$\displaystyle=-\frac{\boldsymbol{\nabla}p_{\text{n}}}{\rho_{\text{n}}}+\nu_{\text{n}}\nabla^{2}\boldsymbol{v}_{\text{n}}-\frac{\rho_{\text{s}}}{\rho_{\text{n}}}\boldsymbol{f}_{\text{ns}},$
(4) $\displaystyle\frac{\partial\boldsymbol{v}_{\text{s}}}{\partial
t}+(\boldsymbol{v}_{\text{s}}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{\text{s}}$
$\displaystyle=-\frac{\boldsymbol{\nabla}p_{\text{s}}}{\rho_{\text{s}}}+\nu_{\text{s}}\nabla^{2}\boldsymbol{v}_{\text{s}}+\boldsymbol{f}_{\text{ns}},$
(5)
$\displaystyle\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}_{\text{s}}$
$\displaystyle=\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}_{\text{n}}=0.$
(6)

Here, $\rho_{\text{s}}$ and $\rho_{\text{n}}$ are the temperature-dependent
densities of the two fluids (the total fluid density is
$\rho=\rho_{\text{n}}+\rho_{\text{s}}$), while $p_{\text{s}}$ and
$p_{\text{n}}$ indicate their respective pressure fields.

The two fluids are coupled by a mutual friction force
$\boldsymbol{f}_{\text{ns}}$. This force is responsible for the transfer of
energy between the components, and it also contributes to the dissipation of
energy of the entire system. As a first approximation,
$\boldsymbol{f}_{\text{ns}}$ increases linearly with the relative velocity
between the components, that is,
$\boldsymbol{f}_{\text{ns}}=\alpha\Omega_{0}(\boldsymbol{v}_{\text{n}}-\boldsymbol{v}_{\text{s}})$,
where $\alpha$ is a non-dimensional mutual friction coefficient – tabulated by
Donnelly & Barenghi (1998) – and $\Omega_{0}$ indicates a mutual friction
frequency, related to the density and orientation of the quantised vortices
embedded in the superfluid. For example, in isotropic superfluid turbulence,
this frequency may be taken as proportional to a characteristic superfluid
vorticity (L’vov et al., 2006). Here, as we consider the case of an isolated
vortex ring, this frequency is taken as a constant control parameter. Note
also that we have included in (5) a superfluid dissipation term, parameterised
by an effective temperature-dependent kinematic viscosity $\nu_{\text{s}}$
(Vinen & Niemela, 2002). This term accounts for the dissipation of superfluid
energy at scales smaller than those resolved by the HVBK model. Such
dissipation mechanisms include quantised vortex reconnections and excitation
of Kelvin waves, which transport energy towards the smallest scales of the
system (see, for example, Švančara & La Mantia, 2019; Villois et al., 2020).
Additionally, in (4) the kinematic viscosity of the normal component
$\nu_{\text{n}}$ is set equal to $\mu_{\text{n}}/\rho_{\text{n}}$, where
$\mu_{\text{n}}$ is the fluid dynamic viscosity, also tabulated by Donnelly &
Barenghi (1998).

In the reference study by Švančara et al. (2020), solid deuterium particles
were used to evaluate the Lagrangian pseudovorticity associated to vortex
rings generated in superfluid 4He. Such particles are not perfect tracers of
the flow, as their density is about $1.4$ times larger than that of the fluid,
and their typical diameter is a few micrometres (Švančara & La Mantia, 2017).
Hence, particle inertia is expected to play some role on pseudovorticity
estimates.

In superfluid 4He, inertial particles are not only subject to the Stokes drag
associated to the normal fluid viscosity, but also respond to pressure
gradient forces from the two fluids (see, for example, Sergeev & Barenghi,
2009). Finite size effects can be neglected, as particles are much smaller
than the core size of the studied vortex ring, as reported below, in §4.1.
Consequently, if one also neglects Basset history terms and assumes that
particles are spherical, the particle equation of motion reads

$\frac{\mathrm{d}\boldsymbol{v}_{\text{p}}}{\mathrm{d}t}=\frac{\boldsymbol{v}_{\text{n}}(\boldsymbol{x}_{\text{p}})-\boldsymbol{v}_{\text{p}}}{\tau_{\text{p}}}+\beta\left(\frac{\rho_{\text{n}}}{\rho}\frac{\mathrm{D}\boldsymbol{v}_{\text{n}}}{\mathrm{D}t}+\frac{\rho_{\text{s}}}{\rho}\frac{\mathrm{D}\boldsymbol{v}_{\text{s}}}{\mathrm{D}t}\right),$
(7)

where $\boldsymbol{x}_{\text{p}}$ and $\boldsymbol{v}_{\text{p}}$ are the
particle position and velocity, respectively, and $\mathrm{D}/\mathrm{D}t$
denotes the material derivative associated to each fluid component. The
density parameter $\beta$ accounts for added mass effects and is given by
$\beta=3\rho/(2\rho_{\text{p}}+\rho)$, where $\rho_{\text{p}}$ is the particle
density. The Stokes time, representing the typical particle response time to
normal fluid fluctuations, is $\tau_{\text{p}}=a_{\text{p}}^{2}/(3\beta\nu)$,
where $a_{\text{p}}$ denotes the particle radius, and
$\nu=(\rho_{\text{n}}/\rho)\nu_{\text{n}}=\mu_{\text{n}}/\rho$ indicates the
temperature-dependent kinematic viscosity of superfluid 4He, also tabulated by
Donnelly & Barenghi (1998). Formally, tracer dynamics is obtained in the limit
$\tau_{\text{p}}\to 0$.

The HVBK equations (4)–(6) are solved by direct numerical simulation in a
three-dimensional periodic domain using a Fourier pseudo-spectral solver
(Homann et al., 2009). Particles are initialised at random locations in the
domain and are advanced in time according to (7). Fluid velocities and
accelerations are interpolated at particle positions using fourth-order
B-splines (van Hinsberg et al., 2012). Time advancement of fields and
particles is performed using a third-order Runge–Kutta scheme.

Figure 10: HVBK simulation of a vortex ring moving downward with three
families of deuterium particles at a relatively early time, approximately 1 s
after the beginning of the simulation. Top left: particles with radius of
$5\,\mu$m. Bottom left: particles with radius of $10\,\mu$m. Right: particles
with radius of $30\,\mu$m. The respective Stokes numbers are $St\approx 0.1$,
$0.3$ and $2.5$ – see table 2 in §4.3. The peculiar shape observed in the
right panel is due to particles being expelled from highly rotating regions as
a result of their high inertia. Additionally, particles initially located in
front of the vortex ring are propelled upwards via the ring centre in a
ballistic motion, resulting in the observed wake pattern.

Figure 10 displays results from a typical simulation, in which we observe the
vortex ring together with spherical particles having the same density of
deuterium particles, at the chosen temperature $T=1.9\,$K (see numerical
details later). All panels correspond to the same time of ring evolution, for
different particle families. From left to right the particle size is
increased, leading to a larger Stokes response time. As a consequence, the
effect of inertia becomes apparent and holes (regions without particles) are
clearly observed. Such structures are similar to those seen in the
experimental pictures displayed in figure 1, but one should keep in mind that
the latter are not Eulerian images, that is, they were not obtained at a fixed
time as the pictures in figure 10.

###  4.1 Simulation setup

We consider superfluid 4He at 1.9 K. At this temperature, the densities of the
normal fluid and superfluid components satisfy the ratio
$\rho_{\text{n}}/\rho_{\text{s}}=0.72$ and the liquid kinematic viscosity is
$\nu\approx 0.01$ mm2 s-1. This value is very close to that used for the FEM
ring – note that, as stated above, the density and viscosity of He II depend
on temperature (see, for example, Donnelly & Barenghi, 1998). Additionally, at
this temperature, the effective superfluid viscosity is close to the normal
fluid one (Vinen & Niemela, 2002) and the mutual friction coefficient is
$\alpha=0.206$. For the numerical simulations we set the mutual friction
parameter to $\Omega_{0}\approx 110$ s-1. We can thus consider that the fluid
components are well locked to each other for the time scales relevant to this
work.

An isolated vortex ring of normal fluid is introduced in the middle of the
periodic box, which has 28 mm sides. The origin of the chosen reference system
is located at the box centre. The ring radius is $R=3.5\,\text{mm}$ and its
initial circulation is $\Gamma_{0}\approx 95$ mm2 s-1. It follows from (2)
that the ring Reynolds number is slightly larger than $10^{4}$. Note that the
superfluid component is initially at rest. However, due to mutual friction,
the normal fluid vortex ring quickly induces the formation of an equivalent
superfluid ring. Such a macroscopic superfluid vortex is typically composed of
a bundle of quantised vortices (see, for example, Wacks et al., 2014) that are
modelled in the HVBK framework as a continuous vorticity field. In this work
we are not interested in this fast transient dynamics.

Note also that, due to the domain periodicity, a vortex ring eventually
returns to its initial position. Although the ring leaves a wake behind it,
its dynamics is almost unaffected by it, since the wake is rapidly dissipated
by viscosity (within one period, lasting about 5 seconds). However, as
particles are not continuously injected in the domain, their spatial
distribution becomes inhomogeneous after the passing of a ring. Hence, the
studied configuration mimics an experiment in which rings are injected
periodically, but one should also account for the fact that the ring size and
velocity change with time. We will not further exploit this analogy in the
present work.

The vortex ring core is initialised with a Gaussian vorticity profile,
$\omega(r)=\omega_{0}\exp\left[-r^{2}/(2r_{0}^{2})\right]$, where $r$ is the
distance from the core centre. The vortex core radius $r_{0}=0.2\,\text{mm}$
and the vorticity at the core centre is set to fix the value of the ring
circulation. Note in passing that the generation of such a thin vortex ring
does not appear straightforward with the apparatus employed by Švančara et al.
(2020), mainly because the experimentally generated rings seem to have a
rather large core size, but, as shown below, the HVBK ring provides a good
setting to study the effect of inertia.

As detailed in §4.2, we track solid hydrogen and deuterium particles – by
setting their density – with sizes similar to those found in 4He experiments.
For comparison, we also follow perfect tracers of the normal fluid. We
specifically consider three families of spherical particles, having radii of
5, 10 and 30 $\mu$m, and we seed $5\times 10^{5}$ particles, homogeneously
distributed over the whole (three-dimensional) computational domain. The
corresponding Stokes times range approximately from 1 to 41 ms. The
characteristic ring time $\tau_{\text{r}}$ is estimated using the initial
circulation $\Gamma_{0}$ and the core radius $r_{0}$. We obtain that
$\tau_{\text{r}}\approx 17$ ms, i.e. relevant values of Stokes number
$St=\tau_{\text{p}}/\tau_{\text{r}}$ are between $0.04$ and $2.47$ – see also
table 2 in §4.3.

###  4.2 Pseudovorticity maps

At the beginning of the simulation, particles are suspended in the cubic
volume. A layer of the latter, parallel to the ring propagation direction and
situated in the centre of the periodic box, is considered for the
pseudovorticity estimate. This volume layer is 0.2 mm thick and contains
between 3100 and 3600 particles at each time step, which lasts approximately
0.09 s (the entire simulation lasts approximately 27 s). The positions and
velocities of these particles were then employed for the calculation of the
pseudovorticity according to (1). This estimation was performed on a square
grid of $101\times 101$ inspection points, covering a $27.6\times
27.6\,\text{mm}^{2}$ area. The chosen annular region has 0.1 mm inner and 3 mm
outer radii, which are the same values used for our FEM analysis in a viscous
fluid, discussed above, in §3. The corresponding time window is equal to the
time step, that is, for the HVBK ring, no temporal averaging was carried out.

Note also that, as in the FEM study, the numerically obtained flow vorticity
values were interpolated on the chosen pseudovorticity grid, for the sake of
comparison. Additionally, in order to follow the ring motion for distances
larger than the box size, the ring is at each time step centred in the box
middle by using the position of the normal fluid vorticity maximum at that
time step (the introduced ring is symmetric). From the latter maximum position
it is then possible to get the ring position as a function of time, in a
chosen reference system not moving with the ring, as for the FEM ring.

Pseudovorticity maps were obtained for fluid tracers and inertial particles.
The latter are characterised by a density different from that of the fluid and
by a finite Stokes response time (a finite radius). Additionally, we chose
particles having features compatible with recent visualisation experiments in
superfluid 4He (see, for example, Švančara et al., 2018). Specifically, we
chose spherical particles made of solid deuterium $D_{2}$ – heavier than
liquid He II, with a density ratio of $1.37$ at $1.9$ K – and solid hydrogen
$H_{2}$ – lighter than superfluid 4He, with a density ratio of $0.60$ at the
same temperature. Similarly, we selected particle radii of the same order of
and larger than the dimensions of particles in experiments – see La Mantia &
Skrbek (2014) and Švančara & La Mantia (2017) for typical distributions of
particle sizes.

Figure 11: Vorticity and pseudovorticity maps obtained 2.69 s after the
beginning of the HVBK simulation. The top left panel displays the Eulerian
vorticity map. The other panels show pseudovorticity maps obtained using
different types of particles, as indicated in the panel titles (the shown
dimension indicates the particle radius). The origin of the reference system,
moving with the ring, is located at the centre of the periodic box. Clockwise
fluid rotation corresponds to positive values of pseudovorticity and vorticity
(the ring is moving downward).

Pseudovorticity maps calculated from the different particle classes are shown
in figure 11. As in the FEM analysis, it is possible to capture the
instantaneous ring location by using tracer particles, but the magnitude of
the computed pseudovorticity is two orders of magnitude smaller than that of
the vorticity; see also the corresponding circulation values in table 2, in
§4.3. Moreover, the pseudovorticity maps obtained using small inertial
particles are quantitatively similar to those obtained using tracers. This is
expected, as the motion of small particles should not be strongly affected by
inertial effects. More practically, it indicates that hydrogen and deuterium
particles having radii up to 10 $\mu$m – corresponding to $St<0.3$ – may be
used to accurately detect the presence of a vortex ring in the present
setting. In our study, it is only for the largest particles, with radius
$a_{\text{p}}=30$ $\mu$m, corresponding to $St>1$, that pseudovorticity
estimates importantly deviate from those of tracer particles, even though
relevant qualitative information may still be extracted from the resulting
maps.

In our simulations, we also observe that, for fluid tracers, the number of
particles does not influence appreciably the obtained results. For example, if
the chosen layer of the cubic volume is five times thicker, the number of
particles used for the pseudovorticity calculation also increases by
approximately five times, but the corresponding magnitude does not change
appreciably, although its behaviour in space and time is possibly smoother
than that obtained for the thinner layer. Note that similar results were
obtained for the FEM ring, that is, increasing the number of particles in a
given area did not significantly improve the agreement between vorticity and
pseudovorticity magnitudes above a certain number of particles, see also the
inset of figure 2. We expect that, for given number of particles, a better
agreement could be obtained for smaller annuli, as discussed in appendix §A.
On the other hand, if one decreases the size of the region used for the
pseudovorticity estimate in actual experiments, the number of particles in
that region also decreases (Švančara et al., 2020). It then follows that, as
mentioned above, the choice we made here for the annulus size is mainly
motivated by the requirement of having enough particles in the region of
interest for the pseudovorticity calculation.

Figure 12: Root-mean-square pseudovorticity and vorticity profiles obtained
2.69 s after the beginning of the HVBK simulation. Profiles are averaged
either in the vertical (top panels) or horizontal (bottom panels) directions.
Left and middle panels display pseudovorticity profiles obtained from
deuterium, $D_{2}$, and hydrogen, $H_{2}$, particles, respectively. Black, red
and blue lines in these panels indicate that corresponding particle radii are
5, 10 and 30 $\mu$m, respectively. On the right panels pseudovorticity
profiles for tracers are plotted as red lines. Black lines on the same panels
indicate corresponding vorticity profiles. Note the different scales on the
vertical axes of the right panels for pseudovorticity (left axes) and
vorticity (right axes). The reference system is as in figure 11.

In order to make a quantitative comparison, we present in figure 12 the root-
mean-square profiles of pseudovorticity for the different particle families.
These are compared with the corresponding vorticity profiles. We clearly
observe that particle inertia becomes more important for the heavier and
larger particles, i.e. for the deuterium particles with 30 $\mu$m radii,
corresponding to the largest Stokes number. Additionally, vorticity and
pseudovorticity profiles qualitatively differ, in agreement with the above
discussion.

###  4.3 Ring position, velocity and radius

Table 2 lists relevant vortex ring properties averaged over the entire HVBK
simulation, for times ranging from 0.09 to 26.93 s. The exception is the ring
propagation velocity, which is averaged between 1.89 and 25.14 s due to the
Gaussian algorithm chosen for its calculation (Švančara et al., 2020). Note
that the little dependence of the ring velocity on the data set is most likely
related to the above mentioned procedure of ring positioning in the box centre
at each time step. Indeed, this centering procedure was performed by using the
normal fluid vorticity maximum, whose magnitude and position do not depend on
the particle type.

data set | $\theta_{0}$ | $v$ | $2R$ | $C$ | $A_{C}$ | $St$
---|---|---|---|---|---|---
vorticity | 1.000 |  $5.24\leavevmode\nobreak\ \pm$ 0.77 | 7.10 $\pm$ 0.11 | 82.86 $\pm$ 5.30 | 11.63 $\pm$ 4.22 | –
tracer | 0.010 |  $5.24\leavevmode\nobreak\ \pm$ 0.78 | 6.53 $\pm$ 0.25 | 1.51 $\pm$ 0.16 | 35.10 $\pm$ 2.96 | –
D2 5 $\mu$m | 0.010 |  $5.24\leavevmode\nobreak\ \pm$ 0.78 | 6.36 $\pm$ 0.21 | 1.58 $\pm$ 0.14 | 34.70 $\pm$ 2.51 | 0.07
D2 10 $\mu$m | 0.010 |  $5.24\leavevmode\nobreak\ \pm$ 0.77 | 5.79 $\pm$ 0.17 | 1.74 $\pm$ 0.10 | 37.37 $\pm$ 3.04 | 0.27
D2 30 $\mu$m | 0.005 |  $5.28\leavevmode\nobreak\ \pm$ 0.84 | 4.99 $\pm$ 0.47 | 0.74 $\pm$ 0.30 | 51.52 $\pm$ 3.34 | 2.47
H2 5 $\mu$m | 0.010 |  $5.24\leavevmode\nobreak\ \pm$ 0.78 | 6.63 $\pm$ 0.28 | 1.49 $\pm$ 0.16 | 36.58 $\pm$ 3.13 | 0.04
H2 10 $\mu$m | 0.020 |  $5.24\leavevmode\nobreak\ \pm$ 0.77 | 7.32 $\pm$ 0.44 | 1.24 $\pm$ 0.18 | 25.18 $\pm$ 1.79 | 0.16
H2 30 $\mu$m | 0.050 |  $5.24\leavevmode\nobreak\ \pm$ 0.77 | 6.34 $\pm$ 1.14 | 1.60 $\pm$ 0.37 | 14.24 $\pm$ 2.63 | 1.45
Table 2: HVBK ring properties. (i) Chosen threshold $\theta_{0}$ for ring
identification in s-1, set to approximately 10 % of the maximum magnitude at
late times. (ii) Ring vertical velocity $v$ in mm s-1; see also the bottom
panels of figure 13. (iii) Ring diameter $2R$ in mm; see also figure 14. (iv)
Ring circulation $C$ in mm2 s-1 from (3). (v) Ring area $A_{C}$ in mm2. (vi)
Stokes number $St=\tau_{\text{p}}/\tau_{\text{r}}$, where
$\tau_{\text{r}}\approx 17$ ms. Results obtained by applying the chosen
threshold values. Symbols as in Švančara et al. (2020).

In figure 13 we present the vortex ring vertical position and velocity,
computed from the pseudovorticity maps. For comparison, we also show the
results directly obtained form the vorticity field, plotted as black lines in
the right panels. Remarkably, the determination of the ring location is robust
and independent of the particle family. Concerning its velocity, the general
trend is well reproduced, but some periodic oscillations are visible for the
heaviest and larger particles – namely the $D_{2}$ $30\,\mu$m family,
corresponding to the largest Stokes number. These oscillations are likely
related to the periodicity of the domain because their period – about 5 s – is
approximately the time it takes for the ring to travel through the periodic
box. Additionally, as mentioned above, when the ring passes, it leaves in its
wake an inhomogeneous distribution of particles that is later encountered in
front of the ring. This effect is more marked for particles with high inertia,
as it is also apparent from figure 10.

Figure 13: Temporal evolution of the ring vertical position (top) and velocity
(bottom) obtained from the HVBK simulation. Panel disposition and line colour
code as in figure 12. Note that here, for the sake of clarity and comparison,
we plot the absolute values of ring position and velocity in a fixed
coordinate system, not moving with the ring and coinciding with the moving
frame of reference at the beginning of the simulation (in the coordinate
system of figure 11 the ring is moving downward, in the negative direction).
The velocity time limits are due to the Gaussian algorithm chosen for the
velocity calculation (Švančara et al., 2020).

The determination of the ring radius, plotted in figure 14, is more sensitive
to particle inertia. It is evident from the figure – see especially the top
panel – that the ring radius estimate does not depend appreciably on particle
size and density for small enough particles, with $St\lesssim 0.1$.
Additionally, for these particles, the expected time increase of the radius,
evident from the vorticity trend, is recovered; see also the related
discussion in §3.3.

Figure 14: Ring radius as a function of the time $t$ elapsed from the
beginning of the HVBK simulation. Note the different scales on the vertical
axes. Reference system as in figure 13.

Another remarkable feature that can be seen in figure 14 is that, for particle
radii of 5 and 10 $\mu$m, the ring radius $R$ is – on average – larger for the
hydrogen particles than for the heavier ones (the ring diameter $2R$ was
defined above, in §3.3, as the distance between the observed counter-rotating
vortices). This may indicate that deuterium particles – heavier than the fluid
– tend not to probe the ring core region, as visualised in experiments (figure
1) and in numerical simulations (figure 10). Moreover, for deuterium particles
with radii equal to 10 and 30 $\mu$m, the ring radius does not change
appreciably with time and is consistently smaller than the tracer one.

The peculiar behaviour observed in the bottom panel of figure 14 for the
largest hydrogen particles, those with $St=1.45$, is at present not entirely
understood. However, from the corresponding pseudovorticity maps – not shown
here – one can notice that, for sufficiently large times, larger than
approximately 10 s, the underlying dipolar ring structure is replaced by two
adjacent dipoles. On the other hand, one may more generally say that such
large particles are not suitable for tracking the ring.

Note finally that, for the sake of completeness, we also list in table 2 the
values of ring circulation and area for the different particle families, in
the same fashion as in table 1, but we do not show here the corresponding time
trends because they do not add any information on our discussion, that is,
they merely confirm the already reported findings.

##  5 Conclusions

In a recent experimental work (Švančara et al., 2020) particle tracking
velocimetry was used to construct Lagrangian pseudovorticity maps that were
found to provide valuable information on the propagation of macroscopic vortex
rings in superfluid 4He. In that work, it was also shown analytically that
such maps are closely related to the Eulerian vorticity. For such a
mathematical proof, it was assumed that particles act as ideal fluid tracers
and that they are homogeneously distributed in space. In practice, such
assumptions are rarely met in experiments.

In this work, we have numerically studied how particles can be used to study
the propagation of vortex rings, in the spirit of Švančara et al. (2020). We
have performed two sets of numerical simulations. In the first set, we have
employed the FEM method to study the behaviour of an isolated vortex ring in a
viscous fluid, in a geometry similar to the experimental setting used by
Švančara et al. (2020). Fluid tracers were specifically added to the flow to
investigate the reliability of pseudovorticity maps. In the second set, we
have performed numerical simulations of an isolated vortex ring in superfluid
4He, in a periodic domain, using the HVBK model. In this case, we have
considered the dynamics of several particle families of different sizes and
densities. We have thus addressed the effect of particle inertia on the
propagation properties of an isolated vortex ring.

Overall, our results confirm that the Lagrangian pseudovorticity is a relevant
estimator for quantifying general features of isolated vortical structures,
such as their position and propagation velocity. This observation is robust
and independent of particle inertia. However, we have seen that the intensity
of pseudovorticity fields can be much weaker than the actual Eulerian
vorticity, especially for sparse particle distributions, and that the ring
size is not correctly captured when highly inertial particles are used.

The outcome is especially important in the absence of Eulerian data, which is
the case of state-of-the-art visualization experiments in superfluid 4He (see,
for example, Švančara et al., 2021). More generally, our work neatly
demonstrates that quantitative comparisons between particle trajectories
obtained in different experimental conditions can be performed by using
pseudovorticity maps, taking also into account particle concentration and
inertia. Additionally, future work could focus on establishing the relevance
of such a measure for the identification of Lagrangian coherent structures,
which is especially relevant for geophysical flows (see, for example,
Hadjighasem & Haller, 2016).

Acknowledgements. We thank P. Švančara for valuable help. HVBK computations
were carried out on the Mésocentre SIGAMM hosted at the Observatoire de la
Côte d’Azur and on the French HPC cluster OCCIGEN through the GENCI allocation
A0072A11003. J. I. Polanco and G. Krstulovic were supported by the Agence
Nationale de la Recherche through the GIANTE ANR-18-CE30-0020-01 project. O.
Outrata, M. Pavelka, J. Hron and M. La Mantia acknowledge the support of the
Czech Science Foundation (GAČR) under grant no. 19-00939S.

Declaration of Interests. The authors report no conflict of interest.

##  Appendix A Analytical results

Švančara et al. (2020) reported that the Lagrangian pseudovorticity $\theta$
is half of the Eulerian vorticity $\omega$. This specifically occurs in the
close vicinity of the inspection point $\boldsymbol{r}$, defined above, (1),
and for homogeneous and isotropic distributions of fluid particles, having
smooth velocity fields. Although these conditions are at present not met in
the case of experiments in He II, the result clearly demonstrates that
$\theta$ and $\omega$ are related, and therefore supports the use of the
Lagrangian pseudovorticity, especially in the absence of Eulerian data.

In this section we further investigate analytically the relation between
pseudovorticity and vorticity in order to explain some results reported in the
main text.

Following Švančara et al. (2020) we first rewrite (1) as

$\theta(\boldsymbol{r},R)=\frac{1}{N}\int_{D(\boldsymbol{r},R)}f(\boldsymbol{r}^{\prime})\frac{\left[\left(\boldsymbol{r}^{\prime}-\boldsymbol{r}\right)\times\boldsymbol{u}(\boldsymbol{r}^{\prime})\right]_{z}}{|\boldsymbol{r}^{\prime}-\boldsymbol{r}|^{2}}\,\mathrm{d}^{2}\boldsymbol{r}^{\prime},$
(8)

where $D(\boldsymbol{r},R)$ is a circle, centred at $\boldsymbol{r}$, with
radius $R$,

$N=\int_{D(\boldsymbol{r},R)}f(\boldsymbol{r}^{\prime})\,\mathrm{d}^{2}\boldsymbol{r}^{\prime},$
(9)

and $f$ denotes the particle distribution function (note that here we
disregard the time dependence of the problem).

We will now estimate the pseudovorticity for a two-dimensional vortex
characterised by a purely azimuthal fluid velocity, having magnitude

$v_{a}=\frac{\Gamma}{2\pi r}g(r^{2}),$ (10)

where $\Gamma$ is the constant fluid circulation associated to the vortex and
$r$ indicates the distance from the vortex centre. We assume here that
$g(r^{2})$ is a smooth, non-negative function, vanishing at the origin,
$g(r^{2})=\mathcal{O}(r^{2})$ for $r\rightarrow 0$, and at most constant at
infinity, $g(r^{2})=\mathcal{O}(1)$ for $r\rightarrow\infty$. The
corresponding flow vorticity magnitude can then be obtained as

$\omega=\frac{1}{r}\frac{\partial rv_{a}}{\partial
r}=\frac{\Gamma}{\pi}g^{\prime}(r^{2}),$ (11)

where $g^{\prime}(r^{2})$ indicates the relevant derivative of $g(r^{2})$.

The pseudovorticity for a circle of radius $R$, centred at the inspection
point $(x_{0},0)$, in the Cartesian reference system having the origin at the
vortex centre, can then be analytically determined, for a given function
$g(r^{2})$, if also the spatial particle distribution $f$ is known. We will
first work with an unknown function $g$ and an uniform particle distribution
$f=N/(\pi R^{2})$, $N$ being the number of fluid particles. By using (10) we
can then rewrite (8) as

$\theta(x_{0};R)=\frac{1}{\pi
R^{2}}\int_{D(x_{0},R)}\frac{\boldsymbol{r}-(x_{0},0)}{|\boldsymbol{r}-(x_{0},0)|^{2}}\times\left[\frac{\Gamma}{2\pi
r}g(r^{2})\boldsymbol{e}_{a}\right]\,\mathrm{d}^{2}\boldsymbol{r},$ (12)

where $\boldsymbol{e}_{a}$ indicates the azimuthal velocity direction in the
chosen Cartesian coordinate system, and $r$ is the magnitude of
$\boldsymbol{r}$.

In the polar reference system having the origin at the inspection point
$(x_{0},0)$ (12) becomes

$\theta(x_{0};R)=\frac{\Gamma}{2\pi^{2}R^{2}}\int_{0}^{R}\int_{0}^{2\pi}\frac{g(r^{2})}{r^{2}}\left(r^{\prime}+x_{0}\cos\varphi^{\prime}\right)\,\mathrm{d}\varphi^{\prime}\,\mathrm{d}r^{\prime}$
(13)

where $r^{\prime}$ and $\varphi^{\prime}$ are the chosen polar coordinates,
and $r^{2}=(r^{\prime})^{2}+2r^{\prime}x_{0}\cos\varphi^{\prime}+x^{2}_{0}$.
Since near the origin
$g(r^{2})=\mathcal{O}(r^{2})=g^{\prime}(0)r^{2}+\mathcal{O}(r^{4})$, the
integrand in (13) has no singularity; $g^{\prime}(0)$ denotes here the
derivative of $g(r^{2})$ at the vortex centre. Additionally, for
$x_{0}\rightarrow\infty$ the integral in (13) goes to zero, that is, the
pseudovorticity vanishes far from the vortex centre, because the velocity
vanishes too.

For $x_{0}=0$, when the inspection point $(x_{0},0)$ coincides with the vortex
centre, (13) becomes

$\theta(0;R)=\frac{\Gamma}{\pi
R^{2}}\int_{0}^{R}\frac{g[(r^{\prime})^{2}]}{r^{\prime}}\,\mathrm{d}r^{\prime}.$
(14)

Since $g$ behaves at infinity at most as a constant, the integral in (14)
behaves at most as $\ln R$. Therefore, for large $R$ values, $\theta(0;R)$
goes to zero. For small $R$ values, we obtain

$\lim_{R\rightarrow 0}\theta(0;R)=\lim_{R\rightarrow 0}\frac{\Gamma}{\pi
R^{2}}\int_{0}^{R}\frac{g^{\prime}(0)(r^{\prime})^{2}+\mathcal{O}[(r^{\prime})^{4}]}{r^{\prime}}\,\mathrm{d}r^{\prime}=\frac{\Gamma}{2\pi}g^{\prime}(0)=\frac{\omega|_{r=0}}{2},$
(15)

which means that, at the origin, the pseudovorticity converges to half of the
vorticity. This is compatible with the general result reported by Švančara et
al. (2020), where it was shown that, for particle distributions radially
symmetric around the inspection point, the pseudovorticity converges to half
of the vorticity, for sufficiently small $R$ values.

Additionally, by using the Hermite-Hadamard inequality (Niculescu & Persson,
2006), it can be shown that $\theta(0;R)$ is a decreasing function of $R$ near
the origin, because the function $g(r^{2})/r$ is there concave. However, for
large $R$ values, the latter must become convex, since it is positive, but, at
the same time, it is expected that, for large $R$ values, the pseudovorticity
goes to zero and therefore its variations are negligible.

Figure 15: Pseudovorticity $\theta(0;R)$ as a function of the circle radius
$R$, (14), for a Lamb-Oseen vortex, (16), with $\Gamma=50$, $t=10$ and
$\nu=0.01$. Note that the pseudovorticity $\theta(0;R)$ approaches half of the
vorticity, $\omega/2\approx 20$, (17), for $R\rightarrow 0$, vanishes for
$R\rightarrow\infty$ and is a decreasing function of $R$. For the sake of
simplicity and generality, we do not explicitly mention in this section
relevant physical units.

In figure 15 we plot $\theta(0;R)$ as a function of $R$, for a Lamb-Oseen
vortex, having the azimuthal velocity magnitude

$v_{LO}=\frac{\Gamma}{2\pi r}g(r^{2})=\frac{\Gamma}{2\pi
r}\left[1-\exp\left(\frac{-r^{2}}{4t\nu}\right)\right],$ (16)

where $r$ denote the distance from the vortex centre and, as in figure 2, we
set $\Gamma=50$, $t=10$ and $\nu=0.01$ (for the sake of simplicity and
generality, we do not explicitly mention in this section relevant physical
units). Note in passing that the corresponding flow vorticity magnitude can be
written as

$\omega_{LO}=\frac{\Gamma}{4\pi t\nu}\exp\left(\frac{-r^{2}}{4t\nu}\right).$
(17)

In summary, the analytical results reported to date – see especially figure 15
– supports the view that the pseudovorticity magnitude becomes smaller if the
circular area chosen for its estimate increases, as it is also mentioned in
the main text.

For $x_{0}\neq 0$ the pseudovorticity is given by (13) and we numerically
calculated the integral for the just introduced Lamb-Oseen vortex, the one
with $\Gamma=50$, $t=10$ and $\nu=0.01$. The integration was carried out by
using the software Wolfram Mathematica and in figure 16 we plot the obtained
result for $R=0.3$ (open orange circles). A remarkable agreement with half of
the actual flow vorticity (red line) is obtained.

In the same figure the green line, showing a pseudovorticity sign change,
corresponds to the green open circles in the main panel of figure 2, that is,
it displays the pseudovorticity for the chosen Lamb-Oseen vortex, estimated
directly from (1), by using 1000 equally spaced grid points, up to a radius of
10. Additionally, 100 equally spaced fluid particles were distributed in the
chosen annular region, having outer radius equal to $0.3$ (the pseudovorticity
was not computed in the region centre and was set to zero at the vortex
centre). Note also that, as mentioned in §2, the particles are in this case
equally spaced along the radial direction, which means that the chosen
particle distribution is not radially symmetric and that the particles are
clustered in the vortex centre vicinity.

Figure 16: Pseudovorticity for a Lamb-Oseen vortex as a function of the
distance from the vortex axis (radius). Vortex parameters as in figures 2 and
15, that is, $\Gamma=50$, $t=10$ and $\nu=0.01$. Black line: vortex azimuthal
velocity $v_{LO}$, (16); red line: half of the vorticity $\omega_{LO}/2$,
(17); green line: pseudovorticity corresponding to the open green circles in
the main panel of figure 2, estimated as described in that figure caption,
that is, by using directly (1); open orange circles: pseudovorticity
$\theta(x_{0};R)$, (13), with $R=0.3$.

We can see from (13) that the integrand is non-negative if

$r^{\prime}+x_{0}\cos\varphi^{\prime}\geq 0,$ (18)

which can be rewritten in the Cartesian coordinate system $(x,y)$ having the
origin at the vortex centre as

$\left(x-\frac{x_{0}}{2}\right)^{2}+y^{2}\geq\left(\frac{x_{0}}{2}\right)^{2}.$
(19)

The above equality is satisfied for the circle with radius $x_{0}/2$, centred
at $(x_{0}/2,0)$. Points outside this circle contribute positively to the
pseudovorticity, while points inside it contribute negatively, and the
intersection of $D(x_{0},R)$ with the positively contributing region is larger
than that with the negatively contributing region. Therefore, if the function
$g(r^{2})/r^{2}$ behaves nearly as a constant within $D(x_{0},R)$, the
contribution of the positive region (outside that circle) will be larger than
that of the negative region (within that circle), and the pseudovorticity will
be positive. However, if the function $g(r^{2})/r^{2}$ is considerably larger
in the negative region, the pseudovorticity can become negative. Since the
function $g(r^{2})/r^{2}$ is here assumed to be smooth, the pseudovorticity
should typically be positive, as indicated in figure 16 (open orange circles).

It then follows that the sign changes of the pseudovorticity displayed in
figures 2 and 15 – see also the middle panel of figure 4 – are most likely an
effect of the discrete particle distributions employed for the $\theta$
estimates based on (1), especially if one also considers that the analytical
results reported here are only valid for uniform particle distributions, that
is, we set $f=N/(\pi R^{2})$ in (8). Indeed, we would like to address
analytically the effect of particle distribution non-uniformities in future
studies.

Another possible line of future research could be based on the following
result, which can be considered a generalisation of the analytical relation
between $\theta$ and $\omega$ reported by Švančara et al. (2020). The latter
was obtained by expanding the fluid velocity to the second order with respect
to the difference $(\boldsymbol{r}^{\prime}-\boldsymbol{r})$. If, in a similar
fashion, we expand it to the fourth order, we get

$\theta(\boldsymbol{r};R)=\frac{\omega(\boldsymbol{r})}{2}+\frac{R^{2}}{32}\Delta\omega(\boldsymbol{r})$
(20)

for a smooth velocity field, in the close vicinity of the inspection point
$\boldsymbol{r}$, for radially symmetric particle distributions. The
pseudovorticity is thus approximately a quadratic function of $R$ and its
behaviour depends on the Laplacian of the vorticity at the inspection point.
It then follows that, if $\theta(\boldsymbol{r};R)$ is known, one could
estimate the flow vorticity $\omega(\boldsymbol{r})$ by using (20). The latter
estimate could be performed by numerical means, or even analytically, and this
could be the focus of future studies, willing to assess in more detail the
effect of $R$ on the relation between $\theta$ and $\omega$.

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