convolution operator does not exist; however, a process called ‘windowing’ can
mitigate the leakage by enforcing periodicity to the signal.

The truncated velocity fields are first multiplied by the window function
before taking the Fourier transform. The optimal choice of the window function
depends on the underlying spectrum. In general, wideband windows are suited
for signals with a high dynamic range like the turbulent energy spectrum but
offer less sensitivity, whereas the narrowband windows such as a rectangular
window have a low dynamic range but high sensitivity. The dynamic range and
sensitivity cannot both be maximized simultaneously. Therefore, it is critical
to test the effect of different windows on the expected spectrum to make an
informed decision. Figure 36a-b shows rectangular (narrowband), hamming
(intermediate), flattop (wideband) windows in the real and Fourier space. The
wider the main lobe becomes, the broader a peak becomes in the Fourier space.
The decay rate of the side lobe peaks determines the strength of the leakage.
When we compute the energy spectrum of the windowed velocity field, we
multiply the spectrum computed using Eq. 43 by the correction factor $\zeta$
(bumping) to compensate for the loss of the signal (velocity) by windowing.

$\displaystyle\zeta=\frac{\int_{\mathcal{V}}u_{i}({\vec{x}})u_{i}({\vec{x}})d{\vec{x}}}{\int_{\mathcal{V}}w^{2}({\vec{x}})u_{i}({\vec{x}})u_{i}({\vec{x}})d{\vec{x}}}$
(47)

Here, $w({\vec{x}})$ is a window function. Figure36(c-e) shows the energy
spectra with no window, a flattop, and a hamming window respectively. The
flattop window enables to resolve the dissipation range thanks to its high
dynamic range.

Figure 36: Windowing mitigates spectral leakage originated from the domain
truncation.(a) Amplitude of the rectangle, hamming, and flattop windows are
shown. Multiplying the rectangle window to the true velocity field results in
a truncated velocity field with width $L_{0}$. (b) The Fourier transforms of
the considered window functions reveal the degree of spectral leakage. The
rectangular window has high spectral resolution but a low dynamic range
(strong leakage), whereas the flattop window has the opposite character. The
hamming window has an intermediate characteristics. (c) Effect of truncating a
velocity field with no windowing (d) Applying the hamming window to the
truncated velocity field mitigates the leakage due to the domain truncation.
The shown spectral density is bumped by a correction factor $\zeta$. (e) Same
as (d) but with the flattop window.

###  VII.5 PIV algorithm and its nature as a low-pass filter

In this section, we demonstrate that it is possible to obtain an energy
spectrum that is consistent with the ground truth via a ‘synthetic PIV’
measurement that mimics ideal PIV conditions. The observed deviations can be
explained by considering spectral leakage, and noise (Figure 35). We further
compare the effects of the PIV interrogation window size to a low pass
filter[79], and find that the effects of PIV interrogation window size are
similar to those of box filtering.

To assess the spectrum of the PIV-generated velocity fields, we created
synthetic images of particles advected by a turbulent field from JHTD data.
The field is resolved spatially and temporarily with a periodic boundary
condition. With the optimized images and PIV settings (particle-per-pixel:
0.012, $L$=2048px, total number of particles per image: $5\times 10^{4}$,
initial interrogation window size: 32px, final interrogation window size
$W=4\eta$=4px), it is indeed possible to extract the accurate three-
dimensional energy spectrum from the PIV-generated field (Figure 37a). In
terms of the image quality, we find that images with high seeding density
(ppp$\approx$ 0.01) is critical. The temporal spacing between the adjacent
frames should be less than 0.5 Kolmogorov time scales. As for the PIV
settings, the smallest interrogation window in the multi-pass algorithm should
be comparable to the Kolmogorov length scale. Further discussion on the
required criteria to resolve a turbulent field can be found in [78, 50]. We
tested two PIV algorithms (1. the standard cross-correlation algorithm called
WIDIM [80] 2\. Pyramid algorithm [51]), and observed that the pyramid
algorithm extracts the small-scale motion better than the WIDIM; the pyramid
algorithm extends the band of accurate measurement by 50% in the wavenumber
space, compared to WIDIM (Figure 37a). The obtained spectra are in agreement
with the reference up to the detection limit.

Any interrogation-based PIV acts as a low-pass filter on the true velocity
field [79]. This is because the fundamental operation of PIV is peak finding
of a cross-correlation map inside the interrogation windows. It assigns
essentially an average displacement vector inside the interrogation window.
Hence, the dimension of the interrogation window ($W_{x}\times
W_{y}\text{px}^{2}$) sets the smallest motion PIV could detect. This effect
could be modeled by a $W_{x}\times W_{y}$ box filter (Figure 37b). The actual
PIV operation is more complex than box filtering; however, we find that the
spectra of the PIV-generated fields are similar to that of the box-filtered
field. If not compensated for, this effect could lead turbulence studies using
PIV to report a narrower inertial subrange than the truth. Understanding this
effect is therefore critical to correctly estimate the $\kappa$ dependence on
the energy spectrum. The same low-pass filtering is expected for 3D PTV if the
Eulerian field is reconstructed by binning the Lagrangian velocities into
voxels.

Figure 37b suggests that the any motion smaller than the interrogation is not
resolved by PIV. Based on this result, we conclude that the energy spectrum
above $\kappa\approx\pi/W$ is significantly underestimated. In main Figure 3f
and Figure 33, unfilled markers indicate the underestimation due to this
effect of PIV.

Figure 37: Energy spectrum of a velocity field extracted by running PIV on
synthetic data (a) Given images with highly seeded particles and zero noise,
the pyramid algorithm (orange) [51] yields a more resolved energy spectrum
than the standard cross-correlational algorithm (WIDIM, green). (b)PIV
operation is comparable to box filtering with its kernel size as same as the
interrogation window $W$. The energy spectrum of the PIV-generated field
(algorithm: pyramid) deviates from the ground truth above $\kappa=\pi/W$.

###  VII.6 Computation of structure function

The structure function $D_{ij}$ possesses equivalent information as the one-
dimensional, energy spectra. With homogeneity and isotropy, it is defined as

$\displaystyle D_{ij}(r)=\langle\delta u_{i}\delta u_{j}\rangle$ (48)

with $\delta u_{i}(\vec{x},\vec{r})=u_{i}(\vec{x}+\vec{r})-u_{i}(x)$. The
longitudinal, structure function is related to the one-dimensional,
longitudinal energy spectrum. For example,
$D_{11}(r\hat{e_{1}})=2\int_{0}^{\infty}E_{11}(\kappa_{1})[1-\cos{(\kappa_{1}r)}]d\kappa_{1}$.

We compute $\delta u_{i}$ at every possible point on the 2D velocity field of
a set of snapshots, then take an average over time and space. The number of
available samples $\delta u_{i}$ with a separation distance $r$ depends on its
position $\vec{x}$. Take a square velocity field as an example. The number of
available samples with $r=$(diagonal distance of the square) is much more
limited than small $r$. To ensure the same statistical weight at every $r$, we
randomly sample $m$ pairs for each $r$ to compute $D_{ij}(r)$. We also make
sure that each snapshot is statistically independent by only considering the
snapshots that are temporally separated by integral time scale
$\tau_{\mathcal{L}}=\mathcal{L}/u^{\prime}$.

By definition, the structure function does not face the undesired effects in
the spectral space as explained in the previous sections. It is, however,
susceptible to the low-pass character of the PIV operation. Hence, we apply
the same criteria as the energy spectrum, and consider that the structure
function values at a scale below $r\approx 2W$ are significantly
underestimated. In main Figure 3g and Figure 33, unfilled markers indicate the
underestimation due to PIV.

###  VII.7 Energy spectrum over a region of interest

Eq. 43 and 46 provide means to compute the energy spectrum over a region of
interest (ROI). The observable band $[\kappa_{min},\kappa_{max}]$ is limited
by the sampling frequency of the data and the size of the ROI. These local
energy spectra do not necessarily match a part of the energy spectra of the
entire flow. To match these two spectra, it is necessary that the turbulent
field is homogeneous.

In the case of inhomogeneous turbulence, the global energy spectra represent a
mixture of different statistical states, and deviate from the results of HIT.
In our experiments, the flow induced by the repeated collisions of vortex
rings consists of a homogeneous, isotropic turbulent core and comparatively
quiescent surroundings. Therefore, we present the local energy spectrum over a
rectangular (or cuboidal for 3D PTV) region of the homogeneous, turbulent core
in Figure 3f.

###  VII.8 3D energy spectrum derived from 3D PTV data

We report the energy spectrum of a fluctuating velocity field measured by 3D
PTV. The extraction of all three components of velocity vectors enables the
computation of the spectrum directly via Eq. 43. Figure 38 shows the energy
spectrum of the central region of our turbulent blob. The temporal average of
the spectrum is shown here as the turbulence inside the blob is statistically
steady with weak dependence on the phase of forcing. We compare it to the
corresponding spectrum from [9] by converting their 1D energy spectrum to
$E(\kappa)$ via
$E(\kappa)=\frac{1}{2}\kappa^{3}\frac{d}{d\kappa}\left(\frac{1}{\kappa}\frac{dE_{11}(\kappa)}{d\kappa}\right)$.
All energy spectra with three $Re_{\lambda}$ exhibit the Kolmogorov scaling in
the inertial subrange. Furthermore, the rescaled spectra obtained by 3D PTV
and 2D PIV are consistent with the universal function reported in [9]. As
discussed in the main text, both 3D PTV and 2D PIV are limited by the image
resolution and the finite field of view. The size of the interrogation window
determines the upper bound of the valid region of the spectrum
($\kappa\eta\leq\pi\eta/W$). If the integral scale were greater than the field
of view, the spectrum could not resolve the energy containing range. This can
be seen with the curve of ($Re_{\lambda}=300$, 3D PTV) in Figure 38. The same
effect is observed for ($Re_{\lambda}=228$, 2D PIV) since the domain of
analysis was restricted to the inner part of the energetic region($r\leq
R_{\rm blob}$).

Figure 38: 3D energy spectra of velocity fields obtained by 3D PTV and 2D PIV.
The blue cubes or square on the right represent the domain that the DFT is
conducted (Eq. 43). The turbulent core ($r=R_{\rm blob}$) is represented by
the purple sphere.

We note that the geometry of the velocity field obtained by 3D PTV is in
general a cuboid and not a cube, reflecting the differences between sensor
dimensions and resolution and the maximum depth that 3D PTV accurately
measures. A difference in the length of the data in each dimension affects the
quality of DFT. This is because three sinc functions with different
frequencies are convoluted in $\kappa$ space. If left uncorrected, this
results in a wavy behavior of the energy spectrum at high $\kappa$. To
mitigate this effect, we pad the velocity field with to make the shape of the
array cubic before taking the DFT. Zero-padding simply extends the signal, and
interpolates it in the wavenumber domain; however, cubing the array allows the
spectral leakage to occur at the same frequencies along all the three
directions, making the spectrum more readily interpretable.

##  VIII Energy balance

In this section we examine the energy balance of a turbulent blob in detail.
We shall quantify how much dissipation occurs inside the blob, and identify
its relation with the injected power. The kinetic energy
$\mathscr{E}=\frac{1}{2}U_{i}U_{i}$ of a viscous flow obeys:

$\displaystyle(\partial_{t}+U_{i}\partial_{i})\mathscr{E}+\partial_{i}T_{i}=2\nu
S_{ij}S_{ij}$ (49)

where $S_{ij}=\frac{1}{2}(\partial_{j}U_{i}+\partial_{i}U_{j})$ is the rate-
of-strain tensor, and $T_{i}=U_{i}p/\rho-2\nu U_{j}S_{ij}$ is the energy flux
[44]. The Reynolds decomposition $U_{i}=\langle{U}_{i}\rangle_{t}+u_{i}$
yields

$\displaystyle(\partial_{t}+\langle
U_{i}\rangle_{t}\partial_{i})\langle\mathscr{E}\rangle_{t}+\partial_{i}(u_{i}\mathscr{E}+\langle
T_{i}\rangle_{t})=-\overline{\epsilon}-\epsilon$ (50)

where
$\overline{\epsilon}=2\nu\langle\overline{S}_{ij}\overline{S}_{ij}\rangle_{s}$
with $\overline{S}_{ij}=\frac{1}{2}(\partial_{j}\langle
U_{i}\rangle_{t}+\partial_{i}\langle U_{j}\rangle_{t})$ is the mean rate of
strain. Eq. 50 reveals the decomposition of the energy dissipation in the mean
and the fluctuating flows. In the case of a turbulent blob, the mean flow
consists of eight vortex rings that repeatedly travel towards the center (See
Figure 3a in the main text). On the $\sigma_{h}$ plane, the enstrophy
measurements (Figure 29f) indicate
$\epsilon/\overline{\epsilon}=\omega^{2}/\overline{\Omega}^{2}=O(10^{2})$.
Below we assume $\overline{\epsilon}\ll\epsilon$ anywhere in the system.

###  VIII.1 Partition of dissipation in a turbulent blob

The energy dissipation within a domain $\mathcal{V}$ that encloses a blob is

$\displaystyle\mathcal{D}(r/R_{\rm
blob})=\rho\int_{\mathcal{V}}\epsilon(\vec{r})dV$ (51)

where $\epsilon(\vec{r})\equiv 2\nu\langle s_{ij}s_{ij}\rangle_{t}$ is a
spatially varying dissipation rate. The radial profile of the local
dissipation rate (or enstrophy density equivalently) shows that
$\epsilon(\vec{r})$ depends only on the radial distance, is constant up to a
radius $R_{\rm blob}$, and decays approximately as $\epsilon(r)\propto r^{-4}$
for $r\geq R_{\rm blob}$.

$\displaystyle\epsilon(r)=\begin{cases}\epsilon_{0}&(r<R_{\rm blob})\\\
\epsilon_{0}(\frac{R_{\rm blob}}{r})^{4}&(r\geq R_{\rm blob})\end{cases}$ (52)

It is useful to compute the dissipation within a sphere of radius $r$ for a
turbulent blob with this profile. The fraction of energy dissipation inside
the sphere to the total is

$\displaystyle p(r/R_{\rm blob})=\begin{cases}\frac{1}{4}\left(\frac{r}{R_{\rm
blob}}\right)^{3}&(r<R_{\rm blob})\\\ \\\ 1-\frac{3R_{\rm blob}}{4r}&(r\geq
R_{\rm blob}).\end{cases}$ (53)

If we include a cutoff in the considered integral because the chamber size is
finite, the expression becomes

$\displaystyle p(r/R_{\rm blob},(L_{box}/2)/R_{\rm
blob})=\begin{cases}\frac{1}{4}\left(\frac{r}{R_{\rm
blob}}\right)^{3}(1-\frac{3R_{\rm blob}}{2L_{box}})^{-1}&(r<R_{\rm blob})\\\
\\\ (1-\frac{3R_{\rm blob}}{4r})(1-\frac{3R_{\rm
blob}}{2L_{box}})^{-1}&(R_{\rm blob}\leq r\leq\frac{L_{box}}{2}).\end{cases}$
(54)

Note that fraction of energy inside the blob to the whole depends not only the
blob radius but also its respective size to the system. The same function
applies to the amount of energy and enstrophy inside the sphere with radius
$r$.

Figure 39(a) shows the model function (Eq. 52) and the scaled cummulative
function $p$ (Eq. 54)for $R_{\rm blob}=\sqrt{6}R$. The value of $R_{\rm blob}$
with respect to the ring radius $R$ is taken from the experimental data
(Figure 3b). There is a geometrical interpretation of $R_{\rm
blob}=\sqrt{6}R_{\rm ring}$. The symmetrical configuration of the eight vortex
rings defines an octahedron as explained in SectX. Since the vortex atmosphere
is ellipsoidal, the majority of the energy is housed inside a sphere as large
as the circumscribed sphere of the octahedron which has a radius of
$\sqrt{6}R_{\rm ring}$. One can also imagine a polyhedron such that eight
spheres with radius $R_{\rm ring}$ fused together on each face of an
octahedron. The radius of such a circumscribed sphere is $(1+\sqrt{2})R_{\rm
ring}\approx 2.41R_{\rm ring}$, sufficiently close to $\sqrt{6}R_{\rm
ring}\approx 2.45R_{\rm ring}$.

Figure 39: Model radial distribution of energy $\mathscr{E}$, enstrophy $Z$,
and dissipation rate $\epsilon$. (a) Radial distribution function (blue) and
scaled cummulative function $p(r)$(orange) of energy/enstrophy/dissipation
rate are shown for a case $R_{\rm blob}=\sqrt{6}R_{\rm ring}$ and $(L_{\rm
box}/2)/R_{\rm blob}=7$. (b)Dependence of $p$ on the blob radius $R_{\rm
blob}$ and the system size $L_{\rm box}$ is shown.

The total dissipation is

$\displaystyle\mathcal{D}_{tot}$
$\displaystyle=\frac{16}{3}\pi\rho\epsilon_{0}R_{\rm
blob}^{3}=\frac{4}{3}\pi\rho\epsilon_{0}R_{\rm eff}^{3}$ (55)

where $R_{\rm eff}=4^{\frac{1}{3}}R_{\rm blob}\approx 1.59R_{\rm blob}$ is the
effective radius of the blob. The dissipation within a sphere of radius $r$ is
given by

$\displaystyle\mathcal{D}(r/R_{\rm blob})$ $\displaystyle=p(r/R_{\rm
blob})\mathcal{D}_{tot}.$ (56)

###  VIII.2 Estimation of dissipation rate

Measuring the dissipation rate robustly with PIV is a challenge [38]. Multiple
methods have been proposed, including direct computation from the rate-of-
strain tensor to fitting to the structure function; however, they are often
susceptible to the choice of interrogation window size and the random noise.
[38] investigated five methods to estimate the dissipation rate from the PIV
data of zero-mean, gaseous turbulence: (1) Direct method via the rate-of-
strain tensor (2) Large eddy method based on the modeling in the subgrid scale
(3) Method based on the scaling $\epsilon_{0}=A\frac{u^{\prime
3}}{\mathcal{L}}$ (4) Fitting the energy spectrum to the Kolmogorov spectrum
(5) Fitting the n-th order structure function to $(\epsilon_{0}r)^{n/3}$.
Their investigation on the sources of errors led to conclude that the method
using the second-order structure function yields the most reliable estimate.
Following their findings, we estimate the dissipation rate using the second-
order structure function. To check the robustness of our measurements, we also
computed the dissipation rate with the direct method and by fitting the energy
spectrum to the universal curve. For the direct method, we compute

$\displaystyle\epsilon_{s_{ij}}=6\nu\langle(\partial_{1}u_{1})^{2}+(\partial_{2}u_{2})^{2}+(\partial_{2}u_{1})(\partial_{1}u_{2})\rangle,$
(57)

assuming isotropy [81]. For the structure function method, we fit the
longitudinal second-order structure function $D_{LL}(r)=\langle\delta
u^{2}\rangle/\langle u_{i}^{2}\rangle$ to $C_{2}(\epsilon_{D_{LL}}r)^{2/3}$
where $\delta u(\vec{x},r)=u_{i}(\vec{x}+r\hat{x}_{i})-u_{i}(\vec{x})$, and
$C_{2}=2.1$ [9]. For the spectral method, we fit the one-dimensional energy
spectrum $E_{11}(\kappa_{1})$ to $C_{1}\epsilon^{2/3}\kappa_{1}^{-5/3}$ with
$C_{1}=0.52$ [75]. Numerically, we look for the plateau of the compensated
structure function $1/r(D_{LL}/C_{2})^{3/2}$ and the compensated energy
spectrum $[E_{11}/(C_{1}\kappa_{1}^{-5/3})]^{3/2}$. This way, we avoid
complications to identify the inertial subrange for each data set to fit. In
addition, before applying the above explained methods, the PIV-extracted field
is convoluted with a median filter (with a kernel size of 3x3 or 5x5 px2) to
remove spurious vectors.

###  VIII.3 Partition of injected energy

The energy balance over a domain $\mathcal{V}$ that encloses a blob is

$\displaystyle\mathcal{I}\approx 8K_{\rm ring}f$ (58)

because we expect only the energy within the vortex atmosphere is fed to
sustain the blob. If there is only one vortex ring in a flow, the ratio of
energy enclosed within the atmosphere to the total $c=K_{\rm ring}/K$ is
completely determined by the shape of the vortex atmosphere (see Box 3), and
remains nearly the same $c\approx 77\%$ for the ellipsoidal atmosphere. For
the two canonical vortex ring models, the injected power $\mathcal{I}$ is
given by

$\displaystyle\mathcal{I}=\begin{cases}4c\rho\Gamma_{\rm ring}^{2}R_{\rm
ring}f\left[\ln{\frac{8R_{\rm ring}}{a}-\alpha}\right]&\text{(thin-cored
vortex ring)}\\\ \frac{64\pi}{105}c\rho\Gamma_{\rm ring}^{2}R_{\rm
ring}f&\text{(Hill's spherical vortex)}.\end{cases}$ (59)

The coefficient represents the difference between the models. The generated
vortex rings in this study has coefficients between these two models(Figure
22); however, it depends on the experimental conditions
($L_{*},v_{eff},D_{o},D_{p}$). Thus we directly measure the energy of the
injected vortex rings from 2D PIV data shown in Figure 26.

###  VIII.4 Balance of energy injection and dissipation

We first turn our attention to the total energy balance ($\mathcal{D}_{\rm
tot}$ vs $\mathcal{I}_{\rm tot}=\mathcal{I}/c$) in the chamber. Regardless of
the methods to estimate the dissipation rate, the total dissipated power
monotonically increases with the injected power over almost two decades
(Figure 40e-g). The method using the rate-of-strain tensor tends to
overestimate the dissipation, compared to the methods involving the energy
spectrum and the structure function. This is consistent with [38] as the noise
in the measured velocity field contributes to this overestimation. Removing
the noise with a median filter (Figure 40a-c) leads the estimates by all three
methods to agree with each other(Figure 40d-f). This gives us more confidence
that the filtered velocity field gives the more accurate estimate of the
dissipation rate than the raw field. There, the dissipated power is indeed
proportional to the injected power. We find $\mathcal{D}_{\rm
tot}/\mathcal{I}_{\rm tot}=1.07\pm 0.09$ (3x3(=$5.6\eta\times 5.6\eta$)),
$0.64\pm 0.05$(5x5=$9.3\eta\times 9.3\eta$) for the median-filtered fields.
The attenuation with a larger kernel is expected as it blurs the velocity
field more.

As discussed in the main text, the far-field contribution in
$\mathcal{I}_{tot}$ should be neglected to assess the energy balance of the
blob. The ratio

$\displaystyle\frac{\mathcal{D}(\frac{r}{R_{\rm
blob}})}{\mathcal{I}}=\frac{\mathcal{D}_{\rm tot}}{\mathcal{I}_{\rm
tot}}\frac{p(\frac{r}{R_{\rm blob}})}{c}$ (60)

represents how much injected power via vortex rings is dissipated as a
function of a (scaled) radius. In a sense, this gives a conversion rate from
coherent vortex motion to turbulence in terms of energy. Considering
$\frac{\mathcal{D}_{\rm tot}}{\mathcal{I}_{\rm tot}}\approx 1$, the power
associated to the vortex ring atmosphere is completely dissipated within a
sphere of radius $r=\frac{3}{4(1-c)}R_{\rm blob}\approx 3.3R_{\rm blob}$ with
$c=0.77$. Of which, 33% is dissipated up to $r=R_{\rm blob}$ and 68% is
dissipated up to $r=R_{\rm eff}=4^{\frac{1}{3}}R_{\rm blob}$.

Figure 40: Dissipated vs injected power inside the chamber. (a)To compute the
dissipation rate, three velocity fields are used: a raw PIV-extracted field,
and two median-filtered fields. (b-d) Median-filtering the velocity fields
attenuates the local dissipation rate $\epsilon_{s_{ij}}(\vec{x})=2\nu
s_{ij}(\vec{x})s_{ij}(\vec{x})$. (e) Dissipated vs injected power inside the
chamber is plotted for the three methods to estimate the dissipation rate
using the raw velocity field: $\epsilon_{sij}$ uses the rate-of-strain tensor.
$\epsilon_{E_{11}}$ is obtained by fitting the energy spectrum to the
Kolmogorov spectrum. $\epsilon_{D_{LL}}$ is obtained by fitting the second-
order structure function to Kolmogorov’s 2/3 law. (f) Same as (e) but with a
median-filtered velocity field (Kernel size: $5.6\eta\times 5.6\eta$). (g)
Same as (e) but with another medial-filtered velocity field (Kernel size:
$9.3\eta\times 9.3\eta$).

###  VIII.5 Dissipation rate according to the energy balance

Equating the dissipation $\mathcal{D}(r/R_{\rm blob})$ (Eq. 56) with the
injected power $\mathcal{I}$ (Eq. 59) gives an expression about the
dissipation rate inside the turbulent blob in terms of the properties of the
vortex rings: $\epsilon_{0}\propto\Gamma_{\rm ring}^{2}f/R_{\rm
ring}^{2}\propto V_{\rm ring}^{2}f$ (main Figure 4b). With $R_{\rm
blob}=\sqrt{6}R_{\rm ring}$, $\epsilon_{0}$ becomes

$\displaystyle\epsilon_{0}=\begin{cases}\frac{c(\log{(8R_{\rm
ring}/a)}-\alpha)}{8\sqrt{6}\pi p}\frac{\Gamma_{\rm ring}^{2}f}{R_{\rm
ring}^{2}}&\text{(thin-cored vortex ring)}\\\
\frac{2c}{105\sqrt{6}p}\frac{\Gamma_{\rm ring}^{2}f}{R_{\rm
ring}^{2}}&\text{(Hill's spherical vortex)}.\end{cases}$ (61)

The prefactor of Eq. 61 differs by the model of a vortex ring and $p(r/R_{\rm
blob})$. Here is a quick calculation of the prefactor for $r/R_{\rm
blob}=2\Leftrightarrow p=5/8$ (unbounded) and $c=0.77$. For the thin-cored
model, we shall use $R_{\rm ring}/a=3$ as observed in the experiments, and
$\alpha=2.04$ [49] for a viscous core. The prefactors of the thin-cored model
and Hill’s spherical vortex model are 0.02 and 0.01 respectively. This
relationship is consistent with $\epsilon_{0}=\alpha_{1}^{2}V_{\rm ring}^{2}f$
in the main Figure 4b.

##  IX Enstrophy flux

As the vortex injection frequency is increased for fixed $V_{\rm ring}/R$ we
observe a transition from a reconnection-and-escape dynamic in which most of
the enstrophy leaves with the outgoing vortices to a blob state in which
little to none of the injected enstrophy escapes the blob. As illustrated in
Figure 4 of the main text we examined this transition by plotting the time
averaged enstrophy in a cross section perpendicular to the incoming rings, as
well as by integrating the enstrophy outflux.

In this section we show the enstrophy flux data in greater detail and
summarize a simple image-moment approach to quantifying the visual information
contained in the enstrophy cross section images (not discussed in the main
text).

The enstrophy balance is

$\partial_{t}\int_{\mathcal{V}}\Omega^{2}dV=2\int_{\mathcal{V}}\Omega_{i}(\partial_{i}U_{j})\Omega_{j}dV-\oint_{\partial\mathcal{V}}\Omega^{2}U_{i}n_{i}dS+\nu\int_{\mathcal{V}}\partial_{j}\partial_{j}\Omega^{2}dV-2\nu\int_{\mathcal{V}}(\partial_{j}\Omega_{j})^{2}dV$

in a 3D incompressible, viscous fluid. The enstrophy is not conserved due to
vortex stretching and viscous dissipation.

Figure 41: Enstrophy flux through a sphere with radius $R=1.2R_{\rm blob}$
reveals the influx always outweighs the outflux throughout a cycle when a
turbulent blob is formed. (a)Enstrophy averaged over a measured volume (Green)
and the (net) enstrophy flux show the arrival of the vortex rings
($\Theta=0.25$) and the ejection event$\Theta=0.35$. The vortex rings are
fired at $f=0.2$Hz. (b) The same as (a) but at $f=4Hz$ at which a turbulent
blob is formed.

To examine the confinement of enstrophy, we consider its normal flux
$\vec{J}_{\Omega^{2}}\cdot\hat{n}=\Omega^{2}\vec{U}\cdot\hat{n}$ through a
sphere with radius $R>R_{\rm blob}$. In the case that coherent reconnections
are dominant reactions ($f<f_{c}$), the enstrophy flux exhibits the inflow due
to the eight incoming vortex rings, followed by outflow by the six outgoing
vortex rings at a later phase (Figure 41a and e). The mass flux also captures
the inflow/outflow due to the mass transported by the incoming/outgoing vortex
rings (Figure 41c). The crucial difference is that the mass is always
balanced, whereas the enstrophy is not. Integrating the enstrophy flux over
the surface and cycles gives the net amount of (escaping) enstrophy at each
phase as shown by a yellow curve in Figure 41(a). The positive value
corresponds to the enstrophy escaping from the considered sphere. On the other
hand, the integrated enstrophy flux remains always negative throughout a cycle
in the state of the turbulent blob (Figure 41b). The enstrophy flux depicts
that the inflow always outweighs the outflow, and the latter is more uniform
than the case of the coherent reconnections(Figure 41f). Because the mass
outflow cannot be zero due to incompressibility (Figure 41d), a small amount
of enstrophy leaves the sphere. Nevertheless, the amount of enstrophy outflow
remains smaller than the inflow.

###  IX.1 Categorization of the flows using the time-averaged enstrophy
distibution

To translate the visible transition from reconnection-and-escape to
confinement dynamics, we performed a simple image moment analysis as detailed
below.

For a 2D image with intensity $I_{\rm img}(x,y)$, the (raw) moment of order
($i,j$) is defined as

$\displaystyle m_{ij}=\sum_{x}\sum_{y}x^{i}y^{j}I_{\rm img}(x,y).$ (62)

The moments sequence for an image uniquely determines an image of finite size;
however, they are not invariant under translation. On the other hand, the
central moments

$\displaystyle\mu_{ij}=\sum_{x}\sum_{y}(x^{i}-\overline{x})(y^{j}-\overline{y})I_{\rm
img}(x,y).$ (63)

are translationally invariant by construction. The moments of inertia is
analogous to the second-order image moments ($i=j=1$) when $I(x,y)$ is
replaced by a mass distribution $\rho(x,y,z)$ of an object. Different images
may have different second-order moments in the same way as the different
objects may have different moments of inertia. To account for the total
intensity, the normalized central moments are also used.

$\displaystyle\nu_{ij}=\frac{\mu_{ij}}{\mu_{00}^{(1+(i+j)/2)}}$ (64)

We characterized the distribution of the time-averaged enstrophy of various
collision experiments by the sum of the second-order moments
$\nu_{02}+\nu_{20}$. This quantity is invariant with respect to translation,
scale, and rotation, and is also known as one of Hu invariants [82]. Figure
42a-c shows that the cross-like pattern takes a higher value than the blob
pattern. This reflects the elongated distribution of the cross-like pattern
created by the secondary rings. Figure 42d plots our measurements on the
parameter space, colored by $\nu_{02}+\nu_{20}$. Once the blob state is
reached, the enstrophy distribution remains roughly the same. Hence, it tends
to approach a fixed value if the blob radius were held constant. Strictly
speaking, the normalized, second-order image moments take a higher value for a
larger blob. As the blob size scales with the ring radius, Figure 42 was
constructed by using only the rings with a certain radius $R_{\rm ring}=18\pm
2$mm. This image analysis is consistent with the the linear trend.

Figure 42: The second-order moments of the time-averaged enstrophy
distribution categorizes the flows into the non-blob and blob states. (a)The
logarithmic, time-averaged enstrophy shows the formation of a turbulent blob
at high forcing frequency. The corresponding second-order moments decrease as
the ejection of enstrophy via secondary vortex structures becomes weak.
(b)Images of thelogarithmic, time-averaged enstrophy are plotted on the
$\nu_{02}-\nu_{20}$ plane. Blob patterns are clustered in the bottom left
region. (c) The same plot as (b) but colored by $\nu_{02}+\nu_{20}$. (d)
$\nu_{02}+\nu_{20}$ categorizes the flows into two groups. The separation is
described by a linear relation $f\sim V_{\rm ring}/R_{\rm ring}$, and is
consistent with the transitional frequencies obtained from the enstrophy flux.

##  X Gross-Pitaevskii Simulation

In this section we report the result of our simulations of symmetric
collisions of multiple quantum vortex rings using the Gross-Pitaevskii
equation (GPE, Eq. 65). In all configurations we tested, we observe that
reconnection events take place, and the secondary rings propagate away from
the center of the collisions.

We simulate the dynamics of the quantum vortex rings through the Gross-
Pitaevskii equation (GPE), a model equation that describes the time evolution
of the superfluid wavefunction
$\psi(x_{i})=\sqrt{\rho(x_{i})}\exp{[i\phi(x_{i})]}$ where $\rho$ and $\phi$
are the spatially varying density and phase. The GPE is given by

$\displaystyle\frac{d\psi}{dt}=-\frac{i}{2}\left(\triangledown^{2}-|\psi|^{2}\right)\psi.$
(65)

The wavefunction can be mapped to classical hydrodynamic fluid velocity and
density by the Madelung transform:
$\vec{u}=\triangledown\phi,\rho=|\psi|^{2}$. The GPE is useful to study vortex
dynamics as vortex reconnections occur without divergencies in physical
quantities, and the topological dynamics was shown to be comparable to real
viscous fluids [84].

All simulations were performed using a grid size of 0.5$\xi$ ($\xi$: healing
length) and time step $\Delta t$=0.02 with periodic boundary conditions unless
otherwise stated.

###  X.1 Symmetric collisions of vortex rings

We study collisions between vortex rings that are configured in a symmetrical
fashion. For simplicity, we restrict ourselves to use only a circular ring
with a fixed circulation and a radius. We associate a configuration to a
polyhedron by the following rules. 1. Consider a plane $P_{i}$ that the $i$-th
ring is embedded at an instant of time. 2. The bounded region by the $n$
planes is the polyhedron associated to the configuration of the rings. By this
construction, the initial configuration of our experiment corresponds to an
octahedron.

We conducted GPE simulations of the symmetrical collisions that correspond to
the platonic solids(a tetrahedron, a cube, an octahedron, a dodecahedron, and
a icosahedron). They are the most symmetrical configurations for 4, 6, 8, 12,
and 20 rings respectively. This is understood by counting the number of
symmetry operations for the corresponding point groups. A tetrahedron belongs
to $T_{d}$. A cube and an octahedron belong to $O_{h}$. A dodecahedron and an
icosahedron belong to $I_{h}$ in the Schönflies notation. The result is
summarized in Figure 43.

Two observations are made: (1)The secondary rings always travel away from the
center of the collision. (2)The polyhedron corresponding to the configuration
of the secondary rings is dual to the initial polyhedron. All adjacent vortex
lines during the collision are anti-parallel so the reconnections occur at
every possible site that correspond to the vertices of the initial
configuration. A symmetric collision of four rings (a tetrahedron) results in
four secondary rings (a flipped tetrahedron). A symmetric collision of six
rings (a cube) result in eight secondary rings (an octahedron). The
propagating directions of the secondary rings are given by the normal vectors
of the faces (of the dual polyhedron). The pattern extends to all platonic
solid configurations. In our experiment (the octahedral configuration), we
observe six secondary rings (a cube) after the reconnection events via bubbles
and Lagrangian trajectories (see SI Movie3).

Figure 43: Gross-Pitaevskii simulation demonstrates the geometric rule of the
symmetric vortex ring collisions. When vortex rings are initially set on the
faces of the platonic solids, the region bounded by the planes that the
secondary rings live forms the dual solids, preserving the symmetry of the
vortex structures before and after the collision.

###  X.2 GPE simulation of a vortex ring train

To generate Figure 1a, the rings with radius of $25\xi$ are initialized at a
distance of 25% away from the edge of the box in each axis, oriented to
propagate towards the center of the box. We add noise ($\sim 5\%$) to the
amplitude of each ring in such a way that its center line remains smooth but
not completely radially symmetric. This formation process repeats every 100
time steps to place a new set of rings. To enforce the wavefunction to be zero
at the boundaries, we impose absorbing boundary conditions by adding a
potential at the shell of the box with the thickness of 4$\xi$.

##  XI Supplementary movies

### Movie 1: Bubble visualization of a single-shot vortex ring collision

This movie displays coherent vortex reconnections result in generation of six
outgoing secondary rings when eight vortex rings collide. The bubbles,
generated by hydrolysis, indicate the location of the cores of the primary and
the secondary rings. The movie plays ten times slower than the real time.

### Movie 2: Bubble visualization of a turbulent blob

This movie displays the irregular motion of the bubbles when the vortex rings
are fired at 5Hz, forming a turbulent blob. The first set of the collisions
resembles a single shot (SI Movie 1). Yet, the bubbles at the center move more
irregularly after a few cycles of vortex ring injection.

### Movie 3: Symmetric collision of vortex rings (Gross-Pitaevskii
simulations)

This movie demonstrates the geometrical rule of vortex ring collisions when
they are initialized on the plane of the platonic solids: the configuration of
the ejecting, secondary rings is dual to the configuration of the primary
rings.

### Movie 4: Ejection pattern of a single-shot vortex ring collision

The pathlines shows that a single-shot collision of eight vortex rings results
in formation of six secondary rings, consistent with the GPE simulation (Movie
3). The trajectories are colored by the vortex rings of origins. Notice that
each secondary vortex ring consists of four colors, reflecting its four
‘parent’ vortex rings.

### Movie 5: Pathline visualization of a turbulent blob

When a turbulent blob forms, the pathlines of particles that are transported
by vortex rings become highly irregular due to turbulence within the blob. The
ejecting pattern becomes more uniform than the case of coherent vortex
reconnections. The colors are assigned by the vortex rings of origins. White
trajectories are found in the turbulent blob, and are of unknown origins.

### Movie 6: Pathline visualization of a turbulent blob, colored by Lagrangian
speeds

This visualization of a turbulent blob shows the formation of a steady,
isolated region with considerably higher energy than its surroundings. Here,
all detected pathlines are shown with opacity weighted by the speed. The slow
particles (purple) are dimmed more than the fast (yellow) ones.

### Movie 7: Mean flow structure and fluctuations of a turbulent blob

This movie shows that the fluctuations dominate inside a turbulent blob (the
blue cloud), whereas the time-averaged (mean) flow (the yellow cloud) is set
up by the injected vortex rings outside the turbulent blob. The volume of
measured field is 120mm$\times$102mm$\times$100mm with a voxel pitch of 2.2mm.

### Movie 8: Normal fluxes of mass and enstrophy in the two different phases:
coherent vortex reconnections and a turbulent blob.

These Mollweide plots show the normal component of each mass and enstrophy
flux at the two phases. In the phase of coherent vortex reconnections (the top
row), the mass and enstrophy fluxes indicate focused ejection. When a
turbulent blob is formed, the ejection is more uniform, and less enstrophy
leaves the central region on average (the bottom row). Here, a spherical
surface with radius $r\approx 1.25R_{\rm blob}=30$mm is considered. The left
column represents the normal mass flux $J_{m}=\rho\vec{U}\cdot\hat{n}$ through
the surface, and the right column is the normal enstrophy flux
$\vec{J}_{\Omega^{2}}=\Omega^{2}\vec{U}\cdot\hat{n}$. The measurements
correspond to ($V_{\rm ring}/R_{\rm ring},f$) = (21 Hz, 0.2 and 4Hz).

### Movie 9: Energy and enstrophy field during a single-shot
collision($L/D,v_{\rm eff},f$) = (3.5, 318mm/s, 1Hz); $V_{\rm ring}/R_{\rm
ring}=26$Hz.

This movie shows the energy and enstrophy fields of vortex ring collisions on
the central slice, obtained by 2D PIV. The vortex reconnections generate
secondary rings, four of which are visible in this movie. Time-averaging these
fields provides equivalent pictures as Figure 2b-c.

### Movie 10: Energy and enstrophy fields of vortex ring train at ($L/D,v_{\rm
eff},f$) = (3.5, 318mm/s, 5Hz); $V_{\rm ring}/R_{\rm ring}=26$Hz.

This movie shows the energy and enstrophy fields of vortex ring collisions in
the state of a turbulent blob on the central slice, obtained by 2D PIV.
Contrary to the single-shot collision (SI Movie 9), energy and enstrophy
remains relatively steady within a spherical blob, and they leave the central
region only intermittently. Time-averaging these fields provides equivalent
figures as Figure 2f-g.

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