Spatial distribution of the quantized vortices' tangle in thermally driven round jets of superfluid helium

Turbulent jets are typical representatives of boundary-free shear flows that can be realized not only in classical Newtonian fluids but also in superfluid ⁴He, e.g., see Refs. 1 and 2. A jet flow is usually obtained by driving a fluid stream through a nozzle into a large reservoir. The interaction between the injection flow, the nozzle, and the fluid inside the reservoir results in a distinct, spatially inhomogeneous structure that can be experimentally studied. In the present work, we focus on turbulent round jets submerged in superfluid ⁴He, also known as He II. This phase of liquid ⁴He exists only at temperatures below approximately 2.2 K, at the saturated vapor pressure, and its behavior is appreciably affected by the quantum order that spontaneously develops in the fluid. Macroscopic consequences of this order are, for example, the possible occurrence of inviscid flows and the coupling between temperature and pressure gradients (the fountain effect)—see Refs. 3 and 4 for reviews on this unique liquid.

At temperatures higher than about 1 K, thermal excitations cannot be neglected and the large-scale hydrodynamics of He II is usually described by the phenomenological two-fluid model.^5,6 The gas of excitations can be seen as a viscous fluid called the normal component, while the quantum-ordered system behaves as an inviscid, superfluid component. The two fluid components are fully miscible, and their density fractions, which depend only on temperature, are plotted in Fig. 1 for the range of temperatures relevant here—note in passing that the He II density $ρ ≈ 145$ kg/m³ depends weakly on temperature.⁷ 

Additionally, the circulation of the superfluid component is quantized.⁸ Specifically, each circulation quantum $κ ≈ 10 − 7$ m²/s is associated with a quantized vortex, which is a line singularity of the quantum order parameter characterized by a narrow core (approximately 0.1 nm) and macroscopic length. When superfluid ⁴He becomes turbulent, quantized vortices tend to interact and eventually form a dynamic tangle, which is the key component of turbulence in He II.^3,4 Here, we employ experimental tools to study the quantized vortices' tangle associated with steady turbulent jets, in view of understanding in what aspects jet flows in superfluid ⁴He may differ from their classical analogs. The quantized vortices' tangle plays an important role in this context because it provides the coupling mechanism between the otherwise independent components of He II. At flow scales larger than the mean distance $ℓ$ between quantized vortices, the magnitude of this interaction—usually called mutual friction force—is proportional to the difference between the velocities of the normal (v_n) and superfluid (v_s) components.^3,4

Here, we take advantage of the just outlined flow regime because our turbulent jets are driven thermally, by dissipating a constant heating power in a small enclosed volume open to the He II bath via a cylindrical nozzle. The resulting flow is usually named thermal counterflow jet—see Ref. 9 for a review on early works—and in the past, it was mainly regarded as a playground system for the study of the mutual friction force in the absence of solid boundaries, which significantly influence the development of turbulence in He II, as recently discussed in Refs. 10–13.

Thermometric measurements carried out along the jet axis—e.g., see Refs. 16 and 17—reveal that a temperature difference $Δ T$ develops inside the nozzle, i.e., the fluid temperature in the bath is smaller than that inside the nozzle. For example, the $Δ T ∝ q 3$ scaling obtained at heat fluxes higher than about 5 kW/m²—and typical of turbulent counterflow in channels¹⁸—confirms that a dense tangle of quantized vortices should be present inside the nozzle in these conditions. However, the temperature gradient seems to disappear about one diameter away from the nozzle, as reported, e.g., in Ref. 16, at least for relatively high heat flux values, i.e., within the just mentioned cubic regime. On the basis of this evidence, it was conjectured that the superfluid component might be coupled to the normal one in the free jet flow, resulting then in a nearly zero counterflow velocity (i.e., coflow of the normal and superfluid components) in the far field. The hypothesis was later supported by acoustic attenuation,¹⁹ velocity-sensitive phase,²⁰ and laser Doppler velocimetry measurements, as close as $0.5 d$ above the nozzle.^21,22 Additionally, it was found that the fluid velocity at the nozzle exit is apparently equal to $p n v n$⁠, which can be explained by the transfer of momentum between the normal component, which exits the nozzle with a velocity v_n, and the surrounding superfluid one, via a process similar to the turbulent entrainment known to occur for classical turbulent jets.^9,23 It then follows that the flow field of the superfluid component must have a stagnation point near the nozzle.^9,22 We schematically sketch this velocity field in Fig. 2.

On the other hand, there is little known about the spatial structure of the quantized vortices' tangle in a thermal counterflow jet. In this work, we aim to fill in this knowledge gap and we present the first spatially resolved measurements of the vortex line density L(r, z) in this kind of flow, where L is defined as the total length of quantized vortices per unit volume. We specifically probe the quantized vortices' tangle using the second sound attenuation technique.^6,28,29 The latter is based on detecting temperature waves, which can be observed in flows of superfluid ⁴He. Within the two-fluid framework, the second sound waves consist of anti-phase oscillations of the normal and superfluid components, while the well-known density (pressure) waves—named first sound waves in He II—correspond instead to in-phase oscillations of the components. The speed of the second sound is temperature dependent—it reaches a flat maximum near 1.65 K and steeply decreases near the superfluid transition temperature, as shown in Fig. 1 (right axis).

In summary, as detailed below, we find that the quantized vortices' tangle spreads in the radial direction, along the jet axis, in a way similar to that observed for the fluid velocity, while its peak density decreases away from the flow source. We then discuss, on the basis of several assumptions, the spatial distribution and the flow forcing dependence of the tangle density, directly obtained from the experimental data.

The counterflow jets here investigated are obtained at the bottom of a standard ⁴He cryostat by supplying constant heating power to a resistive heater located inside a purpose-made enclosure, 3D-printed from copper filled PLA—see Ref. 30 for a detailed description of the apparatus and for preliminary results that are further discussed in the following. The enclosure is terminated by a circular nozzle, with the inner diameter d = 2 mm, as sketched in Fig. 3—the nozzle is machined from brass and is 5d high. The thermally driven jets develop in the open volume above the nozzle—the diameter of this volume is approximately 75d, and it is at least 350d high.

The vortex line density embedded in the flow is measured with a miniature second sound sensor^29,31,32 consisting of two silicon plates placed $δ = ( 2.505 ± 0.002 )$ mm apart. The plates support a heater-thermometer pair, microfabricated at their respective extremities. The sensor's cross section in the direction of the flow at the measurement location is approximately twice 1 mm × 20 μm, and about 14 mm downstream this measurement location, the sensor's cross section enlarges to approximately 2.5 × 3.5 mm². The heater is supplied with a sine-wave voltage signal with frequency f, and, in consequence, second sound waves with frequency 2f are emitted toward the thermometer, which in turn develops weak periodic oscillations of its resistance. To readout this signal, the thermometer is supplied with a small constant current and we collect the corresponding voltage oscillations by a lock-in amplifier.

Note also that the sensitivity of the second sound attenuation technique is further limited by the fact that it is based on the relative measurement of the two second sound amplitudes. In practice, solely amplitudes meeting $A / A 0 ≲ 0.99$ can be distinguished. For the typical value $Δ 0 = 300$ Hz, we estimate from Eq. (6) that at 1.95 K, the minimum density that can be faithfully detected is $L min ≈ 6 × 10 8$ m⁻². The latter corresponds to approximately 1.5 m of vortex lines localized between the heater-thermometer pair, i.e., to a mean distance $ℓ$ between quantized vortices of approximately 40 μm—the volume where L is measured is equal to approximately 2.5 mm³.

To the best of our knowledge, we report here the first use of a mobile second sound sensor to probe a spatially inhomogeneous tangle of quantized vortices. So far, inhomogeneous tangles were investigated by several static sensors placed along a counterflow channel—see Ref. 28 for a few examples—or by multiple sensors located in different loci of the experimental setup, as e.g., in Ref. 32.

Before proceeding, we now mention two effects that might potentially bias the second sound measurements. First, in regions far away from the jet axis, the mean flow encounters the two plates of our sensor at nonzero angles of attack. The mean flow velocity at the location of the sensor, which can be up to approximately 0.1 m/s, as shown e.g., in Fig. 5, is significantly smaller than the velocities of the first and second sound waves—in the investigated temperature range these velocities reach their minima at 2.10 K and are equal to 222 and 13 m/s, respectively.⁷ Thus, no significant supersonic effects are expected to occur when the mean flow changes its direction due to the sensor. Nevertheless, a velocity boundary layer is expected to develop along each plate,^11,33 and quantum vortices localized within these layers may contribute to additional attenuation of the measured signal. Specifically, boundary layer effects were investigated in turbulent flows of He II for similar sensors, at zero mean angle of attack.^29,31 Their contribution was found to be negligible, partly due to the small thickness of the sensor plates, and partly due to the fact that the velocity boundary layer is expected to be small compared to δ (see Ref. 29 for a more detailed discussion of this effect). Additionally, in comparison with Ref. 32, the effect of the boundary layer on the measured signal might be larger, due to the nonzero angles of attack, but, at the same time, the distance between our sensor plates δ is 2.5 times larger than that in the just cited paper, where $δ ≈ 1$ mm. In consequence, the contribution of boundary layers to the total signal attenuation is, in our case, reduced by a factor of roughly 2.5, at the cost of a lower spatial resolution.

The second source of bias is due to the advection of the second sound waves from the resonance cavity, which also accounts for increased signal attenuation. This advection effect is present in all types of the second sound resonators to a greater or lesser degree.²⁹ Specifically, the attenuation attributed to the vortex tangle is more significant, and Eq. (6) represents an accurate estimate of L when the second sound sensor is operated with a low-order resonant mode. In our case, as detailed below, we use the 18th harmonic mode, which means that the second sound beam is more focused and sensitive to advection by the mean flow than it would be for lower order modes. From the comparison with other available sensor geometries, we estimate this bias to be as large as a few tens of percent (a more detailed discussion of this effect is reported in Ref. 29). It is, therefore, more significant than the other bias mentioned above, and, consequently, the additional sensitivity of the employed second sound sensor to the mean flow velocity should be kept in mind.

Aware of these drawbacks, we consolidate our second sound measurements using cryogenic flow visualization. The technique—discussed at length in Ref. 30—is based on following the flow-induced motions of micrometer-sized particles by a digital camera. We specifically visualize solid deuterium particles, which are illuminated by a 1 mm thick and 10 mm high laser sheet crossing the vertical axis of the jet. The acquired camera images are then processed to obtain particle trajectories and velocities. These measurements are carried out in an optical cryostat, with a 12 mm (6d) wide and 8 mm (4d) high field of view located approximately 10 mm (5d) above the nozzle, along the jet axis—see again Ref. 30 for experimental details.

Steady counterflow jets are investigated at three temperatures (approximately 1.65, 1.95, and 2.10 K) corresponding to normal component density fractions p_n of 0.19, 0.48, and 0.74, respectively. Note that the helium bath was kept at constant temperature by maintaining the associated helium vapors at constant pressure, i.e., the temperature standard deviation was less than 12 mK at 1.65 K, and around 1 mK at higher temperatures. Depending on the employed experimental protocol, we split the acquired datasets into four categories, labeled A, B, C, and D—see Table I for relevant experimental conditions.

Datasets A1–A3 were collected near 1.65 K because the temperature dependence of the second sound speed has a plateau here, see Fig. 1 (right axis). The resonance condition given by Eq. (5) then becomes insensitive to small temperature fluctuations experienced in almost every cryogenic setup. Therefore, we continuously excite the sensor at the fixed frequency f = 37.22 kHz, which corresponds to the 18th harmonic mode. By displacing the sensor, we sweep the volume above the nozzle and we then collect the second sound amplitude A for a set of positions given by $( α , z )$⁠. In order to increase the signal-to-noise ratio, multiple (at least 30) sweeps across the jet axis are performed for constant z and the resulting second sound amplitude is calculated as their average. The vortex-free resonance amplitude A₀ and the peak width $Δ 0$ are obtained by fitting the full resonance curves, purposely acquired far away from the jet axis, i.e., far away from the tangle.

An example of the acquired second sound signals is presented in the top panel of Fig. 4. Here, we show two sets of sweeps obtained near the extremities of the accessed axial distance, which are $9.0 d$ and $31.5 d$⁠, respectively. Individual sweeps across the jet axis (points) as well as their ensemble averages (black lines) show a coherent response, with most of the attenuation taking place in a narrow region of sensor's angular deflections. Because this attenuation can be related to the presence of a quantized vortices' tangle, we use the obtained data to calculate the corresponding vortex line density and probe its spatial structure. Additionally, as shown in the bottom panel of Fig. 4, the fluctuations of the second sound amplitude introduce a statistical uncertainty of around 10%, which is comparable to the uncertainty one can associate with Eq. (6), as mentioned above—a similar level of fluctuations is observed for other axial distances and datasets A1–A2. Finally, no systematic error is apparently introduced by the stepper motor that displaces the sensor back and forth across the signal minimum, thanks to its positioning accuracy.

An approach similar to that used above is not possible for datasets in categories B and C because, for the corresponding temperatures, c₂ is strongly temperature dependent. Therefore, for each sensor position, a full frequency sweep across the resonance frequency (approximately 38.5 kHz) is measured and we estimate the corresponding resonance amplitude from the in-phase (A_x) and quadrature (A_y) components of the acquired signal,^28,29 since the plot of A_y as a function of A_x can be fitted with a circle having diameter equal to A. As in the previous case, the values of A₀ and $Δ 0$ are taken from full fits of the resonance curves measured far away from the jet. Following this approach, we obtain a single value of A for each position of the second sound sensor and the statistical fluctuations of the sensor response are, therefore, not available for these datasets.

While sets B1–B4 probe the jet in both radial and axial directions and can then be directly compared to A1–A3, the set C1 only captures the vortex line density along the jet axis, i.e., for a constant angular deflection equal to α₀.

Finally, the dataset D1 refers to the flow visualization study, whose bath temperature and heating power can be directly compared with sets A1 and A2. The dataset comprises approximately 1.88 × 10⁶ particle position-velocity pairs, within the above-mentioned two-dimensional field of view.

For the measured values of T and P, see Table I, one can use Eqs. (1)–(3) to calculate nominal values of v_n, v_ns, and Re, assuming that the heat produced in the heater enclosure is transported toward the helium bath solely through the nozzle. In this idealized case, we obtain normal fluid and counterflow velocities of the order of 1 m/s, corresponding to Re values of the order of 10⁵. These values are reported in Table I.

However, our 3D-printed enclosure is subjected to heat leaks, through its plastic walls and the joint between the enclosure and brass nozzle. The leaks are found to substantially reduce the flow velocity and the Reynolds number of the jets studied below. This claim can be directly verified via the visualization dataset, shown in Fig. 5.

In the left panel, we plot the mean particle velocity in the axial direction (color lines) as a function of the particle horizontal position in the field of view, corresponding to the radial direction. The dependence on the axial coordinate is specifically expressed by five offset lines, i.e., the 4d high field of view is split into five horizontal stripes, $0.8 d$ high. At first glance, we can already say that the particle motions capture a spatially constrained flow structure. Outside the structure, the particles are practically still, while, inside it, they reach peak velocities of the order of 0.1 m/s. This value is in sharp contrast with the nominal values of v_n reported in Table I, which are about one order of magnitude larger than the experimental observation. Therefore, the actual normal fluid and counterflow velocities are at least one order of magnitude smaller than the maximum values estimated with Eqs. (1)–(3).

Heat leaks can be taken into account by assuming a finite efficiency $η ≲ 0.1$ of the heat transfer through the nozzle. In particular, the heat flux through the nozzle q in Eq. (1) should become ηq, which means that the effective heating power is reduced to ηP in comparison with the values reported in Table I. However, providing a clear estimate of η based on the current data is not straightforward, also considering that the visualized region is not directly located above the nozzle, but approximately 5d away from it—note that in a previous study³⁴ it was found that, for an analogous setup, $η ≈ 1 / 30$⁠. We can nevertheless use the upper limits of v_n, v_ns, and Re reported in Table I to compare individual datasets, if we assume that η does not significantly change over the course of the experiment.

In order to investigate heat leaks in more detail, we placed a small germanium thermometer inside the enclosure and measured the temperature difference that develops between the enclosure and the bath after the heater is switched on. Time traces of the recorded temperature are presented in the left panel of Fig. 6 for a constant bath temperature of 1.30 K (note that the temperature difference is more prominent at lower temperatures because the heat capacity of He II is roughly proportional to p_n in the investigated temperature range). Specifically, for a set of rising heating powers, we observe that the heater enclosure warms up and reaches a steady state within a few seconds after the heater is powered. Once the heater is switched off, the enclosure quickly thermalizes with the surrounding bath.

First of all, it is apparent that the steady temperature difference $Δ T$ between the enclosure and the ⁴He bath is significant, of the order of 0.1 K. Additionally, the right panel of Fig. 6 displays $Δ T$ as a function of the heating power P. We observe a neat linear dependence between these quantities, similar to Ref. 17, where it was also reported that, for counterflow jets, $Δ T ∝ q 3$ for $q ≳ 5$ kW/m². The latter is consistent with the data reported in Table I if we take into account the above-mentioned heat leaks, i.e., the results plotted in the right panel of Fig. 6 confirm that $η ≲ 0.1$ in the range of investigated parameters because, otherwise, we should also see here the cubic scaling typical of turbulent counterflow in channels.¹⁸ Nevertheless, the observed temperature difference leads to the formation of a prominent jet that features a dense tangle of quantized vortices. We now investigate this tangle using the second sound sensor as described above.

The observed decrease in the second sound amplitude, indicating the presence of quantized vortices in the flow, depends on the radial and axial positions of the sensor, revealing the spatial structure of the quantized vortices' tangle associated with the counterflow jet. Figure 7 displays the vortex line density L, also named VLD below, calculated from Eq. (6), for datasets A3, B3, and B4. These datasets are characterized by the same heating power (⁠ $P ≈ 776$ mW) and different temperatures. The individual curves correspond to different axial distances, starting from 9d (dark blue) above the nozzle, up to $31.5 d$⁠, 29d, and 34d (dark red), for sets A3, B3, and B4, respectively—the curves are vertically offset for clarity. These radial VLD profiles map the quantized vortices' tangle in unprecedented detail, and we specifically observe the tangle to be spatially confined and growing in the radial direction—considering the errors discussed above, we remind that individual L values are here given with relative uncertainty equal to approximately 10%.

Moreover, the highest L values are obtained for the smallest axial distances and for the lowest temperature. A similar temperature dependence is displayed by the normal fluid velocity v_n for a constant heating power, i.e., v_n increases as T decreases—see Eq. (1). On the other hand, v_ns is not a monotonous function of temperature and, for the displayed data, the smallest v_ns is obtained for 1.95 K—see Eq. (2) and Table I. Specifically, this discrepancy between the temperature dependencies of VLD and v_ns tells us that the physical mechanisms resulting in the formation of counterflow jets are different from those taking place in ordinary channel counterflow, where the scaling $L ∝ v n s 2$ was consistently observed for a rather wide range of temperatures and counterflow velocities.³⁵ In other words, our data support the picture that the quantized vortices' tangle associated with the jet is not simply generated in the nozzle, where channel counterflow takes place, and subsequently ejected into the volume above it. In particular, as discussed below, we find the quantized vortices' tangle to possess properties that are usually attributed to turbulent coflow of He II, i.e., to a configuration where the two fluid components are locked together in a single velocity field.

Additionally, to highlight the effect of our sensor motion on the investigated flow, we compare in Fig. 8 the L₀ values obtained by fitting the full VLD profiles of dataset B4 (red crosses) with the directly measured peak values of dataset C1 (points) as a function of z/d—note that dataset C1 includes two axial strokes of the sensor. The L₀ values detected by the sensor that gradually moves toward the nozzle are marked by cyan points, and the magenta points denote data taken by the sensor that moves in the opposite direction. The curves closely overlap within the full range of investigated distances and within the accuracy of the experimental technique, which neatly illustrates that our observations are fully reproducible.

We now analyze how the peak amplitude L₀ and width w of individual VLD profiles scale with the axial distance z. First of all, the relation between L₀ and z/d is presented in Fig. 9 for datasets A1–A3 and B1–B4, marked by open symbols (we also use this notation in the following figures). Here, no artificial vertical shift is imposed, and the vertical distribution of data lines is given by different experimental conditions, listed in Table I.

The observed $L 0 ( z )$ dependencies are compatible with power laws, with exponents ranging from −2 to $− 3 / 2$⁠, as indicated by the black lines in the figure. Specifically, the former exponent is found to match relatively small peak densities, while the latter one fits relatively dense tangles. Since the centerline velocity $v 0 ∝ z − 1$ holds for both classical and counterflow jets,^23,25 it apparently follows that $L 0 ∝ v 0 3 / 2$ for the largest line densities, while the $L 0 ∝ v 0 2$ dependence holds for the smallest ones observed in our experiment. Two alternative interpretations of this puzzling behavior are discussed in Sec. IV.

Note in passing that our visualization data (see the top right panel of Fig. 5) do not seem to reproduce the expected scaling of the centerline velocity, although the latter does show the tendency to decrease as the axial distance increases. The mismatch can be attributed to the fact that the jet flow was visualized in the near field region, extending up to approximately 7d away from the nozzle,²⁴ where the mentioned scaling law is actually not expected to hold.

The axial dependence of the peak width w is plotted in Fig. 10. The data points characterized by a relatively large VLD amplitude display a neat linear dependence of w with z, which is also independent of experimental parameters, see colored points in Fig. 10. We identify these well-resolved VLD peaks by setting a threshold $L t = 2.2 × 10 9$ m⁻². This value is chosen so that $L min / L t ≈ 0.05$⁠, i.e., it holds that the profiles meeting this threshold density display VLD values larger than $L min$⁠, at least within the interval $[ − 2 w , 2 w ]$⁠, which is required for a well-converged Gaussian fit and a reliable estimation of the fitting parameters. Additionally, the errors associated with the w estimates, which are computed from 95% confidence bounds of the fit, range from 2% to 6% for the well-resolved data, and the similarly obtained errors of L₀ are between 1% and 5%. For the datasets that fail to meet the threshold value L_t, the relative errors of w estimates are as high as 33%—and those of L₀ are as high as 22%. In the following, for the sake of brevity, the colored points in Fig. 10, those with $L 0 > L t$⁠, are named high (data) set, while the others are called low (data) set, because, as already noted, they can be, respectively, associated with high and low values of peak density—see again Fig. 9.

In summary, the emergence of approximately Gaussian profiles for the vortex line density, alongside with the scaling laws outlined above, suggests that the quantized vortices' tangle develops a structure that is tightly coupled to the associated velocity field, at least within the intermediate field region of the jet. In other words, the quantized vortices' tangle grows in size, while decreasing its peak density with z, similarly as the peak flow velocity that decreases with z, due to turbulent entrainment and momentum conservation.

In the following, we outline two interpretations of the just presented experimental results. The first one, detailed in Sec. IV A and proposed by P. Švančara and P.-E. Roche, leads us to conjecture a self-similar functional form of the vortex tangle density across counterflow jets. The other, detailed in Sec. IV B and proposed by M. La Mantia, suggests that the position of the superfluid stagnation point could depend on the flow forcing as well. Both interpretations can be used to explain in part the obtained results.

Both estimates of $L z$ are presented in Fig. 11 as a function of the axial distance z. The left panel represents the right-hand side of Eq. (15), and the right panel displays the result of the numerical integration of the right-hand side of Eq. (11) using the trapezoidal rule. The two methods yield comparable results except for dataset A1 (open blue circles, right panel), where the nonphysical, negative values of $L z$ most likely result from the poor signal-to-noise ratio (a negative vortex line density is obtained from Eq. (6) when $A > A 0$⁠). Note again that Eq. (15) is derived assuming that the studied flows are self-similar.

It is apparent from Fig. 11 that, for a given heating power P and helium bath temperature T, i.e., for a given dataset, the cumulative length $L z$ does not depend on z within the experimental uncertainty and for all datasets but one. Taking Eq. (13) into account, we can therefore say that $L 0 ( z ) ∝ z − 2$ in the range of investigated parameters, excluding dataset A3 (yellow triangles). In this case, $L z$ grows with the axial distance z, which is an unexplained exception. Nevertheless, leaving this dataset aside for the moment, it now seems that the variations of the power-law exponents apparent in Fig. 9 are compensated by suitable changes in the growth of the measured VLD profile width, presented in Fig. 10. This compensation could be explained, at least in part, by stochastic fluctuations of the jet direction. The latter could systematically decrease the measured centerline density of quantized vortices and simultaneously increase the measured jet width. Apparently, the use of the cumulative vortex length allows us to largely discard the data scatter observed in Figs. 9 and 10.

Turbulence along the axis of classical round jets can be locally approximated as homogeneous and isotropic,³⁸ and some local properties of turbulence can be inferred from the local values of $v rms$ and H, as shown, for example, in Refs. 42–44. Specifically, following this reasoning, one can estimate along the jet axis the vorticity magnitude and the energy dissipation rate, appearing in the classical relation analogous to Eq. (16). In other words, we assume here that Eq. (16) holds locally on the counterflow jet axis, with $L c ≡ L 0$ (note that in Ref. 9 the assumptions of local homogeneity and isotropy have been already used to interpret measurements of the second sound waves crossing a counterflow jet).

What now remains is to understand why $F ∝ P 3 / 2$⁠. At any given z, it results from the proportionality between $L 0 2$ and $v 0 3$ in turbulent coflows,^40,41 which is apparent from Eq. (17). Due to the counterflow generation mechanism, Eq. (1), the centerline velocity v₀ scales linearly with P (or ηP to account for heat losses). We, therefore, obtain $F ∝ L 0 ∝ v 0 3 / 2 ∝ P 3 / 2$⁠, consistently with the outcome of Fig. 12.

It is now useful to mention Ref. 19, where an elongated counterflow jet of He II was investigated at a distance of approximately 10 nozzle widths from the flow source using a second sound beam traveling across the jet. For temperature (1.6–2.1 K) and heat flux (3–50 kW/m²) ranges comparable to those of the present work, the attenuation factor, which is proportional to the vortex line density, was reported to scale as P^k with k = 1.3–1.6. The outcome was later explained using geometrical acoustics, assuming that the counterflow jet is turbulent and that the normal and superfluid components are locked in a single velocity field.⁹ 

Additionally, although the determination of the temperature dependence is not possible on the basis of the present data, three physical arguments suggest how to make it explicit. First of all, the spreading rate of classical jets is due to the balance between mixing and axial entrainment, and these processes should remain equally efficient in He II when both components are flowing together. Therefore, we expect that β_L will have a weak dependence on T and P. On the contrary, the virtual origin z₀ can be associated with the crossover from the counterflow to the coflow regime, and we then expect it to explicitly depend on T and P. Finally, the amount of quantized vortex lines generated in a given He II coflow depends on temperature, because the vortex dynamics depends on the normal fluid fraction,¹⁹ which is strongly temperature dependent, see again Fig. 1—one can then embed the latter consideration into an unknown function of temperature, $T$⁠.

In Sec. III B, we reported that the $L 0 ( z )$ dependencies plotted in Fig. 9 are compatible with power laws, with exponents ranging from −2 to $− 3 / 2$⁠. Specifically, the former exponent can be associated with relatively small peak densities, i.e., to the low set, while the latter one is more suitable for relatively denser tangles, i.e., for the high set. Considering that $v 0 ∝ z − 1$ is expected to hold for self-similar round jets, one can then derive $L 0 ∝ v 0 3 / 2$ for the largest line densities, while the $L 0 ∝ v 0 2$ scaling is more adequate for the smallest ones.

As already mentioned, the 3/2-power scaling was reported in previous studies—see, e.g., Refs. 9 and 35—and can be associated with turbulent coflow of He II. More precisely, one can derive it from Eq. (16) assuming that the integral length scale H does not depend locally on other flow scales. This could happen in the region where the hypothesis of locally homogeneous and isotropic flow is expected to hold, i.e., along the jet axis, where no information on the jet width w should be accessible, if the typical size of this region is smaller than w—note again that $H ∝ w$ for self-similar jets. In other words, the assumption made in this section is that the integral length scale H is constant in the region where Eq. (16) holds.

The 2-power scaling is instead often associated with channel counterflow, i.e., $L ∝ v n s 2$⁠, as discussed, e.g., in Ref. 35. However, as already noted in Sec. III A, the normal fluid and counterflow velocities are in general not directly proportional to each other, but this is only true at liquid temperatures higher than approximately 2 K (Ref. 7). It then follows that, at lower temperatures, $v n ∝ v n s$⁠. Additionally, considering that, as reported in Sec. IV A, v₀ should be proportional to v_n, we find that $L 0 ∝ v 0 2$ for our more dilute tangles, i.e., for the low set, collected at temperatures lower than 2 K.

From this discussion, one can then say that, at sufficiently low values of heating power, channel counterflow features could also be observed in counterflow jets. The physical reason could be related to the mutual friction force, which can be seen as an energy sink. Specifically, this dissipation mechanism could become inefficient for counterflow jets at sufficiently high values of heating power. In other words, the position of the stagnation point depicted in Fig. 2 could be a function of the heating power and move closer to the nozzle as the power increases.

The suggestion appears a bit less adventurous if one considers that counterflow jets at relatively low heating power have yet to the thoroughly investigated or at least they have received to date less attention than those at higher heating power. Indeed, we reported in Sec. III that, on the basis of Ref. 17, we expect $Δ T$ to be linearly proportional to q in the range of parameters probed here, while this is not the case of other studies on counterflow jets, such as Refs. 14 and 16, where a neat cubic scaling was observed at larger values of heat flux. Specifically, Refs. 17 and 21 are apparently the only works where some attention is given to the just mentioned linear regime and, more importantly, the claim that the superfluid stagnation point is located in the close proximity of the nozzle is solely based on data collected in the cubic regime, i.e., at relatively high values of heating power.

Finally, it is useful to mention that, from the right-hand side of Eq. (15) and the $L 0 ∝ v 0 3 / 2$ scaling, one can get that $L z$ should be proportional to $z 1 / 2$⁠, at least for the high set. However, in Fig. 11, we observe that $L z$ grows with the axial distance z only for dataset A3, corresponding to the largest Reynolds number, while this is not the case for the other datasets displaying the proportionality between w and z. Once more, further experiments are required to assess the validity of the physical interpretations presented here.

The intermediate field region of round counterflow jets, representing a unique type of thermally driven flow of He II, was studied in the turbulent regime, for Reynolds numbers of the order of 10⁴, using the second sound attenuation and flow visualization techniques. Specifically, a miniature second sound sensor was employed to detect the vortex line density across the jet axis, with a spatial resolution slightly larger than the nozzle diameter, at fluid temperatures between 1.64 and 2.10 K—note, however, that the measured density is biased by an additional sensitivity of the employed second sound sensor to the flow velocity, which cannot be precisely quantified at the moment.

Nevertheless, within the range of investigated axial distances, from 9 to 34 nozzle diameters, we observe a highly inhomogeneous tangle of quantized vortices. In particular, the vortex line density displays a Gaussian-like profile in the radial direction and, for the highest density values, its width w grows linearly with the axial distance z at a rate $d w / d z ≈ 0.14$⁠, which is comparable but not equal to the radial growth rate observed for the classical velocity field—similar remarks apply to the jet virtual origin. Additionally, as shown in Fig. 9, the peak density of the quantized vortices' tangle appears to follow, as a function of the axial distance, power laws with exponents ranging approximately between −2 and $− 3 / 2$⁠. Two physical interpretations of the observed scaling behavior are proposed, and none of them account for all the measured datasets. One interpretation conjectures a general functional form of the vortex line density in counterflow jets, which depends on spatial coordinates and heating power. The other suggests that the position of the superfluid stagnation point along the jet axis could depend on the heating power as well.

In summary, the present work indicates that future studies on counterflow jets should be devoted to extend the ranges of experimental parameters accessed to date, e.g., in order to quantitatively identify the conditions in which jet flows of superfluid ⁴He are appreciably different from their classical analogs.

We thank Jerôme Valentin for microfabricating parts of the employed second sound sensor. We are also thankful to Eric Woillez and the technical staff of Institut Néel for their prompt assistance during the second sound experiments, performed in Grenoble by P. Švančara and P.-E. Roche—the visualization and thermometry experiments were performed in Prague by P. Švančara and M. La Mantia. P.-E. Roche acknowledges support from the ANR QUTE-HPC grant (No. ANR-18-CE46-0013) and the EU Horizon 2020 program The European Microkelvin Platform (EMP 824109). P. Švančara and M. La Mantia were supported in part by the Czech Science Foundation (GAČR) under Grant No. 19-00939S. P. Švančara also acknowledges the Barrande Fellowship Programme supporting doctoral mobility. Additionally, during the manuscript preparation in 2022, M. La Mantia was in part supported by the Czech Science Foundation (GAČR) under Grant No. 20-00918S.

The authors have no conflicts to disclose.

All authors contributed equally to this work.

Patrik Švančara: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Philippe-Emmanuel Roche: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Marco La Mantia: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.