Quantum, or classical turbulence?

Turbulence in viscous fluids is often regarded as the last unsolved problem of classical physics, see, e.g., [1]. Turbulent flow contains eddies of various dimensions, ranging from the smallest ones, of the size of the Kolmogorov dissipation length $\ell_{\rm{K}}$, where viscous dissipation occurs, up to the biggest eddies, where most of the turbulent energy resides. Thanks to advection and stretching processes, this turbulent energy is transferred, via the Richardson cascade, to smaller and smaller length scales, where it is dissipated by viscosity.

Quantum turbulence [2,3] occurs in quantum fluids displaying superfluidity. Their physical properties depend on quantum physics. Flowing quantum fluids, such as liquid ⁴He at temperatures below the λ-line (about 2.17 K at the saturated vapor pressure), also known as He II, do not generally obey the Navier-Stokes equation. On the phenomenological level, He II is viewed as consisting of two interpenetrating fluids, whose density ratio is temperature dependent. The gas of thermal excitations —the normal component of He II— can be considered as a viscous fluid, of dynamic viscosity η and density $\rho_{\text{n}}$, carrying the entire entropy content of the liquid, while its superfluid component, of density $\rho_{\text{s}}$, is inviscid. The quantum restriction to the superfluid motion requires that the circulation of the superfluid velocity is equal to an integer multiple of the quantum of circulation $\kappa=h/m$, where h is the Planck constant and m denotes the atomic mass of ⁴He. Quantized vortices —line singularities where $\rho_{\text{s}}$ vanishes—exist in He II, usually arranged in a tangle (note that only singly quantized vortices exist in He II, as multiply quantized vortices are unstable). In the zero-temperature limit (where there is no normal fluid) the tangle dynamics represents the simplest prototype of turbulence. At finite temperatures, the tangle coexists and interacts with the normal fluid. While in superfluid ³He-B the normal fluid can generally be assumed at rest in the reference frame of the container [4], in He II it may easily become turbulent [5], so quantum turbulence is generally richer than its classical counterpart.

A single quantized vortex, for which the Kelvin theorem is strictly valid, cannot be stretched, so it seems that classical and quantum turbulence ought to be completely different. But is it always the case?

In fact, both quantum and classical features can be observed in the same quantum flow, depending on the scale at which we probe it. At large enough scales, where the "granularity" of the building blocks —singly quantized vortices— is smoothed, quantum turbulence displays classical features, such as the $-3/2$ power-law decay [2,3] and the Kolmogorov form of the turbulent energy spectrum [6]. To date, the crossover from quantum to classical behavior in quantum flows has only been proposed in numerical simulations [6–8]. Here we provide the missing direct experimental evidence, by probing the same quantum flow at various length scales in a single experiment.

The length scale where such a crossover occurs is called the quantum length scale $\ell_{\text{Q}}$ [3]. It can be understood, from a quantum perspective, as the mean distance ℓ between the quantized vortices in the tangle. From a classical perspective, the traditional dimensional reasoning leads to the definition of the dissipation wave number $k_{\text{diss}}\approx(\varepsilon/\nu^3)^{1/4}$, where the energy per unit mass E, injected at the rate $\varepsilon = -{\text{d}}E/{\text{d}}t$ at the outer scale of turbulence, becomes dissipated by viscosity (ν is the kinematic viscosity). The corresponding Kolmogorov dissipation length scale is thus $\ell_{\text{K}}\approx 2\pi/k_{\text{diss}}$. In quantum turbulence (although superflow in He II exists down to the smallest length scale, the size of the vortex core $\xi \approx 1~\mathring{\rm A}$), the quasi-classical Richardson cascade cannot continue, because of quantization, beyond the quantum length scale, defined, in analogy to $\ell_{\text{K}}$, as $\ell_{\text{Q}} \approx 2\pi/k_{\text{Q}}$, where $k_{\text{Q}}\approx(\varepsilon/\kappa^3)^{1/4}$ and the kinematic viscosity ν is replaced by the quantum of circulation κ [3]. Note that the very existence of $\ell_{\text{Q}}$ is a purely quantum-mechanical effect, since in the limit of vanishing Planck constant, $\ell_{\text{Q}} \rightarrow 0$.

It is desirable that the range of experimentally accessible length scales straddles widely across the scale $\ell_{\text{Q}}$ of the flow under study. We have chosen a well-known quantum flow, i.e., steady-state thermal counterflow [2,3] of He II, generated by a flat square heater placed on the bottom of our vertical channel. The superfluid component of He II moves towards the heater where it is converted into the normal component which flows away from it; above a (small) critical velocity $v_{\rm{c}}$, the counterflow velocity $v_{\text{ns}} = q / (\rho_{\text{s}}ST)$ generates a tangle of vortex line density $L=[\gamma(T)(v_{\text{ns}}-v_{\text{c}})]^2$, where the parameter $\gamma(T)$ is known with sufficient accuracy [9] (q is the heat flux per unit area, S indicates the entropy per unit volume and T denotes the temperature). Experimentally, we select the suitable quantum length scale $\ell_{\text{Q}}\approx \ell \approx 1/ \sqrt{L} \approx 75\ \mu \text{m}$ by tuning the heat flux q.

We study the Lagrangian velocities of micrometer-sized deuterium particles, calculated in a planar section of the experimental volume, by using the particle tracking velocimetry (PTV) technique [10,11]. PTV is a very valuable quantitative tool used to study many scientific and industrial problems [12] and allows tracing the motion of small particles suspended in a fluid (the particles reflect the light of a laser beam and their time-dependent positions are captured by a digital camera). In counterflowing He II the motion of the particles is very complex, as they interact with both the normal and superfluid velocity fields simultaneously and may become trapped (and/or de-trapped) onto the cores of quantized vortices.

The size of the particle represents the physically smallest length scale we can access, providing that the images are taken fast enough, so that the particle, between two successively taken images, does not move further than its size. The upper length scale limit is determined by the lengths of the particle trajectories that we can follow. In order to obtain the tracks we use each n-th particle position, evaluated from data sets taken at different frame rates, see fig. 1.

Zoom In Zoom Out Reset image size

Our experimental apparatus [13,14] consists of a custom-built helium bath cryostat, enabling optical access to the experimental volume, a purpose-made seeding system, supplying micrometer-sized solid deuterium particles (generated by mixing helium and deuterium gases at room temperature and injecting the mixture into the helium bath), a continuous-wave laser and cylindrical optics to obtain a laser sheet of about 10 mm high and less than 1 mm thick, and a fast digital camera, situated perpendicularly to the laser sheet, sharply focused on a 12.8 mm by 8 mm field of view by using a macro lens. The PTV technique is then employed for the measurement of Lagrangian quantities in a vertical plane in the middle of the experimental volume, i.e., as far as possible from its boundaries (our channel is about 100 mm long and its square section has 25 mm sides).

Once liquid helium is transferred into the cryostat, a pumping unit is used to gently lower its temperature. The gaseous mixture is injected and the helium bath stabilized at a chosen temperature. As the particles are not neutrally buoyant, images are recorded in order to estimate their settling velocities and dimensions following the procedure described in [13], i.e., the velocities of many particles freely falling in the absence of heat flux are used to calculate their sizes by equating the Stokes drag and buoyancy force. The particle size d is, e.g., $10\pm3\ \mu \text{m}$, at 1.66 K, and $5\pm2\ \mu\text{m}$, at 1.77 K, see fig. 2 (the error corresponds to each distribution standard deviation). The heater is then switched on and images collected in vertical counterflow of superfluid ⁴He at different heat fluxes. Each movie is typically made of a few thousand images and the particle tracks (i.e., the two-dimensional projections of the three-dimensional trajectories on the visualized plane) are obtained by using an open-source algorithm [15]. In each image, there are usually up to a few hundred particles and several thousand trajectories, with up to a few hundred points, are computed for each movie. The tracks obtained from the images are filtered by using a dedicated computer program in order to remove spurious trajectories before calculating the velocities, which are computed by interpolating linearly consecutive position differences. The Lagrangian quantities calculated from several movies obtained under the same conditions are finally combined.

Zoom In Zoom Out Reset image size

The main difference between the present work and our previous publications [13,14] lies in the range of length scales being experimentally investigated. Results on thermal counterflow at scales of the order of ℓ have been reported in [13,14], whereas here the probed length scales straddle two orders of magnitude across ℓ, thanks to the newly implemented data processing scheme, see fig. 1.

Our basic observation is that, within the investigated range of parameters, the character of the observed particle tracks moving upwards and/or downwards in steady-state thermal counterflow appears very similar [14] and that it is not possible to unequivocally identify particles trapped onto quantized vortex lines. We therefore report the statistical investigation of the particle dynamics. In agreement with existing two-fluid flow experiments [13,14,16,17], superflow numerical simulations [7,8,18,19], and a recent theoretical derivation [20], for $\ell_{\exp} \lesssim \ell$, we find strongly non-Gaussian velocity distributions, with power-law tails, while, for $\ell_{\exp}\gtrsim \ell$, they are of nearly Gaussian shape, as in classical turbulent flows.

Note, however, that the observed Gaussian core of the distributions can also be seen as a consequence of the central limit theorem (in the spirit that the sampling time is larger than the correlation time), although, as mentioned above, it is expected that, at large enough length scales, quantum flows display classical-like features. Besides, the reasons why the tails of the velocity distributions obtained experimentally in two-fluid flows [13,14,16,17] are consistent with those computed in the absence of normal-fluid flow [7,8,18,19] are still not entirely understood and further investigations are required to clarify this issue.

The probability density function (PDF) of the non-dimensional instantaneous velocity $(u-\overline u)/u^{\text{sd}}$ in the horizontal direction (i.e., the direction perpendicular to the mean counterflow velocity $v_{\text{ns}}$) is plotted in fig. 3, where $\overline u$ and $u^{\text{sd}}$ are the mean and standard deviation of the dimensional velocity u.

Zoom In Zoom Out Reset image size

The probed length scale $\ell_{\exp}$ is quantified by introducing the non-dimensional time $\tau = t_1 / t_2$, where t₁ is the time interval used for the calculation of the velocities along the tracks and $t_2 = \ell / V_{\text{abs}}$, where $V_{\text{abs}}$ denotes the mean particle velocity obtained in the considered experimental conditions, at the smallest t₁, i.e., $\ell_{\exp}(\tau=1) \approx \ell$. The velocities of particles sufficiently far from vortices can explain the observed Gaussian core of the distributions. The use of even smaller particles, which can probe smaller length scales, would lead to narrower distribution cores.

Figure 4 shows the evolution of the PDF obtained at different τ. As t₁ increases, the PDF changes shape to a nearly Gaussian one. Figure 5 displays this evolution quantitatively. The flatness of the $(u-\overline u)/u^{\text{sd}}$ distribution, calculated as the ensemble average of $[(u - \overline u) / u^{\text{sd}}]^4$, is plotted vs. τ, i.e., as a function of the time t₁ used to compute the velocities. The flatness reaches the value of 3 —that of a Gaussian distribution— for $\tau \approx 2$, in other words, when $\ell_{\exp}\approx 2 \ell$. This may be related to the coherent vortex structures of similar size observed in numerical simulations [21] and may also depend on the fact that particles are not expected to move between vortices along straight lines.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In this study, we concentrate on the horizontal velocity, as the particle velocity distribution in the vertical direction is affected by the imposed vertical counterflow velocities of He II. Still, as shown in fig. 6, where the plots are arranged in the same fashion as those in fig. 4, the evolution of the statistical distributions of the non-dimensional instantaneous velocity $(v-\overline v)/v^{\text{sd}}$ in the vertical direction, evaluated for different τ, qualitatively confirms the crossover ($\overline v$ and $v^{\text{sd}}$ are the mean and standard deviation of the dimensional velocity v). A more quantitative analysis is difficult here, as the vertical velocity distributions are in general skewed and characterized by two peaks, see, e.g., [13]. These two peaks often result in a single (skewed) broader peak, as in the shown case, which partly hides the distribution tails, being wider than the peak observed in the horizontal velocity distributions, see fig. 4. The relatively small sizes of the currently available data sets, see the inset of fig. 5, may also play a role, as it is expected that, once larger data sets become available, the tails of the vertical velocity distributions would show clearer power-law forms, in steady-state counterflow.

Zoom In Zoom Out Reset image size

The velocity PDF in quantum turbulence was observed for decaying thermal counterflow [16] and its power-law shape linked with vortex reconnections [22]. However, this result can also be obtained when vortex reconnections do not occur, as in classical systems of vortex points [18], or by simply assuming that close to a quantized vortex the superfluid velocity $v_{\text{s}} = \kappa/(2\pi r)$, where r is the distance from the vortex core. If the probability $P_{\text{v}}(v)$ of observing a velocity v is assumed proportional to $\int \delta(v - v_{\text{s}}) r{\text{d}}r$, where δ denotes the delta function, it follows that $P_{\text{v}}(v) \propto v^{-3}$, without the need of considering vortex reconnections [23]. As shown in the inset of fig. 3, the power law $P_{\text{v}}(v) \propto v^{-3}$ is indeed consistent with the tails of our experimental PDF.

One can estimate the width of the tails of the velocity PDF (a quantum effect) due to the finite size of the particles. The upper limit results from the assumption that the particle is subject to the tension of a quantized vortex line, which is balanced by the viscous Stokes drag force, leading to a maximum velocity

$$\begin{equation} v_{\max}=\frac{\rho_{\text{s}} \kappa^2}{6 \pi^2 d \eta} \ln \left( \frac{d}{\xi}\right) \end{equation} \tag{ 1 }$$

(a similar formula was derived based on different considerations in [24]). This is in qualitative agreement with the experimental PDF. For example, the data plotted in fig. 4 lead to $v_{\max} \approx 30\ \text{mm/s}$, corresponding to about $15 u^{\text{sd}}$, the experimentally obtained upper limit.

As shown in fig. 5, the flatness values at $\tau < 1\ (\ell_{\exp} < \ell)$ depend on the frame rate used to collect the images. This can be explained by noting that, as the frame rate decreases, the distance between the particle positions becomes larger, see fig. 1. The non-dimensional velocity is thus estimated over a longer interval, leading consequently to the detection of fewer events of large magnitude, meaning that the distribution tails tend to be narrower. Note also that the particle size affects the velocity distributions, as the dynamics of larger particles is expected to lead to narrower tails. Another parameter that strongly influences the distribution appearance, is the number N of data points, see the inset of fig. 5. As N increases, the probability of detecting events of large magnitude increases too, leading to wider distributions. This becomes apparent if the flatness values of distributions having different N, obtained at the same frame rate, are compared. Besides, the statistical distributions of larger data sets are also expected to have tails of clearer power-law shape. There is consequently a clear call to increase the size of the data sets, which are still smaller than those employed for the statistical analysis of classical turbulent flows, and to clarify the effect of particle size and density on the distribution forms. Future studies also aim to extend our recent work on particle accelerations [14] by similarly addressing the probed length scale influence on the corresponding distributions (preliminary results are briefly discussed in [11] and a detailed analysis of the complex behaviour of particle accelerations will be reported elsewhere).

We have provided direct experimental evidence that both quantum and classical characteristics of turbulence can be detected simultaneously in a turbulent quantum flow when probed at small and large length scales, where quantum turbulence mimics the properties of classical turbulence. Coarse graining over a number of quantized building blocks —singly quantized vortices— thus implies classical behavior, in a similar fashion as, for example, many photons or electrons produce classical diffraction or interference patterns. We find it especially remarkable that the quantum flow under study was thermal counterflow of He II, which has no direct classical analogue.

We thank D. Duda and M. Rotter for fruitful discussions and valuable help. We acknowledge the support of GAČR P203/11/0442.