Enhancement of intermittency in superfluid turbulence

Onset of Intermittency and Multiscaling in Active Turbulence

Recent results suggest that highly active, chaotic, nonequilibrium states of living fluids might share much in common with high Reynolds number, inertial turbulence. We now show, by using a hydrodynamical model, the onset of intermittency and the consequent multiscaling of Eulerian and Lagrangian structure functions as a function of the bacterial activity. Our results bridge the worlds of low and high Reynolds number flows as well as open up intriguing possibilities of what makes flows intermittent.

Statistical properties of superfluid turbulence in ^{4}He from the Hall-Vinen-Bekharevich-Khalatnikov model

We obtain the von Kármán-Howarth relation for the stochastically forced three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model of superfluid turbulence in helium (^{4}He) by using the generating-functional approach. We combine direct numerical simulations (DNSs) and analytical studies to show that, in the statistically steady state of homogeneous and isotropic superfluid turbulence, in the 3D HVBK model, the probability distribution function (PDF) P(γ), of the ratio γ of the magnitude of the normal fluid velocity and superfluid velocity, has power-law tails that scale as P(γ)∼γ^{3}, for γ≪1, and P(γ)∼γ^{-3}, for γ≫1. Furthermore, we show that the PDF P(θ) of the angle θ between the normal-fluid velocity and superfluid velocity exhibits the following power-law behaviors: P(θ)∼θ for θ≪θ_{*} and P(θ)∼θ^{-4} for θ_{*}≪θ≪1, where θ_{*} is a crossover angle that we estimate. From our DNSs we obtain energy, energy-flux, and mutual-friction-transfer spectra, as well as the longitudinal-structure-function exponents for the normal fluid and the superfluid, as a function of the temperature T, by using the experimentally determined mutual-friction coefficients for superfluid helium ^{4}He, so our results are of direct relevance to superfluid turbulence in this system.

Dynamic scaling in rotating turbulence: A shell model study

We investigate the scaling form of appropriate timescales extracted from time-dependent correlation functions in rotating turbulent flows. In particular, we obtain precise estimates of the dynamic exponents z_{p}, associated with the timescales, and their relation with the more commonly measured equal-time exponents ζ_{p}. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.

Vortex clustering, polarisation and circulation intermittency in classical and quantum turbulence

The understanding of turbulent flows is one of the biggest current challenges in physics, as no first-principles theory exists to explain their observed spatio-temporal intermittency. Turbulent flows may be regarded as an intricate collection of mutually-interacting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations-associated with intermittency-originate from their non-trivial spatial arrangement. We then link the spatial distribution of vortices in quantum turbulence to the coarse-grained energy dissipation in classical turbulence, enabling the application of existent models of classical turbulence intermittency to the quantum case. Our results provide a connection between the intermittency of quantum and classical turbulence and initiate a promising path to a better understanding of the latter.

Phenomenology of quantum turbulence in superfluid helium

Quantum turbulence-the stochastic motion of quantum fluids such as 4He and 3He-B, which display pure superfluidity at zero temperature and two-fluid behavior at finite but low temperatures-has been a subject of intense experimental, theoretical, and numerical studies over the last half a century. Yet, there does not exist a satisfactory phenomenological framework that captures the rich variety of experimental observations, physical properties, and characteristic features, at the same level of detail as incompressible turbulence in conventional viscous fluids. Here we present such a phenomenology that captures in simple terms many known features and regimes of quantum turbulence, in both the limit of zero temperature and the temperature range of two-fluid behavior.

Mutual-friction-driven turbulent statistics in the hydrodynamic regime of superfluid ^{3}He-B

It is well known that the turbulence that evolves from the tangles of vortices in quantum fluids at scales larger than the typical quantized vortex spacing ℓ has a close resemblance with classical turbulence. The temperature-dependent mutual friction parameter α(T) drives the turbulent statistics in the hydrodynamic regime of quantum fluids that involves a self-similar cascade of energy. From a simple theoretical analysis, here we show that superfluid ^{3}He-B in the presence of mutual damping exhibits a k^{-5/3} Kolmogorov energy spectrum in the entire inertial range ℓ<r<L at temperature T≲0.2T_{c}, while at T≳0.2T_{c} dissipation begins to dominate larger eddies exhibiting a k^{-3} spectrum toward the energy pumping scale L. At T≈0.35T_{c}, eddies of all size, being highly affected by damping, exhibit a k^{-3} spectrum in the entire inertial range. The consistency of this result with the predictions of recent direct numerical simulations indicates that the present theoretical framework is applicable in quantifying the hydrodynamic regime of quantum turbulence.

An apparatus for generation and quantitative measurement of homogeneous isotropic turbulence in He ii

The superfluid phase of helium-4, known as He ii, exhibits extremely small kinematic viscosity and may be a useful tool for economically producing and studying high Reynolds number turbulent flow. Such applications are not currently possible because a comprehensive understanding of the complex two-fluid behavior of He ii is lacking. This situation could be remedied by a systematic investigation of simple, well controlled turbulence that can be directly compared with theoretical models. To this end, we have developed a new apparatus that combines flow visualization with second sound attenuation to study turbulence in the wake of a mesh grid towed through a He ii filled channel. One of three mesh grids (mesh number M = 3, 3.75, or 5 mm) can be pulled at speeds between 0.1 and 60 cm/s through a cast acrylic flow channel which has a 16 mm × 16 mm cross section and measures 330 mm long. The motion of solidified deuterium tracer particles, with diameter of the order 1 μm, in the resulting flow is captured by a high speed camera, and a particle tracking velocimetry algorithm resolves the Lagrangian particle trajectories through the turbulent flow field. A pair of oscillating superleak second sound transducers installed in the channel allows complementary measurement of vortex line density in the superfluid throughout the turbulent decay process. Success in early experiments demonstrates the effectiveness of both probes, and preliminary analysis of the data shows that both measurements strongly correlate with each other. Further investigations will provide comprehensive information that can be used to address open questions about turbulence in He ii and move toward the application of this fluid to high Reynolds number fluid research.

Superfluid Boundary Layer

We model the superfluid flow of liquid helium over the rough surface of a wire (used to experimentally generate turbulence) profiled by atomic force microscopy. Numerical simulations of the Gross-Pitaevskii equation reveal that the sharpest features in the surface induce vortex nucleation both intrinsically (due to the raised local fluid velocity) and extrinsically (providing pinning sites to vortex lines aligned with the flow). Vortex interactions and reconnections contribute to form a dense turbulent layer of vortices with a nonclassical average velocity profile which continually sheds small vortex rings into the bulk. We characterize this layer for various imposed flows. As boundary layers conventionally arise from viscous forces, this result opens up new insight into the nature of superflows.

Dynamic multiscaling in magnetohydrodynamic turbulence

We present a study of the multiscaling of time-dependent velocity and magnetic-field structure functions in homogeneous, isotropic magnetohydrodynamic (MHD) turbulence in three dimensions. We generalize the formalism that has been developed for analogous studies of time-dependent structure functions in fluid turbulence to MHD. By carrying out detailed numerical studies of such time-dependent structure functions in a shell model for three-dimensional MHD turbulence, we obtain both equal-time and dynamic scaling exponents.

Multiscaling in superfluid turbulence: A shell-model study

We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is more complicated than it is in fluid turbulence.

Grid superfluid turbulence and intermittency at very low temperature

Low-temperature grid-generated turbulence is investigated by using numerical simulations of the Gross-Pitaevskii equation. The statistics of regularized velocity increments are studied. Increments of the incompressible velocity are found to be skewed for turbulent states. Results are later confronted with the (quasi) homogeneous and isotropic Taylor-Green flow, revealing the universality of the statistics. For this flow, the statistics are found to be intermittent and a Kolmogorov constant close to the one of classical fluid is found for the second-order structure function.

Vortex clustering and universal scaling laws in two-dimensional quantum turbulence

We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal -5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The -5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a -5/3 exponent.

Experimental, numerical, and analytical velocity spectra in turbulent quantum fluid

Turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, interpenetrating turbulent fluids: the first is inviscid with quantized vorticity, and the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those of ordinary turbulence. We present experimental, numerical, and theoretical results that explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales.