The Statistical Properties of Superfluid Turbulence in ${}^4$He from the Hall-Vinen-Bekharevich-Khalatnikov Model

Abstract

We obtain the von Kármán-Howarth relation for the stochastically forced three-dimensional Hall-Vinen-Bekharvich-Khalatnikov (3D HVBK) model of superfluid turbulence in Helium (${}^4$He) by using the generating-functional approach. We combine direct numerical simulations (DNSs) and analyitcal 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) $P(\gamma )$, of the ratio $\gamma$ of the magnitude of the normal fluid velocity and superfluid velocity, has power-law tails that scale as $P(\gamma )\sim {\gamma }^3$, for $\gamma \ll 1$, and $P(\gamma )\sim {\gamma }^{-3}$, for $\gamma \gg 1$. Furthermore, we show that the PDF $P(\theta )$, of the angle $\theta$ between the normal-fluid velocity and superfluid velocity exhibits the following power-law behaviors: $P(\theta )\sim \theta$ for $\theta \ll {\theta }_{\ast }$ and $P(\theta )\sim {\theta }^{-4}$ for ${\theta }_{\ast }\ll \theta \ll 1$, where ${\theta }_{\ast }$ is a crossover angle that we estimate. From our DNSs we obtain energy, energy-flux, and mutual-friction-transfer spectra, and 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.

13 pages, 3 figures

References in corpus (31)

• Emergence of turbulence in an oscillating Bose-Einstein condensate
• Turbulence without pressure
• Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross-Pitaevskii equation with the small scale dissipation
• Kelvin Wave Cascade and Decay of Superfluid Turbulence
• Bottleneck crossover between classical and quantum superfluid turbulence
• Quantum Turbulence in a Trapped Bose-Einstein Condensate
• Emergence and decay of turbulence in stirred atomic Bose-Einstein condensates
• Visualization study of counterflow in superfluid helium-4 using metastable helium molecules
• Spectrum of Kelvin-wave turbulence in superfluids
• Vortex density spectrum of quantum turbulence
• Twenty-five years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin
• Mean- Field Approximation and a Small Parameter in Turbulence Theory
• Vortex filament method as a tool for computational visualization of quantum turbulence
• Energy cascade with small-scales thermalization, counterflow metastability and anomalous velocity of vortex rings in Fourier-truncated Gross-Pitaevskii equation
• Energy Spectra of Developed Turbulencein Helium Superfluids
• Statistical Properties of Turbulence: An Overview
• Energy cascade and the four-fifths law in superfluid turbulence
• Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence
• Quantum turbulence at finite temperature: the two-fluids cascade
• Turbulence in the two-dimensional Fourier-truncated Gross-Pitaevskii equation
• Extended Self Similarity works for the Burgers equation and why
• Energy and Vorticity Spectra in Turbulent Superfluid ${}^4$He from $T=0$ to $T_{\lambda }$
• Enhancement of intermittency in superfluid turbulence
• Mesoscale Equipartition of kinetic energy in Quantum Turbulence
• Intermittency of quantum turbulence with superfluid fractions from 0% to 96%
• Intermittency enhancement in quantum turbulence
• Shell model of superfluid turbulence
• Turbulent statistics and intermittency enhancement in coflowing superfluid ${}^4$He
• Multiscaling in superfluid turbulence: A shell-model study
• Energy Spectra of Superfluid Turbulence in ${}^3$He
• Homogeneous Isotropic Superfluid Turbulence in Two Dimensions: Inverse and Forward Cascades in the Hall-Vinen-Bekharevich-Khalatnikov model

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