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arXiv:1811.05374 · Physical Review A

Quantitative estimation of effective viscosity in quantum turbulence

Vishwanath Shukla, Pablo D. Mininni, Giorgio Krstulovic, Patricio Clark Di Leoni, Marc Brächet

Published Apr 4, 2019• 28 citations
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Abstract

We study freely decaying quantum turbulence by performing high-resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from ${1024}^{3}$ to ${4096}^{3}$ grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ensuremath{\ell}$, and a second inertial range for scales smaller than $\ensuremath{\ell}$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite-temperature effects in decaying quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions under the GPE. We use finite-temperature GPE simulations to extract mean-free path by measuring the spectral broadening in the Bogoliubov dispersion relation that we obtain from the spatiotemporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, we perform low-Reynolds-number simulations of the Navier-Stokes equations, in order to compare the decay of high-temperature quantum flows with their classical counterparts, and to further calibrate the estimations of the effective viscosity (based on the mean-free-path computations).

DOI
10.1103/physreva.99.043605

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We study freely decaying quantum turbulence by performing high-resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from 10243{1024}^{3}10243 to 40963{4096}^{3}40963 grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale \ensuremathℓ\ensuremath{\ell}\ensuremathℓ, and a second inertial range for scales smaller than \ensuremathℓ\ensuremath{\ell}\ensuremathℓ. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite-temperature effects in decaying quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions under the GPE. We use finite-temperature GPE simulations to extract mean-free path by measuring the spectral broadening in the Bogoliubov dispersion relation that we obtain from the spatiotemporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, we perform low-Reynolds-number simulations of the Navier-Stokes equations, in order to compare the decay of high-temperature quantum flows with their classical counterparts, and to further calibrate the estimations of the effective viscosity (based on the mean-free-path computations).
We perform a study of the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the stochastic Ginzburg-Landau equations, using up to 40963{4096}^{3}40963 grid points with a pseudospectral method. We show that for temperatures close to the critical one, the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence of strong thermal effects can inhibit the development of a proper turbulent cascade. We provide Ans\"atze for the effective viscosity and friction as a function of the temperature.
There is a growing interest in the relation between classical turbulence and quantum turbulence. Classical turbulence arises from complicated dynamics of eddies in a classical fluid. In contrast, quantum turbulence consists of a tangle of stable topological defects called quantized vortices, and thus quantum turbulence provides a simpler prototype of turbulence than classical turbulence. In this paper, we investigate the dynamics and statistics of quantized vortices in quantum turbulence by numerically solving a modified Gross-Pitaevskii equation. First, to make decaying turbulence, we introduce a dissipation term that works only at scales below the healing length. Second, to obtain steady turbulence through the balance between injection and decay, we add energy injection at large scales. The energy spectrum is quantitatively consistent with the Kolmogorov law in both decaying and steady turbulence. Consequently, this is the first study that confirms the inertial range of quantum turbulence.
In this paper, we investigate inertially forced isothermal quantum turbulence (coflow of normal and superfluid components) at temperatures of 1.6 and 2 K. The experiments are carried out in a large optical cryostat, where quasi-isotropic, homogeneous turbulence is generated using a double oscillating grid. Turbulence intensity is tuned by varying the grid stroke and frequency. The flow is probed via 2D and 3D reconstructions of quasi-isodense microsphere trajectories, from which we extract the large-scale properties of the flow in the fully turbulent regime for different Reynolds numbers: the turbulent velocity fluctuations, the energy transfer rate, and the integral length scale. Additionally, we determine the mean vortex line density L\mathcal{L}L via attenuation of a second sound standing wave across the entire measurement volume. Our results confirm with an improved accuracy that the intervortex spacing ℓ=1/L\ell =1/\sqrt{\mathcal{L}}ℓ=1/L​ scales with the Reynolds number Reκ\text{Re}_κReκ​ (based on the quantum of circulation κκκ) as ℓ∝Reκ−3/4\ell\propto \text{Re}_κ^{-3/4}ℓ∝Reκ−3/4​, with a well-defined numerical prefactor and no observed temperature dependence. This scaling recalls that of the dissipative length scale in classical Kolmogorov (K41) turbulence and it lead previous authors to interpret ℓ\ellℓ as an effective dissipation length scale. However, in our temperature range, this interpretation is not consistent with the apparent temperature independence of the prefactor. Based on those arguments, we propose an alternative interpretation that suggests that the inter-vortex distance in coflow turbulence is the consequence of the quantum restricted depth of the superfluid component energy cascade.
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Large scale numerical simulation of quantum turbulence is performed by using 3-D time-dependent Gross-Pitaevskii equation.The energy spectrum obeying Kolmogorov law and large scale self-similar structure of quantum vortex tangle are found in a fully developed dumped turbulent state.Width of the inertial range becomes large depending on the system size of the simulation that is consistent with the result of the normal fluid turbulence.On the other hand, bottleneck effect near coherent length prevents the inertial range from extending to smaller scale.
Quantum turbulence is numerically studied by solving the Gross‐Pitaevskii equation. By introducing both energy dissipation at small scales and energy injection at large scales, we succeed in obtaining steady turbulence sustained by a balance between energy injection and energy dissipation. The energy spectrum in the inertial range takes the form of the Kolmogorov spectrum and the energy flux is consistent with the energy dissipation rate at small scales. These results reveal the properties of quantum turbulence in the inertial range, allowing us to propose a prototype of turbulence much simpler than conventional classical turbulence.