a Numerical experimentation and analysis of quantum turbulence in superfluid helium II

Abstract

The study of turbulence in superfluid Helium II suggests that at least in part the rules of classical turbulence are obeyed. The question posed is, whether the tangles of quantised vorticity that represent turbulence in a superfluid are directly analogous to the swirls and eddies found in turbulent classical fluids. A cornerstone of classical turbulence has been the evidence of the Kolmogorov scaling and this has been observed in some experimental studies of superfluid turbulence. Here we contrast quantum turbulence in various scenarios to further our understanding and confidence in such modelling as well as to search for evidence of any adherence to Kolmogorov. In all numerical simulations presented here turbulence in the superfluid is driven by motions of the normal fluid. My work approaches the superfluid turbulence through three distinct normal fluid models. In most physical experiments with superfluid helium, turbulence is generated in two ways. Firstly, thermally (by applying a heat flux, as in thermal counterflow) and we model this by using a uniform normal fluid. Secondly, mechanically (by stirring the liquid) and we model this in one of two ways; either a synthetic turbulence using a kinematic simulations (KS) flow or with a frozen snapshot from a direct numerical simulation (DNS). We determine the difference between thermally and mechanically driven quantum turbulence. Using the kinematic simulations model we find that in the latter the energy is concentrated at the large scales, the spectrum obeys Kolmogorov scaling, vortex lines have small curvature, and the presence of coherent vortex structures induces vortex reconnections at small angles. In contrast, when we employ our uniform normal fluid we find the energy is concentrated at the mesoscales, the curvature is larger, the vorticity field is featureless and reconnections occur at larger angles. Our results suggest a method to experimentally detect the presence of superfluid vortex bundles. We show that vortex tangles with the same vortex line density have different energy spectra, depending on the driving normal fluid, and identify the spectral signature of two forms of superfluid turbulence: Kolmogorov tangles and Vinen tangles. By decomposing the superfluid velocity field into local and nonlocal contributions, we find that in Vinen tangles the motion of vortex lines depends mainly on the local curvature, whereas in Kolmogorov tangles the long-range vortex interaction is dominant and leads to the formation of clustering of lines, in analogy to the ‘worms’ of classical turbulence. Finally, we compute the frequency spectrum of superfluid vortex density fluctuations for tangles of the same vortex line density, but which are driven by two different normal fluid models. Taking our measurements in a sufficiently small cube to eliminate any filtering effect, we observe the f−5/3 that has been experimentally observed within the Kolmogorov tangles whereas for the Vinen tangles we find a flat and featureless spectrum.