Mesoscale equipartition of kinetic energy in quantum turbulence

Abstract

The turbulence of superfluid helium is investigated numerically at finite temperature. Direct numerical simulations are performed with a "truncated HVBK" model, which combines the continuous description of the Hall-Vinen-Bekeravich-Khalatnikov equations with the additional constraint that this continuous description cannot extend beyond a quantum length scale associated with the mean spacing between individual superfluid vortices. A good agreement is found with experimental measurements of the vortex density. Besides, by varying the turbulence intensity only, it is observed that the inter-vortex spacing varies with the Reynolds number as Re −3/4 , like the viscous length scale in classical turbulence. In the high temperature limit, Kolmogorov's inertial cascade is recovered, as expected from previous numerical and experimental studies. As the temperature decreases, the inertial cascade remains present at large scales while, at small scales, the system evolves towards a statistical equipartition of kinetic energy among spectral modes, with a characteristic k 2 velocity spectrum. The accumulation of superfluid excitations on a range of mesoscales enables the superfluid to keep dissipating kinetic energy through mutual friction with the residual normal fluid, although the later becomes rare at low temperature. It is found that most of the superfluid vorticity can concentrate on these mesoscales at low temperature, while it is concentrated in the inertial range at higher temperature. This observation should have consequences on the interpretation of decaying turbulence experiments, which are often based on vortex line density measurements.

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

“…1A, which can be interpreted as partial thermalization of superfluid excitations. This procedure also leads to the prediction LD 2 = 4Re 3=2 (24), which is consistent with experiments and allows to identify the spectrum of L ðrÞ=κ with the spectrum of the scalar field jω s ðrÞj.…”

“…1A). At smaller scales k > k meso , an intermediate (meso) regime appeared that expands as T is lowered (24). Apparently, superfluid energy, cascading from larger length scales, accumulates beyond k meso .…”

“…2 has been recently obtained in the TOUPIE wind tunnel both above and below T λ in the far wake of a disk. [To normalize the x axis of this plot, the mean intervortex distance ℓ in He-II was estimated using the relation 2ℓ=D = Re −3=4 k (24), where Re κ = DV=κ is a Reynolds number defined using the root-mean-square velocity from the anemometer, and the prefactor 2 was fitted to experimental and numerical data in the range T ' 1.4-1.6 K.] A one-to-one comparison of both datasets allowed to check the validity of the −4/5 KarmanHowarth law (21) below T λ ; this law, sometimes described as the only exact relation known in turbulence, confirms that energy cascades from large to small scales without dissipation within the inertial range where the KO-41 scaling is observed.…”

“…From a computational viewpoint, the HVBK equations are similar to the Navier-Stokes equation (1). Not surprisingly, standard methods of classical turbulence have been adapted to the HVBK equations, e.g., large eddy simulations (58), direct numerical simulations (24,59), and eddy-damped quasinormal Markovian simulations (60).…”

Introduction to quantum turbulence

“…Application of the HVBK equations to turbulence is not justified for randomly oriented vortex lines, as the net superfluid vorticity in each fluid parcel would be zero, yielding zero friction, despite the nonzero vortexline density. Modifications of the HVBK equations have been developed, neglecting the vortex tension and approximating the mutual friction (50,51). Such models probably underestimate friction dissipation, but the coupled motion of both fluids is computed self-consistently.…”