Spreading of superfluid vorticity clouds in normal-fluid turbulence

Abstract

In this paper, we formulate a self-consistent model of thermal superfluid dynamics. By solving it, we analyse the problem of superfluid vorticity cloud propagation in normal-fluid turbulence. We show that superfluid cloud expansion is driven by pattern-forming superfluid vortex instabilities taking place in the interface layer between the cloud's bulk and the outer undisturbed normal-fluid turbulence. The radius of the cloud increases linearly with time. Mutual friction transfers energy from the normal-fluid turbulence to the superfluid cloud, whilst damping the smallest normal-fluid turbulence motions. This damping action is much weaker than viscous dissipation effects in a corresponding pure normal-fluid turbulence. The energy spectrum of superfluid turbulence presents the k −3 scaling that characterizes the spiral superfluid vorticity patterns of normal vortex tube-superfluid vortex interactions. The corresponding k −2 pressure spectrum signifies the singular nature of superfluid vorticity. These two scalings coincide in wavenumber space with the Kolmogorov regime in the normal-fluid turbulence. We compute a fractal dimension d f ≈ 1.652 for superfluid vorticity. Due to simpler underlying superfluid vortex dynamics in relation to the strongly nonlinear classical vortex dynamics, this fractal dimension is smaller than the corresponding dimension of vortex tube centrelines in classical turbulence.

Energy spectra of finite temperature superfluid helium-4 turbulence

“…36 The non-standard element is the smoothing of the delta function singularities in the coupling terms, and this is achieved with the use of Heaviside functions. 1 For the vortices, I renormalize the self-interaction velocity divergence in the Biot-Savart law by employing the velocity of a ring with radius the local radius of curvature, and apply the method of Winckelmans and Leonard for evaluating velocity contributions because of all other points. 37 For the latter, an effective vortex core radius equal to ξ is employed.…”

“…38 and further modified in Ref. 1. Finally, in contrast to the Navier-Stokes part, the algorithms of the vortex dynamical part are difficult exercises in computational geometry.…”

“…The Lundgren force is a method for achieving a controllable injection of energy in a turbulent flow without unnecessary real-life complications. It has been shown 1,27,28 that the Lundgren force is consistent with Kolmogorov's inertial range energy spectrum scaling so it does not "contaminate" the statistical structure of normal-fluid turbulence. This force is introduced in the Navier-Stokes equation for the following reason: the structure of pure normalfluid turbulence is identical to that of turbulence in a simple-fluid, which is a topic of numerous investigations and many of its aspects are very well understood.…”

“…In contrast, numerical computations show that in homogeneous, isotropic turbulence, the actual situation is in between these two important limits. 1 In this work, I model the mesoscopic dynamics of superfluid vortices by developing a corresponding Langevin equation, and I couple together vortex tangle and normal-fluid flow by forming a combined Langevin-Navier-Stokes system. By solving it numerically, I obtain mesoscopic superfluid and normal-fluid velocity fields whose averages correspond to Hall-Vinen and Gorter-Mellink macroscopic formulations.…”

“…Thus, since the other (non-coupling related) forces in the Navier-Stokes equation conserve energy, any normal-fluid energy dynamics in the computations are due entirely to its coupling with the vortices, and the energy transfer between the two subsystems because of their coupling can be explicitly demonstrated. Moreover, since the Lundgren force does not alter spectral scalings, 1,27,28 any normal-fluid turbulence deviation from standard simple-fluid turbulence physics can be safely attributed to the effects of the mutual-friction and lift force couplings.…”