Vortex-Density Fluctuations, Energy Spectra, and Vortical Regions in Superfluid Turbulence

Andrew W. Baggaley, Jason Laurie, Carlo F. Barenghi
2012 Physical Review Letters  
Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the
more » ... tex line density fluctuations. PACS numbers: 67.25.dk, 47.32.C, 47.27.Gs Below a critical temperature, liquid helium becomes a two-fluid system in which an inviscid superfluid component coexists with a viscous normal fluid component. The flow of the superfluid is irrotational: superfluid vorticity is confined to vortex lines of atomic thickness around which the circulation takes a fixed value κ (the quantum of circulation). Superfluid turbulence [1, 2] is easily created by stirring either helium isotope ( 4 He or 3 He-B), and consists of a tangle of reconnecting vortex filaments which interact with each other and with the viscous normal fluid (which may be laminar or turbulent). The most important observable quantity is the vortex line density L (vortex length per unit volume), from which one infers the average distance between vortex lines, ≈ L −1/2 . Our interest is in the properties of superfluid turbulence and their similarities with ordinary turbulence. Experiments [3, 4] have revealed that, if the superfluid turbulence is driven by grids or propellers, the distribution of the turbulent kinetic energy over length scales larger than obeys the celebrated k −5/3 Kolmogorov scaling observed in ordinary (classical) turbulence. Here k is the magnitude of the three-dimensional wavenumber (wavenumber and frequency are related by k = f /v, wherev is the mean flow). Numerical calculations performed using either the vortex filament model [5, 6] or the Gross-Pitaevskii equation [7, 8] confirm the Kolmogorov scaling. It is thought that the effect arises from the partial polarization of the vortex lines [1, 2, 9] , but such effect has never been clearly identified. Another important experimental observation is that in both 4 He [10] and 3 He-B [11], the frequency spectrum of the fluctuations of L has a decreasing f −5/3 scaling typical of passive objects [6, 12] advected by a turbulent flow. This latter result seems to contradict the interpretation of L as a measure of superfluid vorticity, ω = κL which is usually made in the literature [1, 2, 11, [13] [14] [15] . In fact, from dimensional analysis, the vorticity spec- * andrew.baggaley@gla.ac.uk trum corresponding to the Kolmogorov law should increase with f (as f 1/3 ), not decrease. Since the vortex line density is a positive quantity, a better analogy is to the enstrophy spectrum: however in classical turbulence this spectrum is essentially flat [16, 17] , in disagreement with the helium experiments [10, 11] . The aim of this letter is to reconcile these two sets of observations (each separately backed by numerical simulations). We shall show that, at any instant, the vortex tangle can be decomposed into two parts: vortex lines which are locally polarised in the same direction, forming metastable coherent bundles, and vortex lines which are randomly oriented in space. The former is responsible for the Kolmogorov energy spectrum, and the latter for frequency spectrum of the vortex line density. Following Schwarz [18], we model vortex filaments as space curves s(ξ, t) which move according to where t is time, α and α are known temperature dependent friction coefficients [19] , s = ds/dξ is the unit tangent vector at the point s, ξ is arc length, and v n is the normal fluid velocity at the point s. We set the temperature to T = 1.9 K, typical of many finite temperature studies (corresponding to α = 0.206 and α = 0.0083). The self-induced velocity of the vortex line at the point s is given by the Biot-Savart law where κ = 9.97 × 10 −4 cm 2 /s (in 4 He) and the line integral extends over the entire vortex configuration L. The calculation is performed in a periodic cube of size D = 0.1 cm. The numerical techniques to discretize the vortex lines into a number of points s j (j = 1, · · · N ) held at minimum separation ∆ξ/2, compute the time evolution, de-singularize the Biot-Savart integrals, evaluate v s using a tree-method (with critical opening angle 0.4),
doi:10.1103/physrevlett.109.205304 pmid:23215501 fatcat:ar72wwzy35a6tbbzccada7briq