Turbulent statistics and intermittency enhancement in coflowing superfluid ${}^4\mathrm{He}$

The large-scale turbulent statistics of mechanically driven superfluid ${}^4\mathrm{He}$ was shown experimentally to follow the classical counterpart. In this paper, we use direct numerical simulations to study the whole range of scales in a range of temperatures $T\in [1.3,2.1]$ K. The numerics employ self-consistent and nonlinearly coupled normal and superfluid components. The main results are that (i) the velocity fluctuations of normal and super components are well correlated in the inertial range of scales, but decorrelate at small scales. (ii) The energy transfer by mutual friction between components is particulary efficient in the temperature range between 1.8 and 2 K, leading to enhancement of small-scale intermittency for these temperatures. (iii) At low $T$ and close to $T_{\lambda }$, the scaling properties of the energy spectra and structure functions of the two components are approaching those of classical hydrodynamic turbulence.

Figure 1

Temperature dependence of ${}^4\mathrm{He}$ parameters used in the simulations of HVBK Eqs. (1): ratio of the superfluid and normal fluid densities (a), mutual friction parameters $\alpha$ in the superfluid Eq. (1a) and $\alpha {\rho }_{\text{s}}/{\rho }_{\text{n}}$ in the normal fluid Eq. (1b) (b), and kinematic viscosities ${\nu }_{\text{s}}$ and ${\nu }_{\text{n}}$ (c).

Figure 2

(a) Normalized compensated by $k^{5/3}$ energy spectra of normal (solid line) and superfluid (dashed line) components ${\mathcal{E}}_{\text{n,s}}$ [Eq. (19)]. The thick black dashed line, marked $\mathrm{\Omega }=0$, corresponds to the energy spectra in the decoupled case. (b) Normalized cross-correlation functions $R_1\left(k\right)$ (solid lines) and $R_2\left(k\right)$ (dashed lines) [Eq. (21)].

Figure 3

Energy spectra [panels (a)–(c)] and the energy balance [panels (d)–(f)] for the normal (solid lines) and superfluid (dashed lines) components. The spectra and balances are shown for three representative temperatures. Different terms of Eq. (6) are marked in the panels. In panels (d)–(f), absolute values of the terms are shown.

Figure 4

The compensated structure functions $S_3\left(r\right)/r$ of normal (a) and superfluid (b) components as a function of $r$. Here and in other figures, the thick black dashed line, marked $\mathrm{\Omega }=0$, corresponds to the decoupled case. The thin straight solid lines indicate viscous behavior $S_3\propto r^3$. The vertical thin dot-dashed straight lines in panel (b) marks the largest intervortex distance $\ell$.

Figure 5

The compensated structure functions $S_2\left(r\right)/r^{2/3}$ [panels (a) and (b)] and $S_4\left(r\right)/r^{4/3}$ [panels (c) and (d)] as a function of $r$. The thin straight solid lines indicate viscous behavior $S_2\propto r^2$ and $S_4\propto r^4$. The vertical thin dot-dashed straight lines in panels (b) and (d) mark the largest intervortex distance $\ell$.

Figure 6

Flatness of normal (a) and superfluid (b) components as a function of $S_3$. The horizontal dashed line correspond to $F_{\text{v}}=3$. Note log-log scale. Thin straight dashed lines indicate the scaling, corresponding to the She-Leveque model of classical turbulence [36] (marked as “SL”) and the fit for structure function at $T=1.9\mathrm{K}$.

Figure 7

Flatness as a function of temperature for the normal component (a) and for the superfluid component (b) for different separations ${\tilde{r}}_{\text{n,s}}=r/{\eta }_{\text{n,s}}$. The horizontal dashed lines lines correspond to the classical values of flatness for the same separations (marked by the same colors). The classical values are represented by the decoupled case. The lowest black dashed line corresponds to the Gaussian value $F_{\text{v}}=3$. The error bars were obtained by averaging results for different intervals of the time realization.