Fully Coupled Two-Fluid Dynamics in Superfluid ${}^4\mathrm{He}$: Anomalous Anisotropic Velocity Fluctuations in Counterflow

We investigate the thermal counterflow of the superfluid ${}^4\mathrm{He}$ by numerically simulating three-dimensional fully coupled dynamics of the two fluids, namely quantized vortices and a normal fluid. We analyze the velocity fluctuations of the laminar normal fluid arising from the mutual friction with the quantum turbulence of the superfluid component. The streamwise fluctuations exhibit higher intensity and longer-range autocorrelation, as compared to transverse ones. The anomalous fluctuations are consistent with visualization experiments [Mastracci et al., Phys. Rev. Fluids 4, 083305 (2019)], and our results confirm their analysis with simple models on the anisotropic fluctuations. This success validates the model of the fully coupled dynamics and paves the way for solving some outstanding problems in this two-fluid system.

Figure 1

(a) Schematics of counterflow simulation. (b) Averaged values of the vortex line density as a function of the mean relative velocity $V_{ns}$. The slope parameter is $\gamma =165\pm 9\mathrm{s}/{\mathrm{cm}}^2$. (c) Normal-fluid velocity $v_{n,x}$ and superfluid velocity $v_{s,x}$ over the channel cross section in the statistically steady state at $V_n=2.5\mathrm{mm}/\mathrm{s}$.

Figure 2

Three-dimensional structures at $V_n=2.5\mathrm{mm}/\mathrm{s}$. (a) Quantized vortices. The black lines denote vortex filaments. (b) Vortices of the normal fluid. The green surfaces denote the positive isosurfaces of $Q$. (c) Velocity fluctuations of the normal fluid. The red and blue surfaces denote the isosurfaces of ${\delta }_x=0.1$ and $-0.1$, respectively.

Figure 3

(a) Velocity fluctuations $\mathrm{\Delta }{v}_{n,x}$ and $\mathrm{\Delta }{v}_{n,y}$ as a function of time at $V_n=2.5\mathrm{mm}/\mathrm{s}$. (b) Mean values of the velocity fluctuations as a function of $V_n$.

Figure 4

(a) Autocorrelations as a function of distance $r$ at $V_n=2.5\mathrm{mm}/\mathrm{s}$. (b) Distances $r^{*}$ where the autocorrelations decay to 0.1.