Physics of superfluid helium-4 vortex tangles in normal-fluid strain fields

By employing dimensional analysis, we scale the equations of the mesoscopic model of finite temperature superfluid hydrodynamics. Based on this scaling, we set up three problems that depict the effects of kinematic, normal-fluid strain fields on superfluid vortex loops, and characterize small-scale processes in fully developed turbulence. We also develop a formula for the computation of energy spectra corresponding to superfluid vortex tangles in unbounded domains. Employing this formula, we compute energy spectra of superfluid vortex patterns induced by uniaxial, equibiaxial, and simple-shear normal-fluid flows. By comparing the steady-state superfluid spectra and vortex structures, we conclude that normal-flow strain fields do not play an important role in explaining the phenomenology of fully developed superfluid turbulence. This is in sharp contrast with the role of vortical normal-flow fields in offering plausible, structural explanations of superfluid vortex patterns and spectra entailed in numerical turbulent solutions of the mesoscopic model.

Figure 1

Energy spectrum of a single superfluid vortex ring in helium-4. The radius of the ring is $R=0.03\mathrm{cm}$. There are 8192 wave numbers ($k\equiv 1/\ell$) between length scales ${\ell }_{\text{max}}=12\mathrm{cm}$ and ${\ell }_{\text{min}}=9.76\times {10}^{-5}\mathrm{cm}$.

Figure 2

Compensated energy spectrum of a single superfluid vortex ring in helium-4. There is a $k^2$ scaling in the small $k$ regime that corresponds to the linear impulse of the ring (left), and a $k^{-1}$ scaling at large $k$ that corresponds to the energy spectrum of an isolated line vortex (right).

Figure 3

Left: Superfluid vortex tangle length $L$ versus time for equibiaxial extensional flow (left curve), uniaxial extensional flow (middle curve), and simple-shear flow (right curve). Right: Same results for R2 initial conditions.

Figure 4

Top: Vortical impulse $\mathbf{I}$ (left) and angular vortical impulse $\mathbf{L}$ (right) magnitudes versus time. The equibiaxial flow results correspond to the leftmost curves, while the shear-flow curves are marked by circles. Bottom: Same results for R2 initial conditions.

Figure 5

Uniaxial extensional flow. (a)–(e) Evolution of superfluid vortex tangle at times $t=[0,0.31,0.63,0.95,1.29]\mathrm{s}$. The horizontal is coinciding with the extensional axis. All figures are drawn with the same scale. In (f), time is the same as in (e), but the view is along the extensional axis, so that key structural details are better indicated. These vortex configurations are also reproduced in the R2 results.

Figure 6

Top: Energy spectrum of superfluid vortex tangles in uniaxial extension at $t=1.0\mathrm{s}$, after the initial transients has ceased, and the shape has reached a steady state. There are four scaling regimes: $k^2$ at smallest $k$, $k^{10/3}$ at larger but still small $k$ ($k\in [1.0,5.5]$), $k^{-7/3}$ at intermediate $k$ ($k\in [15,75]$), and $k^{-1}$ at high $k$ ($k\in [147,1900]$). The final cutoff is due to the numerical smoothing of the Biot-Savart kernel. The numerical estimate of the end of the $k^2$ regime is $k_t=1.3$. Bottom (left): Demonstration of spectrum convergence. Spectra at five different times are shown. From the lower to the upper curves, the times are $t=[0.47,0.66,0.82,0.92,1.00]\mathrm{s}$. The last three spectrum curves allow identical scaling conclusions. Bottom (right): R2 results for times $t=[1.06,1.10]\mathrm{s}$. The scaling of the wave-number regime before the spectrum peak has switched from $k^{10/3}$ (in the R1 data) to $k^{7/3}$.

Figure 7

Equibiaxial strain flow. (a)–(e) Evolution of superfluid vortex tangle at times $t=[0,0.12,0.25,0.50,0.72]\mathrm{s}$. The vertical is coinciding with the compressive axis. All figures are drawn with the same scale. In (f), time is the same as in (e), but the view is along the compressive axis, so that key structural details are better indicated. These vortex configurations are also reproduced in the R2 results.

Figure 8

Top: Energy spectrum of superfluid vortex tangles in equibiaxial extension at $t=0.54\mathrm{s}$, after the initial transients have ceased, and the shape has reached a steady state. There are four scaling regimes: $k^2$ at smallest $k$, $k^{10/4}$ at larger but still small $k$ ($k\in [3,7.5]$), $k^{-9/5}$ at intermediate $k$ ($k\in [15,70]$), and $k^{-1}$ at high $k$ ($k\in [170,1500]$). The final cutoff is due to the numerical smoothing of the Biot-Savart kernel. The numerical estimate of the end of the $k^2$ regime is $k_t=3.6$. Bottom (left): Demonstration of spectrum convergence. Spectra at five different times are shown. From the lower to the upper curves, the times are $t=[0.21,0.35,0.44,0.50,0.54]\mathrm{s}$. The last three spectrum curves allow identical scaling conclusions. Bottom (right). R2 results for times $t=[0.68,0.64]\mathrm{s}$. The scalings are identical with R1 scalings. However, the $k^{-9/5}$ scaling at intermediate $k$ is now observed over a wider wave-number interval [15,185].

Figure 9

Superfluid vortex sheetlike structure formation ($t=0.72\mathrm{s}$) in equibiaxial extensional flow. The sheets have a fractal structure, composed of a hierarchy of quasicircular loops one within the other.

Figure 10

Simple shear. (a)–(d) Evolution of superfluid vortex tangle at times $t=[0.76,1.14,1.52,1.90]\mathrm{s}$. The graphs show the tangle's projection on the plane normal to the direction of velocity ($x$ direction). All figures are drawn with the same scale. In (e) and (f), time is the same as in (d). The view in (e) is along the axis which is normal to the plane of shear ($z$ direction), and (f) along the direction of velocity gradient ($y$ axis). These vortex configurations are also reproduced in the R2 results.

Figure 11

Top: Energy spectrum of superfluid vortex tangles in simple shear at $t=1.41\mathrm{s}$, after the initial transients have ceased, and the shape has reached a steady state. There are four scaling regimes: $k^2$ at smallest $k$, $k^{12/5}$ at larger but still small $k$ ($k\in [1,8]$), $k^{-1.6}$ at intermediate $k$ ($k\in [7.5,67]$), and $k^{-1}$ at high $k$ ($k\in [192,1640]$). The final cutoff is due to the numerical smoothing of the Biot-Savart kernel. The numerical estimate of the end of the $k^2$ regime is $k_t=1.8$. Bottom (left): Demonstration of spectrum convergence. Spectra at five different times are shown. From the lower to the upper curves, the times are $t=[0.21,0.35,0.44,0.50,0.54]\mathrm{s}$. The last three spectrum curves allow identical scaling conclusions. Bottom (right): R2 results for times $t=[1.59,1.55]\mathrm{s}$. The scalings are identical with R1 scalings.