Active and finite-size particles in decaying quantum turbulence at low temperature

The evolution of a turbulent tangle of quantum vortices in the presence of finite-size active particles is studied by means of numerical simulations of the Gross-Pitaevskii equation. Particles are modeled as potentials depleting the superfluid and described with classical degrees of freedom following a Newtonian dynamics. It is shown that particles do not modify the building-up and the decay of the superfluid Kolmogorov turbulent regime. It is observed that almost the totality of particles remains trapped inside quantum vortices, although they are occasionally detached and recaptured. The statistics of this process is presented and discussed. The particle Lagrangian dynamics is also studied. At large timescales, the velocity spectrum of particles is reminiscent of a classical Lagrangian turbulent behavior. At timescales faster than the turnover time associated with the mean intervortex distance, the particle motion is dominated by oscillations due to the Magnus effect. For light particles, a nonclassical scaling of the spectrum arises. The particle velocity and acceleration probability distribution functions are then studied. The decorrelation time of the particle acceleration is found to be shorter than in classical fluids, and related to the Magnus force experienced by the trapped particles.

Figure 1

Visualizations of the superfluid vortex tangle. Vortices are represented as isosurfaces in red of the density field ($\rho =0.15{\rho }_{\infty }$), sound is rendered in blue, trapped particles in green, and free particles in purple. The upper row is without particles, the lower row is with 200 neutrally buoyant particles of radius $a_{\mathrm{p}}=4\xi$. (a,d) The ABC initial states. (b,e) The most turbulent regime ($t=1.3T_L$). (c,f) A late time ($t=8.1T_L$). $T_L$ denotes the large-eddy-turnover time (see the text).

Figure 2

(a) Time evolution of the superfluid energy components in the cases with no particles (dashed line), 200 small particles (dotted line), 200 large particles (solid line), and 80 large particles (dash-dotted line). (b) Incompressible energy dissipation rate for different numbers of particles with different sizes and different masses (solid lines). Dash-dotted horizontal lines of the corresponding colors indicate the value of the maximum of dissipation, obtained averaging over the shaded region. The dissipation is expressed in units of its maximum ${\epsilon }_{\max }$ in the case without particles.

Figure 3

(a) Time evolution of the mean intervortex distance for different numbers of particles of different sizes and different masses. (b) Incompressible energy spectrum for different numbers of particles of different sizes and different masses. Inset: Compensated incompressible energy spectrum. Solid lines refer to particles of size $a_{\mathrm{p}}=4\xi$, dashed lines refer to particles of size $a_{\mathrm{p}}=10\xi$. The dotted line is the classical scaling ${\epsilon }_{\max }{k}^{-5/3}$. The spectrum is computed averaging over times in the shaded region.

Figure 4

Closeup of the superfluid vortex tangle at the early stage of the simulation (upper row: $t=0.27T_L$) and during the turbulent regime (lower row: $t=1.1T_L$) for the cases with no particles [left column (a),(d)], small particles [central column (b),(e): $a_{\mathrm{p}}=4\xi$], and large particles [right column (c),(f): $a_{\mathrm{p}}=10\xi$]. Vortices are represented as isosurfaces of the density field ($\rho =0.15{\rho }_{\infty }$) and rendered in red, sound is rendered in blue, trapped particles in green, and free particles in purple.

Figure 5

(a) Fraction of trapped particles as a function of time for different numbers of particles of different sizes and different masses. Inset: The same for longer time in the case of 200 neutrally buoyant particles of size $a_{\mathrm{p}}=4\xi$. (b) Comparison between the fraction of multiply trapped particles as a function of time for neutrally buoyant particles. (c) Volume rendering of large particles ($a_{\mathrm{p}}=10\xi$) multiply trapped by quantum vortices. Vortices are rendered in red, sound in blue, particles in green. (d) Probability density function of the continuous time spent by particles inside vortices for different species of particles. The dotted blue line corresponds to the same simulation of blue circles (particles with size $a_{\mathrm{p}}$ and mass $\mathcal{M}=1$) but averaged over the full simulation times). Inset: Absolute value of the circulation around a single particle of size $a_{\mathrm{p}}=4\xi$ and mass $\mathcal{M}=1$ as a function of time. The PDF is computed averaging over times in the shaded region.

Figure 6

Frequency spectrum of the particle velocity for particles of different masses and different sizes, compensated with the prediction for the Lagrangian spectrum in classical turbulence $\propto \epsilon /{\omega }^2$: (a) small particles with $a_{\mathrm{p}}=4\xi$; (b) large particles with $a_{\mathrm{p}}=10\xi$. The dash-dotted gray line is the frequency spectrum of a single small particle trapped in a straight vortex slightly perturbed. Dotted lines of corresponding colors are the prediction for the particle natural frequency ${\mathrm{\Omega }}_{\mathrm{p}}$. The dashed red line is the scaling due to vortex reconnection or Kelvin waves $\propto {\left|\omega \right|}^{-1}$. The dashed golden line is the spectrum evaluated at late times in the simulation ($6T_L<\tau <7T_L$).

Figure 7

(a) Probability density function of the single-component particle velocity, for different species of particles. The dotted golden line is the Eulerian velocity field $\mathbf{\nabla }\varphi$, corresponding to the simulation without particles at the time $1.4T_L$. The data for the particles are averaged in time between $t=1.2T_L$ and $1.6T_L$. Inset: Standard deviation of the particle velocity as a function of the particle mass. (b) The same as (a) but with the velocities normalized by the standard deviation ${\sigma }_v$. Dotted lines are Gaussian, dash-dotted line is a power-law scaling $|v_i{|}^{-3}$.

Figure 8

(a) Probability density functions of the single-component particle acceleration. (b) Probability density functions of the norm of the particle acceleration. The dotted line is a Gaussian, the dashed line is a ${\chi }_3$ distribution, and the dash-dotted line is an exponential tail $e^{-\left|\mathbf{a}\right|/{\sigma }_{\left|\mathbf{a}\right|}}$. Inset: Probability density functions of the natural logarithm of the norm of the particle acceleration. The dashed golden line is a log-normal distribution.

Figure 9

Acceleration two-point correlator, plotted vs time normalized by the dissipation timescale ${\tau }_{\ell }$ (a), and by the Magnus natural frequency $1/{\mathrm{\Omega }}_{\mathrm{p}}$. (b) Markers indicate the time of acceleration decorrelation $t_a$. Inset: $t_a$ normalized by $1/{\mathrm{\Omega }}_{\mathrm{p}}$ as a function of the particle relative mass.

Figure 10

Velocity spectra (a) and the acceleration spectra (b) for particles of size $a_{\mathrm{p}}=10\xi$ and mass $\mathcal{M}=0.13$, evolved using B-spline interpolation (blue lines) and spectral Fourier interpolation (green lines). The spectra are averaged over particles and over the times $1.3T_L<t<1.5T_L$.

Figure 11

Probability density function of the velocity filtered in the frequency window ${\omega }_c<\omega <{\omega }_{\mathrm{noise}}$ for different values of ${\omega }_c$. Data refer to particles of size $a_{\mathrm{p}}=10\xi$ and mass $\mathcal{M}=0.13$. The dotted line is a Gaussian distribution, and the dash-dotted line is a power-law scaling $0.002{\left(v_i-〈v_i〉\right)}^{-3}$. The data are averaged over particles and over the times $1.3T_L<t<1.5T_L$. Different PDFs are shifted for visualization. (a) Particle force interpolated with the B-spline method. (b) Particle force interpolated with the Fourier method.