Intermittency enhancement in quantum turbulence in superfluid ${}^4\mathrm{He}$

Intermittency is a hallmark of turbulence, which exists not only in turbulent flows of classical viscous fluids but also in flows of quantum fluids such as superfluid ${}^4\mathrm{He}$. Despite the established similarity between turbulence in classical fluids and quasiclassical turbulence in superfluid ${}^4\mathrm{He}$, it has been predicted that intermittency in superfluid ${}^4\mathrm{He}$ is temperature dependent and enhanced for certain temperatures, which is in striking contrasts to the nearly flow-independent intermittency in classical turbulence. Experimental verification of this theoretical prediction is challenging since it requires well-controlled generation of quantum turbulence in ${}^4\mathrm{He}$ and flow measurement tools with high spatial and temporal resolution. Here we report an experimental study of quantum turbulence generated by towing a grid through a stationary sample of superfluid ${}^4\mathrm{He}$. The decaying turbulent quantum flow is probed by combining a recently developed ${{\text{He}}_2}^{*}$ molecular tracer-line tagging velocimetry technique and a traditional second-sound attenuation method. We observe quasiclassical decays of turbulent kinetic energy in the normal fluid and of vortex line density in the superfluid component. For several time instants during the decay, we calculate the transverse velocity structure functions. Their scaling exponents, deduced using the extended self-similarity hypothesis, display nonmonotonic temperature-dependent intermittency enhancement, in excellent agreement with a recent theoretical and numerical study [L. Biferale et al., Phys. Rev. Fluids 3, 024605 (2018)].

Figure 1

(a) Schematic diagram of the experimental setup. (b) Sample image of the ${{\text{He}}_2}^{*}$ molecular tracer line. The white dashed line serves to demonstrate the initial location of the trace line for velocity calculations.

Figure 2

(a) and (b) Ensemble-averaged velocity profile $v_y\left(x\right)$ across the channel at different decay times with grid velocities $v_g$ as indicated. (c) and (d) Corresponding velocity variance $\sigma \left(x\right)$ profiles. The shown data are obtained at 1.85 K, at the indicated time instants.

Figure 3

(a) Decaying turbulent kinetic energy of the normal fluid $K\left(t\right)$ and (b) vortex-line density $L\left(t\right)$ originating from towing the grid at 50 mm/s (open symbols and blue line) and 300 mm/s (closed symbols and orange line). The energy decay is shown for temperatures of 1.45 K ($⚫$ and $⚪$), 1.65 K ($\blacksquare$ and $▫$), 1.85 K ($▴$ and $▵$), 2.00 K ($▾$ and $▿$), and 2.15 K ($◆$). The red line corresponds to the early decay of $L\left(t\right)$ for times when the grid is still moving. The decays are quasiclassical in character. The early part of the decay, when the energy containing length scale ${\ell }_{\mathrm{e}}$ grows, displays the characteristic decay exponents $K\left(t\right)\propto t^{-6/5}$ and $L\left(t\right)\propto t^{-11/10}$, while the late universal part of the decay, when ${\ell }_{\mathrm{e}}$ is saturated by the channel size, obeys $K\left(t\right)\propto t^{-2}$ and $L\left(t\right)\propto t^{-3/2}$ [24, 25]. These decay rates are illustrated by thick black lines. For a towed grid velocity of 300 mm/s saturation occurs too early for the early part of the decay to be resolved. The shown data are obtained at 1.85 K.

Figure 4

(a) Calculated second-order transverse velocity structure functions $S_2^{\perp }\left(r\right)$ for $T=1.85$ K at decay times indicated in the legend. The grid velocity is $v_g=300$ mm/s. The gray solid lines represent power-law fits to the data in the range 0.2 mm $\le r\le$ 4 mm. (b) Scaling exponent ${\zeta }_2^{\perp }$ deduced from the power-law fits, such as that shown in (a), for temperatures shown in the legend. The dashed horizontal line shows the Kolmogorov-Obukhov scaling ${\zeta }_2^{\perp }=2/3$.

Figure 5

Third-order transverse velocity structure functions compensated by linear scaling $S_3^{\perp }\left(r\right)/r$ plotted versus the separation distance $r$. Data for 300-mm/s grid velocity and 4-s decay time are shown for temperatures and offsets as indicated.

Figure 6

(a) and (b) Extended self-similarity scaling of $S_2^{\perp }\left(S_3^{\perp }\right)$ for data obtained at $T=1.85$ K and several decay times for the two different grid velocities, as indicated. (c) and (d) Scaling exponent ${\zeta }_2^{\perp }$ extracted using the extended self-similarity hypothesis for the two different grid velocities at a range of temperatures. The dashed horizontal lines show the Kolmogorov-Obukhov scaling ${\zeta }_2^{\perp }=2/3$.

Figure 7

Extended self-similarity. Transverse velocity structure functions $S_n^{\perp }$ for $n=1\text{–}7$ are plotted versus $S_3^{\perp }$. The black lines are linear fits of $log{S}_n^{\perp }$ vs $log{S}_3^{\perp }$ to the data that fall within the shadowed region that corresponds to the shadowed region in Fig. 5 where $S_3^{\perp }/r$ appears to be flat. The particular case shown is for 1.85 K, 300-mm/s grid velocity, and 4-s decay time. Other cases appear qualitatively similar.

Figure 8

Intermittency corrections to the scaling exponents of the transverse structure functions deduced through extended self-similarity for data obtained at 4-s decay time and with grid velocity $v_g=300$ mm/s. The three-dimensional plot shows the temperature-dependent deviation of scaling exponents from Kolmogorov-Obukhov scaling: 1.45 K ($⚫$), 1.65 K ($\blacksquare$), 1.85 K ($▴$), 2.00 K ($◆$), and 2.15 K ($▾$).

Figure 9

(a) Probability distribution function of the velocity increments. The rug plot shows the actual data set used for the KDE. The data shown are for 1.45 K, 300-mm/s grid velocity, and 4-s decay time. (b) Calculation of the sixth moment of the velocity increment distribution. The curves are offset along the $y$ axis with an offset incrementing by 2. The data set is the same as in (a).

Figure 10

Number of velocity increment samples as a function of separation for 4-s data sets and all experimental temperatures.

Figure 11

Structure functions $S_n^{\perp }$ of orders 1–7 as a function $S_3^{\perp }$ at 1.85 K, 4-s decay time, and associated error bars calculated using the scheme described in the Appendix. These plots are analogs of the curves in Fig. 7.

Figure 12

Structure function scaling exponents calculated from PDF-derived structure functions. The error bars are just the standard errors of a total least-squares linear regression, also known as orthogonal distance regression. The inset shows scaling exponents of structure functions calculated using Eq. (3) (see the main text for how the error bars are calculated for the data shown in the inset). All data are for a 4-s decay time and 300-mm/s grid velocity. The temperatures are 1.45 K ($⚫$), 1.65 K ($\blacksquare$), 1.85 K ($▴$), 2.00 K ($◆$), and 2.15 K ($▾$). The black dotted line shows the prediction of the She-Leveque theory [6].