Identifying a Superfluid Reynolds Number via Dynamical Similarity

The Reynolds number provides a characterization of the transition to turbulent flow, with wide application in classical fluid dynamics. Identifying such a parameter in superfluid systems is challenging due to their fundamentally inviscid nature. Performing a systematic study of superfluid cylinder wakes in two dimensions, we observe dynamical similarity of the frequency of vortex shedding by a cylindrical obstacle. The universality of the turbulent wake dynamics is revealed by expressing shedding frequencies in terms of an appropriately defined superfluid Reynolds number, ${\mathrm{Re}}_s$, that accounts for the breakdown of superfluid flow through quantum vortex shedding. For large obstacles, the dimensionless shedding frequency exhibits a universal form that is well-fitted by a classical empirical relation. In this regime the transition to turbulence occurs at ${\mathrm{Re}}_s\approx 0.7$, irrespective of obstacle width.

Figure 1

(Top) A quantum cylinder wake in the quasisteady state. Same-sign vortices aggregate into clusters to form a semiclassical vortex street. Vortices within the fringe region $|x|>w_x$ are unwound in pairs by imprinting opposite-signed vortices on top of them, thus recycling the flow to the uniformly translating state as indicated at the right of the domain. (Bottom) Time series data of the transverse force on the obstacle. The force exhibits a well-defined frequency, which determines the Strouhal number for the flow.

Figure 2

Strouhal number plotted as a function of the superfluid Reynolds number for obstacles of different diameters $D$. The dashed lines indicate the transition between regular and irregular wakes (see text). Solid lines in the left panel indicate regions of oblique dipole (OD) and charge-2 von Kármán ($\mathrm{K}2$) shedding. The solid gray line shows the best-fit curve $\mathrm{St}=0.1402[1-0.1126/({\mathrm{Re}}_s+0.2456)]$. Error bars give an indication of the uncertainty in St due to the Fourier-space resolution $\mathrm{\Delta }f=1/T$. Insets show the original shedding frequency data as a function of velocity. The data for $D/\xi =4$ are truncated as the shedding frequency becomes poorly defined at higher velocities for this particular obstacle. In the Supplemental Material [29], we provide movies showing condensate density and vortex-cluster dynamics in each shedding regime.

Figure 3

Cluster charge probability distribution $\mathcal{P}({\kappa }_c,{\mathrm{Re}}_s)$, which shows the probability that a vortex belongs to a cluster of charge magnitude ${\kappa }_c$. Note that ${\kappa }_c=0$ corresponds to a dipole and ${\kappa }_c=1$ corresponds to a free vortex. The vertical dashed line shows the value ${\mathrm{Re}}_s=0.7$ at which the probability distribution suddenly spreads, indicating that the wake has developed irregularities. The three different shedding regimes observed are labeled for the case $D/\xi =8$.