Dynamic similarity of vortex shedding in a superfluid flowing past a penetrable obstacle

TL;DR

This work demonstrates that wake dynamics in a two-dimensional superfluid flowing past a penetrable obstacle can be organized by a flow-defined length, the effective diameter $D_{ m eff}$, derived from the Mach-1 contour of the irrotational flow. By defining the superfluid Reynolds number $ ext{Re}_{ m s} = (v_0 - v_c) D_{ m eff} / (rac{ar{ abla}^2}{m})$, the authors reveal universal behavior in the wake: a dipole-to-cluster transition occurs near $ ext{Re}_{ m s} oughapprox 2$, and both the Strouhal number $ ext{St}$ and the drag coefficient $C_D$ collapse onto universal curves when plotted against $ ext{Re}_{ m s}$. The effective diameter tracks the lateral width of the vorticity distribution, validating the dynamically relevant length scale as the supersonic region rather than the geometric obstacle size. These results extend dynamic similarity concepts from classical fluids to quantum superfluids and suggest practical experimental pathways to verify the universality across obstacle types and strengths.

Abstract

We numerically investigate wake dynamics in a superfluid flowing past a penetrable obstacle. Unlike an impenetrable object, a penetrable obstacle does not fully deplete the density. We define an effective diameter D_eff from the Mach-1 contour of the time-averaged irrotational flow around the obstacle, which delineates the local supersonic region where quantized vortices nucleate. Using this flow-defined length scale, we construct a superfluid Reynolds number Re_s = (v0 minus vc) times D_eff divided by (hbar over m), where v0 is the flow speed, vc is the critical velocity, and m is the particle mass. We show that Re_s organizes the wake dynamics across obstacle sizes and strengths: the transition from dipole-row emission to alternating vortex cluster shedding occurs at Re_s around 2, and both the Strouhal number and the drag coefficient collapse onto universal curves when plotted as functions of Re_s. These results extend the concept of dynamic similarity in superfluid flows to penetrable obstacles and demonstrate that the dynamically relevant length scale is determined by the supersonic region rather than by the geometric obstacle size.

Figures (10)

• Figure 1: Vortex shedding in a Bose-Einstein condensate flowing past a penetrable obstacle. The condensate flows from right to left (black arrow) at a speed of $v_0$. The Gaussian obstacle has strength $V_0/\mu = 0.9$ and width $\sigma/\xi = 20$. (a)–(c) Superfluid vorticity distributions $\omega_\mathrm{s}(\mathbf{r})$ for $v_0/c_{s0}=0.25$ (a), $0.30$ (b), and $0.40$ (c), where $c_{s0}$ is the speed of sound in the unperturbed condensate. $\omega_\mathrm{s}(\mathbf{r})$ is normalized by $\omega_\mathrm{s0}=\rho_0/\tau$. As $v_0$ increases, shedding evolves from periodic vortex-dipole emission to same-signed vortex-cluster shedding. Dashed circles indicate the obstacle's $1/e^2$ radius, centered at $x=0$. (d)–(f) Time traces of drag force $F_{\mathrm{D}}$ (solid blue) and lift force $F_{\mathrm{L}}$ (dotted red) for the same parameters used in (a)–(c). The force is normalized by $F_{\mathrm{\mu}}=\mu/\xi$. With an increase in $v_0$, $F_\mathrm{D}$ grows and becomes irregular, while $F_\mathrm{L}$ increases in terms of its magnitude and oscillatory strength, reflecting the transition from dipole- to cluster-dominated shedding.
• Figure 2: Drag and lift responses for a penetrable obstacle with $V_0/\mu = 0.9$ and $\sigma/\xi = 20$. (a) Mean drag $\overline{F_{\mathrm{D}}}$ (blue circles) and the root-mean-square of the lift $(\overline{F_\mathrm{L}^2})^{1/2}$ (red squares) versus background flow speed $v_0/c_{s0}$. The critical velocity $v_c=0.18 c_\mathrm{s0}$ and the dipole-to-cluster transition velocity $v_\text{th}= 0.33 c_\mathrm{s0}$ are indicated by the black arrow and the grey dashed line, respectively. Left inset: log--log plot of $F_{\mathrm{D}}$ versus $(v_0 - v_c)$ with guide lines $(v_0-v_c)^{1/2}$ (dotted) and $(v_0-v_c)^2$ (dashed). Right inset: magnified view of $F_\mathrm{L}$ near the transition, showing a rapid rise for $v_0>v_\mathrm{th}$. (b) Drag and lift frequencies $f_\mathrm{D}$ and $f_\mathrm{L}$ as functions of $v_0/c_{s0}$. The frequencies are normalized by $1/\tau = \mu/\hbar$. Reliable determinations were obtained for $f_\mathrm{D}$ when $v_0 < v_{\mathrm{th}}$ and for $f_\mathrm{L}$ when $v_0 > v_{\mathrm{th}}$.
• Figure 3: Time-averaged flow fields around a penetrable obstacle for $V_0/\mu=0.9$, $\sigma/\xi=20$, and $v_0/c_{\mathrm{s0}}=0.4$. (a) Condensate density $\overline{\rho}(\mathbf{r})$; the dashed circle indicates the obstacle's $1/e^2$ radius. (b) Flow speed $|\overline{\bm{v}}(\mathbf{r})|$ and (c) irrotational flow speed $|\overline{\bm{v}_\mathrm{irr}}(\mathbf{r})|$. (d) Difference $|\overline{\bm{v}_\mathrm{irr}}(\mathbf{r})|- \overline{c_{\mathrm{s}}}(\mathbf{r})$, where $\overline{c_{\mathrm{s}}}(\mathbf{r})$ is the local speed of sound as determined by $\overline{\rho}(\mathbf{r})$. The black contour is the Mach-1 locus $|\overline{\bm{v}_\mathrm{irr}}(\mathbf{r})|=\overline{c_{\mathrm{s}}}(\mathbf{r})$.
• Figure 4: Time-averaged vorticity fields $\overline{\omega_\mathrm{s}}(\mathbf{r})$ for various flow speeds $v_0/c_{\rm s0}$; (a-c) $V_0/\mu=0.9$ and $\sigma/\xi=20$, and (d-f) $V_0/\mu=0.9$ and $\sigma/\xi=40$. The black solid contour denotes the Mach-1 locus $\overline{v_{\rm irr}}(\mathbf r)=\overline{c_{\rm s}}(\mathbf r)$, enclosing the locally supersonic region, and the gray dotted circle marks the obstacle's $1/e^2$ radius. The effective distance $D_{\mathrm{eff}}$ is defined as the maximum extent of the Mach-1 region along the $y$ direction. (g) Effective diameter $D_{\rm eff}/\xi$ versus vorticity width $w_{\rm v}/\xi$. The width $w_{\rm v}$ was determined from $\overline{\omega_\mathrm{s}}(\mathbf{r})$ (see text). Data points are shown for various values of $\sigma$ and $0.22\leq v_0/c_{\rm s0} \leq 0.65$ at $V_0/\mu=0.9$. Error bars are smaller than the symbols. The dashed line is a linear fit to the data.
• Figure 5: Dynamic similarity of the dipole-to-cluster transition. Threshold superfluid Reynolds number $\mathrm{Re}_{\rm s}^{\rm th}$ at $v_0=v_\textrm{th}$ for obstacle parameters $V_0/\mu \in \{0.7, 0.9\}$ and $\sigma/\xi \in \{15, 20, 25, 30\}$. Error bars indicate the uncertainties of $\mathrm{Re}_s^{\mathrm{th}}$, arising from the finite numerical resolution in determining $v_c$, $v_{\rm th}$ and $D_\textrm{eff}$. The black dotted line indicates the mean value of all data points.