Superflows around corners

Abstract

We investigate analytically and numerically the dynamics of a two-dimensional superflow governed by the Gross-Pitaevskii equation passing over finite-size rectangular obstacles: an impenetrable wall and an impenetrable rectangular well. Extending classical studies of vortex nucleation around smooth obstacles, we focus on the role of sharp corners in determining the onset of vortex nucleation. Using a combination of analytical techniques based on the Schwarz-Christoffel methods for potential flow and on numerical simulations, we show that local velocity amplification near sharp corners crucially controls the critical flow velocity for vortex nucleation. For both wall and well configurations, we identify analytically and theoretically the critical velocities as a function of the obstacle width and its height or depth, finding an excellent agreement between the theory and our numerical simulations. Our results provide a simple framework for understanding superflow stability past finite-size obstacles with sharp features and are directly relevant to experimentally realizable configurations in atomic Bose-Einstein condensates and related superfluid systems.

Figures (6)

• Figure 1: A wall of height $h$ and a well of depth $h$. Both domains have a width $2a$.
• Figure 2: Numerical simulation of the superflow passing around a wall (top) and a well (bottom) and displaying vortex nucleation. The upper and lower snapshots display the density field, $|\psi|^2$, for three increasing times. In both cases the numerical simulations are for $|\psi_0|^2=1$, $\xi_0=1$, the space and time discretization are $dx= 0.25$ and $dt =0.0025$, and the system size is $64\times 64$. The obstacle is in black, the blue corresponds to $|\psi|^2\approx 1$, and the yellow matches with $|\psi|^2\gtrsim 0.$ Top: The parameters are the Mach number $M=0.5,$ the obstacle width $2a= 7.5,$ and its height $h = 5$. Bottom: The parameters are $M=0.725 ,\, 2a= 7.5, \, h = 3.75$. Notice that in both cases, vortex are nucleated near the right open corners in the sense of the flow, as it can be seen in the snapshots (b) and (d). Subsequently, the vortices are advected in the direction of the flow, as it is shown in the snapshots (c) and (f).
• Figure 3: Conformal map $z=f(\zeta)$ for transforming the plane $\zeta$-plane into a wall in the $z$-plane. Here $f(0)= ih$, $f(\pm k_1)= \pm a+ ih$, and $f(\pm k_2)= \pm a$.
• Figure 4: Conformal map for transforming the plane $\zeta$-plane into a well in the $z$-plane. Here $f(0)= -i h$, $f(\pm k_1) =\pm a-i h$, $f(\pm k_2)= \pm a$.
• Figure 5: Critical Mach number as a function of the width $a/\xi_0$ of the wall (or well) obstacle. The numerical simulations are for $h/\xi_0 =10$, $|\psi_0|^2=1$, $\xi_0=1$, $dx= 0.5$ and $dt=1/100$. The dots correspond to the numerical simulations for the wall and the well for different system size: $L_x = 64$ (green: wall, magenta: well), $L_x = 128$ (red: wall, blue: well), $L_x = 256$ (orange: wall, purple: well). The continuous lines correspond to the theoretical calculations using Schwarz-Christoffel mapping for $h/\xi_0 =10$ and the adjustable parameter $\alpha \simeq 0.5$ for the wall (orange line), and the well (purple line). (See Table \ref{['tab:table1']}). (a) Plot in linear scale. One readily notices an already strong asymmetry for $a< h$. (b) Plot in log-log scale (in base 10). As a guide for the eye, we plot with a segmented gray line a slope $1/6$ for comparisons with the scaling laws given by Eq. \ref{['eq:McAsympWall']} for $h\gg a$, and we plot with a continuous gray line a slope $-1/3$ for comparisons with the scaling laws given by Eq. \ref{['eq:McAsympWell']} for $h\gg a$.