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

This paper demonstrates that dynamic similarity in superfluid flow past a penetrable obstacle is governed by a superfluid Reynolds number based on an effective diameter defined by the Mach-1 contour, rather than the obstacle's geometric size, which successfully unifies wake dynamics, vortex shedding transitions, and drag characteristics across varying obstacle parameters.

The Gist

Imagine a superfluid as a magical, frictionless river where the water is made of atoms that all march in perfect lockstep. Now, imagine you drop a rock into this river. In a normal river, the water flows around the rock, creating a messy wake of swirling eddies behind it. In this magical superfluid, the "eddies" are tiny, quantized whirlpools called vortices.

For a long time, scientists knew how these whirlpools behaved when the rock was solid and impenetrable (like a boulder). But what happens if the "rock" is actually a ghostly, semi-transparent barrier that the fluid can partially flow through? This is the puzzle this paper solves.

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: How do you measure a "Ghost Rock"?

In normal physics, if you want to predict how water flows around an object, you need to know its size. If the object is a solid cylinder, you just measure its diameter.

But in this experiment, the "obstacle" is a laser beam. It's not a solid wall; it's a gentle hill of energy. The superfluid atoms can climb over it or flow through it. Because the fluid penetrates the obstacle, the "size" of the obstacle isn't fixed. It's like trying to measure the size of a cloud; it changes depending on how hard the wind blows.

The researchers realized that simply measuring the laser beam's width didn't work. They needed a new way to define the "size" of the obstacle that made sense for the fluid.

2. The Solution: The "Speed Limit" Zone

The team discovered that vortices (the whirlpools) only appear when the fluid moves faster than the local "speed of sound" in that specific spot.

Think of it like a sonic boom. When a jet breaks the sound barrier, it creates a shockwave. In this superfluid, when the flow gets fast enough to break the local "sound barrier," the fluid gets unstable and spits out a vortex.

The researchers defined a new "effective size" for the obstacle. They didn't measure the laser beam itself; instead, they measured the size of the invisible zone around the obstacle where the fluid is moving fast enough to break the sound barrier.

• The Analogy: Imagine a lighthouse. You can't measure the "size" of the light beam easily. But you can measure the size of the area on the water where the light is so bright it burns your eyes. That "burn zone" is what matters for the fish swimming by. The researchers used this "burn zone" (the supersonic region) as the true size of the obstacle.

3. The Discovery: A Universal Rulebook

Once they used this new "effective size," something magical happened. They could organize all their messy data into a single, clean rulebook, just like classical physics does for normal water.

They found that the behavior of the wake depends on a single number (a "Superfluid Reynolds Number").

• Low Number (Slow Flow): The obstacle spits out pairs of vortices (a positive one and a negative one) in neat, rhythmic rows, like a marching band.
• High Number (Fast Flow): The rhythm breaks. The pairs get crowded, they crash into each other, and they reorganize into chaotic clusters of same-signed vortices.

The paper shows that this transition happens at the exact same "number" regardless of how big the laser beam was or how strong it was. Whether the obstacle was a tiny, weak ghost or a large, strong one, the fluid behaved the same way once you accounted for the "speed limit zone."

4. The Drag and The Rhythm

The researchers also looked at two other things:

• The Drag: How much the obstacle slows down the fluid. They found that if you plot the drag against their new "Superfluid Number," all the different obstacle sizes collapse onto a single, smooth curve.
• The Rhythm (Strouhal Number): How often the vortices are shed. Again, when using their new size measurement, the frequency of the shedding followed a universal pattern, just like the famous "von Kármán vortex street" seen in normal fluids (like smoke rings behind a chimney).

The Bottom Line

The paper claims that even though superfluids are weird quantum things, they still follow the ancient rules of "dynamic similarity" (the idea that small models can predict big flows) IF you measure the obstacle correctly.

You shouldn't measure the physical laser beam. You should measure the region where the fluid gets too fast to stay calm. Once you do that, the chaotic quantum world of superfluids behaves with the same predictable order as a river flowing around a rock.

In short: They found the right "ruler" to measure a ghost, proving that even quantum fluids play by the same universal rules as the water in your bathtub, provided you look at the right part of the flow.

Technical Summary

Technical Summary: Dynamic Similarity of Vortex Shedding in a Superfluid Flowing Past a Penetrable Obstacle

Problem Statement
Dynamic similarity, a cornerstone of classical fluid dynamics, posits that flows with different physical scales exhibit identical behavior when expressed via appropriate dimensionless parameters, most notably the Reynolds number ($Re$). While this universality has been extended to superfluids flowing past impenetrable obstacles (where the fluid density is fully depleted at the boundary), a quantitative framework for penetrable obstacles remains lacking. In penetrable scenarios, typical of atomic Bose-Einstein condensate (BEC) experiments where the obstacle potential is below the chemical potential, the fluid partially flows through the obstacle. This fundamentally alters the vortex nucleation process, preventing the definition of a characteristic geometric length scale (like a cylinder diameter) that is unambiguous and dynamically relevant. Consequently, it is unclear whether dynamic similarity holds for these systems and how to define the effective length scale governing wake dynamics.

Methodology
The authors investigate this problem by numerically solving the two-dimensional Gross-Pitaevskii equations (GPEs) for a BEC flowing past a Gaussian potential obstacle. The obstacle is modeled as penetrable ($V_0 < \mu$), ensuring no fully depleted density core forms.

• Simulation Setup: The system is simulated using a Fourier pseudo-spectral method with a fourth-order Runge-Kutta time integrator. A fringe region technique is employed to absorb density undulations at the domain boundaries, and a vortex-unwinding algorithm prevents vortex re-entry under periodic boundary conditions.
• Flow Characterization: The study analyzes the spatial distribution of superfluid vorticity ($\omega_s$) and the temporal evolution of drag ($F_D$) and lift ($F_L$) forces.
• Defining the Length Scale: To address the lack of a geometric scale, the authors define an effective diameter ($D_{eff}$). This is derived from the time-averaged irrotational flow component ($v_{irr}$) around the obstacle. Specifically, $D_{eff}$ is the transverse width of the region where the irrotational flow speed exceeds the local speed of sound ($|v_{irr}| > c_s$), delineated by the Mach-1 contour. This region represents the zone where compressibility-driven instabilities lead to quantized vortex nucleation.
• Dimensionless Parameters: Using $D_{eff}$, the authors construct a superfluid Reynolds number:
  $$Re_s \equiv \frac{(v_0 - v_c)D_{eff}}{\hbar/m}$$
  where $v_0$ is the flow speed, $v_c$ is the critical velocity, and $m$ is the particle mass. They also compute the Strouhal number ($St = f_s D_{eff}/v_0$) and a superfluid drag coefficient ($C_D$).

Key Results

1. Dipole-to-Cluster Transition: The wake dynamics exhibit a distinct transition as flow velocity increases. Near the critical velocity, the system emits periodic vortex dipoles. As velocity increases, this transitions to the shedding of same-sign vortex clusters. This transition is marked by a sudden increase in the root-mean-square lift force and a change in the drag force scaling from $(v_0 - v_c)^{1/2}$ to $(v_0 - v_c)^2$.
2. Universal Scaling with $Re_s$: When the transition velocity ($v_{th}$) is plotted against the proposed superfluid Reynolds number, the dipole-to-cluster transition consistently occurs at $Re_s \approx 2$, regardless of the obstacle's geometric size ($\sigma$) or strength ($V_0$).
3. Collapse of Dimensionless Quantities:
   • Strouhal Number: The Strouhal number ($St$) collapses onto a universal curve when plotted against $Re_s$. For $Re_s > 2$, $St$ saturates at a value ($St_0 \approx 0.20\text{--}0.30$) remarkably close to the classical value for turbulent vortex shedding.
   • Drag Coefficient: The drag coefficient ($C_D$) also collapses onto a single curve versus $Re_s$. In the low-$Re_s$ regime, $C_D$ follows a power law with an exponent $\alpha \approx 0.6\text{--}0.7$ (analogous to Stokes' law in 2D), while in the high-$Re_s$ regime, the exponent decreases to $\alpha \approx 0.22\text{--}0.27$.
4. Validity of $D_{eff}$: The effective diameter $D_{eff}$ is shown to be linearly proportional to the lateral width of the vorticity distribution ($w_v$), confirming that the supersonic region, rather than the geometric obstacle size, sets the relevant length scale for vortex shedding.

Significance and Claims
The paper claims to extend the notion of dynamic similarity in superfluid flows to the regime of penetrable obstacles. By defining the effective diameter based on the Mach-1 contour of the irrotational flow, the authors demonstrate that the dynamically relevant length scale is determined by the local supersonic region where vortices nucleate, not by the geometric dimensions of the obstacle.

The primary significance lies in the unification of wake dynamics across different obstacle sizes and strengths. The results show that once the flow-defined length scale is incorporated into the Reynolds number, the complex shedding behaviors (transition points, Strouhal numbers, and drag coefficients) exhibit universal behaviors analogous to classical fluids. This framework provides a quantitative tool to characterize the crossover from penetrable to impenetrable regimes and suggests that the Mach-1 contour serves as a physically motivated definition for the relevant obstacle length scale in superfluids. The authors note that this framework breaks down when the flow becomes globally supersonic ($v_0/c_s > 1$), where shock dynamics rather than vortex shedding dominate.