Modelling turbulent flow of superfluid $^4$He past a rough solid wall in the $T = 0$ limit

This paper presents a numerical study using the vortex filament model to demonstrate that superfluid $^4$He flowing past a rough wall at $T=0$ sustains a polarized ultraquantum turbulence state above a critical velocity, characterized by a parabolic velocity profile with wall slip and a friction force proportional to the flow speed.

The Gist

Imagine you have a river, but instead of water, it's made of superfluid helium. This isn't just cold water; it's a magical substance that flows with zero friction, like a ghost slipping through your fingers. Usually, if you push this fluid through a pipe, it slides perfectly smoothly. But if you push it too hard, it gets chaotic, creating tiny, invisible whirlpools called quantum vortices.

This paper is a computer simulation that asks a simple question: What happens when this ghostly fluid rushes past a wall that isn't perfectly smooth, but is actually covered in microscopic "speed bumps"?

Here is the story of what they found, explained without the heavy math.

1. The Setup: A Ghost and a Velcro Wall

The scientists built a virtual channel (a pipe) in a computer.

• The Fluid: Pure superfluid helium at absolute zero (the coldest temperature possible).
• The Wall: They didn't make the wall smooth. They made it "rough" on a tiny scale, like a wall covered in millions of microscopic Velcro hooks.
• The Vortices: These are tiny tornadoes of fluid. In a smooth pipe, they would just spin and float away. But here, the "Velcro" hooks catch the ends of these tornadoes, pinning them to the wall.

2. The "Walking" Dance

This is the coolest part of the discovery. When the fluid starts to flow, it tries to drag these pinned tornadoes along.

• Normally, a tornado stuck to a wall can't move.
• But because the fluid is moving, the tornado stretches.
• Eventually, the stretched tornado snaps and reconnects with its own "mirror image" (a trick of physics near a wall).
• The Result: The tornado breaks free from one Velcro hook and instantly gets caught by the next one a tiny bit further down the wall.

The scientists call this "walking." The vortex ends are literally hopping or walking along the rough wall, dragging the fluid with them. It's like a person trying to run on a floor covered in sticky tape; they can move, but they have to peel their feet off and stick them down again, step by step.

3. The Critical Speed (The "Tipping Point")

The researchers tested different speeds. They found a specific Critical Velocity (about 0.20 cm/s).

• Below this speed: The fluid is too lazy. The vortices get stuck, break free, and then immediately get stuck again without building up enough energy. The chaos dies out, and the flow stays smooth.
• Above this speed: The fluid is moving fast enough that the "walking" vortices create a tangled mess that sustains itself. It becomes a permanent storm of tiny tornadoes, even though the fluid is supposed to be frictionless.

4. The Surprising Friction

You might think, "If it's superfluid, there should be no friction." But because the vortices are constantly snagging and un-snagging from the rough wall, they create friction.

• The faster you push the fluid, the more the vortices "walk," and the harder the wall pushes back.
• The relationship is simple: Double the speed, double the friction.
• This friction acts like a brake, slowing the fluid down, just like air resistance slows a car.

5. The Flow Pattern: A Squashed Parabola

In a normal pipe, water flows fastest in the middle and slowest at the edges, forming a nice curve (like a hill).

• In this superfluid experiment, the flow looked similar—a hill shape.
• BUT, there was a catch: The fluid didn't stop completely at the wall. Because the vortices were "walking," the fluid at the very edge was still sliding a little bit. It was like a car skidding on ice; it wasn't stuck fast, but it wasn't moving at full speed either.

6. The "Ultra-Quantum" Turbulence

Usually, when we think of turbulence (like white water rapids), we think of big, swirling eddies.

• In this experiment, the turbulence was ultra-quantum. It wasn't big swirls; it was a chaotic mess of tiny, short-lived connections and reconnections.
• The "viscosity" (how thick or sticky the fluid feels) was incredibly low, but it was enough to create a specific type of organized chaos near the walls.

The Big Picture Takeaway

This paper shows us that even in a world of "perfect" frictionless fluids, roughness matters.

If you have a super-smooth wall, the fluid flows perfectly. But if the wall is rough, the fluid gets caught in a dance of "sticking and slipping." This creates a new kind of friction and a specific type of turbulence that is driven entirely by the microscopic texture of the wall.

In everyday terms: Imagine trying to slide a heavy box across a floor.

• Smooth floor: It glides forever.
• Rough floor with Velcro: You have to pull it, it gets stuck, you pull harder, it snaps free, gets stuck again. The box moves, but it feels heavy and jerky. That "jerky" feeling is what the scientists found in the superfluid helium.

This helps scientists understand how quantum fluids behave in real-world applications, like sensors or cooling systems for quantum computers, where surfaces are never perfectly smooth.

Technical Summary

1. Problem Statement

The study addresses the behavior of pure superfluid $^4$He flowing through a channel at absolute zero temperature ($T=0$). In this regime, the viscous normal fluid component vanishes, eliminating mutual friction. Consequently, the dissipation of flow momentum and the generation of friction forces depend entirely on the interaction between quantized vortex lines and the channel walls.

While experimental evidence suggests that vortex pinning to rough surfaces weakens at very low temperatures, the specific mechanism of how a vortex tangle sustains itself, generates friction, and transitions into a turbulent state in the presence of microscopically rough walls remains a complex theoretical challenge. The authors aim to model this system to understand:

• The critical velocity ($V_c$) required to sustain a turbulent vortex tangle.
• The nature of the friction force generated by pinned vortices.
• The structure of the velocity profile and the degree of vortex polarization in the absence of a normal fluid component.

2. Methodology

The authors employed the Vortex Filament Model (VFM) to simulate the dynamics of quantized vortices in a channel of width $D = 1$ mm.

• Physical Model:

  • Temperature: $T=0$ limit (no mutual friction with a normal fluid).
  • Boundary Conditions: The channel walls ($z=0$ and $z=D$) are modeled as microscopically rough. This implies that vortex lines terminating at the wall are permanently pinned.
  • Pinning/Unpinning Mechanism: A unique mechanism was implemented where vortices are liberated from pinned ends exclusively through self-reconnection with their image vortices. When a vortex point approaches the wall within a critical distance ($\delta/4$), it reconnects with its mirror image, effectively "walking" along the rough surface by jumping between protuberances spaced at the simulation resolution scale $\delta$.
  • Geometry: A cubic simulation cell ($D \times D \times D$) with periodic boundary conditions in the flow ($x$) and transverse ($y$) directions. The $z$-direction features solid boundaries.
  • Image Method: To satisfy the no-flow-through-wall condition, the method of images was used, creating 27 copies of the simulation volume (reflected and periodic) to calculate non-local velocity contributions.

• Numerical Implementation:

  • Dynamics: Vortex motion is governed by the Biot-Savart law, discretized into a local induction approximation (LIA) and a non-local integral term.
  • Reconnection: A Type-III reconnection algorithm is used when vortex segments approach within a critical distance ($\delta/2$).
  • Driving Force: A constant superfluid velocity $V$ is applied in the $x$-direction.
  • Resolution: Spatial resolution $\delta = 2 \times 10^{-3}$ cm; temporal resolution $dt = 8 \times 10^{-5}$ s.

• Analysis Metrics:

  • Friction Force: Calculated via two methods: (1) The Integral Method (rate of change of total impulse) and (2) The Tension Method (summing line tension components at pinned ends).
  • Velocity Profile: Coarse-grained mean velocity $\langle u_x(z) \rangle$ and vortex line density $\langle L(z) \rangle$.

3. Key Contributions

• Novel Pinning Model: The paper introduces a specific numerical framework where vortex "walking" on rough surfaces is driven by self-reconnection with image vortices. This bridges the gap between idealized smooth walls and macroscopic roughness.
• Characterization of $T=0$ Turbulence: It provides a comprehensive characterization of the "ultra-quantum" (Vinen) turbulence regime in channel flow, distinct from classical turbulence or finite-temperature quantum turbulence.
• Quantitative Friction Laws: The study establishes a linear relationship between friction force and flow velocity in the $T=0$ limit, driven by the density of pinned vortex ends.

4. Key Results

• Critical Velocity ($V_c$):

  • A sustained vortex tangle (lasting $\ge 80$ s) is only observed above a critical velocity $V_c \approx 0.20$ cm s$^{-1}$ (or $V_c D / \kappa \sim 20$).
  • Below $V_c$, vortices detach from the walls and the tangle decays.

• Velocity Profile and Slip:

  • The coarse-grained velocity profile is parabolic, resembling classical laminar Poiseuille flow.
  • However, there is a non-zero slip velocity at the walls ($\sim 0.20$ cm s$^{-1}$), which is comparable to $V_c$. This slip arises from the "walking" motion of pinned vortices.
  • The effective Reynolds number is very low ($Re' < 15$), confirming the flow is not quasi-classical turbulence but rather polarized ultra-quantum turbulence.

• Friction Force:

  • The friction force is proportional to the applied velocity ($F_x \propto V$).
  • The force per pinned vortex end is constant, implying the total friction scales with the number of pinned vortices.
  • The effective kinematic viscosity is found to be $\nu' \sim 0.1\kappa$, consistent with experimental observations of quantum turbulence.

• Vortex Structure and Polarization:

  • Line Density: The vortex line density $\langle L \rangle$ is highest in shear regions approximately $D/4$ from the walls and minimal in the channel center and at the walls.
  • Polarization: The vortex tangle is highly polarized in the shear regions near the walls (up to 60% of the vortex length is polarized), while the center of the channel remains unpolarized (0%).
  • Scaling: The total vortex line length scales approximately with the cube of the velocity ($\Lambda \propto V^3$).

5. Significance

This work is significant because it isolates and models the specific role of surface roughness and vortex pinning in driving dissipation in pure superfluids at $T=0$.

• It challenges the notion that $T=0$ superfluid flow is purely non-dissipative, demonstrating that rough walls can sustain a turbulent state and generate friction comparable to classical viscous flows, albeit through quantum mechanisms.
• The finding of a linear friction-velocity relationship and a parabolic velocity profile with slip provides a theoretical basis for interpreting experimental data in superfluid helium channels where normal fluid effects are negligible.
• The identification of polarized ultra-quantum turbulence driven by short-scale reconnections offers a new perspective on the energy cascade in quantum fluids, suggesting that Kelvin waves generated by wall interactions play a crucial role in energy dissipation.

In summary, the paper successfully models a regime of quantum turbulence where the interplay between vortex pinning, image reconnection, and wall roughness creates a stable, dissipative flow state that mimics classical laminar flow characteristics but is fundamentally quantum in origin.