First observation of the Josephson-Anderson relation in experiments on hydrodynamic drag

This paper reports the first experimental verification of the Josephson-Anderson relation in classical hydrodynamics, demonstrating excellent agreement between the predicted vorticity-flux-based drag force and measurements taken from a flat plate accelerated through water.

The Gist

Imagine you are pushing a heavy, flat paddle through a calm swimming pool. You know from experience that it's hard to get it moving, and once it's moving, it feels different than when you stop. Scientists have long tried to figure out exactly why it feels hard to push things through water.

For a long time, there was a confusing puzzle: if you push a perfect object through a perfect, frictionless fluid, math says you should feel zero resistance (this is called d'Alembert's paradox). But in the real world, with real water and real paddles, you definitely feel resistance.

This paper is about solving that puzzle by looking at the "invisible swirls" (vortices) in the water.

The Big Idea: Two Types of Drag

The researchers discovered that the total force you feel when pushing the paddle is actually made of two distinct parts, like a sandwich:

1. The "Inertia" Sandwich (Potential Flow): When you first start pushing the paddle, you aren't just moving the paddle; you are also dragging a chunk of water along with it. It's like trying to run while wearing a backpack full of water. This resistance is purely because you are accelerating mass. The scientists call this the Added Mass. It happens instantly when you start moving, even before any swirls form.
2. The "Swirl" Sandwich (Vortical Flow): As the paddle moves, it leaves behind a trail of spinning water (vortices), like the wake behind a boat. The paper introduces a special rule (the Josephson-Anderson relation) that says: The resistance you feel from these swirls is directly caused by the swirls crossing the invisible "lanes" of the water that would exist if the water were perfectly smooth.

The "Josephson-Anderson" Rule (The Magic Formula)

This rule was originally thought to only apply to super-cold, frictionless "superfluids" (like liquid helium). But this team proved it works for regular water too!

The Analogy:
Imagine the water has invisible, parallel train tracks (streamlines) laid out perfectly around your paddle.

• Scenario A: If the water flows perfectly along these tracks, you feel no extra drag from the swirls.
• Scenario B: As the paddle moves, it kicks up swirls. When these swirls "jump" across the train tracks, they create a tug-of-war. The more they jump across the tracks, the harder it is to push the paddle.

The paper's "magic formula" calculates the total drag just by counting how many swirls cross these invisible tracks. You don't need to know the pressure or how the water moves over time; you just need a snapshot of the water's speed and swirls at one moment.

The Experiment: The Robotic Paddle

To test this, the scientists built a giant robotic arm that held a flat metal plate and pushed it through a large tank of water.

• The Setup: They used a high-speed camera and a laser sheet to take thousands of pictures of tiny particles floating in the water. This let them see the water's speed and the swirls in real-time.
• The Motion: They accelerated the plate quickly to a specific speed, then let it coast.
• The Surprise: They found that the "Added Mass" (the inertia part) was huge at the very beginning. But as the water started to swirl, the "Swirl" part took over.

The "Aha!" Moment

The most exciting part of the paper is that even after the water became chaotic and full of swirls (long after the initial push), the math still worked perfectly.

Usually, scientists think that once the flow gets messy, the simple rules of "potential flow" (the smooth, invisible tracks) stop making sense. But this experiment showed that the smooth tracks still exist in the math, and the drag is still perfectly explained by how the messy swirls cross those tracks.

It's like saying: "Even though the traffic is a chaotic mess of cars swerving and honking, the total congestion on the highway is still perfectly predicted by how many cars are crossing the imaginary center lines of the lanes."

Why Does This Matter?

1. It's a New Tool: Previously, to calculate how much force a boat or submarine feels, you needed to know the pressure everywhere and how the flow changed over time. This new method says: "Just take a snapshot of the water's speed and swirls, and we can tell you the force instantly."
2. It Connects Worlds: It proves that a rule derived from quantum physics (superfluids) applies perfectly to everyday water.
3. Better Design: Engineers can use this to design better ships, underwater drones, and even wind turbines by understanding exactly how the "swirls" create drag.

In short: The paper proves that the resistance you feel pushing through water is a dance between the water's inertia (the weight you have to move) and the chaotic swirls you leave behind. And remarkably, there is a simple, exact mathematical rule that connects the swirls to the force, even in the messiest of waters.

Technical Summary

1. Problem Statement

The paper addresses the fundamental challenge of quantitatively linking the hydrodynamic drag force on an object moving through a fluid to the underlying velocity and vorticity fields.

• Theoretical Context: While D'Alembert's paradox states that an object in steady, inviscid potential flow experiences zero drag, real fluids exhibit drag due to viscosity and vorticity.
• The Josephson-Anderson (JA) Relation: Recently derived by G. L. Eyink (2021), this relation provides an exact, instantaneous link between drag and the flux of vorticity across the streamlines of the corresponding potential flow. It decomposes the total drag ($F$) into a potential component ($F_\phi$, related to added mass) and a vortical component ($F_\omega$).
• The Gap: While the JA relation is theoretically sound and derived from quantum superfluid descriptions, it had not been experimentally verified in classical fluids. Previous methods to infer force from velocity fields often required pressure measurements, time derivatives, or solving Poisson equations. The authors aim to verify if the JA relation can predict drag using only instantaneous snapshots of the velocity field in a classical, viscous fluid.

2. Methodology

The authors conducted a controlled experiment involving a flat plate accelerated through water, combining high-resolution flow visualization with robotic motion control.

• Experimental Setup:

  • Object: A rectangular flat plate ($300 \text{ mm} \times 60 \text{ mm} \times 6 \text{ mm}$) with an aspect ratio $l/h = 5$.
  • Actuation: A robotic arm accelerated the plate through a large water tank ($2 \text{ m} \times 2 \text{ m}$) from rest to a final velocity of $U = 0.3 \text{ m/s}$.
  • Scenarios: Two acceleration profiles were tested: $a_{max} = 0.8 \text{ m/s}^2$ and $a_{max} = 1.6 \text{ m/s}^2$.
  • Measurement:
    • Force: A high-precision force/torque sensor measured the hydrodynamic drag directly.
    • Velocity Field: Particle Image Velocimetry (PIV) was used at 1200 Hz to map the 2D planar velocity field. Fluorescent tracer particles were illuminated by a laser sheet.
    • Kinematics: The plate's position, velocity, and acceleration were tracked via optical methods, revealing an oscillatory component in the acceleration due to system stiffness.

• Data Processing & JA Relation Application:

  • Velocity Decomposition: The measured velocity field $\mathbf{u}$ was decomposed into a potential part $\mathbf{u}_\phi$ (satisfying the Laplace equation and no-penetration boundary conditions) and a rotational part $\mathbf{u}_\omega$.
  • Potential Flow Calculation: $\mathbf{u}_\phi$ streamlines were computed analytically using the Schwarz-Christoffel transform for the plate's rectangular cross-section.
  • Force Calculation: The vortical drag component $F_\omega$ was calculated using the JA relation (Eq. 1):
    $$F_\omega \cdot V(t) = \int dJ \int (\mathbf{u} \times \boldsymbol{\omega} - \nu \nabla \times \boldsymbol{\omega}) \cdot d\mathbf{l}$$
    where the integration is performed along potential flow streamlines ($d\mathbf{l}$) and weighted by the mass flux ($dJ$).
  • Total Force Prediction: The predicted total drag was the sum of the potential force (added mass force, $F_\phi = -m_a dU/dt$) and the calculated vortical force ($F_\omega$).

3. Key Contributions

• First Experimental Verification: This is the first experimental confirmation of the Josephson-Anderson relation in a classical, viscous fluid.
• Instantaneous Force Prediction: The study demonstrates that drag can be predicted instantaneously from a single snapshot of the velocity field without needing time derivatives or pressure data.
• Persistence of Added Mass: A novel finding is that the concept of "added mass" (a potential flow concept) remains a distinct and dominant component of the drag force even after the flow has fully developed and become highly vortical.
• Mechanism of Drag: The experiment visually and quantitatively confirms that drag arises specifically when vorticity (carried by the real flow) crosses the streamlines of the potential flow.

4. Results

• Agreement between Theory and Experiment: There was excellent agreement between the drag force measured by the sensor and the force predicted by the JA relation ($F = F_\phi + F_\omega$) across both acceleration scenarios.
• Temporal Evolution of Drag:
  • Initial Phase ($t < 0.4 \text{ s}$): The drag force rise is dominated entirely by the potential force ($F_\phi$) due to the plate's acceleration (added mass effect). The vortical contribution ($F_\omega$) is negligible until vortices detach.
  • Peak Drag: The maximum drag force occurs after the added mass force begins to decline. This peak is almost entirely due to the vortical contribution ($F_\omega$), driven by the shedding of vortex pairs from the plate edges.
  • Oscillations: The measured force oscillations correlated perfectly with the oscillatory component of the plate's acceleration, confirming that these modulations are communicated primarily through the potential force term.
• Vorticity Flux Visualization: The analysis of the integrand in the JA equation showed that positive drag contributions occur where vorticity moves outward across potential streamlines (e.g., near the plate edges where vortices shed).
• Uncertainty Analysis:
  • The computed force was sensitive to the resolution of streamlines near the plate edges.
  • Excluding the unresolved boundary layer (approx. $4 \times 10^{-4} \text{ m}$) from the integration introduced a small uncertainty, but the discrepancy between the model and experiment was larger, suggesting other sources of error (likely 3D effects emerging after $t > 0.8 \text{ s}$).

5. Significance

• Validation of Theory: The results validate Eyink's theoretical framework, proving that the quantum-derived Josephson-Anderson relation is universally applicable to classical hydrodynamics.
• New Paradigm for Force Estimation: The study establishes a robust method for inferring hydrodynamic forces on moving bodies using only velocity field data. This is crucial for applications where pressure sensors are impractical (e.g., biological swimming, underwater robotics).
• Physical Insight: It resolves the apparent paradox of how potential flow concepts (added mass) coexist with viscous, vortical flows. The decomposition shows that the "potential" and "vortical" forces are distinct physical mechanisms that sum to the total drag, regardless of the flow's development stage.
• Future Directions: The work highlights the need for 3D velocity field measurements to fully capture drag in complex flows, as the current 2D approximation breaks down when 3D vortex rings form.