Vortex capture dictates efficiency in three-hydrofoil schools

Three-dimensional experiments on a school of three pitching hydrofoils reveal that optimal thrust and efficiency gains are achieved through compact formations where followers directly capture vortices from leaders, driven by specific vortex-body and body-to-body interactions rather than traditional wake drafting, though maintaining such high-performance arrangements likely requires active control due to cross-stream instability.

The Gist

Imagine you are watching a school of fish swimming together. You've probably heard the old saying that swimming in a group saves energy, kind of like how cyclists in a race huddle together to slip through the wind. Scientists have long believed that fish do this by positioning themselves in the "slipstream" or wake of the fish in front, where the water is moving slower, making it easier to push through.

But this new study suggests the real story is much more like a dance with invisible partners than just hiding in the wind.

Here is the breakdown of what the researchers found, using simple analogies:

1. The Setup: A Three-Foil "School"

The researchers didn't use real fish (which are hard to control in a lab). Instead, they built three robotic "tails" (hydrofoils) that flap up and down, mimicking how fish swim.

• Two Leaders: They swim side-by-side, flapping in sync (or sometimes out of sync).
• One Follower: This one moves around behind them, trying to find the "sweet spot" to swim most efficiently.

2. The Big Surprise: It's Not About "Drafting"

The Old Idea (Drafting): Think of a cyclist drafting behind a leader. The leader breaks the wind, creating a calm pocket of air behind them. The follower sits there, saving energy because the air is still.
The New Reality (Vortex Capture): The researchers found that for these swimming tails, the "calm pocket" of slow water actually makes things worse.

• The Analogy: Imagine trying to run on a treadmill that suddenly slows down. You have to work harder to keep your legs moving at the same speed. Similarly, when the follower swam in the slow water between the leaders, it actually lost efficiency.
• The Winner: The follower did best when it swam directly into the fast, swirling jets of water coming off the leaders' tails. It's like a surfer catching a wave. Instead of hiding from the wave, the follower rides the energy of the swirling water to get a free boost.

3. The Secret Sauce: Timing is Everything

The study found that catching that "free boost" isn't just about where you are, but when you move.

• The Analogy: Imagine trying to push a child on a swing. If you push when they are coming toward you, you stop them. If you push when they are moving away, you help them go higher.
• The Discovery: The follower needs to flap its tail at the exact right moment to "grab" the swirling water (vortex) coming from the leader. The researchers created a new rulebook for this timing. They realized that previous scientists had been guessing the speed of the water, but the water was actually moving faster than expected because the leaders were pushing it. By measuring the actual speed of the swirls, they could predict exactly where the follower should stand to get the maximum power boost.

4. The "Domino Effect" (Upstream Influence)

Usually, we think the leader affects the follower. But this study showed the follower can actually affect the leaders, too!

• The Analogy: Imagine a line of people passing a ball. If the person at the back suddenly jumps up and down, it changes the rhythm of the person in the middle, which changes the rhythm of the person at the front.
• The Finding: When the follower gets very close to the leaders, it changes the water flow so much that it actually helps the leaders swim with less effort. It's a two-way street where the whole group helps each other, not just the person in the back.

5. The Catch: Stability vs. Speed

Here is the tricky part. The "sweet spots" where the fish get the most free energy are also the most unstable.

• The Analogy: Think of balancing a broom on your hand. The best place to balance it is right in the middle, but it's also the most sensitive. A tiny breeze knocks it over.
• The Finding: The spots where the follower gets the biggest energy boost are also where the water pushes it sideways the hardest. To stay in that perfect, high-energy spot, the fish (or robot) has to constantly wiggle and adjust its fins to stay on course.
• The Dilemma: Fish have to choose: Do I swim in the "calm" zone where I don't have to work hard to stay straight, but I don't get much free energy? Or do I swim in the "chaos" zone where I get a huge speed boost, but I have to constantly fight to stay on track?

The Bottom Line

This study changes how we understand fish schools. They aren't just hiding in the slow water behind their friends. They are actively surfing the fast, swirling energy created by their friends.

However, to do this, they need perfect timing and they have to be willing to fight a bit of turbulence to stay in the group. It's a high-risk, high-reward strategy that turns a school of fish into a highly efficient, coordinated machine. This could help us design better underwater robots that swim in groups, saving battery life by "dancing" with the water currents rather than just fighting against them.

Technical Summary

1. Problem Statement

The study addresses the hydrodynamic mechanisms governing collective swimming (schooling) in three-dimensional (3D) bio-inspired systems. While previous research has extensively covered pairs of swimmers and theoretical models (such as Weihs' 1973 hypothesis), there is a lack of experimental data on schools larger than two swimmers at biologically relevant Reynolds numbers ($Re > 5,000$).

Specifically, the paper investigates:

• The validity of the drafting hypothesis (swimming in a region of reduced mean flow to reduce drag) versus vortex-body interactions (synchronizing with unsteady vortices to enhance thrust) in a 3D school.
• The interplay between body-body interactions (upstream effects of a follower on leaders) and wake breakdown in compact formations.
• The stability of high-performance formations, specifically whether the hydrodynamic forces naturally maintain the formation or require active control.

2. Methodology

The researchers conducted 3D experiments using a school of three pitching hydrofoils in a recirculating water channel.

• Experimental Setup:
  • Geometry: Three NACA 0012 hydrofoils (chord $c = 0.0488$ m, aspect ratio $A=3$).
  • Configuration: Two side-by-side "leaders" (Leader 1 and Leader 2) with a fixed cross-stream distance ($D^* = 1.1$). A third "follower" foil was traversed in a 2D grid downstream ($1.2 \le X^* \le 4.0$, $-1.6 \le Y^* \le 1.6$).
  • Kinematics: All foils pitched sinusoidally about the leading edge.
    • Leaders: Tested in both in-phase ($\phi=0$) and out-of-phase ($\phi=\pi$) synchronization.
    • Follower: Always in-phase with Leader 1.
  • Flow Conditions: Freestream velocity $U = 0.174$ m/s, Reynolds number $Re = 8,450$, Strouhal number $St = 0.29$.
• Measurement Techniques:
  • Force/Torque: Six-axis transducers measured thrust, lift, and power for each foil.
  • Flow Visualization: Multi-layer phase-averaged stereoscopic Particle Image Velocimetry (sPIV) was used to reconstruct the 3D3C velocity field. The Q-criterion was used to visualize vortex structures.
• Metrics: Performance was normalized against isolated foil baselines to calculate normalized thrust ($C_T^*$), power ($C_P^*$), and efficiency ($\eta^*$).

3. Key Contributions

1. Refutation of Drafting in Oscillatory Schools: The study provides experimental evidence that in 3D oscillatory swimmers, the traditional drafting mechanism (relying on reduced mean flow) does not improve performance. Instead, performance is maximized in regions of accelerated flow.
2. New Spatial Phase Definition: The authors propose a corrected definition for spatial phase ($\Phi$) that utilizes the measured wake wavelength ($\lambda$) rather than the estimated wavelength based on freestream speed ($U/f$). They found the actual wake wavelength is ~29% larger due to the jet-like accelerated flow generated by the thrust-producing foils.
3. Cascading Upstream Effects: The study identifies a "cascading" effect where a follower's position alters the wake of one leader, which subsequently affects the second leader, demonstrating complex upstream propagation of hydrodynamic forces in compact schools.
4. Stability vs. Performance Trade-off: The research maps the relationship between high-performance zones and hydrodynamic stability, revealing that the most efficient formations are inherently unstable and require active control.

4. Key Results

A. Flow Field and Wake Structure

• Wake Breakdown: Coherent vortex structures (reverse von Kármán streets) remained intact for at least three chord lengths downstream. Breakdown and merging of wakes from the two leaders occurred further downstream ($X^* > 3.8$ for in-phase, $X^* > 3.0$ for out-of-phase).
• Mean Flow: While a region of reduced mean flow (up to 17% slower than freestream) existed between the leaders (supporting Weihs' drafting theory), placing the follower there resulted in a thrust penalty.
• Accelerated Jets: The highest thrust regions were located directly behind the leaders where the flow was accelerated (up to 20% faster than freestream).

B. Follower Performance (Vortex Capture)

• Mechanism: The follower achieved maximum thrust (up to 190% higher than an isolated foil) and efficiency (up to 200% higher) when positioned directly in the wake of a leader.
• Vortex-Body Interaction: This enhancement is driven by the synchronization of the follower's pitching motion with the impinging vortices. When the wake vortex impinges on the follower at the correct phase, it increases the effective angle of attack and promotes the formation of a Leading Edge Vortex (LEV) that remains attached longer, boosting lift and thrust.
• Optimal Phase: The optimal spatial phase depends on the measured wake wavelength, not the theoretical freestream-based estimate.

C. Leader Performance (Body-Body Interaction)

• Upstream Influence: The presence of a follower within one chord length of a leader alters the leader's performance.
• Cascading Effect: In compact formations, the follower's proximity to Leader 1 alters Leader 1's circulation, which in turn affects the interaction between Leader 1 and Leader 2. This resulted in measurable changes (up to 10%) in the power consumption of the second leader, even when the follower was not directly behind it.

D. Collective Performance and Stability

• Collective Gains: The school achieved a collective thrust increase of 58% and efficiency increase of 24% compared to three isolated foils.
• Stability Dilemma:
  • High Performance = Low Stability: The regions of highest thrust and efficiency coincided with large lateral (lift) forces.
  • Control Requirement: An unconstrained follower in these high-performance zones would be pushed laterally away from the leaders. Maintaining these formations requires significant active control (e.g., adjusting angle of attack or using control surfaces) to counteract these lift forces.
  • Stable Zones: Regions with low lateral forces (near the centerline) offered little to no performance benefit.

5. Significance

This study fundamentally shifts the understanding of schooling hydrodynamics for 3D oscillatory swimmers:

• Paradigm Shift: It challenges the long-held belief that fish school primarily to draft in low-speed wake regions. Instead, it suggests that vortex capture is the dominant mechanism for energy savings and thrust augmentation in compact schools.
• Bio-inspired Design: The findings provide a blueprint for designing multi-agent underwater vehicles (AUVs). To maximize efficiency, swimmers should be positioned to capture vortices rather than avoid them, but this requires robust active control systems to maintain formation stability.
• Predictive Modeling: The new spatial phase model allows for the design of optimal schools of arbitrary size by progressively calculating wake wavelengths and synchronization, reducing the need for exhaustive $N \times M$ simulations.

In conclusion, the paper demonstrates that in compact 3D schools, vortex-body interactions drive efficiency, while drafting is negligible, and achieving these benefits comes at the cost of hydrodynamic instability that must be managed via active control.