On the stability of an in-line formation of hydrodynamically interacting flapping plates

This paper numerically investigates the stability of in-line flapping plates in an inviscid fluid, identifying quantized equilibrium schooling modes that can destabilize via downstream-propagating oscillations but are successfully stabilized through a simple relative-velocity-based control mechanism that also regularizes the wake vortex pattern.

The Gist

Imagine a group of fish swimming in a straight line, one right behind the other, like a train on a single track. Instead of using muscles to move forward, imagine these fish are just flapping their tails up and down in a rhythmic dance. This paper explores what happens when you have a whole line of these "flapping plates" (our fish stand-ins) trying to swim together in a perfect, straight line.

Here is the story of their journey, told in simple terms:

The Setup: A Dance of Plates

The researchers created a computer simulation of 2 to 4 flat plates in a fluid (like water). They didn't just let them drift; they forced each plate to flap its tail up and down at a specific rhythm. As they flap, they push against the water, which creates a forward thrust, kind of like how a propeller works. However, the water also pushes back (drag), slowing them down.

The goal was to see if these plates could naturally fall into a "schooling mode"—a state where they all swim at the same steady speed and keep a perfect, constant distance from the plate in front of them, just like a well-organized school of fish.

The Discovery: The "Goldilocks" Spacing

The plates did find a way to swim together, but with a very specific rule: The distance between them had to be a multiple of their "flapping wavelength."

Think of it like this: If a plate flaps its tail and moves forward a certain distance in one full cycle of the flap, the next plate needs to be exactly that distance (or twice that distance, or three times) behind the first one to stay in sync. It's like a line of dancers; if the person in front takes a step of a specific length, the person behind them must wait exactly that long before stepping, or they will trip over each other.

The researchers found that the plates naturally settled into these "quantized" distances. If you started them too close or too far apart, they would wiggle and adjust until they found one of these perfect spots.

The Problem: The "Domino Effect" of Instability

Here is where things get tricky. The system is very fragile.

1. Too many plates: When the researchers added more plates to the line (going from 2 to 3 or 4), the system became unstable.
2. Too little flapping: When they made the plates flap with smaller, weaker movements, the system also became unstable.

What happened was a "domino effect." The first plate (the leader) would flap and create a wake (a trail of swirling water). The second plate would try to ride that wake. But because the system was unstable, the second plate would start to speed up and slow down erratically. This erratic motion would then mess up the wake for the third plate, causing it to oscillate even more wildly.

By the time the instability reached the last plate in the line, it was swinging so violently that it would crash into the plate in front of it. The researchers call this a "flow-induced instability." It's like a line of people trying to walk in a straight line while holding hands; if the person in front stumbles, the person behind them stumbles harder, and the person at the back falls over completely.

The Solution: A Simple "Self-Correction" Mechanism

The researchers asked: "Can we teach these plates to stay in line without crashing?"

They programmed a simple rule for the plates: "If you are getting too close to the person in front of you, flap less. If you are falling too far behind, flap harder."

It's like a car using cruise control that automatically adjusts its speed based on the car ahead.

• Without this rule: The plates would eventually crash into each other.
• With this rule: The plates quickly settled into a smooth, steady rhythm. They maintained their perfect distances, and the chaotic, crashing motion disappeared.

The Beautiful Result: Organized Vortex Patterns

When the plates were allowed to crash (unstable), the water behind them was a messy, chaotic soup of swirls. But when the researchers used the simple "self-correction" rule, the water behind the plates formed a stunning, organized pattern.

Imagine the wake of the plates as a trail of smoke. Without the rule, the smoke is a messy cloud. With the rule, the smoke forms perfect, repeating geometric shapes (like a chain of diamonds or loops) that stretch out behind the plates. The simple act of the plates adjusting their flapping created a beautiful, orderly structure in the water.

The "Why": A Simple Explanation

To understand why this happens, the researchers used a simplified math model (like a rough sketch compared to a detailed painting). This model showed that:

• More plates = More chaos: Each new plate adds a layer of complexity that amplifies small errors, making the line harder to keep stable.
• Stronger flapping = More stability: When the plates flap harder, they generate more power, which helps them resist the wobbly forces trying to knock them out of line.

Summary

In short, this paper shows that while nature (or physics) allows flapping objects to naturally fall into a line, that line is very easily broken if the group gets too big or the movement gets too weak. However, a very simple rule—where each object just pays attention to the one in front of it and adjusts its effort accordingly—is enough to keep the whole group stable, organized, and moving smoothly together.

Technical Summary

Technical Summary: On the stability of an in-line formation of hydrodynamically interacting flapping plates

Problem Statement
This study investigates the hydrodynamic interactions within in-line formations of flapping plates in an inviscid, incompressible fluid. Motivated by the Lighthill conjecture, which suggests that orderly animal schooling formations may arise passively from hydrodynamic interactions rather than active control, the paper examines whether stable "schooling modes" exist for multiple plates undergoing prescribed vertical heaving motion. Specifically, the research addresses the stability of these modes as the number of plates increases and the flapping amplitude decreases, phenomena recently observed in experiments on flapping wings in water tanks (Newbolt et al. 2024) where large chains destabilize via oscillations leading to collisions.

Methodology
The authors employ a vortex sheet model to simulate the dynamics of $n$ flat plates ($n=1, 2, 3, 4$) in a two-dimensional flow.

• Fluid Model: The fluid is treated as inviscid and irrotational except for the wake. The plates are represented as bound vortex sheets, and wakes are modeled as free vortex sheets shed from the trailing edges. The Euler equations are solved directly, with viscosity accounted for only to model skin friction drag and to allow vorticity separation.
• Forces: Horizontal thrust is generated by leading-edge suction, calculated from the bound vortex sheet strength. Drag is modeled using a Blasius boundary layer law proportional to $U^{3/2}$.
• Numerical Implementation: The system is solved using a fourth-order Runge-Kutta method. A key technical feature is the careful evaluation of near-singular integrals when a vortex sheet is near a body, utilizing a corrected trapezoid rule to prevent unphysical crossing of vortex sheets through the plates.
• Control Mechanism: To address observed instabilities, a simple feedback control algorithm is implemented where each plate modulates its flapping amplitude based on its velocity relative to the plate directly ahead.
• Reduced Model: A theoretical reduced model based on linear thin-airfoil theory (Wu 1961) is developed to rationalize the simulation results, assuming nearest-neighbor interactions and small-amplitude heaving.

Key Results

1. Steady State and Quantization: For a single plate, the cycle-averaged horizontal velocity reaches a steady state dependent solely on the flapping amplitude $A$, independent of initial velocity. For multiple plates, the system evolves toward equilibrium "schooling modes" where plates translate at a steady velocity with constant separation distances. These separation distances are found to be quantized on the flapping wavelength $\lambda = U/f$, corresponding to integer values of the schooling number $S = d/\lambda$.
2. Instability Mechanisms: As the number of plates ($n$) increases or the flapping amplitude ($A$) decreases, the schooling modes become increasingly unstable. The instability manifests as oscillations in the horizontal separation distances that propagate downstream from the leader. These oscillations grow in amplitude, eventually causing trailing plates to collide with their upstream neighbors. This behavior mirrors the "flonon" instability observed in recent experiments.
3. Stabilization: The proposed control mechanism, where plates adjust their flapping amplitude based on relative velocity to the leader, successfully stabilizes the formation. This control eliminates or significantly reduces the downstream-propagating oscillations, allowing the system to reach steady schooling states even for $n=4$ and low amplitudes ($A=0.1$).
4. Wake Structure: Stabilization has a profound effect on the wake topology. Without stabilization, the wake vorticity is disorganized. With stabilization, the wake forms highly regular, organized patterns (e.g., quadrupoles for $n=2$, sextupoles for $n=3$) that grow in time and space.
5. Reduced Model Validation: The linear reduced model successfully predicts the quantization of equilibrium distances and qualitatively explains the loss of stability. Linear stability analysis reveals that while perturbations decay in time for a single pair, transient growth combined with nonlinear effects leads to amplification of oscillations downstream as $n$ increases, consistent with the simulation results.

Significance and Claims
The paper claims to provide a numerical demonstration that hydrodynamic interactions alone can bind plates into stable schooling modes, but that these modes are inherently susceptible to flow-induced instabilities in larger collectives or at lower amplitudes. The study suggests that while passive hydrodynamic binding is possible, active control mechanisms may be necessary to maintain stability in larger groups, a finding relevant to understanding biological schooling and designing biomimetic underwater vehicles. The authors emphasize that their simple control strategy is sufficient to mitigate these instabilities, offering a potential explanation for how biological collectives might maintain formation without complex decision-making. The work also highlights the importance of accurate numerical treatment of near-singular integrals in vortex sheet simulations to capture the correct thrust and stability characteristics.