Hydrodynamic loads and vortex evolution from a bio-inspired pectoral fin near a solid body

This study investigates the hydrodynamic loads and vortex evolution of a bio-inspired pectoral fin near a solid body in a water tunnel, revealing significant hysteresis in quasi-steady loads, complex vortex behaviors dependent on Strouhal and reduced frequencies, and establishing a data-driven scaling model where hydrodynamic forces are primarily governed by quadratic Strouhal number terms and their nonlinear interactions with reduced frequency.

The Gist

Imagine you are watching a fish swim. Most people focus on the tail, thinking that's the only thing that matters for moving forward. But fish also have side fins (pectoral fins) that they flap like wings. These fins are like the fish's multi-tool: they help it speed up, turn sharply, or even brake.

This paper is like a detective story where scientists try to figure out exactly how those side fins work when they flap next to the fish's body. They built a robot fish fin in a water tunnel to study the invisible forces and swirling water patterns (vortices) that happen when the fin moves.

Here is the breakdown of their findings, explained with some everyday analogies:

1. The Setup: A Flapping Fin in a Stream

The researchers took a flat, rigid fin and attached it to the side of a streamlined body (like a fish). They made it flap back and forth in the water, changing two main things:

• How fast it flapped (Frequency).
• How wide the flap was (Amplitude).

Think of this like a person standing in a river holding a paddle. If they just move the paddle slowly, the water pushes back gently. But if they whip the paddle back and forth quickly and widely, the water reacts violently, creating swirls and sudden pushes.

2. The "Memory" of the Water (Hysteresis)

One of the coolest discoveries was that water has a bit of a "memory."

• The Analogy: Imagine pushing a heavy door open. When you push it open, it feels one way. When you pull it back to close it, it doesn't feel exactly the same, even if you are at the same angle.
• The Finding: When the fin flaps away from the body, the water pushes on it differently than when it flaps back toward the body, even if the angle is identical. The water "remembers" where the fin came from. This creates a lag, or a "hysteresis," where the force depends on the direction of movement, not just the position.

3. The Swirling Dance (Vortices)

When the fin moves, it leaves behind trails of spinning water, called vortices.

• The Main Vortex: As the fin flaps up, it pulls a big, spinning bubble of water (a vortex) off its tip.
• The Orbiting Dancers: In faster, more energetic flapping, something magical happens. Smaller, weaker vortices start to form and orbit around that big main vortex, like moons orbiting a planet.
• The Jet Engine: When the fin snaps back down toward the body, it squeezes the water trapped between the fin and the body. This creates a tiny, high-speed "jet" of water shooting backward.
  • Why it matters: According to Newton's Third Law (for every action, there is an equal and opposite reaction), if the water shoots backward, the fin gets pushed forward. This is how the fin generates thrust (forward motion), acting like a mini-jet engine.

4. The "Suction" Effect

When the fin flaps away quickly, it creates a gap between the fin and the body. The water rushes in to fill that empty space.

• The Analogy: It's like when you pull a suction cup off a window quickly; you feel a strong pull.
• The Finding: This creates a suction force that pulls the fin toward the body. This is very strong when the fin moves fast and creates a lot of "lift" (a force pushing the fin sideways).

5. Cracking the Code with "Data Magic" (SINDy)

The scientists had a mountain of data: thousands of measurements of forces and water speeds. They knew the forces didn't just go up in a straight line as the fin moved faster; it was messy and non-linear.

To make sense of it, they used a computer algorithm called SINDy (Sparse Identification of Nonlinear Dynamics).

• The Analogy: Imagine you have a giant bag of Lego bricks (all the possible math formulas). You want to build a tower that perfectly matches a photo of a real building (the experimental data). SINDy is like a robot that tries different combinations of bricks, throwing away the ones that don't fit, until it finds the simplest set of bricks that builds the perfect tower.
• The Result: The robot found that the most important "bricks" were squares of the speed (Strouhal number) and combinations of speed and frequency. It turned out that the relationship wasn't simple; it was complex and curved, but the math could predict it surprisingly well.

6. Why Does This Matter?

This research isn't just about fish. It's about building better underwater robots.

• Engineers want to build robots that swim like fish because they are quiet, efficient, and agile.
• By understanding exactly how the side fins create thrust and how the water swirls around them, engineers can design robot fins that move more efficiently, saving battery life and moving more smoothly.

In a nutshell:
The paper shows that a fish's side fin is a complex machine that uses spinning water, suction, and tiny jets to move. It doesn't just push water; it dances with it. By using a smart computer algorithm, the scientists figured out the mathematical "dance steps" that describe these forces, which will help us build better underwater robots in the future.

Technical Summary

1. Problem Statement

While the hydrodynamics of fish swimming, particularly regarding the caudal (tail) fin, are well-characterized, the roles of pectoral (side) fins in propulsion and maneuvering remain less understood. Existing studies on pectoral fins often suffer from three limitations:

1. Complexity: Fully 3D and flexible fin geometries obscure fundamental flow mechanisms.
2. Isolation: Many studies analyze fin-induced flows without considering the interaction with the fish body.
3. Lack of Scaling: Specific geometry and motion studies often fail to generate generalizable scaling laws for forces across varying sizes, amplitudes, and frequencies.

This research addresses these gaps by experimentally characterizing the hydrodynamic loads and vortex evolution of an idealized, 2D rigid pectoral fin flapping near a solid body (representing a fish hull). The goal is to establish scaling laws for unsteady lift and drag forces and understand the underlying vortex dynamics.

2. Methodology

The study employed a combination of experimental fluid dynamics and data-driven modeling:

• Experimental Setup:

  • Facility: Free-surface water tunnel at Brown University.
  • Model: A streamlined 2D body (modified NACA 0015 airfoil) with a rigid acrylic fin (chord $c=40$ mm) mounted on one side. The fin pitches about its leading edge.
  • Conditions: Tests were conducted at a freestream velocity $U_\infty = 0.2$ m/s ($Re_c \approx 7,600$).
  • Motion: Sinusoidal flapping defined by $\theta(t) = A_\theta(1 - \cos 2\pi f t)$.
  • Parameters: 12 combinations of amplitude ($A_\theta = 7.5^\circ, 15^\circ, 30^\circ$) and frequency ($f = 0.25, 0.5, 1, 2$ Hz).
  • Dimensionless Numbers: Reduced frequency ($k = \pi f c / U_\infty$) ranged from 0.16 to 1.26; Strouhal number ($St = f A_{tip} / U_\infty$) ranged from 0.013 to 0.419.

• Measurements:

  • Forces: A six-axis force-torque transducer measured lift ($L$) and drag ($D$) at 1,000 Hz. Data were filtered and normalized to coefficients $C_L$ and $C_D$.
  • Flow Visualization: Time-resolved Particle Image Velocimetry (PIV) captured 2D velocity and vorticity fields, synchronized with force measurements.
  • Processing: Data were phase-averaged over multiple cycles (30–120 cycles depending on frequency).

• Data-Driven Scaling:

  • The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm was used to identify scaling relations between the fluctuation amplitudes of lift/drag and the control parameters ($k, St, A_\theta$).
  • A library of candidate terms (linear, quadratic, and nonlinear combinations) was constructed, and sparse regression was applied to select the most significant terms.

3. Key Results

A. Quasi-Steady Baseline

• Hysteresis: Significant hysteresis was observed in lift coefficients between the upstroke and downstroke phases, particularly at low angles.
• Mechanism: At low angles, the shear layer remains attached, creating a cambering effect that increases lift. As the angle increases beyond $\approx 25^\circ$, flow separation occurs, causing a sharp drop in lift. During the downstroke, the shear layer remains separated until the fin closes against the body.

B. Unsteady Flapping and Vortex Evolution

• Lift Fluctuations:
  • Mean lift generally shifts negative (pushing toward the body) as $k$ and $St$ increase.
  • At high $St$($0.42$) and high $k$ ($1.26$), a suction effect occurs during the upstroke, generating positive lift due to fluid continuity filling the space created as the fin moves away from the body.
• Vortex Dynamics:
  • Main Vortex: A primary clockwise-rotating vortex forms at the fin tip during the upstroke and sheds as the fin decelerates at the peak angle.
  • Secondary Vortices: During the downstroke, smaller counter-rotating vortices form at the fin tip. These orbit the main vortex. The number and strength of these secondary vortices increase with $St$.
  • Orbiting Behavior: In high $St$ cases, the secondary vortices exhibit distinct orbiting behavior around the main vortex.
• Drag and Thrust:
  • Drag fluctuations generally increase with frequency and amplitude.
  • Thrust Generation: At higher $St$($\ge 0.209$), negative drag (thrust) is generated during the downstroke.
  • Mechanism: Thrust is caused by a local jet formed between the fin and the body as the fin accelerates downward. This jet pushes the main vortex downstream, creating a reaction force in the forward direction.

C. Data-Driven Scaling

• SINDy Results: The algorithm identified that the fluctuation amplitudes are not linearly correlated with single parameters but are best described by quadratic terms of the Strouhal number ($St^2$) and nonlinear combinations with reduced frequency ($St^2 k$).
• Selected Terms:
  • Lift: Dominated by $St^2$, $A^* St$, and $St^2 k$.
  • Drag: Dominated entirely by nonlinear terms of $St$($St^2$, $A^* St$, $St^2 k$).
• Accuracy: The scaling models showed high accuracy, particularly at higher $k$ and $St$ where load variations are more monotonic and signal-to-noise ratios are higher.
• Phase Shifts: The phase lag between the fin angle and hydrodynamic loads is more strongly dependent on the reduced frequency ($k$) than on the Strouhal number ($St$).

4. Key Contributions

1. Isolation of Mechanisms: By using a 2D rigid fin-body configuration, the study isolates fundamental vortex-body interaction mechanisms that are often obscured in complex 3D flexible fish models.
2. Vortex-Force Correlation: Detailed PIV analysis explicitly links specific flow structures (main vortex shedding, secondary orbiting vortices, and corner jets) to specific force fluctuations (lift hysteresis and thrust generation).
3. Novel Scaling Law: The application of SINDy to this fluid dynamics problem successfully derived a compact, data-driven scaling law. It demonstrates that nonlinear combinations of $St$ and $k$ are superior to linear models for predicting unsteady loads on fin-body configurations.
4. Thrust Mechanism: Clarified the physical origin of thrust generation in pectoral fins at high $St$, attributing it to momentum transfer from a local jet formed during the downstroke.

5. Significance

This work provides a foundational understanding of pectoral fin hydrodynamics, which is critical for the design of bio-inspired underwater robots (AUVs) that rely on pectoral fins for maneuvering and propulsion. The derived scaling laws allow engineers to predict hydrodynamic loads across different sizes and operating frequencies without needing to perform full-scale experiments or complex CFD simulations for every new design. Furthermore, the success of the SINDy approach suggests a powerful pathway for modeling complex unsteady fluid-structure interactions where traditional theoretical derivations are difficult.