Dynamical Characteristics of the Body-Caudal Fin Joint of a Carangiform Swimmer and its Influence on Hydrodynamics

This study demonstrates that a computational model of a carangiform swimmer with a passively pitching caudal fin, regulated by a nonlinear torsional spring, can synchronize with body undulations to generate efficient thrust-producing vortices, offering a biologically inspired strategy for optimizing underwater robotic design through passive kinematics.

The Gist

Imagine a fish swimming through water not just by wiggling its body, but by using a clever, self-correcting tail that acts like a spring-loaded door. This paper explores how a specific type of fish, the Jackfish, uses the mechanics of its tail joint to swim efficiently, and how engineers can copy this trick to build better underwater robots.

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: The "Rigid" vs. The "Springy" Tail

Most underwater robots are built like rigid machines: a motor forces the tail to move back and forth in a perfect, pre-programmed rhythm. It's like a metronome that never misses a beat.

Nature, however, is smarter. A real fish's tail isn't just a stiff paddle; it's attached to the body by a joint (called the peduncle) that acts like a springy hinge. This joint has a special property: it's loose and easy to move when the tail is in the middle of its swing, but it gets stiffer and snaps back harder when the tail reaches the very end of its swing.

The researchers wanted to know: Can we build a robot tail that uses this "springy" trick to move on its own, without needing a motor to force every twist?

2. The Experiment: The "Passive" Tail

The team built a computer simulation of a Jackfish.

• The Body: The main body of the fish wiggles back and forth (like a snake) in a specific rhythm.
• The Tail: The tail is attached to the body with a "virtual joint." This joint has two parts:
  1. A Spring: It tries to pull the tail back to the center.
  2. A Damper: It acts like a shock absorber to stop the tail from wobbling too wildly.
  3. The Secret Sauce: The spring isn't just a normal spring. It's a nonlinear spring. Think of it like a rubber band that is easy to stretch a little bit, but gets incredibly hard to stretch once you pull it far. This mimics the real muscle and tendon in a fish's tail.

They let the water push the tail around. The tail had to "pitch" (tilt up and down) on its own, reacting only to the water pressure and the spring's pull.

3. The Discovery: Finding the "Sweet Spot"

The researchers tested many different settings for the spring and the shock absorber. They found that if you tune them just right, something magical happens: The tail locks into sync with the body.

• The Good Scenario (Synchronized): When the spring and shock absorber are tuned correctly, the tail naturally falls into the perfect rhythm. It tilts at the exact right moment to catch the water.

  • The Analogy: Imagine a child on a swing. If you push at the exact right moment, the swing goes higher and higher with very little effort. The tail does this with the water. It creates a tight, focused stream of water shooting backward, which pushes the fish forward with great speed and efficiency.
  • The Physics: The water forms neat, organized swirls (called "hairpin" and "ring" vortices) that act like a jet engine, boosting the fish forward.

• The Bad Scenario (Out of Sync): If the spring is too loose or the shock absorber is too weak, the tail gets out of rhythm. It flails a bit too early or too late.

  • The Analogy: This is like trying to push a swing when it's coming back toward you. You fight against the motion.
  • The Physics: Instead of a tight jet, the water swirls get messy and spread out sideways. The fish ends up fighting the water (drag) rather than using it for speed. It's like running through a crowd that is pushing you back.

4. The "Recoil" Effect

One of the coolest findings was how the nonlinear spring works.

• When the tail is in the middle of its swing, the spring is soft, allowing the tail to swing wide and fast.
• When the tail reaches the extreme edge of its swing, the spring suddenly gets very stiff. It acts like a rubber band snapping back, forcing the tail to reverse direction quickly.
• This "recoil" is what keeps the tail from spinning out of control and helps it snap back into the perfect rhythm for the next stroke.

5. What This Means for Robots

The paper concludes that you don't need a complex, expensive motor to control every tiny movement of a robot fish's tail. Instead, you can build a tail with the right "springy" joint.

If you get the physics of that joint right, the water itself will help the tail move perfectly. The tail will naturally find the rhythm, create those efficient "jet" swirls, and push the robot forward. It turns the robot from a rigid machine into something that flows with the water, just like a real fish.

In short: By giving a robot tail a "smart spring" that gets stiffer at the edges, the tail learns to dance with the water on its own, creating a powerful push without needing a computer to micromanage every move.

Technical Summary

Technical Summary: Dynamical Characteristics of the Body-Caudal Fin Joint of a Carangiform Swimmer and its Influence on Hydrodynamics

Problem Statement
The hydrodynamic efficiency of carangiform swimmers (e.g., Jackfish) relies heavily on the coordinated interaction between body undulation and caudal fin flapping. While previous research has established the importance of the peduncle (the joint connecting the body to the tail) and the role of passive structures in fish locomotion, the specific fluid-structure interaction (FSI) mechanisms governing a three-dimensional swimmer with a passively pitching caudal fin remain insufficiently understood. Existing studies often focus on two-dimensional models or fail to incorporate the swimmer's body, leaving a gap in understanding how nonlinear peduncle mechanics regulate amplitude, phase, and recoil in a realistic 3D environment. Furthermore, the underlying physical mechanisms linking passive joint dynamics to vortex generation and thrust production require further elucidation.

Methodology
The authors developed a high-fidelity computational framework using OpenFOAM v2312 to simulate unsteady, incompressible Navier-Stokes flows around a 3D Jackfish-inspired swimmer at a Reynolds number ($Re$) of 3000.

• Geometry and Kinematics: The model consists of an undulating trunk and an independently mounted caudal fin connected by a modeled joint. The trunk follows a prescribed carangiform undulation profile. The caudal fin undergoes prescribed heaving (to match the body's undulation) and flow-induced pitching.
• Passive Joint Modeling: The joint is modeled using a torsional spring and a viscous damper. Crucially, the spring incorporates both linear and cubic nonlinear stiffness terms ($K\theta(A\theta_p - B\theta_p^3)$). This formulation mimics biological peduncles, where stiffness increases at larger deflections to provide a stabilizing "recoil" effect, allowing large amplitudes near the neutral position while limiting excessive rotation at extremes.
• Simulation Parameters: The study conducted 45 three-dimensional simulations, varying the damping ratio ($\zeta$) and a synchronizing parameter ($n$) to tune the passive pitching response. A reference case with an actively pitching tail was also simulated to compare hydrodynamic performance.
• Validation: The computational setup was verified through a grid-convergence study (comparing coarse, medium, and fine grids) and validated against previous thrust coefficient data from Khalid et al. [2].

Key Contributions

1. 3D Passive Kinematics: The study extends previous 2D concepts to a realistic 3D carangiform swimmer, explicitly modeling the body-tail joint with nonlinear stiffness.
2. Synchronization Mechanism: It identifies that tuning the damping ratio ($\zeta$) and stiffness parameters allows the passive fin to synchronize with the body's undulation, producing pitching kinematics that closely resemble active pitching without direct actuation.
3. Vortex Dynamics Analysis: The research provides a detailed analysis of how passive pitching influences vortex shedding, specifically the formation of hairpin and ring vortices, and correlates these structures with thrust and drag.

Results

• Synchronization and Stability: The system reaches a steady-state oscillation rapidly (within ~3 cycles) when parameters are tuned correctly. A specific range of damping ratios ($\zeta \approx 0.367$) and tuning parameters ($n \approx 50$) yielded "Mode 5" synchronization, where the passive pitching frequency and phase closely matched the active reference case.
• Hydrodynamic Performance:
  • Synchronized Case (Case 5): When the passive fin is synchronized with the body, it generates coherent hairpin and ring vortices. These structures exhibit inward rotation that reinforces streamwise momentum, resulting in a thrust-dominant wake with a narrow wake-spread angle ($\theta_s \approx 10^\circ$). The thrust performance is comparable to, and in some aspects superior to, the actively pitching configuration.
  • Asynchronized Cases: Deviations from the optimal tuning led to phase mismatches.
    • Phase Lag Synchronization (PLS): Resulted in increased wake spread ($\theta_s \approx 16^\circ$) and reduced thrust efficiency.
    • Phase Advanced Synchronization (PAS): Caused pronounced lateral vortex spreading and a bifurcated wake with a large jet inclination angle ($\approx 54^\circ$). This configuration exhibited drag-dominated behavior and significantly reduced thrust compared to the synchronized case.
• Nonlinear Stiffness Role: The cubic nonlinear stiffness term was found to be critical for the "recoil" mechanism, enabling the fin to snap back from peak amplitudes, which facilitates the formation of the beneficial ring vortices observed in the synchronized case.

Significance and Claims
The paper claims that nonlinear peduncle mechanics serve as a natural regulator for amplitude, phase, and recoil in carangiform swimmers. The primary significance of this work lies in demonstrating that passive kinematics, driven by fluid-structure interaction and nonlinear stiffness, can achieve hydrodynamic performance comparable to active actuation. The study establishes that appropriately tuned passive joints can generate coherent vortex structures (hairpin and ring vortices) that enhance thrust, whereas phase mismatches lead to drag-dominated, laterally spread wakes. These findings offer a biologically inspired pathway for designing next-generation underwater robotic platforms that utilize passive kinematics for efficient propulsion, reducing the need for complex active control systems at the tail. The authors emphasize that their work clarifies the physical mechanisms governing these interactions, which were previously unresolved in the context of 3D bio-inspired swimmers.