Undulatory underwater swimming: Linking vortex dynamics, thrust, and wake structure with a biorobotic fish

This study experimentally investigates the wake dynamics of a biorobotic fish using Particle Image Velocimetry to demonstrate how the Strouhal number governs the relationship between vortex ring characteristics, wake structure, and thrust production, ultimately establishing a universal model for undulatory underwater swimming.

The Gist

Imagine a robotic fish swimming in a giant, clear water tunnel. It's not actually moving forward; instead, it's tied down by its head while the water rushes past it. Its tail wiggles back and forth, just like a real fish. The scientists wanted to understand the invisible "footprints" this tail leaves behind in the water and how those footprints relate to the fish's ability to push itself forward (thrust).

Here is the story of what they found, explained simply:

1. The Invisible Dance of Water

When the fish's tail wiggles, it doesn't just push water back; it spins it into little tornadoes called vortices. Think of these like the swirling rings of smoke you might see from a magician's hat, but made of water.

• The Low-Speed Wiggle: When the tail moves slowly, these water tornadoes line up in a zig-zag pattern, similar to the wake behind a boat moving slowly. This creates a "drag" effect, slowing things down.
• The Fast Wiggle: As the tail wiggles faster and harder, the pattern changes. The water tornadoes start pairing up and shooting out diagonally, forming a V-shape. This is the "thrust" mode, where the fish is effectively pushing itself forward.

The scientists discovered that the key to predicting which pattern appears isn't just about how fast the tail moves, but a specific ratio called the Strouhal number. You can think of this number as a "wiggle recipe" that combines how wide the tail swings, how fast it wiggles, and how fast the water is flowing.

2. The Speed of the Swirls vs. The Speed of the Jet

The researchers used high-speed cameras and lasers to take snapshots of the water's speed. They found a fascinating connection between the speed of the water tornadoes and the speed of the "jet" of water they create.

• The Analogy: Imagine the water tornadoes are like runners on a track. The "jet" is the crowd cheering them on. The scientists found that the speed of the crowd's cheer (the jet) matches the speed of the runners (the vortices) almost perfectly.
• The Discovery: By measuring how fast these water tornadoes move, they could calculate exactly how much "push" (thrust) the fish is generating. If the water tornadoes move faster than the water flowing past the fish, the fish is generating thrust. If they move slower, the fish is being dragged.

3. A Simple Geometric Rule

The most exciting part of the paper is that the scientists found a simple geometric rule that explains the shape of the wake.

• The Metaphor: Imagine the water tornadoes are like cars driving on a road. The road itself is moving forward (the free-stream water speed), but the cars also have their own engine pushing them sideways (the self-propelled speed of the vortex).
• The Result: The angle at which the V-shaped wake opens up is determined by how fast the "road" is moving versus how fast the "cars" are driving sideways. The scientists built a simple math model based on this idea, and it worked perfectly. It predicted the angle of the wake for their robotic fish, and it even matched data from other studies on real fish and different robotic swimmers.

4. Why This Matters (According to the Paper)

The paper concludes that this "wiggle recipe" (the Strouhal number) is a universal rule. Whether it's a robotic fish, a real fish, or a flapping wing, the way the water swirls and the angle of the wake depend almost entirely on this number.

The authors suggest this helps us understand how fish interact with each other. If a fish is swimming behind another, it is swimming through these invisible V-shaped water tunnels. Knowing the angle and speed of these tunnels helps explain how fish might "surf" on the wake of their friends to swim more efficiently, or how they might avoid the "drag" of swimming in the wrong spot.

In short: The paper shows that by watching how water swirls behind a wiggling tail, we can predict exactly how much push the tail is making, using a simple rule based on the speed and angle of those water swirls.

Technical Summary

Problem Statement
Flapping-based propulsive systems, such as those used by undulatory underwater swimmers (fish, cetaceans), rely on fluid-structure interactions to generate thrust. While the dynamics of two-dimensional (2D) wakes and the role of the Strouhal number ($St$) in controlling vortex organization are well-established, the link between vortex dynamics, thrust production, and the specific structure of three-dimensional (3D) wakes remains incompletely understood. Previous studies have identified distinct wake patterns (e.g., 2S and 2P) for low aspect ratio flapping objects, but a comprehensive framework connecting the quantitative evolution of vortex velocities, wake geometry (angles, orientation), and thrust generation across a broad range of Strouhal numbers has been lacking. Specifically, the relationship between the time-averaged jet velocity, the instantaneous vortex velocity, and the transition between drag and thrust regimes in 3D wakes requires further investigation.

Methodology
The authors experimentally investigated the wake produced by a tethered robotic fish immersed in a water tunnel. The robotic fish consists of a rigid head, a servomotor, and a soft, 3D-printed tail with an aspect ratio of approximately 1.7. The fish was fixed in the laboratory frame while the free-stream speed ($U_0$) was varied.

• Parameter Space: The study systematically varied the tail amplitude ($A$), frequency ($f$), and free-stream speed ($U_0$), covering a Strouhal number range ($St_A = fA/U_0$) from 0.1 to 2.2. This range extends beyond the typical biological range (0.2–0.4) to capture the transition between drag and thrust regimes.
• Measurements: Two-dimensional Particle Image Velocimetry (2D PIV) was performed on the horizontal mid-span plane behind the fish to capture vorticity and velocity fields. A force sensor measured the instantaneous net force ($F = F_T - F_D$) to determine the thrust and drag regimes.
• Data Processing: Vortex positions, circulation ($\Gamma$), orientation angles, and velocities were extracted from phase-averaged vorticity fields. Time-averaged velocity fields were used to identify jet structures. The authors compared root-mean-square (RMS) jet velocities with vortex velocities to account for fluctuating components.

Key Contributions and Results

1. Vortex Velocity and Thrust Production:

   • The study establishes a strong coupling between vortex velocities and thrust. It demonstrates that the RMS of the jet velocities in the time-averaged field is approximately equal to the vortex velocities ($U_{vortex} \approx \sqrt{\langle U_{jet}^2 \rangle}$).
   • Using a momentum balance equation on a control volume surrounding the fish, the authors modeled the streamwise vortex velocity ($U_{vortex}$) as a function of $St_A$. The model successfully predicts that $U_{vortex}/U_0$ is nearly constant at low $St_A$ and increases at higher $St_A$.
   • The analysis reveals that the "jet inversion" (where the time-averaged jet velocity equals the free-stream speed, $U_{vortex} = U_0$) occurs at a Strouhal number ($St_{inv} \approx 0.36$) slightly lower than the critical Strouhal number for the drag-thrust transition ($St^* \approx 0.48$). This shift is attributed to the transverse component of the vortex velocity, which contributes to the pressure term in the momentum balance.

2. Universal Wake Structure Model:

   • The authors propose a simple geometrical framework where the vortex pair velocity is the vector sum of the free-stream speed and the pair's self-induced (dipole) speed.
   • This model links the wake angle ($\theta$) and the vortex pair orientation angle ($\alpha$) directly to the vortex velocities.
   • Experimental data for wake angles and orientation angles from the robotic fish, combined with literature data from various configurations (numerical simulations, other robotic fish, pitching panels), collapse onto universal master curves when plotted against the Strouhal number. This suggests that the wake structure of 3D low-aspect-ratio swimmers is dominated by $St_A$, largely independent of Reynolds number, exact body shape, or swimming mode.
   • The model predicts that the vortex pair orientation ($\alpha$) changes from negative to positive values as $St_A$ increases, crossing zero at the jet inversion point ($St_{inv}$).

3. Circulation and Vortex Diameter Scaling:

   • The study investigates the scaling of vortex ring circulation ($\Gamma$). While previous work on pitching panels suggested a kinematic scaling ($\Gamma \sim fA^2$), the authors find that for the undulatory robotic fish, circulation is better described by a combination of kinematic and stationary scalings: $\Gamma \sim fA^2 + U_0 A$.
   • A model for the pair diameter ($\xi$) is derived based on the circulation and self-induced velocity. The experimental data for $\xi/A$ versus $St_A$ aligns well with this model, showing a decreasing trend at low $St_A$ and saturation at high $St_A$.

4. Wake Classification:

   • Four wake types were observed: 2S (horseshoe vortices), 2P (vortex rings), 2S+2P, and 4P. Despite structural differences, the trends in vortex velocities and wake angles remained consistent across these types, with the 2P wake being the most prevalent in the parameter space.

Significance and Claims
The paper claims to provide a comprehensive understanding of the relationship between thrust production, vortex dynamics, and wake structure in 3D undulatory swimming. By validating a simple geometrical model against extensive experimental data and literature, the authors highlight a universal behavior governed primarily by the Strouhal number.
The work establishes that:

• The transition between drag and thrust is closely linked to, but distinct from, the vortex/jet inversion, with the difference explained by transverse velocity components.
• Wake geometry (angle and orientation) can be predicted solely from the Strouhal number and vortex velocities.
• These findings offer a quantitative basis for understanding fish-fish hydrodynamic interactions, such as schooling and predator-prey dynamics, by characterizing the specific vortical structures (angles and velocities) emitted by swimmers. The authors note that for fish cruising at typical biological Strouhal numbers (0.2–0.4), the wake angle is predicted to be between $11^\circ$ and $17^\circ$.