A minimal model of pump foil dynamics

This paper presents a minimal mechanical model of pump foil propulsion that explains how periodic leg pumping converts vertical oscillations into sustained forward motion, revealing that the front wing primarily generates lift while the rear wing is essential for pitch stability.

The Gist

Imagine you are standing on a surfboard in the middle of a calm lake. There are no waves, no wind, and no motor. Yet, you are gliding forward at a steady pace, hovering above the water on invisible wings. How is this possible?

This is the magic of Pump Foiling. And this paper is like a "user manual" written by physicists to explain the secret mechanics behind this trick. They built a simplified computer model to figure out exactly how a rider can turn up-and-down leg movements into forward speed.

Here is the breakdown of their discovery, using everyday analogies:

1. The Big Idea: The "Human Engine"

Think of the rider and the board as a single machine. The rider isn't just sitting there; they are the engine. By rhythmically pumping their legs up and down, they aren't just bouncing; they are tricking the water into pushing them forward.

The authors created a minimal model. Imagine you are trying to explain how a car works. You could write a 1,000-page manual about every bolt and wire, or you could say: "It has an engine, wheels, and a steering wheel." This paper is the "short version." It strips away the messy details (like tiny ripples on the water or the rider's flailing arms) to focus on the three main things that matter:

• Moving forward (Thrust).
• Moving up and down (Lift).
• Tilting the board (Pitch).

2. The Two Wings: The "Heavy Lifter" and the "Steering Wheel"

The pump foil has two underwater wings: a big one in the front and a smaller one in the back. The paper reveals a surprising division of labor:

• The Front Wing (The Heavy Lifter): This is the muscle. It does about 80% of the work to keep the rider out of the water. It's like the main engine of a plane. When you pump, this wing generates the lift that holds you up and the thrust that pushes you forward.
• The Rear Wing (The Stabilizer): This is the trick. Even though it's small and doesn't lift much weight, it is crucial. Imagine trying to balance a broom on your hand. If you only have one point of contact, it wobbles. The rear wing acts like a long lever arm. Because it is far away from the pivot point (the mast), even a tiny force from it creates a huge turning effect. It acts like the rudder on a ship or the tail feathers on an arrow, keeping the board from flipping over. Without it, the front wing would push the nose up too hard, and you'd flip backward.

3. The Secret Sauce: The "Rider's Rhythm"

How does the rider control this? The model treats the rider's legs like a robot with a very specific program:

• The Legs: The rider pumps up and down. The model assumes the rider pushes down with a force roughly equal to half their body weight.
• The Tilt: The rider doesn't just pump; they subtly shift their weight. The model suggests the rider uses their front foot to push the nose down whenever it starts to tilt up too much. It's like a self-correcting system. If the board tries to flip, the rider's foot pushes it back down, keeping the ride smooth.

4. The "Sweet Spot"

The researchers found a "Goldilocks zone" for this activity.

• Too weak: If you don't pump hard enough, gravity wins, and you sink.
• Too strong: If you pump too hard, you shoot out of the water and lose your grip on the surface.
• Just right: At about 45% of your body weight in pumping force, and a rhythm of about 1.65 pumps per second, the system finds a stable balance. You hover about 50cm above the water, gliding at a steady 3.5 meters per second (about 8 mph).

5. Why This Matters

Before this paper, pump foiling was mostly learned by trial and error. You just "felt" the water. This model gives us a mechanical map.

• It explains why the rear wing is so important (it's not for lifting; it's for balance).
• It tells us that the rider's body acts as a feedback loop, constantly adjusting to keep the board stable.
• It suggests that if you want to design better boards or teach people faster, you should focus on the relationship between the front wing's power and the rear wing's leverage.

The Bottom Line

Think of pump foiling as a dance between the rider and the water. The rider provides the rhythm (the pump), the front wing provides the power (the lift), and the rear wing provides the balance (the stability). This paper is the first time someone has written down the "choreography" of that dance in simple math, proving that you don't need a motor to fly over water—you just need the right rhythm and a little bit of physics.

Technical Summary

1. Problem Statement

Pump foiling is an emerging water sport where a rider on a hydrofoil board sustains forward motion on flat water solely through periodic vertical leg pumping. Unlike traditional propulsion, this system converts vertical oscillations into hydrodynamic lift and thrust without external engines. While the phenomenon is observed, a rigorous, simplified mechanical understanding of the coupled fluid-structure dynamics is lacking. Existing models often rely on complex unsteady potential flow or prescribe specific kinematics. The authors aim to develop a minimal mechanical model that captures the core physics of self-propulsion in pump foiling, treating the rider as an active control input rather than a passive load, to explain how stable forward motion is achieved.

2. Methodology

A. System Formulation

The authors model the rider-board-foil system as a coupled second-order dynamical system with three degrees of freedom (DOF) in an inertial frame:

• Horizontal translation ($x$)
• Vertical translation ($y$)
• Pitch rotation ($\theta$)

The equations of motion are derived from Newton's second law ($F=ma$ and $\tau=I\alpha$):
$$m\ddot{x} = \sum F_x, \quad m\ddot{y} = \sum F_y, \quad I\ddot{\theta} = \sum \tau$$

B. Modeling Assumptions

To maintain a "minimal" framework, several simplifications are made:

1. Quasi-steady Hydrodynamics: Lift and drag are treated as functions of instantaneous kinematics (velocity and angle of attack), neglecting unsteady effects like added mass (shown to be negligible in Appendix A) and vortex shedding history.
2. Geometry: The system consists of a front wing (Fw) and a rear wing (Rw) connected by a mast. The rider's mass is concentrated at the body center, and the foil mass is assigned to the mast-linker pivot ($O$).
3. Forces:
   • Lift/Drag: Calculated using standard quasi-steady coefficients ($C_L, C_D$) dependent on the angle of attack ($\alpha$).
   • Buoyancy: Included for the submerged volume of the wings.
   • Rotational Effects: Includes a Magnus-type rotational lift and nonlinear rotational drag (proportional to $\dot{\theta}|\dot{\theta}|$).
4. Rider Control (The Core Innovation): Instead of prescribing the foil's motion, the rider's input is modeled as a control force.
   • Total Pumping Force: Prescribed as a sinusoidal function $F_{pump}(t) = A_{pump} \sin(\omega t)$.
   • Torque Control: The rider adjusts the distribution of force between the front and rear feet to control pitch. The rear foot is fixed at the pivot (no torque), while the front foot generates a nose-down torque proportional to the pitch angle deviation (a proportional control law). The rider is assumed to adjust their center of mass to eliminate weight-induced torque.

C. Numerical Implementation

The coupled ordinary differential equations (ODEs) are solved numerically using MATLAB's ode45. Parameters (mass, geometry, wing areas) are derived from standard pump foil specifications and a rider mass of ~70 kg. The pumping frequency is fixed at 1.65 Hz based on empirical observations.

3. Key Contributions

1. Minimal Dynamical Model: The paper presents the first reduced-order model that treats the rider as an internal actuator within the dynamical system, rather than an external forcing function. This allows for the prediction of self-sustained stable regimes.
2. Decoupling of Roles: The model explicitly distinguishes the aerodynamic roles of the front and rear wings, providing a mechanistic explanation for their distinct functions.
3. Control Strategy Simplification: By reducing the rider's complex biomechanics to a single effective force amplitude and a proportional pitch-control law, the model isolates the critical parameters governing stability.
4. Validation of Quasi-Steady Assumption: The study demonstrates that despite the unsteady nature of the motion, a quasi-steady approach is sufficient to capture the essential dynamics of pump foiling, provided the rider's control strategy is correctly modeled.

4. Results

A. Stable Propulsion Regimes

The model successfully predicts a stable forward-propulsion regime when the pumping amplitude is approximately 45% of the rider's body weight ($0.45mg$).

• Forward Velocity: Stabilizes around 3.5 m/s, which is roughly 20 times the vertical heaving velocity.
• Vertical Position: The pivot point oscillates stably about 50 cm below the free surface, keeping the board above water.
• Pitch Angle: The system naturally stabilizes at a small positive pitch angle ($\theta \approx 2.3^\circ$), justifying small-angle approximations.

B. Force and Torque Balance Analysis

Detailed force decomposition reveals the specific roles of the components:

• Front Wing (Lift Generator): Acts as the primary lifting surface, generating ~80% of the total vertical lift (due to its larger area). It is the main source of thrust.
• Rear Wing (Stabilizer): Contributes minimally to vertical lift (~20%) but is essential for pitch stability. Because it has a much longer moment arm ($l_{Rw} \approx 4.3 \times l_{Fw}$), it generates a counter-torque that balances the large nose-up moment produced by the front wing.
• Rider Control: The rider's front foot provides a nose-down torque to regulate the net moment, keeping the pitch angle small. The net torque oscillates near zero, maintaining equilibrium.
• Rotational Lift: The Magnus-type lift contribution is found to be negligible in this configuration compared to translational lift.

C. Sensitivity to Frequency

Appendix B shows that stable motion exists only within a specific frequency band ($f \in [0.27, 1.65]$ Hz) for a fixed pumping force.

• Too Low: The rider sinks.
• Too High: The pumping force becomes insufficient to maintain depth, leading to sinking.
• Optimal: The system finds a stable submerged oscillation where energy input matches hydrodynamic losses.

5. Significance and Conclusion

This work provides a mechanistic interpretation of pump foil propulsion, moving beyond empirical observation to a predictive physical framework.

• Scientific Insight: It clarifies that pump foiling is a bifurcation phenomenon where symmetry breaking (via the rider's control and wing geometry) leads to self-propulsion. It highlights that the rear wing's primary function is stabilization via torque balance, not lift generation.
• Practical Application: The model identifies measurable quantities (e.g., force distribution between feet, pitch angle sensitivity) that can guide future field experiments.
• Future Directions: The authors suggest that while the current model is idealized, it serves as a foundation for refining rider control strategies (open-loop vs. feedback) and improving hydrodynamic loading laws through targeted laboratory and field measurements.

In summary, the paper demonstrates that a simple quasi-steady model, coupled with a realistic representation of rider control, can accurately reproduce the complex dynamics of pump foiling, revealing the critical interplay between the front wing's lift, the rear wing's stabilizing moment arm, and the rider's active pitch control.