A minimal model of pump foil dynamics

Abstract

Pump foiling enables a hydrofoil surfboard to sustain forward motion on flat water using only periodic leg pumping, converting vertical oscillations into hydrodynamic lift and thrust. We present a minimal mechanical model of pump foil propulsion, formulated as a coupled second-order system for horizontal and vertical translation and pitch in an inertial frame. The rider-board system is modeled in reduced form, with the rider mass concentrated at the body center and the foil mass assigned to the mast-linker pivot, about which the front and rear wing dynamics are written. Hydrodynamic loading includes quasi-steady lift and drag, buoyancy, and rotational effects, including rotational lift and nonlinear rotational drag. Under simplified control assumptions in which the pumping frequency is fixed and the net rider pumping input is represented by a single effective force-amplitude parameter, the model predicts sustained stable forward-propulsion regimes at modest forcing amplitude and small positive pitch angle. In this regime, the front wing acts primarily as the main lifting surface, while the rear wing, although contributing less to vertical support, is essential for pitch stability because of its longer moment arm. These results provide a mechanistic interpretation of pump foil propulsion and identify measurable quantities and parameter sensitivities that can guide targeted field and laboratory experiments and help refine assumptions on rider control inputs and hydrodynamic loading.

Figures (16)

• Figure 1: Aerodynamic force representation in (a) inertial reference frame ($xy$) and (b) non-inertial frame ($x'y'$): $L$ stands for lift force, $D$ the drag force, $\theta$ the pitch angle, $\phi$ the relative inflow angle (null in $x'y'$ frame), and $\alpha$ the angle of attack (AoA) satisfying the relation $\alpha = \theta-\phi$. It is negative value in this example. The forces $F_x$ and $F_y$ (or $F_{x'}, F_{y'})$ are the projected total forces by lift and drag onto the reference frame $xy$ (or $x'y'$). The incoming velocity $V=\sqrt{\dot{x}^2+\dot{y}^2} = \sqrt{\dot{x}^{'2}+\dot{y}^{'2}}$.
• Figure 2: (a) Typical pump foil shape and the names of components (image source HipHop98). (b) Schematic representation of pump foil and rider and relevant dimensions. The pivot point, the rotation center ($\mathbf{O}$), is marked with red cross. The positive directions are chosen as $x$ from right to left, $y$ from bottom to top and $\theta$ the nose up direction.
• Figure 3: Forces due to the weight and pumping, and due to the buoyancy of the front and rear wings.
• Figure 4: Velocities at the front and rear wings. The local velocities at each point are modified by the rotational velocity induced by the pitching motion.
• Figure 5: Lift ($L$) and drag ($D$) forces acting at $\mathbf{Fw}$ and $\mathbf{Rw}$ and their projections $F_{x,\cdot}$ and $F_{y,\cdot}$ in the $xy$ frame.