Windsurf-mimetic study about unsteady propulsion

Abstract

We study experimentally a a three-dimensional reduced model of a sail shape performing pitching oscillations around a mean incidence angle ($α_{m}$) with respect to an incoming flow in a hydrodynamic channel at a constant velocity where the Reynolds number based on the mean chord of the sail is Re$_{c} = ρU_{\infty} c / μ= 11900$. The problem is inspired by the "pumping" maneuver used by windsurf athletes. At the start of a race or in light winds, to get or keep the board in foiling mode, for example after a tack change, athletes use intermittent propulsion by "pumping" the sail, i.e. periodically changing the angle of incidence of the sail relative to the wind. The flapping or pitching parameters and position of the sail according to the flow (incidence angle) influence the aerodynamic forces acting on the sail by destabilising the flow and generating unsteady forces. We experimentally characterise the aerodynamic forces of the sail. We compare the sailing ($C_{drive}, \ C_{drift}$) and aerodynamic ($C_{drag}, \ C_{lift}$) coefficients between a static and an oscillating sail for different flapping parameters and different mean incidence angles of the sail and angles of attack of the boat. Thanks to the use of "pumping", we observe that it is possible to generate a drive force greater than the one generated without oscillation. Furthermore, "pumping" increases the range of mean incidence angle in which the drive force is positive. However, this increase inevitably comes with an increase in drift force, which is often detrimental. These data can be used to improve the Velocity Prediction Programme (VPP) associated with windsurfing and to help athletes optimise their "pumping".

Figures (6)

• Figure 1: a) Chronophotographic sequence of a one-period "pumping" maneuver performed by a French athlete during a training session in Quiberon, 2021. From left to right, the decomposition of the athlete's movement shows how their center of mass motion enables sail oscillations. The athlete's arms are not the main cause of the movement. b) Sketch of sailboat dynamics in upwind conditions with the associated sailing speed triangle, showing forces applied to the sail and foil. The true wind angle (TWA) is defined as the angle between the true wind and the boat's longitudinal axis (blue dotted-dashed line). The aerodynamic force (green) is decomposed in both the boat reference frame (blue) and the flow reference frame (red). The apparent wind angle (AWA) is the angle between the apparent wind speed (AWS, black dashed line) and the boat speed (BS, blue dashed line) directions. $\alpha_{m}$ is the mean angle of attack: the angle between the sail's centerline (red dashed line) and the apparent wind speed direction (AWS, black dashed line). (Figure adapted from bertrand2025).
• Figure 2: Sketches of the setup and technical drawings of the sail on a scale of 1/30, where we added a perforated cylinder to the base of the mast, which allows us to attach the sail to our force measurement system. a) Closed loop water channel with a length of $1.80$ m and a cross-section with water of $0.2 \times 0.2$ m$^{2}$auregan2023scaling. b) Control and acquisition setup to measure forces and kinematics data. c) Side view for defining the chord $c$ and the span $s$ of the sail. d) Isometric rear view for defining the twist angle $\tau$ between the top and bottom of the sail, which is $20^{\circ}$ in this case. e) Sketch in top-view of an experiment of pitching sail.
• Figure 3: Raw (points) and filtered (lines) data of drag (top) and lift (bottom) forces versus time, for an experiment where $\alpha_{m}$ = 5$^\circ$ , $\theta_{0}$ = 14$^\circ$ and $f = 1.5$ Hz ($St_{A}=0.18$). We have smoothed here with a low-pass filter (Butterworth filter). We can observe the phase between the raw signal and the filtered signal for each case. The lift force is periodic with $f_{Lift} = f$. The drag force signal is periodic too with $f_{Drag} = 2f$.
• Figure 4: Lift and drag coefficients as a function of $\alpha_{m}$ for a range of Strouhal numbers (colorbar) such as $St_{A} = [0, 0.06, 0.12, 0.18, 0.22]$ (Table \ref{['tabvoile']}). a) Lift coefficients. A vertical line crosses $\alpha_{m} =$ 20$^\circ$ where the static stall appears. In insert, we show the value of $\alpha_{m}$ where the maximum of lift coefficient is reached as a function of St$_{A}$. b) Drag coefficients.
• Figure 5: Aerodynamic force coefficients versus the Strouhal number for 4 values of mean incidence angle. Left: $C_{D}$ vs St$_A$. Right: $C_{L}$ vs St$_A$. For each value of St$_A$, we performed experiments with different pairs ($f$, $A$).