Lift reversal from vortex-surface phase coupling in a heaving foil near a free surface

TL;DR

This work reveals that a deformable free surface can act as a phase clock that modulates vertical loading on a heaving foil via vortex–surface phase coupling. By combining force measurements, wake visualization, and 2D potential-flow simulations, the authors show that the cycle-averaged lift reverses from repulsion to suction as the unsteady number $\tau = \frac{2\pi f U}{g}$ increases within $0.2 \lesssim \tau \lesssim 0.4$ for moderate to deep submergence, due to phase-driven vertical advection that reorients trailing-edge vortex (TEV) pairing and redirects wake momentum. A force decomposition demonstrates that the reversal arises from a coordinated shift between the quasi-steady pressure loading $\overline{C}_L^{\mathrm{QS}}$ and wake-induced force $\overline{C}_L^{\mathrm{WI}}$, with a negligible added-mass contribution to the mean. The results highlight how deformable boundaries introduce a dynamical clock into vortex-dominated flows, enabling lift control through surface-phase coupling with potential implications for near-surface biology and robotics.

Abstract

Classical descriptions of flapping propulsion near a free surface emphasize the energetic penalties of wave generation, treating the interface primarily as an energy sink. Here, we show that the same deformable boundary can also act as a phase-dependent kinematic constraint on vertical force generation. Using force measurements, particle image velocimetry and potential-flow simulations, we characterize how a free surface reorganizes vortex shedding for a heaving hydrofoil at moderate Reynolds number (O(10^4)). For moderate to deep submergence, the cycle-averaged lift undergoes a systematic transition from repulsion to suction as the unsteady number increases. The reversal occurs within a narrow band of unsteady numbers, where the phase-shifted surface motion generates vertical advection that alters the pairing of trailing-edge vortices and redirects the wake momentum flux. A force decomposition shows that the reversal arises from a coordinated change in quasi-steady pressure loading and wake-induced force. These results identify the phase of the free-surface response, organized by unsteady number, as a key parameter governing near-surface lift and illustrate how deformable boundaries can reconfigure unsteady loading through vortex-surface phase coupling.

Figures (5)

• Figure 1: Experimental setup. (a) Hydrofoil mounted along the z-axis traverse with end-plates to ensure two-dimensional flow. (b) Schematic of the heaving kinematics ($U, A$) and geometric parameters ($d, c$) relative to the free surface.
• Figure 2: Time-averaged lift coefficient ($\overline{C}_L$) regimes defined by the unsteady number $\tau$, Strouhal number $St$, and submergence $d/c$. (a) Lift polarity reversal at $St=0.5$, showing the transition from repulsion (negative) to suction (positive) with increasing $\tau$. (b--g) Contours of $\overline{C}_L$ in the $(St, d/c)$ domain for fixed $\tau$. The dashed vertical lines in (a--g) delineate the near-surface and far-field regions. Blue is repulsion and red is suction.
• Figure 3: Cycle-resolved lift ($C_L(t)$) dynamics governing the mean-lift reversal. (a) $\overline{C}_L$ versus $\tau$ across varying $St$ and submergence depths. Dashed vertical lines mark the transition band; square markers indicate the specific cases analyzed in (b) and (c). (b) Instantaneous lift waveforms $C_L(t)$ for representative cases ($St=0.7$), highlighting the distinct bias shift and phase shift mechanisms. (c) Peak-to-peak statistics showing the maximum, minimum, and cycle-averaged (square marker) lift.
• Figure 4: The unsteady number $\tau$ modulates the interaction between the free surface and trailing-edge vortices (TEVs), leading to distinct deflected wakes. Panels (a--d) correspond to $\tau = 0.1$ and $St = 0.7$; panels (e--h) correspond to $\tau = 0.6$ and $St = 0.7$, both at $d/c = 0.52$. Spanwise vorticity $\omega_y$ is shown in colour (red: CCW; blue: CW). Annotated vectors isolate distinct transport mechanisms: Light-blue arrows denote surface-wave--induced velocity; green arrows indicate pressure-gradient forcing from the foil--surface geometry; and purple arrows represent the self-induced motion of the TEV pair.
• Figure 5: Simulation wake fields and cycle-averaged lift decomposition. (a,b) Simulated wake topology at phase $\pi$ reproducing the shift from a downward dipole (low $\tau$) to an upward jet (high $\tau$). (c) Decomposition of mean lift. The theoretical reconstruction (black lines) captures the experimental lift reversal (grey lines), driven by the concerted shift of quasi-steady ($\overline{C}_L^{QS}$) and wake-induced ($\overline{C}_L^{WI}$) components.