Propulsion of a flexible foil in a wavy flow: resonance, antiresonance, and destructive self-interference

TL;DR

The paper develops a linear inviscid theory for a torsionally anchored, flexible foil driven by a prescribed heave in a uniform flow with an additive traveling-wave disturbance. By coupling elastohydrodynamics with Theodorsen’s function, it reveals resonance, antiresonance, and destructive self-interference as principal mechanisms in wavily driven propulsion, and shows that waviness generally enhances thrust and efficiency when frequencies differ, while equal-frequency interactions depend on wavenumber and phase. The analysis identifies a low-pass, oscillatory waviness response due to destructive interference of wave-induced pressure along the foil, and demonstrates the possibility of simultaneous propulsion and energy extraction from the flow under certain conditions. These insights extend beyond the specific setup to broad classes of spatially and temporally heterogeneous flows, with implications for bio-inspired propulsion and design of flexible propulsion systems in complex environments.

Abstract

Swimming and flying animals demonstrate remarkable adaptations to diverse flow conditions in their environments. In this study, we aim to advance the fundamental understanding of the interaction between flexible bodies and heterogeneous flow conditions. We develop a linear inviscid model of an elastically mounted foil that passively pitches in response to a prescribed heaving motion and an incoming flow that consists of a traveling wave disturbance superposed on a uniform flow. In addition to the well-known resonant response, the wavy flow induces an antiresonant response for non-dimensional phase velocities near unity due to the emergence of non-circulatory forces that oppose circulatory forces. We also find that the wavy flow destructively interferes with itself, effectively rendering the foil a low-pass filter. The net result is that the waviness of the flow always improves thrust and efficiency when the wavy flow is of a different frequency than the prescribed heaving motion. Such a simple statement cannot be made when the wavy flow and heaving motion have the same frequency. Depending on the wavenumber and relative phase, the two may work in concert or in opposition, but they do open the possibility of simultaneous propulsion and net energy extraction from the flow, which, according to our model, is impossible in a uniform flow.

Figures (19)

• Figure 1: Schematic of the problem
• Figure 2: (a) Magnitude of Theodorsen's function, (b) phase of Theodorsen's function, (c) natural frequency, and (d) damping coefficient as a function of the reduced frequency, for $R = 0.01$.
• Figure 3: (a) Magnitude and (b) phase of the response to heaving, for $R = 0.01$. (c) Magnitude and (d) phase of the response to waves, for $R = 0.01$ and $k^* = 0.1$. The zero's location is marked by a vertical short-dashed black line, and the natural frequencies' locations with vertical long-dashed colored lines.
• Figure 4: Magnitude of the approximate zero of the wave transfer function as a function of wavenumber.
• Figure 5: Amplitude of the moment induced by the wavy flow as a function of frequency for $k^* = 8.55$. Insets show the circulatory (blue), non-circulatory (red), and total wavy moments (black) as a function of time at the marked frequencies.