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Hydrodynamics of Flapping Foils Undergoing Irregular Motion with Application to Wave-Assisted Propulsion

Harshal S. Raut, Jung-Hee Seo, Rajat Mittal

TL;DR

The paper investigates how irregular ocean waves affect the hydrodynamics and propulsion of flapping foils used in wave-assisted propulsion (WAP). Using time-resolved 2D CFD with fluid–structure interaction, the authors model an elliptic submerged foil driven by irregular heave from a Bretschneider spectrum and two pitch-constraining mechanisms (spring-limiter and angle-limiter), supported by a leading-edge vortex (LEV)–based thrust model. They show that irregular forcing can yield higher mean thrust than energy-equivalent regular waves, with the spring-limiter consistently outperforming the angle-limiter due to favorable heave–pitch phase relationships; the LEVBM captures the observed thrust enhancements and links them to a higher effective Strouhal number in irregular seas. The results offer practical design guidance for WAP systems, including adaptive stiffness strategies and a robust option using a fixed modest pitch amplitude, and demonstrate the importance of pitch compliance for efficient propulsion in stochastic environments, with potential implications for bioinspired flyers and swimmers as well as engineered ocean vehicles. $ ar{C}_T = K \, \bar{\Lambda}_{LEV}(\theta_s) , \Lambda_{LEV} = \sin(\alpha_{eff}) \sin(\theta_{LE}) , \alpha_{eff} = \tan^{-1}(V_{LE}/U_\infty) - \theta_{LE} $. The optimal pitching amplitude satisfies $\theta_0^{opt} = 0.5 \tan^{-1}(\pi \text{St}_w) - \theta_s$. $S(f) = \frac{5}{16} H_s^2 \frac{f_p^4}{f^5} \exp(-\frac{5}{4}(\frac{f_p}{f})^4)$. For irregular waves, $\text{St}_w^{irr*} = \sqrt{2}\dot{h}_{rms}/(\pi U_\infty)$.

Abstract

Flapping foils are widely studied as bioinspired propulsors, yet most investigations have focused on regular, sinusoidal kinematics. In realistic environments, however, irregular motions arise naturally due to environmental disturbances, fluid-structure interactions, and control inputs, but their hydrodynamic consequences remain largely unexplored. One system where response to irregular forcing is particularly relevant is wave-assisted propulsion (WAP) systems where free-to-pitch submerged foils generate thrust due to wave-induced heaving. We employ time-accurate flow simulations of elliptic WAP foils subjected to irregular waves at three different sea-states to gain insights into this system. Our results demonstrate that irregular heaving and pitching can generate greater mean thrust than energetically equivalent sinusoidal heaving. Moreover, a spring-based pitch-limiting mechanism yields higher thrust than an angle-limiter under the same conditions. By leveraging a previously developed leading-edge vortex (LEV) model, we uncover the mechanisms driving this thrust enhancement and highlight the critical interactions between unsteady flow structures and foil dynamics. These findings provide new insights into flapping-foil propulsion in irregular environments and have direct implications for the design and optimization of WAP systems.

Hydrodynamics of Flapping Foils Undergoing Irregular Motion with Application to Wave-Assisted Propulsion

TL;DR

The paper investigates how irregular ocean waves affect the hydrodynamics and propulsion of flapping foils used in wave-assisted propulsion (WAP). Using time-resolved 2D CFD with fluid–structure interaction, the authors model an elliptic submerged foil driven by irregular heave from a Bretschneider spectrum and two pitch-constraining mechanisms (spring-limiter and angle-limiter), supported by a leading-edge vortex (LEV)–based thrust model. They show that irregular forcing can yield higher mean thrust than energy-equivalent regular waves, with the spring-limiter consistently outperforming the angle-limiter due to favorable heave–pitch phase relationships; the LEVBM captures the observed thrust enhancements and links them to a higher effective Strouhal number in irregular seas. The results offer practical design guidance for WAP systems, including adaptive stiffness strategies and a robust option using a fixed modest pitch amplitude, and demonstrate the importance of pitch compliance for efficient propulsion in stochastic environments, with potential implications for bioinspired flyers and swimmers as well as engineered ocean vehicles. . The optimal pitching amplitude satisfies . . For irregular waves, .

Abstract

Flapping foils are widely studied as bioinspired propulsors, yet most investigations have focused on regular, sinusoidal kinematics. In realistic environments, however, irregular motions arise naturally due to environmental disturbances, fluid-structure interactions, and control inputs, but their hydrodynamic consequences remain largely unexplored. One system where response to irregular forcing is particularly relevant is wave-assisted propulsion (WAP) systems where free-to-pitch submerged foils generate thrust due to wave-induced heaving. We employ time-accurate flow simulations of elliptic WAP foils subjected to irregular waves at three different sea-states to gain insights into this system. Our results demonstrate that irregular heaving and pitching can generate greater mean thrust than energetically equivalent sinusoidal heaving. Moreover, a spring-based pitch-limiting mechanism yields higher thrust than an angle-limiter under the same conditions. By leveraging a previously developed leading-edge vortex (LEV) model, we uncover the mechanisms driving this thrust enhancement and highlight the critical interactions between unsteady flow structures and foil dynamics. These findings provide new insights into flapping-foil propulsion in irregular environments and have direct implications for the design and optimization of WAP systems.
Paper Structure (16 sections, 27 equations, 13 figures, 2 tables)

This paper contains 16 sections, 27 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem Set-up
  4. Computational Method
  5. Results
  6. Foil Kinematics for Regular and Irregular Waves
  7. Performance of Angle-limiter Versus Spring-limiter for Different Sea-States
  8. Partitioning of the Thrust Forces on the Foil
  9. Thrust analysis using the LEV-based model (LEVBM)
  10. Coherence between Heaving and Pitching
  11. Enhanced Performance of Spring-Limiter in Irregular Waves
  12. Enhanced Performance of WAP in Irregular Waves
  13. Guidance for WAP Design
  14. Conclusions
  15. Force-Partitioning Method
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Schematic of surface craft with a wave-assisted propulsion (WAP) system. The schematic is not drawn to scale.
  • Figure 2: (a) Non-dimensional Bretschneider wave energy spectrum for irregular waves. (b) Three realizations (R1, R2 and R3) of irregular waves generated from the Bretschneider specturm for $H_s^*=0.62$ and St$_C=0.12$.
  • Figure 3: Schematic of the hydroelastic system used in this study for (a) spring-limiter and (b) angle-limiter. (c) Computational domain and close-up of the Cartesian computational grid.
  • Figure 4: Comparison between the regular and the irregular wave conditions for the (a) heaving motion, (b) pitching motion and (c) thrust coefficient for sea-state 2 with spring-limiter ($f_\theta/f_h=9$) and angle-limiter ($\theta_0=7.5$) as pitch constraining mechanism. (d) shows the power spectral density for the heaving and pitching motion in regular and irregular wave condition corresponding to (a) and (b). (IW: Irregular Wave, RW: Regular Wave, SL: Spring Limiter and AL: Angle Limiter)
  • Figure 5: Vorticity contour plots for (a) regular wave with spring-limiter ($f_\theta/f_h=9$), (b) irregular wave with spring-limiter ($f_\theta/f_h=9$) and (c) irregular wave with angle-limiter ($\theta_0=7.5^\circ$) for sea-state 2.
  • ...and 8 more figures