Dynamic response and stability of a flapping foil in a dense and viscous fluid

Abstract

It is important to understand and accurately predict the static and dynamic response and stability of flexible hydro/aero lifting bodies to ensure their structural safety, to facilitate the design/optimization of new/existing concepts, and to test the feasibility of using advanced materials. The present study investigates the influence of solid-to-fluid added mass ratio (\documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}$\end{document}μb) and viscous effects on the fluid-structure interaction (FSI) response and stability of a flapping foil in incompressible and turbulent flows using a recently presented efficient and stable numerical algorithm in time-domain, which couples an unsteady Reynolds Average Navier-Stokes solver with a two degrees-of freedom structural model. The new numerical coupling method is able to stably and accurately simulate the FSI behavior of light foils in dense fluids: a limit which is known to be numerically difficult to study with classical FSI coupling methods. The studied FSI responses include static/dynamic divergence and flutter instabilities, which are compared with inviscid, linear potential theory predictions obtained with both time and frequency domain formulations, as well as with several published experimental data. In general, the results show that the critical reduced flutter velocities and reduced divergence velocities both decrease as \documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}$\end{document}μb decreases, and are captured with good accuracy using the viscous FSI solver for a wide range of relative mass ratios that are typical to air/hydrofoils. The comparative analyses showed that the classic frequency-domain linear potential theory is severely unconservative for predicting the flutter velocity for cases with \documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}<3$\end{document}μb<3: this includes the typical operating conditions of most marine and biomedical lifting devices, where the fluid forces are comparable to the solid forces, and strong nonlinear interactions may develop. In addition, the viscous FSI solver is shown to correctly predict the experimentally reported critical divergence speed of a light foil in a dense fluid for a case where the classical potential theory predicts an infinite divergence speed as the foil's elastic axis (E.A.) coincided with the aerodynamic center. The results show that static/dynamic divergence will occur before flutter for light hydrofoils with an E.A. downstream of the center of pressure. However, for high solid-to-fluid added mass ratios (\documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}>2$\end{document}μb>2), flutter tends to occur prior to divergence. In addition, in between the regions governed by static divergence (\documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}<1$\end{document}μb<1) and flutter (\documentclass[12pt]{minimal}\begin{document}$\sqrt{\mu _b}>2$\end{document}μb>2), there is a dynamic divergence region, where the foil deformations oscillate with an increasing mean amplitude, and the oscillation frequency decreases toward zero as the deformation increases; this region could only be captured by using a viscous FSI solver.