Wake Interference Effects on Flapping Dynamics of Elastic Inverted Foil

TL;DR

The paper addresses how wake interference from an upstream cylinder alters the self-induced flapping of a downstream inverted elastic foil. Using high-fidelity 2D fluid-structure simulations at $Re=1000$, it reveals a critical stiffness $K_{B,Cr}=0.25$ that triggers lock-in to the cylinder wake, with additional SAF and MAF regimes appearing before a transition to large-amplitude flapping. The dynamics depend on the mass ratio $m^*$, displaying regimes of high and low sensitivity to inertia under wake influence. A new non-dimensional parameter $K^{*}_{eff}=K_B - f^{*2}m^*$ with $f^{*}=f_{A_z}/U_ abla$ collapses the behavior into three zones and provides a unified scaling for predicting flapping under wake interference, offering guidance for optimized inverted-foil energy-harvesting arrays.

Abstract

Using high-fidelity simulations, we study the self-induced flapping dynamics of an inverted elastic foil when it is placed in tandem with a stationary circular cylinder. The effect of wake interference on the inverted foil's coupled dynamics is examined at a fixed Reynolds number ($Re$) as a function of non-dimensional bending rigidity ($K_{B}$) and the structure to fluid mass ratio ($m^{*}$). Our results show that there exists a critical $K_{B, Cr} = 0.25$, above which the downstream foil is synchronized with the unsteady wake, and the cylinder controls the flapping response and the wake vortex dynamics. During synchronization, two additional flapping modes namely, small and moderate amplitude flapping mode are observed as a function of decreasing $K_{B}$. Below $K_{B, Cr}$, the downstream foil undergoes self-induced large-amplitude flapping (LAF) similar to an isolated foil counterpart. When the dynamics of the downstream foil are analyzed for a range of $m^{*}$, we can characterize the response dynamics into two regions, namely low and high sensitivity. The high sensitivity region is observed when the dynamics are controlled by the cylinder oscillations, i.e., for foils with high stiffness. In this regime, the foil dynamics negatively correlate to $K_{B}$ and $m^{*}$. The low sensitivity region is observed when the downstream foil is no longer synchronized with the wake and undergoes an LAF response, with dynamics that are weakly correlated to $K_{B}$. A new non-dimensional parameter is proposed that combines the effect of the foil's inertia and elastic forces and can capture the foil's response when it is subjected to wake interference effects. The findings from this study aim to generalize our understanding of the self-induced flapping dynamics of inverted foils in an array configuration and have relevance to the development of inverted foil-based renewable energy harvesters.

Figures (19)

• Figure 1: Schematic of an inverted foil placed downstream of a rigid cylinder
• Figure 2: Schematic of the computational domain of an inverted foil placed downstream of a stationary circular cylinder
• Figure 3: Response characteristics of the inverted foil at $Re = 1000$ and $m^{*} = 0.1$: (a) variation of maximum transverse tip displacement for the isolated and downstream foil as a function of $K_{B}$.
• Figure 4: Full body profiles of the downstream inverted foil at $Re = 1000$ and $m^{*} = 0.1$: (a) small-amplitude flapping, (b) moderate-amplitude flapping, and (c) large-amplitude flapping regimes
• Figure 5: Variation of fluid loading for the isolated and downstream foil as a function of $K_{B}$