Self-propulsion of a flapping flexible plate near the ground

The self-propulsion of a three-dimensional flapping flexible plate near the ground is studied using an immersed boundary-lattice Boltzmann method for fluid flow and a finite-element method for plate motion. When the leading edge of the flexible plate is forced into a vertical oscillation near the ground, the entire plate moves freely due to the fluid-structure interaction. The mechanisms underlying the dynamics of the plate near the ground are elucidated. Based on the propulsive behaviors of the flapping plate, three distinct regimes due to the ground effect can be qualitatively identified. These regimes can be described briefly as the expensive, benefited, and uninfluenced propulsion regimes. The analysis of unsteady dynamics and plate deformation indicates that the ground effect becomes weaker for a more flexible plate. We have found that a suitable degree of flexibility can improve propulsion near the ground. The vortical structure around the plate and the pressure distribution on the plate are analyzed to understand propulsive behaviors. The results obtained in this study can provide some physical insights into the propulsive mechanisms of a flapping flexible plate near the ground.

Figure 1

Schematic of a flapping flexible plate near the ground.

Figure 2

Time-dependent propulsive velocity of a flapping flexible plate near the ground for three grid spacings with the parameters $D=0.5,F=0.6,H=2.0,A=0.25,M=0.5$, and ${\text{Re}}_f=100$.

Figure 3

Comparison of the present results and previous data [27] for the time history of the lateral displacement of the lower corner on the trailing edge of a flag with the parameters $H=1.0,M=1.0,K=0.0001$, and $S=1000$: (a) $\text{Re}=100$ and (b) $\text{Re}=500$.

Figure 4

Comparison of the metrics of propulsion with and without the ground effect: (a) mean propulsive speed, (b) input work during one period, (c) propulsive efficiency, and (d) mean bending energy for several aspect ratios and the 2D case. In (a) and (c), the line marked by circles represents the net increments of propulsive speed $▵U$ and propulsive efficiency $▵\eta$ for $H=2$, respectively.

Figure 5

The influence of ground proximity on the metrics of propulsion for $H=2$: (a) mean propulsive speed, (b) input work in one flapping cycle, (c) propulsive efficiency, and (d) mean lift coefficient for several frequency ratios.

Figure 6

Time-dependent drag force (left column) and input power (right column) in one flapping cycle for $H=2$: (a,b) $F=0.3$; (c,d) 0.6; and (e,f) 1.2.

Figure 7

Plate deformation shape on the spanwise symmetry plane at eight instances during one flapping cycle with and without the ground effect for $F=0.6$ and $H=2.0$. The solid and dashed lines represent the shapes at $D=0.5$ and $\infty$, respectively. The red and blue lines correspond to the downstroke and upstroke period.

Figure 8

Time-dependent deformation and bending energy during one flapping cycle for $H=2$: (a) vertical displacement of the midpoint on the trailing edge; (b) bending energy.

Figure 9

Vortical structures visualized by an isosurface of the $Q$ criterion with $Q=1$ (left column) and spanwise vorticity contours on the spanwise symmetry plane (right column) during one flapping cycle for $F=0.6,H=2$, and $D=0.5$: (a,b) $t/T=0$, (c,d) 0.25, (e,f) 0.5, and (g,h) 0.75.

Figure 10

Pressure distribution along the two side surfaces of a plate on the spanwise symmetry plane during one flapping cycle for $F=0.6,H=2$, and $D=0.5$ and $\infty$: (a) $t/T=0$, (b) 0.25, (c) 0.5, and (d) 0.75. Here, the solid and dashed lines denote the results for $D=0.5$ and $\infty$, respectively, and the red and blue lines represent the distribution on the upper and lower surface of the plate.