Propulsion performance of tandem flapping foils with chordwise prescribed deflection from linear potential theory

The effect of prescribed flexural deflection on the propulsive performance of tandem-arranged flapping foils is analyzed by using the vortex impulse formulation in the limit of two-dimensional, linear potential flow theory. Analytical expressions are given for a general configuration of tandem foils undergoing a quadratic flexural deflection coupled with heaving and pitching motions about arbitrary pitching and deflection axes, although quantitative results are focused to the cases pivoting about their leading edges. Flexural deflection may augment not only the thrust and the propulsive efficiency of the individual foils in relation to an otherwise identical rigid-foil configuration, it also modifies the wake behind the front foil, changing the wake interaction with the trailing foil and, consequently, modifying the tandem propulsive performance. The effect of the upstream wake on the thrust force of the trailing foil is analyzed by using the vorticity distributions obtained analytically. Patterns of propulsive performance enhancement in both the spacing-frequency and the spacing-phase shift planes are analyzed and discussed in relation to previous available experimental and numerical results. The present theoretical results provide some new insights for the design of small aerial or aquatic vehicles using biomimetic tandem propulsors.

Figure 1

Schematic of the oscillating airfoils for heaving and flexural motions (dimensionless).

Figure 2

Contours of the normalized propulsive efficiency in the $\tilde{d}/c\text{−}\phi$ plane for tandem heaving foils with $k=1.25$ and $p=-1$. (a) $\widehat{\eta }$ for a pair of rigid foils. (b) $\tilde{\eta }$ for a pair of flexible foils with $d_m/h_0=1/4$ and $\psi ={90}^{\circ }$. The symbols correspond to equilibrium gap distances obtained by Ryu et al. [45] in their numerical simulations for two self-propelled flexible plates in tandem for the same values of $k$ and $p$, and $h_0=0.4$ [i.e., their tandem flapping (TF) modes in Fig. 2(d) of Ref. [45]; note that their ${\overline{G}}_x$ is our $\tilde{d}/c+1$, while their $\mathrm{\Delta }\varphi$ is our $\phi$].

Figure 3

Contour values of $\mathcal{M}/h_0^2$ in the schooling number-frequency plane for purely heaving (a) rigid and (b) flexible foils with $d_m/h_0=1/4$, $\psi =\phi =0$, and $p=-1$.

Figure 4

Contour values of $\mathcal{M}/h_0^2$ in the schooling number-phase shift plane for heaving plates with flexural deflection amplitude $d_m/h_0=1/4$. $\phi =0$, $p=-1$, and $k=2.75$.

Figure 5

Contours of normalized thrust (left panels) and normalized propulsive efficiency (right panels) of the tandem in the $\tilde{d}/c\text{−}k$ plane for pure heave with $\phi =0$ and $p=-1$. Panels (a) and (b) are for a tandem of rigid foils, with quantities normalized by those of an otherwise identical single foil. Panels (c) and (d) are for a tandem of flexible foils with chordwise deflection when $d_m/h_0=1/4$ and $\psi ={270}^{\circ }$, now normalized by the corresponding values for the tandem of rigid foils.

Figure 6

Maxima in (a) ${\tilde{C}}_T$ and (b) $\tilde{\eta }$ within the first amplification modes (with lower frequencies) plotted in Figs. 5 and 2(d), respectively, as a function of the phase shift $\phi$ and the corresponding optimal values of $k$, $\tilde{d}/c$, and $\psi$.

Figure 7

Contours of the tandem normalized thrust (left panels) and the tandem normalized propulsive efficiency (right panels) in the $\tilde{d}/c\text{−}\phi$ plane for purely pitching motions with $k=\pi$ and $a=p=-1$. Panels (a) and (b) are for a tandem of rigid foils, with quantities normalized by those of an otherwise identical single foil, while panels (c) and (d) are for a tandem of flexible foils with trailing-edge deflection amplitude $d_m/a_0=1/4$ and $\psi ={180}^{\circ }$, now normalized by the corresponding values for the tandem of rigid foils.

Figure 8

Maxima in (a) ${\tilde{C}}_T$ and (b) $\tilde{\eta }$ in the first amplification modes plotted in Figs. 7 and 7, respectively, as a function of the deflection phase shift $\psi$ and the corresponding optimal values of $\phi$ and $\tilde{d}/c$. $k=\pi$.