Vortical patterns generated by flapping foils of variable ratio chord-to-thickness

Abstract

Depending upon the flapping amplitude and frequency, different types of vortex streets are shed downstream of a pitching foil. In the present experimental work, the foil aspect ratio chord-to-thickness is varied ($L/D=5;\,4;\,3$ and 2) additionally to the frequency, while the amplitude is kept constant. Dye visualizations allow to find that the number of vortices shed by oscillation cycle slightly depends upon the foil aspect ratio. Moreover, whereas the von Kármán street can reverse for the longest two foils, this change is not observed for the shortest two foils. Finally, when the frequency is high enough, the formation mechanism of the asymmetric reverse Kármán street, which is significative of a positive thrust, is shown to be the same as the $P+S$ pattern that develops for an oscillating cylinder. Detailed visualizations show that a jet is produced for the longest two foils because the foil tip vorticity is maximum just when a counter-rotating vortex is ejected.

Graphical abstract

Appendix

The crank–rocker system is described by McCarthy and Soh (2011) and is schematized in Fig. 11. In this system, the crank (center $O_1$, radius $r_1$) is connected to a motor such that it performs complete rotations. A connecting rod is fixed at one end to the point $A_1$ of the crank and at the other end to the point $A_2$ of the rocker. In this way, point $A_2$ oscillates between two extreme angles ($\theta _1$ and $\theta _2$ in Fig. 11) with the same frequency as the crank. The rocker has a center $O_2$ and the distance $\ell =O_1O_2$ is fixed. If we call $r_1=O_1A_1$ and $r_2=O_2A_2$, the angle $\theta =(\overrightarrow{O_1O_2},\overrightarrow{O_2A_2})$ is given by the cosine triangle law:

Fig. 11

Crank–rocker mechanism. The crank center is $O_1$. At radius $r_1$, an end of the connecting rod is fixed. The rocker has center $O_2$ such that $O_1O_2=\ell$. The other end of the connecting rod is fixed at a radius $r_2$ on the rocker, such that, while the crank is performing complete rotations with an angular velocity $\omega$ the rocker is performing oscillations limited by two angles $\theta _1$ and $\theta _2$

Fig. 12

Temporal evolution of angle $\theta$ in the crank–rocker system for the $L/D=4$ foil configuration: $\ell =15.4$ cm, $d=17.6$ cm, $r_1=1.9$ cm, and $r_2=6.6$ cm

$$\begin{aligned} \theta =\pi -\cos ^{-1}\left( \frac{r_2^2+s^2-d^2}{2r_2s}\right) -\cos ^{-1}\left( \frac{\ell ^2+s^2-r_1^2}{2\ell s}\right) , \end{aligned}$$

(4)

where $\displaystyle s=A_1O_2=\sqrt{\ell ^2+r_1^2-2\ell r_1\cos (\omega t)}$. An example of the temporal evolution of $\theta$ is shown in the left-hand plot of Fig. 12 (for $\ell =15.4$ cm, $d=17.6$ cm, $r_1=1.9$ cm and $r_2=6.6$ cm), which is the configuration of the $L/D=4$ foil. The corresponding Fourier spectrum (determined along 20 periods) is shown in the right-hand plot of Fig. 12. To characterize the asymmetry of the signal, two parameters are used. The first parameter S is defined as the time proportion spent by the system in a cycle to reach the maximum amplitude from the minimum amplitude, as in the study of Koochesfahani (1989). In Fig. 12, the pitched-up time is noted as $T_{\mathrm{up}}$, and therefore, $S=T_{\mathrm{up}}/T$. The second parameter $c(2f_0)/c(f_0)$ is the ratio between the second harmonic amplitude and the amplitude of the first harmonic.

Table 1 shows the different values taken by the angle $2\theta _0$ for each foil. For each one of the four configurations, the asymmetry parameters are determined as $48\%\le S\le 52\%$ and $c(2f_0)/c(f_0)\le 0.5\%$, which are very low values.

Table 1 Oscillation characteristics for each foil