Collective behavior and hydrodynamic advantage of side-by-side self-propelled flapping foils

Fish schools and their potential hydrodynamic advantages are intriguing problems and many underlying mechanisms are unclear due to the complexity of the system, especially for large schools. Here large schools containing four, six, and eight self-propelled foils in a side-by-side configuration are numerically studied. The effect of different combinations out of phase and in phase between two neighboring foils is studied. The results show that the multiple abreast self-propelled foils driven by synchronized harmonic flapping motions can spontaneously form stable side-by-side configurations. When compared with a single foil flapping alone, for cases in which any two neighboring foils are in an out-of-phase state, foils consume more energy with a specific cruising speed. For cases where any two neighboring foils are in an in-phase state, foils propel at a lower speed for a specific flapping frequency. Interestingly, the foils in hybrid states in which both out of phase and in phase coexist are preferred to enhance speed and save power. Further analysis indicates that the stability of the configuration and the lower cost of transport are attributed to the synchronized collaborative wake vortex structure and bow configuration formed by any three neighboring foils in a hybrid state. The collaborative vortices in the wake help the foils move forward alternatively during one flapping cycle. The bow configuration prevents the wake from spreading laterally and enhances the performance. Our paper sheds some light on understanding the self-organized collective behavior and hydrodynamic advantages of large schools.

Figure 1

Multiple self-propelled flapping foils (a) and zoom-in view of two adjacent flapping foils (b). The dashed lines show the superimposed body outlines over one period. In our paper, $N=4,6,8$ and NACA0012 foil is used.

Figure 2

Schematic diagram of Eulerian grids and Lagrangian points in the immersed boundary method. The red dashed line denotes the action range of the delta function.

Figure 3

Schematic diagram of a flapping elliptical wing. $x\left(t\right)$ and $y\left(t\right)$ are the center of the wing, and $\alpha \left(t\right)$ is the angle between the long axis and x axis. $c$ and $0.1c$ are the length of the long and short axes of the wing, respectively.

Figure 4

(a) Drag coefficient for a flapping elliptical wing as functions of time. (b) Cruising speed as functions of time with different mesh sizes.

Figure 5

The instantaneous vorticity contours at $t/T=0$. (a)–(f) are the contours of cases A1 to A6, respectively. The colors red and blue correspond to positive and negative vorticity, respectively.

Figure 6

The evolution of horizontal distances $G_x$ for case A3 with different initial horizontal distances: (a) $G_{x0}^{12}=c$, $G_{x0}^{23}=-2c$, $G_{x0}^{34}=1.5c$; ($\mathrm{b})G_{x0}^{12}=0$, $G_{x0}^{23}=0$, $G_{x0}^{34}=4c$. The insets show the initial and final relative position of the foils.

Figure 7

The flapping Reynolds number ${\text{Re}}_f$ (a) and dimensionless cost of transport ${\text{COT}}^{*}$ (b) as functions of ${\text{Re}}_c$. The insert in (a) shows the tail-flapping frequency $f$ (HZ) as a function of the cruising speed $U$ (one body length per second) in Ref. [20].

Figure 8

Horizontal force $\left(F_x\right)$ (a) and cruising velocity $\left(u_c\right)$ (b) of the foils as functions of time in one period for case A1 (dashed lines) and A3 (solid lines) with ${\text{Re}}_f=1000$. I–IV with the black arrows in (a) denote $t/T=0.15,0.3,0.65$, and 0.8, respectively. The horizontal force is normalized by $F_{\text{ref}}=1/2{\rho }_f{u}_f^{2}c$.

Figure 9

The instantaneous vorticity contours (a)–(d) and pressure contours (e)–(h) for case A1 at $t/T=0.15$ (a), (e); $t/T=0.3$ (d), (f); $t/T=0.65$ (c), (g); and $t/T=0.8$ (d), (h).

Figure 10

The instantaneous vorticity contours (a)–(d) and pressure contours (e)–(h) for case A3 at $t/T=0.15$ (a), (e); $t/T=0.3$ (d), (f); $t/T=0.65$ (c), (g)l and $t/T=0.8$ (d), (h). Black arrows in (a)–(d) denote the directions of local flows induced by the vortices.

Figure 11

Time-averaged jet in the wake for case A1 (a) and case A3 (b) with ${\text{Re}}_f=1000$. The inclination angles are approximately ${12}^{\circ }$ for case A1 and $3^{\circ }$ for case A3. The contours show the magnitude of adjusted horizontal velocity $\overline{u^{\prime }}=\overline{u}+u_c$. Dashed lines are drawn to guide the eye.

Figure 12

The flapping Reynolds number ${\text{Re}}_f$ (a) and dimensionless cost of transport ${\text{COT}}^{*}$ (b) as functions of ${\text{Re}}_c$ for six and eight foils.

Figure 13

The instantaneous vorticity contours for eight foils in AOP state (a)–(d) and hybrid state (e)–(h) at $t/T=0$ (a), (e); $t/T=0.25$ (b), (f); $t/T=0.5$ (c), (g0; and $t/T=0.75$ (d), (h). Black arrows in (e) and (g) denote the directions of local flows induced by the vortices.

Figure 14

Time-averaged jet in the wake for eight foils in AOP state (a) and hybrid state (b). The inclination angles are approximately $10.7^{\circ }$ for AOP state and $2.6^{\circ }$ for hybrid state. The contours show the magnitude of adjusted horizontal velocity $\overline{u^{\prime }}=\overline{u}+u_c$. Dashed lines are drawn to guide the eye.