Flow control by means of a traveling curvature wave in fishlike escape responses

Fish usually bend their bodies into a ‘‘C’’ shape and then beat their tails one or more times to escape from predators (in nature) or stimuli (in experiments). The maneuvering behavior, i.e., the C-shape bending and the return flapping, is called C-start. In this paper, the escaping performance of fishlike C-start motions has been numerically investigated for a flow physics study by the use of a two-dimensional deformable foil bending and stretching quickly. The C-start motions, performed in the quiescent water and based on prescribed deforming modes, are predicted by a numerical method coupling the two-dimensional incompressible Navier-Stokes equations and the deforming body dynamic equations. It has been found earlier that a typical C-start motion consists of (1) a main C-shape bending and (2) a rearward travelling curvature wave which was seldom mentioned in previous studies. In order to reveal the flow control mechanism of the traveling curvature wave in a fish's C-start motion, two kinds of C-start flows with different deforming modes, namely the integrated mode (IM, a C-shape bending plus a travelling curvature wave) and the basic mode (BM, a C-shape bending only) are analyzed and compared in detail. According to the numerical results, it shows that if proper values of the travelling curvature wave parameters are chosen, the foil's escaping maneuverability presented in the IM is much better than that in the BM, i.e. the turn angle and the speed of the center of mass at the end of a C-start in the IM is almost twice as large as those in the BM. Further study shows that the travelling curvature wave not only can enhance the thrust and the centripetal force but also increase the propulsive efficiency. These results suggest that an efficient travelling curvature wave is of great significance in the flow control of a C-start motion. Finally, a parametric study finds that the phase difference between the C-shape bending and the travelling curvature wave (i.e., the initial phase angle in the travelling curvature wave of the deforming model) is a key parameter in the flow control. To achieve the desirable turn angle, escaping speed, and propulsive efficiency in the C-start motions, the initial phase angles must be ranged within specific magnitudes. It is found that for optimum values of the initial phase angle, the foil's flexible deforming process is qualitatively consistent with that of a fish body in nature. The results obtained in this study provide a new physical insight into the understanding of swimming mechanisms of fish's C-start maneuvers.

Figure 1

Midlines of the foil's C-start in head coordinates: (a) IM and (b) BM. Here solid lines are in the period of bending, and dashed-dotted lines represent the return flapping; $x$ and $y$ are normalized by the body length $L$. Parameters of the IM are shown in Table ; $\beta$ is the angle between the tangential lines of the tail and the head.

Figure 2

Contours of fish body and the COM trajectories in (a) the IM (case 1) and (b) the BM (case 2).

Figure 3

Time-dependent nondimensional COM speed $V_C$ and body angular velocity ${\omega }_b$ in the IM (case 1) and the BM (case 2).

Figure 4

Coefficients of thrust, lateral force, and torque vs time. The IM and BM represent cases 1 and 2, respectively. Superscripts $p$ and $f$ denote the pressure and the friction components, respectively, of the force coefficients.

Figure 5

Coefficients of the total power, propulsive power, and rotational power vs time: (a) cases 1 and (b) 2.

Figure 6

Contours of voticity in the flow field of (a)–(c) the IM (case 1) and (d)–(f) BM (case 2) at some typical moments. At the initial moment, the foil is stretched; the leading edge is in the direction opposite to $x$ axis; both water and foil are at rest. At the end of the C-start, the circulations of the concentrated vortices are ${\Gamma }_1=-0.14,{\Gamma }_2=0.32,\mathrm{and}{\Gamma }_3=-0.11$ in the IM, and ${\Gamma }_1=-0.12,{\Gamma }_2=0.26,\mathrm{and}{\Gamma }_3=-0.08$ in the BM.

Figure 7

${\alpha }_e$, $u_e$, and $\eta$ vs $A_w$. Deforming parameters except $A_w$ in these cases are the same as those shown in Table , and $\mathrm{Re}=4.25\times {10}^5$.

Figure 8

${\alpha }_e$, $u_e$, and $\eta$ in cases with different $\phi$. Other deforming parameters are the same as those shown in Table , and $\mathrm{Re}=4.25\times {10}^5$.

Figure 9

(a) ${\alpha }_e$, (b) $u_e$, and (c) $\eta$ in cases with different $\phi$ and $c$. Using the reference speed $U=2L/T_c$ to nondimensionalize the wave velocity, e.g., when $T_w=T_c$ and $\lambda =L$, the nondimensional $c$ is equal to 0.5, i.e., the solid lines with diamond symbols in these figures. Other deforming parameters are the same as those shown in Table , and $\mathrm{Re}=4.25\times {10}^5$.

Figure 10

Midline shapes in the IM when the initial phase angle (a) $\phi =0.4\pi$ which much enhances the performance and (b) $\phi =1.0\pi$ which leads to a poor maneuvering performance. Solid lines, stage 1; dashed-dotted lines, stage 2.

Figure 11

${\alpha }_e$, $u_e$, and $\eta$ vs $\mathrm{Re}$ (horizontal abscissa is the logarithm of $\mathrm{Re}$). Other deforming parameters in these cases are the same as those shown in Table .

Figure 12

Overset grid; the whole domain is $20L\times 20L$, where L is the body length.

Figure 13

Deforming shapes of the foil midlines in the COM reference system. In this local frame $x^{\prime }c{y}^{\prime }$, the foil COM is located at the origin $c$, and the translational and rotational momenta of the deforming motion are separately conserved.

Figure 14

Schematic diagram of the dynamic iteration. $A_G$ denotes the guessed acceleration (including the angular acceleration) of the foil, and $A_C$ represents the calculated one. The superscripts $n$ and $m$ mean the time step and the iteration step, respectively.

Figure 15

Time-dependent COM speed of the IM (deforming parameters and Reynolds number are the same as case 1 in the main text). Grid 1: grid number $1.8\times {10}^4$, minimum scale of element $0.002$, and time step $\Delta t=2.0\times {10}^{-4}$; grid 2: grid number $1.85\times {10}^4$, minimum scale of element $0.001$, and time step $\Delta t=2.0\times {10}^{-4}$; grid 3: grid number $3.4\times {10}^4$, minimum scale of element $0.0007$, and time step $\Delta t=1.0\times {10}^{-4}$. The domain sizes of these grids are the same $20L\times 20L$. In the present study, the grid 2 is adopted, and the residual of the velocity divergence is ${\varepsilon }_{\mathrm{div}}=1.0\times {10}^{-4}$.

Figure 16

(a) Mean thrust coefficient of a pitching foil in a uniform flow vs Strouhal number Sr. (b) Time-dependent coefficients of thrust $C_T$ and lateral force $C_L$ of a uniform flow past a wavy foil; T is the period of the undulatory wave; $\mathrm{Re}=5000$.