Self-propelled plate in wakes behind tandem cylinders

Fish may take advantage of environmental vortices to save the cost of locomotion. The complex hydrodynamics shed from multiple physical objects may significantly affect fish refuging (holding stationary). Taking a model of a self-propelled flapping plate, we numerically studied the locomotion of the plate in wakes of two tandem cylinders. In most simulations, the plate heaves at its initial position $G_0$ before the flow comes (releasing Style I). In the typical wake patterns, the plate may hold stationary, drift upstream, or drift downstream. The phase diagrams of these modes in the $G_0-A$ plane for the vortex shedding patterns were obtained, where $A$ is the flapping amplitude. It is observed that the plate is able to hold stationary at multiple equilibrium locations after it is released. Meanwhile, the minimum amplitude and the input power required for the plate seem inversely proportional to the shedding vortex strength. The effect of releasing style was also investigated. If the plate keeps stationary and does not flap until the vortex shedding is fully developed (releasing Style II), then the plate is able to hold stationary at some equilibrium locations but the flapping plate has a very minor effect on the shedding vortices. However, in Style I, the released plate is able to achieve more equilibrium locations through adjusting the phase of vortex shedding. The effort of the preflapping in Style I is not in vain, because although it consumes more energy, it becomes easier to hold stationary later. The relevant mechanism is explored.

Figure 1

Schematic diagram of a self-propelled flexible plate behind two tandem cylinders.

Figure 2

(a) Time histories of the drag and lift coefficients of the cylinder without the flexible plate and (b) the locomotion of the leading-edge and trailing-edge of the flapping plate in the $y$ direction as functions of time in the wake of a cylinder.

Figure 3

The grid independence (a) and time step independence (b) studies for the case of a flexible plate flapping behind the wake of two tandem cylinders ($D_x=1.1, G_0=10, A=0.3, f=0.185, K=0.5$, and $\text{Re}=200$). The streamwise force experienced by the plate as a function of time is presented.

Figure 4

Vorticity contours. (a) SBB pattern ($D_x=1.1$), (b) SLR pattern ($D_x=3.0$), and (c) SVS pattern ($D_x=4.5$).

Figure 5

(a) Streamwise trajectory of the leading edge for the three movement modes. The initial locations and flapping amplitudes are listed in the following: the DU mode (curve I) $G_0=3, A=0.1$, the DU mode (curve II) $G_0=10, A=0.9$, the DD mode $G_0=10, A=0.1$, and the HS mode $G_0=10, A=0.6$. (b) The trajectory of the leading edge in one period for the HS mode.

Figure 6

(a) Schematic phase diagram for the three motion modes. Approximate critical points $G_{c1}, G_{c2}, A_{c1}$, and $A_{c2}$ are able to define the motion mode distribution in the $G_0-A$ plane. For different flow patterns, these values are listed in Table 3. (b) Phase diagram for the three motion modes for SVS flow pattern. Symbols $\circ , ▹$, and $\diamond$ represent the HS, DU, and DD modes, respectively.

Figure 7

Instantaneous vorticity contours at (a) $t=\frac{1}{8}T$, (b) $\frac{3}{8}T$, (c) $\frac{5}{8}T$, and (d) $\frac{7}{8}T$ in the SBB pattern. Blue and red colors denote negative and positive vorticity, respectively.

Figure 8

Instantaneous pressure contours at (a) $t=\frac{1}{8}T$, (b) $t=\frac{3}{8}T$, (c) $t=\frac{5}{8}T$, (d) $t=\frac{7}{8}T$ in the SBB pattern ($G_0=10$ and $A=0.6$). The inset in (b) represents an instantaneous longitudinal force $F_x$ of the plate. Solid and dashed lines denote the positive and negative normalized pressure contours, respectively.

Figure 9

(a) The schematic diagram of the vortex-induced velocity. (b) The minimum amplitude required for the HS mode in the three flow patterns as a function of $y$ component of vortex-induced velocity $V_{\mathrm{\Gamma },\perp }$ (the case in the wake behind a single cylinder is also shown). Here, $\alpha$ is the orientation angle between the dipole-induced velocity $V_{\mathrm{\Gamma }}$ and the $x$ axis, $d$ is the distance between two vortex centers, and $\mathrm{\Gamma }$ is the circulation of the vortex.

Figure 10

(a) Input power $P$ as functions of $A$; (b) vertical force experienced by the plate ($A=0.6$) as functions of times, where the black, blue and red lines represent the SBB, SLR, and SVS flow patterns, respectively. It is noted that $T=\frac{1}{f}$ of the SLR is larger than those of the SBB and SVS patterns (see Table 1).

Figure 11

Streamwise trajectories of the leading edge in cases with different $G_0$ for the (a) SBB pattern, (b) SLR pattern, and (c) SVS pattern. Each curve represents a case we simulated.

Figure 12

Streamwise location of the leading edge as a function of time for cases with different $G_0$. In the cases, the releasing style in Ref. [21] was adopted. (a) The plate is self-propulsive behind a single cylinder. The key parameters Re, $M, K, A, f$ are identical to those in Ref. [21]. (b) The plate is self-propulsive in the SBB pattern and (c) the SVS pattern of two tandem cylinders.

Figure 13

Instantaneous vorticity contours at the releasing moment for four cases with different releasing styles, (a) the case of $G_0=9$ with releasing Style II [21], (b) the case of $G_0=14$ with releasing Style II, (c) the case of $G_0=9$ with releasing Style I, (d) the case of $G_0=12$ with releasing Style I. The inset in (a) represents an instantaneous vorticity contour without the plate. Blue and red colors denote negative and positive vorticity, respectively.

Figure 14

The phases of $v$ at the releasing moment ($t/T=0$) for fixed points with different releasing styles, (a) the phase at (6, 0) and $(G_0-0.5,0)$ with releasing Style II, (b) the phase at (6, 0) and $(G_0-0.5,0)$ with releasing Style I, respectively. Every two points in a vertical dashed line represents a case we simulated.

Figure 15

Instantaneous vertical velocity $v$ before the releasing moment ($t/T=0$) at fixed points (releasing Style I). The left and right columns represent the time history of $v$ at (6,0), and ($G_0-0.5$,0), respectively. The upper, middle, and lower rows represent cases in the SBB, SLR, and SVS wake patterns, respectively. For all cases, $A=0.6$.