Study on the self-propulsion of the rigid-flexible composite plate

The governing equations of the fluid-plate problem are solved numerically by the immersed boundary method (IBM) for the fluid flow and the finite element method for the motion of the flexible plate. When the IBM is used to treat flow-structure interaction, a reactive force is used to represent the effect of the structure on the fluid dynamics (Peskin 2002), namely a body force f_b is added into the right hand of equation (11). The grids around the plate are uniform to use IBM, while the other grids are nonuniform to save the computing resources, as shown in figure 5(b). Equation (11) can be rewritten as

$$\begin{equation}\frac{{\partial {\boldsymbol{U}}}}{{\partial t}}{{\ +\ }}{\boldsymbol{U}} \cdot \nabla \left( {{\boldsymbol{U}} - {\boldsymbol{\tilde U}}} \right) = - \frac{1}{\rho }\nabla p + v\Delta {\boldsymbol{U}} + {{\boldsymbol{f}}_b}\end{equation} \tag{ 14 }$$

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where ${\boldsymbol{\tilde U}}$ is the velocity vector of the grid, which is calculated using scaling method. A center point and two radii R₁ and R₂ are defined to implement the dynamic mesh computation. The design formulas are

$$\begin{equation}S{{\ =\ }}\min \left\{ {\max \left[ {\frac{{{R_2} - \left| {{\boldsymbol{P}} - {{\boldsymbol{P}}_0}} \right|}}{{{R_2} - {R_1}}},0} \right],1} \right\}\end{equation} \tag{ 15 }$$

$$\begin{equation}{{\boldsymbol{d}}_g}{{\ =\ }}\min \left\{ {\max \left[ {0.5 - 0.5\cos \left( {\pi S} \right),0} \right],1} \right\} \cdot {{\boldsymbol{d}}_o}.\end{equation} \tag{ 16 }$$

The displacement of the grids is denoted as d _g , and d _o is the displacement of the head point of the plate. P is the initial position of the grid, and P ₀ is the position of the center point of the circular computational domain.

The IBM is embedded into OpenFOAM (Jasak 2009) following the procedure proposed by Pinlli et al (2010) and Constant et al (2017). In Pinlli et al (2010), the hydrodynamic force in the right hand side of equation (1) is calculated as

$$\begin{equation}{{\boldsymbol{F}}_f}\left( {s,t} \right) = \frac{{{\boldsymbol{\dot r}}\left( {s,t} \right) - {\boldsymbol{u}}\left( {s,t} \right)}}{{\Delta t}}\end{equation} \tag{ 17 }$$

where ${\boldsymbol{\dot r}}\left( s \right)$ is the velocity of the plate. ${\boldsymbol{u}}\left( s \right)$ is the velocity interpolated from the predicted flow velocity, namely Lagrangian–Euler interpolation

$$\begin{equation}{\boldsymbol{u}}\left( {s,t} \right) = \varphi \left( {{{\boldsymbol{U}}^*}} \right)\end{equation} \tag{ 18 }$$

ϕ is the interpolation function, U ^* is the predicted velocity of the flow without the disturbance form the plate

$$\begin{equation}{{\boldsymbol{U}}^*} = {\left[ { - \frac{1}{\rho }\nabla p + v\Delta {\boldsymbol{U}} - {\boldsymbol{U}} \cdot \nabla \left( {{\boldsymbol{U}} - {\boldsymbol{\tilde U}}} \right)} \right]^n} \cdot \Delta t + {{\boldsymbol{U}}^n}.\end{equation} \tag{ 19 }$$

The superscript n denote the value in previous time step.

After the Lagrangian force F _f has been achieved, it can be transformed by a spreading operator C into the body force f _b

$$\begin{equation}{{\mathbf{f}}_b} = C\left[ {{{\mathbf{F}}_f}} \right].\end{equation} \tag{ 20 }$$

Here, we just give a brief explanation of implementation process of the immersed boundary method. The detailed description can be consulted from the literature of Pinlli et al (2010). A second-order Gauss integration scheme with a linear interpolation for the face-centered value of the unknown is used for the divergence, gradient, and Laplacian terms in the governing equations. The second-order backward Euler method is adopted for time integration. Thus, the numerical discretization scheme gives second order accuracy in space and time.

The beam element is used to simulate the dynamics of the plate, and the element is uniformly distributed along the axis of the plate. After the dissociation of the finite element method, equation (1) can be expressed as a set of algebraic equations

$$\begin{equation}{\boldsymbol{r}}\left( {s,t} \right) = {{\boldsymbol{r}}_0}\left( s \right) + {\boldsymbol{N}}\left( s \right){\boldsymbol{q}}\left( t \right)\end{equation} \tag{ 21 }$$

$$\begin{equation}{{\boldsymbol{r}}_{\text{0}}}\left( s \right) = {\left( {s,0,0} \right)^T}\end{equation} \tag{ 22 }$$

$$\begin{equation}{\boldsymbol{M\ddot q}} + {\boldsymbol{Kq}} = {\boldsymbol{F}}\end{equation} \tag{ 23 }$$

$$\begin{equation}{\boldsymbol{K}} = EI\int_0^l {{{{\boldsymbol{N^{{\prime\prime}}}}}^T}{\boldsymbol{N^{{\prime\prime}}}}{\text{d}}s} + EA\int_0^l {\left[ \begin{gathered} \frac{1}{{{T^{\text{3}}}}}{{{\boldsymbol{N^{{\prime}}}}}^T}\left( {{\boldsymbol{N^{{\prime}}q}}{{\ +\ }}{{{\boldsymbol{r^{{\prime}}}}}_{\text{0}}}} \right){\left( {{\boldsymbol{N^{{\prime}}q}} + {{{\boldsymbol{r^{{\prime}}}}}_{\text{0}}}} \right)^T}{\boldsymbol{N^{{\prime}}}} \hfill \\ + \left( {1 - \frac{1}{T}} \right){{{\boldsymbol{N^{{\prime}}}}}^T}{\boldsymbol{N^{{\prime}}}} \hfill \\ \end{gathered} \right]{\text{d}}s} \end{equation} \tag{ 24 }$$

$$\begin{equation}T{{\ =\ }}\sqrt {{{\left( {{\boldsymbol{N^{{\prime}}}}{{\boldsymbol{q}}_{}} + {{{\boldsymbol{r^{{\prime}}}}}_{\text{0}}}} \right)}^T}\left( {{\boldsymbol{N^{{\prime}}q}} + {{{\boldsymbol{r^{{\prime}}}}}_{\text{0}}}} \right)} \end{equation} \tag{ 25 }$$

where M is the mass matrix, K is the stiffness matrix, F is the generalized force of the hydrodynamic force. r ₀ is the original position of the point, which is added to the deformation of the plate to get the absolute position of the point. N is the shape function. q is the generalized displacement.

The Newmark-β method and the Newton–Raphson method are used to solve equation (23). The acceleration ${{\boldsymbol{\ddot q}}_{t + \Delta t}}$ and velocity ${{\boldsymbol{\dot q}}_{t + \Delta t}}$ at time t +Δt can be expressed as

$$\begin{equation}{{\boldsymbol{\ddot q}}_{t + \Delta t}} = {a_0}\left( {{{\boldsymbol{q}}_{t{{\ +\ }}\Delta t}} - {{\boldsymbol{q}}_t}} \right) - {a_2}{{\boldsymbol{\dot q}}_t} - \left( {{a_3} - 1} \right){{\boldsymbol{\ddot q}}_t}\end{equation} \tag{ 26 }$$

$$\begin{equation}{{\boldsymbol{\dot q}}_{t + \Delta t}} = {a_1}\left( {{{\boldsymbol{q}}_{t{{\ +\ }}\Delta t}} - {{\boldsymbol{q}}_t}} \right) - {a_5}{{\boldsymbol{\ddot q}}_t} - \left( {{a_4} - 1} \right){{\boldsymbol{\dot q}}_t}.\end{equation} \tag{ 27 }$$

The subscript t denotes the values at time t. Δt is the time step. The expressions of a₀, a₁, a₂, a₃, a₄ and a₅ are

$$\begin{equation}{a_0} = \frac{1}{{\beta \Delta {t^2}}},{a_1} = \frac{\delta }{{\beta \Delta t}},{a_2} = \frac{1}{{\beta \Delta t}},{a_3} = \frac{1}{{2\beta }},{a_4} = \frac{\delta }{\beta },{a_5} = \left( {\frac{\delta }{{2\beta }} - 1} \right)\Delta t\end{equation} \tag{ 28 }$$

where δ = 0.505 and β = 0.2525 are the Newmark coefficients. Equation (23) can be expressed as the non-linearity system of equations

$$\begin{equation}{\boldsymbol{f}}\left( {{{\boldsymbol{q}}_{t + \Delta t}}} \right) = 0.\end{equation} \tag{ 29 }$$

According to Newton–Raphson method, equation (29) is solved iteratively

$$\begin{equation}{\boldsymbol{q}}_{t + \Delta t}^{\,j + 1} = {\boldsymbol{q}}_{t + \Delta t}^j - \frac{{{\boldsymbol{f}}\left( {{\boldsymbol{q}}_{t + \Delta t}^j} \right)}}{{{\boldsymbol{f^{\,{\prime}}}}\left( {{\boldsymbol{q}}_{t + \Delta t}^j} \right)}}\end{equation} \tag{ 30 }$$

${\boldsymbol{f^{\,{\prime}}}}$is the first derivative of ${\boldsymbol{f}}$ respect to ${{\boldsymbol{q}}_{t + \Delta t}}$. The initial value of the iteration is ${{\boldsymbol{q}}_t}$. As ${{\boldsymbol{q}}_{t + \Delta t}}$ has been achieved, ${{\boldsymbol{\ddot q}}_{t + \Delta t}}$ and ${{\boldsymbol{\dot q}}_{t + \Delta t}}$ are calculated using equations (26) and (27).

As shown in figure 2, the structure solver is validated by calculating the large deflection of a cantilever beam with the length l and the bending stiffness of EI under a point force load F transversally at the free end. The results of the axial displacement u of the cantilever beam, the lateral displacement w and the deflection angle θ in three dimensionless form $u/l$, $w/l$ and θ are compared with the results of Mattiasson (2010) at different dimensionless force $F{l^2}/EI$ are shown in figure 3. The excellent agreement can be found.

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The locomotion of the flexible plate has been simulated in Hua et al (2013). The leading-edge is forced to heave sinusoidally with amplitude A₀ and frequency f in the vertical direction, which can be described as $h\left( t \right) = {A_0}\cos \left( {2\pi ft} \right)$. The frequency Reynolds number is 100, the heave amplitude of the leading edge ${A_0}{{\ =\ 0}}{\text{.25}}$. The propulsive velocity and efficiency of the flexible plate are used to compare with those in Hua et al (2013). The dimensionless propulsive velocity is defined as ${\tilde U_p} = \frac{{{U_p}}}{{lf}}$, where U_p is the propulsive velocity of the plate. The propulsive efficiency for the locomotion of flexible plate is expressed as (Kern and Koumoutsakos 2006)

$$\begin{equation}\eta = \frac{{0.5mU_p^2}}{{\int_{{t_0}}^{{t_0} + T} {\int_0^l {{{\boldsymbol{f}}_b} \cdot \frac{{\partial {\boldsymbol{r}}}}{{\partial t}}{\text{d}}s{\text{d}}t} } }}.\end{equation} \tag{ 31 }$$

As shown in figure 4, the propulsive velocity and efficiency achieved in this paper match well with the results of Hua et al (2013). It is expected that the model and method in this paper can accurately simulate the locomotion of the flexible plate.

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A circular computational domain has been used here, whose radius is R= 250 l. The division of the computational domain is shown in figure 5(a). The length of the sides of the rectangle is l_g = 3 l, and the grids inside the rectangle domain is uniform, as shown in figure 5(b). The outer domain is divided using an O-type mesh. The size of the grids in the rectangle domain is d_g , while the sizes of the other grids adjust following these sizes smoothly. To employ the IBM proposed in Pinlli et al (2010), the size of the grid on the plate used to solve the structure dynamics is as the same as that of the gird in the rectangle domain. Five different grids are used for the grid testing. The parameters of the case are l^* = 0.5 and k= 30. The results are shown in table 1. It can be found that, the propulsive velocity and the propulsive efficiency at grid3 are accurate enough, namely, the effect of the grid refinement can be ignored. In this paper, the parameters for grid3 are used for all cases.

Table 1. The result for different grids.

	Grid1	Grid2	Grid3	Grid4	Grid5
dg	0.005	0.0067	0.01	0.0133	0.02
${\tilde U_p}$	−1.204	−1.211	−1.214	−1.214	−1.223
η	40.27%	40.74%	41.35%	41.33%	42.61%

Figure 6 shows the propulsive velocity of the composite plate. For l*= 0.3 ∼ 0.6, the propulsive velocity increases following the stiffness and tends to be stable finally. For l*= 0.7 ∼ 1, the propulsive velocity increases following stiffness in the range of K< 60, then slightly decreases in the range of K> 60 and finally tends to be stable. For arbitrary K, the propulsive velocity increases following increasing l*, which means that the anterior rigid part of the composite plate actually produces smaller thrust than the flexible plate. It is worth mentioning that the resonance effect is not observed in the locomotion of the composite plate driving by pitch motion, which has been mentioned in Hua et al (2013) for a flexible plate driving by heave motion of the leading edge. It may be due to the pitching motion make the plate have an attack angle, which induce a large enough gradient at the tip (Lighthill 1960). Figure 7 shows the Froude efficiency of the composite plate. For arbitrary l*, the Froude efficiency increases following the increasing stiffness and finally tends to be stable. For arbitrary stiffness K, the Froude efficiency increases following the increasing l*.

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The Froude efficiency discussed above only relates thrust to the total kinetic energy transmitted to the fluid, while the elastic potential energy stored in the flexible plate has been ignored. Here we want to investigate the efficiency of the external force used to drive its locomotion. In practice, the live fish or the bionic fish is driven by the muscle or servo actuator. It is supposed that the forces exerted on the junction point only. We can achieve the forces to maintain the motion of the composite plate as

$$\begin{equation}{F_{ey}} = {\rho _r}{l_r}{\ddot y_c} - \frac{1}{2}{\rho _r}l_r^2\frac{{{\partial ^{\text{3}}}{y_c}}}{{\partial {s_r}\partial {t^{\text{2}}}}} - \int_0^{{l_r}} {{p_y}{\text{d}}{s_r}} + EI{\left. {\frac{{{\partial ^3}y}}{{\partial {s^3}}}} \right|_{s = 0}}\end{equation} \tag{ 32 }$$

$$\begin{equation}{M_{ex}} = - \frac{1}{{\text{2}}}{\rho _r}l_r^{\text{2}}{\ddot x_c}{{\ +\ }}\frac{1}{{\text{3}}}{\rho _r}l_r^{\text{3}}\frac{{{\partial ^{\text{3}}}{x_c}}}{{\partial {s_r}\partial {t^{\text{2}}}}} - \int_0^{{l_r}} {{p_x}\left( {{s_r} - {l_r}} \right){\text{d}}{s_r}} - EI{\left. {\frac{{{\partial ^2}x}}{{\partial {s^2}}}} \right|_{s = 0}}\end{equation} \tag{ 33 }$$

$$\begin{equation}{M_{ey}} = - \frac{1}{{\text{2}}}{\rho _r}l_r^{\text{2}}{\ddot y_c}{{\ +\ }}\frac{1}{{\text{3}}}{\rho _r}l_r^{\text{3}}\frac{{{\partial ^{\text{3}}}{y_c}}}{{\partial {s_r}\partial {t^{\text{2}}}}} - \int_0^{{l_r}} {{p_y}\left( {{s_r} - {l_r}} \right){\text{d}}{s_r}} - EI{\left. {\frac{{{\partial ^2}y}}{{\partial {s^2}}}} \right|_{s = 0}}.\end{equation} \tag{ 34 }$$

In equations (32)–(34), the terms on the right side can be calculated at every moment. So we can achieve the work done by the external forces in a period as

$$\begin{equation}{W_e} = \int_{{t_0}}^{{t_0} + T} {\left( {{F_{ey}} \cdot {{\left. {\frac{{\partial y}}{{\partial t}}} \right|}_{s = 0}}{{\ +\ }}{M_{ex}} \cdot {{\left. {\frac{{{\partial ^2}x}}{{\partial s\partial t}}} \right|}_{s = 0}}{{\ +\ }}{M_{ey}} \cdot {{\left. {\frac{{{\partial ^2}y}}{{\partial s\partial t}}} \right|}_{s = 0}}} \right){\text{d}}t} .\end{equation} \tag{ 35 }$$

We define a locomotor efficiency here, which represents the ratio of the kinetic energy to the work done by the external force

$$\begin{equation}{\eta _e} = \frac{{0.5mU_p^2}}{{{W_e}}}.\end{equation} \tag{ 36 }$$

It is expected that the efficiency can give a more intuitive description on the energy cost of the locomotion of fish than Froude efficiency.

Figure 7 shows the locomotor efficiency of the composite plate. Due to the external forces do not exert on the nose of the live fish or bionic fish, the results of the onefold flexible plate are not included in the figure. It can be found that a lot of work is used to maintain the elastic deformation of the flexible plate and the locomotor efficiency has an order of 10⁻⁴, so the Froude efficiency maybe not enough to describe the efficiency of the locomotion of the composite plate. For l*= 0.3 ∼ 0.5, the locomotor efficiency first increases sharply then decreases with increasing stiffness, and finally tends to be stable. For l*= 0.6 ∼ 0.9, the locomotor efficiency decreases with increasing stiffness and finally tends to be stable. As the stiffness is small enough (K< 20), the locomotor efficiency increases with increasing l*. As the stiffness is large (K> 30), the locomotor efficiency first decreases sharply then increases slowly with increasing l*. The small figure at the up-right of figure 7 shows the tendency of the locomotor efficiency following l* at K = 10, 50, 100 and 500. The larger the stiffness is, the less variation of locomotor efficiency following l*. For small stiffness, K = 10, the locomotor efficiency increases following increasing l*. For large stiffness, K > 10, the locomotor efficiency first decreases and then increases following increasing l*.

From above discussion, we find that the tendency of the locomotor efficiency is very different from that of the Froude efficiency. So we think that it is necessary to analyze the locomotor efficiency to study the locomotion of fish thoroughly. As the stiffness is small, such as K = 10, larger flexible portion (l* = 0.9) results in larger locomotor efficiency. While the stiffness is large, such as K = 50, smaller flexible portion (l* = 0.3) gives a larger locomotor efficiency. This happens to coincide with the real swimming behavior of live fish, although the plate may be simple to model the whole fish body. In the nature, the fish with soft body, such as the eel and the larva of fish, often swim in anguilliform mode. While the fish with stiff body, such as Clupe and Scomber, often swim in carangiform mode.

The morphology response of the composite plate is an important issue for its self-propulsion. A thrust can be caused by the backward propagating wave on the composite plate, which drives the plate swim under the water. A 3D surface is constructed, where the axis x is the position on the plate. The axis y is the time, and the axis z is the deformation of the plate. It is expected that the evolution of the deformation of the plate can be demonstrated more clearly than the traditional envelope chart.

As that derived by Lighthill (1960) based on potential flow theory, the negative correlation between the gradient of the deformation at the tip ${\left. {\frac{{\partial y}}{{\partial s}}} \right|_l}$ and the lateral velocity ${\left. {\frac{{\partial y}}{{\partial t}}} \right|_l}$ can leads to a negative propulsive velocity and Froude efficiency. The large amplitude and the small gradient of the deformation at the tip are beneficial to improve the propulsion of the plate. For all l* and K discussed in this paper, the correlation between ${\left. {\frac{{\partial y}}{{\partial s}}} \right|_l}$ and ${\left. {\frac{{\partial y}}{{\partial t}}} \right|_l}$ is positive. The lateral amplitude and the gradient of the deformation at the tip affect the propulsive velocity and Froude efficiency simultaneously. For a specified l*, both the lateral amplitude and the deformation gradient of the tip decrease following the increasing stiffness K, as shown in figure 8. Due to the effect of the deformation gradient is predominant, so the propulsive velocity and the Froude efficiency increase with stiffness for arbitrary l*, as shown in figure 6. For a specified K, both the lateral amplitude and the deformation gradient of the tip decrease following the increasing stiffness l*, so the propulsive velocity and the Froude efficiency increase with stiffness for arbitrary K, as shown in figure 6.

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The discussion in this section matches well with the result mentioned in Lighthill (1960), which proves the validity of the hydrodynamic study in this paper.

To investigate the effects of the rigid portion and the flexible portion on the locomotion of the composite plate, the average longitudinal hydrodynamic forces ${\bar p_s}$ distributed on the plate are shown in figure 9. The black dotted line represents the location of the junction. The left is the rigid portion, while the right is the flexible portion. The definition of ${\bar p_s}$ is

$$\begin{equation}{\bar p_s} = \frac{1}{N}\sum\limits_{n = 1}^N {{p_s}} \end{equation} \tag{ 37 }$$

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where, ${p_s}$ is the longitudinal hydrodynamic forces distributed on the plate, N is the number of the data in integral multiple periods. To reduce the error, the data in eight periods are used in this paper.

An unexpected result is that the hydrodynamic force on the tail of the plate is positive for arbitrary l* and K, which is hindering the propulsion of the plate. Lighthill (1960) stated that the average thrust of the plate only rely on the motion of the tail point. So we conclude that the motion of the tail point plays a decisive role on the force distribution on the plate, rather than the thrust only produced at tail point. From figure 9(a), it can be found that the hydrodynamic force on the tail point is small as l*= 0.3 and K= 10. This may be the reason why most fish has a short ultra-soft membrane at the tail, which can reduce the resistance effectively. Although the positive values on the tail point of the flexible plate is large, but the range of its action is so small, whose effect on the resultant hydrodynamic force on the plate is limited.

The average thrust distribution on the rigid portion of the plate is smooth, while it fluctuates on the flexible portion. For small stiffness, such as K= 10 at l*= 0.3, the hydrodynamic force on the rigid portion is propelling the plate, while the force on the flexible portion is hindering the propulsion of the plate. For large stiffness, such as K> 50, the reverse is true. For other l*, the force on the rigid plate is hindering the propulsion, while the force on the flexible portion is fluctuating but repelling the plate overall as the stiffness is small (such as K= 10). As the stiffness is large enough, such as K> 50, the longitudinal force on whole composite plate varies gently with the location and repel the plate forward overall. The larger l* is, the greater the volatility of the force on the flexible plate is.

To investigate the significances of the pressure force and shear stress, figure 10 shows the distribution of the average pressure force and shear stress along the plate at K = 10. The pressure force on the plate is almost distributing uniformly for arbitrary l*. For l* = 0.3, the shear stress on the rigid plate is thrusting the plate forward, while the shear stress on the flexible plate is hindering the self-propulsion. For other l*, the shear stress on the rigid plate is hindering its propulsion, while the shear stress on the flexible plate is thrusting the plate.

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Figure 11 shows the external forces, see equations (32)–(34), exerted on the junction point to maintain the self-propulsion of the plate, the moment of force M is calculated by $M = \sqrt {M_{ex}^{\,2} + M_{ey}^{\,2}}$. For arbitrary l*, both the moment of force and the lateral force increase linearly following increasing stiffness K. For arbitrary stiffness, both the moment of force and the lateral force first increase from l* = 0.3 to l* = 0.8 then decrease from l* = 0.8 to l* = 0.9. These results echo the variation tendency of locomotor efficiency shown in figure 7. The flexible deformation of the plate needs large external force and energy to maintain, which reduces the energy utilization in the propulsion of the plate.

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In this subsection, we want to investigate the flow structure around the composite plate, as shown in figure 12. The cases for four typical l* and four typical K are selected, and all vorticity contours shown here is grabbed as the rigid plate reaches its lowest location.

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For all cases, the reverse Karman vortices can be observed in the wake of the plate, which is emblematic for the locomotion of the fish. The larger the stiffness is, the smaller the difference between the vortex contours at different l*. Although the lateral motion of the head point is confined, the vortices pair shedding from it can also be observed. In most cases, except l*= 0.3, 0.5 and K= 10, the vortices pair can merge into the vortices shedding from the tail.

As l* and K are both small, such as l*= 0.3, 0.5 and K= 10, the vortices around the plate is tanglesome. The vortices shedding from both the head and tail point of the plate deposit in a limited region due to the low propulsive velocity of the plate. These vortices interact with each other drastically, which induces the unsteady motion, as the IM state shown in Hua et al (2013). The tail of the plate is so soft that the strength of the vortex can curl it, for example in l*= 0.3 and K= 10, which cause a positive (hindering) force on the flexible plate, see figure 9(a). As l*= 0.3 and K= 50, the wake behind the plate is asymmetry, this is due to the inequality of the propulsive velocity during different half period (Hua et al 2013). For other cases, 2P mode vortex shedding mode from the tail can be observed. Due to the participation of the vortex shedding from the head, 2T vortices can be seen in the wake very close to the tail.