Coupled states of dual side-by-side inverted flags in a uniform flow

Journal of Fluids and Structures 91 (2019) 102768

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Coupled states of dual side-by-side inverted flags in a uniform flow Kang Jia a,b , Le Fang a , Wei-Xi Huang b , a b

∗

LMP, Ecole Centrale de Pékin, Beihang University, Beijing 100191, PR China AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China

article

info

Article history: Received 12 April 2019 Received in revised form 3 September 2019 Accepted 9 October 2019 Available online xxxx Keywords: Inverted flag Side-by-side Dynamic regime Bistability

a b s t r a c t The dynamics of dual side-by-side inverted flags with free leading edges and clamped trailing edges in a uniform flow are simulated using the immersed boundary method. The dynamics of the two inverted flags are systematically analyzed in terms of the bending rigidity, the gap distance between flags and the six initial conditions which are set by adjusting the initial angle of attack of flags. Four distinct dynamic regimes are observed: straight, flapping, chaotic and deflected regimes. According to the relative bending directions in which the inverted flags are bent, the deflected regime is further divided into three sub-regimes: outside deflected, inside deflected and one-side subregimes. It is found that the initial conditions exert an influence on which sub-regime to occur. Bistable state occurs in transition from the straight regime to the flapping regime, from the chaotic regime to the flapping regime, and from the flapping regime to the deflected regime. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The coupling of a flexible sheet-like structure and an impinging flow is an important subject in the field of fluid– structure interaction (FSI). Such FSI system is widely spread in medical and industrial applications, such as the phenomenon of snoring (Huang, 1995), the paper flutter in paper processing (Watanabe et al., 2002), the mixing enhancement by elastic free flaps in process industries (Ali et al., 2015) and the heat transfer enhancement by clamped flexible flags in a channel flow (Park et al., 2016; Lee et al., 2018). These structures exhibit various behaviors depending on the flow conditions and the boundary conditions of the flexible sheet. As a typical example, the conventional flag configuration where a flexible sheet or filament with a fixed leading edge and a free trailing edge in the fluid flow, has received considerable attention. The dynamic behaviors of this structure have been studied using experimental (Zhang et al., 2000; Shelley et al., 2005; Eloy et al., 2008) and numerical methods (Connell and Yue, 2007; Huang et al., 2007; Alben and Shelley, 2008; Michelin et al., 2008; Huang and Sung, 2010). Allen and Smits (2001) and Michelin and Doaré (2013) proposed an energy harvesting system using the flexible sheetlike structure with a piezoelectric material attached to its surface. This energy harvesting process consists of two steps. First, the kinetic energy of incoming flow is transferred to strain energy of the flexible structure. Then, the strain energy is converted to electrical energy with piezoelectric material. To improve the energy harvesting of the first step, Kim et al. (2013) suggested a configuration of an inverted flexible flag with a free leading edge and a clamped trailing edge. It was found that the critical flow velocity required for flapping initiation of the inverted flag was much lower than that of ∗ Corresponding author. E-mail address: [email protected] (W.-X. Huang). https://doi.org/10.1016/j.jfluidstructs.2019.102768 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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K. Jia, L. Fang and W.-X. Huang / Journal of Fluids and Structures 91 (2019) 102768

the conventional flag. Kim et al. (2013) also observed three distinct dynamic modes of the inverted flag: straight mode, flapping mode and deflected mode. Inspired by Kim et al. (2013), various numerical and theoretical studies of the couplings between the inverted flag and the fluid flow were carried out. Ryu et al. (2015) numerically explored the flapping dynamics of the inverted flag and analyzed the vortical structures in the wake. Tang et al. (2015) showed that the dynamics of an inverted flag depends mainly on the bending stiffness and the aspect ratio of the flag. Furthermore, Gurugubelli and Jaiman (2015) and Sader et al. (2016) identified the flapping of the inverted flag as a vortex-induced vibration. Meanwhile, Shoele and Mittal (2016) examined the performance of the energy harvesting by flow-induced fluttering of an inverted piezoelectric flag. Goza et al. (2018) carried out a global stability analysis of the inverted flag. More recently, Gurugubelli and Jaiman (2019b) performed a three-dimensional numerical study on the mechanism of large-amplitude flapping of an inverted flag. Insights into the interaction between multiple inverted flags can further promote the design of energy harvesting plants. Parallel and tandem inverted flags are elementary building blocks of more complex configurations. HuertasCerdeira et al. (2018) experimentally explored the interaction between dual inverted flags placed in a side-by-side arrangement. They found the enhancement of flapping amplitude and frequency by placing dual inverted flags. Coupling between flags that had different lengths was also observed to occur. Ryu et al. (2018) conducted a numerical simulation of two side-by-side inverted flags in a uniform flow. They focused on the effects of bending rigidity and gap distance between the dual inverted flags on system energy harvesting performance. In-phase and out-of-phase flapping motions were observed by varying the gap distance and the initial angle of attack (AOA). Huang et al. (2018) explored numerically the harvesting performance of tandem inverted flags. The drag force and bending strain energy of the tandem inverted flags as functions of spacing distance were presented. They also simulated the cases of infinite tandem inverted flags with a periodic boundary conditions in which two periodic arrangements of flags were investigated: in-line and staggered. Besides, the dynamic behavior of dual inverted flags can be a considerable issue to be investigated. On the other hand, a similar problem, the coupling of two/three side-by-side traditional flags or filaments in the flow has been studied extensively, and various dynamics have been discovered (Zhang et al., 2000; Zhu and Peskin, 2003; Farnell et al., 2004; Jia et al., 2007; Alben, 2009; Tian et al., 2011; Dong et al., 2016; Sun et al., 2016; Gurugubelli and Jaiman, 2019a). The stability and coupled dynamics of dual inverted flags were explored experimentally and theoretically by Kim and Kim (2019). They also observed the synchronized in-phase, out-of-phase modes, and an attached mode in dual side-by-side inverted flags. It was found that the critical velocity for the transition from the straight regime to the flapping regime was affected by the height of flag and distance between dual flags. This property is critical for energy harvesting performance of inverted flags, since it is generally considered that the strain energy can be effectively achieved in the flapping mode (Ryu et al., 2015; Tang et al., 2015). As seen above, the previous researches (Huertas-Cerdeira et al., 2018; Ryu et al., 2018; Kim and Kim, 2019) mainly focused on the flapping regime and the stability boundary of dual side-by-side inverted flags. However, the transition from the flapping regime to the deflected regime of dual inverted flags, which can further clarify the available zone of energy harvesting, has not been systematically studied. Besides, according to the previous studies (Kim et al., 2013; Ryu et al., 2015; Tang et al., 2015), the deflected regime is defined when the inverted flag bends in one direction and remains a highly curved shape. This bending direction may not be important for the dynamic of a single inverted flag. But for dual inverted flags placed side-by-side, different relative bending directions may produce completely different dynamics. A compromise on this issue has not been reached yet. Huertas-Cerdeira et al. (2018) discovered experimentally the flags perform inside, outside and asymmetric deflected (one flag inside and one outside) modes. Kim and Kim (2019) observed that each flag could be deflected outward or both flags could be deflected to the same side in their experiment. In the numerical study of Ryu et al. (2018), the flags only display deflected outwards. In present study, we simulate the coupled states of two side-by-side inverted flags in a uniform flow using the immersed boundary method, with the aim to achieve an improved understanding of the issues not yet clarified so far. We systematically summarize the coupled states of dual parallel inverted flags and explore the effects of bending rigidity, gap distance between the clamped edges and initial AOA on the coupled states. The deflected regime is further divided into three sub-regimes according to the relative bending directions. The dynamic characteristics of the three sub-regimes are analyzed systematically. Moreover, we also try to search the bistable behavior at the boundaries of regime transition of dual inverted flags. The paper is organized as follows: the problem formation and computational method are described in Section 2; the results and discussion are presented in Section 3; finally, the conclusions are drawn in Section 4. 2. Problem formulation and computational method A schematic diagram of two inverted flags system with the coordinates is shown in Fig. 1a. We consider dual sideby-side inverted flags with free leading edges and clamped trailing edges in a uniform flow of freestream velocity U∞ . Both two flags have the same length L. The gap distance G is the vertical spacing between the clamped edges of the flags. In this paper, the dynamics of the inverted flag is described by the parameter θ which is the angle between the velocity vector of incoming flow and the line joining the free edge and the clamped edge of the flag (see Fig. 1b). This parameter was also used in Huertas-Cerdeira et al. (2018). The previous studies of a single inverted flag (Kim et al., 2013; Ryu et al., 2015; Tang et al., 2015) and dual inverted flags (Kim and Kim, 2019) adopted the y-coordinate of the free edge to describe the motion of inverted flag. In the case where the deformation of the flag is not too large, both methods are completely

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Fig. 1. (a) Schematic of two side-by-side inverted flags in a uniform flow with a freestream velocity U∞ . The gap distance between the clamped edges of the flags is G. (b) Definition of the parameter θ used in this article.

feasible. When the deformation of flag surpasses 90◦ angle, the y-coordinate of the free edge is slightly decreased as the deformation increases (as shown in Fig. 2(d) of Kim et al. (2013)). This non-monotonic relationship may have some misleading effects on the research. In the present study, the incompressible Navier–Stokes equations and the continuity equation govern the fluid motion, i.e.

ρf

(

) ∂u + u · ∇ u = −∇ p + µ∆u + f , ∂t

∇ · u = 0,

(1)

(2)

where ρf is the density of fluid, u is the velocity vector, p is the pressure, µ is the dynamic viscosity of the fluid, and f is the momentum forcing which is used to enforce the no-slip boundary condition along the immersed boundary, i.e., the flag surface. The deformation of inverted flag is governed by structure motion equation and the inextensibility condition, i.e.,

ρ1

( ) ( 2 ) ∂ ∂X ∂2 ∂ X ∂ 2X = T − γ 2 − F + Fc, ∂t2 ∂s ∂s ∂ s2 ∂s

∂X ∂X · = 1, ∂s ∂s

(3)

(4)

where s is the arc length, X is the position of the flag, T is the tension force along the flag, γ is the bending rigidity of the flag, ρ1 is the density difference between the fluid and the flag, F is the Lagrangian force exerted on the flag by the surrounding fluid, F c is the repulsive force between two adjacent flags. The boundary conditions of the free edge of flag can be expressed as T = 0,

∂ 2X ∂ 3X = (0, 0) , 3 = (0, 0) , 2 ∂s ∂s

(5)

which indicate that the tension force, the bending moment and the shearing force are zero at the free edge. On the other hand, the boundary conditions of clamped edge are X = X O,

∂X = (−1, 0) , ∂s

(6)

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Fig. 2. Results for computational domain independence test: time histories of the angle of the upper flag in three different computation domains with D = 3, γ = 0.28, Re = 200 and ρ = 0.1.

where X O is located at (0, G/2) and (0, −G/2) for two inverted flags, respectively. As for the repulsive force between the flags, we use the strategy proposed by Huang et al. (2007), i.e., F c (s, t ) =

L

∫

( ( )) X − X ′ ′ ⏐ ds . δ X (s, t ) − X ′ s′ , t ⏐ ⏐X − X ′ ⏐

0

(7)

For normalization of Eqs. (1)–(4), we introduce the following characteristic scales: the reference flag length L for length, 2 2 the fluid density ρf for density, the freestream velocity U∞ for velocity, L/U∞ for time, ρf U∞ for pressure, ρ1 U∞ for the 2 2 2 2 tension force T , ρ1 U∞ L for the bending rigidity γ , ρ1 U∞ /L for the Lagrangian force F and the repulsive force F c , ρf U∞ /L for the momentum force f . The dimensionless forms of Eqs. (1) and (3) are

∂u 1 + u · ∇ u = −∇ p + ∆u + f , ∂t Re

(8)

( 2 ) ( ) ∂ X ∂ 2X ∂ ∂X ∂2 = T − 2 γ 2 − F + Fc, ∂t2 ∂s ∂s ∂s ∂s

(9)

where Re represents the Reynolds number defined by Re = ρf U∞ L/µ. Eqs. (2) and (4) remain the same form after non-dimensionalization. The interaction force between the fluid and the immersed boundary can be formulated by the feedback law: F (s, t ) = α

t

∫

(U ib − U s ) dt ′ + β (U ib − U s ) ,

(10)

0

where α and β are large negative free constants, U s is the velocity of the flag expressed by U s = dX /dt, U ib is the fluid velocity obtained by interpolation at the immersed boundary, i.e., U ib (s, t ) =

∫ Ω

u (x, t ) δ (X (s, t ) − x) dx.

(11)

In addition, the Lagrangian forcing can be distributed to the nearby Eulerian grids by utilizing the smoothed delta function (Peskin, 2002), f (x, t ) = ρ

∫ Γ

F (s, t ) δ (x − X (s, t )) ds,

(12)

where ρ denotes the density ratio, i.e. ρ = ρ1 /(ρ0 L). In the present study, the equations of fluid motion are solved by the fractional step method (Kim et al., 2002), and the equations of structure motion are solved by the finite difference method (Huang et al., 2007). The detailed descriptions of the immersed boundary method and the computational procedure can be found in Huang et al. (2007). As depicted in Fig. 1(a), the Dirichlet boundary condition (ux = U∞ , uy = 0) is used at the inflow and far-field boundaries, and the convective boundary condition is applied at the outflow. The inlet velocity is started instantaneously at the beginning of

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Fig. 3. Phase plot of the cross-stream and streamwise displacements of the free edge of an inverted flag in a uniform flow for (a) γ = 0.1 and (b)

γ = 2 at Re = 1000 and ρ = 0.1.

each simulation. There are 65 Lagrangian nodes along each inverted flag. For dual side-by-side inverted flags with a spacing of less than 2L, the computational domain size is set to be [−3L, 5L] × [−4L, 4L]with a grid number of 1024 × 1024. While for a spacing larger than 2L, a bigger computational domain of [−3L, 5L] × [−8L, 8L]with a grid number of 1024 × 2048 is used. Several trial computations were carried out to verify the independence of the results on the computational domain size and grid number. The results of computational domain independence test for inverted flags with a large spacing (D = 3) is presented in Fig. 2. It is seen that the results are converged as the domain size increases. Furthermore, as a numerical validation, the typical case of an isolated inverted flag in a uniform flow is simulated. Fig. 3 shows the phase plot of the cross-stream and streamwise displacements of the free edge for (a) γ = 0.1 and (b) γ = 2 at Re = 1000 and ρ = 0.1. It is seen that the present numerical results agree well with those of Gurugubelli and Jaiman (2015). There are 4 dimensionless parameters in this problem: bending rigidity γ , gap distance between the two flags D = G/L, density ratio ρ , and Reynolds number Re. In this study, we give priority to explore the effects of the gap distance D and the bending rigidity γ on the dynamics of the system. Similar research strategies were also adopted in Huertas-Cerdeira et al. (2018), Ryu et al. (2018), and Kim and Kim (2019). However, it is worth noting that for multiple inverted flag systems, the effects of Reynolds number and mass ratio are still not completely clear. Tang et al. (2015) and Ryu et al. (2015) explored the effect of Reynolds number on a single inverted flag. Previous studies (Tang et al., 2015; Gurugubelli and Jaiman, 2015; Goza et al., 2018) also demonstrated that the mass ratio might affect the flapping dynamics of the inverted flag. Here we set the density ratio ρ = 1, the Reynolds number Re = 200, the bending rigidity γ ranging from 0.04 to 0.52, and the gap distance D ranging from 0.05 to 3.00. Specifically, six initial conditions are set by adjusting the initial AOAs of the two inverted flags, as plotted in Fig. 4. According to the initial bending directions, the initial conditions of each flag can be identified as straight, outward, and inward (S, O and I) states. Thus, the six initial conditions in Fig. 4 are sequentially denoted as SS, OS, OO, OI, IS and II. 3. Results and discussions 3.1. Four main dynamic regimes The three typical dynamic regimes of a single inverted flag (straight, flapping, deflected) as well as the chaotic regime described by Sader et al. (2016) and Goza et al. (2018) are also observed in the dual parallel inverted flags, consistent with the experimental results of Huertas-Cerdeira et al. (2018). The straight regime (see Figs. 5a and 6a) is occurring when the flags oscillate with small amplitudes relative to the length of flag. If both flags flap from side to side with large amplitude, the dynamic state is classified as the flapping regime (see Fig. 6b). The inverted flags swipe a larger angle towards inside (see Fig. 5b), which is consistent with the experimental result of Huertas-Cerdeira et al. (2018). The deflected regime is formed when both flags are bent in one direction (inside or outside) with a large deformation. According to the relative bending directions in which the inverted flags are deflected, the deflected regime is further divided into three sub-regimes: outside deflected, one-side deflected and inside deflected regimes (see Fig. 5d, e, and f, respectively), which will be discussed in more details in Section 3.2. The time histories of the angle θ for both the upper and lower flags of the three sub-regimes are showed in Fig. 6d, e, and f, respectively. The chaotic dynamic regime (see

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Fig. 4. Schematic diagram of six initial conditions in this study: (a) initial condition SS, (b) initial condition OS, (c) initial condition OO, (d) initial condition OI, (e) initial condition IS, (f) initial condition II, corresponding to (θ1initial , θ2initial ) = (0◦ , 0◦ ), (1.8◦ , 0◦ ), (1.8◦ , −1.8◦ ), (1.8◦ , 1.8◦ ), (−1.8◦ , 0◦ ), (−1.8◦ , 1.8◦ ), respectively. In order to present more clearly, the degree of deformation of the flag in the figure is larger than the actual situation.

Fig. 5. The dynamic regimes in terms of superimposed views: (a) γ = 0.52, D = 1.8 with initial condition SS, (b) γ = 0.24, D = 2.2 with initial condition OS, (c) γ = 0.20, D = 1.8 with initial condition OS, (d) γ = 0.08, D = 1.8 with initial condition SS, (e) γ = 0.08, D = 1.8 with initial condition OI, (f) γ = 0.08, D = 1.8 with initial condition II.

Figs. 5c and 6c) can be considered as a transition regime between the flapping regime and the deflected regime, in which flapping with large amplitude and bending in one direction of large curvature can be observed irregularly. Fig. 7 shows an overview of the four dynamic regimes on the plane of the bending rigidity and the gap distance. The deflected, chaotic, flapping and straight regimes, whose regions are represented respectively by red, yellow, green and blue zones in Fig. 7, occur in turn as the bending rigidity increases. It is identified from Fig. 7 that the critical value of bending rigidity dividing the straight and flapping regimes obviously increases as the gap distance decreases, i.e., a system of dual parallel inverted flags with a smaller gap distance tends to lose its stability for initiation of flapping. Meanwhile, as the gap distance decreases, the critical value of bending rigidity differentiating the chaotic and deflected regimes also increases. As a result, because both the lower and upper boundaries of bending rigidity for the flapping regime increase, it is conducive to the energy harvesting performance of the dual side-by-side inverted flags with smaller gap distances since stiffer flags can store greater strain energy. The characteristics that facilitates energy harvesting are consistent with the previous researches of Ryu et al. (2018) and Kim and Kim (2019). Besides, as shown in Fig. 7, there is no chaotic regime between the flapping and deflected regimes when the gap distance is within the range D ≤ 0.2. In addition, it is noted that the six-initial conditions perform limited influence on the stability boundary and the dynamics of flapping regimes, consistent with the experimental results of Kim and Kim (2019). However, this small initial deflection exerts

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Fig. 6. Time histories of the angles for the upper flag (solid black line) and the lower flag (dashed red line) corresponding to the six cases showed in Fig. 5.

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Fig. 7. Overview of the four dynamic regimes on the γ − D plane for Re = 200 and ρ = 1, where the symbols □, △, ◦, ♢ represent the deflected, chaotic, flapping, and straight regimes, respectively. The red symbols denote the bistable state which will be discussed in Section 3.3.

non-negligible impact on the deflected regime and promotes the discovery of bistable states, which will be discussed in Sections 3.2 and 3.3 respectively. 3.2. Three sub-regimes of deflected state According on the relative bending direction of dual flags, the deflected regime can be further divided into three subregimes: outside deflected, one-side deflected, and inside deflected (see Fig. 5d, e and f), which is consistent with the experimental results of Huertas-Cerdeira et al. (2018). Kim and Kim (2019) also observed the outside deflected and oneside deflected sub-regimes in their experiment. However, in these previous studies, the transition of the three sub-regimes has not been fully explored. When we simulated the flapping dynamic of a single inverted flag to validate the numerical method, it was found that the initial AOAs may affect the final bending direction of the inverted flag, which inspired our exploration of the effect of initial AOAs on the three sub-regimes. In this section, we first systematically analyze the transition of the sub-regimes, and subsequently, the dynamic characteristics of the three sub-regimes are analyzed separately. 3.2.1. Formation of the sub-regimes To explore the transition of the three sub-regimes, in Fig. 8 we classify all the cases according to their deflected states on the γ − D plane, with the bending rigidity ranging from 0.04 to 0.16 under the six initial conditions as depicted in Fig. 4. When both flags are clamped along the freestream direction (initial condition SS), the outside deflected regime are formed in most of cases (see Fig. 8a), which is consistent with the experimental results of Kim and Kim (2019). It can be found that in the bottom right area of Fig. 8a–f, which is labeled as region B in Fig. 8g, the outside deflected regime is always observed regardless the setting of initial conditions. It means that for small gap distance and increasing bending rigidity, the two parallel inverted flags display outside deflected regime, regardless the initial perturbations. As the gap distance increases and the bending rigidity decreases, the other sub-regimes manifest and the effect of initial conditions can be observed. When both flags are initially bent to the same side (initial condition OI), Fig. 8d shows the one-side deflected regime by increasing the gap distance and reducing the bending rigidity. Similarly, when both flags are initially bent outward (initial condition OO) or inward (initial condition II), the corresponding outward or inward deflected sub-regimes can be found, as shown in Fig. 8c and f, respectively. To explain the formation of the three sub-regimes on the γ − D plane, a discussion on the effects of initial condition is first carried out. According to the previous studies on the stability of inverted flag (Sader et al., 2016; Goza et al., 2018), the dynamic state of inverted flag without deflection exhibits a divergence instability, which was postulated by Kim et al. (2013) and also derived by Gurugubelli and Jaiman (2015) with the aid of a simplified analytical model. This instability is known to not involve flapping and to therefore be associated with departure in the direction of the initial deflection. After

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Fig. 8. Overview of the three sub-regimes on the γ − D plane with different initial conditions depicted in Fig. 4: (a) SS; (b) OS; (c) OO; (d) OI; (e) IS; (f) II; (g) analysis diagram. The symbols □, ◦, △ represent the outside, one-side and inside deflected sub-regimes, respectively. The characters A and B in (g) represent the regimes where the states are affected by the initial conditions, and where the outside deflected regime is always observed regardless the setting of initial conditions, respectively.

Fig. 9. Time history of the angular change relative to the initial position for the upper flag (solid line) and the lower flag (dashed line) with the same γ = 0.12 and different gap distances for initial conditions SS (a) and OS (b).

the inverted flag loses its stability, its bending continuously increases due to the formation of leading edge vortex (LEV). Hence, the initial bending setting facilitates the formation of the corresponding sub-regime. Moreover, as the bending rigidity increases, the growth rate of the instability decreases, indicating that the effect of the initial AOA is weakened. Effect of the side-by-side arrangement on the final bending direction can be further analyzed by using the potential flow approach, which was applied to obtain the stability boundary of two-dimensional dual parallel inverted flags in a uniform flow (Kim and Kim, 2019). The potential flow approach does not consider the viscous effect and the boundary conditions of dual edges of the flags, however, it enables us to derive a useful conclusion: regardless of the gap distance, the dual parallel inverted flags perform the out-of-phase state initially when the flags lose its stability. The coupling effect is weakened as the gap distance increases. If the fluid viscosity is considered, the fluid near the flag surface is decelerated by viscous shear force. When the dual inverted flags are placed close enough, the deceleration effect of the fluid between the flags is more significant. As a result, the pressure between the dual flags is stronger than that of the outside, which causes the dual inverted flags to move outward in this case. Combining the results from the potential flow approach and the viscous effect, it can be expected that the dual parallel inverted flags tend to display a symmetrical and outward state when the instability occurs, i.e. the formation of the outside deflected regime, and this effect is enhanced as the gap distance decreases. To verify our predictions, we test seven cases of different gap distances for dual parallel inverted flags with the same γ = 0.12 and the same initial condition SS. The time history of the angular change relative to the horizontal position for

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Fig. 10. Instantaneous pressure field around the inverted flags for γ = 0.04 and D = 0.1 with the initial condition SS at (a) t = 0.4 and (b) t = 1.6.

Fig. 11. Time histories of the angles of dual flags for the case of γ = 0.16 and D = 3.0 with (a) initial condition OS and (b) initial condition IS.

the dual flags is shown in Fig. 9a. It can be seen that, regardless of the gap distance, the dual inverted flags are always symmetric, and both flags are outward deformed. As the gap distance decreases, the degree of deformation of flags in an equal time is much greater. Eventually, the seven cases are developed into the outside deflected regime. Another set of cases with different gap distances for dual parallel inverted flags with the same γ = 0.12 and initial condition OS are tested to exhibit the influence of the initial AOA, as shown in Fig. 9b. It is seen that with the same gap distance, the flag with an initial inclination (solid line) performs more deformation than the flag placed straight (dashed line) during the same time period. Furthermore, the instantaneous pressure field around the inverted flags is examined to elucidate the fluid forces exerted on the flags, as shown in Fig. 10, for the case of γ = 0.04 and D = 0.1 with the initial condition SS. It can be found that during the process of inverted flags being deflected from the initial stage, the pressure between the dual inverted flags is significantly larger than that of the outside region, which causes the formation of the outside deflected regime. Moreover, as seen in Fig. 8, some abnormalities occur in the cases with γ = 0.16 and high D values, where the formed sub-regime does not exactly correspond to the initial conditions. These cases are close to the boundary between the deflected and chaotic regimes (see Fig. 7). There is a more complicated development stage before the formation of the sub-regime. Two examples are provided to explain this phenomena. The time histories of the angles of dual flags for the case of γ = 0.16 and D = 3.0 with initial conditions OS and IS are plotted in Fig. 11. It can be seen that during the process of inverted flags from initial state to being deflected, the flags do move in the direction of the initial AOA. After that, the flags flap with large amplitudes irregularly. Finally, the inside and outside deflected regimes can be observed for the two cases, respectively. The similar phenomenon occurs in most cases with γ = 0.16 and the spacing D larger than 2.0.

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In general, formation of the three sub-regimes of dual parallel inverted flags under different initial conditions can be considered as a combined effect of the coupling between flags and the initial AOAs. Juxtaposition develops the outside deflected regime and the influence fades as the gap distance increases, which can be regarded as an inherent property of the dual parallel inverted flags. On the other hand, the effect of the initial AOA can be considered as an external interference. The smaller the bending rigidity, the more sensitive the inverted flags are to the interference. Therefore, with high bending rigidity and low gap distance, because of the strong coupling and the insensitivity to interference, the flags tend to perform an outside deflected regime. On the contrary, for dual flags that have low bending rigidity and are placed far apart, the initial AOA setting dominates the bending directions of flags. In brief, through the above analysis and numerical verification, an improved understanding of the deflected regime has been achieved. The dual parallel inverted flags inherently perform the outside deflected regime. The other sub-regimes can be implemented by adjusting the initial AOA setting, which is especially effective for low bending rigidity and large gap distance. In the following, more detailed dynamics of each sub-regime are examined. 3.2.2. Outside deflected regime Fig. 12 shows the time history of the angular change relative to the averaged position of both flags, superimposed views and the corresponding phase diagrams for the outside deflected regime, where four different dynamic states can be observed. As depicted in Fig. 12a, for D = 0.2, we obtain an in-phase state, where the dual inverted flags flap antisymmetrically, i.e., δθ1 = δθ2 . The corresponding vortical structures are plotted in Fig. 13a, where a pair of positive vortices and a pair of negative vortices (2P mode) are shed alternatively from the outside of dual inverted flags, and the vortices shed from the inside of dual flags decay quickly. The erratic state is formed when both flags flap with irregular changes in the flapping period and amplitude (see Fig. 12b). The out-of-phase is obtained when the inverted flags flap symmetrically, i.e. δθ1 = −δθ2 (see Fig. 12c and d). As shown in Fig. 12c, initially, the two inverted flags are not symmetrical (initial condition OS), but under the coupling between the dual flags they eventually become completely symmetric. It can be seen more intuitively in the phase diagram that the curve eventually converges to δθ1 = −δθ2 as time progresses. As shown in Fig. 12d, when the initial condition is set to be symmetric, the flags are always symmetric, and the curve in the phase diagram is maintained on the δθ1 = −δθ2 line. The vortical structures corresponding to the case of Fig. 12d are plotted in Fig. 13b. The vortices shedding from the inverted flags are symmetric about the midline. As the gap distance increases, the coupling between the dual flags is weakened, and both flags perform periodic, constant-amplitude vibrations after a transient phase (see Fig. 12e and f). The phase difference is constant and depends on the initial conditions. As shown in Fig. 12e, when the initial condition OS is set, the constant phase difference is between 0 and π , and the curve in the phase diagram is presented as a ring accordingly. This dynamic state is classified as the staggered state. If the two flags are initially set to be symmetrical, the phase difference is kept to be zero (see Fig. 12f). The relationship among bending rigidity, gap distance and the above four dynamic states under different initial conditions OS, SS and OO is presented in Fig. 14. The results of the other three initial conditions are not shown because they lead to the one-side deflected and inside deflected regimes, as seen in Fig. 8. For small gap distances (indicated as region D in Fig. 14d), the dual inverted flags perform in-phase state. The erratic state appears when the gap distance increases to about 0.4 (indicated as region C in Fig. 14d). Regardless the initial conditions, the out-of-phase state is observed in most of the cases of the region B in Fig. 14d. As the gap distance increases further, the dynamic state depends on the initial conditions. The inverted flags with symmetrical and asymmetric initial conditions perform out-of-phase and staggered states, respectively. The emergence of influence of the initial conditions indicates the weakening of the coupling between the two inverted flags, consistent with the numerical results of Ryu et al. (2018). 3.2.3. Inside deflected regime In the previous studies on parallel inverted flags, the inside deflected regime was only observed by Huertas-Cerdeira et al. (2018). According to our simulation, this sub-regime only appears when the gap distance is more than 1.4, and it is highly dependent on the initial condition. As shown in Fig. 8, inside deflected cases can be obtained when both flags are initially bent inward (initial condition II). The inside deflected regime was not recorded in the experimental study of Kim and Kim (2019) and the numerical work of Ryu et al. (2018), which may be due to the strict forming condition of this dynamic sub-regime. Two synchronization states and an erratic state are observed for the inside deflected regime as shown in Fig. 15. The in-phase state is formed when the inverted flags flap antisymmetrically (see Fig. 15a). The corresponding vortical structures are plotted in Fig. 16a. It can be seen that two pairs of counter-rotating vortices are alternatively shed from both the leading and trailing edges of the inverted flags. If the flags flap without synchronization as shown in Fig. 15b, the dynamic state is classified into the erratic state. When the inverted flags flap with a zero-phase difference as depicted in Fig. 15c, the out-of-phase state is obtained. For the out-of-phase state (Fig. 16b), two leading edge vortices (LEV) and two trailing edge vortices (TEV) are fully symmetric about the midline. With the same bending rigidity and gap distance, the flapping amplitude of the inside deflected regime is greater than that of the outside and one-side deflected regimes (see Fig. 5d, e and f). For further comparison, the time history of bending strain energy of the upper flag for these three cases is plotted in Fig. 17. The bending strain energy is defined as EB =

1 2

γ

∫ L( 0

∂ 2X ∂ s2

)2

ds.

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Fig. 12. Time history of δθ for the upper flag (solid black line) and the lower flag (dashed red line) on the left, superimposed views in the middle and phase diagram on the right: (a) γ = 0.16, D = 0.2 with initial conditions OS; (b) γ = 0.08, D = 0.4 with initial conditions SS; (c) γ = 0.04, D = 1.0 with initial conditions OS; (d) γ = 0.04, D = 1.0 with initial conditions SS; (e) γ = 0.04, D = 2.8 with initial conditions OS; (f) γ = 0.04, D = 2.8 with initial conditions SS. In phase diagrams, lines of different colors are used to represent the time evolution.

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Fig. 13. Vortical structures of the outside deflected inverted flags: (a) in-phase state; (b) out-of-phase state.

Fig. 14. Overview of the state regions on the γ − D plane for the outside deflected sub-regime with different initial conditions: (a) OS; (b) SS; (c) OO; (d) analysis diagram. The symbols ♢, □, ◦, △ and × represent the in-phase, erratic, out-of-phase, staggered and non-outside-deflected states, respectively. The characters A, B, C and D in (d) represent the regions of staggered, out-of-phase, erratic, and in-phase regimes, respectively.

The bending strain energies of the upper and lower flags are of the same order, so only that of the upper flag is plotted for clarity. We use the maximum fluctuation ∆EBmax to quantify the energy harvesting performance from the surrounding fluid flows. As shown in Fig. 17, ∆EBmax of the inside deflected regime is significantly greater than that of the other sub-regimes, indicating the better performance of energy harvesting for the inside deflected regime. 3.2.4. One-side deflected regime Formation of the one-side deflected regime also depends on the initial conditions. As shown in Fig. 8, most cases for the one-side deflected regime are under the initial conditions OI and IS. For the other sub-regimes, the mean flapping amplitude of the upper flag is roughly the same as that of the lower one. However, for the one-side deflected regime, when the gap distance is less than 1.0, the lower flag cannot be fully stretched due to the blocking of the upper flag (see Fig. 18a). In this case, the flapping amplitude of the upper flag is several times of that of the lower one. At D = 1.0, which allows the lower flag to be fully bent, the lower flag has a larger amplitude than the upper flag (see Fig. 18b). As the spacing increases further, the difference between the amplitudes of the two flags is gradually decreased (see Fig. 18c). When the gap distance is larger than 2.4, the coupling between the dual flags is mitigated, and the amplitudes of the two flags are approximately the same (see Fig. 18d). The relationship between the gap distance and the ratio of the mean flapping amplitudes of the dual flags is clearly shown in Fig. 19. 3.3. Bistable state In the traditional flapping flag problem, the bistability phenomena has been identified in the previous researches using experimental (Zhang et al., 2000; Watanabe et al., 2002; Shelley et al., 2005) and numerical methods (Zhu and Peskin,

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Fig. 15. Time history of δθ for the upper flag (solid black line) and the lower flag (dashed red line) on the left, superimposed views in the middle and phase diagram on the right with initial condition II: (a) γ = 0.12, D = 2.2; (b) γ = 0.08, D = 1.6; (c) γ = 0.04, D = 1.8. In phase diagrams, we use lines of different colors to present the time evolution.

Fig. 16. Vortical structures of inside deflected inverted flags for two synchronizations: (a) in-phase state and (b) out-of-phase state.

2002; Lee et al., 2014). Interestingly, for a single inverted flag (Tang et al., 2015), bistability was also observed by slightly adjusting the initial perturbation of the transverse displacement of flag in the transition zone between two dynamic regimes. In the present work, we use a similar strategy to explore the existence of bistability for dual parallel inverted flags. The bistable state is identified if different dynamic regimes, i.e. straight, flapping, deflected and chaotic regimes, can be induced by adjusting initial perturbations applied to inverted flags with the bending rigidity and spacing unchanged. By comparing the results under the six initial conditions, the bistable state is observed in the transition of the four main dynamic regimes, as represented by red symbols in Fig. 7. For the case of γ = 0.2 and D = 0.8, when both flags are

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Fig. 17. Time history of the bending strain energy of the upper flag for the outside, one-side and inside deflected regimes with the same bending rigidity γ = 0.08 and gap distance D = 1.8.

Fig. 18. Superimposed views of the one-side deflected regime with the same initial condition OI and bending rigidity γ = 0.04 but different gap distances: (a) D = 0.6; (b) D = 1.0; (c) D = 1.4; (d) D = 2.4.

Fig. 19. Variations of the ratio of the amplitude of the upper flag to that of the lower flag with the gap distance for the one-side deflected regime.

initially set to be straight (initial condition SS), the deflected regime is formed (see Fig. 20a). If a small declination is initially applied to the upper flag (initial condition OS), the chaotic regime can be observed as shown in Fig. 20b, which

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Fig. 20. Time history of δθ for the upper flag (solid black line) and the lower flag (dashed red line) on the left, and superimposed views on the right: (a) γ = 0.20, D = 0.8 with initial condition SS; (b) γ = 0.20, D = 0.8 with initial condition OS; (c) γ = 0.24, D = 2.2 with initial condition OI; (d) γ = 0.24, D = 2.2 with initial condition OS; (e) γ = 0.44, D = 2.6 with initial conditions OS; (f) γ = 0.44, D = 2.6 with initial condition OI.

indicates the bistable state between the deflected and chaotic regimes. As shown in Fig. 20c and d, respectively, the chaotic

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and flapping regimes can be observed by adjusting the initial conditions while all other parameters remain unchanged. Similarly, from the behaviors in Fig. 20e and f, the bistability property can be identified for the flapping and straight regimes. 4. Conclusions In this paper, the dual side-by-side inverted flags with free leading edges and fixed trailing edges in a uniform flow was simulated using the immersed boundary method. The four main dynamic regimes of dual flags were found, including the straight, flapping, chaotic and deflected regimes. Effects of the bending rigidity, the gap distance and the six initial conditions were analyzed systematically. According to relative bending directions in which the inverted flags are bent, the deflected regime was further divided into three sub-regimes: outside deflected, inside deflected and one-side deflected. It was found that in the case of small gap distance and high bending rigidity, the two parallel inverted flags display the outside deflected regime, independent of the initial conditions. In the case of large gap distance and low bending rigidity, different dynamic sub-regimes can be obtained by adjusting the initial conditions. The dynamic characteristics of the three sub-regimes were analyzed separately. For the outside deflected regime, the in-phase, erratic, out-of-phase, and staggered states appear in sequence as the gap distance increases. The in-phase and out-of-phase states were observed in the inside deflected regime. The energy harvesting performance of the inside deflected regime is better than the other two sub-regimes at the same gap distance and bending rigidity. The inequality of flapping amplitude of the dual flags was found in the one-side deflected regime. 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