Stability and coupled dynamics of three-dimensional dual inverted flags

Journal of Fluids and Structures 84 (2019) 18–35

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Stability and coupled dynamics of three-dimensional dual inverted flags Hyeonseong Kim, Daegyoum Kim

∗

Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

article

info

Article history: Received 12 May 2018 Received in revised form 15 August 2018 Accepted 16 October 2018 Available online xxxx Keywords: Inverted flag Dual flags Stability Coupled dynamics Flapping

a b s t r a c t Fluid–structure interaction of an inverted flag, which has a free leading edge and a clamped trailing edge, has drawn attention recently because of its novel properties such as divergence stability, a low stability threshold, and large-amplitude flapping motion. In this study, the stability and flapping behaviors of dual inverted flags with finite height are investigated for a side-by-side arrangement, and their noticeable characteristics are compared to those of dual conventional flags. The critical velocity at which the inverted flags break the equilibrium of a straight configuration reduces monotonically when a gap distance between the two flags becomes smaller and an aspect ratio becomes larger, which is also predicted by our linear stability analysis using simple theoretical models of twodimensional flags and slender flags. After bifurcation, in addition to the synchronized inphase and out-of-phase modes commonly observed in dual conventional flags, a novel attached mode appears which is mainly observed for small gap distance and small aspect ratio. In this non-linear mode, the leading edges of the two inverted flags touch each other on a midline, and the deformed inverted flags maintain static equilibrium. In a non-linear flapping regime, a new mechanism of a mode transition from an out-of-phase mode to an in-phase mode is identified, which is allowed by the collision of the two flags flapping with large amplitude. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In spite of its simple geometry, a flag exhibits complicated dynamic behaviors when it interacts with a fluid flow. The instability and flapping dynamics of a single flag have been studied as one of the classical problems in fluid–elastic structure interactions (e.g., Watanabe et al., 2002; Shelley et al., 2005; Connell and Yue, 2007; Eloy et al., 2008, 2011; Shelley and Zhang, 2011). Recently, these fundamental flag studies have been extended in an attempt to evaluate the potential use of flags in novel energy harvesting systems (e.g., Dunnmon et al., 2011; Giacomello and Porfiri, 2011; Michelin and Doaré, 2013; Howell and Lucey, 2015; Xia et al., 2016; Yu and Liu, 2016). Several flapping modes have been identified for multiple flags in a side-by-side arrangement which involve dynamic fluid coupling between the flags. The parameters that determine these interactions include the mass ratio of fluid to flags, and the distance separating the flags. To better understand the collective behaviors of flags, numerous studies have been conducted, including experimental measurements (Zhang et al., 2000; Schouveiler and Eloy, 2009), numerical simulations (Zhu and Peskin, 2003; Farnell et al., 2004; Tian et al., 2011), and theoretical analyses (Jia et al., 2007; Alben, 2009; Michelin and Llewellyn Smith, 2009; Mougel et al., 2016). ∗ Corresponding author. E-mail address: [email protected] (D. Kim). https://doi.org/10.1016/j.jfluidstructs.2018.10.005 0889-9746/© 2018 Elsevier Ltd. All rights reserved.

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Among these studies, research with dual flags in soap-film experiments found that they exhibited two synchronized modes, an antisymmetric in-phase mode and a symmetric out-of-phase mode (Zhang et al., 2000). The in-phase mode was observed when there was a small gap between the two filaments, and the out-of-phase mode was observed for a larger gap. Zhang et al. (2000) suggested that the synchronization was induced by an energy balance or mass balance between the filaments and the surrounding fluid. Later, the gap-dependent mode behaviors of the two filaments reported by Zhang et al. (2000) were theoretically analyzed by Jia et al. (2007) in relation to the density ratio of the filament to the fluid. Alben (2009) also theoretically investigated the synchronization of two flags, both in tandem and in side-by-side configurations. When three flags are arranged in a side-by-side configuration, they exhibit six coupling modes, depending on gap distance. Four types of vortical structures are associated with these coupling modes (Tian et al., 2011). The collective dynamics of a system with more than two flags were generalized by Schouveiler and Eloy (2009) and Michelin and Llewellyn Smith (2009). All of the aforementioned studies were conducted with two-dimensional flags. In contrast, Mougel et al. (2016) investigated the coupling of dual slender flags with small height using the elongated body theory (EBT) and the large-amplitude elongated body theory (LAEBT). Recently, unlike the conventional flag whose trailing edge is free to flap, an inverted flag whose leading edge is free to flap was introduced (Kim et al., 2013). An inverted flag exhibits three distinct modes, a straight mode, a periodic large-amplitude flapping mode, and a deflected mode, and those dynamic behaviors were found to be independent of the mass ratio of the flag to the surrounding fluid (Kim et al., 2013, 2017). Kim et al. (2013) also demonstrated that an inverted configuration becomes unstable at a lower critical velocity than a conventional configuration. Based on the work of Kornecki et al. (1976), Sader et al. (2016b) theoretically determined the critical free-stream velocity at which an initially straight inverted flag becomes unstable. That theoretical estimation of the critical velocity was in good agreement with experiments for an inverted flag with a high aspect ratio; however, the theory overestimated the critical velocity of a slender flag with small height. By adopting the slender body theory, Sader et al. (2016a) suggested a modified model which drastically reduced the discrepancy between theoretical estimation and experimental data. Numerical studies about the inverted flag have also been conducted to investigate the characteristics of its non-linear dynamics (e.g., Ryu et al., 2015; Tang et al., 2015; Gilmanov et al., 2015; Gurugubelli and Jaiman, 2015). While the dynamics of multiple conventional flags have been reported in many studies as mentioned above, there are few experimental and numerical studies on the coupled dynamics of multiple inverted flags. Cerdeira et al. (2017) experimentally explored the coupling motions of dual inverted flags in a side-by-side arrangement and found the enhancement of flapping amplitude and frequency by virtue of their interaction, compared to the case of a single inverted flag. In their study, the dynamic mode of the two flags depends on separation distance between them and dimensionless free-stream velocity. However, the distance was so large that direct interaction by contact was not allowed. Ryu et al. (2018) and Huang et al. (2018) conducted numerical simulations for dual inverted flags in side-by-side and tandem arrangements, respectively. They focused on evaluating energy harvesting performance of the flags in a flapping regime. In this study, we experimentally and theoretically investigate the behavior of dual inverted flags in a side-by-side configuration. The study was conducted in a wide range of aspect ratio and separated distance between the flags, from a short distance at which the flags can collide each other to a long distance beyond maximum peak-to-peak flapping amplitude. The previous studies of dual inverted flags have been limited to nonlinear flapping dynamics and have not addressed the stability of the two flags in great detail. Hence, in Section 3, we analyze a critical free-stream velocity by varying the aspect ratio of the dual inverted flags and the gap distance between them. Theoretical prediction of the critical free-stream velocity obtained by the linear stability theory is compared with experimental measurements. In Section 4, we discuss non-linear dynamics of the dual inverted flags after bifurcation and introduce a novel dynamic mode and mode transition process which is not exhibited by dual conventional flags. 2. Experimental setup The overall schematic of experiments is illustrated in Fig. 1. We used an open-loop suction-type wind tunnel with a cross section 0.6 m high and 0.6 m wide. The free-stream velocity in our experiments ranged from 1.5 m/s to 10.4 m/s. In order to check the spatial uniformity of the free-stream velocity, the velocity was measured using a grid of 3 by 5 equally-spaced points across a cross section with seven different velocities. The spatial standard deviation of the free-stream velocity was within 2% of the mean free-stream velocity at each velocity considered in this study. The two inverted flags were placed side by side in the middle of the test section, and the trailing edge of each flag was clamped with two vertical aluminum plates of thickness 1 cm and width 2 cm. Our model setup is similar to that of the previous studies (e.g., Schouveiler and Eloy, 2009; Eloy et al., 2011). In these previous studies, the cross section of the clamping plate was streamlined while it is rectangular in our setup. However, the total thickness of the clamping plates is 2 cm, which is much smaller than the length of a flag model, 15 cm, and the clamping plates are positioned at the trailing edge of the flag. Hence, we assume that the effect of the cross-section shape of the clamping plates on flapping dynamics is negligible in our experiments. Additionally, a closed-loop free-surface water tunnel with a cross section 0.4 m high and 0.5 m wide was used in order to investigate the effect of mass ratio (fluid density). The same flags used for wind-tunnel experiments were clamped vertically in a way similar to the wind-tunnel setup. Because of the difference in fluid density, the mass ratio of the water-tunnel experiments was several orders lower than that of the wind-tunnel experiments. The detailed results and discussion of the water-tunnel experiments will be presented in Sections 3.2.4 and 4.4.

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Fig. 1. Schematics of dual inverted flags: (a) isometric view and (b) top view.

Fig. 2. Schematic description of the physical model of the dual inverted flags. U is the free-stream velocity, L is the length of the flags, and η1 and η2 are the displacements in the transverse direction of the flags. Regions 1 and 3 are flow regions outside the upper and lower flags, respectively, and region 2 is the flow region between the two flags.

A polycarbonate sheet (Young’s modulus E = 2.38 × 109 N/m2 , Poisson’s ratio ν = 0.38, and density ρs = 1.2 × 103 kg/m3 ) of length L = 15 cm was used as the flag model. The thickness h of the sheet was set to 0.5 mm, which was large enough to prevent sagging of the flag by gravity. We considered several different heights H = 1, 2.5, 5, 10, 15, and 20 cm in order to have an aspect ratio H ∗ (= H /L) ranging from 0.07 to 1.33. In addition, we used a flag with H = 55 cm spanning the top and bottom surfaces of the test section to simulate a two-dimensional model. The gap distance G between the two flags was 5, 10, 15, and 20 cm, providing non-dimensional gap distance G∗ (= G/L) of 0.33, 0.67, 1.00, and 1.33. When clamping the flags, initial deflection along the length of the flag was observed due to the defect of the material itself and the error generated by manual setup. For the initial condition, initial transverse displacement of the leading edge from the midline ∆ was adjusted to be smaller than 0.02L for each flag. From our tests, we confirmed that non-zero initial deflection within this threshold (∆/L < 0.02) had negligible effects on stability and flapping dynamics. To capture the movements of the flags, a white plastic tape thin enough to avoid influencing the flag rigidity was attached along the bottom edge of the flags, and captured by a high-speed camera (MINI UX 160K M1, Photron, Inc.) mounted below the test section. The bottom edges of the sheets were illuminated by halogen lamps from both sides of the test section, and the upper side of the test section was covered with black paper to highlight the white lines on the flags. Images were recorded at 50 frames per second and processed with MATLAB (The Mathworks, Inc.) to obtain the coordinates of the deforming flags. 3. Stability analysis of dual inverted flags 3.1. Two-dimensional dual inverted flags (H ∗ = ∞) 3.1.1. Theoretical model For stability analysis, we will use the approach of Jia et al. (2007) applied to two-dimensional conventional flags of an infinite aspect ratio (H ∗ = ∞). Two-dimensional dual inverted flags with length L were immersed in a uniform flow, and their clamped trailing edges were positioned at y = G/2 and y = −G/2, respectively (Fig. 2). Transverse displacements from the flags’ initially straight positions are denoted as η1 and η2 , respectively. Here, we simply summarize the approach of Jia

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et al. (2007). For detailed procedures, refer to Jia et al. (2007), Schouveiler and Eloy (2009) and Tian et al. (2011). A governing equation is the Laplace equation for the velocity potential of perturbation ϕj (u′j = ∇ϕj ) with a potential flow assumption. The subscript j indicates regions 1−3 (Fig. 2). The linearized Euler–Bernoulli equation for transverse displacements η1 and η2 of the flags and a pressure difference across the flags and the linearized Bernoulli equation for a pressure difference and a fluid velocity are used. For boundary conditions, ϕj is bounded at infinity, and the linearized kinematic boundary conditions for ϕj are imposed at the initial positions of the flags. By employing the method of normal mode, two equations are obtained for η01 and η02 , respectively (ηn = η0n ei(ωt +kx) , n = 1, 2): (−ml1 ω2 + B1 k4 )η01 ei(ωt +kx) = ρ (−ml2 ω2 + B2 k4 )η02 ei(ωt +kx) = ρ

(ω + kU)2 csch(kG) k (ω + kU)2 csch(kG) k

[η01 ekG − η02 ]ei(ωt +kx) ,

(1a)

[−η01 + η02 ekG ]ei(ωt +kx) ,

(1b)

where B(= Eh3 /12(1 − ν 2 )) is the flexural rigidity of a flag, ml (= ρs h) is the mass of a flag per unit length. For non-trivial solutions of Eq. (1), the determinant of the coefficient matrix with respect to η01 and η02 must be zero. Given that the two flags have the same geometry and material properties (ml1 = ml2 = ml and B1 = B2 = B), two equations ¯ ω¯ , and k: ¯ are obtained in terms of U ∗ , m∗ , G,

( (

−m ω¯ + 2

∗

−m∗ ω¯ 2 +

k¯ 4 U ∗2 k¯ 4 U ∗2

) ( ) ¯ 2 csch(k¯ G) ¯ (ω ¯ + k) 2 ¯ − (ω ¯ + k) − = 0, k¯ k¯ ) ( ) ¯ ¯2 ¯¯ 1 + coth(k¯ G) ¯ 2 + (ω¯ + k) csch(kG) = 0, − (ω ¯ + k) k¯ k¯ ¯ 1 + coth(k¯ G)

(2a) (2b)

¯ = ωL/U, and G¯ = G/L. The non-dimensional free-stream where U ∗ = U ρf L3 /B, m∗ = ml /ρf L(= ρs h/ρf L), k¯ = kL, ω velocity U ∗ represents the ratio of fluid inertial force to bending force of a flag; instead of U ∗ , several previous studies have used non-dimensional bending stiffness β (= 1/(U ∗ )2 ) or κ (= (U ∗ )2 ) (e.g. Connell and Yue, 2007; Alben and Shelley, 2008; Kim et al., 2013; Sader et al., 2016b). Mass ratio m∗ is the ratio of flag inertial force to fluid inertial force. Eq. (2a) corresponds to a symmetrical out-of-phase mode (η01 = −η02 ), and Eq. (2b) is for an antisymmetrical in-phase mode (η01 = η02 ).

√

3.1.2. Effects of mass ratio and wave number Eq. (2) can be used for both conventional flags and inverted flags because boundary conditions at a leading edge and a trailing edge were not required when deriving the equations. Thus, to obtain a stability boundary for the dual inverted flags from Eq. (2), the effect of a mass ratio should be considered, and a wave number at bifurcation, which is appropriate for the dual inverted flags, should be adopted. Unlike a conventional flag which shows an oscillating mode by flutter instability, an inverted flag becomes statically unstable from an initial straight position due to divergence instability. It was found that the critical velocity of the inverted flag is not affected by mass ratio m∗ (Kim et al., 2013; Sader et al., 2016b). Hence, at the bifurcation, ω should be zero because the mass ratio is multiplied by ω in the dispersion relation of a single inverted flag; ω = 0 at the bifurcation indicates divergence instability. For the dual inverted flags, here we also assume the effect of the mass ratio on the critical velocity is negligible, which will be justified by experimental observation in Section 3.2.4. The term including the mass ratio in Eq. (2) should be zero in order to have negligible mass-ratio effect on the critical velocity. In other words, for any arbitrary m∗ , ω should be zero:

√ Uc∗,out =

√ Uc∗,in =

k¯ 3

¯ + csch(k¯ G) ¯ 1 + coth(k¯ G) k¯ 3

¯ − csch(k¯ G) ¯ 1 + coth(k¯ G)

,

,

(3a)

(3b)

where Eq. (3a) is for an out-of-phase mode and Eq. (3b) for an in-phase mode. For conventional flags, a wave number in Eq. (2) is determined by a harmonic mode predicted at bifurcation and boundary conditions at two edges of a flag. For the fixed-free boundary condition, the conventional flag is known to display a second harmonic mode mainly when stability is broken (e.g. Guo and Païdoussis, 2000; Eloy et al., 2007). In contrast to the conventional flag, the inverted flag exhibits flapping of a fundamental mode after instability (Kim et al., 2013). Therefore, the dimensionless wave number corresponding to the fundamental harmonic vibration with a clamped-free boundary condition (k¯ = 1.875) should be used in Eq. (3) (Meirovitch, 1997). 3.1.3. Comparison with two-dimensional dual conventional flags In Eq. (3) and Fig. 3, we notice that, for the two-dimensional dual inverted flags, Uc∗,out is always smaller than Uc∗,in for any distance G∗ although the difference between Uc∗,out and Uc∗,in becomes narrower with increasing G∗ . This result means that, regardless of G∗ , the two-dimensional dual inverted flags are predicted to exhibit an out-of-phase mode first when

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Fig. 3. Stability boundary of two-dimensional dual inverted flags obtained by Eqs. (3a) and (3b); out-of-phase mode (solid line) and in-phase mode (dashed line).

Fig. 4. Schematic of gap flow patterns for various aspect ratios. A white arrow is an incoming free stream between the flags. Blue arrows indicate the leakage of a gap flow around upper and lower edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

instability occurs from an initial equilibrium. This trend is also observed in the case of dual conventional flags (Jia et al., 2007; Schouveiler and Eloy, 2009; Mougel et al., 2016). In the derivation of Eq. (3), we assumed negligible mass-ratio effect. Thus, at a given G∗ , the critical velocity of dual inverted flags is constant regardless of m∗ . When compared to the stability boundary of two-dimensional dual conventional flags (Jia et al., 2007; Schouveiler and Eloy, 2009), the dual inverted flags establish the theoretical stability boundary at a lower Uc∗ in general; 1.0 < Uc∗ < 2.0 in Fig. 3. Since the stability boundary of the dual conventional flags is dependent on the mass ratio, a direct comparison between the inverted flags and conventional flags is not possible. However, in a reasonable range of the mass ratio, Uc∗ is O(1) − O(10) for the dual conventional flags, and even can be beyond Uc∗ = O(102 ) for an extreme mass ratio; for example, Uc∗ = 5.7 − 9.1 for G∗ = 0 − 0.8 and m∗ = 0.70 according to Schouveiler and Eloy (2009). 3.2. Dual inverted flags with an arbitrary aspect ratio 3.2.1. Stability of slender dual inverted flags (H ∗ → 0) Sader et al. (2016a) suggested a stability boundary for a slender single inverted flag in the limit of zero aspect ratio (H ∗ → 0) by using the static equilibrium theory: Uc∗ = 2.8. Mutual interaction of two flags by a gap flow between them affects the stability boundary as demonstrated for two-dimensional flags in Section 3.1. However, as H ∗ becomes small, an incoming free stream between the two flags may leak through the upper and lower edges of the flags (see Fig. 4 for schematic illustration). As a result, interaction between the two flags would become negligible as H ∗ approaches to zero; a flag cannot induce additional hydrodynamic force on another flag because most of the gap flow leaks through the upper and lower edges. Thus, we assume that the critical velocity of the slender dual inverted flags (H ∗ → 0) is not changed by G∗ and has the same constant value as a slender single inverted flag: Uc∗ = 2.8. 3.2.2. Stability of dual inverted flags with an arbitrary aspect ratio The critical velocity of dual inverted flags with an arbitrary aspect ratio can be estimated from the results of twodimensional flags (H ∗ = ∞) (Section 3.1.2) and slender flags (H ∗ → 0) (Section 3.2.1). For H ∗ = ∞, the stability boundary of the out-of-phase mode is chosen over that of the in-phase mode because it is theoretically predicted that the out-of-phase mode occurs in a lower U ∗ . When bifurcation occurs in the experiment, the flags initially show out-of-phase mode in all

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Fig. 5. Stability boundary of dual inverted flags obtained by theoretical analysis and experimental measurements in (a)[H ∗ , U ∗ ] space and (b) [G∗ , U ∗ ] space. (a) Critical velocity obtained by Eq. (4) (left) and experimental measurements (right) for each G∗ . The experimental data of the single inverted flag (filled black circle) is included in the right subfigure. Note that symbols corresponding to the two-dimensional flags (H ∗ = ∞) are placed at H ∗ = 5 instead in the right subfigure. (b) Theoretical prediction (left) and experimental measurements (right) for each H ∗ . The lower and upper dashed black lines in the left are the critical velocities for the two-dimensional flags (H ∗ = ∞, Section 3.1.2) and the slender flags (H ∗ → 0, Section 3.2.1), respectively.

cases although mutual interaction between the flags may cause another mode after a few flapping cycles. Thus, the critical velocity obtained experimentally corresponds to the out-of-phase mode solution of the linear stability analysis. For a single inverted flag, Sader et al. (2016a) employed Padè approximation to obtain the critical velocity of an arbitrary aspect ratio with the results of H ∗ = ∞ and H ∗ → 0. Here, we use the same approach for the dual inverted flags (Eq. (4)).

  H  Uc∗,slender 2 + (Uc∗,slender 2 − Uc∗,inf 2 ) 2L ∗ Uc = √Uc∗,inf 2 , H Uc∗,inf 2 + (Uc∗,slender 2 − Uc∗,inf 2 ) 2L

(4)

where Uc∗,inf for H ∗ = ∞ is a lower limit of Uc∗ and Uc∗,slender for H ∗ → 0 is a upper limit of Uc∗ . For a finite H ∗ , tip vortices generated by both upper and lower edges of the flag reduce the angle of attack between the incoming free stream and the flag slightly deviated from a straight equilibrium position. The decreased angle of attack attenuates a hydrodynamic force (lift force) per unit height acting on a flag. Meanwhile, a restoring bending force of a flag per unit height remains unchanged. Thus, the critical velocity of a finite H ∗ should increase compared with a two-dimensional flag (Sader et al., 2016b). As suggested by Sader et al. (2016a,b), a scaling factor 1 + 2L/H is applied to the critical velocity of H ∗ = ∞ (Uc∗,inf ) to approximate the critical velocity of a large H ∗ close to H ∗ = ∞. 3.2.3. Effects of aspect ratio and gap distance We investigate the dependence of a critical velocity Uc∗ on an aspect ratio H ∗ and a dimensionless gap distance G∗ using theoretically prediction (Eq. (4)) and experimental measurements (Fig. 5). For experimental measurements, we define Uc∗ as the free-stream velocity at which either of the two flags has a peak-to-peak amplitude of a free leading edge A/L larger than 0.1. Some discrepancy is found in Fig. 5 between the theoretical and experimental results. In the theoretical analysis, critical velocities of low and high aspect ratios are under- and overestimated, respectively. For the linear stability analysis with two-dimensional flags, we used a very simple inviscid model and did not consider vortex shedding from the leading and trailing edges and viscous effect on the flag surface. The theory of a slender inverted flag did not consider unsteady hydrodynamic force and three-dimensional twisting motion. Nevertheless, the simplified theories enable us to estimate the order of magnitude of the critical velocity and qualitatively predict the noticeable trends of U ∗ : monotonic increase of U ∗ with G∗ and monotonic decrease with H ∗ (Fig. 5). The effect of H ∗ on the stability of dual inverted flags is similar to that of a single inverted flag for any G∗ ; Uc∗ decreases monotonically with increasing H ∗ (Fig. 5a). As mentioned in the derivation of Eq. (4), when H ∗ decreases, the angle of attack reduces because of the stronger effect of tip vortices, and the net lift force acting on the flag decreases accordingly. Thus, a higher free-stream velocity is required to break the stability by complementing the decreased lift force. In the experiment, we also considered two-dimensional flags by extending the height of the flags to be close to the height of the test section (see Section 2). For the two-dimensional flags (H ∗ = ∞) in Fig. 5b, Uc∗ is around 1.0 − 1.5 for the G∗ range covered in this study. Thus, in the curves shown in Fig. 5a, we can expect that U ∗ approaches asymptotically to 1.0 − 1.5 as H ∗ → ∞.

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Sader et al. (2016a) theoretically obtained Uc∗ (κ in their study) for a slender single inverted flag with an infinitesimal height: Uc∗ → 2.8 as H ∗ → 0. They also experimentally demonstrated that Uc∗ approaches to 2.8 asymptotically with H ∗ → 0. However, in our experimental study, for both a single flag (filled black circles in Fig. 5a) and dual flags, Uc∗ increases sharply as H ∗ falls below 0.2 instead of approaching to 2.8. Since our experiment and the experiment in Sader et al. (2016a) have a similar Uc∗ in the high aspect-ratio range, this noticeable discrepancy in a small H ∗ is not due to errors in experimental procedure. As mentioned in Sader et al. (2016a), because of its small height, a slender flag shows torsional twist due to gravitational loading when stability is broken. For a given value of H ∗ , this torsional twist is expected to have a larger effect on instability as the flag length L becomes larger. We tested two cases (L = 15 cm and 30 cm) for a single inverted flag with the same H ∗ = 0.033; in the study of Sader et al. (2016a), two different lengths (L = 19 and 30 cm) were examined, and H ∗ = 0.033 was chosen as the smallest aspect ratio. In our test, instability did not occur within the maximum U ∗ available ∗ in our setup (Umax = 4.9) for L = 15 cm, while Uc∗ was 3.8 for L = 30 cm. When a flag becomes unstable, even though it shows only two-dimensional bending at L = 15 cm, it displays both bending and three-dimensional torsional twisting at L = 30 cm. These discrepancies in Uc∗ and dynamics between L = 15 cm and 30 cm indicate that the three-dimensional deformation of a slender flag by gravitational loading induces a significant reduction in critical velocity. As H ∗ decreases, at a given position along flag length L, the hydrodynamic (lift) force per unit height H continues to decrease accordingly. Therefore, as H approaches to zero, the hydrodynamic force will not be strong enough to bend the flag; note that restoring bending force per unit height remains unchanged as H ∗ decreases. Therefore, Uc∗ will continue to increase drastically rather than converging to a specific value as depicted in Fig. 5a. For reference, slender conventional flags exhibit a sharp increase in Uc∗ as H ∗ approaches to zero, like the slender inverted flags in our experiment (Fig. 6a of Mougel et al., 2016). Further investigation will be necessary to fully resolve this inconsistent result for Uc∗ in the small aspect-ratio regime. In a side-by-side configuration, a free stream deflected by one flag imposes a fluid force on the other flag. Because of fluid loading by the gap flow between the flags, the two flags bend slightly outward from their initially straight positions, toward region 1 for the upper flag and region 3 for the lower flag in Fig. 2. The distance between the leading edges of the two flags becomes wider. As the free-stream velocity (and the hydrodynamic force on the flags) increases further, instability eventually occurs, and the flags start to bend outward with a noticeable displacement. This phenomenon is in agreement with the linear stability analysis, which predicts that an out-of-phase (outward bending) mode would occur at the instant when stability breaks. In Section 3.2.1, we assumed that the critical velocity is independent of G∗ for dual flags with H ∗ → 0. This assumption was confirmed by our experimental measurements with a small aspect ratio, H ∗ = 0.07 (Fig. 5b). The difference in Uc∗ between G∗ = 0.33 and G∗ = 1.33 is ∆Uc∗ = 0.32 for H ∗ = 1.33. However, this difference is reduced to ∆Uc∗ = 0.09 for H ∗ = 0.07. Since this tendency is due to the slenderness of the flags, we can also expect that slender conventional flags at small H ∗ will show a negligible variation in Uc∗ with the change in G∗ . This was indeed demonstrated in Fig. 6b of Mougel et al. (2016). For the given height H ∗ of flags, hydrodynamic loading by a gap flow between the two flags becomes larger as the gap distance G∗ decreases. This strong interaction in small G∗ causes instability in the lower free-stream velocity. Consequently, Uc∗ increases monotonically with G∗ (Fig. 5b). At large G∗ , additional fluid loading caused by the existence of the other flag becomes weak, and Uc∗ approaches to the value observed for a single inverted flag as presented in the right subfigure of Fig. 5a. 3.2.4. Experimental verification of mass-ratio effect It is well known that the stability of dual conventional flags is strongly affected by the mass ratio m∗ for either twodimensional flags or slender flags (Jia et al., 2007; Michelin and Llewellyn Smith, 2009; Mougel et al., 2016). The dominant synchronized mode is also affected by m∗ , resulting in an out-of-phase mode for large m∗ and an in-phase mode for small m∗ . Note that our definition of mass ratio (m∗ = ρs h/ρf L) is different than in the works of Michelin and Llewellyn Smith (2009) and Mougel et al. (2016) (M ∗ = ρf L/ρs h). Meanwhile, it was reported both experimentally (Kim et al., 2013) and theoretically (Sader et al., 2016b) that the critical velocity of a single inverted flag is independent of m∗ . To determine whether the stability boundary of dual inverted flags is also independent of m∗ and justify our assumption to set ω = 0 in the linear stability analysis of Section 3.1.2, an experiment with the same model was conducted in a water tunnel for low m∗ = 0.004 (m∗ = 3.33 for air). In water, Uc∗,water = 1.77 for [G∗ , H ∗ ] = [0.67, 1.33] and Uc∗,water = 1.82 for [G∗ , H ∗ ] = [1.00, 1.33]. These Uc∗ values corresponded well to the Uc∗ values obtained in the wind tunnel experiments for the same configuration; Uc∗,air = 1.85 and 1.89, respectively (Fig. 5). In other cases not reported here, we verified little difference between Uc∗,air and Uc∗,water , which implies that Eq. (4) and the data in Fig. 5 can be used as an approximation of the stability boundary for any mass ratio. 4. Non-linear behaviors of dual inverted flags 4.1. Classification of non-linear modes The state that occurs after instability, U ∗ > Uc∗ , is defined as a non-linear regime while the state of U ∗ < Uc∗ is a linear regime (Jia et al., 2007; Mougel et al., 2016). As reported by Kim et al. (2013), for a single inverted flag, a straight mode is

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Fig. 6. Time sequence of non-linear modes: (a) in-phase mode, (b) out-of-phase mode, (c) erratic mode, (d) deflected mode, and (e) attached mode. See supplementary movies for (a), (b), and (e). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

observed in the linear regime, and a flapping mode with large amplitude and a deflected mode are observed in the non-linear regime. For dual inverted flags, due to the interaction between the two flags, a pre-instability state is additionally observed after an initial straight mode, at which the two flags are bent outward slightly with the transverse displacement of the leading edge smaller than 0.1L and maintain their deformed shapes statically. From the straight mode to the pre-instability state, the transverse displacement, no matter how small it is, continues to increase monotonically as U ∗ increases, which is followed by divergence instability at U ∗ = Uc∗ . Note that this pre-instability state is quite similar to the deformed steady state mentioned in Gurugubelli and Jaiman (2015), but the two flags bent only outward regardless of the initial deflection direction of leading edges. In the non-linear regime, five modes are observed: in-phase mode, out-of-phase mode, erratic mode, deflected mode, and attached mode (Fig. 6). The first two synchronization modes (in-phase mode and out-of-phase mode) are divided into I1 and I2 for the in-phase mode O1 and O2 for the out-of-phase mode, based on a free-stream velocity range and a flapping amplitude. In addition, we also observe another interesting coupling mode which has not been reported in the studies of dual conventional flags: attached mode (Fig. 6e). The coupling modes addressed in this section are consistently observed at a given flow velocity. A detailed explanation of the synchronization modes is given in Section 4.2. The attached mode is discussed in Section 4.3. In addition, an erratic mode is observed in a specific range of a free-stream velocity when a single coupling mode cannot be established (Fig. 6c). When the hydrodynamic force applied to a flag becomes larger than the restoring bending force of the flag, the flag cannot rebound once it is deflected, and instead maintains the deflected shape statically or quasi-statically. This is defined as the deflected mode (Kim et al., 2013). For dual inverted flags, the deflected mode displays two different types of deformation. Each flag can be deflected outward, or both flags can be deflected to the same side (Fig. 6d). In most cases of our experiments, the first deflected mode in Fig. 6d was observed, and the second deflected mode in Fig. 6d was observed in the limited cases where the distance between the two flags is small and the aspect ratio is small in general. Depending on the magnitudes of G∗ and H ∗ , dual inverted flags can exhibit various combinations of coupling modes: an attached mode, two in-phase modes (I1 and I2 ), and two out-of-phase modes (O1 and O2 ) (Table 1). When two flags are close to each other, a stronger interaction between them can be expected. For this reason, we will mainly focus on the dynamics of the coupling modes and mode transition processes at small G∗ = 0.33 and 0.67 in the following sections (Figs. 10 and 11).

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H. Kim, D. Kim / Journal of Fluids and Structures 84 (2019) 18–35 Table 1 Classification of coupling modes observed in each case of G∗ and H ∗ : ‘A’ is the attached mode, ‘I1 ’ and ‘I2 ’ are the first and second in-phase modes, ‘O1 ’ and ‘O2 ’ are the first and second out-of-phase modes. Note that straight mode, erratic mode, and deflected mode are excluded in this table. In each case above, observed coupling modes are listed in the order of increasing U ∗ . The symbol ‘-’ in the second column means that the inverted flags display only straight and deflected modes without attached mode, in-phase mode, and out-of-phase mode; see also Section 4.3.3. H ∗ (= H /L) G∗ (= G/L)

0.07

0.17

0.33

0.67

1.33

∞

0.33 0.67 1.00 1.33

– – – –

A A A A

O2 , A, I2 A I2 I2

I1 , A, I2 I1 , O2 , A, I2 I 1 , O2 , I 2 O2 , I 2

O1 , I1 , I2 I1 , O2 , I2 I1 , O2 , I2 O2 , I2

O1 , I1 , I2 O1 , I1 , I2 , O2 O1 , I1 , O2 O1 , I1 , O2

4.2. Synchronization modes 4.2.1. Characteristics of synchronization modes In Section 3, we showed that the stability boundary of dual inverted flags Uc∗,dual is lower than that of a single inverted flag Uc∗,single because of the interaction between the two flags (Fig. 5). Interestingly, two distinct regions exist for the synchronization modes at U ∗ > Uc∗,dual . Between Uc∗,dual and Uc∗,single (region I of red color in Fig. 7a), dual flags exhibit coupling modes whereas a single flag remains as a straight mode, which means an additional hydrodynamic force originated by mutual interaction of the two flags plays a critical role in inducing a non-linear behavior in this region. The hydrodynamic force is not enough to lead each flag to flap with large amplitude in the range of Uc∗,dual < U ∗ < Uc∗,single . Instead, the synchronized flags in this region flap with relatively small amplitude. We define the synchronization modes in Uc∗,dual < U ∗ < Uc∗,single as I1 for the in-phase mode and O1 for the out-of-phase mode, respectively. The peak-to-peak flapping amplitude A/L of the I1 mode is around 1.0, and A/L = 0.2 − 0.6 for the O1 mode. Here, A is the peak-to-peak flapping amplitude of a leading edge averaged for the two flags (Fig. 1). Meanwhile, synchronization modes at U ∗ > Uc∗,single (region II of blue color in Fig. 7a) generally exhibit large-amplitude flapping because of sufficient hydrodynamic force. Note that a single flag can flap with large amplitude in this range without an additional hydrodynamic force (mutual interaction) by another flag. In this region, the synchronization modes are defined as I2 for the in-phase mode and O2 for the out-of-phase mode, respectively. A/L of the I2 mode is larger than 1.5, and A/L = 1.0 − 1.3 for the O2 mode. In general, the out-of-phase mode has lower amplitude than the in-phase mode because of the symmetric motion of the two flags and resultant interference by the opposite flag near a midline. When gap distance G∗ between the two flags increases, Uc∗,dual approaches to Uc∗,single , and the region I (I1 and O1 modes) tends to disappear. In Table 1, G∗ = 1.33 cases do not show either I1 or O1 mode except H ∗ = ∞. Meanwhile, as H ∗ decreases (e.g., H ∗ = 0.17 and 0.33), Uc∗,dual becomes larger and approaches to Uc∗,single . Then, I1 and O1 modes are not observed. Instead, a mode transition occurs directly from the linear regime to the synchronization mode with large amplitude (I2 and O2 modes) or to the attached mode (Table 1). The four synchronization modes (I1 , I2 , O1 , O2 ) are clearly distinguished by a phase diagram for the angle of attack of the leading edge (tangential angle at the leading edge), which provides the information of a phase difference between the two flags and a flapping amplitude of each flag indirectly (Fig. 7b−d). Refer to Fig. 7e for the definition of a tangential angle at the leading edge. In our experiment, the in-phase and out-of-phase modes could be easily identified although the amplitudes of the two flags differed slightly in many cases and the phase difference between them was not exactly 0◦ for the in-phase mode or 180◦ for the out-of-phase mode. Although the strict definition of the in-phase and out-of-phase modes does not allow such deviation, here we use the definition loosely to determine whether the flags are in a coupled mode; larger deviation is found for two-dimensional flags (Section 4.2.3). In Fig. 7b,c, most points are collected near the θ2 = −θ1 line for the out-of-phase mode (Fig. 7b) and near the θ2 = θ1 for the in-phase mode (Fig. 7c). The points are spread in a wider range of the angle of attack for the I2 and O2 modes (Fig. 7bii,cii) compared with the I1 and O1 modes (Fig. 7bi,ci). Also note that the phase of the erratic mode is broadly distributed and its pattern is distinctly different from those of the synchronization modes (Fig. 7d). 4.2.2. Transition from O2 mode to I2 mode In-phase and out-of-phase modes are mainly observed at G∗ ≥ 0.67 and H ∗ = 0.67 and 1.33 where the attached mode disappears, except in one case, [G∗ , H ∗ ] = [0.67, 0.67] (Table 1). In these cases, we found a clear trend in the flapping dynamics. Both the O2 and I2 modes are always observed in these cases, and there is a direct transition from the O2 mode to the I2 mode. In the O2 mode, the leading edges of the two flags periodically clap each other near the midline. When U ∗ continues to increase, the out-of-phase mode becomes unstable due to the increased impact, and the two flags eventually bend together in one direction; the transition from the O2 mode to the I2 mode occurs (Fig. 8). Meanwhile, at small G∗ (G∗ = 0.33), the O2 mode is not observed, and the periodic strong impact of the two flags does not occur because of short gap distance between them.

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Fig. 7. (a) Two distinct regions for synchronization modes and (b−d) phase map of four synchronization modes and erratic mode. θ1 and θ2 are tangential angles at the leading edges of the upper and lower flags, respectively (see subfigure (e)) . (bi) O1 mode ([G∗ , H ∗ , U ∗ ] = [0.33, 1.33, 1.67]), (bii) O2 mode ([G∗ , H ∗ , U ∗ ] = [0.67, 1.00, 2.30]), (ci) I1 mode ([G∗ , H ∗ , U ∗ ] = [0.33, 1.33, 1.95]), (cii) I2 mode ([G∗ , H ∗ , U ∗ ] = [0.33, 1.33, 3.20]), and (d) erratic mode ([G∗ , H ∗ , U ∗ ] = [0.33, 1.33, 1.97]).

The change between in-phase and out-of-phase modes has previously been investigated for two-dimensional dual conventional flags (Jia et al., 2007; Schouveiler and Eloy, 2009; Alben, 2009). However, it has not been reported that this mode change is initiated by periodic collision between the two flags. Two conventional flags in the out-of-phase mode may contact each other at small G∗ . However, compared to inverted flags undergoing a large-amplitude flapping motion, the impact of conventional flags may not be strong enough to cause a transition from the out-of-phase mode to the in-phase mode. That mode transition, produced by periodic clapping, may be a unique behavior of an inverted configuration exhibiting a large-amplitude oscillation.

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Fig. 8. Transition from O2 mode to I2 mode; from the left image on the first row to the right image on the second row. [G∗ , H ∗ , U ∗ ] = [0.67, 1.33, 2.52]. See a supplementary movie.

Fig. 9. Dimensional frequency drop shown in the O2 -I2 mode transition: O2 mode (filled circle) and I2 mode (empty circle). (a) [G∗ , H ∗ ] = [0.67, 1.33], (b) [G∗ , H ∗ ] = [1.00, 0.67], (c) [G∗ , H ∗ ] = [1.00, 1.33], (d) [G∗ , H ∗ ] = [1.33, 1.33].

A sudden drop in a frequency occurs when the mode changes directly from the O2 mode to the I2 mode at a given G∗ (Fig. 9). A frequency jump was also observed during the change from the in-phase mode to the out-of-phase mode in conventional flags, but it was investigated by varying the distance G∗ between the flags instead of varying U ∗ (Zhang et al., 2000; Farnell et al., 2004; Jia et al., 2007). For dual inverted flags, the frequency drop ratio is at most 18%, as shown in Fig. 9, which is lower than the jump ratio (20%−40%) reported for the conventional flags. If G∗ is small, the dual inverted flags can flap with a much larger amplitude in the in-phase mode as opposed to the out-of-phase mode at which the two flags may come into contact on the midline. For this reason, the increase in amplitude is observed with the O2 -I2 mode transition as shown in Fig. 11e, which is accompanied by a frequency drop. 4.2.3. Synchronization modes in large aspect ratio According to the stability analysis in Section 3.1, an out-of-phase mode first occurs after an initial equilibrium breaks. We observed consistently that the dual flags initially opened the gap (the gap distance became larger) as the mode changed from the pre-instability state to a non-linear mode for all cases. For two-dimensional flags (H ∗ = ∞), as the theory predicted, an out-of-phase (O1 ) mode is consistently observed after the initial equilibrium for all gap distances considered in this study (Table 1). However, although the flags initially open the gap at the critical velocity, because of the interaction between the flags, the in-phase mode is eventually established in many cases instead of the out-of-phase mode. In other words, the in-phase mode may occur as the first synchronization mode observed after bifurcation. In the experiment with two-dimensional flags, the deviation in a phase and a flapping amplitude between the two flags was generally larger than the other flags with finite H ∗ . For two-dimensional flags, asymmetric flapping is frequently observed in the O2 mode. Due to asymmetric flapping, the impact between the two flags may not be strong enough to initiate the O2 -I2 transition. Instead of the I2 mode, an erratic mode appears after the O2 mode (Table 1 and Fig. 11f). Note that, in this

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Fig. 10. Dimensionless peak-to-peak amplitude A/L of dual inverted flags and coupling modes exhibited at G∗ = 0.33. (a) H ∗ = 0.07, (b) H ∗ = 0.17, (c) H ∗ = 0.33, (d) H ∗ = 0.67, (e) H ∗ = 1.33, and (f) H ∗ = ∞. The vertical dashed lines in each graph divide the ranges of each mode, and the corresponding mode in each range is presented above each graph. ‘S’ : straight mode, ‘E’: erratic mode, and ‘D’: deflected mode. For the other modes, see the caption of Table 1.

study, we used flags with H = 55 cm as an alternative to using two-dimensional flags. A small degree of torsion was observed between the upper edge and bottom edge of the flags, although such torsion should not exist in ideal two-dimensional flags. The asymmetric flapping caused by the torsion may also be responsible for the O2 -erratic mode transition. When H ∗ becomes larger, Uc∗ decreases inversely because of the increase in a hydrodynamic force per unit flag height, as mentioned in Section 3.2.3 (Fig. 5). In the deflected mode, the fluid force and bending force are balanced statically or quasi-statically. In the deflected mode, for higher H ∗ , a larger fluid force per unit flag height is imposed on the flags. Thus, as H ∗ increases, U ∗ at which the flags have a transition to the deflected mode is expected to be lower. This was confirmed by our experimental results (Figs. 10 and 11). 4.3. Attached mode 4.3.1. Characteristics of attached mode For dual conventional flags, leading edges are fixed, and trailing edges are free to move. A gap flow between the flags prohibits the flags from touching each other periodically or statically on the midline. However, for dual inverted flags, since a free stream impinges on free leading edges, both leading edges can contact on the midline. Accordingly, symmetric clapping motion can occur in the O2 mode. Besides, in some conditions, the flags will maintain their deformed shape statically or quasi-statically after the contact. This novel attached mode exists in a certain U ∗ range, mainly after the pre-instability state. The deformed shape of the attached mode is determined by the fundamental harmonic mode of the flag. From the linearized Euler–Bernoulli equation with a free-clamped boundary condition, the fundamental mode is as follows (Meirovitch, 1997): y1 = cosh(k1 x) − cos(k1 x) +

cosh(k1 L) + cos(k1 L) sin(k1 L) + sinh(k1 L)

(sin(k1 x) − sinh(k1 x)),

(5)

where k1 L is 1.875. The scaled shape of the fundamental mode shows little discrepancy from the actual deformed shape of the attached mode as illustrated in Fig. 12. Although not presented here, in addition to the cases of Fig. 12, the deformed shape of any attached mode and other coupling modes can be constructed with good accuracy by the fundamental mode without

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Fig. 11. Dimensionless peak-to-peak amplitude A/L of dual inverted flags and coupling modes exhibited at G∗ = 0.67. (a) H ∗ = 0.07, (b) H ∗ = 0.17, (c) H ∗ = 0.33, (d) H ∗ = 0.67, (e) H ∗ = 1.33, and (f) H ∗ = ∞. The vertical dashed lines in each graph divide the ranges of each mode, and the corresponding mode in each range is presented above each graph. ‘S’ : straight mode, ‘E’: erratic mode, and ‘D’: deflected mode. For the other modes, see the caption of Table 1.

Fig. 12. Comparison between the actual shape (solid line) and scaled fundamental harmonic mode (dashed line) of the flags in the attached mode. (a)[G∗ , H ∗ ] = [0.33, 0.17], U ∗ = 2.88 and (b)[G∗ , H ∗ ] = [0.33, 0.33], U ∗ = 3.16.

incorporating harmonic modes of the higher order, which is one of the characteristics of the inverted flag dynamics (Kim et al., 2013). The attached mode is typically observed when G∗ and H ∗ are small (Table 1). Noticeably, regardless of G∗ , for the ∗ H = 0.17 cases, only the attached mode is observed out of the three coupled modes (attached mode, in-phase mode, and out-of-phase mode) (see also Figs. 10b and 11b). Since A is defined as the peak-to-peak flapping amplitude, A is zero in the attached mode of Figs. 10 and 11. When stability breaks at Uc∗ ≈ 2.8 − 3.0, the two flags deform outward at first. As the two flags rebound and approach symmetrically to the midline between the two flags, the leading edges of the flags meet and flap together with small amplitude. Shortly, the flags form the attached mode where the fluid loading of free-stream flow, the bending force of the flags, and the reaction force produced by the other flag in contact are statically balanced (Fig. 13a,c). For H ∗ = 0.17, the transition occurs directly from the straight mode to the attached mode. However, for H ∗ ≥ 0.33, the attached mode generally occurs between the in-phase mode, the out-of-phase mode or the erratic mode instead of a direct transition from the straight mode, or to the deflected mode (Figs. 10c,d and 11c,d). Before the transition to the attached mode, the two flags intermittently contact each other in the erratic mode, and contact regularly in the in-phase or out-of-phase mode. With increasing U ∗ , a static or quasi-static state of the attached mode is eventually

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Fig. 13. Transition to the attached mode. (a) [G∗ , H ∗ ] = [0.33, 0.17] and U ∗ = 2.88. (b) [G∗ , H ∗ ] = [0.33, 0.33] and U ∗ = 2.67. (c, d) Time history of transverse displacements of two leading edges during the transition to the attached mode; (c) and (d) correspond to (a) and (b), respectively. A solid line is for the lower flag, and a dashed line is for the upper flag in Fig. 2. See supplementary movies for (a) and (b).

established. See Fig. 13b,d for the transition from the erratic mode to the attached mode. In the attached mode, the leading edges of the two flags are forced to touch each other by the free stream. When the attached mode is formed, the contact point of the two flags is not positioned exactly at the leading edges of the two flags. In addition, there is slight difference in material properties between the two flags. Thus, as U ∗ increases further, because of increasing fluid loading, the two flags move off the centerline and start to bend together to either side. Then, they eventually begin an in-phase mode or an erratic mode. The out-of-phase mode is not observed right after the attached mode in our cases (Table 1, Figs. 10 and 11). 4.3.2. Effect of gap distance The U ∗ range and configuration of the attached mode is strongly influenced by G∗ . For small G∗ (G∗ = 0.33), the two flags bend symmetrically and contact each other exactly on the midline in the attached mode. In the symmetric attached mode, the deflection displacement of the free leading edge with respect to the x-axis, ∆y/L = 0.165 is smaller than that of the deformed mode, ∆y/L > 0.7. That is, to maintain the static equilibrium in the attached mode, the y-directional reaction force by another flag is imposed at the leading edge. At [G∗ , H ∗ ] = [0.33, 0.17], the deformed configuration is streamlined and stable without any oscillation (Fig. 14a). Because of this stable force balance, the flags maintain their deformed shapes in spite of increasing U ∗ , and the transition to another mode is delayed. Under these conditions, the transition from the attached mode to another mode did not occur in the U ∗ range considered in this study; the attached mode was still observed at U ∗ = 4.3, the maximum U ∗ we tested. At [G∗ , H ∗ ] = [0.33, 0.33], the flags are symmetric with respect to the midline in the attached mode, at the relatively low U ∗ = 2.67 (Fig. 14b). However, as U ∗ increases, the leading edges of the two flags begin to move off the midline together, and the configuration becomes asymmetric. The equilibrium of the attached mode eventually breaks at U ∗ = 3.36, which is followed by the in-phase mode. On the other hand, for G∗ = 0.67, although two flags are static, they are not symmetric in the attached mode, and the leading edge of one flag is ahead of the other flag (Fig. 14c,d). When U ∗ increases further, the fluid force on the more deflected flag becomes larger than it is on the less deflected flag, and the contact point between the two flags moves gradually inside the less deflected flag. The flags maintains the equilibrium state of the attached mode until the less deflected flag bends outward. Since the shape of the flags can be easily altered by an increase in U ∗ due to the asymmetric configuration at G∗ = 0.67, the U ∗ range of the attached mode is relatively shorter, and the transition to another mode can occur at lower U ∗ , compared ∗ to G∗ = 0.33, at given H ∗ . The velocity when the transition to another mode from the attached mode occurs is Utran > 4.9 ∗ ∗ (H ∗ = 0.17, not observed in the experiment), Utran = 3.4 (H ∗ = 0.33), 3.1 (H ∗ = 0.67) for G∗ = 0.33 while Utran is 3.2

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Fig. 14. Deformed shapes of inverted flags in attached mode. (a) [G∗ , H ∗ ] = [0.33, 0.17]. U ∗ = 2.88 (solid), 2.98 (dotted), 3.35 (dashed), 3.72 (dash-dot). (b) [G∗ , H ∗ ] = [0.33, 0.33], U ∗ = 2.67 (solid), 2.87 (dotted), 3.02 (dashed), 3.16 (dash-dot). (c) [G∗ , H ∗ ] = [0.67, 0.17], U ∗ = 2.96 (solid), 3.09 (dotted). (d) [G∗ , H ∗ ] = [1.00, 0.17], U ∗ = 3.24 (solid), 3.48 (dotted), 3.87 (dashed).

Fig. 15. Stable equilibrium mode of the dual inverted flags for H ∗ = 0.07 and 0.17 (G∗ = 0.33 and 0.67). The stable equilibrium mode in the parenthesis was obtained manually.

(H ∗ = 0.17), 2.9 (H ∗ = 0.33), 2.8 (H ∗ = 0.67) for G∗ = 0.67 (Figs. 10b−d and 11b−d). Even when G∗ increases (G∗ ≥ 1.00), an attached mode can be formed, but only at small H ∗ = 0.17. As indicated in Fig. 14d, the attached mode shows a different behavior than that of G∗ = 0.33 and 0.67; the asymmetric flags continue to bend inward with increasing U ∗ . With an increase in U ∗ , the inwardly deformed configuration is sustained statically or quasi-statically, rather than changing to the two types of deflected mode shown in Fig. 6d. Such inward deflection of both flags was only observed for [G∗ , H ∗ ] = [1.00, 0.17] and [1.33, 0.17] among all the cases investigated in this study. 4.3.3. Stable equilibrium states of small aspect-ratio flags For very small height H ∗ , a single inverted flag displays only straight and deflected modes without a large-amplitude flapping mode (Sader et al., 2016a,b). The same trend is also observed with dual inverted flags. At low H ∗ , because of the small fluid force acting on the flag per unit height, Uc∗ becomes large as explained in Section 3.2.3. Thus, when the initial equilibrium breaks and the flags start to bend outward, the fluid force causing the bending is so large the flag may not rebound to form the attached mode. At H ∗ = 0.07, Uc∗ > 3.5 (Fig. 5), and no coupling modes are observed (Table 1, Figs. 10a and 11a). At H ∗ = 0.07, as U ∗ crosses Uc∗ , the equilibrium of the straight mode becomes unstable, and the flags reach directly to another stable equilibrium of the deflected mode for all of the G∗ cases considered in this study. However, this deflected mode is not the only stable equilibrium mode in U ∗ > Uc∗ . In the experiment with G∗ = 0.33 and 0.67, we manipulated the flags to form the symmetric attached mode by imposing external force manually when they were in the deflected mode at U ∗ > Uc∗ . With the external force removed, the flags maintained the attached mode without returning to the deflected mode (Fig. 15). In other words, at U ∗ > Uc∗ , the flags have two stable equilibriums: the deflected mode and the attached mode hidden in the original experiment conducted without imposing manual external force.

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Fig. 16. Hysteresis of amplitude in the mode transition for [G∗ , H ∗ ] = [0.33, 0.33]. Amplitudes with the solid line and a dashed line were obtained when the free-stream velocity U increased and decreased, respectively. The notations of the modes are presented above the graph as Figs. 10 and 11. Note that additional experimental points are included to clearly show hysteresis, compared to Fig. 10c.

The H ∗ = 0.17 cases exhibit the attached mode first after the instability occurs (Table 1). In the attached mode regime, when the flags were manipulated to form the deflected mode with the manual external force, the flags returned to the attached mode instantly with the external force removed (Fig. 15). The deflected mode is not at equilibrium, and the attached mode is the only stable mode. With a further increase in U ∗ , the flags continue to maintain the attached mode for G∗ = 0.33 in the U ∗ range tested in this study (Fig. 10b) and have the transition to the deflected mode for G∗ = 0.67 (Fig. 11b). When the flags displayed the deflected mode, we were able to realize the symmetric attached mode manually like the H ∗ = 0.07 cases. That is, the attached mode is the stable equilibrium mode for all U ∗ larger than Uc∗ . Nevertheless, the mode transition naturally occurs from the stable attached mode to the stable deflected mode when U ∗ crosses a specific value. The transition from the attached mode to the deflected mode is influenced by the symmetry of the flags in the attached mode as mentioned above (see Fig. 14). When the attached mode is formed naturally by increasing U ∗ , the leading edges of the flags may not be aligned symmetrically, but be off the midline, which eventually causes the transition to the deflected mode with the increase in U ∗ . If the symmetric attached mode is built manually, then the transition to the deflected mode is suppressed. Theoretical investigation is necessary in future to address if the symmetric attached mode remains stable at any U ∗ beyond the U ∗ range of our experiment. 4.4. Hysteresis and mass-ratio effect Hysteresis in flapping amplitude has been reported in the studies of dual conventional flags and a single inverted flag (Zhang et al., 2000; Kim et al., 2013). A single inverted flag exhibits hysteresis in two transitions with a drastic change in the amplitude: a transition from the straight mode to the flapping mode and a transition from the flapping mode to the deflected mode. The straight and deflected modes are static or quasi-static with negligible amplitude, and the flapping mode has a peak-to-peak amplitude A/L larger than 1.6. Like the single inverted flag, hysteresis is also found in dual inverted flags when there is a sudden change in amplitude between two successive modes, such as the straight to flapping mode transition, the O1 or I1 to the attached mode transition, the attached to the in-phase (I2 ) mode transition, and the in-phase (I2 ) to the deflected mode transition. For an example, see Fig. 16 for [G∗ , H ∗ ] = [0.33, 0.33]. When U ∗ decreases (when the free-steam velocity U decreases in the experiment, shown by the solid line in the graph), each mode transition occurs at a lower U ∗ compared to when U ∗ increases (the dashed line in the graph). In Fig. 16, the transition between the I2 and deflected modes with decreasing U ∗ occurs even at a U ∗ similar to the U ∗ observed for the attached to I2 mode transition with increasing U ∗ . In the mode transition between O2 and I2 , although the difference in amplitude between the two modes is relatively small, ∆A/L ≈ 0.3, hysteresis is also observed. For example, for [G∗ , H ∗ ] = [0.67, 1.33], with increasing U ∗ , the transition between O2 and I2 occurs at U ∗ = 2.52, while it occurs at U ∗ = 2.43 with decreasing U ∗ . In Section 3.2.4, we revealed that the stability boundary of the dual inverted flags was independent of the mass ratio m∗ . However, the coupled dynamics of the flags in the non-linear regime are affected by m∗ . In the water tunnel experiments, the two inverted flags exhibited only the erratic mode until they reached the deflected mode, and did not exhibit any coupled mode such as the attached mode, the in-phase mode, or the out-of-phase mode. When the flags are immersed in water, the high density of the water acts as a damper, reducing the amplitude and frequency of the flags. The interaction of two flags in a fluid – in other words, the transmission of force from one passive object to another passive object – weakens accordingly, which makes the appearance of synchronized modes or the attached mode more difficult in a fluid medium of high density. 5. Concluding remarks In this study, the stability and coupled dynamics of dual inverted flags were investigated theoretically and experimentally, and noticeable distinctions were identified in comparison to dual conventional flags. The critical velocity for the transition

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from a static linear regime to a non-linear regime is affected by both flag height and gap distance between the flags. The critical velocity is reduced monotonically when the flag height increases and the gap distance decreases. For the flags with small height, the critical velocity is less influenced by variation in the gap distance. In addition to the observation of the synchronized modes common to dual conventional flags, we found a novel attached mode, where the leading edges of the two flags meet and the flags maintain their deformed shapes statically. The attached mode is mainly observed when the flag height and the gap distance are small. We also identified a new mechanism of a mode transition, from the out-of-phase mode to the in-phase mode, caused by the periodic impact of two flags flapping with large amplitude. The inverted flag configuration has been of interest because of several interesting properties including divergence instability, low critical velocity, and large-amplitude flapping. From the perspective of engineering applications, research on the stability and non-linear behaviors of the inverted configuration of multiple objects will provide insight for the efficient design of novel systems to harvest energy from a fluid flow with detailed analysis on energy transfer (i.e., Tang and Païdoussis, 2009; Howell et al., 2009; Doaré and Michelin, 2011; Howell and Lucey, 2015; Xia et al., 2016; Shoele and Mittal, 2016) and to enhance convective heat transfer (Shoele and Mittal, 2014; Park et al., 2016). 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