Interaction modes of multiple flexible flags in a ... - koasas - KAIST

J. Fluid Mech. (2013), vol. 729, pp. 563–583. doi:10.1017/jfm.2013.314

c Cambridge University Press 2013

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Interaction modes of multiple flexible flags in a uniform flow Emad Uddin1 , Wei-Xi Huang2 and Hyung Jin Sung1 , † 1 Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Korea 2 Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

(Received 22 June 2012; revised 4 June 2013; accepted 12 June 2013; first published online 24 July 2013)

Fish schooling is not merely a social behaviour; it also improves the efficiency of movement within a fluid environment. Inspired by the hydrodynamics of schooling, a group of flexible bodies was modelled as a collection of individuals arranged in a combination of tandem and side-by-side formations. The downstream bodies were found to be strongly influenced by the vortices shed from an upstream body, as revealed in the vortex–vortex and vortex–body interactions. To investigate the interactions between flexible bodies and vortices, the present study examined flexible flags in a viscous flow by using an improved version of the immersed boundary method. Three different flag formations were modelled to cover the basic structures involved in fish schooling: triangular, diamond and conical formations. The drag coefficients of the downstream flags could be decreased below the value for a single flag by adjusting the streamwise and spanwise gap distances and the flag bending coefficient. The drag variations were influenced by the interactions between vortices shed from the upstream flexible flags and those surrounding the downstream flags. The interactions between the flexible flags were investigated as a function of both the gap distance between the flags and the bending coefficients. The maximum drag reduction and the trailing flag position were calculated for different sets of conditions. Singlefrequency and multifrequency modes were identified and were found to correspond to constructive and destructive vortex interaction modes, which explained the variations in the drag on the downstream flags. Key words: flow-structure interactions, swimming/flying, wakes/jets

1. Introduction The swimming behaviour of fish schools is a familiar but complicated problem that has enticed both biologists and hydrodynamicists during the last several decades. Since both social factors and hydrodynamic influences are entangled in fish schooling, the questions surrounding which factors contribute dominantly to the formation of fish schools remain controversial (Weihs 1973; Partridge & Pitcher 1979). From a hydrodynamic point of view, experimental evidence has supported the assertion that the appropriate synchronization of movements and individual positioning in a † Email address for correspondence: [email protected]

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fish school can be advantageous by decreasing the hydrodynamic resistance due to wake interactions and introducing significant energy savings (Belyayev & Zuyev 1969; Weihs 1975; Breder 1976; Abrahams & Colgan 1985). Nevertheless, the properties of the hydrodynamic interactions within a fish school are not yet well-understood, and a more detailed understanding of the mechanisms would be desirable (Fish 1999). A swimming fish is typically modelled as a flapping rigid foil (Streitlien, Triantafyllou & Triantafyllou 1996; Zhu & Peng 2009) or a travelling wavy foil (Dong & Lu 2007; Deng, Shao & Yu 2007), and the fluid–flexible-body interactions have not generally been considered; thus, the key advantages of schooling behaviour among flexible bodies in a fluid flow might be missed. Recently, flexible flags (or filaments in two dimensions) immersed in a surrounding fluid flow have been considered to be simplified heuristic models for understanding the swimming of fish (Zhang et al. 2000). Although swimming fish are expected to experience a combination of active and passive responses to the surrounding environment, systems comprising a purely passive flapping flag in a fluid flow can be described intrinsically in terms of the fluid–body coupling physics that apply to fish swimming. Such systems can lend insight into how the flexibility of a fish (flag) affects the swimming efficiency (Huber 2000; Shelley & Zhang 2011). Indeed, experimental observations have shown that a downstream fish may benefit from energy savings due to reduced muscle activity by swimming in the wake of an upstream obstacle in a manner that resembles passive flapping (Liao et al. 2003; Muller 2003). Hence, investigations of vortex–vortex and vortex–body interactions among multiple flexible flags appropriately arranged and immersed in a fluid flow can shed light on the possible mechanisms intrinsic to fish schooling. The possibility of gaining these types of insights has motivated the present study. Using flexible filaments in a flowing soap film, Zhang et al. (2000) observed two distinct stable states for a single filament: a stretched-straight state and a selfsustained flapping state. These states were regarded as analogous to the gliding and swimming states, respectively, of fish. Under certain circumstances, a filament can exhibit bistability, that is, it will assume either of the above two states, depending on the initial conditions. Inspired by this work, extensive studies have sought to identify the different motion modes of the flag-in-flow problem and to evaluate the effects of certain key parameters: the mass ratio, the non-dimensional bending rigidity and the Reynolds number (Zhu & Peskin 2002; Shelley, Vandenberghe & Zhang 2005; Connell & Yue 2007; Alben & Shelley 2008; Eloy et al. 2008). Flow past multiple flags placed in tandem or side-by-side arrangements has been examined toward furthering our understanding of schooling behaviour. Using the same experimental set-up, Zhang et al. (2000) showed that two side-by-side filaments will flap in phase for sufficiently small spanwise gap distances, whereas the flapping transits to an out-of-phase mode as the gap distance increases beyond a critical value. The experimental findings were later confirmed in a serious of experimental, theoretical, and numerical studies (Zhu & Peskin 2003; Farnell, David & Barton 2004; Jia et al. 2007; Tang & Paidoussis 2009). Using a vortex method to model the flow, Alben (2009) observed additional irregular flapping modes in a two-flag system. These modes were characterized by abrupt switching between the in-phase and out-of-phase modes. Experimental studies of the in-phase swimming of fish in a diamond formation were also conducted (Partridge & Pitcher 1979). The distance between two fish was set to a sufficiently small value such that the fish could push on each other to improve their overall energy savings. Systems comprising more than two flags in a side-by-side arrangement in a uniform flow displayed different coupling modes in both experimental and analytical investigations

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(Schouveiler & Eloy 2009; Michelin & Llewellyn Smith 2009). Taking the three evenly spaced side-by-side flags as an example, three flapping modes, i.e. symmetric, in phase and out of phase, were described. More recently, a detailed numerical study by Tian, Luo & Zhu (2011a) revealed three additional coupling modes in the nonlinear regime, including a half-frequency mode, an irrational frequency mode and erratic flapping. In systems of flags placed in tandem, vortices shed from the upstream flag encounter the vortices shed from the downstream flag and produce complicate interactions. Interestingly, an experiment involving two tandem filaments in a flowing soap film revealed that the downstream filament experienced a higher drag force than was experienced by the upstream flag or a single-filament system (Ristroph & Zhang 2008). An arrangement characterized by a higher drag force was described as displaying inverted drafting, in contrast with the conventional drafting forces experienced by rigid bodies arranged such that the downstream body benefits from drag reduction. This finding was significant in the sense that the flapping form varied with the position in a fish school, thus casting doubt on previous models that described the swimming motions of all fish in a school using the same form as that of an isolated fish (Fish 1999). Zhu (2009) simulated a system comprising two tandem flexible flags in a viscous flow characterized by a much lower Reynolds number, and they discussed the effect of the Reynolds number on the flapping motions. Normal drafting was found to be present at Reynolds numbers sufficiently small that both flags remained in the static state. Inverted drafting was observed once the Reynolds number reached sufficiently large values that both flags flapped steadily. The system within the transitional range of Reynolds numbers displayed both the static and flapping states, depending on the gap distance and the non-dimensional bending rigidity. In the potential flow system, Alben (2009) revealed that inverted drafting resulted from situations in which the wake of the upstream flag added constructively to the wake of the downstream flag. On the other hand, normal drafting was observed at certain gap distances, under which circumstances the flags’ wakes interacted destructively and caused erratic flapping of the downstream flag. At intermediate Reynolds number values, Kim, Huang & Sung (2010) clearly observed both constructive and destructive modes of vortex interactions in the flag wakes, corresponding to the inverted and normal drafting, respectively, at different streamwise gap distances (Gopalkrishnan et al. 1994). Furthermore, the spanwise gap distance or the bending rigidity was also varied to favour either of these two vortex interaction modes (Kim et al. 2010). In the present study, the dynamics of coupled flexible flags arranged in different formations and subjected to a two-dimensional uniform viscous flow were explored, including the triangular, diamond and conical arrangements, inspired by the problem of fish schooling. Although the tandem and side-by-side cases have been studied extensively, the introduction of both arrangements in a single assembly imposes more complicated interaction conditions. It is unknown whether the flapping forms pertinent to the tandem or side-by-side flags will be applicable to larger assemblies or whether new flapping modes will be formed. Thus, studies of the assemblies of flags in triangular, diamond and conical formations in a fluid flow provide a necessary step toward a better understanding of schooling behaviour. To this end, the dependence of the drag coefficients on the streamwise and spanwise gap distances of the flags were examined in detail. Different flapping modes were identified and were related to the modes observed previously in the tandem and side-by-side cases. The next section introduces the formulation of the problem, and the numerical method is briefly described. In § 3, the numerical results and discussion are presented, with

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F IGURE 1. Schematic diagram showing the three flags in a triangular formation subjected to a uniform viscous flow.

three subsections devoted to the triangular, diamond and conical formations. Finally, conclusions are drawn in § 4. 2. Problem formulation Here, a group of flexible fish was modelled using three, four or six flexible flags in triangular, diamond or conical arrangements in a two-dimensional viscous flow. A schematic diagram of the triangular configuration and the coordinate system is shown in figure 1. The relative positions of the three flexible flags were varied by adjusting the streamwise gap distance Gx between the upstream and downstream flags and the spanwise gap distance Gy between the downstream flags. The heads of both the upstream and downstream flags were fixed under a simply supported boundary condition. The boundary conditions for the free ends were considered at the tails. The fluid domain was defined by an Eulerian coordinate system, and a separate Lagrangian coordinate system was applied to each flexible flag. The multiple flexible flags in a viscous flow were modelled using an improved immersed boundary method (Huang, Shin & Sung 2007), in which the governing equations of the fluid flow and the multiple flexible flags were solved in each coordinate system, and the interactions among components were calculated using a feedback law. Moreover, a repulsive force term was added to the flag motion equation to model collisions of downstream flags, which were important to the out-of-phase flapping modes. The fluid motion was governed by the incompressible Navier–Stokes and continuity equations,

∂u 1 2 + u · ∇u = −∇p + ∇ u+f, ∂t Re ∇ · u = 0,

(2.1) (2.2)

where u is the velocity vector, p is the pressure, f is the momentum force applied to enforce the no-slip conditions along the immersed boundary and the Reynolds number Re is defined by Re = ρ0 U∞ L/µ, with ρ0 the fluid density, U∞ the free-stream velocity, L the flag length and µ the dynamic viscosity. The flag motion was governed by 2 ∂ 2X ∂ ∂X ∂2 ∂ X = T − 2 γ 2 − F + Fc , (2.3) 2 ∂t ∂s ∂s ∂s ∂s where s denotes the arc length, X = X(s, t) the position, T the tension force, γ the bending rigidity, F the Lagrangian force exerted on the flexible flag by the fluid and Fc the repulsive force. Equation (2.3) was non-dimensionalized by the flag density ρ1 ,


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the flag length L and the free-stream velocity U∞ . It should be pointed out that the tension force can be determined from the inextensibility condition of flag (Huang et al. 2007), and the bending rigidity is usually defined as EI, with E the Young’s modulus and I the second moment of area, which gives the non-dimensional value γ . The boundary conditions applied at the fixed end and the free end were ∂ 2X = (0, 0) for the fixed end ∂s2 ∂ 2X ∂ 3X = (0, 0), = (0, 0) for the free end. ∂s2 ∂s3

X = X0 , T = 0,

(2.4) (2.5)

The Lagrangian force F in (2.3) can be expressed by the feedback law, i.e. F(s, t) = −κ[(Xib − X) + 1t(Uib − U)],

(2.6)

where κ is a large constant, and 1t denotes the computational time step. In (2.6), X and U denote, respectively, the position and velocity of the material points with mass. Here X was solved using the solid motion equation and U = ∂X/∂t. On the other hand, Xib and Uib denote, respectively, the position and velocity of the immersed boundary, which were determined using the local Eulerian fluid velocity, as expressed by Z t 0 Xib = Xib + Uib dt, (2.7) 0 Z Uib (s, t) = u(x, t)δ(X(s, t) − x) dx (2.8) Ωf

where δ( ) denotes a smoothed approximation of the Dirac delta function and Ωf is the fluid region. Equation (2.8) provides an interpolation of the Eulerian fluid velocity at the Lagrangian points. In physics, (2.6)–(2.8) represent a stiff spring system, where the stiffness is denoted by κ. This system connected the Lagrangian points on the immersed boundary to the surrounding fluid particles. To ensure that the system would provide an accurate model of fluid–structure interactions, the free constant κ in (2.6) should be selected so as to be sufficiently large. In our simulations, we found that the numerical results become independent of κ as κ > 106 . After obtaining the Lagrangian force F, the expression was transformed into the Eulerian form using the smoothed Dirac delta function, Z f (x, t) = ρ F(s, t)δ(x − X(s, t)) ds, (2.9) Ωs

where ρ = ρ1 /(ρ0 L), based on the non-dimensionalization steps, and Ωs denotes the structure region. After discretization, the force was applied over a width of several grids, which supported the smoothed delta function. In the present study, an additional repulsive force was introduced to handle the collisions between the two side-by-side flags flapping in an out-of-phase mode in a viscous flow. Owing to the fluid lubrication, the two flags did not actually collide, but rather interacted repulsively via the intervening fluid when in close proximity. In the numerical simulation, an artificial repulsive force was activated at close range, as proposed by Glowinski et al. (1999). Among flexible flags, this short-range repulsive

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force can be formulated using the smoothed delta function (Huang et al. 2007), Z L X − X0 ds0 , (2.10) Fc (s, t) = δ(X(s, t) − X 0 (s0 , t)) 0| |X − X 0

where X(s, t) and X 0 (s0 , t) are the position vectors along the two flags, respectively. Equation (2.10) indicates that the repulsive force becomes non-zero for conditions in which the two flags are separated by a small distance, i.e. the width of the support of the smoothed delta function. In the present study, the repulsive force was not significant because the out-of-phase flag flapping motions were not generally stable due to the synchronizing effects of the upstream shedding vortices, as will be shown in the next section. The overall computational process, in which the present numerical algorithm was used to simulate the fluid–flexible-body interactions, is summarized as follows. (i) Initialize the computation parameters, the meshes and the fluid and solid motions; set Xib0 = X 0 . (ii) At the nth time step, the fluid velocity un and the solid position X n and velocity U n are known. Interpolate the fluid velocity at the Lagrangian points to obtain Uibn and calculate the positions of the immersed boundary points Xibn . Then calculate the Lagrangian momentum force Fn . (iii) Map the Lagrangian momentum force onto the Eulerian grid. Obtain the updated fluid velocity field and pressure field. (iv) Substitute Fn into the flag motion equation and obtain the flag position at the new time step X n+1 , as well as the velocity U n+1 = (X n+1 − X n )/1t. Return to step 2 and march to the next time step. The computational domain for the fluid flow ranged from −2 to 6 in the streamwise (x) direction and from −4 to 4 in the spanwise (y) direction, both of which directions were normalized by the flag length. The Eulerian grid size for the fluid was 512 × 350 in the streamwise and spanwise directions, respectively, and the Lagrangian grid size for each flag was 64. The Eulerian grid was uniformly distributed along the x direction, and it was uniform in the y direction for −2 6 y 6 2, but stretched otherwise. The far-flow field was applied at the top and bottom boundaries as well as the inlet of the fluid domain, whereas the convective boundary condition was used at the outlet. The computational time step was set to 0.0005, which resulted in a CFL number of 0.1. The validation of the flow solver employing the immersed boundary method, as well as the structure solver for the flag motion, has been provided in Huang et al. (2007). In the present study, we also verified our method by simulating a single flag and two side-by-side flags in a uniform flow. The computations were p performed at Re = 100 √ and Us = 100, as adopted in Tian et al. (2011b), where US = ρ0 U02 L3 /EI = 1/ ργ denotes the ratio of the fluid inertial force to the bending force. The results were compared with those reported in previous studies (Tian et al. 2011b; Zhang et al. 2000) and good agreements were found, as listed in table 1, including the critical mass ratio (ρcr ) for a single flag to start flapping, the critical gap distance (Gycr ) of transition from the in-phase flapping to out-of-phase flapping and the induced Strouhal number difference (1St) for two side-by-side flags with ρ = 0.3. Figure 2 shows the time history of the tail position of a single flag flapping in a uniform flow. The data from Zhang, Zhu & He (2013), who used the direct-forcing immersed boundary approach based on the discrete stream function method, was included for comparison. In this

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F IGURE 2. Time history of the tail position of a single flag flapping in a uniform flow.

Present Tian et al. (2011b) Zhang et al. (2000)

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TABLE 1. Comparison of the critical mass ratio (ρcr ) for a single flag in a uniform flow, and the critical gap distance (Gycr ) and the Strouhal number difference (1St) for two side-by-side flags with ρ = 0.3.

simulation the gravity force was also taken into account by adding an additional term Fr g/g at the right-hand side of (2.3), where g denotes the gravity acceleration vector 2 with g = |g|, and Fr = gL/U∞ is the Froude number. The simulation was carried out with Re = 200, ρ = 1.5, γ = 0 and Fr = 0.5. From figure 2 we can see that the present results agreed well with those of Zhang et al. (2013) on both the flapping amplitude and frequency. 3. Results and discussion 3.1. Triangular formation

A schematic diagram of the problem set-up and the coordinate system for the triangular formation of flags in a uniform flow is plotted in figure 1. The computational configurations were set to be the same as described in the previous section. The initial position of the upstream flag was parallel to the streamwise direction with its leading edge located at the origin and the downstream flags were inclined at an angle of 0.1π relative to the streamwise direction. Even if the inclination angle was changed to −0.1π, the results obtained after several transient flapping periods remained unchanged. The flow patterns were investigated by conducting simulations over a long time period, between 100 and 200 flapping periods, and the flow patterns obtained after a minimum of 20 flapping periods were analysed. The flow patterns and interaction forces were characterized in terms of the mean drag coefficient Cd . A quantitative characterization of the interaction behaviour was supplemented by qualitative evaluations of the flag deformations and the vorticity contours within the flow, which provided an intuitive understanding of the vortex–flexible-body interactions. Three parameters were chosen for optimization within the following

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F IGURE 3. The drag coefficient as a function of (a) the streamwise (Gy = 0.2) and (b) the spanwise (Gx = 0.5) gap distances. The inset of figure 3(a) is from Kim et al. (2010), for two tandem flags in a uniform flow.

ranges: the streamwise (0.1 6 Gx 6 2.0) and spanwise (0.05 6 Gy 6 2.2) gap distances between flags, and the bending coefficient of the flags (0.005 6 γ 6 0.018). The drag coefficients Cd of the three flags as a function of the streamwise and spanwise gap distances, Gx and Gy , respectively, are presented in figure 3. In this simulation, Re = 400, ρ = 1 and γ = 0.005, and these values remained unchanged unless otherwise stated. For a fixed spanwise gap distance of Gy = 0.2 (figure 3a), Cd of the upstream flag, labelled the ‘first flag’, remained constant except for a very small Gx (Gx 6 0.3), indicating that the upstream flag was almost independent of the downstream flags. On the other hand, Cd of the downstream flags, labelled the ‘second’ and ‘third’ flags, decreased to a greater extent as Gx increased and became much smaller than the value for the upstream flag. Similarly, for a fixed streamwise gap of Gx = 0.5 (figure 3b), Cd of the first flag changed slightly with Gy , whereas the values for the second and third flags varied significantly with Gy , reaching a minimum at about Gy = 0.2 and a maximum at about Gy = 1.2. For sufficiently large streamwise and spanwise gap distances, Cd converged gradually. A two-dimensional contour plot of Cd for the third flag as a function of Gx and Gy is shown in figure 4. The low- and high-drag regions may be clearly observed. The underlying mechanics were revealed in a detailed analysis of four cases in which the spanwise distance was fixed at Gy = 0.2 and four cases in which the streamwise distance was fixed at Gx = 0.5, as marked in figure 4. Figure 5 shows the time histories of the tail positions of the three flags for Gx = 0.2, 0.9, 1.3 and 1.9, for fixed Gy = 0.2. The corresponding frequency spectra are shown in figure 6. The first flag always flapped regularly with a single dominant frequency, whereas the motions of the downstream flags varied, depending on Gx . For Gx = 0.2, the downstream flags flapped regularly with the same frequency as the upstream flag; however, for larger Gx , i.e. 0.9, 1.3 and 1.9, the flapping of the downstream flags became more complicated and two dominant frequency peaks appeared, indicating the presence of interactions between the downstream flags and the wake of the upstream flag. Instantaneous vorticity contours at the instants of the minimum and maximum transverse positions of the first flag are displayed in figure 7 for the above four cases. For Gx = 0.2, the vortices did not detach from the first flag; rather, the vortices merged with the shear layers on the downstream flags and shed as a whole. As a result, the three flags behaved as a single


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F IGURE 4. Contours of the drag coefficient of the third flag as a function of the streamwise and spanwise gap distances.

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flag. As Gx increased, the vortices were shed from the first flag and interacted with the vortices from the downstream flags. Recall that for two tandem flags in a uniform flow (Alben 2009; Kim et al. 2010), the interactions between vortices assumed two different modes depending on the streamwise gap distance, i.e. the constructive mode and the destructive mode. As a result, the drag coefficient of the downstream flag decreased significantly in the destructive mode and remained high in the constructive mode, as Gx was varied, as shown in the inset of figure 3(a). Here, the constructive mode was observed for Gx = 0.2 (figure 7a), whereas the destructive mode formed for larger Gx values (figure 7b–d), resulting a significant drag reduction of the downstream flags, as shown in figure 3(a).

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F IGURE 6. Power spectra of the three flags for Gy = 0.2 and (a) Gx = 0.2, (b) Gx = 0.9, (c) Gx = 1.3 and (d) Gx = 1.9.

Note that the drag reduction on the downstream flag was observed only within a small range of Gx for the system comprising two tandem flags, as indicated in the inset of figure 3(a). In the present study, the destructive mode always occurred for larger Gx values, which distinguished the properties of systems comprising a single downstream flag or two side-by-side downstream flags that supported each other hydrodynamically (Partridge & Pitcher 1979). The broadening of the low-drag region may indicate an important advantage for fish schooling over swimming individually or in tandem. Note that the presence of a single dominant peak in the frequency spectrum corresponded to the constructive mode, whereas a double dominant peak corresponded to the destructive mode. We concluded that the dominant frequencies resulted from vortex interactions. Figure 4 shows the classification of various cases into single-frequency or multifrequency modes, marked by an ‘s’ or ‘m’, respectively, according to the properties of the frequency spectra. The effects of the spanwise gap distance were next studied in detail. Figure 8 shows the time histories of the tail positions of the three flags for Gy = 0.2, 0.6, 1.2 and 2.2, fixed at Gx = 0.5. The corresponding frequency spectra are shown in figure 9. The multifrequency mode was observed only for Gy = 0.2, whereas the single-frequency mode was always observed for larger Gy . The instantaneous vorticity contours are presented in figure 10. The vortex interaction was characterized as a destructive mode for Gy = 0.2 (figure 10a), which corresponded to the multifrequency mode, whereas the constructive mode was observed as Gy increased (figure 10b,c). For Gy sufficiently large, i.e. Gy = 2.2 (figure 10d), the vortices shed from the upstream flag passed between the downstream flags, and the interaction strength decreased as Gy increased. The drag coefficients of the downstream flags reached a minimum

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F IGURE 7. Vorticity contours around the three flags for Gy = 0.2 and (a) Gx = 0.2, (b) Gx = 0.9, (c) Gx = 1.3 and (d) Gx = 1.9.

at Gy = 0.2 (figure 3b) due to the destructive vortex interactions. The coefficients then increased significantly as Gy increased until Gy = 1.2, which was attributed partially to the constructive vortex interactions and partially to the exposition to the free stream upon leaving the wake of the upstream flag. As Gy increased further, constructive interactions no longer formed and the drag coefficients of the downstream flags decreased gradually and converged to a single flag value because the downstream flags were mainly subjected to the uniform free stream. These four cases are marked in figure 4. A more detailed examination showed that the single-frequency and multifrequency modes corresponded well to the high- and low-drag regions in figure 4. A dashed line is plotted to roughly separate these two regions. An interesting finding was discovered by comparing with the problem of the two side-by-side flags in a uniform flow, where in-phase flapping was observed for small gap distances and out-of-phase flapping was always observed for Gy > 0.2 (Zhang et al. 2000; Zhu & Peskin 2003).

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F IGURE 8. Tail positions of the three flags for Gx = 0.5 and (a) Gy = 0.2, (b) Gy = 0.6, (c) Gy = 1.2, (d) Gy = 2.2. (a) 1.2

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F IGURE 9. Power spectrum of the three flags for Gx = 0.5 and (a) Gy = 0.2, (b) Gy = 0.6, (c) Gy = 1.2 and (d) Gy = 2.2.


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F IGURE 10. Vorticity contours around the three flags for Gx = 0.5 and (a) Gy = 0.2, (b) Gy = 0.6, (c) Gy = 1.2 and (d) Gy = 2.2.

Out-of-phase flapping was not observed in the triangular formation, however, for any gap distances, as shown in figure 8. This result indicated that the downstream flags were always synchronized with the vortices shed from the upstream flags, resulting in in-phase flapping in both the single-frequency and multifrequency modes. Thus, the dynamics of the triangular formation differed significantly from the dynamics of the side-by-side flags subjected to a uniform flow, indicating the necessity of studying complex formations of flags rather than simple formations, such as the two tandem or side-by-side arrangements, as basic models for understanding the dynamics of fish schooling. After studying the effects of the gap distances, the influence of the bending rigidity of the flag was considered. The gap distances optimal for minimizing the drag forces on the downstream flags were selected, as described above (Gx = 0.5 and Gy = 0.2), and the Cd for the three flags was calculated as a function of the bending coefficient, as plotted in figure 11. In general, Cd of the downstream flags was much smaller

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F IGURE 11. The drag coefficient as a function of the bending coefficient.

than the value of the upstream flag, and it decreased as γ increased, indicating that the flag became more rigid. The instantaneous vorticity contours showed that the destructive vortex interactions always formed within the range of γ values considered in the present study, as shown in figure 12. A local minimum drag coefficient on the downstream flags was observed at γ = 0.007 (figure 11). This value was considered to be the optimal value because a low drag was obtained with an appropriate flexibility. The flags became too rigid as γ increased, as shown in figure 12. 3.2. Diamond formation The basic building block configuration in fish schooling is the diamond formation, which consists of four fish. The diamond formation was simulated by adding a fourth flag to the triangular formation in a downstream position. Sufficient space for the downstream evolution of the wake was ensured by doubling the streamwise length of the computational domain from −2 to 14, using 1024 grids. Here and in the following discussion, the bending rigidity of γ = 0.007 was adopted. The streamwise gap distance between the fourth flag and the second/third flag was denoted G0x . The drag coefficient of the fourth flag was calculated as a function of the streamwise gap distance (G0x ), as shown in figure 13. A low-drag region was observed over the range 0.1 6 G0x 6 0.7, with a local minimum value at about G0x = 0.5. Within the range of 0.7 6 G0x 6 1.1, the drag coefficient gradually increased, reaching a stable range over G0x > 1.1. The vorticity contours, together with the time histories of the tail positions of the flags, are shown in figure 14. Although the tail motions of the second and third flags were synchronized by the wake of the first flag, the interactions between the vortices of the second and third flags and the fourth flag varied with G0x . For G0x = 0.1 and 0.5 (figure 14a,b), the destructive vortex pair shed from the leading edge of the second row of flags impacted the fourth flag, and the small streamwise gap distance between the second and third rows prevented detachment of the vortices from the second row, which interacted with the fourth flag in a destructive mode. As a result, Cd of the fourth flag was low, although the flapping amplitude was not small compared with the amplitudes of the second and third flags. At streamwise gap distances of G0x = 1.0 and 1.8 (figure 14c,d), the vortices shed from the upstream flags interacted with the fourth flag in a more complicated manner, which could not be identified as a single

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destructive or constructive mode. As a result, the Cd of the fourth flag gradually increased and converged as G0x increased. According to the drag force as a function of the gap distance presented above, an optimal diamond formation was obtained, as shown in figure 15 that assumed a symmetric shape with a small expansion angle of about 12◦ . The formation was validated by comparing with the experimental results of Partridge & Pitcher (1979), who reported the experimentally measured distance between fish in a school. 3.3. Conical formation The conical formation was examined by modelling three rows of fish in a school, one in the first row, two in the second row, and three in the third row. The streamwise distance between the rows was fixed by the optimal conditions for the diamond formation, based on figure 15. Here, emphasis was placed on the third row of flags, which were assumed to have negligible influence on the upstream ones. The spanwise distance between the outer flags (fourth and sixth) in the third row was varied while the leading edge of the middle flag (fifth) was fixed at

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F IGURE 13. The drag coefficient of the fourth flag as a function of the streamwise gap distance.

the centreline. The drag coefficients of the downstream flags, i.e. the fourth to sixth flags, were calculated for different spanwise gap distances (G0y ) between the downstream flags, as shown in figure 16. In general, the drag coefficients of the three flags increased as G0y increased. The drag coefficients of the outer two flags were higher than the coefficients of the middle flag because the outer flags were more exposed to the free stream, and coupling between the vortices shed from the fourth and sixth flags and those shed from the second row flags restricted the motion of the fifth flag. These results were consistent with those of Tian et al. (2011a), who calculated the tail positions of three side-by-side flags in a uniform flow. The vorticity contours of the conical formation are shown in figure 17. The tail positions of the three flags in the third row are also plotted. For G0y = 0.1 and 0.5 (figure 17a,b), the downstream three flags flapped in phase, and the upstream flow did not significantly penetrate between the downstream flags. Recall that for the three side-by-side flags in a uniform flow, the in-phase mode formed for a gap distance of less than 0.2 (Tian et al. 2011a). Here, the in-phase mode was observed over a much wider range, i.e. 0.1 6 G0y 6 0.6, due to the synchronizing effects of the upstream vortices. As the gap distance increased to G0y = 0.9 and 1.6 (figure 17c,d), the upstream vortices flowed between the flags and the flags behaved differently. The fifth flag flapped with half the frequency of the fourth and sixth flags, consistent with the half-frequency mode reported by Tian et al. (2011a). Consequently, Cd of the fourth and sixth flags increased rapidly at around G0y = 0.6 (figure 16) as the flapping mode switched from the in-phase mode to the half-frequency mode. Note that the flapping frequency of the fifth flag was actually half the flapping frequencies of the upstream flags for both the in-phase and half-frequency modes, whereas the flapping frequencies of the fourth and sixth flags doubled due to the mode change. Interestingly, the flapping frequency of the in-phase mode of the third row flags was halved compared with the frequencies observed in a uniform flow, possibly due to a mechanism similar to the mechanism that led to the half-frequency mode. For the in-phase mode, the third rows of flags were close, supported each other and were affected by the vortices shed from the upstream flags on the both top and bottom sides,

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F IGURE 15. Optimal configuration for the diamond formation.

a necessary condition for the half-frequency mode. Other different modes found in the three side-by-side flag system in a uniform flow were not observed in the present study, so the flapping dynamics appeared to be simplified by the wake of the upstream

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flags. Because the vortex interactions were more complicated than the interactions observed in the three side-by-side flag system, some flapping modes were observed with perturbations, whereas other flapping modes were no longer observed. 4. Conclusions

In the present study, we examined the dynamics of coupled flexible flags in three different formations, triangular, diamond and conical formations, subjected to a uniform flow by using the immersed boundary method, as basic models of fish schooling. The analysis was carried out under different conditions by varying the streamwise and spanwise gap distances and the bending coefficient of the flags. Single-frequency and multifrequency modes were identified in the triangular formation, and these modes were generally attributed to the constructive and destructive vortex interaction modes, respectively, and corresponded to the high- and low-drag regions. The downstream flags were synchronized by the vortices shed from the upstream flag, and out-of-phase flapping was not observed. The diamond formation was studied by varying the streamwise gap distances based on the drag information from the triangular case. The predicted optimum form agreed well with the experimental data. The conical formation was analysed to study the effects of widening the conical base. The drag on the outer flags in the third row was higher than the drag on the middle flag. In-phase and half-frequency modes were observed in the third row of flags, whereas other modes reported for the three side-by-side flags in a uniform flow were not observed. The relationship between the drag force and the interaction modes of the multiple-flag-in-fluid system suggested a mechanism by which advantages may be gained via the vortex–flexible-body interactions in schools of fish. Acknowledgements

This work was supported by the Creative Research Initiatives program (grant no. 2013-003364) of the National Research Foundation of Korea (MSIP).

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