Flapping dynamics of coupled flexible flags in a uniform viscous flow
Introduction
In biology, any group of fish that stays together for social reasons are shoaling, and if the group is swimming in the same direction in a coordinated manner with a tighter organization, they are schooling. Schooling fish are usually of the same species and the same age/size and move with members precisely spaced relative to each other (Weihs, 1973). The collective behavior of fish is not only for social interaction but also for numerous other benefits, such as defending against predators, foraging, and greater success in finding a mate. From a hydrodynamic point of view, the movement of fish in a school results in an increase in the hydrodynamic efficiency due to wake interactions (Weihs, 1975). The hydrodynamics of groups of fish has been studied extensively while also considering tandem and side-by-side arrangements of flags (, ). However, fish within a school are often arranged in a staggered manner in nature, for example, an inverted triangular formation and a diamond-shaped formation (Alexander, 2004), and the dynamics of fish in a formation differ significantly from the dynamics of simple formations, suggesting the necessity of studying complex formations of flags for a better understanding of fish schooling (Fish, 1999).
Fish and flag hydrodynamics have been simply modeled by flexible flags (or filaments in two dimensions). Although actual fish swim both actively and passively, studies of purely passive flapping flags in a fluid flow can provide insight into how some fish in a school take advantage of passive flapping for better swimming efficiency. Zhang et al. (2000) examined the flapping dynamics of a filament in a flowing soap film and showed that a flexible filament exhibits a bistable response depending on the filament length, showing the presence of a stretched-straight state and a flapping state. Inspired by this work, significant efforts toward a better understanding of the interactions between a flexible flag and a viscous fluid flow have been made (Alben and Shelley, 2008, , , , , ). In systems of flags arranged in tandem, the vortices shed from the upstream flags encounter the vortices shed from the downstream flags, producing complicated vortex-to-vortex interactions. Ristroph and Zhang (2008) conducted an experiment in a flow tunnel with two flexible filaments aligned in the streamwise direction and showed that the drag of the downstream flag is greater than that of the upstream flag. Flapping motion with higher drag force in the downstream has been characterized as inverted drafting, in contrast to tandem rigid bodies described as conventional normal drafting in which the downstream body undergoes relatively low drag force as compared to that of a single flag (, ). Using an inviscid vortex sheet model, Alben (2009) revealed that inverted drafting occurs when the wake of the upstream flag merges constructively with that of the downstream flag. However, normal drafting was identified when the wakes from the upstream and downstream locations interacted destructively. These two different flapping motions of downstream flags were determined by the streamwise gap distance between the flags when they were in tandem. Based on a direct fluid-structure interaction solver in a viscous flow, Kim et al. (2010) demonstrated two vortex interaction modes in flag wakes corresponding to inverted and normal drafting, respectively, as the streamwise gap distance varies. They also examined the effects of the spanwise gap distance between tandem flags on flapping motions.
Studies involving two or three flags placed in a side-by-side arrangement have also shown complicated interaction modes between filaments and surrounding fluid flows in schools of fish. Zhang et al. (2000) reported that when two identical filaments are positioned in a side-by-side manner with a sufficiently small gap distance, they tend to flap in phase with each other. However, the flapping state enters an out-of-phase state when the gap distance exceeds a critical value. The experimental observation of Zhang et al. (2000) was further scrutinized in depth through experimental, theoretical, and numerical studies later (, , ). Furthermore, systems consisting of three flags in a side-by-side arrangement in a uniform flow have been characterized with distinct coupling modes by means of experimental and analytical analyses (, , ).
As an alignment of fish in a staggered manner, a diamond formation of swimming fish has been studied with regard to locomotion efficiency (Partridge and Pitcher, 1979), although previous studies of such patterns have focused on prescribed fish motions with assumption that all of the fish undulate with the same amplitude, frequency, wavenumber and phase (, Breder, 1976). Recently, three different arrangements of flags for fish schooling have been considered, involving triangular, diamond and conical formations (Uddin et al., 2013). By focusing on the flapping motion of downstream flags with respect to the streamwise and spanwise gap distances in each case, they found two regions of gap distances for larger and smaller drag coefficients compared to that of a single flag. The drag variations were explained by interactions between vortices in the upstream and downstream flexible flags. Tian et al. (2011b) also simulated a system consisting of ten identical flexible flags organized in a diamond formation. With fixed streamwise and spanwise gap distances, they showed that the amplitude of the downstream flags is amplified compared to that of a single flag due to vortex-to-vortex and vortex-to-flag interactions.
In the present study, the flapping dynamics of multiple flexible flags in a two-dimensional uniform viscous flow are analyzed in an effort to study the collective behavior of coupled flexible bodies arranged in a staggered manner, inspired by fish schooling. The well-known staggered alignment is the diamond formation of flags with a single flag placed in the first row. However, when two flags are located in the most upstream location, a staggered alignment of the flags creates an inverted triangular formation for three flags and an X-shaped formation for five flags respectively (Alexander, 2004), which have distinctive flapping dynamics compared to the diamond formation. Although tandem, side-by-side, triangular and diamond arrangements have been studied extensively, the flapping dynamics of flags in the inverted triangular formation and X-shaped formation have not been examined as much. Thus, studies of an aggregation of flags in these two formations can provide a better understanding of schooling behavior. To this end, the flapping dynamics of an individual flag and the interaction modes between flags are examined based on flag deformations with corresponding frequency and vorticity contours within the flows. Furthermore, in order to explain the complex flapping dynamics of flags in each formation, the flapping motions of basic building block configurations creating inverted and X-shaped flag formations, in this case a single flag, flags in a side-by-side arrangement, flags in tandem, and flags in triangular and rectangular formations, are scrutinized based on independent simulations to provide an intuitive understanding of vortex-flexible body interactions. This paper is organized as follows. The numerical method used here is briefly described in Section 2, and the results and discussion are presented in Section 3 with two subsections for the inverted triangular and the X-shaped formations. Finally, the summary and conclusion are given in Section 4.
Section snippets
Numerical method
The incompressible viscous fluid motion is described using the Navier-Stokes and continuity equations as follows:
The variables are non-dimensionalized by the free-stream velocity U∞, the length of the flag L, and the fluid density ρf, which are combined to yield the scales for the time L/U∞, the pressure , and the momentum force . The Reynolds number Re is defined as Re=U∞L/ν, where ν is the kinematic viscosity. Eqs. (1), (2) are integrated in time
Inverted triangular formation
A two-dimensional contour plot of the drag coefficient CD of a single flag in the downstream position in the inverted triangular formation is shown in Fig. 2 to examine the hydrodynamic benefit as a function of the streamwise and spanwise gap distances, Gx1 and Gy1, respectively. In the present study, Re=200, ρ=1 and γ=0.005, and these values remain unchanged. The high- and low-drag regions are displayed in black and white, respectively. The minimum value of CD for the single flag is found at G
Summary and conclusions
In the present study, we examined the flapping dynamics of coupled flexible flags arranged in two different formations using a direct fluid-structure interaction solver in a viscous flow with improved immersed boundary method. An analysis of the flapping motion of three flags placed in an inverted triangular formation shows that the drag of the downstream flag is largely influenced by the spanwise gap distance of the two upstream flags in a side-by-side alignment. The significant drag reduction
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2014R1A1A2057031).
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