On the Sub-Critical Bifurcation of Anti-Phase and In-Phase Synchronized Vortex Shedding Forms

Transition of flows past a pair of side-by-side circular cylinders are investigated by numerical simulations and the bifurcation analysis of the numerical results. Various flow patterns behind the cylinder-pair have been identified by the gap ratio ( G ) and Reynolds number ( Re ). This study focus on transition of in-phase and anti-phase vortex shedding synchronized forms. A nested Cartesian-grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure, has been adopted to simulate the flows. Numerical results reveal that the in-phase and anti-phase vortex shedding flows at Re = 100 can co-exist at 2.08 Hysteresis loop with in-creasing/decreasing G at constant Reynolds number Re = 100 is reported.


Introduction
Because of its fundamental importance and engineering significance, unstable flow interferences across bluff bodies have been investigated extensively. Flow interference with a pair of cylinders is complex in which both relative-gap between the cylinders and arrangements of relative position (in tandem, side-by-side, or in staggered) play crucial roles in the physical transition. Among flows around a pair of cylinders in various configurations, those behind circular cylinders in a side-by-side arrangement have been mostly extensively studied [1][2][3][4][5][6][7][8][9][10][11]. It is now well established that the flow patterns behind a pair of side-by-side cylinders can be classified [3,12] by the gapratio G (surface-to-surface distance divided by cylinderdiameter) and the Reynolds number (Reynolds number, Re, is defined as where o U is the freestream velocity, D is the cylinder diameter, and  is the kinematic viscosity). At very small Gap ratios, the two cylinders may behave in a similar fashion to a single bluffbody [1]. At intermediate Gap ratios, the two side-byside cylinders is known [2] to exhibit a deflected or biased flow patterns, which are bi-stable in nature. The deflected flow pattern is characterized by a gap flow biased towards one of the two cylinders. The gap flow in this regime switches spontaneously from one side to the other and thus corresponds to the flip-flopped regime [5]. For weak coupling (with relatively large G, 1  G  5) many of the past researchers report occurrences of both anti-phase and in-phase synchronized vortex shedding behind a pair of circular cylinders. Findings of Williamson [4] confirm that the shedding vortices (for 40  Re  160) remain synchronized either in phase or in antiphase. Anti-phase streets often preserve the phase-locked identity of the vortices even at a far downstream location. However, for strong coupling (with relatively lower G) only in-phase vortex shedding has been reported, and the associated physical process eventually leads to formation of a complex asymmetrically evolving Benard von Karman streets.
Although there are varieties of flow patterns behind a pair of side-by-side cylinders have been identified (which include semi-single and twin vortex street formations, symmetric and deflected flows, stationary, biased and flip-flopped-type vortex shedding, periodic, quasi-periodic, and weakly-chaotic flows, and in-phase and antiphase vortex synchronizations), very restriction results have been reported on transition of those various flow patterns. Examples can be found in the studies of Mizushima & Ino [13] and Peng et al. [14]. In [13], parameter space of the gap ratio and the critical Reynolds number of symmetry/deflected vortex shedding flows and inphase/anti-phase vortex shedding flows behind a pair of circular cylinders at low Reynolds number have been reported by results of numerical simulations and linear stability analyses. While in the study [14], transition from semi-single symmetry vortex shedding flow to semisingle deflected vortex shedding flow as well as transition from twin symmetry vortex shedding to twin deflected vortex shedding and then transition to the flipped-flopped vortex shedding regime behind a pair of elliptical cylinders at low Reynolds number are shown by results of direct numerical simulations. Although there are a lot of experimental and numerical studies on the side-byside cylinders, the important transition phenomenon from in-phase to anti-phase vortex shedding synchronization and vs. are still unclear. Purpose of this study is then focus on transition of in-phase/anti-phase vortex shedding synchronized forms behind a pair of side-by-side circular cylinders. High resolution numerical methods based on a nested Cartesian grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure, have been adopted to simulate flows past a pair of side-by-side circular cylinders. Hysteresis loop with increasing/decreasing Re at constant gap ratio describing the hysteresis phenomenon of in-phase/ anti-phase vortex shedding synchronized forms are reported.

Numerical Methods and Validations
In this study, transitions of in-phase and anti-phase vortex shedding flows behind a pair of circular cylinders in the side-by-side arrangement are numerically investigated. A nested Cartesian-grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure, has been adopted to simulate the flows. Extensive related details of the discretization schemes consisting of inside fine/coarse grid-areas and the associated immersed boundary method may be found in Peng et al. [14][15].
Here we briefly describe the implemented numerical method. Governing equations used are unsteady incompressible Navier-Stokes equations in primitive variables. In integral forms, the dimensionless governing equations (with lengths normalized by the diameter D of cylinders, velocities normalized by the uniform inflow velocity U 0 , and time by D /U 0 ) appear as the following.
The mass conservation equation and the momentum conservation equation Ni n inner fine-gri (2) CS and CV in Eqs. (1) and (2) denote the control-surface and control-volume, respectively, and is a unit vector normal to the control-surface. While advancing in time, a second order accurate two-step fractional-step method is used. A second-order Adams-Bashforth scheme is employed for discretizing the convective terms, and diffusion terms are discretized using an implicit Crank n  colson scheme. In the iterations of the discretized equations, a local grid refinement technique is adopted through the introduction of two nested blocks in the computational domain. The implemented nested-block finite-volume based Cartesian-grid method is noted to facilitate effective/accurate simulation of the presently investigated unsteady viscous incompressible flows past multiple immersed boundaries. The procedure adopted here allows systematic simulation of flows past the cylinder-pair, and preserves global second-order accuracy [15]. A sketch for the computational domain with implemented boundary conditions and a side-by-side arrangement of cylinder-pair is provided in Figure 1. Various domain-lengths used for simulations under the present study are defined as: L 1 = 5D, L 2 = 50D, L 3 = 12D, L 4 = 4D, L 5 = 8D, and L 6 = 5.5D, facilitating generation of a physical domain of size 55D × 24D, consisting a d (Grid 2) area 12D × 11D. Upon the outer Grid1 (the coarse grid), and 0.05 for the inner Grid 2, the total grid size became 168,000, with Grid 1 = 120,000, an ted body force in the Navier-Stokes equation, i.e., d Grid 2 = 48,000. The simple concept of immersed boundary (IB) method adopted here helps to simulate effectively the wake evolutions past the cylinders. The virtual presence of the cylinders within the flow domain is facilitated by introducing a locally active distribu , and ϕ represents the volume-fraction of the solid body within a computational cell. For a cell entirely occupied by the cylinders, ϕ = 1 is used; and for a cell fully occupied by fluid, ϕ = 0 is taken. However, for an interface-cell, partially occupied by a cylinder and partially by fluid, 0 < ϕ < 1 is devoted.
Thereafter, the governing equations are solved everywhere in the computational domain, including cells which are ccupied by the elliptic cylinders. o It is noted that an extensive validation of the underlying method has been well-documented in [14][15]. In [14], computations of important critical Reynolds numbers (Re cr,v ) that correspond to onsets of vortex shedding for uniform flows past a circular cylinder, an elliptical cylinder, and two side-by-side attached elliptical cylinders were performed. As listed on Table 1, for a circular cylinder, our previous study observed this Re cr,v to be 47.2, which compares quite well with the experimentally predicted values 46.9 -47.9 [16][17][18], and the theoretical values 46.1 -47.3, as obtained by those linear stability analysis [19][20] and the bifurcation analysis of Dusek et al. [21].

Side-by-Side Circular Cylinders
This study begins with the investigation of critical transition characteristics in the narrow gap range, and extracts the underlying bifurcation patterns. For this, first, flow properties past two side-by-side circular cylinders in the gap-ratio range 0.2 ≤ G ≤ 3.0 and 40 ≤ Re ≤ 100 are extensively simulated. The observed distinctive physical properties of these flows sequentially characterized in Table 2, in which muted data, the readers may note that at all Reynolds.  Table 2) are classified based on the following three special characteristics. The first letter in the abbreviated flow-type, "S," corresponds to the semi-single flow, and "T" corresponds to twin flow. For the second letter, "S" represents symmetric flow, and "D" denotes the deflected flow. In the third and fourth places, "SS" indicates steady-state flow, "VP" represents periodic vortex shedding flow, "VQ" stands for quasi-periodic vortex shedding flow, and "VC" denotes the chaotic vortex shedding. Various vortex shedding regimes, including flip-flopped, in-phase, and anti-phase vortex shedding flows are also denoted by superscripts in last column. From the si la   In-phase vortex shedding is detected at 0.4 ≤ G ≤ 1.8, and anti-phase vortex shedding is founded at large G (G ≥ 2.2). Notably, while Table 2 presents the explicit physical details of various flows, for the sake of facilitating immediate comprehension of a trend, the parameter space diagram is suitably summarized in Figure 2. As shown in Figure 2, it is clear that flows are semi-single at G < 0.5. At intermediate Gap ratios the two side-by-side cylinders is known to exhibit a deflected or biased flow patterns, which are bi-stable in nature. The deflected jets through circular gap are further affected by the shedding vortices at higher Re and flows become flip-flopped consequently. The range of Gap ratios where the flip-flopped flow pattern is observed extends from approximately Gap ratios between 0.4 -1.8 depending on the Reynolds number. At higher Gap ratios, i.e., the cylinders are spaced sufficiently far apart, the pair of cylinders may behave as two independent bluff bodies. Proximity interference effects, however, lead to various modes of synchronization, anti-phase and in-phase, in the vortex formation and shedding processes and the resulting parallel vortex streets. For example, in-phase and anti-phase vortex shedding flows at Re = 100 are clearly revealed at G = 2.0 and G = 3.0, respectively.

Hysteresis Scenario of In-phase and Anti-phase Synchronized Forms
To investigated the transition of in-phase and anti-phase synchronized forms in flows past a pair of side-by-side circular cylinders, two sets of computations including anti-phase and in-phase branches are carried out. The anti-phase branch started from the anti-phase vortex  shedding flow at G = 3.0 and Re = 100 and are calculated by progressively decreasing G in very small steps. In the meanwhile, the in-phase branch started from the in-phase vortex shedding flow at G = 2.0 and Re = 100 and are calculated by progressively increasing G in small steps. The observed distinctive physical properties of these flows are sequentially characterized in Table 3. It is noted that in anti-phase branch, the solution at a higher G is used as the initial condition for the next lower G. Similarly, in in-phase branch, the solution at a lower G is used as the initial condition for the next higher G.
The flow past cylinder-pair at constant Reynolds number (Re = 100) retains anti-phase vortex shedding synchronized forms at high gap ratio ( ), as indicated by our extracted data on Table 3 9,9]. The continuation of zero central-line velocity (Figure 3(d)), and perfectly anti-phase synchronized growth of the lift (C L,1 , C L,2 ) coefficients (Figures 3(b) and (c)) ensure the inherent anti-phase characteristic of vortex shedding (Figure 3(a)) in the wake. Note that, unlike for in-phase vortex shedding flow (Figure 4), the equal but opposite natured variations of C L,1 and C L,2 in the present case contribute to the continued vanishing of C L,1+2 (Figure 3(d)) during the entire time-evolution.

2.08
3.0 G   Upon maintaining anti-phase synchronized vortex shedding forms with the past findings related to side-by-side cylinder-pair (Re = 100, ), once the gapratio was subsequently increased (from 2.08 3.0 G   2.0 G  ), the in-phase synchronized vortex shedding were encountered behind the cylinder-pair (Re = 100, ). The distinguishable physical characteristics associated with the in-phase vortex shedding flows past the cylinder-pair at Re = 100 and again G = 2.32 is extracted in Figure 4. It can be noted from the figure that the gap-flow quickly lost stability; however, the shedding vortices appeared clearly in-phase synchronized at least up to x ≤ 12. Thereafter, instability is seen to quickly grow, leading to the development of a combined binary vortex street within 12 < x <26, and beyond that there occurred an irregular flow pattern over a conversion point. Physical details of this in-phase flow in terms of enhanced C L,1+2 , and significantly modulated transient evolution (in-phase) of individual lift (C L,1 and C L,2 ) coefficients are extracted in Figures 4(d), (b) and (c).

Hysteresis Loop
From the simulated data on Table 3, the reader may note that at constant Reynolds number (Re = 100), the anti-phase branch ranges between , and the in-phase branch ranges between . In the other word, the anti-phase and in-phase synchronized forms behind a pair of circular cylinders (Re = 100) can co-exist at . The co-existences of the anti-phase and in-phase vortex shedding flow patterns (at Re = 100 and G = 2.32) have been shown in Figures 3  and 4, respectively. Figures 3(a) and 4(a) (Figures 3(d) and 4(d)). However, an enlarged view of individual lift-coefficients ( , ) for the upper and the lower  G Re  cylinders (Figures 3(b) and (c)/Figures 4(b) and (c)) reveals occurrence of anti-phase/in-phase vortex shedding vortices of the upper and lower vortex streets. Readers may note that the corresponding lift-amplitudes ( ,1 2 ) in anti-phase vortex shedding flows become neutralized as shown in Figure 3(d), on the contrary, the corresponding lift-amplitudes ( ) in in-phase vortex shedding flows become enhanced as shown in Figure  4(d), respectively. Since the value of stands as a characteristic value of anti-phase/in-phase synchronized vortex shedding forms, it is worthy to show the

Conclusions
Numerical results have been presented for the in-phase and anti-phase vortex shedding synchronized forms of flows behind a pair of side-by-side circular cylinders. Flows are restricted in low-Reynolds-number (Re  100) laminar regime for various small/middle gap ratio ( 0.2 3.0 G   ). The computations have been carried out in two-dimensional, using a high resolution numerical method based on a nested Cartesian grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure.
Hysteresis phenomenon of the in-phase/anti-phase vortex shedding synchronized forms of flows has been studied in detail. For flows behind a pair of side-by-side circular cylinders at Re = 100, the hysteresis loop with width ranges between is clearly found. 2.08 2.58 G  

Acknowledgements
This work was supported in part by the National Science Council of the Republic of China (Taiwan) under Contract No. NSC 101-2221-E-260-038-.