Flow-induced Vibration of Two Tandem Oscillating Square Cylinders in Reattachment Regime

To improve our understanding of structural stability, two vibrating tandem square cylinders is numerically investigated in the present study. In reattachment regime (L* = 2), mass ratios (m* = 3), Reynolds numbers (Re = 100) are considered. Different branches are identified. For the first cylinder, the maximum A* is about 0.10, which is one-third of a single cylinder. For the second cylinder, the maximum A* is about 0.42, which is 1.45 times of a single cylinder. Wake mode is observed, and the existence of the second cylinder restrains the vortex shedding from the first cylinder.


Introduction
In recent years, the phenomenon of flow induced vibration has received widespread attention, and the vibration event of the Humen Bridge in the past few years has sparked widespread discussion throughout society.Among them, the phenomenon of blunt body turbulence is the most common phenomenon in flow induced vibration.In real life, blunt body turbulence is also widely used.In the field of engineering, the construction of offshore bridge piers and building clusters precisely utilize the principle of blunt body turbulence.As fluid passes through a blunt body at a specific speed, vortices are generated from the structure surface called Kamen vortex street.Resonance will influence the safety of the structure and even destroy the building.Singh and Biswas studied the phase change at subcritical Reynolds numbers [1].Luo et al. found the wake transition experimentally [2].In the study of the galloping phenomenon of square cylinders, Barrero Gil et al. used numerical simulation methods to simulate lateral galloping [3].Parkinson&Smith predicted a theory that produces the phenomenon of square cylinder galloping through research [4].The phenomenon of single column flow can help us better understand the phenomenon and principle of flow induced vibration.In our real life, there are more than one blunt body, such as antennas, building clusters, pipeline systems, cooling tower clusters, offshore oil platforms, etc.Therefore, studying multiple objects can help us further understand complex phenomena.When there are multiple objects, interference occurs between them.The mutual interference between objects results in more complex flow characteristics when fluid flows through them than when only a single object is present.The study of flow around multiple cylinders usually starts from two cylinders.Two square cylinders with different L * and Re is numerically studied [5][6].Kondo [11].The results observed 2S, 2P, P+S, and 2T modes, providing a reference for our study of flow induced vibration phenomena.At present, research on convection induced vibration has been widely conducted on a single square column, and research on two square columns mainly focuses on the vibration of a certain cylinder.There is relatively little analysis and research on two oscillating square columns.Therefore, this article studied the FLV of two oscillating square cylinders.

Equations
The research mainly utilized fluid mechanics formulas, there are continuity equations and Navier-Stokes equations: The quantities ρ, μ, represent the density and kinematic viscosity, respectively.The u, v shows the speed component of the x axis and the y axis velocity components.
The cylinders can oscillate freely.The lateral motion of a cylinder is controlled by equation: In the equations, the  ̈,  ̇ and  are the acceleration, velocity and displacement in the direction of the cross flow.The CL is the lift coefficient.m * is mass ratio defined by m * = m/(ρD 2 ).Fn is normalized natural frequency defined by Fn = fnD/U.To get the highest amplitude of the cylinders, ξ=0.So, Eq. ( 4) is reduced to  In the study, the structured mesh is employed for the whole computational domain.Figure .2 displays the distribution and zoom-in view of mesh at L * = 2. Table 1.shows four meshes with different nodes.The number of nodes varies from 27000 to 109000.

Mesh and time-step validation
For the experiment, Re= 100 and m *=3.Four grids were tested, with a total of 27000, 51750, 83880, 109000 grids.The three time steps were 0.001s, 0.002s, and 0.005s, respectively.Table 2 summarizes the relevant data for Re=100 grid validation.Parameter validation and comparison were conducted on each grid at different time intervals.Taking into account both saving computational resources and evaluating computational accuracy, it can be seen from the table data that computing resources can be saved in the 51750 grid, and the calculation parameters begin to converge at a time step of 0.002.From the results, M2 and Δ T=0.002 is appropriate.The chosen mesh (M2) was compared with published studies in Table 3.The comparison displays small differences 4.1%, 5.0%, 5.6% in D C , CL ' and St respectively, which states a good agreement.

Dynamic validation
After the mesh-independent test of a single stationary cylinder, simulation grid selection (M2) and time step size (Δ T=0.002s), verify the parameters of a free square vibrating cylinder.Figure .3 shows the response curve of the dimensionless amplitude at Re = 100.When Ur is small, the amplitude value is very small.As Ur increases, it suddenly jumps to a larger value, and finally the amplitude begins to decrease.Finally, as Ur increases, the amplitude remains almost unchanged.The numerical values and variation trends of the simulation results are close to those in the literature, indicating that this grid can be used for further simulation research.

Discussion
The vibration amplitude A* increases slightly at Ur = 1 -6, corresponding to the initial branch (IB) for both cylinders.At Ur = 7, A* jumps to the maximum, which is delayed for both cylinders compared with the single cylinder counterpart.For the upstream cylinder, the value of maximum A* is about 0.10 A/D, which is one-third of a single cylinder.For the first cylinder, the maximum A* is about 0.42 A/D.The A* is decreased at Ur = 7 -14 in the lower branch (LB).The amplitude for two cylinders is larger than that of a single at the same Ur.The maximum A* in the desynchronize branch (DB) changes little and reaches a small, steady value.For L* = 2, the distance between two cylinders is small, resulting in more complex disturbances.The vortex shedding forms become more complex.The formation of vortices in the first cylinder is influenced by the second cylinder.The first cylinder cannot form a complete vortex shedding, so the shear layer separated from the first square cylinder is reattached to the second square cylinder.
Figure 6 shows the vortex shedding forms at different Ur.The selected Ur represents the square column in the upper branch, maximum amplitude value, lower branch, and desynchronized branch.As shown in the figure, due to the small Reynolds number, the shear layer of the upstream column always adheres to the downstream column in each branch, and the first column does not generate independent vortex shedding.Two cylinders can be considered as a whole, with vortex shedding occurring only behind the downstream square cylinder.When Ur=5 in IB, the effect of vortex shedding on the cylinder is minimal.At Ur=7, a stable eddy current mode 2S is formed.The upstream column does not generate vortex shedding, and its shear layer still adheres to the downstream column.The presence of downstream cylinders suppresses vortex shedding from upstream cylinders.Therefore, the amplitude of the upstream cylinder is still smaller than that of the rear cylinder.

Figure 1
is a schematic diagram of two cylinders.The side length of the cylinder is set to D, and the dimensions of the two cylinders are the same.The positional relationship between two cylinders is arranged in series.The origin is set at the center of the upstream cylinder, perpendicular in the y-direction and horizontal in the x-direction.The boundary condition on the surface of a cylinder is no slip.The inlet boundary condition is set as u * = 1, v * = 0.The outlet boundary condition is defined as pressures outlet, which is given by ∂u * /∂x = 0, ∂v * /∂x * = 0.The symmetry conditions are set as ∂u * /∂y * = 0, v * = 0.

Figure 1 .
Figure 1.Geometric model of two square cylinders

Figure 2 .
Figure 2. (a) Global view of the grid.(b) Mesh view around the cylinder

Figure 3 .
Figure 3.Comparison of the amplitude A* with Zhao et al[15]

Figure 5 .
Figure 5. Variation of CL' at different Ur for L* = 2, m* = 3, Re = 100For L* = 2, the distance between two cylinders is small, resulting in more complex disturbances.The vortex shedding forms become more complex.The formation of vortices in the first cylinder is influenced by the second cylinder.The first cylinder cannot form a complete vortex shedding, so the shear layer separated from the first square cylinder is reattached to the second square cylinder.Figure6shows the vortex shedding forms at different Ur.The selected Ur represents the square column in the upper branch, maximum amplitude value, lower branch, and desynchronized branch.As shown in the figure, due to the small Reynolds number, the shear layer of the upstream column always adheres to the downstream column in each branch, and the first column does not generate independent vortex shedding.Two cylinders can be considered as a whole, with vortex shedding occurring only behind the downstream square cylinder.When Ur=5 in IB, the effect of vortex shedding on the cylinder is minimal.At Ur=7, a stable eddy current mode 2S is formed.The upstream column does not generate vortex shedding, and its shear layer still adheres to the downstream column.The presence of downstream cylinders suppresses vortex shedding from upstream cylinders.Therefore, the amplitude of the upstream cylinder is still smaller than that of the rear cylinder.

Figure 6 .
Figure 6.Instantaneous vorticity at different Ur for L* = 2, m* = 3, Re = 100 4. Summary Flow-induced vibration of two tandem square cylinders in the laminar (Re = 100) is investigated.The vibration response for two cylinders has the same tendency.At Ur = 7, A* jumps to the maximum, which is delayed for both cylinders compared with the single cylinder counterpart.The fluctuating lift coefficients C L ' stays at a small value until Ur = 7.Then C L ' increases rapidly at Ur = 7 -12, decreases at Ur > 12, and finally stays at 0.05 and 0.19 for first and second cylinders, respectively.The vortex shedding modes 2S is observed.The exist of the downstream cylinder restrains the vortex shedding from the upstream cylinder.
et al. numerically simulated the variation of two cylinders in different arrangements [7].Sun et al. experimentally studied the L* of flow induced vibration of two cylinders and determined four states [8].More et al. experimentally studied the vortex shedding process and flow structure [9].Different parameters will have different effects on the induced vibration phenomenon.Adeeb et al. studied the parameter changes under different L* conditions [10], including drag coefficient, lift coefficient, and Strouhal number.The flow induced vibration phenomenon will generate different vortex shedding phenomena in different application scenarios.

Table 1 .
Several meshes of a single cylinder

Table 3 .
Comparison of simulation parameters and literature ( D C , CL ' , St at Re = 100)