Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study

Abstract

A numerical investigation was conducted in this study utilizing Force Element Analysis to explore the vortex-induced vibration (VIV) mechanism of side-by-side dual cylinders under the conditions of Reynolds number Re = 100, mass ratio m* = 10, and spacing ratios L/D ranging from 3 to 6. The hydrodynamic forces by force element formulas were incorporated into the vibration response calculations of elastically supported rigid cylinders using a User-Defined Function (UDF) and the fourth-order Runge–Kutta method. A comprehensive analysis was performed to elucidate the combined effects of the spacing ratio L/D and reduced velocity U_r on the vibration responses, quantifying the hydrodynamic forces involved in the mutual interaction during VIV for side-by-side dual cylinders. The influence mechanisms of inter-cylinder interaction and their effects on the resultant hydrodynamic phenomena were discussed. It was revealed that for side-by-side arranged dual cylinders outside the “lock-in region”, the lift and drag forces are predominantly supplied by the volume vorticity forces in conjunction with surface vortices (including frictional) forces. However, within the “lock-in region”, the surface acceleration lift forces provide greater force contributions, and the volume vorticity lift force contributes significantly to negative values. Notably, alterations to the spacing ratio do not change the proportion of force element components. The amplitudes of the cylinders’ mutual interaction forces are identical in magnitude but opposite in phase. Additionally, the “slapping” phenomenon near the “lock-in region” leads to “bounded” trajectories of cylinders.

1. Introduction

Vortex-induced vibration (VIV), a significant topic, that plays a pivotal role in engineering applications involving bluff body structures, such as bridge stay cables, marine risers, and heat exchanger tube bundles, profoundly impacts structural stability, fatigue damage, and overall performance [1]. Under specific reduced velocities, contingent upon the system’s mass ratio and damping ratio, the frequency of vortex shedding matches the cylinder’s natural frequency ($f_n=(1/2\pi )\sqrt{k/m}$, where k is the system stiffness and m is the mass of the cylinder), leading to large-amplitude oscillations, a phenomenon known in the literature as “lock-in” [2].

The presence of two closely spaced cylinders in a fluid flow has garnered considerable attention and has been extensively investigated. As a prototypical configuration of multi-cylinder arrangements, side-by-side dual cylinders exhibit pronounced interference effects on the wake vortex shedding flow [3,4,5]. The VIV characteristics of such configurations are intricately influenced by a complex interplay of vortex-shedding modes, flow field interference effects, and geometric parameters [1,6]. A thorough exploration of these factors is critical for gaining insight into and controlling the VIV behavior of such structures.

Early research predominantly focused on stationary dual cylinders (Alam and Zhou [7], Zhao and Cheng [8], Bai et al. [9]), and single-cylinder vortex-induced vibration (Feng [10], Bearman [11], Williamson [1,6], Williamson and Govardhan [12], Sarpkaya [13], Gabbai and Benaroya [14], Zhao and Cheng [15], Wu et al. [16]). Zdravkovich [4] explored the VIV of two cylinders, revealing that the amplitude of oscillation is largely contingent on the relative positioning of the two cylinders. Williamson [17] observed that within a certain gap range, the wakes of side-by-side dual cylinders synchronize, in-phase or out-of-phase. However, when the distance between two cylinders is less than 2.2D (D is the diameter), only a single wake forms [17,18,19]. Cui et al. [20] investigated the VIV of elastically coupled side-by-side dual cylinders in a cross-flow direction. Chen et al. [21] comprehensively summarized six discernible wake patterns for VIV of two side-by-side cylinders at a Reynolds number Re = 100 and spacing ratios in the range of 2 ≤ L/D ≤ 5 (L is the center-to-center spacing of cylinders). These patterns encompassed irregular, in-phase and out-of-phase flapping, in-phase and anti-synchronous, and biased anti-synchronous modes. Rahmanian [22] studied the VIV of side-by-side dual cylinders with different diameters, delving into the effects of their interaction during the VIV process. Xu et al. [23] systematically examined the impact of the spacing ratio L/D on the two-degrees-of-freedom (2DOF) VIV of side-by-side dual cylinders at Re = 1470 to 10320. They found that even at high spacing ratios of up to 10, the interference between cylinders remained non-negligible for the vibration response; at the same time, six wake patterns, similar to Chen et al. [21], were identified. Chen et al. [24] investigated the VIV of 2DOF side-by-side dual cylinders at Re = 100 and 2 ≤ L/D ≤ 5, exploring the characteristics of hydrodynamic forces, trajectories, phase lags, and wakes in depth. They categorized the response of two cylinders into four distinct branches. Subsequently, Chen et al. [25] examined the effect of unilateral hysteresis of two laterally vibrating cylinders at Reynolds numbers of 60 to 200.

In previous studies, whether dealing with rigidly coupled or individually oscillating elastically mounted cylinders, limitations existed in that they could only provide macroscopic insights into the variations in lift and drag forces across the entire flow field, lacking a comprehensive understanding of the specific impacts of the flow field on object forces. For dual cylinders, for instance, there was no way to ascertain the magnitude of the mutual influence between cylinders. To address this, our group [26] introduced a force theory for incompressible flows around multiple solids, enabling us to examine the force contributions of individual fluid elements to each object. This methodology stems from the Force Element Theory (FET) proposed by Chang [27], which originates from d’Alembert’s principle and represents a discretization-based fluid–structure interaction theory. By dividing the cylinder surfaces into a series of discrete elements (force elements), the hydrodynamic forces exerted on each element by the flow field can be simulated, and when combined with structural dynamics models, the full VIV process can be effectively simulated.

The Force Element Theory has been widely employed in various dynamic-related applications and has gained recognition among scholars (Lee et al. [28], Lin [29], Menon and Mittal [30,31], Yin et al. [32], Luo et al. [33], Xu et al. [34], Moriche et al. [35]). Lin et al. [29] employed the Force Element Theory to analyze the relationship between vorticity in the wake region and total thrust contribution across different regions for a three-dimensional heaving flexible plate. This approach allowed for a deeper understanding of the mechanism behind thrust generation in flexible plates. The Force Element Theory was applied to investigate the forces acting on cylinders and spheres in Stokes accelerating flows at low Reynolds numbers by Xu et al. [34]. They discovered that at extremely low Reynolds numbers, the vorticity within the flow domain contributes almost nothing to the drag force. It is worth noting that both studies utilized the Force Element Analysis to gain insights into the dynamics of fluid–structure interactions at varying scales. In this study, we will leverage the Force Element Analysis to elucidate the VIV mechanisms of side-by-side dual cylinders under different L/D and reduced velocities U_r (here, U_r = U_∞/f_nD, U_∞ is the free-stream velocity). This will deepen our understanding of their dynamic response characteristics and provide theoretical foundations for structural design, vibration control, and safety assessments in relevant engineering domains.

This study initiates with an introduction to the fluid control equations, which include the Navier–Stokes equations, force element formula, and control equations for VIV systems, complemented by details on the simulation model and fundamental parameters. A grid independence study is then performed. Subsequently, to validate the accuracy of the vortex force formula, pertinent literature verification and analysis are conducted, concentrating on the VIV phenomenon of side-by-side dual cylinders with 2DOF. Frequency and amplitude responses, hydrodynamic coefficients, motion paths, and wake patterns specific to each cylinder are discussed. Ultimately, conclusions with principal findings are presented.

2. Control Equations and Numerical Methods

2.1. Flow Field Setting

The computational domain and boundary conditions are established as depicted in Figure 1. The outer boundary of the computational domain is rectangular, and the streamwise and cross-flow directions are L_x = 150D and L_y = 100D. The distance from the cylinder centers to the inlet boundary is 50D and to outlet boundary is 100D. The two cylinders are symmetrically positioned with respect to each other, with the center-to-center distance between them and the top and bottom boundaries being 50D. This arrangement ensures that the influence of the nearby walls is minimized, allowing a well-developed wake behind the cylinders and small computational error [36]. For the simulation setup, the inlet is specified as a velocity inlet where Dirichlet boundary conditions are applied, setting the velocity components as u = U_∞ (streamwise component) and v = 0 (cross-flow component). The outlet is set as Neumann boundary conditions, representing a zero-pressure condition. The top and bottom boundaries are set as symmetry boundaries. No-slip conditions are enforced on the surfaces of both cylinders. The fluid medium within the domain is water.

2.2. Control Equations

In this study, the flow is assumed to be two-dimensional (2D), incompressible, and laminar. Thus, the flow is governed by the non-dimensionalized Navier–Stokes equations:

$${\displaystyle \frac{\partial \mathit{v}}{\partial t}}+(\mathit{v}\cdot \nabla )\mathit{v}=-\nabla P+{\displaystyle \frac{1}{Re}}{\nabla }^2\mathit{v}$$

(1)

and the continuity equation as:

$$\nabla \cdot \mathit{v}=0$$

(2)

where $\mathit{v}$ represents the non-dimensional velocity vector; P = P*/ρU_∞² is the non-dimensional pressure, where P* is the pressure and ρ is the density; Re = ρU_∞D/μ is the Reynolds number, where μ is the dynamic viscosity of water; and t = τU_∞/D is non-dimensional time, where τ is flow time.

Consider a viscous flow field of an incompressible and uniform incoming flow passing through two bodies, designated as C₁ and C₂. The surfaces of these two bodies are represented by S₁ and S₂, respectively, and S_R denotes the boundary of the far-field domain.

In the Pressure Method (PM), it is typically necessary to first determine the pressure distribution exerted by the fluid and the frictional forces resulting from viscous effects on the surface of the object. These forces distributed over the surface of the object are then integrated to obtain the total force, which can be subsequently decomposed into a component parallel to the direction of motion, representing drag or thrust, and a component perpendicular to the direction of motion, representing lift. As an example, the force coefficient on cylinder C₁ surface S₁ in i-direction can be expressed as follows:

$$C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}={\int }_{S_1}\mathit{P}(\mathit{n}\cdot \mathbf{i})dA+{\displaystyle \frac{1}{Re}}{\int }_{S_1}\mathit{n}\times \mathit{\omega } \cdot \mathbf{i}dA$$

(3)

Here, $C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}$ represent the total hydrodynamic force coefficient in i-direction, and $C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}=F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}/0.5\rho {U_{\mathrm{\infty }}}^{2}D$, $F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}$ is the total hydrodynamic force.

On the other hand, we may take another viewpoint from the perspective of the vorticity force. This is particularly useful for separated flow, mainly featured by its distribution of vorticity in the flow field. Based on Chang et al. [26], it is necessary to introduce an auxiliary potential flow $\varphi$ in the moving direction of the object in the flow field, where $\varphi$ rapidly diminishes as it moves away from the object concerning the force exerted on cylinder C₁ in the i-direction. The parameter $\varphi$ is defined as follows:

$$\begin{gathered} {\nabla }^2\varphi =0\to \mathrm{o}\mathrm{n} V_R \\ \mathit{n}\cdot \nabla \varphi =-\mathit{n}\cdot \mathbf{i}\to \mathrm{o}\mathrm{n} S_1 \\ \mathit{n}\cdot \nabla \varphi =0\to \mathrm{o}\mathrm{n} S_2 \\ \varphi =0\to \mathrm{a}\mathrm{t} \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{y} \end{gathered}$$

(4)

The force element equation that quantifies the hydrodynamic forces acting on C₁ in the i-direction can be deduced from Equation (1) by taking inner products with $\nabla \varphi$:

$$\begin{array}{cc}\hfill C_{total1}=& {\int }_{S_1}\varphi {\displaystyle \frac{\partial \mathit{v}}{\partial t}} \cdot \mathit{n}dA+{\displaystyle \frac{1}{2}}{\int }_{S_1}\left|v^2\right| \nabla \varphi \cdot \mathit{n}dA-{\int }_{V_R}\mathit{v} \times \mathit{\omega }\cdot \nabla \varphi dV\hfill \\ & +{\displaystyle \frac{1}{Re}}{\int }_{S_1}\mathit{n}\times \mathit{\omega } \cdot \left(\nabla \varphi +\mathit{i}\right)dA+{\int }_{S_2}\varphi {\displaystyle \frac{\partial \mathit{v}}{\partial t}}\cdot \mathit{n}dA+{\displaystyle \frac{1}{R\mathit{e}}}{\int }_{S_2}\mathit{n} \times \mathit{\omega }\cdot \nabla \varphi dA\hfill \end{array}$$

(5)

here, the first term on the right-hand side of the equation signifies the magnitude of the acceleration forces acting on the surface S₁ of C₁. The second term denotes the contribution of the velocity on the surface S₁ of C₁. Please note that in this study, the cylinders are symmetric, and this term is zero. The third term denotes the contribution of the volume vorticity in the flow field to the forces on C₁. The fourth term accounts for the combined effect of surface vorticity (including frictional) forces on the surface S₁. The fifth term reflects the influence of the acceleration at the surface S₂ of C₂ on the forces experienced by C₁. Lastly, the sixth term captures the impact of the surface vorticity at the surface S₂ of C₂ on the forces acting on C₁.

In this case, i denotes the drag direction, $C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}1}$ becomes $C_{d1}$, and the five terms on the right-hand side of the Equation (5) become C_da1, C_dv1, C_ds1 C_da21, and C_dv21, so Equation (5) can be written as:

$$C_{d1}=C_{da1}+C_{d\nu 1}+C_{ds1}+C_{da21}+C_{d\nu 21}$$

(6)

where C_da1, C_dv1, and C_ds1 denote force coefficients of surface acceleration, the volume vorticity, and the surface vorticity (including frictional) acting on cylinder C₁ in the drag direction, respectively. The subscript ‘1’ signifies forces pertinent to cylinder C₁. C_da21 and C_dv21 represent force coefficients of surface acceleration and surface vorticity acting on cylinder C₁ in the drag direction, which represent the influence of cylinder C₂ on cylinder C₁.

Similarly, for forces acting in the lift direction, Equation (5) can be written as:

$$C_{l1}=C_{la1}+C_{l\nu 1}+C_{ls1}+C_{la21}+C_{l\nu 21}$$

(7)

2.3. Vibration Response System Model

In this study, the VIV model of cylinders is simplified to a mass-spring-damper system, permitting the cylinder to vibrate freely in both streamwise and cross-flow directions. The vibration governing equations of the cylinder are as follows:

$${\ddot{x}}_i={\displaystyle \frac{F_{D_i}\left(t\right)}{m}}-{\displaystyle \frac{c}{m}}{\dot{x}}_i-{\displaystyle \frac{k}{m}}{x}_i$$

(8)

$${\ddot{y}}_i={\displaystyle \frac{F_{L_i}\left(t\right)}{m}}-{\displaystyle \frac{c}{m}}{\dot{y}}_i-{\displaystyle \frac{k}{m}}{y}_i$$

(9)

where m is the mass of the cylinder (m^∗ = m/m_f, where m_f is the mass of the displaced fluid), c = 2mω₀ζ is damping coefficient, where ω₀ is the natural circular frequency, ζ is damping ratio, x is the displacement of the cylinder in the streamwise direction, $\dot{x}$ is the velocity of the cylinder in the streamwise direction, and $\ddot{x}$ is the acceleration of the cylinder in the streamwise direction. Similarly, $y$, $\dot{y}$, and $\ddot{y}$ are the displacement, velocity, and acceleration in the cross-flow direction, respectively. As a function of time, $F_{D_i}\left(t\right)$ and $F_{L_i}\left(t\right)$ are the drag force in the streamwise direction and the lift force in the cross-flow direction, respectively.

The computational process for the VIV response using the Force Element Analysis involves several critical steps. Initially, the Force Element Analysis is coupled with the system vibration equations through User-Defined Functions (UDFs), which are then integrated into the ANSYS-FLUENT computational platform by the Finite Volume Method (FVM). The hydrodynamic force elements are computed. Subsequently, these hydrodynamic forces, which stem from the solutions of Equation (5), representing the forces in streamwise and cross-flow directions, are transferred into Equations (8) and (9) for vibration response calculations. A fourth-order Runge–Kutta method [15] is employed to discretize and solve Equations (8) and (9), effectively capturing the dynamic behavior of the system under consideration. The utilization of the CG_MOTION macro is to relay the calculated vibration response data back to the flow field for dynamic mesh updates. The motion of the cylindrical structure’s boundary is facilitated through the dynamic meshing technique. The movement of the grid is governed by diffusion equations, ensuring that the grid adapts to the boundary motion while preserving its quality. For detailed insights into the boundary motion diffusion method, one may refer to Han et al. [37].

This comprehensive approach ensures an accurate representation of the complex interaction between the fluid flow and the structural motion of cylinders, providing a reliable framework for studying VIV phenomena at the specified parameter ranges. The UDF and the dynamic mesh technique are essential approaches in achieving high-fidelity VIV simulations of cylinders under varying flow conditions.

2.4. Numerical Verification

2.4.1. Flow Meshing

As shown in Figure 2, the grid refinement around cylinders follows a “C”-type pattern to capture the complex flow phenomena accurately. Local grid refinement zones are placed around each cylinder to better observe the vortex-induced vibration effects and the vortex shedding behavior. Each cylinder’s surface is discretized with 240 nodes [38]. The grid expansion ratio is set below 1.05, employing an exponential growth method. The smallest grid size, which is adjacent to the cylinder surface, is characterized by a nondimensionalized height h* = h_first/D = 0.005, where h_first is near-wall first grid height and y+ < 1. This grid configuration and boundary condition setup are designed to ensure that the simulations accurately resolve the flow physics while maintaining computational efficiency at the same time.

2.4.2. Independence Verification

As shown in

Table 1. Grid and time step independence test: the results of side-by-side dual circular cylinders at Re = 100, m* = 2, L/D = 3, U_r = 6, ζ = 0.

Mesh	Cell Number	h*	∆tU∞/D	${\overline{\mathit{C}}}_{\mathit{d}}$	${\overline{\mathit{C}}}_{\mathit{l}}$	${\mathit{A}}_{\mathit{x}}^{\mathit{*}}$	${\mathit{A}}_{\mathit{y}}^{\mathit{*}}$
M1	56,490	0.01	0.0025	1.901	0.191	0.069	0.514
M2	80,550	0.005	0.0025	1.872 (1.5%)	0.186 (2.7%)	0.066 (4.5%)	0.511 (0.6%)
M3	146,400	0.0025	0.0025	1.863 (0.5%)	0.184 (1.1%)	0.065 (1.5%)	0.503 (1.6%)
M2	80,550	0.005	0.00125	1.860 (0.6%)	0.187 (0.5%)	0.066 (0%)	0.508 (0.6%)
Chen et al. [24]	-	-	-	1.828	0.187	0.060	0.536

, a grid independence study was conducted in the VIV of side-by-side cylinders with 2DOF. The computational parameters were set as Re = 100, m* = 2, L/D = 3, U_r = 6, and ζ = 0. Here, $A_x^{*}=A_x/D$ denotes the dimensionless amplitude in the x-direction, where $A_x$ is the streamwise amplitude and $A_y$ is the cross-flow amplitude, and $A_y^{*}=A_y/D$ represents the dimensionless amplitude in the y-direction.

The results reveal that the outcomes obtained with a first-layer grid spacing of h* = h_first/D = 0.005 were in excellent agreement with those from a finer grid spacing of h* = 0.0025, with a maximum discrepancy of merely 1.6%. This suggests that the simulation accuracy was satisfactory with a grid spacing of h* = 0.005. Furthermore, the independence of the simulation with respect to the time step was confirmed, where results with a dimensionless time step of ∆tU_∞/D = 0.0025 matched very closely with those from a finer time step of ∆tU_∞/D = 0.00125, with a maximum difference of only 0.6%. Consequently, a dimensionless time step of ∆tU_∞/D = 0.0025 was selected for the simulations.

Comparisons with results reported by Chen et al. [24] have also been made, showing a favorable agreement, which further validates the reliability of the present numerical approach. These analyses ensure that the computational parameters chosen provide accurate and robust solutions for the VIV problem under investigation.

2.4.3. Model and Method Validation

In this section, the force element formulation was utilized to validate the numerical methodology for the VIV of side-by-side cylinders with 2DOF. The simulation parameters were consistent with those used by Chen et al. [24], including Re = 100, m* = 2, ζ = 0, and L/D = 3. A comparison of the computed results with those from the literature, as illustrated in Figure 3, reveals that the trends in the mean lift coefficient ${\overline{C}}_l$ and the mean drag coefficient ${\overline{C}}_d$ of the two cylinders as functions of the reduced velocity U_r were in good agreement. This indicates that the numerical approach adopted could be capable of providing accurate predictions of VIV responses for side-by-side cylinders. Consequently, Force Element Analysis and the vibration model employed herein were validated as suitable tools for studying such phenomena.

Concurrently, we present the simulation results for VIV of a single cylinder at Re = 100 and U_r = 5, as depicted in Figure 4. It is observed that the lift and drag coefficients obtained by solving the Navier–Stokes equations using the Force Element Analysis (FEA) align almost perfectly with those derived from the Pressure Method (PM), which further proves the accuracy of our computational approaches in capturing the hydrodynamic forces associated with VIV phenomena.

3. Results and Discussion

This section focuses on the numerical simulation of VIV for side-by-side cylinders oscillating independently in the streamwise and cross-flow directions at Re = 100, m* = 10, and ζ = 0 to gain higher amplitude responses. The computations encompass a range of spacing ratios L/D is 3 to 6 and reduced velocities U_r spanning from 2 to 12.

3.1. Frequency and Amplitude Response

3.1.1. Frequency Response

The structural response of the cylinder is dominated by its natural frequency and vortex shedding frequency. When the system’s natural frequency f_n is close to the vortex shedding frequency f_s, a phenomenon known as “lock-in” occurs, characterized by a range of U_r. Within this region, the dimensionless frequency f* = f_s/f_n ≈ 1. This involves a synchronization mechanism or locking mechanism that leads to large-amplitude oscillations of each cylinder, as described by De Lange [39] and Mittal and Kumar [40]. This synchronization is a critical aspect of VIV phenomena, where the fluid–structure interaction results in enhanced vibrations when the frequencies align. Understanding the “lock-in” region is crucial for predicting and mitigating excessive vibrations in structures subject to fluid flows.

Under the parameters of the VIV response of cylinders for low Reynolds number, following the classification by Chen et al. [24], the vibration response is divided into three phases before, during, and after the “lock-in” region: the initial branch (IB), the lower branch (LB), and the desynchronization branch (DB). The maximum amplitude occurs on LB. The classification of vibration response into these branches is critical for understanding the complex dynamics of VIV. The initial branch typically corresponds to a region where the vibration frequency is close to the natural frequency of the system, while the lower branch is marked by significant amplitude amplification due to the “lock-in” phenomenon. The desynchronization branch, after the “lock-in” region, is characterized by a departure from resonance, resulting in decreased amplitudes and often complex, multi-frequency behavior.

Figure 5 depicts the dimensionless vibration frequency trend of side-by-side cylinders as a function of U_r. The dashed line represents vortex shedding frequency (Strouhal number St = 0.185) for a single cylinder (SC) in flow, as calculated by Jiang et al. [41]. According to the definition of reduced velocity U_r = u/f_nD, Strouhal number St = f_sD/u, and frequency ratio f* = f_s/f_n, we have U_r = u/f_nD = f*u/f_sD = f*/St. Prior to the “lock-in” region (on the IB), the dimensionless frequency curve closely follows the Strouhal line. After the “lock-in” region (on the DB), as L/D increases, the dimensionless frequency curve diverges from the Strouhal line. Given the symmetrical arrangement of the cylinders, the frequencies of cylinders C₁ and C₂ are identical; hence, only the variation in the VIV frequency of C₁ with U_r is shown in Figure 5. When L/D ≤ 4, the “lock-in” region spans approximately U_r = 5 to 6. As the spacing ratio increases, for L/D ≥ 5, the range of the “lock-in” region expands to approximately U_r = 5 to 7.

3.1.2. Amplitude Response

This subsection will analyze the variations in amplitude, which correspond to the periods before (on the IB), during (on the LB), and after (on the DB) the “lock-in” region. Figure 6 illustrates the trends in the dimensionless cross-flow amplitude ($A_y^{*}$) and the dimensionless streamwise amplitude ($A_x^{*}$) of the cylinders with different L/D varying from 3 to 6, under 2DOF. For a clearer exposition of the cylinders’ vibration patterns, the amplitude variations for a single cylinder are also presented for comparison. Since the amplitudes of vibration for both cylinders are nearly identical when experiencing VIV at low Reynolds numbers, which is same as Chen et al. [24,25]; this study exclusively reports the amplitude characteristics of cylinder C₁ for each spacing ratio.

From Figure 6, it is evident that the maximum amplitude responses for side-by-side cylinders with L/D from 3 to 6 occur at approximately U_r = 5. Specifically, the greatest dimensionless cross-flow amplitude ($A_y^{*}$) is approximately 0.56, while the largest dimensionless stream-wise amplitude ($A_x^{*}$) is at its peak when L/D = 3, reaching a value of 0.12. As L/D increases, the resonance range for larger spacing ratios visibly broadens, a fact also observable in Figure 5. At this point, the dimensionless amplitudes $A_y^{*}$ and $A_x^{*}$ gradually approach a single cylinder as U_r varies. Notably, amplitudes $A_y^{*}$ for different spacing ratios are significantly less than those of a single cylinder at U_r = 8.

These observations reveal the complex interactions between cylinders under different spacing ratios L/D and reduced velocities U_r. As L/D increases and U_r deviates from the “lock-in” region, the amplitudes of cylinders gradually converge towards those of a single cylinder.

3.2. Lift and Drag Coefficients

As depicted in Figure 7a,b, at reduced velocities U_r = 2 and 3, the lift and drag coefficients of cylinder C₁ are out of phase, and the phase differences between the individual force element components and the total lift and drag forces are less than π/2, indicating positive contributions to the total forces. During this stage, cylinder C₁’s surface acceleration force, and the forces exerted by cylinder C₂ on cylinder C₁ are negligible. Cylinder C₂ exhibits force element components that are oppositely symmetric to those of cylinder C₁ in lift direction and identical in drag direction. Due to the surface interaction forces, the lift curve shifts, causing its equilibrium position to no longer be proximate to the x-axis. This offset is observable in subfigures (I) and (II) of Figure 7 at lower reduced velocities. Subfigure (III) of Figure 7 elucidates the magnitude of the surface interaction forces, with solid lines representing forces on cylinder C₁ and dashed lines for cylinder C₂. Surface vortices and friction forces seem to make a more substantial contribution here. Notably, in the drag direction, the surface interaction forces are of equal magnitude and phase, causing the curves to overlap. In lift direction, the surface vorticity lift components C_lv12 and C_lv21 of the two cylinders have equal amplitudes but opposite phases. As U_r increases, the amplitudes of all lift and drag coefficients rise.

As shown in Figure 7c, when U_r is approximately 4, nearing the end of the IB and the proximity of the “lock-in” region, within the IB, the phase differences between the force element components and the total lift and drag forces remain less than π/2, signifying positive contributions. Figure 7d,e illustrates the scenario at U_r = 5 and 6 when the system enters the “lock-in” region. The amplitudes of lift and drag force element components are markedly heightened, peaking at U_r = 5. At this point, in the lift direction, the volume vorticity lift coefficient C_lv1 of cylinder C₁ has phase differences greater than π/2 relative to the total lift coefficient C_l1, making negative contributions and inhibiting lift variations. Similarly, the surface acceleration force coefficient C_da1 in the drag direction for cylinder C₁ exhibits a “large cyclic” behavior, with an overall phase difference greater than π/2 compared to the total drag coefficient C_d1, suppressing total drag variations. The acceleration force C_la21 in the drag direction produced by cylinder C₂ on cylinder C₁ noticeably increases. Given the stable periodicity of each force component, the total lift and drag coefficients of the cylinders manifest as steady non-sinusoidal periodicities. Cylinder C₂ mirrors the phenomena observed in cylinder C₁.

When U_r = 7 to 12, the vibration exits the “lock-in” region and enters the desynchronization zone. As observed in Figure 7f, when U_r = 7, akin to the situation at U_r = 4, the curves of the lift and drag force element components display a “multi-frequency” phenomenon. At this juncture, the phase difference between the surface acceleration force coefficient C_la1 in lift direction and the total lift coefficient C_l1 is greater than π/2, inhibiting total lift variations. Other force element components have phase differences less than π/2 relative to them, promoting the total lift variations. Moreover, within the desynchronization zone, the curves of lift and drag force element components gradually stabilize as U_r increases. As shown in Figure 7g, when U_r = 8, the “multi-frequency” phenomenon disappears.

Observations from Figure 8 reveal that when U_r is small, within the initial branch and as U_r increases, the amplitudes of lift and drag force element components show a growing trend. Particularly noticeable is the increase in the surface acceleration lift coefficient C_la21 exerted by cylinder C₂ on cylinder C₁. The phase differences between the force element component curves and the total lift and drag forces are minimal, with all force components contributing positively to the total lift and drag forces. Among these, the volume vorticity force and surface friction force of the cylinder play a dominant role in the variations in the total lift and drag forces, whereas the contributions of the force element components due to inter-cylinder interactions are relatively smaller, having a limited impact on the total lift and drag forces of the cylinders. When U_r = 5, entering the “lock-in” region, the amplitudes of lift and drag forces reach their maxima, with the amplitudes of all force element component curves notably higher than in the initial branch, a phenomenon consistent with the case for L/D = 3.

At U_r = 5 and 6, the phase difference between the vortex lift coefficient C_lv1 of cylinder C₁ and the total lift coefficient C_l1 exceeds π/2, making a negative contribution to the lift. The phase difference between the surface acceleration force coefficient C_da1 and the total drag coefficient C_d1 is also greater than π/2, suppressing the variations in total drag. The acceleration force coefficient produced by cylinder C₂ on cylinder C₁ in the drag direction shows a significant increase. Within this interval, the surface friction force and surface acceleration force coefficient curves exerted by cylinder C₂ on cylinder C₁ are out of phase by more than π/2 with the total lift coefficient curve of C₁, inhibiting the variations in lift. At U_r = 7, the force element components of lift and drag coefficients exhibit a “multi-frequency” phenomenon, similar to that observed for a spacing ratio L/D = 3 (please note that no multi-frequency phenomena occurred when L/D = 3 and U_r = 4). At this point, the phase difference between the surface acceleration force coefficient C_la1 and the total lift coefficient C_l1 exceeds π/2, suppressing the variations in lift. The phase difference between the surface acceleration force coefficient C_da1 exerted on cylinder C₁ and the total drag coefficient C_d1 is also greater than π/2, restraining the variations in drag. After U_r > 7, the curves of the force element components of lift and drag coefficients tend to stabilize.

As illustrated in Figure 9, the behavior of the force element components of the lift and drag coefficients for the cylinders with L/D = 5, as a function of reduced velocity, is similar to that observed for L/D = 3 and 4. However, at each reduced velocity, the amplitudes of the surface friction and surface acceleration force coefficients in the lift and drag directions are notably diminished compared to those at L/D = 3 and 4, indicating a significant reduction in the surface interaction forces. Furthermore, the total lift and drag coefficient curves progressively converge towards those of a single cylinder. An additional observation is that when the L/D = 5, the “multi-frequency” vibration phenomenon, which is presented at L/D = 3 and 4 at U_r = 7, is absent. This suggests that at a spacing ratio L/D = 5, the mutual influence between cylinders is considerably reduced, to the extent that it can be largely disregarded.

As depicted in Figure 10, when L/D is further increased to 6, the phase and amplitude fluctuation patterns of the force element component curves of lift and drag force as functions of U_r are similar to those observed at L/D = 5. The “multi-frequency” phenomena in the coefficient curves are attenuated. The amplitudes of surface acceleration forces and surface vorticity forces in lift and drag directions, generated by the inter-cylinder interactions, are further reduced. The lift, drag, and associated force element components of the cylinders converge even closer to those of a single cylinder.

3.3. Motion Trajectory

Tu et al. [42] classified the vibration trajectories into seven patterns, including “figure-eight”, “teardrop”, “elliptical”, “double-teardrop”, “double-figure-eight”, “double-elliptical”, and irregular shapes. In the present study, additional trajectory patterns emerge beyond those identified by Tu et al. [42], including “bounded elliptical”, “bounded teardrop”, and “bounded figure-eight”. As the motion trajectories of the two cylinders are completely symmetrical in this study, only the trajectory of cylinder C₁ is presented in this section.

3.3.1. Trajectory Modes

As shown in Figure 11a,b, when U_r = 2 and 3, the cylinder C₁ with L/D = 3 is in the initial branch of its vibration responses. During this phase, the cylinder motion exhibits a small amplitude, and the trajectory is relatively stable, taking on the appearance of an “asymmetric figure-eight”. This trajectory is characterized by being narrow on one side and wider on another side. According to Tu et al. [42], this is attributed to the asymmetric wake distribution around the cylinders at this stage. This study further analyzes the cause of this phenomenon from the perspective of force elements, as detailed below in Section 3.3.2. When U_r = 4, approaching the end of the initial branch, the trajectory transforms into an irregular shape, as seen in Figure 11c. This is characterized by a series of nested “elliptical” curves, which can be considered a form of “bounded elliptical”. When U_r = 5 and 6, the vibration enters the “lock-in” region, where the trajectories are stable elliptical curves throughout this phase, as depicted in Figure 11d,e. The cylinder vibration amplitude increases but remains stable. At U_r = 7, exiting “lock-in” region and entering the desynchronization zone, Figure 11f reveals a trajectory consisting of nested “elliptical” curves, also interpretable as a “bounded elliptical”. The cause of this phenomenon will be analyzed later in Section 3.3.2. As shown in Figure 11g–k, once entering the desynchronization region (for U_r > 7), the trajectories consistently manifest as “elliptical” curves.

The motion trajectories for cylinders with L/D = 4 are depicted in Figure 12. At a reduced velocity U_r = 2, the motion traces an “asymmetric figure-eight” pattern that is narrow at the top and wider at the bottom. When U_r = 3, this “figure-eight” pattern transitions to one that is broader at the top and narrower at the bottom. At U_r = 4, the trajectory shows a “teardrop” shape, which is relatively stable. Within the “lock-in” region, the trajectories consistently exhibit a “teardrop” shape that is broader at the top and narrower at the bottom, remaining relatively stable. At U_r = 7, once exiting the “lock-in” region and entering the desynchronization zone, the trajectory becomes a regular bounded motion curve, which can be described as a “bounded teardrop”. The causes behind this phenomenon will be discussed later in Section 3.3.2. For U_r = 8 and 9, the trajectory evolves into a “double-teardrop” shape. When U_r > 9, the trajectory once again reverts to a “teardrop” form.

Figure 13 displays the displacement trajectories of cylinders at various reduced velocities for a spacing ratio of L/D = 5. When U_r ranges from 2 to 4, the cylinders are in the initial branch, tracing “figure-eight” paths. As U_r increases, the “figure-eight” curves gradually become narrower. When U_r spans from 5 to 7 and the cylinders are within the “lock-in” region, the trajectories take on a “teardrop” shape that is broader at the top and pointed at the bottom. As U_r continues to increase and the cylinders exit the “lock-in” region into the desynchronization zone, the motion trajectories evolve, with the curves increasingly resembling a “figure-eight” pattern as U_r grows. At U_r = 8, it can be observed that over the course of the simulation, the trajectory transitions from a “teardrop” to a “figure-eight” shape. For U_r ≥ 9, the cylinder motion trajectories shift from dual “figure-eight” to “bounded figure-eight”. This transformation might be attributed to the presence of unstable frequencies in the streamwise vibration, a detailed explanation for which will be provided in Section 3.3.2.

Figure 14 presents the motion trajectories of cylinders at various U_r for a spacing ratio L/D = 6. Under this spacing condition, the trajectories resemble those of a single cylinder, predominantly a “figure-eight” pattern. Within the initial branch, relatively regular “figure-eight” shapes are observed, with the “figure-eight” becoming increasingly narrow as U_r increases, with the ends approaching near-coincidence. When in the “lock-in” region, the curves adopt “figure-eight” forms that are nearly “teardrop”-like, owing to the stronger transverse vibrations and the amplified effect of surface interaction forces, as observable in Figure 10d,e.

Once exiting the “lock-in” region at U_r = 8 and entering desynchronization zone, the trajectories initially manifest as disordered “figure-eight” curves. However, as reduced velocity continues to rise, the trajectories gradually settle into a more distinct “figure-eight” pattern. At U_r = 12, the cylinder motion traces a double “figure-eight” trajectory. According to Tu et al. [42], this is due to the second-order frequency in the power spectral density (PSD) of x/D and C_d being twice that of the primary frequency in y/D and C_l, generating an asymmetrical double “figure-eight” in the trajectory.

To have a clearer comparison of amplitude, the cylindrical motion trajectories at different U_r and L/D = 3, 4, 5, 6 are shown in Figure 15 under the same axis ranges for all subfigures in Figure 11, Figure 12, Figure 13 and Figure 14.

3.3.2. Trajectory Patterns Analyses

In this section, further analysis will be conducted on the motion trajectories of interest discussed in the previous section.

For gaining deeper insight into the genesis of the asymmetric “figure-eight” pattern, time-history plots of the surface acceleration force coefficient C_da1 and streamwise displacement relative to x/D have been delineated for a cylinder with a spacing ratio L/D = 3 at reduced velocities U_r = 2 and 3. These plots, presented in Figure 16, reveal a distinct “modulation period” within the curves, serving as the principal mechanism behind the formation of the “asymmetric figure-eight” trajectory. It is noteworthy that the side characterized by elevated peak values manifests a constricted “figure-eight” configuration.

At L/D = 3 and U_r = 7, the time-history curves of the streamwise and cross-flow positions of the cylinder is depicted in Figure 17. Here, the phenomenon of “flutter” is evident in each direction, characterized by periodic simultaneous amplification and diminution of displacements in the x and y directions, respectively. This results in the formation of superimposed “bounded elliptical” trajectories in the motion paths. Similarly, at L/D = 4 and U_r = 7, the time-history curves for the streamwise and cross-flow positions of the cylinder are illustrated in Figure 18. In this case, the displacements in the x and y directions exhibit a “multi-periodic” behavior, leading to “bounded teardrop-shaped” trajectories. It is noteworthy that these “bounded” trajectories predominantly occur at the boundaries of branches for spacing ratios L/D ≤ 4, particularly when transitioning into or out of the “lock-in” region; this is consistent with the conclusion reached in the previous section.

To understand the underlying mechanism of the specific trajectory “bounded figure-eight” at L/D = 5, U_r = 10 in Figure 13i, as depicted in Figure 19a, the Fast Fourier Transform (FFT) analysis of drag coefficient C_d unveils a multitude of sub-harmonics within the PSD, contributing to a subtle transverse displacement of “figure-eight” trajectory. In conjunction, the time-series plot of x/D and C_da1 are shown in Figure 19b, the component of the acceleration force coefficient associated with the motion. It is evident that C_da1 exhibits a “modulation period” correlating with the development of a “figure-eight” trajectory, which is distinctly constricted at the lower portion and expanded at the upper section. This observation aligns seamlessly with the conclusions drawn earlier in the study.

These observations provide profound insights into the complex dynamics of VIV for cylinders in side-by-side configurations under varying spacing ratios and reduced velocities. The identification of distinct patterns and the underlying mechanisms driving the transition between different vibration modes is essential for the development of predictive models and mitigation strategies in engineering applications involving VIV phenomena.

3.4. Wake-Stream Pattern

In the flow past side-by-side dual circular cylinders, the interaction between them varies with spacing ratio L/D, leading to several distinct wake patterns [43]. Investigations by Singh and Mittal [44] and Prasanth and Mittal [45] revealed that the vortex shedding modes corresponding to IB and LB are denoted as 2S and C(2S), respectively. The C(2S) mode shares similarities with the 2S mode, but it is characterized by the coalescence of vortices in the wake.

For L/D = 3, the vortex shedding patterns around cylinders are illustrated in Figure 20. When U_r ≤ 4, within the primary branch, the vortex shed from both cylinders exhibits a uniform single-type pattern, with small amplitudes and minimal influence on the generation of vortices. Consequently, the 2S shedding mode, akin to that observed in the flow around an isolated cylinder, where pairs of vortices of opposite rotation directions are shed alternately from the upper and lower sides of the cylinder during each oscillation cycle, is prevalent. As U_r increases to five, the VIV enters the lower branch’s “lock-in” region. With enhanced transverse vibrations, the shedding pattern transforms; pairs of counter-rotating vortices form on either side of the cylinders’ wake. During each oscillation cycle, two rows of vortices are shed, one stronger than the other, resulting in a symmetrically opposed double-row C(2S) shedding mode. The vortices dissipate energy more rapidly in this configuration. At U_r = 6, the wake transitions from the initially antiphase synchronous 2S mode towards the symmetrically opposed C(2S) mode. Within this stage, the shedding length shortens, vortex shedding accelerates, and the shedding frequency significantly increases. The cylinders exhibit larger transverse amplitudes, with vigorous vortex shedding and a pronounced lateral displacement of the vortices. The wake widens, and the vortices detach further from the cylinder surface. When U_r ≥ 7, the vortex shedding pattern reverts to 2S, similar to that at U_r = 3. It can also be observed from Figure 6 that the amplitudes of side-by-side cylinders are suppressed in the primary branch U_r = 2 to 4 and non-synchronous range U_r = 7 to 12, dominated by 2S mode throughout.

When L/D = 4, the vortex shedding patterns around cylinders are depicted in Figure 21. Both within the primary branch U_r ≤ 4 and in the non-synchronous regime U_r ≥ 7, the wake exhibits a pair of antiphase synchronized 2S mode vortices. Once entering the “lock-in” zone, at U_r = 5, the vortices shed follows a counter-rotating double-row C(2S) mode. At U_r = 6, the near-wake region demonstrates an antiphase synchronized 2S mode, while in the far-wake region, the 2S mode vortices merge and weaken, gradually transitioning toward C(2S) mode. For an increased spacing ratio of L/D = 5, the vortex shedding behavior of cylinders across different reduced velocities adheres to the same pattern as observed for L/D = 4; thus, these observations will not be reiterated here.

As L/D between the cylinders further increases to six, the vortex shedding patterns around cylinders are illustrated in Figure 22. When the reduced velocity U_r ≤ 4, the vortices shed from both cylinders display an anti-phase synchronized shedding of two sets in 2S mode. As U_r increases and the system’s vibration frequency reaches the “lock-in” region, the cylinder amplitudes are substantial, and the vortex shedding pattern assumes a double-row C(2S) mode. At a reduced velocity U_r = 6, despite the increase in U_r, the significant spacing between cylinders causes a reduction of inter-cylinder interactions, and as a result, no vortex merging is observable in the wake. When U_r ≥ 7 and the system enters the non-synchronous zone, the vibration response amplitudes further decrease, and the vortex shedding pattern returns to the 2S mode, similar to that in the primary branch.

It is noteworthy that, in the present study, all spacing conditions exhibited anti-phase synchronized vortex streets, even when the spacing ratio was further increased to L/D = 6. No in-phase synchronized vortex streets, as mentioned by Williamson [17] and Sumner [43], were observed in this investigation. This finding highlights the dominance of anti-phase synchronization in the vortex shedding patterns for the examined range of spacing ratios, suggesting that other factors may be influential in achieving in-phase synchronization reported in previous studies.

4. Conclusions

This study presents a force mechanism, from the perspective of the multi-body force element formulations, of the vortex-induced vibration (VIV) phenomenon of the side-by-side dual cylinders at the Reynolds number Re = 100. The main findings are summarized as follows:

(1)

The VIV phenomenon for side-by-side dual cylinders exhibits three vibration regions with different spacing ratios, similar to those observed for a single cylinder, namely the initial branch (IB), the lower branch (LB), and the desynchronization branch (DB). When the spacing ratio L/D is greater than or equal to five, the range of reduced velocities U_r corresponding to the lower branch is wider and more closely resembles that of a single cylinder. For side-by-side cylinders at different spacing ratios, the maximum displacements in the streamwise and cross-flow directions occur at a reduced velocity U_r = 5. Cross-flow displacements are comparable to those of a single cylinder. In the streamwise direction, the amplitude decreases gradually with increasing spacing ratio, approaching the behavior of a single cylinder. Notably, there is no sudden increase in streamwise amplitude at a U_r = 8, unlike the case for a single cylinder.

(2)

During the VIV phenomenon, the lift and drag forces acting on side-by-side cylinders are primarily due to the volume vorticity forces associated with the cylinders themselves, combined with surface vortices (including friction) forces. When entering the “lock-in” zone, as cylinder vibrations intensify increase, the acceleration forces on the cylinder surface also increase. The volume vorticity lift force phase opposes the total lift force, thus inhibiting (negatively contributing) to the lift force. As the reduced velocity increases and the cylinder amplitude reaches its maximum, the surface acceleration forces begin to suppress changes in lift force, decreasing with further increases in reduced velocity. At lower reduced velocities, side-by-side cylinders are mainly influenced by the surface vorticity forces from the surface of the other cylinder. As reduced velocity increases, the influence of inter-cylinder surface acceleration forces grows, leading to increased amplitudes. The effect of surface vorticity forces in lift direction stabilizes over time. As the spacing ratio between cylinders increases, the influence between cylinders diminishes, and the force variation trends approach those of a single cylinder.

(3)

Once in the “lock-in” zone and experiencing significant vibrations, the wake vortices of side-by-side cylinders transition from a 2S mode to a C(2S) mode. The smaller the spacing ratio, the more pronounced this transformation becomes. At the onset and end of the wake mode transition, the displacement trajectories of the cylinders exhibit irregular “bounded” motions. During this period, the lift and drag coefficients display a “multi-frequency” behavior. Simultaneously, the vibration trajectories of side-by-side cylinders form “bounded elliptical”, and “bounded teardrop” shapes.