Open AccessArticle

Numerical Study on Hydrodynamic Performance and Vortex Dynamics of Multiple Cylinders Under Forced Vibration at Low Reynolds Number

Fulong Shi

^1,2,3

Chuanzhong Ou

^1,*,

Jianjian Xin

⁴,

Wenjie Li

¹,

Qiu Jin

⁵,

Yu Tian

¹ and

Wen Zhang

School of Shipping and Naval Architecture, Chongqing Jiaotong University, Chongqing 400074, China

State Key Laboratory of Coastal and Offshore Engineering, Dalian 116000, China

CSSC Haizhuang Windpower Co., Ltd., Chongqing 400074, China

⁴

Institute of Naval Architecture and Ocean Engineering, Ningbo University, Ningbo 315211, China

⁵

School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China

Author to whom correspondence should be addressed.

J. Mar. Sci. Eng. 2025, 13(2), 214; https://doi.org/10.3390/jmse13020214

Submission received: 22 December 2024 / Revised: 17 January 2025 / Accepted: 21 January 2025 / Published: 23 January 2025

(This article belongs to the Section Ocean Engineering)

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Abstract

Flow around clustered cylinders is widely encountered in engineering applications such as wind energy systems, pipeline transport, and marine engineering. To investigate the hydrodynamic performance and vortex dynamics of multiple cylinders under forced vibration at low Reynolds numbers, with a focus on understanding the interference characteristics in various configurations, this study is based on a self-developed radial basis function iso-surface ghost cell computing platform, which improves the implicit iso-surface interface representation method to track the moving boundaries of multiple cylinders, and employs a self-constructed CPU/GPU heterogeneous parallel acceleration technique for efficient numerical simulations. This study systematically investigates the interference characteristics of multiple cylinder configurations across various parameter domains, including spacing ratios, geometric arrangements, and oscillation modes. A quantitative analysis of key parameters, such as aerodynamic coefficients, dimensionless frequency characteristics, and vorticity field evolution, is performed. This study reveals that, for a dual-cylinder system, there exists a critical gap ratio between X/D = 2.5 and 3, which leads to an increase in the lift and drag coefficients of both cylinders, a reduction in the vortex shedding periodicity, and a disruption of the wake structure. For a three-cylinder system, the lift and drag coefficients of the two upstream cylinders decrease with increasing spacing. On the other hand, this increased spacing results in a rise in the drag of the downstream cylinder. In the case of a four-cylinder system, the drag coefficients of the cylinders located on either side of the flow direction are relatively high. A significant increase in the lift coefficient occurs when the spacing ratio is less than 2.0, while the drag coefficient of the downstream cylinder is minimized. The findings establish a comprehensive theoretical framework for the optimal configuration design and structural optimization of multicylinder systems, while also providing practical guidelines for engineering applications.

Keywords:

ghost cell method; flow around multiple cylinders; low Reynolds number; forced vibration

1. Introduction

The flow around clustered structures is a fundamental fluid mechanics phenomenon with wide-ranging engineering applications, including aerospace design, wind energy systems, bridge aerodynamics, pipeline transport, and marine engineering. For instance, in aerospace design, understanding the interaction of vortices between wing and tail surfaces can enhance the aerodynamic efficiency of aircraft wings. In wind energy systems, optimizing the layout of wind turbines can maximize the energy capture by minimizing the wake effects. Similarly, in marine engineering, the design of offshore structures such as bridge piers must consider vortex-induced vibrations to ensure structural safety. The interference between clustered structures significantly affects the fluid velocity and pressure distribution, while also triggering the generation and shedding of vortices. Despite the extensive research on cylindrical flows across various scales and conditions, fundamental questions remain regarding the mechanisms and evolution of wake interference phenomena. Studying these phenomena is crucial for advancing our understanding of nonlinear fluid dynamics, boundary layer behavior, and the development of complex flow structures.

Extensive research has been conducted on the flow characteristics of cylindrical structures [1,2,3,4]. These investigations have addressed the flow characteristics, forced vibrations, pressure distributions, vortex-shedding phenomena, and aerodynamic force variations [5,6,7]. These studies aim to elucidate the effects of cylindrical arrangements, the incoming flow conditions, and the mass ratio on the flow field structures and characteristics, facilitating the control of vortex-induced vibration (VIV) states in cylindrical structures [8,9,10]. Under oscillatory conditions, the cylinders exhibit characteristic oscillation frequencies. When the vortex shedding frequency coincides with this natural frequency, a significant increase in the lift coefficient occurs [11,12], resulting in a “lock-in” phenomenon. Tang [13] examined the relationship between the phase and energy transfer within the “lock-in” region, demonstrating that, for small amplitudes (A* ≤ 0.4), the energy transfer coefficient remains positive, indicating fluid-to-cylinder energy transfer, while, for larger amplitudes (A* ≥ 0.6), the coefficient becomes negative. Taheri [14] performed numerical investigations on cylinder responses to lateral oscillatory flow at varying vibration angles, revealing a negative correlation between the maximum amplitude and vibration angle, while characterizing novel flow patterns. Soufi [15] investigated the effects of a steady flow on cylinder forces and vorticity fields, and demonstrated that the velocity ratio substantially influences the transition period and lock-in state of cylinder oscillations. Cylinder cross-sectional geometry represents a significant research focus because it fundamentally affects flow characteristics, including the vorticity distribution, circulation patterns, jet angles, and wake flow behavior [16,17]. Dekhatawala [18] conducted numerical simulations of two-dimensional laminar flow around circular, square, and elliptical cylinders using computational fluid dynamics with user-defined functions, demonstrating the superior aerodynamic performance of elliptical cylinders, characterized by the lowest time-averaged drag coefficient. Singh [19] demonstrated that elevated vibration frequencies induce a phase lag between the near and far wake regions of square cylinders, resulting in the formation of multipole vortices. Additionally, modifications to the corner radius have negligible effects on the flow field characteristics [20]. Luo [21,22,23] examined the influence of the square cylinder tilt angle on the vortex characteristics, shedding frequency, and drag coefficient, establishing that increased tilt angles enhance the vortex shedding intensity. Boundary effects significantly influence the extent and characteristics of the lock-in region [24]. Shah [25] investigated the effects of cylinder wall spacing on the flow characteristics, revealing that the flow periodicity is dominated by the outer shear layer detachment and oscillation frequency at small spacing ratios, whereas the mean flow governs the periodicity at larger spacing ratios.

Clustered cylindrical structures are extensively employed in marine engineering applications, including offshore platforms, wind turbine foundations, and marine facilities. The distinct geometric configurations of these structures generate complex flow characteristics in fluid dynamic environments, particularly with respect to vortex-induced vibrations. Inter-cylinder spacing and arrangement configurations significantly influence vortex shedding patterns and flow field structures [26,27,28], potentially inducing sudden variations in the structural forces. Additionally, the oscillation frequency predominantly determines the lift fluctuations [29]. For tandem cylinder configurations, the vibration mode of an oscillating upstream cylinder significantly influences the coupled wake dynamics and vortex-shedding behavior of the system [30,31]. Under sinusoidal flow conditions, the flow field structure demonstrates distinctive patterns governed by the incoming flow frequency [32,33]. The drag coefficient exhibits a nonlinear variation with the Reynolds number (Re), characterized by reduced sensitivity at Re > 10⁴ [34]. Despite variations in the force magnitude across different Reynolds numbers, the temporal force patterns remain consistent within Re = 500–32,000, suggesting invariant fundamental fluid dynamics mechanisms [35,36]. This phenomenon justifies the use of low-Reynolds-number conditions to investigate the real fluid flows.

While extensive research exists on individual cylinder vortex-induced vibrations, limited attention has been paid to the interference phenomena in clustered cylindrical arrangements. This study aims to address this knowledge gap by investigating the hydrodynamic performance of clustered cylinders in various arrangements, thereby establishing theoretical foundations and design guidelines for marine engineering applications. The investigation utilizes a radial basis function ghost cell immersed boundary method solver to characterize the hydrodynamic behavior of grouped cylinders under low-Reynolds-number conditions. The analysis encompasses variations in the aerodynamic coefficients, vortex shedding characteristics, and wake flow patterns. A systematic investigation of the spacing ratios, geometric configurations, and motion parameters provided comprehensive insights into the cylinder interaction mechanisms. This study aims to investigate the hydrodynamic performance and vortex dynamics of multiple cylinders under forced vibration at low Reynolds numbers, and to explore the following questions: How does the spacing ratio (X/D) affect the hydrodynamic forces, vortex shedding characteristics, and wake patterns of a dual-cylinder system under forced vibration in the lock-in regime? For a three-cylinder configuration, how do the streamwise (X/D) and transverse (Y/D) spacing ratios influence the lift and drag forces acting on each cylinder, and how does the wake interference modify the vortex shedding dynamics? In the four-cylinder arrangement, what are the effects of in-phase and anti-phase forced vibrations on the aerodynamic forces experienced by each cylinder, and how do these different oscillation modes affect the resulting wake structures? What are the dominant vortex shedding frequencies for each cylinder configuration (dual, three, and four) under forced vibration, and how do these frequencies relate to the imposed vibration frequency and the spacing ratios? What insights can be gained from the vorticity field analysis to understand the complex flow interactions and wake evolution in the different multicylinder arrangements?

The results contribute to marine engineering design optimization and advance the understanding of multicylinder flow dynamics.

2. Overview of Ghost Cell Method

2.1. Numerical Model

In computational fluid dynamics, most of the flow scenarios can be classified as incompressible viscous fluid problems, except for specific cases, such as underwater explosions and high-speed projectiles entering water. The dimensionless governing equations for two-dimensional unsteady incompressible viscous flows are expressed as [37]

\nabla \cdot u = 0

(1)

\frac{\partial u}{\partial t} + (u \cdot \nabla) u = - \nabla p + \frac{1}{R e} \nabla^{2} u

(2)

Equations (1) and (2) represent the continuity and momentum equations, respectively, which collectively ensure the conservation of mass and momentum in the fluid system, where u represents the dimensionless velocity field; t is the dimensionless time; ∇ is the dimensionless gradient operator, defined based on the characteristic length L; p is the dimensionless pressure, representing the actual pressure p divided by the product of fluid density ρ and the square of the characteristic velocity; Re is the Reynolds number, which reflects flow characteristics, defined as the ratio of characteristic velocity to characteristic length divided by the kinematic viscosity ν; ∇² represents the dimensionless Laplacian operator, describing the diffusion effects in the velocity field.

The Reynolds number is an important parameter in fluid mechanics used to describe the flow state of fluids [38]. It is the ratio of inertial forces to viscous forces, indicating the relative significance of inertia and viscosity within the fluid. The magnitude of the Reynolds number determines the nature of the fluid flow. When the Reynolds number is low, viscous forces dominate, resulting in laminar flow, which is stable and orderly. Conversely, when the Reynolds number is high, inertial forces prevail, leading to turbulent flow, which is chaotic and unpredictable. The formula for calculating the Reynolds number is given by

R e = \frac{ρ U D}{μ} = \frac{U D}{υ}

(3)

In the formula, U represents the incoming flow velocity, D is the characteristic length (such as the diameter of a cylinder), and ν = u/ρ denotes the kinematic viscosity, where u is the dynamic viscosity and ρ is the fluid density.

The Strouhal number (St) is a dimensionless parameter in fluid mechanics used to describe periodic oscillations or wave phenomena in fluids [38]. It is employed to characterize phenomena such as vortex shedding and self-excited oscillations in airflow. The Strouhal number has significant applications in fields such as aerodynamics, hydrodynamics, and acoustics. For specific fluids and structures, when the Strouhal number reaches a certain range, it may lead to resonance or other stable vibrational modes. The formula for calculating the Strouhal number is given by

S t = \frac{f_{s} D}{U}

(4)

In the formula, f_s represents the natural vortex shedding frequency of a stationary single cylinder.

In fluid mechanics, the lift coefficient (C_l) and drag coefficient (C_d) are dimensionless parameters used to describe the lift and drag forces experienced by an object in a fluid [39]. By utilizing the lift and drag coefficients, one can compare and analyze the lift and drag forces on an object under varying conditions of speed, density, and shape, without the need to consider specific physical scales. The formulae for calculating the lift and drag coefficients are as follows:

C_{d} = \frac{F_{d}}{\frac{1}{2} ρ D U^{2}}

(5)

C_{l} = \frac{F_{l}}{\frac{1}{2} ρ D U^{2}}

(6)

In these formulae, F_d and F_l represent the drag force and lift force acting on the cylinder, respectively.

2.2. Immersed Boundary Method

The immersed boundary method (IBM) is an advanced numerical approach for handling boundary conditions at fluid–fluid and fluid–solid interfaces. This method has been extensively applied to biological fluid dynamics and fluid–structure interaction (FSI) problems, as demonstrated in Figure 1. In contrast to conventional body-fitted mesh approaches, the IBM significantly simplifies mesh generation for complex geometries, effectively resolving the challenges associated with mesh movement and regeneration. Consequently, this method proves particularly advantageous for analyzing flow problems involving moving boundaries and fluid–structure interactions. Additionally, the IBM effectively handles topological changes in the computational domain while maintaining a high computational efficiency.

In this investigation, the ghost cell method is implemented to characterize and track the moving boundaries of structures [40], with a particular emphasis on large boundary deformations [41]. This approach effectively facilitates the extension from two-dimensional to three-dimensional computational domains. Velocity–pressure decoupling is accomplished through a sequential solution strategy, implementing an explicit temporal scheme based on the third-order TVD-RK3 formulation [42]. For temporal discretization, explicit schemes are applied to convective terms, whereas viscous terms are treated semi-implicitly through the Crank–Nicolson scheme. Spatial discretization employs a high-order TVD MUSCL scheme [43] for convective terms and a central difference scheme for diffusive terms. The resulting pressure Poisson equation is resolved using a bi-conjugate gradient-stabilized (BICGSTAB) algorithm, enhanced by Jacobi preconditioning.

The detailed numerical calculation methods have been presented in previously published articles, such as the simulation of the swimming of biomimetic fish and the immersion of objects in water [44,45], which effectively validate the reliability of the numerical method.

2.3. CPU/GPU Heterogeneous Parallel Computing Platform

The implemented ghost cell solver [46] is developed using Fortran 90 and integrated within the CUDA framework, enabling efficient CPU/GPU heterogeneous parallel computation. Through systematic optimization and architectural refinement, the solver achieves significant computational acceleration. The CPU host orchestrates the entire computational workflow, encompassing task allocation, thread scheduling, and transaction management throughout the execution process. The GPU device handles computationally intensive operations, while the CPU host manages lightweight pre- and post-processing tasks during time advancement [47].

The computational platform employs a GeForce RTX 3080 Ti GPU, featuring 10,240 CUDA cores and 80 Streaming Multiprocessor (SM) units. The GPU operates at 1.67 GHz with 12 GB of global memory capacity. The system utilizes an Intel 12th generation i7-12700KF processor, equipped with 12 cores, operating at a base frequency of 3.6 GHz and capable of 5.0 GHz turbo boost. The computational environment consists of the Ubuntu 20.04 operating system, incorporating NVIDIA driver version 470.86 and CUDA toolkit version 11.4. A detailed schematic of the computational workflow and GPU heterogeneous parallel computing architecture is presented in Figure 2.

3. Numerical Model and Validation

3.1. Boundary Condition and Mesh Generation

This investigation examines the flow dynamics surrounding a two-dimensional circular cylinder. The computational domain configuration is presented in Figure 3, with corresponding boundary conditions detailed in Figure 4. The computational domain extends from the inlet at X = −6 to the outlet at X = 24. The domain spans 12 units in width, incorporating a refined mesh region bounded by X = [−1.5, 1.5] and Y = [−2, 2]. The total number of grid points is 289,009, and the time step is set to satisfy the CFL criterion [48]. The simulation parameters include the following: inlet velocity, U = 1 m/s; cylinder diameter, D = 1 m, with the cylinder center positioned at coordinates (0, 0). The fluid kinematic viscosity is specified as ν = 0.01 m²/s, yielding a Reynolds number of 100.

3.2. Validation of Low Around a Single Cylinder

Figure 5 illustrates the relationship between the frequency ratio (f₀/f_s) and aerodynamic coefficients (maximum lift coefficient C_l and mean drag coefficient C_d) for a single cylinder under forced vibration at Re = 100, where the motion equation for cylinder is given by Y = −Asin(2πf₀t). The analysis reveals that the maximum lift coefficient C_l exhibits a non-monotonic behavior, characterized by an initial decrease followed by an increase, as the forced vibration frequency f₀ increases. The mean drag coefficient C_d demonstrates the following distinct pattern: a gradual increase with f₀, followed by an abrupt decrease at f₀ = 1.2f_s, and, subsequently, a gradual recovery phase. The observed aerodynamic behavior aligns well with previous numerical investigations by Zhao [49] and Placzek [50].

The periodic vortex shedding phenomenon induces oscillatory lift forces, with the lift coefficient frequency directly corresponding to the vortex shedding frequency. The vortex shedding frequency is quantified through the Fast Fourier Transform (FFT) analysis of the temporal lift coefficient data. The FFT analysis transforms the time-domain lift coefficient signal into a frequency-domain spectrum, revealing the constituent harmonic components. Given the flow parameters U = 1 m/s and cylinder diameter D = 1 m, the characteristic Strouhal number (St) is determined by the shedding frequency f_s. Figure 6 depicts the lock-in region of the cylinder under forced vibration. The forced vibration amplitude is characterized by parameter A = 0.25. The numerical results demonstrate a strong correlation with Koopmann’s findings [51], with minor deviations attributed to variations in the numerical methodology and mesh resolution.

Table 1 shows the hydrodynamic coefficients for the flow around a stationary cylinder at Re = 100. The current numerical results exhibit reasonable agreement with previously published data. However, due to the differences in the numerical methods or grid resolutions, particularly at low Reynolds numbers, where the flow is highly sensitive to these factors, some discrepancies are observed, which fall within the acceptable error range. This comparison validates the effectiveness of the methodology used in this study.

Figure 7 presents the simulated vorticity field evolution over one complete cycle at the frequency ratio f₀/f_s = 1.0. The vortex shedding period T characterizes the temporal evolution, during which the cylinder exhibits harmonic oscillations about its equilibrium position Y = 0. The analysis reveals that, within each cycle, the oscillating cylinder generates two distinct counter-rotating vortices, characteristic of the classical 2S vortex shedding mode.

The feasibility of the virtual grid numerical method was validated by conducting a thorough investigation of the flow characteristics around a stationary single cylinder at varying Reynolds numbers, as well as the flow behavior around a forced vibrating cylinder.

4. Flow Characteristics Around Clustered Cylinders

The numerical simulations were conducted under consistent computational parameters and boundary conditions. This study focuses on clustered cylinders subjected to forced vibrations in a uniform flow field at a Reynolds number Re = 185. The computational domain specifications and boundary conditions remain identical to those employed in the single-cylinder forced vibration analysis. The vibration frequency is set as f₀ = f_s, with the vortex shedding frequency taken as f_s = 0.195 Hz, the diameter of the cylinder D = 1 m, and an amplitude of A = 0.2 m [55]. To ensure the robustness and statistical significance of the frequency analysis, as well as to clearly represent the dominant frequencies and their harmonics, this study selected 50 motion cycles with a relatively stable lift coefficient for the Fast Fourier Transform (FFT).

4.1. Forced Vibration of Dual Cylinders

The arrangement of the dual cylinders is illustrated in Figure 8. The center of cylinder 1 is located at (0, 0), while the center of cylinder 2 is positioned at (2.0, 0). The spacing between the cylinders is characterized by X/D = 1.2, 1.5, 2.0, 2.5, 3.0, 4.0. The motion equation for cylinder 1 is given by Y = −Asin(2πf₀t), with its motion directed in the negative Y-axis, perpendicular to the incoming flow direction. Conversely, the motion equation for cylinder 2 is expressed as Y = Asin(2πf₀t), with its motion directed in the positive Y-axis, which is also perpendicular to the incoming flow direction.

4.1.1. Analysis of Lift and Drag Coefficient

Figure 9a illustrates the relationship between the mean drag coefficient and the spacing ratio (X/D) for the tandem vibrating cylinders. For cylinder 1, the mean drag coefficient exhibited a monotonic decrease until it reached its minimum value of 1.20514 at X/D = 2.5, followed by a subsequent increase. Similarly, the mean drag coefficient of cylinder 2 reached its minimum at an identical spacing ratio (X/D = 2.5), notably achieving a negative value of −0.0213. This specific spacing ratio corresponds to the minimum drag coefficient fluctuation, suggesting the optimal drag reduction characteristics for both cylinders. For all other investigated spacing ratios, the drag coefficients maintained positive values within the range of 0.20 to 0.43.

Figure 9b shows the variation in the maximum lift coefficient with respect to the spacing ratio X/D. The maximum lift coefficient of cylinder 1 demonstrates non-monotonic behavior (a decrease–increase–decrease pattern), indicating complex bidirectional interference effects between the cylinders. For cylinder 2, the maximum lift coefficient exhibits an overall increasing trend with a pronounced peak at X/D = 3, suggesting the existence of a critical spacing ratio in the range of 2.5 ≤ X/D ≤ 3.0. A significant observation reveals that the lift coefficient amplitude of cylinder 1 surpasses that of cylinder 2 exclusively at X/D = 1.2, suggesting enhanced vortex shedding from cylinder 2 at larger spacing ratios. At X/D = 4, a concurrent reduction in the lift coefficient amplitudes for both cylinders indicates diminished cylinder–cylinder interactions.

Figure 10 shows the variation curves of the lift and drag coefficients of two tandem vibrating cylinders at different spacing ratios. From the figure, it can be observed that the drag coefficient of cylinder 1 is greater than that of cylinder 2, and the lift coefficient C_l of both cylinders oscillates periodically around zero. When X/D < 3 (as shown in Figure 10a–d), the curves exhibit significant periodicity. As the spacing ratio increases, the amplitude of the lift coefficient of cylinder 2 increases, indicating that cylinder 1 has a notable effect on the vortex shedding of cylinder 2, suppressing it at small spacings and promoting it at moderate spacings. The drag coefficient of cylinder 2 oscillates and gradually stabilizes with increasing spacing. When X/D = 3 (as shown in Figure 10e,f), the curves exhibit a sudden change, showing a certain periodicity with pronounced amplitude oscillations, indicating that the interaction between the two cylinders has increased at this point.

4.1.2. Analysis of Vortex Shedding Frequency and Strouhal Number

The dominant vortex shedding frequencies were extracted through a Fast Fourier Transform (FFT) analysis of the lift coefficient time series, as illustrated in Figure 11. Spectral analysis presents the frequency characteristics of the vortex shedding for tandem vibrating cylinders across spacing ratios X/D ranging from 1.2 to 4.0. The abscissa denotes the normalized vortex shedding frequency (f_v), and the ordinate represents the amplitude spectrum of the lift coefficient C_l.

Spectral analysis revealed a dominant frequency of 0.192 Hz, which corresponds precisely to the imposed forced vibration frequency f₀. Within the spacing ratio range of 1.2 ≤ X/D ≤ 2.5, the primary vortex shedding frequency maintains a constant value of 0.192 Hz, as exemplified by the spectral distribution at X/D = 1.2 (Figure 11a). The spectral distributions demonstrate consistent patterns with the amplitude variations, suggesting a pronounced periodicity in the vortex shedding and well-organized vortex streets in the wake region. At X/D = 3 (Figure 11b), the emergence of multiple dominant frequencies indicates reduced periodicity and the deterioration of the vortex shedding coherence. Moreover, cylinder 2 exhibited a substantially enhanced spectral amplitude at the primary frequency, indicating intensified wake interference from the upstream cylinder. At X/D = 4 (Figure 11c), the spectral analysis revealed multiple dominant frequencies with comparable amplitudes for both cylinders. Significantly, cylinder 2 displayed prominent secondary frequency components, manifesting as characteristic beating phenomena in its lift coefficient temporal evolution.

4.1.3. Analysis of Vortex

Figure 12 illustrates the instantaneous vorticity fields of the tandem cylinder configuration at various spacing ratios during equivalent simulation periods. For the spacing ratios X/D < 3, the vorticity patterns exhibit consistent characteristics with the representative case at X/D = 1.2, as depicted in Figure 12a. The flow field manifests as a single von Kármán vortex street configuration, analogous to the wake structure behind an isolated cylinder. The shear layer emanating from cylinder 1 remained attached and extended to cylinder 2, whereas the shear layer of cylinder 2 underwent distinct separation, generating coherent vortical structures with evident periodic characteristics. At X/D = 3 (Figure 12b), the negative vorticity shear layer of cylinder 1 exhibited incomplete separation, whereas its positive vorticity shear layer coalesced with the synchronized shear layer of cylinder 2. This shear layer interaction modified the separation characteristics of cylinder 2, thereby expanding the wake width. At X/D = 4 (Figure 12c), the partially separated shear layer from cylinder 1 impinged directly on cylinder 2, inducing asymmetric separation patterns in the shear layer of cylinder 2. This phenomenon resulted in a degraded vortex shedding coherence and substantial wake expansion.

4.2. Forced Vibration of Three Cylinders

The arrangement of the three cylinders is illustrated in Figure 13. The center of cylinder 1 is located at (0.0, −1.0), the center of cylinder 2 is located at (0.0, 1.0), and the center of cylinder 3 is located at (3.0, 0.0). The spacing ratios in the X-direction are X/D = 2.0, 3.0, and 4.0, while the spacing ratios in the Y-direction are Y/D = 2.0, 3.0, 4.0, and 5.0. The motion equation for cylinders 1 and 3 is given by Y = −Asin(2πf₀t), indicating motion in the negative Y-direction, which is perpendicular to the incoming flow direction. Conversely, the motion equation for cylinder 2 is Y = Asin(2πf₀t), indicating motion in the positive Y-direction, which is also perpendicular to the incoming flow direction.

4.2.1. Analysis of Lift and Drag Coefficient

Figure 14a shows the evolution of the mean drag coefficients for the three forced-oscillating cylinders as a function of the transverse spacing ratio Y/D at a fixed streamwise spacing ratio X/D = 3. With the increasing Y/D, the mean drag coefficients of cylinders 1 and 2 exhibited a monotonic decrease while maintaining nearly identical values, suggesting a symmetric aerodynamic force distribution. Conversely, the mean drag coefficient of cylinder 3 demonstrated a progressive increase, exceeding the values observed for cylinders 1 and 2 at Y/D = 5.0. This phenomenon indicates that the wake sheltering effect provided by the upstream cylinders 1 and 2 on the downstream cylinder 3 diminishes with increasing transverse spacing, thereby resulting in enhanced drag forces on cylinder 3.

Figure 14b shows the variation in the mean lift coefficients for the three forced-oscillating cylinders as a function of Y/D at a fixed streamwise spacing ratio of X/D = 3. As the transverse spacing ratio Y/D increases, cylinder 1 exhibits a monotonic decrease in the mean lift coefficient, whereas cylinder 2 demonstrates comparable behavior but with opposite directionality. The mean lift coefficients of cylinders 1 and 2 maintain equal magnitudes but opposite signs throughout the investigated range, indicating an antisymmetric aerodynamic behavior. The mean lift coefficient of cylinder 3 exhibited non-monotonic behavior, characterized by a transition from negative to positive values, followed by a decrease and subsequent sign reversal, demonstrating the oscillatory characteristics of the zero value.

Figure 15 and Figure 16 illustrate the dependencies of the mean drag and lift coefficients on the transverse spacing ratio Y/D at the streamwise spacing ratios X/D = 2 and X/D = 4, respectively. The qualitative behavior of the mean drag coefficients is consistent with the patterns observed at X/D = 3, suggesting similar flow physics mechanisms. For the configuration with X/D = 2, the mean drag coefficient of cylinder 3 surpassed those of cylinders 1 and 2 at Y/D = 4, indicating enhanced wake interference effects. The antisymmetric behavior of the mean lift coefficients for cylinders 1 and 2 remained consistent with the patterns observed at X/D = 3, demonstrating the stability of their aerodynamic interactions. At X/D = 2, cylinder 3 exhibited a monotonic decrease in the mean lift coefficient, ultimately transitioning to negative values, suggesting altered wake interactions. Conversely, at X/D = 4, cylinder 3 consistently maintains negative mean lift coefficients, displaying non-monotonic behavior characterized by an initial increase, followed by a gradual decay toward zero. These observations suggest that variations in the streamwise spacing X/D significantly influence the aerodynamic forces acting on cylinder 3, whereas cylinders 1 and 2 exhibit relatively robust force characteristics, presumably because of their upstream positioning.

Figure 17 shows the variation curves of the lift and drag coefficients for three cylinders at X/D = 3 as the spacing ratio Y/D changes. From the figure, it can be observed that the oscillation amplitudes of the lift and drag coefficients for cylinders 1 and 2 are smaller and more regular compared to cylinder 3. When Y/D = 2.0, the drag coefficients of cylinders 1 and 2 are close in value and in phase, with two peaks in one cycle. The drag coefficient of cylinder 3 is lower than that of cylinders 1 and 2, exhibiting a more periodic oscillation. The lift coefficients of cylinders 1 and 2 are out of phase, while cylinder 3 is in phase with cylinder 2. When Y/D = 3.0, the drag coefficient of cylinder 3 increases, and the lift coefficient exhibits amplitude variations, while the other coefficients are similar to those at Y/D = 2.0. When Y/D = 5.0, after a period of calculation, the coefficients of cylinder 3 stabilize into periodic oscillations. The lift coefficients of cylinders 1 and 2 remain out of phase, while cylinder 3 is close in phase to cylinder 1.

4.2.2. Analysis of Vortex Shedding Frequency and Strouhal Number

Figure 18 presents the frequency spectra obtained through the FFT analysis of the lift coefficient time histories for the three forced-oscillating cylinders at X/D = 3, with transverse spacing ratios Y/D varying from 2.0 to 5.0. Spectral analysis revealed that cylinders 1 and 2 exhibited nearly identical frequency signatures, characterized by a dominant shedding frequency of 0.192 Hz and a secondary harmonic at 0.392 Hz. This spectral similarity indicates synchronized vortex shedding dynamics between cylinders 1 and 2, demonstrating coherent periodic behavior. With increasing transverse spacing, the spectral amplitude progressively diminishes, suggesting attenuated cylinder-to-cylinder interference and more regular vortex shedding characteristics. Conversely, cylinder 3 demonstrated a broader frequency spectrum with multiple peaks, indicating more complex vortex shedding dynamics. The presence of the 0.192 Hz component in the spectrum of cylinder 3 suggests wake-induced synchronization with the upstream cylinders. The spectral amplitudes associated with cylinder 3 consistently exceed those of cylinders 1 and 2, indicating enhanced vortex shedding intensity owing to the upstream wake interactions. The spectral energy of cylinder 3 exhibits monotonic decay with increasing transverse spacing, which is consistent with reduced wake interference effects.

4.2.3. Analysis of Vortex

Figure 19 depicts the instantaneous vorticity fields at X/D = 3 for various transverse spacing ratios Y/D, captured at equivalent nondimensional time steps. Figure 19a reveals that the inner shear layers emanating from cylinders 1 and 2 remained attached prior to their impingement on cylinder 3, resulting in complex shear layer interactions and accelerated vorticity dissipation in the near-wake region. The outer shear layers of cylinders 1 and 2 exhibit periodic vortex shedding, although their evolution is modulated by both the presence of cylinder 3 and self-induced rotation, leading to coherent structural coalescence in the wake. Figure 19b shows the incipient separation of the inner shear layers from cylinders 1 and 2, which interact with the counter-rotating shear layer of cylinder 3. This shear layer interaction mechanism induces prolonged vorticity mixing, which manifests as increased wake turbulence and spatial disorder. Figure 19c illustrates the complete shear layer separation from cylinders 1 and 2, which facilitates distinct vortex formation and shedding processes. Furthermore, the wake dynamics exhibit the partial entrainment of the shed vortices of cylinder 3 into the upstream cylinder wakes, whereas residual vortical structures undergo gradual diffusion and dissipation.

4.3. Forced Vibration of Four Cylinders

The arrangement of the four cylinders is illustrated in Figure 20. The initial positions are as follows: the center of cylinder 1 is located at (−1.0, 0.0), cylinder 2 at (0.0, −1.0), cylinder 3 at (1.0, 0.0), and cylinder 4 at (0.0, 1.0). In the case of the in-phase motion, the spacing ratios L/D = X/D = Y/D = 2.0, 2.5, 3.0, 3.5, and 4.0 are employed, with the motion equations for the four cylinders expressed as Y = −Asin(2πf₀t), indicating movement in the negative Y-direction. For the out-of-phase motion, the spacing ratios L/D = X/D = Y/D = 2.0, 2.5, 3.0, 3.5, and 4.0 are utilized. The motion equations for cylinders 1 and 3 are given by Y = −Asin(2πf₀t), indicating movement in the negative Y-direction, perpendicular to the incoming flow. Conversely, cylinders 2 and 4 have motion equations defined as Y = Asin(2πf₀t), indicating movement in the positive Y-direction.

4.3.1. Analysis of Lift and Drag Coefficient

(a): In phase

Figure 21 illustrates the variations in the mean drag and lift coefficients for a rhombic configuration of four cylinders undergoing synchronized in-phase vibrations. Figure 21a demonstrates that cylinders 2 and 4 exhibit nearly identical mean drag coefficients, maintaining the highest magnitudes among all of the cylinders. The upstream cylinder 1 experiences intermediate drag forces, whereas the downstream cylinder 3 exhibits the minimum drag owing to the wake interference effects. With increasing cylinder spacing, the mean drag coefficients of cylinders 2 and 4 remained relatively stable within the range of 1.38–1.49, indicating minimal sensitivity to spatial separation. The mean drag coefficient of cylinder 1 remained approximately constant for L/D < 3.0, followed by a substantial increase beyond this critical spacing ratio. This trend suggests asymptotic behavior, where the mean drag coefficient of cylinder 1 is expected to converge toward the values observed for cylinders 2 and 4. Cylinder 3 demonstrated a non-monotonic variation in its mean drag coefficient, characterized by an initial increase, followed by a subsequent decrease. As shown in Figure 21b, the mean lift coefficients of cylinders 1 and 3 fluctuated around zero, suggesting symmetric flow conditions. At L/D = 2.5, cylinder 2 exhibited a pronounced minimum in its mean lift coefficient, indicating an optimal spacing ratio for lift reduction. Beyond this optimal spacing, the mean lift coefficient demonstrated a non-monotonic behavior, with a secondary minimum occurring at L/D = 4.0. Cylinder 4 displayed antisymmetric behavior relative to cylinder 2, exhibiting comparable magnitudes but opposite signs in the mean lift coefficient.

Figure 22 shows the lift and drag coefficient curves for four cylinders vibrating in phase as the spacing ratio L/D varies. From the figure, it can be observed that, as the number of cylinders increases, the complexity of the curves also increases. However, cylinders 1 and 3 exhibit some similarity in their curves, while cylinders 2 and 4 show similar characteristics. When L/D = 2.0, the drag coefficient of cylinder 1 is greater than that of cylinder 3. The lift coefficients of cylinders 1 and 3 are in phase, influenced by the upstream cylinder, resulting in a larger and more irregular oscillation for cylinder 3′s lift coefficient. The drag coefficients of cylinders 2 and 4 are close in value and exhibit similar periodicity, but are out of phase, while their lift coefficients are nearly in phase. When L/D = 2.5, the lift coefficients of cylinders 2 and 4 have consistent periods and magnitudes, and are in phase, while the other coefficients are similar to those at L/D = 2.0. As the spacing ratio increases, the fluctuations in the lift coefficients significantly decrease, and the force coefficient variations become more stable.

(b) Anti-phase

Figure 23 illustrates the variations in the mean drag and lift coefficients for a staggered rhombic arrangement of four cylinders. The mean drag coefficients of cylinders 2 and 4, as depicted in Figure 23a, demonstrated consistent behavior with the highest magnitudes, exhibiting a positive correlation with the spacing ratio. Cylinder 1 exhibited a non-monotonic variation in its mean drag coefficient, characterized by an initial decrease, followed by a subsequent increase. Owing to its downstream position and wake interference effects, cylinder 3 experienced the minimum drag force among all of the cylinders. A critical phenomenon occurred at L/D = 3, where cylinder 1 underwent a substantial reduction in the mean drag coefficient, concurrent with an abrupt increase in the drag coefficient of cylinder 3. This behavior suggests that the aerodynamic interference effects from cylinders 2 and 4 on adjacent cylinders exhibit a strong dependence on the spatial separation. At L/D = 3.5, cylinders 1 and 3 maintained relatively constant mean drag coefficients, whereas cylinders 2 and 4 exhibited an increasing trend. At L/D = 4, a significant transition occurred, where cylinders 1, 2, and 4 experienced a marked increase in the mean drag coefficients, whereas cylinder 3 exhibited a contrasting decrease. As illustrated in Figure 23b, the mean lift coefficients of cylinders 1 and 3 fluctuated by approximately zero, indicating symmetric flow patterns. With an increasing spacing ratio, cylinder 2 experienced a rapid reduction in the mean lift coefficient, accompanied by a sign inversion from negative to positive values, followed by a gradual monotonic increase. At the critical spacing ratio of L/D = 2.5, cylinder 2 underwent a fundamental change in the lift direction, corresponding to a global minimum in the mean lift magnitude. Cylinder 4 demonstrated antisymmetric behavior relative to cylinder 2, maintaining equivalent magnitudes but opposite signs in the mean lift coefficient.

Figure 24 shows the lift and drag coefficient curves for four cylinders vibrating anti-phase as the spacing ratio L/D varies. From the figure, it can be observed that, when cylinders 1 and 3 move in a different direction from cylinders 2 and 4, some of the trends in the lift and drag coefficient curves still align with those observed during codirectional vibrations, although there are variations. When L/D = 2.0, compared to the case of codirectional movement, the oscillation amplitude of the drag coefficient for cylinder 1 significantly increases. Due to the influence of the small spacing ratio, the lift coefficients of cylinders 2, 3, and 4 exhibit irregular oscillations in amplitude. When L/D = 2.5, the oscillation amplitude of the drag coefficient for cylinder 1 decreases, and the amplitude of the periodic oscillation of the lift coefficient stabilizes. The lift coefficients of cylinders 2 and 4 are approximately equal in magnitude, and their phase is the same, resulting in nearly overlapping curves.

4.3.2. Analysis of Vortex Shedding Frequency and Strouhal Number

(a): In phase

The vortex shedding frequency curves are shown in Figure 25. For cylinder 1, the dominant vortex shedding frequency was identified at 0.192 Hz, whereas a secondary frequency component of 0.083 Hz emerged exclusively at L/D = 2, as illustrated in Figure 25a. Consequently, the spectral analysis focused on two representative spacing ratios, L/D = 2 and L/D = 4. At other spacing ratios, the secondary frequency component become negligible, whereas the amplitude of the primary frequency exhibited a monotonic decrease with increasing spacing. Cylinder 3, under the influence of cylinder 1′s wake, demonstrated an identical vortex shedding frequency, but with enhanced spectral intensity. With varying spacing ratios, cylinder 3 exhibited multiple frequency components, suggesting complex and irregular vortex shedding patterns. Cylinders 2 and 4 demonstrated similar spectral characteristics, featuring a dominant frequency of 0.192 Hz and a secondary component at 0.392 Hz. At the minimal spacing ratio of L/D = 2.0 (Figure 25a), the close proximity of the cylinders induced a characteristic frequency of 0.083 Hz, owing to the strong interference effects. At L/D = 4.0 (Figure 25b), cylinders 2 and 4 manifested dual dominant frequencies, indicating bifurcated vortex shedding characteristics.

(b) Anti-phase

Figure 26 illustrates the spectral characteristics of the vortex shedding frequencies in a cylindrical array configuration. At L/D = 2.0 (Figure 26a), the spectral analysis revealed multiple frequency components, suggesting complex vortex shedding dynamics with significant irregularity. Cylinders 1 and 3 exhibited dominant shedding frequencies at 0.192 Hz, whereas cylinders 2 and 4 exhibited a higher characteristic frequency of 0.392 Hz. Furthermore, cylinder 3 manifested a secondary frequency component at 0.075 Hz, which is attributed to wake interference effects from the upstream cylinders. The dominant shedding frequencies exhibited a monotonic increase with an increasing spacing ratio, indicating a systematic evolution of the flow structure. For L/D = 4.0 (Figure 26b), cylinders 1 and 3 displayed single-mode vortex shedding characterized by a distinct dominant frequency, whereas cylinders 2 and 4 maintained a high-amplitude primary frequency with substantially suppressed secondary modes. These spectral characteristics suggest well-organized vortex shedding patterns with enhanced coherence and vortical strength.

4.3.3. Analysis of Vortex

(a): In phase

Figure 27 presents the instantaneous vorticity fields for the four-cylinder array undergoing synchronous vibrations at various spacing ratios L/D while maintaining consistent simulation parameters. As illustrated in Figure 27a, the wake structures behind the four cylinders exhibit rapid coalescence, evolving into a single-row vortex street configuration similar to the classical von Kármán vortex street of an isolated cylinder. Figure 27b shows the deflection of cylinder 1′s shear layer toward cylinders 3 and 4, initiating complex shear layer interactions and subsequent vorticity amalgamation. Concurrently, the shear layer emanating from cylinder 2 underwent mutual interaction and coalescence with the shear layer of cylinder 3. The increased cylinder spacing enhanced the gap flow intensity, consequently decelerating the vortex street merger process, owing to strengthened interwake interactions. Figure 27c reveals that the incipient separation of the shear layer of cylinder 1 coincides with its interaction with the co-rotating shear layer from cylinder 3, resulting in vorticity amalgamation. The enhanced gap flow intensity at larger spacing ratios promotes extended vortex persistence, leading to delayed merger dynamics and subsequent wake destabilization, manifesting as increased flow disorder in the downstream region.

(b) Anti-phase

Figure 28 illustrates the instantaneous vorticity field evolution for the four-cylinder configuration experiencing anti-phase oscillations. As shown in Figure 28a, the wake structures exhibited rapid coalescence, resulting in a highly chaotic vortex street characterized by significant spatial disorder. The phase opposition between cylinders (1,3) and (2,4) induces irregular vortex shedding patterns despite the confined spacing configuration. As depicted in Figure 28b, the negative vorticity shear layer from cylinder 1 deflects toward cylinders 3 and 4, initiating complex shear layer interactions and subsequent vorticity amalgamation. Concurrently, the positive-vorticity shear layer from cylinder 1 undergoes collision and coalescence with the corresponding shear layer of cylinder 3. Furthermore, the shear layer of cylinder 2 exhibits complex interactions, merging simultaneously with the shear layers emanating from cylinders 3 and 1. The enhanced gap flow intensity results in delayed vortex street coalescence dynamics. Figure 28c reveals that the incipient separation of the shear layer of cylinder 1 synchronizes with its interaction with the shear layer of cylinder 3, culminating in the vorticity merger. Cylinders 2 and 4 maintain relatively autonomous shear layer dynamics, generating distinct vortical structures. The shear layer emanating from cylinder 3 undergoes complex interactions with the shed vortices from cylinders 2 and 4, leading to vorticity diffusion and dissipation. The intensified gap flow promotes extended vortex persistence, resulting in delayed merger dynamics and contributing to the development of a distinct wake structure.

5. Conclusions

The present investigation employed a self-developed ghost cell immersed boundary methodology for the numerical simulation of the flow characteristics around staggered dual-cylinder configurations subjected to prescribed oscillatory motion. Based on a comprehensive analysis of the numerical results, the following key findings were obtained:

In the dual-cylinder configuration, the variation in the spacing ratio X/D significantly affected the flow field characteristics and wake dynamics. We found that a critical transition region occurred when 2.5 ≤ X/D ≤ 3.0, characterized by an increase in the hydrodynamic coefficients, disturbances in vortex shedding, and a significant interference in the wake. This is consistent with the studies of Zhao [9] and Williamson [4]. Zhao’s research also found that, as the X/D value changed, the vortex shedding behavior underwent dramatic changes, leading to instability in the wake structure. Williamson noted that an increase in the spacing ratio significantly altered the vortex shedding patterns in the wake, thereby affecting the hydrodynamic characteristics.
In the three-cylinder configuration, we observed that increasing the streamwise spacing (X/D) led to a reduction in the lift and drag of the upstream cylinder, while the drag on the downstream cylinder increased with the increasing spacing of the two upstream cylinders. This phenomenon was consistent with the study by Shah [10], which indicated that, in a three-cylinder system, increasing the spacing of the upstream cylinders reduced the drag on the upstream cylinder, while the drag on the downstream cylinder increased due to the influence of the upstream wake.
In the four-cylinder array, both the in-phase and out-of-phase vibrations exhibited a consistent trend. The middle cylinder experienced the highest drag, while the downstream cylinder had the lowest drag. The chaotic interference of the wake led to an increase in the hydrodynamic loading when L/D = 2.0. This is consistent with the studies by Kumar [3] and Bao [30], where Kumar noted that the hydrodynamic characteristics in multiple cylinder configurations were often significantly affected by strong wake interference, resulting in larger hydrodynamic loads. Similarly, Bao’s research found that wake interactions had a significant impact on the drag distribution among the cylinders in the array.
The dominant vortex shedding frequency was typically consistent with the forced vibration frequency (0.192 Hz), indicating that the system was in the “locked-in” region. However, under certain conditions, secondary frequencies and multiple peaks appeared in the spectrum, suggesting complex flow interactions and wake interference. This phenomenon aligned with the studies by Koopmann [51] and Tang [14], where Koopmann found that, when the system entered the locked-in region, the vortex shedding frequency coincided with the vibration frequency, while Tang noted that, at larger spacing ratios, multiple frequency components emerged in the spectrum, reflecting complex wake interference.
The analysis of the vorticity field revealed complex shear layer interactions, vortex merging, and wake evolution patterns. These patterns changed significantly with variations in the spacing ratio and vibration mode, providing a deeper understanding of the physical mechanisms of the flow. This analysis aligned with the studies by Zhu [6] and Soufi [16], which also emphasized the impact of shear layer interactions on vortex structures at different spacing ratios, and suggested that different vibration modes can significantly alter the formation and development of vortex structures.

These findings provide fundamental insights into the complex flow interactions and wake dynamics of multicylinder configurations under forced vibration. They also offer practical guidelines for engineering design, particularly in minimizing adverse fluid–structure interactions.

This study investigated the two-dimensional hydrodynamic performance of multiple cylinders under forced vibration at low Reynolds numbers. However, it is important to recognize that three-dimensional effects may play a significant role in influencing the flow dynamics and vortex interactions. Future research will prioritize the development of high-fidelity three-dimensional numerical models to explore the complex flow phenomena associated with both single- and multiple-cylinder configurations, and will extend the study to higher Reynolds numbers to enhance its relevance to practical marine engineering applications.

Author Contributions

Conceptualization, J.X. and F.S.; methodology, C.O. and J.X.; software, J.X. and F.S.; formal analysis, J.X. and F.S.; investigation, Y.T., Q.J. and C.O.; resources, Y.T., Q.J. and W.L.; writing—original draft preparation, C.O. and F.S.; writing—review and editing, Y.T., Q.J., W.Z. and F.S.; visualization, W.L.; funding acquisition, Y.T., Q.J., W.Z. and F.S. All authors have read and agreed to the published version of the manuscript.

Funding

This project was supported by the National key R&D Program of China (No. 2023YFE0102000), the National Science Foundation of China (No. 51909124 and 12301495), the State Key laboratory of Aerodynamics (No. RAL202404-3), the Scientific and Technological Program of Chongqing Municipal Education Commission (Nos. KJQN202400746 and KJQN202200737), the State Key Laboratory of Maritime Technology and Safety (No. W24CG000048), the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (No. LP2401), and the Hubei Provincial Natural Science Foundation of China (No. 20231J0208).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article.

Conflicts of Interest

Author Fulong Shi was employed by the company CSSC Haizhuang Windpower Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Nomenclature

D (m)	Diameter of a cylinder	Re	Reynolds number
X (m)	Streamwise distance	St	Strouhal number
Y (m)	Lateral distance	f₀ (Hz)	Forced vibration frequency
L (m)	Streamwise/lateral distance	f_s (Hz)	Vortex shedding frequency of stationary cylinder
U (m/s)	Free-stream velocity	F_l (N)	Lift force
ρ (kg/m³)	Fluid density	F_d (N)	Drag force
ν (Pa·s)	Kinematic viscosity	C_l	Maximum lift coefficient
u (m²/s)	Dynamic viscosity	C_d	Mean drag coefficient
A (m)	Vibration amplitude of cylinder

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Figure 1. Schematic diagram of an immersed body in a Cartesian grid.

Figure 2. Computation flowchart of virtual grid method based on CPU/GPU heterogeneous parallelism.

Figure 3. Mesh generation for a single cylinder computational domain.

Figure 4. Boundary condition diagram or schematic.

Figure 5. Lift and drag coefficients: (a) Mean drag coefficient; (b) Amplitude of lift coefficient [49,50].

Figure 6. Lock-in range of cylinder-induced forced vibration [51].

Figure 7. The vorticity distribution over one vortex shedding cycle when f₀/f_s = 1.0.

Figure 8. Arrangement of the dual cylinders.

Figure 9. Lift and drag coefficients: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 10. Temporal variation in the drag and lift coefficient: (a,c,e) Drag coefficient; (b,d,f) Amplitude of the lift coefficient.

Figure 11. Spectrum diagrams of lift coefficients for double cylinders: (a) X/D = 1.2; (b) X/D = 3.0; (c) X/D = 4.0.

Figure 12. Vorticity distribution diagrams for the double cylinders: (a) X/D = 1.2; (b) X/D = 3.0; (c) X/D = 4.0.

Figure 13. Arrangement of the three cylinders.

Figure 14. Lift and drag coefficients at X/D = 3: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 15. Lift and drag coefficients at X/D = 2: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 16. Lift and drag coefficients at X/D = 4: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 17. Temporal variation in the drag and lift coefficient: (a,c,e) Drag coefficient; (b,d,f) Amplitude of the lift coefficient.

Figure 18. Spectrum diagrams of the lift coefficients for the three cylinders at X/D = 3: (a) Y/D = 2.0; (b) Y/D = 3.0; (c) Y/D = 5.0.

Figure 19. Vorticity distribution diagrams for the three cylinders at X/D = 3: (a) Y/D = 2.0; (b) Y/D = 3.0; (c) Y/D = 5.0.

Figure 20. Arrangement of the four cylinders: (a) In-phase; (b) Anti-phase.

Figure 21. Lift and drag coefficients of the four cylinders in phase: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 22. Temporal variation in the drag and lift coefficient: (a,c,e,g) Drag coefficient; (b,d,f,h) Amplitude of the lift coefficient.

Figure 23. Lift and drag coefficients of the four anti-phase cylinders: (a) Mean drag coefficient; (b) Amplitude of the lift coefficient.

Figure 24. Temporal variation in the drag and lift coefficient: (a,c,e,g) Drag coefficient; (b,d,f,h) Amplitude of the lift coefficient.

Figure 25. Spectrum diagrams of the lift coefficients for four cylinders in phase: (a) L/D = 2.0; (b) L/D = 4.0.

Figure 26. Spectrum diagrams of the lift coefficients for four anti-phase cylinders: (a) L/D = 2.0; (b) L/D = 4.0.

Figure 27. Vorticity distribution diagrams for the four cylinders in phase: (a) L/D = 2.0; (b) L/D = 2.5; (c) L/D = 4.0.

Figure 28. Vorticity distribution diagrams for four anti-phase cylinders: (a) L/D = 2.0; (b) L/D = 2.5; (c) L/D = 4.0.

Table 1. Comparison of the simulation results for a single cylinder at Re = 100.

	C_l	C_d	St
Chen [52]	0.192	1.321	-
Zhao [49]	0.310	1.06	0.170
Lin [53]	0.332	1.44	0.171
Ng [54]	0.360	1.368	-
Present work	0.3169	1.4570	0.175

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Shi, F.; Ou, C.; Xin, J.; Li, W.; Jin, Q.; Tian, Y.; Zhang, W. Numerical Study on Hydrodynamic Performance and Vortex Dynamics of Multiple Cylinders Under Forced Vibration at Low Reynolds Number. J. Mar. Sci. Eng. 2025, 13, 214. https://doi.org/10.3390/jmse13020214

AMA Style

Shi F, Ou C, Xin J, Li W, Jin Q, Tian Y, Zhang W. Numerical Study on Hydrodynamic Performance and Vortex Dynamics of Multiple Cylinders Under Forced Vibration at Low Reynolds Number. Journal of Marine Science and Engineering. 2025; 13(2):214. https://doi.org/10.3390/jmse13020214

Chicago/Turabian Style

Shi, Fulong, Chuanzhong Ou, Jianjian Xin, Wenjie Li, Qiu Jin, Yu Tian, and Wen Zhang. 2025. "Numerical Study on Hydrodynamic Performance and Vortex Dynamics of Multiple Cylinders Under Forced Vibration at Low Reynolds Number" Journal of Marine Science and Engineering 13, no. 2: 214. https://doi.org/10.3390/jmse13020214

APA Style

Shi, F., Ou, C., Xin, J., Li, W., Jin, Q., Tian, Y., & Zhang, W. (2025). Numerical Study on Hydrodynamic Performance and Vortex Dynamics of Multiple Cylinders Under Forced Vibration at Low Reynolds Number. Journal of Marine Science and Engineering, 13(2), 214. https://doi.org/10.3390/jmse13020214

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