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.

It is known that Stokes' paradox exhibits in various flow conditions, most notably, in flow past a two-dimensional (2D) circular cylinder. In this study, we provide an example through detailed analytical solution that Stokes' paradox can be lifted by accelerating flow past a stationary cylinder. The analysis is also extended to the case of the accelerating flow past a stationary sphere although in this case, there is no Stokes' paradox. The effects of the acceleration parameter on the flow streamlines, the pressure, and the vorticity distributions, as well as on the drag coefficient, are investigated. The drag comprises the potential component and vorticity component, which are further due to form drag and frictional drag receiving a separate investigation. However, the drag decomposition is also examined the viewpoint of the force decomposition: the total drag = the potential component + surface vorticity component + volume vorticity component.

It is difficult to find a valid high-order (of Reynolds number) analytical approximation to the uniform viscous flow past an infinite circular cylinder by applying perturbation techniques. In this study, we show that by accelerating flow past a circular cylinder with taking unsteady Stokes' solution as an initial approximation, higher-order approximation solutions to unsteady Oseen's equation can be obtained by an iteration scheme of perturbation, which exactly satisfy the boundary conditions. As a matter of fact, the nonlinear (convective) term linearized by Oseen's approximation is overweighted especially near the body surface. To eliminate the overweight, a multi-region analysis is proposed in this study to improve analytical unsteady Oseen's solution so as to extend the valid range of Reynolds numbers. In other words, the flow region is hypothetically divided into several annular regions, and the linearized convective term in each region is modified by multiplying with a Carrier's coefficient c (0 < c ≤ 1). The results show that, when the accelerating parameter in the range of 0.5 ≤ a ≤ 4, the maximum effective Reynolds number Reeff of the fourth-order five-region solution is both much larger than that of Stokes' and Oseen's ones. For example, when a = 0.5, Reeff = 22.26 is roughly eight times that of Stokes' solution (Reeff = 2.67), and nine times that of Oseen's solution (Reeff = 2.41). Moreover, when the instantaneous Reynolds number Re(t) is smaller than the Reeff, the flow separation angle and the wake length are both consistent with the numerical results obtained by accurately solving the full Navier-Stokes equations. In addition, the flow properties, including the drag coefficients, streamline patterns, and the pressure coefficients, as well as the vorticity distributions also agree well with the numerical results. This study represents a significant extension of our previous study for unsteady Stokes' equation [Xu et al., Phys. Fluids 35, 033608 (2023)] to unsteady Oseen's equation and its generalization.

This work numerically investigates the two-degree-of-freedom flow-induced vibration (FIV) of rigidly coupled square-circular cylinders at Reynolds number Re = 150, mass ratio m* = 2, and reduced velocity Ur = 2–16 under different configurations. The aim of this work is to understand the complex dynamic interactions between fluid and the structure. Force Element Theory (FET) is employed to analyze the underlying mechanisms of FIV, focusing on the influence of geometric parameters and configurations on the vibration characteristics of the coupled system. By systematically varying the spacing ratios and arrangements of square-circular cylinders, the force element contributions under various FIV conditions are quantified to identify optimal configurations. The results indicate that different configurations can alter the phase relationship between force elements and total hydrodynamic forces, thereby changing contributions of force elements. Instead of being placed upstream, the square cylinder placed downstream results in more chaotic vortex shedding patterns. Among the configurations, 4D-US and 6D-DS (where 4D and 6D is the center-to-center distance between the two cylinders, D is the cylinder diameter, US and DS are the different arrangements that square cylinder placed upstream and downstream, respectively) exhibit the highest drag and lift coefficients, respectively. The diagnostic analysis using FET helps to explain the mechanism of how the spacing and arrangement affect the vortex structure of whole flow field and phase of forces, thus determining the fact that the total drag and lift has been revealed. In general, the forces under different configurations can be inferred from the flow field morphology.

This study employs a force element analysis to investigate vortex-induced vibrations (VIV) of three side-by-side circular cylinders at Reynolds number Re = 100, mass ratio m* = 10, spacing ratios S/D = 3–6, and reduced velocities Ur = 2–14. The lift and drag forces are decomposed into three physical components: volume vorticity force, surface vorticity force, and surface acceleration force. The present work systematically examines varying S/D and Ur effects on vibration amplitudes, frequencies, phase relationships, and transitions between distinct vortex-shedding patterns. By quantitative force decomposition, underlying physical mechanisms governing VIV in the triple-cylinder system are elucidated, including vortex dynamics, inter-cylinder interference, and flow structures. Results indicate that when S/D < 4, cylinders exhibit “multi-frequency” vibration responses. When S/D > 4, the “lock-in” region broadens, and the wake structure approaches the patterns of an isolated single cylinder; in addition, the trajectories of cylinders become more regularized. The forces acting on the central cylinder present characteristics of stochastic synchronization, significantly different from those observed in two-cylinder systems. The results can advance the understanding of complex interactions between hydrodynamic and structural dynamic forces under different geometric parameters that govern VIV response characteristics of marine structures.