In this paper, we present a numerical study on the transverse flow-induced vibration (FIV) of an elastically mounted sphere in the vicinity of a free surface at subcritical Reynolds numbers. We assess the interaction dynamics and the vibration characteristics of fully submerged and piercing spheres that are free to vibrate in the transverse direction. We employ the recently developed three-dimensional two-phase flow–structure interaction solver to investigate fully and partially submerged configurations of an elastically mounted sphere. To begin, we examine the vortex-induced vibration (VIV) phenomenon and the vortex-shedding modes of a fully-submerged sphere vibrating freely in all three spatial directions. We systematically verify and analyze the mode transitions and the motion trajectories in the three degrees-of-freedom (3-DOF) for the Reynolds number up to $30000$. We next simulate the transversely vibrating (1-DOF) full-submerged sphere over a wide range of reduced velocities $3\le U^{\ast }\le 20$, whereby the reduced velocity is adjusted by changing the freestream Reynolds number. The VIV response amplitude and the topology of the wake structure are compared with the measurements for the mode I and mode II response branches. We further look into the effect of the free surface on the FIV response of a transversely vibrating sphere in the proximity of a free surface. The response dynamics of the sphere is studied for three representative values of normalized immersion ratio ($h^{\ast }=h∕D$, where $h$ is the distance from the top of the sphere to undisturbed free-surface level and $D$ is the sphere diameter), at $h^{\ast }=1$ (fully submerged sphere with no free-surface effect), $h^{\ast }=0$ (where the top of the sphere touches the free surface) and $h^{\ast }=-0.25$ (where the sphere pierces the free surface). At the lock-in range, we observe that the amplitude response of the sphere at $h^{\ast }=0$ is decreased significantly compared to the case at $h^{\ast }=1$. It is found that the vorticity flux is diffused due to the free-surface boundary and the free surface acts as a sink of energy that leads to a reduction in the transverse force and amplitude response. When the sphere pierces the free surface at $h^{\ast }=-0.25$, the amplitude response at the lock-in state is found to be greater than all the submerged cases studied with the maximum peak-to-peak amplitude of $\sim 2D$. We find that the interaction of the piercing sphere with the air–water interface causes a relatively large surface deformation and has a significant impact on the synchronization of the vortex shedding and the vibration frequency. The streamwise vorticity contours and pressure distribution are employed to understand the VIV characteristics and wake dynamics. Increased streamwise vorticity gives rise to a relatively larger transverse force to the piercing sphere at $h^{\ast }=-0.25$, resulting in greater positive energy transfer per cycle to sustain the large-amplitude vibration. Lastly, we study the sensitivity of large-amplitude vibration on the mass ratio, $m^{\ast }$, and, Froude number, $Fr$, at the lock-in state.

在本文中，我们对在亚临界雷诺数下自由表面附近的弹性安装球体的横向流致振动（FIV）进行了数值研究。我们评估了相互作用动力学和完全沿横向自由振动的完全浸入和刺穿球体的振动特性。我们使用最近开发的三维两相流-结构相互作用求解器来研究弹性安装球体的全部和部分浸没配置。首先，我们研究了在所有三个空间方向上自由振动的完全浸没球体的涡激振动（VIV）现象和涡流脱落模式。$30000$。接下来，我们在减小的速度范围内模拟横向振动（1-DOF）全浸没球$3\le {ü}^{\ast }\le 20$，从而通过更改自由流雷诺数来调整降低的速度。将VIV响应幅度和唤醒结构的拓扑与模式I和模式II响应分支的测量结果进行比较。我们进一步研究了自由表面对自由表面附近横向振动球的FIV响应的影响。研究了球体的响应动力学，以归一化浸入率的三个代表值（$H^{\ast }=H∕d$，在哪里 $H$ 是从球体顶部到不受干扰的自由表面水平的距离，并且 $d$ 是球体直径），在 $H^{\ast }=\mathrm{1个}$ （完全淹没的球体，没有自由表面效果）， $H^{\ast }=0$ （球体顶部接触自由表面的位置）和 $H^{\ast }=-0。25$（球体穿透自由表面的位置）。在锁定范围内，我们观察到球体的振幅响应为$H^{\ast }=0$ 相较于 $H^{\ast }=\mathrm{1个}$。已经发现，涡旋通量由于自由表面边界而扩散，并且自由表面充当能量的吸收器，从而导致横向力和幅度响应的减小。当球体穿透自由表面时$H^{\ast }=-0。25$，发现在锁定状态下的幅度响应要大于研究的所有淹没情况下的最大峰峰值幅度。 $〜2d$。我们发现，穿刺球与空气-水界面的相互作用会引起较大的表面变形，并对涡旋脱落和振动频率的同步产生重大影响。沿流的涡度轮廓和压力分布被用来了解VIV特性和尾流动力学。沿流方向的涡流增加，则在以下位置对穿刺球产生较大的横向力$H^{\ast }=-0。25$，导致每个周期更大的正能量传递，以维持大振幅振动。最后，我们研究了大振幅振动对质量比的敏感性，${米}^{\ast }$和Froude号 $F\mathrm{[R}$，处于锁定状态。