Occurrence of orbital cylinder motion for flow around freely vibrating circular cylinder in uniform stream

In this study flow around a freely vibrating circular cylinder in two-degrees-of-freedom is investigated using two-dimensional numerical approach. The equations of motion, the continuity and the pressure Poisson equations are solved for incompressible constant property Newtonian fluid using the finite difference method. Systematic computations are carried out to investigate the effect of the natural frequency fN of the cylinder on the oscillation amplitudes and aerodynamic force coefficients. The mass ratio and the structural damping coefficient values are fixed at m*=10 and ζ=0, respectively. In addition to the typical distorted figure-eight motion, a raindrop-shaped path is also found whose range widens with increasing fN. Within this range the streamwise oscillation amplitude reaches higher values compared to figure-eight motions and the curves shift upwards with increasing fN. Different vortex structures belong to each cylinder path: 2S or C(2S) modes are observed for distorted figure-eight and P+S mode is found for raindrop-shaped trajectories. The time-mean values of lift jump abruptly between two solutions. Pre- and post-jump analysis reveals that these solutions are mirror images of each other. The root-mean-square (rms) values of lift plotted against U*St shift upwards in the raindrop-shaped motion range and downwards in the figure-eight motion range, while the rms of drag shifts to higher values in both regimes with increasing fN. Chaotic cylinder paths are observed when increasing the natural frequency over a critical value.本研究采用二维数值方法，针对两自由度（two-degrees-of-freedom）自由振动圆柱的绕流问题展开研究。针对不可压缩常物性牛顿流体，采用有限差分法（finite difference method）求解运动方程、连续性方程与压力泊松方程。通过系统性数值计算，探究圆柱固有频率（natural frequency）$f_N$对振动幅值及气动力系数（aerodynamic force coefficients）的影响规律。本研究中质量比（mass ratio）与结构阻尼系数（structural damping coefficient）分别固定为$m^*=10$和$zeta=0$。 除典型的畸变八字形运动外，本研究还发现雨滴状运动轨迹，且该轨迹的覆盖范围随固有频率$f_N$的增大而拓宽。在此范围内，相较于八字形运动，圆柱的流向振动幅值（streamwise oscillation amplitude）更高，相关曲线亦随$f_N$增大而上移。两种运动轨迹对应不同的涡结构（vortex structures）：畸变八字形运动对应2S或C(2S)模态，雨滴状轨迹则对应P+S模态。升力的时均值在两种解之间发生突变，突变前后的分析表明，二者互为镜像。以$U^*St$为横坐标绘制的升力均方根（root-mean-square, rms）值，在雨滴状运动范围内随$f_N$增大而上移，在八字形运动范围内则下移；而阻力的均方根值在两种运动区域内均随$f_N$增大而升高。当固有频率超过临界值时，可观测到混沌圆柱运动轨迹。