The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

Abstract

AbstractThis study concerns the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter $d$ and thickness $h$) falling in a vertical tube of diameter $D$. We investigated experimentally the influence of the confinement ratio ($S= d/ D\lt 0. 8$) on the motion of the body, for different aspect ratios ($\chi = d/ h= 3$, $6$ and $10$), Reynolds numbers ($80\lt Re\lt 320$) and a density ratio between the fluid and the body close to unity. For a given body, the Reynolds number based on its mean vertical velocity is observed to decrease when $S$ increases. The critical Reynolds number for the onset of the periodic motion decreases with $S$ in the case of thin bodies ($\chi = 10$), whereas it appears unaffected by $S$ for thicker bodies ($\chi = 3$ and $6$). The characteristics of the periodic motion are also strongly modified by the confinement ratio. A thick body ($\chi = 3$) tends to go back to a rectilinear path when $S$ increases, while a thin body ($\chi = 10$) displays oscillations of growing amplitude with $S$ until it touches the tube (at about $S= 0. 5$). For a given aspect ratio, however, the amplitudes of the oscillations follow a unique curve for all $S$, which depends only on the relative distance of the Reynolds number to the threshold of path instability. In parallel, numerical simulations of the wake of a body held fixed in a uniform confined flow were carried out. The simulations allowed us to determine in this configuration the effect of the confinement ratio on the thresholds for wake instability (loss of axial symmetry at $R{e}_{c 1} $ and loss of stationarity at $R{e}_{c 2} $) and on the maximal velocity ${V}_{w} $ in the recirculating region of the stationary axisymmetric wake. The evolution with $\chi $ and $S$ of ${V}_{w} $ at $R{e}_{c 1} $ was used to define a Reynolds number $R{e}^{\ast} $. Remarkably, for a freely moving body, $R{e}^{\ast} $ remains almost constant when $S$ varies, regardless of the nature of the path.