  In this study, a series of simulations are conducted to investigate the
motion of a small cylinder in an expansion tube, focusing on two-dimensional
dynamics. These simulations are performed on the FLUENT platform employing the
Overset function. The collision between the cylinder and the tube wall is
modeled as a positive rigid body collision without losing energy. Two key
parameters, the dimensionless gravity (Mg*) and the Reynolds number (Re), were
explored within the ranges of 4.9-79.4 and 1-300, respectively. Two types of
inflow are considered: the invariable inflow or superimposed a sinusoidal
inflow of periodical fluctuation. For invariable inflow, three motion modes
(drainage mode, balance mode and vibration mode) are found in the phase
diagrams of (Mg*, Re). For an invariable inflow, the high-amplitude vibrations
of a cylinder is proved to be widespread in the Reynolds number range from 5 to
40. Additionally, the scenario involving Re =300, Mg* =39.25 with sinusoidal
periodic fluctuation incoming flow demonstrated intense vibrations of the
cylinder. The behavior of the two-dimensional cylinder can be approximated as a
simplified version of a three-dimensional sphere when the spherical motion is
disregarded. Our research may help to understand this ancient flow problem.
