Dynamics of a Tubular Cantilever Simultaneously Subjected to Internal and Partially-Confined External Axial Flows

In this paper the dynamics of a tubular cantilever, simultaneously subjected to internal and external axial flows, is examined theoretically. The tube is discharging fluid downwards which then flows upwards through an annular region surrounding the tube. Thus, the internal and external flows are interdependent and in opposite directions. Also, the external flow is confined over a certain range of the cantilever length and unconfined over the rest. The Heaviside step function has been used in the literature, for such a system, to model the discontinuity in the external flow velocity occurring when the flow enters the annular region. A more accurate way to model this discontinuity is introduced in this study, in which the logistic function is used instead of the Heaviside step function. The stability of the system is investigated by analysis of the system eigenfrequencies, and the effects of varying the length of the confined region are theoretically studied. The obtained results are compared to theoretical predictions and experimental data from the literature having the same system parameters. The proposed theory captures the same dynamical behaviour as observed experimentally, and has a better estimation for the onset of instability and the frequency of oscillations compared to the theory in the literature.