On the forced oscillations in a precessing open cylindrical channel

H E Alshoufi

doi:10.1063/5.0046782

Abstract

In this paper, theoretical and experimental results of the forced oscillations in an open precessed cylindrical channel are reported. The theoretical part treats the problem using a linear inviscid irrotational approximation, where different inertial modes, such as resonance ones, are presented. A real-channel flume model was constructed for the experimental part, where three different control parameters were considered: the nutation angle, the rotation rate, and the average water depth. The experiments focused on tracking the different responses toward the provided forces to the system with comparison with the assumed theory where this was possible, as other nonlinear aspects appeared. The experimental observations were tracked using a charge-coupled device sensor type camera, which enabled the extraction of both quantitative and qualitative results. The measurement of the azimuthal velocity involved the use of an acoustic Doppler velocimeter, an approach that closely aligned with the theoretical linear model; this measured velocity was also compared with the three different control parameters. The system shows instabilities in the form of resonance collapse and triadic resonance; in addition, an experimental diagram involving variation in both Reynolds and Rossby numbers is provided.

Highlights

The effect of rotation on wave motion, in general, is an interesting topic that has been discussed intensively in previous research work; what if rotation and tilt is added to the problem? Such conditions of rotation and tilt bring about precession conditions, an interesting topic that has many applications in many domains, for instance, in geophysics such as the liquid in the Earth’s core,1–4 or in aeronautics such as the instability of payload liquids in spacecrafts,5,6 or in engineering such as chemical reactors or combustion chambers
Carrying out the experiments in the lab, changing the control parameters such as the water depth and the rotation rate leads to the results in Table III, where the condition Ω ≤ 2Ωnm is valid for all cases, as the disturbances may propagate through the fluid without losing intensity by means of inertial waves, which have frequencies in the range (0, 2Ωnm), the computations for the relative errors for the case of computing the axial wave number show relatively good approximation, just the cases where resonance exists; the cases of single wave mode appear in the channel where the theory fails totally in this situation, and the nonlinear effect should be included
The theoretical part focused on linearization of Euler equations under precession conditions, which leads to a forced Bernoulli equation with the forcing term being the gravity force radial projection

Summary

Introduction

The effect of rotation on wave motion, in general, is an interesting topic that has been discussed intensively in previous research work; what if rotation and tilt is added to the problem? Such conditions of rotation and tilt bring about precession conditions, an interesting topic that has many applications in many domains, for instance, in geophysics such as the liquid in the Earth’s core, or in aeronautics such as the instability of payload liquids in spacecrafts, or in engineering such as chemical reactors or combustion chambers.7,8The oscillations that appear in such a system are called inertial waves, which are the natural consequences of systems under rotation due to the restorative nature of detuning the Coriolis effect, as long as the forcing frequency is less than twice the base rotational frequency 2Ω or, as Fultz called it, the inertia circle frequency. The case of sphere for instance is old enough to Poincaré (1910) celebrated work of uniform vorticity flow, this was derived a century ago as an inviscid and steady state in a precessing spheroid from the velocity fields that have a linear dependence of the spatial variables in the Cartesian coordinate system.. The case of sphere for instance is old enough to Poincaré (1910) celebrated work of uniform vorticity flow, this was derived a century ago as an inviscid and steady state in a precessing spheroid from the velocity fields that have a linear dependence of the spatial variables in the Cartesian coordinate system.18 This linear mathematical treatment for those waves showed strong correspondence with the real observations. A recent study by Meunier et al. showed that resonance occurs when the extracted Kelvin modes have the wavelength

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