Vortex evolution in a rotating tank with an off-axis drain

Significance 

Concentrated vortices are prevalent in both natural and artificial fluid motions, and have been observed on widely differing scales. To date, several concentrated vortices, such as Batchelor and Burgers vortices, have been extensively studied, which are among the several exact solutions known. In many cases, these vortices arise in rotating systems with dominant Coriolis forces, and can have complicated core structures. For a better understanding of these vortices, recent research has focused on studying the vortex that forms above a constant-discharge drain hole in a rotating tank.

For axisymmetric cases, where the location of the drain is on the rotation axis at the tank center, the concentrated cyclonic vortex forms above the drain, where it remains positioned. Provided the flow-rate-dependent Rossby number remains sufficiently small, a steady state is established in the spin-up time. The flow is driven by fluid entering the tank from its outer rim, and then flowing radially inward in the lid and base Ekman layers, causing a vortex to form in the interior of the fluid. When the fluid moving inward arrives at the center of the tank, it erupts out of the Ekman layers and moves further inward into a narrow, cylindrical vortex core, then flowing downward into the drain. These studies notwithstanding, axial symmetry is rare in any real-world setting.

Instead, Dr. Richard Munro from the University of Nottingham and Dr. Michael Foster from Ohio State University conducted several experiments to study the temporal evolution of the vortex that arises with an off-axis drain in the base of a rotating, cylindrical tank. The details of the vortex motion, and its relation to an inviscid ‘image vortex’ model were examined and reported to provide more insights into the vortex evolution. The research work has been recently published in the Journal of Fluid Mechanics.

The authors observed that when the fluid entering through the periphery of a steadily rotating cylindrical tank exits through an off-axis drain hole, located at the half radius, in the base of the tank, a concentrated cyclonic vortex initially forms above the drain hole. However, shortly after forming, the vortex was found to migrate off the drain and move around the tank in what begins as a cyclonic circular path. Although the inviscid vortex dynamics predicts continued motion around this circular path, the experimental results show instead that the vortex gradually slows and moves slightly inwards from the circular path, ultimately coming to a rest with a steady structure.

The final steady-state vortex was found to have a hollow core, with vorticity concentrated in a thin shear layer surrounding the stagnant vortex center, whose central axis was found to be located about 50° from the drain. The broadening vortex structure and eventual formation of the steady-state are likely due to the effects of the growing boundary layer on the outer vertical wall. The final position of the vortex is not sensitive to the Rossby number if it is sufficiently small.

The study by Richard Munro and Michael Foster is the first to report the details of a drain-hole-vortex flow under asymmetrical conditions. Their observations are unexpected, even remarkable, as illustrated by the dye-visualization images shown in Fig. 1, which show fluorescent dye, introduced at the perimeter of the tank base, used to mark the flow. Initially, the dye spirals inwards through the base Ekman layer and flows out through the drain (Fig. 1a). As the vortex subsequently moves off the drain, the downward flow through the vortex core is redirected horizontally by the tank’s base, resulting in a local flow around the vortex core and vertically upwards, which is marked by the dye (Fig. 1b). This feature of the flow is not well understood and is the subject of future work.

Vortex evolution in a rotating tank with an off-axis drain - Advances in Engineering
Figure 1: Images showing fluorescent dye introduced at the perimeter of the tank base to mark the flow. (a) Initially, the dye marks the inward radial flow through the base Ekman layer. (b) As the vortex moves off the drain, the dye marks the local flow that forms around the vortex and vertically upwards.

Reference

Munro, R., & Foster, M. (2021). Vortex evolution in a rotating tank with an off-axis drainJournal Of Fluid Mechanics, 933(R2), R2-12.

Go To Journal of Fluid Mechanics

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