Study on the spin-up of fluid in a rectangular container using Ekman pumping models

Abstract

Spin-up in a rectangular container with a free surface is investigated numerically and experimentally. In the formulation of two-dimensional numerical computation, we use a potential-like function in addition to the stream function to deal with the first-order Ekman pumping model. It is shown that our numerical results are in good agreement with those obtained by the experiment when either the leading-order or first-order pumping model is used. On the other hand, when no pumping effect is considered the numerical results show, except in the initial development, a considerable discrepancy from those of the experiment. Our attention in this study is focused on clarifying the physical mechanism of cyclonic vortex merging. At low Reynolds numbers and/or liquid depths the Ekman pumping damps the vortical flows fast, resulting in non-merging. At moderate Reynolds numbers, it enhances merging because the cyclonic vortices expand due to the Ekman pumping. We discuss the influence of various parameters, including Reynolds number, Rossby number, and dimensionless liquid depth, on the evolution of the vortical flows.