Spin-Up From Rest of a Two-Layer Liquid in a Cylinder

Abstract

A numerical and analytical study is made of spin-up from rest of a two-layer liquid in a rapidly rotating cylinder. The overall system Ekman number is small. The density of the top layer is smaller than that of the bottom layer (ρ1 /ρ2 <1.0), but the ratio of the individual layer kinematic viscosities is arbitrary (v1 /v2 <1.0 or v1 /v2 >1.0). The highlights of the analytical model, which is based on amended formulations of the Wedemeyer-Gerber-Homicz flow configurations, are briefly recapitulated. Comprehensive numerical solutions are secured to the time-dependent Navier–Stokes equations. The numerical solutions are validated by comparing the maximum interface displacements with the available experimental data as well as the analytical model predictions. Descriptions are made of the prominent characteristics of the interface shape for the two regimes of v1 /v2 <1.0 and v1 /v2 > 1.0. Details of the azimuthal and meridional flow structures are illustrated by exploiting the numerical solutions. The computed meridional flows are compatible with the basic assumptions embedded in the development of the analytical model. Sequential plots of the radial profiles of azimuthal velocities are presented. These show that the global spin-up process is substantially accomplished over (En −1/2 Ω−1 ), where En denotes the value of the smaller Ekman number of the two layers. The numerical study gives credence to the reliability and accuracy of the simplified analytical model.