Numerical investigations of flow of a two-layer viscous fluid system in a rotating 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. Comprehensive numerical solutions are secured to the time-dependent Navier-Stokes equations. VOF method suggested by Hirt et al. was utilized after some modifications to prevent smoothing of free boundary (interface) occurs when a conventional scheme is used. 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 $v_1/v_2<1.0$ and $v_1/v_2>1.0$. These show that the global spin-up process is substantially accomplished over ($E_n^{1-/2}\Omega^{-1}$), where E$_n$ denotes the value of the smaller Ekman number of the two layers. Taking advantage of the wealth of numerical data, scrutiny is made of the detailed profiles of azimuthal and meridional velocity component. The computed transient meridional flow fields are supportive of the basic contentions of the modified analytic model. As limit case of free boundary the study on spin-up from rest of a fluid within free surface presented. The free surface sustain parabolic contour during the whole spin-up time span, which is different from two-layer fluid. The present numerical results predict well the flow characteristics. Columnar flow approximation assumed in Homicz and Gerber``s study is satisfied substantial region of fluid except endwall and boundary layer. With a knowledge of detailed flow structure of meridional flow the extension of analytic model is made to large Froude number case. Large Fr number implies large interface deformation for given density ratio. Previous model is restricted to a small value. A weighting factor was introduced to adjust the flux induced by Ekman pumping at the endwall disks when the interface touches top either/or bot...