Spin-Up of a Stratified Fluid in a Cylinder Rotating with a Finite Acceleration

Abstract

A numerical study is made to delineate the time-dependent spin-up flow of a stratified fluid in a cylinder. The time-dependent process of a rotating and stratified fluid, in response to the alteration of the rotation rate of the container, is studied. The change in rotation rate of the cylinder is described as an exponential function, with time constant tc. The classical instantaneous spin-up is modeled as a special case when tc is very small. Numerical solutions were acquired to the governing time-dependent, Boussinesq-fluid, Navier-Stokes equations. Detailed evolution patterns of flow are scrutinized to reveal both the angular velocity and meridional flows, as the value of tc covers a wide range. The numerically-obtained relative sizes of the dynamic effects are compared. The direct effect of fluid stratification on the global adjustment process is depicted. The role of viscous diffusion is discussed especially when the fluid stratification is substantial. In the case of a homogeneous fluid, the principal dynamic effects are confined to the boundary layers. However, for a stratified fluid, the viscous diffusion effect is not insignificant in much of the interior region.

Keywords

• spin-up
• stratified flow
• viscous-diffusion
• ekman number
• ekman layer