The Onset of Taylor-Like Vortex in the Flow Induced by an Impulsively Started Rotating Cylinder

The onset of instability induced by impulsively started rotating cylinder was first investigated experimentally by Chen and Christensen [1]. The initial laminar flow evolves into a secondary flow pattern which consists of a series of Taylor-like vortices. In this transient boundary-layer system the critical time tc to mark the onset of secondary motion becomes an important question. This problem may be called an extension of Taylor instability. The related instability analysis has been conducted by using amplification theory [2], the frozen-time model [2], and the maximum-Taylor-number criterion [3]. The first model requires the initial conditions and the criterion to define manifest convection. The second model is based on linear theory and yields the critical time as the parameter. These models take advantages of the similarity between Taylor instability and Rayleigh-Bénard instability. Here we will extend propagation theory [4], which has been employed to analyze time-dependent Rayleigh-Bénard problem, into the instability of flow induced by an impulsively started rotating cylinder. The system considered here is a Newtonian fluid confined between the two concentric cylinders of radii Ri and Ro(> Ri). Let the axis of inner cylinder be along the z′ axis of a cylindrical coordinate system (r′, θ, z′). At the time t = 0, the inner cylinder is impulsively started and maintained at a constant surface speed V ′ 0(= RiΩi) and outer cylinder is kept stationary Ωo = 0. Here Ωi and Ωo are the angular velocities of inner and outer cylinder, respectively. The schematic diagram of the basic system is shown in Figure 1. For a high V ′ 0 , secondary motion will set at a certain time before the flow becomes fully developed. The governing equations of the present flow field is expressed by ∇ ·U = 0, (1) { ∂ ∂t + U · ∇ } U = − ρ ∇P + ν∇2U (2)