Laminar Spirals in the Outer Stationary Cylinder Couette-Taylor System

We present numerical simulations to demonstrate the existence of laminar spiral flows between both finite and infinite length concentric cylinders and finite truncated cones where only the inner wall rotates. The velocities and pressure are calculated by a spectral element/Fourier method. Different gap ratios are investigated. Convergence of the numerical results is shown with reference to flows between infinite cylinders. The presence of top and bottom endplates results in vortex dislocations that are observed at the frontiers between the Ekman vortices present at each end and the spiral vortices. Introduction The last century has seen hundreds of works devoted to the flow confined between circular cylinders since the pioneering investigation of Taylor [1], but even now, new flow behaviours are reported both experimentally and theoretically [2-4]. Laminar spirals have been studied extensively and were reported when both cylinders were oppositely rotated [5-10]. With such an experimental system, Andereck et al. [5] obtained a stability diagram where laminar spirals occupied a very narrow region. In the Couette system the basic flow is naturally unstable to nonaxisymmetric spiral vortices near the inner cylinder when the outer cylinder rotates in the opposite direction as reported by Coles [11], Krueger et al. [12] and Snyder [13-14]. Spiral vortices are strongly dependent on initial conditions, as they exist in different flow modes, and may even co-exist with steady Taylor vortices [5]. Laminar spirals have been also observed in the system of conical cylinders. Wimmer [15] found that helical flow structures could be generated between the conical cylinders when only the inner body was rotated. These structures were very sensitive to initial conditions. Noui-Mehidi et al. [16] have reported that the helical structure between conical cylinders could be generated only for very slow acceleration rates of the inner body. Andereck et al. [5], working in a system with counter-rotating finite cylinders, related the presence of laminar spirals to the presence of top and bottom boundaries. However, Antonijoan et al. [4] predicted via numerical simulation that spiral regimes may exist in infinite Taylor-Couette flow. They reported that spiral regimes (which in general are temporally periodic in a fixed frame of reference) arose through a Hopf bifurcation from the circular Couette flow in a system with counter-rotating cylinders. Their results also included solutions with laminar spirals in a system where only the inner cylinder could rotate, but this possibility was not dealt with in detail. The present paper reports on numerical simulations of laminar spirals in circular Couette flow. Motivation for this work initially emanated from an investigation of the helical flow between finite conical cylinders of same apex angle, and where only the inner cylinder rotates. Study of the effect of the conical apex angle on the stability of the helical structure showed that spiral vortices were maintained even at very small apex angle. Then, the numerical projection of the solution obtained between conical cylinders into a cylindrical annular gap showed that spiral vortices were also maintained in a configuration of circular Couette flow where the outer cylinder is stationary, and where endwalls are present. In agreement with the results of Antonijoan et al. [4], spiral vortices are also found to be stable when the cylinders are infinite in length for both counter-rotating cylinders, and when the outer cylinder is fixed. Thus the results show that laminar spirals can exist between both infinite and finite circular cylinders when the outer cylinder is stationary and that these spirals are stable over a range of Reynolds numbers. Numerical method The direct numerical simulation formulation used here has been described in detail by Blackburn & Sherwin [17] and is reported briefly in the following. In the cylindrical geometry the velocity u (z, r, θ) is projected by Fourier transformation in the azimuthal direction onto a set of two-dimensional complex modes ŭk (z, r). This Fourier basis is coupled with a spectral element discretization in the meridional plane. Time integration is carried out with a second-order semi-implicit scheme. The meridional computational domain was discretized into 231 spectral elements with Gauss-Lobatto-Legendre (GLL) nodal basis functions. Through numerical experiments, it was found that 16 planes of data in the azimuthal direction were sufficient to generate a stable and smooth solution. The accuracy of the results is also checked with regards of the number of element np as shown in Table 1. A tensor-product polynomial order of 7 within each spectral element was found sufficient to achieve a good accuracy.