Three-Dimensional Study of Flow between Concentric Rotating Cylinders

Abstract

The wavy vortex mode of flow between concentric rotating cylinders is studied using a combination of Fourier analysis and finite differencing. Series expansions of the form u(r, z, θ, t) = u0(r, z, t) + u1(r, z, t) sin nθ + u¯1(r, z, t) cos nθ + . . . are introduced into the full Navier-Stokes equations. The resulting time-dependent partial differential equations for u0, u1, u¯1, etc., obtained by separating the various harmonics are solved by finite differencing in the r, z plane. Numerical results were obtained at Reynolds numbers of 750, 1000, and 2000 at a radius ratio (of outer to inner cylinder) of 1.2. The growth rates of small three-dimensional disturbances are obtained for a laminar base flow and a Taylor cell base flow. Azimuthal wave velocities for small disturbance cases and the fully established wavy mode case are obtained and agree with linear stability calculations and experiment. The effect of the three-dimensional motion (at constant axial wavenumber) on the torque is examined and appears to be small. The effect, on torque, of varying axial wavenumber is reconsidered, and it is suggested that this is the predominant mechanism for affecting the torque in the wavy mode regime.