Schematic drawing of the flow geometry, consisting of two parallel, infinite disks. The lower disk rotates with an angular velocity Ω, while the upper disk is stationary. The distance between the disks is h.
A random disturbance δ i is added to the azimuthal velocity profile G(z). In this example figure, the order of the disturbance is exaggerated. The vector δ i normally has n z random numbers between −0.005 and 0.005. However, for visualization reasons, in this figure δ i ranges from −0.05 to 0.05.
The Lyapunov coefficient λ is determined via a log-linear fit of Δ(t) for all stable solution branches at Re h = 500, 750, and 1000.
A fluid which is initially at rest (i.e., s = 0) rapidly develops a flow structure described by Stewartson-type of flow. This intermediate flow field, however, is unstable and evolves into a Batchelor-type of flow. The transition from a Batchelor-type to a Stewartson-type flow occurs around t = 150. The azimuthal and radial components of the velocity vector during this transition are given in panels (a) and (b), respectively. (Re h = 1000, t f = 505, δt = t f /200 = 25.3).
A fluid which is initially in mild counter-rotation (s = −0.1) with respect to the bottom disk develops a flow structure which retains most of the negative azimuthal velocity, but is very similar to a Batchelor-type of flow. The flow field converges via a damped oscillation towards the steady situation. The oscillatory nature of the flow makes a graphical representation of the temporal evolution of the flow field less clear. For clarity reasons, the steady velocity profiles have been given as well. In panels (a) and (b), the temporal evolution of the azimuthal and radial velocity profiles is depicted, respectively. The ultimate, steady azimuthal, and radial velocity profiles are shown in panels (c) and (d), respectively. (Re h = 1000, t f = 250, δt = t f /200 = 12.6).
A fluid which is initially in counter-rotation (s = −1) with respect to the bottom disk develops a flow structure which retains most of the negative azimuthal velocity. Panels (a) and (b) show the azimuthal and radial velocity components, respectively. (Re h = 1000, t f = 251.3, δt = t f /200 = 1.26).
The Lyapunov coefficients for the three stable solution branches. Branch 4 is more stable than the Batchelor branch, while Branch 3 is the least stable of these solutions.
(a) Branch 3 hosts an oscillation in the flow field which amplitude exponentially decays in line with the decay of the magnitude of the disturbance propagation itself. (b) The frequency of oscillation f as a function of the Reynolds number Re h is determined via a Discrete Fourier transform.
During the transition from Stewartson to Batchelor-flow, the (a) azimuthal velocity increases from near-zero to 0.313 Ωr, in line with earlier findings by Dijkstra and Van Heijst (1983), 9 whereas the (b) radial velocity profile develops a Bödewadt boundary layer.
(a) The dimensionless magnitude of the different transport terms for the transport of angular momentum are plotted versus time during the transition from a fluid at rest, via a Stewartson-type of flow, to a Batchelor-type of flow. In panel (b), only the period of time is considered in which a fluid at rest develops into an unstable intermediate type of flow.
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