Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments

Abstract

The stability of flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.

Numerical studies of viscous incompressible flow between two rotating concentric spheres

“…Contour plots of the stream function for Re = 5000 are given in Figure 4.4. Our results are consistent with the flow visualization patterns obtained from experiments by Munson and Menguturk [9] as well as those of Wimmer [16].…”

Instabilities in magnetized spherical Couette flow

“…For higher Re, there is a transition to an m = 2 mode. It is numerically expensive to conduct simulations at higher Reynolds numbers, but it is expected that successive bifurcations involving higher azimuthal wavenumbers eventually yield a turbulent state [28]. It has been recently shown that these non-axisymmetric instabilities can trigger dynamo action, but only when P m > 1 [29], which will not be considered here.…”

The spherical Taylor–Couette flow

“…Wimmer [12] showed that the flow modes could be produced by different acceleration histories of the inner sphere. Another study on torque measurements as a function of flow regimes was done by Menguturk and Munson [3]. They found a good agreement between the experiment and perturbation theory for narrow gap values.…”

“…Kinematic viscosity (m 2 s −1 ) θ i Angle measured from the sphere axis (°) S Wall velocity gradient (s −1 ) s ′ Fluctuation intensity of S (s −1 ) Tc 1 Critical value of start-up of Taylor vortices Tc 2 Critical value of spiral mode Tc 3 Onset of spiral mode & wavy mode Tc 4 Critical value of spiral wavy mode Tc 5 Critical value of azimuthal waves Tc 6 The near-turbulence regime Tc 7 Onset of chaos Tc 8 Developed turbulence…”