Incompressible, viscous flows in the spherical gap between a rotating inner-sphere and a stationary outer-shell, Spherical Couette Flows (SCF), are studied via direct numerical simulations. The investigation covers both “small-gap” and “large-gap” geometries, and is concerned primarily with the first occurrence of transition in those flows. Strong emphasis is put on the physical understanding of the basic flows and their transition mechanisms.

An alias-free spectral method, based on divergence-free vector expansions for the 3-D velocity field in spherical coordinates, is developed. The vector expansions are constructed with Chebyshev polynomials in the radial direction and Vector Spherical Harmonics for the two angular directions. Accuracy and spectral convergence of the resulting initial-value code are thoroughly tested. Three-dimensional transitional flows in both narrow-gaps and large-gaps as well as axisymmetric transitions in moderate-gaps are simulated.

For small-gap SCF’s, this study shows that the formation of Taylor-vortices at transition is a deterministic process and not the result of the instability of initial perturbations. The formation process involves the sub-critical appearance of a saddle-stagnation point within the meridional circulation cell in each hemisphere. A minimum length-scale ratio is shown necessary, and for a given inner-sphere radius, this leads to a theoretical prediction of the largest gap-width in which Taylor-vortices may form.

This investigation confirms that the first transition in large-gap SCF’s is caused by a 3-D instability of a linear nature. It is found that the process is characterized by very small growth-rates of the disturbance and by the absence of a “jump” in the friction torque. The supercritical flow is a complex-structured, laminar, time-periodic flow that exhibits traveling azimuthal-waves. The physical mechanism responsible for the large-gap transition is shown to be related to a shear instability of the “radial-azimuthal jet” that develops at the equator of the basic flow. A physical model is proposed in which that jet is viewed as a sequence of adjacent “fan-spreading quasi-2-D plane jets”. Predictions from the model are presented and verified from the computed unstable disturbance field. Extension of the model to the transition toward waviness in the Taylor-Couette flow, the Gôrtler-vortex flow and the Dean-vortex flow is proposed.