Flow Instability: Transition and Non Linearity

We numerically investigate the transitional and turbulent flow dynamics in spherical Couette flow (SCF) with a narrow gap ratio (β= 0.24), using a spectral solver built on the open-source Dedalus framework. The study systematically explores both co-rotating and counter-rotating regimes over a range of Reynolds numbers (Re_i up to 4000), and four distinct outer-to-inner Reynolds number ratios. The incompressible Navier--Stokes equations are solved in spherical coordinates using a combination of Jacobi and Chebyshev polynomials with implicit-explicit Runge--Kutta time integration.

In the co-rotation regime (Re_o / Re_i > 0), the flow remains steady and axisymmetric across all Re_i for ratios 0.5 and 1.0. For a rotation ratio of 1.5, increasing Re_i leads to the development of equatorial recirculation regions near the outer sphere, which eventually detach and migrate inward. At a ratio of 2.0, this transition is initiated at lower Re_i (from 1600), and fully detached bubbles form by Re_i = 3600. The emergence and migration of these recirculation regions are captured via analysis of the radial velocity component in the meridional plane.

In the counter-rotation regime (Re_o / Re_i < 0), the flow exhibits secondary circulations, vortex pinching, and progressive transition to non-axisymmetric and unsteady states. Spatio-temporal diagnostics, including temporal signals, FFT spectra, and phase portraits, highlight the bifurcation behavior leading to chaotic flow with broadband energy spectra. Modal decompositions using Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) reveal the loss of symmetry and coherent structures, especially in equatorial and polar regions, as rotation ratio increases.

The results offer insights into how rotation ratio governs the onset and evolution of flow instabilities in narrow-gap SCF, relevant to geophysical and astrophysical contexts.