Bifurcation aspect of wide-gap spherical Couette flow emphasizing polygonal coherence and wave numbers observed over transitional Reynolds numbers

TL;DR

This paper numerically explores the bifurcation behavior and wave number competition in wide-gap spherical Couette flow, revealing how different spiral states emerge and coexist over transitional Reynolds numbers, explaining experimental bistability.

Contribution

It provides a detailed numerical analysis of bifurcation phenomena and wave number competition in wide-gap spherical Couette flow, highlighting the coexistence and transition of spiral states.

Findings

4-fold spiral state bifurcates at the first transitional Reynolds number

3-fold spiral state bifurcates at a higher Reynolds number

Periodic-like state combines 3- and 4-fold spiral states, resembling a beat phenomenon

Abstract

This study numerically investigates the bifurcation aspect of the wide-gap spherical Couette flow (SCF), with an emphasis on the competition among polygonal coherence with different wave numbers observed over transitional Reynolds numbers. Focusing on a representative case, the half-radius ratio, we confirm that the axisymmetric state becomes unstable over the first transitional Reynolds number at which the 4-fold spiral state bifurcates, using the continuation method based on the Newton-Raphson algorithm. The Galerkin-spectral method was employed to numerically solve the governing equations. It is found that the 3-fold spiral state bifurcates from the axisymmetric state at a slightly higher Reynolds number than the first transitional Reynolds number. The attraction of the 3-fold spiral state expands rapidly with an increase in the Reynolds number, which is determined by verifying the distance of the unstable periodic-like state to both spiral states in the state space. This aspect of the state space explains the experimentally bistable realization of different equilibrium states over the first transitional Reynolds number. This study also found that the periodic-like state is composed of the 3- and 4-fold spiral states, similar to a beat with two different frequencies.