Stability and visualization mechanism for spherical Couette flow 2-fold spiral state*

Numerous studies have explored the flow of fluid within a sphere. Notably, spherical Couette flow (SCF), generated by the rotation of the inner sphere at a constant angular velocity and the outer sphere at rest within a fluid-filled concentric double spherical gap, has been linked to phenomena observed in celestial bodies and atmospheric dynamics. The primary parameters influencing SCF transitions include the radius ratio of the inner and outer spheres and the Reynolds number, which are known to determine transition route to various flow states, such as laminar basic flow, Taylor vortex flow, and turbulent flow. Among these transitional flows is the m-fold spiral state, characterized by an m-arm-like structure extending from the poles to the equatorial plane. Despite extensive research on these parameters, accurately mapping and reproducing the flow states has been remained a challenging issue due to the dependency of the transition on the hysteresis.

In this study, we focus on the 2-fold spiral state, a rare occurrence in our experiments. By employing direct numerical simulation (DNS), we successfully solved an exact 2-fold spiral state of incompressible Navier-Stokes equation by imposing a symmetry on the flow. Additionally, we performed a bifurcation analysis of the solution and evaluated its stability, that leads to specify the range of Reynolds number where the 2-fold spiral state is stable. Furthermore, we simulated the translational and rotational motion of flakes for flow visualization suspended in the obtained 2-fold spiral state and calculated how the flakes reflected the light from the light source and incident on the camera. We succeeded in obtaining the same light and dark patterns as obtained in the visualization experiments conducted in the laboratory.