Numerical simulation of multiple steady and unsteady flow modes in a medium-gap spherical Couette flow

We study the multiple steady and unsteady flow modes in a medium-gap spherical Couette flow (SCF) by solving the three-dimensional incompressible Navier-Stokes equations. We have used an artificial compressibility method with an implicit line Gauss-Seidel scheme. The simulations are performed in SCF with only the inner sphere rotating. A medium-gap clearance ratio, sigma=(R-2 - R-1)/R-1=0.25, has been used to investigate various flow states in a range of Reynolds numbers, Re is an element of [400,6500]. First, we compute the 0-vortex basic flow directly from the Stokes flow as an initial condition. This flow exists up to Re=4900 after which it evolves into spiral 0-vortex flows with wavenumber s(p)=3,4 in the range Re[4900,6000], and then the flows become turbulent when Re>6000. Second, we obtain the steady 1-vortex flow by using the 1-vortex flow at Re=700 for sigma=0.18 as the initial conditions and found that it exists for Re[480,4300]. The 1-vortex flow becomes wavy 1-vortex in the range Re[4400,5000]. Further increasing the Reynolds number, we obtain new spiral waves of wavenumber sp=3 for Re[5000,6000]. The flow becomes turbulent when Re>6000. Third, we obtain the steady 2-vortex flow by using the 2-vortex flow at Re=900 for sigma=0.18 as the initial conditions and found that it exists for Re[700,1900]. With increasing Reynolds number the 2-vortex flow becomes partially wavy 2-vortex in the small range Re[1900,2100]. We obtain distorted spiral wavy 2-vortex in the range Re[4000,5000]. when Re>6000 the flow evolves into spiral 0-vortex flow and becomes turbulent. The present flow scenarios with increasing Re agree well with the experimental results and further we obtain new flow states for the 1-vortex and 2-vortex flows.