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, σ= (R ₂ - R ₁ ) / R ₁ = 0.25 , has been used to investigate various flow states in a range of Reynolds numbers, Re∈ [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 σ= 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 s _p = 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 σ= 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.