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

Abstract

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, $$\sigma =\left( R_{2}-R_{1}\right) /R_{1}=0.25,$$ has been used to investigate various flow states in a range of Reynolds numbers, $${Re}\in [400,6500]$$ . 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 $${Re} \in [4900,6000]$$ , 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 $${Re} \in [480,4300]$$ . The 1-vortex flow becomes wavy 1-vortex in the range $${Re} \in [4400,5000]$$ . Further increasing the Reynolds number, we obtain new spiral waves of wavenumber $$s_p=3$$ for $${Re}\in [5000, 6000]$$ . 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 $${Re} \in [700,1900]$$ . With increasing Reynolds number the 2-vortex flow becomes partially wavy 2-vortex in the small range $${Re} \in [1900,2100]$$ . We obtain distorted spiral wavy 2-vortex in the range $${Re} \in [4000,5000]$$ . 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.