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=R2-R1/R1=0.25,\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =\left( R_{2}-R_{1}\right) /R_{1}=0.25,$$\end{document} has been used to investigate various flow states in a range of Reynolds numbers, Re∈[400,6500]\documentclass[12pt]{minimal}
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\begin{document}$${Re}\in [400,6500]$$\end{document}. First, we compute the 0-vortex basic flow directly from the Stokes flow as an initial condition. This flow exists up to Re=4900\documentclass[12pt]{minimal}
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\begin{document}$${Re}=4900$$\end{document} after which it evolves into spiral 0-vortex flows with wavenumber sp=3,4\documentclass[12pt]{minimal}
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\begin{document}$$s_p=3,4$$\end{document} in the range Re∈[4900,6000]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [4900,6000]$$\end{document}, and then the flows become turbulent when Re>6000\documentclass[12pt]{minimal}
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\begin{document}$${Re}>6000$$\end{document}. Second, we obtain the steady 1-vortex flow by using the 1-vortex flow at Re=700\documentclass[12pt]{minimal}
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\begin{document}$${Re} =700$$\end{document} for σ=0.18\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =0.18$$\end{document} as the initial conditions and found that it exists for Re∈[480,4300]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [480,4300]$$\end{document}. The 1-vortex flow becomes wavy 1-vortex in the range Re∈[4400,5000]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [4400,5000]$$\end{document}. Further increasing the Reynolds number, we obtain new spiral waves of wavenumber sp=3\documentclass[12pt]{minimal}
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\begin{document}$$s_p=3$$\end{document} for Re∈[5000,6000]\documentclass[12pt]{minimal}
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\begin{document}$${Re}\in [5000, 6000]$$\end{document}. The flow becomes turbulent when Re>6000\documentclass[12pt]{minimal}
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\begin{document}$${Re}>6000$$\end{document}. Third, we obtain the steady 2-vortex flow by using the 2-vortex flow at Re=900\documentclass[12pt]{minimal}
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\begin{document}$${Re} =900$$\end{document} for σ=0.18\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =0.18$$\end{document} as the initial conditions and found that it exists for Re∈[700,1900]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [700,1900]$$\end{document}. With increasing Reynolds number the 2-vortex flow becomes partially wavy 2-vortex in the small range Re∈[1900,2100]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [1900,2100]$$\end{document}. We obtain distorted spiral wavy 2-vortex in the range Re∈[4000,5000]\documentclass[12pt]{minimal}
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\begin{document}$${Re} \in [4000,5000]$$\end{document}. when Re>6000\documentclass[12pt]{minimal}
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\begin{document}$${Re}>6000$$\end{document} 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.
