Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:49 https://doi.org/10.1007/s40430-018-1556-z TECHNICAL PAPER Numerical simulation of multiple steady and unsteady fow modes in a medium‑gap spherical Couette fow Suhail Abbas 1,2,3  · Li Yuan 2,3  · Abdullah Shah 4 Received: 5 May 2018 / Accepted: 25 December 2018 © The Brazilian Society of Mechanical Sciences and Engineering 2019 Abstract We study the multiple steady and unsteady fow modes in a medium-gap spherical Couette fow (SCF) by solving the three-dimensional incompressible Navier–Stokes equations. We have used an artifcial 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,  = ( R 2 − R 1 ) ∕R 1 = 0.25, has been used to investigate various fow states in a range of Reynolds numbers, Re ∈[400, 6500] . First, we compute the 0-vortex basic fow directly from the Stokes fow as an initial condition. This fow exists up to Re = 4900 after which it evolves into spiral 0-vortex fows with wavenumber s p = 3, 4 in the range Re ∈[4900, 6000], and then the fows become turbulent when Re > 6000. Second, we obtain the steady 1-vortex fow by using the 1-vortex fow at Re = 700 for  = 0.18 as the initial conditions and found that it exists for Re ∈[480, 4300] . The 1-vortex fow 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 fow becomes turbulent when Re > 6000. Third, we obtain the steady 2-vortex fow by using the 2-vortex fow at Re = 900 for  = 0.18 as the initial conditions and found that it exists for Re ∈[700, 1900] . With increasing Reynolds number the 2-vortex fow 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 fow evolves into spiral 0-vortex fow and becomes turbulent. The present fow scenarios with increasing Re agree well with the experimental results and further we obtain new fow states for the 1-vortex and 2-vortex fows. Keywords Incompressible Navier–Stokes equation · WENO scheme · Line Gauss–Seidel scheme · Spherical Couette fow · Spiral wavy Taylor vortex List of symbols J Determinant of coordinate transfor- mation Jacobian p Pressure n Physical time level m Pseudo-time level I Identity matrix R 1 Radius of inner sphere R 2 Radius of outer sphere r, ,  Spherical coordinates l Gauss–Seidel sweeps Re =ΩR 2 1 ∕ Reynolds number Re c Critical Reynolds number t Physical time U, V, W Contra-variant velocity components  Artifcial compressibility factor  = ( R 2 − R 1 ) ∕R 1 Clearance ratio  Kinematic viscosity  Pseudo-time   Azimuthal vorticity component Ω Angular velocity s p Spiral waves Technical Editor: Jader Barbosa Jr., Ph.D. * Suhail Abbas suhailkiu156@gmail.com 1 Department of Mathematical Sciences, Karakorum International University, Gilgit, Pakistan 2 LSEC, Institute of Computational Mathematics and Scientifc/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 4 Department of Mathematics, COMSATS University, Islamabad, Pakistan