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

  • Suhail AbbasEmail author
  • Li Yuan
  • Abdullah Shah
Technical Paper


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.


Incompressible Navier–Stokes equation WENO scheme Line Gauss–Seidel scheme Spherical Couette flow Spiral wavy Taylor vortex 

List of symbols


Determinant of coordinate transformation Jacobian




Physical time level


Pseudo-time level


Identity matrix


Radius of inner sphere


Radius of outer sphere

\(r, \theta , \phi\)

Spherical coordinates


Gauss–Seidel sweeps

\({Re}=\Omega R_{1}^{2}/\nu\)

Reynolds number


Critical Reynolds number


Physical time


Contra-variant velocity components


Artificial compressibility factor

\(\sigma = \left( R_{2}-R_{1}\right) /R_{1}\)

Clearance ratio


Kinematic viscosity



\(\omega _\phi\)

Azimuthal vorticity component


Angular velocity


Spiral waves



This work is supported by Natural Science Foundation of China (11321061, 11261160486, and 91641107) and Fundamental Research of Civil Aircraft (MJ-F-2012-04). Suhail Abbas thanks the financial support of CAS-TWAS President’s Fellowship Program during his PhD study in University of Chinese Academy of Sciences, Beijing, China.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKarakorum International UniversityGilgitPakistan
  2. 2.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.Department of MathematicsCOMSATS UniversityIslamabadPakistan

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