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

• Suhail Abbas
• Li Yuan
• Abdullah Shah
Technical Paper
• 34 Downloads

## 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.

## Keywords

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

## List of symbols

J

Determinant of coordinate transformation 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

$$r, \theta , \phi$$

Spherical coordinates

l

Gauss–Seidel sweeps

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

Reynolds number

$${{Re}}_\mathrm{c}$$

Critical Reynolds number

t

Physical time

UVW

Contra-variant velocity components

$$\beta$$

Artificial compressibility factor

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

Clearance ratio

$$\nu$$

Kinematic viscosity

$$\tau$$

Pseudo-time

$$\omega _\phi$$

Azimuthal vorticity component

$$\Omega$$

Angular velocity

$$s_{p}$$

Spiral waves

## Notes

### Acknowledgements

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