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Existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow

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

Abstract

We study the existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow by time marching the three-dimensional incompressible Navier–Stokes equations numerically. Three wide-gap clearance ratios, \(\beta =\left( R_{2}-R_{1}\right) /R_{1}=0.33\), 0.38 and 0.42 are investigated for a range of Reynolds numbers respectively. Using the 1-vortex flow for clearance ratio \(\beta =0.18\) at Reynolds number \({Re}=700\) as the initial conditions and suddenly increasing \(\beta\) to the target value, we can compute Taylor vortices for the three wide gaps. For \(\beta =0.33\), Taylor vortices exist in the range \(450\le {Re}\le 2050\). With increasing Re the steady symmetric 1-vortex flow becomes steady asymmetric at \({Re}=1850\), and then become periodic at \({Re}=2000\). When \({Re}>2050\) the flow returns back to the steady basic flow state with no Taylor vortices. For \(\beta =0.38\), Taylor vortices can exist in the range \(500\le {Re}\le 1400\). With increasing Re, the steady symmetric 1-vortex flow become steady asymmetric at \({Re}=1200\), and then the flow evolves into the steady basic flow for \({Re}>1400\). For \(\beta =0.42\), Taylor vortices can exist in the range \(650\le {Re}\le 1300\). With increasing Re, steady asymmetric Taylor vortices occur at \({Re}=1150\), and then the flow evolves into the steady basic flow for \({Re}>1300\). The present numerical results are in good agreement with available numerical and experimental results. Furthermore, the existence regime of Taylor vortices in the \((\beta ,{Re})\) plane for \(\beta \ge 0.33\) and the three-dimensional transition process from periodic asymmetric vortex flow to steady basic flow with increasing Re are presented for the first time.

Keywords

Spherical Couette flow Wide gap Symmetric Taylor vortices Asymmetric Taylor vortices 

List of symbols

J

 Determinant of coordinate transformation Jacobian

p

 Pressure

\(R_{1}\)

 Radius of inner sphere

\(R_{2}\)

 Radius of outer sphere

\(r, \theta , \phi\)

 Spherical coordinates

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

 Reynolds number

\({Re}_{\mathrm{c}}\)

 Critical Reynolds number

t

 Physical time

UVW

 Contra-variant velocity components

\(\alpha\)

 Artificial compressibility factor

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

 Clearance ratio

\(\beta _{\text{W}}\)

 Lower bound value for wide-gap clearance ratio

\(\nu\)

 Kinematic viscosity

\(\tau\)

 Pseudo time

\(\omega _\phi\)

 Azimuthal vorticity component

\(\Omega\)

 Angular velocity

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 support of CAS-TWAS President’s Fellowship Program to finance his PhD in University of Chinese Academy of Sciences, Beijing, China.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.LSEC and Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.School of Mathematical Sciences, University of Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Department of MathematicsCOMSATS Institute of Information TechnologyIslamabadPakistan

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