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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
  • 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.

References

  1. 1.
    Harlow F, Welch J (1965) Numerical calculation of time-dependent viscous incompressible flow for fluid with free surface. Phys Fluids 8:21–82MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chorin A (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22:745–762MathSciNetCrossRefGoogle Scholar
  3. 3.
    Patanker S (1980) Numerical heat transfer and fluid flow. Hemisphere, Washington, DCGoogle Scholar
  4. 4.
    Van KJ (1986) A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J Sci Stat Comput 7(3):870–891MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kim J, Moin P (1985) Application of a fractional time-step method to incompressible Navier–Stokes equations. J Comput Phys 59:308–323MathSciNetCrossRefGoogle Scholar
  6. 6.
    Rosenfeld M, Kwak D, Vinokur M (1991) A fractional step solution method for the unsteady incompressible Navier–Stokes equations in generalized coordinate systems. J Comput Phys 94:102–137CrossRefGoogle Scholar
  7. 7.
    Jordan SA (1996) An efficient fractional-step technique for unsteady incompressible flows using a semi-staggered grid strategy. J Comput Phys 127:218–225CrossRefGoogle Scholar
  8. 8.
    Rogers S, Kwak D (1990) Upwind differencing scheme for the time-accurate incompressible Navier–Stokes equations. AIAA J 28(2):253–262CrossRefGoogle Scholar
  9. 9.
    Rogers S, Kwak D, Kiris C (1991) Steady and unsteady solutions of the incompressible Navier–Stokes equations. AIAA J 29:603–610CrossRefGoogle Scholar
  10. 10.
    Huang L (2000) Numerical solution of the unsteady incompressible Navier–Stokes equations on the curvilinear half-staggered mesh. J Comput Math 18(5):521–530MathSciNetzbMATHGoogle Scholar
  11. 11.
    Briley W, Neerarambam S, Whitfield D (1996) Implicit lower–upper/approximate-factorization schemes for incompressible flows. J Comput Phys 128(1):32–42MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu H, Kawachi K (1998) A numerical study of insect flight. J Comput Phys 146(1):124–156CrossRefGoogle Scholar
  13. 13.
    Hartwich P, Hsu C (1987) High-resolution upwind schemes for the three-dimensional incompressible Navier–Stokes equations. AIAA paper 87–0547. AIAA Press, Washington, DCGoogle Scholar
  14. 14.
    Shah A, Yuan L (2011) Numerical solution of a phase-field model for incompressible two-phase flows based on artificial compressibility. Comput Fluids 42:54–61MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yang JY, Yang SC, Chen YN, Hsu CA (1998) Implicit weighted ENO schemes for three-dimensional incompressible Navier–Stokes equations. J Comput Phys 146:464–487CrossRefGoogle Scholar
  16. 16.
    Bernardo C, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J Comput Phys 141:199–224MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kwak D, Chang J, Shanks S, Chakravarthy S (1986) A three-dimensional incompressible Navier–Stokes flow solver using primitive variables. AIAA J 24(3):390–396CrossRefGoogle Scholar
  18. 18.
    Rogers S, Chang J, Kwak D (1987) A diagonal algorithm for the method of pseudocompressibility. J Comput Phys 73(2):364–379CrossRefGoogle Scholar
  19. 19.
    Yoon S, Kwak D (1991) Three-dimensional incompressible Navier–Stokes solver using lower–upper symmetric-Gauss–Seidel algorithm. AIAA J 29(6):874–875CrossRefGoogle Scholar
  20. 20.
    Rogers S (1995) Comparison of implicit schemes for the incompressible Navier–Stokes equations. AIAA J 33(11):2066–2072CrossRefGoogle Scholar
  21. 21.
    Yuan L (2002) Comparison of implicit multigrid schemes for three-dimensional incompressible flows. Comput Phys 77:134–155MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nakabayashi K, Tsuchida Y, Zheng Z (2002) Characteristics of disturbances in the laminar-turbulent transition of spherical Couette flow, 1. Spiral Taylor–Görtler vortices and traveling waves for narrow gaps. Phys Fluids 14(11):3963–3972MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yavorskaya I, Belyaev Y, Monakhov A, Astaf N, Scherbakov S, Vvedenskaya N (1980) Stability, nonuniqueness and transition to turbulence in the flow between two rotating spheres. Report No. 595, Space Research Institute of the Academy of Science, USSRGoogle Scholar
  24. 24.
    Marcus P, Tuckerman L (1986) Simulation of flow between two concentric rotating spheres Part 1: Steady states. Fluid Mech 185:1–30 (Simulations of flow between two concentric rotating spheres. Part 2: Transitions. ibid. 185:31–65)CrossRefGoogle Scholar
  25. 25.
    Yuan L, Fu DX, Ma YW (1996) Numerical study of bifurcation solutions of spherical Taylor–Couette flow. Sci China Ser A 39(2):187–196zbMATHGoogle Scholar
  26. 26.
    Yuan L (2004) Numerical study of multiple periodic flow states in spherical Taylor-Couette flow. Sci China Ser A 47:81–91MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yuan L (2012) Numerical investigation of wavy and spiral Taylor–Gortler vortices in medium spherical gaps. Phys Fluids 24:104–124CrossRefGoogle Scholar
  28. 28.
    Junk M, Egbers C (2000) Isothermal spherical Couette flow. In: Egbers C, Pfister G (eds) Physics of rotating fluids. Lecture notes in physics, vol 549. Springer, Berlin, pp 215–235CrossRefGoogle Scholar
  29. 29.
    Sawatzki O, Zierep J (1970) Das Stromfeld im Spalt zwichen zwei konzentrischen Kulgelflachen, von denen die innere rotiert. Acta Mech 9:13–35CrossRefGoogle Scholar
  30. 30.
    Munson B, Menguturk M (1975) Viscous incompressible flow between concentric rotating spheres. Part 3. Linear Stab J Fluid Mech 69:281–318zbMATHGoogle Scholar
  31. 31.
    Wimmer M (1976) Experiments on a viscous fluid flow between concentric rotating spheres. J Fluid Mech 78:317–335CrossRefGoogle Scholar
  32. 32.
    Bartels F (1982) Taylor vortices between two-concentric rotating spheres. J Fluid Mech 119:1–65CrossRefGoogle Scholar
  33. 33.
    Nakabayashi K (1983) Transition of Taylor–Gortler vortex flow in spherical Couette flow. J Fluid Mech 132:209–230CrossRefGoogle Scholar
  34. 34.
    Schrauf G (1986) The first instability in spherical Couette flow. J Fluid Mech 166:287–303CrossRefGoogle Scholar
  35. 35.
    Nakabayashi K, Tsuchida Y (1995) Flow-history effect on higher modes in the spherical Coutte flow. J Fluid Mech 295:43–60CrossRefGoogle Scholar
  36. 36.
    Shah A, Yuan L, Islam S (2012) Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes. Comput Math Appl 63:1548–1556MathSciNetCrossRefGoogle Scholar
  37. 37.
    Suhail A, Yuan L, Shah A (2018) Simulation of spiral instabilities in wide-gap spherical Couette flow. Fluid Dyn Res 50:025507MathSciNetCrossRefGoogle Scholar
  38. 38.
    Suhail A, Yuan L, Shah A (2018) Existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow. J Braz Soc Mech Sci Eng. 40:156CrossRefGoogle Scholar
  39. 39.
    Liu M, Blohm C, Egbers C, Wulf P, Rath HJ (1996) Taylor vortices in wide spherical shells. Phys Rev Lett 77:286–289CrossRefGoogle Scholar
  40. 40.
    Hollerbach R (1998) Time-dependent Taylor vortices in wide-gap spherical Couette flow. Phys Rev Lett 81:3132–3135CrossRefGoogle Scholar
  41. 41.
    Hollerbach R, Junk M, Egbers C (2006) Non-axisymmetric instabilities in basic state spherical Couette flow. Fluid Dyn Res 38:257–273MathSciNetCrossRefGoogle Scholar

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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