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

• Suhail Abbas
• Li Yuan
• Abdullah Shah
Technical Paper

## 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

$$R_{1}$$

$$R_{2}$$

$$r, \theta , \phi$$

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 8:21–82
2. 2.
Chorin A (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22:745–762
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):870–891
5. 5.
Kim J, Moin P (1985) Application of a fractional time-step method to incompressible Navier–Stokes equations. J Comput Phys 59:308–323
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 94:102–137
7. 7.
Jordan SA (1996) An efficient fractional-step technique for unsteady incompressible flows using a semi-staggered grid strategy. J Comput Phys 127:218–225
8. 8.
Rogers S, Kwak D (1990) Upwind differencing scheme for the time-accurate incompressible Navier–Stokes equations. AIAA J 28(2):253–262
9. 9.
Rogers S, Kwak D, Kiris C (1991) Steady and unsteady solutions of the incompressible Navier–Stokes equations. AIAA J 29:603–610
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):521–530
11. 11.
Briley W, Neerarambam S, Whitfield D (1996) Implicit lower–upper/approximate-factorization schemes for incompressible flows. J Comput Phys 128(1):32–42
12. 12.
Liu H, Kawachi K (1998) A numerical study of insect flight. J Comput Phys 146(1):124–156
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 42:54–61
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 146:464–487
16. 16.
Bernardo C, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J Comput Phys 141:199–224
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):390–396
18. 18.
Rogers S, Chang J, Kwak D (1987) A diagonal algorithm for the method of pseudocompressibility. J Comput Phys 73(2):364–379
19. 19.
Yoon S, Kwak D (1991) Three-dimensional incompressible Navier–Stokes solver using lower–upper symmetric-Gauss–Seidel algorithm. AIAA J 29(6):874–875
20. 20.
Rogers S (1995) Comparison of implicit schemes for the incompressible Navier–Stokes equations. AIAA J 33(11):2066–2072
21. 21.
Yuan L (2002) Comparison of implicit multigrid schemes for three-dimensional incompressible flows. Comput Phys 77:134–155
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):3963–3972
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)
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):187–196
26. 26.
Yuan L (2004) Numerical study of multiple periodic flow states in spherical Taylor-Couette flow. Sci China Ser A 47:81–91
27. 27.
Yuan L (2012) Numerical investigation of wavy and spiral Taylor–Gortler vortices in medium spherical gaps. Phys Fluids 24:104–124
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 215–235
29. 29.
Sawatzki O, Zierep J (1970) Das Stromfeld im Spalt zwichen zwei konzentrischen Kulgelflachen, von denen die innere rotiert. Acta Mech 9:13–35
30. 30.
Munson B, Menguturk M (1975) Viscous incompressible flow between concentric rotating spheres. Part 3. Linear Stab J Fluid Mech 69:281–318
31. 31.
Wimmer M (1976) Experiments on a viscous fluid flow between concentric rotating spheres. J Fluid Mech 78:317–335
32. 32.
Bartels F (1982) Taylor vortices between two-concentric rotating spheres. J Fluid Mech 119:1–65
33. 33.
Nakabayashi K (1983) Transition of Taylor–Gortler vortex flow in spherical Couette flow. J Fluid Mech 132:209–230
34. 34.
Schrauf G (1986) The first instability in spherical Couette flow. J Fluid Mech 166:287–303
35. 35.
Nakabayashi K, Tsuchida Y (1995) Flow-history effect on higher modes in the spherical Coutte flow. J Fluid Mech 295:43–60
36. 36.
Shah A, Yuan L, Islam S (2012) Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes. Comput Math Appl 63:1548–1556
37. 37.
Suhail A, Yuan L, Shah A (2018) Simulation of spiral instabilities in wide-gap spherical Couette flow. Fluid Dyn Res 50:025507
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. 40:156
39. 39.
Liu M, Blohm C, Egbers C, Wulf P, Rath HJ (1996) Taylor vortices in wide spherical shells. Phys Rev Lett 77:286–289
40. 40.
Hollerbach R (1998) Time-dependent Taylor vortices in wide-gap spherical Couette flow. Phys Rev Lett 81:3132–3135
41. 41.
Hollerbach R, Junk M, Egbers C (2006) Non-axisymmetric instabilities in basic state spherical Couette flow. Fluid Dyn Res 38:257–273