^{1,a)}

### Abstract

The spherical Couette flow between two concentric spheres with only the inner sphere rotating is simulated by solving the 3D incompressible Navier-Stokes equations with a fifth order upwind compact finite difference method. Two moderate clearance ratios, β = (*R* _{2} − *R* _{1})/*R* _{1} = 0.14 and 0.18, respectively, are chosen for comparison with previous experimental and numerical results. First, the spiral Taylor-Görtler (TG) vortexflow at Re = 1110 for β = 0.14 [W. M. Sha and K. Nakabayashi, “On the structure and formation of spiral Taylor-Görtler vortices in spherical Couette flow,” J. Fluid Mech.431, 323–345 (2001)10.1017/S0022112000003128] is found to develop traveling waves at Re = 1800. A wavy TG vortexflow formed at low Re numbers can return to steady TG vortex as Re number is increased to a critical value Re = 6600, thus confirming the occurrence of a reverse Hopf bifurcation from limit cycle to fixed point [K. Nakabayashi, W. M. Sha, and Y. Tsuchida, “Relaminarization phenomena and external-disturbance effects in spherical Couette flow,” J. Fluid Mech. **534**, 327–350 (2005)]. Second, multiple supercritical flows for β = 0.18 [M. Wimmer, “Experiments on a viscousfluidflow between concentric rotating spheres,” J. Fluid Mech.78, 317–335 (1976)10.1017/S0022112076002462] are simulated for a wide range of Re numbers from the first instability (Re ≈ 655) up to the proximity of transition to turbulence (Re ≈ 8000). The simulation confirms Wimmer's experimental observation that a periodic 2-vortex flow coexists with the steady 0- and 1-vortex flows in certain low Re range. There is also a reverse Hopf bifurcation for this periodic wavy 2-vortex flow at Re = 2270. As Re number is further increased, the steady 0- and 2-vortex flows begin to form spiral waves in the secondary flow region for Re ⩾ 6500, while the 1-vortex flow has similar spiral disturbances for Re = 8000. Multiple higher modes with different numbers of spiral waves can be generated by using different wavenumbers in the imposed perturbation. Detailed description of these multiple higher modes is given in terms of rotational frequency, wavenumber, and spatial structure.

This work was supported by state key program for developing basic sciences (Grant No. 2010CB731505) and National Natural Science Foundation of China (Grant Nos. 10972230 and 11021101).

I. INTRODUCTION

II. NUMERICAL METHOD AND COMPUTATIONAL SETUP

A. Numerical method

B. Computational setup

III. RESULTS

A. β = 0.14

B. Medium gap β = 0.18

IV. CONCLUSIONS

### Key Topics

- Rotating flows
- 95.0
- Bifurcations
- 22.0
- Flow instabilities
- 21.0
- Stokes flows
- 20.0
- Turbulent flows
- 18.0

## Figures

(ϕ, θ)-plane distributions of the azimuthal vorticity component at four different times in the formation process of the spiral TG vortex flow IITS(*T* = 1, *S* _{ P } = 3) for β = 0.14, Re = 1110. The quantity is integrated along the radial direction over the gap. The contour levels range from −0.12 to 0.12 in (a) and (b), and from −0.14 to 0.14 in (c) and (d), in step of 0.02. Solid lines show positive values while dashed lines show negative values.

(ϕ, θ)-plane distributions of the azimuthal vorticity component at four different times in the formation process of the spiral TG vortex flow IITS(*T* = 1, *S* _{ P } = 3) for β = 0.14, Re = 1110. The quantity is integrated along the radial direction over the gap. The contour levels range from −0.12 to 0.12 in (a) and (b), and from −0.14 to 0.14 in (c) and (d), in step of 0.02. Solid lines show positive values while dashed lines show negative values.

(ϕ, θ)-plane distributions of the azimuthal vorticity component as in Figure 3, plus velocity vectors in one meridional plane (ϕ = 360°) for the wavy spiral TG vortex flow IIWTS(*T* = 1, *S* _{ P } = 3, *m* = 6) for Re = 1800, β = 0.14. The contour levels range from −0.28 to 0.28 in step of 0.04.

(ϕ, θ)-plane distributions of the azimuthal vorticity component as in Figure 3, plus velocity vectors in one meridional plane (ϕ = 360°) for the wavy spiral TG vortex flow IIWTS(*T* = 1, *S* _{ P } = 3, *m* = 6) for Re = 1800, β = 0.14. The contour levels range from −0.28 to 0.28 in step of 0.04.

Time history of the circumferential velocity component at a point (*r*, θ, ϕ) = (1 + 0.5β, 0.5π, 0). The fundamental period of the spiral TG vortex (Re = 1110) is simply counted between every solid peak, and its rotational period is simply *T* _{ rot, s } = 3*T* _{ s } due to *S* _{ P } = 3. The fundamental periods *T* _{ w } and *T* _{ s } of the wavy spiral TG vortex flow (Re = 1800) are indicated in the graph, and the rotational period *T* _{ rot, w } of the flow is counted between every six peak intervals due to *m* = 6. Clearance ratio β = 0.14.

Time history of the circumferential velocity component at a point (*r*, θ, ϕ) = (1 + 0.5β, 0.5π, 0). The fundamental period of the spiral TG vortex (Re = 1110) is simply counted between every solid peak, and its rotational period is simply *T* _{ rot, s } = 3*T* _{ s } due to *S* _{ P } = 3. The fundamental periods *T* _{ w } and *T* _{ s } of the wavy spiral TG vortex flow (Re = 1800) are indicated in the graph, and the rotational period *T* _{ rot, w } of the flow is counted between every six peak intervals due to *m* = 6. Clearance ratio β = 0.14.

Variations of the wavy TG vortex flow IITW(*T* = 2, *m* = 5) with increasing Re number. For each frame, the left graph is instantaneous iso-values of the azimuthal angular velocity quantity (ω = *v* _{ϕ}/*r*sin θ) in the unwrapped middle spherical surface *r* = (1 + β)/2, 0 ⩽ θ ⩽ π, 0 ⩽ ϕ ⩽ 2π, the middle graph is that in the meridional plane at ϕ = 2π, and the right graph is velocity vectors (*v* _{ r }, *v* _{θ}) in the same meridional plane. The clearance ratio β = 0.14.

Variations of the wavy TG vortex flow IITW(*T* = 2, *m* = 5) with increasing Re number. For each frame, the left graph is instantaneous iso-values of the azimuthal angular velocity quantity (ω = *v* _{ϕ}/*r*sin θ) in the unwrapped middle spherical surface *r* = (1 + β)/2, 0 ⩽ θ ⩽ π, 0 ⩽ ϕ ⩽ 2π, the middle graph is that in the meridional plane at ϕ = 2π, and the right graph is velocity vectors (*v* _{ r }, *v* _{θ}) in the same meridional plane. The clearance ratio β = 0.14.

The same as Figure 4 but for clearance ratio β = 0.18.

The same as Figure 4 but for clearance ratio β = 0.18.

Multiple 0-vortex flows with different numbers of shear waves at Re = 7200 for clearance ratio β = 0.18. For each frame, the left graph is instantaneous iso-values of the circumferential velocity component *v* _{θ} in the unwrapped spherical surface *r* = 1 + 0.7β, 0 ⩽ θ ⩽ π, 0 ⩽ ϕ ⩽ 2π, the middle graph is that in the meridional plane at ϕ = 2π, and the right graph is velocity vectors (*v* _{ r }, *v* _{θ}) in the same meridional plane. (a) 7 shear waves; (b) 8 shear waves; (c) 9 shear waves.

Multiple 0-vortex flows with different numbers of shear waves at Re = 7200 for clearance ratio β = 0.18. For each frame, the left graph is instantaneous iso-values of the circumferential velocity component *v* _{θ} in the unwrapped spherical surface *r* = 1 + 0.7β, 0 ⩽ θ ⩽ π, 0 ⩽ ϕ ⩽ 2π, the middle graph is that in the meridional plane at ϕ = 2π, and the right graph is velocity vectors (*v* _{ r }, *v* _{θ}) in the same meridional plane. (a) 7 shear waves; (b) 8 shear waves; (c) 9 shear waves.

The same as Figure 6 but for multiple 2-vortex flows with different numbers of shear waves. (a) 8 shear waves; (b) 9 shear waves; (c) 10 shear waves.

The same as Figure 6 but for multiple 2-vortex flows with different numbers of shear waves. (a) 8 shear waves; (b) 9 shear waves; (c) 10 shear waves.

The same as Figure 6 but for multiple 1-vortex flows with different numbers of shear waves at Re = 8000. (a) 5 shear waves; (b) 6 shear waves; (c) 6:5 shear waves.

The same as Figure 6 but for multiple 1-vortex flows with different numbers of shear waves at Re = 8000. (a) 5 shear waves; (b) 6 shear waves; (c) 6:5 shear waves.

## Tables

Nondimensional rotational frequencies of TG vortex flows with shear waves for β = 0.18.

Nondimensional rotational frequencies of TG vortex flows with shear waves for β = 0.18.

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