Spherical Couette flow experiments were conducted according to the work of Egbers and Rath [Acta Mech. 111, 125–140 (1995)]. While the value of the critical Reynolds number obtained by the previous experiments was in good agreement with the numerical prediction, it has remained a question why a spiral wave bifurcating over the critical Reynolds number can be visualized even by a classical flow visualization technique like the mixing of a small amount of aluminum flakes to the working fluid. In the present study, through visualization using aluminum flakes drifting on a horizontal plane illuminated by a laser sheet, the flow was identified as a spiral wave with azimuthal wavenumber m = 3, using the experimentally obtained and numerically deduced comparison between phase velocities. By solving the equation of motion for the infinitesimal planar particles advecting in the flow field of the spiral wave, a visual distribution of reflected light was virtually reproduced, which is in good agreement with the experimentally obtained picture.
References
1.
C. M. R.
Fowler
, The Solid Earth
(Cambridge University Press
, 2004
).2.
F.
Feudel
, K.
Bergemann
, L.
Tuckerman
, C.
Egbers
, B.
Futterer
, M.
Gellert
, and R.
Hollerbach
, “Convection patterns in a spherical fluid shell
,” Phys. Rev. E
83
, 046304
(2011
).3.
S.
Chandrasekhar
, Hydrodynamic and Hydromagnetic Stability
(Clerendon Press
, Oxford
, 1961
).4.
F. H.
Busse
, “Patterns of convection in spherical shells
,” J. Fluid Mech.
72
(1
), 67
–85
(1975
).5.
A.
Zebib
, G.
Schubert
, J. L.
Dein
, and R. C.
Pariwal
, “Character and stability of axisymmetric thermal convection in spheres and spherical shells
,” Geophys. Astrophys. Fluid Dyn.
23
, 1
–42
(1983
).6.
S.
Kida
, K.
Araki
, and H.
Kitauchi
, “Periodic reversals of magnetic field generated by thermal convection in a rotating spherical shell
,” J. Phys. Soc. Jpn.
66
(7
), 2194
–2201
(1997
).7.
A.
Sakuraba
and M.
Kono
, “Effect of the inner core on the numerical solution of the magnetohydrodynamic dynamo
,” Phys. Earth Planet. Inter.
1111
, 105
–121
(1999
).8.
F.
Garcia
, M.
Seilmayer
, A.
Giesecke
, and F.
Stefani
, “Four-frequency solution in a magnetohydrodynamic Couette flow as a consequence of azimuthal symmetry breaking
,” Phys. Rev. Lett.
125
, 264501
(2020
).9.
F.
Feudel
and U.
Feudel
, “Bifurcations in rotating spherical shell convection under the influence of differential rotation
,” Chaos
31
, 113112
(2021
).10.
X.
Song
and P. G.
Richards
, “Seismological evidence for differential rotation of the Earth's inner core
,” Nature
382
, 221
–224
(1996
).11.
B. R.
Munson
and M.
Menguturk
, “Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments
,” J. Fluid Mech.
69
, 705
–719
(1975
).12.
Y. N.
Belyaev
, A. A.
Monakhov
, S. A.
Scherbakov
, and I. M.
Yavorskaya
, “Some routes to turbulence in spherical Couette flow
,” in Proceedings of the Laminar-Turbulent Transition IUTAM-Symposium
, edited by V. V.
Kozlov
(Springer-Verlag, Berlin Heidelberg
, New-York, Tokyo
, 1984
), pp. 669
–676
.13.
Yu. N.
Belyaev
and I. M.
Yavorskaya
, “Transition to stochasticity of viscous flow between rotating spheres
,” in Nonlinear Dynamics and Turbulence
, edited by G. I.
Barenblatt
, G.
Iooss
, and D. D.
Joseph
(Pitman Adwallced Publislling Program, Boston, 1983) [“Spherical Couette flow: Transitions and onset of chaos,” translated from Izvestiya Akademii Nauk SSSR, Mekh. Zhidk. Gaza 10 (1991)].14.
C.
Egbers
and H. J.
Rath
, “The existence of Taylor vortices and wide-gap instabilities in spherical Couette flow
,” Acta Mech.
111
, 125
–140
(1995
).15.
K.
Nakabayashi
, Y.
Tsuchida
, and Z.
Zheng
, “Characteristics of disturbances in the larminar-turbulent transition of spherical Couette flow. 1. Spiral Taylor–Görtler vortices and traveling waves for narrow gaps
,” Phys. Fluids
14
, 3963
–3972
(2002
).16.
K.
Finke
and A.
Tilgner
, “Simulations of the kinematic dynamo onset of spherical Couette flows with smooth and rough boundaries
,” Phys. Rev. E
86
, 016310
(2012
).17.
J.
Wicht
, “Flow instabilities in the wide-gap spherical Couette system
,” J. Fluid Mech.
738
, 184
–221
(2014
).18.
A.
Barik
, S. A.
Triana
, M.
Hoff
, and J.
Wicht
, “Triadic resonances in the wide-gap spherical Couette system
,” J. Fluid Mech.
843
, 211
–243
(2018
).19.
D. A.
Godfrey
, “A hexagonal feature around Saturn's north pole
,” Icarus
76
, 335
–356
(1988
).20.
G.
Dumas
and A.
Leonard
, “A divergence-free spectral expansions method for three dimensional flows in spherical-gap geometries
,” J. Comput. Phys.
111
, 205
–219
(1994
).21.
K.
Araki
, J.
Mizushima
, and S.
Yanase
, “The nonaxisymmetric instability of the wide-gap spherical Couette flow
,” Phys. Fluids
9
, 1197
–1199
(1997
).22.
R.
Hollerbach
, M.
Junk
, and C.
Egbers
, “Non-axisymmetric instabilities in basic state spherical Couette flow
,” Fluid Dyn. Res.
38
, 257
–273
(2006
).23.
S.
Abbas
, L.
Yuan
, and A.
Shah
, “Simulation of spiral instabilities in wide-gap spherical Couette flow
,” Fluid Dyn. Res.
50
, 025507
(2018
).24.
S.
Abbas
, L.
Yuan
, and A.
Shah
, “Existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow
,” J. Braz. Soc. Mech. Sci. Eng.
11
, 154
(2018
).25.
F.
Goto
, T.
Itano
, M.
Sugihara-Seki
, and T.
Adachi
, “Bifurcation aspect of polygonal coherence over transitional Reynolds numbers in wide-gap spherical Couette flow
,” Phys. Rev. Fluids
6
, 113903
(2021
).26.
S.
Abbas
and A.
Shah
, “Simulation of different flow regimes in a narrow-gap spherical Couette flow
,” Appl. Math. Comput.
421
, 126929
(2022
).27.
P.
Wulf
, C.
Egbers
, and H. J.
Rath
, “Routes to chaos in wide-gap spherical Couette flow
,” Phys. Fluids
11
, 1359
–1372
(1999
).28.
M.
Hoff
and U.
Harlander
, “Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence
,” J. Fluid Mech.
878
, 522
–543
(2019
).29.
M.
Junk
and C.
Egbers
, “Isothermal spherical Couette flow
,” in Physics of Rotating Fluids, Lecture Notes in Physics
Vol. 549
, edited by C.
Egbers
and G.
Pfister
(Springer
, 2000
), pp. 215
–233
.30.
N.-S.
Cheng
, “Formula for the viscosity of a glycerol-water mixture
,” Ind. Eng. Chem. Res.
47
, 3285
–3288
(2008
).31.
S.
Goto
, S.
Kida
, and S.
Fujiwara
, “Flow visualization using reflective flakes
,” J. Fluid Mech.
683
, 417
–429
(2011
).32.
T.
Inagaki
, T.
Itano
, and M.
Sugihara-Seki
, “Numerical study on the axisymmetric state in spherical Couette flow under unstable thermal stratification
,” Acta Mech.
230
, 3499
–3509
(2019
).33.
N.
Schaeffer
, “Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations
,” Geochem., Geophys., Geosyst.
14
(3
), 751
–758
, (2013
).34.
M.
Frigo
and S. G.
Johnson
, “The design and implementation of FFTW3
,” Proc. IEEE
93
(2
), 216
–231
(2005
).35.
T.
Itano
and S. C.
Generalis
, “Hairpin vortex solution in planar Couette flow: A tapestry of knotted vortices
,” Phys. Rev. Lett
102
, 114501
(2009
).36.
E.
Anderson
, Z.
Bai
, C.
Bischof
, S.
Blackford
, J.
Demmel
, J.
Dongarra
, J.
Du Croz
, A.
Greenbaum
, S.
Hammarling
, A.
McKenney
, and D.
Sorensen
, LAPACK Users' Guide
, 3rd ed. (Society for Industrial and Applied Mathematics
, Philadelphia, PA
, 1999
).37.
S.
Subbotin
, N.
Shmakova
, E.
Ermanyuk
, and V.
Kozlov
, “Stewartson layer instability and triadic resonances in rotating sphere with oscillating inner core
,” Phys. Fluids
34
, 064103
(2022
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
