The phase dynamics of flows in the Taylor–Couette system have been studied by applying a forced modulation to the upper boundary of a large aspect ratio concentric cylinder system. In a one phase variable case, the Taylor vortex flow, the perturbations diffuse along the axial direction and the pattern’s response is well described by a simple diffusion model. In a two phase variable case, the wavy vortex flow, the perturbations either propagate as traveling waves or diffuse, depending on the coupling between the axial and azimuthal phase variables. In the turbulent Taylor vortex flow, where spatial coherence coexists with the turbulent flow, the phase dynamics of the coherent structure are described by a diffusion model with a diffusion coefficient an order of magnitude larger than for the laminar Taylor vortex flow.
REFERENCES
1.
H. L. Swinney and R. DiPrima, “Instabilities and transitions in flow between concentric rotating cylinders,” in Hydrodynamic Instabilities and the Transition to Turbulence, edited by H. L. Swinney and G. P. Gollub, 2nd ed. (Springer-Verlag, Berlin, 1985), p. 139.
2.
H.
Bénard
, “Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en regime permanent
,” Ann. Chim. Phys.
7
, Ser. 23, 62
(1901
).For recent reviews, see F. H. Busse, “Transition to turbulence in Rayleigh-Bénard convection,” in Ref. 1, p. 97.
3.
Y. Couder, “Growth patterns: from stable curved fronts to fractal structures,” in Chaos, Order and Patterns, edited by R. Artuso, P. Cvitanović, and G. Casati (Plenum, New York, 1991), p. 203.
4.
C.
Ho
and P.
Huerre
, “Perturbed free shear layers
,” Annu. Rev. Fluid Mech.
16
, 365
(1984
).5.
G. P.
Smith
and A. A.
Townsend
, “Turbulent Couette flow between concentric cylinders at large Taylor numbers
,” J. Fluid Mech.
123
, 187
(1982
).6.
P. S.
Marcus
, “Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment
,” J. Fluid Mech.
146
, 45
(1984
).7.
P. S.
Marcus
, “Simulation of Taylor-Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave
,” J. Fluid Mech.
146
, 65
(1984
).8.
A.
Davey
, R. C.
DiPrima
, and J. T.
Stuart
, “On the instability of Taylor vortices
,” J. Fluid Mech.
31
, 17
(1968
).9.
M.
Nagata
, “Bifurcations in Couette flow between almost corotating cylinders
,” J. Fluid Mech.
169
, 229
(1986
);M.
Nagata
, “On wavy instabilities of the Taylor-vortex flow between corotating cylinders
,” J. Fluid Mech.
188
, 585
(1988
)., J. Fluid Mech.
10.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981).
11.
H.
Yahata
, “Temporal development of the Taylor vortices in a rotating fluid
,” Prog. Theor. Phys. Suppl.
64
, 176
(1978
);“
Temporal development of the Taylor vortices in a rotating fluid II
,” Prog. Theor. Phys.
61
, 791
(1979
).12.
A. C.
Newell
and J. A.
Whitehead
, “Finite bandwidth, finite amplitude convection
,” J. Fluid Mech.
38
, 279
(1969
).For application to the Taylor-Couette system see
R.
Graham
and J. A.
Domaradzki
, “Local amplitude equation of Taylor vortices and its boundary condition
,” Phys. Rev. A
26
, 1572
(1982
) and references therein.13.
P.
Hall
, “Evolution equations for Taylor vortices in the small-gap limit
,” Phys. Rev. A
29
, 2921
(1984
).14.
H.
Yahata
, “Slowly varying amplitude of the Taylor vortices near the instability point
,” Prog. Theor. Phys.
57
, 347
(1977
).15.
G.
Pfister
and I.
Rehberg
, “Space-dependent order parameter in circular Couette flow transitions
,” Phys. Lett. A
83
, 19
(1981
).16.
G.
Ahlers
, D. S.
Cannell
, M. A.
Dominguez-Lerma
, and R.
Heinrichs
, “Wavenumber selection and Eckhaus instability in Couette-Taylor flow
,” Physica D
23
, 202
(1986
).17.
Y.
Pomeau
and P.
Manneville
, “Stability and fluctuations of a spatially periodic convective flow
,” J. Phys. Lett. (Paris)
40
, L609
(1979
).18.
Y.
Kuramoto
, “Phase dynamics of weakly unstable periodic structures
,” Prog. Theor. Phys.
71
, 1182
(1984
).19.
J. E.
Wesfreid
and V.
Croquette
, “Forced phase diffusion in Rayleigh-Bénard convection
,” Phys. Rev. Lett.
45
, 634
(1980
).20.
V.
Croquette
and F.
Schosseler
, “Diffusive modes in Rayleigh-Bénard structures
,” J. Phys. (Paris)
43
, 1183
(1982
).21.
P.
Tabeling
, “Dynamics of the phase variable in the Taylor vortex system
,” J. Phys. Lett. (Paris)
44
, L665
(1983
).22.
H. A.
Snyder
, “Wave-number selection at finite amplitude in rotating Couette flow
,” J. Fluid Mech.
35
, 273
(1969
).23.
M.
Lücke
and D.
Roth
, “Structure and dynamics of Taylor vortex flow and the effect of subcritical driving ramps
,” Z. Phys. B
78
, 147
(1990
).24.
H.
Paap
and H.
Riecke
, “Drifting vortices in ramped Taylor vortex flow. Quantitative results from phase equation
,” Phys. Fluids A
3
, 1519
(1991
).25.
U. Gerdts, Ph.D. thesis, Universit Kiel, 1985.
26.
M.
Wu
and C. D.
Andereck
, “Phase modulation of Taylor vortex flow
,” Phys. Rev. A
43
, 2074
(1991
).27.
H. R.
Brand
and M. C.
Cross
, “Phase dynamics for the wavy vortex state of the Taylor instability
,” Phys. Rev. A
27
, 1237
(1983
).28.
H. R.
Brand
, “Nonlinear phasedynamics for the spatially periodic states of the Taylor instability
,” Prog. Theor. Phys.
71
, 1096
(1984
).29.
H. R. Brand, “Phase dynamics—a review and perspective,” in Propagation in Systems Far from Equilibrium, edited by J. E. Wesfreid, H. R. Brand, P. Manneville, G. Albinet, and N. Boccara (Springer-Verlag, New York, 1988), p. 206.
30.
M.
Wu
and C. D.
Andereck
, “Phase dynamics of wavy vortex flow
,” Phys. Rev. Lett.
67
, 1258
(1991
).31.
M. Wu and C. D. Andereck, “Phase dynamics in the Taylor-Couette system,” in Ordered and Turbulent Patterns in Taylor-Couette Flow, edited by C. D. Andereck and F. Hayot (Plenum, New York, 1992).
32.
M. Wu, C. D. Andereck, and H. R. Brand, “The phase dynamics of turbulent Taylor vortex flow,” to appear in Europhys. Lett. (1992).
33.
W. Eckhaus, Studies in Nonlinear Stability Theory (Springer-Verlag, New York, 1965).
34.
D. S.
Cannell
, M. A.
Dominguez-Lerma
, and G.
Ahlers
, “Experiments on wave number selection in rotating Couette-Taylor flow
,” Phys. Rev. Lett.
50
, 1365
(1983
).35.
L.
Ning
, G.
Ahlers
, and D. S.
Cannell
, “Wave-number selection and traveling vortex waves in spatially ramped Taylor-Couette flow
,” Phys. Rev. Lett.
64
, 1235
(1990
).36.
P. G. De Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).
37.
J. Hegseth, Ph.D. thesis, The Ohio State University, 1990.
38.
M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (U. S. Government Printing Office, Washington, DC, 1964), p. 297.
39.
M. Lücke and D. Roth (private communication, 1991).
40.
M. A.
Dominguez-Lerma
, G.
Ahlers
, and D. S.
Cannell
, “Marginal stability curve and linear growth rate for rotating Couette-Taylor flow and Rayleigh-Benard convection
,” Phys. Fluids
27
, 856
(1984
).41.
H. Riecke (private communication, 1990). The method of computation is described in Ref. 24.
42.
H. R. Brand (private communication, 1991).
43.
R. W.
Walden
and R. J.
Donnelly
, “Reemergent order of chaotic circular Couette flow
,” Phys. Rev. Lett.
42
, 301
(1979
).44.
H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).
45.
W. Y.
Tam
and H. L.
Swinney
, “Mass transport in turbulent Couette-Taylor flow
,” Phys. Rev. A
36
, 1374
(1987
).
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