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Supercritical transition to turbulence in an inertially driven von Kármán closed flow

Published online by Cambridge University Press:  25 April 2008

FLORENT RAVELET ,
ARNAUD CHIFFAUDEL  and
FRANÇOIS DAVIAUD
FLORENT RAVELET
Affiliation:
Service de Physique de l'Etat Condensé, Direction des Sciences de la Matière, CEA-Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
ARNAUD CHIFFAUDEL
Affiliation:
Service de Physique de l'Etat Condensé, Direction des Sciences de la Matière, CEA-Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
FRANÇOIS DAVIAUD
Affiliation:
Service de Physique de l'Etat Condensé, Direction des Sciences de la Matière, CEA-Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
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Abstract

We study the transition from laminar flow to fully developed turbulence for an inertially driven von Kármán flow between two counter-rotating large impellers fitted with curved blades over a wide range of Reynolds number (102–106). The transition is driven by the destabilization of the azimuthal shear layer, i.e. Kelvin–Helmholtz instability, which exhibits travelling/drifting waves, modulated travelling waves and chaos before the emergence of a turbulent spectrum. A local quantity – the energy of the velocity fluctuations at a given point – and a global quantity – the applied torque – are used to monitor the dynamics. The local quantity defines a critical Reynolds number Rec for the onset of time-dependence in the flow, and an upper threshold/crossover Ret for the saturation of the energy cascade. The dimensionless drag coefficient, i.e. the turbulent dissipation, reaches a plateau above this finite Ret, as expected for ‘Kolmogorov’-like turbulence for Re→∞. Our observations suggest that the transition to turbulence in this closed flow is globally supercritical: the energy of the velocity fluctuations can be considered as an order parameter characterizing the dynamics from the first laminar time-dependence to the fully developed turbulence. Spectral analysis in the temporal domain, moreover, reveals that almost all of the fluctuation energy is stored in time scales one or two orders of magnitude slower than the time scale based on impeller frequency.

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Papers
Information
Journal of Fluid Mechanics , Volume 601 , 25 April 2008 , pp. 339 - 364
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Batchelor, G. K. 1951 Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow. Q. J. Mech. App. Maths 4, 29.CrossRefGoogle Scholar
Benzi, R. 2005 Flow reversal in a simple dynamical model of turbulence. Phys. Rev. Lett. 95, 024502.CrossRefGoogle Scholar
Bouchet, F. & Sommeria, J. 2002 Emergence of intense jets and Jupiter's great red spot as maximum-entropy structures. J. Fluid Mech. 464, 165.Google Scholar
Bourgoin, M., Marié, L., Pétrélis, F., Gasquet, C., Guigon, A., Luciani, J.-B., Moulin, M., Namer, F., Burguete, J., Chiffaudel, A., Daviaud, F., Fauve, S., Odier, P. & Pinton, J.-F. 2002 MHD measurements in the von Kármán sodium experiment. Phys. Fluids 14, 3046.CrossRefGoogle Scholar
Buchhave, P., George, W. K. & Lumley, J. L. 1979 The measurement of turbulence with the Laser-Doppler anemometer. Annu. Rev. Fluid Mech. 11, 443.CrossRefGoogle Scholar
Cadot, O., Couder, Y., Daerr, A., Douady, S. & Tsinober, A. 1997 Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring. Phys. Rev. E 56, 427.Google Scholar
Cadot, O., Douady, S. & Couder, Y. 1995 Characterization of the low-pressure filaments in a three-dimensional turbulent shear flow. Phys. Fluids 7, 630.CrossRefGoogle Scholar
Chavanis, P.-H. & Sommeria, J. 1998 Classification of robust isolated vortices in two-dimensional hydrodynamics. J. Fluid Mech. 356, 259.CrossRefGoogle Scholar
Chossat, P. 1993 Forced reflexional symmetry breaking of an o(2)-symmetric homoclinic cycle. Nonlinearity 6, 723.Google Scholar
Coullet, P. & Iooss, G. 1990 Instabilities of one-dimensional cellular patterns. Phys. Rev. Lett. 64, 866.CrossRefGoogle ScholarPubMed
Douady, S., Couder, Y. & Brachet, M. E. 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983.CrossRefGoogle ScholarPubMed
Dutta, P. & Horn, P. M. 1981 Low-frequency fluctuations in solid: 1/f noise. Rev. Mod. Phys. 53, 497.CrossRefGoogle Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exps. Fluids 2, 189.CrossRefGoogle Scholar
Falkovich, G. 1994 Bottleneck phenomenon in developed turbulence. Phys. Fluids 6, 1411.CrossRefGoogle Scholar
Fauve, S., Douady, S. & Thual, O. 1991 Drift instabilities of cellular patterns. J. Phys. II 1, 311.Google Scholar
Fauve, S., Laroche, C. & Castaing, B. 1993 Pressure fluctuations in swirling turbulent flows. J. Phys. II 3, 271.Google Scholar
Frisch, U. 1995 Turbulence – The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Gauthier, G., Gondret, P. & Rabaud, M. 1999 Axisymmetric propagating vortices in the flow between a stationary and a rotating disk enclosed by a cylinder. J. Fluid Mech. 386, 105.CrossRefGoogle Scholar
Gelfgat, A. Y., Bar-Yoseph, P. Z. & Solan, A. 1996 Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom. Phys. Fluids 8, 2614.CrossRefGoogle Scholar
Gioia, G. & Chakraborty, P. 2006 Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory. Phys. Rev. Lett. 96, 044502.CrossRefGoogle ScholarPubMed
Goldenfeld, N. 2006 Roughness-induced critical phenomena in a turbulent flow. Phys. Rev. Lett. 96, 044503.CrossRefGoogle Scholar
Harriott, G. M. & Brown, R. A. 1984 Flow in a differentially rotated cylindrical drop at moderate Reynolds number. J. Fluid Mech. 144, 403.Google Scholar
Hodgman, C. D. (Ed.) 1947 Handbook of Chemistry and Physics, 30th Edn. Chemical Rubber Publishing Co.Google Scholar
Jung, S. W., Morrison, P. J. & Swinney, H. L. 2006 Statistical mechanics of two-dimensional turbulence. J. Fluid Mech. 554, 433.Google Scholar
von Kármán, T. 1921 Über laminäre und turbulente Reibung. Z. Angew. Math. Mech. 1, 233.CrossRefGoogle Scholar
Knobloch, E. 1996 Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows. Phys. Fluids 8, 1446.CrossRefGoogle Scholar
Labbé, R., Pinton, J.-F. & Fauve, S. 1996 Power fluctuations in turbulent swirling flows. J. Phys II 6, 1099.Google Scholar
La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle acceleration in fully developed turbulence. Nature 409, 1017.Google Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. L. 1992 Transition to shear-driven turbulence in Couette-Taylor flow. Phys. Rev. A 46, 6390.Google Scholar
Leprovost, N., Dubrulle, B. & Chavanis, P.-H. 2006 Dynamics and thermodynamics of axisymmetric flows: Theory. Phys. Rev. E 73, 046308.Google Scholar
Leprovost, N., Marié, L. & Dubrulle, B. 2004 A stochastic model of torque in von Kármán swirling flow. Euro. Phys. J. B 39, 121.Google Scholar
Lesieur, M. 1990 Turbulence in Fluids, 2nd Revised Edn. Kluwer.Google Scholar
Lohse, D. & Müller-Groeling, A. 1995 Bottleneck effects in turbulence: Scaling phenomena in r versus p space. Phys. Rev. Lett. 74, 1747.Google Scholar
Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic.Google Scholar
Marié, L. 2003 Transport de moment cinétique et de champ magnétique par un écoulement tourbillonaire turbulent: influence de la rotation. PhD thesis, Université Paris VII.Google Scholar
Marié, L., Burguete, J., Daviaud, F. & Léorat, J. 2003 Numerical study of homogeneous dynamo based on experimental von Kármán type flows. Euro. Phys. J. B 33, 469.CrossRefGoogle Scholar
Marié, L. & Daviaud, F. 2004 Experimental measurement of the scale-by-scale momentum transport budget in a turbulent shear flow. Phys. Fluids 16, 457.CrossRefGoogle Scholar
Mellor, G. L., Chapple, P. J. & Stokes, V. K. 1968 On the flow between a rotating and a stationary disk. J. Fluid Mech. 31, 95.CrossRefGoogle Scholar
Moisy, F., Willaime, H., Andersen, J. S. & Tabeling, P. 2001 Passive scalar intermittency in low temperature helium flows. Phys. Rev. Lett. 86, 4827.Google Scholar
Monchaux, R., Berhanu, M., Bourgoin, M., Moulin, M., Odier, Ph., Pinton, J.-F., Volk, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Gasquet, C., Marié, L. & Ravelet, F. 2007 Generation of magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98, 044502.CrossRefGoogle Scholar
Monchaux, R., Ravelet, F., Dubrulle, B., Chiffaudel, A. & Daviaud, F. 2006 Properties of steady states in turbulent axisymmetric flows. Phys. Rev. Lett. 96, 124502.Google Scholar
Nikuradse, J. 1932 Gesetzmassigkeiten der turbulenten Stromungen in glatten Rohren. vDI Forshungsheft 356, (English transl. NASA TT F-10 (1966), p. 359).Google Scholar
Nikuradse, J. 1933 Stromungsgesetz in rauhren Rohren. vDI Forshungsheft 361, (English transl. Tech. Mem. 1292, NACA (1950)).Google Scholar
Nore, C., Moisy, F. & Quartier, L. 2005 Experimental observation of near-heteroclinic cycles in the von Kármán swirling flow. Phys. Fluids 17, 064103.CrossRefGoogle Scholar
Nore, C., Tartar, M., Daube, O. & Tuckerman, L. S. 2004 Survey of instability thresholds of flow between exactly counter-rotating disks. J. Fluid Mech. 511, 45.CrossRefGoogle Scholar
Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1:2 mode interaction in exactly counter-rotating von Kármán swirling flow. J. Fluid Mech. 477, 51.Google Scholar
Ouellette, N. T., Xu, H., Bourgoin, M. & Bodenschatz, E. 2006 Small-scale anisotropy in Lagrangian turbulence. New J. Phys. 8, 102.CrossRefGoogle Scholar
Pinton, J.-F. & Labbé, R. 1994 Correction to the Taylor hypothesis in swirling flows. J. Phys. II 4, 1461.Google Scholar
Porter, J. & Knobloch, E. 2005 Dynamics in the 1:2 spatial resonance with broken reflection symmetry. Physica D 201, 318–344.Google Scholar
Ravelet, F. 2005 Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de von Kármán turbulent. PhD thesis, Ecole Polytechnique, France.Google Scholar
Ravelet, F., Chiffaudel, A., Daviaud, F. & Leorat, J. 2005 Toward an experimental von Kármán dynamo: Numerical studies for an optimized design. Phys. Fluids 17, 117104.Google Scholar
Ravelet, F., Marié, L., Chiffaudel, A. & Daviaud, F. 2004 Multistability and memory effect in a highly turbulent flow: Experimental evidence for a global bifurcation. Phys. Rev. Lett. 93, 164501.Google Scholar
Ravelet, F., Volk, R., Behranu, M., Chiffaudel, A., Daviaud, F., Dubrulle, B., Fauve, S., Monchaux, R., Mordant, N., Odier, Ph., Pétrélis, F. & Pinton, J.-F. 2007 Magnetic induction in a turbulent flow of liquid sodium: mean behaviour and slow fluctuations. Phys. Fluids (submitted).Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equililbrium states for two-dimensional flows. J. Fluid Mech. 229, 291.Google Scholar
Schlichting, H. 1979 Boundary-Layer Theory, 7th Edn. McGraw-Hill.Google Scholar
Schouveiler, L., Le Gal, P. & Chauve, M.-P. 2001 Instabilities of the flow between a rotating and a stationary disk. J. Fluid Mech. 443, 329.Google Scholar
Sørensen, J. B. & Christensen, E. A. 1995 Direct numerical simulation of rotating fluid flow in a closed cylinder. Phys. Fluids 7, 764.CrossRefGoogle Scholar
Spohn, A., Mory, M. & Hopfinger, E. J 1998 Experiments on vortex breakdown in a confined flow generated by a rotating disc. J. Fluid Mech. 370, 73.CrossRefGoogle Scholar
Stewartson, K. 1953 On the flow between two rotating coaxial disks. Proc. Camb. Phil. Soc. 49, 333.CrossRefGoogle Scholar
Tabeling, P., Zocchi, G., Belin, F., Maurer, J. & Willaime, H. 1996 Probability density functions, skewness and flatness in large Reynolds number turbulence. Phys. Rev. E 53, 1613.Google ScholarPubMed
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Titon, J.-H. & Cadot, O. 2003 The statistics of power injected in a closed turbulent flow: Constant torque forcing versus constant velocity forcing. Phys. Fluids 15, 625.CrossRefGoogle Scholar
de la Torre, A. & Burguete, J. 2007 Slow dynamics in a turbulent von Kármán swirling flow. Phys. Rev. Lett. 99, 054101.CrossRefGoogle Scholar
Volk, R., Odier, P. & Pinton, J.-F. 2006 Fluctuation of magnetic induction in von Kármán swirling flows. Phys. Fluids 18, 085105.CrossRefGoogle Scholar
Zandbergen, P. J. & Dijkstra, D. 1987 von Kármán swirling flows. Annu. Rev. Fluid Mech. 19, 465.Google Scholar
Zocchi, G., Tabeling, P., Maurer, J. & Willaime, H. 1994 Measurement of the scaling of the dissipation at high Reynolds numbers. Phys. Rev. E 50, 3693.Google Scholar