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Rotating Taylor–Green flow

Published online by Cambridge University Press:  13 March 2015

A. Alexakis
A. Alexakis*
Affiliation:
Laboratoire de Physique Statistique, Ecole Normale Supérieure, CNRS UMR 8550, Université Pierre et Marie Curie and Université Paris Diderot, 24 rue Lhomond, Paris, 75005, France
*
†Email address for correspondence: [email protected]
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Abstract

The steady state of a forced Taylor–Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers $\mathit{Re}_{F}$ and Rossby numbers $\mathit{Ro}_{F}$. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space ($\mathit{Re}_{F},\mathit{Ro}_{F}$), and the examination of different limiting procedures for approaching the large $\mathit{Re}_{F}$ small $\mathit{Ro}_{F}$ limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-two-dimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries $\mathit{Ro}_{F}\propto \mathit{Re}_{F}^{-{\it\gamma}}$ in the small $\mathit{Ro}_{F}$ limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.

JFM classification

Geophysical and Geological Flows: Rotating flows Turbulent Flows: Rotating turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 769 , 25 April 2015 , pp. 46 - 78
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© 2015 Cambridge University Press 

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References

Babin, A., Mahalov, A. & Nicolaenko, B. 1969 Global splitting, integrability and regularity of three-dimensional Euler and Navier–Stokes equations for uniformly rotating fluids. Eur. J. Mech. (B/Fluids) 15, 291–300.Google Scholar
Bardina, J., Ferziger, J. H. & Rogallo, R. S. 1985 Effect of rotation on isotropic turbulence – computation and modelling. J. Fluid Mech. 154, 321–336.CrossRefGoogle Scholar
Bartello, P., Metais, O. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 1–29.CrossRefGoogle Scholar
Bewley, G. P., Lathrop, D. P., Maas, L. R. M. & Sreenivasan, K. R. 2007 Inertial waves in rotating grid turbulence. Phys. Fluids 19 (7), 071701.CrossRefGoogle Scholar
van Bokhoven, L. J. A., Clercx, H. J. H., van Heijst, G. J. F. & Trieling, R. R. 2009 Experiments on rapidly rotating turbulent flows. Phys. Fluids 21 (9), 096601.CrossRefGoogle Scholar
Boubnov, B. M. & Golitsyn, G. S. 1986 Experimental study of convective structures in rotating fluids. J. Fluid Mech. 167, 503–531.CrossRefGoogle Scholar
Brachet, M. E., Meneguzzi, M., Vincent, A., Politano, H. & Sulem, P. L. 1992 Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids 4, 2845–2854.CrossRefGoogle Scholar
Bustamante, M. D. & Hayat, U. 2013 Complete classification of discrete resonant Rossby/drift wave triads on periodic domains. Commun. Nonlinear Sci. Numer. Simul. 18, 2402–2419.CrossRefGoogle Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.CrossRefGoogle Scholar
Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2005 Resonant interactions in rotating homogeneous three-dimensional turbulence. J. Fluid Mech. 542, 139–164.CrossRefGoogle Scholar
Cortet, P.-P., Chiffaudel, A., Daviaud, F. & Dubrulle, B. 2010 Experimental evidence of a phase transition in a closed turbulent flow. Phys. Rev. Lett. 105 (21), 214501.CrossRefGoogle Scholar
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135–144.CrossRefGoogle Scholar
Doering, C. R. & Gibbon, J. D. 1995 Applied Analysis of the Navier–Stokes Equations. Cambridge University Press.CrossRefGoogle Scholar
Gallet, B., Campagne, A., Cortet, P.-P. & Moisy, F. 2014 Scale-dependent cyclone–anticyclone asymmetry in a forced rotating turbulence experiment. Phys. Fluids 26 (3), 035108.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301.Google ScholarPubMed
Galtier, S. 2014 Theory for helical turbulence under fast rotation. Phys. Rev. E 89, 041001(R).Google ScholarPubMed
Gence, J.-N. & Frick, C. 2001 Naissance des corrélations triples de vorticité dans une turbulence statistiquement homogène soumise à une rotation. C. R. Acad. Sci. Paris B 329, 351–356.Google Scholar
Godeferd, F. S. & Lollini, L. 1999 Direct numerical simulations of turbulence with confinement and rotation. J. Fluid Mech. 393, 257–308.CrossRefGoogle Scholar
Gómez, D. O., Mininni, P. D. & Dmitruk, P. 2005 Parallel simulations in turbulent MHD. Phys. Scr. T 116, 123–127.CrossRefGoogle Scholar
Greenspan, H. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hopfinger, E. J., Gagne, Y. & Browand, F. K. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.CrossRefGoogle Scholar
Hopfinger, E. J. & van Heijst, G. J. F. 1993 Vortices in rotating fluids. Annu. Rev. Fluid Mech. 25, 241–289.CrossRefGoogle Scholar
Hossain, M. 1994 Reduction in the dimensionality of turbulence due to a strong rotation. Phys. Fluids 6, 1077–1080.CrossRefGoogle Scholar
Julien, K., Knobloch, E., Milliff, R. & Werne, J. 2006a Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274.CrossRefGoogle Scholar
Julien, K., Knobloch, E., Rubio, A. M. & Vasil, G. M. 2006b Heat transport in low-Rossby-number Rayleigh–Benard convection. Phys. Rev. Lett. 109, 254503.Google Scholar
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11, 251–261.CrossRefGoogle Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, L21–L24.CrossRefGoogle Scholar
Kolvin, I., Cohen, K., Vardi, Y. & Sharon, E. 2009 Energy transfer by inertial waves during the buildup of turbulence in a rotating system. Phys. Rev. Lett. 102 (1), 014503.CrossRefGoogle Scholar
Lamriben, C., Cortet, P.-P. & Moisy, F. 2011 Anisotropic energy transfers in rotating turbulence. J. Phys.: Conf. Ser. 318 (4), 042005.Google Scholar
Mansour, N. N., Gambon, C. & Speziale, C. G. 1992 Theoretical and Computational Study of Rotating Isotropic Turbulence. pp. 59–75. Springer.Google Scholar
Maurer, J. & Tabeling, P. 1998 Local investigation of superfluid turbulence. Eur. Phys. Lett. 43, 29–34.CrossRefGoogle Scholar
Mininni, P. D., Alexakis, A. & Pouquet, A. 2006 Large-scale flow effects, energy transfer, and self-similarity on turbulence. Phys. Rev. E 74 (1), 016303.Google ScholarPubMed
Mininni, P. D., Alexakis, A. & Pouquet, A. 2009 Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers. Phys. Fluids 21 (1), 015108.CrossRefGoogle Scholar
Mininni, P. D. & Pouquet, A. 2009 Helicity cascades in rotating turbulence. Phys. Rev. E 79 (2), 026304.Google ScholarPubMed
Mininni, P. D. & Pouquet, A. 2010 Rotating helical turbulence. I. Global evolution and spectral behavior. Phys. Fluids 22, 035105.CrossRefGoogle Scholar
Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Isotropization at small scales of rotating helically driven turbulence. J. Fluid Mech. 699, 263–279.CrossRefGoogle Scholar
Monchaux, R., Berhanu, M., Aumaître, S., Chiffaudel, A., Daviaud, F., Dubrulle, B., Ravelet, F., Fauve, S., Mordant, N., Pétrélis, F., Bourgoin, M., Odier, P., Pinton, J.-F., Plihon, N. & Volk, R. 2009 The von Kármán sodium experiment: turbulent dynamical dynamos. Phys. Fluids 21 (3), 035108.Google Scholar
Monchaux, R., Berhanu, M., Bourgoin, M., Moulin, M., Odier, P., 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 a magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98 (4), 044502.CrossRefGoogle Scholar
Morinishi, Y., Nakabayashi, K. & Ren, S. Q. 2001 Dynamics of anisotropy on decaying homogeneous turbulence subjected to system rotation. Phys. Fluids 13, 2912–2922.CrossRefGoogle Scholar
Morize, C. & Moisy, F. 2006 Energy decay of rotating turbulence with confinement effects. Phys. Fluids 18 (6), 065107.CrossRefGoogle Scholar
Morize, C., Moisy, F. & Rabaud, M. 2005 Decaying grid-generated turbulence in a rotating tank. Phys. Fluids 17 (9), 095105.CrossRefGoogle Scholar
Müller, W.-C. & Thiele, M. 2007 Scaling and energy transfer in rotating turbulence. Eur. Phys. Lett. 77, 34003.CrossRefGoogle Scholar
Nazarenko, S. V. 2011 Wave Turbulence. Springer.CrossRefGoogle Scholar
Newell, A. C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35, 255–271.CrossRefGoogle Scholar
Ponty, Y., Mininni, P. D., Laval, J.-P., Alexakis, A., Baerenzung, J., Daviaud, F., Dubrulle, B., Pinton, J.-F., Politano, H. & Pouquet, A. 2008 Linear and non-linear features of the Taylor Green dynamo. C. R. Phys. 9, 749–756.Google Scholar
Ruppert-Felsot, J. E., Praud, O., Sharon, E. & Swinney, H. L. 2005 Extraction of coherent structures in a rotating turbulent flow experiment. Phys. Rev. E 72 (1), 016311.Google Scholar
Salort, J., Baudet, C., Castaing, B., Chabaud, B., Daviaud, F., Didelot, T., Diribarne, P., Dubrulle, B., Gagne, Y., Gauthier, F., Girard, A., Hébral, B., Rousset, B., Thibault, P. & Roche, P.-E. 2010 Turbulent velocity spectra in superfluid flows. Phys. Fluids 22 (12), 125102.CrossRefGoogle Scholar
Scott, J. F. 2014 Wave turbulence in a rotating channel. J. Fluid Mech. 741, 316–349.CrossRefGoogle Scholar
Sen, A., Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence. Phys. Rev. E 86, 036319.Google ScholarPubMed
Seshasayanan, K., Benavides, S. J. & Alexakis, A. 2014 On the edge of an inverse cascade. Phys. Rev. E 90, 051003(R).Google ScholarPubMed
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11, 1608–1622.CrossRefGoogle Scholar
Squires, K. D., Chasnov, J. R., Mansour, N. N. & Cambon, C.(Eds) 1994 The asymptotic state of rotating homogeneous turbulence at high Reynolds numbers. In Application of Direct and Large Eddy Simulation to Transition and Turbulence, NASA Ames Reseach Center.Google Scholar
Sreenivasan, B. & Davidson, P. A. 2008 On the formation of cyclones and anticyclones in a rotating fluid. Phys. Fluids 20 (8), 085104.CrossRefGoogle Scholar
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81–105.CrossRefGoogle Scholar
Sugihara, Y., Migita, M. & Honji, H. 2005 Orderly flow structures in grid-generated turbulence with background rotation. Fluid Dyn. Res. 36, 23–34.CrossRefGoogle Scholar
Teitelbaum, T. & Mininni, P. D. 2009 Effect of helicity and rotation on the free decay of turbulent flows. Phys. Rev. Lett. 103 (1), 014501.CrossRefGoogle ScholarPubMed
Teitelbaum, T. & Mininni, P. D. 2010 Large-scale effects on the decay of rotating helical and non-helical turbulence. Phys. Scr. T 142 (1), 014003.Google Scholar
Thiele, M. & Müller, W.-C. 2009 Structure and decay of rotating homogeneous turbulence. J. Fluid Mech. 637, 425–442.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids 4, 350–363.CrossRefGoogle Scholar
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids 5, 677–685.CrossRefGoogle Scholar
Yarom, E., Vardi, Y. & Sharon, E. 2013 Experimental quantification of inverse energy cascade in deep rotating turbulence. Phys. Fluids 25 (8), 085105.CrossRefGoogle Scholar
Yeung, P. K. & Zhou, Y. 1998 Numerical study of rotating turbulence with external forcing. Phys. Fluids 10, 2895–2909.CrossRefGoogle Scholar
Yoshimatsu, K., Midorikawa, M. & Kaneda, Y. 2011 Columnar eddy formation in freely decaying homogeneous rotating turbulence. J. Fluid Mech. 677, 154–178.CrossRefGoogle Scholar