Skip to main content Accessibility help
×
Home

Exploring the phase diagram of fully turbulent Taylor–Couette flow

  • Rodolfo Ostilla-Mónico (a1), Erwin P. van der Poel (a1), Roberto Verzicco (a1) (a2), Siegfried Grossmann (a3) and Detlef Lohse (a1)...

Abstract

Direct numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to $3\times 10^{5}$ , corresponding to Taylor numbers of $\mathit{Ta}=4.6\times 10^{10}$ , were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

Copyright

Corresponding author

Email address for correspondence: r.ostillamonico@utwente.nl

References

Hide All
Ahlers, G., He, X., Funfschilling, D. & Bodenschatz, E. 2012 Heat transport by turbulent Rayleigh–Bénard convection for $Pr=0.8$ and $3\times 10^{12}\lesssim Ra\lesssim 10^{15}$ : aspect ratio ${\rm\Gamma}=0.50$ . New J. Phys. 14, 103012.
Andereck, C. D., Dickman, R. & Swinney, H. L. 1983 New flows in a circular Couette system with corotating cylinders. Phys. Fluids 26 (1395).
Andereck, C. D., Liu, S. S. & Swinney, H. L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155183.
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous liquid. Proc. R. Soc. Lond. A 359, 143.
Brauckmann, H. & Eckhardt, B. 2013a Direct numerical simulations of local and global torque in Taylor–Couette Flow up to $\mathit{Re}=30.000$ . J. Fluid Mech. 718, 398427.
Brauckmann, H. J. & Eckhardt, B. 2013b Intermittent boundary layers and torque maxima in Taylor–Couette flow. Phys. Rev. E 87 (3), 033004.
Couette, M. 1890 Études sur le frottement des liquides. Gauthier-Villars et fils.
Donnelly, R. 1991 Taylor–Couette flow: the early days. Phys. Today 3239.
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221250.
van Gils, D. P. M., Huisman, S. G., Bruggert, G. W., Sun, C. & Lohse, D. 2011 Torque scaling in turbulent Taylor–Couette flow with co- and counter-rotating cylinders. Phys. Rev. Lett. 106, 024502.
van Gils, D. P. M., Huisman, S. G., Grossmann, S., Sun, C. & Lohse, D. 2012 Optimal Taylor–Couette turbulence. J. Fluid Mech. 706, 118149.
Görtler, H. 1940a Über den Einflusss der Wandkrümmung auf die Entstehung der Turbulenz. Z. Angew. Math. Mech. 20, 138147.
Görtler, H. 1940b Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden. Z. Angew. Math. Mech. 21, 250252.
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 2756.
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
Grossmann, S., Lohse, D. & Sun, C. 2014 Velocity profiles in strongly turbulent Taylor–Couette flow. Phys. Fluids 26, 025114.
He, X., Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2012a Heat transport by turbulent Rayleigh–Bénard convection for $Pr=0.8$ and $4\times 10^{11}\lesssim Ra\lesssim 2\times 10^{14}$ : ultimate-state transition for aspect ratio ${\rm\Gamma}=100$ . New J. Phys. 14 (6), 063030.
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012b Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.
Hoffmann, P. H., Muck, K. C. & Bradshaw, P. 1985 The effect of concave surface curvature on turbulent boundary layers. J. Fluid Mech. 161, 371403.
Huisman, S. G., van Gils, D. P. M., Grossmann, S., Sun, C. & Lohse, D. 2012 Ultimate turbulent Taylor–Couette flow. Phys. Rev. Lett. 108, 024501.
Huisman, S. G., Scharnowski, S., Cierpka, C., Kähler, C., Lohse, D. & Sun, C. 2013 Logarithmic boundary layers in strong Taylor–Couette turbulence. Phys. Rev. Lett. 110, 264501.
Huisman, S. G., van der Veen, R. C. A., Sun, C. & Lohse, D. 2014 Multiple states in ultimate Taylor–Couette turbulence. Nature Commun. 5, 3820.
Kraichnan, R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 13741389.
Lathrop, D. P., Fineberg, Jay & Swinney, H. S. 1992a Transition to shear-driven turbulence in Couette–Taylor flow. Phys. Rev. A 46, 63906405.
Lathrop, D. P., Fineberg, Jay & Swinney, H. S. 1992b Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 15151518.
Lewis, G. S. & Swinney, H. L. 1999 Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette–Taylor flow. Phys. Rev. E 59, 54575467.
Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196212.
Mallock, A. 1896 Experiments on fluid viscosity. Phil. Trans. R. Soc. Lond. A 187, 4156.
Manna, M. & Vacca, A. 2009 Torque reduction in Taylor–Couette flows subject to an axial pressure gradient. J. Fluid Mech. 639, 373401.
Martinez-Arias, B., Peixinho, J., Crumeyrolle, O. & Mutabazi, I. 2014 Effect of the number of vortices on the torque scaling in Taylor–Couette flow. J. Fluid Mech. 748, 756767.
Merbold, S., Brauckmann, H. & Egbers, C. 2013 Torque measurements and numerical determination in differentially rotating wide gap Taylor–Couette flow. Phys. Rev. E 87, 023014.
Muck, K. C., Hoffmann, P. H. & Bradshaw, P. 1985 The effect of convex surface curvature on turbulent boundary layers. J. Fluid Mech. 161, 347369.
Ostilla-Mónico, R., Stevens, R. J. A. M., Grossmann, S., Verzicco, R. & Lohse, D. 2013 Optimal Taylor–Couette flow: direct numerical simulations. J. Fluid Mech. 719, 1446.
Ostilla-Mónico, R., Huisman, S. G., Jannink, T. J. G., van Gils, D. P. M., Verzicco, R., Grossmann, S., Sun, C. & Lohse, D. 2014a Optimal Taylor–Couette flow: radius ratio dependence. J. Fluid Mech. 747, 129.
Ostilla-Mónico, R., van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. 2014b Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor–Couette flow. Phys. Fluids 26, 015114.
Ostilla-Mónico, R., Verzicco, R., Grossmann, S. & Lohse, D. 2014c Turbulence decay towards the linearly-stable regime of Taylor–Couette flow. J. Fluid Mech. 747, 129.
Paoletti, M. S. & Lathrop, D. P. 2011 Angular momentum transport in turbulent flow between independently rotating cylinders. Phys. Rev. Lett. 106, 024501.
Prandtl, L. 1933 Neuere ergebnisse der turbulenzforschung. Z. Verein. Deutsch. Ing. 77 (5), 105114.
Ravelet, F., Delfos, R. & Westerweel, J. 2010 Influence of global rotation and Reynolds number on the large-scale features of a turbulent Taylor–Couette flow. Phys. Fluids 22 (5), 055103.
Roche, P. E., Gauthier, G., Kaiser, R. & Salort, J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12, 085014.
Spiegel, E. A. 1971 Convection in stars. Annu. Rev. Astron. Astrophys. 9, 323352.
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213218.
Taylor, G. I. 1936 Fluid friction between rotating cylinders. II. Distribution of velocity between concentric cylinders when outer one is rotating and inner one is at rest. Proc. R. Soc. Lond. A 157, 565578.
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402413.
Wendt, F. 1933 Turbulente Strömungen zwischen zwei rotierenden Zylindern. Ingenieurs-Archiv 4, 577595.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Exploring the phase diagram of fully turbulent Taylor–Couette flow

  • Rodolfo Ostilla-Mónico (a1), Erwin P. van der Poel (a1), Roberto Verzicco (a1) (a2), Siegfried Grossmann (a3) and Detlef Lohse (a1)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed