Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T21:43:38.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Taylor–Couette flow: radius ratio dependence

Published online by Cambridge University Press:  10 April 2014

Rodolfo Ostilla-Mónico ,
Sander G. Huisman ,
Tim J. G. Jannink ,
Dennis P. M. Van Gils ,
Roberto Verzicco ,
Siegfried Grossmann ,
Chao Sun  and
Detlef Lohse
Rodolfo Ostilla-Mónico*
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Sander G. Huisman
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Tim J. G. Jannink
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Dennis P. M. Van Gils
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Roberto Verzicco
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Dipartimento di Ingegneria Meccanica, University of Rome ‘Tor Vergata’, via del Politecnico 1, 00133 Roma, Italy
Siegfried Grossmann
Affiliation:
Department of Physics, University of Marburg, Renthof 6, 35032 Marburg, Germany
Chao Sun
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Detlef Lohse
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: R.Ostillamonico@utwente.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

Taylor–Couette flow with independently rotating inner ($i$) and outer ($o$) cylinders is explored numerically and experimentally to determine the effects of the radius ratio $\eta $ on the system response. Numerical simulations reach Reynolds numbers of up to $\mathit{Re}_i=9.5\times 10^3$ and $\mathit{Re}_o=5\times 10^3$, corresponding to Taylor numbers of up to $\mathit{Ta}=10^8$ for four different radius ratios $\eta =r_i/r_o$ between 0.5 and 0.909. The experiments, performed in the Twente Turbulent Taylor–Couette ($\mathrm{T^3C}$) set-up, reach Reynolds numbers of up to $\mathit{Re}_i=2\times 10^6$ and $\mathit{Re}_o=1.5\times 10^6$, corresponding to $\mathit{Ta}=5\times 10^{12}$ for $\eta =0.714\mbox{--}0.909$. Effective scaling laws for the torque $J^{\omega }(\mathit{Ta})$ are found, which for sufficiently large driving $\mathit{Ta}$ are independent of the radius ratio $\eta $. As previously reported for $\eta =0.714$, optimum transport at a non-zero Rossby number $\mathit{Ro}=r_i |\omega _i-\omega _o |/[2(r_o-r_i)\omega _o]$ is found in both experiments and numerics. Here $\mathit{Ro}_{opt}$ is found to depend on the radius ratio and the driving of the system. At a driving in the range between $\mathit{Ta}\sim 3\times 10^{8}$ and $\mathit{Ta}\sim 10^{10}$, $\mathit{Ro}_{opt}$ saturates to an asymptotic $\eta $-dependent value. Theoretical predictions for the asymptotic value of $\mathit{Ro}_{opt}$ are compared to the experimental results, and found to differ notably. Furthermore, the local angular velocity profiles from experiments and numerics are compared, and a link between a flat bulk profile and optimum transport for all radius ratios is reported.

JFM classification

Convection: Convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 1 - 29
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlers, G. 1974 Low temperature studies of the Rayleigh–Bénard instability and turbulence. Phys. Rev. Lett. 33, 1185–1188.CrossRefGoogle Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.Google Scholar
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, 155–183.CrossRefGoogle Scholar
Behringer, R. P. 1985 Rayleigh–Bénard convection and turbulence in liquid–helium. Rev. Mod. Phys. 57, 657–687.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous liquid. Proc. R. Soc. Lond. A 359, 1–43.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Brauckmann, H. & Eckhardt, B. 2013a Direct numerical simulations of local and global torque in Taylor–Couette flow up to ${\rm Re} = 30\, 000$ . J. Fluid Mech. 718, 398–427.CrossRefGoogle Scholar
Brauckmann, H. & Eckhardt, B. 2013b Intermittent boundary layers and torque maxima in Taylor–Couette flow. Phys. Rev. E 87, 033004.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Couette, M. 1890 Études sur le Frottement des Liquides. Gauthier-Villars et Fils.Google Scholar
Coughlin, K. & Marcus, P. S. 1996 Turbulent bursts in Couette–Taylor flow. Phys. Rev. Lett. 77 (11), 2214–2217.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.CrossRefGoogle Scholar
Dominguez-Lerma, M. A., Cannell, D. S. & Ahlers, G. 1986 Eckhaus boundary and wavenumber selection in rotating Couette–Taylor flow. Phys. Rev. A 34, 4956–4970.Google Scholar
Dong, S. 2007 Direct numerical simulation of turbulent Taylor–Couette flow. J. Fluid Mech. 587, 373–393.Google Scholar
Dong, S. 2008 Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study. J. Fluid Mech. 615, 371–399.Google Scholar
Donnelly, R. 1991 Taylor–Couette flow: the early days. Phys. Today 44 (November), 32–39.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221–250.CrossRefGoogle Scholar
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids 8, 1814–1819.Google Scholar
Fasel, H. & Booz, O. 1984 Numerical investigation of supercritical Taylor-vortex flow for a wide gap. J. Fluid Mech. 138, 21–52.CrossRefGoogle Scholar
Gebhardt, Th. & Grossmann, S. 1993 The Taylor–Couette eigenvalue problem with independently rotating cylinders. Z. Phys. B 90 (4), 475–490.CrossRefGoogle Scholar
van Gils, D. P. M., Bruggert, G. W., Lathrop, D. P., Sun, C. & Lohse, D. 2011a The Twente Turbulent Taylor–Couette ( $\mathrm{T^3C}$ ) facility: strongly turbulent (multi-phase) flow between independently rotating cylinders. Rev. Sci. Instrum. 82, 025105.Google Scholar
van Gils, D. P. M., Huisman, S. G., Bruggert, G. W., Sun, C. & Lohse, D. 2011b Torque scaling in turbulent Taylor–Couette flow with co- and counter-rotating cylinders. Phys. Rev. Lett. 106, 024502.Google Scholar
van Gils, D. P. M., Huisman, S. G., Grossmann, S., Sun, C. & Lohse, D. 2012 Optimal Taylor–Couette turbulence. J. Fluid Mech. 706, 118–149.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.Google Scholar
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.Google Scholar
Grossmann, S. & Lohse, D. 2012 Logarithmic temperature profiles in the ultimate regime of thermal convection. Phys. Fluids 24, 125103.CrossRefGoogle Scholar
Huisman, S. G., Scharnowski, S., Cierpka, C., Kaehler, C., Lohse, D. & Sun, C. 2013 Logarithmic boundary layers in highly turbulent Taylor–Couette flow. Phys. Rev. Lett. 110, 264501.CrossRefGoogle Scholar
Huisman, S. G., van Gils, D. P. M. & Sun, C. 2012a Applying laser Doppler anemometry inside a Taylor–Couette geometry – using a ray-tracer to correct for curvature effects. Eur. J. Mech. (B/Fluids) 36, 115–119.Google Scholar
Huisman, S. G., van Gils, D. P. M., Grossmann, S., Sun, C. & Lohse, D. 2012b Ultimate turbulent Taylor–Couette flow. Phys. Rev. Lett. 108, 024501.Google Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54 (8), 34–39.Google Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992a Transition to shear-driven turbulence in Couette–Taylor flow. Phys. Rev. A 46, 6390–6405.Google Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. S. 1992b Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 1515–1518.Google Scholar
Lewis, G. S. & Swinney, H. L. 1999 Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette–Taylor flow. Phys. Rev. E 59, 5457–5467.CrossRefGoogle ScholarPubMed
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Mallock, A. 1896 Experiments on fluid viscosity. Phil. Trans. R. Soc. Lond. A 187, 41–56.Google Scholar
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.Google Scholar
Ostilla, R., Stevens, R. J. A. M., Grossmann, S., Verzicco, R. & Lohse, D. 2013 Optimal Taylor–Couette flow: direct numerical simulations. J. Fluid Mech. 719, 14–46.Google Scholar
Ostilla-Mónico, R., van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. 2014 Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor–Couette flow. Phys. Fluids 26, 015114.Google Scholar
Paoletti, M. S. & Lathrop, D. P. 2011 Angular momentum transport in turbulent flow between independently rotating cylinders. Phys. Rev. Lett. 106, 024501.Google Scholar
Pfister, G. & Rehberg, I. 1981 Space dependent order parameter in circular Couette flow transitions. Phys. Lett. 83, 19–22.CrossRefGoogle Scholar
Pfister, G., Schmidt, H., Cliffe, K. A. & Mullin, T. 1988 Bifurcation phenomena in Taylor–Couette flow in a very short annulus. J. Fluid Mech. 191, 1–18.Google Scholar
Pirro, D. & Quadrio, M. 2008 Direct numerical simulation of turbulent Taylor–Couette flow. Eur. J. Mech. (B/Fluids) 27, 552–566.Google Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.Google Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187–217.Google Scholar
Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31–43.Google Scholar
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495–507.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Perseus Press.Google Scholar
Swinney, H. L. & Gollub, J. P. 1981 Hydrodynamic Instabilities and the Transition to Turbulence., Topics in Applied Physics, vol. 45. Springer.Google Scholar
Taylor, G. I. 1936 Fluid friction between rotating cylinders. Proc. R. Soc. Lond. A 157, 546–564.Google Scholar
Tong, P., Goldburg, W. I., Huang, J. S. & Witten, T. A. 1990 Anisotropy in turbulent drag reduction. Phys. Rev. Lett. 65, 2780–2783.Google Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–413.CrossRefGoogle Scholar
Wendt, F. 1933 Turbulente Strömungen zwischen zwei rotierenden Zylindern. Ing.-Arch. 4, 577–595.Google Scholar
Xia, K.-Q., Lam, S. & Zhou, S. Q. 2002 Heat-flux measurement in high-Prandtl-number turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 88, 064501.Google Scholar