Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T16:31:50.688Z Has data issue: false hasContentIssue false
  • English
  • Français

The near-wall region of highly turbulent Taylor–Couette flow

Published online by Cambridge University Press:  22 December 2015

Rodolfo Ostilla-Mónico ,
Roberto Verzicco ,
Siegfried Grossmann  and
Detlef Lohse
Rodolfo Ostilla-Mónico*
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Roberto Verzicco
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Dipartimento di Ingegneria Industriale, University of Rome ‘Tor Vergata’, Via del Politecnico 1, Roma 00133, Italy
Siegfried Grossmann
Affiliation:
Department of Physics, University of Marburg, Renthof 6, D-35032 Marburg, Germany
Detlef Lohse
Affiliation:
Physics of Fluids, Mesa+ Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Direct numerical simulations of the Taylor–Couette (TC) problem, the flow between two coaxial and independently rotating cylinders, have been performed. The study focuses on TC flow with mild curvature (small gap) with a radius ratio of ${\it\eta}=r_{i}/r_{o}=0.909$, an aspect ratio of ${\it\Gamma}=L/d=2{\rm\pi}/3$, and a stationary outer cylinder. Three inner cylinder Reynolds numbers of $1\times 10^{5}$, $2\times 10^{5}$ and $3\times 10^{5}$ were simulated, corresponding to frictional Reynolds numbers between $Re_{{\it\tau}}\approx 1400$ and $Re_{{\it\tau}}\approx 4000$. An additional case with a large gap, ${\it\eta}=0.5$ and driving of $Re=2\times 10^{5}$ was also investigated. Small-gap TC was found to be dominated by spatially fixed large-scale structures, known as Taylor rolls (TRs). TRs are attached to the boundary layer, and are active, i.e. they transport angular velocity through Reynolds stresses. An additional simulation was also conducted with inner cylinder Reynolds number of $Re=1\times 10^{5}$ and fixed outer cylinder with an externally imposed axial flow of comparable strength to the wind of the TRs. The axial flow was found to convect the TRs without any weakening effect. For small-gap TC flow, evidence was found for the existence of logarithmic velocity fluctuations, and of an overlap layer, in which the velocity fluctuations collapse in outer units. Profiles consistent with a logarithmic dependence were also found for the angular velocity in large-gap TC flow, albeit in a very reduced range of scales. Finally, the behaviour of both small- and large-gap TC flow was compared to other canonical flows. Small-gap TC flow has similar behaviour in the near-wall region to other canonical flows, while large-gap TC flow displays very different behaviour.

JFM classification

Turbulent Flows: Rotating turbulence Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 95 - 117
Check for updates
Copyright
© 2016 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

Afzal, N. & Yajnik, K. 1973 Analysis of turbulent pipe and channel flows at moderately large Reynolds number. J. Fluid Mech. 61, 23–31.Google 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
Avsarkisov, V., Hoyas, S., Oberlack, M. & García-Galache, J. P. 2014 Turbulent plane Couette flow at moderately high Reynolds number. J. Fluid Mech. 751, R1–R8.Google Scholar
Bailey, S. C. C., Vallikivi, M., Hultmark, M. & Smits, A. J. 2014 Estimating the value of von Kármáns constant in turbulent pipe flow. J. Fluid Mech. 749, 79–98.CrossRefGoogle Scholar
Bech, K. H., Tillmark, N., Alfredsson, P. H. & Andersson, H. I. 1995 An investigation of turbulent plane Couette flow at low Reynolds numbers. J. Fluid Mech. 286, 291–325.Google Scholar
van den Berg, T. H., Luther, S., Lathrop, D. P. & Lohse, D. 2005 Drag reduction in bubbly Taylor–Couette turbulence. Phys. Rev. Lett. 94, 044501.Google Scholar
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to $Re_{{\it\tau}}=4000$ . J. Fluid Mech. 742, 171–191.Google Scholar
Bradshaw, P.1973 The effects of streamline curvature on turbulent flow Tech. Rep. no. 169. AGARDograph.Google Scholar
Brauckmann, H. & Eckhardt, B. 2013 Direct numerical simulations of local and global torque in Taylor–Couette flow up to $Re=30.000$ . J. Fluid Mech. 718, 398–427.Google Scholar
Brauckmann, H., Salewski, M. & Eckhardt, B.2015 Momentum transport in Taylor–Couette flow with vanishing curvature (under review), arXiv:1505.06278.Google Scholar
Chandrasekhar, S. 1960a The hydrodynamic stability of inviscid flow between coaxial cylinders. Proc. Natl Acad. Sci. USA 46 (1), 137–141.Google Scholar
Chandrasekhar, S. 1960b The hydrodynamic stability of viscid flow between coaxial cylinders. Proc. Natl Acad. Sci. USA 46 (1), 141–143.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503–539.Google Scholar
Dong, S. 2007 Direct numerical simulation of turbulent Taylor–Couette flow. J. Fluid Mech. 587, 373–393.CrossRefGoogle Scholar
Dong, S. 2008 Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study. J. Fluid Mech. 615, 371–399.Google Scholar
Faisst, H. & Eckhardt, B. 2000 Transition from the Couette–Taylor system to the plane Couette system. Phys. Rev. E 61, 7227–7230.Google Scholar
van Gils, D. P. M., Narezo-Guzman, D., Sun, C. & Lohse, D. 2013 The importance of bubble deformability for strong drag reduction in bubbly turbulent Taylor–Couette flow. J. Fluid Mech. 722, 317–347.Google Scholar
Grossmann, S., Lohse, D. & Sun, C. 2014 Velocity profiles in strongly turbulent Taylor–Couette flow. Phys. Fluids 26, 025114.Google Scholar
Grossmann, S., Lohse, D. & Sun, C. 2016 High-Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 53–80.Google Scholar
Hoffmann, P. H., Muck, K. C. & Bradshaw, P. 1985 The effect of concave surface curvature on turbulent boundary layers. J. Fluid Mech. 161, 371–403.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $Re_{{\it\tau}}=2003$ . Phys. Fluids 18, 011702.Google Scholar
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.Google Scholar
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.Google Scholar
Huisman, S. G., van der Veen, R. C. A., Sun, C. & Lohse, D. 2014 Multiple states in highly turbulent Taylor–Couette turbulence. Nat. Commun. 5, 3820.Google Scholar
Hunt, I. A. & Joubert, P. N. 1979 Effects of small streamline curvature on turbulent duct flow. J. Fluid Mech. 91, 633–659.Google Scholar
Hwang, J.-Y. & Yang, K.-S. 2004 Numerical study of Taylor–Couette flow with an axial flow. Comput. Fluids 33, 97–118.Google Scholar
Jimenez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.Google Scholar
Jiménez, J. & Moser, R. 2007 What are we learning from simulating wall turbulence? Phil. Trans. R. Soc. Lond. A 365, 715–732.Google Scholar
Kim, H. T., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–160.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_{{\it\tau}}=5200$ . J. Fluid Mech. 774, 395–415.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
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100–130.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to $Re_{{\it\tau}}=4200$ . Phys. Fluids 26, 011702.CrossRefGoogle Scholar
Lu, J., Fernández, A. & Tryggvason, G. 2005 The effect of bubbles on the wall drag in a turbulent channel flow. Phys. Fluids 17, 095102.CrossRefGoogle Scholar
Lueptow, R. M., Docter, A. & Min, K. 1992 Stability of axial flow in an annulus with a rotating inner cylinder. Phys. Fluids 4 (11), 2446–2455.Google Scholar
Marcus, P. S. 1984 Simulation of Taylor–Couette flow. Part 2. Numerical results for wavy-vortex flow with one traveling wave. J. Fluid Mech. 146, 65–113.Google Scholar
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, 756–767.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
Moser, R. D. & Moin, P. 1986 The effects of curvature in wall-bounded flow. J. Fluid Mech. 175, 479–510.Google Scholar
Ng, B. S. & Turner, E. R. 1982 On the linear stability of spiral flow between rotating cylinders. Proc. R. Soc. Lond. A 382 (1782), 83–102.Google Scholar
Ostilla-Monico, 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, 1–29.Google Scholar
Ostilla-Monico, 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.Google Scholar
Ostilla-Monico, R., van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. 2014c Exploring the phase diagram of fully turbulent Taylor–Couette flow. J. Fluid Mech. 761, 1–26.Google Scholar
Ostilla-Mónico, R., Verzicco, R. & Lohse, D. 2015 Effects of the computational domain size on DNS of Taylor–Couette turbulence with stationary outer cylinder. Phys. Fluids 27, 025110.Google Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.Google Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.Google Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2014 Turbulence statistics in Couette flow at high Reynolds number. J. Fluid Mech. 758, 327–343.Google Scholar
van der Poel, E. P., Ostilla-Monico, R., Donners, J. & Verzicco, R. 2015 A pencil distributed finite difference code for strongly turbulent wall-bounded flows. Comput. Fluids 116, 10–16.Google Scholar
Pope, S. B. 2000 Turbulent Flow. Cambridge University Press.Google Scholar
Schwarz, K. W., Pringett, B. E. & Donnelly, R. J. 1964 Modes of instability in spiral flow between rotating cylinders. J. Fluid Mech. 20, 281–289.Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to ${\it\delta}+\approx 2000$ . Phys. Fluids 25, 105102.Google Scholar
Snyder, H. A. 1962 Experiments on the stability of spiral flow at low axial Reynolds numbers. Proc. R. Soc. Lond. A 265 (1321), 198–214.Google Scholar
Takeuchi, D. I. & Jankowski, D. F. 1981 A numerical and experimental investigation of the stability of spiral Poiseuille flow. J. Fluid Mech. 102, 101–126.Google Scholar
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–218.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Tsameret, A. & Steinberg, V. 1994 Competing states in a Couette–Taylor system with an axial flow. Phys. Rev. E 49 (5), 4077–4087.CrossRefGoogle Scholar
Tsukahara, T., Kawamura, H. & Shingai, K. 2006 DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. J. Turbul. 7, 19.Google Scholar
van der Veen, R. C. A., Huisman, S., Merbold, S., Harlander, U., Egbers, C., Lohse, D. & Sun, C.2015 Taylor–Couette turbulence at radius ratio ${\it\eta}=0.5$ : scaling, flow structures and plumes (under review), arXiv:1508.05802.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
Wereley, S. T. & Lueptow, R. M. 1994 Azimuthal velocity in supercritical circular Couette flow. Exp. Fluids 18 (1–2), 1–9.Google Scholar
Wereley, S. T. & Lueptow, R. M. 1999 Velocity field for Taylor–Couette flow with an axial flow. Phys. Fluids 11 (12), 3637–3649.Google Scholar
Wu, X. & Moin, P. 2008 A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112.Google Scholar
Zhu, X., Ostilla-Mónico, R., Verzicco, R. & Lohse, D.2015 Direct numerical simulations of Taylor–Couette flow with grooved cylinders (under review), arXiv:1510.01608.Google Scholar