Skip to main content Accessibility help
×
Home

On the stages of vortex decay in an impulsively stopped, rotating cylinder

  • Frieder Kaiser (a1), Bettina Frohnapfel (a1), Rodolfo Ostilla-Mónico (a2), Jochen Kriegseis (a1), David E. Rival (a3) and Davide Gatti (a1)...

Abstract

The flow within an infinitely long cylinder exhibiting solid-body rotation (SBR) is impulsively stopped. The complete decay of the initial SBR is captured by means of direct numerical simulations for a wide range of Reynolds numbers ( $Re$ ). Five distinct stages are identified during the decay process according to their flow structure and their underlying mechanisms of kinetic-energy dissipation. Initially, the laminar boundary layer undergoes a primary centrifugal instability, which causes the formation of coherent Taylor rolls. The flow then becomes turbulent, once the Taylor rolls are corrupted by secondary instabilities. Within the turbulent stage, two phases are distinguished. In the first turbulent phase, the SBR core is still intact and turbulence is sustained. The mean velocity profile is well described by the superposition of a near-wall region, a retracting SBR core and an intermediate region of constant angular momentum. In the latter region, the magnitude of angular momentum in viscous units $l^{+}(Re)$ is approximately constant in time. In the second turbulent phase, the SBR core breaks down, turbulence starts to decay exponentially and the kinetic energy of the mean flow decays logarithmically. Eventually, the flow relaminarises and the velocity profile of the analytical solution for purely laminar decay is recovered, albeit at an earlier temporal instant due to the net effect of transition and turbulence.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the stages of vortex decay in an impulsively stopped, rotating cylinder
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the stages of vortex decay in an impulsively stopped, rotating cylinder
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the stages of vortex decay in an impulsively stopped, rotating cylinder
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email addresses for correspondence: frieder.kaiser@kit.edu, davide.gatti@kit.edu

References

Hide All
Barlow, R. S. & Johnston, J. P. 1988 Structure of a turbulent boundary layer on a concave surface. J. Fluid Mech. 191, 137176.
Bilson, M. & Bremhorst, K. 2007 Direct numerical simulation of turbulent Taylor–Couette flow. J. Fluid Mech. 579, 227270.
Bippes, H. 1972 Experimentelle Untersuchung des laminar-turbulenten Umschlags an einer parallel angestroemten konkaven Wand. In Sitzungsberichte der Heidelberger Akademie der Wissenschaften Mathematisch naturwissenschaftliche Klasse, pp. 103180. Springer.
Brauckmann, H. J. & Eckhardt, B. 2013 Direct numerical simulations of local and global torque in Taylor–Couette flow up to Re = 30 000. J. Fluid Mech. 718, 398427.
Brunton, S. L. & Rowley, C. W. 2010 Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20 (1), 017503.
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121 (787), 402420.
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221250.
Euteneuer, G.-.A. 1969 Störwellenlängen-Messung bei Längswirbeln in laminaren Grenzschichten an konkav gekrümmten Wänden. Acta Mech. 7 (2–3), 161168.
Euteneuer, G.-A. 1972 Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden. Acta Mech. 13 (3–4), 215223.
Fabbiane, N.2011 An innovative DNS code for high-$Re$ turbulent pipe flow. Master’s thesis, Politecnico di Milano, http://hdl.handle.net/10589/33202.
Fardin, M. A., Perge, C. & Taberlet, N. 2014 The hydrogen atom of fluid dynamics–introduction to the Taylor–Couette flow for soft matter scientists. Soft Matt. 10 (20), 35233535.
Floryan, J. J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316324.
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28 (3), 235271.
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21 (4), 250252.
Green, M. A., Rowley, C. W. & Haller, G. 2007 Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111120.
Grossmann, S., Lohse, D. & Sun, C. 2016 High–Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 5380.
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357375.
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137162.
Hoffmann, P. H., Muck, K. C. & Bradshaw, P. 1985 The effect of concave surface curvature on turbulent boundary layers. J. Fluid Mech. 161, 371403.
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20 (10), 101511.
Hunt, I. A . & Joubert, P. N. 1979 Effects of small streamline curvature on turbulent duct flow. J. Fluid Mech. 91 (4), 633659.
Jiménez, J., Hoyas, S., Simens, M. P. & Mizuno, Y. 2010 Turbulent boundary layers and channels at moderate Reynolds numbers. J. Fluid Mech. 657, 335360.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kim, M. C. & Choi, C. K. 2006 The onset of Taylor–Görtler vortices during impulsive spin-down to rest. Chem. Engng Sci. 61 (19), 64786485.
Kim, M. C., Song, K. H. & Choi, K. C. 2008 Energy stability analysis for impulsively decelerating swirl flows. Phys. Fluids 20 (6), 064101.
Kitoh, O. 1991 Experimental study of turbulent swirling flow in a straight pipe. J. Fluid Mech. 225, 445479.
Kozul, M., Chung, D. & Monty, J. P. 2016 Direct numerical simulation of the incompressible temporally developing turbulent boundary layer. J. Fluid Mech. 796, 437472.
Lathrop, D. P., Fineberg, J. & Swinney, H. L. 1991 Transition to shear-driven turbulence in Couette–Taylor flow. Phys. Rev. A 46, 63906405.
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103 (1), 1642.
Lewis, H. & Bellan, P. 1990 Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates. J. Math. Phys. 31 (11), 25922596.
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77100.
Luchini, P. & Bottaro, A. 2001 Linear stability and receptivity analyses of the Stokes layer produced by an impulsively started plate. Phys. Fluids 13 (6), 16681678.
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 1544.
Marcus, P. S. 1984 Simulation of Taylor–Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave. J. Fluid Mech. 146, 65113.
Mascotelli, L.2016 Direct numerical simulations of a turbulent pipe flow at high Reynolds numbers. Master’s thesis, Politecnico di Milano, http://hdl.handle.net/10589/123696.
Mathis, D. M. & Neitzel, G. P. 1985 Experiments on impulsive spin-down to rest. Phys. Fluids 28 (2), 449454.
Maxworthy, T. 1971 A simple observational technique for the investigation of boundary layer stability and turbulence. In Turbulence Measurements in Liquids (ed. Paterson, G. K. & Zakin, J. L.), University of Missouri.
Meroney, R. N. & Bradshaw, P. 1975 Turbulent boundary-layer growth over a longitudinally curved surface. AIAA J. 13 (11), 14481453.
Moser, R. D. & Moin, P. 1987 The effects of curvature in wall-bounded turbulent flows. J. Fluid Mech. 175, 479510.
Neitzel, G. P. 1982 Marginal stability of impulsively initiated Couette flow and spin-decay. Phys. Fluids 25 (2), 226232.
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., Verzicco, R., Grossmann, S. & Lohse, D. 2016 The near-wall region of highly turbulent Taylor–Görtler flow. J. Fluid Mech. 788, 95117.
Ostilla-Mónico, R., Zhu, X., Spandan, V., Verzicco, R. & Lohse, D. 2017 Life stages of wall-bounded decay of Taylor–Görtler turbulence. Phys. Rev. Fluids 2 (11), 114601.
Panton, R. L. 1991 Scaling laws for the angular momentum of a completely turbulent Couette flow. C. R. Acad. Sci. Paris II 315 (12), 14671473.
Rayleigh, L. 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93 (648), 148154.
Ren, J. & Fu, S. 2015 Secondary instabilities of Görtler vortices in high-speed boundary layer flows. J. Fluid Mech. 781, 388421.
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379409.
Schlatter, P., Örlü, R., Li, Q., Brethouwer, G., Fransson, J. H. M., Johansson, A. V., Alfredsson, P. H. & Henningson, D. S. 2009 Turbulent boundary layers up to Re𝜃 = 2500 studied through simulation and experiment. Phys. Fluids 21 (5), 051702.
Schlichting, H. 1979 Boundary Layer Theory. McGraw-Hill.
Schrader, L. U., Brandt, L. & Zaki, T. A. 2011 Receptivity, instability and breakdown of Görtler flow. J. Fluid Mech. 682, 362396.
Sescu, A. & Afsar, M. Z. 2018 Hampering Görtler vortices via optimal control in the framework of nonlinear boundary region equations. J. Fluid Mech. 848, 541.
Shadden, S. C., Lekien, F. & Marsden, J. E. 2005 Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212 (3–4), 271304.
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 25 (10), 105102.
da Silva, C. B., Hunt, J. C. R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46, 567590.
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255290.
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289343.
Verschoof, R. A., Huisman, S. G., van der Veen, R., Sun, C. & Lohse, D. 2016 Self-similar decay of high Reynolds number Taylor–Couette turbulence. Phys. Rev. Fluids 1 (6), 062402.
Verschoof, R. A., te Nijenhuis, A. K., Huisman, S. G., Sun, C. & Lohse, D. 2018 Periodically driven Taylor–Couette turbulence. J. Fluid Mech. 846, 834845.
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 541.
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66100.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Type Description Title
VIDEO
Movies

Kaiser et al. supplementary movie 1
Transition to turbulence for Re=12000: (a) boundary-layer thickness and wall-shear stress; (b) turbulent and mean kinetic energy; (c) energy distribution inside the boundary layer, red dashed line marks the critical wavenumber. The purple line in (a,b,c) depicts the time instance visualized in (d,e). The vertical dashed lines are described in the manuscript in figure 7; (d) one-dimensional spectrum inside the boundary layer. The black dashed-dotted line marks the critical wavenumber for Re=12000, the orange line marks the wavenumber, which would be the most energetic wavenumber if the streamwise vortices were circular and would extend over the boundary-layer thickness; and (e) velocity fluctuations. The orange line marks the boundary-layer thickness, red line (appears after transition to turbulence) marks the inner limit of the region of constant angular momentum.

 Video (13.4 MB)
13.4 MB
VIDEO
Movies

Kaiser et al. supplementary movie 2
FTLE visualisations of the transition process to turbulence at Re = 12000.

 Video (5.2 MB)
5.2 MB
VIDEO
Movies

Kaiser et al. supplementary movie 3
(a,b) Mean velocity profile and Reynolds stresses in outer (a) and wall-based (b) units. (c,d) Terms of the mean (c) and turbulent (d) kinetic energy budget equations.

 Video (3.3 MB)
3.3 MB

On the stages of vortex decay in an impulsively stopped, rotating cylinder

  • Frieder Kaiser (a1), Bettina Frohnapfel (a1), Rodolfo Ostilla-Mónico (a2), Jochen Kriegseis (a1), David E. Rival (a3) and Davide Gatti (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