Hostname: page-component-6bf8c574d5-nvqbz Total loading time: 0 Render date: 2025-03-08T01:53:26.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular flow in a partially filled rotating drum: regular and chaotic advection

Published online by Cambridge University Press:  21 July 2017

Francesco Romanò Open the ORCID record for Francesco Romanò [Opens in a new window] ,
Arash Hajisharifi  and
Hendrik C. Kuhlmann
Francesco Romanò*
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria
Arash Hajisharifi
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria Department of Chemical Engineering, University of Pisa, via Diotisalvi 2, 56126 Pisa, Italy
Hendrik C. Kuhlmann
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

JFM classification

Nonlinear Dynamical Systems: Chaos Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 631 - 650
Check for updates
Copyright
© 2017 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

Aref, H. 2002 The development of chaotic advection. Phys. Fluids 14 (4), 1315–1325.Google Scholar
Arneodo, A., Coullet, P. & Tresser, C. 1982 Oscillators with chaotic behavior: an illustration of a theorem by Shil’nikov. J. Stat. Phys. 27 (1), 171–182.Google Scholar
Arter, W. 1983 Ergodic stream-lines in steady convection. Phys. Lett. 97A, 171–174.CrossRefGoogle Scholar
Bajer, K. 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos Solitons Fractals 4 (6), 895–911.Google Scholar
Balmer, R. T. 1970 The hygrocyst – a stability phenomenon in continuum mechanics. Nature 227, 600–601.Google Scholar
Biemond, J. J. B., de Moura, A. P. S., Károlyi, G., Grebogi, C. & Nijmeijer, H. 2008 Onset of chaotic advection in open flows. Phys. Rev. E 78, 016317.Google Scholar
Brøns, M., Voigt, L. K. & Sørensen, J. N. 1999 Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers. J. Fluid Mech. 401, 275–292.CrossRefGoogle Scholar
Broomhead, D. S. & Ryrie, S. C. 1988 Particle paths in wavy vortices. Nonlinearity 1 (3), 409–434.Google Scholar
Chen, P.-J., Tsai, Y.-T., Liu, T.-J. & Wu, P.-Y. 2007 Low volume fraction rimming flow in a rotating horizontal cylinder. Phys. Fluids 19 (12), 128107.Google Scholar
Chernikov, A. A. & Schmidt, G. 1992 Chaotic streamlines in convective cells. Phys. Lett. A 169, 51–56.Google Scholar
Contreras, P. S., de la Cruz, L. M. & Ramos, E. 2016 Topological analysis of a mixing flow generated by natural convection. Phys. Fluids 28 (1), 013602.Google Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Dormand, J. R. & Prince, P. J. 1980 A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6 (1), 19–26.Google Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exp. Fluids 2, 189–196.Google Scholar
Gaspard, P. & Nicolis, G. 1983 What can we learn from homoclinic orbits in chaotic dynamics? J. Stat. Phys. 31 (3), 499–518.Google Scholar
Hajisharifi, A.2016 Topology of three-dimensional steady cellular flow in a partially liquid-filled rotating drum. Master’s thesis, TU Wien and University of Pisa.Google Scholar
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4), 248–277.Google Scholar
Hofmann, E. & Kuhlmann, H. C. 2011 Particle accumulation on periodic orbits by repeated free surface collisions. Phys. Fluids 23 (7), 072106.Google Scholar
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11 (1), 97–106.Google Scholar
Kaitna, R. & Rickenmann, D. 2007a Flow of different material mixtures in a rotating drum. In Debris-Flow Hazards Mitigation, Fourth International DFHM Conference: Mechanics, Prediction and Assessment, pp. 10–13. IOS Press, STM Publishing House.Google Scholar
Kaitna, R. & Rickenmann, D. 2007b A new experimental facility for laboratory debris flow investigation. J. Hydraul. Res. 45 (6), 797–810.Google Scholar
Kroujiline, D. & Stone, H. A. 1999 Chaotic streamlines in steady bounded three-dimensional Stokes flows. Physica D 130, 105–132.Google Scholar
Kuhlmann, H. C., Romanò, F., Wu, H. & Albensoeder, S. 2016 Particle-motion attractors due to particle–boundary interaction in incompressible steady three-dimensional flows. In The 20th Australasian Fluid Mechanics Conference (ed. Ivey, G., Zhou, T., Jones, N. & Draper, S.), pp. 102‐1–102‐4.Google Scholar
Lin, Y.-Y. 1986 Numerical solutions for flow in a partially filled, rotating cylinder. SIAM J. Sci. Comput. 7 (2), 560–570.Google Scholar
Lugt, H. J. 1996 Introduction to Vortex Theory. Vortex Flow Press.Google Scholar
Malkin, B. A. 1937 The behaviour of condensate in paper machine dryers. The Dominion Engineer 4 (4), 8.Google Scholar
Marsh, A. J., Stuart, D. M., Mitchell, D. A. & Howes, T. 2000 Characterizing mixing in a rotating drum bioreactor for solid-state fermentation. Biotechnol. Lett. 22 (6), 473–477.Google Scholar
Matson, W. R., Ackerson, B. J. & Tong, P. 2003 Pattern formation in a rotating suspension of non-Brownian settling particles. Phys. Rev. E 67 (5), 050301.Google Scholar
Melo, F. 1993 Localized states in a film-dragging experiment. Phys. Rev. E 48 (4), 2704.Google Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Méc. 16 (5), 651–673.Google Scholar
Mukin, R. V. & Kuhlmann, H. C. 2013 Topology of hydrothermal waves in liquid bridges and dissipative structures of transported particles. Phys. Rev. E 88 (5), 053016.Google Scholar
Muldoon, F. H. & Kuhlmann, H. C. 2016 Origin of particle accumulation structures in liquid bridges: particle–boundary interactions versus inertia. Phys. Fluids 28, 073305.Google Scholar
Neitzel, G. P. 1984 Numerical computation of time-dependent Taylor-vortex flow in finite length geometries. J. Fluid Mech. 141, 51–66.Google Scholar
Orr, F. M. & Scriven, L. E. 1978 Rimming flow: numerical simulation of steady, viscous, free-surface flow with surface tension. J. Fluid Mech. 84 (01), 145–165.Google Scholar
Oteski, L., Duguet, Y., Pastur, L. & Quéré, P. L. 2015 Quasiperiodic routes to chaos in confined two-dimensional differential convection. Phys. Rev. E 92, 043020.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Ottino, J. M. & Khakhar, D. V. 2000 Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32 (1), 55–91.Google Scholar
Phillips, O. M. 1960 Centrifugal waves. J. Fluid Mech. 7 (03), 340–352.Google Scholar
Romanò, F., Albensoeder, S. & Kuhlmann, H. C.2017 Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity. J. Fluid Mech. (to appear).Google Scholar
Rudman, M. 1998 Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette flow. AIChE J. 44, 1015–1026.Google Scholar
Schuster, H. G. 2005 Deterministic Chaos: An Introduction. Wiley-VCH.Google Scholar
Schwabe, D., Hintz, P. & Frank, S. 1996 New features of thermocapillary convection in floating zones revealed by tracer particle accumulation structures (PAS). Microgravity Sci. Technol. 9, 163–168.Google Scholar
Seiden, G. & Thomas, P. J. 2011 Complexity, segregation, and pattern formation in rotating-drum flows. Rev. Mod. Phys. 83 (4), 1323–1365.Google Scholar
Shil’nikov, L. P. 1965 A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163–166.Google Scholar
Siegmann-Hegerfeld, T., Albensoeder, S. & Kuhlmann, H. C. 2008 Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls. Exp. Fluids 45, 781–796.Google Scholar
Sotiropoulos, F., Ventikos, Y. & Lackey, T. C. 2001 Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: Šil’nikov’s chaos and the devil’s staircase. J. Fluid Mech. 444, 257–297.Google Scholar
Speetjens, M. F. M., Clercx, H. J. H. & Van Heijst, G. J. F. 2004 A numerical and experimental study on advection in three-dimensional Stokes flows. J. Fluid Mech. 514, 77–105.Google Scholar
Speetjens, M. F. M., Demissie, E. A., Metcalfe, G. & Clercx, H. J. H. 2014 Lagrangian transport characteristics of a class of three-dimensional inline-mixing flows with fluid inertia. Phys. Fluids 26 (11), 113601.Google Scholar
Thoroddsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in a partially-filled horizontally rotating cylinder. Exp. Fluids 23 (1), 1–13.Google Scholar
Thoroddsen, S. T. & Tan, Y.-K. 2004 Free-surface entrainment into a rimming flow containing surfactants. Phys. Fluids 16 (2), L13–L16.Google Scholar
Vallette, D. P., Jacobs, G. & Gollub, J. P. 1997 Oscillations and spatio-temporal chaos of one-dimensional fluid fronts. Phys. Rev. E 55 (4), 4274–4287.Google Scholar
Weiss, J. B. 1991 Transport and mixing in traveling waves. Phys. Fluids 3 (5), 1379–1384.Google Scholar
White, R. E. 1956 Residual condensate, condensate behavior, and siphoning in paper driers. Tech. Assoc. Pulp. Paper Ind. 39, 228–233.Google Scholar
White, R. E. & Higgins, T. W. 1958 Effect of fluid properties on condensate behavior. TAPPI J. 41 (2), 71–76.Google Scholar
Yih, C.-S. & Kingman, J. F. C. 1960 Instability of a rotating liquid film with a free surface. Proc. R. Soc. Lond. A 258, 63–89.Google Scholar