Advertisement

Formation of Coherent Structures in a Class of Realistic 3D Unsteady Flows

  • Michel F. M. Speetjens
  • Herman J. H. Clercx
Chapter
Part of the Environmental Science and Engineering book series (ESE)

Abstract

The formation of coherent structures in three-dimensional (3D) unsteady laminar flows in a cylindrical cavity is reviewed. The discussion concentrates on two main topics: the role of symmetries and fluid inertia in the formation of coherent structures and the ramifications for the Lagrangian transport properties of passive tracers. We consider a number of time-periodic flows that each capture a basic dynamic state of 3D flows: 1D motion on closed trajectories, (quasi-)2D motion within (approximately) 2D subregions of the flow domain and truly 3D chaotic advection. It is shown that these states and their corresponding coherent structures are inextricably linked to symmetries (or absence thereof) in the flow. Symmetry breaking by fluid inertia and the resulting formation of intricate coherent structures and (local) onset of 3D chaos is demonstrated. Finally, first experimental analyses on coherent structures and the underlying role of symmetries are discussed.

Keywords

Coherent Structure Periodic Point Passive Tracer Flow Topology Invariant Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Alexandroff P (1961) Elementary concepts of topology. Dover, New YorkGoogle Scholar
  2. Anderson PA, Galaktionov OS, Peters GWM, van de Vosse FN, Meijer HEH (1999) Analysis of mixing in three-dimensional time-periodic cavity flows. J Fluid Mech 386:149CrossRefGoogle Scholar
  3. Anderson PA, Ternet TJ, Peters GWM, Meijer HEH (2006) Experimental/numerical analysis of chaotic advection in a three-dimensional cavity flow. Int Polym Process 4:412Google Scholar
  4. Arnol’d VI (1978) Mathematical methods of classical mechanics. Springer, New YorkGoogle Scholar
  5. Arnol’d VI, Khesin BA (1991) Topological methods in hydrodynamics. Springer, New YorkGoogle Scholar
  6. Bennet A (2006) Lagrangian fluid dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  7. Biskamp D (1993) Nonlinear magnetohydrodynamics. Cambridge University Press, CambridgeGoogle Scholar
  8. Cartwright JHE, Feingold M, Piro O (1996) Chaotic advection in three-dimensional unsteady incompressible laminar flow. J Fluid Mech 316:259CrossRefGoogle Scholar
  9. Dombre T, Frisch U, Greene JM, Hénon M, Mehr A, Soward AM (1986) Chaotic streamlines in the ABC flows. J Fluid Mech 167:353CrossRefGoogle Scholar
  10. Feingold M, Kadanoff LP, Piro O (1987) A way to connect fluid dynamics to dynamical systems: passive scalars. In: Hurd AJ, Weitz DA, Mandelbrot BB (eds) Fractal aspects of materials: disordered systems. Materials Research Society, Pittsburgh, pp 203–205Google Scholar
  11. Feingold M, Kadanoff LP, Piro O (1988) Passive scalars, three-dimensional volume-preserving maps and chaos. J Stat Phys 50:529CrossRefGoogle Scholar
  12. Franjione JG, Leong C-W, Ottino JM (1989) Symmetries within chaos: a route to effective mixing. Phys Fluids A 11:1772CrossRefGoogle Scholar
  13. Gómez A, Meiss JD (2002) Volume-preserving maps with an invariant. Chaos 12:289CrossRefGoogle Scholar
  14. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkGoogle Scholar
  15. Haller G, Mezić I (1998) Reduction of three-dimensional, volume-preserving flows by symmetry. Nonlinearity 11:319CrossRefGoogle Scholar
  16. Luethi B, Tsinober A, Kinzelbach W (2005) Lagrangian measurement of vorticity dynamics in turbulent flow. J Fluid Mech 528:87CrossRefGoogle Scholar
  17. Malyuga VS, Meleshko VV, Speetjens M (2002) Mixing in the Stokes flow in a cylindrical container. Proc R Soc Lond A 458:1867CrossRefGoogle Scholar
  18. MacKay RS (1994) Transport in 3D volume-preserving flows. J Nonlinear Sci 4:329CrossRefGoogle Scholar
  19. Meier SW, Lueptow RM, Ottino JM (2007) A dynamical systems approach to mixing and segregation of granular materials in tumblers. Adv Phys 56:757CrossRefGoogle Scholar
  20. Meleshko VV, Peters GWM (1996) Periodic points for two-dimensional Stokes flow in a rectangular cavity. Phys Lett A 216:87CrossRefGoogle Scholar
  21. Mezić I, Wiggins S (1994) On the integrability and perturbation of three-dimensional fluid flows with symmetry. J Nonlinear Sci 4:157CrossRefGoogle Scholar
  22. Mezić I (2001) Break-up of invariant surfaces in action-angle-angle maps and flows. Physica D 154:51CrossRefGoogle Scholar
  23. Moffatt HK, Zaslavsky GM, Comte P, Tabor M (1992) Topological aspects of the dynamics of fluids and plasmas. Kluwer Academic Publishers, DordrechtGoogle Scholar
  24. Mullowney P, Julien K, Meiss JD (2008) Blinking rolls: chaotic advection in a three-dimensional flow with an invariant. SIAM J Appl Dyn Sys 4:159186Google Scholar
  25. Mullowney P, Julien K, Meiss JD (2008) Chaotic advection and the emergence of tori in the Küppers–Lortz state. Chaos 18:033104CrossRefGoogle Scholar
  26. Ottino JM (1989) The kinematics of mixing: stretching, chaos and transport. Cambridge University Press, CambridgeGoogle Scholar
  27. Ottino JM, Jana SC, Chakravarthy VS (1994) From Reynolds stretching and folding to mixing studies using horseshoe maps. Phys Fluids 6:685CrossRefGoogle Scholar
  28. Pouransari Z, Speetjens MFM, Clercx HJH (2010) Formation of coherent structures by fluid inertia in three-dimensional laminar flows. J Fluid Mech 654:5CrossRefGoogle Scholar
  29. Shankar PN (1997) Three-dimensional eddy structure in a cylindrical container. J Fluid Mech 342:97CrossRefGoogle Scholar
  30. Speetjens MFM (2001) Three-Dimensional chaotic advection in a cylindrical domain. PhD thesis, Eindhoven University of Technology, The NetherlandsGoogle Scholar
  31. Speetjens MFM, Clercx HJH, van Heijst GJF (2004) A numerical and experimental study on advection in three-dimensional Stokes flows. J Fluid Mech 514:77CrossRefGoogle Scholar
  32. Speetjens MFM, Clercx HJH, van Heijst GJF (2006) Inertia-induced coherent structures in a time-periodic viscous mixing flow. Phys Fluids 18:083603CrossRefGoogle Scholar
  33. Speetjens MFM, Clercx HJH, van Heijst GJF (2006) Merger of coherent structures in time-periodic viscous flows. Chaos 16:043104CrossRefGoogle Scholar
  34. Sturman R, Ottino JM, Wiggins S (2006) The mathematical foundation of mixing. Cambridge University Press, CambridgeGoogle Scholar
  35. Sturman R, Meier SW, Ottino JM, Wiggins S (2008) Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows. J Fluid Mech 602:129CrossRefGoogle Scholar
  36. Voth GA, Haller G, Gollub JP (2002) Experimental measurements of stretching fields in fluid mixing. Phys Rev Lett 88:254501CrossRefGoogle Scholar
  37. Wiggins S (2010) Coherent structures and chaotic advection in three dimensions. J Fluid Mech 654:1CrossRefGoogle Scholar
  38. Znaien JG, Speetjens MFM, Trieling RR, Clercx HJH (2012) On the observability of periodic lines in 3D lid-driven cylindrical cavity flows. Phys Rev E 85(6):066320–1/14CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michel F. M. Speetjens
    • 1
  • Herman J. H. Clercx
    • 2
  1. 1.Department of Mechanical EngineeringEnergy Technology and J.M. Burgers Center for Fluid DynamicsEindhovenThe Netherlands
  2. 2.Department of Applied PhysicsFluid Dynamics Laboratory and J.M. Burgers Center for Fluid DynamicsEindhovenThe Netherlands

Personalised recommendations