Research Papers

Topological Chaos by Pseudo-Anosov Map in Cavity Laminar Mixing

[+] Author and Article Information
Baiping Xu

Technology Development Center
for Polymer Processing Engineering
of Guangdong Colleges and Universities,
Guangdong Industry Technical College,
Guangzhou 510641, China
e-mail: xubaiping2003@163.com

Lih-Sheng Turng

Polymer Engineering Center,
University of Wisconsin–Madison,
Madison, WI 53706
e-mail: turng@engr.wisc.edu

Huiwen Yu, Meigui Wang

Technology Development Center
for Polymer Processing Engineering
of Guangdong Colleges and Universities,
Guangdong Industry Technical College,
Guangzhou 510641, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received December 2, 2013; final manuscript received January 27, 2014; published online January 12, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(2), 021013 (Mar 01, 2015) (8 pages) Paper No: CND-13-1309; doi: 10.1115/1.4026634 History: Received December 02, 2013; Revised January 27, 2014; Online January 12, 2015

A numerical investigation was carried out to study the mixing behavior of Stokes flows in a rectangular cavity stirred by three square rods. The square loops of the rods move in such a way that a pseudo-Anosov map can be built in the flow domain in the augmented phase space. The finite volume method was used, and the flow domain was meshed by staggered grids with the periodic boundary conditions of the rod motion being imposed by the mesh supposition technique. Fluid particle tracking was carried out by a fourth-order Runge–Kutta scheme. Tracer stretches from different initial positions were used to evaluate interface prediction by a pseudo-Anosov map. The colored short period Poincaré section was obtained to reveal the size of the domain in which the pseudo-Anosov map was in effect. Dye advection patterns were used to analyze chaotic advection of passive tracer particles using statistical concepts such as “variances” and “complete spatial randomness.” For the fluid in the core region of the cavity, tracer interface stretches experienced exponential increases and had the same power index as that predicted by the pseudo-Anosov map matrix.

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Aref, H., 1984, “Stirring by Chaotic Advection,” J. Fluid Mech., 143, pp. 1–21. [CrossRef]
Aref, H., 2002, “The Development of Chaotic Advection,” Phys. Fluids, 14, pp. 1315–1325. [CrossRef]
Finn, M. D., Cox, S. M., and Byrne, H. M., 2003, “Topological Chaos in Inviscid and Viscous Mixers,” J. Fluid Mech., 493, pp. 345–361. [CrossRef]
Jana, S. C., Metcalfe, G., and Ottino, J. M., 1994, “Experimental and Computational Studies of Mixing in Complex Stokes Flows: The Vortex Mixing Flow and Multicellular Cavity Flows,” J. Fluid Mech., 269, pp. 199–246. [CrossRef]
Kusch, H. A., and Ottino, J. M., 1992, “Experiments on Mixing in Continuous Chaotic Flows,” J. Fluid Mech., 236, pp. 319–348. [CrossRef]
Meleshko, V. V., Galaktionov, O. S., Peters, G. W. M., and Meijer, H. E. H., 1999, “Three Dimensional Mixing in Stokes Flows: The Partitioned Pipe Mixer Problem Revised,” Eur. J. Mech. Fluids, 18, pp. 783–792. [CrossRef]
Sivasamy, J., Che, Z., Wong, T. N., Nguyen, N.-T., and Yobas, L., 2010, “A Simple Method for Evaluating and Predicting Chaotic Advection in Microfluidic Slugs,” Chem. Eng. Sci.65, pp. 5382–5391. [CrossRef]
Ottino, J. M., 1989, The Kinematics of Mixing: Stretching, Chaos, and Transport, 1st ed., Cambridge University, Cambridge, UK.
Vikhansky, A., 2002, “Enhancement of Laminar Mixing by Optimal Control Methods,” Chem. Eng. Sci., 57, pp. 2719–2725. [CrossRef]
Sturman, R., Ottino, J. M., and Wiggins, S., 2006, The Mathematical Foundations of Mixing, 1st ed., Cambridge University, Cambridge, UK.
Boyland, P. L., Aref, H., and Stremler, M. A., 2000, “Topological Fluid Mechanics of Stirring,” J. Fluid Mech., 403, pp. 277–304. [CrossRef]
Boyland, P. L., Stremler, M. A., and Aref, H., 2003, “Topological Fluid Mechanics of Point Vortex Motions,” Physica D, 175, pp. 69–95. [CrossRef]
Thurston, W., 1988, “On the Geometry and Dynamics of Diffeomorphisms of Surfaces,” Bull. Ser., Am. Math. Soc., 19, pp. 417–431. [CrossRef]
Vikhansky, A., 2003, “Simulation of Topological Chaos in Laminar Flows,” Chaos, 14(1), pp. 14–22. [CrossRef]
Clifford, M. J., and Cox, S. M., 2006, “Smart Baffle Placement for Chaotic Mixing,” Nonlinear Dyn., 43, pp. 117–126. [CrossRef]
Finn, M. D., Thiffeault, J.-L., and Gouillart, E., 2006, “Topological Chaos in Spatially Periodic Mixers,” Physica D, 221, pp. 92–100. [CrossRef]
Stremler, M. A., and Chen, J., 2007, “Generating Topological Chaos in Lid-Driven Cavity Flow,” Phys. Fluids, 19, pp. 1–6. [CrossRef]
Kang, T. G., and Kwon, T. H., 2004, “Colored Particle Tracking Method for Mixing Analysis of Chaotic Micromixer,” J. Micromec. Microeng., 14, pp. 891–899. [CrossRef]
Phelps, J. H. L., and Tucker, C., III, 2006, “Lagrangian Particle Calculations of Distributive Mixing: Limitations and Applications,” Chem. Eng. Sci., 61, pp. 6826–6836. [CrossRef]


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Fig. 1

Cavity flow driven by moving rods

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Fig. 2

Location of the rods and variables on the finite volume grid at time t = 0

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Fig. 3

Contrast of x-directional velocity distributions at t = 0 along the vertical line x = 1.5 for different grids

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Fig. 4

Streamline plot of the velocity field at (a) t = 0, (b) t = 0.4, (c) t = 0.8, and (d) t = 1.2

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Fig. 5

Comparisons of particle tracing precision for different integration time intervals. (a) x coordinates of the p1 particle and (b) x coordinates of the p2 particle.

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Fig. 6

Numerical tracer s1 advection simulation for different full periods of rod motion: (a) t = T, (b) t = 1.5T, (c) t = 2T, and (d) local view for t = 2T

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Fig. 7

Contrast of different tracer evolutions for different full periods of rod motions: (a) tracers s0, s2, s3, and s4t = 0,T, (b) tracer s2 t = 2T, (c) tracer s3t = 2T, (d) tracer s4t = 2T, (e) tracer s4t = 5T, and (f) tracer s0t = 1.5T

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Fig. 8

Contrast of growth of tracer stretching and topology prediction

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Fig. 9

Colored short period Poincaré sections for t = 8T

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Fig. 10

Pseudoperiodic motions of particles in the upper right corner

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Fig. 11

Dye advection pattern for squares centered at p1 for four full periods: (a) t = 0, (b) t = 2T, (c) t = 3T, and (d) t = 4T

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Fig. 12

Dye advection pattern for the square slug centered at p2: (a) t = 0 and (b) t = 3T

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Fig. 13

Variance indexes for two sets of particles




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