0
Research Papers: Fundamental Issues and Canonical Flows

Mixing by Time-Dependent Orbits in Spatiotemporal Chaotic Advection

[+] Author and Article Information
Mohammad Karami

Department of Mechanical Engineering,
Isfahan University of Technology,
Isfahan 84156-83111, Iran

Ebrahim Shirani

Foolad Institute of Technology,
Fooladshahr, Isfahan 84916-63763, Iran

Mojtaba Jarrahi

University Paris-Sud,
LIMSI-CNRS UPR 3251,
BP 133, Orsay Cedex 91403, France

Hassan Peerhossaini

University Paris Diderot,
Sorbonne Paris Cité,
Laboratoire Interdisciplinaire des Energies de
Demain (LIED)–CNRS FRE 3597–BP 7040,
Paris Cedex 13 75205, France
e-mail: hassan.peerhossaini@univ-paris-diderot.fr

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 16, 2013; final manuscript received May 1, 2014; published online September 10, 2014. Assoc. Editor: Ye Zhou.

J. Fluids Eng 137(1), 011201 (Sep 10, 2014) (13 pages) Paper No: FE-13-1669; doi: 10.1115/1.4027588 History: Received November 16, 2013; Revised May 01, 2014

The simultaneous effects of flow pulsation and geometrical perturbation on laminar mixing in curved ducts have been numerically studied by three different metrics: analysis of the secondary flow patterns, Lyapunov exponents and vorticity vector analysis. The mixer that creates the flow pulsation and geometrical perturbations in these simulations is a twisted duct consisting of three bends; the angle between the curvature planes of successive bends is 90 deg. Both steady and pulsating flows are considered. In the steady case, analysis of secondary flow patterns showed that homoclinic connections appear and become prominent at large Reynolds numbers. In the pulsatile flow, homoclinic and heteroclinic connections appear by increasing β, the ratio of the peak oscillatory velocity component of the mean flow velocity. Moreover, sharp variations in the secondary flow structure are observed over an oscillation cycle for high values of β. These variations are reduced and the homoclinic connections disappear at high Womersley numbers. We show that small and moderate values of the Womersley number (6 ≤ α ≤ 10) and high values of velocity amplitude ratio (β ≥ 2) provide a better mixing than that in the steady flow. These results correlate closely with those obtained using two other metrics, analysis of the Lyapunov exponents and vorticity vector. It is shown that the increase in the Lyapunov exponents, and thus mixing enhancement, is due to the formation of homoclinic and heteroclinic connections.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Dean, W. R., 1927, “Note on the Motion of Fluid in a Curved Pipe,” Philos. Mag., 4(20), pp. 208–223. [CrossRef]
Mutabazi, I., Normand, C., Peerhossaini, H., and Wesfreid, J. E., 1989, “Oscillatory Modes Between Two Horizontal Corotating Cylinders With a Partially Filled Gap,” Phys. Rev. A, 39(2), pp. 763–771. [CrossRef] [PubMed]
Le Guer, Y., and Peerhossaini, H., 1991, “Order Breaking in Dean Flow,” Phys. Fluids, 3(5), pp. 1029–1032. [CrossRef]
Duchêne, C., Peerhossaini, H., and Michard, P. J., 1995, “On the Flow Velocity and Tracer Patterns in a Twisted Duct Flow,” Phys. Fluids, 7(6), pp. 1307–1317. [CrossRef]
Lyne, W. H., 1971, “Unsteady Viscous Flow in a Curved Pipe,” J. Fluid Mech., 45(1), pp. 13–31. [CrossRef]
Zalosh, R. G., and Nelson, W. G., 1973, “Pulsating Flow in a Curved Tube,” J. Fluid Mech., 59(4), pp. 693–705. [CrossRef]
Bertelsen, A. F., 1975, “An Experimental Investigation of Low Reynolds Number Secondary Streaming Effects Associated With an Oscillating Viscous Flow in a Curved Pipe,” J. Fluid Mech., 70(3), pp. 519–527. [CrossRef]
Sudo, K., Sumida, M., and Yamane, R., 1992, “Secondary Motion of Fully Developed Oscillatory Flow in a Curved Pipe,” J. Fluid Mech., 237, pp. 189–208. [CrossRef]
Timité, B., Castelain, C., and Peerhossaini, H., 2010, “Pulsatile Viscous Flow in a Curved Pipe: Effects of Pulsation on the Development of Secondary Flow,” Int. J. Heat Fluid Flow, 31(5), pp. 879–896. [CrossRef]
Achrya, N., Sen, M., and Chang, H. C., 1992, “Heat Transfer Enhancement in Coiled Tubes by Chaotic Mixing,” Int. J. Heat Mass Transfer, 35(10), pp. 2475–2489. [CrossRef]
Achrya, N., Sen, M., and Chang, H. C., 1994, “Thermal Entrance Length and Nusselt Numbers in Coiled Tubes,” Int. J. Heat Mass Transfer, 37(2), pp. 336–340. [CrossRef]
Peerhossaini, H., Castelain, C., and Le Guer, Y., 1993, “Heat Exchanger Design Based on Chaotic Advection,” Exp. Therm. Fluid Sci., 7(4), pp. 333–344. [CrossRef]
Mokrani, A., Castelain, C., and Peerhossaini, H., 1997, “The Effects of Chaotic Advection on Heat Transfer,” Int. J. Heat Mass Transfer, 40(13), pp. 3089–3104. [CrossRef]
Lasbet, Y., Auvity, B., Castelain, C., and Peerhossaini, H., 2006, “A Chaotic Heat Exchanger for PEMFC Cooling Applications,” J. Power Sources, 156(1), pp. 114–118. [CrossRef]
Lemenand, T., and Peerhossaini, H., 2002, “A Thermal Model for Prediction of the Nusselt Number in a Pipe With Chaotic Flow,” Appl. Therm. Eng., 22(15), pp. 1717–1730. [CrossRef]
Habchi, C., Lemenand, T., Della Valle, D., and Peerhossaini, H., 2010, “Liquid/Liquid Dispersion in a Chaotic Advection Flow,” Int. J. Multiphase Flow, 35(6), pp. 485–497. [CrossRef]
Nguyen, N., 2008, Micromixers: Fundamentals, Design and Fabrication, William Andrew, New York, pp. 42–52.
Siginer, D. A., 1991, “Multiple Integral Constitutive Equations in Unsteady Motions and Rheometry,” ASME Appl. Mech. Rev., 44(11S), pp. 232–245. [CrossRef]
Wang, G. R., 2003, “A Rapid Mixing Process in Continuous Operation Under Periodic Forcing,” Chem. Eng. Sci., 58(22), pp. 4953–4963. [CrossRef]
Timité, B., Jarrahi, M.Castelain, C. and Peerhossaini, H., 2009, “Pulsating Flow for Mixing Intensification in a Twisted Curved Pipe,” ASME J. Fluids Eng., 131(12), p. 121104. [CrossRef]
Jarrahi, M., Castelain, C., and Peerhossaini, H., 2011, “Secondary Flow Patterns and Mixing in Laminar Pulsating Flow Through a Curved Pipe,” Exp. Fluids, 50(6), pp. 1539–1558. [CrossRef]
Timité, B., Castelain, C., and Peerhossaini, H., 2011, “Mass Transfer and Mixing by Pulsatile Three-Dimensional Chaotic Flow in Alternating Curved Pipes,” Int. J. Heat Mass Transfer, 54(17–18), pp. 3933–3950. [CrossRef]
Ansari, M. A., and Kim, K. Y., 2009, “Parametric Study on Mixing of Two Fluids in a Three-Dimensional Serpentine Microchannel,” Chem. Eng. J., 146(3), pp. 439–448. [CrossRef]
Yamagishi, A., Inaba, T., and Ymaguchi, Y., 2007, “Chaotic Analysis of Mixing Enhancement in Steady Laminar Flows Through Multiple Pipe Bends,” Int. J. Heat Mass Transfer, 50(7–8), pp. 1237–1248. [CrossRef]
Jones, S. W., Thomas, O. M., and Aref, H., 1989, “Chaotic Advection by Laminar Flow in a Twisted Pipe,” J. Fluid Mech., 209, pp. 335–357. [CrossRef]
Fluent, Inc. Fluent 6.3, 2006, User's Guide.
Strasser, W., 2006, “CFD Investigation of Gear Pump Mixing Using Deforming/Agglomerating Mesh,” ASME J. Fluids Eng., 129, pp. 476–484. [CrossRef]
Jarrahi, M., Castelain, C., and Peerhossaini, H., 2011, “Laminar Sinusoidal and Pulsatile Flows in a Curved Pipe,” J. Appl. Fluid Mech., 4(2SI), pp. 21–26.
Ottino, J. M., 1989, The Kinematic of Mixing, Cambridge University Press, Cambridge, UK, Ch. 8.
Mancho, A. M., Small, D., and Wiggins, S., 2006, “A Tutorial on Dynamical Systems Concepts Applied to Lagrangian Transport in Oceanic Flows Defined as Finite Time Data Sets: Theoretical and Computational Issues,” Phys. Rep., 437(3–4), pp. 55–124. [CrossRef]
Castelain, C., Mokrani, A., Le Guer, Y., and Peerhossaini, H., 2001, “Experimental Study of Chaotic Advection Regime in a Twisted Duct Flow,” Eur. J. Mech., 20(2), pp. 205–232. [CrossRef]
Mokrani, A., Castelain, C., and Peerhossaini, H., 1998, “A Quantitative Measure of the Chaotic Behavior in Conservative Systems: A Study of Transport Phenomena in Open Systems,” Revue Generale de Thermique, 37(6), pp. 459–474. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Helically coiled pipe and twisted pipe [17]

Grahic Jump Location
Fig. 2

Geometry of the twisted duct

Grahic Jump Location
Fig. 3

Mixing enhancement by vorticity [21]

Grahic Jump Location
Fig. 4

Directions of the vorticity vector components

Grahic Jump Location
Fig. 5

(a) Grid independency test in the steady flow and (b) the effects of time step in the pulsating flow

Grahic Jump Location
Fig. 6

Comparison of axial velocity profiles in steady flow (Re = 830) with experimental results [31]: (a) in median plane at outlet of first bend, (b) in plane perpendicular to the median plane at outlet of second bend, and (c) in plane perpendicular to the median plane at outlet of third bend

Grahic Jump Location
Fig. 7

Axial velocity profiles at the outlet of different bends in steady flow (a) Re = 400, (b) Re = 600, and (c) Re = 830

Grahic Jump Location
Fig. 8

Secondary flow patterns at the outlet of different bends in steady flow. Secondary flow structures that contain homoclinic connections are bordered in gray.

Grahic Jump Location
Fig. 9

Divergence of particle trajectories in streamwise direction in steady flow (400 ≤ Re ≤ 830)

Grahic Jump Location
Fig. 10

Effect of Reynolds number on the total vorticity vector components in steady flow

Grahic Jump Location
Fig. 11

Axial velocity profiles at outlet of third bend in pulsating flow for Re = 400 and (a) β = 1, α = 6, (b) β = 1, α = 20, (c) β = 2, α = 6, and (d) β = 2, α = 20

Grahic Jump Location
Fig. 12

Singularity points in the secondary flow structure at outlet of third bend for β = 1, α = 10, Re = 600, ω.t = 270 deg. E, H, WS, and WU denote, respectively, the elliptic points, hyperbolic points, stable manifolds, and unstable manifolds. Dark gray and light gray lines represent, respectively, the homoclinic and heteroclinic connections.

Grahic Jump Location
Fig. 13

Secondary flow structures in the pulsating flow at the outlet of the third bend for β = 1, 400 ≤ Re ≤ 830, 6 ≤ α ≤ 20. Patterns consisting of homoclinic and heteroclinic connections are marked, respectively, with dark gray and light gray borders.

Grahic Jump Location
Fig. 14

Secondary flow structures at the outlet of different bends for Re = 400, 6 ≤ α ≤ 20 and 2 ≤ β ≤ 3

Grahic Jump Location
Fig. 15

Effects of variations in β on the divergence of fluid particle trajectories for Re = 400, α = 10

Grahic Jump Location
Fig. 16

Effects of variations in α on the divergence of fluid particle trajectories for Re = 400: (a) β = 1 and (b) β = 2

Grahic Jump Location
Fig. 17

Effects of variations in Re on the divergence of fluid particle trajectories for β = 1 and α = 10

Grahic Jump Location
Fig. 18

Effects of pulsation on the vorticity vector components for (a) β = 1 and (b) Re = 400

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In