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,
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.

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Fig. 1

Helically coiled pipe and twisted pipe [17]

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Fig. 2

Geometry of the twisted duct

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Fig. 3

Mixing enhancement by vorticity [21]

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Fig. 4

Directions of the vorticity vector components

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Fig. 5

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

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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

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Fig. 7

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

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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.

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Fig. 9

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

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Fig. 10

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

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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

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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.

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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.

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Fig. 14

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

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Fig. 15

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

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Fig. 16

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

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Fig. 17

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

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Fig. 18

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



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