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The motion of a single and multiple neutrally buoyant elliptical cylinders in plane Poiseuille flow
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10.1063/1.4757387
/content/aip/journal/pof2/24/10/10.1063/1.4757387
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/10/10.1063/1.4757387

Figures

Image of FIG. 1.
FIG. 1.

(a) Model for validation: Schematic of single rigid elliptical cylinder in plane Couette flow. (b) Focus problem of this study: Schematic of multiple rigid elliptical cylinders in plane Poiseuille flow with the domain of computation D * = (L * × 2H * ).

Image of FIG. 2.
FIG. 2.

Validation of the computational results: The figure shows comparisons in both angle of inclination and angular velocity of an elliptical cylinder. Orientation θ/π versus time t: (a) our result (solid line) at Re p = 0.08; (b) Ding's result (dashed line) at Re p = 0.08; (c) our result at Re p = 1; (d) Ding's result at Re p = 1. Angular velocity ω/π versus time t: (e) our result at Re p = 0.08; (f) Ding's result at Re p = 0.08; (g) our result at Re p = 1; (h) Ding's result at Re p = 1. Jeffery's solutions for θ p versus t: J θ = θ/π, and for ω versus t: J ω = ω/π at Re p = 0 are also plotted for comparisons. The domain of computation is D = (5 × 1); the size ratio K = 0.2 for this study and K = 0.1 for Ding's work; the aspect ratio A = 2.

Image of FIG. 3.
FIG. 3.

Normalized minimum angular velocity ω min /π (straight line) and period of rotation T (curve) versus Re p of an elliptical cylinder. It is noted that T increases to infinity as Re p approaches the critical value Re p,c ∼ 29. The domain of computation is D = (5 × 1); the size ratio K = 0.2 for this study and K = 0.1 for Ding's work; the aspect ratio A = 2.

Image of FIG. 4.
FIG. 4.

Temporal development of lateral migration of an elliptical cylinder with different A and K at Re = 10.

Image of FIG. 5.
FIG. 5.

(a) Translational velocity U p versus time t of an elliptical cylinder for different A and K at Re = 10. (b) The local plot at 2400 ≤ t ≤ 2450.

Image of FIG. 6.
FIG. 6.

Average translational velocity U p, avg and its amplitude of oscillation U p, osc versus the aspect ratio A at Re = 10 when it fluctuates about the averaged equilibrium position. Amplitude of oscillation is marked as the error bar.

Image of FIG. 7.
FIG. 7.

(a) Angular velocity ω/π versus time t between 2490 < t < 2500; (b) averaged angular velocity ω s /π versus time t of an elliptical cylinders for different aspect ratios A at Re = 10.

Image of FIG. 8.
FIG. 8.

Maximum angular velocity ω max/π and minimum angular velocity ω min/π for an elliptical cylinder of A = 3 at Re = 10 for different K = 0.2, 0.4, 0.5, 0.6. Jeffery's solutions are also plotted for comparisons. The notation [θ/π] denotes the decimal part of θ/π calculated in the present method.

Image of FIG. 9.
FIG. 9.

(a) Equilibrium position Y eq versus Reynolds number Re for the elliptical cylinder with A = 3.333, K = 0.2 (▲), and A = 1.875, K = 0.15 (■); and the circular cylinder with K = 0.11 (●). (b) The present results compared to a sphere by Schonberg and Hinch17 (□), a circular cylinder with K = 0.1 (○) and a sphere with K = 0.1 (⋄) by Yang et al.,32,37 and a lager circular cylinder with K = 0.25 (Δ) by Feng et al. 36

Image of FIG. 10.
FIG. 10.

Temporal development of lateral migration Y p of an elliptical cylinder with A = 1.875, K = 0.15 at different Re. The curves at t = 2500 from the bottom to the top correspond to Re = 10, 20, 40, 100, 200, 1000, respectively.

Image of FIG. 11.
FIG. 11.

The amplitude of oscillation Y osc versus Re of an elliptical cylinder when it fluctuates about the averaged equilibrium position for (1) K = 0.15 and A = 1.875 (○), (2) K = 0.2 and A = 3.333 (□), and (3) K = 0.4 and A = 3 (⋄).

Image of FIG. 12.
FIG. 12.

Angular velocity ω/π versus time t of an elliptical cylinder with (a) small size ratio K = 0.4 and aspect ratio A = 3, (b) large size ratio K = 0.76 and aspect ratio A = 2 at different Re p . Note that for the smaller K = 0.4, the elliptical cylinder keeps rotating at Re p = 4–40, but oscillates in orientation at Re p = 80. For the larger K = 0.76, the elliptical cylinder oscillates in orientation at Re p = 30–380, but becomes stationary in orientation with θ = 0 at Re p = 418.

Image of FIG. 13.
FIG. 13.

The maximum and minimum normalized angles of inclination θ max/π, θ min/π versus Re for the elliptical cylinders with K = 0.4, A = 3 and K = 0.76, A = 2, respectively.

Image of FIG. 14.
FIG. 14.

Lateral migration of an elliptical cylinder with (a) small size ratio K = 0.4 and aspect ratio A = 3, (b) large size ratio K = 0.76 and aspect ratio A = 2 at different Re p . Note that for the smaller K = 0.4, the center-of-mass is oscillating mildly about an equilibrium position for Re p = 4–40, but will eventually oscillate about the channel center at the higher Re p = 80. For the larger K = 0.76, the elliptical cylinder oscillates about the channel center for Re p = 30–380, and eventually moves to the channel center at the higher Re p = 418.

Image of FIG. 15.
FIG. 15.

(a) The initial positions of ND = 16 (ϕ T = 3.77%) cylinders with A = 1.875 and K = 0.15, the initial setup is s x = 0.25, s y = 0.25, and random θ 0. Temporal development of lateral migration Y p of ND = 16 cylinders at (b) Re = 10, (c) Re = 100, and (d) Re = 1000.

Image of FIG. 16.
FIG. 16.

(a) The initial positions of ND = 16 (ϕ T = 3.77%) cylinders with A = 1.875 and K = 0.15, the initial setup is s x = 0.5, s y = 0.2, and θ 0 = π/4. (b) Temporal development of lateral migration Y p of ND = 16 cylinders at Re = 10.

Image of FIG. 17.
FIG. 17.

The behaviors of 72 cylinders (ϕ T = 16.9%) with A = 3.333, K = 0.2 at Re = 1000: (a) The cylinder positions at t = 1000; (b) the distribution of the translational velocities of cylinders U p (●), the averaged flow velocity profile u (solid line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution ϕ during t = 900 and 1000.

Image of FIG. 18.
FIG. 18.

The behaviors of 72 cylinders (ϕ T = 16.9%) with A = 1.875, K = 0.15 at Re = 1000: (a) The cylinder positions at t = 1000; (b) the distribution of the translational velocities of cylinders U p (●), the averaged flow velocity profile u (solid line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution ϕ during t = 900 and 1000.

Image of FIG. 19.
FIG. 19.

The behaviors of 72 cylinders (ϕ T = 16.9%) with A = 1, K = 0.11 at Re = 1000: (a) The cylinder positions at t = 1000; (b) the distribution of the translational velocities of cylinders U p (●), the averaged flow velocity profile u (solid line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution ϕ during t = 900 and 1000.

Image of FIG. 20.
FIG. 20.

The distribution of the time-averaged angular velocities ω s /π during t = 900 and 1000 for the 72 cylinders (ϕ T = 16.9%) with A = 3.333 and K = 0.2 (●), A = 1.875 and K = 0.15 (○), A = 1 and K = 0.11 (×).

Image of FIG. 21.
FIG. 21.

The normalized flow velocity profile u/Re α (solid line) and the normalized distributions of the translational velocities U p /Re α with α = 1.068 of 72 cylinders (ϕ T = 16.9%, A = 1.875, K = 0.15) at Re = 500 (●), 1000 (■), 2000 (▲) during t = 900 and 1000. The normalized Poiseuille flow velocity (without cylinders) u/Re is denoted by the dashed line for comparison.

Image of FIG. 22.
FIG. 22.

The time-averaged solid area distribution ϕ for 72 cylinders (ϕ T = 16.9%) with A = 1.875, K = 0.15 at Re = (a) 500, (b) 1000, (c) 2000 during t = 900 and 1000.

Image of FIG. 23.
FIG. 23.

The thickness of the particle-free layer y f versus the total solid area fraction ϕ T of the cylinders with A = 3.333 and K = 0.2 (●), A = 1.875 and K = 0.15 (○), A = 1 and K = 0.11 (Δ) at Re = 100 (dotted line), 500 (dashed line), 1000 (solid line).

Image of FIG. 24.
FIG. 24.

(a) The time-averaged distribution of the translational velocities of cylinder U p for ND = 54 (●), 72 (■), 108 (▲) circular cylinders with K = 0.11 (ϕ T = 0.12, 0.16, 0.25, respectively), and averaged flow velocity profile u (solid line) during t = 100 and 200 at Re = 10 (Re p = 1.1). The flow velocity profiles without particles are denoted by dashed line. (b) The time-averaged solid area distribution ϕ.

Tables

Generic image for table
Table I.

The number of equilibrium positions (if it is 2) or the averaged equilibrium positions (if it is greater than 2) and the thickness of the particle-free layers y f .

Generic image for table
Table II.

The relations between the thickness of the particle-free layer and the total solid area fraction of the cylinders, where 8.48% < ϕ T < 25.45%.

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/content/aip/journal/pof2/24/10/10.1063/1.4757387
2012-10-12
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The motion of a single and multiple neutrally buoyant elliptical cylinders in plane Poiseuille flow
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/10/10.1063/1.4757387
10.1063/1.4757387
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