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Flow mediated interactions between two cylinders at finite Re numbers
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10.1063/1.4704195
/content/aip/journal/pof2/24/4/10.1063/1.4704195
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/4/10.1063/1.4704195
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Case A1: The set up consists of a master of diameter D m , forced to move forward at the constant velocity U m , and a slave of diameter D s , initially located behind the master at the separation distance s. Case A2: The set up consists of a master of diameter D m , forced to move forward at the constant velocity U m , and a slave of diameter D s , initially located above the master at the separation distance s. Case B: The set up consists of a master of diameter D m and a slave of diameter D s , initially located beside the master at a distance d between the centers of mass. The master is forced to oscillate horizontally according to , x m being the master's location of its center of mass and A, , and ϕ, respectively, the oscillation amplitude, period, and phase.

Image of FIG. 2.
FIG. 2.

Drag coefficient (C d ) evolution against dimensionless time T = 2|U T |t/D for flow past impulsively started cylinder at Re = 1000. The dotted (red) line was obtained setting ER = 8192 × 8192, LCFL = 0.01, and . Solid (black and blue) lines are, respectively, reference solutions of Refs. 25 and 37. Since the discrepancy is negligible, the three solutions overlap and the two solid lines are nearly indistinguishable.

Image of FIG. 3.
FIG. 3.

Inline impact of two cylinders without free stream. (a)–(d) Evolution of the force coefficient C d against the dimensionless time T* for, respectively, Re = 25, 50, 100, and 200. Dotted (red) lines and symbols represent, respectively, the solutions of the present method and Bampalas and Graham.10 (e) Given from left to right and top to bottom is the time sequence of the vorticity field (−0.66 ⩽ T* ⩽ 2, ΔT* = 0.33).

Image of FIG. 4.
FIG. 4.

Inline impact of two cylinders in uniform free stream. (a) and (b) Evolution of the force coefficient C d against the dimensionless time T* at Re = 100 before (a) and after (b) collision. Circles and triangles correspond, respectively, to upstream and downstream cylinders according to Bampalas and Graham.10 Dashed lines (red and blue) correspond, respectively, to upstream and downstream cylinders in the present computations. (c) From left to right and top to bottom, vorticity fields at times T* = −13.3, −6.7, 0, 1.7, 3.3, 5, 6.7, 8.3, 10, 11.7, 18.3, 25.

Image of FIG. 5.
FIG. 5.

Convergence study for case B. The master oscillates horizontally according to Fig. 1. The amplitude was set to A = D m , period physical time unit, ϕ = 0, and simulations were carried out up to physical time . (a) Space convergence (mollification length fixed based on ER, : L (e) (blue), L 1(e) (black), and L 2(e) (red) are plotted against ER. LCFL was set to 0.01. (b) Space convergence (ratio ε/h e fixed to ): L (e) (blue), L 1(e) (black), and L 2(e) (red) are plotted against ER. LCFL was set to 0.01. (c) Time convergence: L (e) (blue), L 1(e) (black), and L 2(e) (red) are plotted against LCFL. ER was set to 4096 × 4096. For all studies we used λ = 104. Dashed lines (blue, black, red, green) represent (respectively) first, second, third, and fourth order slopes.

Image of FIG. 6.
FIG. 6.

Case A1. (a) Test case D s = D m at dimensionless time T = 24: for several Re, master and slave (full grey aft cylinder, where the initial position is represented by the dashed cylinder) are superimposed to the corresponding vorticity field. (b) Test case D s = D m /2 at dimensionless time T = 24: vorticity fields. (c) Normalized separation distance s/D m as function of dimensionless time T for test case D s = D m . (d) Normalized separation distance s/D m as function of dimensionless time T for test case D s = D m /2. In (c) and (d) solid lines corresponds to Re = 10, 50, 100, 200, and 500 (blue, red, back, green, and orange, respectively) and the dotted lines refer to the inviscid solution.

Image of FIG. 7.
FIG. 7.

Case A1. (a) Test case D s = D m /4 at dimensionless time T = 24: for several Re, master and slave (full grey aft cylinder, where the initial position is represented by the dashed cylinder) are superimposed to the corresponding vorticity field. (b) Normalized separation distance s/D m as function of dimensionless time T. The Reynolds number is between 10 and 500. Solid lines (of varying color) denote differing Reynolds numbers while the dotted line indicates the inviscid solution. (c) Enlargement of panel (b).

Image of FIG. 8.
FIG. 8.

The distance traveled to the right by a slave cylinder of diameter D s /D m = 0.25, 0.5, 0.25 (dotted lines, black, blue, red, respectively) with s/D m = 0.1 and a passive tracer (solid lines) initially located at the center of the slave cylinder for (a) Re = 50 and (b) Re = 500. The passive tracer is initially placed at a distance x/D m = −1.1, − 0.85, − 0.725 (black, blue, red) with respect to the center of the master cylinder.

Image of FIG. 9.
FIG. 9.

Case A2. (a) Test case D s = D m at dimensionless time T = 32 for several Re. Master (initial position is represented by the orange dashed cylinder) and slave (full grey aft cylinder, where the initial position is represented by the black dashed cylinder) are superimposed to the corresponding vorticity field. (b) Test case D s = D m /2 at dimensionless time T = 32: vorticity fields. (c) Normalized separation distance in the horizontal direction s/D m as function of dimensionless time T for test case D s = D m . (d) Normalized separation distance in the horizontal direction s/D m as function of dimensionless time T for test case D s = D m /2. In (c) and (d) solid (blue, green, red, black, and orange) lines correspond to Re = 10, 30, 50, 100, and 500 whereas the dashed line corresponds to the inviscid case.

Image of FIG. 10.
FIG. 10.

Case A2. (a) Test case D s = D m /4 at dimensionless time T = 32 for several Re. The master (initial position is represented by the orange dashed cylinder) and slave (full grey aft cylinder, where the initial position is represented by the black dashed cylinder) cylinders are superimposed to the corresponding vorticity field. (b) Normalized separation distance in the horizontal direction s x /D m as function of dimensionless time T. Solid lines correspond to Reynolds number ranging between 10 and 500. Dashed line corresponds to the inviscid case. (c) Enlargement of panel (b).

Image of FIG. 11.
FIG. 11.

The advection of a passive tracer up to T = 32 initialized above the forced cylinder for (a) Re = 50 and (b) Re = 500. The passive tracer is initially placed at a distance y/D m = 1.2, 0.95, 0.825 corresponding to the center of the slave cylinders for cases D s /D m = 0.25, 0.5, 0.25 (dotted line, black, blue, red, respectively) with s/D m = 0.2. Grey dots indicate the starting and ending positions of the tracers. Vorticity field is given for reference.

Image of FIG. 12.
FIG. 12.

The distance traveled to the right by a slave cylinder of diameter D s /D m = 0.25, 0.5, 0.25 (dotted line, black, blue, red, respectively) with s/D m = 0.2 and a passive tracer (solid line) initially located at the position corresponding to the center of the slave cylinder for (a) Re = 50 and (b) Re = 500. The passive tracer is initially placed at a distance y/D m = 1.2, 0.95, 0.825 (black, blue, red) with respect to the center of the master cylinder. Note that oscillatory vortex shedding is not visible in (b) as compared to Fig. 11 because the flow remains symmetric.

Image of FIG. 13.
FIG. 13.

Case B. (a) Slave's normalized displacement (Δx/D m ) versus dimensionless time T, for several Reynolds numbers (from top to bottom Re = 100, 80, 60, 50, 40, 30, 27, 25, 20, 15) and ϕ = 0, A = D m , D s = D m , d = 2.5D m . (b) Threshold Reynolds number (Re th ) as function of D s /D m and d/D m . (c)–(e) Evolution in time of the vorticity fields for the cases Re = 80, 27, and 20 reported in panel (a). The dashed cylinder represents the initial slave's location.

Image of FIG. 14.
FIG. 14.

Case B. Vorticity fields at physical time t = 6, corresponding to oscillation amplitudes A = 1.5D m , A = 1D m , A = 0.5D m , A = 0.25D m , and ϕ = 0. The dashed cylinder represents the initial slave's location. (b) Slave's normalized displacement (Δx/D m ) versus physical time t for A = 1.5D m (blue), A = 1D m (green), A = 0.5D m (red), A = 0.25D m (black). Solid and dashed lines correspond, respectively, to ϕ = 0 and ϕ = π.

Image of FIG. 15.
FIG. 15.

Vorticity field at physical time t = 6, with oscillation amplitude A = D m , Re = 500, and ϕ = 0. The slave cylinder on the right is free to move. Shades are over saturated to clearly show how the vortex pair created from the motion of the master cylinder impinges on the slave cylinder, imparting it with positive x −momentum.

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/content/aip/journal/pof2/24/4/10.1063/1.4704195
2012-04-23
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Flow mediated interactions between two cylinders at finite Re numbers
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/4/10.1063/1.4704195
10.1063/1.4704195
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