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Research Papers: Flows in Complex Systems

Effects of Damping on Flow-Mediated Interaction Between Two Cylinders

[+] Author and Article Information
Zhonglu Lin

State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China;
Collaborative Innovation Center for Advanced
Ship and Deep-Sea Exploration (CISSE),
Shanghai 200240, China
e-mail: zl352@eng.cam.ac.uk

Dongfang Liang

Professor
State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China;
Collaborative Innovation Center for Advanced
Ship and Deep-Sea Exploration (CISSE),
Shanghai 200240, China
e-mail: d.liang@sjtu.edu.cn

Ming Zhao

School of Computing,
Engineering and Mathematics,
Western Sydney University,
Locked Bag 1797,
Penrith, NSW 2751, Australia
e-mail: M.Zhao@westernsydney.edu.au

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 15, 2017; final manuscript received March 8, 2018; published online May 2, 2018. Assoc. Editor: Moran Wang.

J. Fluids Eng 140(9), 091106 (May 02, 2018) (12 pages) Paper No: FE-17-1503; doi: 10.1115/1.4039712 History: Received August 15, 2017; Revised March 08, 2018

This study investigates the flow-mediated interaction between two vibrating cylinders of the same size immersed in an otherwise still fluid. The master cylinder carries out forced vibration, while the slave cylinder is elastically mounted with one degree-of-freedom along the centerline between the two cylinders. We examined the stabilized vibration of the slave cylinder. In total, 6269 two-dimensional (2D) cases were simulated to cover the parameter space, with a fixed Reynolds number of 100, the structural damping factor of the slave cylinder ranging from 0 to 1.4, the mass ratio of the slave cylinder ranging from 1.5 to 2.5, the initial gap ratio ranging from 0.2 to 1.0, the vibration amplitude ratio of the master cylinder ranging from 0.025 to 0.1, and the vibration frequency ratio ranging from 0.05 to 2.4. We found that the vibration amplitude of the slave cylinder is highly sensitive to damping when the damping coefficient is small. The two cylinders' vibration is in antiphase at low frequencies but in phase at high frequencies. The phase of the slave cylinder changes abruptly at resonance when it has little damping, but the phase change with the frequency becomes increasingly gradual with increasing damping. With a nonzero damping factor, the maximum vibration amplitude of the slave cylinder is inversely correlated with its mass ratio. The response of the slave cylinder is explained by examining the pressure distribution and velocity field adjacent to it.

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Figures

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Fig. 1

Sketch of interaction between two cylinders, where the master cylinder undergoes harmonic forced vibration, and the slave cylinder is elastically mounted with a dashpot and can vibrate passively along the y-axis

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Fig. 2

A typical computational mesh for simulating the flow mediated interaction between two cylinders, with G/D = 0.2

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Fig. 3

(a) Variation in magnification factor of the slave A2/(F0/k) with f1/fn, and (b) variation in force–displacement phase difference for the slave (for frequency components with f/fn = f1/fn) with f1/fn at G/D = 0.2, A1/D = 0.1, m* = 2.0, and ζ = 0–1.4. The dashed line is the locus of maxima by assuming harmonic force input.

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Fig. 4

(a) Variation of A2/A1 and (b) variation of slave-master phase difference Δϕ21 with f1/fn (for the frequency components with f/fn = f1/fn) with f1/fn at G/D = 0.2, A1/D = 0.1, m* = 2.0, and ζ = 0–1.4. A2/A1 is negatively correlated with ζ, particularly within the regime of resonance. The phase difference curve converges at Δϕ21 ≈ 80 deg, f1/fn ≈ 0.775. The increase in damping slows down the 180 deg phase shift of the slave cylinder as f1/fn reaches its immersed natural frequency.

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Fig. 5

Amplitude spectra showing the responding displacement of the slave with various f1/fn at ζ = 0.2, G/D = 0.2, A1/D = 0.1, and m* = 2.0. (a) An overview of the FFT spectra and (b) the FFT spectra at f1 = fw/2 ≈ 0.375. The dashed line tracks the dominant frequencies. At f1 = fw/2, the high frequency component is always smaller than the fundamental component, and the secondary resonance does not occur.

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Fig. 6

(a) Variation in A2/A1 with f1/fn, and (b) variation in displacement phase difference Δϕ21 between the slave and the master (for the frequency components with f/fn = f1/fn) with f1/fn at G/D=0.2,A1/D=0.1,m*=1.5−2.5, and ζ = 0–0.2. The line types denote the different values of ζ, whereas the symbol types denote the different values of m*. With a nonzero damping, the peak A2/A1 decreases with m*. The curve of Δϕ21 disperses at resonance, which follows the same pattern as the undamped cases.

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Fig. 7

(a) Variation in A2/A1 with f1/fn, and (b) variation in displacement phase difference Δϕ21 between the slave and the master (for the frequency components with f/fn = f1/fn) with f1/fn at G/D=0.4,A1/D=0.025−0.1,m*=1.5, and ζ = 0.2. The curves largely overlap each other, which means A2 and A1 are linearly correlated. This is different from the undamped cases, where nonlinearity is found at resonance. In addition, A1/D is negatively correlated with Δϕ21.

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Fig. 8

(a) Zoom-in at peaks for variation of A2/A1 at ζ = 0.2, and (b) zoom-in at peaks for variation of A2/A1 at ζ = 0.4 for G/D = 0.4, A1/D = 0.025–0.1, and m* = 1.5. We discovered a threshold damping ratio of the slave cylinder beyond which its peak relative amplitude decreases with the master vibration amplitude and under which the peak increases with the amplitude. In this scenario, the threshold lies in ζ = 0.2–0.4.

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Fig. 9

(a) Variation of A2/A1 with f1/fn, and (b) variation of displacement phase difference Δϕ21 between the slave and the master (for frequency components with f/fn = f1/fn) with f1/fn at G/D = 0.3–0.9, A1/D = 0.05, m* = 2.5, and ζ = 0.2. A2/A1 decreases with G/D, but the initial gap distance has little effect on Δϕ21. This pattern is exactly the same as the undamped cases.

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Fig. 10

(a)–(d) Nondimensional pressure contours p*=p/ρfn2D2 at G/D=0.4, A1/D=0.075, m*=1.5, f1/fn=0.74,ζ=0, andΔϕ21 = 90.673 deg. The solid and dashed lines indicate positive and negative values of the nondimensional pressure, respectively. The velocity vectors are plotted every 20 points. The vector scale factor is 0.3 grid units/magnitude. (e) The y-direction force coefficient upon the slave and its shear and pressure components were examined. Pressure rather than viscosity is the main contributor to the force upon the slave.

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Fig. 11

Nondimensional pressure contours p*=p/ρfn2D2 at G/D=0.2; A1/D=0.1; m*=1.5; f1/fn=0.78; and ϕ1 = 0 deg ((a), (e), and (i)), 90 deg ((b), (f), and (j)), 180 deg ((c), (g), and (k)); and 270 deg ((d), (h), and (l)). ((a)–(d)) ζ = 0, Δϕ21 = 46.6 deg; ((e)–(h)) ζ = 0.1, Δϕ21 = 50.0928 deg; and ((i)–(l)) ζ = 0.8, Δϕ21 = 68.309 deg. The dashed lines indicate negative values of nondimensional pressure, with solid lines indicating positive values. With the increase in damping, the main driver of the slave changes from the pressure fluctuation at the far side of the slave to that within the gap.

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Fig. 12

Velocity vectors on the left of the slave cylinder at G/D=0.2, A1/D=0.1, m*=1.5,  and f1/fn = 0.75, with (a) ζ = 0, Δϕ21 = 108.4 deg; (b) ζ = 0.1, Δϕ21 = 95.62 deg; (c) ζ = 0.8, Δϕ21 = 68.309 deg; and (1) ϕ1 = 90 deg; (2) ϕ1 = 180 deg. The velocity vector is drawn on every grid point and the vector scale factors are 0.15 grid units/magnitude. The lifespan and the strength of vortices both decrease with damping.

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Fig. 13

Velocity vectors in the gap at G/D = 0.2, A1/D = 0.1, m* = 1.5, and f1/fn = 0.75, with (a) ζ = 0, Δϕ21 = 108.4 deg; (b) ζ = 0.1, Δϕ21 = 95.62 deg; and (1) ϕ1 = 158 deg; (2) ϕ1 = 189 deg. The velocity vector is drawn on every grid point and the vector scale factors are 0.011 grid units/magnitude. The zero-velocity point (see dashed circle in (b1)) travels from the bottom of the slave to the top of the master. For the damped case with ζ = 0.1, the stagnant flow point appears earlier than the undamped case and has a longer lifespan. For both the damped and undamped cases, the stagnant flow point disappears at ϕ1 = 189 deg.

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