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Research Papers: Fluid-Structure Interaction

A Semi-Analytical Model for Fluid-Elastic Instability in Tube Arrays Including the Effects of Nonuniform Flow Velocity and Tube Displacement

[+] Author and Article Information
M. Utsumi

Senior Researcher
Machine Element Department,
Technical Research Laboratory,
IHI Corporation,
1 Shin-Nakaharacho, Isogo Ward,
Yokohama, Kanagawa 235-8501, Japan
e-mail: masahiko_utsumi@ihi.co.jp

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received June 4, 2015; final manuscript received February 17, 2016; published online April 29, 2016. Assoc. Editor: Jong Chull Jo.

J. Pressure Vessel Technol 138(5), 051302 (Apr 29, 2016) Paper No: PVT-15-1115; doi: 10.1115/1.4032853 History: Received June 04, 2015; Revised February 17, 2016

A semi-analytical method to examine the influences of the axial variations in the tube vibration amplitude and flow velocity on the critical flow velocity is investigated. We illustrate that neglecting the axial variation in the tube vibration amplitude can result in an overestimation of the critical flow velocity (nonconservative estimate) when the flow velocity is nonuniform. A condition under which such overestimation arises is derived by the transformation of the eigenvalue problem that is made to take into account the axial variations in the tube vibration amplitude and flow velocity. This condition is the existence of a positive correlation between the deviations of two functions: one representing the axial variation in the flow velocity and the other square of the function representing the nonuniformity of the tube vibration amplitude. The case with marked partial admission is investigated through physical consideration for this flow-induced vibration problem. We also study cases where the difference between tube eigenfrequencies in the flow and transverse directions results in a transition in the instability direction, from the transverse direction to that of flow.

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References

Figures

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Fig. 1

Cross sections of tube arrays: (a) square tube array and (b) rotated triangular tube array

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Fig. 2

Coordinate systems for each regular prism cell (the case of a square tube array is shown as an example)

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Fig. 3

Unstable modes in case of Fig. 12: (a) ωy/ωx = 1, Vreduc = 7.3; (b) ωy/ωx  = 1.3, Vreduc = 9.9 (ωy/ωx is increased by increasing ωy and holding ωx constant, and the reduced velocity is defined by Vreduc=Vc/(ωxd/2π))

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Fig. 4

Critical reduced velocity for the square tube array (T/d = 1.33, ωy/ωx = 1, L/d = 66.7, water flow, mass-damping parameter mb2πζb/ρ d2 is increased by raising ζb for a fixed mass parameter mb/ρ d2 = 1.85, L = 2 m). Black line: present analysis, case 1, 2, and 4; black dotted line: present analysis, case 3; and filled circle: experiment (Ref. [7]), case 2.

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Fig. 5

Additional result for the previous study [26] on case 2 (square tube array, T/d = 1.33, ωy/ωx = 1, air flow, mass-damping parameter mb2πζb/ρ d2 is increased by raising ζb for a fixed mass parameter mb/ρ d2 = 1480). Black line: present analysis; filled circle: experiment (Ref. [7]); and black dashed line: 2.35(mb2πζb/ρ d2)1/2 (Refs. [4] and [5]).

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Fig. 6

Critical reduced velocity for uniform and nonuniform flow velocity: (a) uniform tube vibration amplitude ϕ(z)=1 and (b) nonuniform tube vibration amplitude ϕ(z)=2 sin(π z/L). Black line: nonuniform flow velocity F(z)=2 sin(π z/L); black dotted line: uniform flow velocity F(z)=1; and filled circle: experiment (Ref. [7]).

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Fig. 7

Damping ratio −λR/(λR2+λI2)1/2 for the eigenvalue λ=λR+iλI subject to destabilization (solid line, vr∂Vφ/∂r is retained; dotted line, vr∂Vφ/∂r is ignored; the case of mb2πζb/ρ d2 = 0.0015 and F(z)=ϕ(z)=2 sin(π z/L) in Fig.6(b))

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Fig. 8

Critical reduced velocity for the rotated triangular tube array (T/d = 1.375, ωy/ωx = 1, L/d = 66.7, airflow, mass-damping parameter mb2πζb/ρ d2 is increased by raising ζb, logarithmic decrement 2πζb is 0.007 for mass-damping parameter 1.3, L = 2 m). Black line: present analysis, case 1, 2, and 4; black dotted line: present analysis, case 3; filled circle: experiment (Ref. [18]), case 2; and black dashed line: design guide line 2.8(mb2πζb/ρ d2)1/2 (Refs. [4] and [5]), case 2.

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Fig. 9

Critical reduced velocity for the rotated triangular tube array for smaller mass-damping parameters (T/d = 1.364, ωy/ωx = 1, mass parameter mb/ρ d2 is given in the text; black circle shows experimental results [25]; case 2)

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Fig. 10

Critical reduced velocity for uniform and nonuniform flow velocity: (a) uniform tube vibration amplitude ϕ(z)=1 and (b) nonuniform tube vibration amplitude ϕ(z)=2 sin(π z/L). Black line, nonuniform flow velocity F(z)=2 sin(π z/L); black dashed line, uniform flow velocity F(z)=1; and filled circle, experiment (Ref. [18]).

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Fig. 11

Results corresponding to Fig. 10 for the case with larger pitch-diameter ratio T/d = 1.43: (a) uniform tube vibration amplitude ϕ(z)=1 and (b) nonuniform tube vibration amplitude ϕ(z)=2 sin(π z/L)

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Fig. 12

Critical reduced velocity versus tube eigenfrequency ratio ωy/ωx (mass-damping parameter mb2πζb/ρ d2 = 2.33, ζb = 0.00201; ωy/ωx is increased by increasing ωy and holding ωx constant, and the reduced velocity is defined by Vreduc=Vc/(ωxd/2π))

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Fig. 13

Critical reduced velocity versus tube eigenfrequency ratio ωy/ωx (mass-damping parameter mb2πζb/ρ d2 = 1.12, ζb = 0.00096; ωy/ωx is increased by increasing ωy and holding ωx constant, and the reduced velocity is defined by Vreduc=Vc/(ωxd/2π))

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Fig. 14

Critical reduced velocity versus tube eigenfrequency ratio ωy/ωx (ωy/ωx is increased by reducing ωx and holding ωy constant, and the reduced velocity is defined by Vreduc = Vc/(ωyd/2π) using constant ωy). Black line: mass-damping parameter mb2πζb/ρ d2 = 2.33, ζb = 0.00201 and dark black line: mass-damping parameter mb2πζb/ρ d2 = 1.12, ζb = 0.00096.

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Fig. 15

Critical reduced velocity versus tube eigenfrequency ratio ωy/ωx for various mass-damping parameters: mb2πζb/ρd2  = 1.39, 2.84, 3.59, 5.55, and 8.26. (Larger mass-damping parameter results in higher critical reduced velocity for a fixed ωy/ωx; the mass-damping parameter mb2πζb/ρd2 is increased by raising ζb; ζb is 0.0057 when mb2πζb/ρd2 = 3.59; the case where ωx is made relatively lower than ωy is considered by increasing ωy and holding ωx constant, and the reduced velocity is defined by Vreduc=Vc/(ωxd/2π).)

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Fig. 16

Critical reduced velocity for large ωy/ωx versus mass-damping parameter (the case where ωx is made relatively lower than ωy is considered by increasing ωy and holding ωxconstant, and the reduced velocity is defined by Vreduc=Vc/(ωxd/2π)). Black line, present analysis and filled circle, experiment (Ref. [19]).

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Fig. 17

Left- and right-hand sides of inequality (28) and correlation C1 defined by Eq. (33) for the case where F(z) and ϕ(z) are given by Eq. (39) (b1 = 0) and Eq. (40), respectively. Black dashed line, (S0/S1)case A/(S0/S1)case B; black line, (S0/S1)case C/(S0/S1)case D; black dotted line, C1 (Eq. (33)); and unfilled circle, experiment [22] for the ratio of critical reduced velocity for b2 > 1 to that for b2 = 1.

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Fig. 18

Parameters for the case where F(z) and ϕ(z) are given by Eq. (39) (b1 = 0) and Eq. (40), respectively. Dark black line, r1 (Eq. (42)); dark black dashed line, (S2/L)1/2; black line, S1/L; and black dotted line, C1 (Eq. (33)).

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Fig. 19

The ratio of critical reduced velocity for b2<1 to that for b2 = 1 (circle, experiment [22]; dotted line, S̃)

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Fig. 20

Mode shapes and flow distributions used for numerical examples for partial admission [23] (flow is present over the range shown in broken line, mb/ρd2 = 445, T/d = 1.47, d = 0.0126 m, thickness and Young's modulus are 0.00025 m and 100 GPa, respectively). (a) S̃ = 1, mb2πζb/ρd2 = 17.8; (b) S̃ = 0.971, mb2πζb/ρd2 = 10.2; (c) S̃ = 0.010, mb2πζb/ρd2 = 16; and (d) S̃  = 0.0035, mb2πζb/ρd2 = 4.

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Fig. 21

Mode shapes and flow distributions used for numerical examples for partial admission [23] (the parameters except S̃ and mb2πζb/ρd2 are the same as in Fig. 20). (a) S̃ = 0.046, mb2πζb/ρd2 = 13.3; (b) S̃ = 0.203, mb2πζb/ρd2 = 13.3; (c) S̃ = 0.0029, mb2πζb/ρd2 = 3.6.

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Fig. 22

Critical upstream velocity versus energy fraction S̃ for the cases of Figs. 20 and 21 (white circle, experiment [23]; black circle, present method)

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Fig. 23

Critical upstream velocity versus energy fraction S̃ for the cases of Figs. 20 and 21 (circle, experiment [23]; triangle, Eq. (50))

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Fig. 24

Critical upstream velocity versus energy fraction S̃ for the case of Figs. 20 and 21 (circle, experiment [23]; square, Eq.(51))

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Fig. 25

Flow pattern and cross-sectional mean flow velocity in the circumferential direction near the clearance φ = 270 deg in the midcell

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Fig. 26

Circumferential variations around the tube in radially integrated residual components. (a) Solid and broken lines show Er1 and −Er2, respectively. (b) Solid and broken lines show Eφ1 and −Eφ2, respectively.

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Fig. 27

Normalized residuals at various positions (a) |Er|/Ech, (b) |Eφ|/Ech (their maximum values are shown in the figure)

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Fig. 29

Coordinate systems for each regular prism cell in the case of rotated triangular tube array

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Fig. 30

Axial lines i1−i6 for each cell in the case of staggered arrays (i1−i6 for each cell are written in its square)

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Fig. 31

Example of mass parameter dependence of critical reduced velocity for a fixed mass-damping parameter (square tube array, T/d = 1.33, ωy/ωx = 1, water flow, mb2πζb/ρ d2 = 0.0862, case 2)

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