Analysis of flow-induced vibration of steam generator tubes subjected to cross flow

Nuclear Engineering and Design 275 (2014) 375–381

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Analysis of flow-induced vibration of steam generator tubes subjected to cross flow Wensheng Zhao a,b,∗ , Fei Xue b , Guogang Shu c , Meiqing Liu a , Lei Lin b , Zhaoxi Wang d , Zhihuai Xiao a a

School of Power and Mechanical Engineering, Wuhan University, 8 Donghu St. S., Wuhan 430072, Hubei, PR China Suzhou Nuclear Power Research Institute, 1788 Xihuan St., Suzhou 215004, Jiangsu, PR China c China Nuclear Power Engineering Co., Ltd., 1001 Shangbu Rd., Shenzhen 518031, Guangdong, PR China d CPI Nuclear Power Institute, 18 Xizhimen St., Beijing 100044, PR China b

h i g h l i g h t s • • • •

The dynamics of tubes subjected to cross flow is investigated. A three-dimensional CFD model, coupled with a structural model is obtained. The relationship between pitch and displacement for two tubes is analyzed. The mutual influence of two contiguous tubes in cross flow is discussed.

a r t i c l e

i n f o

Article history: Received 7 September 2013 Accepted 22 May 2014

a b s t r a c t Flow-induced vibration of PWR steam generator tubes is one of the key problems to be considered in nuclear engineering. In this paper, the dynamics of tubes subjected to cross flow is investigated. A fully coupled model for fluid dynamics and structure is used to analyze this fluid–structure problem. The study begins with an analysis of a single tube subjected to cross flow, and the dynamics and flow patterns of the system are investigated. Time trace, power spectral density (PSD), phase-plane plot and Poincaré map are used to characterize the motion of the tube. Then, the dynamical behaviours of two tubes with in-line and parallel configuration are studied, and the mutual influence of the two tubes is analyzed. The relationship between pitch-to-diameter ratio and non-dimensional displacement for two contiguous tubes is investigated. Finally, the interactions within a 3 × 3 tube bundle in cross flow, and the vibration responses of the 9 tubes are investigated. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Fluid–structure interaction (FSI) is one of the important problems of steam generators in nuclear power plants. The tubes of steam generators, which undergo a cross flow, are highly susceptible to flow-induced vibration (FIV). The vibration may lead to fretting wear on the contact surface of tubes with supporting plates. The evaluation of this flow-induced vibration is involving on the fluid dynamics calculation for the flow characteristics and the structural analysis.

∗ Corresponding author at: School of Power and Mechanical Engineering, Wuhan University, 8 Donghu St. S., Wuhan 430072, Hubei, PR China. Tel.: +86 27 68774699; fax: +86 27 87865615. E-mail address: [email protected] (W. Zhao). http://dx.doi.org/10.1016/j.nucengdes.2014.05.029 0029-5493/© 2014 Elsevier B.V. All rights reserved.

The conditions leading to vibration of tubes subjected to cross flow have been investigated extensively in the past years. A bulk of experimental and theoretical studies have been carried out to investigate the vibration characteristics of tubes with various configurations subjected to cross flow. Connors (1970) proposed the earliest model to evaluate the flow-induced vibration of the steam generator tubes. Cheng and Finnie (1996) derived an equation of motion of tubes subjected to an uniform cross flow, and then predicted the vibration response of tubes and fluid dynamic forces. Fournier et al. (2007) simulated fluid flow in a pressurized water reactor (PWR) core. Cui et al. (2008) adopted direct method to investigate fundamental characteristics of flow-induced vibration and stabilities of parallel-plate fuel assemblies in cross flow. Hassan and Hayder (2008) conducted a time-domain model to calculate the critical flow velocity and tube-support interaction force. Sigrist and Broc (2008) and Kuehlert et al. (2008) used Homogenization

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flow. Borazjani et al. (2008) adopted a strong coupled model to analyze vibration of tubes undergoing arbitrarily large deformation in cross flow. Lee et al. (2010) developed a modified immersed finite element method (IFEM) to assess vibration characteristics of tubes in cross flow. Simoneau et al. (2010) used LES, coupled with subgrid approach to investigate FSI model. Simoneau et al. (2011) used a fully coupled approach to further improve the assessment of fluid structure interaction problems, and analyzed the vibration response of tubes subjected to cross flow. In the present paper, a three dimensional fully coupled model is adopted to analyze the fluid flow through tubes, and the vibration characteristics of tubes. The dynamics of a single tube subjected cross flow, and the velocity contours are first investigated. Time trace, power spectral density (PSD), phase-plane plot, and Poincaré map are used to analyze the dynamical behaviour of the tube. Then, the dynamical behaviours of two in-line tubes and two parallel tubes, the velocity contour and the mutual influence of two tubes are studied; the relationship between the pitch-to-diameter ratio and the non-dimensional displacement for two contiguous tubes is investigated. Finally, the interactions within a 3 × 3 tube bundle, the dynamical behaviours of 9 tubes, and the velocity contour are investigated.

Nomenclature C D Di Do E F K L M P Px Py Re t u, v, w U y

damping matrix tube diameter tube inner diameter tube outer diameter Young’s modulus fluid load stiffness matrix tube length mass matrix pressure X-directional pitch Y-directional pitch Reynolds number time velocity components inlet fluid velocity Y-directional displacement of tube

Greek symbols  fluid viscosity fluid density f s solid density

2. Mathematical models In this study, a three-dimensional CFD model with a moving grid is proposed, and the fluid force acting on tubes is numerically simulated using the CFD code CFX-14.0, which utilizes a cell-vertex pressure based finite volume formulation. The CFD code employs an Arbitrary Lagrangian–Eulerian (ALE) formulation and a moving/deforming mesh model to simulate the boundary motion. In the CFX model, diffusion transport equations are used to smoothly propagate motion at boundaries into the solution domain, and preserve the existing boundary layer meshes.

method to investigate FSI effects on steam generator tube bundles. Prakash et al. (2009) analyzed modal frequencies of tube bundle induced by cross flow in heat exchangers. Park et al. (2009) suggested an indirect input force estimation theory to calculate turbulence flow induced force exerted on fuel rods in a PWR. Owing to the improvement in computational theory, various numerical approaches have been adopted to evaluate FSI effects of tubes in cross flow. Eisinger et al. (1995) created a numerical model of tube bundle, coupled with a finite element solver to simulate the flow-induced vibration. Longatte et al. (2003) suggested a fully coupled method with periodic boundary conditions to simulate the dynamical behaviour of a tubes in cross flow. Kim and No (2004) used Large-Eddy Simulation (LES) to investigate the characteristics of a realistic 4 × 4 and 5 × 5 cylinders in cross flow. Kim and Mohan (2005) validated LES model, and calculated the forces on a tube in high Reynolds number cross flow. Oakley et al. (2005) used Detached-Eddy Simulation (DES) to simulate a full FSI of single tube in high Reynolds number cross flow. Ahn and Kallinderis (2006) adopted arbitrary Lagrangian–Eulerian (ALE) to investigate characteristics of a tube in cross flow. Kim and Choi (2006) conducted a new immersed boundary method (IBM) to simulate the flow pattern and dynamical behaviour of a tube subjected to cross

2.1. The CFD model The basic transport equations for fluid flow are the continuity and the momentum equations. For incompressible viscous flows, the continuity equation is defined as

∇ · U = 0,

(1)

and the momentum equation is defined as



f

∂U + U · ∇U ∂t



− ∇ 2 U + ∇ P = F,

(2)

where  is the divergence operator, f is the fluid density, U is the velocity vector (u, v, w), t is the time,  is the fluid viscosity, P is the

Pitch P x U U

Upper tube

U

Single tube

Upstream tube

Downstream tube Lower tube

(a)

(b)

(c)

Fig. 1. Schematics of tubes subjected to cross flow: (a) single tube, (b) two in-line tubes, (c) two parallel tubes.

Pitch P y

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Fig. 2. Computational grids of a tube in cross flow.

pressure, F is the fluid load acting on structure. More details on the mathematical formulation are well described by Lee et al. (2008), where the directly coupled Euler–Lagrange method (DCELM) is adopted to calculate the FSI force between fluid and structure. 2.2. The structural analysis model For case where an elastically mounted tube in fluid flow vibrates freely, the tube is assumed to be a rigid body and the equation of motion of the tube is generated as M

∂2 y ∂y + Ky = F, +C ∂t ∂t 2

(3)

where y is the Y-directional displacement of the tube, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, and F is the fluid load acting on the tube. 2.3. The coupling model The ANSYS Multifield solver is used to couple the fluid and structure solution field for the FSI simulation. During the fluid and structure interaction calculations, the numerical procedure in each iteration can be summarized as follows: • • • •

solving the fluid velocity u, v, w, pressure P in the fluid field, calculating the fluid load F acting on structure, solving the displacement y via the structural equation of motion, calculating the new fluid grid after structure movement.

3. Results and discussion In order to investigate the dynamical behaviours of elastically mounted tubes due to fluid–structure interaction, a single tube, two in-line tubes, two parallel tubes and 3 × 3 tubes subjected to cross flow are simulated in the following three section. Schematics of the simulation models are presented in Fig. 1. Fig. 3. Contour plots of velocities for a tube subjected to cross flow.

3.1. A single tube subjected to cross flow In this section, a tube with inner diameter Di = 16.87 mm, outer diameter Do = 19.05 mm and length L = 400 mm subjected to cross

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(a)

(b)

0.2

Power Spectral density (db/Hz)

20

0.15 0.1

y/D

0.05 0 -0.05 -0.1 -0.15 -0.2

0

2

4

6

8

10

12

14

16

18

0 -20 -40 -60 -80 -100 -120 -140

20

0

1

2

3

4

t

5

6

7

8

9

10

0.15

0.2

f

(c)

(d)

1.5

2 1.5

1

1

d(y/D)/dt

d(y/D)/dt

0.5

0

0.5 0 -0.5

-0.5 -1 -1

-1.5 -1.5 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

y/D

-2 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

y/D

Fig. 4. The dynamics of a tube subjected to cross flow at U = 0.2 m/s: (a) time trace, (b) power spectral density (PSD), (c) phase-plane plot, (d) Poincaré map.

Table 1 Geometrical and numerical parameters of the simulation. Tube

Fluid

Solid density (s ) Young’s modulus (E) Inner diameter (Di ) Outer diameter (Do ) Length (L) Reynolds (Re) Fluid density (f ) Velocity (U)

8200 kg/m3 2.18 × 1011 Pa 16.87 × 10−3 m 19.05 × 10−3 m 0.4 m 4267 1000 kg/m3 0.1–2.5 m/s

flow is investigated, as shown in Fig. 1(a). The geometrical and the numerical parameters are listed in Table 1. The computational grids of the model in the simulation are presented in Fig. 2. The grids are constructed using ICEM CFD, refined meshes are set in square section encompassing the tube, and relatively coarse meshes are set elsewhere. The fluid model employs a mesh of 111,452 hexahedral elements for this simulation. In the simulation, in order to secure sufficient region for fluid development, the tube is positioned at one fifth of a whole computational domain, the size of which is set to 25Do × 10Do . Velocity inlet boundary is applied on the left side of the computational domain, pressure outlet boundary is applied on the

right side of computational domain, free-slip wall boundary is applied on the top and bottom sides of computational domain, and fluid–structure interface is applied on the tube outer surface. The computational geometry is fixed, whereas the inlet flow velocity is varied, which ensures that the Reynolds number remains constant, Re = 4267. Contour plots of velocities for this typical tube subjected to cross flow at U = 0.2 m/s are shown in Fig. 3, the results for each 0.1 s increment of time from 4.0 s to 4.5 s, which illustrate the vortex shedding pattern as the time elapse. Analysis of the dynamics of this tube subjected to cross flow at U = 0.2 m/s is shown in Fig. 4. The time trace in Fig. 4(a) shows that the oscillation is periodic and almost harmonic, and the maximum non-dimensional Y-directional displacement y/D = 0.14. In Fig. 4(b), the main oscillation frequency is found to be f = 1.99. Fig. 4(c) shows that the oscillation is indeed periodic, as confirmed by the single point in Fig. 4(d) for the Poincaré map. 3.2. Two in-line tubes subjected to cross flow In this section, two in-line tubes subjected to cross flow are investigated, as shown in Fig. 1(b). The geometrical and the

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379

0.4 Upstream tube Downstream tube Single tube

0.35 0.3

y/D

0.25

Fig. 5. Contour plot of velocity for two in-line tubes with Px /D = 2.8 subjected to cross flow at t = 4 s.

0.2 0.15 0.1

numerical parameters of tubes are the same as the former one in Section 3.1, and the X-directional distance between the upstream tube and the downstream tube is defined as pitch Px . The mutual influence of the two in-line tubes is investigated. Contour plot of velocity for the two in-line tubes with Px /D = 2.8 subjected to cross flow at a given time is shown in Fig. 5. It is seen that the downstream tube is in the wake of the upstream tube. The time traces of the two in-line tubes with pitch-to-diameter ratio Px /D = 2.8 subjected to cross flow at U = 0.2 m/s are shown in Fig. 6; it can be seen in Fig. 6 that, the maximum non-dimensional displacement of the upstream tube is y/D = 0.15, and the maximum non-dimensional displacement of the downstream tube is y/D = 0.24; the difference between the displacement of the upstream tube and the displacement of the downstream tube is due to the wake effect of the upstream tube. It is seen that the two tubes undergo periodic limit-cycle oscillations, and vibrate in same phase for every step. The relationship between the pitch-to-diameter ratio Px /D and the non-dimensional displacement y/D for two in-line tubes subjected to cross flow at U = 0.2 m/s is shown in Fig. 7. The values of pitch-to-diameter ratio Px /D are between 1.2 and 2.8. It can be seen in Fig. 7 that, for small pitch-to-diameter ratio, i.e., 1.2 < Px /D < 1.8, with increasing Px /D, the vibration amplitudes of the two tubes increase. When the two tubes are very close to each other, i.e., Px /D = 1.2, the vibration amplitudes of the two tubes are small. When Px /D >1.5, the vibration amplitude of the upstream tube is close to the amplitude of a single tube subjected to cross flow, the upstream tube is almost not disturbed by the downstream one. For middle pitch-to-diameter ratio i.e., 1.8 < Px /D < 2.4, with increasing

0.4 Upstream tube Downstream tube

0.3 0.2

y/D

0.1

0.05 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

P /D x

Fig. 7. The relationship between pitch-to-diameter ratio Px /D and non-dimensional displacement y/D for two in-line tubes subjected to cross flow at U = 0.2 m/s.

Px /D, the vibration amplitudes of the two tubes decrease. For large pitch-to-diameter ratio, i.e., 2.4 < Px /D < 2.8, with increasing Px /D, the vibration amplitudes of the two tubes again increase. When Px /D = 1.8 and 2.8, the difference between the vibration amplitude of the upstream tube and the vibration amplitude of the downstream tube is large, up to 0.1, which is considered to be due to the wake effect of the upstream tube. 3.3. Two parallel tubes subjected to cross flow In this section, two parallel tubes subjected to cross flow are investigated, as shown in Fig. 1(c). The geometrical and the numerical parameters of tubes are the same as the former one in Section 3.1, and the Y-directional distance between the upper tube and the lower tube is defined as pitch Py . Contour plot of velocity for the two parallel tubes with Py /D = 2.8 subjected to cross flow at a given time is shown in Fig. 8. It is seen that the wakes of the two tubes are in the opposite phase. The time traces of the two parallel tubes with pitch-to-diameter ratio Py /D = 2.8 subjected to cross flow at U = 0.2 m/s are shown in Fig. 9; it is seen that, the maximum positive non-dimensional displacement of the upper tube is y/D = 0.16, and the maximum negative non-dimensional displacement of the upper tube is y/D = 0.17; the maximum positive non-dimensional displacement of the lower tube is y/D = 0.17, and the maximum negative non-dimensional displacement of the lower tube is y/D = 0.15; that is to say the displacements of the two tubes in the inward direction are slightly greater than the displacements in the outward direction; the mutual influence of the two parallel tubes is small. It is seen that

0

-0.1 -0.2 -0.3 0

2

4

6

8

10

12

14

16

18

20

t Fig. 6. Time traces of two in-line tubes with Px /D = 2.8 subjected to cross flow at U = 0.2 m/s.

Fig. 8. Contour plot of velocity for two parallel tubes with Py /D = 2.8 subjected to cross flow at t = 4 s.

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0.3

Pitch P x

Upper tube Lower tube

0.1

y/D

1

2

3

4

5

6

7

8

9

Pitch P y

0.2

0

U

-0.1

-0.2

-0.3

0

1

2

3

4

5

6

7

8

9

10

t Fig. 10. Schematic of a 3 × 3 tube bundle subjected to cross flow. Fig. 9. Time traces of two parallel tubes with Py /D = 2.8 subjected to cross flow at U = 0.2 m/s.

the two tubes undergo periodic limit-cycle oscillations, and vibrate in opposite phase for every step. 3.4. 3 × 3 tube bundle subjected to cross flow In this section, a bundle of 9 tubes (3 × 3) in a square array subjected to cross flow are investigated. The 9 tubes are indicated respectively by tube 1, 2, 3, 4, 5, 6, 7, 8 and 9, as shown in Fig. 10. The geometrical and the numerical parameters of the tubes are the same as the former one in Section 3.1, and the X-directional and Ydirectional pitch-to-diameter ratios between the contiguous tubes are Px /D = Py /D = 2.8. Contour plot of velocity for the 3 × 3 tube bundle subjected to cross flow at a given time is shown in Fig. 11. It is seen that the downstream tubes are in the wake of the upstream tubes. The time traces of the 3 × 3 tubes subjected to cross flow at U = 0.2 m/s are shown in Fig. 12. It is seen that the maximum nondimensional displacement of tube 1, tube 2, tube 3, tube 4, tube 5, tube 6, tube 7, tube 8 and tube 9 are respectively y/D = 0.13, 0.19,

Fig. 11. Contour plot of velocity for a 3 × 3 tube bundle subjected to cross flow at t = 4 s.

0.20, 0.12, 0.21, 0.19, 0.10, 0.23 and 0.21; the vibration amplitudes of downstream tubes are larger than those of upstream tubes. The oscillations of the 9 tubes are quite irregular, the non-dimensional displacements of the tubes are varied compared to the single tube and two tubes, which is considered to be due to the mutual influence of the tubes.

0.3 tube 1

tube 2

tube 3

tube 4

tube 5

tube 6

tube 7

tube 8

tube 9

0.2

y/D

0.1

0

-0.1

-0.2

-0.3 0

1

2

3

4

5

6

7

t Fig. 12. Time traces of 3 × 3 tubes subjected to cross flow at U = 0.2 m/s.

8

9

10

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4. Conclusions In this paper, the main work presented is focused on the tubes subjected to cross flow. A three-dimensional CFD model, coupled with a structural analysis model is used to investigate the dynamical behaviours and flow patterns of this fluid–structure system. The single tube subjected to cross flow undergoes periodic limitcycle oscillation when the flow velocity exceeds the critical flow velocity, which is due to the periodic vortex shedding. The analysis of the two in-line tubes subjected to cross flow shows that the dynamical behaviour of the downstream tube is affected by the wake effect of the upstream tube, the vibration amplitude of the downstream tube is larger than the amplitude of the upstream tube; the two tubes undergo periodic limit-cycle oscillations, and vibrate in same phase. For small pitch-to-diameter ratio Px /D, the vibration amplitudes of the two tubes increase as Px /D increases; for middle pitch-to-diameter ratio Px /D, the vibration amplitudes of the two tubes decrease as Px /D increases; for large pitch-to-diameter ratio Px /D, the vibration amplitudes of the two tubes again increase as Px /D increases; when Px /D > 1.5, the vibration amplitude of the upstream tube is close to the vibration amplitude of a single tube, the upstream tube is almost not disturbed by the downstream one. The analysis of the two parallel tubes subjected to cross flow shows that the displacements of two tubes in the inward direction are slightly greater than the displacements in the outward direction, the mutual influence of the two parallel tubes is small; the two tubes undergo periodic limit-cycle oscillations, and vibrate in opposite phase. The analysis of the 3 × 3 tubes subjected to cross flow shows that the vibration amplitudes of downstream tubes are larger than those of upstream tubes, and the oscillations of the 9 tubes are quite irregular, the displacements of the tubes are varied because of the mutual influence of tubes. Acknowledgements The authors gratefully acknowledge the support by the National Natural Science Foundation of China (grant no. 51179135) and the National Science and Technology Major Project (grant no. 2011ZX06004-002). References Ahn, H., Kallinderis, Y., 2006. Strongly coupled flow/structure interactions with a geometrically conservative ALE scheme on general hybrid meshes. J. Comput. Phys. 219, 671–696.

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