# Vibration Response and Stress Analysis of Planar Elastic Tube Bundle Induced by Fluid Flow

- De-Rong Duan
^{1}, - Pei-Qi Ge
^{1, 2}Email author, - Wen-Bo Bi
^{1}and - Yan-Ying Dong
^{1}

**31**:32

https://doi.org/10.1186/s10033-018-0240-7

© The Author(s) 2018

**Received: **25 June 2017

**Accepted: **16 April 2018

**Published: **26 April 2018

## Abstract

Flow-induced vibration plays a positive role on heat transfer enhancement. Meanwhile, it is also a negative factor for fatigue strength. Satisfying the fatigue strength is the primary prerequisite for heat transfer enhancement. This paper numerically studied the flow-induced vibration of planar elastic tube bundle based on a two-way fluid–structure interaction (FSI) calculation. The numerical calculation involved the unsteady, three-dimensional incompressible governing equations solved with finite volume approach and the dynamic balance equation of planar elastic tube bundle solved with finite element method combined with dynamic mesh scheme. The numerical approach was verified by comparing with the published experimental results. Then the vibration trajectory, deformation and stress contour of planar elastic tube bundle were all studied. Results show that the combined movement of planar elastic tube bundle represents the agitation from inside to outside. The vibration of out-of-plane is the main vibration form with the typically sinusoidal behavior because the magnitude of displacement along the out-of-plane direction is the 100 times than the value of in-plane direction. The dangerous point locates in the innermost tube where the equivalent stress can be utilized to study the multiaxial fatigue of planar elastic tube bundle due to the alternating stress concentration. In the velocity range of 0.2‒3 m/s, it is inferred that the vibration amplitude plays a role on the stress response and the stress amplitude is susceptible to the fluid velocity. This research paves a way for studying the fatigue strength of planar elastic tube bundle by flow-induced vibration.

## Keywords

## 1 Introduction

Flow-induced vibration, considered as a complex phenomenon of fluid–structure interaction, may occur in many engineering fields, such as heat exchanger, nuclear engineering, ocean engineering [1–3]. This vibration will lead to the fatigue of structure due to the periodic alternating force. Thus, many scholars pay their attention on the fatigue failure by flow-induced vibration.

Ji et al. [4] analyzed the flow-induced vibration of heat transfer tube in evaporator heating chamber. The shedding frequency of vortex street, natural frequency and critical flow velocity were all calculated. Results show that the Karman vortex and turbulence buffeting occur at the entrance with high velocity of water vapor. While, the resonance of heat transfer tube occur because the shedding frequency is similar to the natural frequency of tube. The coupling effect of vibration and temperature difference stress lead to the fatigue damage of tube. In ocean engineering, fatigue failure of marine risers due to flow-induced vibration is one of the most significant failure modes. Xue et al. [5] proposed a new prediction model of FIV fatigue damage for riser considering both cross-flow and in-line vibrations based on the energy equilibrium theory. The effect of top velocity of sheared current, top tension, fluid density and the material of structure on the fatigue damage were discussed. Results reveal that the displacement of in-line vibration is much less than the value of cross-flow. However, there is a same important level for both fluid flows because high vibration frequency occurs in in-line vibration. The top velocity of sheared current plays a great role on the fatigue damage. The fatigue damage may be reduced 50% owing to the 10% reduction in top velocity of sheared current. Sun et al. [6] investigated the fatigue damage of riser undergoing in-line and cross-flow flow-induced vibration using pseudo-excitation method. Results show that the position near the top of the riser Z/LR = 0.9 is the dangerous point. The fatigue damage is mainly subjected to the cross-flow vibration. Wang et al. [7] experimentally studied the fatigue damage caused by vortex-induced vibration based on rain-flow counting and a standard *S*–*N* curve in an ocean basin. Results show that amplitude modulation and mode transition dominate the fatigue of flexible cylinder. Ei et al. [8] studied the effect of tube support clearance on the flow-induced vibration of steam generators. Special emphasis was placed on crack propagation and leakage rates. The developed model was established based on the stress-time history due to the flow-induced vibration. Results show that clearances below 0.3 mm are crucial to restricting crack growth rates to acceptable levels.

Every coin has two sides. This vibration is a negative factor for fatigue failure. However, Flow-induced vibration has been proved to be a positive role on heat transfer enhancement [9–11]. Planar elastic tube bundle is a novel heat transfer device to enhance heat transfer using flow-induced vibration [12]. The heat transfer may be enhanced by two times than the value of fixed tube bundle [13]. Some emphasis has been placed on the heat transfer characteristics and the vibration characteristics [14–16]. For the fatigue failure of planar elastic tube bundle, there is very less researches in the open literatures. Yan et al. [17] pointed out that the cross section diameter of tube bundle and the mass block played major role on the stress of tube bundle.

Satisfying the fatigue strength is the primary prerequisite for heat transfer enhancement. In order to enrich the study of planar elastic tube bundle, the vibration response and stress analysis are studied in this paper to pave the way for fatigue analysis of planar elastic tube bundle. Thus, a two-way fluid–structure interaction is conducted for the flow-induced vibration of planar elastic tube bundle. The vibration trajectory, deformation and stress contour of planar elastic tube bundle are all studied. The conclusions are the basis for investigating the fatigue failure of planar elastic tube bundle.

## 2 Numerical Approach

### 2.1 Physical Model

*H*

_{ s }and diameter

*D*

_{ s }of heat exchanger are 780 mm and 300 mm, respectively. The diameter of inlet

*D*

_{ in }and outlet

*D*

_{ out }are 63 mm and 65 mm, respectively. The planar elastic tube bundle is installed in the middle of shell-side. The installation height

*H*

_{ t }is 430 mm. The planar elastic tube bundle is shown in Figure 1(b) including four curved copper tube and two mass-blocks. The radius of innermost tube

*R*

_{1}is 70 mm and the tube spacing is 20 mm. In the present work, the diameter of tube is 10 mm and the tube wall thickness is 1.5 mm.

In the heat transfer process, the fluid of shell-side flows into heat exchanger from the bottom inlet and the planar elastic tube bundle is heated by the fluid of tube-side. Then the fluid of shell-side is heated and flows out from the top outlet. Meanwhile, the tube vibration occurs because of the periodic alternating force. It is verified that this vibration is beneficial to heat transfer [18–20]. In this paper, the flow-induced vibration by shell-side is studied because the effect of tube-side on the vibration of planar elastic tube bundle can be ignored [21].

### 2.2 Governing Equations

*u*

_{ f }−

*u*

_{ mf }) is the convective velocity in shell-side domain.

*u*

_{ f }is the fluid velocity and

*u*

_{ mf }is the mesh velocity.

*T*,

*p*,

*ρ*,

*ν*,

*C*

_{ p }and

*λ*represent the temperature, pressure, density, kinematic viscosity, specific heat and thermal conductivity of fluid.

*d*and

*σ*represent the displacement and stress on the fluid–structure interface.

*n*is the normal vector.

### 2.3 Meshing and Boundary Conditions

^{−4}and growth ratio 1.2 were established around the planar elastic tube bundle. The inlet section was discretized by tetrahedral mesh. At last, three sections were connected by Interface 1 and Interface 2. The inlet was set to velocity-inlet boundary with the constant temperature. The outlet was set to pressure boundary condition with 0 (gauge). The wall of planar elastic tube bundle was set to no-slip boundary condition with constant wall temperature and all surface of planar elastic tube bundle were set to FSI interface for data transfer. Others wall of shell-side were set to adiabatic. In Figure 3(b), the four curved were discretized by hexahedral mesh and two mass-blocks were discretized by tetrahedral mesh. The innermost and outermost tubes were fixed. All of the surfaces of planar elastic tube bundle were set to FSI surface with the constant wall temperature. The earth gravity is applied along the

*z*direction.

### 2.4 Validation and Grid Independence Test

Validation of the calculation in flow-induced vibration

Inlet velocity (m/s) | Monitoring points | Vibration frequency | Relative error (%) | |
---|---|---|---|---|

Numerical results | Experimental results [28] | |||

0.2 | Big MB | 28.57 | 28.0 | 2.04 |

Small MB | 29.7 | 29.0 | 2.41 | |

0.4 | Big MB | 28.37 | 28.0 | 1.32 |

Small MB | 29.63 | 29.0 | 2.17 |

Meanwhile, three kinds of grid number in shell-side were established to conduct the grid independence test, 316723, 323123 and 329523. It is derived that the relative error of convective heat transfer coefficient between 316723 and 323123 is 0.23% and the relative error of convective heat transfer coefficient between 323123 and 329523 is 0.16%. Almost equal vibration frequency of planar elastic tube bundle occurs for these three kinds of grid number. Therefore, the shell-side was discretized to 323123 grids to solve the flow-induced vibration calculation in view of saving computer resource and keeping a balance between computational economics and accuracy.

## 3 Results and Discussion

### 3.1 Vibration Trajectory

*x*–

*y*and

*x*–

*z*plane. The counterclockwise movement occurs on the

*y*–

*z*plane. It means that the combined movement of planar elastic tube bundle is the agitation from inside to outside, which is beneficial to reducing the thickness of thermal boundary layer to enhance heat transfer [29].

It is found that there is the same magnitude of displacement in *x* and *y* direction. The magnitude of displacement in *z* direction is 100 times than the value in *x* and *y* direction. It means that the vibration direction is perpendicular to the horizontal plane of planar elastic tube bundle, which is agreed with the conclusion of out-of-plane vibration [15].

### 3.2 Deformation

It is found that that the direction of vibration is along the *z* direction. The vibration balance position is below the horizontal plane of planar elastic tube bundle subjected to the earth gravity. The red arrow represents the instantaneous vibration direction. At the time of *τ*/4 and 3*τ*/4, the planar elastic tube bundle reaches the position of peak and valley, respectively. For the velocity 0.2 m/s, the planar elastic tube bundle is still below the horizontal plane at the position of peak. While the planar elastic tube bundle is almost over the horizontal plane at the position of peak in the velocity 0.5 m/s. The reason behind this is that the fluid force generated by increasing velocity overcomes the effect of earth gravity. It means that the vibration balance position will be raised because of the increasing fluid force.

### 3.3 Stress Analysis

Stress concentration of planar elastic tube bundle will occur in the vibration process as shown in Figure 6. Usually, the dangerous point locates in the innermost tube because the innermost tube is fixed with the greatest curvature. In Figure 6, it is clear that the maximum stress occurs at the time of 3*τ*/4, where is the valley of vibration. The value of stress at dangerous point is various with the vibration of planar elastic tube bundle. As a result, the fatigue failure will occur due to the alternating stress concentration in planar elastic tube bundle. It is verified that the planar elastic tube bundle suffers from bending stress and torsional stress. Thus, the equivalent stress can be utilized to study the multiaxial fatigue of planar elastic tube bundle, which is established based on the Von Mises stress criteria [30].

### 3.4 Vibration Response in High Velocity

In Figure 8(a), the maximum vibration amplitude 0.48 mm occurs in the velocity 2 m/s, which is the 5.3 times than the value in the velocity 0.2 m/s. The vibration balance position is over the horizontal plane when the value of velocity exceeds 2 m/s. Then, the vibration amplitude decreases with the increases in the velocity when the velocity exceeds 2 m/s. The minimum vibration amplitude 0.4 mm is obtained in the velocity 3 m/s, which is the 4.4 times than the value in the velocity 0.2 m/s. It is derived that the fluctuation 20% occurs for the vibration amplitude in the velocity range of 1‒3 m/s. For the vibration frequency, the maximum vibration frequency 23.69 Hz occurs in the velocity 2.5 m/s and the minimum vibration frequency 23.33 Hz occurs in the velocity 1 m/s. Comprehensive analysis are conducted that the fluctuation 2% occurs for the vibration frequency in the velocity range of 1‒3 m/s. It means that the vibration frequency exhibits independence on the inlet velocity. As a result, the vibration amplitude is more susceptible to the fluid velocity compared to the vibration frequency.

In Figure 8(b), the stress response is changed with the various in the vibration balance position and vibration amplitude. As a result, the mean stress decreases first and then increases. The maximum mean stress 9.0 MPa occurs in the velocity 1 m/s. The stress amplitude increases first and then decreases. The maximum stress amplitude 7.7 MPa occurs in the velocity 2 m/s. There is a larger fluctuation in stress amplitude compared the value in the velocity range 0.2‒0.5 m/s. Comprehensive analysis is conducted for the vibration and stress response. It implies that the vibration amplitude plays a role on the stress response and the stress amplitude is susceptible to the fluid velocity.

## 4 Conclusions

- (1)
The combined movement of planar elastic tube bundle represents the agitation from inside to outside. The magnitude of displacement along the out-of-plane direction is the 100 times than the value of in-plane direction, which means that the vibration of out-of-plane is the main vibration form of planar elastic tube bundle.

- (2)
The vibration of planar elastic tube bundle represents typically sinusoidal behavior. The vibration balance position is lifted with the increases in fluid force due to the increasing fluid velocity. Compared to the vibration frequency, the vibration amplitude is more susceptible to the fluid velocity.

- (3)
The dangerous point locates in the innermost tube because the innermost tube is fixed with the greatest curvature. The equivalent stress can be utilized to study the multiaxial fatigue of planar elastic tube bundle due to the alternating stress concentration.

- (4)
Comprehensive analysis is conducted for the vibration and stress response in the velocity range of 0.2‒3 m/s. It is inferred that the vibration amplitude plays a role on the stress response and the stress amplitude is susceptible to the fluid velocity.

## Declarations

### Authors’ Contributions

P-QG was in charge of the whole trial; D-RD wrote the manuscript; W-BB and Y-YD assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

De-Rong Duan, born in 1988, is currently a PhD candidate at *School of Mechanical Engineering, Shandong University, China*. He received his bachelor degree from *University of Jinan, China*, in 2010. His research interests include flow-induced vibration. Tel: +86-531-88399245; E-mail: dr_duan@126.com.

Pei-Qi Ge, born in 1963, is currently a professor at *Shandong University, China*. He received his PhD degree from *Harbin Institute of Technology, China*, in 1997. His research interests include flow-induced vibration, crystal slicing processing and diamond wire saw. Tel: +86-531-88399277; E-mail: pqge@sdu.edu.cn.

Wen-Bo Bi, born in 1979, is currently a lecturer at *School of Mechanical Engineering, Shandong University, China*. He received his PhD degree from *Shandong University, China*, in 2015. His main research interests include mechachonics engineering, crystal slicing processing and diamond wire saw. Tel: +86-531-88399277; E-mail: biwenbo@sdu.edu.cn.

Yan-Ying Dong, born in 1991, is currently a master candidate at *School of Mechanical Engineering, Shandong University, China*. She received her bachelor degree from *Harbin University of Commerce, China*, in 2015. Her research interests include flow-induced vibration. Tel: +86-531-88399245; E-mail: dongyanying7@163.com.

### Competing Interests

The authors declare that they have no competing interests.

### Ethics Approval and Consent to Participate

Not applicable.

### Funding

Supported by National Natural Science Foundation of China (Grant No. 51475268), National Basic Research Program of China (973 Program, Grant No. 2007CB206903).

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## Authors’ Affiliations

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