# Fluid-Structure Interaction Response of a Water Conveyance System with a Surge Chamber during Water Hammer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

_{z}denotes axial pipeline stress, u

_{z}denotes axial pipeline velocity, Q

_{y}denotes transverse shear force, u

_{y}denotes transverse pipeline velocity, M

_{x}denotes the bending moment of the pipeline, θ

_{x}denotes angular velocity, E is the Young’s modulus of elasticity of pipe materials, ν denotes Poisson’s ratio, e denotes wall thickness, A

_{t}and A

_{f}represent the sectional area of the pipe wall and internal sectional area of the pipeline, respectively, and ρ

_{t}and ρ

_{f}are density of pipe and fluid respectively, shear modulus of elasticity of pipe materials is G = E/2 (1 + ν). R

_{f}and R

_{t}are fluid damp coefficient and structure damp coefficient, separately, R

_{f}= f|V|/4R, R

_{t}= R

_{f}ρ

_{t}A

_{t}/ρ

_{f}A

_{f}. In this study, f = F(Re), Re is Reynolds number, When Re < 2100, f = 16/Re, when 3000 < Re < 10000, f = 0.079/Re

^{0.25}When Re > 105, f = 0.046/Re

^{0.2}[25,26] Here, Re = ρ

_{f}|V|d/ν, v is Kinematic viscosity of liquid.

#### 2.2. Surge Chamber

_{L}denotes the flow at the upstream end adjacent to the surge chamber, Q

_{R}denotes the flow across the section shortly downstream the chamber and Q

_{s}denotes the exchanged flow between the surge chamber and the pipeline. As a result of the FSI response, the velocity of the fluid in the surge chamber is actually the one relative to the lateral vibration velocity of the surge chamber. The motion and continuity equation of the surge chamber can be defined as in Equation (10)

_{s}and H

_{s}, separately, denote the section area and dynamic water level of the surge chamber. In consideration of the FSI, the velocities corresponding to Q

_{L}and Q

_{R}separately are taken as the relative ones to the pipe vibration speed. The following equation is then obtained

_{V}represents the mass of the surge chamber and J represents the rotating inertia of the lumped mass block around the x–axis.

#### 2.3. Elbow Tube

_{1}= P

_{2}

_{1}− u

_{z1}= V

_{2}− u

_{z2}

_{f}P

_{1}− A

_{s}σ

_{z1}= (A

_{f}P

_{2}− A

_{s}σ

_{z2}) cosα + Q

_{y2}sinα

_{y1}= (A

_{f}P

_{2}− A

_{s}σ

_{z2}) sinα + Q

_{y2}cosα

_{x1}= M

_{x2}

#### 2.4. Outlet and Inlet Boundary

_{0}

_{0}denotes the pressure provided by the water level in the reservoir. The pipeline is rigidly connected to the water tank. When the inlet or outlet is connected rigidly, continuity equations of axial, lateral, and bending directions are as follows [31]

_{z}= 0

_{y}= 0

_{x}= 0

_{z}, Δu

_{y}and Δθ

_{x}represent the variations in the axial and lateral velocities at the outlet of pipeline and the variation in the turning angle during a time step, respectively.

## 3. Solution Technique

#### 3.1. Finite Volume Method

**A**,

**B**and

**S**are the matrices decided by Equations (1)–(8), and they are the matrices composed by constant. Δt is used to calculate the time step, Δz refers to the spacing between the calculation nodes of adjacent pipelines, n refers to the nth time step and i refers to the number of pipes.

**C**

_{i}refers to the state vector of calculation node in the ith pipe at the nth time step. When calculating the state vector at the (n + 1)th time step,

**C**

_{i}is a known parameter. Vector

**Q**

_{i}denotes the calculation node state vector at the (n + 1)th time step in the ith pipe as obtained from the iteration of Equation (34) in the nth time step state,

**Q**

_{i}is an unknown vector. In these equations, i = 1 to N, where 1 and N refer to the calculation nodes at the inlet and outlet of the pipeline, respectively. When the pipeline or fluid between adjacent pipe structure is discontinuous, then this pipeline or fluid will be processed as an interior boundary condition.

**A**

_{i}and

**C**

_{i}are included in the boundary condition’s equation.

#### 3.2. Boundary Condition

**A**

_{i},

**Q**

_{i}and

**C**

_{i}, then

**C**

_{i}according to Equation (34), the following equation is obtained.

**T**

_{1}and

**T**

_{2}denote the coefficient matrices of the boundary condition’s equations at the inlet and outlet, respectively. It is notable that Equation (34) is an iterative formula of a water conveyance system without elbows, auxiliary buildings, or supports. In the method used in this study, elbows, auxiliary buildings, or supports are taken as boundary conditions, and the relevant iterative formula is obtained by substituting Equation (39) into Equation (34). The iterative formula is complex, especially the first term of Equation (34). Thus, a computing code was programmed by the first author to perform the complex matrix calculations.

## 4. Experimental Verification

_{f}= 1128 m/s, the frequency is computed as f =c

_{f}/4 L = 25.41 Hz, the stress wave velocity was c

_{s}= 2557 m/s and the time cycle was f =c

_{s}/4 L = 53.19 Hz. A data acquisition frequency of 200 Hz could reflect the stress and pressure waves. The pipeline vibration frequency was far less than the calibration frequency of the pressure sensor and did not cause resonance. Therefore, the pressure sensor in this experiment was considered to be suitable.

_{p}= c

_{f}/4d = 72.63 Hz (d = 2 m) and the frequency of stress wave f

_{s}= c

_{s}/4 L = 38.37 Hz. The time domain of the FSI response in the whole process is shown in Figure 5a,b. The results of fast Fourier transformation are displayed in Figure 6a,b. The pressure fluctuation was composed of the stress wave, pressure wave, and gravity wave (surge). A comparison of Figure 6a,b reveals that, for the pipe closer to the surge chamber, the gravity wave contributed more to the pressure pulsation amplitude because energy losses occur during the propagation of the stress and water hammer waves.

## 5. Numerical Results and Discussion

^{3}/s and 0.4 MPa, separately. The cross–section areas of the surge chamber and impedance hole were 11 m

^{2}and 3 m

^{2}. The schematic of the water conveyance system is shown in Figure 7. The governing equation was built by using a universal FVM. Given the very small frequencies of the stress and water hammer waves, a relatively large calculation time step could be selected. However, to ensure the reliability of the numerical results, the calculation time step could not be too large.

_{f}V and R

_{t}u

_{z}in Equations (1) and (3), the original vibrations of pipe wall are shown in Figure 10.

_{f}V and R

_{t}u

_{z}in Equations (1) and (3). Without fractional coupling, R

_{t}u

_{z}equals zero. The numerical results of pressure with and without fractional coupling are shown as the red solid line and the black solid line respectively in Figure 13. With fractional coupling, the disturbance mainly comes from lower axial displacement. So the fluid fluctuations with frequencies of stress wave and pressure wave have a smaller amplitude.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{s}, reduced by Equation (A2), Equation (A3) is obtained

## References

- Leishear, R.A. Fluid Mechanics, Water Hammer, Dynamic Stresses, and Piping Design; ASME: New York, NY, USA, 2012. [Google Scholar] [CrossRef] [Green Version]
- Olalla, G.; SasIrene, D.; Begoña, G. Liquid-liquid extraction of phenolic compounds from water using ionic liquids: Literature review and new experimental data using C2mim FSI. J. Environ. Manag.
**2018**, 78, 475–482. [Google Scholar] [CrossRef] - Osama, M.; Theofilis, V.; Ahmed, E. Fluid Structure Interaction (FSI) Simulation Of the human eye under the air puff tonometry using CFD. In Proceedings of the Tenth International Conference on Computational Fluid Dynamics (ICCFD10), Barcelona, Spain, 9–13 July 2018. [Google Scholar]
- Bazilevs, Y.; Yan, J.; Deng, X. Simulating Free-Surface FSI and Fatigue Damage in Wind-Turbine Structural Systems. In Frontiers in Computational Fluid-Structure Interaction and Flow Simulation: Research from Lead Investigators under Forty; Springer: Berlin, Germany, 2018. [Google Scholar] [CrossRef]
- Tijsseling, A.S.; Vardy, A.E. Time scales and FSI in unsteady liquid-filled pipe flow. In The 9th International on Pressure Surges; Chester, UK, 2004. [Google Scholar] [CrossRef]
- Wiggert, D.C.; Hatfield, F.J.; Stuckenbruck, S. Analysis of Liquid and Structural Transients in Piping by the Method of Characteristics. J. Fluids Eng.
**1987**, 109, 161–165. [Google Scholar] [CrossRef] - Lavooij, C.S.W.; Tijsseling, A.S. Fluid–structure interaction in liquid-filled piping systems. J. Fluids Struct.
**1991**, 5, 573–595. [Google Scholar] [CrossRef] - Ahmadi, A.; Keramat, A. Investigation of fluid–structure interaction with various types of junction coupling. J. Fluids Struct.
**2010**, 26, 1123–1141. [Google Scholar] [CrossRef] - Liu, Y.; He, X.; Zhi, Y. Dynamical strength and design optimization of pipe-joint system under pressure impact load. Proceedings of the Institution of Mechanical Engineers, Part G. J. Aerosp. Eng.
**2011**, 226, 1029–1040. [Google Scholar] [CrossRef] - Huang, S.; Zhou, B.; Bu, S.; Li, C.; Zhang, C.; Wang, H.; Wang, T. Robust fixed-time sliding mode control for fractional-order nonlinear hydro-turbine governing system. Renew. Energy
**2019**, 139, 447–458. [Google Scholar] [CrossRef] - Zhai, H.; Wu, Z.; Liu, Y.; Yue, Z. In-plane dynamic response analysis of curved pipe conveying fluid subjected to random excitation. Nucl. Eng. Des.
**2013**, 256, 214–226. [Google Scholar] [CrossRef] - Tijsseling, A.S.; Vardy, A.E.; Fan, D. Fluid-Structure Interaction and Cavitation in a Single-elbow Pipe System. J. Fluid Struct.
**1996**, 10, 395–420. [Google Scholar] [CrossRef] - Mu, L.Z.; Li, X.Y.; Chi, Q.Z.; Yang, S.Q.; Zhang, P.D.; Ji, C.J.; He, Y.; Gao, G. Experimental and numerical study of the effect of pulsatile flow on wall displacement oscillation in a flexible lateral aneurysm model. Acta Mech. Sin.
**2019**, 39, 1120–1129. [Google Scholar] [CrossRef] - Forbes, T.B.; Stephen, C.T. Dynamic Behavior of Complex Fluid-Filled Tubing Systems-Part ii: System Analysis. J. Dyn. Syst. Meas.
**2001**, 123, 78–84. [Google Scholar] [CrossRef] - Altstadt, E.; Carl, H.; Prasser, H.M.; Weis, R. Fluid-structure interaction during artificially induced water hammers in a tube with a bend—Experiments and analyses. Multiph. Sci. Technol.
**2008**, 20, 213–238. [Google Scholar] [CrossRef] - Wang, Y.G. Efficient prediction of wave energy converters power output considering bottom effects. Ocean Eng.
**2019**, 181, 89–97. [Google Scholar] [CrossRef] - Zhang, Y.L.; Miao, M.F.; Ma, J.M. Analytical study on water hammer pressure in pressurized conduits with a throttled surge chamber for slow closure. Water Sci. Technol.
**2010**, 3, 174–189. [Google Scholar] [CrossRef] - Chen, S.; Zhang, J.; Yu, X.D. Characterization of surge superposition following 2-stage load rejection in hydroelectric power plant. Can. J. Civ. Eng.
**2016**, 43, 844–850. [Google Scholar] [CrossRef] - Guo, W.C.; Yang, J.D. Combined effect of upstream surge chamber and sloping ceiling tailrace tunnel on dynamic performance of turbine regulating system of hydroelectric power plant. Chaos Solitons Fractals
**2017**, 99, 243–255. [Google Scholar] [CrossRef] - Zhang, Y.L.; Niao, M.F. Explicit formulas for calculating surges in throttled surge chamber. J. Hydraul. Eng.
**2012**, 4, 467–472. (In Chinese) [Google Scholar] - Wiggert, D.C.; Otwell, R.S.; Hatfield, F.J. The effect of elbow restraint on pressure transients. J. Fluid Eng.
**1985**, 107, 402–406. [Google Scholar] [CrossRef] - Okosun, F.; Cahill1, P.; Hazra, B.; Pakrash, V. Vibration-based leak detection and monitoring of water pipes using output-only piezoelectric sensors. Eur. Phys. J. Spec. Top.
**2019**, 228, 1659–1675. [Google Scholar] [CrossRef] - Tijsseling, A.S.; Wiggert, D.C. Fluid transients and fluid-structure interaction in flexible liquid-filled pipeing. Appl. Mech. Rev.
**2001**, 54, 455–481. [Google Scholar] [CrossRef] - Wang, Y.H.; Zhang, J.H.; Ma, Z.X. Experimental determination of single-phase pressure drop and heat transfer in a horizontal internal helically-finned tube. Int. J. Heat Mass Transf.
**2017**, 104, 240–246. [Google Scholar] [CrossRef] - Taitel, Y.; Dukler, A.E. A model for predicting flow regime transition in horizontal and near horizontal gas-liquid flow. AIChE J.
**1976**, 22, 47–55. [Google Scholar] [CrossRef] - Meindlhumer, M.; Pechstein, A. 3D mixed finite elements for curved, flat piezoelectric structures. Int. J. Smart Nano Mater.
**2019**, 10, 249–267. [Google Scholar] [CrossRef] [Green Version] - Wei, X.; Sun, B. Study on fluid–structure interaction in liquid oxygen feeding pipe systems using finite volume method. Acta Mech Sin.
**2011**, 27, 706–712. [Google Scholar] [CrossRef] - Ferras, D.; Manso, P.A.; Covas, D.I.C. Fluid–structure interaction in pipe coils during hydraulic transients. J. Hydraul. Res.
**2017**, 55, 491–505. [Google Scholar] [CrossRef] - David, F.; Pedro, A.M.; Anton, J.S. Fluid-structure interaction in straight pipelines with different anchoring conditions. J. Sound Vib.
**2017**, 394, 348–365. [Google Scholar] [CrossRef] - Banks, J.W.; Henshaw, W.D.; Schwen, D.W. A stable partitioned FSI algorithm for rigid bodies and incompressible flow in three dimensions. J. Comput. Phys.
**2018**, 73, 455–492. [Google Scholar] [CrossRef] [Green Version] - Li, S.J.; Karney, B.W.; Liu, G.M. FSI research in pipeline systems-A review of the literature. J. Fluids Struct.
**2015**, 57, 277–297. [Google Scholar] [CrossRef] - Ramírez, L.; Nogueira, X.; Ouro, P. A Higher-Order Chimera Method for Finite Volume Schemes. Arch. Comput. Methods Eng.
**2018**, 25, 691–706. [Google Scholar] [CrossRef] [Green Version] - Hwang, Y.H.; Chung, N.M. A fast Godunov method for the water-hammer problem. Int. J. Numer. Methods Fluids
**2002**, 40, 799–819. [Google Scholar] [CrossRef] - Sreejith, B.; Jayaraj, K.; Ganesan, N. Finite element analysis of fluid-structure interaction in pipeline systems. Nucl. Eng. Des.
**2004**, 227, 313–322. [Google Scholar] [CrossRef] - Zhang, L.; Tijsseling, A.S.; Vardy, E.A. FSI analysis of liquid-filled pipes. J. Sound Vib.
**1999**, 224, 69–99. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**The experimental equipment. (

**a**) is the graph of piping system and (

**b**) is the graph of pressure sensor.

**Figure 5.**Pressure fluctuation history with fluid–structure interaction (FSI) taking into account (

**a**) closer outlet; (

**b**) near surge chamber. The red solid line refers to numerical result, the black solid line refers to experimental result.

**Figure 6.**Pressure in frequency domain with FSI taking into account (

**a**) closer outlet; (

**b**) near surge chamber. The red solid line refers to numerical result, the black solid line refers to experimental result.

**Figure 10.**Amplitude of original modes of axial displacement of pipe produced by surge and closed valve.

**Figure 11.**Time history and frequency responses of dynamic pressure. (

**a**,

**c**,

**e**) are time histories of dynamic pressure near surge chamber, at the outlet and at the elbow. (

**b**,

**d**,

**f**) are frequency responses of dynamic pressure near surge chamber, at the outlet and at the elbow.

**Figure 13.**Time history of pressure at the bottom of surge chamber with and without frictional coupling.

**Figure 14.**Time history of pressure at the bottom of surge chamber with and without Poisson coupling.

ν | E | L | e | D | ρ_{t} |
---|---|---|---|---|---|

0.3 | 3.88 Gpa | 11.1 m | 0.005 m | 0.05 m | 1378.57 kg/m^{3} |

Mode | ν = 0.3 | ν = 0 |
---|---|---|

1 | 2.1 Hz | 2.1 Hz |

2 | 5.8 Hz | 5.6 Hz |

3 | 9.3 Hz | 9.0 Hz |

4 | 13.1 Hz | 12.7 Hz |

5 | 17.0 Hz | 16.1 Hz |

6 | 20.5 Hz | 19.7 Hz |

7 | 21.2 Hz | 23.1 Hz |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guo, Q.; Zhou, J.; Li, Y.; Guan, X.; Liu, D.; Zhang, J.
Fluid-Structure Interaction Response of a Water Conveyance System with a Surge Chamber during Water Hammer. *Water* **2020**, *12*, 1025.
https://doi.org/10.3390/w12041025

**AMA Style**

Guo Q, Zhou J, Li Y, Guan X, Liu D, Zhang J.
Fluid-Structure Interaction Response of a Water Conveyance System with a Surge Chamber during Water Hammer. *Water*. 2020; 12(4):1025.
https://doi.org/10.3390/w12041025

**Chicago/Turabian Style**

Guo, Qiang, Jianxu Zhou, Yongfa Li, Xiaolin Guan, Daohua Liu, and Jian Zhang.
2020. "Fluid-Structure Interaction Response of a Water Conveyance System with a Surge Chamber during Water Hammer" *Water* 12, no. 4: 1025.
https://doi.org/10.3390/w12041025