Fluid–structure interaction with pipe-wall viscoelasticity during water hammer

Abstract

Fluid–structure interaction (FSI) due to water hammer in a pipeline which has viscoelastic wall behaviour is studied. Appropriate governing equations are derived and numerically solved. In the numerical implementation of the hydraulic and structural equations, viscoelasticity is incorporated using the Kelvin–Voigt mechanical model. The equations are solved by two different approaches, namely the Method of Characteristics–Finite Element Method (MOC-FEM) and full MOC. In both approaches two important effects of FSI in fluid-filled pipes, namely Poisson and junction coupling, are taken into account. The study proposes a more comprehensive model for studying fluid transients in pipelines as compared to previous works, which take into account either FSI or viscoelasticity. To verify the proposed mathematical model and its numerical solutions, the following problems are investigated: axial vibration of a viscoelastic bar subjected to a step uniaxial loading, FSI in an elastic pipe, and hydraulic transients in a pressurised polyethylene pipe without FSI. The results of each case are checked with available exact and experimental results. Then, to study the simultaneous effects of FSI and viscoelasticity, which is the new element of the present research, one problem is solved by the two different numerical approaches. Both numerical methods give the same results, thus confirming the correctness of the solutions.

Introduction

There are four important items, which may affect classical water-hammer results: unsteady friction (UF), column separation (CS), fluid–structure interaction (FSI) and viscoelasticity (VE), each of which has been separately investigated and verified in various researches. With the inclusion of two or more of these items in the analysis, eleven possibilities are offered from which some combinations already have been studied and some have not. The combinations of VE and UF (Covas et al., 2004a, Covas et al., 2004b, Covas et al., 2005, Duan et al., 2010, Soares et al., 2008), CS and UF (Bergant et al., 2010, Bergant et al., 2008a, Bergant et al., 2008b, Bughazem and Anderson, 2000), FSI and UF (Elansary et al., 1994), FSI and CS (Fan and Tijsseling, 1992, Tijsseling and Vardy, 2005, Tijsseling et al., 1996) and VE and CS (Hadj-Taïeb and Hadj-Taïeb, 2009, Keramat et al., 2010, Soares et al., 2009) have already been investigated. The remaining combination of two, namely FSI and VE, is the scope of this article. Combinations of three were modelled by Neuhaus and Dudlik (2006) (CS, UF and FSI) and Warda and Elashry (2010) (CS, UF and VE).

Fluid–structure interaction deals herein with the transfer of momentum and forces between a pipeline and its contained fluid. This matter has been investigated widely for elastic pipes and various experimental and numerical researches have been reported (Tijsseling, 1996, Wiggert and Tijsseling, 2001). In the numerical researches (most of which are in the time domain as opposed to the frequency domain), solutions based on the Method of Characteristics (MOC), the Finite Element Method (FEM), or a combination of these, are predominant. Lavooij and Tijsseling (1991) presented two different procedures for computing FSI effects: full MOC uses MOC for both hydraulic and structural equations and in MOC–FEM the hydraulic equations are solved by the MOC and the structural equations by the FEM. Using the MOC–FEM approach, Ahmadi and Keramat (2010) investigated various types of junction coupling. Heinsbroek (1997) compared MOC and FEM for solving the structural beam equations for the pipes and his conclusion for axial vibration was that both full MOC and MOC–FEM are valid methods that give equivalent results. In the current research, these two approaches were selected and developed for transients in pipes with viscoelastic walls.

For pipes made of plastic such as polyethylene (PE), polyvinyl chloride (PVC) and acrylonitrile butadiene styrene (ABS), viscoelasticity is a crucial mechanical property that changes the hydraulic and structural transient responses. Covas et al., 2004a, Covas et al., 2005 presented a model that deals with the dynamic effects of pipe-wall viscoelasticity for hydraulic transients. The model included an additional term in the continuity equation to describe the retarded radial wall deformation based on a creep function fitted to experimental data. The governing equations were solved using MOC and it was said that, unlike the classical water-hammer model, only a model that includes viscoelasticity can predict accurately transient pressures. A more detailed research in this field by Soares et al. (2008) gave a general algorithm to include viscoelasticity and unsteady friction within the conventional MOC solution procedure. Their final conclusion that unsteady friction effects are negligible when compared to pipe-wall viscoelasticity was partially confirmed by Duan et al. (2010). Frequency-dependent pressure wave propagation in viscoelastic pipes was studied by Prek, 2004, Prek, 2007 using wavelets and transfer functions, respectively.

Papers dealing with both viscoelasticity and fluid–structure interaction are relatively rare. Williams (1977) performed FSI experiments in a 12 m long ABS pipe (27 mm inner diameter, 3.2 mm wall thickness). He stated that ABS is not significantly viscoelastic, and therefore not the main mechanism for dispersion (degradation of wave front). A thorough theoretical analysis and illustration of the degradation of wave fronts, as caused by FSI and viscoelasticity, was given by Bahrar et al. (1998). Weijde (1985) did experiments on a 50 mm diameter PVC pipeline containing a 24 m long U-shaped test section. An adjustable spring was used to vary the stiffness of the system. Rachid and Stuckenbruck (1989) presented a model for FSI transients in plastic pipes assuming that the viscoelastic behaviour of the pipe walls is solely due to pure shear. They tested their model for straight and Z-shaped pipe systems and concluded that the high frequencies caused by FSI virtually vanished as a result of viscoelasticity by the end of the first water-hammer cycle. In later works the classical water-hammer equations were extended to include plastic circumferential (hoop) deformations, offering models for elasto-plastic (Rachid and Costa Mattos, 1995, Rachid and Costa Mattos, 1998b) or elasto-viscoplastic (Rachid et al., 1994) pipe materials. These models were developed with the aim of estimating the damage accumulation and lifetime prediction of industrial pipe systems. In addition, the FSI research by Rachid and Costa Mattos (1998a) provided a formulation and corresponding numerical solution for assessing structural failure due to fluid transients. Tijsseling and Vardy (1996) studied the effects of suppression devices on water hammer and pipe vibration. It was demonstrated that a short plastic extension significantly influenced the axial vibration of a water-filled steel pipe. With longer plastic sections a significant reduction in the amplitude of vibration was predicted. In another experiment on the damping of water hammer due to viscoelastic effects, Tijsseling et al. (1999) investigated water hammer in a steel pipeline fitted with an internal air-filled plastic tube of rectangular cross-section. Hachem and Schleiss (2011) modelled viscoelasticity and FSI to determine wave speeds in steel-lined rock-bored tunnels. Most recently, Achouyab and Bahrar (2011) published a numerical study on water hammer with FSI and viscoelasticity – similar to ours – in which the MOC–FEM approach was used to solve the equations. All cited simulations of water hammer in viscoelastic pipes did not consider the issue of support and elbow motion, something that can hardly be avoided in practice (Hambric et al., 2010, Heinsbroek and Tijsseling, 1994, Tijsseling and Heinsbroek, 1999). The present research aims at providing a mathematical model and its numerical solution for the inclusion of this matter.

The contribution of the current research compared to previous studies on FSI or viscoelasticity in fluid-filled pipes is that by taking into account both effects simultaneously, more extensive governing equations for the hydraulics and the structure are obtained. In addition, appropriate numerical solutions for them are proposed and verified. Validation of the developed mathematical model and its numerical solutions is carried out from different angles: axial vibration of a viscoelastic bar (or empty pipe), where the numerical solutions are validated against the available analytical solution; FSI in an elastic pipe, where the results are checked against exact results; water hammer in a viscoelastic pipe, where the experimental results of two case studies are used; FSI in a viscoelastic pipe – the new element of this research – where two different numerical solutions are compared.