A new finite element formulation based on the velocity of flow for water hammer problems

Abstract

The primary objective of this paper is to develop a simulation model for the fluid–structure interactions (FSI) that occur in pipeline systems mainly due to transient events such as rapid valve closing. The mathematical formulation is based on waterhammer equations, traditionally used in the literature, coupled with a standard beam formulation for the structure. A new finite element formulation, based on flow velocity, has been developed to deal with the valve closure transient excitation problems. It is shown that depending on the relative time-scales associated with the structure, fluid and excitation forces, there are situations where the structural vibration response increases with FSIs. This is in contrast to what is normally accepted in the literature, i.e. FSI reduces the structural displacements.

Introduction

Even though many researchers have used hybrid models for waterhammer problems, with the method of characteristics (MOC) modeling the waterhammer equations and the finite element method (FEM) modeling the structure, few have used the wave equation resulting from the elimination of one of the variables from the waterhammer equation in FEM. The wave equation can be formed with flow velocity as the fluid variable, which is appropriate for the valve closure excitation. This equation is elliptical in nature and hence can be readily modeled using FEM. In this investigation, this feature is exploited to develop a coupled FEM formulation of both the structure and the fluid. Effects such as junction coupling and Poisson coupling are included while friction coupling has been neglected due to the short time-scales associated with the excitation. Model reduction, based on the structural and fluid vibration modes, has been used to reduce the size of the problem and care has been exercised to include axial mode shapes since the interaction occurs through the axial equations of the beam.

Tijsseling [1] presented a very detailed review of transient phenomena in liquid-filled pipe systems. He dealt with waterhammer, cavitation, structural dynamics and fluid–structure interaction (FSI). The main focus was on the history of FSI research in the time-domain. One-dimensional FSI models were classified based on the equations used. The two-equation (one-mode) model refers to classical waterhammer theory, where the liquid pressure and velocity are the only unknowns, the four-equation (two-mode) model allows for the axial motion of straight pipes; axial stress and axial pipe-wall velocity are additional variables. The six-equation model is necessary if radial inertia forces are to be taken into account; hoop stress and radial pipe-wall velocity are the additional unknowns. The state-of-the-art fourteen equation model describes axial motion (liquid and pipes), in and out-of-plane flexure, and torsional motion of three-dimensional pipe systems. Wiggert et al. [2] used the MOC to study transients in pipeline systems. They identified seven wave components, coupled axial compression of liquid and pipe material, coupled transverse shear and bending of the pipe elements in two principal directions and torsion of the pipe wall. The fourteen characteristic hyperbolic partial differential equations were converted to ordinary differential equations by the MOC transformation. The formulation was applied to two systems of three mutually perpendicular pipes.

Heinsbroek [3] reported an application of FSI in the nuclear industry. His analysis was based on a combination of MOC and FEM. His conclusion was that while the MOC technique was superior for axial dynamics, FEM was more robust for transverse/lateral dynamics. The investigation also highlighted the fact that FSIs do take place and a model based only on the fluid gives erroneous results. This is corroborated by data from experiments. Lee and Kim [4] used a finite element formulation for the fully coupled dynamic equations of motion and applied it to several pipeline systems. Wang and Tan [5] combined MOC and FEM to study the vibration and pressure fluctuation in a flexible hydraulic power system on an aircraft. Casadei et al. [6] presented a method for the numerical simulation of FSI in fast transient dynamic applications. They had used both finite element and finite volume discretization of the fluid domain and the peculiarities of each with respect to the interaction process were highlighted.

An earlier study carried out by Kellner et al. [7] showed that FSI reduced displacements and the corresponding loads on the snubber below the elbow by a factor of almost four. In this investigation junction coupling was considered whereas Poisson coupling was neglected. Lavooij and Tijsseling [8] suggested a provisional guideline to judge when the FSI is important. This guideline is based on the characteristic time-scales of the system under consideration. One of the objectives of this study is to re-examine those proposed guidelines using the new finite element formulation based on flow velocity.