Numerical simulation of water-hammer in tapered pipes using an implicit least-squares approach

Highlights

• A technique is presented for numerical simulation of the water-hammer problem in pipes with variable cross-sectional area.

• The method formulation is so that all of the solution procedure consists of some sparse matrix algebra.

• The effects of localized clogged and swollen parts on pressure wave propagation due to water-hammer are investigated.

Abstract

A technique using radial-basis functions and least-squares optimisation is applied for the numerical solution of one-dimensional equations governing transient pipe flow. The method can deal with geometrically non-uniform pipes by employing an arbitrary distribution of scattering nodes in the space domain. The formulation is implicit in time with a sparse and symmetric matrix equation to be solved at each time step. One tapered-pipe problem with available analytical solution is used for verification and one straight-pipe problem with available experimental data is used for validation. The effect of gradually clogged and swollen pipe sections on pressure wave propagation is investigated. The latter is of importance for transient-based fault-detection techniques.

Introduction

Any rapid change in the momentum of a fluid flow in a pipeline system produces a pressure wave that propagates through the pipes. This phenomenon is called water-hammer. One of the important parameters that affects the frequency and shape of the pressure wave is the variation of the cross section of the pipe. Diffuser junctions in pipelines, the presence of locally clogged or swollen sections, and blood flow in tapered vessels, are some examples of cross-sectional area variation. In these kinds of problems, the conventional governing equations of water-hammer, which can be found in Ref. [1], and its standard numerical solution, Method of Characteristics (MOC) [1], should be rewritten. Streeter et al. [2] were among the first researchers who dealt with wave propagation in tapering pipes. They derived the governing equations for pulsating pressure in tapered distensible vessels and provided a modified formulation of the MOC to solve them numerically. They evaluated their numerical simulation with experimental results for a tapering vessel. The comparison was not satisfactory and the authors presented a useful discussion about the sources of discrepancies between numerical and experimental results. Rouleau and Young [3,4] studied the distortion of short pulses in tapered tube pulse transformers analytically, using Laplace transforms, for both inviscid and viscous liquids. They concluded that tapering geometry along with viscous shear can severely distort pulses that propagate through a tapered transmission line. Tarantine and Rouleau [5] calculated the transient behavior of a tapered tube using a step-line impedance technique. They continued their research by performing laboratory experiments to show the validity of their theoretical research. They showed that the pressure surge resulting from quick closing-reopening or opening-reclosing of a valve can be effectively reduced by a tapered section at the end of a long uniform line [6]. Yoshizawa and Ando [7] mentioned that the step-line impedance technique, that was used by Tarantine and Rouleau [5], is inefficient computationally. They presented an analytical solution based on the Laplace transform and produced solutions for two shapes of tapered pipes (linear and exponential). Washio et al. [8] carried out a frequency response test on a linearly tapered line filled with oil. They also compared the results of Zielke's method of characteristics [9] and conventional step-line approach with their experimental results and concluded that the step-line approach is not efficient for the solution of tapering pipes due to its complexity. Tanahashi et al. [10] studied travelling pressure pulses in tapered pneumatic pipelines with an end reservoir to investigate the effect of tapering on the shape and peak of the pressure history, both experimentally and analytically. They concluded that the characteristics of the pressure waves are largely influenced by the tapering geometry. Wu and Ferng [11] reviewed the fundamental equations for the study of water-hammer in tapered pipes and solved the governing equations numerically using the MOC method. They investigated numerically the effect of a mild expansion or contraction in the pipe on the pressure history at the valve. There have been more researchers who used modified versions of the MOC to solve the water-hammer equations in tapering pipes. They can be found for instance in Ref. [[12], [13], [14]]. The effect of tapering on a pressure wave travelling through a distensible tube is an important topic in hemodynamics. Evans studied the effects of tapering geometry on the pulsating flow in the artery, analytically [15]. Belardinelli and Cavalcanti [16] developed a nonlinear two-dimensional model of blood flow in tapered and elastic vessels. They concluded, numerically, that in the tapered tube the axial velocity profile becomes flatter, the flow decreases and consequently the flow resistance increases considerably. Stergiopulos et al. [17] solved one-dimensional blood flow equations numerically and investigated the effect of arterial and aortic stenoses on the pulsating flow. They showed that the presence of severe arterial stenoses significantly affects the flow pulses. Numerical simulation in this field recently is going toward practical medical usage. For instance, Sazonov et al. [18] presented a method for non-invasively detecting the severity and location of aortic aneurysms using numerical simulation of pulsating flow passing through tapered vessels. Similarly, the effects of blockage in water pipelines on a travelling pressure wave studied by Duan et al. [19] and Che et al. [20]. They provided a promising tool for blockage detection in pipes.

Minimizing the sum of squared residuals using least-squares techniques for the solution of PDEs has been used broadly in the finite element method (for instances [[21], [22], [23]]). The same technique was applied to minimize the squared residuals which came from moving least-squares interpolation method [24,25] and radial point interpolation methods [26,27]. In the present paper, we used radial point interpolation and least-squares techniques for the numerical simulation of the water-hammer problem in pipes with variable cross-sectional area. In this method, the space domain is discretized uniformly, randomly or purposefully using scatter nodes. The point-wise character of this method enables it to deal with non-uniform parameters (speed of the sound in the fluid, diameter, modulus of elasticity and thickness of the pipe wall) without any extra complication in the solution procedure. Here, the governing equations of one-dimensional water-hammer are derived by averaging the continuity and momentum equations over the pipe's cross-section. Frequency-dependent laminar viscous losses are included via Schohl's method [28]. The implicit Euler method is used in the time stepping to give the method stability. We presented a vectorized formulation for the numerical method so that all of the solution procedure consists of some sparse matrix algebra which makes it efficient in computation. After verification of the scheme, analytically and experimentally, the effect of clogged and swollen sections in a pipeline is investigated numerically.