Fixed-grid Method Of Characteristics Research Articles

Water Hammer in Steel–Plastic Pipes Connected in Series

This paper experimentally and numerically investigates the water hammer phenomenon in serially connected steel and HDPE pipes with different diameters. The aim of the laboratory tests was to obtain the time history of the pressure head at the downstream end of the pipeline system. Transient tests were conducted on seven different pipeline system configurations. The experimental results show that despite the significantly smaller diameter of the HDPE pipe compared to the steel pipe, introducing an HDPE section makes it possible to suppress the valve-induced pressure surge. By referring to the results of the experimental tests conducted, the comparative numerical calculations were performed using the fixed-grid method of characteristics. To reproduce pressure wave attenuation in a steel pipe, Brunone-Vitkovský instant acceleration-based model of unsteady friction was used. To include the viscoelastic behavior of the HDPE pipe wall, the one-element Kelvin–Voigt model was applied. By calibrating the unsteady friction coefficient and creep parameters, satisfactory agreement between the calculated and observed data was obtained. The calibrated values of parameters for a single experimental test were introduced in a numerical model to simulate the remaining water hammer runs. It was demonstrated that using the same unsteady friction coefficient and creep parameters in slightly different configurations of pipe lengths can be effective. However, this approach fails to reliably reproduce the pressure oscillations in pipeline systems with sections of significantly different lengths.

Water-hammer control in pressurized-pipe flow using an in-line polymeric short-section

Water-hammer control strategies constitute an essential and critical task for both hydraulic designers and manufacturers to ensure the global economic efficiency and safety operations of hydraulic utilities. The primary objective of this paper is to present an alternative strategy to control water-hammer up- and down-surges, induced into a steel piping system. The proposed technique is based on replacing a short-section of the transient sensitive regions of the existing piping system by another one made of polymeric material. Two types of polymeric materials, used for the short-section and including high- or low-density polyethylene (HDPE) or (LDPE), are addressed in this study. The 1-D pressurized-pipe flow model is used to describe the hydraulic system, along with the Ramos formulation, based on two decay coefficients being used for considering the pipe-wall viscoelastic behavior and unsteady friction effects. Numerical computations were performed using the fixed-grid method of characteristics. The efficiency of the numerical model is first verified against experimental data available from the literature. Thereafter, critical flow scenarios relating to water-hammer up- and down-surges, including a cavitating flow, are revealed and discussed to point out the efficiency of the used protection technique. From the case studied, it is found that such a technique could mitigate critical water-hammer surges and, hence, might greatly enhance the reliability of the industrial hydraulic systems and urban water utilities, while safeguarding operators. Despite the available protection measures, the utilized technique can substantially soften both up- and down-surge waves induced by severe water-hammer events. It is also found that the amortization of pressure rise and pressure drop is slightly more important for the case of a short-section made of LDPE polymeric material than that using an HDPE polymeric material. It is also observed that other factors contributing to the damping rate depended upon the short-section length and diameter. In fact, the examination of the pressure peak magnitude sensitivity, with the short-section length and diameter being the controlling variables, provides optimum values of these parameters for sizing the replaced polymeric short-section.

Nonlinear Interaction of Friction and Interpolation Errors in Unsteady Flow Analyses

AbstractExplicit analytical assessments of numerical errors in solutions of decaying standing waves in pipes are obtained using analyses based on the fixed-grid method of characteristics and are used as a quantitative basis for comparing alternative grid types, sizes, and shapes. It is shown that the influences of friction and interpolation combine in a nonlinear manner, but that the combination has the form of a product so that the two effects may be considered independently to a certain extent. The analysis is presented for finite pipes, thus creating both upper and lower limits for the range of possible standing wave frequencies. It is found that frequencies in the upper half of the range are strongly damped. The performances of diamond and rectangular grids are compared, but there is no clear winner because, for equivalent computer resources, each has both advantages and disadvantages. The overall purposes of the paper are to move toward methods of (1) relieving untrained software users from tasks rel...

Efficient Second-Order Accurate Shock-Capturing Scheme for Modeling One- and Two-Phase Water Hammer Flows

This paper focuses on the formulation and assessment of a second-order accurate finite volume (FV) shock-capturing scheme for simulating one- and two-phase water hammer flows. The two-phase flow model is based on the single-equivalent fluid concept. The proposed scheme for one- and two-phase flows is the same, except for the Riemann solvers used to evaluate fluxes between computational cells. For one-phase flows, the accuracy and numerical efficiency of the proposed scheme is contrasted against the fixed-grid method of characteristics (MOC) and a recently proposed FV scheme. For two-phase flows, the accuracy and numerical efficiency of the proposed scheme are compared to the fixed-grid MOC scheme. The results for one-phase flows show that, when a Courant number (Cr) very close to 1.0 (around 0.99 or higher) is used, the MOC scheme is more efficient than the proposed scheme and the other FV scheme. In this case, the latter two schemes have similar numerical efficiency. When Cr drops below about 0.95, the proposed scheme is more efficient than the MOC scheme and the other FV scheme, especially for smooth transient flows (no discontinuities). For two-phase water hammer flows, all the simulations were carried out using a maximum Courant number of 0.95 to avoid numerical instability problems. The results for two-phase flows show that the proposed scheme is much more efficient than the fixed-grid MOC scheme. The fixed-grid MOC and the proposed scheme are also used to reproduce a set of two-phase flow experiments reported in the literature. Good agreement between simulated and experimental data is found.

Transient flow in pipe networks

In order to apply the fixed-grid method of characteristics to transient flow in multi-pipe systems, some pipe reaches may not satisfy the Courant condition. Various numerical techniques can be applied for these reaches. In the present study, the finite difference or interpolation techniques are coupled with the method of characteristics. The location of the disproportionate pipe, which does not satisfy the Courant condition, is changed along the pipe in different schemes. The accuracy of the developed schemes has been checked by comparing their results to available exact analytical solutions. The comparison indicated that the scheme in which the finite difference method is employed for an intermediate location of the disproportionate pipe reach is the best. A general computer program based on this method has been developed. This program is capable of analyzing pipe networks including pumps, valves, surge tanks, air chambers, etc.

Energy Estimates for Discretization Errors in Water Hammer Problems

An integrated energy approach is developed for transient problems in pipelines using the fixed-grid method of characteristics. The goal is to understand how discretization errors associated with common interpolation schemes arise and how these errors can be controlled. Specifically, new analytical energy expressions demonstrate that both time-line and space-line interpolation attenuate the total energy of the system; in contrast, the wave-speed adjustment approach preserves the total energy but distorts the partitioning between kinetic and internal energy. The analytic results are confirmed through numerical studies of several series pipe systems. Both the numerical experiments and the mathematical expressions show that the discretization errors are small and can be ignored as long as the work in the system is continuous and substantial. When the work is small or negligible, however, a Courant number Cr value close to 1 is required to control the numerical dissipation.

Flexible Discretization Algorithm for Fixed-Grid MOC in Pipelines

The problem of selecting a time step that exactly satisfies the Courant condition arises most commonly when the fixed-grid method of characteristics (MOC) is applied to multipipe systems. Traditionally, this problem has been solved by employing one of a variety of interpolation techniques or by allowing small adjustments in the value of wave speed. In this paper, the number of available strategies is increased substantially by allowing several new kinds of interpolation as well as blended combinations of these interpolation approaches with the wave-speed–adjustment technique. The new hybrid approaches include interpolation along a secondary characteristic line and minimum-point interpolation, which minimizes the distance from the interpolated point to the primary characteristic. An intuitively appealing technique involving direct adjustment of the wave path is rejected because it is potentially unstable. The entire flexible algorithm is implemented as a preprocessor step, uses memory efficiently, executes quickly, and provides a flexible tool for investigating the importance of discretization errors in pipeline systems. The properties of the new algorithm are analyzed theoretically and illustrated by example.