Water Hammer Problems Research Articles

New Advances in Water Hammer Problems

When the flow within pressurized pipes experiences abrupt stoppages, initiation, or directional alteration, it gives rise to the phenomenon of water hammer, characterized by the propagation of waves [...]

Water hammer protection for diversion systems in front of pumps in long-distance water supply projects

For a water supply system with long-distance diversion pipelines, in addition to the water hammer problems that occur beyond pumps, the safety of the water diversion pipeline in front of pumps also deserves attention. In this study, a water hammer protection scheme combined with an overflow surge tank and a regulating valve was developed. A mathematical model of the overflow surge tank was developed, and an analytical formula for the height of the overflow surge tank was derived. Furthermore, a practical water supply project was used to evaluate the feasibility of the combined protection scheme and analyze the sensitivity of valve regulation rules. The results showed that the combined protection scheme effectively reduced the height of the surge tank, lessened the difficulties related to construction, and reduced the necessary financial investment for the project. The two-stage closing rule articulated as fast first and then slow could minimize the overflow volume of the surge tank when the power failure occurred, while the two-stage opening rule articulated as slow first and then fast could be more conducive to the safety of the water supply system when the pump started up.

Alternative methods for solving a hyperbolic system of equations in a simplified water hammer problem

The paper addresses a mathematical model of the transient process in a pipeline, namely, a water hammer model. Based on this model, a mixed problem for a system of partial differential equations of hyperbolic type is set. In this paper, we obtain formulas for the exact solution of the Cauchy problem for a linear homogeneous hyperbolic system. We compare the effectiveness of explicit and implicit numerical methods for solving a simplified system of water hammer equations.

Numerical studies of two-phase water hammer flows using Godunov methods

In this paper, an explicit finite volume Godunov-type scheme is applied for a two-phase homogeneous water hammer flow model. The developed finite volume method is based on an exact Riemann solver for a gas–liquid two-phase flow model. This paper investigates the case of a water hammer problem caused by a rapid closing valve located in upstream of the pipeline. The performance of the proposed numerical method is validated against experimental data from literature. The good agreement found between the numerical and experimental results confirms the accuracy and the robustness of both the numerical methods and the proposed exact Riemann solver.

Dimensionless impedance method for the transient response of pressurized pipeline system

Modeling an unsteady flow is an important problem in pressurized pipeline systems. Depending on the dimensions and properties of the fluid and pipeline material, the generation of hydraulic transient issues and their spatiotemporal variation patterns can be different. To address the water hammer problem without scale issues in pipeline systems, this study developed a dimensionless impedance method (DIM). The existing partial differential equations of mass, momentum, and energy conservation were converted into corresponding dimensionless equations. The solutions of proposed equations were analytically developed in terms of several operators in the dimensionless frequency domain. A dimensionless water hammer number was introduced to address the similarity in hydraulic transients between different systems. Both numerical and experimental comparisons showed that the DIM well matched the existing methods. Based on comprehensive simulations of water hammer phenomena for ranges of water hammer numbers, a vectorized chart was formulated for a reservoir pipeline valve system. Both simulations and calibrations of the experimental hydraulic transients achieved well-matched results between the DIM computations and chart interpolation results. The proposed DIM method can provide substantial efficiency for both simulation and calibration, as well as generality in transient solutions for pipeline systems.

Analytical Solution for Laminar Water Hammer With Frequency-Dependent Friction

Abstract An analytic solution for the classical water hammer problem for unsteady laminar pipe flow involving Zielke's frequency-dependent friction model is derived. By means of separation of variables, the solution of the initial boundary value problem is compiled in the form of an infinite series representing its modal expansion. In this way, solving the damped wave equation reduces to root-finding of scalar algebraic equations for the complex eigenfrequencies. An analysis of this dispersion relation underlines the frequency-dependent nature of the solution in comparison with less sophisticated friction models.

Strong Stability Preserving Runge-Kutta Methods Applied to Water Hammer Problem

The characteristic method of lines is the most used numerical method applied to the water hammer problem. It transforms a system of partial differential equations involving the independent variables time and space in two ordinary differential equations along the characteristics curves and then solve it numerically. This approach, although showing great stability and quick execution time, creates ∆x-∆t dependency to properly model the phenomenon. In this article we test a different approach, using the method of lines in the usual form, without the characteristics curves and then applying strong stability preserving Runge-Kutta Methods aiming to get stability with greater ∆t.

Numerical simulation of water hammer and cavitation phenomena including the convective term in pipeline problems

Water hammer problems occurs where restrictions flow suddenly change in pipelines, usually the closing or opening of valves. Presence of a high velocity pressure wave traveling though the fluid could cause damages in the pipeline structures due to the implosion of gas cavities formed by a physical phenomenon called cavitation. On this work, it is studied the water hammer physical problem considering the influence of the convective terms in the momentum and continuity equations, the cavitation problem has been evaluated by the discrete vapour cavity physical model. A MATLAB code is developed to solve the transient problem and find the hydraulic head evolution in some points along the pipeline. The method of characteristics is used to find a numerical solution of the coupled partial differential equation system. Results shows good agreement with result presented in literature reviewed, also, it is found that the influence of the convective term is small compared with a simple model where those terms are neglected, the maximum difference value of 2.4x10−4 where found, considering and neglecting the convective term on the physical model.

Proper orthogonal decomposition–Galerkin projection method for quasi-two-dimensional laminar hydraulic transient flow

The proper orthogonal decomposition (POD)-Galerkin projection method was applied to the quasi-two-dimensional (quasi-2D) water-hammer problem for achieving a model order reduction. The computation of the axial velocity profile requires a large number of numerical grids that is equal to the number of degrees of freedom (DOFs) in the quasi-2D water-hammer problem. To alleviate the large computational burden, POD was employed to reduce the number of DOFs of the axial velocity. The possibility of expression of the state variable using global basis vectors obtained with POD and the expression of the behavioural characteristics of the POD basis as a function of time were analysed. To verify its accuracy, the proposed method was validated using experimental data. The computational cost was compared for test cases. To estimate the efficiency, the computational complexities of the original model and the reduced-order model were evaluated and compared. Therefore, the overall performance – including the accuracy and efficiency – was proven.

GPU Acceleration of Hydraulic Transient Simulations of Large-Scale Water Supply Systems

Simulating hydraulic transients in ultra-long water (oil, gas) transmission or large-scale distribution systems are time-consuming, and exploring ways to improve the simulation efficiency is an essential research direction. The parallel implementation of the method of characteristics (MOC) on graphics processing unit (GPU) chips is a promising approach for accelerating the simulations, because GPU has a great parallelization ability for massive but simple computations, and the explicit and local features of MOC meet the features of GPU quite well. In this paper, we propose and verify a GPU implementation of MOC on a single chip for more efficient simulations of hydraulic transients. Details of GPU-MOC parallel strategies are introduced, and the accuracy and efficiency of the proposed method are verified by simulating the benchmark single pipe water hammer problem. The transient processes of a large scale water distribution system and a long-distance water transmission system are simulated to investigate the computing capability of the proposed method. The results show that GPU-MOC method can achieve significant performance gains, and the speedup ratios are up to hundreds compared to the traditional method. This preliminary work demonstrates that GPU-MOC parallel computing has great prospects in practical applications with large computing load.

Mesoscopic-based finite volume solutions for waterhammer flows

ABSTRACTIt is becoming evident that high resolution finite volume (FV) numerical schemes for multi-dimensional waterhammer problems are needed for the development of an accurate transient-based condition assessment of pipelines. As a prerequisite, FV methods for one-dimensional waterhammer flows need to be developed. Such models can then be applied to multi-dimensional problems via directional splitting. In this paper, two FV schemes (one uses Bhatnagar–Gross–Krook Boltzmann (BGK) and the other uses kinetic flux vector splitting (KFVS) in the flux approximation) for one-dimensional waterhammer problems are formulated and applied. It is found that the KFVS and BGK schemes correctly capture discontinuity fronts in a classical reservoir-pipe-valve system. An oscillation-free collision time formulation has been proposed, tested and found to be robust. The stability of the proposed schemes is guaranteed when . Comparison between the BGK, KFVS, fixed-grid MOC, and first and second order Godunov schemes reveals that the second-order Godunov performs best, followed closely by the BGK scheme. However, the BGK should not be quickly dismissed since it is known that its real power becomes evident when dealing with complex physics, multi-scale and multi-dimensional problems.

Formulation of consistent finite volume schemes for hydraulic transients

ABSTRACTIntentionally generated transient waves are used in the detection of defects in pipelines. Experience to date in these applications shows that the wider the band of injected wave frequencies the better is the detection effectiveness. Thus, as these technologies develop further, models that can handle high frequency waves will be required. Finite volume (FV) methods are well suited for high frequency wave phenomena, but their use in the water hammer field has been limited to date compared with other fields such as open channel flow. A major reason for this is that FV methods are formulated for simple boundary conditions such as no-flux or no-slip, but have not been well developed for the typically more complex boundary conditions found in pressurized pipeline systems (e.g. junctions, control valves, orifices, tanks and reservoirs). In those few instances in the literature where FV has been applied to water hammer problems, the approach has been to use FV for internal sections and method of characteristics (MOC) at the boundaries. The global order of accuracy of FV–MOC is governed by the MOC solution. To our knowledge, this paper constitutes a first attempt at handling boundary conditions within the FV framework. The approach places the boundary element within a FV to enforce mass and momentum conservation within this volume. The fluxes between the FV and the adjacent elements are then formulated in the usual manner. The approach is illustrated for the case of a valve, a reservoir and a junction. The finite volume method used is the Boltzmann-type scheme. The accuracy and efficiency of all schemes with the proposed non-iterative FV formulation of the boundary conditions are demonstrated through the following test cases: (i) water hammer wave interactions with a junction boundary characterized by a geometric discontinuity, (ii) water hammer wave interactions with a junction boundary characterized by a discontinuity in the value of wave speed, and (iii) water hammer wave interactions with a junction characterized by a flow rate discontinuity. The pure FV formulation guarantees mass and momentum conservation. The Boltzmann-based FV schemes capture discontinuity as well as wave interactions with boundary elements accurately. The stability of the proposed FV schemes is satisfied when .

Numerical modelling of one- and two-dimensional water hammer problems using a modified wave propagation algorithm and turbulence model

ABSTRACTIn this paper a modified Godunov-type wave propagation method is presented for modelling one- and two-dimensional water hammer problems. The proposed numerical solver is well-balanced and treats the friction terms within the flux-differencing of the finite volume computational cells. To investigate the effect of turbulence behaviour during the water hammer process, a turbulence model that provides precise prediction for the near wall-region is utilized. First, the one-dimensional water hammer problem without the friction term is considered and the obtained results are compared with the analytical solution. Then, the suitability of the proposed scheme is investigated for two-dimensional transient problems for a range of Reynolds numbers and the results are validated with experimental data. It is found that the proposed numerical solver with a new choice of turbulence model produces satisfactory agreement with the exact solution and available experiment data for the given ranges of Reynolds numbers.

On the Approximation of Two-Dimensional Transient Pipe Flow Using a Modified Wave Propagation Algorithm

In this study, a second-order accurate Godunov-type finite volume method is used for the solution of the two-dimensional (2D) water hammer problem. The numerical scheme applied here is well balanced and is able to treat the unsteady friction terms, together with the convective terms, within the differences between fluxes of neighboring computational cells. In order to consider the effect of unsteady friction terms during the water hammer process, k−ε and k−ω turbulence models are employed. The performance of the proposed method with the choice of different turbulence models is evaluated using experimental data obtained from one low and one high Reynolds-number turbulent test cases. In addition to velocity and pressure distributions, the turbulence characteristics of each variant of the model, including eddy viscosity, dissipation rate, and turbulent kinetic energy during the water hammer process are fully analyzed. It is found that the inclusion of the convective inertia terms leads to more accurate pressure profiles. The results also show that using a relatively high Courant–Friedrichs–Lewy (CFL) number close to unity, the introduced numerical solver with both choices of turbulence models provides reasonable and acceptable predictions for the studied flows.

SimHydraulics approach to water hammer problems with the development of new formulas for static pressure diameter coefficient

Water hammer analysis and prediction is of overriding importance for operation safety of hydraulic systems. Therefore, the verification of computer codes capability to model and simulate water hammer is a very interesting topic for the analysis of different fluid-filled pipeline systems. Until now, transient approach of SimHydraulics is not developed, and the validity of this approach against benchmark examples, has never been investigated in scientific literature. In this paper, a numerical model of a reservoir-pipelines-valve system is created using Matlab SimHydraulics software. The main component of this model is the Segmented Pipeline with wall compliance parameters: the viscoelastic process time constant τ and the static pressure diameter coefficient K P for which new formulas were derived and validated. The numerical model is performed to solve transient flow problems caused by closure of a valve and the obtained results are presented and compared with those using the method of characteristic. Also, the effectiveness of the transient model of SimHydraulics is evaluated by comparing the obtained numerical results with the most pertinent experimental results found in the literature. The proposed transient model was adapted for the investigation of more complex piping systems. The results of this study show the ability of the SimHydraulics model to predict the pressure oscillation behavior accurately in all cases considered. This study provides useful information for future research in pump transient simulation using SimHydraulics.

Singular Value Decomposition–Based Collocation Spectral Method for Quasi-Two-Dimensional Laminar Water Hammer Problems

AbstractThe radial distributions of velocity components need to be resolved in quasi-two-dimensional laminar water hammer problems. In a collocation spectra method, the radial distributions are app...

Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves

A two-layer long-wave approximation of the homogeneous Euler equations for a free-surface flow evolving over mild slopes is derived. The upper layer is turbulent and is described by depth-averaged equations for the layer thickness, average fluid velocity and fluid turbulent energy. The lower layer is almost potential and can be described by Serre–Su–Gardner–Green–Naghdi equations (a second-order shallow water approximation with respect to the parameter $H/L$, where $H$ is a characteristic water depth and $L$ is a characteristic wavelength). A simple model for vertical turbulent mixing is proposed governing the interaction between these layers. Stationary supercritical solutions to this model are first constructed, containing, in particular, a local turbulent subcritical zone at the forward slope of the wave. The non-stationary model was then numerically solved and compared with experimental data for the following two problems. The first one is the study of surface waves resulting from the interaction of a uniform free-surface flow with an immobile wall (the water hammer problem with a free surface). These waves are sometimes called ‘Favre waves’ in homage to Henry Favre and his contribution to the study of this phenomenon. When the Froude number is between 1 and approximately 1.3, an undular bore appears. The characteristics of the leading wave in an undular bore are in good agreement with experimental data by Favre (Ondes de Translation dans les Canaux Découverts, 1935, Dunod) and Treske (J. Hydraul Res., vol. 32 (3), 1994, pp. 355–370). When the Froude number is between 1.3 and 1.4, the transition from an undular bore to a breaking (monotone) bore occurs. The shoaling and breaking of a solitary wave propagating in a long channel (300 m) of mild slope (1/60) was then studied. Good agreement with experimental data by Hsiao et al. (Coast. Engng, vol. 55, 2008, pp. 975–988) for the wave profile evolution was found.

An exact multiphase Riemann solver for compressible cavitating flows

This paper presents a new one-fluid method for simulating formation and the collapse of cavitation regions in water during an isothermal process. In this method, the fluid phase changes are included into the wave pattern of an exact Riemann solver. The model behavior is assessed by comparing the numerical results with the other numerical models for several 1D Riemann problems. One-dimensional water hammer problems with vast creation and collapsing of cavitation zones are simulated as well—and the numerical results are compared to experimental results. The new model results are in very good agreement with accepted results reported in the literature. The presented results clearly show that the new model is able to capture the various behaviors of water during the phase change in the saturation dome and the vapor state, which was neglected in previous studies. Finally, the new model is adopted to an ALE method on an adaptive triangular grid to simulate an underwater explosion phenomenon inside a rigid cylinder—and the results are compared with other simulations.

Investigation of Waterhammer Problems in Wind-Hydro Hybrid Power Plants

Sudden flow changes in confined pipe systems generate transient flow (waterhammer) conditions accompanied with very high or sometimes very low pressures traveling back and forth in the system. Depending on the magnitude of those pressures, pipe collapse or pipe bursting may take place at the weakest points of the system, resulting in costly damages as well as loss of human lives in some cases. In general, it may be encountered in the penstocks of hydropower plants, water transmission lines and water networks, etc. It is essential that the safe operation guidelines of the hydropower plants be defined accurately, in advance. To achieve that, detailed numerical simulations should be done with reliable software. In some cases, even physical modeling may also be necessary. In the present study, waterhammer problems in the penstocks of Yahyali Wind-Hydro Hybrid Plant are investigated (the construction of the plant has not started yet). Time-dependent flow conditions in the penstocks are simulated by the help of computer software. It solves nonlinear differential equations by using the method of characteristics that uses the principles of finite difference method. Firstly, hydraulic transients for various operational cases are investigated using some scenarios. Then, a surge tank, protective device, for waterhammer, is added to the system, and for the same operational cases, hydraulic transients are studied again.

Numerical solutions of quasi-two-dimensional models for laminar water hammer problems

ABSTRACTNumerical solutions of two quasi-two-dimensional models for water hammer problems under laminar flow conditions are proposed. The solutions are based on modified Chebyshev polynomial expansion of radial distribution of velocities. The collocation method is used to solve for the expansion coefficients. Spatial variation in the axial direction and time advancement are treated using the method of characteristics. One model includes the radial velocity components in the continuity equation while the other is focused on the cross-sectional mean velocity without the radial component. The effect of including the radial velocity component in the flow system is evaluated. The Chebyshev polynomials constitute an orthogonal basis for approximating numerical solutions and show suitable behaviour of the expansion coefficients.