Transient Pipe Research Articles

A Lagrangian particle model for one-dimensional transient pipe flow with moving boundary

For simulating fluid transients in pipelines with moving water-air interface, a one-dimensional Lagrangian particle model with second-order accuracy in both space and time is proposed. In this model, the meshless smoothed particle hydrodynamics (SPH) is used to approximate the spatial derivatives in the waterhammer equations, and the symplectic leap frog scheme is used for time integration. The nonlinear convective terms – which are mostly neglected in classical water hammer – are taken into account. The details of the Lagrangian particle method, boundary condition treatment and artificial viscosity for remedy of numerical oscillations due to shocks are presented. Two typical cases including transient flow with entrapped air pocket and rapid pipe filling are simulated and the results are validated against available experimental and numerical solutions, which has wide applications for flow simulation in drainage networks. To test the shock capturing ability of the developed model, the classical water hammer problem is also simulated and good agreement with the theoretical solution is obtained. It is shown that the proposed Lagrangian particle model is capable of solving waterhammer equations with moving boundaries and that it has high potential for multiphase transient flows.

Impact of Interpolation on Numerical Properties of the Method of Characteristics Used for Solution of the Transient Pipe Flow Equations

Impact of Interpolation on Numerical Properties of the Method of Characteristics Used for Solution of the Transient Pipe Flow Equations

Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids

Modulated pressure waves propagating in the drilling fluids inside the drill string are a reliable real-time communication technology that transmit data from downhole to the surface during oil and gas drilling. In the analysis of pressure waves’ propagation characteristics, the modeling of transient friction in non-Newtonian fluids remains a great challenge. This paper establishes a numerical model for transient pipe flow of power-law non-Newtonian fluids by using the weighted residual collocation method. Then, the Newton–Raphson method is applied to solve the nonlinear equations. The numerical method is validated by using the theoretical solution of Newtonian fluids and is proven to converge reliably with larger time steps. Finally, the influencing factors of the wall shear stress are analyzed using this numerical method. For shear-thinning fluids, the friction loss of periodic flow decreases with the increase in flow rate, which is opposite to the variation law of friction with the flow rate for stable pipe flow. Keeping the amplitude of pressure pulsation unchanged, an increase in frequency leads to a decrease in velocity fluctuations; therefore, the friction loss decreases with the increase in frequency.

Godunov-type scheme for air–water transient pipe flow considering variable heat transfer and laboratorial validation

A robust comprehensive energy dissipation model is developed to investigate the rapid filling process of a T-shaped bifurcated pipeline with entrapped air pocket, while an experimental system is designed to validate the numerical model. In this work, a self-adapting heat transfer model is proposed to describe the energy exchange during transient event, fully considering the heat transfer of entrapped air pocket and hydraulic losses caused by steady friction and unsteady friction in the section of the filling water column. Importantly, the related heat transfer coefficient is variable and determined by mechanism formula of media characteristics, which is obviously different from the constant heat transfer coefficient in conventional model relying on the trial-and-error method and experimental data. Moreover, the second-order Godunov-type scheme is introduced to solve the governing equations of filling water column, while a virtual plug approach is proposed to track the air–water interface. The resulting predictions are compared to those obtained via a conventional empirical polytropic model and constant coefficient heat transfer model, and to experimental data. The proposed model accurately reproduces the experimental pressure oscillations. The results display that when an air pocket content exceeds a certain threshold (>1.7%), the comprehensive model using forced convection can reproduce the pressure fluctuations. When the air content is lower (<0.9%), the comprehensive model using natural convection can simulate pressure changes more accurately. The conventional empirical polytropic model underestimates the pressure damping. The constant coefficient heat transfer model previously cannot be employed if there is no experimental data to calibrate the constant coefficient, but now the constant coefficient can be obtained from the results of the proposed model. Significantly, the comprehensive model can directly and accurately simulate the energy dissipation during the rapid filling process.

Modeling and simulation of xenon storage and supply system for electric propulsion system based on fundamental equation of state

The electric propulsion system (EPS) xenon storage and supply system (SSS) is a complex pipeline network system, which involves the multi-physics coupled fields of transcritical flow, heat transfer, rigid body motion, electromagnetics, and control. Based on the fundamental equation of state, a one-dimensional finite volume model system is proposed for compressible transcritical transient pipe flow. Six typical components of the SSS are modularization modeled by using an object-oriented programming method in Fortran 2018. To evaluate the performance of the algorithm, a simulation model consisting of 77 components is developed for a proportional controlled EPS xenon SSS, and a system-level simulation is performed. Compared with the standard case constructed on the Amesim platform, the model has a pressure rise time error lower than 6%, and the restrictors mass-flow-rate errors of the anode, cathode, and neutralizer are less than 2.8% of the rated flow rate after stabilization. Moreover, the proposed model yields more accurate results at certain operating points. The model system provides an efficient reference for the design and optimization of EPS SSSs, and offers a new approach to simulate complex cryogenic pipe flow systems.

Coupled Second-Order GTS-MOC Scheme for Transient Pipe Flows with an Entrapped Air Pocket

Coupled Second-Order GTS-MOC Scheme for Transient Pipe Flows with an Entrapped Air Pocket

Finite Volume Method for Transient Pipe Flow with an Air Cushion Surge Chamber Considering Unsteady Friction and Experimental Validation

In various water transmission systems such as long-distance water transfer projects and hydropower stations, accurate simulation of water hammer is extremely important for safe and stable operation and the realization of intelligent operations. Previous water hammer calculations usually consider only steady-state friction, underestimating the decay of transient pressure. A second-order Finite Volume Method (FVM) considering the effect of unsteady friction factor is developed to simulate the water hammer and the dynamic behavior of air cushion surge chamber in a water pipeline system, while an experimental pipe system is conducted to validate the proposed numerical model. Two unsteady friction models, Brunone and TVB models, were incorporated into the water hammer equations, which are solved by the MUSCL–Hancock method. One virtual boundary method was proposed to realize the FVM simulation of Air Cushion Surge Chamber. Comparisons with water hammer experimental results show that, while the steady friction model only accurately predicts the first pressure peak, it seriously underestimates pressure attenuation in later stages. Incorporating an unsteady friction factor can better predict the entire pressure attenuation process; in particular, the TVB unsteady friction model more accurately reproduces the pressure peaks and the whole pressure oscillation periods. For water pipeline systems with an air cushion surge chamber, energy attenuation of the elastic pipe water hammer is primarily due to pipe friction and the air cushion. The experimental results with the air cushion surge chamber demonstrate that the proposed FVM model with the TVB unsteady friction model and the air chamber polytropic exponent near 1.0 can well reproduce the experimental pressure oscillations.

Numerical simulation of transient pipe flow with entrapped air and wet bed effects

In this study, the evolution of transient pipe flow along the wet bed is numerically investigated. In the investigation, the shear stress transport k-ω model is used and the volume of fluid method is employed to track the surface of air and water. Two key parameters in the flow as upstream head H and initial water depth h in the pipe are examined. It is found that the bottom stress is significantly affected by the two parameters. The upstream head H determines the magnitude of the shear force, and the downstream water depth in the pipe affects the stability of the shear force. The boundary layer separation and flow pattern are the essential causes of shear instability. By analyzing the simulation results, an empirical equation with the average flow velocity is obtained to estimate the overflow capacity of the pipe by just the upstream water level and the depth of the wet bed.

Developments in analytical wall shear stress modelling for water hammer phenomena

New developments in analytical modelling of wall shear stresses during water hammer events are presented and discussed in this paper. These include further development of models based on the quasi-steady and unsteady hypothesis of hydraulic resistances. All the analyzed solutions have been improved so that they have extended ranges of applicability and the simplest possible mathematical form. Exemplary comparative studies were carried out, which showed their advantages and disadvantages. Explicit analytical formulas of wall shear stress were derived for the first time. They will be particularly useful at the stage of design and modernization of the pressurized piping systems (water supply, oil-hydraulics etc.) and enable effective analysis of leakages and other damages (ruptures, fatigue related failures) occurring in industrial pipeline systems. With their help, it will be easier to prevent the increased pulsating pressures occurring during water hammer events (protect against failures), and to verify numerical models used for years in simulations of transient pipe flows.

Energy Analysis of a Quasi-Two-Dimensional Friction Model for Simulation of Transient Flows in Viscoelastic Pipes

Quasi-two-dimensional (quasi-2D) friction models have been widely investigated in transient pipe flows. In the case of viscoelastic pipes, however, the effect of different values of the Reynolds number (Re) on pressure fluctuations (which can lead to water hammer) have not been considered in detail. This study establishes a quasi-2D friction model employing an integral total energy method and investigates the work due to frictional and viscoelastic terms at different Re values. The results show that viscoelastic work (WP) and frictional work (Df) increase with an increase in Re. However, when the initial Re values are high, the Df values are much larger than the WP values. In addition, for Re &lt; 3 × 105, the 1D model underestimated the viscoelastic terms. There was no significant difference between the two models for Re &gt; 3 × 105. In the case of different initial Re values, the two models produced almost the same values for WP. This study provides a theoretical basis for investigating transient flow from the perspective of energy analysis.

Energy relationships in transient pipe flow with fluid–structural interaction

During water hammer wave propagation, pressure and velocity fields vary over time due to transformations between strain and kinetic energies. In the literature, some energy expressions have been applied to explore energy transformations during water hammers, which account for various effects including friction stress, leaks, blockages, air pockets, and side flows. However, there are no energy expressions for water hammers that consider the fluid–structural interaction (FSI) effect, such as a buffer blocked riser subjected to heave motion. Therefore, in this study, the energy expressions for transient flow with FSI are obtained from the governing equations of the extended water hammer model. Moreover, based on continuum mechanics theory, general energy expressions are developed for the classical, the extended, and the partitioned shell–based water hammer models. Using the partitioned shell–based water hammer model, a buffer–blocked riser subjected to heave motion was investigated for energy transformation. Accounting for the fluid–structural interaction, there are three resonances in the dynamic response of riser while the heave motion frequency varies in the range of ocean wave frequencies. These three resonant frequencies are correlated to the frequency of pressure wave, the fundamental frequencies of fluid–filled riser that only accounted for fluid mass and the empty riser.

A new model for fluid transients in piping systems taking into account the fluid–structure interaction

The present work extends a recently developed quasi-2D flow model for fluid transients in elastic pipes to accommodate fluid–structure interaction mechanisms. In this context, the primary goal of the proposed approach is to analyze energy transfer and dissipative effects in the fluid–pipe system. The fluid–structure interaction couples the flow dynamics with the axial movement of the tube, giving rise to friction, Poisson, and junction coupling mechanisms to take place. The mechanical model mathematical structure forms a quasi-linear hyperbolic system of partial differential equations for which approximated solutions are sought by employing the method of characteristics. The proposed model reproduces classical benchmark solutions and presents a good agreement with experimental data. In addition, the whole thermomechanical consistent framework in which the model is established allows an accurate description of the flow’s internal structure. This feature allows a better comprehension of the phenomenon when compared to the classic frictionless and quasi-steady-friction-based four-equation models. The present approach allows a better description of the friction coupling mechanism and unveils uneven shear stress distributions in the unsteady flow. As a result, the energy dissipation in the fluid is captured with accuracy. Due to the absence or limited ability to describe these dispersive and dissipative effects, the traditional approaches are proved to lose their accuracy as the fluid transient goes. The coupled and uncoupled models are also analyzed and are proven to respond differently due to localized pipe–fluid interface effects in addition to the bulk mechanisms of transfer of energy that occur differently in both approaches.

On Boussinesq and Coriolis coefficients and their derivatives in transient pipe flows

A rigorous analytical justification is developed for a simplification that is widely used in one-dimensional simulations of steady and unsteady flows in pipes, namely treating the flow as a 'plug flow' in which cross-sectional variations in axial velocity are neglected except for consequential shear forces on pipe walls. The proof is obtained without assuming that the flow is nearly incompressible flow and, indeed, it is found that the plug flow approximation remains good even for compressible flows at moderate, subsonic speeds. In more general analyses, explicit allowance is sometimes made for the influence of (i) the Boussinesq coefficient β and (ii) its axial rate of change ∂β/∂x. Typically, such analyses discard the terms in ∂β/∂x in the basic equations and proceed using β alone. This is done on the grounds that no method of evaluating ∂β/∂x is available. It is shown in this paper that discarding ∂β/∂x is not only unnecessary, but that it actually leads to less accurate outcomes than simply assuming plug flow. The process used to derive the analytical justification has a spin-off benefit of shedding light on alternative methods of integrating source terms over the pipe cross-section. Although the primary purpose of the paper is to demonstrate that the use of plug flow approximations can be justified rigorously for most flows, brief attention is also paid to cases where it has potential to mislead.

Parametric analysis of discrete multiple-cavity models with the quasi-two-dimensional friction model for transient cavitating pipe flows

Abstract Discrete multiple-cavity models coupled with quasi-two-dimensional (quasi-2D) friction models are effective solutions to simulating transient cavitation pipe flows. The simulation accuracy of such models hinges upon the understanding of key parameters of the models, which remains elusive so far. To address such an open issue, this paper employs the discrete vapor cavity model (DVCM) and the discrete gas cavity model (DGCM), combined with the quasi-2D friction model, with a particular focus on revealing the sensitivity of these models to the key parameters such as grid number and weighting parameters. Based on the quantitative analysis and pressure fluctuation history, a method is developed to evaluate the accuracy of numerical results. Results show that the inclusion of the quasi-2D friction model improves the accuracy of predicting time of cavity formation and collapse; however, it does not affect the selection of grid number. Meanwhile, numerical results are sensitive to the weighting parameter of the viscous term in the quasi-2D friction model except for the case of low-intensity cavitation and its value of 1 is suggested for all cases. From the practical point of view, our finding is helpful to understand the feature of discrete multiple-cavity models and improve the simulating accuracy of transient cavitation pipe flows.

Numerical analysis of transient cavitating pipe flow by Quasi 2D and 1D models

Quasi two dimensional (2D) and one dimensional (1D) numerical simulations of transient flow with column separation are conducted using the discrete vaporous and gaseous cavitation models (DVCM and DGCM) in a simple reservoir-pipeline-valve system. In the quasi 2D transient model, the governing equations are solved by the method of characteristics along the pipe and by finite difference across the pipe cross section to consider the velocity profile, and the final model is coupled with the cavitating models. The comparison between the numerical and experimental results shows that the quasi 2D DGCM is more successful in predicting the maximum head pressures and reproducing the pressure spikes at different water hammer cycles. The quasi 2D model of DGCM correlates better with the experimental data than the quasi 2D DVCM, 1D DVCM and 1D DGCM in terms of pressure magnitude. In sensitivity analysis, the 2D DGCM shows more grid convergence than the 1D models of DGCM and DVCM and 2D DVCM by increasing the number of longitudinal and radial intervals.

Unsteady friction model modified with compression–expansion effects in transient pipe flow

Abstract This paper aims to modify the conventional one-coefficient instantaneous acceleration-based (IAB) model for better prediction of unsteady friction behavior. In this work, the energy dissipation caused by viscous stress during fluid volume compression–expansion (CE) was derived from the compressible Navier–Stokes equation. It is found that the energy dissipation term can be expressed by the product of the second-order partial derivative of velocity in space and the second viscosity coefficient. On this basis, a modified IAB-CE model was developed with the energy dissipation term and solved by the method of characteristic (MOC). The numerical results obtained from the modified model showed a good agreement with the four test cases, where the relative errors are improved by 0.26, 2.03, 9.56, and 36.67%, compared with the results from the original IAB model. The estimation for wave peak and valley is improved as well. Furthermore, the Bradley equation can be applied to establish the relationship between the dissipation coefficient and the Reynolds number. The modified model developed in this study takes into account the fluid CE effects and improves the prediction accuracy of wave amplitude of unsteady flow.

Modeling Transient Pipe Flow in Plastic Pipes with Modified Discrete Bubble Cavitation Model

Most of today’s water supply systems are based on plastic pipes. They are characterized by the retarded strain (RS) that takes place in the walls of these pipes. The occurrence of RS increases energy losses and leads to a different form of the basic equations describing the transient pipe flow. In this paper, the RS is calculated with the use of convolution integral of the local derivative of pressure and creep function that describes the viscoelastic behavior of the pipe-wall material. The main equations of a discrete bubble cavity model (DBCM) are based on a momentum equation of two-phase vaporous cavitating flow and continuity equations written initially separately for the gas and liquid phase. In transient flows, another important source of pressure damping is skin friction. Accordingly, the wall shear stress model also required necessary modifications. The final partial derivative set of equations was solved with the use of the method of characteristics (MOC), which transforms the original set of partial differential equations (PDE) into a set of ordinary differential equations (ODE). The developed numerical solutions along with the appropriate boundary conditions formed a basis to write a computer program that was used in comparison analysis. The comparisons between computed and measured results showed that the novel modified DBCM predicts pressure and velocity waveforms including cavitation and retarded strain effects with an acceptable accuracy. It was noticed that the influence of unsteady friction on damping of pressure waves was much smaller than the influence of retarded strain.

On the Hierarchy of Models for Pipe Transients: From Quasi-Two-Dimensional Water Hammer Models to Full Three-Dimensional Computational Fluid Dynamics Models

Abstract A hierarchy of models exists in the literature for the simulation of pipe transients. One-dimensional (1D) water hammer models provide a cost-effective tool for the analysis of such transients. Traditional 1D models implement a quasi-steady approximation of the frictional term, which results in poor modeling of the attenuation of the transient. To improve the modeling of the attenuation phenomenon, alternative unsteady friction models were developed for the 1D water hammer formulation. Moreover, quasi-two-dimensional (quasi-2D) water hammer models were introduced, which allow the computation of the unsteady velocity profile and hence provide improved modeling of the attenuation phenomenon. Recently, interest has developed in the use of computational fluid dynamics (CFD) models based on the Navier–Stokes equations in the simulation of fluid transients. Both axisymmetric and full three-dimensional (3D) CFD models are used in this regard. The aim of the current paper is to carry out a comparative study between the performance of quasi-2D water hammer models, axisymmetric CFD models, and full 3D CFD models. Numerical computations using the three models are performed for both laminar and turbulent flow cases. Present results show that the quasi-2D water hammer model and the axisymmetric CFD model provide near identical results in terms of computing the magnitude, phase, and attenuation of the transient. Reported results also demonstrate the computational efficiency of the quasi-2D model, which provides results that agree reasonably well with the full 3D CFD model results while using a grid density, which is an order of magnitude lower than the grid requirements for the full 3D CFD model.

An accurate and efficient scheme involving unsteady friction for transient pipe flow

Abstract A robust prediction system should monitor all possible hydraulic transients, which is significant for the appropriate and safe operation of pipe systems. A second-order finite volume method (FVM) Godunov-type scheme (GTS) considering unsteady friction factors is introduced to simulate hydraulic transients, which was rarely involved in previous work. One explicit-solution source item approach developed in this work is crucial for the proposed GTS to easily incorporate various forms of the existing unsteady friction models, including original convolution-based models (Zielke model and Vardy–Brown model), simplified convolution-based model (Trikha–Vardy–Brown (TVB) model), and Brunone instantaneous acceleration-based model. Results achieved by the proposed models are compared with experimental data as well as predictions by the classic Method of Characteristics (MOC). Results show that the MOC scheme may produce severe numerical attenuation in the case of a low Courant number. The proposed second-order GTS unsteady friction models are accurate, efficient, and stable even for Courant numbers less than one and sparse grid, and only need much less grid number and computation time to reach the same numerical accuracy. The TVB convolution-based model and Brunone model in the second-order GTS are suggested for further applications in hydraulic transients due to their high accuracy and efficiency.

Godunov-Type Solutions for Transient Pipe Flow Implicitly Incorporating Brunone Unsteady Friction

AbstractAn approach combining the Brunone unsteady friction model and first- and second-order Godunov-type scheme (GTS) is developed to simulate transient pipe flow. The exact solution to the Riema...