Systematic Evaluation of One-Dimensional Unsteady Friction Models in Simple Pipelines

In this paper, basic unsteady flow types and transient event types are categorized, and then unsteady friction models are tested for each type of transient event. One important feature of any unsteady friction model is its ability to correctly model frictional dissipation in unsteady flow conditions under a wide a range of possible transient event types. This is of importance to the simulation of transients in pipe networks or pipelines with various devices in which a complex series of unsteady flow types are common. Two common one-dimensional unsteady friction models are considered, namely, the constant coefficient instantaneous acceleration-based model and the convolution-based model. The modified instantaneous acceleration-based model, although an improvement, is shown to fail for certain transient event types. Additionally, numerical errors arising from the approximate implementation of the instantaneous acceleration-based model are determined, suggesting some previous good fits with experimental data are due to numerical error rather than the unsteady friction model. The convolution-based model is successful for all transient event types. Both approaches are tested against experimental data from a laboratory pipeline.     

Efficient Treatment of the Vardy–Brown Unsteady Shear in Pipe Transients

An accurate, simple, and efficient approximation to the Vardy–Brown unsteady friction equation is derived and shown to be easily implemented within a one-dimensional characteristics solution for unsteady pipe flow. For comparison, the exact Vardy–Brown unsteady friction equation is used to model shear stresses in transient turbulent pipe flows and the resulting waterhammer equations are solved by the method of characteristics. The approximate Vardy–Brown model is more computationally efficient (i.e., requires one-sixth the execution time and much less memory storage) than the exact Vardy–Brown model. Both models are compared with measured data from different research groups and with numerical data produced by a two-dimensional turbulence waterhammer model. The results show that the exact Vardy–Brown model and the approximate Vardy–Brown model are in good agreement with both laboratory and numerical experiments over a wide range of Reynolds number and wave frequencies. The proposed approximate model only requires the storage of flow variables from a single time step while the exact Vardy–Brown model requires the storage of flow variables at all previous time steps and the two-dimensional model requires the storage of flow variables at all radial nodes.     

Nonconservative Formulation of Unsteady Pipe Flow Model

A new approach to numerical modeling of water hammer is proposed. An unsteady pipe flow model incorporating Brunone’s unsteady friction model is used, but in contrast to the standard treatment of the unsteady friction term as a source term, the writers propose a nonconservative formulation of source term. Second-order flux limited and high order weighted essentially nonoscillating numerical schemes were applied to the proposed formulation, and results are in better agreement with measurements when compared with results obtained with standard form.     

Efficient Approximation of Unsteady Friction Weighting Functions

A simple method is presented for evaluating wall shear stresses from known flow histories in unsteady pipe flows. The method builds on previous work by Trikha, but has two important differences. One of these enables the method to be used with much larger integration time steps than are acceptable with Trikha’s method. The other, a general procedure for determining approximations to weighting functions, enables it to be used at indefinitely small times (high frequencies). The method is applicable to both laminar and turbulent flows.     

Unsteady Flow in Hydraulic Networks with Polymeric Additional Pipe

This paper presents results of an experimental and theoretical study on the use of a high density polyethylene (HDPE) additional pipe inserted downstream of a pump in a hydraulic network as a surge suppressor. The experiments consistently show a reduction of the oscillations with respect to the case without a HDPE device, while in the case of a single pumping pipeline the oscillations can be amplified for small volumes of the additional pipe. Previously calibrated mechanical parameters are considered in the mathematical models whose results are compared with experimental results. Both linear elastic and Kelvin-Voigt viscoelastic behavior of the pipe material and both one-dimensional (1D) and quasi-2D flow models are taken into account. The numerical results show that the viscoelastic model better describes the phenomenon, but the elastic model adequately estimates the maximum and minimum oscillations. Furthermore, the results of the quasi-2D model are in better agreement with the experimental maximum and minimum oscillations than those of the 1D model, but the differences are less important than in the case of networks without a HDPE device.     

Head- and Flow-Based Formulations for Frequency Domain Analysis of Fluid Transients in Arbitrary Pipe Networks

Applications of frequency-domain analysis in pipelines and pipe networks include resonance analysis, time-domain simulation, and fault detection. Current frequency-domain analysis methods are restricted to series pipelines, single-branching pipelines, and single-loop networks and are not suited to complex networks. This paper presents a number of formulations for the frequency-domain solution in pipe networks of arbitrary topology and size. The formulations focus on the topology of arbitrary networks and do not consider any complex network devices or boundary conditions other than head and flow boundaries. The frequency-domain equations are presented for node elements and pipe elements, which correspond to the continuity of flow at a node and the unsteady flow in a pipe, respectively. Additionally, a pipe-node-pipe and reservoir-pipe pair set of equations are derived. A matrix-based approach is used to display the solution to entire networks in a systematic and powerful way. Three different formulations are derived based on the unknown variables of interest that are to be solved: head-formulation, flow-formulation, and head-flow-formulation. These hold significant analogies to different steady-state network solutions. The frequency-domain models are tested against the method of characteristics (a commonly used time-domain model) with good result. The computational efficiency of each formulation is discussed with the most efficient formulation being the head-formulation.     

Numerical Simulation of Swirling Flow in Complex Hydroturbine Draft Tube Using Unsteady Statistical Turbulence Models

A numerical method is developed for carrying out unsteady Reynolds-averaged Navier-Stokes (URANS) simulations and detached-eddy simulations (DESs) in complex 3D geometries. The method is applied to simulate incompressible swirling flow in a typical hydroturbine draft tube, which consists of a strongly curved 90° elbow and two piers. The governing equations are solved with a second-order-accurate, finite-volume, dual-time-stepping artificial compressibility approach for a Reynolds number of 1.1 million on a mesh with 1.8 million nodes. The geometrical complexities of the draft tube are handled using domain decomposition with overset (chimera) grids. Numerical simulations show that unsteady statistical turbulence models can capture very complex 3D flow phenomena dominated by geometry-induced, large-scale instabilities and unsteady coherent structures such as the onset of vortex breakdown and the formation of the unsteady rope vortex downstream of the turbine runner. Both URANS and DES appear to yield the general shape and magnitude of mean velocity profiles in reasonable agreement with measurements. Significant discrepancies among the DES and URANS predictions of the turbulence statistics are also observed in the straight downstream diffuser.     

One-Dimensional Numerical Model for Nonuniform Sediment Transport under Unsteady Flows in Channel Networks

In this study, the proposed one-dimensional model simulates the nonequilibrium transport of nonuniform total load under unsteady flow conditions in dendritic channel networks with hydraulic structures. The equations of sediment transport, bed changes, and bed-material sorting are solved in a coupling procedure with a direct solution technique, while still decoupled from the flow model. This coupled model for sediment calculation is more stable and less likely to produce negative values for bed-material gradation than the traditional fully decoupled model. The sediment transport capacity is calculated by one of four formulas, which have taken into consideration the hiding and exposure mechanism of nonuniform sediment transport. The fluvial erosion at bank toes and the mass failure of banks are simulated to complement the modeling of bed morphological changes in channels. The tests in several cases show that the present model is capable of predicting sediment transport, bed changes, and bed-material sorting in various situations, with reasonable accuracy and reliability.     

Use of Steady-State Pump Head-Discharge Curve for Unsteady Pipe Flow Applications

An experimental pipeline system with a multistage centrifugal pump was used to study the effect of transient operations on the hydrodynamic performance of a centrifugal pump. Several transient flow operations were considered, ranging from very mild to severe transients. The dynamic relationship of total pressure rise across the pump to the flow rate was compared with that of the steady state. Deviation between the dynamic pump head and the value given by the steady-state curve at the same instantaneous discharge was established and found to be a function of the severity of the transient. It was found that severe flow conditions could cause this deviation to exceed 30% of the steady-state value. The use of the steady-state pump head-discharge relationship in the solution of transient pipe flow by the method of characteristics (MOC) is discussed. It was found that the steady-state pump head-discharge curve was not accurate enough to support the solution of unsteady pipe flow application by the MOC.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.     

Time-Line Interpolation Errors in Pipe Networks

An exact method of assessing numerical errors in analyses of unsteady flows in pipe networks is introduced. The assessment is valid for fixed-grid method of characteristics analyses using time-line interpolations. A pipe polynomial transfer matrix is developed and is analogous to transfer function matrices used in free oscillation theory. The influence of reachback is assessed by comparing exact numerical predictions using a polynomial transfer matrix with exact analytical predictions obtained using free oscillation theory. The investigation is part of a long-term project aimed at automating the selection of numerical grid sizes in unsteady flow analyses. The eventual goal is to enable users of unsteady flow software to prescribe required degrees of accuracy instead of specifying the numerical grid itself. This paper is only a first step toward the long-term aim, but it is a big step toward an intermediate objective of providing exact benchmarking data for the assessment of approximate methods of automatic grid selection.     

Wall Shear Stress in the Early Stage of Unsteady Turbulent Pipe Flow

Conventionally, wall shear stress in an unsteady turbulent pipe flow is decomposed into a quasi-steady component and an “unsteady wall shear stress” component. Whereas the former is evaluated by using “standard” steady flow correlations, extensive research has been carried out to develop methods to predict the latter leading to various unsteady friction models. A different approach of decomposition is used in the present paper whereby the wall shear in an unsteady flow is split into the initial steady value and perturbations from it. It is shown that in the early stages of an unsteady turbulent pipe flow, these perturbations are well described by a laminar-flow formulation. This allows simple expressions to be derived for unsteady friction predictions, which are in good agreement with experimental and computational results.     

Experimental and Numerical Investigation of Bed-Load Transport under Unsteady Flows

The dynamic behavior of bed-load sediment transport under unsteady flow conditions is experimentally and numerically investigated. A series of experiments are conducted in a rectangular flume (18?m in length, 0.80?m in width) with various triangular and trapezoidal shaped hydrographs. The flume bed of 8?cm in height consists of scraped uniform small gravel of D50 = 4.8??mm. Analysis of the experimental results showed that bed-load transport rates followed the temporal variation of the triangular and trapezoidal hydrographs with a time lag on the average of 11 and 30?s, respectively. The experimental data were also qualitatively investigated employing the unsteady-flow parameter and total flow work index. The analysis results revealed that total yield increased exponentially with the total flow work. An original expression which is based on the net acceleration concept was proposed for the unsteadiness parameter. Analysis of the results then revealed that the total yield increased exponentially with the increase in the value of the proposed unsteadiness parameter. Further analysis of the experimental results revealed that total flow work has an inverse exponential variation relation with the lag time. A one-dimensional numerical model that employs the governing equations for the conservation of mass for water and sediment and the momentum was also developed to simulate the experimental results. The momentum equation was approximated by the diffusion wave approach, and the kinematic wave theory approach was employed to relate the bed sediment flux to the sediment concentration. The model successfully simulated measured sedimentographs. It predicted sediment yield, on the average, with errors of 7% and 15% of peak loads for the triangular and trapezoidal hydrograph experiments, respectively.     

Numerical Calculation of Turbulence Structure in Depth-Varying Unsteady Open-Channel Flows

Unsteady depth-varying open-channel flows are really observed in flood rivers. Owing to highly accurate laser Doppler anemometers (LDA), some valuable experimental databases of depth-varying unsteady open-channel flows are now available. However, these LDA measurements are more difficult to conduct in open-channel flows at higher unsteadiness, in comparison with unsteady wall-bounded flows such as oscillatory boundary layers and duct flows. Therefore, in this study, a low-Reynolds-number k–ε model involved with a function of unsteadiness effect was developed and some numerical calculations were conducted using the volume of fluid method as a free-surface condition. The present calculated values were in good agreement with the existing LDA data in the whole flow depth from the wall to the time-dependent free surface. These values were also compared with those of unsteady wall-bounded flows. The present calculations were able to predict the distributions of turbulence generation and its dissipation, and consequently the unsteadiness effect on turbulence structure was discussed on the basis of the outer-variable unsteadiness parameter α, which is correlated with the inner-variable unsteadiness parameter ω+ in unsteady wall-bounded flows.     

Numerical Sensitivity Study of Unsteady Friction in Simple Systems with External Flows

This paper investigates the importance of unsteady friction effects when performing water hammer analyses for pipe systems with external fluxes due to demands, leaks, and other system elements. The transient energy equation for a system containing an orifice-type external flow is derived from the two-dimensional, axial momentum equation. A quasi-two-dimensional flow model is used to evaluate the relative energy contribution of total friction, unsteady friction, and the external flow, in a 1,500?m pipeline, with orifice flows ranging from steady-state flows of 2–70% of the mean pipe flow, and a Reynolds number of 600,000. It is found that for initial lateral flows larger than around 30% of the mean flow, unsteady friction effects can probably be neglected, whereas for external flows smaller than this, unsteady friction should generally be considered. Overall, the relative role of unsteady friction is found to diminish as the external flux increases, implying that unsteady friction is not critical for systems with large external flows. These results imply that unsteady friction may have a significant impact on the validity of transient leak detection techniques that have been derived assuming quasi-steady friction. To demonstrate this point, an existing transient leak detection method, originally derived under quasi-steady conditions, is tested with unsteady friction included.     

Timescale Behavior of the Wall Shear Stress in Unsteady Laminar Pipe Flows

Based on two-dimensional (2D) flow model simulations, the effects of the radial structure of the flow (e.g., the nonuniformity of the velocity profile) on the pipe wall shear stress, τw, are determined in terms of bulk parameters such as to allow improved 1D modeling of unsteady contribution of τw. An unsteady generalization, for both laminar and turbulent flows, of the quasi-stationary relationship between τw and the friction slope, J, decomposes the additional unsteady contribution into an instantaneous energy dissipation term and an inertial term (that is, based on the local average acceleration-deceleration effects). The relative importance of these two effects is investigated in a transient laminar flow and an analysis of the range of applicability of this kind of approach of representing unsteady friction is presented. Finally, the relation between the additional inertial term and Boussinesq momentum coefficient, is clarified. Although laminar pipe flows are a special case in engineering practice, solutions in this flow regime can provide some insight into the behavior of the transient wall shear stress, and serve as a preliminary step to the solutions of unsteady turbulent pipe flows.     

Numerical Oscillations in Pipe-Filling Bore Predictions by Shock-Capturing Models

The introduction of nonlinear, shock-capturing schemes has improved numerical predictions of hydraulic bores, but significant numerical oscillations have been reported in the predictions of pipe-filling bore fronts associated with the transition between open-channel and pressurized flow regimes. These oscillations can compromise the stability of numerical models. A study of these oscillations indicates that the strength of the numerical oscillations is associated with the sharp discontinuities in the flow parameters across the jump, particularly the wave celerity. Approaches to attenuate oscillations by artificially reducing acoustic wave speeds may result in the loss of simulation accuracy. Two new techniques to attenuate the oscillation amplitudes are presented, the first based on numerical filtering of the oscillations and the second based on a new flux function that judiciously introduces numerical diffusion only in the vicinity of the bore front. Both approaches are effective in decreasing the strength of the numerical oscillations.     

Steady and Unsteady Simulations of Turbulent Flow and Transport in Ultraviolet Disinfection Channels

The effects of unsteadiness in the turbulent flow through a staggered array of circular cylinders, modeling an ultraviolet disinfection system, are studied by means of solutions of the two-dimensional Reynolds-averaged Navier–Stokes equations incorporating the standard k–? turbulence model. Time averaging is applied to the unsteady solution, and the time-averaged characteristics are compared with a solution where a steady flow is a priori assumed, as well as with time-averaged measurements. Differences between the predictions of time-averaged and the steady-flow models are found to be largest in the entrance region of the array, and to decline in importance in the downstream direction. Comparison with measurements indicate that, while the time-averaged unsteady model predictions exhibited better agreement in some respects, the turbulent kinetic energy remained substantially underpredicted. Predictions of head losses through the array are also discussed.     

Simplified Numerical and Analytical Approach for Solutes in Turbulent Flow Reacting with Smooth Pipe Walls

A large group of reactions that affect water quality in distribution networks occur on the pipe wall surface. Existing simulation models are usually based on cross-sectionally averaged variables that use mass-transfer coefficients derived for constant-concentration (Dirichlet) boundary conditions to account for cross-sectional variations. In the case of a first-order wall-demand problem, the boundary condition is however of Robin type. We derive a simple one-dimensional (1D) model for the radial concentration profile of a solute of arbitrary Schmidt number (Sc) reacting with pipe walls in a fully developed turbulent flow. A modified van Driest mixing length model was used to approximate the Reynolds-averaged velocity and eddy diffusivity. Numerical solutions of the 1D model agree well with a two-dimensional mass transport model and experimental data. An asymptotic solution for high Sc is derived, which is in excellent agreement with the 1D model for Sc>100. A comparison with the mass-transfer coefficients for constant-concentration boundary conditions shows that the differences between the two boundary conditions are small.     

Numerical Modeling of Bed Evolution in Channel Bends

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data than those obtained by previous numerical models. The model is used to investigate the process of evolution and stability of bed deformation in circular bends.