Wall Shear Stress in Transient Turbulent Pipe Flow by Local Velocity Measurement

The modeling of unsteady wall shear stress plays a crucial role in the analysis of fast transients in pressurized pipe systems, since it allows to evaluate transient energy dissipation properly. The main aim of this paper is to give a contribution to the understanding of transient pressurized flow dynamics in turbulent regime by measuring not only pressure but also the instantaneous axial velocity profile at two sections of the laboratory pipe. Specifically, by means of ultrasonic Doppler velocimetry—a completely nonintrusive technique—instantaneous velocity gradients at pipe wall are measured allowing to evaluate the time history of the actual wall shear stress by coupling velocity measurements to a two-zone stress model. As a result, the behavior of accelerating and decelerating flows with respect to the corresponding steady ones, i.e., with the same value of the discharge, is pointed out. Due to the characteristics of the laboratory pipe—a 352-m long high density polyethylene pipe—transients phenomena are investigated both at short and long time scales.     

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.     

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.     

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.     

Transient Friction in Pressurized Pipes. I: Investigation of Zielke’s Model

This paper investigates the well-known model for unsteady friction developed by Zielke in 1968. The model is based on weights of past local bulk accelerations and is analytically correct for laminar flow, but computationally demanding. Different models have been proposed using dynamic properties, typically based on instantaneous accelerations (IAB) that are more rapid in computational schemes. Unfortunately, they are not as accurate as Zielke’s model and fail to model certain types of transients. This paper points out that the water hammer transient is dominated by a periodicity varying along the pipe. Because of this, the unsteady friction calculated by the Zielke model is distributed nonuniformly along the pipe, and changes in the pipe length change the local unsteady friction. This phenomenon may explain why IAB models using calibrated coefficients to match experimental results have a large span in value for the reported coefficients. This paper will hopefully contribute to further work to find highly accurate and rapid models. The subject deserves to be brought up for discussion as a part of a total understanding of the problem.     

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.     

Numerical Error in Weighting Function-Based Unsteady Friction Models for Pipe Transients

The accurate simulation of pressure transients in pipelines and pipe networks is becoming increasingly important in water engineering. Applications such as inverse transient analysis for condition assessment, leak detection, and pipe roughness calibration require accurate modeling of transients for longer simulation periods that, in many situations, requires improved modeling of unsteady frictional behavior. In addition, the numerical algorithm used for unsteady friction should be highly efficient, as inverse analysis requires the transient model to be run many times. A popular model of unsteady friction that is applicable to a short-duration transient event type is the weighting function-based type, as first derived by Zielke in 1968. Approximation of the weighting function with a sum of exponential terms allows for a considerable increase in computation speed using recursive algorithms. A neglected topic in the application of such models is evaluation of numerical error. This paper presents a discussion and quantification of the numerical errors that occur when using weighting function-based models for the simulation of unsteady friction in pipe transients. Comparisons of numerical error arising from approximations are made in the Fourier domain where exact solutions can be determined. Additionally, the relative importance of error in unsteady friction modeling and unsteady friction itself in the context of general simulation is discussed.     

Dynamic Memory Computation of Impedance Matrix Method

The computational efficiency of the impedance matrix method has been greatly improved for large pipe networks with various dimensions and complexity. Several numerical methods for solving linear system were modified to deal with the complex domain operation and used into impedance evaluation. Two different memory reduction schemes were developed based on one-dimensional storage and implemented with the biconjugate gradient method and the Gaussian elimination scheme, respectively. A new implementation of the impedance matrix method, namely, the dynamic memory allocation scheme, was introduced to efficiently model hydraulic transients in pipeline systems that have large topological structures. Three hypothetical pipe networks, the multiseries system, the multilooped system, and the multiblock system, were used to test the performance of the developed schemes. The impact of randomizing pipeline parameters, i.e., friction factor, length, and wave speed, on computation efficiency was evaluated and compared. The dynamic memory allocation scheme not only reduces costs substantially in CPU execution time and memory space compared to other schemes but also shows significant potential as a real-time unsteady flow predictor for large pipe networks.     

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.     

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.     

Reynolds Stress and Bed Shear in Nonuniform Unsteady Open-Channel Flow

Expressions for the Reynolds stress and bed shear stress are developed for nonuniform unsteady flow in open channels with streamwise sloping beds, assuming universal (logarithmic) velocity distribution law and using the Reynolds and continuity equations of two-dimensional open-channel flow. The computed Reynolds stress distributions are in agreement with experimental data.     

Explicit Power Formula for the Darcy–Weisbach Pipe Flow Equation: Application in Optimal Pipeline Design

Although the Darcy–Weisbach equation combined with the Colebrook–White semitheoretical formula for calculating the friction coefficient is a highly accurate generalized pipe-water flow resistance equation, most users prefer the use of simple, explicit power law form formulas. Because of their simplicity (despite their limitations) the purely empirical power formulas of Hazen–Williams and Manning remain the most popular pipe flow resistance equations used in routine hydraulic engineering applications. In this paper, a new simple power law form formula is derived to approximate the generalized Darcy–Weisbach combined with the Colebrook–White equation. The two main pipe flow parameters, such as the discharge (or velocity) and the diameter, appeared explicitly in the proposed formula. The suggested power-form formula compared with the Darcy–Weisbach and Coolbrook–White equation yields a maximum relative error of about ±4.5%. The power-form suggested formula is dimensionally homogeneous and its accuracy is sufficient for practical engineering applications. A correction factor is introduced for the variation of kinematic viscosity with temperature. The usefulness of the formula is demonstrated in an application concerning the optimal design of a delivery pipeline with pumping. The power form of the friction formula facilitates the formulation of the problem leading to the derivation of a simple equation from which the economic diameter is explicitly calculated.     

Approximation of Turbulent Wall Shear Stresses in Highly Transient Pipe Flows

Theoretical predictions of wall shear stresses in unsteady turbulent flows in pipes are developed for all flow conditions from fully smooth to fully rough and for Reynolds numbers from 103 to 108. A weighting function approach is used, based on a two-region viscosity distribution in the pipe cross section that is consistent with the Colebrook–White expression for steady-state wall friction. The basic model is developed in an analytical form and the resulting weighting function is then approximated as a sum of exponentials using a modified form of an approximation due to Trikha. A straightforward method is presented for the determination of appropriate values of coefficients for any particular Reynolds number and pipe roughness ratio. The end result is a method that can be used relatively easily by analysts seeking to model unsteady flows in pipes and ducts.     

Shear Strength of Municipal Solid Waste

A comprehensive large-scale laboratory testing program using direct shear (DS), triaxial (TX), and simple shear tests was performed on municipal solid waste (MSW) retrieved from a landfill in the San Francisco Bay area to develop insights about and a framework for interpretation of the shear strength of MSW. Stability analyses of MSW landfills require characterization of the shear strength of MSW. Although MSW is variable and a difficult material to test, its shear strength can be evaluated rationally to develop reasonable estimates. The effects of waste composition, fibrous particle orientation, confining stress, rate of loading, stress path, stress-strain compatibility, and unit weight on the shear strength of MSW were evaluated in the testing program described herein. The results of this testing program indicate that the DS test is appropriate to evaluate the shear strength of MSW along its weakest orientation (i.e., on a plane parallel to the preferred orientation of the larger fibrous particles within MSW). These laboratory results and the results of more than 100 large-scale laboratory tests from other studies indicate that the DS static shear strength of MSW is best characterized by a cohesion of 15?kPa and a friction angle of 36° at normal stress of 1?atm with the friction angle decreasing by 5° for every log cycle increase in normal stress. Other shearing modes that engage the fibrous materials within MSW (e.g., TX) produce higher friction angles. The dynamic shear strength of MSW can be estimated conservatively to be 20% greater than its static strength. These recommendations are based on tests of MSW with a moisture content below its field capacity; therefore, cyclic degradation due to pore pressure generation has not been considered in its development.     

Local Balance Unsteady Friction Model

The paper proposes the evaluation of unsteady friction by a one-dimensional local balance model. The model is applied for the case of water hammer in a single pipeline for both the downstream end and upstream end valve, and for both rapid valve closure and opening. The model is based on local balance of the friction force. Comparisons with experimental results show that the model correctly predicts the extreme values of pressure head oscillation, as well as its shape for both rapid valve closure and opening, and then overcomes the limits of previous unsteady friction models based on instantaneous acceleration. As the comparisons with experimental results can be made easily only for pressure oscillations and can be affected by dissipation mechanisms other than friction, the performance of the model is examined also by comparison with the results of a two-dimensional low-Reynolds number k–ε model.     

工程流体力学常用公式电脑计算程序

提供了工程流体力学常用公式电脑计算程序,利用这些程序可以快速、准确地计算管道的压力损失、摩擦压力损失系数、流速、雷诺数、求管道直径,或者求紊流过渡管区摩擦压力损失系数,并指出流体的流动性质（层流、紊流光滑管区、紊流过渡管区、紊流粗糙管区）。提供了2个简化计算程序,其目的是：在部分条件（如压力、重度、动力粘度、温度…等）不变的情况下,加快多次计算中的进度。     

Prediction of hemolysis in turbulent shear orifice flow

This study proposes a method of predicting hemolysis induced by turbulent shear stress (Reynolds stress) in a simplified orifice pipe flow. In developing centrifugal blood pumps, there has been a serious problem with hemolysis at the impeller or casing edge; because of flow separation and turbulence in these regions. In the present study, hemolysis caused by turbulent shear stress must occur at high shear stress levels in regions near the edge of an orifice pipe flow. We have computed turbulent shear flow using the low-Reynolds number k-epsilon model. We found that the computed turbulent shear stress near the edge was several hundreds times that of the laminar shear stress (molecular shear stress). The peak turbulent shear stress is much greater than that obtained in conventional hemolysis testing using a viscometer apparatus. Thus, these high turbulent shear stresses should not be ignored in estimating hemolysis in this blood flow. Using an integrated power by shear force, it is optimal to determine the threshold of the turbulent shear stress by comparing computed stress levels with those of hemolysis experiments or pipe orifice blood flow.     

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.     

Quantifying Linearization Error When Modeling Fluid Pipeline Transients Using the Frequency Response Method

The unsteady mass and momentum equations for pipe flow can be solved in the frequency domain and provides additional insight into the behavior of fluid transients. Additionally, this approach has significant computational advantages compared to the method of characteristics because it is not based on a rigid time-space grid. Despite its advantages, the frequency domain approach must be used with care as it uses linearized forms of the steady friction and orifice equations—which can deviate significantly from the true nonlinear solution. The conditions in which the frequency response method can be accurately used are currently unknown. This paper investigates and quantifies the error in the frequency-domain method, via comparison to a highly discretized time-domain model that uses the method of characteristics, and describes situations where the frequency response method can be used with accurate results. A reservoir-pipe-valve system was used in this study with transients generated by perturbation of the valve. The error consists of errors from two sources: the linear approximations of the steady friction and the steady orifice equations. The frequency response method was shown to produce identical results to the method of characteristics when these two sources of error are minimized. The error in the frequency-domain model was quantified as functions of the perturbation magnitude, frequency, and system parameters. The results indicate that errors are significant when the perturbation size is more than 25% of the steady-state condition and this error is frequency dependent with the largest errors occurring at the harmonic peaks of the system.