Godunov-Type Solutions for Water Hammer Flows

First- and second-order explicit finite volume (FV) Godunov-type schemes for water hammer problems are formulated, applied, and analyzed. The FV formulation ensures that both schemes conserve mass and momentum and produce physically realizable shock fronts. The exact solution of the Riemann problem provides the fluxes at the cell interfaces. It is through the exact Riemann solution that the physics of water hammer waves is incorporated into the proposed schemes. The implementation of boundary conditions, such as valves, pipe junctions, and reservoirs, within the Godunov approach is similar to that of the method of characteristics (MOC) approach. The schemes are applied to a system consisting of a reservoir, a pipe, and a valve and to a system consisting of a reservoir, two pipes in series, and a valve. The computations are carried out for various Courant numbers and the energy norm is used to evaluate the numerical accuracy of the schemes. Numerical tests and theoretical analysis show that the first-order Godunov scheme is identical to the MOC scheme with space-line interpolation. It is also found that, for a given level of accuracy and using the same computer, the second-order scheme requires much less memory storage and execution time than either the first-order scheme or the MOC scheme with space-line interpolation. Overall, the second-order Godunov scheme is simple to implement, accurate, efficient, conservative, and stable for Courant number less than or equal to one.     

Stability Analysis of Velocity Profiles in Water-Hammer Flows

This paper performs linear stability analysis of base flow velocity profiles for laminar and turbulent water-hammer flows. These base flow velocity profiles are determined analytically, where the transient is generated by an instantaneous reduction in flow rate at the downstream end of a simple pipe system. The presence of inflection points in the base flow velocity profile and the large velocity gradient near the pipe wall are the sources of flow instability. The main parameters that govern the stability behavior of transient flows are the Reynolds number and dimensionless timescale. The stability of the base flow velocity profiles with respect to axisymmetric and asymmetric modes is studied and its results are plotted in the Reynolds number∕timescale parameter space. It is found that the asymmetric mode with azimuthal wave number 1 is the least stable. In addition, the results indicate that the decrease of the velocity gradient at the inflection point with time is a stabilizing mechanism whereas the migration of the inflection point from the pipe wall with time is a destabilizing mechanism. Moreover, it is shown that a higher reduction in flow rate, which results in a larger velocity gradient at the inflection point, promotes flow instability. Furthermore, it is found that the stability results of the laminar and the turbulent velocity profiles are consistent with published experimental data and successfully explain controversial conclusions in the literature. The consistency between stability analysis and experiments provide further confirmation that (1) water-hammer flows can become unstable; (2) the instability is asymmetric; (3) instabilities develop in a short (water-hammer) timescale; and (4) the Reynolds number and the wave timescale are important in the characterization of the stability of water-hammer flows. Physically, flow instabilities change the structure and strength of the turbulence in a pipe, result in strong flow asymmetry, and induce significant fluctuations in wall shear stress. These effects of flow instability are not represented in existing water-hammer models.     

Applicability of Quasisteady and Axisymmetric Turbulence Models in Water Hammer

Two of the existing turbulence water hammer models, namely the two-layer and the five-layer eddy viscosity models, are implemented and analyzed and the accuracy of their quasi-steady and axisymmetric assumptions evaluated. In addition, a dimensionless parameter P (ratio of the time scale of radial diffusion of shear to the time scale of wave propagation) for assessing the accuracy of quasi-steady turbulence modeling in water hammer problems is developed and applied. It is found that the results of both models are in reasonable agreement, confirming that the turbulence modeling of water hammer flows is insensitive to the magnitude and distribution of the eddy viscosity within the pipe core. Comparison of model results with available data shows that the quasi-steady assumption becomes more accurate as the dimensionless parameter P increases. Furthermore, the analysis shows that the quasi-steady assumption is highly accurate as long as the simulation time is below the diffusion time scale and that this assumption causes an almost linear increase in the difference between model results and data with time. The accuracy of the flow axisymmetry assumption is evaluated by applying both models to a water hammer problem where flow asymmetry has been observed experimentally. It is found that the difference between models and data grows exponentially and reaches 100% after six wave periods.     

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.     

Practical Formulas for the Dimensioning of Air Valves

On the basis of theoretical considerations, the technical note presents two practical formulas for the dimensioning of air valves when filling a pipe with water. One is to be used for designing air valves on the basis of the maximum allowed water hammer overpressures; the other when the maximum in pipe water velocity is set. The reliability of these formulas was tested with a numerical model based on the same hypothesis, which was in turn verified with experimental tests.     

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.     

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.     

Explicit Schemes for Dam-Break Simulations

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.     

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.     

Semianalytical Model for Solid Concentration in Turbulent Pipe Flows with Dispersed Particles

This paper presents a semianalytical model for the radial distribution of the solid concentration in a fully developed vertical turbulent pipe two-phase flow. A simplified momentum equation in the radial direction for solid phase in a two-phase flow with dilute suspended particles was first obtained. A linear empirical closure relation for the mean gas and solid velocities along the pipe direction was constructed using published experimental data. By incorporating the closure relation, an analytical solution to the simplified solid momentum equation with the appropriate boundary conditions at the pipe center and wall was obtained. The results from this semianalytical model are able to describe the core-annulus phenomenon commonly occurring in two-phase turbulent pipe flows. Very good agreements were found between the model predictions and published experimental data.     

3D Numerical Simulation for Water Flows and Sediment Deposition in Dam Areas of the Three Gorges Project

This paper presents a three-dimensional (3D) mathematical model for suspended load transport in turbulent flows. Based on the stochastic theory of turbulent flow proposed by Dou, numerical schemes of Reynolds stresses for anisotropic turbulent flows are obtained. Instead of a logarithmic law, a specific wall function is used to describe the velocity profile close to wall boundaries. The equations for two-dimensional suspended load motion and sorting of bed material have been improved for a 3D case. Numerical results are in good agreement with the measured data of the Gezhouba Project. The present method has been employed to simulate sediment erosion and deposition in the vicinity of the Three Gorges Dam. The size distribution of the deposits and bed material, and flow and sediment concentration at different times and elevations, are predicted. The results agree well with the observations in physical experiments. Thus, a new method is established for 3D simulation of sediment motion in the vicinity of dams.     

Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flows

A total variation diminishing Runge Kutta discontinuous Galerkin finite-element method for two-dimensional depth-averaged shallow water equations has been developed. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally as efficient as comparable finite-volume schemes. For smooth parts of the solution, the scheme is second order for linear elements and third order for quadratic shape functions both in time and space. Shocks are usually captured within only two elements. Several steady transcritical and transient flows are investigated to confirm the accuracy and convergence of the scheme. The results show excellent agreement with analytical solutions. For investigating a flume experiment of supercritical open-channel flow, the method allows very good decoupling of the numerical and mathematical model, resulting in a nearly grid-independent solution. The simulation of an actual dam break shows the applicability of the scheme to nontrivial bathymetry and wave propagation on a dry bed.     

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 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.     

长距离输水工程瞬态水力计算及处理措施设计

运用Hammer软件对长距离输水工程瞬态进行动态模拟,清晰地呈现了瞬态发生时,随着压力波的推移,管道系统因为管道负压而引起弥合水锤的过程.针对模拟情况,采取在输水管道系统的合理位置增设三功能排气阀、抗水锤泄压阀,在泵站出口设置抗水锤气压罐等联合措施,有效地消除瞬态条件下在管道系统内产生的水锤,保证输水管道系统瞬态的安全...     

Application of Three-Dimensional Hydrodynamic Model for Lake Okeechobee

A 3D hydrodynamic and heat transport model was developed for Lake Okeechobee. Continuity, momentum, and temperature transport equations were solved. Dynamically coupled transport equations for turbulent kinetic energy and turbulent scale also were solved. The numerical scheme used spatial finite differencing and a three-time-level, external-internal mode splitting procedure. A 28-day calibration was conducted, using measured bathymetry, rainfall, relative humidity, total solar radiation, wind velocity, inflow, and outflow data. During the calibration period, little rainfall occurred, and lake water levels receded. Water surface elevation, horizontal velocities, and temperature were computed. Agreement between observed and simulated values was based on graphical comparisons, minimizing mean absolute and root-mean-square errors, and spectral analysis. Comparisons showed that the model reproduced general observed trends and short-term fluctuations. The model's heat transport and turbulence closure schemes behaved as expected with regard to water column stratification and mixing. Simulation accuracy may potentially be improved by adding wind-wave and vegetation resistance algorithms to the model.     

Computer and Experimental Models of Transient Flow in a Pipe Involving Backflow Preventers

Transient flow in a pipe was studied using both experimental and computer models. In the present study, three different numerical models: The method of characteristics model, the axisymmetrical model, and the implicit scheme model are utilized and compared. Experiments for transient flow in a simple pipeline have been conducted to verify the results from the computer models. It was found that head loss coefficient for the 1D models, such as the method of characteristics model and the implicit scheme model, should be much bigger than the Darcy-Weisbach frictional coefficient. Experiments for transient flow with the backflow preventer in a pipe were conducted. Results show that backflow preventer serves as a strong damper to the water hammer generated by the hydraulic transients. Numerical investigation simulating a backflow preventer in transient flow has been performed in this study. It was found that different values of head loss coefficient should be applied for the upstream and downstream of backflow preventer. All of the numerical models were compared with the experiments. The results of different computer models developed in the present study agree well with the experimental data.     

海水绕流扬矿管边界层分析

介绍不同雷诺数下海水绕流深海扬矿管的流动情况,分析管面形成层流及湍流边界层的分离过程,比较二者分离点的位置及压差阻力情况,分析绕流阻力和举力的形成过程、计算方法、影响因素及相应的减阻措施。     

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.     

Quasi-2D Model for Unsteady Flow in Pipe Networks

A quasi-two-dimensional model for unsteady-flow analysis in pipes and pipe networks is presented. The turbulence model is based on the mixing length hypothesis in the turbulent zone and on Newton's law in the viscous sublayer. An expression of the mixing length in terms of the Reynolds number and an expression of the parameter of logarithmic law of the wall in terms of the friction Reynolds number are found from Nikuradse's experimental data. An implicit numerical scheme for the integration of the equations is proposed to overcome the limitations of the explicit schemes. Uniqueness of the head and continuity of discharge are considered at the junctions. The results of both a quasi-steady 1D model and a quasi-2D model are compared with results from a laboratory network. For these experimental runs, the comparisons show that the average relative errors on the maximum head oscillations are 19.1% with the 1D model and 8.6% with the quasi-2D model; those on the minimum oscillations are 19.2% with the 1D model and 5.3% with the quasi-2D model. The latter model is in better agreement because it takes into account the velocity profile, thus allowing for a more accurate evaluation of the shear stress.