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.     

Global Predictive Real-Time Control of Sewers Allowing Surcharged Flows

A global predictive real-time control strategy minimizing overflow volumes from combined sewers during rainfalls is presented. For an optimal use of controlled sewer transport and storage capacities, the proposed strategy allows surcharged flows. Flows and piezometric heads in the sewer are computed according to flow inputs by a hydraulic simulation model. The optimal operation of the regulators controlling these flow inputs is determined on a finite control horizon using the generalized reduced gradient optimization algorithm. The control strategy was applied to the 23 rain events that occurred during the summer of 1989 on the urban area drained by the Marigot interceptor in Laval, Canada. In this application, the admitted intensity of surcharges was varied to assess this parameter impact on total overflow volumes. A comparison between performances of the proposed strategy and a local reactive control was also carried out. Results obtained indicate that the global predictive control can reduce overflow volumes during rainstorms and that this reduction may be improved by allowing surcharged flows.     

Self-Cleansing Sewer Design Based on Sediment Transport Principles

The need for sewers to carry sediment has been recognized for many years. Traditionally, a minimum “self-cleansing” velocity was specified and, although this approach had been successful in many cases, it was appreciated that a minimum velocity—unrelated to the characteristics and concentration of the sediment or to other aspects of the hydraulic behavior of the sewer—could not properly represent the ability of sewer flows to transport sediments. During the 1980’s, sediment transport theory had been increasingly applied to the design of sewers, particularly in major interceptor sewer schemes. But, in the absence of any universally recognized guidance, the design methodologies and criteria adopted were developed on a project-by-project basis, building on the increasing experience and understanding of the subject of the designer. In recognition of this, a research project was initiated by the U.K.’s Construction Industry Research and Information Association to develop a new design methodology for sewers, which would take advantage of the available knowledge (mostly laboratory derived) on sediment mobility and the effects of sediment deposition on the hydraulic performance of sewers. This paper describes the main findings of the project and summarizes the recommended design guidance.     

Junction and Drop-Shaft Boundary Conditions for Modeling Free-Surface, Pressurized, and Mixed Free-Surface Pressurized Transient Flows

A junction and drop-shaft boundary conditions (BCs) for one-dimensional modeling of transient flows in single-phase conditions (pure liquid) are formulated, implemented and their accuracy are evaluated using two computational fluid dynamics (CFD) models. The BCs are formulated in the case when mixed flows are simulated using two sets of governing equations, the Saint-Venant equations for the free-surface regions and the compressible water hammer equations for the pressurized regions. The proposed BCs handle all possible flow regimes and their combinations. The flow in each pipe can range from free surface to pressurized flow and the water depth at the junction or drop shaft can take on all possible levels. The BCs are applied to the following three cases: (1) a three-way merging flow; (2) a three-way dividing flow; and (3) a drop shaft connected to a single-horizontal pipe subjected to a rapid variation of the water surface level in the drop shaft. The flow regime for the first two cases range from free surface to pressurized flows, while for the third case, the flow regime is pure pressurized flow. For the third case, laboratory results as well as CFD results were used for evaluating its accuracy. The results suggest that the junction and drop-shaft BCs can be used for modeling transient free-surface, pressurized, and mixed flow conditions with good accuracy.     

Improved Simulation of Flow Regime Transition in Sewers: Two-Component Pressure Approach

Operational problems and system damage have been linked to the flow regime transition between free surface and pressurized flow in rapidly filling stormwater and combined sewer systems. In response, emphasis has been placed on the development of numerical models to describe hydraulic bores and other flow phenomena that may occur in these systems. Current numerical models are based on rigid column analyses, shock-fitting techniques, or shock-capturing procedures employing the Preissmann slot concept. The latter approach is appealing due to the comparative simplicity, but suffers from the inability to realistically describe subatmospheric full-pipe flows. A new modeling framework is proposed for describing the flow regime transition utilizing a shock-capturing technique that decouples the hydrostatic pressure from surcharged pressures occurring only in pressurized conditions, effectively overcoming the cited Preissmann slot limitation. This new approach exploits the identity between the unsteady incompressible flow equations for elastic pipe walls and the unsteady open-channel flow equations, and the resulting numerical implementation is straightforward with only minor modifications to standard free surface flow models required. A comparison is made between the model predictions and experimental data; good agreement is achieved.     

Supercritical Flow across Sewer Manholes

The hydraulics of supercritical flow across manholes in sewers is explored using systematic experimentation. Due to the expansion at the manhole entrance an in-manhole wave is generated. Further, at the downstream manhole end, flows with a sufficiently large filling ratio impinge on the wall and lead to a so-called swell. In addition, due to shock wave generation in the downstream sewer, a sewer wave is generated. The heights and locations of these three waves were determined in terms of basic hydraulic quantities. More importantly, the capacity of the manhole and the downstream sewer under wave action was quantified. It was found that in order for free surface flow to be maintained the common design standard for sewers with a supercritical approach flow have to be revised. These implications have to be accounted for in future designs.     

Accurate Two-Dimensional Simulation of Advective-Diffusive-Reactive Transport

The present paper presents an accurate numerical algorithm for the simulation of 2D solute∕heat transport by unsteady advection-diffusion-reaction. The model was specifically developed for the study of convective exchange processes in a cross section of lakes and ponds, when the currents are predominantly driven by density (temperature) gradients. The numerical scheme is based on the split-operator approach, in which advection and diffusion with chemical∕biological kinetic processes are calculated separately at each time step. Special attention is given to the advection operator in order to avoid excessive numerical damping or oscillations, as well as to the source∕sink term, which may cause numerical instability and inaccuracy if improperly treated. The model has been verified on standard test problems for a wide range of Courant, Fourier, Péclet, and Thiele numbers, and found to produce stable results of high accuracy.     

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.     

Channel Routing in Open-Channel Flows with Surges

Using numerical models for the purpose of channel-routing calculation has been well accepted in engineering practice. However, most traditional models fail to predict the transcritical flows because of numerical instability. This paper presents two high-resolution, shock-capturing schemes for the simulation of 1D, rapidly varied open-channel flows. The present schemes incorporate the method of characteristics to deal with the unsteady boundary conditions. Also, the Strang-type splitting operator is used to include the effects of bottom slope and friction terms. To assess the performance of the proposed algorithms, several steady and unsteady problems are simulated to verify the accuracy and robustness in capturing strong shocks in open-channel flows. Furthermore, the results of dynamic flood routing and steady routing are compared to demonstrate the risk of using steady routing for flood mitigation.     

Stability and Accuracy of Weighted Four-Point Implicit Finite Difference Schemes for Open Channel Flow

The unsteady motion of streams, with a free surface, has been described by a system of equations that were proposed by Saint-Venant in 1871. These are universally known as “Saint-Venant’s equations.” Weighted four-point implicit finite difference schemes are largely used for their numerical solution in the case of one-dimensional flow. For these models, stability, dissipation, and dispersion are first investigated by looking at the truncation error and then by using Fourier’s classic linear analysis. Variations of the space and time weighting coefficients, the Courant number, the Froude number, and the frictional resistance term are examined in this paper. In particular, instabilities are analyzed that are due to the progressive accumulation of dispersion errors, which are the consequence of an increasing frictional resistance term. However, in this case the numerical scheme requires a dissipation mechanism. Computational suggestions are given for this mechanism.     

Evaluation of Advective Schemes for Estuarine Salinity Simulations

Several advective transport schemes are considered in the context of two-dimensional scalar transport. To review the properties of these transport schemes, results are presented for simple advective test cases. Wide variation in accuracy and computational cost is found. The schemes are then applied to simulate salinity fields in South San Francisco Bay using a depth-averaged approach. Our evaluation of the schemes in the salinity simulation leads to some different conclusions than those for the simple test cases. First, testing of a stable, but nonconservative Eulerian-Lagrangian scheme does not produce accurate results, showing the importance of mass conservation. Second, the conservative schemes that are stable in the simulation reproduce salinity data accurately independent of the order of accuracy of each scheme. Third, the leapfrog-central scheme was stable for the model problems but not stable in the unsteady, free surface computations. Thus, for the simulation of salinity in a strongly dispersive setting, the most important properties of a scalar advection scheme are stability and mass conservation.     

Longitudinal Dispersion due to Surcharged Manhole

New data describing the longitudinal dispersion of a solute tracer due to a surcharged manhole structure are presented. The increase in both travel time and dispersion caused by this structure is quantified over ranges of discharge and surcharge. Discharge variations exhibit clear trends, whereas surcharge variations are less evident. The results are presented as both Taylor dispersion coefficients and values of parameters used in aggregated dead zone modeling. The limitations of using surcharged averaged parameters with each technique for describing mixing due to a surcharged manhole structure are illustrated. The study shows that the presence of manhole structures within urban drainage schemes has a significant effect on both the travel time and mixing processes of a solute. It is suggested that these processes should be included for the accurate modeling of urban drainage schemes.     

High-Resolution Method for Modeling Hydraulic Regime Changes at Canal Gate Structures

A method for modeling flow regime changes at gate structures in canal reaches is presented. The methodology consists of using an approximate Riemann solver at the internal computational nodes, along with the simultaneous solution of the characteristic equations with a gate structure equation at the upstream and downstream boundaries of each reach. The conservative form of the unsteady shallow-water equations is solved in the one-dimensional form using an explicit second-order weighted-average—flux upwind total variation diminishing (TVD) method and a Preissmann implicit scheme method. Four types of TVD limiters are integrated into the explicit solution of the governing hydraulic equations, and the results of the different schemes were compared. Twelve possible cases of flow regime change in a two-reach canal with a gate downstream of the first reach and a weir downstream of the second reach, were considered. While the implicit method gave smoother results, the high-resolution scheme—characteristic method coupling approach at the gate structure was found to be robust in terms of minimizing oscillations generated during changing flow regimes. The complete method developed in this study was able to successfully resolve numerical instabilities due to intersecting shock waves.     

Three-Dimensional Unsteady RANS Modeling of Discontinuous Gravity Currents in Rectangular Domains

Discontinuous gravity currents in rectangular channels are modeled numerically by solving the 3D unsteady Reynolds-averaged Navier-Stokes (URANS) equations closed with a buoyancy corrected low-Reynolds number (LRN) k-ε model using second-order accurate finite-volume numerics. It is shown that, on moderately fine computational meshes (with ～ 106 grid nodes) and with careful modeling of the near-wall flow, the URANS model can capture the essential large-scale 3D dynamics of gravity current flows, which previously had been resolved only by DNS and/or large-eddy simulation (LES) on very fine computational meshes. These 3D dynamics include the onset of the well-known lobe-and-cleft instability at the current head, the onset of large-scale Kelvin-Helmholtz billows at the head of the gravity current, and the breakdown of the interfacial billows in the rear part of the current head due to intense three-dimensional mixing. The computed results underscore the importance of careful modeling of the near wall flow in URANS simulations. The standard k-ε model with wall functions fails to capture the aforementioned complex 3D dynamics, which are only resolved by the LRN k-ε model on grids that resolve the near-wall region. Furthermore, numerical experiments show that including in the simulation the lateral no-slip end walls of the channel has a profound effect on the accuracy of the computed solutions. End-wall effects enhance the three-dimensionality of the flow, result in increased mixing of the dense and the ambient fluids behind the head of gravity currents, and yield results in good agreement with measurements. On the other hand, when end-wall effects are omitted, by imposing periodicity in the spanwise direction, three-dimensional mixing is suppressed and the breakdown of interfacial billows is significantly underestimated. Grid sensitivity studies are also carried out using three successively refined meshes and show that the URANS LRN model yields grid-converged solutions at affordable computational resources. As URANS modeling requires only a fraction of the computational cost of DNS or LES with near-wall resolution, the present results underscore the potential of unsteady statistical turbulence models for predicting and elucidating the physics of gravity current flows in complex geometries and at Reynolds numbers of engineering relevance.     

Numerical Morphological Modeling of Open-Check Dams

Open-check dams are built in mountain streams to control sediment transport during a flood. Sediment passes through them at the lowest discharges, whereas deposition occurs during the highest discharges. Open-check dams are currently designed based mainly on construction experience. Modeling of hydraulics and bed morphology in check dams involves mixed flows (supercritical and subcritical) as well as discontinuities such as hydraulic jumps. In this paper an unsteady coupled numerical mobile-bed model that can tackle rapid varying flows and discontinuities is applied. The numerical technique is based on the classical staggered grids and implicit integration schemes, together with a proper mass and momentum balance. The 1D numerical model is successfully verified with experimental data of slit-check dams. The applicability of the model in the design of open-check dams is also illustrated.     

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.     

Automated Sewer and Drainage Flushing Systems in Cambridge, Massachusetts

This paper summarizes the design of passive automatic flushing systems installed in the City of Cambridge’s storm and sanitary sewer system tributary to the Alewife Brook as part of a $75 million sewer separation program. Grit and debris deposition is severe in the existing combined sewers, storm drains, and sanitary trunk sewers due to the flat topography of the area. This condition is exacerbated by hydraulic constraints imposed on the system’s outlet by the Alewife Brook (shallow stream) and downstream sanitary siphons (again because of the Alewife Brook). The use of pumps to lift flows from sewers and drains to permit self-scouring velocities is prohibitively expensive. To overcome this problem, five automated flushing systems using quick opening (hydraulic operated) gates discharging collected stormwater were constructed in conjunction with downstream collector grit pits covering a distance of 1,604 m for storm drain pipes ranging from 1.4 m circular to 1.2 m by 1.8 m rectangular. New 450 and 600 mm sanitary trunk sewers, 561 m long, will be flushed daily by two flushing systems using spent filtrate water from Cambridge’s water treatment plant recently constructed nearby. The flushing systems are sized to achieve wave velocity of 1 m/s at the end of the flushing segment. The flush vault volumes range from 11 to 40?m3 for the storm drain systems and 6?m3 for the sanitary system. Construction was completed in May 2002 and functional testing of the flushing systems is in progress. Partial test results are reported.     

Shallow Water Wave Propagation in Convectively Accelerating Open-Channel Flow Induced by the Tailwater Effect

Propagation of shallow water waves in viscous open-channel flows that are convectively accelerating or decelerating under gradually varying water surface profiles is theoretically investigated. Issues related to the hydrodynamics of wave propagation in a rectangular open channel are studied: the effect of viscosity in terms of the Manning coefficient; the effect of gravity in terms of the Froude number; wave translation and attenuation characteristics; nonlinearity and wave shock; the role of tailwater in wave propagation; and free surface instability. A uniformly valid nonlinear solution to describe the unsteady gradually varying flow throughout the complete wave propagation domain at and away from the kinematic wave shock as well as near the downstream boundary that exhibits the tailwater effect is derived by employing the matched asymptotic method. Different scenarios of hydraulically spatially varying surface profiles such as M1, M2, and S1 type profiles are discussed. Results from the nonlinear wave analysis are further interpreted and the influence of the tailwater effect is identified. In addition to the nonlinear wave analysis, a linear stability analysis is introduced to quantify the impact from such water surface profiles on the free surface instability. It is shown that the asymptotic flow structure is composed of three distinct regions: an outer region that is driven by gravity and channel resistance; a near wave shock region dominated by the convective inertia, pressure gradient, gravity and channel resistance; and a downstream boundary impact region where the convective inertia, pressure gradient, gravity and channel resistance terms are of importance. The tailwater effect is demonstrated influential to the flow structure, free surface stability, wave transmission mechanism, and hydrostatic pressure gradient in flow.     

Modeling Hydrogen Sulfide Emission Rates in Gravity Sewage Collection Systems

The factors affecting sulfide buildup in gravity sewers are complex, consisting of biological and physical processes, both in the aqueous and the gas phases of the sewer. The rate of each of these processes varies (among other parameters) according to flow characteristics, temperature, and pH. Under fast and turbulent flow conditions, the stripping of hydrogen sulfide into the gas phase may become the dominant process. The paper presents a semiempirical approach to the problem of quantifying hydrogen sulfide emission rates in sewers. Kinetics of hydrogen sulfide emission as a function of hydraulic parameters was measured in the laboratory using methods adopted from flocculation theory. A flocculation unit was used to impart a selected velocity gradient (G) into the water, and sulfide concentration was measured with time. The process was repeated for a number of G values. Regression analysis was then used to fit the rate of hydrogen sulfide emission equation against G. An equation was developed linking G to HL (head loss) in sewers assuming plug flow conditions. The hydraulic model and the kinetic model were linked (via G) to give the desired rate equation for hydrogen sulfide emission along a sewer line. The model was used to predict H2S emission from a uniform flow sewer and the effect of parameters such as pH, sewer slope and degree of fullness was studied. As expected, results show that low pH, high slope, and low degree of fullness enhance emission rates. Reasonable agreement was attained when the model output was compared with measured results from a field test sewer in Virginia, South Africa, under conditions where sulfide stripping was the rate-dominant process.     

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.