Flow Resistance and Momentum Flux in Compound Open Channels

New formulations are presented for flow resistance and momentum flux in compound open channels. As implemented in the St. Venant equations, these formulations facilitate a physically enhanced approach to evaluating conveyance, roughness, stage-discharge relationship, and unsteady flood routing in compound open channels. An analysis using steady flow data from the well-controlled experiments at the large-scale Flood Channel Facility, HR Wallingford, demonstrates the ability of the present approach to properly resolve the discontinuity of overall roughness across the main-channel bankfull level. Also, the proposed formulations are shown to be conducive to obviating the long-standing computational difficulty in unsteady flood routing due to small flow depths over flat and wide floodplains. The present work should find general applications in one-dimensional computation of river flows.     

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.     

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.     

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.     

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.     

Evaluation of Unsteady Wall Shear Stress by Zielke’s Method

The method used in the classical paper by Zielke to estimate the unsteady component of shear stress in unsteady pipe flows is revisited. It is found that the method is undesirably sensitive to the size of the integration time step. The sensitivity is shown to be caused dominantly by the first term in the integration when inadequate allowance is made for the infinite value of the weighting function. A simple method of avoiding the error without requiring the use of small grid sizes is presented.     

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.     

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.     

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 of Periodically Unsteady Axisymmetric Jet

To date, the majority of studies on stability of axisymmetric jets have been completed under the assumption of steady mean flow. Yet, many of the natural and man-made flows that are modeled by these jets can have an inherent unsteadiness; the effects of which on the stability and transition have not been determined. Moreover, controlled unsteadiness can be used to control stability and possibly the transition to turbulence. In this note, the effects of periodic variations of the mean flow on the stability of axisymmetric jets are examined. The problem is treated analytically. The results show that the governing equations and dispersion relation for the unsteady jet can be reduced to those governing the steady jet with a time transformation. It is shown that the periodic variations in the mean flow cause amplitude and phase modulations of the unstable modes. The implications of the modulations on the subsequent transition stages are discussed.     

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.     

Numerical Model for Channel Flow and Morphological Change Studies

In this paper a depth-integrated 2D hydrodynamic and sediment transport model, CCHE2D, is presented. It can be used to study steady and unsteady free surface flow, sediment transport, and morphological processes in natural rivers. The efficient element method is applied to discretize the governing equations, and the time marching technique is used for temporal variations. The moving boundaries were treated by locating the wet and dry nodes automatically in the cases of simulating unsteady flows with changing free surface elevation in channels with irregular bed and bank topography. Two eddy viscosity models, a depth-averaged parabolic model and a depth-averaged mixing length model, are used as turbulent closures. Channel morphological changes are computed with considerations of the effects of bed slope and the secondary flow in curved channels. Physical model data have been used to verify this model with satisfactory results. The feasibility studies of simulating morphological formation in meandering channels and flows in natural streams with in-stream structures have been conducted to demonstrate its applicability to hydraulic engineering research∕design studies of stream stabilization and ecological quality among other problems.     

Two-Dimensional Depth-Averaged Flow Modeling with an Unstructured Hybrid Mesh

An unstructured hybrid mesh numerical method is developed to simulate open channel flows. The method is applicable to arbitrarily shaped mesh cells and offers a framework to unify many mesh topologies into a single formulation. A finite-volume discretization is applied to the two-dimensional depth-averaged equations such that mass conservation is satisfied both locally and globally. An automatic wetting-drying procedure is incorporated in conjunction with a segregated solution procedure that chooses the water surface elevation as the main variable. The method is applicable to both steady and unsteady flows and covers the entire flow range: subcritical, transcritical, and supercritical. The proposed numerical method is well suited to natural river flows with a combination of main channels, side channels, bars, floodplains, and in-stream structures. Technical details of the method are presented, verification studies are performed using a number of simple flows, and a practical natural river is modeled to illustrate issues of calibration and validation.     

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.     

Conservative Scheme for Numerical Modeling of Flow in Natural Geometry

A numerical model is proposed to compute one-dimensional open channel flows in natural streams involving steep, nonrectangular, and nonprismatic channels and including subcritical, supercritical, and transcritical flows. The Saint-Venant equations, written in a conservative form, are solved by employing a predictor-corrector finite volume method. A recently proposed reformulation of the source terms related to the channel topography allows the mass and momentum fluxes to be precisely balanced. Conceptually and algorithmically simple, the present model requires neither the solution of the Riemann problem at each cell interface nor any special additional correction to capture discontinuities in the solution such as artificial viscosity or shock-capturing techniques. The resulting scheme has been extensively tested under steady and unsteady flow conditions by reproducing various open channel geometries, both ideal and real, with nonuniform grids and without any interpolation of topographic survey data. The proposed model provides a versatile, stable, and robust tool for simulating transcritical sections and conserving mass.     

Role of Artificial Dissipation Scaling and Multigrid Acceleration in Numerical Solutions of the Depth-Averaged Free-Surface Flow Equations

A numerical model is developed for solving the depth-averaged, open-channel flow equations in generalized curvilinear coordinates. The equations are discretized in space in strong conservation form using a space-centered, second-order accurate finite-volume method. A nonlinear blend of first- and third-order accurate artificial dissipation terms is introduced into the discrete equations to accurately model all flow regimes. Scalar- and matrix-valued scaling of the artificial dissipation terms are considered and their effect on the accuracy of the solutions is evaluated. The discrete equations are integrated in time using a four-stage explicit Runge–Kutta method. For the steady-state computations, local time stepping, implicit residual smoothing, and multigrid acceleration are used to enhance the efficiency of the scheme. The numerical model is validated by applying it to calculate steady and unsteady open-channel flows. Extensive grid sensitivity studies are carried out and the potential of multigrid acceleration for steady depth-averaged computations is demonstrated.     

Shallow Turbulent Flows by Video Imaging Method

Shallow turbulent flows were produced in a tank of small thickness to study the friction effects on large-scale turbulent motion of small depth. The tank was constructed of two parallel walls. The space between the parallel walls (4.4, 1.57, and 0.59 cm) was small compared with the height (107 cm) and the width (212 cm) of the tank, and was varied during the experiments for different friction effects. Turbulent flows were produced by the injection of water in the form of starting jets into the tank filled with water. The large-scale turbulent flow in the small space between the walls of the tank is confined to essentially two-dimensional motion, and the motion is retarded by the force of friction. Dye was injected with the source fluid as the tracer for the highly unsteady and quasitwo-dimensional turbulent motion. From the initiation of the turbulent motion at the source to the final interaction of the jets with the tank bottom, the entire sequence of events was recorded by a pair of video cameras. The depth-averaged concentration of the dye was analyzed using the recorded video images.     

Analysis of the Friction Term in the One-Dimensional Shallow-Water Model

The numerical simulation of unsteady open channel flows is very commonly performed using the one-dimensional shallow-water model. Friction is one of the relevant forces included in the momentum equation. In this work, a generalization of the Gauckler-Manning friction model is proposed to improve the modeling approach in cases of dominant roughness, unsteady flow, and distorted cross-sectional shapes. The numerical stability conditions are revisited in cases of dominant friction terms and a new condition, complementary to the basic Courant-Friedrichs-Lewy condition, is proposed. Some test cases with measured data are used to validate the quality of the approaches.     

Modeling of One-Dimensional, Unsteady, Free-Surface, and Pressurized Flows

The utilization of mathematical models in hydraulic engineering for the analysis of one-dimensional, unsteady free-surface, and pressurized flows is discussed, with an emphasis when the models performed well as well as when they did not. For illustration purposes, the applications to a number of real-life projects are presented, outlining limitations, successes, and failures.