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.     

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.     

Transcritical Flow due to Channel Contraction

The singular point is a physical phenomenon consistent with critical flow conditions, and, consequently, the real control section of a water surface profile. A general method to study the location, type, and water surface slope of a singular point is described. This method, in addition to improving the classic gradually-varied flow theory, can be used for the design of channel contractions involving transcritical flows, as in the case of chute spillways and inlets to river diversion tunnels. The method is explained in the case of a chute spillway and verified against experimental data recorded in a Venturi channel.     

Well-Balanced Scheme between Flux and Source Terms for Computation of Shallow-Water Equations over Irregular Bathymetry

Although many numerical techniques such as approximate Riemann solvers can be used to analyze subcritical and supercritical flows modeled by hyperbolic-type shallow-water equations, there are some difficulties in practical applications due to the numerical unbalance between source and flux terms. In this study, a revised surface gradient method is proposed that balances source and flux terms. The new numerical model employs the MUSCL–Hancock scheme and the HLLC approximate Riemann solver. Several verifications are conducted, including analyses of transcritical steady-state flows, unsteady dam break flows on a wet and dry bed, and flows over an irregular bathymetry. The model consistently returns accurate and reasonable results comparable to those obtained through analytical methods and laboratory experiments. The revised surface gradient method may be a simple but robust numerical scheme appropriate for solving hyperbolic-type shallow-water equations over an irregular bathymetry.     

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.     

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.     

Upwind Conservative Scheme for the Saint Venant Equations

An upwind conservative scheme with a weighted average water-surface-gradient approach is proposed to compute one-dimensional open channel flows. The numerical scheme is based on the control volume method. The intercell flux is computed by the one-sided upwind method. The water surface gradient is evaluated by the weighted average of both upwind and downwind gradients. The scheme is tested with various examples, including dam-break problems in channels with rectangular and triangular cross-sections, hydraulic jump, partial dam-break problem, overtopping flow, a steady flow over bump with hydraulic jump, and a dam-break flood case in a natural river valley. Comparisons between numerical and exact solutions or experimental data demonstrated that the proposed scheme is capable of accurately reproducing various open channel flows, including subcritical, supercritical, and transcritical flows. The scheme is inherently robust, stable, and monotone. The scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients or discontinuities in the solution.     

Evaluation of Flow Resistance in Smooth Rectangular Open Channels with Modified Prandtl Friction Law

Flow resistance in open channels is usually estimated by applying the approach that is developed originally for pipe flows. Such estimates may be useful for engineering applications but always differ from measurements to some extent. This paper first summarizes empirical approaches that have been proposed in the literature to reconcile the resistance difference. These include various modifications of the pipe friction for applications to rectangular ducts and open channel flows. An improved friction equation is then derived for evaluating flow resistance of smooth rectangular open channels. Comparisons are made with experimental data reported by previous researchers and those collected in the present study. It is shown that the new proposed equation is applicable for both narrow and wide channels and is more accurate than those available in the literature.     

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.     

Hydraulic Elements Chart for Pipe Flow Using New Definition of Hydraulic Radius

The Manning formula is used routinely to calculate the mean velocity of uniform flow. Although this empirical formula is effective when applied to uniform flow in simple rectangular or trapezoidal cross sections, the roughness coefficient of the formula is variable when examining flow in a pipe that is partially full. Thus, the coefficient must be altered depending on the relative depth of fluid in the pipe. As this seems to be due to the definition of the hydraulic radius, a new definition of hydraulic radius is proposed here that was used to calculate a hydraulic elements chart for flow in pipes with a constant roughness coefficient. The results of the calculations showed very good agreement with Camp’s chart. Furthermore, with adjustment of the “free-surface weight factor,” this method was also capable of expressing other hydraulic elements charts reported previously. This new definition of hydraulic radius can also be applied to flow in simple cross sections and may be developed further for use with compound channel flows in future studies.     

Depth-Averaged Shear Stress and Velocity in Open-Channel Flows

Turbulent momentum and velocity always have the greatest gradient along wall-normal direction in straight channel flows; this has led to the hypothesis that surplus energy within any control volume in a three-dimensional flow will be transferred toward its nearest boundary to dissipate. Starting from this, the boundary shear stress, the Reynolds shear stress, and the velocity profiles along normal lines of smooth boundary may be determined. This paper is a continuous effort to investigate depth-average shear stress and velocity in rough channels. Equations of the depth-averaged shear stress in typical open channels have been derived based on a theoretical relation between the depth-averaged shear stress and boundary shear stress. Equation of depth mean velocity in a rough channel is also obtained and the effects of water surface (or dip phenomenon) and roughness are included. Experimental data available in the literature have been used for verification that shows that the model reasonably agrees with the measured data.     

Multilayer Depth-Averaged Flow Model with Implicit Interfaces

Using a depth-averaged model to obtain the velocity and pressure distributions in the vertical direction is difficult. A multilayer model is an option that can be used to improve on the depth-averaged model. However, the unknown flow depth needs to be predicted first and then divided into layers as an input for the multilayer model. An improved multilayer model is proposed here by introducing an implicit layer dividing interfaces that are associated with the flow velocity and pressure distribution. The formulation of interfaces also applies to boundary faces: Free surface and channel beds. Therefore, each flow layer behaves like that in the classical depth-averaged model. Subsequently the governing equations are also simplified due to the vanishing terms related to interfacial flow exchanges. This improved model has been satisfactorily applied to steady flow simulations in three cases: Flow over a slope transition from mild to steep, from steep to mild, and over a trapezoidal weir. The results demonstrate the efficiency and validity of the proposed models to simulate open channel flows with bed slope changes.     

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.     

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.     

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.     

Fuzzy Analysis of Slope-Area Discharge Estimates

When velocity estimates of channel flows are not available, the slope-area method is widely used to make discharge estimates. The accuracy of such estimates depends on many factors but is generally believed to be low. In this study, fuzzy set analyses were used to compute the distribution of slope-area discharge estimates, with the distribution very similar to that assessed using statistical methods used in hydrology. The effects of errors in channel roughness, channel width, channel side-slopes, and flow depth on the accuracy of discharges were assessed. The large errors possible in channel roughness are a primary source of error in discharge estimates. Channel incision and vegetal growth are shown to significantly influence the accuracy of slope-area discharge estimates and the length of time before a slope-area rating curve should be recalculated. Records as short as 10 years can be seriously biased due to a failure to account for the effects of incision or vegetal growth.     

Efficient Quasi-Two-Dimensional Model for Water Hammer Problems

Quasi-two-dimensional models for turbulent flows in water hammer are necessary for advancing the understanding of flow behavior in pipe transient; conducting detailed investigation of the fate of transient-induced contamination; and validating one-dimensional water hammer models. An existing quasi-two dimensional numerical model for turbulent water hammer flows has the attributes of being robust, consistent with the physics of wave motion and turbulent diffusion, and free from the inconsistency associated with the enforcement of the no slip condition while neglecting the radial velocity at boundary elements, such as valves and reservoirs. However, this scheme is computationally intensive making it unsuitable for practical pipe systems or for conducting numerical experiments. This paper addresses the efficiency and stability of this existing scheme. In particular, algebraic manipulations show that the original scheme can be decoupled into two tridiagonal systems, one for piezometric head and radial flux and another for axial velocity. This decoupling is the reason for the high efficiency of the modified scheme. The original and proposed schemes are applied to a pipe–reservoir–valve system. It is found that, for the same spatial and temporal discretization, both schemes are of equal accuracy. However, significant saving in computer execution time is achieved by using the modified scheme. Application of the modified scheme to pipes of realistic dimensions and wavespeeds (length 35.2 km, diameter 200 mm, and wave speed 1000 m/s) takes only a few minutes to execute. This small execution time requirement makes the current quasi-two-dimensional model suitable for application to practical water hammer problems. The stability domain of the proposed scheme is established using the Von Neumann method.     

Solution for the Closed-Loop Problem in Pressurized Multipipe Systems

The fundamental background of the solution for the steady-state flow in pressurized water closed-loop pipe systems, without any reservoirs in-between, is presented in this paper. The use of the steady-state mass balance and energy equations to calculate discharges and heads in this type of hydraulic system leads to an undetermined problem. The way to solve this indeterminacy is to consider an additional continuity equation associated with the difference between initial and final conditions, taking into account fluid compressibility and pipe-wall deformability. A complete formulation is derived considering pressure and temperature changes in the hydraulic system. Simplified formulae are presented for isothermal flows in simple systems and multiple closed-loops with pipes in series and in parallel. This problem can also be solved by a pseudotransient analysis technique applied to steady-state conditions. Proposed solutions for this problem are applied to steady-state flows and tested for different system configurations.     

Formulas for Friction Factor in Transitional Regimes

This note concerns variations of the friction factor in the two transitional regimes, one between laminar and turbulent flows and the other between fully smooth and fully rough turbulent flows. An interpolation approach is developed to derive a single explicit formula for computing the friction factor in all flow regimes. The results obtained for pipe flows give a better representation of Nikuradse’s experimental data, in comparison with other implicit formulas available in the literature. Certain modifications are also made for applying the obtained friction formula to open-channel flows.     

Experimental Study of a Right-Angled End Junction between a Pipe and an Open Channel

This paper presents an experimental study of a junction between a closed conduit and an open channel. This study was undertaken to explore hydraulic properties of outlets of subsurface drainage or sewage networks into an open air stream during flood events. Experiments were conducted in a laboratory flume, with a main rectangular channel joined at right angle to a lateral circular pipe. Both branches were supplied with independent flow rates and downstream water level was controlled by an adjustable weir. Several flow patterns were identified, combining free-surface and pressurized flows. Transitions between these flow patterns, as well as changes in water level or energy, in response to the modifications of experimental variables, were studied and could be linked to known properties of single channels, single pipes, and homogeneous junctions. Transitions between free-surface and pressurized pipe flow appeared to be strongly dependent on the whole set of experimental variables and the pipe longitudinal slope. This work contributes to a better knowledge of hydraulic and hydrologic key processes for point source discharging.