Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects

Simulating the transient processes in complex water transmission system is time-consuming, and improving computational efficiency by means of parallelization on CPU clusters or even faster GPU platform is demanded. This paper proposes an approach to accelerate the transient simulations of open channel and pipe combined flows on single GPU chip. The Saint-Venant equations for open channel flows is solved by using the method of characteristics (MOC), whose inherent parallelism can be well exploited by GPU implementations in the thread-level parallelism structure of Compute Unified Device Architecture (CUDA). The sub-processes, including open channel computation, pipe flow computation and connecting boundary treatment, are implemented by different kernels. The procedures are first verified by analyzing the parallel computation efficiency of hydraulic transient processes in an open channel. Then the transient processes of a practical engineering project, which involves both open channel flow and pressurized pipe flow, are simulated. The GPU kernels are found to be memory bandwidth bounded, and the proposed single chip GPU parallel can achieve up to hundreds of speedup ratios compared to the sequential counterpart on single CPU chip.

A one-dimensional mathematical model of water and sediment flow in open channels is proposed based on the forces influencing a water stream and bottom and stream sediments. The equations of water and sediment flow are closed by the equations of continuity of stream, velocity of particle motion in a stream, and the equation of balance of kinetic energy and moving particles: joint calculation of stream hydraulic characteristics and sediment discharge is performed. Hydraulic resistances are retrieved not from the Chezi formula, but based on the balance of forces and kinetic energy. A sediment discharge is calculated from velocities and the number of moving solid particles instead of empirical relationships whose accuracy is usually low. The model is verified against the data of experiments in hydraulic flumes with glass and sand bottom. Comparison with independent data in the glass flume in a wide range of water discharges and bottom inclinations showed a high accuracy of calculation of hydraulic characteristics (relative error is less than 4%). The experimental data showed that the accuracy of sediment discharge calculated by the model exceeds the accuracy of the calculation using traditional empirical formulas.

SUMMARY The Saint Venant equations for modelling flow in open channels are solved in this paper, using a variety of total variation diminishing (TVD) schemes. The performance of second- and third-order-accurate TVD schemes is investigated for the computation of free-surface flows, in predicting dam-breaks and extreme flow conditions created by the river bed topography. Convergence of the schemes is quantified by comparing error norms between subsequent iterations. Automatically calculated time steps and entropy corrections allow high CFL numbers and smooth transition between different conditions. In order to compare different approaches with TVD schemes, the most accurate of each type was chosen. All four schemes chosen proved acceptably accurate. However, there are important differences between the schemes in the occurrence of clipping, overshooting and oscillating behaviour and in the highest CFL numbers allowed by a scheme. These variations in behaviour stem from the different orders and inherent properties of the four schemes. © 1998 John Wiley & Sons, Ltd. In this paper, numerical schemes for solving the Saint Venant equations, which model bulk channel flow, are presented. This set of hyperbolic equations yields discontinuous solutions, which can be difficult to represent accurately without the use of a modern shock-capturing method. There are problems with the high level of truncation errors when using first-order upwind schemes and with the oscillatory behaviour of most higher-order schemes. However, coupling a total variation diminishing (TVD) interpolation with an appropriate Riemann solver, yields a high-order-accurate scheme which numerically captures discontinuities with sharp corners and avoids unrealistic oscillations. The myriad of TVD schemes may be categorised into algebraic and geometric approaches. Further subdivisions of the algebraic schemes are symmetric, upwind and predictor-corrector. One scheme has been chosen from each of these forms of TVD schemes. The schemes used were a second-order symmetric, an upwind scheme called the modified flux, the two step TVD‐McCormack and the third-order MUSCL representing the geometric approach. Comparison of numerical schemes has a long tradition; here we summarise our experience with these numerical schemes and suggestions for further work are provided.

In this study the fractional governing equations for diffusion wave and kinematic wave approximations to unsteady open-channel flow in prismatic channels in fractional time-space were developed. The governing fractional equations were developed from the mass and motion conservation equations in order to provide a physical basis to these equations. A fractional form of the resistance formula for open-channel flow was also developed. Detailed dimensional analyses of the derived equations were then performed in order to ensure dimensional consistency of the derivations. It is shown that these fractional equations of unsteady open-channel flow are fundamentally nonlocal in terms of nonlocal fluxes. The derived fractional governing equations of diffusion wave and kinematic wave open-channel flow can accommodate both the long-memory nonlocal behavior of open-channel flow as well as its local, finite memory behavior, as is numerically demonstrated in the accompanying paper by the authors.

Open-channel flows are very common and important flows occurring in both natural and artificial watercourses such as rivers, canals and drainages, and irrigation and navigation channels. Correct prediction and simulation of the flows play a crucial role in channel design, river management and flood control. In theory, the flows can be efficiently and accurately described by the shallow-water equations in most cases. In fact, the solution to these equations has turned out to be a very successful tool to model different free-surface flows such as tides, tsunami, dam breaks, dyke breaks, bores and broken waves. Although there are many numerical methods available for solving the shallow-water equations (e.g. the finite-element method), they are often complicated and computationally demanding. Over the last decade, the lattice Boltzmann method (LBM) has been developed as a very attractive numerical method in computational fluid dynamics. The method is extremely simple and efficient with its very easy treatment of boundary conditions and natural parallel processing, and is suitable for complex flows within complicated geometries. Consequently, the LBM has been developed for shallow-water equations with or without turbulence modelling (LABSWETM and LABSWE, respectively). This paper presents both methods and their application to typical open-channel flows. The results demonstrate the power, potential, applicability and accuracy of the LBM in simulations of many open-channel flows and also indicate that the method needs to be improved for some open-channel flow problems.

The Saint Venant equations are the equations governing open channel flow when a flood propagates. To obtain an analytical solution, the Saint Venant equations are approximately transformed to a nonlinear diffusion equation. The numerical result of the nonlinear diffusion equation explains the nonlinearity of the open channel flow and overland flow and the resulting asymmetry of the water surface profile. When the Saint Venant equations are numerically applied to flood routing in a finite region and finite time, a downstream boundary condition is necessary. Analytical solution of the appropriate nonlinear diffusion equation does not require the downstream boundary condition. The numerical result of the analytical solution of the approximate nonlinear diffusion equation is found to be suitable for simulation of the original nonlinear differential equation and the Saint Venant equations. The linearized diffusion model is found to be restricted in practical applications, and it cannot express nonlinea...

The Reynolds equations of mean turbulent flow in a two-dimensional open channel of arbitrary cross-section have been analyzed. An integral equation for the turbulent hydraulic jump is proposed. In the closure model adopted, the depth-averaged effective normal Reynolds stress is taken proportional to the product of constant eddy viscosity and depth-averaged axial velocity gradient, and the constant of proportionality that is independent of channel geometry. The general theory has been applied to the flows in circular and U-shaped channels. The solutions for sequent depth, hydraulic jump length, roller length and aeration length have been estimated. The comparison of the theory with experimental data of Hager and Stahl for circular and U-shaped channels give very encouraging results.

Modeling rainfall-runoff processes using smoothed particle hydrodynamics with mass-varied particles

In order to simulate the bed scouring around piers, many researchers have developed sophisticated numerical algorithms. In this work, the numerical modeling of souring around cylindrical piers are performed by solution of the shallow water equations flow and depth averaged turbulent model equations coupled with the empirical relations for cohesive and granular beds. The shallow water flow equations include to the depth average continuity and motion and depth average turbulent model equations in horizontal Cartesian coordinate systems. For computation of bed erosion at each time step, the numerical results of turbulent flow model are combined with SRICOS empirical bed erosion empirical relations, which are derived from laboratory experiments. The governing partial differential equations of depth average turbulent flow are converted to discrete form using a vertex base finite volume method suitable for triangular unstructured meshes. Therefore, scouring due to flow around piers with arbitrary shape can be modeled. Several test cases of scouring on the cohesive and sandy beds around circular piers are modeled using the developed numerical model and the computed results are compared with the laboratory measurements reported for scouring in cohesive and granular bed around circular piers by the previous experimental workers. Despite of the simplicity and light computational work load of the introduced algorithm, the computed results show promising agreements with the anticipated trends.

Measurements of the mean and turbulence characteristics in nonuniform open-channel flows were carried out using a 3D acoustic Doppler velocimeter. Both accelerating and decelerating flows were investigated. Analyses based on the Reynolds equation and the continuity equation of 2D open-channel flow show that a flow will be in equilibrium if the pressure-gradient parameter β is a constant at different sections along the flow direction. The experimental data show that all the flows investigated are in equilibrium. The effect of the flow nonuniformity on the mean velocity and turbulence profiles was also examined. The study shows that (1) the log law is still valid for both accelerating and decelerating flows in the inner region. The Coles law can be used for the entire region, but the wake-strength parameter Π depends on the β-value; and (2) the turbulence intensities and the Reynolds stress decrease in accelerating flow and increase in decelerating flow, when compared with those in uniform flow. Finally, using the Reynolds equation and the continuity equation of 2D open-channel flow, the theoretical expressions for the distribution of vertical velocity and the Reynolds stress have been developed. The measured vertical velocity and the Reynolds stress profile compare well with the derived expressions.

A new numerical technique by Chang is described and used to solve the one-dimensional (1D) and two-dimensional (2D) Saint Venant equations. This new technique differs from traditional numerical methods (i.e., finite-difference, finite-element, finite-volume, spectral methods, etc.). Chang's method treats space and time on the same footing, so that space and time are unified—a key characteristic that distinguishes the new method from other techniques. This method is explicit, uses a staggered grid, enforces flux conservation in space and time, and does not require upwinding, flux-splitting, flux limiters, evaluation of eigenvalues, or the addition of artificial viscosity. Furthermore, the scheme is simple, easy to implement (see Appendix I), and can be extended to higher dimensions. First, the new technique, as developed by Chang, is introduced, explained, and applied to the 1D Saint Venant equations. To illustrate its effectiveness, an idealized dam-break and a hydraulic jump in a straight rectangular channel are simulated. The numerical results obtained using the new method are compared with results from other, more conventional techniques of similar complexity, experimental data, and an analytical solution. Next, Chang's scheme is extended for solution of the 2D Saint Venant (i.e., shallow water or depth-averaged) equations. The writers do not follow Chang's 2D development but instead employ Strang's method of fractional steps, which transforms the 2D problem into two 1D problems. The new 2D scheme is applied to an oblique hydraulic jump, and the results are compared with an analytical solution.

A new lattice Boltzmann method to simulate open channel flows with complex geometry described by a conservative form of Saint-Venant equations is developed. The Saint-Venant equations include an original treatment of the momentum equation source term. Concrete hydrostatic pressure thrust expressions are provided for rectangular, trapezoidal and irregular cross-section shapes. A D1Q3 lattice arrangement is adopted. External forces, such as bed friction and the static term, are discretized with a centred scheme. Bounce back and imposed boundary conditions are considered. To verify the proposed model, four cases are carried out: tidal flow over a regular bed in a rectangular cross-section, steady flow in a channel with horizontal and vertical contractions, steady flow over a bump in a trapezoidal channel and steady flow in a non-prismatic channel with friction. Results indicate that the proposed scheme is simple and can provide accurate predictions for open channel flows.

Most existing analytical and mathematical models for sediment-laden flows are based on the governing equations for single-phase flows, and are valid only for low sediment concentration situations. This paper presents an analysis, on the basis of the fundamental equations for fluid-solid two-phase flows, of the velocity and sediment concentration profiles in open-channel flows. A new diffusion equation is established for suspended sediment concentration from a rigorous derivation of the water-sediment mixture's normal velocity in the sense of mass flux conservation. The differences between this new equation and Schmidt's as well as Hunt's are shown to be attributable to the different approximations of sediment velocity. Previous formulations for velocity and sediment concentration distributions are special cases of the present model for low sediment concentration flows. The developed model is extensively tested against available measurements, and satisfactory or fairly good agreement is obtained.

This flow in rivers is concerned with unsteady flow in open channel and it mathematically governed by the Saint Venant equation, using a four-point implicit finite difference scheme. For a one-dimensional applications, the relevant flow parameters are functions of time, and longitudinal positions. Considering the equations for the conservation of mass (continuity) and conservation of momentum. The mathematical method is empirical with the computer revolution, numerical methods are now effective to develop the hydraulic model. Conventional methods used to compute discharge, such as stage-discharge and stagefall discharge relationships has been inadequate. Further research is recommended to include tributaries in rivers.

High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry