Simple analytical model for depth-averaged velocity in meandering compound channels

A simple but applicable analytical model is presented to predict the lat- eral distribution of the depth-averaged velocity in meandering compound channels. The governing equation with curvilinear coordinates is derived from the momentum equation and the flow continuity equation under the condition of quasi-uniform flow. A series of experiments are conducted in a large-scale meandering compound channel. Based on the experimental data, a magnitude analysis is carried out for the governing equation, and two lower-order shear stress terms are ignored. Four groups of experimental data from different sources are used to verify the predictive capability of this model, and good predictions are obtained. Finally, the determination of the velocity parameter and the limitation of this model are discussed.

Numerical research on flow and thermal transport in cooling pool of electrical power station using three depth-averaged turbulence models

This paper describes a numerical simulation of thermal discharge in the cooling pool of an electrical power station, aiming to develop general-purpose computational programs for grid generation and flow/pollutant transport in the complex domains of natural and artificial waterways. Three depth-averaged two-equation closure turbulence models, k-ε, k- w, and k- w, were used to close the quasi three-dimensional hydrodynamic model. The k- w model was recently established by the authors and is still in the testing process. The general-purpose computational programs and turbulence models will be involved in a software that is under development. The SIMPLE （Semi-Implicit Method for Pressure-Linked Equation） algorithm and multi-grid iterative method are used to solve the hydrodynamic fundamental governing equations, which are discretized on non-orthogonal boundary-fitted grids with a variable collocated arrangement. The results calculated with the three turbulence models were compared with one another. In addition to the steady flow and thermal transport simulation, the unsteady process of waste heat inpouring and development in the cooling pool was also investigated.

Comparison of depth-averaged concentration and bed load flux sediment transport models of dam-break flow

This paper presents numerical simulations of dam-break flow over a movable bed. Two different mathematical models were compared: a fully coupled formulation of shallow water equations with erosion and deposition terms(a depth-averaged concentration flux model), and shallow water equations with a fully coupled Exner equation(a bed load flux model). Both models were discretized using the cell-centered finite volume method, and a second-order Godunov-type scheme was used to solve the equations. The numerical flux was calculated using a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored(HLLC). A novel slope source term treatment that considers the density change was introduced to the depth-averaged concentration flux model to obtain higher-order accuracy. A source term that accounts for the sediment flux was added to the bed load flux model to reflect the influence of sediment movement on the momentum of the water. In a onedimensional test case, a sensitivity study on different model parameters was carried out. For the depth-averaged concentration flux model,Manning's coefficient and sediment porosity values showed an almost linear relationship with the bottom change, and for the bed load flux model, the sediment porosity was identified as the most sensitive parameter. The capabilities and limitations of both model concepts are demonstrated in a benchmark experimental test case dealing with dam-break flow over variable bed topography.

A hybrid forecasting model for depth-averaged current velocities of underwater gliders

In this paper,we propose a hybrid forecasting model to improve the forecasting accuracy for depth-averaged current velocities(DACVs) of underwater gliders.The hybrid model is based on a discrete wavelet transform(DWT),a deep belief network(DBN),and a least squares support vector machine(LSSVM).The original DACV series are first decomposed into several high-and one low-frequency subseries by DWT.Then,DBN is used for high-frequency component forecasting,and the LSSVM model is adopted for low-frequency subseries.The effectiveness of the proposed model is verified by two groups of DACV data from sea trials in the South China Sea.Based on four general error criteria,the forecast performance of the proposed model is demonstrated.The comparison models include some well-recognized single models and some related hybrid models.The performance of the proposed model outperformed those of the other methods indicated above.

Flow and transport simulation of Madeira River using three depth-averaged two-equation turbulence closure models

This paper describes a numerical simulation in the Amazon water system, aiming to develop a quasi-three-dimensional numerical tool for refined modeling of turbulent flow and passive transport of mass in natural waters. Three depth-averaged two-equation turbulence closure models, k-ε,k-w, and k-w, were used to close the non-simplified quasi-three-dimensional hydrodynamic fundamental governing equations. The discretized equations were solved with the advanced multi-grid iterative method using non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, have also been numerically investigated. The three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence models such as the k-w model, with a higher order of magnitude of the turbulence parameter, provide a possibility for improving computational precision.

THE EQUATIONS OF COMPLETE DEPTH-AVERAGED TURBULENCE MODEL IN GENERAL ORTHOGONAL COORDINATES

For shallow water flow, the depth-averaged governing equations are derived by depth-averaging of the mean equations for three-dimensional turbulent flows. The influences of free water surface and of topography of river bed are taken into account.The depth-averaged equations of k-εturbulence model are also obtained. Because it Accounts for the three-dimensional effect, this model is named as the complete Depth-averaged model.The boundaries of natural water bodies are usually curved.In this work, the derived equations in Cartesian coordinates are transformed into orthogonal coordinates. The obtained equations can be applied directly to numerical computation of practical problems.

A NEW DEPTH-AVERAGED TWO-EQUATION (K-) TURBULENT CLOSURE MODEL AND ITS APPLICATION TO NUMERICAL SIMULATION FOR A RIVER

The paper mainly focuses on describing the modification made to a new depth-averaged two-equation turbulent closure model based on the revised κ-ω model recently. In the case of side discharged jets with tempera- ture difference and transverse current, the new model has been investigated numerically in detail. As a practical example of application to use the new model, the side discharge of the cooling water from three outlets into a natu- ral river on one bank has been simulated, and the geomorphic variation under water has been treated suitably. Two depth-averaged models, and have been used, the later was the unique one up to the present. Emphasis is placed on the comparative research with different models under the same computational conditions. It has been verified that if the discharged flow rates are relatively small, when the pollutant plume in the near and transitional zons is predicted, the agreement with experimental and field data simulated by the model is better than by the model or other methods commonly used in engineering.

A NEW DEPTH-AVERAGED TWO-EQUATION(?) TURBULENT CLOSURE MODEL

Based on the revised turbulent k-w model, a new depth-averaged two-equation closure model is offered in this paper. Through numerical investigation, it is shown that the - model has the same order of accuracy as the - model in predicting the eddy regions produced by side temprature jets with transverse currents, but the newly developed model is better than the - model in predicting the jet width,

Study of pollutant transport in depth-averaged flows using random walk method

The random walk particle tracking(RWPT)method is compared with the Eulerian methods in investigating pollutant transport in depth-averaged flows.As a typical representative of the Eulerian model with high performance,the MacCormack scheme with Total Variation Diminishing modification(TVD-Mac)is selected for comparison.Solute concentration is simulated in four case studies.First,both numerical models have been tested in two idealized cases and compared against analytical solutions.Numerical dissipation is observed for TVD-Mac model where the concentration changes abruptly,especially under the circumstances of low resolution and misalignment between the flow direction and grid orientation.On the contrary,simulations by the random walk model achieve higher accuracy in both cases and are free of fictitious oscillations in the vicinity of sharp concentration gradients.Then,the solute oscillation along a one-dimensional hypothetical tidal estuary is simulated,with the RWPT accurately con serving mass and suffering less rm merical diffusion compared with the Eulerian method.Finally,the process of pollutant transport in a Yangzte River reach is predicted by the RWPT.The longitudinal dispersion coefficient DL is calculated accordingly.It is compared favorably with the theoretical/empirical formulae,indicating the validity of the RWPT in solving complex natural problems.

Computing Depth-averaged Flows Using Boundary-fitted Coordinates and Staggered Grids

A depth averaged nonlinear k ε model for turbulent flows in complex geometries has been developed in a boundary fitted coordinate system. The SIMPLEC procedure is used to develop an economical discrete method for staggered grids to analyze flows in a 90° bend. This paper describes how to change a program in rectangular coordinate into a boundary fitted coordinate. The results compare well with experimental data for flow in a meandering channel showing the efficiency of the model and the discrete method.

THE INTERACTIONS BETWEEN WAVE-CURRENTS AND OFFSHORE STRUCTURES WITH CONSIDERATION OF FLUID VISCOSITY

Study of the how held around the large scale offshore structures under the action of waves and viscous currents is of primary importance for the scouring estimation and protection in the vicinity of the structures. But very little has been known in its mechanism when the viscous effects is taken into consideration. As a part of the efforts to tackle the problem, a numerical model is presented for the simulation of the how held around a fixed vertical truncated circular cylinder subjected to waves and viscous currents based on the depth-averaged Reynolds equations and depth-averaged k-epsilon turbulence model. Finite difference method with a suitable iteration defect correct method and an artificial open boundary condition are adopted in the numerical process. Numerical results presented relate to the interactions of a pure incident viscous current with Reynolds number Re = 10(5), a pure incident regular sinusoidal wave, and the coexisting of viscous current and wave with a circular cylinder, respectively. Flow fields associated with the hydrodynamic coefficients of the fixed cylinder, as well as corresponding free surface profiles and wave amplitudes, are discussed. The present method is found to be relatively straightforward, computationally effective and numerically stable for treating the problem of interactions among waves, viscous currents and bodies.

Numerical Simulation of Residual Circulation due to Bottom Roughness Variability Under Tidal Flows in A Semi-Enclosed Bay

Nowadays there are some chronic serious environmental problems, such as eutrophication, blue tide and so on, in a complicated coastal zone or a semi-enclosed bay, because the water exchanges between an inner bay and an outer sea is weak compared with the supply of contaminant. Under this situation, a method to improve the water quality by 3-dimensional small unsymmetrical structures has been proposed by Komatsu et al. In this paper, several numerical simulations of the tidal current and concentration for various arrangements of bottom roughness in a semi-enclosed model bay are carfled out with a depth-averaged 2-D numerical model. The model is solved by the hybrid finite analytic method with nonstaggered grid. And the SIMPLES algorithm with Rhie and Chow＇ s momentum interpolation technique is used for the simulation. The effect of Komatsu＇ s method for water purification is examined by numerical simulation. The result of numerical experiment indicates that it is possible to generate a new tidal residual current and to activate a tidal exchange by bottom roughness arrangement only.

Two dimensional analytical solution for a partially vegetated compound channel flow

The theory of an eddy viscosity model is applied to the study of the flow in a compound channel which is partially vegetated. The governing equation is constituted by analyzing the longitudinal forces acting on the unit volume where the effect of the vegetation on the flow is considered as a drag force item, The compound channel is divided into 3 sub-regions in the transverse direction, and the coefficients in every region＇s differential equations were solved simultaneously. Thus, the analytical solution of the transverse distribution of the depth-averaged velocity for uniform flow in a partially vegetated compound channel was obtained. The results can be used to predict the transverse distribution of bed shear stress, which has an important effect on the transportation of sediment. By comparing the analytical results with the measured data, the analytical solution in this paper is shown to be sufficiently accurate to predict most hydraulic features for engineering design purposes.

Prediction of velocity distribution in straight open-channel flow with partial vegetation by singular perturbation method

A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation and the momentum exchange between the vegetation and non-vegetation zones are considered. The depth-averaged streamwise velocity is solved by the singular perturbation method, while the Reynolds stress is calculated based on the results of the streamwise velocity. Comparisons with the experimental data indicate that the accuracy of prediction is significantly improved by introducing a term for the secondary current in the model. A sensitivity analysis shows that a sound choice of the secondary current intensity coefficient is important for an accurate prediction of the depth-averaged streamwise velocity near the vegetation and non-vegetation interfaces, and the drag force coefficient is crucial for predictions in the vegetation zone.

Numerical modeling of bed deposition in rapid flow-like landslides:a case study of the Gaolou landslide in Shaanxi Province,China

Some flow-like landslides tend to lose materials while moving on a relatively dry sliding surface.This phenomenon is called bed deposition.In contrast to the bed entrainment phenomenon,bed deposition is relatively poorly understood.Therefore,an improved depth-averaged model is proposed to quantify this phenomenon.The deposition depth is calculated according to the momentum conservation of the deposited mass,and the rheological property of the sliding mass on the bottom is modified considering an abrupt increase in the depth of the sliding surface after deposition.Utilizing the proposed model,the Gaolou landslide,a typical flowlike landside occurring on October 6 th,2006 with an obvious bed deposition phenomenon in Shaanxi Province of China,is simulated to investigate the influence of bed deposition on its propagation process.The results indicate that the proposed model can effectively depict the bed deposition phenomenon in the Gaolou landslide.Bed deposition dissipates part of the kinetic energy of this landslide;thus,the simulated debris inundation area would likely be overestimated when this effect is neglected.On the other hand,the thin liquefied layer formed by the bed deposition process reduces the friction energy dissipation,contributing to the high mobility of this landslide.

Numerical simulation of transport and dispersion of side-discharged sewage in tidal flow

Based on N - S equations, the depth-averaged stress - flux algebraic model is used to simu late the anisotropic transport and dispersion when hot water or pollutants are side-discharged into large water they. In this model the depth-averaged continuity, momentum and concentration equations are employed, and the tide is asymmetric. The results show the changes of velocity field and pollutant con centration with time and space during one tidal period.

Analytical solutions for transverse distributions of stream-wise velocity in turbulent flow in rectangular channel with partial vegetation

The theory of poroelasticity is introduced to study the hydraulic properties of the steady uniform turbulent flow in a partially vegetated rectangular channel. Plants are assumed as immovable media. The resistance caused by vegetation is expressed by the theory of poroelasticity. Considering the influence of a secondary flow, the momentum equation can be simplified. The momentum equation is nondimensionalized to obtain a smooth solution for the lateral distribution of the longitudinal velocity. To verify the model, an acoustic Doppler velocimeter （ADV） is used to measure the velocity field in a rectangular open channel partially with emergent artificial rigid vegetation. Comparisons between the measured data and the computed results show that the method can predict the transverse distributions of stream-wise velocities in turbulent flows in a rectangular channel with partial vegetation.

Simulation of Artificial Grain Feeding in Order to Reach Dynamical Bed Stabilisation Along the River Rhine

The Wesel-Xanten stretch of the fiver Rhine between km-812.5 and km-821.5 is one of the reaches where strong erosion leads to high maintenance efforts conceming navigability.In order to improve the naviga- tion conditions without aggravating the flood protection,but also ensuring that the ecological system of the river is not damaged,investigations of the morphodynamical processes in connection with artificial grain feeding ac- tivities have to be carried out by Federal Waterways Engineering and Research...

Saint-Venant Equations and Friction Law for Modelling Self-Channeling Granular Flows: From Analogue to Numerical Simulation

Rock avalanches are catastrophic events involving important granular rock masses (>106 m3) and traveling long distances. In exceptional cases, the runout can reach up to tens of kilometers. Even if they are highly destructive and uncontrollable events, they give important insights to understand interactions between the displaced masses and landscape conditions. However, those events are not frequent. Therefore, the analogue and numerical modelling gives fundamental inputs to better understand their behavior. The objective of the research is to understand the propagation and spreading of granular mass released at the top of a simple geometry. The flow is unconfined, spreading freely along a 45° slope and deposit on a horizontal surface. The evolution of this analogue rock avalanche was measured from the initiation to its deposition with high speed camera. To simulate the analogue granular flow, a numerical model based on the continuum mechanics approach and the solving of the shallow water equations was used. In this model, the avalanche is described from a eulerian point of view within a continuum framework as single phase of incompressible granular material. The interaction of the flowing layer with the substratum follows a Mohr-Coulomb friction law. Within same initial conditions (slope, volume, basal friction, height of fall and initial velocity), results obtained with the numerical model are similar to those observed in the analogue. In both cases, the runout of the mass is comparable and the size of both deposits matches well. Moreover, both analogue and numerical modeling gave same magnitude of velocities. In this study, we highlighted the importance of the friction on a flowing mass and the influence of the numerical resolution on the propagation. The combination of the fluid dynamic equation with the frictional law enables the self-channelization and the stop of the granular mass.

Effects of vegetation patch density on flow velocity characteristics in an open channel

The aquatic plants are often found in natural rivers,and they affect the channel flow structure significantly.To study the effects of the vegetation density a(the frontal area per volume)on the flow velocity characteristics,rigid bamboo circular cylinders are chosen as the model emergent vegetation.In the experiments,the density of the vegetation takes various values while all other flow parameters are kept constant.A 3-D acoustic Doppler velocimeter(ADV)is used to measure the local flow velocities for different vegetation densities.The results show that the existence of the vegetation patch leads to an increase of the depth-averaged velocity on the right and left sides behind the vegetation patch,and it increases monotonically with the vegetation density.For different vegetation densities,the lateral distribution of the stream-wise velocity behind the vegetation patch follows approximately an S-shaped profile when a≤60 m-1,and a logarithmic profile if a>60 m-1.The vertical distribution of the stream-wise velocity along the channel varies with the vegetation density and also follows an S-shaped distribution in a certain range,with the locations and the ranges being affected by the vegetation density.