Steady Open Channel Flow Research Articles

Suspended Sediments in Environmental Flows: Interpretation of Concentration Profiles Shapes

In environmental flows, field and laboratory measurements of suspended sediments show two kinds of concentration profiles. For coarse sediments, a near-bed upward convex profile is observed beneath the main upward concave profile. In this study, we consider two 1-DV models, namely, the classical advection–diffusion equation (ADE) based on the gradient diffusion model, and the kinetic model. Both need sediment diffusivity, which is related to the eddy viscosity, and an y-dependent β-function (i.e., the inverse of the turbulent Schmidt number). Our study shows that the kinetic model reverts to the classical ADE with an “apparent” settling velocity or sediment diffusivity. For the numerical resolution of the ADE, simple and accurate tools are provided for both the sediment diffusivity and hindered settling. The results for the concentration profiles show good agreement with the experimental data. An interpretation of the concentration profiles is provided by two “criteria” for shapes. The main for steady open-channel flows shows that the shape of the concentration profiles in the Cartesian coordinate depends on the vertical distribution of the derivative of R (the ratio between the sediment diffusivity and the settling velocity of the sediments): dR/dy > −1 for the upward concave concentration profile while dR/dy < −1 for the near-bed upward convex profile. A generalization is proposed for oscillatory flows over sand ripples, where the time-averaged concentration profiles in the semi-log plots are interpreted by a relation between the second derivative of the logarithm of the concentration and the derivative of the product between the sediment diffusivity and an additional parameter related to the convective sediment entrainment process.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Entropy Wake Law for Streamwise Velocity Profiles in Smooth Rectangular Open Channels.

In narrow open channels, the three-dimensional nature of the flow and the transport momentum from the sidewalls to the central region cause the maximum longitudinal velocity to occur below the water surface. The entropy model is unable to accurately describe the velocities near the free surface when the dip phenomenon exists. The present paper proposes a new dip-modified entropy law for steady open channel flows, which consists of three additional terms: the first one similar to Coles’ function; the second one linearly proportional to the logarithmic distance from the free surface; and the third one depending on the cubic correction near the maximum velocity. The validity of the new model was tested on a set of laboratory measurements carried out in a straight rectangular flume with smooth boundaries and for different values of water discharge, bottom slope, and aspect ratio. A detailed error analysis showed good agreement with the data measured through the present research and a more accurate prediction of the velocity-dip-position compared with the one evaluated through the original entropy model. In addition, the modified entropy wake law matched very well with other literature data collected in rectangular cross-sections with different flow conditions.

Predicting the vertical low suspended sediment concentration in vegetated flow using a random displacement model

Based on the Lagrangian approach, this study proposes a random displacement model (RDM) to predict the concentration of suspended sediment in vegetated steady open channel flow. Validation of the method was conducted by comparing the simulated results by using the RDM with available experimental measurements for uniform open-channel flows. The method is further validated with the classical Rouse formula. To simulate the important vertical dispersion caused by vegetation in the sediment-laden open channel flow, a new integrated sediment diffusion coefficient is introduced in this study, which is equal to a coefficient β multiplying the turbulent diffusion coefficient. As such, the RDM approach for sandy flow with vegetation was established for predicting the suspended sediment concentration in low-sediment-concentration flow with both the emergent and submerged vegetation. The study shows that the value of β for submerged vegetation flow is larger than that for emergent vegetation flow. The simulated result using the RDM is in good agreement with the available experimental data, indicating that the proposed sediment diffusion coefficient model can be accurately used to investigate the sediment concentration in vegetated steady open channel flow.

Wavelet Coherency Structure in Open Channel Flow

Many studies based on single-point measurement data have demonstrated the impressive ability of wavelet coherency analysis to catch the coherent structures in the wall-bounded flows; however, the question of how the events found by the wavelet coherency analysis relate to the features of the coherent structures remains open. Time series of velocity fields in x–y plane of the steady open channel flow was obtained from a time-resolved particle imaging velocimetry system. The local wavelet spectrum found shows that one of the main energetic scales in open channel flows is 3h motions. The wavelet coherent coefficients of u and v series from the same measurement points successfully detected the occurrence and the scale of these 3h motions, and the phase angle indicates their inside velocity structure is organized by the Q2 and Q4 events. The wavelet coherency analysis between different measurement points further reveals the incline feature of the 3h scale motions. All the features of this 3h motion found by the wavelet coherency analysis coincide well with the flow field that is created by the passing of hairpin packets.

Simulation of laboratory experiments for vortex dynamics at shallow tidal inlets using the fine resolution environmental hydrodynamics (Frehd) model

In this paper, we present the results of numerical simulations matched to laboratory experiments of tidal, starting-jet vortices forming at idealized, barotropic inlets. The laboratory experiments are for sinusoidally varying inflow/outflow along a wide, flat basin. Inlet configurations include simple inlets with negligible channel length as well as a longer inlet channels through a long barrier island and jettied inlet. Laboratory observations are made using particle image velocimetry, and we analyze the laboratory and numerical model velocity fields to compare starting-jet trajectory, diameter, and total circulation. The numerical simulations use the Fine Resolution Environmental Hydrodynamic model, solving the three-dimensional, hydrostatic, barotropic equations of motion. Our goal is to identify the minimum model attributes necessary to match the vortex properties. This is accomplished using a third-order upwind scheme for advection, a constant bottom friction drag coefficient, and a one-equation turbulence model for transport of turbulent kinetic energy. The optimal value of the bottom drag coefficient matches the time-average value at the basin inflow/outflow boundaries, and the optimal dissipation coefficient, $$C_{me}$$, was an order of magnitude smaller than literature values for a steady open-channel flow. We show that the drag coefficient mainly affects the advection speeds of the outer coastal waters, which control the penetration of the tidal vortices away from the inlet, and the lower turbulence dissipation rate results from the low turbulent dissipation in the large, shallow starting-jet vortices.

Practical application of the coupled DDA-SPH method in dynamic modeling for the formation of landslide dam

Landslides along river margins can cause permanent or temporal landslide dams and dammed-lakes, threatening people’s life and properties. Investigation of the formation process for the landslide dam is crucial for emergency response and mitigation planning. However, the formation process modeling for a practical case usually involves large scale and complicates geometry. To overcome the computation complexity and promote efficiency, a series of practical techniques have been proposed. Firstly, an Open Channel Model with Steady Flow (OCMSF) has been developed to naturally produce a river flow. Then, a three-stage simulation strategy has been proposed to fulfill the large-scale practical modeling. In specific, stage 1 generated a steady open channel flow using SPH method. In stage 2, the DDA method is used to simulate landslide movement until the mass reaches the river. In stage 3, the formation process with landslide-river interaction is realized using a coupled DDA-SPH method. The formation process of the Yangjiagou landslide dam was selected as the practical application. Simulation results showed the Yangjiagou landslide reached the river with a front velocity of 22 m/s in around 8 s and formed a dam with estimated volume of 500,000 m3, which is consistent with the site investigation. It is thus demonstrated the applicability and performance of the coupled method and numerical techniques in modeling practical landslide dam case.

On the steady two-dimensional open channel flow

This note concerns the open channel flow problem. The flow is assumed to be steady (with constant discharge), two-dimensional (with non-hydrostatic pressure distribution), without both wave breaking and sediment transport. The flow problem consists in finding the location of the free surface, and the velocity and pressure fields. The problem solution is obtained by a hybrid model which incorporates the dissipation in potential flows. In order to set up the model, this note proposes a new Picard iteration for the potential velocity field. By coupling approximate expressions for the velocity field to dynamic equations, the flow problem is reduced to one of solving a nonlinear ordinary differential equation. To illustrate the model validation, numerical simulations are performed for open channel flow over curved bottom, and the free overfall problem. In terms of free surface location and bottom pressure profile, the obtained results are in good agreement with data in literature.

A general mixture model for sediment laden flows

A mixture model for general description of sediment-laden flows is developed based on an Eulerian-Eulerian two-phase flow theory, with the aim at gaining computational speed in the prediction, but preserving the accuracy of the complete two-fluid model. The basic equations of the model include the mass and momentum conservation equations for the sediment-water mixture, and the mass conservation equation for sediment. However, a newly-obtained expression for the slip velocity between phases allows for the computation of the sediment motion, without the need of solving the momentum equation for sediment. The turbulent motion is represented for both the fluid and the particulate phases. A modified k-ε model is used to describe the fluid turbulence while an algebraic model is adopted for turbulent motion of particles. A two-dimensional finite difference method based on the SMAC scheme was used to numerically solve the mathematical model. The model is validated through simulations of fluid and suspended sediment motion in steady open-channel flows, both in equilibrium and non-equilibrium states, as well as in oscillatory flows. The computed sediment concentrations, horizontal velocity and turbulent kinetic energy of the mixture are all shown to be in good agreement with available experimental data, and importantly, this is done at a fraction of the computational efforts required by the complete two-fluid model.

L’analyse isotopique du plomb : un outil utile en santé au travail en cas de multi-expositions

A mixture model for general description of sediment-laden flows is developed based on an Eulerian-Eulerian two-phase flow theory, with the aim at gaining computational speed in the prediction, but preserving the accuracy of the complete two-fluid model. The basic equations of the model include the mass and momentum conservation equations for the sediment-water mixture, and the mass conservation equation for sediment. However, a newly-obtained expression for the slip velocity between phases allows for the computation of the sediment motion, without the need of solving the momentum equation for sediment. The turbulent motion is represented for both the fluid and the particulate phases. A modified k-ε model is used to describe the fluid turbulence while an algebraic model is adopted for turbulent motion of particles. A two-dimensional finite difference method based on the SMAC scheme was used to numerically solve the mathematical model. The model is validated through simulations of fluid and suspended sediment motion in steady open-channel flows, both in equilibrium and non-equilibrium states, as well as in oscillatory flows. The computed sediment concentrations, horizontal velocity and turbulent kinetic energy of the mixture are all shown to be in good agreement with available experimental data, and importantly, this is done at a fraction of the computational efforts required by the complete two-fluid model.

Theoretical analysis and numerical simulation of mechanical energy loss and wall resistance of steady open channel flow

The mechanical energy loss and the wall resistance are very important in practical engineering. These problems are investigated through theoretical analysis and numerical simulation in this paper. The results are as follows. (1) A new mechanical energy equation for the total flow is obtained, and a general formula for the calculation of the mechanical energy loss is proposed. (2) The general relationship between the wall resistance and the mechanical energy loss for the steady channel flow is obtained, the simplified form of which for the steady uniform channel flow is in consistent with the formula used in Hydraulics deduced by π theorem and dimensional analysis. (3) The steady channel flow over a backward facing step with a small expansion ratio is numerically simulated, and the mechanical energy loss, the wall resistance as well as the relationship between the wall resistance and the mechanical energy loss are calculated and analyzed.

一方向流中で渦励振動する水平円柱による発電力特性

The characteristics of power generation using non-linear vortex-excited vibration of a circular cylinder in steady open channel flow were estimated with considering the variation of a) the amplification of hydrodynamic forces acting on the vortex-excited vibrating cylinder in comparison with the hydrodynamic forces acting on the test cylinder stationary mounted and b) the amplitudes of transverse vibration of the test cylinder were evaluated quantitatively. The maximum values of power generation using vortex-excited vibration of the test cylinder were appeared in the range of 2< Ksa <4.

Form-induced stress in non-uniform steady and unsteady open channel flow over a static rough bed

This paper presents a series of 18 steady and 45 unsteady flow experiments over an artificial, rough surface, obtained by a novel molding technique for gravel beds. Velocity data are obtained by a Particle Image Velocimetry (PIV) system. The vertical distributions of the form-induced stresses 〈u˜2〉 and −〈u˜w˜〉 are compared for all experiments. Furthermore, quadrant maps are drawn to analyze the near bed spatial velocity fluctuations u˜ and w˜ in a vertical plane parallel to the flow direction. All experiments were done under non-uniform flow. It is shown that (1) the qualitative shape of the vertical form-induced stress distributions 〈u˜2〉 and −〈u˜w˜〉 is independent from the discharge, as stated in the literature, as well as from the unsteadiness of the flow. (2) The magnitude of the form-induced stress 〈u˜2〉 and −〈u˜w˜〉 in the roughness layer increases with increasing discharge for non-uniform flow, while uniform flow experiments in the literature report the form-induced stresses to be independent from the discharge. A hypothesis is, therefore, proposed, that the independence of the form-induced stress magnitude from the discharge is only valid for uniform flow. (3) Quadrant maps, describing the distribution of the spatial fluctuations u˜ and w˜ in a two-dimensional plane are independent from discharge and unsteadiness of the flow.

One-dimensional modelling of curvilinear free surface flow: generalized Matthew theory

ABSTRACTThe potential flow theory is a good mathematical approximation to various problems in the fields of free surface open channel and groundwater flows. In both fields, second-order equations are commonly derived using perturbation methods to account for the non-hydrostatic pressure distribution. However, an alternative technique is Picard's iteration approach, used by Matthew for steady open channel flows. Regretfully, this technique was not expanded to unsteady potential flow, limiting its full impact. The purpose of this work is to generalize the theory of Matthew to unsteady potential free surface flow, both for open channels and groundwater. The new development is an alternative to perturbation techniques, highlighting the relevance of Matthew's work. To illustrate the widespread problems to which the theory applies, simulations are made for a dambreak wave and for flow in sloping and curved aquifers. In both cases one-dimensional results are in good agreement with two-dimensional data.

Manning's Roughness through the Entropy Parameter for Steady Open Channel Flows in Low Submergence

Entropy model application to river presents relevant aspects for theoretical and practical issues useful for cross section velocity distribution. The ratio between mean and maximum velocities is dependent on local branch morphology and remains quite uniform among similar location sections. Recent studies propose the formulation of Manning's roughness, n, based on such ratio and the ratio between the positions where the velocity is zero and maximum, y0/ymax, but falling for low depth. Basing on steady flows experiences, the paper analyses the dependence of n on entropy parameter, M, for low depths, proposing an equation for y0/ymax knowing slope and submergence.

Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

SUMMARYA new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles.To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved.Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Fine sediment dynamics in unsteady open-channel flow studied with acoustic and optical systems

In order to simulate fine sediment dynamics over an armored bed in a tidal river, unsteady accelerating, then steady open-channel flow over a movable (but not moving) coarse gravel bed (D50=5.5mm) was studied. A layer of fine sediment (D50=120μm) was placed on the coarse gravel bed. The thickness of the fine sediment layer on the gravel bed was varied between 4 and 6mm, but it was found that the thickness of the layer had no effect on the results. Quasi-instantaneous profiles of velocity and sediment concentration were taken simultaneously and co-located. An Acoustic Doppler Velocity Profiler (ADVP) was combined with Particle Tracking Velocimetry (PTV) for suspended sediment particle tracking. Measurements resolved turbulence scales. During the final phase of the accelerating flow range, fine sediment suspension from the bed started in packets and rapidly created a ripple pattern that remained nearly stationary. Thereafter, vortex shedding produced most of the sediment suspension into the water column in the form of events or packets, making suspension intermittent. Simultaneously, sediment particles rolled along the bed following the ripple structure, thus slowly advancing the ripple pattern in the direction of the flow without altering ripple geometry. Fine sediment particles and hydrogen bubbles were used individually or combined as flow tracers in the acoustic measurements. When used individually, hydrogen bubbles provided full depth flow and backscattering information, whereas sediment particles traced only the lower layers of the flow, indicating sediment suspension. When both tracers were combined, hydrogen bubbles could only be distinguished from sediment particles when results at two different acoustic carrier frequencies were compared. The intermittency was observed in the backscattering of the acoustic system. The event structure in fine sediment suspension is seen by the PTV method. PTV velocity vectors varied in speed and orientation, were organized in large coherent packets, mainly in the near bed layers, but also extended well above the bed. The two methods provide complementary information. ADVP measurements allow long time series analysis, whereas most of the spatial details seen in the PTV results cannot be resolved in the ADVP measurements.

Steady free-surface flow in porous media: generalized Dupuit–Fawer equations

Steady free-surface flow in open channels may be analysed by the reduction of the two-dimensional (2D) potential flow problem to a quasi-2D potential flow approach by inclusion of streamline curvature and inclination effects following Fawer's theory. Flow of water in open channels and groundwater flow with a free surface can be characterized by potential functions and stream functions both obeying Laplace's 2D equation, provided there is isotropy and homogeneity. It is thus natural to expect the transfer of ideas from the open channel to the groundwater flow. A novel approximate model based on Fawer's theory for steady open channel flows with curved streamlines, coupled with Darcy's law, is proposed for groundwater flow with a free seepage surface. The seepage surface is described by a second-order differential equation which is applied to flow through both rectangular and trapezoidal dams, resulting in good agreement with the full 2D potential flow methods and test data.

Jean-Baptiste Bélanger: hydraulic engineer and academic

Jean-Baptiste Bélanger (1790–1874) worked as a hydraulic engineer at the beginning of his career. He developed the backwater equation to calculate the free-surface profile of gradually varied, steady open channel flow. He also introduced the concept of critical flow and the numerical technique called the direct step method. Later, as an academic staff member at the leading French engineering schools (Ecole Centrale des Arts et Manufactures, Ecole des Ponts et Chaussées and Ecole Polytechnique), he developed a new university curriculum in mechanics and wrote several textbooks including a seminal text in hydraulic engineering. His influence on his contemporaries was considerable, and his name is written on the border of one of the four façades of the Eiffel Tower. Bélanger's leading role demonstrated the dynamism of practising engineers at the time, and his contributions paved the way to many significant works in hydraulics and fluid dynamics during the late nineteenth and early twentieth centuries.

Reply to comment on “steady open channel flows with curved streamlines: the Fawer approach revised”

In this reply to a comment to (Environ Fluid Mech 10(3):297–310 [1]), the comparison between the Dressler curved streamline equations and Fawer’s theory is further expanded by momentum considerations. The importance of Fawer’s theory for steady curved open channel flow is thereby highlighted, both from energy and momentum considerations.