Lateral Distribution Of Depth-averaged Velocity Research Articles

Flow resistance in straight gravel bed inbank flow with analytical solution for velocity and boundary shear distribution

ABSTRACT In the present investigation, the roughness characteristics of the gravel bed is tested and modelled for the no-load condition. A straight trapezoidal channel having gravel bed surface of grain size D50 as 13.5 mm for no-load condition is used to carry out the investigation. Depth averaged velocity (DAV) and boundary shear stress (BSS) have been measured for five different depths. An improved model for prediction of friction factor is derived as the function of corresponding hydraulic radius and average diameter of the gravel. This improved model is validated with the models of past investigators. Furthermore, BSS is then calculated using Shiono and Knight Method (SKM), which is dependent on the following hydraulic parameters, such as friction factor f, coefficient of eddy viscosity λ and secondary flow term Г. Since friction factor is modelled for the given channel, same model is used to calculate the value of f over the bed, which tests the reliability of the model while using numerical method. The lateral distribution of DAV and BSS is shown for five different flow depths, which elaborate the bed friction due to the no-load condition. Finally, friction factor model is corroborated on two reaches of the natural river.

Application of the Shiono and Knight Method in asymmetric compound channels with different side slopes of the internal wall

The Shiono and Knight Method (SKM) is widely used to predict the lateral distribution of depth-averaged velocity and boundary shear stress for flows in compound channels. Three calibrating coefficients need to be estimated for applying the SKM, namely eddy viscosity coefficient (λ), friction factor (f) and secondary flow coefficient (k). There are several tested methods which can satisfactorily be used to estimate λ, f. However, the calibration of secondary flow coefficients k to account for secondary flow effects correctly is still problematic. In this paper, the calibration of secondary flow coefficients is established by employing two approaches to estimate correct values of k for simulating asymmetric compound channel with different side slopes of the internal wall. The first approach is based on Abril and Knight (2004) who suggest fixed values for main channel and floodplain regions. In the second approach, the equations developed by Devi and Khatua (2017) that relate the variation of the secondary flow coefficients with the relative depth (β) and width ratio (α) are used. The results indicate that the calibration method developed by Devi and Khatua (2017) is a better choice for calibrating the secondary flow coefficients than using the first approach which assumes a fixed value of k for different flow depths. The results also indicate that the boundary condition based on the shear force continuity can successfully be used for simulating rectangular compound channels, while the continuity of depth-averaged velocity and its gradient is accepted boundary condition in simulations of trapezoidal compound channels. However, the SKM performance for predicting the boundary shear stress over the shear layer region may not be improved by only imposing the suitable calibrated values of secondary flow coefficients. This is because difficulties of modelling the complex interaction that develops between the flows in the main channel and on the floodplain in this region.

Depth-averaged velocity and bed shear stress in unsteady open channel flow over rough bed

Very few studies have been carried out in the past in estimating depth-averaged velocity and bed shear stress in unsteady flow over rough beds. An experiment is thus conducted to investigate the vertical and lateral velocity profiles under unsteady flow conditions in a rough open channel for various flow depths. One hydrogram is repeatedly passed through the rectangular flume with a fixed rigid grass bed. Using micro Pitot tube and Acoustic Doppler Velocimeter (ADV), the flow patterns are investigated at both lateral and longitudinal positions over different cross-sections. For two typical flow depths, the velocities in both the rising limb and falling limb are observed. Hysteresis effect between stage-discharge (h ~ Q) rating curve between rising and falling limbs is illustrated. Lateral distribution of depth-averaged velocity and bed shear stress are plotted at three different cross sections and compared with the steady flow conditions. In falling limb of an unsteady flow case, both depth-averaged velocity and bed shear stress distribution in the central region is higher than that of steady flow case. However, in the rising limb, the bed shear stress of unsteady flow is less than that of steady flow case. Further, in an unsteady flow, the magnitude of depth-averaged velocity is found to increase towards the downstream sections. Along the downstream positions, bed shear stress values increase for lower flow depths and decrease for higher flow depth cases.

Transverse distribution of streamwise velocity in open-channel flow with artificial emergent vegetation

Artificial emergent vegetation can increase flow resistance and water depth to improve navigation in an open channel. An innovative and eco-friendly artificial vegetation system is analyzed in this study. First, a new method is presented to modify the depth-averaged Navier–Stokes equation by incorporating the vegetation drag force term. Two analytical formulas based on theoretical analysis are proposed to predict the lateral distribution of depth-averaged velocity in an open channel with fully and partially artificial vegetation. Then, the analytical formulas are verified by laboratory experiments. The results of the analytical model are consistent with experimental data, thereby proving that the analytical formulas can be used to predict the lateral distribution of streamwise velocity in open-channel flow with artificial emergent vegetation. The presented model can provide a significant theoretical reference for research on open channels with artificial emergent vegetation.

Analytical models for velocity distributions in compound channels with emerged and submerged vegetated floodplains

The lateral distributions of depth-averaged velocity in open compound channels with emerged and submerged vegetated floodplains were analyzed based on the analytical solution of the depth-integrated Reynolds-Averaged Navier-Stokes equation with a term to account for the effects of vegetation. The three cases considered for open channels were two-stage rectangular channel with emerged vegetated floodplain, rectangular channel with submerged vegetated corner, and two-stage rectangular channel with submerged vegetated floodplain, respectively. To predict the depth-averaged velocity with submerged vegetated floodplains, we proposed a new method based on a two-layer approach where flow above and through the vegetation layer was described separately. Moreover, further experiments in the two-stage rectangular channel with submerged vegetated floodplain were carried out to verify the results. The analytical solutions of the cases indicated that the corresponding analytical depth-averaged velocity distributions agree well with the simulated and experimental prediction. The analytical solutions of the cases with theoretical foundation and without programming calculation were reasonable and applicable, which were more convenient than numerical simulations. The analytical solutions provided a way for future researches to solve the problems of submerged vegetation and discontinuous phenomenon of depth-averaged velocity at the stage point for compound channels. Understanding the hydraulics of flow in compound channels with vegetated floodplains is very important for supporting the management of fluvial processes.

The lateral distribution of depth-averaged velocity in a channel flow bend

This paper proposes an analytical model to predict the lateral distribution of streamwise velocity for flow in a curved channel with vertical sides, based on the depth-integrated Navier–Stokes equations. The model includes the effects of bed friction, lateral turbulence and secondary flows, where the additional secondary flow is approximated by a linear function of the lateral distance, as demonstrated by the limited data which are available. Two analytical solutions for the depth-averaged velocity are obtained, one for a flat bed and another for a bed with a transverse slope. Two parameters (denoted by m and n herein), which define the secondary flow, have been examined to analyze how they affect the velocity distribution in these two cases. Comparison of the analytical results with the limited experimental data available shows that the proposed model predicts the lateral distributions of depth-averaged velocity well. Further studies are needed to validate the values of the model parameters (m and n) for bends with different geometric properties.

Lateral Distributions of Depth-Averaged Velocity in Compound Channels with Submerged Vegetated Floodplains

Lateral distributions of depth-averaged velocity in open compound channels with submerged vegetated floodplains are analyzed, based on an analytical solution to the depth-integrated Reynolds-Averaged Navier-Stokes equation with a term included to account for the effects of vegetation. The cases of open channels are: rectangular channel with submerged vegetated corner, and compound channel with submerged vegetated floodplain. The present paper proposes a method for predicting lateral distribution of the depth-averaged velocity with submerged vegetated floodplains. The method is based on a two-layer approach where flow above and through the vegetation layer is described separately. An experiment in compound channel with submerged vegetated floodplain is carried out for the present research. The analytical solutions of the three cases are compared with experimental data. The corresponding analytical depth-averaged velocity distributions show good agreement with the experimental data.

Modeling depth-averaged velocity and bed shear stress in compound channels with emergent and submerged vegetation

This paper presents an approach to modeling the depth-averaged velocity and bed shear stress in compound channels with emergent and submerged vegetation. The depth-averaged equation of vegetated compound channel flow is given by considering the drag force and the blockage effect of vegetation, based on the Shiono and Knight method (1991) [40]. The analytical solution to the transverse variation of depth-averaged velocity is presented, including the effects of bed friction, lateral momentum transfer, secondary flows and drag force due to vegetation. The model is then applied to compound channels with completely vegetated floodplains and with one-line vegetation along the floodplain edge. The modeled results agree well with the available experimental data, indicating that the proposed model is capable of accurately predicting the lateral distributions of depth-averaged velocity and bed shear stress in vegetated compound channels with secondary flows. The secondary flow parameter and dimensionless eddy viscosity are also discussed and analyzed. The study shows that the sign of the secondary flow parameter is determined by the rotational direction of secondary current cells and its value is dependent on the flow depth. In the application of the model, ignoring the secondary flow leads to a large computational error, especially in the non-vegetated main channel.

Analytical model for streamwise velocity in vegetated channels

This paper presents a method for predicting the lateral distribution of depth-averaged streamwise velocity in straight prismatic open channels with emergent vegetation. The approach is fully consistent with the original Shiono and Knight method of analysis, and extends it to include the drag force due to vegetation, modelled as an additional momentum ‘sink' term, with the blockage effect of the vegetation modelled by way of a porosity analogy. An analytical solution to the depth-integrated Reynolds-averaged Navier–Stokes equation is given which includes the effects of bed friction, drag force, lateral turbulence and secondary flows, governed by four coefficients f, CD, λ and Γ respectively. The analytical results are shown to predict the lateral distributions of depth-averaged velocity well for three sets of experimental data: one for inbank flow in a partially vegetated channel and two for overbank flows in compound channels with partially vegetated floodplains. The analytical solution is also shown to give identical results to numerical simulations of the same governing Reynolds-averaged Navier–Stokes equation applied to some experiments undertaken in the UK Flood Channel Facility. The lateral distribution of bed shear stress is also obtained through the Darcy–Weisbach friction factor, thus providing a basis for modelling flood conveyance, fluvial processes and eco-environmental impacts in natural rivers.

Lateral distributions of streamwise velocity in compound channels with partially vegetated floodplains

Natural rivers are commonly characterized by a main channel for primary flow conveyance and a floodplain, often partially covered with vegetation such as shrubs or trees, to carry extra flow during floods. The hydraulic resistance due to vegetation on the floodplain typically causes a further reduction of flow velocity and increases the velocity difference between the main channel and the floodplain. As a consequence a strong lateral shear layer leads to the exchange of mass and momentum between the main channel and floodplain, which in turn affects the overall channel conveyance and certain fluvial processes. The prediction of the lateral velocity distribution is important for many flood alleviation schemes, as well as for studies on sediment transport and dispersion in such channels. The present paper proposes a method for predicting the depth-averaged velocity in compound channels with partially vegetated floodplains, based on an analytical solution to the depth-integrated Reynolds-Averaged Navier-Stokes equation with a term included to account for the effects of vegetation. The vegetation is modelled via an additional term in the momentum equation to account for the additional drag force. The method includes the effects of bed friction, drag force, lateral turbulence and secondary flows, via four coefficients f, CD, λ & Γ respectively. The predicted lateral distributions of depth-averaged velocity agree well with the experimental data. The analytical solutions can also be used to predict the distribution of boundary shear stresses, which adds additional weight to the method proposed.

A novel application of a multi-objective evolutionary algorithm in open channel flow modelling

The Shiono and Knight method (SKM) is a simple depth-averaged flow model, based on the RANS equations which can be used to estimate the lateral distributions of depth-averaged velocity and boundary shear stress for flows in straight prismatic channels with the minimum of computational effort. However, in order to apply the SKM, detailed knowledge relating to the lateral variation of the friction factor (f), dimensionless eddy viscosity (λ) and a sink term representing the effects of secondary flow (Γ) are required. In this paper a multi-objective evolutionary algorithm is used to study the lateral variation and value of these parameters for simple trapezoidal channels over a wide range of aspect ratios through the model calibration process. Based on the available experimental data, four objectives are selected and the NSGA-II algorithm is applied to several datasets. The best answer for each set is then selected based on a proposed methodology. Rules relating f, λ and Γ to the wetted parameter ratio (Pb/Pw) for a variety of situations have been developed which provide practical guidance for the engineer on choosing the appropriate parameters in the SKM model.

A general model of lateral depth-averaged velocity distributions for open channel flows

This paper reviews a model, developed by Shiono and Knight [Shiono K, Knight DW. Two-dimensional analytical solution for a compound channel. In: Proceedings of the 3rd international symposium on refined flow modelling and turbulence measurements, Tokyo, Japan, July 1988. p. 503–10; Shiono K, Knight DW. Turbulent open channel flows with variable depth across the channel. J Fluid Mech 1991;222:617–46 [231:693]], which yields analytical solutions to the depth-integrated Navier–Stokes equations, and includes the effects of bed friction, lateral turbulence and secondary flows. Some issues about the original model developed by Shiono and Knight (1988, 1991) are highlighted and discussed. Based on the experimental data concerning the secondary flow, two assumptions are proposed to describe the contribution of the streamwise vorticity to the flow. Two new analytical solutions are compared with the conventional solution for three simple channel shapes and one trapezoidal compound channel to highlight their differences and the importance of the secondary flow and planform vorticity term. Comparison of the analytical results with the experimental data shows that the general SKM predicts the lateral distributions of depth-averaged velocity well.

Modeling Depth-Averaged Velocity and Boundary Shear in Trapezoidal Channels with Secondary Flows

The Shiono and Knight method (SKM) offers a new approach to calculating the lateral distributions of depth-averaged velocity and boundary shear stress for flows in straight prismatic channels. It accounts for bed shear, lateral shear, and secondary flow effects via 3 coefficients—f,λ, and Γ—thus incorporating some key 3D flow feature into a lateral distribution model for streamwise motion. The SKM incorporates the effects of secondary flows by specifying an appropriate value for the Γ parameter depending on the sense of direction of the secondary flows, commensurate with the derivative of the term Hρ(UV)d. The values of the transverse velocities, V, have been shown to be consistent with observation. A wide range of boundary shear stress data for trapezoidal channels from different sources has been used to validate the model. The accuracy of the predictions is good, despite the simplicity of the model, although some calibration problems remain. The SKM thus offers an alternative methodology to the more traditional computational fluid dynamics (CFD) approach, giving velocities and boundary shear stress for practical problems, but at much less computational effort than CFD.

Stage-discharge prediction for rivers in flood applying a depth-averaged model

The prediction of the stage-discharge relationship for rivers in flood is described by a finite element model of depth-averaged turbulent flow, calibrated using (hree hydraulic coefficients governing local bed friction, lateral eddy viscosity and depth-averaged secondary flow. The resulting lateral distributions of depth-averaged velocity are subsequently integrated to yield the stage-discharge relationship. The calibration of the model involves the establishment of simplifying hypotheses for certain coefficients in order to give the correct depth-mean velocity and boundary shear, both across the channel and with stage. Comparisons against some experimental data from the UK Flood Channel Facility, for channels with trapezoidal and compound cross-sections, help develop the calibration philosophy for both inbank and overbank flows. Numerical experiments with the coherence method for a hypothetical river are used to extend the model calibration to rivers with homogeneous and heterogeneous roughness. Applications of the model to simulating the flow in a number of natural valley and mountain rivers serve to test hypotheses and results obtained at a real scale.

Application and validation of numerical codes in the prediction of compound channel flows

This paper attempts to compare the accuracy of a two-dimensional (2D) depth-averaged code (TELEMAC-2D) and a three-dimensional (3D) code (SSIIM) in the numerical simulation of free surface flows. Assessment was conducted using benchmark data collected from a meandering compound channel at the UK EPSRC Flood Channel Facility. For both codes a standard k–ε turbulence model was employed to simulate the flow field. Assessment of both codes was based on the prediction of depth-averaged velocity traverses around a meander bend, flow features inferred by velocity vectors in the horizontal plane, and water surface elevations for both low and high relative flow depths. In addition, predictions with the 3D code of velocities parallel and perpendicular to the main channel direction are compared with those observed in the experimental programme. Both codes predict the lateral distribution of depth-averaged velocity in the main channel with reasonable accuracy at low relative flow depth, with the 3D code predicting better than the 2D code (maximum percentage errors in depth-averaged velocity are 16% and 22% respectively). At a higher flow depth no extra accuracy in the prediction of depth-averaged velocity is to be gained from the 3D model than from the 2D model, although the 3D model can predict the correct location and direction of secondary currents. For all flow conditions better accuracy was achieved for both codes at the apex section where the mean flow direction is more aligned with the local main channel direction. An accuracy of within 20% can be expected for predictions of depth-averaged velocity magnitude from a 3D model using a standard isotropic representation of turbulence.

Application and validation of numerical codes in the prediction of compound channel flows

This paper attempts to compare the accuracy of a two-dimensional (2D) depth-averaged code (TELEMAC-2D) and a three-dimensional (3D) code (SSIIM) in the numerical simulation of free surface flows. Assessment was conducted using benchmark data collected from a meandering compound channel at the UK EPSRC Flood Channel Facility. For both codes a standard k–ε turbulence model was employed to simulate the flow field. Assessment of both codes was based on the prediction of depth-averaged velocity traverses around a meander bend, flow features inferred by velocity vectors in the horizontal plane, and water surface elevations for both low and high relative flow depths. In addition, predictions with the 3D code of velocities parallel and perpendicular to the main channel direction are compared with those observed in the experimental programme. Both codes predict the lateral distribution of depth-averaged velocity in the main channel with reasonable accuracy at low relative flow depth, with the 3D code predicting better than the 2D code (maximum percentage errors in depth-averaged velocity are 16% and 22% respectively). At a higher flow depth no extra accuracy in the prediction of depth-averaged velocity is to be gained from the 3D model than from the 2D model, although the 3D model can predict the correct location and direction of secondary currents. For all flow conditions better accuracy was achieved for both codes at the apex section where the mean flow direction is more aligned with the local main channel direction. An accuracy of within 20% can be expected for predictions of depth-averaged velocity magnitude from a 3D model using a standard isotropic representation of turbulence.

Application and validation of numerical codes in the prediction of compound channel flows

This paper attempts to compare the accuracy of a two-dimensional (2D) depth-averaged code (TELEMAC-2D) and a three-dimensional (3D) code (SSIIM) in the numerical simulation of free surface flows. Assessment was conducted using benchmark data collected from a meandering compound channel at the UK EPSRC Flood Channel Facility. For both codes a standard k–ε turbulence model was employed to simulate the flow field. Assessment of both codes was based on the prediction of depth-averaged velocity traverses around a meander bend, flow features inferred by velocity vectors in the horizontal plane, and water surface elevations for both low and high relative flow depths. In addition, predictions with the 3D code of velocities parallel and perpendicular to the main channel direction are compared with those observed in the experimental programme. Both codes predict the lateral distribution of depth-averaged velocity in the main channel with reasonable accuracy at low relative flow depth, with the 3D code predicting better than the 2D code (maximum percentage errors in depth-averaged velocity are 16% and 22% respectively). At a higher flow depth no extra accuracy in the prediction of depth-averaged velocity is to be gained from the 3D model than from the 2D model, although the 3D model can predict the correct location and direction of secondary currents. For all flow conditions better accuracy was achieved for both codes at the apex section where the mean flow direction is more aligned with the local main channel direction. An accuracy of within 20% can be expected for predictions of depth-averaged velocity magnitude from a 3D model using a standard isotropic representation of turbulence.

Comparison of turbulence models for stage-discharge rating curve prediction in reach-scale compound channel flows using two-dimensional finite element methods

This paper attempts to assess the accuracy of constant eddy viscosity, Elder and k–ϵ turbulence models in the numerical simulation of reach-scale compound channel flows using two-dimensional (2D) finite element methods. Assessment was conducted using benchmark stage-discharge data collected from straight and meandering compound channel configurations at the UK Engineering and Physical Science Research Council (EPSRC) Flood Channel Facility. For mesh resolutions and topologies used in reach-scale studies, all models were found to be adequate predictors (<5% error in predicted flow depth) of the stage-discharge relationship at moderate overbank flows (Figs. 1 and 2) . However, at inbank and low overbank flows the Elder and k–ϵ turbulence models can reproduce stage-discharge points with much greater accuracy than the constant eddy viscosity model. Hence, for an unsteady simulation where low flows are relevant a constant eddy viscosity turbulence closure may prove problematic. In terms of computed lateral distributions of depth-averaged velocity for both channel configurations, at higher depths (Relative depth=0.666) all turbulence models predict the velocity with greater accuracy than at a lower depth (Relative depth=0.333). At this latter depth, all turbulence models predict the depth-averaged longitudinal velocity distribution with poor accuracy (>20% error). Also, sensitivity of the turbulence parameter calibration with respect to the predicted flow depth showed that the constant eddy viscosity model's performance can be highly dependent on the choice of turbulence parameter value.